Warehousing in the Global Supply Chain: Advanced Models, Tools and Applications for Storage Systems

Warehousing in the Global Supply Chain

Riccardo Manzini Editor

Warehousing in the Global Supply Chain Advanced Models, Tools and Applications for Storage Systems

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Riccardo Manzini Department of Industrial and Mechanical Plants University of Bologna Viale Risorgimento 2 40136 Bologna Italy e-mail: [email protected]

ISBN 978-1-4471-2273-9 DOI 10.1007/978-1-4471-2274-6

e-ISBN 978-1-4471-2274-6

Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011941668  Springer-Verlag London Limited 2012 GoogleTM and Google Scholar are registered trademarks of Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA http://www.google.com CNNmoney  2011 Cable News Network. Turner Broadcasting System, Inc. Turner House, 16 Great Marlborough Street, London+ comScore  11465 Sunset Hills Road, Suite 200, Reston, Virgi+ BBC news  BBC Worldwide Limited, Media Centre, 201 Wood Lane, London, W127TQ Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Valentina and our beautiful children

Foreword

Warehousing is one of the most important and critical logistic activities in industrial and service systems. A few production philosophies, e.g. just in time (JIT) and lean manufacturing, propose and support the so-called ‘‘zero stock’’ as basic and strategic pillar. Also manufacturing requirement planning (MRP), the well known and widely adopted ‘‘push-’’ based fulfillment technique, theoretically guarantees no storage quantities when the ‘‘lot for lot’’ reorder policy is adopted. Nevertheless, these special production systems do not operate in absence of warehousing systems that support and smooth the discontinuity of flow materials, products and components, at the input and at the bottom of a generic production stage. Warehousing activities and storage systems are necessary! This is true in many industrial and not industrial sectors: from automotive to tile industry passing from food industry, health care production systems, service sectors (e.g. banks, universities, hospitals), etc. Obviously, warehousing is the core activity of logistic providers, usually specialized in distribution activities including storage and transportation issues. In special sectors, like the food industry and the health care supply chains, warehousing means storage systems in critical operating conditions, e.g. controlled temperature and/or humidity levels, by the management of fresh and perishable products. The storage systems significantly affect the level of quality of products, the customer’s service level, and the global logistic cost. Just an example: the food industry. Warehousing and transportation issues significantly affect the level of quality of foodstuffs at the consumer’s location, especially when production plants and final points of demand (consumers’ locations) are far away and frequently located in different countries (e.g. wine produced in Italy and used in Taiwan), and the distribution system is very complex including many actors, e.g. distribution centers, wholesalers, dealers, etc. The mission of warehousing is the same of the discipline ‘‘logistics’’: to effectively ship products in the right place, at the right time, and in the right quantity (i.e. in any configuration) without any damages or alterations. Important keywords in warehousing and storage systems are: safety, quality, availability, cost vii

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saving, customer service level, traceability, picking, automation, fulfillment, travel time, etc. They animate the researches and discussions illustrated in this manuscript. With increased globalization and offshore sourcing, global supply chain management is becoming an important issue for many businesses. Global supply chain management involves a company’s worldwide interests and suppliers rather than simply a local or national orientation. This is the operational arena of warehouses in most complex production systems. The generic warehouse plays a critical role in supporting the success of a global supply chain especially in presence of many products, many facilities (eventually located in a wide geographical area, e.g. worldwide), many decision makers and actors [e.g. sources, production plants, central distribution systems (CDCs), regional distribution systems (RDCs), wholesalers, dealers, customers, etc.] and limited resources in terms of people, equipment and space. Literature on storage systems was very popular during 1990s. Many papers were published on several issues, mainly based on mathematical models as supporting decision-making activities. A few papers are recently published on warehousing and storage systems for industry: the largest part of them does not adopt a systematic approach to the whole production–distribution system and supply chain. The challenge of current and future studies and applications is to develop and apply effective models, methods and tools for the whole logistic system optimization supporting the activities of planning, design, management and control in order to find global optima and renouncing to local sub optimizations. Logistic managers and professionals of production systems need effective tools for the global supply chain planning and management. Aim of this book is to present a collection of professional and research chapters on main problems and challenges in warehousing activities, i.e. warehouse operations, and issues for storage industrial and service systems as parts of global supply chains. This book is useful to managers and practitioners of industry and service sectors: it is the basis for the determination and modeling of the most critical issues concerning warehousing systems planning and design. Advanced and effective solving methods are also illustrated and the discussed case studies help the reader to quickly apply the proposed models and techniques/algorithms. The book is also useful for students and researchers of academic institutions who are searching for advanced modeling approaches and solving techniques to complex logistic decision making problems. This book presents and discusses a set of models, tools and real applications, also including a few case studies rarely presented with a sufficient detail by the literature. It is made of 17 chapters organized in four parts: the first presents main issues, models and policies on warehousing systems and picking activities; the second, named Part 1—Manual Storage System, collects basic and advanced contributions on the so-called ‘‘manual’’ storage systems, i.e. not supported by automation; the third part, named Part 2—Automated Storage Systems, presents six original contributions on models and tools for the design, management and

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control of automated storage systems (e.g. AS/RSs automated storage & retrieval systems, carousels, etc.); finally the last part, named Part 3—Applications and Case Studies, illustrates significant applications of the proposed models and methods very useful to managers and practitioners of industry and service sector. The authors who accepted to contribute to this book are important researchers on supply chain systems and warehousing systems. They published significant studies and papers on academic and professional referenced journals. The included chapters have been subjected to a double blind review process that significantly improve the level of quality of the original versions as submitted at the beginning of the publishing project. Further editions of this book including new chapters contributions is expected: to this purpose new authors are welcome for further contributions and are invited to contact me. I would like to thank my colleagues and students of Bologna University, particularly those who dealing with warehousing systems and global supply chains are interested in quantitative approaches and optimization. Special thanks to Yavuz Bozer, University of Michigan, who accepted to write the preface of this book: it was a great honour. He published important researches on main topics and issues of this book significantly contributing to the progress of Logistics and Operations in industrial research and applications. Thanks to all the authors contributing to the publication of this book, and to Springer (especially to the editorial assistants). Bologna, June 15, 2011

Riccardo Manzini Professor of Logistics Department of Industrial Mechanical Plants and Logistics Bologna University [email protected]

Preface

During the 1960s through the 1980s, back when manufacturing dominated the agenda for many private and public sector companies, warehousing was generally viewed as a back-office operation that added little or no value, and in many cases it was treated as an undesirable but necessary cost-center. As the power of lean thinking and just-in-time inventory management spread through the manufacturing sector, one might even say that many ‘‘eyes looking for waste’’ turned to the warehouse, where plenty of excess inventories (considered one of the primary sources of ‘‘waste’’ in lean thinking) were kept in endless rows of racks, reaching heights of 40 ft (12 m) and more. Many individuals, who embarked on applying lean thinking to manufacturing, started to view warehouse design/management as a dead-end career. However, two fundamental changes started to take shape, and it is not clear that either one of the changes were anticipated by the research or the industrial community. At least, it seems no one anticipated how fast the changes would occur, and how deep their impact would be. First, competition, which traditionally was mostly a regional phenomenon, confined primarily to continents or limited by geography, became global. Well-established, multi-national corporations as well as new companies just entering the market began to compete on a global scale never seen before. And the competition was not only for reaching large numbers of customers in existing and new markets, but it was also for identifying and utilizing sources of low-cost labor in every corner of the world. As a result, the manufacturing sector underwent a tremendous transformation, and in the process, global supply chain design and management emerged as a top-priority topic for many companies as well as the research community. In fact, a quick Google search on ‘‘global supply chain’’ generates over 750,000 hits on the web. Based only on articles, and excluding patents, a search of the same subject in Google Scholar generates over one million results! A global supply chain, however, is not just an abstract entity composed of policies, supplier contracts, purchasing agreements, etc. that exist on a computer network or database. Rather, a global supply chain, in order to function properly, is an entity supported by a logistics system that makes it possible to move a variety xi

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of goods through the system in a timely and cost-effective manner. When the supply chain is viewed as a network, transportation systems such as trucking, railways and shipping/air lines represent the ‘‘arcs’’ in the network, while the facilities that handle the goods through the network represent the ‘‘nodes.’’ Such facilities include: • • • • • • •

Manufacturing facilities Warehouses and distribution centers Container terminals (or seaports) Consolidation/deconsolidation centers Rail yards Crossdocks Airports (handling freight)

When viewed in the above context, warehousing—which takes place in all of the above facilities, to one extent or another—suddenly becomes a critical component of the global supply chain. In other words, as their supply chains extended around the globe, companies began to recognize that where and how raw materials, components and (semi) finished goods are stored in the network have a major impact on their operations, their agility/flexibility, their service levels and responsiveness to their customers, and their overall costs. The above recognition by no means implied that storing excess inventories became acceptable or desirable. Rather, as companies watched their supply networks get longer and more complex, and as they learned to apply lean thinking to their supply networks, they realized that warehousing the right amount/type of inventory in the right/most strategic locations can be a major competitive advantage. In the process, many companies also learned that warehousing is not only a point of storage in the network but also a point of critical information. That is, as supply chains became leaner, timely and correct information, especially inventory visibility and accuracy, which is a key function in warehousing, became essential for success. The second fundamental change that impacted warehousing is the emergence of online shopping or e-tailing, especially in the United States and other developed/ developing countries around the world. For example, according to CNNMoney.com and Virginia-based comScore, consumers in the United States spent a record 30.8 billion dollors shopping online during the month of December 2010. This figure represented a 13% increase from the same period in 2009, despite the recession. The growth in e-tail in Europe has been equally impressive. According to a BBC News article, for example, under the heading ‘‘online shopping defies slowdown,’’ it is reported that the e-tail growth rate is about ten times that of the growth rate of the traditional retail market in the United Kingdom as a whole, and it is forecast that online retail sales would reach almost 45 billion pounds by 2012, which represents roughly 14% of the total spending. Other projections on e-tail are equally startling. According to Forrester Research, web-based sales in the United States will reach 249 billion dollors by 2014, and in western Europe, online retail

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sales are projected to increase by 68% from 68 billion euros in 2009 to 114 billion euros by 2014! The phenomenal growth in e-tailing has had (and will continue to have) a major impact on warehousing and parcel/package delivery systems from the warehouse to the customer. That is, stocking the items that the customers need/want, picking, packing and shipping these items to the customers on a timely basis, and doing so with maximum accuracy, has now become a big and very viable business model, which transformed the warehouse from a back-office cost-center to a front-office profit-center! As a result, designing and operating a warehouse, and its various functions such as storage, order picking, sortation and so on, in the most efficient and effective manner has become a front-and-center concern for companies that are competing for a slice of the growing e-tail market. Given the above changes, and the renewed focus on warehousing, a book concerned with advanced models, tools and applications for storage systems could not have been more timely. In the pages of this book, the reader will find a rich collection of insightful and practical models and algorithms presented in a series of chapters arranged in a logical, easy-to-follow manner. The models and algorithms presented cover a wide range of topics, such as order picking/batching and warehouse layout, and a wide range of systems, including automated systems (such as Automated Storage/Retrieval Systems), semi-automated systems (such as carousels) and manual systems (such as walk-and-pick systems). The book is further enriched with a section devoted to applications and case studies that show how theory is applied in practice. I congratulate the editor and the authors for their hard work and for putting together a book that is going to serve as an excellent reference for researchers and warehousing professionals alike. Professor Yavuz A. Bozer Industrial and Operations Engineering The University of Michigan Ann Arbor, MI USA

Contents

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Order Picking: Issues, Systems and Models . . . . . . . . . . . . . . . . . Byung Chun Park

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Storage Systems and Policies. . . . . . . . . . . . . . . . . . . . . . . . . . . . Marc Goetschalckx

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Part I

Manual Storage Systems

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Warehouse Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Goran Dukic and Tihomir Opetuk

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Order-Picking by Cellular Bucket Brigades: A Case Study . . . . . Yun Fong Lim

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A Sequential Order Picking Policy for Shipping Large Numbers of Small Quantities of Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiun-Yan Shiau

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Order Batching in Order Picking Warehouses: A Survey of Solution Approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Henn, Sören Koch and Gerhard Wäscher

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Storage Assignment for Order Picking in Multiple-Block Warehouses . . . . . . . . . . . . . . . . . . . . . . . . . . Kees Jan Roodbergen

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Part II 8

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Automated Storage and Retrieval Systems: A Review on Travel Time Models and Control Policies . . . . . . . . . . . . . . . . . . M. R. Vasili, Sai Hong Tang and Mehdi Vasili

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Designing Unit Load Automated Storage and Retrieval Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tone Lerher and Matjazˇ Šraml

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Warehouse Management: Productivity Improvement in Automated Storage and Retrieval Systems. . . . . . . . . . . . . . . . Yaghoub Khojasteh-Ghamari

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Analytical and Numerical Modeling of AS/RS Cycle Time in Class-Based Storage Warehousing. . . . . . . . . . . . . . . . . . . . . . Mauro Gamberi, Riccardo Manzini and Alberto Regattieri

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A New Technology For Unit-Load Automated Storage System: Autonomous Vehicle Storage and Retrieval System . . . . . . . . . . . Banu Yetkin Ekren and Sunderesh Sesharanga Heragu

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Intelligent Optimization Methods for Industrial Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirko Ficko, Simon Klancnik, Simon Brezovnik, Joze Balic, Miran Brezocnik and Tone Lerher

Part III 14

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Automated Storage Systems

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Applications and Case Studies

Correlated Storage Assignment and Iso-Time Mapping Adopting Tri-Later Stackers. A Case Study from Tile Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riccardo Manzini, Filippo Bindi, Emilio Ferrari and Arrigo Pareschi

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Design and Optimization of Picking in the Case of Multi-Item Multi-Location Multi-Pallet Customer Orders . . . . R. Gamberini, B. Rimini, M. Dell’Amico, F. Lolli and M. Bianchi

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The Logistics Reengineering Process in a Warehouse/Order Fulfillment System: A Case Study . . . . . . . . . . . . . . . . . . . . . . . . Alberto Regattieri, Riccardo Manzini and Mauro Gamberi

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Warehouse Assessment in a Single Tour . . . . . . . . . . . . . . . . . . . M. B. M. De Koster

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About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors

Byung Chun Park Department of Industrial and Systems Engineering, Keimyung University, Dalseo-gu Shindang-dong 1000, Daegu, South Korea, e-mail: bcpark@ kmu.ac.kr Marc Goetschalckx Georgia Institute of Technology, 765 Ferst Drive, Georgia 30332-0205, USA, e-mail: [email protected] Goran Dukic Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 1, 10000 Zagreb, Croatia, e-mail: goran. [email protected] Tihomir Opetuk Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 1, 10000 Zagreb, Croatia Yun Fong Lim Lee Kong Chian School of Business, Singapore Management University, 50 Stamford Road, # 04-01, Singapore 178899, Singapore, e-mail: [email protected] Jiun-Yan Shiau Department of Logistics Management, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan, e-mail: sho@ nkfust.edu.tw Sebastian Henn Faculty of Economics and Management, Otto-von-Guericke University, Postbox 4120, 39016 Magdeburg, Germany, e-mail: sebastian.henn@ ovgu.de Sören Koch Faculty of Economics and Management, Otto-von-Guericke University, Postbox 4120, 39016 Magdeburg, Germany, e-mail: [email protected] Gerhard Wäscher Faculty of Economics and Management, Otto-vonGuericke University, Postbox 4120, 39016 Magdeburg, Germany, e-mail: gerhard. [email protected] Kees Jan Roodbergen University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, e-mail: [email protected]

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M. R. Vasili Department of Industrial Engineering, Lenjan Branch, Islamic Azad University, Esfahan, Iran, e-mail: [email protected] Sai Hong Tang Department of Industrial Engineering, Lenjan Branch, Islamic Azad University, Esfahan, Iran Mehdi Vasili Department of Industrial Engineering, Lenjan Branch, Islamic Azad University, Esfahan, Iran Tone Lerher Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia, e-mail: [email protected] Matjazˇ Šraml Faculty of Civil Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia, e-mail: [email protected] Yaghoub Khojasteh-Ghamari Temple University, Japan Campus, 4-1-27 Mita, Minato-ku, Tokyo 108-0073, Japan, e-mail: [email protected] Mauro Gamberi Department of Management and Engineering—DTG, Padova University, Padova, Italy Riccardo Manzini Department of Industrial Mechanical Plants—DIEM, Bologna University, Bologna, Italy, e-mail: [email protected] Alberto Regattieri Department of Industrial Mechanical Plants—DIEM, Bologna University, Bologna, Italy Banu Yetkin Ekren Deptartment of Industrial Engineering, Pamukkale University, Denizli, Turkey Sunderesh Sesharanga Heragu Department of Industrial Engineering, University of Louisville, Louisville, LY 40292, USA, e-mail: [email protected] Mirko Ficko Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia, e-mail: [email protected] Simon Klancnik Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Simon Brezovnik Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Joze Balic Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Miran Brezocnik Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Tone Lerher Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia

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Riccardo Manzini Department of Industrial Mechanical Plants, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy, e-mail: riccardo.manzini@ unibo.it Filippo Bindi Department of Industrial Mechanical Plants, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Emilio Ferrari Department of Industrial Mechanical Plants, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Arrigo Pareschi Department of Industrial Mechanical Plants, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy R. Gamberini University of Modena and Reggio Emilia, Via Amendola, 2, Padiglione Morselli, 42100 Reggio Emilia, Italy, e-mail: [email protected] B. Rimini University of Modena and Reggio Emilia, Via Amendola, 2, Padiglione Morselli, 42100 Reggio Emilia, Italy M. Dell’Amico University of Modena and Reggio Emilia, Via Amendola, 2, Padiglione Morselli, 42100 Reggio Emilia, Italy F. Lolli University of Modena and Reggio Emilia, Via Amendola, 2, Padiglione Morselli, 42100 Reggio Emilia, Italy M. Bianchi University of Modena and Reggio Emilia, Via Amendola, 2, Padiglione Morselli, 42100 Reggio Emilia, Italy Alberto Regattieri DIEM—Department of Industrial and Mechanical Plants, Bologna University, v.le Risorgimento 2, 40136 Bologna, Italy, e-mail: alberto. [email protected] Riccardo Manzini DIEM—Department of Industrial and Mechanical Plants, Bologna University, v.le Risorgimento 2, 40136 Bologna, Italy Mauro Gamberi Department of Management and Engineering, DTG University, Padova, Italy M. B. M. De Koster Rotterdam School of Management, Erasmus University, Burg. Oudlaan 50, 3062 PA, Rotterdam, The Netherlands, e-mail: [email protected]

Chapter 1

Order Picking: Issues, Systems and Models Byung Chun Park

Abstract Order picking is the process of retrieving items from storage to meet a specific customer order, which is known to be the most labor-intensive and costly function among all the warehouse functions. This function is also important in that it has a critical impact on downstream customer service. For understanding the background of order picking and related performance issues, we will briefly introduce warehousing functions. Then we will introduce order picking strategies, and discuss about performance issues and measures in the domain of order picking operations and systems. Productive and quality measures will be reviewed in more detail. Then, we will introduce order picking systems. In picker-to-stock systems, we will address three most popular storage equipments: bin shelving, modular storage drawer cabinet, and gravity flow rack. We will also introduce three popular retrieval equipments: picking carts, picking conveyors, and order picker trucks. In stock-to-picker systems, we will introduce three popular storage/retrieval systems: carousel systems, miniload systems, and A-frame dispenser systems. Then, the remaining part of the chapter will be devoted to performance models for carousel and miniload systems. A modeling issue here is the interaction between the storage/retrieval system and the picker, robot or human.

B. C. Park (&) Department of Industrial and Systems Engineering, Keimyung University, Dalseo-gu Shindang-dong 1000, Daegu, South Korea e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_1,  Springer-Verlag London Limited 2012

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1.1 Introduction In modern business environments, warehousing functions are becoming important more than ever. Yet, market pressures such as smaller shipments, faster delivery, and rapid proliferation of products make warehouse management difficult for meeting the demand for high service quality with low costs. Material in a usual warehouse flows through the following functions: • • • • •

Receiving Put-away Storage Order picking Shipping.

Receiving is the function involved in receipts of materials coming into the warehouse. Put-away is the act of moving and placing materials in storage. The important decision in put-away is determining where and how much to store materials as there may be several storage modes or systems in most warehouses. By storage mode, we mean a region of storage area or a piece of storage equipment where the costs of picking from and restocking any location are all approximately equal (Bartholdi and Hackman 2010). Example storage modes include pallet rack used for pallet storage, carton flow rack typically used for storing active cartons, and bin shelving typically used for storing slower, small loads. Storage is to physically house material until requested. Storage systems can be classified into two types: unit load systems and small load systems. Unit load storage systems are used for housing large loads such as full pallets or large boxes. Block stacking or rack storage is a typical storage system used. Small load storage systems house small loads such as tote pans or small boxes, where the maximum capacity per storage location is typically less than 500 lbs (Tomkins et al. 1996). Usual order picking is involved in picking from small load storage systems. Order picking is the process of retrieving items from storage to meet a specific customer order. Order picking function may include packaging of individual items or assortments. When an order consists of more than one item, it may be required to sort and/or accumulate items into individual orders, depending on the picking strategy employed. Shipping is involved in the activity of packing and accumulating orders by outbound carrier for loading. Material within a warehouse may be handled in pallet loads, in cases or cartons, in tote pans or bins, in inner packs, or even in individual pieces. Cases, cartons, tote pans, and bins are all containers that are used for holding items. A case is a container that is specially designed to hold or protect items. A carton refers to a rectangular cardboard box in use for storing items. A bin is a generic term referring to a container typically made of plastic, though metal bins are available. A tote pan in general refers to more standardized, reusable containers, typically used for inprocess handling and stackable on top of one another. Both metal and plastic tote

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pans are available. So, we will not distinguish between bins and tote pans. Further, we will not distinguish between cases and cartons, if not in case of need. Depending on the types of retrieval units, types of picks can be classified into pallet pick, case pick, and broken-case pick. It is called a pallet pick when the picking quantity is a multiple of a full pallet load. We call it a case pick when the picking quantity is a multiple of a case quantity but less than a full pallet load. A broken-case pick, sometimes called a piece pick, is termed to represent an order pick where the picking quantity is less than a full case or in pieces. A pallet pick is typically done in unit load storage systems. Case pick can be done from pallet storage, bulk storage, or such case storage systems as carton flow racks. Usual broken-case pick is done from small load storage systems, where items are commonly housed in cartons or bins. General characteristics of broken-case picks include a large number of item types, small quantities per pick, and short cycle times. Accordingly, broken-case picks are most costly and difficult to manage. Typically, order picking is involved in broken-case picks or case picks. Order picking is known to be the most labor-intensive and costly function among all the warehouse functions, typically accounting for 55% of warehouse operating costs (Tomkins et al. 1996). Order picking is also important in that it has a critical impact on downstream customer service. Customers expect quick and accurate processes of their orders, which is directly dependent on the efficiency of order picking operations. Order picking systems can be classified largely into two types: picker-to-stock systems and stock-to-picker systems. In picker-to-stock systems, the picker travels by walking or riding to storage locations to retrieve items. In these systems, the travelling activity is the most expensive activity, typically accounting for 55% of order picker’s time (Frazelle 2002). An alternative solution to eliminate or reduce picker’s travel is the stock-to-picker system. In typical stock-to-picker systems, the container or the storage location housing requested item is mechanically brought to the picker for retrieval. Thus, the picker’s travel is replaced by the container’s travel in stock-to-picker systems. The organization of the chapter is as follows. After the introduction to order picking strategies, we will discuss about performance issues and measures in the domain of order picking operations and systems. Productive measures and quality measures will be reviewed in more detail. Then, we will introduce order picking systems. In picker-to-stock systems, we will address three most popular storage equipments: bin shelving, modular storage drawer cabinet, and gravity flow rack. We will also introduce three popular retrieval equipments: picking carts, picking conveyors, and order picker trucks. In stock-to-picker systems, we will introduce three popular storage/retrieval systems: carousel systems, miniload systems, and A-frame dispenser systems. Then, the remaining part of the chapter will be devoted to performance models for carousel and miniload systems. A modeling issue here is the interaction between the storage/retrieval system and the picker, robot or human.

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1.2 Order Picking and Performance 1.2.1 Order Picking Strategies An order is simply a list of one or more order lines going to a specific customer or destination. An order line typically consists of a line item and its quantity requested. A line item is a separate item of supply on a transaction document. A stock keeping unit (sku) represents one unique inventory item in the smallest physical unit handled in a warehouse. Thus, line items and requested quantities should be expressed in terms of skus. Otherwise, communication error may arise. Customer orders should be converted into pick lists for actual picking. A pick list is a list of one or more pick lines. A pick line typically consists of a line item, its quantity to be picked, and the storage location to be picked from. A pick list can be established from either only a single order or several orders. On the other hand, a single order can be broken up such that its line items appear in several pick lists. It is usual that the pick lines in a pick list are cleverly sequenced in order to reduce travel time, thereby improving the efficiency of picking operations. It is called the single order picking if customer orders are completed one at a time. It is called the batch picking if orders are batched together so that all the line items requested in the batch of orders are picked from a location as it is visited by or presented to the picker. In other words, the order picker picks all orders within the batch using a consolidated pick list in a cycle. Usual batch sizes are from 4 to 12 orders, depending on the number of order lines per order. In general, the single order picking is efficient for large orders with low demand, or emergency orders. By large order, we mean an order consisting of a large number of order lines, i.e., greater than ten order lines. Batch picking is efficient for small orders with high demand, let us say, high demand for orders of less than five-line items. The advantage of batch picking is that it can reduce travel time between picking locations by increasing pick density, the number of picks per unit of area (or per unit of travel length in one-dimensional systems). The disadvantage is that it requires additional work needed for order integrity, that is to say, sorting the picked items into individual customer orders. The two typical approaches used for order integrity are sort-while-picking and sort-after-picking. In sort-while-picking, the order picker sorts the line items into individual orders while picking in a cycle. Separate containers and/or special containers with several compartments may be used for order sortation/accumulation. In sort-after-picking, all the line items requested in the batch of orders are picked together during a picking cycle. The picked items are then sorted into individual orders in a downstream process. Batch picking is mostly used for broken-case picks. It can be used for case picks with sort-after-picking. It is rarely used in case picks with sort-while-picking, primarily because of its large physical size. For the efficiency of picking operations, the picking area might be exhaustively divided into several picking zones. In picker-to-stock systems, a picking zone may

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consist of either a portion of a storage aisle, one aisle, or several aisles. In stockto-picker systems, it could be a pod of retrieval equipments. It is called the zone picking if pickers are dedicated to a picking zone and responsible for picking items stored in that zone. Typically, zones are sized such that one or two pickers are assigned to a zone. It is called the free-form picking if pickers are free to pick from any location in the entire picking area. Zone pickings are frequently used along with batch pickings. The primary objective of configuring zones in picker-to-stock systems is to reduce travel times of pickers. A well-designed zone picking system can significantly reduce the picker’s travel time. But, reduction in travel time can be offset by additional work required for order integrity. In stock-to-picker systems, the primary objective is to balance workload between the picker and retrieval equipments. In both systems, order integrity is a matter of concern. The two popular methods used for order integrity in zone picking are progressive assembly and downstream sortation. In zone picking with progressive assembly, also called the pick-and-pass system, the contents of an order are passed from zone to zone in such containers as tote pans or cartons along a conveyor or on a cart until the order is completely assembled. Progressive assembly is most effective for systems with a large number of skus and for small to moderate orders with high demand. In zone picking with downstream sortation, also called wave picking, the order pickers work in their picking zones in parallel during a wave. In other words, picking operations are done in all zones at the same time. Typically, the order picker applies a bar code label to each unit picked. The picked item is placed into a large cart or onto a conveyor. It is then induced into a sortation system, where each customer order is completed. Wave picking requires a substantial investment on downstream sortation systems, and hence is suitable for systems with a high number of skus and heavy demand for moderate to large orders. For zone picking to be successful, it is critical to balance the workload among zones, which is not an easy task. An alternative solution to progressive assembly with fixed zone sizes is to configure the picking area as a bucket brigade, a kind of progressive assembly with varying zone sizes. In a bucket brigade, zone sizes are automatically adjusted in order to keep balancing the workload among zones. For more detail on bucket brigades in a warehouse, see Bartholdi and Hackman (2010).

1.2.2 Performance Issues 1.2.2.1 Warehouse Activity Profiling In short, the mission of a warehouse is to meet the customer demand on higher service quality with lower costs. Thus, understanding the customer demand is the first step to be taken for the proper design and/or operation of order picking systems. This can be done by establishing warehouse activity profiling, which is

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the systematic analysis of warehouse activities. Example activity profiles include customer order profile, item activity profile, purchase order profile, and inventory profile. Most information on customer demand can be identified by establishing the customer order profile. A typical customer order profile includes such information on order activity as number of orders per day, number of order lines per order, number of units (pallets, cases, or pieces) per order line, number of order units per day, etc. The customer order profile can serve as a valuable input to establishing other warehouse activity profiles such as item activity profile. The statistical distribution of order activity is extremely useful not only for designing or reconfiguring a warehouse but also for improving warehouse activities. As an example, it can be used for encouraging customers to order in larger units, thereby reducing handling costs while improving service quality. It can also be used in receiving effectively for making orders and scheduling receipt. An example is in use for implementing cross-docking within a warehouse, where items move directly from receiving to shipping. With this approach, costly putaway, storage, and order picking can be eliminated. The item activity profile shows activities of individual items in a warehouse, typically including popularity and volume in units handled for each item, handling characteristics, daily demand variation and seasonality, and demand correlation among line items. The popularity of an item can be measured in several ways, but the most important one to be used for warehouse management is the number of retrieval requests per unit time for the item. Handling characteristics of an item includes physical characteristics such as shape and weight, units to be handled, and environmental characteristics such as frozen, flammable, or hazardous material. The item activity profile can be used effectively for selecting proper storage or handling methods for each item.

1.2.2.2 Slotting and Operating Policies By slotting, we mean the careful determination of storage place for each item. Specifically, it involves determining proper storage mode, space volume to be allocated, and specific storage location within a storage mode for each item. The whole quantity of the same item type can be stored in a specific storage mode, or distributed over several storage modes. The most information in use for slotting or reslotting the warehouse properly can be provided from the item activity profile. Slotting has a significant impact on put-away and order picking activities. Performance issues closely related to slotting are layout configuration, pickand-reserve problem, and restocking policy, among others. Layout configuration is involved in determining storage space requirements, dimension of handling units, number and dimension of aisles, layout of aisles, number and locations of input/ output points, and takeaway/delivery systems. In most distribution warehouses, the forward pick area and the reserve storage area are separated. A forward pick area is a small subregion of a warehouse, specially configured for efficient picking

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operations, where items are usually stored in small units. The reserve storage area is a large area for use in storing items in large units, which are primarily used for restocking forward areas. The restocking policy is involved in determining what, when, where, and how much to restock the forward area from the reserve area. All these have a great impact on the efficiency of picking operations. Performance issues regarding operating policies include storage and retrieval policies. The storage policy is involved in determining the storage location within a storage mode for each item. Storage policies can be classified into random storage, dedicated storage, and class-based storage. Random storage is a generic term to represent the storage policy where the storage location is assigned to the item without considering its activity or turnover. When modeling random storage, each available location is assumed to be equally likely selected for storage, resulting in uniformly distributed activity over the locations. Selecting available locations in random, however, is cumbersome, and sometimes even costly. In practice, the closest open location rule is used. Under the closest open location rule, the closest, available location to the input/output point is selected for storage. Park and Lee (2007) showed that the assumption of uniform activity can be used fruitfully for modeling storage systems operated under the closest open location rule. In dedicated storage, each item has a dedicated location or a fixed slot. Some items may be requested more frequently than others, thus having higher activity. We call the turnover-based storage if the storage location for each item is determined based on its turnover rate. In class based storage, the items are partitioned into several classes, typically based on their turnover rates. Then, a set of storage locations is assigned to each class. Random storage is used within a class. The retrieval policy is involved in actual retrieval operations within a storage mode, and includes issues of retrieval time window, interleaving policy, and pick sequencing and path problem. A retrieval time window is the period of time during which requested orders are batched together to reduce travel time between picking locations. An example is the retrieval time window with zero time length, which results in the single order picking with first-come-first-serve basis. The interleaving policy is concerned with interleaving retrieval operations with storage operations. In some systems, retrieval requests are interleaved with storage requests such that both storage and retrieval operations are performed in a cycle. With this approach, the overall efficiency of an order picking system can be greatly improved. A miniload system is a typical example system where interleaving is commonly employed. Given a set of pick lines, the pick sequencing problem is to determine the proper sequence of the picking locations to reduce travel time. In picker-to-stock systems, this problem is a kind of a travelling salesman problem, the problem of visiting a given set of locations as quickly as possible during a tour. The travelling salesman problem is extensively studied in the literature, yet no efficient, exact solution is reported (NP-hard). Consequently, heuristic procedures are commonly used. Clever pick sequencing can also reduce travel time dramatically in stockto-picker systems (Bartholdi and Platzman 1986; van den Berg 1996; Litvak 2006).

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The pick path problem is involved in determining the actual path to follow. Given a set of locations with a visiting sequence, there may be a number of different paths to visit the locations. Two most popular pick paths used in picker-to-stock systems are the serpentine pick path and the mainline path with side trips. A serpentine path is the travel path that goes through in each storage aisle in a one-way direction. No turning back is allowed in the aisle. Clearly, travelling along the serpentine path, in the literal sense of the word, can result in a huge waste of time, unless the picker is required to visit every storage aisle during a tour. In mainline path with side trips, the picker travels along the main path that crosses the storage aisles, and detours into the aisle when necessary for picking. The advantage of this approach is that it can reduce travel time by storing more active items in locations close to the main path, thereby being able to decrease the number and length of the side trips. For more on pick-path optimization problems, see Ratliff and Rosenthal (1983) and Bartholdi and Hackman (2010). Besides performance issues addressed here, there are other design and operational issues resolved for the fast, accurate, and efficient handling of customer orders. But, probably the most general principle of utmost importance would be that put-away, storage, and order picking should be considered simultaneously in both design and operation. Clever put-away can greatly improve not only its effciency but also order picking efficiency. As mentioned, slotting significantly impacts the efficiency of both put-away and order picking activities. In turn, innovative order picking strategy and system can eliminate or reduce well the need for put-away or storage. An example is picking from the reserve storage area. Frazelle (2002) reports the case of Ford’s service parts distribution center, where the 54 carousels act as the reserve storage area while achieving forward picking rates by bringing reserve storage locations to picking stations.

1.2.2.3 Automation Automation also greatly impacts order picking performance. In general, there are two kinds of automation issues in a warehouse: one for physical handling systems and the other for information handling systems. Automation issues in physical handling systems include automation and integration of storage/retrieval operations and material flows together with information. They are involved in determining the type and technology of equipments and the degree of automation and integration. In the domain of order picking operations, a usual approach to automation regarding physical handling systems is to facilitate automatic presentation of materials or information to the picker. High speed and accuracy of picking operations are essential elements for success in today’s market environment, which demands fast delivery without error. In that context, this technology is becoming important more than ever, since it can significantly improve both picking productivity and accuracy.

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Automatic presentation of materials to the picker is increasingly popular. A system may be configured such that the requested locations or containers are automatically brought to the picker. Example systems are miniload and carousel systems. The primary objective of employing this technology is to reduce or eliminate the picker’s search and travel, the two most costly work elements of picking operation. This technology can also help the picker extract items without error. Automatic presentation of information to the picker is also increasingly popular. The main direction is toward facilitating hand-free and paperless operation. Pick-to-light systems, voice-directed picking systems, and virtual displays are example technologies. In the context of picking operations, the primary advantage of employing these technologies is that these technologies can greatly facilitate efficient searching for and extracting items from the locations with less error. The pick-to-light system is a light-directed picking system, typically consisting of indicator lights and displays. It is used for presenting the picking location and quantity to be picked to the picker. It has a wide variety of applications. Example application systems include bin shelving, gravity flow racks, and carousels. In a typical application, a light display is placed at the front of each location. The light is illuminated if a pick is required from that location. In some systems, light displays can be integrated even into cartons or tote pans where picked items are placed, thereby being able to improve both picking efficiency and accuracy. The pick-to-light system is most popular in high volume picking. In voice-directed picking systems, synthesized voice is used for giving instructions to the picker in real time. Voice technology is rapidly growing in recent years and is now a very viable solution for piece pick, case pick, or pallet pick operations. Lower cost than pick-to-light system, voice picking can deliver similar number of picks per hour. This system is also used effectively in high volume picking. In a virtual display-directed picking system, virtual display technology is used for directing the picking tour and/or to give instructions to the picker. Information handling systems in a warehouse include the warehouse management system, communication system, and automatic identification systems. A warehouse management system is a kind of warehouse information system that tracks materials from receiving to shipping, manages warehouse activities and inventory, and processes or generates business data and information, and so on. Communication systems are interfaces that connect the warehouse management system to physical and information handling systems. Local area networks, radio frequency data communication, and voice-based systems are example communication systems. Automatic identification systems are used for collecting relevant data automatically, which are then used for real-time control of materials or operations. Example systems include bar coding, radio frequency, and vision systems. Proper automation has several advantages. It can greatly reduce labor costs, increase inventory and picking accuracy, and shorten order cycle time. But, it usually requires a substantial investment. Furthermore, it is inflexible, i.e., it is

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more difficult to reconfigure the system to adapt to new business environments. Thus, to make it function properly and reliably, it should be well designed and integrated. To work effectively, it should also be accompanied with work simplification and standardization.

1.2.3 Performance Measures System throughput capacity, system sojourn time, and picking accuracy are considered to be the three most important performance measures for order picking systems. Throughput capacity is the typical measure used for comparing performances of order picking systems. System sojourn time and picking accuracy have a direct impact on downstream customer service.

1.2.3.1 Financial and Productive Measures The performance of order picking systems can be measured from the financial point of view or the operational point of view. Financial measures are commonly used when needed to measure the system’s performance in financial terms at the higher level. Usual financial measures are based on such data as costs, volume in units handled, and sales. Cost data can be collected for each of the warehouse activities and for each of the major resources consumed. Usual activity-based costing/management and benchmarking are very useful when measuring financial performances. Since we are more interested in efficient operation of the order picking system, we focus on operational measures. A common measure used in measuring how well a system operates is productivity. Productivity is a kind of efficiency measure, and is defined to be the ratio of outputs divided by the inputs used. In the context of order picking operations, the outputs may be the number of completed orders, line items, transactions, etc. Usual inputs are such resources consumed as time, labor, space, energy, and money. An example productive measure is labor productivity, the ratio of the number of line items or orders completed to the number of labor hours spent. A productive measure that is conveniently used in a warehouse is the storage density, the ratio of the amount of storage capacity available to the square footage used. The utilization of such key resources as machines or order pickers is another example of productivity measures. Since operation is mainly concerned about making the best use of resources available, productivity is a fundamental, operation-oriented performance measure. In regard to performance of order picking operations, probably the most important productivity measure is the system throughput capacity or maximum throughput. By system throughput capacity, we will mean the ‘maximum’ long run average rate at which the system can process requests. Here, the input is the period

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of time spent and the output is the number of requests completed. For the system operating under the stability condition, the system throughput is simply the demand rate. Thus, a matter of concern is to estimate the system throughput capacity. An approach for use in determining the throughput capacity is to model the system under the assumption that there exist an infinite number of requests to the system. The efficient operation of an order picking system is critical under peak demand. Otherwise, it may become a bottleneck in an integrated warehouse system. Consequently, system designers are interested in the realization of maximum throughput, and consider the throughput capacity as the most important design criterion. In order picking systems, throughput is typically measured in terms of pick rate. For broken-case pick, the pick rate is usually expressed in the number of pick lines picked per unit time (hour). For case pick, the pick rate can be expressed in either or both the number of cases and the number of pick lines picked per unit time. For pallet pick, the pick rate is usually expressed in the number of pallets picked per unit time. Throughput can also be measured in terms of number of orders completed per unit time. But, it must be careful to use this measure as a basis for comparison between two different systems, since it depends on the distribution of order sizes requested to the system.

1.2.3.2 Quality Measures Resources consumed in the warehouse are important for warehouse management. But, it is not true for outside customers. Usual customers are not concerned about is resources consumed to meet their expectations. What they are concerned about are how well the system performs, how well it supports their requirements, etc. Quality measures are to evaluate how well a system functions. Today’s market pressure in a globally competitive environment makes quality measures increasingly important. From the progressive point of view, the quality of a system refers to the perception of the degree to which the system meets the customer’s expectations. Thus, the quality of a system is determined by the customer, based upon the evaluation of his or her own entire experience. By system quality measures, we will mean the performance measures that are directly relevant to customer service. The two most important quality measures are system sojourn time and picking accuracy. Other quality measures include travel time and system queue size. System sojourn time, also called system cycle time, is the elapsed time from when a request is released to the system until it is completed. Here, the request can be a pick line, a customer order, or a batch of customer orders. Again, it must be careful to use system sojourn time of an order as a basis for comparing performances of different order picking systems, because the performance depends on the distribution of order sizes. Furthermore, in some systems, the customer order may be completed in downstream sortation/accumulation system. Thus, it is very

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critical to clearly define the system boundary when comparing performances of different order picking systems. System sojourn time is becoming more important under today’s business environment that frequently demands the same day shipment. System sojourn time is also important from the operational point of view, since failure to correctly estimate the sojourn time of a request can cause starving or blocking at downstream operations, which will significantly affect customer service. In regard to warehouse performance, warehouse order cycle time measures the elapsed time from when an order is released to the warehouse until it is ready for shipping. Picking accuracy is the percent of pick lines picked without error. Picking accuracy is a key measure of great concern to usual customers, since it directly impacts the customer service. There are a variety of factors that impact the picking accuracy in a warehouse, including human, equipment, material, method, and environment. A popular approach to improving picking accuracy is to employ such technologies as automatic identification and presentation. Travel time is one of the issues extensively studied in the literature. For pickerto-stock systems, a matter of concern is the picker’s travel time to the picking location. For stock-to-picker systems, the matter is the travel time of the storage location or container to the picking station. Travel time and throughput are closely related. To increase throughput, travel time should be minimized. The usual approach to minimizing travel time is to sequence a set of the pick lines such that the sum of the individual travel times is minimized. But, care must be exercised: for order picking systems, simply minimizing the travel time cannot necessarily guarantee maximizing the throughput, as it depends upon the interaction with the picker. In fact, Park et al. (2006) showed that for miniload systems, minimizing the expected single or dual command travel time did not result in maximizing system throughput. System queue size is the number of requests waiting for system service. System queue size and system sojourn time are also closely related. Large queue typically implies long sojourn time, and vice versa. Furthermore, large queues may indicate heavy congestion at interfaces or serious delays at downstream operations.

1.2.3.3 Benchmarking Many warehouses use key performance indicators (KPI) to manage warehouse operations and to improve productivity. The KPIs are simply a set of several key performance indicators, typically representing operation costs, productivity, cycle time, or order accuracy. Based on the KPIs, warehouse performance gap analysis can be carried out, which is the graphical representation of a warehouse’s performance in each performance dimension, compared to that of the industry best. The performance gap analysis can be used to identify weak or strong points in the performance of warehouse operations (Frazelle 2002). Benchmarking is the process of comparing one’s products, business processes, or even strategies to those of the most successful firms in the same industry or from

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other industries. Dimensions typically measured are quality, time, and cost. Benchmarking gives a single efficiency score that can be used for comparing performance. There are several types of benchmarking. Metric benchmarking involves using more aggregative cost or operation information to identify strong and weak performing units. The two most common forms of quantitative analysis used in metric benchmarking are data envelope analysis and regression analysis. Hackman et al. (2001) conducted a data envelope analysis to compare warehousing and distribution operations. The conclusions from their research are as follows. There was no definite difference in performance between union and nonunion warehouses, and between smaller warehouse and larger warehouse, respectively. Warehouses with low capital investment tend to outperform those with high capital investment.

1.3 Order Picking Systems 1.3.1 Picker-to-Stock Systems In picker-to-stock systems, sometimes referred to as in-the-aisle systems, the picker walks or rides to the picking location to retrieve items. The two most important components of the system that determine the throughput capacity are storage equipments and retrieval equipments. The throughput capacity depends more on retrieval equipments employed. In general, the selection of storage equipments and that of retrieval equipments can be done separately.

1.3.1.1 Storage Equipments Here, we will briefly introduce the three most popular storage equipments in use for small load storage: static shelving, modular drawer cabinet, and gravity flow rack. For space savings, these equipments can be placed on a mezzanine, or mobilized. Both static shelving and gravity flow rack can be used for either borken-case picks or case picks. Modular drawer cabinets are typically used for broken-case picks. Pallet racks can be used for case picks, but are not dealt with here. (1) Static Shelving Static shelving is a set of static shelves with usual depths from 12 inches to 24 inches. Items can be placed either directly on the shelf or in containers such as cartons or bins. More deep shelves are available for storing large cartons. Both picking and restocking are done from the same picking aisle. To extract items, the picker must reach into the shelving unit. This system characteristic not only necessitates providing space over each shelving unit for the picker’s hand and forearm, but also makes it difficult to utilize fully the inside of

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Fig. 1.1 Application of bin shelving systems (Courtesy of Omnilift, Inc.)

the shelving unit. It also impairs the efficiency of reaching and extracting tasks, which are the essential work elements of picking operations in static shelving. Bin shelving is simply a static shelving that is used for housing bins. Each bin typically contains small items of the same item type. In some applications, the system is configured such that the picker pulls out the bin into the aisle for picking. Yet, some clearance is required for pulling out the bin. Figure 1.1 shows an application of bin shelving systems. Bin shelving is the most popular equipment among the storage equipments used for small load storage. It is inexpensive to install and maintain, and easily reconfigured. The drawback is that in bin shelving, space tends to be underutilized, which leads to low sku density, the number of skus available per unit of area on the pick face. By pick face, we mean the 2-dimensional surface along the picking aisle, from which items are extracted. It results in increase in both of travel times and labor costs. Another drawback is the item security problem, because bin shelving is open to the pick face. Bin shelving is used for broken-case picks, and suitable for storing small items with low demand. (2) Modular Drawer Cabinet A modular drawer cabinet is the cabinet that houses a number of modular storage drawers. By subdividing into compartments, each storage drawer can contain small items of several item types, which increases the sku density. Again, both picking and restocking are done from the same picking aisle. Modular drawer

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Fig. 1.2 Example modular drawer cabinets (Courtesy of Ellis Systems Corp.)

cabinets can be easily reconfigured as storage requirements for items change. Figure 1.2 shows modular storage drawers in cabinets. For picking items, the picker pulls out the drawer into the aisle, which provides such advantages over bin shelving as improved space utilization, picking accuracy, and item protection. It is, however, more expensive than bin shelving. Modular storage drawer cabinets are suitable for storing items that are smaller but more valuable. (3) Gravity Flow Rack Gravity flow rack consists of a set of sections used for storing items in cartons, tote pans, or bins. Each section is equiped with rollers or rails, built on a slight incline so that the containers move forward when they are loaded. The containers are placed at the back of the sections from the replenishment aisle, and move toward the pick face as containers are depleted, which ensures first-in-first-out turnover.

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Fig. 1.3 Application of gravity flow rack (Courtesy of Conveyor & Storage Solutions, Inc.)

There is a wide variety of gravity flow racks available. Consequently, gravity flow racks can be used for storing cases, broken-cases or, even pallet loads. Broken-case pick, case pick, and pallet pick are all possible. Figure 1.3 shows an application of gravity flow racks in use for broken-case picks. Carton flow racks are gravity flow racks used for housing cartons, open or closed. In carton flow racks, only one carton of each sku is located on the pick face. Consequently, the sku density is very high in carton flow racks, compared to other carton storage systems. Thus, well-designed carton flow rack can significantly decrease both travel times and labor costs. Carton flow racks are most useful for storing items with high demand.

1.3.1.2 Retrieval Equipments The picker’s work in picker-to-stock systems can be broken down into the following work elements: searching for and travelling to picking locations, reaching into and extracting items from locations, sorting and packing items, documenting picking transactions, and others. Among them, travelling and searching tasks are the most costly elements, typically accounting for 55 and 15% of picker’s working time, respectively. The next costly work element is extracting, accounting for 10%

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Fig. 1.4 Application of picking carts (Courtesy of Creform Corp.)

of picker’s working time (Frazelle 2002). Thus, performances of retrieval methods or equipments are highly dependent on how well they can reduce or eliminate these work elements. Here, we will briefly introduce three popular retrieval equipments in use for order picking operations: picking carts, picking conveyors, and order picker trucks. Picking carts and conveyors are primarily used for broken-case picks or case picks at floor level. Order picker trucks are typically in use for case picks from high-rise pallet racks. (1) Picking Carts A picking cart is simply a cart used for broken-case or case picks. Carts are widely used for pick-to-cart, order picking operations not only in warehouses but also in assembly, kitting, and other manufacturing applications. It could be used either for single order picking, batch picking, or zone picking. A variety of picking carts are available for order accumulation and/or sortation during a picking tour. Batch picking carts are designed to pick and sort multiple orders into individual orders during a tour. More sophisticated picking carts are equipped with such technologies as mobile computing system, scanner, and label printer to inspect, make price tag, label, etc. Pick-to-light systems or voice directed systems, integrated into cart picking systems, can even increase the picking efficiency. The right picking cart can greatly increase the order picking efficiency while lowering accidents and fatigue of the picker. Figure 1.4 shows an application of picking carts used for both broken-case and case picks simultaneously. Pick-to-cart systems are highly flexible and inexpensive to install. The drawback of the system is low pick rate due to longer travel time between picking locations as the picker walks. It is commonly used for the picking area having a large number of skus with low demand.

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Fig. 1.5 Application of picking conveyors (Courtesy of Cornerstone Automation Systems, Inc.)

(2) Picking Conveyors Picking conveyors are conveyors used for moving totes or cartons in order picking area. It can be used for broken-case or case picks, sometimes even for pallet picks. Pick-to-conveyor systems can greatly improve the picking productivity by eliminating much of walking or lifting associated with pick-to-cart systems. Typically, a bar code label is put on each unit, which is then used for identification and downstream sortation. Other technologies are often used together with bar code systems to improve picking productivity even more. Example technologies include radio frequency terminals, light directed picking, and voice directed picking. Figure 1.5 shows an application of picking conveyor systems. Pick-to-conveyor systems are popular for use in zone picking with either progressive assembly or downstream sortation. In application to zone picking with progressive assembly, the conveyor moves cartons or tote pans to the first zone. They either stop or are diverted for picking. If there is nothing to pick, they continue forward to the next zone. The primary advantage of pick-to-conveyor systems is a significant increase in picking productivity due to less travel of the picker between picking locations. The disadvantage is the need for a substantial investment on downstream sortation system, particularly in zone picking with downstream sortation. It could be effectively used in the picking area with heavy demand.

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(3) Order Picker Trucks An order picker truck is an industrial truck, typically for use in narrow aisle order picking. It is composed of an extendable mast, a long load platform, a stand-up operator platform, and a load former. The operator platform moves vertically up and down along with the load platform, which facilitates manual loading and unloading directly from the storage rack. Order picker trucks in a warehouse are typically used for case picks from pallet racks in order to fill a customer order. Figure 1.6 shows an application of order picker trucks. An advantage of the order picker truck is that the picker can travel to pick locations well above floor level. It is able to lift loads up to 40 feet, but becomes unstable at the maximum height. Further, continued movement of the load platform, such as vertically to a different elevation or horizontally as the order picker truck travels down an aisle, can cause an unstable load to fall off the load platform, which may damage or destroy the load. Special care must be taken in positioning the order picker truck in front of pick location. Consequently, the picking productivity is relatively low. Order picker trucks are usually used where high storage density is required but demand to the system is low. An automated version of the order picker truck is the person-aboard automated storage/retrieval machine. The difference is that the storage/retrieval machine is running on the fixed path, mounted on rails along the top and bottom of each storage aisle. Thus, the unstability and positioning problems associated with order picker trucks are greatly mitigated. Typically, it is aisle-captive and driven by automatic control. It could be used for retrieval from static shelving, storage cabinets, or pallet racks. Again, the storage/retrieval machine travel is simultaneous in the horizontal and vertical directions. There are a wide variety of applications for person-aboard automated storage/ retrieval systems. Consequently, the picking productivity covers a wide range, depending on the application. A person-aboard automated storage/retrieval system can offer significant floor-space savings. But, it is fairly expensive and inflexible. Its speed is also limited since the picker still rides on board. The system might be used where high storage density is required but demand to the system is relatively low.

1.3.2 Stock-to-Picker Systems In stock-to-picker systems, sometimes referred to as end-of-aisle systems, the retrieval equipments are typically integrated with storage equipments to become an automated storage/retrieval system. Consequently, usual stock-to-picker systems act as a modular subsystem of the whole warehouse system. Yet, the picker, human or robot, interacts with the system in order to complete orders. Here, we will describe three most popular stock-to-picker systems: carousels, miniload systems, and A-frame systems.

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Fig. 1.6 Application of order picker trucks (Courtesy of Hyster Company)

(1) Carousel Systems Carousel systems are becoming popular in use for order picking and workin-process storage/retrieval applications. Combined with the state-of-art technology such as computer control and auto-inserter/extractor mechanism of bins to/from the storage rack, carousel systems meet well the industry’s need for

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Fig. 1.7 Application of horizintal carousels (Courtesy of Integrated Solutions)

automated handling of lean inventory and high throughput. They are substantially less expensive but more flexible than other highly productive alternatives such as miniload systems. There are several types of carousels used in the industry. A horizontal carousel is simply a linked series of tiered shelves that house bins. The drive mechanism rotates the carousel horizontally in a closed loop such that the shelving section housing the requested bin is automatically lined up against the picker at the input/ output station. A vertical carousel, instead, rotates vertically. A rotary rack is a carousel with independent rotating shelves and much like multiple one-level horizontal carousels stacked on the top of one another, each level rotating independently. Figure 1.7 shows an application of horizontal carousels. Carousels can be configured to accommodate storage bins of various sizes. Typically, a picker runs a pod of 2 to 4 carousels at a time in order to keep the picker busy. Pick-to-light systems are often integrated into carousels. Picking is usually performed in batches with orders downloaded from the host system. Carousels provide very high pick rates as well as high storage density. Horizontal

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Fig. 1.8 Miniload system (Courtesy of Daifuku Co., Ltd.)

carousels are most common in picking operations with very high number of orders, low to moderate picks per order, and low to moderate picks per sku. (2) Miniload Systems Miniload automated storage/retrieval systems are used to store/retrieve such small loads as small parts and tools that can be stored in a storage bin. A typical miniload system comprises of multiple aisles of storage racks, a storage/retrieval machine operating in each aisle, numerous modular storage bins for housing items, and two load stands (or pick positions) at the end of each aisle to facilitate order picking. The load stands are arranged such that each aisle has a ‘left’ and a ‘right’ pick position. While the order picker is extracting items from the bin in one pick position, the storage/retrieval machine returns the bin in the other pick position to a location in the rack, and returns with the next bin to be processed. Figure 1.8 shows a miniload system. In miniload systems, an automated storage/retrieval machine is mounted on rails along the top and bottom of each storage aisle. The mast of the storage/ retrieval machine moves horizontally along the aisle, while a carriage vertically up

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Fig. 1.9 A-frame dispenser systems (Courtesy of Cornerstone Automation Systems, Inc.)

and down the mast. Since there are separate motors for the horizontal and vertical directions, the travel time between any two rack locations is the maximum of horizontal and vertical travel times, which is sometimes called the Tchebychev metric. A miniload system can provide excellent space utilization, accurate picking and security for facilities that store and handle many thousands of small-parts. Doubledeep racks can increase space utilization even more. But, it is a highly sophisticated system. Consequently, it is inflexible, and requires extremely high initial investment. High maintenance requirements are another drawback of this system. To be effective, this system should be well designed to be able to keep the balance between space utilization and throughput capacity. (3) A-Frame Dispenser Systems An A-frame dispenser system is an automated order picking system used for high-speed, high throughput order picking of small and well packaged individual items with uniform size and shape. It consists of a set of dispensers in two rows configured as an A-frame. Typically, a conveyor runs through under the A-frame. Figure 1.9 shows A-frame dispenser systems. In general, there are two approaches to filling an order. In belt conveyor application, the conveyor is divided into sections, each of which corresponds to an

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order. To fill an order, the dispensing mechanism under computer control removes the bottom item from the dispencer onto a corresponding section of the conveyor as it runs through. The complete order is picked when the conveyor section exits the A-frame. In roller conveyor application, storage bins onto a roller conveyor are used to fill orders as they pass beneath the A-frame. For higher volume items, multiple dispensers are assigned to each item, while a single dispenser is assigned to each of the slower moving items. Each dispenser is adjustable for a vast array of item sizes. The vertical stack design allows the dispensers to be replenished from the back while the systems operates. Typically, each dispenser is equipped with low level and out of stock sensors as well as with a replenishment indicator to show dispensers most in need of replenishment. An A-frame dispenser system can provide the highest picking rate and accuracy among all the order picking alternatives. However, it is expensive, inflexible, and requires high maintenance. The system is most popular in high volume facilities with a small number of skus in high volume, such as pharmaceutical distribution centers.

1.4 Performance Model Here, we will introduce a basic performance model for use in determining the system throughput capacity. The model is developed for carousel systems with remote picking stations and miniload systems with two load stands, described in Sect. 1.3.2. The determination of system performances is critical, especially in the early design phase of a system. Given the basic data on system components and cost factors, estimating system performances without expensive analysis can greatly help the system designer quickly narrow down alternative system configurations. Here, we develop a simple, analytical tool for use in determining system throughput capacity, which could be used as a first-cut solution in the system design. It could also enable the system designer to perform easy sensitivity analysis.

1.4.1 Carousel Systems with Remote Picking Stations 1.4.1.1 System Description Consider a ‘‘remote’’ order picking system where several carousels are linked to remote picking stations by a powered circulating conveyor. The carousel consists of several tiered, open shelving sections that house individual containers (bins or tote pans). This section moves in unison whenever the carousel rotates. Containers are bar code labeled for easy identification. A computerized inventory control

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system keeps track of where each container is located on the carousel and how many items are stored in each container. In this system, the processed container at the picking station returns to the carousel for storage. Thus, the whole system forms a closed loop. In this system, an auto-inserter/extractor located at the input/output station performs storage and retrieval operations. A shuttle is mounted on the mast of the auto-inserter/extractor, and moves vertically up and down the mast between the tiers of the carousel and the level of the conveyor. Actual extraction/insertion of the container in the rack and drop-off/pick-up of the container at the input/output station are performed by this shuttle that can hold two containers. Thus, the shuttle is the storage/retrieval machine in this system. The operations of a storage/retrieval machine can be classified by the operating mode. In carousel systems with remote picking stations, the operating mode can be classified into single and dual command cycle. A single command cycle consists of either a single storage or a single retrieval transaction in a trip away from and back to the input/output station. A dual command cycle involves both a storage and a retrieval transaction in a trip away from and back to the input/output station. During busy period, the shuttle executes both a retrieval and a storage in a cycle (dual command cycle) for maximum efficiency.

1.4.1.2 Assumptions We will limit the analysis to one carousel served by one auto-inserter/extractor, since a carousel integrated with the auto-inserter/extractor operates almost independently. Note that the circulating conveyor acts as a buffer that separates a carousel system from the rest of the system. For convenience, it is assumed that the drop-off and pick-up at the input/output station are performed at the ground level, i.e., at the lowest tier level of the carousel. We assume that the returned container takes the location of the container just extracted from. With this strategy, each container floats over the rack as time goes by. Consequently, all tiers are evenly utilized, and the resulting, long-run activity distribution is uniform over the rack face. In some systems, however, locations in tiers closer to the level of the conveyor can be utilized more by assigning fast moving containers to those locations (class-based storage). In other systems, a dedicated location can be assigned to each container (dedicated storage). But these strategies will require unnecessary travels between retrieval and storage locations: horizontal travels of the carousel and vertical travels of the shuttle. It is typical to assume that the carousel is bidirectional, i.e., can rotate in either the clockwise or the counterclockwise direction. The direction of rotation is typically selected such that the rotation time for the next request is minimized. It is also typical to approximate the carousel as a continuous loop rather than consisting of a large number of discrete locations. It is very convenient to normalize time data such that the maximum rotation time between any two locations on a carousel is one unit of time. All units of time data are relative to this

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maximum rotation time. We say that time data are expressed in normalized time unit. Note that it takes two normalized time units to do a full rotation for a bidirectional carousel. In modeling, system throughput is measured in terms of pick rate. In this system, the system‘s pick rate is expressed as the number of containers picked per unit time (hour). Thus, requests are formed on individual container basis. The last assumption is that all requests are processed on first-come-first-serve basis. This is rather an unusual assumption for picker-to-stock systems, since it is common in those systems that pick lines in a pick list are cleverly sequenced in order to reduce travel times. For carousel systems, it is relatively easy to sequence the pick lines in order to reduce carousel rotation times only. It is also easy to sequence the pick lines in order to reduce shuttle travel times only. But, sequencing the pick lines to reduce both carousel rotation times and shuttle travel times simultaneously is a complicated problem. Because of this, we will simply assume that all requests are processed on first-come-first-serve basis.

1.4.1.3 Throughput Capacity The system throughput capacity can be defined as the system throughput when there exist an infinite number of requests to the system. Consider a carousel system with an infinite number of retrieval requests to be processed. Suppose that the shuttle extracts a container from the carousel, rotates itself 180 degree, and inserts the returned container into the location just extracted from. Then, the carousel will start to rotate to line up the next container requested against the auto-inserter/extractor at the input/output station, during which the shuttle will travel down the mast to the level of the conveyor, drop off the container on the takeaway conveyor, pick up the container returned from the picking stations, and travel up the mast to the tier level housing the next requested container. If the carousel finishes its rotation first, it waits idle until the shuttle finishes its dual command travel. If the shuttle finishes first, it waits idle until the carousel finishes its rotation. When both the carousel and the shuttle finish, a new cycle begins. Let R denote the carousel rotation time, T1 the time for the shuttle to travel down the mast for drop-off/pick-up (retrieval travel), T2 the time for the shuttle to travel up the mast for extraction/insertion (storage travel), s the constant drop-off/ pick-up time at the input/output station, and d the constant extraction/rotation/ insertion time at the location in the carousel. Let T denote the dual command cycle time of the shuttle, i.e., T = T1 ? T2 ? s. Then, it is not difficult to show that the expected system cycle time E[Sc] is determined by E½Sc  ¼ E½maxfR; Tg þ d:

ð1:1Þ

The system throughput capacity is simply 1/E[Sc]. To determine the throughput capacity, we have to compute E[max{R, T}]. Let b be the longest shuttle travel time in normalized time unit between any two tiers

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on a carousel. Note that R and T are independent, R is uniformly distributed in [0, 1], and Ti, i = 1,2, is uniformly distributed in [0, b]. Hence, it is not difficult to determine the throughput capacity. Here is an example from Park (2008). Suppose we are going to design a new horizontal carousel system, where the carousel is served by an insertion/extraction robot and linked to remote picking stations by a powered circulating conveyor. An alternative system configuration under consideration is a horizontal carousel with length Lv ¼ 40 m, horizontal rotation speed Sh ¼ 1 m= s, height Lv ¼ 6 m, vertical shuttle speed Sv ¼ 1:2 m= s, the constant drop-off/pick-up time s = 1.5 s, and the constant extraction/rotation/insertion time d = 1 s. The longest shuttle travel time between any two tiers on a carousel is 6/1.2 = 5 s. In this system, the time-normalizing factor TN  Lh =2sh ¼ 20 s. By dividing by TN we can represent all the time data in normalized time unit. Thus, we have b = 5/20 = 0.25 normalized time units, s = 0.075 normalized time units, and d = 0.05 normalized time units. Then, we can show that E[Sc] = 0.608 normalized time units. The system capacity in terms of retrieval request is 1=E½Sc   1:645 transactions per normalized time unit (20 s). It is equivalent to 4.934 transactions per minute or 296 transactions per hour.

1.4.2 Miniload Systems with Two Load Stands 1.4.2.1 Throughput Capacity Expression The throughput capacity expression for carousel systems with remote picking stations is still valid for miniload systems with two load stands, but it must be used with some modifications. In miniload systems, a matter of concern is the interaction between the picker and the storage/retrieval machine, while the interaction between the carousel and the auto-inserter/extractor in carousel systems with remote picking stations. In miniload systems, a storage/retrieval machine performs dual command cycles during busy period. We will limit the analysis to one miniload aisle with two racks served by a dedicated picker. We assume that retrievals are processed on a first-come-firstserved basis and there is an infinite queue of requests. We also assume that requested bin locations are independently and identically distributed from each other. Therefore we are not allowing dependence induced by storage/retrieval sequencing, or by the fact that certain bins tend to be requested successively. While the order picker is extracting items from the bin in one pick position, the storage/retrieval machine picks up the bin from the other pick position, returns the bin to a location in the rack, travels empty to the next bin location, retrieves the bin, travels to the input/output point, and deposits the bin on the empty table. If the picker finishes before the storage/retrieval machine returns with the next bin, the picker waits idle. If the storage/retrieval machine returns with the next bin before the picker is finished, the storage/retrieval machine waits idle.

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Let P denote the pick time. The pick time is the length of time needed by the picker to process the bin. The processing activities may include extracting items, documenting tasks, counting and weighing items, interruptions, and restocking. Let D denote the variable portion of the dual command cycle time and c the constant portion of the dual command cycle time which includes pick-up and deposit time in pick positions. Let C denote the dual command cycle time such that C = D ? c. Then, the expected system cycle time is determined by E½Sc  ¼ E½maxfP; Cg;

ð1:2Þ

and the throughput capacity is 1/E[Sc] (Foley and Frazelle 1991). This expression is valid, irrespective of storage policy or rack shape, as long as the sequence of bin locations visited are independently and identically distributed over the rack face. To compute E[Sc], we should know both the pick time and dual command cycle time distribution.

1.4.2.2 Throughput Capacity Approximation The expression for the expected system cycle time is very simple, but the actual computation is fairly lengthy and cumbersome in almost all cases, if not impossible. It is not simple to develop the closed-form expression for the dual command cycle time, even for a uniform activity distribution over the rack face. Furthermore, exact expressions are not available in many cases, including turnover-based storage. An alternative approach is to use accurate bounds on or approximations to the throughput capacity. Here, we will present an accurate approximation to the throughput capacity of a miniload system. The following results are from Foley et al. (2004). Let k denote the pick rate of the picker, or k = 1/E[P]. Then, the expected cycle time E[Sc] can be approximated by m  E½C þ eE½C=k : _

ð1:3Þ _

Now the approximation to the throughput capacity is simply 1/m. Approximations _ to the picker utilization and the storage/retrieval machine utilization are 1/km and _ E[C]/m, respectively. These approximations are fairly simple to use, once the value of E[C] is known. The approximations developed perform very well for most actual pick time distributions. In particular, they are very accurate for exponential pick time dis_ tributions. In fact, the approximation 1/m is a tight upper bound on throughput capacity for exponential pick time distributions. Foley et al. (2002) showed that among the class of NBUE (new better than used in expectation) with a given mean, the exponential pick time distribution gives the lowest throughput capacity, while the deterministic pick time distribution gives the highest throughput capacity. A pick time distribution is NBUE if the expected

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remaining pick time is smaller than the expected pick time. Thus, all reasonable pick time distributions are among the class of NBUE. Note that assuming exponentially distributed pick times provides a conservative bound on the throughput capacity. Exponentially distributed pick times are reasonable for modeling a human picker, while deterministic pick time distributions are often used for modeling a robot picker. Here is an example from Foley et al. (2004) that shows how Eq. 1.3 can be used for determining throughput capacity. Assume the following miniload system. The rack is 24 feet high and 150 feet long. The storage/retrieval machine travels 240 FPM in the vertical direction and 600 FPM in the horizontal direction. The input/ output point is located at the lower left-hand corner of the rack. The (total) constant pickup/deposit time is 0.4 min. Finally, the picker’s average pick time per container is 3/4 min, i.e., the pick rate is 4/3 containers per minute or 80 containers per hour. For convenience, we will express all the time data in normailized time unit. To do this, let TV denote the time it takes the storage/retrieval machine to travel from the top to the bottom of the rack, let TH denote the time it takes the storage/ retrieval machine to travel from the front to the back of the aisle. Define timenormalizing factor TN such as TN = max{TV, TH}. The rack shape factor b is defined as b = min{TV, TH}/TN. When b = 1, the rack is said to be squarein-time; otherwise, it is called non-square-in-time. Note that a square-in-time rack does not necessarily mean a physically square rack. Since max{150/600, 24/240} = 0.25, the time-normalizing factor TN = 0.25 min while the rack shape factor b = (24/240)/(150/600) = 0.4. The pick rate of the picker k = (TN)(4/3) = 1/3 per normalized time unit, and c = 0.4/ TN = 1.6 normalized time units. For simplicity, let us consider a uniformly distributed rack. For a uniformly distributed, time-normalized rack, it is well known from Bozer & White (1984) that the variable portion of the expected dual command cycle time is E½D ¼  b3 =30 þ b2 =2 þ 4=3:

ð1:4Þ

Since b = 0.4, E[D] = 1.4112 normalized time units. Hence, E[C] = 1.4112 ? 1.6 = 3.0112 normalized time units. Substituting this value in Eq. 1.3, we have _ the throughput capacity 1/m = 0.2433 containers per normalized time unit. By dividing this value by TN, the throughput capacity is 0.9733 containers per minute or 58.39 containers per hour. Since the average pick rate of the picker is 80 containers per hour, the picker utilization is 58.39/80 * 0.73 or 73%. Since E[C] = (3.0112)(0.25) = 0.7528 min, which is equivalent to 79.70 dual command cycles per hour, the utilization of the storage/retrieval machine is 58.39/79.70 = 0.73 or 73%. In this example, we consider only a uniformly distributed rack. To obtain E[D] for other storage policies, see Park et al. (2003) for non-square-in-time racks with turnover-based storage, and Park (2006) for non-square-in-time racks with twoclass storage. Miniload system configurations other than miniload systems with

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two load stands are also possible. See Bozer and White (1996) for uniformly distributed miniload systems with configuration of a horse-shoe front end or two or more aisles per picker. See Park et al. (1999) for the effect of buffer on system performance in miniload systems with horse-shoe front-end configuration, where the distribution of dual command cycle time is not specified so that the results are valid, irrespective of storage policy or rack shape.

References Bartholdi JJ III, Hackman ST (2010) Warehouse & distribution science: release 0.92, The Supply Chain and Logistics Institute, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Bartholdi JJ III, Platzman LK (1986) Retrieval strategies for a carousel conveyor. IIE Trans 18(2): 166–173 Bozer YA, White JA (1984) Travel-time models for automated storage/retrieval systems. IIE Trans 16(4):329–338 Bozer YA, White JA (1996) A generalized design and performance analysis model for endof-aisle order picking systems. IIE Trans 28(4):271–280 Foley RD, Frazelle EH (1991) Analytical results for miniload throughput and the distribution of dual command travel time. IIE Trans 23(3):273–281 Foley RD, Frazelle EH, Park BC (2002) Throughput bounds for miniload automated storage/ retrieval systems. IIE Trans 34(10):915–920 Foley RD, Hackman ST, Park BC (2004) Back-of-the-envelope miniload throughput bounds and approximations. IIE Trans 36(3):279–285 Frazelle EH (2002) World-class warehousing and material handling. McGraw-Hill, New York Hackman ST, Frazelle EH, Griffin PM, Griffin SO, Vlatsa DA (2001) Benchmarking warehousing and distribution operations: an input-output approach. J Prod Anal 16:241–261 Litvak N (2006) Optimal picking of large orders in carousel systems. Oper Res Lett 34(2):219–227 Park BC (2006) Performance of automated atorage/retrieval systems with non-square-in-time racks and 2-class storage. Int J Prod Res 44(6):1107–1123 Park BC (2008) Performance of carousel systems with remote picking stations. Progress in material handling: 2008. The Material Handling Institute, Charlotte, pp 458–466 Park BC, Lee MK (2007) Closest open location rule under stochastic demand. Int J Prod Res 45(7):1695–1705 Park BC, Frazelle EH, White JA (1999) Buffer sizing models for end-of-aisle order picking systems. IIE Trans 31(1):31–38 Park BC, Foley RD, White JA, Frazelle EH (2003) Dual command travel times and miniload system throughput with turnover based storage. IIE Trans 35(4):343–355 Park BC, Foley RD, Frazelle EH (2006) Performance of miniload systems with 2-class storage. Eur J Oper Res 170(1):144–155 Ratliff HD, Rosenthal AS (1983) Order picking in a rectangular warehouse: a slovable case of the traveling salesman problem. Oper Res 31(3):507–521 Tomkins JJ, White JA, Bozer YA, Frazelle EH, Tanchoco JMA, Trevino J (1996) Facilities planning, 2nd edn. John Wiley & Sons, Inc., New York van den Berg JP (1996) Multiple order pick sequencing in a carousel system: a solvable case of the rural postman problem. J Oper Res Soc 47(12):1504–1515

Chapter 2

Storage Systems and Policies Marc Goetschalckx

Abstract A storage system is an engineered system with the function to store materials, in other words to hold materials until they are needed. These materials come in an enormous variety from consumer products such as TVs in local distribution centers, emergency drug doses for battling a biological attack on a city, vertical storage silos for grain, to the strategic reserve of main battle tanks parked in the dessert. Storage systems are an essential component of virtually every supply chain. While most storage systems are stationary, some are moveable such as the truck of a repair man that holds a ready inventory of service parts, a bunker ship that resupplies a navy fleet, an intercity long-haul truck, or an ocean-going intermodal container ship. Irrespective of the type of material, the geographical location of the storage system, or the size of the storage system, storage systems have three main processes: put-away, holding, and retrieval. The first process put materials into storage; the second process holds the materials in a stationary position inside the storage system, and the third process removes materials from storage and is often also called order picking.

2.1 Introduction As any engineered system storage systems have structure and behavior. The structure concerns the physical form of the storage system, while the behavior is concerned with the management of the storage system. The performance of a storage system in its physical environment is based in part on the efficiency with which it executes its basic function. The performance is dependent on its design,

M. Goetschalckx (&) Georgia Institute of Technology, 765 Ferst Drive, Georgia 30332-0205, USA e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_2, Ó Springer-Verlag London Limited 2012

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which determines its structure, and on the management policies, which determine its behavior. Given the complexity and importance of storage systems, a large body of literature exists that describes engineering methodology to either determine or optimize the structure or behavior of storage systems. The performance of a storage system depends on four internal characteristics and their interrelations: (1) storage capacity or equivalently storage density; (2) ease of access to storage locations; (3) complexity of the internal structure; and (4) level of information technology. The performance of a storage system also depends on external characteristics such as number of products, type of products, total inventory to be stored, and type and balance of the input and output flows. External characteristics are always assumed to be given parameters in the following methodology. An example of how external product characteristics influence the storage system is a grocery warehouse with products in three temperature classes: ambient, refrigerated, and frozen. The products in one temperature section of the warehouse cannot be mixed with products in the other sections. An example of where the design methodology partitions the warehouse is a forward-reserve distribution center where the most commonly retrieved products are stored together in the forward area and the relatively rarely retrieved products are picked from the reserve area. Splitting the products in those two classes and the warehouse into two sections allows both sections of the warehouse to perform better than a joint warehouse, but a cost of increased internal complexity. In general, determining the number of sub warehouses under one roof of the overall warehouse is part of the warehouse design process. Such partitioning inside a single warehousing system is a common occurrence in practice. In the remainder of this research it is also assumed that all the products to be stored are compatible with respect to their physical characteristics such as temperature, weight, chemical type, and hazardousness. The focus of this paper is two fold. The first emphasis is on the design of the structure of a storage system, specifically determining the size or holding capacity of the system. This approach ignores many other important characteristics of the storage system such as safety, structural soundness, energy efficiency, and environmental impact. The second focus is on storage policies, which are the management policies that determine where an incoming item is placed in the warehouse. Even for this restricted problem, a single paper can only touch on one or more examples of storage systems. This paper focuses on unit load storage systems since the simplicity of their physical form allows for a clearer illustration of the underlying principles. The methodology is based on the construction of mathematical models of the storage system.

2.2 Literature Review Providing a comprehensive literature review of warehousing systems, storage systems, and storage policies is not possible given the large variety of systems and the large body of research. Earlier research reviews were compiled by Cormier and Gunn (1992) and Rouwenhorst et al. (2000). Some of the most recent reviews are

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given in Gu et al. (2005, 2010). Most of the results are concerned with modelling or optimizing the behavior of warehousing systems. A comprehensive, modelingbased engineering methodology for warehouse design does not yet appear to exist. It should be noted that the methodology presented here is independent of how the expected travel time to a storage location is computed, be it by physical observation, simulation, mathematical model, or closed form equation. There exists a large variety of research results to determine these travel times for various hardware and storage policy combinations.

2.3 Unit Load Storage Systems and Policies 2.3.1 Introduction In unit load warehouses it is assumed that all the items in the warehouse are aggregated into units of the same dimensions that can be moved, stored, and controlled as a single entity. Typical examples of unit loads are pallets, intermodal containers, and wire baskets. Furthermore, it is assumed that no incompatibilities exist between the materials to be stored. It is also assumed that all the storage locations are the same size and each location can hold any unit load. Hence unit load storage system eliminates many of the material, material-to-location, and location specifics and complications. One of the most common examples of a unit load storage system is the singledeep pallet rack. In this storage system the warehouse contains a number of parallel travel aisles; on each side of an aisle is a steel rack capable of holding a single pallet in each of its storage locations. The storage locations, also called pallet openings, are conceptually arranged in horizontal rows and vertical stacks or columns. The storage locations are completely inert, i.e. their location cannot be changed and the locations cannot move the unit loads stored in them. The putaway and retrieval operations are performed by a material handling truck such as a forklift truck or a turret truck. A single-deep unit load pallet rack storage system and its material handling truck are shown in the next three figures. Unit load systems can be further divided based on the access allowed to individual stored loads. Random access unit load systems allow access to all unit loads in all storage locations with the same amount of effort except for the travel to the storage location. Examples are the single-deep pallet rack, person-aboard tote stacker, and side-loaded intercity trucks (Europen model). If a unit load system is not random access, then the effort of accessing a load is different (again excluding the travel to reach the storage location). Examples are double-deep pallet rack, deep lane storage, block stacking, intermodal container storage yard, an intermodal container ship, or a rear-loaded intercity truck (United States model). Reaching the unit load at the back end of a lane or at the bottom of stack requires much more effort than reaching the most accessible unit load since other unit loads may block the access.

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A second important characteristic of storage system is their command cycle. A warehouse is said to operate under single command when on each trip of the material handling truck or device a single operation is performed, be it either storing a load or retrieving a load. During a dual command cylce first a unit load is put away in the system and then a unit load is retrieved from the system. Finally, individual items such as individual cartons can also be retrieved from the unit load using a order picking truck. The order picking truck is said to operate under multicommand. In the latter storage systems, typically the unit load is stored by a different material handling truck operating in single command mode. Unit load systems exhibit the standard advantages and disadvantages of standardization and unitization. Some of the major advantages are the standardized material handling and storage operations and equipment, the reduced effort of control, and the high macro storage volume utilization. Macro storage space utilization is high because a storage location is never left unused or empty because a storage load does not fit. If the unit load storage system does not have hardware that holds the loads, then a loss of storage volume occurs when only a few unit loads remain in a lane or a stack but these few unit loads still consume the full space for that lane or stack. This loss is also called the honeycombing loss, see White and Francis (1971). Some of the major disadvantages of unit load storage systems are the effort required to assemble and disassemble the unit load, the cost of the containers to hold the unit load and the empty container movement, storage, and management, and low micro storage volume utilization. The low micro storage volume utilization is illustrated by the top pallet on the right-hand side of the aisle in Fig. 2.1; only a few boxes remain stored on the pallet and most of the volume of this storage location is empty (Figs. 2.2 and 2.3).

2.3.2 Single Command Operation Cycles Because of the elegant simplicity of the problem and because such unit load systems have been implemented frequently in practice, the problems of designing and managing unit load storage systems have been studied extensively in the literature, especially when the storage system hardware implementation is an automated storage and retrieval system (ASRS) for unit loads such as pallets. Most of the early results were for storage systems operating in a single command cycle mode, where on each trip of the material handling truck a single operation is performed, be it either storing a load or retrieving a load.

2.3.3 Individual Unit Load Storage Model A (unit load) storage model is a mathematical optimization model that determines in which storage location the (unit) loads are to be held in order to minimize the material handling effort and cost of put-way and retrieval, while observing the

2 Storage Systems and Policies

35

Fig. 2.1 Single-deep pallet rack (photos courtesy of LogisticsCAD, used with permission)

storage capacity of the storage system. A storage policy is a management rule that makes the same determination. A storage policy may not generate the optimal storage plan, but it is easier to implement since it does not require the solution of an optimization model. Some of the earliest results were developed by Heskett (1963, 1964) and some the earliest mathematical models were developed in

36

M. Goetschalckx

Fig. 2.2 Parallel aisles of a pallet rack (photos courtesy of LogisticsCAD, used with permission)

Malette and Francis (1972) and White and Francis (1971). Hausman et al. (1976) studied the performance of storage system under dual command cycles, where on each trip of the material handling truck first a storage and then a retrieval operation are performed. Malette and Francis (1972) and Hausman et al. (1976) proved that for single command product turnover-based dedicated storage minimizes the total travel time required to execute all the tasks. In a series of articles Malmborg together with several co-authors, starting with Malmborg and Deutsch (1988), extended these results to dual command cycles.

2.3.3.1 Individual Load Storage Model Notation H

arri, depi

bi

The number of equal-sized periods in the planning horizon. The length of a period may range from minutes to months depending on the application. Arrival and departure period of individual unit load i in the storage system; the unit load occupies a storage location starting with the arrival period and up to and including the departure period. Residence vector of individual unit load i in the storage system, this vector has elements equal to one from the arrival to the departure

2 Storage Systems and Policies

37

Fig. 2.3 Turret truck accessing a unit load (photo courtesy of LogisticsCAD, used with permission)

cij xij

period of the unit load i and elements zero anywhere else. Note that the residence vector exhibits the consecutive ones property, i.e. all nonzero elements are equal to one and they appear in a single contiguous section of the vector. The dimension of the vector is H. This vector is sometimes also called occupancy vector. Expected one-way travel time or cost for a unit load i stored in location j. Decision variable equal to one if unit load i is stored in storage location j.

2.3.3.2 Factoring Condition If the cost of the material handling operation for a location is independent of the stock keeping unit (SKU) stored in that location, then the cost elements in the objective function can be simplified and can be computed in advance of the solution of the model. If this travel independence or factoring condition is satisfied, then it is assumed that all the items in the warehouse have the same probability mass function for selection of a dock or input/output point. The travel independence condition is equivalent to

38

M. Goetschalckx

ppk ¼ pk 8p or ej ¼ epj

8 p:

ð2:1Þ

The expected one-way distance for each location can then be computed as X pk  cjk ð2:2Þ ej ¼ k

The travel independence condition was first specified by Malette and Francis (1972) under the name factoring condition, since, if this condition is satisfied, the expected travel time to a particular location holding a particular product can be factored or computed as the product of the expected travel time of the location and the frequency of access of the product. In practice it is most often satisfied if the storage system has only one input/output point, e.g. a single aisle in an automated storage and retrieval system, or if all SKUs follow the same path through the warehouse, e.g. the receiving dock is on one side of the facility and the shipping dock is at the opposite side.

2.3.3.3 Individual Load Storage Model M X N X

min

ð2:3Þ

4cij xij

i¼1 j¼1 N X

s.t.

xij ¼ 1

8i

ð2:4Þ

8j

ð2:5Þ

j¼1 M X

bi xij  1

i¼1

xij 2 f0; 1g

ð2:6Þ

The objective is to minimize the access cost associated with the storage of an item in particular storage location. Constraint (4) ensures that every unit load is stored exactly once. Constraint (5) ensures that the residences of unit loads do not overlap for a particular location since the location has a capacity of a single unit, i.e. multiple items can be stored subsequently in the same storage location but not at the same time. The individual load storage model is a vector assignment problem (VAP). The VAP is an extension of the standard assignment problem (AP), where the constraints that prohibit sharing of the storage location are not for a single time period but for a vector of time periods. Since the occupancy vector for every item exhibits the consecutive ones property, the vector of constraints (5) can be converted into network flow constraints by the following transformation.

2 Storage Systems and Policies

39

The transformation has two steps. Recall that there are H time periods in the planning horizon. 1. Add a row of zeroes corresponding to the flow balance constraint of node or period H ? 1. Add a zero element for row H ? 1 to the right-hand size vector. 2. Execute iteratively the following linear row operation (including the right-hand size column) for r = H down to 1 row½r þ 1 ¼ row½r þ 1  row½r : Each transformed column of the occupancy vector now contains a single positive and negative one, indicating that this column corresponds to a directed arc in a network with (H+1) nodes. The positive one occurs at the arrival period and the negative one occurs at the period following the departure. The positive one corresponds to the start node of the arc in a network with (H+1) nodes and the negative one corresponds to the terminal node of the arc. The right-hand side vector has been transformed into a vector with a positive one in the first period (row) and a negative one in the last row, which corresponds to period H ? 1. The VAP has been transformed into problem with the block diagonal structure shown in the following figure and where each block column corresponds to a storage location and matrix B corresponds to the directed arcs in the network. 2

I 6B 6 A¼6 60 4 :: 0

I 0 B :: 0

I 0 0 :: 0

:: :: :: :: ::

3 I 07 7 07 7; :: 5 B

Unlike the AP the VAP does not posses the integrality property, so constraint (6) must be explicitly enforced and the individual load storage model requires a mixed integer programming solver. But its linear relaxation is very close to the binary formulation and many contemporary solvers recognize the special block diagonal structure of the problem and the network structure of each block row. Large instances can be solved relatively quickly. For even larger problem instances Lagrangean relaxation or other decomposition techniques can be used to reduce the problem size and the required memory for the problems that have to solved as a whole by the solver, but this increases the solution time. The model is thus useful for storage systems where a modest number of unit loads are to be stored during the planning horizon. However, for standard warehouses, the instance size makes solving the model with mathematical programing solvers impossible. Consider the example where there are 20,000 storage locations and 100,000 unit loads need to be stored during the planning horizon, the number of variables for this instance would be 2,000,000,000. The storage problem needs to be aggregated to yield a smaller and solvable problem instance.

40

M. Goetschalckx

2.3.3.4 Individual Load Storage Policy A heuristic storage policy for individual unit loads can be determined if the expected travel time for a storage location is independent of the unit load that is stored in that location. A warehouse that satisfies this condition is said to satisfy the factoring condition. The storage policy sorts all the unit loads to held during the planning horizon by increasing departure time and sorts all storage locations by increasing expected travel time. The storage policy then assigns the unit loads by increasing index to the storage location with the lowest index that feasibly hold the unit load, i.e. the storage location that is empty during the residence time of the unit load. Since the storage policy only requires two sorting operations it can be applied to large problem instances. Since it is a heuristic policy no information on its performance for a particular case is known, either with respect to the required warehouse size or with respect to the travel cost.

2.3.4 Aggregate Unit Load Storage Model Aggregation of the individual load storage model is based on the assumption that the requirements on the storage system are stationary. The most common assumption is that each SKU has a constant demand rate over the planning horizon and is replenished periodically in batches of identical quantities. Stochastic behavior that differs from the deterministic mean values is assumed to be absorbed by the safety stock for that SKU. The most common aggregation of individual unit loads is to consider the unit loads as inventory of a particular product or SKU. This is called product based aggregation. Two unit loads of the same SKU are always treated identically. The corresponding storage model is then based on product characteristics and the resulting solution yields product based storage policies. This product based aggregation is shown next. However, the individual unit loads can also be aggregated based on how long they stay or reside in the system, i.e. how long they occupy their storage location. This duration-of-stay based aggregation is developed later on. One particular duration-of-stay based storage policy that divides the unit loads into two classes, i.e. the short-term-residents and the long-term-residents, is better known as cross docking. In cross docking the short-term-resident unit loads are never placed or held in the main storage area but moved directly from receiving to shipping or moved from receiving into a short term order picking area and from there to shipping. 2.3.4.1 Storage Model Notation Ipt Npol

On-hand inventory of product p during time period t expressed in unit loads. Required number of storage locations in the warehouse when using a particular storage policy such as dedicated, shared, or maximum, which is indicated by the subscript.

2 Storage Systems and Policies

qp sp rp rpk ppk

fp ej cjk epj Tp

41

Replenishment quantity of product p in unit loads, also called the cycle inventory of product p. Safety inventory quantity of product p in unit loads. Demand rate for product p. Demand rate for product p entering or leaving through warehouse dock k. Probability for a unit of product p to enter or leave through warehouse dock k. If the product enters through one dock on one side of the warehouse and departs through another dock at the opposite site, then each of those docks would have a probability equal to 0.5. Frequency of access of a location in zone p or assigned to product p. Expected one-way travel cost to location j. Travel cost to location j from warehouse dock k Expected one-way travel time or cost for a unit load of product p stored in location j. Total travel time or cost for product p or zone p during the planning horizon.

By definition, the following relationships exist since the storage of one unit load requires two material handling operations. rp ¼

1X rpk rpk and ppk ¼ 2rp 2 k epj ¼

X

ppk  cjk :

ð2:7Þ ð2:8Þ

k

2.3.4.2 Warehouse Sizing Problem The storage policies partly determine the behavior of the storage system but also impact the required warehouse size, which is an element of the structure of the storage system. In other words the storage policy which is a management policy must be taken in consideration when designing the structure of the warehouse. Consider the class of product dedicated storage policies where each product is assigned a dedicated section of the warehouse. Such policies are often implemented because of their simplicity. Under such policies a particular SKU will always be located in the same section of the warehouse which facilitates put-away and order picking operations and inventory management. However, each section dedicated to a product must be large enough to hold the maximum inventory of that product during the planning horizon. The required warehouse size for this policy is the largest of all the warehouse sizes required by any policy. X  X   NDED ¼ sq þ qp ¼ max Ipt ¼ NMAX : ð2:9Þ p

p

t

42

M. Goetschalckx

Now consider a storage policy which assigns an arriving unit load randomly to one of the open storage locations of the warehouse. This policy is called random storage and denoted by RAN. A similar policy assigns an arriving unit load to the open location with the lowest expected travel cost for that product. This policy is called closest open location and denoted by closest open location (COL). Since no internal restriction is placed on the location of a unit load, the warehouse must be large enough to hold the maximum aggregate inventory over the time horizon. This required warehouse size is the smallest warehouse size among all the storage policies. ( ) X Ipt ¼ NMIN : ð2:10Þ NRAN ¼ max t

p

A warehouse is said to be balanced if the aggregate number of arriving unit loads is equal to the aggregate number of departing unit loads for every time period. In a balanced warehouse the aggregate number of unit loads stored remains constant. The smallest possible warehouse size for a balanced warehouse is the average aggregate inventory. If the warehouse were to operate without safety inventory for any product and if the classic triangular inventory pattern is assumed then the average aggregate inventory is equal to half the sum of the maximum product inventories. The policy warehouse size ratio of the required warehouse size to the maximum required size under dedicated storage is denoted by a. Alpha has a range of [0.5, 1]. The upper bound is achieved by any product dedicated storage policy since they all require the maximum warehouse size. The lower bound is achieved by any storage policy that fully shares the storage locations among the unit loads without any restrictions and when the warehouse is balanced and when no safety inventory is stored. Both random storage and closest open location are examples of such fully sharing policies. The real value of a can most easily be determined by simulation. It depends on the function of the warehouse and daily or seasonal material flow patterns. A conservative choice for the value of a is 0.85 or higher. a¼

N NDED

2 ½0:5; 1:

ð2:11Þ

The warehouse balance b can then be computed as b ¼ 2ð1  aÞ a¼1

b 2

ð2:12Þ ð2:13Þ

The warehouse balance indicates how well balanced the input and output flows of the warehouse are. A value of a ¼ 1 or b ¼ 0 indicates that the flows are not balanced at all. A value of a ¼ 0:5 or b ¼ 1 indicates that the flows are perfectly balanced.

2 Storage Systems and Policies

43

Fig. 2.4 Inventory patterns for the warehouse example with four products

Consider the following warehouse example. The warehouse stores four products A, B, C, and D with replenishment quantities equal to 5, 4, 3, and 2, respectively. Each product has a safety inventory of one unit load and a demand rate of one unit load per time period. The inventory patterns for the individual products are shown in Fig. 2.4. The aggregate inventory is indicated by All. The required warehouse size by a perfectly sharing storage policy is equal to the maximum of the aggregate inventory, which is equal to 16 and indicated by Sha. The required warehouse size by a product based dedicated storage policy is the sum of the maximums of the product inventories, i.e. the sum of 6, 5, 4, and 3, respectively, which is equal to 18 and indicated by Ded. The warehouse size ratio for this example is then 16 ¼ 0:889 a¼ 18 b ¼ 2ð1  0:889Þ ¼ 0:22 Observe that for this particular example the average inventory is 11 and the minimum inventory during the time horizon shown in the figure is five units (Fig. 2.4).

2.3.5 Product-Based Storage Model Product based storage policies determine the storage location of an arriving unit load based on the characteristics of the product or SKU this unit load holds.

44

M. Goetschalckx

Product characteristics that are commonly used are the demand rate, the maximum inventory over the planning horizon, or the turnover ratio. Product based storage policies also partition the storage locations into a number of sections each of which is dedicated to hold the unit loads of a group of SKUs. The numbers of sections that are commonly used are two, three, or the number of products when each section hold unit loads belonging to a single SKU. A very common product based storage policy is to divide the products in fast, medium, and slow movers depending on their demand rate. 2.3.5.1 Product Dedicated-Based Storage Model If the warehouse is partitioned in sections that each are dedicated to hold unit loads of a single SKU then the following storage model determines the optimal product dedicated storage policy. Note that e indicates the one-way travel cost, so it has to be doubled for the trip to either store or retrieve the unit load. Each unit load that is held in a particular storage location requires two material handling operations, one to store the load and one to retrieve the load. The frequency of access f of a storage location is thus twice the number of times this location is used to hold a unit load. Most warehouses operate on a first-in first-out (FIFO) basis with respect to the unit loads of a particular SKU to avoid spoilage and obsolescence of the product held in the unit load. This implies that there is no difference between the storage locations that hold safety stock and cycle stock for that SKU since in the long run they all will be accessed the same number of times. The frequency of access of a storage location can then be approximated by the following expression. fi ¼

2ri qi þ s i

ð2:14Þ

M X N  X  2fi eij xij

Min

ð2:15Þ

i¼1 j¼1

s:t:

N X

xij ¼ ðqi þ si Þ

8i

ð2:16Þ

j¼1 M X

xij  1

8j

ð2:17Þ

i¼1

xij  0

ð2:18Þ

In the product dedicated storage model, the cost of a material handling operation depends both on the storage location and on which SKU is stored in that location. In practice this often occurs when the different SKUs have different interface patterns with the storage systems docks. For example, some SKUs may

2 Storage Systems and Policies

45

arrive by truck while other SKUs arrive by rail and all SKUs depart through the truck shipping dock. 2.3.5.2 Product Turnover Storage Policy If the storage system satisfies the travel independence or factoring condition, then the travel cost to or from a storage location can be computed without knowledge of which unit load will be stored in this location. This computation can occur outside the model. Min

M X N  X

 2fi ej xij :

ð2:19Þ

i¼1 j¼1

This objective can be minimized by sorting the products by decreasing frequency of access and the storage locations by increasing expected access cost. The storage policy then iteratively assigns a number of storage locations equal to the maximum inventory of the next product to the next storage locations. If in the frequency of access vector each product is represented by a number of elements equal to its maximum inventory, then the objective value is equal to the inner product of the frequency of access and expected roundtrip travel cost vectors. This storage policy is known as product dedicated storage and also denoted as the cubeper-order index (COI) by Heskett (1963, 1964). It has been studied extensively since then. It is the optimal storage policy among the class of product dedicated storage policies with respect to the travel time or cost provided the factoring condition is satisfied. But because it belongs to the class of product dedicated storage policies it also requires the maximum warehouse size to store the unit loads. The conditions for the optimality of the product turnover-based storage or COI have not always been listed explicitly, especially in the material handling trade literature. This has led to some confusion on the optimality of this policy. 2.3.5.3 Two Class Product Turnover Storage Policy If the storage policy allows sharing of the storage locations among the different SKUs then the required warehouse size can be reduced. This reduction may outweigh the increase in travel times because the unit loads are no longer perfectly ordered. This trade-off also has been repeatedly studied, mostly by simulation based analysis; see for instance the original paper by Hausman et al. (1976) and Goetschalckx and Ratliff (1990). The warehouse is divided into a number of sections dedicated to a group of SKUs. Each section is called a (storage) zone. The group of SKUs assigned to a single zone is called a (product) class. Most often SKUs are assigned to a product class based on their frequency of access, but assignment based on the demand rate is also commonly used. The most common numbers of classes used in practice range from two to four. Inside a storage zone, the unit loads of the different SKUs that belong to this class are assumed to be stored randomly.

46

M. Goetschalckx

A fundamental problem is to determine the size of each zone in function of the products that are to be stored in this zone. If the replenishment patterns of the SKUs are correlated then sharing of the storage locations may generate only a limited reduction of the required warehouse size. Typically, it is assumed that the replenishment patterns of the SKUs are independent processes. In that case, sharing the storage locations can take advantage of the fact that during a particular time period some SKUs will have a high inventory will other SKUs will have a low inventory. If the replenishment patterns are independent, statistical analysis can be used to determine the probability that the storage zone size will be sufficiently large to hold the unit loads of the product class. When more unit loads of product class are to be stored than fit in the corresponding storage zone, it is assumed that the excess unit loads will be stored in the next storage zone. Assume initially that no safety inventory of the SKU is stored. A constant demand rate and an instantaneous replenishment batch are also assumed. The cycle inventory for that SKU is then uniformly distributed over time between q down to one unit. The inventory pattern is shown in the next figure. Note that the expected value and variance of a uniformly distributed random variable between the boundary values a and b is equal to bþa 2

ð2:20Þ

ðb  aÞ2 : 12

ð2:21Þ

x ¼ r2 ¼

Assuming there is no safety inventory present, the mean and standard deviation of the inventory of product p is then (Fig. 2.5) Ip ¼ qp þ 1 ð2:22Þ 2   qp  1 ð2:23Þ rp ¼ pffiffiffiffiffi 12 Based on the central limit theorem, if the class K contains a reasonable large number of products, the total inventory in class K is normally distributed with mean and standard deviation equal to X IK ¼ Ip ð2:24Þ p2K

rK ¼

sffiffiffiffiffiffiffiffiffiffiffiffi X ffi r2p :

ð2:25Þ

p2K

The required zone size dedicated to a particular class can then be determined given the acceptable probability that the zone will be full when a unit load of the class arrives. Let a be the maximum acceptable probability that a zone will be full, then

2 Storage Systems and Policies

q

Inventory

Fig. 2.5 Inventory patterns for a constant demand rate

47

1

Time

  x  x z a r

ð2:26Þ

ZK ¼ IK þ z  rK :

ð2:27Þ

P

If the size of the storage zone is computed with the formulas above, then a percent of the accesses is executed with the expected travel of zone k and (1 - a) percent is executed with the expected travel of zone k ? 1. It is assumed that no unit load will have to be located in a zone more than one removed from its home zone. The performance of the storage system depends on the interactions of the number of classes, the size of the corresponding zones, the number of SKUs in the warehouse, the size of the replenishment batch sizes of the SKUs, and the balance or correlation of the input and output flows of different SKUs. Simulation appears the only analysis method than can accommodate all of these factors to predict the warehouse performance. Travel time models for specific hardware configurations such as an ASRS rack with I/O station in one corner have been developed.

2.3.6 Residence-Based Storage Model The second major type of aggregation for individual unit loads is based on the duration of their residence in the storage system, which is also called their duration of stay. Again it is assumed that the warehousing processes are stationary.

48

M. Goetschalckx

The aggregation based on residence time is based on the observation that the first and last unit loads of a replenishment batch for a particular SKU have a very different behavior in the warehouse. Assume again initially that there is no safety stock for this SKU. The first unit is put away and almost immediately removed from the storage system and thus spends most of its residence time in the warehouse in motion. The last unit is put away and remains in storage until the whole replenishment batch has been removed through the demand process and then finally is removed. This unit spends most of its time in the warehouse immobilized in storage. To exploit this difference, the storage policy for the first unit should emphasize movement costs while the storage policy for the last unit should emphasize storage efficiency. Two external factors impact the performance of duration-of-stay-based policies. First, the larger the replenishment batch size, the more different the first and last unit load of a batch are and the more impact duration-of-stay policies will have. Second, the more safety inventory for an SKU is held, the more similar the first and last unit load of batch will behave because of the FIFO requirement and the least impact duration of stay storage policies will have. Two cases can be further distinguished. In the first case each storage zone is exclusively occupied by unit loads that the same duration-of-stay. In the second case storage zones are assigned to a group of duration of stays.

2.3.6.1 Duration-of-Stay Storage Policy for Perfectly Balanced Warehouses nDOS ðtÞ zDOS

Number of unit loads arriving during time period t that have a durationof-stay equal to DOS periods. Size of the zone, expressed as a number of storage locations, reserved for the storage of unit loads with duration of stay equal to DOS periods

Previously it was shown that a balanced warehouse will require the smallest possible warehouse size to hold all the unit loads during the planning horizon. nin ðtÞ ¼ nout ðtÞ

8t

ð2:28Þ

For a perfectly balanced warehouse, the dedicated duration-of-stay storage policy will both required the smallest possible warehouse size and simultaneously minimize the transportation time or cost. However, the satisfaction of the requirement that the warehouse is perfectly balanced is exceedingly rare in real warehouse operations, so the dedicated duration-of-stay storage policy and its corresponding performance is more to be considered as a theoretical limit rather than an actually attainable performance. Finally, it should be observed that a perfectly balanced warehouse implies that the warehouse is also balanced. nDOS ðtÞ ¼ nDOS ðt þ DOSÞ

8 t; 8 DOS

ð2:29Þ

2 Storage Systems and Policies

49

zDOS ¼

DOS X

nDOS ðiÞ

ð2:30Þ

i¼1

0 P TDOS

ej

1

1 4 4 Bj2ZDOS C zDOS tDOS ¼ zDOS @ ¼4 A¼ zDOS DOS DOS DOS

X

! ej

ð2:31Þ

j2ZDOS

The total required warehouse size and total transportation cost per period is compute by the following expression. NDOS ¼

X

zDOS

ð2:32Þ

DOS

T¼

X

TDOS

ð2:33Þ

DOS

The optimal storage policy among duration-of-stay based storage policies is to sort the storage locations by increasing expected travel time. The policy then assigns the storage zones by increasing duration of stay to storage locations by increasing travel time. Since this is again equivalent to the inner product of two vectors, this policy minimizes the travel cost. Since the warehouse is balanced and the policy maintains a constant level of inventory, the policy also minimizes the required warehouse size.

2.3.6.2 Duration-of-Stay Storage Policy Virtually no real world warehouse will satisfy the perfectly balanced condition. A more practical storage policy is to assign all unit loads within a range of duration of stay to a zone in the warehouse. This is similar to class turnover-based storage, but the criterion to divide the unit loads is their residence time rather than the frequency of access of their product. Again the performance of this storage policy depends on the interactions of the number of classes, the size of the corresponding zones, the number of SKUs in the warehouse, the skewness of the Pareto curve of the duration of stays of the unit loads, the size of the replenishment batch sizes of the SKUs, and the balance or correlation of the input and output flows of different SKUs. Simulation appears the only analysis method than can accommodate all of these factors to predict the warehouse performance. There is some evidence presented in Goetschalckx and Ratliff (1990) that policies with few zones and based on the product frequency of access perform slightly better than policies with few zones and based on duration of stay of unit loads.

50

M. Goetschalckx

2.3.7 Random Storage and Closest Open Location Storage Policies The simplest storage policy is the random storage policy since it uses no information about the unit load. It ignores both the product characteristics of the SKU to which the unit load belongs or the residence time characteristics of the unit load. Since no internal structure or partitioning of the storage locations is imposed, the random storage policy requires the smallest possible warehouse size of all storage policies. The COL storage policy is equivalent to the pure random storage policy if all locations in the rack are used. ! N X X 1 X 4fi ðsi þ qi Þ ej ¼ 4rit: ð2:34Þ T¼ N j¼1 i i Random storage and COL require the smallest storage size. A smaller storage size in turn reduces the travel cost to a particular location. However, under these policies the location of unit loads belonging to different SKUs is constantly changing in the warehouse. The reductions in travel associated with COL may be completely negated if the put-away or retrieval operations have to search for either an open storage location or a unit load belonging to a particular SKU. Maintaining an accurate inventory map that is accessible in real time to the material handling operators is an absolute requirement for the efficient operation of the storage system under these policies. This almost always requires a fully automated system such as an ASRS or a rack system with automated stacker cranes.

2.4 Conclusion Storage systems are ubiquitous elements of supply chains. Storage models and storage policies are essential components during the design of a warehouse in order to ensure it has the required capacity and during the operation of the warehouse to maximize its operational performance. The characteristics of storage models or storage policies are the clearest exposed in storage systems for unit loads, since many of the practical complexities of other storage systems are eliminated. This clarification is even stronger for random access unit load storage systems. For these systems an individual unit load storage model has been developed. But the problem instances of this model can only be solved for small storage systems. For practical sizes of storage systems, aggregation has to be applied so that the storage problem can be solved. This aggregation requires that the storage system has stationary process. However, the storage model is also useful to compute the relative performance of other policies.

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The two most prominent aggregation methods are either based on the product or residence time characteristics of the unit load. The optimal policy for full product dedicated storage policies is shown to be based on the product frequency of access also known as COI. However, the optimality of this policy requires the factoring condition and maximum warehouse size. If storage locations can be shared based on product characteristics, the required warehouse size is reduced. For random storage, closest open location, or product class-based storage (with few classes), this space reduction may be sufficient to overcome mixing the products assigned to a class. For aggregation based on duration of stay, in the extreme case of a perfectly balanced warehouse, the duration of stay dedicated storage policy is optimal with respect to both warehouse size and cost of the material handling operations. However, practical warehouse operations are far from perfectly balanced. When unit loads with similar durations of stay share a zone dedicated to them, the performance of the warehouse system is similar to the class based product turnover storage policies. There exist many directions for future research, but two appear to be particularly promising. The first direction extends the unit load storage policies to cases where the locations in the warehouse cannot be randomly accessed. Container yards and container ships for intermodal containers are very important examples of such system. The second direction is to extend analysis to conditions that incorporate more stochastic conditions than the current research which is mostly based on mean value analysis. Current methodology for such stochastic systems is almost exclusively based on simulation.

References Cormier G, Gunn E (1992) A review of warehouse models. Eur J Oper Res 58:1–13 Goetschalckx M, Ratliff HD (1990) Shared versus dedicated storage policies. Manage Sci 36(9):1120–1132 Gu J, Goetschalckx M, McGinnis L (2005) A comprehensive review of warehouse operation. Eur J Oper Res 177(1):1–21 Gu J, Goetschalckx M, McGinnis LF (2010) Research on warehouse design and performance evaluation: a comprehensive review. Eur J Oper Res 203(3):539–549 Hausman WH, Schwarz LB, Graves SC (1976) Optimal storage assignment in automatic warehousing systems. Manag Sci 22(6):629–638 Heskett JL (1963) Cube-per-order index: a key to warehouse stock location. Trans Distrib Manag 3:27–31 April Heskett JL (1964) Putting the cube-per-order index to work in warehouse layout. Trans Distrib Manag 4:23–30 Malette AJ, Francis RL (1972) Generalized assignment approach to optimal facility layout. AIIE Trans 4(2):144–147 Malmborg CJ, Deutsch SJ (1988) A stock location model for dual address order picking systems. IIE Trans 20(1):44–52 Rouwenhorst B, Reuter B, Stockrahm V, Van Houtum GJ, Mantel RJ, Zijm WHM (2000) Warehouse design and control: framework and literature review. Eur J Oper Res 122:515–533 White JA, Francis RL (1971) Normative models for some warehouse sizing problems. AIIE Trans 9(3):185–190

Part I

Manual Storage Systems

Chapter 3

Warehouse Layouts Goran Dukic and Tihomir Opetuk

Abstract Warehouse layouts, due to their influence on total warehousing costs, are of interest to the theory and practice of warehouse design. While the layout problem of unit-load storage area of conventional warehouses has quite a long history, the layout of conventional systems with manual order-picking from multiple aisles has been the topic of a number of research papers only in the previous decade. The research has resulted, among other things, in various models for optimal layout design. Moreover, some new innovative layouts for storage area have been proposed recently. These layouts result in a reduced travel distance needed to store or retrieve a single pallet, thus improving the efficiency in the storage area. However, the question of whether these layouts could perform better than traditional layouts in manual order-picking operations has remained unanswered. This chapter provides a short overview of optimal traditional layouts of the storage and order-picking area as well as new innovative storage area layouts, followed by results of the analysis of order-picking in these new innovative layouts and relevant conclusions.

3.1 Introduction It is well known that logistics costs have an important influence on the business success of any company. According to the logistics cost and service studies, these costs represent on average around 10% of sales in western companies. Constituting

G. Dukic (&)  T. Opetuk Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 1, 10000 Zagreb, Croatia e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_3, Ó Springer-Verlag London Limited 2012

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an important part of overall production cost, costs of logistics operations in industrial systems can play a vital role in determining the competitiveness of a company. The efficiency and effectiveness of the supply chain of a company are largely determined by operations performed in the nodes of such chains. Warehousing, along with transportation and inventory carrying, is one of the three major drivers of logistics costs. Since warehouses are in most cases non-avoidable places within the production site of industrial companies, and are also nodes in the distribution network towards final customers, proper warehouse planning and control have drawn full attention in the literature (Van den Berg 1999; Rouwenhorst et al. 2000; Gu et al. 2010). Warehousing costs are to a large extent already determined during the design phase. Unfortunately, warehouse design is a highly complex task with many tradeoffs between conflicting objectives and a large number of feasible designs. In (Rouwenhorst et al. 2000), the warehouse design is defined as a structured approach to decision making at the strategic, tactical and operational levels. Decisions that have a long-term impact, mostly related to high investments, such as the process flow design and the selection of the type of warehouse systems, are put on the strategic level. Based on the outcome of strategic decisions, mediumterm decisions, such as the dimensioning of storage systems, the layout design and the selection of equipment, are to be made on the tactical level. At the operational level, short- term decisions, which are mainly related to control policies, are made within the constraints of higher decision levels. In (Gu et al. 2010) a framework of warehouse design and operation is proposed, classifying the warehouse design problems as overall structure, sizing and dimensioning, department layout, equipment selection and operation strategy. Both contributions have a common conclusion that multiple decisions are interrelated and have to be solved simultaneously, while, unfortunately, the majority of papers listed in their literature reviews are focused on the analysis of an isolated problem rather than on the synthesis. According to Gu et al. (2010) a researcher addressing one decision would require a research infrastructure which would integrate all other decisions. The authors think that the scope and scale of that infrastructure appear to be too great a challenge for individual researchers. To properly evaluate the impact of changing one of the design decisions requires estimating changes in the operation of the warehouse. The authors conclude that the most important future direction for the warehouse design research community is to find ways to overcome these hurdles. What is good about that approach is that even partial problem solving and analysis can lead us to the previously mentioned overall goal since new research papers should and mostly do tend to enlarge the scope and to combine interrelated decisions. Clearly, decisions regarding warehouse layouts are an integral part of the warehouse design process. There are two types of layout decision problems that can be distinguished (De Koster et al. 2007). The first problem concerns the decision of where to place various departments (receiving, picking, storage, sorting, shipping, etc.). This problem is usually called the facility layout problem. Using the activity relationship between the departments, a warehouse block layout

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is derived. The common objective is either to minimize the handling cost (travel distances) or it is based on ‘‘closeness ratings’’. The second layout decision problem is concerned with the placement of equipment, storage space, paths, etc. within departments. It is usually called internal layout design or aisle configuration problem. In most papers, the warehouse layout problem is defined as finding an optimal (or at least good) layout of storage or order-picking area. In most cases, the criterion is travel distance (or travel time). This type of layout problem is also the focus of attention in this paper. The layout problem of unit-load storage area of conventional warehouses has quite a long history ever since the 1960s. On the other hand, in the 1980s and 1990s, the layout of unit-load AS/RS received a lot of attention. The literature on the order-picking area layout design is not extensive. While the systems with order-picking restricted to a single aisle in person-on-board AS/RS were analyzed in several papers in the late 1980s and 1990s, the layout of conventional systems with manual order-picking from multiple aisles has been the topic of several papers only in the previous decade. Conventional warehouse layouts, both for the storage area and order-picking area, imply a traditional warehouse layout. This is the layout found today in the majority of warehouses. The basic form is rectangular, with parallel straight aisles. There are two possibilities for changing aisles at the front and at the rear of the warehouse. These aisles are also straight and meet the main aisles at right angles. Modifications of this basic form are usually done by adding one or more additional cross-aisles, creating the so-called multiple-block layout. Therefore, the term ‘‘a conventional storage area layout’’ refers to the layout with unit-load operations within a selective pallet racks system, while the term ‘‘conventional order-picking area layout’’ refers to the one with manual order-picking operations within a selective pallet rack system or shelving system. Layouts of conventional warehouses with other types of storage systems, such as carousels, flow rack system, drive-in/through pallet racks, mobile racks, etc. are not considered here. Recently, some radically new, innovative warehouse layouts, which do not include traditional assumptions, have been proposed in (Gue and Meller 2009a). These layouts result in reduced travelling needed to store or retrieve a single pallet, and consequently in improved efficiency in the storage area. The question is whether these new layouts can be used in order-picking areas, which would result in shortened picking paths compared to those in traditional layouts. Seeking an answer to that question, a simulation of routing pickers in two new layouts was done and compared to the performances of routing in traditional layouts. In the following two sections of this paper, the main ideas and findings in the optimal traditional layout design of the storage and the order-picking area are presented respectively. Then, the previously mentioned new layouts are presented. Finally, results of the analysis of order-picking in new layouts are given, followed by relevant conclusions.

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3.2 Optimal Layouts of Storage Area The layout of storage area considered in this paper is one with parallel pallet racks, as illustrated in Fig. 3.1 (left). Aisle-based pallet floor storage has the same characteristics as the pallet rack storage and can be considered in the same way. Operations within such a layout are single cycles, either to store a single unit-load or to retrieve (pick) a single unit-load. Since for a given capacity of storage area (number of storage locations) one could design various layouts (altering the number of aisles and the length of aisles), the problem is which layout is optimal regarding the design objective. For example, a design objective could be costs (investment costs and operational costs). For a given capacity, different layouts have slightly different costs of required area due to the length (and therefore space) of front aisle. Please note that differences are in most cases negligible and it could be assumed that a given capacity defines the required total storage space. They also have different perimeter costs if walls should be built around the area. Operational costs are the costs of cycles in a considered layout. Most models in the literature optimize the layout minimizing the expected travel distance to store/retrieve an item. The theoretical background to warehouse layout can be found in (Francis and White 1974), with derived expressions for optimal warehouse designs represented as continuous storage areas both for non-rectangular and rectangular designs. A simple model for optimal storage layout, modified from the model presented in (Bauer 1985), is presented below. It minimizes the expected travel in the rectangular storage area with parallel aisles. Figure. 3.2 illustrates a general storage area with a capacity of Q storage locations per layer, for given dimensions of storage location l1 9 b1, with the width of main aisles b2 and the width of front aisle b3. Thus, dimensions of the layout can be represented by the length of aisles Lr and the width of area Br as a function of the number of aisles n1 as follows: Lr ¼

Q  b1 2  n1

Br ¼ n1  2  l1 þ n1  b2

ð3:1Þ ð3:2Þ

If random storage is used (in other words, any item can be stored/retrieved from any location with the same probability) and a single dock (depot, pickup and delivery point—P&D) is located in the middle of the front aisle, the expected travel to storage locations can be simply represented, according to findings in (Francis and White 1974), as: s¼

Lr Br þ þ b3 2 4

ð3:3Þ

A single dock can be located in any place along the front aisle or in the corner. However, from (Francis and White 1974) and several other papers published after 1974, it is known that the location of single dock in the middle is optimal. Inserting

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Fig. 3.1 Traditional warehouse layouts (Pohl et al. 2009a)

Fig. 3.2 Average travel to storage locations

(3.1) and (3.2) into (3.3) leads to an expression where the expected travel is a function of only one variable—the number of aisles; therefore s = f(n1). Finding the minimum yields the optimal number of aisles as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q  b1 n1 ¼ ð3:4Þ 2  l1 þ b2 Corresponding Lr and Br can be then calculated using (3.1) and (3.2). Theoretically, the shape of the resulting optimal layout is rectangular with the proportion of Lr : Br = 1 : 2, while in practice it should usually be slightly modified by rounding the number of aisles to integer. From the model of optimal storage layout it is obvious that the rear aisle in the basic layout is not even needed. Furthermore, adding one or more cross-aisles is not beneficial as this leads only to an increased total space required and to

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increased expected travel of single-command. On the other hand, using a different operational policy, i.e. combining one storage command with one retrieval command in a dual-command (interleaving) will not only improve the efficiency in the existing layout but will create an opportunity for a further improvement by altering the layout. Dual-command operations in common warehouses are analyzed in (Pohl et al. 2009a). The paper demonstrates the efficiency of dual-command with respect to single-command, with savings in the range of 16–33% over a variety of shapes and sizes of the basic traditional layout. The efficiency of dual-command with respect to single-command was consistently higher in the layout with an additional cross-aisle in the middle, except for very small warehouses, as illustrated in Fig. 3.1 (middle). Travelling from the storage location to the retrieval location in dual-command (travel-between) is much more efficient in the layout with a middle cross-aisle. The authors developed a model for an optimal layout with dual-command and showed that the warehouse optimized for a dualcommand travel distance has a smaller number of aisles (it is narrower and taller) than the warehouse optimized for a single-command travel. They also analyzed the layout with a cross-aisle in the middle of aisles parallel to the front wall, as illustrated in Fig. 3.1 (right). This layout, as well as the previous one, has higher efficiency than the basic layout (in some cases even for a single-command cycle). Both layouts minimize the dual-command travel with a shape factor (height/width) of approximately 0.6. The layout in Fig. 3.1 (right) is superior to the layout in Fig. 3.1 (middle) in a wide range of parameters although the latter one is more common in practice. According to the authors, the cause of that could be less dependence on one central P&D. In addition, it was shown that optimal placement of middle cross-aisle is moved from the center to the rear cross-aisle, but improvements are in most cases \1%. In conclusion, the authors pointed out some unanswered questions. How would the inserting of more cross-aisles and more P&D locations influence the travel distance and optimal layout? From analysis and comparison of single-command and dual-command operations it can be concluded that the optimal layout is affected by them. What is the influence of other operational policies, such as using the turnover-based storage instead of the assumed random storage, or using sequencing storage/retrieval requests instead of the firstcome first-served policy?

3.3 Optimal Layouts of Order-Picking Area Designing the layout of order-picking area can have even greater influence on the efficiency of warehouse operations. The order-picking process, defined as the process of retrieving items from storage locations in response to a specific customer request, is the most laborious and the most costly activity in a typical warehouse, making up to 55% of the total operating costs of a warehouse (Tompkins et al. 1996). Besides the pressure to reduce costs, additional pressure put on companies is to deliver their products faster than before. A crucial link

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between order-picking and delivery speed is the fact that the faster an order can be retrieved, the sooner it is available for shipping to the customer. Therefore, it is very important to put in some effort to reduce order-picking costs, i.e. to improve the order-picking efficiency. One way to improve the order-picking process is to redesign it, using new equipment, new layout or/and automation and computerization of the process. The other way is to improve the operational efficiency of order-picking using appropriate operating policies. The time to pick an order can be divided into three components: time for travelling between items, time for picking the items and time for remaining activities. The fact that about 50% of the total order-picking time is spent on travelling (Tompkins et al. 1996) has the potential for improving the order-picking efficiency by reducing travel distances. Most methods of improving the operational efficiency of order-picking focus on reducing travel times, and can be categorized as one of three groups of operating policies: routing, storage and batching. Routing methods (policies) determine the picking sequences and routes of travelling, trying to minimize the total travel distances. Storage methods or assigning items to storage locations based on some rules could also reduce travel distances with respect to random assignment. Order batching methods, or grouping two or more customer orders in one picking order, are also very efficient in reducing total travel distances. All the methods mentioned above are well known and proven in improving the order-picking efficiency. However, the performances depend greatly on the layout and size of the warehouse, the size and characteristics of orders and the order-picker capacity. The performance of a particular method also depends on the other methods used; therefore, it is important to understand their interactions. Extensive research in this area has been undertaken recently, and a growing body of literature exists on various methods of picking an order as efficiently as possible. For a most comprehensive overview of the literature regarding the methods in order-picking systems we refer to (De Koster et al. 2007). For a given layout of the picking area, characteristics of orders, and other influencing factors, the right combination of order-picking methods can be implemented. However, the analysis of methods has shown a non-negligible influence of the layout on performances. Therefore, finding the optimal layout of order-picking area will lead to a minimized expected travel in routes, and consequently to reduced costs and an increased speed of response. Order-picking area layouts that can be found today in the majority of warehouses are the same as for the storage area. The basic form has parallel aisles, a central depot (pick up/delivery point) and two possibilities for changing aisles (i.e. two cross-aisles), at the front and at the rear of the warehouse, as already shown in Fig. 3.1. A nonlinear programming model for optimal order-picking layout is presented in (Roodbergen 2001). The model aims at finding the minimum average travel distance expressed as a function of a number of layout variables and parameters (number of aisles, length of aisles, depot location, width of aisles including storage racks, width of a cross-aisle), under defined conditions. The expression for the average travel distance in a picking area is derived for one simple routing policy, S-shape, while in (Roodbergen and Vis 2006) the research

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also included the largest gap routing policy. Both routing policies are rather simple heuristics. With the S-shape routing policy, any aisle containing at least one item is traversed through the entire length. Aisles where nothing has to be picked are skipped. Using the largest gap routing policy, the picker enters the first aisle and traverses this aisle to the back part of the warehouse. Each subsequent aisle is entered and traversed up to the ‘‘largest gap’’ and left from the same side as it was entered. A gap represents the distance between any two adjacent items, or between a cross-aisle and the nearest item. The last aisle is traversed entirely and the picker returns to the depot along the front aisle, traversing again each aisle up to the largest gap. Thus, the largest gap is the part of the aisle that is not traversed. Conclusions drawn from (Roodbergen 2001) and (Roodbergen and Vis 2006) are that for high picking densities, the S-shape routing is best employed in a layout with an even number of aisles instead of an odd number of aisles. From the viewpoint of strict travel distance minimization, a very high pick density is best dealt with in a picking area where each picking zone consists of exactly two aisles. For practical implementations, in addition to travel distance, other considerations can also be taken into account easily since there are generally several layouts that have an average travel distance that is close to the optimum. Furthermore, the authors found that the optimal layout is sensitive to the routing policy used in the optimization. Optimizing the layout for a routing policy other than that used for the actual operation showed efficiency losses of up to 18%. Therefore, it is advisable to perform the optimization for more than one routing method before deciding on the layout. For an overview on routing policies we refer to, among others (Petersen 1997; Roodbergen 2001; De Koster et al. 2007). The modification of the basic form of layout is usually carried out by adding one or more additional cross-isles. In that case, we refer to a multiple block layout with multiple crossaisles shown in Fig. 3.3. In papers (Roodbergen and De Koster 2001a, b; Vaughan and Petersen 1999) it is shown that adding one or more additional cross-aisles can improve the total travel distances, and that it is also possible to find an optimal number of cross-aisles. In (Roodbergen 2001) the author also presented a nonlinear programming model for optimal order-picking layout with multiple blocks. In this case, the expression for average travel distance, derived for the S-shape routing policy, includes an additional variable—the number of blocks. In (Roodbergen et al. 2008) it is concluded that apart from special cases with very high pick density, it is always better to have a multiple-block layout than a one-block layout. Additional testing showed that the layouts generated by the model are also fairly adequate for the case in which the actual operation of the warehouse uses a routing method other than the S-shape policy. This is contrary to the findings in (Roodbergen and Vis 2006), where large differences were found between optimizations with the S-shape and largest gap routing policies for one-block layouts. Therefore, optimal layouts with multiple blocks are much less sensitive to the routing operational policy. The papers mentioned so far, as well as the optimization software and an interactive warehouse tool can be found on the valuable web site www.roodbergen.com.

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Fig. 3.3 Multiple block warehouse layout (Roodbergen et al. 2008)

The order-picker routing in the layout with a cross-aisle in the middle of aisles parallel to the front wall, Fig. 3.1 (right), was analyzed in (Caron et al. 1998), while the optimal layout was considered in (Caron et al. 2000).

3.4 Non-traditional Warehouse Layouts The traditional design of warehouse layout is based on a number of unspoken, and unnecessary, assumptions. The two most restrictive are that cross-aisles are straight and must meet picking aisles only at right angles, and that picking aisles are straight and are oriented in the same direction. In (Gue and Meller 2009a) the

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Fig. 3.4 Innovative warehouse layouts (Gue and Meller 2009b)

authors show that these design assumptions, which are unnecessary from a construction point of view, limit efficiency and productivity because they require workers to travel longer distances and to take less-direct routes to retrieve products from racks and deliver them to designated pickup-and-deposit points. In the layout that maintains parallel picking aisles, but allows the cross-aisle to take a different shape, the expected distance to be travelled to retrieve a single pallet is 10% shorter than that in an equivalent traditional design. Such a layout, named the Flying-V layout, is shown in Fig. 3.4 (left). Disapproving the second assumption that picking aisles must be parallel, they derived the so-called fishbone layout shown in Fig. 3.4 (middle). The fishbone layout also incorporates the V-shaped cross-aisles, with the V extending across the entire warehouse. The picking aisles below the V are horizontal, while the aisles above the V are vertical. The expected distance to a pick in the fishbone design is approximately 20% shorter than that in a traditional warehouse. Similar to traditional layouts with cross-aisles, these alternative layouts also require a facility 3–5% larger than that of the traditional layout, which was designed to minimize the footprint of a warehouse. In (Pohl et al. 2009b), an analytical expression for travel-between locations in the fishbone layout was developed. A comparison of fishbone warehouses that have been optimized for dual-command with traditional warehouses that have been optimized in the same manner has shown that an optimal fishbone design reduces the dual-command travel by 10–15%. A drawback of the fishbone design is limited access to the storage space due to the single, central P&D point. Therefore, the authors also proposed a third design and named it the ‘‘chevron aisles’’ layout, illustrated in Fig. 3.4 (right). Expected distances to store or to retrieve a single pallet in this layout are very close to those in the fishbone layout (Gue and Meller 2009b).

3.5 Order-Picking in Non-Traditional Layouts Despite the great potential of new, innovative unit-load warehouse designs for reducing the travel distance in pallet operations (single command and dual command), the question is how these layouts perform as layouts in picking

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Fig. 3.5 Picking route example (S-shape policy) in the basic traditional layout

operations (multiple command), compared to the traditional layouts. To address this question, we tried to analyze the routing of order-pickers in the fishbone and the chevron layout by means of simulation. The analysis was restricted to the already mentioned S-shape routing policy and the composite routing policy. The composite routing policy is advanced heuristics that minimizes the travel in individual aisles by deciding whether picking in the aisle is done by traversing it entirely or by making a return route. The simulation was conducted on four warehouse layouts with 576 locations per layer: basic traditional, traditional with one (middle) cross-aisle, fishbone layout, and chevron layout. Due to the simplicity of distance calculation, dimensions of a location are 191 m, and the width of all aisles is 2 m. The basic traditional layout had 12 main aisles (total width across aisles is 48 m) and the length of main aisles was 24 m (24 locations per row). Figure. 3.5 shows this basic traditional layout with an example of a picking route. With the location of a depot in the middle, it is the optimal layout for single command. The layout with an added cross-aisle had the cross-aisle positioned exactly in the middle, as shown in Fig. 3.6. A comparable fishbone design is shown in Fig. 3.7 and a comparable chevron layout in Fig. 3.8. Two situations were considered regarding the order size (10 and 30 locations per route). Pick locations were generated randomly according to assumed random storage policy. The first problem encountered was how to define the routing algorithms in the fishbone and the chevron layout. A simplified explanation of S-shape routing policy in the layouts with multiple blocks defined in (Roodbergen 2001) could be that picking is done first in the farthest block, then repeated in other blocks and finished in the closest block. It is impossible to say which block in the fishbone or the chevron layout is the farthest from the depot, and which is the closest to the depot. The algorithm for a 2-block traditional warehouse was modified in a way

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Fig. 3.6 Picking route example (S-shape policy) in a traditional layout with a middle cross-aisle

Fig. 3.7 Picking route example (S-shape policy) in a fishbone layout

that the layout is considered as a 3-block warehouse. The order-picker starts at the depot and visits the blocks in a clockwise manner. In each block the route is done according to the routing policy for a single block, as illustrated with the example of route in Fig. 3.6. Such an algorithm is easily applicable to the fishbone design, as illustrated in Fig. 3.7, and to the chevron layout considered as a 2-block layout illustrated in

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Fig. 3.8 Picking route example (S-shape policy) in a chevron layout

Table 3.1 Simulation results of average picking travel distance (in meters) S-shape routing policy Order size Composite routing policy 10 Warehouse layout

Traditional (basic) Traditional (middle cross-aisle) Fishbone Chevron

30

258.7 375.8 Warehouse layout 193.9 329.0 227.5 351.9 268.5 397.2

Order size 10

Traditional (basic) Traditional (middle cross-aisle) Fishbone Chevron

30

228.2 363.9 182.8 309 213.1 317.3 233.2 370.2

Fig. 3.8. The same idea was also used for the composite routing policy. The simulation results showing the average picking travel distance are given in Table 3.1. As it was expected for the examined cases, adding a middle cross-aisle in the traditional layout decreases average routes compared to those in the basic traditional layout. For both order sizes the density of pick locations in 12 main aisles is not high, and adding a middle cross-aisle will eliminate some unnecessary travel in the main aisles without pick locations. But it should be also noted that the percentage of reduction for the order size 30 (12.5%) is smaller than the reduction for the order size 10 (25%). As the order size increases (i.e. the pick density increases—average distance between picks decreases) there will be a point where adding a middle cross-aisle is not beneficial. Although the fishbone layout will give a shorter travel distance compared to the basic traditional layout (between 6 and 12% in conducted simulations, depending on the routing method and order size), it is still outperformed by the layout with a middle cross-aisle. It seems that adding a V-shaped cross-aisle has smaller potential than adding a straight middle aisle. In the chevron layout average travel distances are even longer than in a comparative basic traditional layout. Longer

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average travel distances in the fishbone and the chevron layout compared to the traditional layout with a middle cross-aisle could be explained as follows. The fishbone and chevron layouts create blocks of aisles with different lengths, with a higher probability that a pick location is in longer aisles than in shorter aisles. This is especially a disadvantage of the S-shape routing policy where the picker should traverse the entire aisle, and not as much of the composite routing policy where the picker can make return trips. The superiority of composite routing policy over the S-shape policy is confirmed in all the considered situations.

3.6 Conclusion The fishbone layout is without doubt an excellent layout for pallet storing and picking (single or double command), already being implemented in real warehouses. However, the presented analysis indicates the conclusion that in the order-picking area with picking from multiple locations (item and case picking), the fishbone layout results in longer routes than the traditional layout with a straight, right angled cross-aisle in the middle. The same holds for the chevron layout, with picking routes even longer than the ones in a comparable basic traditional layout. However, a further analysis and a wide-range of experiments are needed in order to draw fully reliable conclusions. Despite all the presented contributions regarding optimal layouts and the intensive pioneering work on innovative layouts, the research on warehouse layouts is not over. The presented analysis was limited to only one warehouse size with a fixed shape, while the future work should include more cases. Furthermore, according to the previous research presented in Sects. 3.2 and 3.3, optimal order-picking layouts are different with respect to having cross-aisle(s) or not. The investigated shape is optimal only for a single command. Therefore, order-picking routes from different, optimally shaped, warehouses should also be analyzed. The research on the optimal fishbone and the chevron order-picking layout is yet to be done. In addition, the presented analysis assumed random storage. The reduction in the order-picking travel distance with the turnover-based storage could differ with respect to different layouts and patterns used. Another assumption was the location of dock in the middle (optimal for traditional layouts and for new layouts under a single or a dual command regime). Using the idea of fishbone layout while having dock in the corner creates a new layout in which one diagonal, straight cross-aisle creates only two blocks. Some preliminary research results indicated a possible reduction in average order-picking travel in such a layout compared to the fishbone layout illustrated in Fig. 3.7. On the other hand, the travel distance might not be the only decisive factor. If picking equipment requires additional time to change aisle, performance measure should be the time required to make a route instead of the distance travelled in the route. The required area of traditional layout with a middle cross-aisle is obviously larger than the area of basic traditional layout. The required area of the fishbone or the chevron layout is also increased in comparison

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with traditional layouts. Warehouse designers should be aware of all advantages and disadvantages of different layouts and, depending on a given situation and objectives, should choose the most appropriate one.

References Bauer P (1985) Planung und Auslegung von Palettenlagern. Springer, Berlin Caron F, Marchet G, Perego A (1998) Routing policies and COI-based storage policies in pickerto-part systems. Int J Prod Res 36(3):713–732 Caron F, Marchet G, Perego A (2000) Optimal layout in low-level picker-to-part systems. Int J Prod Res 38(1):101–107 De Koster R, Le-Duc T, Roodbergen KJ (2007) Design and control of warehouse order-picking: a literature review. Eur J Oper Res 182:481–501 Francis RL, White JA (1974) Facility layout and location: an analytical approach. Prentice-Hall, NJ Gu J, Goetschalckx M, McGinnis L (2010) Research on warehouse design and performance evaluation: a comprehensive review. Eur J Oper Res 203:539–549 Gue KR, Meller RD (2009a) Aisle configurations for unit-load warehouses. IIE Trans 41(3):171–182 Gue KR, Meller RD (2009b) The application of new aisle designs for unit-load warehouses. In: Proceedings of 2009 NSF engineering research and innovation conference. Honolulu Petersen CG (1997) An evaluation of order picking routeing policies. Int J Oper Prod Man 17(11):1098–1111 Pohl LM, Meller RD, Gue KR (2009a) An analysis of dual command operations in common warehouse designs. Transport Res E 45(3):367–379 Pohl LM, Meller RD, Gue KR (2009b) Optimizing the fishbone aisle design for dual-command operations in a warehouse. Nav Res Log 56(5):389–403 Roodbergen KJ (2001) Layout and routing methods for warehouses. ERIM, Rotterdam Roodbergen KJ, De Koster R (2001a) Routing order pickers in a warehouse with a middle aisle. Eur J Oper Res 133:32–43 Roodbergen KJ, De Koster R (2001b) Routing methods for warehouses with multiple cross aisles. Int J Prod Res 39(9):1865–1883 Roodbergen KJ, Vis IFA (2006) A model for warehouse layout. IIE Trans 38(10):799–811 Roodbergen KJ, Sharp GP, Vis IFA (2008) Designing the layout structure of manual order picking areas in warehouses. IIE Trans 40(11):1032–1045 Rouwenhorst B, Reuter B, Stockrahm V, Van Houtum GJ, Mantel RJ, Zijm WHM (2000) Warehouse design and control: framework and literature review. Eur J Oper Res 122:515–533 Tompkins JA, White JA, Bozer YA, Frazelle EH, Tanchoco JMA, Trevino J (1996) Facilities planning, 2nd edn. Wiley, NY Van den Berg J (1999) A literature survey on planning and control of warehousing systems. IIE Trans 31:751–762 Vaughan TS, Petersen CG (1999) The effect of warehouse cross aisles on order picking efficiency. Int J Prod Res 37(4):881–897

Chapter 4

Order-Picking by Cellular Bucket Brigades: A Case Study Yun Fong Lim

Abstract Workers in bucket brigade order-picking perform unproductive travel when they walk back to receive work from their colleagues. The loss in productivity due to this unproductive travel can be significant especially if the orderpicking line is long. We present a novel design alternative that may provide substantial improvement to the productivity of a bucket brigade. Under the new design, products are stored on both sides of an aisle. Each worker picks products to fulfill customer orders from one side of the aisle when he proceeds in one direction. The worker picks products, possibly for other customer orders, from the other side of the aisle when he proceeds in the reverse direction. We introduce new rules for workers to share work because they are not only required to pick products in both directions, but also to cross the aisle. In this study, we perform computer simulations based on data from a distributor of service parts in North America. Our results suggest that bucket brigade order-picking under the new design can be significantly more productive than a traditional bucket brigade, even if the latter is equipped with wireless technology to reduce travel.

4.1 Introduction Order-picking typically accounts for about 55% of the total operating cost of a distribution center (Frazelle 2002) and is considered one of the most critical functions of a supply chain. As a result, the management of distribution centers

Y. F. Lim (&) Lee Kong Chian School of Business, Singapore Management University, 50 Stamford Road, # 04-01, Singapore 178899, Singapore e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_4,  Springer-Verlag London Limited 2012

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often gives the highest priority to order-picking for productivity improvement. In an order-picking process, stock-keeping units (SKUs) are retrieved from storage to fulfill specific customer orders. Some critical issues in this process include: 1. Allocating work for workers (or order-pickers) such that their workload is balanced. 2. Reducing workers’ travel as it accounts for more than 50% of the total orderpicking time (Frazelle 2002). One way to address the first issue is to coordinate workers by forming a bucket brigade (Bartholdi and Eisenstein 1996a; Bartholdi et al. 2001). In a bucket brigade, every worker follows a simple rule: Continue to pick SKUs along the order-picking line until either your work is taken over by your colleague downstream or you finish it if you are the last worker of the line; then you walk back to receive more work, either from your colleague upstream or from the start of the line if you are the first worker. Under certain assumptions, Bartholdi and Eisenstein (1996a) show that if workers are sequenced from slowest to fastest according to their work velocities in the direction of work flow, then a bucket brigade will self-balance such that the hand-offs between any two neighboring workers will converge to a fixed location. As a result, every worker will repeatedly work on a fixed segment of the line. Furthermore, the system’s throughput attains its largest possible value. Bucket brigades are especially effective in coordinating workers for orderpicking due to the following reasons (Bartholdi and Eisenstein 1996b; Bartholdi et al. 2001): 1. The rule is easy for workers to remember and follow. 2. Neither a work-content model nor computation for work balance is required. Both are necessary for any static work-allocation policy. 3. Workers dynamically and constantly balance their work and thus, the system can restore balance spontaneously from temporary disruptions and is adaptive to SKUs’ demand seasonality. Bucket brigades are also used to produce garments, to package cellular phones, and to assemble tractors, large-screen televisions, and automotive electrical harnesses (Bartholdi and Eisenstein 1996a, b, 2005; Villalobos et al. 1999a, b). Despite their impressive performance, there is a way to remarkably improve the throughput of bucket brigades. To distinguish from this new way of coordinating workers, we call the traditional bucket brigades introduced by Bartholdi and Eisenstein (1996a) serial bucket brigades. Recall that the serial bucket brigade rule requires each worker to walk back to receive more work after he relinquishes his work to his colleague downstream or after he completes his work. The travel to get more work either from his colleague upstream or from the start of the line is unproductive. This waste of capacity is especially significant for a long orderpicking line. For example, it is common to have an aisle covering more than a hundred of racks for order-picking in large distribution centers. The throughput

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of serial bucket brigades will be compromised in such environments because of workers’ unproductive travel to get more work. In this chapter, we discuss a new design of bucket brigades introduced by Lim (2011) that totally eliminates this unproductive travel. This is also to address the second issue in order-picking mentioned above. Our basic idea is to store SKUs on both sides of an aisle. A worker picks SKUs to fulfill customer orders from one side of the aisle while he proceeds in one direction. He picks SKUs, possibly for other customer orders, from the other side of the aisle while he proceeds in the reverse direction. Since the new design requires workers to pick SKUs in both directions and to cross the aisle, we need new rules for workers to share work. Although the new design totally eliminates the unproductive travel inherent in serial bucket brigades, it introduces a new type of unproductive travel: Workers are required to walk from one side of the aisle to the other. Since this cross-aisle travel is small if the aisle is narrow, we expect the system to perform well under this new design for narrow aisles. We perform computer simulations based on data from a distributor of service parts in North America. Our results suggest that bucket brigade orderpicking under the new design can be significantly more productive than a serial bucket brigade, even if the latter is equipped with wireless technology to reduce travel. We outline the chapter as follows: After reviewing the literature in Sect. 4.2, we describe the new design of bucket brigades and discuss rules for workers to share work in order-picking in Sect. 4.3. We then perform simulations to compare the average throughput of a system under the new design with that of a serial bucket brigade based on data from a distribution center in Sect. 4.4. Finally, we give some concluding remarks in Sect. 4.5.

4.2 Related Literature Bartholdi and Eisenstein (1996a) provide the first theoretical analysis on serial bucket brigade assembly lines. In their model, the product has deterministic work content. Workers cannot overtake each other and so they remain in a fixed sequence along the line. Each worker has a work velocity representing his familiarity with work content. These work velocities are deterministic and finite. They assume workers walk back to receive work with an infinite velocity because the time to walk the entire line is negligible compared with the time required to assemble an item (an instance of the product). The most interesting result of Bartholdi and Eisenstein (1996a) is that if workers are sequenced from slowest to fastest in the direction of production flow according to their work velocities, the hand-off points between any two neighboring workers will converge to a fixed location. Eventually, every worker will repeatedly work on a fixed portion of work content on each item produced. The system is said to self-balance. Mathematically, the system converges to a fixed

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point (see, for example, Alligood et al. 1996). Another important result is that if work content is continuously and uniformly distributed on the assembly line, then the throughput of the line on the fixed point is the sum of work velocities of all workers, which is the maximum possible for the system. The self-balancing property of bucket brigades is attractive to managers because it creates several positive effects. For example, the skills of workers are reinforced by learning through repetition, each worker is constantly busy, and the output is regular, which simplifies the coordination of downstream processes. All these effects are created without the intervention of management or engineering. Bartholdi et al. (1999) describe all possible dynamics of two- and three-worker bucket brigades. They analyze the system with workers not necessarily sequenced from slowest to fastest. Bartholdi et al. (2006) generalize the ideas of bucket brigades to a network of subassembly lines so that all subassembly lines are synchronized to produce at the same rate. Bartholdi et al. (2001) investigate the behavior of bucket brigades when work content on each work station is stochastic. They find that the dynamics and throughput of the stochastic system will be similar to that of the deterministic system if there is sufficient work distributed among sufficiently many stations. They also report the effectiveness of bucket brigades in order-picking in a distribution center, which experienced a 34% increase in productivity after the workers began picking orders by bucket brigades. Armbruster and Gel (2006) analyze a two-worker bucket brigade in which the work velocities of workers do not dominate each other along the entire line. Armbruster et al. (2007) study the behavior of bucket brigades when workers improve their work velocities as they learn. They observe that if workers are allowed to change their order along the line, the self-balancing property of bucket brigades is typically preserved. Lim and Yang (2009) study policies that maximize the throughput of bucket brigades on discrete work stations. For the three-station, two-worker line they show that fully cross-training the workers and sequencing them from slowest to fastest outperforms all other policies for most work-content distributions on the stations. They also find that the policy that only partially cross-trains the workers and sequences them from slowest to fastest performs equally well in many situations. In contrast to the normative model studied by Bartholdi and Eisenstein (1996a), some work has been done based on the assumption that workers spend significant time to walk back for more work. Bartholdi and Eisenstein (2005) consider the case where each worker spends a constant walk-back time and a constant hand-off time to get a new item from his colleague upstream. They assume different workers may have different walk-back times and different hand-off times. Bartholdi et al. (2009) assume each worker has a constant, finite walk-back velocity. Workers are allowed to overtake or pass each other. The authors show that the system may behave chaotically, according to rigorous mathematical definitions, if it is not configured properly. This chaotic behavior causes the inter-completion times of items to be effectively random, even though the model is purely deterministic.

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Lim (2011) first introduces the ideas of cellular bucket brigades to totally eliminate the unproductive travel inherent in serial bucket brigades. Based on a model where work content is continuously and uniformly distributed, the author analyzes the dynamic behavior and the throughput of a cellular bucket brigade. The author shows that a cellular bucket brigade can be substantially more productive than a serial bucket brigade. Inspired by the ideas of Lim (2011), Lim and Wu (2011) introduce simple rules for workers to share work on U-shaped lines with discrete work stations. The authors study the dynamics and determine the long-run average throughput of the system. This chapter complements the theoretical work of Lim (2011) with a case study on the order-picking operation of a distribution center. We describe how the ideas of cellular bucket brigades proposed by Lim (2011) can be implemented in order-picking and introduce simple rules for workers to share work under this new design in practice. We evaluate the system’s performance under the new design using computer simulations based on real data from the distribution center. Eisenstein (2008) analyzes the optimal design of discrete order-picking technologies. Gue et al. (2006) study the effects of pick density on order-picking in narrow aisles. For a recent review of the order-picking literature, see de Koster et al. (2007). van den Berg (1999) provides a nice survey on planning and control of warehousing systems.

4.3 Order-Picking by Cellular Bucket Brigades 4.3.1 A Basic Model Our method can be used in carton-picking or piece-picking (Bartholdi and Hackman 2010) that occurs along an aisle, where jobs are released from one end of the aisle. Workers may or may not travel with handling equipment such as orderpicker trucks or carts. To facilitate the analysis, we consider an order-picking line that consists of s sections of shelving shown in Fig. 4.1. Each section of shelving is represented by a pick point that is located at the center of the section. We assume all SKUs in the section are retrieved from this pick point. An order is a list of pick lines for a customer. Each pick line requests a piece of a SKU from a specific pick point. If multiple pieces of the same SKU are requested by the customer then multiple pick lines are printed. We assume b orders are batched into a job. The average number of pick lines per job increases with the job size b. Every job is released from the start of the order-picking line (left end of Fig. 4.1). SKUs requested by a job are progressively picked along the line until the job is completed.

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Fig. 4.1 An order-picking line

Fig. 4.2 A new design

Each worker carries a single job and walks with a finite, constant velocity w along the order-picking line. He stops at the next pick point to pick SKUs for his job. The time required to retrieve a SKU from a pick point depends on the worker’s experience and dexterity. We assume the kth slowest worker in a team takes Tk seconds to retrieve a piece of a SKU from a pick point, where Tk ¼ T ebðk1Þ ;

ð4:1Þ

k = 1,…, n, and T, b [ 0. We also try other functions where Tk decreases with k. The simulation results are generally similar. Thus, we only report the results based on Eq. 4.1 in this chapter. After all orders of a job are picked, they are brought to the shipping department where they are sent to the customers. The objective is to maximize the average number of pick lines (or, equivalently, the average number of orders) picked per unit time.

4.3.2 A New Design Consider a new design of bucket-brigade order-picking to reduce unproductive travel of workers inherent in serial bucket brigades. The idea is to fold the entire order-picking line at its midpoint so that the two halves of the line form an aisle in between. Figure 4.2 shows such a design. The work flow of the two halves runs in opposite directions. Each job is initiated at the start of the first half. SKUs requested by the job are picked in the forward direction until the end of the first half, where the job is transferred to the second half of the line. SKUs from the second half of the line are then picked for the job until the job is completed. Thus, the first half is called the forward line. The second half, which continues the jobs in the reverse direction, is called the backward line. The aisle width between the forward and the backward lines is a. Each job is initiated at the start of the forward line. The SKUs requested by the job are progressively picked until the end of the forward line, where the job is

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transferred to the backward line. The SKUs for the job are then picked in the backward direction until the job is completed. The width of the aisle between the two lines is a. To uniquely determine the coordinate of each worker along the aisle, we conceptualize the aisle as a real line and define its left end as the origin. Let xi denote the coordinate of worker i along the aisle, for i = 1,…, n. We require workers to remain in a fixed ordering along the aisle from 1 to n so that x1 B x2 B _ B xn. This requirement is reasonable for narrow aisles where workers cannot overtake or pass each other. This will not be a limitation of our model as the new design is effective only if the aisle is sufficiently narrow. As a result, only worker 1 can initiate and finish a job at the start of the aisle, and only worker n can transfer a job from the forward line to the backward line at the end of the aisle. We call workers i - 1 and i ? 1 the predecessor and the successor, respectively, of worker i. Since workers work in both directions along the aisle, a hand-off between any two workers now becomes an exchange of work. A worker i continues to work on the forward line until a hand-off occurs at point p along the aisle where he meets his successor, who is working on the backward line. A hand-off is not instantaneous now: The two workers first relinquish their jobs at point p and then walk across the aisle with velocity w, which takes time a/w. After they exchange their work, worker i works on the backward line with the job from his successor, while his successor works on the forward line with the job from worker i. Note that if the picked SKUs are carried by trucks (or carts), then each truck (or cart) follows its associated job. In this case, the two workers exchange not only their jobs, but also their trucks (or carts). When worker n finishes his work at the end of the forward line, the system resets itself: Worker n transfers his job from the forward line to the backward line. He continues to work on his job on the backward line until he meets worker n - 1, who is working forward. After a hand-off, worker n - 1 works backward on the job that he receives from worker n. Worker n - 1 then meets and exchanges work with worker n - 2, who in turn meets and exchanges work with worker n - 3, and so on until worker 1 completes the job at the end of the backward line. Worker 1 relinquishes the completed job, crosses the aisle, and initiates a new job. Each reset triggers a job completion and each job completion is followed by a job initiation. Thus, the work-in-process is bounded by the number of workers in the system.

4.3.3 Extended Rules A worker’s coordinate along the aisle remains unchanged when he is crossing the aisle. To keep workers in the same ordering along the aisle, we need to introduce additional rules to handle situations in which a worker, working on the forward (backward) line, catches up with his successor (predecessor) while the latter is crossing the aisle. If worker i, who is working forward, catches up with

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Fig. 4.3 Movement of workers in a cellular bucket brigade

worker i ? 1, who is crossing the aisle, then worker i must wait for worker i ? 1. We set xi = xi+1. If worker i, who is working backward, catches up with worker i - 1, who is crossing the aisle, then worker i must wait for worker i - 1. We set xi = xi-1. Under the new design each worker independently follows these rules: Work forward. Continue to pick SKUs for your job on the forward line until 1. you exchange work with your successor, then work backward; or 2. you reach the end of the forward line if you are the last worker, then transfer your job to the backward line and work backward; or 3. you catch up with your successor, who is crossing the aisle, then wait. Work backward. Continue to pick SKUs for your job on the backward line until 1. you exchange work with your predecessor, then work forward; or 2. you complete your job at the end of the backward line if you are the first worker, then initiate a new job and work forward; or 3. you catch up with your predecessor, who is crossing the aisle, then wait. Wait. Stay with your job, 1. if you are on the forward line, remain idle until your successor finishes crossing the aisle, then work forward; or 2. if you are on the backward line, remain idle until your predecessor finishes crossing the aisle, then work backward. Figure 4.3 illustrates the movement of workers in the new design. Let xti be the hand-off point along the aisle between worker i and his successor due to the tth reset. The two workers exchange their work and cross the aisle with velocity w. After the hand-off, worker i works backward until he meets worker i - 1, who is t . After a hand-off between the two workers at point working forward, at point xi-1 t xi-1, worker i works forward again. Meanwhile, worker i ? 1 picks SKUs for the job that he receives from worker i on the forward line. He meets worker i ? 2, who is working backward, at point xt+1 i+1. After a hand-off, worker i ? 1 works backward again. The next hand-off between workers i and i ? 1 occurs at point xt+1 i . We call this a cellular bucket brigade because the workers move in cells. This figure illustrates the movement of worker i (represented by the solid arrows) and worker i ? 1 (represented by the dotted arrows) between two

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successive hand-offs that occur at points xti and xt+1 between the two workers in a i cellular bucket brigade. The start and the end of a worker’s path are represented by a circle and a square respectively. A worker is blocked by his successor (predecessor) on the forward (backward) line if the former catches up with the latter while the latter is picking SKUs at a pick point. When a worker is blocked, he remains idle until his successor (predecessor) works forward (backward) again.

4.4 Comparing With Serial Bucket Brigades: A Case Study We use data from a distribution center to test the performance of cellular bucket brigades under realistic order-picking settings. Our simulation results suggest that cellular bucket brigades can be significantly more productive than their traditional counterparts even if the latter are equipped with wireless technology to reduce travel.

4.4.1 Order-Picking in a Distribution Center We perform simulations based on data from a major distributor of service parts in North America. In their order-picking facility, there are 214 pick points. The pick frequency of a pick point is the number of times SKUs are retrieved from the pick point. Different SKUs face different demands and therefore, different pick points may have different pick frequencies. Figure 4.4a shows the daily pick frequency of each pick point. The figure shows that some pick points are visited over 120 times daily and some are visited no more than twice a day. Figure 4.4b shows that the number of pick lines per order is typically small for the distribution center. Eighty percent of the orders contain only three pick lines or fewer. The average is 2.18 pick lines per order. Thus, it makes sense to batch many orders into a job to reduce travel. Since most of the SKUs in the distribution center are small parts, each worker carries a tote while they pick the SKUs. A job is passed together with its associated tote from one worker to another in the order-picking process. No other handling equipment is needed in this case.

4.4.2 Order-Picking Policies Several factors may impact the productivity of bucket brigades in this distribution center. For example, the distribution of SKUs along the order-picking line will affect the throughput of bucket brigades. Furthermore, the distribution center may

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consider using wireless technology to reduce travel of workers. With wireless technology, each job is transmitted to electronic devices held by workers so that workers can start picking SKUs for the job without traveling to the start of the line. We consider the following four different order-picking policies: 1. Serial bucket brigades without wireless technology. Pick points are distributed from highest to lowest pick frequencies on one side of an aisle such that the pick point with the highest frequency is located nearest to the start of the aisle. All jobs are initiated at the start of the aisle. Workers follow the serial bucket brigade rule and are sequenced from slowest to fastest in the direction of work flow along the aisle. Figure 4.5a illustrates this policy. 2. Serial bucket brigades with wireless technology. Pick points are distributed from lowest to highest pick frequencies on one side of an aisle such that the pick point with the lowest frequency is located nearest to the start of the aisle. We assume the facility is equipped with wireless technology such that all jobs are initiated at their first pick points along the aisle. For example, if the first SKU requested by a job is located at section 23, then we start the job from section 23. Workers follow the serial bucket brigade rule and are sequenced from slowest to fastest in the direction of work flow along the aisle. See Fig. 4.5b for an illustration. 3. Cellular bucket brigades with slowest-to-fastest ordering of workers. Pick points are distributed on both sides of an aisle. A forward line is on one side of the aisle and a backward line is on the other side (see Fig. 4.5c). Pick points are distributed, from highest to lowest pick frequencies, first on the forward line and then on the backward line such that both pick points with the highest and the lowest frequencies are located nearest to the start of the aisle. All jobs are initiated at the start of the forward line. Workers follow the cellular bucket brigade rules and are sequenced from slowest to fastest in the direction of the forward line.

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Fig. 4.5 The four different order-picking policies. a Policy 1. b Policy 2. c Policy 3. d Policy 4

4. Cellular bucket brigades with fastest-to-slowest ordering of workers. Pick points are distributed, from highest to lowest pick frequencies, alternatively on the backward and the forward lines: The pick points with the highest frequency, third-highest frequency, fifth-highest frequency, and so on are distributed on the backward line. They are sequenced in this ordering in the reverse direction of the backward line such that the pick point with the highest frequency is located nearest to the start of the aisle (see Fig. 4.5d). On the other side of the aisle, the pick points with the second-highest frequency, fourth-highest frequency, sixthhighest frequency, and so on are sequenced in this ordering along the forward line such that the one with the second-highest frequency is located nearest to the start of the aisle. All jobs are initiated at the start of the forward line. Workers follow the cellular bucket brigade rules and are sequenced from fastest to slowest in the direction of the forward line. The disadvantage of Policy 1 is that the faster workers near the end of the aisle spend a lot of time walking rather than picking SKUs. This does not fully utilize system capacity as we want fast workers to spend more time picking rather than walking. Policy 2 overcomes this problem by distributing pick points with high pick frequencies near the end of the aisle. Thus, SKUs with high demand are picked by faster workers. Furthermore, it uses wireless technology to reduce

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unproductive travel of workers. We expect Policy 2 to be more productive than Policy 1. Policy 3 distributes SKUs in a way to reduce the probability of blocking in both directions. It corresponds to a cellular bucket brigade with more work content on the forward line than the backward line. Following the results of Lim (2011), workers should be sequenced from slowest to fastest in the forward direction. On the other hand, the backward line has more work content than the forward line under Policy 4. According to Lim (2011), workers should be sequenced from slowest to fastest in the backward direction (fastest to slowest in the forward direction). Policy 4 will keep the faster workers busy because pick points with high pick frequencies are located near the start of the aisle. In Fig. 4.5 the heights of the shaded bars represent the pick frequencies of the corresponding pick points. (a) SKUs are distributed on one side of an aisle. Workers are sequenced from slowest to fastest along the aisle. (b) Same as (a), but pick points are distributed from lowest to highest pick frequencies. (c) SKUs are distributed on both sides of an aisle. Workers are sequenced from slowest to fastest along the aisle. (d) Pick points with high pick frequencies are close to the start of the aisle. Workers are sequenced from fastest to slowest along the aisle.

4.4.3 Performance of Different Policies We compare the performance of all policies through simulations. Figure 4.6a shows the average throughput (number of pick lines picked per second) under different policies. For all policies, the throughput increases with the number of workers n. Policy 2 outperforms Policy 1, while Policy 4 is more productive than Policy 3. The results suggest that the throughput of cellular bucket brigades is significantly higher than that of serial bucket brigades. The improvement in throughput by cellular bucket brigades can be as large as 25% (for n = 20). Note that a cellular bucket brigade with fewer and slower workers can be significantly more productive than a serial bucket brigade, even if the latter is equipped with wireless technology to reduce travel. For example, the distribution center currently employs 14 workers for order-picking. Figure 4.6a suggests that a team of 12 workers (slower on average) under Policy 4 is about 6% more productive than a team of 14 workers (faster on average) under Policy 2. This implies that even with 14% reduction in labor, the throughput can still be improved by about 6% using cellular bucket brigades. Thus, cellular bucket brigades not only fulfill customer orders more efficiently, but also save the costs of labor and wireless technology. Figure 4.6b shows the average throughput of Policy 4 with different job sizes. As the job size b gets larger, there are more pick lines per job and so there is less unproductive travel. This results in higher average throughput. We observe similar results under other policies.

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Fig. 4.6 Throughput of different policies. a The average throughput of each policy increases with the number of workers. Policy 4 outperforms all other policies. For this graph, we set b = 8. b The average throughput of Policy 4 increases with the job size b. For both graphs, a = 3 (ft), w = 3 (ft/s), T = 15 (s), and b = 0.1

Since workers pick SKUs in both forward and backward directions in a cellular bucket brigade, a worker may be blocked by his successor (when the worker works forward) or by his predecessor (when the worker works backward). The chance of blocking increases with the time required to retrieve a SKU from a pick point. To simulate situations in which blocking occurs more frequently, we increase the value of T in Eq. 4.1. Figure 4.7a shows that when T = 30 s, Policy 3 outperforms Policy 4 as the team size n increases. This is because under Policy 4, pick points with high frequencies are located near the end of the backward line. As the time to retrieve a SKU from a pick point increases, blocking occurs more frequently near the end of the backward line. Furthermore, blocking is more likely to occur when there are more workers in the system. As a result, Policy 4 becomes less productive than Policy 3 as the team size n increases. However, cellular bucket brigades (Policies 3 and 4) are still more productive than serial bucket brigades (Policies 1 and 2). Figure 4.7b shows the average throughput of all policies when T = 60 s. Policy 3 remains the most productive among all policies. However, the problem of blocking under Policy 4 becomes so severe that the policy is outperformed by Policy 2 as the team size gets larger. Figures 4.6 and 4.7 suggest that Policy 4 is preferred if the time required to retrieve a SKU from a pick point is short. Otherwise, Policy 3 should be used. The results of this case study suggest that the ideas of cellular bucket brigades are promising for boosting productivity and reducing the costs of an order-picking system. In Fig. 4.7: (a) When T increases to 30 s, cellular bucket brigades are still more productive than serial bucket brigades. (b) When T increases to 60 s, Policy 3 gives the highest throughput. Policy 2 outperforms Policy 4 when the team size is large. For both graphs, b = 8, a = 3 (ft), w = 3 (ft/s), and b = 0.1.

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12

n

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

n

Fig. 4.7 Cellular bucket brigades remain productive when pick time increases

4.5 Conclusion The main contribution of this chapter is to investigate the effectiveness of cellular bucket brigades in a realistic order-picking setting. Under the design of cellular bucket brigades, SKUs are stored on both sides of an aisle. Workers pick SKUs to fulfill customer orders from one side of the aisle as they proceed in one direction. They pick SKUs, possibly for other customer orders, as they proceed in the reverse direction. This totally eliminates workers’ unproductive travel to receive more work inherent in traditional (serial) bucket brigades. Since workers work in both directions and they are required to walk across the aisle in a cellular bucket brigade, we discuss extended rules to coordinate the workers so that they can share work in an effective way. The extended rules remain easy for workers to remember and follow. Thus, it is straightforward to implement them in practice. We compare the average throughput of cellular and serial bucket brigades through computer simulations based on data from a distributor of service parts in North America. Our results suggest that cellular bucket brigades generally attain higher throughput than serial bucket brigades. The improvement in throughput by a cellular bucket brigade can be as high as 25% over a serial bucket brigade for a team of 20 workers. Even with fewer and slower workers, cellular bucket brigades can be more productive than serial bucket brigades that are equipped with wireless technology to reduce travel. For example, the throughput of a cellular bucket brigade with 12 workers can be 6% higher than that of a serial bucket brigade with 14 workers equipped with wireless technology. This suggests that cellular bucket brigades can not only boost productivity, but can also save the costs of labor and wireless technology. Our method of order-picking can be extended to a setting with multiple aisles, where SKUs are stored on both sides of each aisle. A global pick-path that traverses through all the aisles can be constructed. Workers pick SKUs from one

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side of the path when they travel in one direction, and they pick from the other side of the path when they travel in the reverse direction.

References Alligood KT, Sauer TD, Yorke JA (1996) Chaos: an introduction to dynamical systems. Springer, New York. ISBN 0-387-94677-2 Armbruster D, Gel ES (2006) Bucket brigades revisited: are they always effective? Eur J Oper Res 172(1):213–229 Armbruster D, Gel ES, Murakami J (2007) Bucket brigades with worker learning. Eur J Oper Res 176(1):264–274 Bartholdi JJ III, Eisenstein DD (1996a) A production line that balances itself. Oper Res 44(1): 21–34 Bartholdi JJ III, Eisenstein DD (1996b) The bucket brigade web page. http://www.BucketBrigades. com. Accessed 10 April 2011 Bartholdi JJ III, Eisenstein DD (2005) Using bucket brigades to migrate from craft manufacturing to assembly lines. Manuf Serv Oper Manag 7(2):121–129 Bartholdi JJ III, Hackman ST (2010) Warehouse and distribution science. http://www.warehousescience.com. Accessed 12 April 2011 Bartholdi JJ III, Bunimovich LA, Eisenstein DD (1999) Dynamics of two- and three-worker ‘‘bucket brigade’’ production lines. Oper Res 47(3):488–491 Bartholdi JJ III, Eisenstein DD, Foley RD (2001) Performance of bucket brigades when work is stochastic. Oper Res 49(5):710–719 Bartholdi JJ III, Eisenstein DD, Lim YF (2006) Bucket brigades on in-tree assembly networks. Eur J Oper Res 168(3):870–879 Bartholdi JJ III, Eisenstein DD, Lim YF (2009) Deterministic chaos in a model of discrete manufacturing. Nav Res Logist 56(4):293–299 de Koster R, Le-Duc T, Roodbergen KJ (2007) Design and control of warehouse order picking: a literature review. Eur J Oper Res 182(2):481–501 Eisenstein DD (2008) Analysis and optimal design of discrete order picking technologies along a line. Nav Res Logist 55(4):350–362 Frazelle EH (2002) World-class warehousing and material handling. McGraw-Hill, New York. ISBN 0-07-122686-9 Gue KR, Meller RD, Skufca JD (2006) The effects of pick density on order picking areas with narrow aisles. IIE Trans 38(10):859–868 Lim YF (2011) Cellular bucket brigades. Oper Res Forthcoming Lim YF, Wu Y (2011). Sharing work dynamically on U-lines with special skill requirements. Working paper, Lee Kong Chian School of Business, Singapore Management University, Singapore. Lim YF, Yang KK (2009) Maximizing throughput of bucket brigades on discrete work stations. Prod Oper Manag 18(1):48–59 van den Berg JP (1999) A literature survey on planning and control of warehousing systems. IIE Trans 31:751–762 Villalobos JR, Munoz LF, Mar L (1999a) Assembly line designs that reduce the impact of personnel turnover. In: Proceedings of IIE Solutions Conference, Phoenix, Arizona, USA Villalobos JR, Estrada F, Munoz LF, Mar L (1999b) Bucket brigade: a new way to boost production. Twin Plant News 14(12):57–61

Chapter 5

A Sequential Order Picking Policy for Shipping Large Numbers of Small Quantities of Goods Jiun-Yan Shiau

Abstract The recent growth in electronic commerce has brought new attention to order picking with many firms caught unprepared for the tactical requirements of their new retail markets. The internet has had an obvious influence on warehouse management. The internet-based stores are less likely to require physical place for selling. However, warehouses are still required to store the goods. On the Internet, direct contact with customers means many small orders; this is a different situation from stores which would normally order large numbers of goods. Simply stated, warehouses change from shipping large quantities of goods to shipping large numbers of small quantities of goods. The aim of this chapter is to describe a sequential order picking policy for the distribution center which ships large numbers of small quantities of goods. The order picking policy executes order picking and packing process synchronously. A nonlinear programming decision model is formed to minimize the remaining space of delivering boxes and the picking path of each order. A generic warehouse management system with the proposed sequential order picking function has been conceptually implemented to demonstrate the elimination of storage buffers and the reduction of operation time.

5.1 Introduction Studies on order picking of a warehouse management system (WMS) can be categorized into four main areas including layout design of storage facilities, product storage policy, pickers’ routing policy, and picking policy. For layout

J.-Y. Shiau (&) Department of Logistics Management, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_5,  Springer-Verlag London Limited 2012

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88 Table 5.1 Traditional commerce versus internet based electronic commerce

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Number of orders Order size

Traditional commerce

Internet based electronic commerce

Small Large

Large Small

design problem, two main issues have received the most attention. One is facility layout problem that deals with where to locate various areas such as receiving, storage, picking, and shipping. The other is internal layout design that determines numbers of blocks, numbers of aisles between blocks in picking area, length and width of aisles, etc. Product storage policies are usually classified as the following five types: random storage, closest open location storage, dedicated storage, full turnover storage, and class-based storage. Picker’s routing policy is to determine the shortest routes for pickers. There are five types of heuristic route policies in general: traversal (also called S-Shaped), return, mid-point, largest gap, and composite heuristic. Picking policy includes discrete, batch, zone, bucket brigade, and ware picking. Different from the four-order picking research areas above, there are also studies on different performance indexes, such as order fill rate (Rim and Park 2008), cost and throughput modelling (Russel and Meller 2003), and picking and deliveries (Shiau and Lee 2010) for order picking. Most existing research addresses various design and operating issues with an objective to reduce order fulfillment costs or to improve overall system performance in terms of three process decisions considered most often: (1) how to pick the stock-keeping units (SKUs), (2) how to store the SKUs, and (3) how to route the pickers in warehouses. The internet has had an obvious influence on warehouse management. Although internet-based stores are less likely to require physical place for selling, warehouses are still required to store the goods. On the internet, direct contact with customers means many small orders; this is a different situation from stores which would normally order large numbers of goods. In other words, warehouses change from shipping large quantities of goods to shipping large numbers of small quantities of goods. Table 5.1 indicates the comparison of these two cases. Take PChome 24 h shopping online in Taiwan as an example, where any order made between 8 o’clock in the morning to 10 o’clock in the evening will arrive to the customer in 24 h. To avoid delays, such a marketing strategy requires a highly supporting and efficient warehouse operation and management system. The particularity of order picking operations in the 24 h online-shopping industry is that the warehouse personnel choose different numbers and different sizes of mailing boxes according to the total volume of picked items in the buffering space. To optimizing space utilization, warehouse personnel will try to arrange the products into the mailing boxes with appropriate location and orientation. Shiau and Lee (2010) proposed a sequential order routing policy, which is one type of put systems, for a warehouse management system with such purpose. It is an order picking policy based on optimizing the usages of delivering boxes for customer orders. Put system, introduced by De Koster et al. (2007), is particularly popular in

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cases where a large number of customer orders have to be picked in a short time period. The put system consists of two major activities, order picking and packing process for distribution. Generally, picking is a process for physical dissemination of order information, locations, and quantities of products. Packing process is for dissemination of box size or loading configurations (Ackerman 1997). Normally, a put system can result in about 500 picks or even up to 1000 picks on average per order picker hour (De Koster et al. 2007). For considering order picking and packing processes simultaneously may be of crucial importance for contemporary warehouse management in internet-based electronic commerce. In our previous research (Shiau and Lee 2010), it only considered optimal usages of delivering boxes for order picking. Sometimes, it may generate a picking sequence with asks a picker to travel back and forth in a warehouse. This chapter describes an extensive research on sequential order picking policy, which is to reduce more order fulfillment costs by considering both optimizing the usages of delivering boxes and optimizing the routing of pickers. An optimal order picking sequence would be generated by considering locations and orientations of SKUs in a mailing box and storage locations of SKUs in a warehouse. The performance indexes are to maximize mailing boxes’ space utilization and to reduce travelling distances of a picker simultaneously.

5.2 Warehouse and Order Picking Policy In this section, we review the background concepts and related techniques that we will refer to in our proposed methodology. According to Lambert et al. (1998), the missions of a warehouse system could be to (1) achieve transportation economies, (2) achieve production economies, (3) take advantage of quality purchase discounts and forward buys, (4) support the firm’s customer service policies, (5) meet changing market conditions and uncertainties, (6) overcome the time and space differences that exist between producers and customers, (7) accomplish the least total cost logistics commensurate with a desired level of customer service, (8) support the just-in-time programs of suppliers and customers, (9) provide customers with a mix of products instead of a single product on each order, (10) provide temporary storage of material to be disposed or recycled, (11) provide a buffer location for trans-shipment. The pickup point and the delivery point in a warehouse are shortened as P/D point. There are 6 types of P/D points: 1. Type I. The pickup and delivery points are in the same location, but are not located in the corner of a warehouse (Fig. 5.1). 2. Type II. The pickup and delivery points are in the same location, and are located in the corner of a warehouse (Fig. 5.2). 3. Type III. The pickup and delivery points are on the same side, but not located in the same corner of a warehouse (Fig. 5.3).

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Fig. 5.1 Type I P/D Points

Fig. 5.2 Type II P/D Points

Fig. 5.3 Type III P/D Points

4. Type IV. The pickup and delivery points are on adjacent sides of a warehouse (Fig. 5.4). 5. Type V. The pickup and delivery points are on opposite sides of a warehouse (Fig. 5.5). 6. Type VI. The pickup and delivery points are in the diagonal corners of a warehouse (Fig. 5.6).

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Fig. 5.4 Type IV P/D Points

Fig. 5.5 Type V P/D Points

Fig. 5.6 Type VI P/D Points

Order picking can be defined as the activity by which a small number of goods are extracted from a warehousing system, to satisfy a number of independent customer orders (Molnar and Lipovszki 2005). Picking policies specify how operations are organized to retrieve orders. In terms of the picking policies, Ackerman (1990) divided order pickings into strict, batch and zone picking and proposed policies tailored to each case. Studies on order-related picking policies include (Petersen and Aase 2004):

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• How to pick SKUs • How to load SKUs • What are picking routs Operation time of order picking policies is effected by: • Processing time of searching SKUs • Processing time of retrieving SKUs • Moving time in a warehouse In strict picking, one single order is assigned at a time during a picking route, leading to lower service times and higher customer satisfaction. This policy is ideal when the order to be picked is quite small and easy to be found. Drawbacks of this policy include an increase in the overall transportation time and a cost penalty. Alternatively, the batch picking policy assigns to a picker more than one orders during a picking tour (Gibson and Sharp 1992; De Koster et al. 1999; Petersen 2000). The batch scheme may significantly reduce the total picking time, but introduces an additional cost for monitoring and separating the orders at a later stage. Zone picking assigns a picker to a designated picking zone, where the picker is responsible for those products that are in his/her picking zone in the warehouse. This scheme decreases the chances for destructions and mistakes, but a possible delay in one zone is then a threshold for the entire picking procedure for a big order. Frazelle and Apple (1994) further divided zone picking into: sequential zone, batch zone, and wave order pickings. Petersen (2000) suggested that in the sequential zone scheme the order integrity is maintained, in batch zone the orders are batched together and each picker collects the products within a zone, and in wave picking a group of orders is programmed in a precise time period. The routing that a picker traveled around aisles in a warehouse for picking all SKUs in an order list is called picking path. The planning of picking path is known as routing policies, which determine the picking sequences of SKUs (Petersen 1997). In general cases, a shorter picking path is better. A shorter picking path also means a shorter order picking processing time, which is an important performance index, in general. Petersen (1997) evaluated the following five different strategies of routing policies (as shown in Fig. 5.7): • Traversal strategy. This strategy leads to a route in which the aisles, which are to be visited, are totally traversed. Aisles where nothing has to be picked are skipped. Thus aisles are visited in the shape of an S. The picker enters an aisle from one end and leaves the aisle from the other end, starting at the left side of the warehouse. • Return strategy. The order picker travels in the aisle but always returns to the aisle head where he started. • Midpoint strategy. The order picker travels in the aisle but always returns to the aisle head where he started. The point of return in the aisle is determined by the aisle midpoint.

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Fig. 5.7 Five routing strategies (Petersen 1997)

• Largest gap strategy. The order picker travels in the aisle but always returns to the aisle head where he started. The point of return in the aisle is determined by the largest gap between two sequential item locations. • Composite strategy. A strategy which combines two or more strategies above. Locations of P/D points also have a considerable influence on order picking performance. Normally, type I and type V P/D points are better designs if a warehouse is wide and relatively only few SKUs have to be picked in one routing. Warehouse design (including decisions on where the stock keeping units (SKUs) are to be located within the warehouse) and efficient policies for picking (allocation of items to pickers) and routing (determination of the route of a single picker) within warehouses hold a large potential for cost savings. Petersen and Aase (2004) showed that picking and storage strategies are closely interrelated, implying that decisions on the storage policy have a major influence on order picking performance. Molnar and Lipovszki (2005) propose a two-stage scheduling model. The model aims at minimizing the labor cost and picking routing.

5.3 Sequential Order Picking Policy The proposed sequential order picking policy consists of three elements: mailing box selection, loading configuration, and picking routing (as shown in Fig. 5.8). First, mailing box selection, which is a mathematic model used to determine the required sizes and numbers of boxes. The second element is called loading configuration. Based on customer orders, another mathematic model is built to simulate the position of each unit box within a mailing box. The third element is called

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Fig. 5.8 Sequential order picking policy

picking routing. Type I P/D points warehouse with a TSP (travelling salesman problem)-based model is utilized to plan the shorter routing distances for a picker. The proposed model integrated those three elements as a nonlinear programming model to generate an order picking sequence. The picking sequence is then deployed to a mobile barcode reader. With the sequence in the barcode reader, an operator can pick and pack the order sequentially. The merit of the proposed algorithm is that it allows picking process and packing process to be done simultaneously.

5.3.1 Terminology Parameters N Number of unpacked items m Number of boxes M A very large number The length of item i pi The width of item i qi ri The height of item i Lj The length of box j Wj The width of box j Hj The height of box j tik The storage distance between item i and item k

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Variables sij 1 if item i is put into box j; 0 otherwise 1 if box j is used; 0 otherwise nj xi x-axis position of the front-left bottom corner of item i be assigned y-axis position of the front-left bottom corner of item i be assigned yi zi z-axis position of the front-left bottom corner of item i be assigned 1 if the length of item i is parallel to x-axis of the box; 0 otherwise lxi lyi 1 if the length of item i is parallel to y-axis of the box; 0 otherwise lzi 1 if the length of item i is parallel to z-axis of the box; 0 otherwise wxi 1 if the width of item i is parallel to x-axis of the box; 0 otherwise wyi 1 if the width of item i is parallel to y-axis of the box; 0 otherwise wzi 1 if the width of item i is parallel to z-axis of the box; 0 otherwise hxi 1 if the height of item i is parallel to x-axis of the box; 0 otherwise hyi 1 if the height of item i is parallel to y-axis of the box; 0 otherwise hzi 1 if the height of item i is parallel to z-axis of the box; 0 otherwise aik 1 if item i is on the left side of item k bik 1 if item i is on the right side of item k cik 1 if item i is behind item k dik 1 if item i is in front of item k eik 1 if item i is below item k 1 if item i is above item k fik ui picking sequence of item i gik 1 if item i is picked preceding to item k; 0 otherwise

5.3.2 The Nonlinear Programming Model The objective function: The objective of this model is to minimize the total unused space of the mailing boxes selected and to minimize the picker routing distance (i.e., the third segment in Eq. 5.1). The total unused space is equal to the sum of the total volumes of boxes (i.e., the first segment in Eq. 5.1) minus the total volumes of items (i.e., the second segment in Eq. 5.1). $ % m N N X N X X X Lj  Wj  Hj  nj  pi  q i  r i þ tik  gik ð5:1Þ Minimize j¼1

i¼1

i¼0 k¼1

The constraints: Constraints (5.2–5.7) ensure that items do not overlap each other. For instance, if item i is on the left side of item k, x-axis position of the front-left bottom corner of item k (i.e., the right-hand side of Eq. 5.2) must be larger than x-axis position of the front-left bottom corner of item i plus its size of item i is parallel to x-axis of the box (i.e., the left-hand side of Eq. 5.2).

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xi þ pi  lxi þ qi  wxi þ ri  hxi  xk þ ð1  aik Þ  M xk þ pk  lxk þ qk  wxk þ rk  hxk  xi þ ð1  bik Þ  M yi þ qi  wyi þ pi  lyi þ ri  hyi  yk þ ð1  cik Þ  M yk þ qk  wyk þ pk  lyk þ rk  hyk  yi þ ð1  dik Þ  M zi þ ri  hzi þ qi  wzi þ pi  lzi  zk þ ð1  eik Þ  M zk þ rk  hzk þ qk  wzk þ pk  lzk  zi þ ð1  fik Þ  M

8i; k; i 6¼ k 8i; k; i 6¼ k 8i; k; i 6¼ k 8i; k; i 6¼ k 8i; k; i 6¼ k 8i; k; i 6¼ k

ð5:2Þ ð5:3Þ ð5:4Þ ð5:5Þ ð5:6Þ ð5:7Þ

Checking for overlap is necessary only if a pair of items is placed in the same mailing box, and checking step is taken care of by constraint (5.8). Constraints (5.9–5.10) guarantee that each item will be placed in exactly one mailing box. aik þ bik þ cik þ dik þ eik þ fik  sij þ skj  1 m X

sij ¼ 1

8i; k; j; i 6¼ k

8i

ð5:8Þ ð5:9Þ

j¼1 N X

sij  M  nj

8j

ð5:10Þ

j¼1

Constraints (5.11–5.13) ensure that all the items placed in a mailing box fit within the physical dimensions of the box. For instance, if item i is placed in mailing box j, the size of item i is parallel to x-axis of the box (i.e., the left-hand side of Eq. 5.11) must be smaller than the size of length of box j (i.e., the righthand side of Eq. 5.11). xi þ pi  lxi þ qi  wxi þ ri  hxi  Lj þ ð1  sij Þ  M

8i; j

ð5:11Þ

yi þ pi  lyi þ qi  wyi þ ri  hyi  Wj þ ð1  sij Þ  M

8i; j

ð5:12Þ

zi þ pi  lzi þ qi  wzi þ ri  hzi  Hj þ ð1  sij Þ  M

8i; j

ð5:13Þ

Constraints (5.14–5.19) ensure that the orientation of each item inside a mailing box is proper. For instance, the length of item i is only parallel to x-axis, y-axis, or z-axis of the box (i.e., Eq. 5.14), whereas lxi , lyi , and lzi are binary variables.

.

lxi þ lyi þ lzi ¼ 1

ð5:14Þ

wxi þ wyi þ wzi ¼ 1

ð5:15Þ

hxi þ hyi þ hzi ¼ 1

ð5:16Þ

lxi þ wxi þ hxi ¼ 1

ð5:17Þ

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lyi þ wyi þ hyi ¼ 1

ð5:18Þ

lzi þ wzi þ hzi ¼ 1

ð5:19Þ

lxi ; lyi ; lzi ; wxi ; wyi ; wzi ; hxi ; hyi ; hzi ; aik ; bik ; cik ; dik ; eik ; fik ; sij ¼ 0 _ 1 xi ; yi ; zi  0

ð5:20Þ ð5:21Þ

Constraints (5.22–5.23) ensure that location of each item is visited only once, so that only one item will be picked after a participated item i (i.e., Eq. 5.22), and also only one item will be picked before a participated item k (i.e., Eq. 5.23). Constraint (5.24) guarantees that no cycle routing is planned. For instance, if item 1 is picked before item 2, and item 2 is picked before item 3, then item 3 is not possible to be picked before item 1. N X gik ¼ 1 8k; k 6¼ i ð5:22Þ i¼1 N X

gik ¼ 1

8i; i 6¼ k

ð5:23Þ

k¼1

ui  uk þ ðN þ 1Þ  gik  N

8i; k; i 6¼ k

ð5:24Þ

ui  N

8i

ð5:25Þ

uk  N

8k

ð5:26Þ

ui  0 8i

ð5:27Þ

uk  0 8k

ð5:28Þ

Constraints (5.29–5.32) ensure that items placed in the lower layer of a mailing box are visited and picked in advance. If item i is placed in lower layer to item k (i.e., eik = 1 and fik ¼ 0), then Eqs. 5.22 and 5.23 hold. If there are some items placed below item i (i.e., sum of fik is larger than 0), then z-axis position of the front-left bottom corner of item i to be assigned must be larger than 0 (i.e., Eq. 5.31 and 5.32). ð5:29Þ eik  ðui  uk Þ  0 8i; k; i; i 6¼ k fik  ðui  uk Þ  0 M

N X

8i; k; i; i 6¼ k

fik  zi  0

ð5:30Þ

8i; i 6¼ k

ð5:31Þ

8i; i 6¼ k

ð5:32Þ

k¼1

M  zi 

N X k¼1

fik  0

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Fig. 5.9 Picked items versus their complexities

The solution The order picking sequence is equal to the solutions of ui’s in ascending order. For example, if u1 = 2, u2 = 4, u3 = 3, u4 = 1, and u5 = 5, it means the order picking sequence is 0(P/D) ? item 4 ? item 1 ? item 3 ? item 2 ? item 5 ? 0(P/D). Complexity of the model Complexity of the proposed model is effected by number of picked items and number of mailing boxes. The more the items picked in a route, the more the mailing boxes expected and the more complex the proposed model will be. The number of variables and number of constraints of the proposed model are increasing exponentially when number of picked items are increasing. Fig. 5.9 illustrates the trends of picked items versus their complexities in terms of number of variables and number of constraints. The currently proposed model is a nonlinear programming model. Sometimes, a nonconvex solution set will be generated by a given nonlinear programming model. In that case, without an appropriate search algorithm, it might only be able to find a local optimal solution set instead of a global optimal solution set for the nonlinear programming model. To date, we are still in the conceptual deisgn stage on this research project. We have solved several generated cases to conclude the correctness of the proposed model in terms of maximizing mailing boxes’ space utilization and reducing travelling distances of a picker simultaneously. Lingo 11.0 API was utilized to solve the nonlinear programming models we generated. Sometimes, we have to solve a model a number of times in order to get a global solution set. A simplified example is described in the next section to demonstrate the merit of the proposed model. A more efficient algorithm, which is necessary in practice for solving this proposed model, is still under development.

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Table 5.2 Customer orders Item # Required qty

Volume (cm3) Length (cm)

Width (cm)

Height (cm)

1 2 3 4 5 6

15 20 20 15 25 10

10 25 20 10 25 25

5 8 10 6 6 10

A7 C3 D13 F9 G6 K5

1 1 1 1 1 1

Storage

Table 5.3 Available mailing boxes

Box #

Length (cm)

Width (cm)

Height (cm)

1 2 3 4 5 6

25 30 20 30 25 25

35 30 20 30 25 25

30 30 20 20 20 20

Table 5.4 Item numbers versus picking sequence numbers

Itemi

–

1

2

3

4

5

6

Storage ui

P/D 0

A7 1

C3 2

D13 3

F9 4

G6 5

K5 6

5.4 Numerical Example A customer order is received by a 24-h online shopping store. Assume that there are 6 items ordered. The 6 items are identified as item 1–6. The aim of distribution center (DC) of the online shopping store is to deliver items to customers within 24 h. Therefore, an efficient order picking mechanism for its warehouse management system (WMS) is desired. Table 5.2 illustrates the customer orders. There are six different sizes of mailing boxes available (as shown in Table 5.3). Each item in the customer orders is assigned with an item number and a picking sequence number (as shown in Table 5.4). The P/D point of the warehouse is assigned with a picking sequence number only. Figure 5.10 illustrates layout of the warehouse, and the storage locations of the ordered items and related aisles are illustrated in Fig. 5.11. The distance matrix between storage locations of items are as shown in Table 5.5. The proposed nonlinear programming model is implemented by Lingo11.0. A global optimal solution for the demonstration above is generated. The objective is 8417. Mailing boxes 1 and 3 are selected. Items 2, 5, and 6 are placed in mailing

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Fig. 5.10 Warehouse layout

Fig. 5.11 Storage location and aisles

Table 5.5 Distance matrix of items ðtik Þ(unit: meter) Picking point 0(P/D) 1(A7) 2(C3) 3(D13)

4(F9)

5(G6)

6(K5)

0(P/D) 1(A7) 2(C3) 3(D13) 4(F9) 5(G6) 6(K5)

6 17 16 8 0 3 18

9 19 13 11 3 0 15

14 22 16 16 18 15 0

0 14 16 4 6 9 14

14 0 12 11 17 19 22

16 12 0 14 16 13 16

4 11 14 0 8 11 16

Table 5.6 Solutions of loading configuration Boxðsij Þ Position of the front-left bottom corner Itemi 1 2 3 4 5 6

3 1 3 3 1 1

xðxi Þ

yðyi Þ

zðzi Þ

0 0 0 5 0 0

0 0 10 0 29 0

0 0 0 0 0 20

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Itemi

x-axis

y-axis

z-axis

1 2 3 4 5 6

5 25 20 10 25 10

10 28 10 6 6 25

15 20 20 15 25 10

Fig. 5.12 3D perspective figure of loading configuration

Fig. 5.13 The travelling and picking sequence

box 1. Items 1, 3, and 4 are placed in mailing box 3. The loading configuration and orientations of each item in a mailing box are as shown in Tables 5.6 and 5.7. A 3D perspective figure of loading configuration is illustrated in Fig. 5.12. The travelling and picking sequence is 0(P/D) ? 3(D13) ? 1(A7) ? 2(C3) ? 6(K5) ? 4(F9) ? 5(G6) ? 0(P/D) (as shown in Fig. 5.13).

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5.5 Conclusion Today, a warehouse management system normally consists of separated modules for order picking functions and packing functions. It would cause the need for extra storage buffers if such a WMS was utilized in a factory that required a put system. A put system is a kind of order picking system of a warehouse management system for a large numbers of customer orders that have to be picked in a short time window. Such a system not only involves the process of clustering and scheduling customer orders but also the process of distribution. In this chapter, a sequential picking policy is proposed to replace independent picking and packing functions of a system. It is also implemented as a nonlinear programming model. The model consists of three elements: container selection, loading configuration, and routing sequence. A generic warehouse management system with the proposed sequential order picking function is implemented to demonstrate the merits of our idea. As the simplified example shown in the previous section, the system not only helps eliminate the storage buffer but also reduces the operation time. For further research, different types of P/D points can be considered in the model or include other concerns in picking operation such as expiration date of SKUs and weight restriction of packed containers.

References Ackerman KB (1990) Practical handbook of warehousing, 3rd edn. Van Nostrand Reinhold, New York. ISBN 0442005571 Ackerman KB (1997) Practical handbook of warehousing, 4th edn. Chapman & Hall, New York. ISBN 0412125110 De Koster MBM, Van der Poort ES, Wolters M (1999) Efficient order batching methods in warehouses. Int J Prod Res 37(7):1479–1504 De Koster R, Le-Duc T, Roodbergen KJ (2007) Design and control of warehouse order picking: a literature review. Eur J Oper Res 182:481–501 Frazelle EH, Apple JM (1994) Warehouse operations. In: Tompkins JA, Harmelink DA (eds) The distribution management handbook. McGraw-Hill, New York, pp 22.1–22.36 Gibson DR, Sharp GP (1992) Order batching procedures. Eur J Oper Res 58:57–67 Lambert DM, Stock JR, Ellram LM (eds) (1998) Fundamentals of logistics management. McGraw-Hill, Singapore Molnar B, Lipovszki G (2005) Multi-objective routing and scheduling of order pickers in a warehouse. Int J Simul 6(5):22–32 Petersen CG (1997) An evaluation of order picking routing policies. Int J Oper Prod Manag 17(1):1096–1111 Petersen CG (2000) An evaluation of order picking policies for mail order companies. Prod Oper Manag 9(4):319–335 Petersen CG, Aase G (2004) A comparison of picking, storage, and routing policies in manual order picking. Int J Prod Econ 92:11–19 Rim SC, Park IS (2008) Order picking plan to maximize the order fill rate. Comput Ind Eng 55:557–566

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Russel ML, Meller RD (2003) Cost and throughput modeling of manual and automated order fulfilment systems. IIE Trans 35:589–603 Shiau JY, Lee MC (2010) A warehouse management system with sequential picking for multi-container deliveries. Comput Ind Eng 58(3):382–392

Chapter 6

Order Batching in Order Picking Warehouses: A Survey of Solution Approaches Sebastian Henn, Sören Koch and Gerhard Wäscher

Abstract Order picking is a warehouse function dealing with the retrieval of articles from their storage locations in order to satisfy a given demand specified by customer orders. Of all warehouse operations, order picking is considered to include the most cost-intensive ones. Even though there have been various attempts to automate the picking process, manual order picking systems are still prevalent in practice. This chapter will focus on order batching, a central planning issue in order picking systems. Order batching has been proven to be pivotal for the efficiency of order picking operations. Improved order batching reduces the total picking time required to collect the requested articles. According to experience from practice, this can result in significant savings of labour cost and in a reduction of the customer order’s delivery lead time. The aim of this contribution is to provide comprehensive insights into order batching by giving a detailed state-of-the-art overview of the different solution approaches which have been suggested in the literature. Corresponding to the available publications, the emphasis will be on static order batching. In addition to this, the chapter will also review the existing literature for variants and extensions of static order batching. Furthermore, solution approaches for dynamic order batching problems will be presented.

S. Henn (&)  S. Koch  G. Wäscher Faculty of Economics and Management, Otto-von-Guericke University, Postbox 4120, 39016 Magdeburg, Germany e-mail: [email protected] S. Koch e-mail: [email protected] G. Wäscher e-mail: [email protected]

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6.1 Introduction Order picking is a warehouse function dealing with the retrieval of articles from their storage locations in order to satisfy a given demand specified by customer requests (Petersen and Schmenner 1999). Order picking arises because incoming articles are received and stored in (large-volume) unit loads while (internal or external) customers order small volumes (less-than-unit loads) of different articles. It is a function critical to each supply chain, since underperformance results in an unsatisfactory customer service (long processing and delivery times, incorrect shipments) and high costs (labour cost, cost of additional and/or emergency shipments). Of all warehouse operations, order picking is considered to include the most cost-intensive ones. According to Frazelle (2002) up to 50% of the total warehouse operating costs can be attributed to order picking. Drury (1988, also see Tompkins et al. 2003) and Coyle et al. (1996) even estimate these costs up to 60 and 65%, respectively. The large proportion of order picking (operations) costs originates from the fact that order picking systems still involve the employment of human operators on a large scale, despite that there have been various attempts to automate the picking process. Among such manual order picking systems, picker-to-parts systems can be considered as the most important ones, where order pickers move through the warehouse and collect the requested articles (Wäscher 2004). For an efficient organization of the corresponding picking operations, order batching, i.e. the grouping of customer orders into picking orders, has been proven to be pivotal (de Koster et al. 1999a). The aim of this contribution is to provide a comprehensive state-of-the-art review of solution approaches for order batching in picker-to-parts systems. The focus of the paper will be on static order batching, which is based on the assumption that all customer orders are known in advance. Furthermore, this paper will also review the existing literature for variants and extensions of static order batching (e.g. wave picking, batching and sequencing problems). Finally, solution approaches for dynamic order batching problems (e.g. time window batching) will be presented. The remainder of this chapter is organized as follows: In the next section we give a brief introduction into order picking systems, the corresponding planning issues and objectives. Section 6.3 describes the (static) order batching problem. Furthermore, a mathematical model for this problem and an exact solution approach are presented. Since the exact solution approach is limited to problems of small size, the application of heuristic solution approaches is necessary. The Sects. 6.4 and 6.5 are dedicated to these heuristic approaches, Sect. 6.4 to constructive solution approaches and Sect. 6.5 to the application of metaheuristics to the order batching problem. The performance of these algorithms is discussed in Sect. 6.6. In addition, several variants of the static order batching problem will be presented. In Sect. 6.7 wave picking will be discussed, whereas Sect. 6.8 reviews solution approaches for batching and sequencing problems. Dynamic order batching is described in Sect. 6.9. The chapter concludes with a summary.

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Fig. 6.1 Classification of order picking systems (based on de Koster et al. 2007)

6.2 Fundamentals 6.2.1 A Classification of Order Picking Systems Two kinds of order picking systems can be distinguished in practice (cf. Fig. 6.1), namely manual order picking systems, which employ human operators, and technical systems, in which the process of retrieving articles from the warehouse is completely automated. The first group of systems can be further differentiated into picker-to-parts systems and parts-to-picker systems. In picker-to-parts systems the order picker walks or rides (e.g. on an automated guided vehicle) through the picking area, stops at the storage locations of the respective articles, and removes the required number of article units/items. In low-level picker-to-parts systems the items can be removed from pallets or bins placed on the warehouse floor, or from low-level racks which are directly accessible by the order picker. In high-level (or man-aboard) systems the picking area consists of high storage racks and a crane or a vehicle with a hoisting platform moves the order picker to the storage locations from which items have to be picked. In parts-to-picker systems automated storage and retrieval systems (AS/RS) retrieve unit loads (pallets or bins) from the warehouse and deliver them at a transfer site (depot) where one or several stationary order pickers are located. The order pickers remove the requested items, and the AS/RS returns the unit load to its location in the warehouse. According to de Koster et al. (2007), more than 80% of all order picking systems in Western Europe are low-level picker-to-parts systems. Therefore, in this chapter we will concentrate on this type of order picking system.

6.2.2 Standard Layouts of Manual Order Picking Systems In a standard layout of a (low-level) picker-to-parts system, the storage locations (bays) are of identical size. The bays are arranged on both sides of straight picking aisles of equal length and width, which run in parallel to each other and

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Fig. 6.2 Single block layout

perpendicular to the front of the picking area. The depot or input/output (I/O) point is the place where the order pickers enter the picking area and where they return to in order to deposit the picked items. Order pickers are enabled to change from one picking aisle to another by means of two cross aisles, one at the front and one at the rear of the picking area. The conditions described here constitute a so-called single block layout which is depicted in Fig. 6.2. Depending on the types of articles, their sizes, weights, demands etc. the introduction of additional cross aisles might increase the efficiency of the system, resulting in a multi-block layout. Such layouts and other, non-standard layout types will not be discussed here any further; instead, we refer to the literature (Pohl et al. 2009).

6.2.3 Picking Devices and Pick Lists When covering the distances within the picking area on foot, order pickers typically utilize devices like roll pallets or carts, which they pull or push along with them through the warehouse and on which they deposit the picked items until they finally return to the depot. Likewise, when travelling on a vehicle, there will be space available on the vehicle for intermediate storage where the picked items can be placed. Consequently, the required items are collected on tours through the warehouse, where the number of stops on each tour is limited by the available space of the picking device on the one hand and by the capacity requirements of the items to be picked on the other. On their tours through the warehouse, order pickers are guided by pick lists. A pick list comprises a set of order lines, each one identifying a particular article, the quantity of this article requested by a customer and the respective storage location. The order lines are already sorted into the sequence according to which the order picker is meant to collect the items.

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6.2.4 Operative Planning Issues in Manual Order Picking Systems Given a picking area with a fixed layout, the following essential planning issues can be identified (de Koster et al. 2007): • storage assignment, i.e. the assignment of articles to storage locations; • zoning, i.e. the establishment of (work) zones, to which order pickers are restricted in their picking operations; • order consolidation, i.e. the transformation of customer orders into picking orders; • picker routing, i.e. the determination of sequences according to which the items have to be picked and the identification of the corresponding paths in the warehouse. Storage assignment and zoning represent medium-term planning issues and rather are part of the tactical planning level, while order consolidation and picker routing refer to the operative level in the first place. Order consolidation can be organized in two different ways. As for discrete order picking (pick-by-order), each tour comprises the items of a single customer order only; while for batch picking (pick-by-batch) items of several customer orders can be collected simultaneously on a single tour. In the latter case, the process of grouping a set of customer orders into picking orders is referred to as order batching. With respect to the availability of information concerning customer orders, order batching can be distinguished into static (off-line) batching and dynamic (on-line) batching (Yu and de Koster 2009). In the static case it is assumed that the set of customer orders is self-contained and complete information about its composition (i.e. for each customer order the corresponding order lines are known) is available when the batching decision is taken. In the dynamic case customer orders arrive at different points in time while the picking process is already being executed. Picker routing deals with the determination of the sequence in which the items of a given picking order are to be picked and the identification of the corresponding (shortest) tour for the order picker which connects the respective article locations among each other and with the depot. Ratliff and Rosenthal (1983) have presented an exact (polynomial-time) algorithm for this problem, which is hardly ever used in practice, though. Order pickers do not seem to be willing to follow the routes provided by the algorithm, because they are not always straightforward and sometimes even confusing (de Koster et al. 1999a). Instead, so-called routing strategies are applied, which may be looked upon as heuristic solution methods, not necessarily giving tours of minimal length but of plausible patterns, easy to memorize and easy to follow. In this way, the risk of missing an item to be picked is reduced, which may be an aspect more important than a small reduction of the tour length.

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Fig. 6.3 Routing strategies in a single block layout (Hall 1993; Petersen and Schmenner 1999; Roodbergen and de Koster 2001)

In Fig. 6.3, several customary routing strategies (Return, S-Shape, Largest Gap and Combined) are depicted. The black rectangles symbolize the storage locations from which items have to be picked (pick locations). When proceeding according to the Return strategy, the order picker enters each aisle in which an item has to be picked from the front cross aisle, walks up to the most distant pick location in this aisle and then returns to the front cross aisle. As for S-Shape routing, the order picker successively traverses each aisle entirely if it contains at least one pick location. Correspondingly, the first aisle is entered from the front cross aisle, the second one from the rear cross aisles, etc. With Largest Gap routing the aisles are entered from front and back aisle in such a way that the non-traversed distance between two adjacent pick locations is maximal. Only the leftmost and the rightmost aisle which contain items to be picked are traversed entirely. The Combined strategy integrates elements of the S-Shape and Return strategy. Aisles may be traversed entirely or may be entered and left from the same cross aisle. The respective solutions are usually provided by application of dynamic programming. We note that—from a planning-theoretical point of view—it would be desirable to solve the order consolidation problem and the picker routing problem simultaneously. Such an approach, however, does not appear to be very realistic for practical purposes, due to the complexity and the size of the problem which would have to be solved (Wäscher 2004). In practice, decisions are made sequentially

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(de Koster et al. 2007). Here, we will assume that a decision has already been made in favour of a particular routing strategy, which will serve as input for a subsequent order batching decision.

6.2.5 Planning Objectives of Operative Planning Involving a large proportion of time-consuming manual operations, order picking is considered to be the most labour-cost-intensive function in a warehouse (Drury 1988). According to experience from practice, a reduction of picking times results in significant savings of labour cost, since it does not only allow for reducing the necessary regular working hours of the pickers, but also for reducing expensive overtime or even for downsizing the (picker) workforce. Furthermore, since the picking time is an integral part of the delivery lead time, a reduction of the picking times may also immediately result in an improvement of the customer service provided by the warehouse (Henn et al. 2010). Consequently, the minimization of picking times is of vital importance for controlling the picking processes in a warehouse efficiently. Given a particular picking order, the time the order picker takes for the completion of a tour on which the respective items are collected, will be called (picking) order processing time. Essentially, it consists of the following components: • travel time, i.e. the time the order picker spends travelling from the depot to the first pick location, between the pick locations and from the last pick location to the depot; • search time, i.e. the time required for the identification of articles; • pick time, i.e. the time needed for moving the items from the corresponding article location onto the picking device; • setup time, i.e. the time consumed by administrative and setup tasks at the beginning and end of each tour (Chew and Tang 1999), including the receipt of the pick list and of an empty picking device at the beginning of a tour and the return of the picking device at the depot at the end of the tour (van Nieuwenhuyse and de Koster 2009). Figure 6.4 depicts how the order processing time is typically composed of these time elements in practice. Among these components, the travel time consumes the major proportion. The other components can either be looked upon as constants (search times and pick times) or as negligible (setup times). In other words, the travel time makes up for the major influencing factor of the order processing time. Furthermore, assuming that the order pickers travel at a constant speed, minimization of the total travel time is equivalent to (and can be achieved by) the minimization of the total length of all picker tours necessary to collect all items of a given set of customer orders (Jarvis and McDowell 1991).

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Fig. 6.4 Typical composition of the order processing time (Tompkins et al. 2003)

6.3 Order Batching 6.3.1 Problem Definition and Analysis As has been mentioned before, the required items are collected on tours through the warehouse, where the number of stops on each tour is limited by the available capacity of the picking vehicle/picking device on the one hand and by the capacity requirements of the items to be picked on the other. Customer orders can be combined (batched) into picking orders (batches) until the capacity of the device is exhausted. For the definition of the capacity of the picking device various criteria are used in the literature, e.g. the capacity can be expressed in the number of customer orders (Le-Duc and de Koster 2007), or in the number of items (Bozer and Kile 2008; Henn et al. 2010). Splitting of customer orders, i.e. the inclusion of items from the same customer order in several picking orders, is usually prohibited since it would result in an additional, not-acceptable sorting effort. Based on these conclusions, the (static) order batching problem can be defined as follows: How, given routing strategy and capacity of the picking device, can a given set of customer orders with known storage locations be grouped (batched) into picking orders such that the total length of all picker tours is minimized? (Wäscher 2004). For a simple example, namely the case of two customer orders and Largest Gap routing, Fig. 6.5 demonstrates that benefits may arise, indeed, from collecting the items requested by two customer orders on a single tour instead of collecting them on two separate tours. Figures 6.5a and b depict the two tours based on separate picking, while Fig. 6.5c illustrates the single tour resulting from the batching of both customer orders. The length of the resulting tour is obviously shorter than the total length of the two separate tours. As can easily be concluded further, the benefits will be particularly large whenever the items of the batched customer orders have nearby locations. The order batching problem as described above is known to be NP-hard (in the strong sense) if the number of orders per batch is larger than two. This result can be proven by showing that the partition-into-triangles problem (Garey and Johnson 1979)

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Fig. 6.5 Benefits arising from the combination of two customer orders

is polynomial reducible to the order batching problem (Gademann and van de Velde 2005). The order batching problem features some similarities with the capacitated vehicle routing problem, however differs from that with respect to the (customer) order integrity condition, i.e. all items of a customer order must be picked on the same tour. Thus, traditional solution approaches to the capacitated vehicle routing problem cannot be applied directly to the order batching problem (Bozer and Kile 2008).

6.3.2 Model Formulation A straightforward optimization model of the order batching problem has been introduced by Gademann and van de Velde (2005). In this model, all feasible batches (i.e. all batches which do not exceed the capacity of the picking device) are considered explicitly. For the presentation of the model, the following parameters are introduced: J set of customer orders, where J = {1,. . ., n}; C capacity of the picking device; cj capacity utilization required by customer order j ðj 2 J Þ; I set of all feasible batches; di length of a picking tour in which all orders of batch i ði 2 I Þ are collected.

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Let ai = (ai1,. . ., ain) be a vector of binary entries aij, stating whether customer  order j ðj 2 J Þ is included in batch i aij ¼ 1 or not aij ¼ 0 . Then the set of feasible batches is characterized by the fact that the capacity constraint is not violated, therefore X cj aij  C; 8 i 2 I ð6:1Þ j2J

holds. Furthermore, the following decision variables are used: xi

binary decision variable, xi = 1, if batch i ði 2 IÞ is chosen, or xi = 0, otherwise. The optimization model can then be formulated as follows: X d i xi min

ð6:2Þ

i2I

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aij xi ¼ 1;

8 j 2 J;

ð6:3Þ

8i 2 I:

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xi 2 f0; 1g;

The sets of constraints (6.3) and (6.4) ensure that a set of batches is chosen in such a way that each customer order is included in exactly one of these batches. Another optimization model for order batching in a single block layout has been introduced by Bozer and Kile (2008). However, the model is only applicable for the special case of S-Shape routing in narrow aisles where reversing is not permitted. With respect to these special conditions we do not discuss any further details here.

6.3.3 Solution Approaches It is important to note that the number of possible batches and, consequently, the number of binary decision variables in the optimization system (6.2–6.4) grows exponentially with the number of customer orders. Henn et al. (2010), e.g., report problem instances consisting of 40 customer orders in which the number of feasible batches is larger than 350,000 for a warehouse with 900 storage locations. Thus, a solution approach which involves the application of commercial LP/IP solvers to an explicitly formulated optimization system (6.2–6.4) covers only a limited range of problem instances. Henn et al. (2010) were only able to solve problem instances with at most 50 customer orders to optimality. For a large number of instances, the LP/IP solver was only able to generate feasible solutions but was unable to prove their optimality, since memory restrictions of the used PC were violated.

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Based on the model formulation (6.2–6.4), Gademann and van de Velde (2005) present a column generation approach which starts from an initial set of batches and consecutively adds new batches as long as they improve the solution. The initial set of batches is determined by an iterated descent algorithm. Then—limited to these batches—the linear relaxation of the optimization system (6.1–6.4) is solved. In each iteration, a pricing algorithm generates one or several batches whose reduced costs are minimal. These batches are added to the (relaxed) model and the model is solved again. These steps are repeated until no further batch can be identified which would improve the current solution. If the obtained solution is not integer, a branch-and-price algorithm is used in order to generate an optimal integer solution. Gademann and van de Velde (2005) carried out an extensive numerical study on their exact solution approach. In their experiments they focused on a warehouse with 400 storage locations. They investigated several problem classes with up to 30 customer orders and 10 customer orders as a maximal capacity of the picking device. The algorithm was able to provide optimal solutions for almost all considered instances within a few minutes. However, the computing times increased significantly with an increasing number of customer orders; therefore the algorithm is not applicable for larger problem instances. Due to the fact that the existing exact solution approaches to the order batching problem have only a limited applicability, real-world problems will have to be solved by means of heuristic approaches. These heuristic approaches can be classified into two groups, namely constructive solution approaches and metaheuristics.

6.4 Constructive Solution Approaches Constructive solution approaches can be distinguished into priority rule-based algorithms, seed algorithms, savings algorithms and data mining approaches.

6.4.1 Priority Rule-Based Algorithms Priority rule-based algorithms consist of a two-step procedure: In the first step, priorities are assigned to the customer orders. In the second step, in accordance with these previously assigned priorities, the customer orders are assigned successively to batches ensuring that the capacity constraint is not violated. A pseudocode for this type of algorithm is presented in Fig. 6.6. Several rules have been suggested in the literature for the determination of priorities. The probably most straightforward one consists of assigning the priorities to the customer orders as they come in, i.e. according to the First-Come-FirstServed (FCFS) rule. We note that this will practically result in a random sequence

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Fig. 6.6 General principle of priority rule-based algorithms

of the customer orders. Thus one cannot really expect that good-quality solutions will be provided by a procedure of this kind. However, the results from the application of the FCFS heuristic are commonly used in numerical experiments as reference points for the determination of the solution quality of other algorithms. Gibson and Sharp (1992) introduced the application of two-dimensional and four-dimensional space-filling curves, Pan and Liu (1995) suggested the application of six-dimensional space-filling curves for the determination of priorities. The coordinates of the storage locations of the demanded items in a customer order are mapped onto a theta-value on the unit circle. This value defines the priority of the customer order. The order with the largest value receives the highest priority. Ruben and Jacobs (1999) propose that the customer orders are sorted according to a so-called order envelope. The envelop of a customer order is defined as the pair of aisle numbers where the first number corresponds to the leftmost and the second number corresponds to the rightmost aisle from which an item has to be picked. Batches can be generated sequentially (Next-Fit Rule) or simultaneously (First-Fit Rule, Best-Fit Rule). As for the Next-Fit Rule, customer orders are added to a batch until the capacity of the picking device is exhausted; in this case a new batch is opened. According to the First-Fit Rule batches are numbered in the order in which they were started (opened); the next customer order is allocated to a batch which possesses the smallest number and still provides sufficient capacity for the accommodation of the customer order (Ruben and Jacobs 1999). The Best-Fit Rule assigns an order to the batch with the least remaining capacity (Wäscher 2004).

6.4.2 Seed Algorithms Seed algorithms, introduced by Elsayed (1981), generate batches sequentially by means of a two-phase procedure: a seed selection phase and an order congruency phase. During the seed selection phase, an initial order (seed) is chosen for a batch

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Table 6.1 Examples of seed selection rules Name of the rule From the set of unassigned customer orders select one order. . . Random Seed (Gibson and Sharp 1992) Smallest (largest) Number of Items (Elsayed and Stern 1983) Smallest (largest) Number of Picking Locations (Elsayed and Stern 1983; Elsayed 1981) Smallest (largest) Number of Picking Aisles (Ho and Tseng 2006; de Koster et al. 1999b) Smallest (greatest) Location-aisle Ratio (Ho and Tseng, 2006) Smallest (greatest) Aisle-simple-weight Sum (Ho et al. 2008)

Smallest Aisle-exponential-weight Sum (Ho and Tseng 2006) Smallest (greatest) Rectangular-covering Area (Ho et al. 2008)

Shortest Average Rectangular (Euclidean) Distance to the Depot (Ho et al. 2008) Shortest Average Aisle Distance to the Depot (Ho et al. 2008) Farthest Storage Location (de Koster et al. 1999b) Longest Travel Time (de Koster et al. 1999b) Largest Aisle Range (de Koster et al. 1999b)

. . . randomly. . . . which consists of the smallest (largest) number of items. . . . which possesses the smallest (largest) number of locations the order picker will have to visit in order to collect the items of this order. . . . which possesses the smallest (largest) number of aisles the order picker will have to enter in order to collect the items of this order. . . . with the smallest (largest) ratio of the number of storage locations to be visited over the number of aisles to be entered. . . . for which the sum of the weights of the aisles to be entered is minimal (maximal). The weight of an aisle is equal to its index, i.e. the weight of an aisle increases with its distance from the depot. . . . for which the sum of the weights of the aisles to be entered is minimal. The weight of an aisle depends exponentially on the index of the aisle. . . . for which the smallest rectangle is minimal (maximal) by which the storage locations of all required items of this order can be covered. . . . for which the average rectilinear (Euclidean) distance between the depot and the storage locations to be visited is minimal. . . . for which the average distance along the cross aisle between the depot and the storage locations to be visited is minimal. . . . which requires the collection of an item located farthest from the depot. . . . which requires the longest travel time. . . . with the largest aisle range, i.e. the absolute difference between the number of the leftmost aisle and the number of the rightmost aisle to be entered.

which has just been opened. A large variety of rules is available for the selection of the seed (for some examples see Table 6.1). Furthermore, the seed can be determined in a single mode (where only the first order in the batch defines the seed) or in cumulative mode (where all orders included in the batch define the seed). Afterwards, in the order congruency phase, unassigned customer orders are added to the seed according to an order-congruency rule (see Table 6.2), which measures

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Table 6.2 Examples of order-congruency rules Name of the Rule Add an order. . . Smallest Number of Additional Picking . . . such that the number of additional picking Locations (aisles) (Rosenwein 1996; locations (aisles) to be visited is minimal. Ho and Tseng 2006) Smallest (greatest) Overlapping Covering . . . such that the overlapping area between the Area (Ho et al. 2008) smallest rectangle covering the storage locations of the items in this order (order covering rectangle) and the smallest rectangle covering the locations of the items already in the batch (batch covering rectangle) is minimal (maximal). Smallest (greatest) Additional Covering . . . such that the difference in size between the Area (Ho et al. 2008) original batch covering rectangle and the new batch covering rectangle (i.e. the one also including the additional order) is minimal (maximal). Shortest Average Mutual-nearest. . . which minimizes the minimal average mutualrectangular (Euclidean) Distance nearest-rectangular (Euclidean) distance. This (Ho et al. 2008) distance is determined as follows: At first, the sum of the rectangular (Euclidean) distances from each pick location of the order to the respective closest pick locations of the batch is divided by the number of items in the order. Secondly, the sum of the rectangular (Euclidean) distances from each pick location of the batch to the respective closest pick location of the order is divided by the number of items in the batch. The mean of these two values gives the average mutual-nearest-rectangular (Euclidean) distance. Shortest Average Mutual-nearest-aisle . . . which minimizes the average mutual-nearestDistance (Ho et al. 2008) aisle distance. This distance can be determined similar to the average mutual-nearest-rectangular distance, whereas the distance between two pick locations is defined according to the distance on the cross aisle between the two pick locations. . . . for which the sum of the distances between the Smallest Sum I/II (Gibson and Sharp storage locations of the batch (order) and the 1992; closest item of the order (batch) is minimal. Pan and Liu 1995) Smallest Center of Gravity (Rosenwein . . . for which the absolute difference between the 1996) centre of gravity of the order and the centre of gravity of the batch is minimal. The centre of gravity is defined as the average number of aisles which have to be entered by the order picker. Time Saving (de Koster et al. 1999b) . . . for which the travel time reduction is maximal when being added to the batch, i.e. when batch and order are picked on one tour instead of two separate tours. (continued)

6 Order Batching in Order Picking Warehouses Table 6.2 (continued) Name of the Rule

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Add an order. . .

. . . which has the largest number of identical pick Greatest Number of Identical Picking locations (aisles) in common with the batch. Locations (aisles) (Elsayed and Stern 1983; Ho and Tseng 2006) Greatest Picking Location (aisles) . . . which maximizes the ratio of the number of Similarity Ratio (Ho and Tseng 2006) picking locations (aisles) batch and order have in common over the number of picking locations (aisles) of batch plus order. Greatest Picking Location (Aisles) . . . which maximizes the ratio of the number of Covering Ratio (Ho and Tseng 2006) picking locations (aisles) batch and order have in common over the number of picking locations (aisles) of the order.

Fig. 6.7 General principle of seed algorithms

the ‘‘distance’’ from a customer order not yet allocated to the seed of the batch. A pseudocode for seed algorithms is depicted in Fig. 6.7.

6.4.3 Savings Algorithms Savings algorithms are based on the Clarke-and-Wright Algorithm for the vehicle routing problem (Clarke and Wright 1964) which has been adapted in several ways for the order batching problem. In the initial version (C&W(i)) of the algorithm for the order batching problem, for each combination of customer orders savings are computed which can be obtained in terms of tour length reduction by assigning the items of the customer orders to one (large) batch instead of collecting them separately. Starting with the pair of orders that provides the highest savings, the pairs are considered one after another in a non-ascending order and checked with respect to the following three situations: (1) None of the two orders has been

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Fig. 6.8 Savings algorithm C&W(ii)

assigned to a batch yet; in this case a new batch will be opened and the orders will be assigned to it. (2) One of the orders has already been assigned to a batch; the other one will then be added to the batch if the remaining capacity is sufficient; otherwise the next pair of orders will be considered. (3) Both orders have already been assigned; then the next pair of orders is considered. All orders which are left unallocated at the end of the process will be assigned to an individual batch each (Elsayed and Unal 1989). In the second variant of the algorithm, denoted by C&W(ii), savings are recalculated each time a new assignment of customer orders to batches has been made. The (elementary) orders which have already been combined into a batch are excluded from these calculations, in which, on the other hand, the new batch is treated as a new ‘‘large’’ order (Elsayed and Unal 1989). A pseudocode of this algorithm is presented in Fig. 6.8. Referring to C&W(ii), Bozer and Kile (2008) suggest a normalized savings algorithm in which normalized instead of absolute savings are used, i.e. the travel time which can be saved by picking two orders on the same tour is divided by the time necessary if both orders are picked on separated tours. A drawback of C&W(i) can be seen in the fact that the algorithm may generate solutions with a large number of batches. In order to control the number of batches, C&W(i) can be modified in the following way: The initial savings matrix is modified each time customer orders have been assigned, i.e. for those pairs of customer orders of which both orders have not yet been assigned to a batch, the savings are reduced by a constant value. Therefore, the algorithm tends to select pairs of customer orders where at least one order is already assigned to a batch (C&W(iii); Elsayed and Unal 1989). The EQUAL (Elsayed and Unal 1989) algorithm generates batches sequentially and uses the seed algorithm principle. The pair of customer orders with the highest saving is allocated to a batch and considered as an initial seed. Then a single order

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is assigned to the batch which does not violate the capacity constraint and results in the highest savings in combination with the seed. The seed and the assigned order form the new seed. If none of the remaining unassigned orders fits into the batch, then a new batch is opened. In the Small-and-Large Algorithm (Elsayed and Unal 1989) two subsets are defined, namely a set of large customer orders whose number of requested items exceeds a predefined number and a set of small ones made up by the remaining ones. The set of large orders is assigned to batches by application of the EQUAL algorithm. The small customer orders are sorted in a non-ascending order of their size; then—in the sequence given by this sorting—each order is allocated to a batch where it does not violate the capacity constraint and generates the highest savings. If during the procedure a ‘‘small’’ order cannot be added to any of the existing batches, this order is assigned to a new batch. This batch is included in the set of batches and considered in the following iterations.

6.4.4 Data Mining Approach Finally, Chen and Wu (2005) describe an order batching approach based on data mining and integer programming. In this approach, at first similarities of customer orders are determined by means of an association rule. For each pair of orders a support value (or order correlation measure) is obtained. A 0–1 integer programming approach is then used to cluster customer orders into batches. The objective is to maximize the sum of all support values between the customer orders and the median of the batch each customer order is assigned to. The batch median is an order which serves as basis for each batch.

6.5 Metaheuristics 6.5.1 Local Search The general principle of local search-based heuristics consists of exploring the neighbourhood of a solution in order to identify a new solution with a smaller objective function value. For a solution S a solution is called neighbour solution if it can be obtained by applying a single local transformation (move) to S. Classic local search generates a sequence of solutions S0, S1, S2, . . ., where each member of the sequence is a neighbour of its predecessor. Each element possesses a smaller objective function value than the previous one. Classic local search stops at a local minimum, i.e. when no neighbour solution can be found that has a smaller objective function value than the incumbent solution. Classic local search suffers from the fact that it gets stuck in a local minimum which is probably far from the global minimum. Therefore, the local search principle has been modified in several ways in order to overcome such local minima.

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The first local search-based approach for the order batching problem was suggested by Gademann and van de Velde (2005). Their method starts from an initial solution generated by means of the FCFS rule. Neighbour solutions are obtained by so-called SWAP moves, in which two orders from different batches are interchanged. Since Gademann and van de Velde (2005) consider a problem in which the capacity of the picking device is defined with respect to the number of customer orders, the algorithm operates only on feasible neighbour solutions. The authors have implemented a first-improvement strategy, i.e. the first solution from the neighbourhood which has been identified to possess a better (smaller) objective function value is accepted and taken as the new incumbent solution. When a local minimum has been reached, the solution is perturbed by a sequence of three operations. In each operation three customer orders from three different batches are interchanged at random. The solution obtained from these perturbations is taken as a new incumbent solution from which the improvement phase is started again. This series of operations is repeated for a predefined number of iterations. Henn et al. (2010) propose the application of Iterated Local Search (ILS) to the order batching problem. ILS tries to intensify the search for improved solutions in the vicinity of local minima. It consists of two phases, a perturbation and a local search phase. In the perturbation phase, the incumbent solution is partially modified (perturbed). In the local search phase, proceeding from this solution, one tries to identify an improved solution. The solution stemming from the local search phase has to pass an acceptance criterion in order to become the new incumbent solution; otherwise the previous solution remains the incumbent solution to which another perturbation is applied. The two phases are repeated in turn until a termination condition is met. With respect to the order batching problem, Henn et al. (2010) suggest the following: An initial solution is generated by means of the FCFS rule. The local search phase includes SWAP moves (as defined above) and SHIFT moves (in which one customer order is selected and assigned to a different batch). One starts with a series of SWAP moves, until no further improvement can be obtained by this type of moves. Then one switches to a series of SHIFT moves, again until no further improvement can be achieved. Then one goes back to a series of SWAP moves etc. This sequence of alternating stages of SWAP and SHIFT moves is repeated until no further improvement can be obtained. In the perturbation phase two different batches are selected at random and a randomly generated number of orders from the first batch are moved to the second one, and vice versa. Orders whose addition to a batch would result in a violation of the capacity constraint will be assigned to a new batch. A new solution is accepted as an incumbent solution if its objective function value is lower than the one of the currently best known solution. Furthermore, a few deteriorating steps are allowed if a sequence of perturbation phases and local search phases applied to a particular incumbent solution does not lead to a new global best solution within a certain time limit. Albareda-Sambola et al. (2009) apply Variable Neighbourhood Search (VNS) to the order batching problem. They define three different kinds of neighbourhoods, based on the following moves: (1) Assignment of one order to a different

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batch (the SHIFT move as defined above). (2) Assignment of up to two orders from one batch to other batches; this includes (i) the simple SHIFT move, (ii) the assignment of two orders from one batch to another batch, and (iii) the assignment of two orders of one batch to two different batches. (3) Assignment of up to two orders from one or two batches to one or two other batches; this set includes (i) the moves from neighbourhoods (1) and (2), (ii) the SWAP move as defined above, (iii) the assignment of two orders from different batches to one single batch, and (iv) the transfer from one order to a another batch while an order from that batch is transferred to a third batch. In all three cases, only moves to feasible solutions are accepted. Starting from neighbourhood (1), the algorithm explores the three neighbourhoods successively. Whenever no improvement can be identified within one neighbourhood, it proceeds to the next (more extensively-defined) one. Having accepted an improved solution within neighbourhood (3), the algorithm returns to exploring neighbourhood (1), etc. The algorithm terminates if the incumbent solution is optimal for all three neighbourhoods.

6.5.2 Tabu Search Tabu Search—developed by Glover (1986)—aims at simulating human memory processes by means of a so-called tabu list. This list records moves applied in previous iterations. In order to avoid cycling and to diversify the search, the application of these moves is set forbidden (tabu) for a particular number of iterations. In each iteration Tabu Search considers only those elements of the neighbourhood which can be obtained by a non-tabu move. From this solution the one with smallest objective function value, which may not be smaller than the objective function value of the incumbent solution, is chosen as the next incumbent solution. Henn and Wäscher (2010) explore several variants of Tabu Search for the order batching problem. Options for the generation of initial solutions include the FCFS rule and C&W(ii). Three neighbourhoods are investigated, namely those based on (1) SWAP moves, (2) SHIFT moves, and (3) SWAP or SHIFT moves. The neighbourhoods may either be explored completely or partially, only. Henn and Wäscher (2010) also introduce the Attribute-based Hill Climber (ABHC) heuristic for the order batching problem. ABHC is an almost parameter-free heuristic based on a simple Tabu Search principle which can be described as follows: For each problem, a set of attributes is introduced. An attribute can be any specific solution feature. During the local search phase a solution can be accepted if and only if it possesses the smallest objective function value found so far for at least one attribute. The algorithm stops if the current neighbourhood contains no solution that represents a best solution for at least one attribute. The advantage of ABHC can be seen in the fact that—related to the design of the algorithm—only three decisions have to be taken, namely with respect to the choice of the initial solution, the neighbourhood structure, and the set of attributes (Whittley and Smith 2004).

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For the determination of the initial solution and for the neighbourhood structure Henn and Wäscher (2010) make use of the same options as for Tabu Search. For the attributes, two sets are proposed: The first attribute set characterizes each solution by pairs of customer orders which are assigned to the same batch. The second set of attributes is related to the assignment of customer orders to batches.

6.5.3 Population-Based Approaches The Rank-Based Ant System (RBAS) is a population-based solution approach in which each ant presents a single solution (Bullnheimer et al. 1999). Henn et al. (2010) modify the RBAS for the order batching problem. In their method a specified number of ants is observed during a period of several iterations. For each ant the algorithm starts from a solution in which each order forms a single batch. In the subsequent steps, the batches are combined as long as the capacity constraint for the picking device is not violated. For each possible combination of two batches into one, the savings (cf. Sect. 6.4.3) and the pheromone intensity are computed. The pheromone intensity of a batch combination is determined as a relative expression, namely as the sum of the pheromones of all order combinations (i.e. of all pairs of customer orders, where one order is in the first batch and one order is in the second batch) over the number of possible order combinations between the two batches. Savings and pheromone intensity define the probability according to which two batches will be combined. If no further feasible combinations of batches can be identified, an attempt is made to improve the obtained solution by applying an elementary local search function. The process is repeated for each ant that is used. After the last ant of an iteration has been taken care of, the pheromones for all order combinations are updated: in general, a fraction of the pheromone ‘‘evaporates’’, while the ‘‘good’’ solutions receive an additional amount of pheromone. Genetic algorithms iteratively generate a large number of possible solutions and select the best ones. These solutions are modified (mutation) and combined (crossover) in order to generate new solutions. Hsu et al. (2005) have modified this general approach for the order batching problem. Each solution is represented by a string of integers which groups each customer order into a particular batch. The fitness of a solution is calculated as the difference between the length of the longest tour in the population and the tour length of this solution. The authors also propose several crossover and mutation mechanisms for their algorithm.

6.6 Performance of Heuristic Algorithms An up-to-date study which considers a comprehensive set of problem parameters and provides an in-depth analysis of the performance of a representative set of the solution methods for the order batching problem does not exist in the literature so far (Gu et al. 2007). Published results of numerical experiments are usually based on very diverse settings (e.g. size and dimensions of the warehouse, number and

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size of customer orders, capacity of the picking device, demand structure, or the used routing policy). Thus, general conclusions, e.g. concerning the superiority of one method over another, are hardly possible at this stage. Gibson and Sharp (1992) analyze the performance of priority rule-based algorithms using space-filling curves, and that of a basic seed algorithm which combines Random Seed as a seed selection rule and Smallest Sum I as an order congruency rule. Their numerical experiments are based on a fixed warehouse with 800 storage locations. It is shown that the seed algorithm in which distances between locations are measured according to an aisle metric provides the shortest tour lengths. Rosenwein (1996) analyzes the performance of the following order congruency rules: Smallest Center of Gravity, Smallest Sum I, and Smallest Number of Additional Picking Aisles. For a warehouse with 750 storage locations, the Smallest Number of Additional Picking Aisles generates smaller average tour lengths than the two other approaches. De Koster et al. (1999b) have carried out extensive numerical experiments in order to investigate the performance of seed and savings algorithms. In their experiments the heuristics are evaluated with respect to parameters like the size of the warehouse (240, 400, 1250 storage locations), the demand structure (uniformly, class-based), and the number of orders per batch. Their experiments show that for seed algorithms the cumulative mode outperforms the single mode. Furthermore, Largest Number of Picking Aisles, Longest Travel Time, and Largest Aisle Range are the seed selection rules by which the shortest total tour lengths are obtained. In comparison to FCFS, the best seed-algorithms improve the tour lengths by 19.7% (small warehouse size) to 7.5% (large warehouse size). Ho and Tseng (2006) and Ho et al. (2008), in two comprehensive numerical studies, analyze the performance of seed algorithms. They compare various combinations of seed selection and order congruency rules to each other. In both articles an identical test setting for a warehouse with 384 storage locations is used. In the experiments of Ho and Tseng (2006) a combination of Smallest Number of Picking Aisles (seed selection rule) and Smallest Number of Additional Picking Aisles (order-congruency rule) gives the smallest tour lengths. Ho et al. (2008) observed that these results can be improved by combination of Smallest Number of Picking Aisles and Shortest Average Mutual-Nearest-Aisle Distance. In their analysis of savings algorithms, de Koster et al. (1999b) show that C&W(ii) generates the smallest tour lengths among the described variants. For a small warehouse, the tour lengths obtained by application of C&W(ii) are more than 20% shorter than the ones obtained by FCFS. In de Koster et al. (1999b), several variants of savings and seed algorithms are compared to each other. For a small warehouse, C&W(ii) provides smaller tour lengths than the best seed algorithm, whereas for a larger warehouse the best seed algorithm outperforms C&W(ii) in terms of solution quality. However, C&W(ii) consumes about ten times the computing time of the other savings algorithms and about 100–200 times the computing time required for the seed algorithms.

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Bozer and Kile (2008) compare the lower and upper bound from their model (cf. Section 6.3.2) to the results obtained by their Normalized Savings approach. In total, results obtained by the Normalized Savings approach are 11% above the lower bound and 6–7.5% above the upper bound. In their experiments the number of customer orders is limited to 25. In the numerical experiments of Chen and Wu (2005) different warehouse sizes (ranging from 40 up to 300 storage locations) are considered. The authors demonstrate that their approach which combines data mining and integer programming may reduce the tour lengths provided by FCFS significantly. Henn et al. (2010) benchmark their versions of ILS and RBAS against C&W(ii). Extensive numerical experiments have been carried out in which a warehouse with 900 storage locations and a class-based demand structure have been assumed. In these experiments the picker routing problem was solved by means of the S-Shape and Largest Gap Heuristics. The authors find that—in comparison to FCFS—C&W(ii) reduces the total tour length by 17%, while the improvements of ILS and RBAS approximately amounts to 20%. The results from ILS on one hand and RBAS on the other differ by \1%. For a similar test setting, Henn and Wäscher (2010) demonstrate that the best performing Tabu Search and ABHC variants manage to improve the solutions obtained by C&W(ii) by 4.1 and 4.6%, respectively. The results provided by the iterated descent algorithm of Gademann and van de Velde (2005) differ from the optimal total tour length by a 1% on average and up to 6% as the maximum. Albareda-Sambola et al. (2009) perform numerical experiments of their VNS approach using the warehouse layouts described in de Koster et al. (1999b) and Ho and Tseng (2006). In their experiments they benchmark VNS against FCFS, and several seed and savings approaches. It can be concluded that among these heuristics VNS was able to find the best solutions in most cases and improves the results obtained by FCFS by 19% on average. Hsu et al. (2005) benchmark their genetic algorithm against the results obtained by FCFS, considering warehouse sizes differing between 40 and 400 locations. They obtain improvements of up to 31%. As stated above, it is difficult to compare the presented results with each other due to the different parameter settings used in the numerical experiments. In order to make results more comparable, it would be useful for future research to design experiments similar to already existing ones. Furthermore, a systematic and comprehensive study is desirable in order to evaluate the performance of existing solution approaches.

6.7 Wave Picking Wave picking is a variant of order picking in which a large set of customer orders (called wave) for a joint destination is released simultaneously. Different to the approaches described before, the workload can be spread over several order

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pickers. Wave picking aims at collecting the items of the set of orders as fast as possible, even if this would result in a non-minimal tour length. Typically, this situation arises when this set of orders belongs to the same truck load and the truck cannot leave until all orders are completed. The next wave can only be started when the previous one is completed (Gademann et al. 2001). In order to control the picking and shipping process, the following optimization problem can be stated: How, given routing strategy, capacity of the picking device and a fixed number of order pickers, can a given set of customer orders with known storage locations be grouped (batched) into picking orders such that the maximal processing time of all batches is minimized? It is usually assumed that the order pickers do not block each other when entering the aisles or travelling through them. Thus, the impact of congestions is not explicitly considered in the corresponding planning models and solution approaches. A similar model to the one described in Sect. 6.3.2 can also be used for a representation of this problem. The following additional notation is used: pti m0

processing time of batch i ði 2 IÞ, i.e. the time which is necessary to collect all customer orders of batch i; number of order pickers, i.e. the number of batches which can be processed in parallel.

min max pti xi i2I

subject to

X

aij xi ¼ 1;

8 j 2 J;

ð6:5Þ

ð6:6Þ

i2I

X

xi  m0 ;

ð6:7Þ

i2I

xi 2 f0; 1g;

8i 2 I:

ð6:8Þ

Again, constraints (6.6) and (6.8) ensure that a set of batches is chosen in such a way that each customer order is included in exactly one of the chosen batches. Equation (6.7) guarantees that at most m0 batches are selected. The objective function (6.5) minimizes the maximal processing time of the batches. Bozer and Kile (2008) point out that solutions of this optimization problem may include situations in which order pickers may be idle while remaining orders are still being collected. By reducing the partition problem (Garey and Johnson 1979) to the wave picking problem, Gademann et al. (2001) show that the corresponding decision variant is NP-complete. For the solution of the above-stated problem Gademann et al. (2001) propose a branch-and-bound algorithm and a 2-opt heuristic. They evaluate their algorithms for several problem classes. For problem classes with up to 24 customer orders their branch-and-bound algorithm was able to solve all instances. For larger

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problem classes only a subset of instances could be solved within a time limit of 600 s. From extensive numerical experiments the authors conclude that the number of order pickers and the number of customer orders have a significant impact on the computing time of the algorithm. They also demonstrate that maximal processing times provided by the 2-opt heuristic are very close to the optimal ones.

6.8 Order Batching and Batch Scheduling In order picking systems it is not uncommon that the customer orders have to be completed and provided by certain due dates. In distribution warehouses due dates have to be met in order to guarantee the scheduled departure of trucks (Gademann et al. 2001). In material warehouses which provide the input to a production system (internal customers), on-time retrievals from the warehouse are vital in order to avoid production delays. In a Just-in-Time environment, in addition to not allowing orders to be late, it is also not acceptable that the items are provided a long time ahead of the due date, since that would result in an unnecessary accumulation of material or work-in-progress. In such cases, instead of measuring the quality of a solution by means of the total picking time or the total length of the picking tours, the batching of customer orders into picking orders will have to be evaluated with respect to both earliness and tardiness of the orders. The weighted sum of the (total) earliness and the (total) tardiness of all customer orders may be minimized in order to model these aspects (Elsayed et al. 1993). The composition of the batches, but also the sequence according to which the batches are processed and the corresponding release times (i.e. the points in time when the various batches and/or customer orders are started) determine whether and how this goal is met. With respect to the described situation, the following problem can be stated: How, given routing strategy and capacity of the picking device, can a given set of customer orders with known due dates and storage locations be grouped (batched) into picking orders such that the weighted sum of the total earliness and the total tardiness of all customer orders is minimized? Elsayed and Lee (1996) have suggested an optimization model for order batching in an AS/RS which minimizes the total tardiness. We adopt this model here for the above described, more general goal and introduce the following index sets and constants in addition to those which have already been used in the abovedescribed models: ddj due date of customer order j (j 2 J); Ji set of customer orders which are included in batch i (i 2 I), i.e. Ji = {j 2 J|aij = 1}; M a sufficiently large (positive) number; a (relative) weight for the total earliness of all customer orders; b (relative) weight for the total tardiness of all customer orders.

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Furthermore, the following variables are introduced: completion time of batch i ði 2 I Þ; point in time when the order picker returns to the depot after having collected all items of batch i; binary decision variable which describes whether batch i is released directly vik before batch k (vik = 1) or not (vik = 0) (i, k 2 I); taij tardiness of customer order j (j 2 J) in batch i (i 2 I); eaij earliness of customer order j (j 2 J) in batch i (i 2 I). cti

Then the following mixed integer programming model can be formulated: XX XX min a aij eaij þ b aij taij ð6:9Þ i2I j2J

i2I j2J

subject to: X

aij xj ¼ 1;

8 j 2 J;

ð6:10Þ

i2I

pti xi  cti ; cti  ctk þ Mvik  ptj xj ;

8 i 2 I; 8 i; k 2 I

ctk  cti þ Mð1  vik Þ  ptk xk ;

ð6:11Þ and i\k;

8 i; k 2 I

and i\k;

ð6:12Þ ð6:13Þ

cti  taij  ddj xi ;

8 j 2 Ji ; 8 i 2 I;

ð6:14Þ

eaij  cti  ddj xi ;

8j 2 Ji ; 8 i 2 I;

ð6:15Þ

cti  0;

8 i 2 I;

ð6:16Þ

taij  0;

8 i 2 I; j 2 J;

ð6:17Þ

eaij  0;

8 i 2 I; j 2 J;

ð6:18Þ

xi 2 f0; 1g; vik 2 f0; 1g;

8 i 2 I;

ð6:19Þ

8 i; k 2 I; i\k:

ð6:20Þ

In this model, objective function (6.9) represents the weighted sum of the total earliness and the total tardiness of all customer orders. As in the previously described models, it is guaranteed by constraints (6.10) and (6.19) that each customer order is assigned to exactly one batch. Inequalities (6.11) imply that the completion time of any batch is always greater than or equal to its processing time. For any pair (i, k) of batches i and k (i, k 2 I) constraints (6.12) and (6.13) imply that the completion time of either batch i or batch k must be greater than or equal to the completion time of the immediately preceding batch plus the corresponding processing time. Constraints (6.14) determine the tardiness of each customer order i (i 2 I); it is defined by the number of time units according to which the due date

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of i is exceeded by the completion time of the batch to which it has been assigned to. Analogously, constraints (6.15) determine the earliness of each customer order. Constraints (6.16–6.18) ensure that the variables assume non-negative values which represent the completion times, the earliness and the tardiness of the customer orders. The variables vik which describe whether batch i is released directly before batch k (i, k 2 I) are binary by definition; this is guaranteed by constraints (6.20). Elsayed et al. (1993) suggest a solution method for the above problem which consists of three steps. In step 1, priorities are determined for the customer orders. Each customer order is considered as a single batch and its priority value is defined by the weighted sum of its due date and the corresponding processing time. Customer orders are ranked in ascending order of their priority index. Based on this sequence the objective function value is determined. This sequence is updated, if a pairwise exchange of two adjacent customer orders can be identified which would improve the objective function value. In step 2, customer orders are combined into batches according to the sequence established in step 1: For each customer order it is determined whether it is favourable to pick it separately or if it is better to add the customer order in one of the already existing batches. In the latter case the customer order is assigned to the first batch which results in a smaller objective function value. For each of the obtained batches, release times are determined (step 3). Now an idle time may occur between the release time of a batch and the completion time of its predecessor. As a consequence the algorithm shifts batches along the time axes in order to obtain improved objective function values by reducing the total earliness, since earliness is penalized as well in the objective function. Elsayed and Lee (1996) propose a solution approach for the generation and sequencing of batches when only the total tardiness has to be minimized. At first, the customer orders are sequenced according to their due dates and the times that would be needed if each customer order would be processed in a single batch. A first bound on the optimal total tardiness can be obtained by assuming that the items of each customer order would be collected on a single tour and sequencing the tours according to the order’s position in the sequence. In order to improve this bound, three decision rules for the generation of batches are proposed and evaluated: (1) The Nearest Schedule Rule assigns the first customer order of the obtained sequence as a seed to the first batch. Those customer orders of the sequence are assigned to the batch if the inclusion does not increase the total tardiness and does not violate capacity restrictions. When no further customer order can be added, the first remaining customer order in the sequence will serve as seed order for the next batch. (2) The Shortest Service Time Rule adds orders to the seed which would possess the shortest processing time, if the customer order was processed in a separate tour. (3) In the Most Common Location Rule, a customer order is added to the seed which has the largest number of pick locations in common with the seed order. In numerical experiments it is shown that the Nearest Schedule Rule outperforms the two other rules and achieves results which are close to optimality. Additionally, the authors show how storage orders (orders

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where items have to be carried to the storage locations) can be included in the obtained sequence. Tsai et al. (2008) consider an order batching and sequencing problem where the total travel costs (depending on the total travel time) have to be minimized and earliness and tardiness are penalized. Unlike in the approaches previously discussed, here the authors permit the splitting of orders; therefore, the items required by a single customer order may be collected on different tours. The batching problem is solved by means of a genetic algorithm. In this algorithm, a solution is represented by a sequence of integers. This sequence contains the indices of the batches to which the items requested in the customer order have been assigned. In order to determine the fitness of the generated solutions, a travelling salesman problem is solved by another genetic algorithm. Won and Olafsson (2005) discuss a joint batching and picking problem where customer orders arrive at (known) different points in time. They formulate an optimization model where the objective function is a weighted sum of the travel distance and the time period between the arrival of a customer order and the time when the order picker starts to collect its items. The authors observe a tradeoff between the necessary travel time and the length of the time period for which the customer order stays in the system. In order to solve the problem the authors propose a two step heuristic. In this heuristic, batches are formed by application of the FCFS rule. The subsequent routing problem is then solved by a 2-opt procedure. Additionally, they present an algorithm which solves the batching and routing problem simultaneously.

6.9 Dynamic Order Batching 6.9.1 Problem Description Whereas in the static case of order batching the characteristics of each customer order (i.e. the requested articles and the corresponding quantities) are known in advance, the dynamic case (also referred to as on-line batching; cf. Yu and de Koster 2009) can be characterized as an order picking environment in which the customer orders arrive stochastically over time, and only when an order has arrived the information becomes available of which articles and respective quantities the order is composed of. Under such conditions time window batching is prevalent, which can be carried out in two different ways, namely variable time window batching and fixed time window batching (van Nieuwenhuyse and de Koster 2009). In variable time window batching it is usually assumed that the capacity of the picking device is defined in the number of customer orders which can be accommodated. The order picker waits until a particular number of customer orders (smaller or equal to the capacity) has arrived and then collects the items of these orders on a single tour.

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In fixed time window batching all customer orders arriving during a particular time interval are assigned to batches. It has to be noted that all batching decisions, i.e. how the orders should be assigned to batches and how the batches should be released in time, are to be based on the known customer orders only. The point in time when a customer order becomes available is called arrival time of the order. The waiting time of a customer order is defined by the length of the time interval between the arrival time of the order and the point in time where the order picker starts to collect the requested items of the customer order. The turnover time (also called throughput time or response time) of a customer order is equal to the length of the time period for which the order stays in the system, i.e. the time period between the arrival time of the customer order and its completion time. The performance of a dynamic order picking system can be measured by the (average) turnover time of the customer orders. This measure can be seen as an indicator of the service level on the one hand, but also as an expression for the available capacity on the other, i.e. the number of orders which can be processed in a given period of time. Reduced turnover times result in improved service levels and increase the capacity of the warehouse. Dynamic order batching deals with the following question: In a dynamic order picking environment, in which customer orders arrive over time, how should the customer orders (with given storage locations and given routing strategy) be grouped into picking orders such that the average turnover time of the customer orders is minimized?

6.9.2 Time Window Batching for the Minimization of Turnover Times The time an order picker is expected to spend collecting the items of a batch is estimated by travel time models. These estimations depend on several parameters, e.g. the average number of items to be picked on a tour, the demand distribution, or the layout of the warehouse and its dimensions (Caron et al. 2000). In the literature, the order picking system is regarded as a continuous system with an infinite number of arrivals of customer orders. Chew and Tang (1999) present a travel time model of a single block order picking warehouse with variable time window batching. Based on this model, an estimation for the determination of the number of customer orders is developed which should be assigned to a single batch in order to minimize the average turnover time of a customer order. On the basis of S-Shape routing, they carry out a theoretical analysis of travel and processing times for the first customer order in a batch. The system is designed as a queuing network with two queues. In the first one, customer orders arrive according to a Poisson-process and batches are generated by means of the FCFS rule. If a particular number of customer orders is in the first queue, these

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Fig. 6.9 Average turnover time against batch size for variable time window batching (based on Chew and Tang 1999)

customer orders are assigned to a batch and are moved to the second queue. The batches in the second queue are released successively according to the availability of order pickers. In numerical experiments the authors focus on the optimal number of customer orders which should be assigned to a batch such that the average turnover time is minimized. The optimal number depends on the storage policy and also on the pick times. A similar investigation of the average turnover time of a random customer order for a two block layout is carried out by Le-Duc and de Koster (2007). A corresponding model for fixed time window batching in a two block layout is presented by van Nieuwenhuyse and de Koster (2009). All studies show that for variable time window batching the average turnover time of customer orders is a convex function of the number of orders per batch (batch size). A large batch size results in a small average processing time of each customer order, but also in a large average waiting time. On the other hand, the average (order) processing time is large for a small batch size, whereas the average waiting time is small. For fixed time window batching a similar convex function for the average turnover time of a customer order can be observed which is dependent on the length of the fixed time window (van Nieuwenhuyse and de Koster 2009). Therefore, it can be concluded that an optimal batch size (or an optimal length of the fixed time window) exists which minimizes the average turnover time of a customer order. The function of the average turnover time in variable time window batching is depicted in Fig. 6.9. In numerical experiments, van Nieuwenhuysen and de Koster (2009) demonstrate that the application of fixed time window batching can lead to slightly smaller average turnover times than the application of variable time window batching. Extensions to the above-described travel time models may include multiple order pickers which can be modelled by additional queues (Le-Duc and de Koster 2007). For time window batching, van Nieuwenhuyse and de Koster (2009) present a combined analysis of a pick-and-sort system, in which the collected items are sorted and packed after the picking process. It is assumed that the time an order spends in the picking process is also stochastic. Turnover times now include the time in the picking process and the time in the sorting process. The authors concentrate on the allocation of workforce to both processes and aim at a minimization of the average turnover time. Yu and de Koster (2009) describe an order picking area which is divided into several zones of identical size. Batches of customer orders are formed according to

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variable time window batching. The items of each batch are collected sequentially by zones, i.e. at first only items located in the first zone are picked, then all items located in the second zone, etc. The batch is complete when the items from the last zone have been picked. For this kind of systems, Yu and de Koster (2009) give an estimation of the average turnover times and observe that an optimal batch size exists.

6.9.3 Alternative Objective Functions and Approaches Kamin (1998) describes a batching problem from practice, in which greeting cards have to be retrieved from a warehouse. Order pickers use automated guided vehicles on a fixed course collecting the items according to given customer orders. These customer orders arrive dynamically over time. Kamin focuses on the minimization of average turnover times. Therefore, the number of pick stops on the course is minimized. By means of a competitive analysis it is shown that every online algorithm for this problem is at least 2-competitive. Furthermore, the system is simulated and evaluated for several (simple) batching algorithms according to different evaluation criteria. Henn (2010) describes an online order batching problem in a walk-and-pick warehouse in which the total completion time (makespan) of all (dynamically arriving) customer orders is to be minimized. A competitive analysis reveals that any on-line algorithm for this problem is at least 2-competitive. The author also presents modifications of solution approaches for static order batching (FCFS, C&W(ii) and ILS) in order to deal with the dynamic situation.

6.10 Summary and Outlook on Future Research In this chapter we reviewed the literature dedicated to order batching in pickerto-parts order picking systems. We pointed out, that order batching is pivotal for the efficient management and control of order picking systems in distribution warehouses, since—due to the large amount of manual labour—order picking is the most cost-intensive function in a warehouse. It has been shown that a large range of order batching methods exists and that research has evolved into several directions, of which static and dynamic batching are the predominant ones. Goals which may be used for the evaluation of solutions do not only include the minimization of processing times but also the earliness and tardiness of the customer orders, in particular when these orders have to be shipped at specific points in time. We provided comprehensive insight into order batching by giving a detailed state-of-the-art overview of the different solution approaches which have been suggested in the literature. Application of these methods can contribute to a significant improvement of the performance of the picking operations in warehouses.

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As further research opportunities, research on static order batching could concentrate on the minimization of the total processing time for problems involving due dates. From a practical point of view it would also be desirable to intensify research on dynamic order batching and incorporate due dates in dynamic situations. Finally, we conclude that the interaction of order batching with the other related planning issues (layout design, item location, zoning, picker routing) has not been considered sufficiently in the literature so far. Thus, it might be worthwhile to provide corresponding simultaneous solution approaches in order to obtain a ‘‘global’’ optimum.

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Chapter 7

Storage Assignment for Order Picking in Multiple-Block Warehouses Kees Jan Roodbergen

Abstract Order picking is a warehousing activity that is concerned with the retrieval of products from storage locations to meet customer demands. In this paper, we provide insights into the interactions between layout, routing and storage assignment, with an emphasis on the effectiveness of various storage assignment methods. To this end, we performed an extensive simulation study. It appears that a selection of appropriate storage assignment policies is possible based only on pick list sizes, so without considering the actual layout of the area.

7.1 Introduction Warehouses are increasingly faced with a situation where orders occur more frequently and in smaller quantities. A change in quantities to handle per order can have significant consequences for a warehouse; other systems may be needed or the operation of the existing systems may need to be adapted. Order picking is the warehousing process where small quantities are retrieved from storage to meet demand. Typically, in manual operations an order picker travels through the storage area and visits several locations to retrieve products. Due to the time required to handle items individually and to travel from location to location, order picking is often one of the most laborious of all warehousing processes (Tompkins et al. 2003). The manual order-picking activity can roughly be divided into three components: traveling between items, picking of items and remaining activities. Traveling between items refers to movements of the order picker and his vehicle

K. J. Roodbergen (&) University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands e-mail: [email protected]

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between the various locations that must be visited. Picking of items consists of a series of actions ranging from positioning the vehicle to putting the picked items onto a product carrier such as a pallet or into a box. The remaining activities include picking up an empty product carrier, the acquisition of information and dropping off the full product carrier at some point after picking is completed. To obtain an efficient order picking process, several factors must be considered. When designing the warehouse, the type of racks and vehicles must be chosen, and the layout must be determined. To operate the system, certain operating policies must be formulated to control the sequence of actions to take, given a set of received orders. For existing warehouses, it may not be economical to change the physical aspects of the order-picking process; however, past case study research (Dekker et al. 2004) has shown that a change in operating policies may give ample opportunity for efficiency improvements. In this paper, we aim at reducing travel times in the order-picking operation by means of the operating policies. As noted, operating policies can be an effective means to improve efficiency and they can be applied in both new and existing facilities. The focus on travel times stems from the fact that the other order-picking activities, i.e., picking of items and remaining activities, are strongly influenced by the racking and equipment and are hence difficult to change in existing facilities. Furthermore, travel time often accounts for more than half of an order picker’s time (Tompkins et al. 2003) and is therefore an important factor to consider per se. The objectives of this paper are as follows: • To introduce a series of storage assignment policies for multiple-block warehouses; • To compare the efficiency, expressed as average travel time for the order pickers, of these storage assignment policies in multiple-block warehouses; • To present simple guidelines for applying storage assignment policies based on as few system characteristics as possible. The first two objectives are considered because even though many studies have been performed concerning storage assignment policies in single-block warehouses, research concerning multiple-block warehouses is very limited. More specifically, little research has so far been performed on storage assignment policies that can be applied to general multiple-block layouts. The third objective aims at providing warehouse managers with an opportunity to quickly make a first selection of potentially good control policies without the need to build a simulation model.

7.2 Situation Description A graphical sketch of the warehouse layout considered in this paper is given in Fig. 7.1. The warehouse is rectangular with no unused space and consists of a number of parallel pick aisles. The warehouse is divided into a number of blocks,

7 Storage Assignment for Order Picking in Multiple-Block Warehouses Fig. 7.1 A typical layout of an order-picking area

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Pick aisle

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each of which contains a number of subaisles. A subaisle is that part of a pick aisle that is within one block. At the front and back of the warehouse and between each pair of blocks, there is a cross aisle. Cross aisles do not contain storage locations, but can be used to change aisles. Every block has a front cross aisle and a back cross aisle; the front cross aisle of one block is the back cross aisle of another block, except for the first block. The number of cross aisles equals the number of blocks plus one. This layout is a generalization of the ladder-structure layout, which consists of only one block. Many warehouse operations are such that an order picker works in only one block. Such a warehouse may have more blocks for order picking, but an order picker’s task is limited to one block only. Situations where an order picker retrieves orders from more than one block in the same tour, also occur frequently in practice (Dekker et al. 2004). Furthermore, significant efficiency gains can be achieved by using these layouts instead of ladder-structure layouts (Roodbergen and De Koster 2001). Order pickers are assumed to be able to traverse the aisles in both directions and to be able to change direction within the aisles. The aisles are narrow enough to allow picking from both sides of the aisle without changing position. Each order consists of a number of items that are usually spread out over a number of

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subaisles. If customer orders are too large to be retrieved in a single route, or if it is desirable to pick multiple orders in a single route, then items from customer orders may be regrouped into pick orders. This regrouping of items, called batching, is not considered explicitly in this paper. All orders are assumed to be pick orders after regrouping. Various articles discuss batching issues (Gibson and Sharp 1992; Ruben and Jacobs 1999). Aisle changes are possible in any of the cross aisles. Picked orders have to be deposited at the depot, where the picker also receives the instructions for the next route. The depot is located at the head of the first pick aisle in the front cross aisle.

7.3 Routing of Order Pickers Given a set of storage locations that must be visited, a routing policy determines the sequence to visit these locations. Efficient optimal algorithms exist for a limited number of warehouse layouts. Ratliff and Rosenthal (1983) developed an algorithm to calculate shortest routes for single-block layouts. For most other layouts, non-polynomial algorithms or heuristics must be used. Even in situations where an efficient optimal algorithm exists, a heuristic is often preferred in practice because it is easier to use and understand. Hall (1993) presents a number of common routing heuristics, which were extended for situations with multiple blocks by Roodbergen and De Koster (2001). Routing heuristics specifically developed for multiple-block situations are the aisle-by-aisle heuristic presented in Vaughan and Petersen (1999) and the combined and combined+ heuristics presented in Roodbergen and De Koster (2001). Each of the routing heuristics is described briefly here and illustrated in Fig. 7.2. Detailed descriptions can be found in the original papers. The simplest way to route order pickers is by using the S-shape or traversal heuristic. The order picker traverses the left-most pick aisle that contains items to the back of the warehouse and then returns to the front while picking items blockby-block. Any subaisle containing at least one item is traversed through the entire length. Subaisles with no picks are skipped. After picking the last item in a block, the order picker returns to the front of that block and continues with the next block. This policy is likely to be the most frequently used routing heuristic in practice. With the largest gap heuristic, the picker again starts by moving to the back of the warehouse. The picker travels through the cross aisles and each subaisle is entered as far as the ‘largest gap’ and left from the same side that it was entered. A gap represents the distance between any two adjacent items, or between a cross aisle and the nearest item. The last subaisle of a block is traversed entirely to allow the picker to enter the subaisles from the other side of the block as well. Thus, the largest gap is the part of the aisle that is not traversed. Hall (1993) showed that this policy is good for situations with random storage in a single-block layout if the number of picks is less than four per aisle.

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Fig. 7.2 Illustration of six routing methods

The combined heuristic follows the same sequence of visiting subaisles as the S-shape heuristic. However, this policy has the ability to choose between traversing a subaisle and making a turn in that subaisle. These choices are made by dynamic programing, which—in a way—makes it possible to look one subaisle ahead. For example, suppose a subaisle for which it would be shortest to make a return. The combined heuristic could choose to traverse this subaisle anyway because this gives a better starting point for the next subaisle. The combined+ heuristic is an improved (but more complex) version of the combined heuristic. It differs from the basic version by the fact that the warehouse is now divided into two parts. The dynamic programing method is used twice. Once to route the order picker to the back of the warehouse through the left part of the warehouse and once to return to the front through the right part. The division of the warehouse into two parts is optimized.

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The aisle-by-aisle heuristic is based on the notion that every pick aisle will be visited only once, starting with the left-most pick aisle. For every change from one pick aisle to the next, the heuristic selects the most suitable cross aisle by means of dynamic programing.

7.4 Storage Assignment Policies Before orders can be retrieved from the warehouse, products must be allocated to the various storage locations. A storage assignment policy is a rule to determine this allocation. Heskett (1963) was one of the first to formalize a relation between product location and product characteristics, based on the volume that needs to be stored of each product and the demand frequency for each product. Several approaches exists to storage assignment (Hausman et al. 1976) including dedicated storage assignment, random storage assignment, full-turnover-based storage assignment and class-based storage assignment. We will first briefly introduce each method. For random storage, incoming products are assigned to a location in the warehouse that is selected randomly from all eligible empty locations with equal probability. If all products from a location have been retrieved, that location may be used for another product. This way, in the long run, the probability that an item is needed from any location is equal. This storage policy is used often in practice; for example, in situation where insufficient data is available to determine demand frequencies of individual products, for example, for rapidly changing product assortments (De Koster et al. 1999). Another method is the full-turnover-based storage. All products are ranked from most frequently requested to least frequently requested and all locations are ranked from best (generally closest to the depot) to worst. Then products are assigned to locations by matching these two rankings. This method is often combined with dedicated storage, where each product has a fixed location in the warehouse. Timely relocation of products may be required if significant demand shifts occur. A disadvantage of dedicated storage is that a location is reserved even for products that are out of stock. Moreover, for every product sufficient space has to be reserved such that the maximum inventory level can be stored. Thus the space utilization is much lower than for random storage. The concept of class-based storage combines the best of two worlds. This method divides the products into a number of classes. Each class is then assigned to a dedicated area of the warehouse. Storage within an area is random. Classes are determined by some measure of demand frequency of the products. Fast moving items are called A-items. The next fastest moving category of products are called B-items and so on. Generally, the number of classes equals three, which may give about 85% of the potential efficiency gains of full-turnover-based storage (Hausman et al. 1976). The advantage of this way of storing is that fast moving

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products can be stored close to the depot and simultaneously most of the flexibility and low-storage space requirements of random storage are applicable. Most research on class-based storage has been performed in the context of automated storage and retrieval systems (AS/RSs). Hausman et al. (1976) and Graves et al. (1977) analyze class shapes and sizes for AS/RS. Many other authors have extended this work in later years, all related to AS/RSs. Research on storage assignment for manual picking environments is less advanced. This may be due to its complexity. AS/RS research typically assumes a single unit-load crane moving in a single aisle. Therefore, in such AS/RSs system there are no aisle changes and a route has at most two stops (one storage job and one retrieval job) before returning to the depot. In this paper, we consider storage assignment for a situation where there are no restrictions on number of aisles, the number of blocks, or the number of stops per route. Research concerning multiple-block warehouses has mainly focused on random storage (Roodbergen and De Koster 2001; Vaughan and Petersen 1999). For single-block layouts, Jarvis and McDowell (1991) suggest that each aisle should contain only one class, resulting in the within-aisle storage policy. Petersen (1999, 2002), Petersen and Schmenner (1999), Petersen and Aase (2004) and Petersen et al. (2004) compare multiple configurations for single-block layouts. Caron et al. (1998) and Le-Duc and De Koster (2005) consider a maximum of two blocks and are not focused on a full comparison of routing and storage assignment policies combinations. In this paper we consider five storage policies: random storage and four variations of class-based storage. Each of the class-based storage policies is described briefly and illustrated in Fig. 7.3. Across-aisle storage is an ABC-storage policy, where the A-items are assigned to the front-most locations of each pick aisle. C-items are stored at the back of each pick aisle and B-locations are stored in-between. The method comes out favorably in Le-Duc and De Koster (2005). For within-aisle storage all products in a pick aisle belong to the same class (A, B, or C). The pick aisles nearest to the depot contain the A-items. This policy appears to be good according to Jarvis and McDowell (1991). For nearest-subaisle storage it is assumed that all items in a subaisle belong to the same class. The subaisles with their center closest to the depot contain the A-items. This is actually a variation on the within-aisle storage rule, and as such could also have been called within-subaisle storage. This policy is specifically meant for layouts with multiple blocks; for one-block warehouses it will give exactly the same results as within-aisle storage. With nearest-location storage, the A-items are assigned to the locations that are closest to the depot. One (sub)aisle may, therefore, contain items from different classes. This policy is closely related to the method of diagonal storage (Petersen 1999), which defines class boundaries based on diagonal lines in the picking area such that the A-items are closest to the depot. Nearest-location storage minimizes expected travel time if all orders consist of one item only.

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Depot

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Fig. 7.3 Four ABC-storage assignment methods. Black locations indicate A-items, dark gray locations indicate B-items and light gray locations indicate C-items

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7.5 Experiments This section evaluates the various storage assignment policies and routing policies for a number of layouts. We consider a shelf area where order pickers walk through the warehouse to pick small items, using a small pick cart. This environment can be considered as one of the prime candidates for improvements through routing and storage assignment (Dekker et al. 2004; Petersen and Aase 2004). The following assumptions are made. The average walking speed in both cross aisles and pick aisles is 0.6 m/s. The center-to-center distance between two neighboring pick aisles is 2.5 m and the cross aisle width is also 2.5 m. Picking of items can be performed simultaneously from both sides of a pick aisle since the aisles are fairly narrow. Order pickers are assumed to walk through the middle of the pick aisles and cross aisles. There are two important characteristics of demand to consider; the lines per order (i.e., the number of locations to visit in a single route) and the demand skewness. To have a single parameter for demand skewness, we make use of the following formula: FðxÞ ¼

ð1 þ sÞ  x ; sþx

FðxÞ  0 and x  1; s  0 and s þ x 6¼ 0;

where x denotes the percentage of the storage space and F(x) gives the percentage of picks resulting from this part of the storage space. The parameter s indicates the skewness of the demand. For example, for s = 0.067 it holds that 80% of the picks are generated by 20% of the items. For large values of s, demand is hardly skewed and approaches random storage. This formula obviously can only approximate the shape of an actual demand curve, but the approximation is appropriate (Caron et al. 1998) and quite helpful for the type of mass experiments we perform in this paper. Furthermore, there are four important design factors: the number of aisles, the number of cross aisles, the size of zone A (expressed as the percentage of total storage space allocated to the A-items) and the size of zone B. The size of zone C can be calculated by subtracting the sizes of zone A and B from 100% because we consider situations with exactly three zones. The height of the storage racks is generally not a consideration in this type of environments, since the height is restricted by the physical capabilities of the order pickers. In total we now have six characteristics which can be taken in different combinations to obtain test instances for comparisons of the operating policies. Inspired by practical observations, we decided on boundaries for the six parameters, which will be used in the experiments. Furthermore, one or more points between the boundaries will be included. This gives us the following set of parameter values: • the number of pick aisles equals 4, 8, 12, 16 or 20; • the number of cross aisles equals 2, 4, 6, or 8;

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the size of the zone A equals 5, 10, 15 or 20% of total storage space; the size of the zone B equals 10, 20, 30 or 40% of total storage space; the lines per order equal 5, 10, 15 or 20; demand skewness (s) equals 0.05, 0.15 or 0.25.

This gives us a total of 3,840 different situations. For each of these situations, we determine the average travel time by means of simulation. The number of replications is determined separately for each experiment such that the 99% confidence interval half width is at most 1% of the average value. We repeated the simulations for all combinations of five storage assignment policies (random, across-aisle, within-aisle, nearest-subaisle and nearest-location) and five routing policies (S-shape, largest gap, combined, combined+ and aisle-by-aisle). In total, this results in 5*5*3,840 = 96,000 situations to simulate. Below, summaries of these results are given in Tables 7.1, 7.2, 7.3 and 7.4. Considering the massive amount of data, not all individual results are displayed in this chapter.

7.5.1 General Results Table 7.1 summarizes the results grouped by the number of cross aisles. For a given value of the number of cross aisles, the average travel time is calculated over all instances for each combination of control policies (routing heuristic and storage policy). For each value of the number of cross aisles the best combination of control policies is indicated in Table 7.1. It appears that for layouts with two cross aisles, the largest gap routing heuristic has the best performance, whereas the combined+ routing heuristic has the best performance for layouts with more than two cross aisles. These findings are consistent with the results of Roodbergen and De Koster (2001), who evaluated random storage only. It appears that all four ABC-storage policies give a significant improvement over random storage for most of the situations. As it seems, the adoption of ABC storage as opposed to random storage appears to be a more important decision, than to select the best ABC-storage policy. To investigate the validity of this idea, we reworked the data as follows. Table 7.2 contains the percentage travel time improvement when using the worst ABC-storage policy when compared to random storage. Note that we, ex post, select the worst ABC-storage policy for every instance individually and then compare these results with random storage. As can be seen from Table 7.2, significant reductions in travel time can be achieved by implementing a ‘‘bad’’ ABC-storage policy, because even the worst of all ABCstorage policies strongly outperforms random storage. The additional gains from selecting the best ABC-storage policy instead of the worst policy are given in Table 7.3. It appears that the gains from introducing an ABC-storage policy (Table 7.2) are much higher than the gains from selecting the best ABC-storage policy (Table 7.3). To be precise, if we take the average of all values in Table 7.2, we find an average travel time improvement of 16.6%; if we

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Table 7.1 Average travel time (in seconds per route) over 3,840 instances for various combinations of operating policies Random Across-aisle Within-aisle Nearest-subaisle Nearest-location 2 cross aisles S-shape Largest gap Combined Combined+ Aisle-by-aisle 4 cross aisles S-shape Largest gap Combined Combined+ Aisle-by-aisle 6 cross aisles S-shape Largest gap Combined Combined+ Aisle-by-aisle 8 cross aisles S-shape Largest gap Combined Combined+ Aisle-by-aisle

446.5 386.5 393.2 393.2 393.2

434.4 330.2 304.5 304.5 304.5

327.9 289.6 294.7 294.7 294.7

327.9 289.6 294.7 294.7 294.7

377.5 312.6 298.2 298.2 298.2

361.0 360.6 333.7 309.7 349.0

295.6 303.1 275.2 255.8 280.1

262.5 264.1 248.9 237.7 269.4

265.1 266.2 250.0 236.6 262.8

271.3 273.5 255.8 239.9 265.5

351.7 367.5 336.0 304.9 369.3

288.0 308.5 277.3 255.3 290.8

261.4 269.7 254.2 240.0 288.0

266.4 274.8 256.5 238.6 277.7

268.4 278.7 259.4 240.5 279.3

358.9 377.4 348.9 312.1 396.9

296.2 321.7 288.3 263.1 307.7

271.3 280.7 266.9 250.2 311.4

276.2 287.8 269.3 248.1 297.8

276.8 289.5 270.3 248.4 298.3

Table 7.2 Travel time reduction when going from random storage to the worst of four ABCstorage assignment policies Cross aisles 2 (%) 4 (%) 6 (%) 8 (%) S-shape Largest gap Combined Combined+ Aisle-by-aisle

3 14 22 22 22

17 15 17 16 19

17 15 17 15 19

17 14 16 14 20

Table 7.3 Travel time reduction when going from the worst to the best ABC-storage assignment policy (as a percentage of random travel time) Cross aisles 2 (%) 4 (%) 6 (%) 8 (%) S-shape Largest gap Combined Combined+ Aisle-by-aisle

23.8 11.1 4.2 4.2 4.2

10.5 12.3 9.4 8.1 6.6

9.5 12.4 9.1 8.2 6.1

9.3 12.8 8.9 8.0 6.0

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Table 7.4 Overview of characteristics for the best combinations of operating policies Average Across aisle Within aisle Nearest Nearest subaisle location Largest gap Number of observations Number of aisles Number of cross aisles Size of zone A (%) Size of zone B (%) Demand skewness Lines per order Combined Number of observations Number of aisles Number of cross aisles Size of zone A (%) Size of zone B (%) Demand skewness Lines per order Combined+ Number of observations Number of aisles Number of cross aisles Size of zone A (%) Size of zone B (%) Demand skewness Lines per order Aisle-by-aisle Number of observations Number of aisles Number of cross aisles Size of zone A (%) Size of zone B (%) Demand skewness Lines per order

6 18.7 2.0 13.3 10.0 0.25 18.3

588 12.7 2.0 12.1 24.3 0.17 14.4

596 12.5 2.0 12.0 24.4 0.17 14.4

0

12.0 5.0 12.5 25.0 0.15 12.5

12.0 5.0 12.5 25.0 0.15 12.5

63 6.5 2.0 14.4 24.4 0.17 5.8

160 12.4 2.0 14.7 27.9 0.08 14.3

166 12.0 2.1 14.8 28.0 0.08 14.4

143 11.0 2.0 11.0 25.7 0.14 5.3

12.0 5.0 12.5 25.0 0.15 12.5

84 5.9 3.3 15.2 24.6 0.15 6.5

1633 12.0 5.6 12.9 26.1 0.14 16.0

1049 12.0 5.2 12.0 24.6 0.14 10.0

640 12.6 5.4 12.7 24.2 0.16 6.9

12.0 5.0 12.5 25.0 0.15 12.5

63 6.5 2.0 14.4 24.4 0.17 5.8

160 12.4 2.0 14.7 27.9 0.08 14.3

160 12.4 2.0 14.7 27.9 0.08 14.3

143 11.0 2.0 11.0 25.7 0.14 5.3

take the average of all values in Table 7.3, we find an improvement of 9.2%. Or stated differently: on average, selecting the worst ABC-storage policy already gave 64% of the potential gains in our experiments.

7.5.2 Influence of Layout and Demand Characteristics Another summary of the same experiments is given in Table 7.4. The table consists of four partial tables. Each partial table concerns one routing policy. Four columns in each partial table refer to a specific storage assignment policy as

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indicated in the first cell of the column. The first row of each partial table contains the number of times a certain combination was the best. For example, the combination of largest gap routing and across-aisle storage was the best possible set of operating policies in six of the evaluated situations; the combination of largest gap routing and within-aisle storage was the best combination in 588 situations. Note that several combinations may be tied for first place, thus explaining the fact that the sum of the observations exceeds 3,840. The large difference here is almost exclusively due to the fact that combined, combined+ and aisle-by-aisle all give identical routes for warehouses with two cross aisles. No partial table has been included for S-shape routing because this policy was never in the best combination of operating policies for any of the situations we considered. The remaining six rows in each partial table give specifics about the situations for which that combination of operating policies was the best. The column ‘‘average’’ gives the average value over all situations in the sample. By comparing the sample average with the average of a subset, it is possible to identify characteristics of situations that favor a certain combination of operating policies. For example, the average number of aisles in the sample equals 12. The average number of aisles in situations for which combined routing and across-aisle storage are best, equals 6.5 indicating a signal that this combination usually turns out better in warehouses with fewer aisles. One of the most striking point of Table 7.4 is that three routing policies, largest gap, combined and aisle-by-aisle have an average value of two for the number of cross aisles (i.e., a layout with one block). This means that these three policies almost only turn out to be the best policy in one-block warehouses (the full data files reveal that there are only 14 exceptions). It must be noted that for one-block warehouses, within-aisle storage and nearest-subaisle storage give identical results. Furthermore, routes generated by combined, combined+ and aisle-by-aisle are identical. Five hundred and eighty eight of the 960 instances with two cross aisles favor largest gap routing and within-aisle storage. This shows that largest gap remains a strong candidate for routing in one-block situations. For extended comparisons for one-block layouts refer to, for example, Petersen and Schmenner (1999).

7.5.3 Probability of Correct Selection Considering this fairly strong division between situations with two cross aisles and situations with more than two cross aisles, we will now look into the latter situation in more depth. Since it appears from Table 7.4 that in situations with more than two cross aisles, combined+ routing seems to strongly outperform all other routing policies, we will focus on combined+ routing. The main question then remaining to answer is in which situation to apply which storage policy. We will disregard across-aisle storage because this policy only turns out best in a small fraction of the situations. Looking at the partial table of combined+ routing in Table 7.4, we see that all factors, except for the lines per order, give fairly similar

152 100 nearest location

90

within aisle

80 70

% be st

Fig. 7.4 Percentage of all evaluated instances for which a given storage assignment policy (as indicated next to the curves) has the lowest average travel time. All routes are calculated with the combined+ routing method

K. J. Roodbergen

60 50 40 30 20

nearest subaisle

10 0 0

1

2 3

4

5

6 7

8

9 10 11 12 13 14 15 16 17 18 19 20

lines per order

averages for the three storage assignment policies nearest-location, nearestsubaisle and within-aisle. In the following experiment we will therefore add extra values for the lines per order, while keeping the other factors as they were. Specifically, we will for three combinations of operating policies (namely, combined+ routing, with either within-aisle storage, nearest-subaisle storage or nearest-location storage) determine average travel times for the same set of parameters as described at the start of Sect. 7.5, with the exception that the lines per order equal 1, 2, 3,…, 20. Next, for each situation we determine which of the three storage policies gives the lowest travel time. We count for each value of the lines per order how often each storage policy gives the lowest travel time (ties are possible) from all 960 situations and express this as a percentage. For example, if there are six lines per order, within-aisle storage gives the lowest travel time in ten situations, nearestsubaisle storage in 489 situations and nearest location in 463 situations, or in percentages 1, 51 and 48%,respectively. We depict these percentages in Fig. 7.4. The x-axis displays the different values for the lines per order from 1 to 20. The y-axis gives the percentage of instances for which the storage policy is best. From Fig. 7.4, it can quickly be seen that nearest-location storage is an excellent choice for pick lists consisting of one pick only, but the quality of the policy quickly decreases if the pick list size increases. Within-aisle storage performs badly for small pick list sizes, but very well for large pick list sizes. Looking at the figure, it seems to be possible to formulate some cut-off points to decide on the storage policy based on the lines per order.

7.5.4 Maximum Deviation from the Best Choice The previous experiment compared the three storage assignment policies nearest-location, nearest-subaisle and within-aisle based on the percentage of the

Fig. 7.5 This figure depicts for three storage methods (as indicated next to the curves) and for different pick list sizes (x-axis) the maximum difference in travel time between the chosen storage method and the best storage method, as encountered in the set of test instances. All routes are calculated with the combined+ routing method

maximum percentage deviation from best

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153

120

within aisle

100

80

60

40 nearest subaisle 20 nearest location 0 0

1 2

3

4

5 6

7

8 9 10 11 12 13 14 15 16 17 18 19 20

lines per order

instances for which the storage assignment policies were the best. However, it is also interesting to analyze the magnitude of the errors if the wrong storage policy is chosen that is, we would like to know the difference between the travel time using a specific storage policy and the travel time using the best possible storage policy. As our performance measure, we use the maximum percentage difference from all test instances. If we use the same set test instances as before, we obtain Fig. 7.5. The most striking feature of Fig. 7.5 concerns the high values for within-aisle storage for small pick lists. For example, for pick lists with one item, within-aisle storage can result in travel times that are more than two times higher than with the best choice. Nearest-location and nearest-subaisle storage both seem to be more robust. The maximum deviation from the best option never exceeded 20% for nearest-subaisle or 15% for nearest-location in any of the test instances (for all pick list sizes).

7.5.5 Guidelines for Storage Policy Selection As is evident from the above experiments concerning the probability of correct selection and the maximum deviation, each storage policy has some areas with good performance and some areas with bad performance. It would therefore be interesting to search for a combination of policies such that there is a good performance for all pick list sizes. Using the intersections of curves in Fig. 7.4, we can formulate the following storage policy selection strategy: Strategy 1: use nearest-location storage for pick list sizes of 1, 2, 3, or 4 picks; use nearest-subaisle storage for pick list sizes ranging from 5 to 10; use within-aisle storage for pick list sizes of 11 or more picks.

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For obvious reasons, this will lead to a maximization of the probability of correct selection (in 75.2% of the instances, the best storage policy will be selected). However, in the worst situation encountered the travel time is 22.8% over the best choice. To find the lowest possible deviation, i.e., to minimize the maximum deviation, we can formulate a strategy based on the intersections in Fig. 7.5. Strategy 2: use nearest-location storage for pick list sizes of 1, 2, 3, or 4 picks; use nearest-subaisle storage for pick list sizes ranging from 5 to 17; use within-aisle storage for pick list sizes of 18 or more picks. Strategy 2 has a 68% probability of correct selection, which is somewhat lower than for strategy 1, however, the maximum error of strategy 2 is only 10.8%. The average performance of the two strategies is comparable (an average difference with the best of 0.2%, respectively 0.3%). Finally, we note that it appears from these experiments that we can select the best storage policy with a fairly high accuracy based only on the number of lines per order. All other characteristics (number of aisles, number of cross aisles, zone sizes, and demand skewness) can only be added to try and improve the last few percent.

7.6 Conclusion In this paper we investigated the effect of storage assignment policies for warehouses with multiple cross aisles. It has appeared that ABC storage can give significant improvements over random storage in warehouses with multiple cross aisles. The main findings of this paper are: • Largest gap routing is mostly appropriate for warehouse layouts with two cross aisles. For warehouses with more than two cross aisles, the combined+ routing policy has the best average performance. This result was known for random storage warehouses, but is now also confirmed for warehouses with ABC storage. • It is more important to introduce any ABC policy than to choose the right ABC policy. Even the worst ABC policy gives, on average, almost two-third of the achievable gains. • The main criterion for selecting a storage assignment policy is the number of picks per route (i.e., number of lines per order). The effects of physical layout of the area and demand skewness on correctly selecting a storage policy are much smaller. • Across-aisle storage is not a good option with regard to travel time minimization. Hence, its main application should be restricted to situation where significant congestion may occur. • Nearest-location storage should be used in environments with few picks per route (typically four or less). Furthermore, nearest-location storage is a fairly

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robust storage policy; in our experiments it never gave a deviation of more than 15% from the best storage policy. • Within-aisle storage should be used in environments with many picks per route, typically at least 11 picks per route. Within-aisle storage can give bad results if employed with few picks per route, therefore it should be applied with great care.

References Caron F, Marchet G, Perego A (1998) Routing policies and COI-based storage policies in pickerto-part systems. Int J Prod Res 36(3):713–732 De Koster R, Roodbergen KJ, Van Voorden R (1999) Reduction of walking time in the distribution center of De Bijenkorf. In: Speranza MG, Stähly P (eds) New trends in distribution logistics. Springer, Berlin, pp 215–234 (ISBN 3540666176) Dekker R, De Koster MBM, Roodbergen KJ, Van Kalleveen H (2004) Improving order-picking response time at Ankor’s warehouse. Interfaces 34(4):303–313 Gibson DR, Sharp GP (1992) Order batching procedures. Eur J Oper Res 58(1):57–67 Graves SC, Hausman WH, Schwarz LB (1977) Storage-retrieval interleaving in automatic warehousing systems. Manag Sci 23(9):935–945 Hall RW (1993) Distance approximations for routing manual pickers in a warehouse. IIE Trans 25(4):76–87 Hausman WH, Schwarz LB, Graves SC (1976) Optimal storage assignment in automatic warehousing systems. Manag Sci 22(6):629–638 Heskett JL (1963) Cube-per-order index—a key to warehouse stock location. Transp Distrib Manag 3:27–31 Jarvis JM, McDowell ED (1991) Optimal product layout in an order picking warehouse. IIE Trans 23(1):93–102 Le-Duc T, De Koster R (2005) Travel distance estimation and storage zone optimization in a 2-block class-based storage strategy warehouse. Int J Prod Res 43(17):3561–3581 Petersen CG (1999) The impact of routing and storage policies on warehouse efficiency. Int J Oper Prod Manag 19(10):1053–1064 Petersen CG (2002) Considerations in order picking zone configuration. Int J Oper Prod Manag 22(7):793–805 Petersen CG, Aase G (2004) A comparison of picking, storage, and routing policies in manual order picking. Int J Prod Econ 92(1):11–19 Petersen CG, Schmenner RW (1999) An evaluation of routing and volume-based storage policies in an order picking operation. Decision Sci 30(2):481–501 Petersen CG, Aase GR, Heiser DR (2004) Improving order-picking performance through the implementation of class-based storage. Int J Phys Distrib Logist Manag 34(7):534–544 Ratliff HD, Rosenthal AS (1983) Orderpicking in a rectangular warehouse: a solvable case of the traveling salesman problem. Oper Res 31(3):507–521 Roodbergen KJ, De Koster R (2001) Routing methods for warehouses with multiple cross aisles. Int J Prod Res 39(9):1865–1883 Ruben RA, Jacobs FR (1999) Batch construction heuristics and storage assignment strategies for walk/ride and pick systems. Manag Sci 45(4):575–596 Tompkins JA, White JA, Bozer YA, Tanchoco JMA (2003) Facilities planning. Wiley, New York (ISBN 9812530142) Vaughan TS, Petersen CG (1999) The effect of warehouse cross aisles on order picking efficiency. Int J Prod Res 37(4):881–897

Part II

Automated Storage Systems

Chapter 8

Automated Storage and Retrieval Systems: A Review on Travel Time Models and Control Policies M. R. Vasili, Sai Hong Tang and Mehdi Vasili

Abstract Automated storage and retrieval system (AS/RS) is one of the major material handling systems, which is widely used in distribution centers and automated production environments. AS/RSs have been utilized not only as alternatives to traditional warehouses but also as a part of advanced manufacturing systems. AS/RSs can play an essential role in modern factories for work-in-process storage and offer the advantages of improved inventory control and cost-effective utilization of time, space and equipment. Many issues and approaches related to the efficiency improvement of AS/RSs have been addressed in the literature. This chapter presents an overview of this literature from the past 40 years. It presents a comprehensive description of the current state-of-the-art in AS/RSs and discusses future prospects. The focus is principally on travel time estimates and different control policies such as dwell-point of the stacker crane, storage assignment, request sequencing and so on. In particular, this chapter will provide researchers and decision makers with an understanding of how to apply existing approaches effectively.

8.1 Introduction The chapter is presented in four sections. The current section and Sect. 8.2 provide brief background information on facilities planning and design, material handling, material handling equipment and Automated Storage and Retrieval System

M. R. Vasili (&)  S. H. Tang  M. Vasili Department of Industrial Engineering, Lenjan Branch, Islamic Azad University, Esfahan, Iran e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_8,  Springer-Verlag London Limited 2012

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Plant facility system

Facility system design Facilities design

Layout design

Apply to a manufacturing plant

Plant design

Handling systems design

Plant Layout

Material handling

Fig. 8.1 Facilities design hierarchy for a manufacturing plant (Modified after Tompkins et al. 1996)

(AS/RS). Section 8.3 comprehensively reviews existing travel time models on different aspects of the AS/RS, especially its control policies. Finally, Sect. 8.4 presents conclusions and promising areas for further research.

8.1.1 Facilities Planning and Design Manufacturing and service firms spend a considerable amount of time and money on planning or re-planning of their facilities. In broad terms, facilities planning determines how tangible fixed assets of an activity best support achieving the activity’s objective. For a manufacturing firm, facility planning involves the determination of how the manufacturing facility best supports production (Tompkins et al. 1996). Facilities planning can be divided into its location and its design components. In this regard, facilities design is an extremely important function, which must be addressed before products are produced or services are rendered. A poor facility design can be costly and may result in poor-quality products, low employee morale and customer dissatisfaction. Facilities design is the arrangement of the company’s physical facilities to promote the efficient use of the company’s resources such as equipment, material, energy and people. Facilities design in a manufacturing plant includes not only plant facility system and plant layout but also material handling (Fig. 8.1) (Heragu 1997; Meyers and Stephens 2005).

8.1.2 Definition and Scope of Material Handling Material handling is defined simply as moving material. The current widely used definition of material handling was presented by Tompkins et al. (1996) as the function of ‘‘providing the right amount of the right material, in the right

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condition, at the right place, at the right time, in the right position, in the right sequence, and for the right cost, by using the right method(s)’’. The American Society of Mechanical Engineers (ASME) defines material handling as ‘‘the art and science of moving, packaging, and storing of substances in any form’’. However, in recent years it has taken on broader connotations. Material handling may be thought as having five distinct dimensions: movement, quantity, time, space and control (Meyers and Stephens 2005). Raw material and parts must be delivered to the automated work cell, and the finished parts must be removed. Material handling systems are responsible for this transfer activity (Rehg 2003). Material handling is also defined by the Material Handling Industry of America as: ‘‘The movement, storage, protection and control of materials throughout the manufacturing and distribution process including their consumption and disposal’’ (Groover 2001). To begin with, any definition of material handling should include the concept of time and place utility. Material handling should also be investigated within a system context. In addition to these, a thorough definition of material handling must include the human aspect. Moreover the facility or space in which operations are housed should be considered as a part of the system. Finally, the definition of material handling must contain an economic consideration. Considering all the factors, a more complete definition might be the following (Kulwiec 1985): ‘‘Material handling is a system or combination of methods, facilities, labor, and equipment for moving, packaging, and storing of materials to specific objectives’’. It is important to note the factors that are not part of definition, as well as those that are. For instance, size and degree of mechanization are not parts of the definition. Material handling operation can either be simple and small, and involve only a few pieces of basic equipment, or it may be large, complex, or automated.

8.1.3 Material Handling Equipment A wide variety of material handling equipment is available commercially. Material handling equipment includes (Groover 2001): (1) transport equipment, (2) storage systems, (3) unitizing equipment and (4) identification and tracking systems. Traditionally, material handling equipment has been grouped into four general categories (Table 8.1). The first category includes the fixed-path or point-to-point equipment such as automated guided vehicles (AGVs). Fixed path material handling systems are also referred to as continuous-flow systems. The second category is the fixed-area equipment such as AS/RSs. The third category is variable-pass variable-area equipment such as all manual carts and the fourth category consists of all auxiliary tools and equipment (Meyers and Stephens 2005).

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Table 8.1 Four general categories of material handling equipment (Adapted from Meyers and Stephens 2005) Category Description Example Fixed-path or point-to-point equipment, (or continuous-flow systems)

Fixed-area equipment

This class of equipment serves the material handling need along a predetermined, or a fixed path This class of equipment can serve any point within a 3D area or cube

Variable-path variable-area equipment

This class of equipment can move to any area of the facility

Auxiliary tools and equipment

This class of equipment consists of all auxiliary tools and equipment

• • • •

Train and railroad track Conveyor systems Gravity-fed AGVs

• • • • • • • • • •

Jib cranes AS/RSs Bridge cranes All manual carts Motorized vehicles Fork trucks Pallets Skids Containers Automated data collection systems

8.2 Automated Storage and Retrieval System 8.2.1 Definitions of AS/RS AS/RS has been one of the major tools used for warehouse material handling and inventory control, since its introduction in 1950s. AS/RSs are widely used in automated production and distribution centers and can play an essential role in integrated manufacturing systems, as well as in modern factories for workin-process (WIP) storage. AS/RSs offer the advantages of improved inventory control and cost-effective utilization of time, space and equipment (Hur et al. 2004; Manzini et al. 2006; Van den Berg and Gademann 1999). In the broadest sense, AS/RSs (Fig. 8.2) can be defined as a combination of equipment and controls which automatically handle, store and retrieve materials with great speed and accuracy, without direct handling by a human worker (Linn and Wysk 1990b; Manzini et al. 2006; Lee et al. 1996). This definition covers a wide variety of systems with varying degrees of complexity and size. However, the term automated storage and retrieval system has come to mean a single type of system comprising one or multiple parallel aisles with multi-tiered racks; stacker crane (also referred to as storage/retrieval machine or S/R machine); input/output (I/O) stations (pickup/delivery stations, P/D stations or docks); accumulating conveyors and a central supervisory computer and communication system (Lee et al. 1996; Van den Berg and Gademann 2000).

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Fig. 8.2 Automated storage and retrieval systems (Courtesy of Stöcklin Logistik AG)

Racks are typically steel or extruded aluminum structures with storage cells that can accommodate loads which need to be stored. Stacker cranes are the fully automated storage and retrieval machines that can autonomously move, pick up and drop-off loads. Aisles are formed by the empty spaces between the racks, where the stacker cranes can move. An I/O station is a location where retrieved loads are dropped off, and where incoming loads are picked up for storage. Pick positions (if any) are locations where human workers remove individual items from a retrieved load before the load is sent back into the system (Roodbergen and Vis 2009). Figure 8.14 (see Appendix to this chapter) illustrates the generic structure and principal constituents of an AS/RS. The AS/RS will automatically put away the product or parts, or take out the product, move it to where required and adjust the inventory level at both ends of the move (Meyers and Stephens 2005). AS/RSs are automated versions of the standard warehouses and come in a wide variety of sizes. Some are very large and some are no longer than a vertical file cabinet (Rehg 2003). Briefly, a conventional AS/RS operates as follows: the incoming items are first sorted and assigned to the pallets or boxes. The loads are then routed through weighing station to ensure that those are within the load weight limit. For the pallet loads, their sizes should also be within the load size limit. Those accepted are transported to I/O station(s), with the contents of the loads being communicated to the central computer. This computer assigns the load a storage location in the rack, and stores the location in its memory. The load is moved from the I/O station to storage by stacker crane. Upon receipt of a request for an item, the computer will search its memory for the storage location and direct

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the stacker crane to retrieve the load. The supporting transportation will transport the loads from the I/O station to its final destination (Linn and Wysk 1987).

8.2.2 Types and Applications of AS/RS Several types of the AS/RS can be distinguished according to size and volume of items to be handled, storage and retrieval methods and interaction of a stacker crane and a human worker. The following are the principal types (Groover 2001; Automated Storage Retrieval Systems Production Section of the Material Handling Industry of America 2009): 1. Unit-load AS/RS. The unit-load AS/RS is typically a large automated system designed to handle, unit-loads stored on pallets or in other standard containers. The system is computer controlled, and the stacker cranes are automated and designed to handle unit-load containers. The unit-load system is the generic AS/RS. Other systems described below represent variations of the unit-load AS/RS. 2. Deep-lane AS/RS. The deep-lane AS/RS is a high density unit-load system that is appropriate when large quantities of stock are stored, but the number of separate stock types is relatively small. The loads can be stored to greater depths in the storage rack and the storage depth is greater than two loads deep on one or both sides of the aisle. 3. Miniload AS/RS. This storage system is generally smaller than a unit-load AS/ RS and it is used to handle small loads (individual parts or supplies) that are contained in small standard containers, bins or drawers in the storage system. A miniload AS/RS works like a unit-load system, except that the insertion/ extraction devices are designed to handle standard containers, totes or trays that store pieces, components and tools instead of unitized loads. 4. Man-on-board AS/RS. A man-on-board (also called man aboard) storage and retrieval system represents an alternative approach to the problem of retrieving individual items, from storage. In this system, a human operator rides on the stacker crane’s carriage. 5. Automated item-retrieval system. These storage systems are also designed for retrieval of individual items or system product cartons; however, the items are stored in lanes rather than bins or drawers. 6. Vertical lift storage modules (VLSM). These are also called vertical lift automated storage/retrieval system (VL-AS/RS). All of the preceding AS/RS types are designed around a horizontal aisle. The same principle of using a center aisle to access loads is used except that the aisle is vertical. Vertical lift modules, some with height of 10 m (30 foot) or more, are capable of holding large inventories while saving valuable floor space in the factory. Since in the material handling industry the carousel-based storage systems are distinguished from AS/RSs, they are not included in the above classification. A carousel storage system consists of a series of bins or baskets suspended from on

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AS/RSs

Stacker Crane

Motion

Aisle captive

Aisle changing

Handling

Shuttle

Single

Dual

Man-on-board

Picking

Triple

Single deep

Loads

Pallets

End-of-Aisle

Rack

Bins

Double deep

Using flow rack

Deep Lane

Using shuttle car

Unit load

Fig. 8.3 Various system concepts for AS/RSs (Modified after Roodbergen and Vis 2009)

overhead chain conveyor that revolves around a long oval rail system. A general comparison between an AS/RS and a carousel storage system can be found in (Groover 2001). Based on the rack structure, stacker crane capabilities and its interaction with the worker and the product handling and picking methods, a large number of system options can be found for the AS/RSs. The most basic version of an AS/RS has in each aisle one stacker crane, which cannot leave its designated aisle (aisle-captive) and which can transport only one unit-load at a time (single shuttle). Product handling in this case is by unit-load (any load configuration handled as a single item, e.g., full pallet quantities) only; no people are involved to handle individual products. The racks in the basic version are stationary and single-deep (see Fig. 8.15 in Appendix to this chapter), which means that every load is directly accessible by the stacker crane. This AS/RS type is referred to as a single unit-load aisle-captive AS/RS. Numerous variations exist from this basic AS/RS. An overview of the main concepts is presented in Fig. 8.3. Recall that carousel storage systems with rotating racks are not considered in this overview. Often an AS/RS is used for handling unit-loads only. If the unit-loads are bins, then the system is generally called a miniload AS/RS. Unit-loads arrive at the I/O station of the AS/RS from other parts of the warehouse by means of automated guided vehicles, conveyors and so on. The AS/RS stores the unit-loads and retrieves them again after a period of time. In some cases only part of the unit-load may be required to fulfill a customer’s order. This can be resolved by having a separate picking area in the warehouse; in which case the AS/RS serves to replenish the picking area. Alternatively, the picking operation can be integrated with the AS/RS. One option is to design the crane such that a person can ride along (man-on-board). Instead of retrieving a full pallet automatically from the location, the person can pick one item from the location. Another option to integrate item

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picking is when the AS/RS drops off the retrieved unit-loads at a workstation. A picker at this workstation takes the required amount of products from the unitload after which the AS/RS moves the remainder of the load back into the rack. This system is often referred to as an end-of-aisle (EOA) system (Roodbergen and Vis 2009). The AS/RSs are typically used in the applications where there is a very high volume of loads being moved into and out of the storage locations; storage density is important due to the space constraints; no value adding content is present in this process, and where the accuracy is critical in order to prevent potentially costly damages to the loads (ASAP Automation 2008). Under such circumstances, most applications of AS/RS technology have been associated with warehousing and distribution operations. An AS/RS can also be used to store raw material and WIP in manufacturing. Three application areas can be distinguished for AS/RSs (Groover 2001): (1) unit-load storage and handling, (2) order picking, and (3) WIP storage systems.

8.2.3 Types of Stacker Crane in AS/RS In an AS/RS, the stacker crane (storage/retrieval, S/R machine) is a rectangular geometry robot and it is used to store and retrieve loads into/from the storage cells. This autonomous vehicle is equipped with a vertical drive, a horizontal drive and typically one or two shuttle drives. The vertical drive raises and lowers the load. The horizontal drive moves the load back-and-forth along the aisle. The shuttle drives transfer the loads between the stacker crane’s carriages and the storage cells in the AS/RS rack (carriage is that part of a stacker crane by which a load is moved in the vertical direction). For greater efficiency, the vertical and horizontal drives are capable of simultaneous operations (Hu et al. 2005). Figure 8.4 shows some common types of stacker crane in AS/RSs.

8.2.4 Automatic Identification System in AS/RS Load identification is the primary role of automatic identification in AS/RSs. The scanners are located at the induction or transfer location, to scan a product identification code. The data are sent to AS/RS computer, which upon receipt of load identifications, assigns and directs the load to the storage location. Working this sequence in reverse can effectively update inventory file based on transaction configuration. Scanners also play an important role in integrating AS/RSs, AGVs, conveyors and robotics in the automated factory by providing discrete load or product information to the appropriate controllers/computers as transfers occur (Kulwiec 1985).

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Fig. 8.4 Some common types of stacker cranes in AS/RSs (Courtesy of Stöcklin Logistik AG)

8.2.5 AS/RS Design Decisions In the last decades there have been several studies which present general overviews of warehouse design and control include Van den Berg (1999), Rouwenhorst et al. (2000), De Koster et al. (2007), Gu et al. (2007) and Baker and Canessa (2009). These papers discuss only a fraction of the AS/RS issues and the literature, due to their broad scope. More specifically, Roodbergen and Vis (2009) presented an extensive explanation of the current state-of-the-art in AS/RS design for a range of related issues. This paper seems to be the first review paper over last 10 years devoted exclusively to AS/RSs, and the first ever to give a broad overview of all design issues in AS/RSs. Therefore some part of this paper related to AS/RS design is investigated in the following. Due to the complexity and enormous cost associated with automated material handling systems, it is crucial to design an AS/RS in such a way that it can efficiently handle current and future demand requirements, while avoiding overcapacity and bottlenecks. Furthermore, due to the inflexibility of the physical layout and the equipment, it is essential to design it right at once. Figure 8.5 presents a schematic view of design issues and their interdependence for AS/RSs and provides an overview of all design decision problems that may need to be selected. These policies will be discussed later in the Sect. 8.3. It is important to realize that the AS/RS is usually just one of the several systems to be found in a warehouse. The true performance of the AS/RS is typically influenced by the other systems as are the other systems’ performances influenced by the AS/RS. As depicted in Fig. 8.5, part of the actual design of an

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Physical design and related decisions System Choice

System Configuration

(Section: 8.2.2)

(Sections: 8.3.3, 8.3.4 & 8.3.6)

• Unit load AS/RS • Deep -lane AS/RS • Miniload AS/RS • Man-on-board AS/RS • Automated item-retrieval system • Vertical lift storage modules (VLSM)

• Number of aisles • Height of the storage racks • Length of the aisles • Equally sized, unequally sized or modular cells • Number and location of the I/O stations • Buffer capacity at the I/O stations • Number of stacker cranes per aisle • Number of order pickers per aisle (if any)

Performance measurement Examples of performance measures: • Travel time estimates • Throughput estimates

• Utilization of rack and stacker crane

Control policies and related decisions Dwell Point (Section: 8.3.5) • Type of positioning (static or dynamic) • Location where idle stacker cranes will be placed

(Section: 8.3.7) • Storage assignment method • Number of storage classes • Positioning of the storage classes

Batching

Sequencing

(Section: 8.4.9) • Type of batching (static or dynamic) • Batch size (capacity or time based) • Selection rule for assignment of orders to batches

(Sections: 8.3.2 & 8.3.8) • Sequencing restrictions (e.g., due dates) • Type of operation (single or dual command) • Scheduling approach (block or dynamic) • Sequencing method

Load Shuffling (Section: 8.3.10) • Selection rule for shuffling of loads

Design Storage Assignmentand control of other material handling systems in the facility

Fig. 8.5 Design of an AS/RS and related decisions (Modified after Roodbergen and Vis 2009)

AS/RS consists of determining its physical appearance. The physical design consists of two aspects which together determines the physical manifestation of the system. First is the choice of the AS/RS type (system choice). Second, the selected

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system must be configured, for instance, by deciding on the number of aisles and the rack dimensions (system configuration). These interrelated choices can be made based on, among others, historical and forecasted data, product characteristics, the available budget, required throughput, required storage space and available land space. Various concepts for AS/RS types were displayed in Fig. 8.3; however, little research can be found to support the selection of the best type of system from the available concepts. Control policies are methods which determine the actions performed by the AS/RS. Typically, the operation of an AS/RS is administrated by a coherent set of such control policies, which each take care of a specific subset of the activities. The position where an idle crane (i.e., a crane that has no jobs to perform) waits is determined by a dwellpoint policy. The dwell-point is best chosen to minimize the expected time to travel to the next (still unknown) request. A storage assignment policy serves to determine which products are assigned to which locations. Meanwhile, updating and shuffling of items and reconsidering storage assignment decisions can be vital in current dynamic environments to meet the fluctuating, short-term throughput requirements imposed on the AS/RSs. The objective of load-shuffling strategy is to shuffle (i.e., pre-sort, relocate or rearrange) the loads to specified locations during the slacker crane idleness, in order to minimize the response time of retrieval. A tour of an AS/RS consists of a sequence of requests, starting at the origin of the first request and ending at the destination of the last request. Sequencing rules can be used to create tours such that the total time to handle all request is minimized or the due times are least violated. As another control policy of AS/RS, batching considers how one can combine different customer orders into a single tour of the crane. This policy is mainly applicable to man-on-board AS/RS. For a typical design problem, total capacity is given beforehand. This essentially means that the mathematical product of the number of aisles, rack height, and rack length is constant. Increasing the number of aisles thus implies reducing rack length and/or height to maintain the desired storage capacity. Because of this relation, having more aisles indirectly results in shorter response times, due to the decreased rack length and height. Furthermore, design changes often have an impact in multiple ways at the same time. In a standard system with one crane per aisle, having more aisles also means having more cranes, which in turn results in a higher throughput and higher investment costs.

8.3 Existing Travel Time Models on Different Aspects of AS/RS 8.3.1 AS/RS Travel Time Interpretations Travel time for an AS/RS is the service time for a transaction including both stacker crane travel time and pick up/deposit time. The pick up/deposit time is generally independent of the rack shape and travel velocity of the slacker crane.

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Hence, in order to simplify the derivations, in analytical approaches the pick up/ deposit time are often ignored without affecting the relative performance of the control policies (Hausman et al. 1976; Bozer and White 1984; Hu et al. 2005; Sari et al. 2005 and so on). Therefore the travel time for an AS/RS is the time used by stacker crane to move from its dwell-point to the location of requested item and lastly return to its dwell-point position. Due to the fact that the stacker crane has independent drives for horizontal and vertical travel, the travel time of the stacker crane may be measured by the Chebyshev metric (i.e., the travel time of the stacker crane is the maximum of the isolated horizontal and vertical travel times). Thus if Dx and Dy denote the translations in horizontal and vertical direction, respectively, and vx and vy denote the maximum speeds in the horizontal and vertical direction, respectively, then the associated travel time is max{Dx/vx, Dy/vy}. The Chebyshev metric is also known as the maximum metric or the L?-norm (Van den Berg 1999). The AS/RS travel time models are based on either the discrete-approach or continuous-approach. In the discrete-approach travel time models, the AS/RS rack face is considered as a discrete set of locations. However using a continuous-approach to represent the rack, the rack is normalized to a continuous pick face. In practice, there is no significant difference between the results obtained from the continuous-approach-based expressions and the ones from the discrete-approach-based solutions (Sari et al. 2005). Discrete representation of the rack, for example, was investigated by Egbelu (1991), Thonemann and Brandeau (1998), Ashayeri et al. (2002), Sari et al. (2005) and so on. Continuous representation of the rack has received considerable interests since the study of Hausman et al. (1976) and these literatures can be classified into two groups according to the shape of the AS/RS: (1) square-in-time and (2) rectangular-in-time. In a square-in-time AS/RS, the dimensions of the rack and the vertical and horizontal speeds of the stacker crane are such that the time to reach the most distant row (tier) from the I/O station equals the time to reach the most distant bay (column) (Sarker and Babu 1995). Any rack that is not square-in-time is called rectangular-in-time. Based on a continuous rack approximation approach, Bozer and White (1984) presented expressions for the expected cycle times of an AS/RS performing singlecommand (SC) and dual-command (DC) cycles. They normalized the rack as a continuous rectangular pick face with length of 1.0 and height of b in terms of time. By definition, Tv = H/sv and Th = L/sh. Let T = max{Tv, Th} and b = min{Tv/T, Th/T}, which implies that 0 B b B 1, where L is length of the rack, H is height of the rack, sh and sv are the speed of stacker crane in the horizontal and vertical directions, respectively, Th represents the horizontal travel time required to go the farthest column from I/O station and Tv denotes the vertical travel time required to go to the farthest row (level). As the value of b may represent the shape of a rack in terms of time, b was referred to as the ‘‘shape factor’’. An illustration of the continuous, normalized rack face is shown in Fig. 8.6. As illustrated in Fig. 8.6, to analyze the expected travel time between two points, any storage (or retrieval) point is represented as (x, y) in time, where 0 B x B 1 and

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Fig. 8.6 Illustration of AS/RS continuous rack face (Modified after Peters et al. 1996)

0 B y B b. Hence, the normalized rack is b time units long in vertical direction and 1.0 time units long in the horizontal direction. Example (Bozer and White 1984) Suppose that rack dimensions and the stacker crane speed in such that L = 348 ft, H = 88 ft, sh = 356 fpm, and sv = 100 fpm. Using the approach explained earlier, so Th = L/sh = 348/356 = 0.9775 min, and Tv = H/sv = 88/100 = 0.8800 min and T = max{Tv, Th} = Th. Therefore the shape factor is b = Tv/Th = 0.8800/0.9775 = 0.90. Hence, the normalized rack is 0.90 time units long in the vertical direction and 1.0 time units long in the horizontal direction

8.3.2 Different Command Cycles of the Stacker Crane In the single-shuttle AS/RSs, the stacker crane can operate under SC cycle and/or DC cycle. In a SC, only one operation of storage or retrieval of item is conducted. However, in a DC both storage and retrieval of items are conducted during one cycle of the stacker crane (Lee et al. 2005). In multi-shuttle system with two transport unit-load (TUL), (i.e., twin-shuttle system) the stacker crane can perform up to two storages and two retrievals in a cycle, which is called a quadruple command (QC) cycle (Meller and Mungwattana 1997; Potrc et al. 2004). A QC cycle transports two storages and two retrievals at the same AS/RS cycle. The first transaction must always be a storage transaction and the last transaction must always be a retrieval one. The second and the third transaction must be storage transaction and the retrieval transaction (Sarker and Babu 1995; Meller and Mungwattana 1997; Potrc et al. 2004). Likewise, in multi-shuttle system with three TUL (i.e., triple-shuttle systems) the stacker crane can perform up to three storages and three retrievals in a cycle, which is called a sextuple command (STC or SxC) cycle (Meller and Mungwattana 1997; Potrc et al. 2004). However, the stacker cranes capable of transporting more than two loads are still rarely seen and it is believed that there are no systems in practice with more than three shuttles (Meller

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Fig. 8.7 Different command cycles of the stacker crane

and Mungwattana 1997; Roodbergen and Vis 2009). The storage (S) and retrieval (R) operations in an AS/RS rack for different command cycles are shown in Fig. 8.7. Hausman et al. (1976) analyzed the travel time of AS/RS only for the SC cycle in a single-shuttle system. Graves et al. (1977), Bozer and White (1984) and Pan and Wang (1996) studied the single- and dual-operating modes together with other control policies for AS/RS. Bozer and White (1984) developed analytical models for calculating SC and DC cycles under a single-shuttle system. By assuming uniformly distributed coordinate locations for random storage, they used a statistical approach to develop expressions for travel time. For discrete rack model, the expected travel times were computed using the expressions, N   1X 2t0i E SC ¼ N i¼1

  E DC ¼

N1 X N  X  2 t0i þ tij þ t0j NðN  1Þ i¼1 j¼iþ1

ð8:1Þ

ð8:2Þ

where, E(SC) is the expected SC cycle travel time, E(DC) is the expected DC cycle travel time, N is the total number of openings in the rack, t0i is the one-way travel time between the I/O station (which is located at the lower left-hand corner of the rack) and the ith opening (t0i = ti0), and tij is the one-way travel time between the ith opening and the jth opening (tij = tji) and t0j is the travel time from jth opening to I/O station (t0j = tj0). Using the method explained in Sect. 8.3.1,

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Bozer and White (1984) derived expected travel times models for both SC and DC cycles based on a continuous rack approximation approach. The expressions are, 1 EðSC Þ ¼ b2 þ 1; 3 EðDC Þ ¼

4 1 2 1 þ b  b3 : 3 2 30

ð8:3Þ ð8:4Þ

Note that the above expressions provide results corresponding to the normalized continuous rack. In order to obtain the results corresponding to the original rack, the above travel times should be denormalized to obtain:     1 2 ð8:5Þ E SC ¼ EðSCÞ  T ¼ b þ 1  T; 3     4 1 2 1 þ b  b3  T: E DC ¼ EðDC Þ  T ¼ ð8:6Þ 3 2 30 Referring to the example in Sect. 8.3.1, when T = 0.9775 and b = 0.90, thus E(SC) = 1.27 time units and, E(DC) = 1.7140 time units. To obtain the results corresponding to the original rack, the above travel times are denormalized to obtain EðSCÞ ¼ EðSC Þ  T ¼ 1:2414 min and EðDCÞ ¼ EðDC Þ  T ¼ 1:6754 min: Sarker et al. (1991) analyzed the travel time and the performance of a doubleshuttle AS/RS operating on a QC cycle under nearest-neighbor (NN) and classbased storage scheduling techniques. It was observed that a dual-shuttle AS/RS operating under the proposed scheduling techniques would significantly improve system throughput performance over a single-load shuttle system. Since the majority of researchers investigated single-shuttle system, throughput capacity is thus limited with maximal technical characteristics of stacker crane and optimal geometry of high storage racks. In general, the throughput capacity of an AS/RS increases as the number of shuttles increases, since the amount of empty travel decreases correspondingly. Hence, in order to increase the throughput capacity, it is necessary to employ the stacker crane that can store and retrieve several TUL at the same time. Analytical models under multi-shuttle system were presented by Meller and Mungwatana (1997). Within storage operation of QC and STC cycles, modified NN storage strategy was used. Storage in single- and multi-shuttle systems were investigated by Potrc et al. (2004). Comparison of the single-shuttle system and multi-shuttle system showed large improvements in throughput capacities of multi-shuttle system. Foley and Frazelle (1991) considered EOA miniload AS/RSs and derived the distribution of the DC cycle time for uniformly distributed activity in a square-in-time rack. Using this distribution, they obtained closed form expressions for the maximum throughput of minload systems with deterministic or exponentially distributed pick times. In order to handle extra heavy loads (loads above 20 tons, such as sea container cargo) at high speed, a new kind of S/R mechanism in split-platform AS/RS, or

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A. Handover stations B. Horizontal platforms C. Vertical platforms D. I/O stations E. Storage locations (Cells)

Fig. 8.8 A schematic view of split-platform AS/RS (SP-AS/RS) (Modified after Hu et al. 2005)

SP-AS/RS in short, was presented by Chen et al. (2003) and Hu et al. (2005). They developed discrete (Chen et al. 2003) and continuous (Hu et al. 2005) travel time models for the proposed system under SC cycle. In the SP-AS/RS transports of the load within individual storage aisles are separated into vertical and horizontal movements and handled by different devices, namely the vertical platform and the horizontal platform, respectively. Figure 8.8 gives a schematic view of a standard aisle in the SP-AS/RS. By separating the mechanisms for vertical and horizontal movements, the proposed system can handle heavier loads at a higher speed. High lifting capacity enables the SP-AS/RS to deal with all the different types and sizes of containers which pass through the container terminals (Hu et al. 2005). A container terminal in a port is the place where container vessels dock on berths and unload inbound (import) containers (empty or filled with cargo) and load outbound (export) containers. The terminals have storage yards for the temporary storage of these containers (Murty et al. 2005). From the literature survey in this section, it is concluded that most of the literature assumes single-shuttle systems that the stacker crane performs only either SC or DC at each operation. However, the throughput capacity of an AS/RS increases as the number of shuttles increases, since the amount of empty travel decreases correspondingly.

8.3.3 Operating Characteristics of the Stacker Crane The majority of studies have assumed a constant stacker crane velocity and instantaneous acceleration. Gudehus (1973) proposed a method to adjust the previous results when the acceleration and deceleration of the stacker crane are taken into account. Guenov and Raeside (1989) observed in their experiments that an optimum tour with respect to Chebyshev travel may be up to 3% above the

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optimum for travel times with acceleration/deceleration. Hwang and Lee (1990) presented continuous analytical travel time models which integrate the operating characteristics of the stacker crane. Using a randomized assignment policy, travel times are determined for both SC and DC cycles and the models are validated through discrete evaluation procedures. They defined the acceleration/deceleration rate and maximum velocities in the horizontal and vertical directions as three important elements in the travel time model. Considering these three elements which describe the capabilities of the stacker crane, they derived the travel time of the stacker crane as,

EðSCÞ ¼ 2

max Zðth ;tl Þ

zgk ðzÞdz; k ¼ 1; 2; 3:

ð8:7Þ

0

where E(SC) is SC travel time, G(z) is the probability that travel time to the point (x, y) on the aisle is less than or equal to z and gk(z) is the probability density function. They used the same relation for the DC cycle time, but calculated the E(TB) using a different expression as,

EðTBÞ ¼

max Zðth ;tl Þ

zbk ðzÞdz; k ¼ 1; 2; 3:

ð8:8Þ

0

where E(TB) is the expected travel time between to randomly selected points, bk(z) is the probability density function of travel time between. Their model gives values very close to those obtained by the discrete rack methods. Chang et al. (1995) proposed a travel time model for AS/RS by considering the speed profiles that exist in real-world applications. Compact forms of expected travel-times under randomized storage conditions are determined for both SC and DC cycles. An extension of Chang et al. (1995) was proposed by Chang and Wen (1997) to investigate the impact on rack configuration on the speed profile of the stacker crane. The results demonstrate that the optimal rack configuration of the SC is square-in-time whereas the DC cycle may not be. Furthermore, the travel times for both SC and DC cycles are quite insensitive to the deviation in the length of the rack configuration. As another extension of Chang et al. (1995), Wen et al. (2001) proposed travel time models that consider various travel speeds with known acceleration and deceleration rates. Compact forms of expected travel time under class-based and full-turnover storage assignments were determined. Their results show that both the proposed exponential travel time model and the adjusted exponential model perform satisfactorily and could be useful tools for designing an AS/RS in real-world applications.

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Fig. 8.9 Structure of the rack with a Equal sized, b Unequal sized and c Modular cells

8.3.4 Storage Cells in a Rack, Design of the Rack Structure and Physical Layout 8.3.4.1 Storage Cells in a Rack The storage cells in an AS/RS rack may be considered homogeneous or may be partitioned into several areas called classes (Hu et al. 2005). There are various types of AS/RS with equally sized cells according to the size and volume of items to be handled, storage and retrieval methods and interaction of a stacker crane with the worker such as unit-load AS/RS, mini-load AS/RS, man-on-board AS/RS, automated item-retrieval system and deep-lane AS/RS (Groover 2001). Many researchers have studied the optimal design of AS/RS with the rack of equally sized cells for using the concept of unit load (Fig. 8.9a). However, in terms of the flexibility of storage capability, the existing rack configuration using the concept of unit load is inefficient and inadequate for the storage of various types and various sizes of customers’ demands. Moreover, if the various sizes of products are to be stored in existing systems, the space utilization will be considerably decreased due to the increase of lost space in each cell (Lee et al. 2005). Lee et al. (1999) proposed a model for AS/RS with the rack of unequally sized cells. In this model, the cells within a zone have the same size, but the sizes of cells in the different zones are different in height such that the rack can hold various types of the load (Fig. 8.9b). This model will be a good alternative for coping with those problems described above. However, if the quantity of the storage demands for different sized products fluctuates in large, even the model proposed by Lee et al. (1999) will not basically be able to overcome inflexibility and low-space utilization problems in the existing AS/RS rack structures. Lee et al. (2005) presented the model of AS/RS with the rack of modular cells (Fig. 8.9c). They determined the best size of modular cell as a decision variable, and presented the effectiveness of the model. This type of AS/RS is more flexible to the size and has higher space utilization than those of existing rack structure, and could be a useful alternative for the storage of different unit-load sizes.

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8.3.4.2 Design of Rack Structure and Physical Layout Design in an AS/RS AS/RSs are very expensive investments. Once installed, the technical characteristics are difficult to modify. Therefore a formalized decision model should be available in the design process (Ashayeri et al. 1985). The design of AS/RS involves the determination of the number of stacker cranes, their horizontal/vertical velocities and travel times, the physical configuration of the storage racks, etc. Only a few researches address the design of AS/RSs in combination with the design of other material handling systems in the facility. Most of these researches consider manufacturing environments. The design of warehouses has been studied basically with two approaches: (i) analytical optimization methods; and (ii) simulation. The studies which cover such approaches are investigated in the following. (i)

Analytical Optimization Methods

As for the analytical methods, Roberts and Reed (1972) presented an optimization model to determine the warehouse bay configuration that minimizes the cost of handling and construction, ignoring the constraints on handling capacity of equipment and building sites. It was assumed that storage space is available in units of identical bays, and the optimal bay configuration was determined to minimize the construction and material handling cost. According to De Koster et al. (2007), one of the first publications in the subject of optimizing the warehouses was presented by Bassan et al. (1980). The optimum dimensions of the warehouse were analyzed, considering the chosen volume of the warehouse in dependence on the various storage strategies. Two configurations of racks in a homogeneous or a zoned warehouse were compared, considering handling costs as well as costs associated with the warehouse area and perimeter. From these, expressions for optimal design parameters were developed. It was shown that, depending on ratios between the relevant costs, some general preference rules for the two layouts examined can be laid down. A Design package based on a cost model for AS/RS was developed by Zollinger (1975). According to Zollinger’s cost model, the mathematical properties of the cost functions were defined corresponding to various elements in the system. Subsequently, the minimum-cost design was determined by performing a Fibonacci search over the number of aisles in the system. Hodgson and Lowe (1982) studied a layout problem involving the placement of items in a storage rack serviced by a stacker crane. The analysis was restricted to the case of dedicated storage and SC cycles. Karasawa et al. (1980) developed a non-linear mixed-integer programming (MIP) for a deterministic model of an AS/RS to minimize the total cost. The objective function included three main decision variables: the number of stacker cranes, the height and the length of the rack. Constant values involved were cost of the land, cost of the warehouse, cost of the rack construction and cost of stacker cranes. Optimization was performed as a function of sufficient storage volume for all items and sufficient number of cranes to serve all storage and retrieval requests. The main disadvantage of this model is that it refers only to the single-aisle AS/RS and the warehousing operation of only the SC cycle. Ashayeri et al. (1985)

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described a model which allows the determination of the major design characteristics of the warehouse. The objective of the model was to minimize the investment and operating costs over the project lifetime. They presented this mathematical model for the calculation of the optimal number of cranes and the optimal width and length of the warehouse subject to constraints on the constant crane velocities, the throughput and the length and width of building site. Park and Webster (1989) investigated the design of warehouses by proposing an approach that simultaneously selects the used storage equipment, that might be an AS/RS, and the overall size and shape of the storage area. The objective was to develop an optimization procedure to aid a warehouse planner in the design of selected threedimensional (3D), palletized storage systems. All alternatives were compared in the overall model while simultaneously considering the following factors: control procedures, handling equipment movement in an aisle, storage rules, alternative handling equipment, input and output patterns for product flow, storage rack structure, component costs and the economics of each storage system. Bozer and White (1990) addressed the design of EOA order picking systems by focusing on a miniload AS/RS. Performance models and a design algorithm were developed and presented. The objective of the design algorithm was the minimization of the number of storage aisles subject to two types of capacity constraints: throughput and storage space. Although the system with two pick positions can be modeled directly as a closed queuing network with two servers and two customers, its special structure led to an alternative approach in developing the performance model. For two and more pick positions, the results obtained were compared with those obtainable using simulation and a diffusion approximation. However, since the analysis assumes that the requests are always available, it represents an overestimation of the system throughput. In a later study, Bozer and White (1996) presented an analytical design algorithm to determine the near-minimum number of pickers required in a same EOA miniload AS/RS. The algorithm was for general system configurations with two or more pick positions per aisle and/or two or more aisles per picker. Moreover, for systems with two pick positions, the possibility of improving the picker utilization by sequencing container retrievals within each order was investigated. In many man-on-board AS/RSs, some very typical, recurrent orders have to be retrieved. Van Oudheusden and Zhu (1992) presented a straightforward methodology to design the storage layout of a rack when such recurrent orders represent a high percentage of total turnover. The approach makes use of sorting, assignment, and traveling salesman like algorithms. The resulting layouts were compared against more classical arrangements. Based on numerical simulations it was observed that, in specific situations, more than a significant saving in travel time of the retrieval crane can be expected. Malmborg (2001) modified a well-known rule of thumb for evaluating storage rack configurations in AS/RSs to avoid the need for two key assumptions. These assumptions are the proportion of SC and DC order picking cycles used in operating a system and the total storage capacity requirements when randomized versus dedicated storage is used. Procedures for generating AS/RS cost estimates were also directly coupled with models for estimating the utilization of stacker cranes.

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The modified rules of thumb were also designed for implementation on PC-level hardware, but with adequate computational efficiency for analyzing a broad range of rack design alternatives in large-scale applications. Hwang et al. (2002) investigated the design of miniload AS/RSs in combination with AGVs. Both nonlinear model and heuristics have been proposed to determine the optimal number of loads to be transferred by each AGV to machines in combination with an optimal design of the AS/RS. Bozer and Cho (2005) derived the results which can be used in the design or evaluation of new/proposed systems. Assuming a particular dwell-point strategy for the storage/retrieval machine, they derive closed-form analytical results to evaluate the performance of an AS/RS under stochastic demand and determine whether or not it meets throughput. Design of a new compact 3D AS/RS was proposed by Le-Duc et al. (2006) and De Koster et al. (2006). The research objective was to analyze the system performance and optimally dimension the system. Under SC cycle a closed-form expression was developed for the expected retrieval travel time of the system. From the expected travel time, the optimal ratio between 3D that minimizes the travel time for a random storage strategy was calculated. In addition, an approximate travel time expression for the DC cycle was derived for the system with powered and gravity conveyors, respectively, and it was used to optimize the system dimensions. Kuo et al. (2007) proposed computationally efficient design conceptualization models for unit-load AS/RSs based on autonomous vehicle technology (AVS/RS). Vehicle and lift travel times and the probability distribution for twelve service scenarios occurring under realistic operating assumptions were formulated and used to generate expected transaction service times. Additional measures of system performance including transaction waiting time and vehicle utilization are formulated for systems using random storage and point-of-servicecompletion dwell-point rules. The models provide a practical means of predicting key aspects of system performance based on five design variables that drive the majority of system costs. (ii)

Simulation

Owing to the complexity and enormous cost involved in automated material handling systems, there is a growing need to use computer simulation in both the physical aspect and control software design of such systems. Simulation models can be developed to test not only the final system configuration, but also each installation phase (Raghunath et al. 1986). Simulations are mandatory to adequately model all operational features of the AS/RSs, since existing analytical models only apply to special instances (Van den Berg and Gademann 2000). As for the simulation methods, Bafna and Reed (1972) developed a design package where the optimum configuration is determined by using simulation in conjunction with a search procedure. A similar approach was presented by Koenig (1980), where the search for the optimum configuration was limited to certain values of the design variables specified by the user. Perry et al. (1984) presented an optimum-seeking approach to the design of AS/RS. The method was developed to improve the effectiveness with which simulation models of such systems can be used as design

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aids. The system modeled consists of several aisles of storage bins, storageretrieval devices (stacker cranes), closed loop conveyor, work stations, and input/ output buffers to interface with the conveyor. Optimum-seeking rules or heuristics were used in conjunction with the simulation model to reach a local optimum solution. Rosenblatt and Roll (1984) presented a search procedure for finding a global optimal solution for a specific formulation of the warehouse design problem. In this formulation three types of costs were considered: costs associated with the initial investment (construction and handling facilities), a shortage cost and costs associated with the storage policy. The search procedure for finding the optimal storage design was developed, comprising analytical optimization and simulation techniques. Raghunath et al. (1986) described the development of an interactive and flexible simulation software for AS/RS of the miniload variety. A modular approach was taken in the development of the simulation software so that the user, through an interactive menu, has the capability to model an AS/RS by selecting a combination of modules that define the AS/RS. The user then enters the values of the system variables specified for each module. This user-defined simulation model is translated into a simulation language source code and then executed. The study of AS/RS in warehouses has developed along two main lines: One seeks to minimize the total cost of an AS/RS, while the other explores the dynamic behavior of such a system. Rosenblatt et al. (1993) addressed the two issues simultaneously and presented a combined optimization and simulation approach for designing AS/RSs. A heuristic recursive optimization/simulation procedure was developed and applied to several situations, and converged within a few iterations. This model finds the physical characteristics of the AS/RS, however the relationship between dimension of rack and capacity of stacker crane that could affect its performance was not considered in this model. Randhawa and Shroff (1995) performed the most extensive simulation study. They examined the effect of different sequencing rules on six layout configurations (with a varying I/O-point, item distribution over racks, rack configuration and rack dimensions). Based on a limited number of experiments they concluded, among other things, that locating the I/O-point at the middle of the aisle, instead of at the end of the aisle, results in a higher throughput. Manzini et al. (2006) presented a multi-parametric dynamic model of a product-to-picker storage system with class-based storage allocation of products. Thousands of what-if scenarios were simulated in order to measure the impact of alternative design and operating configurations on the expected system performance and to identify the most critical factors and combinations of factors affecting the response of the system. Class-based storage was found to be a very effective way of both reducing the picking cycle time and maximizing the throughput of the system. The rapid effectiveness of visual interactive simulation (VIS) in supporting the design and control of new warehouses emerges, responding to the need for flexibility which modern companies need in order to adapt to strongly changing operating conditions quickly. Based on examination of the literature, it can be concluded that the strength of simulation could be better exploited in AS/RS researches to compare numerous

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designs, while taking into account more design aspects, especially in combination with control policies. Sensitivity analyses on input factors should also be performed such that a design can be obtained which can perform well in all applicable scenarios. As a result more general information could be obtained on good design practices (Roodbergen and Vis 2009).

8.3.5 Dwell-Point Policy of the Stacker Crane The dwell-point in an AS/RS is the position where the stacker crane resides, or dwell, when the system is idle (Van den Berg 1999). Hu et al. (2005) defined the dwell-point policy as the policy to decide where the stacker crane will stay when it becomes idle. The dwell-point is selected such that the expected travel time to the position of the first transaction after the idle period is minimized (Van den Berg 1999). There is extensive research in the area of dwell-points for stacker cranes. The dwell-point of a stacker crane is the rest or home position of the machine when it becomes idle. A machine is said to be idle if it is functional but has no assignment in progress. Machine idleness occurs when a stacker crane completes a task and there is no immediate other storage or retrieval request task to reassign the machine. Machine idleness is not a continuous process: idle periods are broken up by periods of busy activity by the machine. Thus every instance of a machine idleness involves a time during which the machine has no assignment. Strategic pre-positioning of stacker cranes when they become idle, in anticipation of incoming requests for order storage and retrieval, is one method of improving the system response time (Chang and Egbelu 1997a). Graves et al. (1977) selected the dwell-point of the stacker crane at the I/O station. They introduced the design, planning and control of warehousing systems as new research topics. Bozer and White (1984) and Linn and Wysk (1987) investigated various dwell-point policies. For the dwell-point specification problem, the following static dwell-point rules were outlined by Bozer and White (1984), although they provided no quantitative comparison of their performance: 1. Return to the input station following the completion of a SC storage; remain at the output station following the completion of either a SC retrieval or DC cycle; 2. Remain at the storage location following the completion of a SC storage; remain at the output station following the completion of either a SC retrieval or a DC cycle; 3. Travel to a midpoint location in the rack following the completion of any cycle; 4. Travel to the input station following the completion of any cycle. Linn and Wysk (1987) investigated two dwell-point policies for AS/RS: (1) single addressing; and (2) pursuit mode. Under single addressing mode, the storage or retrieval command is initiated at the I/O station, which is the stacker crane home base. Given a storage request, the stacker crane picks up the load, stores it in its assigned location, then, returns empty to home base; given a retrieval request, the

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stacker crane goes, from the home base, to retrieve the pallet, and bring it back to I/O station. However, under pursuit mode, the stacker has no fixed home base, and remains in the position of the last completed command. Depending on the next command issued, it may go on to retrieve a pallet, or return to I/O station to pick up a pallet for storage. The results show that the pursuit mode appears to be better than single addressing mode, and should always be used for AS/RS control strategy when both types of requests are available. Egbelu (1991) showed that the expected travel time of stacker crane could be obtained by summing the expected travel time to each location in the rack from an unknown dwell-point and then obtaining the expected travel time of stacker crane as a linear program. For this purpose, a linear programing methodology was developed which minimizes the service response time in an AS/RS through the optimal selection of the dwell-point of the stacker crane. A framework for selecting the dwell-point location of the stacker crane was proposed and two formulations based on the relative likelihood that the next request was a storage or a retrieval request were developed. The first formulation uses an objective of minimizing the expected response time and the second one uses an objective of minimizing the maximum response time for an AS/RS. He then transformed these nonlinear programing formulations into linear programing problems that can be solved optimally. Egbelu and Wu (1993) presented the comparison of six dwellpoint rules under randomized and dedicated storage policies by means of simulation. They compared the two formulations presented by Egbelu (1991) and the four rules proposed by Bozer and White (1984). It was found that the solution from the minimum expected response time formulation performed well, as did the dwell-point strategy of Bozer and White (1984) to always return to the input point. Hwang and Lim (1993) showed that the two formulations of Egbelu (1991) could be transformed to the single-facility location problem with Chebyshev distance, and the Chebyshev minimax facility location problem, respectively, in order to reduce the computational time. These transformations reduced the required computational times by two orders of magnitude. Peters et al. (1996) proposed analytical models using continuous rack approximation for determining the optimal dwell-point locations for the stacker crane. These models provide closed-form expressions for the dwell-point location in an AS/RS. Extensions are made to consider AS/RS with a variety of configurations including multiple input and output stations. The models not only provide solutions to the dwell-point location problem, but also provide considerable insight into the nature of dwell-point positioning problem, which is particularly valuable when the requirements facing the AS/RS are uncertain. However, a computational study of the effectiveness of the optimal dwell-point strategy is not provided in Peters et al. (1996). Chang and Egbelu (1997a, b) presented formulations for prepositioning of stacker cranes to minimize the expected system response time (Chang and Egbelu 1997a) and minimize the maximum system response time (Chang and Egbelu 1997b) for multi-aisle AS/RS. Park (1999, 2001) developed two models to obtain optimal dwell-point under square-in-time rack with dedicated storage (Park 1999) and uniformly distributed rectangular racks (Park 2001).

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A closed-form solution was presented for the optimal dwell-point in terms of the probability of the next transaction demand type, storage or retrieval in a nonsquare-in-time rack. He also introduced various return paths to the dwell-point for the efficient operation of the stacker crane. Van den Berg (2002) determined a dwell-point position which minimizes the expected travel time to the position of the first operation after the idle period. He referred to this problem as the dwellpoint problem (DPP) and demonstrated that the DPP may be modeled as a facility location problem with rectilinear distances (FLPrd). He considered the continuous situation and derived analytic expressions for the optimal dwell-point position under the randomized and class-based storage policies, respectively. The expressions may be incorporated in a design framework for estimating the system performance. Hu et al. (2005) developed a reliable travel time model for SP-AS/RS (see Fig. 8.8) under stay dwell-point policy (i.e., the platforms remain where they are after completing a storage or retrieval operation). The travel time model is validated by means of simulation. Vasili et al. (2006) extended the study of Hu et al. (2005) and developed two reliable travel time models for the SP-AS/RS under return to middle and return to start, dwell-point policies. Under return to middle dwell-point policy the horizontal platform returns to middle of tier and the vertical platform returns to middle of handover station upon finishing a job. However, under return to start dwell-point policy the horizontal platform returns to the handover station and the vertical platform returns to the I/O station upon finishing a job. Based on examination of the literature, although many dwell-point strategies have been suggested, and an optimal strategy defined, there does not appear to exist a computational study that illustrates the benefits of using the optimal dwell point over the more simple rules suggested by Bozer and White (1984). Moreover, for AS/RSs with high system utilizations, it is not clear what opportunity exists in a practical sense to take advantage of the dwell-point strategies since the stacker crane will not be idle very often (Meller and Mungwattana 2005).

8.3.6 Position of the I/O Station(s) The position of the I/O station(s) is also a factor that affects the AS/RS operation. Bozer and White (1984) analyzed and derived the expected travel time of the following alternative configurations for the I/O station: 1. 2. 3. 4.

Input Input Input Input

and and and and

Output Output Output Output

at opposite ends of the aisle; at the same end of the aisle, but at different elevations; at the same elevation, but at a midpoint in the aisle; elevated at the end of the aisle.

For the travel time models which were investigated in Sect. 8.3.2 (Eqs. 8.3 and 8.4) it had been assumed that the I/O station is located at the lower left-hand corner

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of the rack and every trip originated and terminated at the I/O station. Bozer and White (1984) then relaxed this assumption and analyzed the above four alternative configurations. In the followings, these four configurations and their corresponding expected travel time expressions are reviewed. 8.3.6.1 Input and Output at Opposite Ends of the Aisle For this configuration, first, assuming the dwell-point strategy (1) (see Sect. 8.3.5), the expected travel time model per operation E1(T) was shown to be, 1 1 h ai E1 ðT Þ ¼ EðV Þð1 þ aÞ þ EðTBÞð1  aÞ þ K 1  ð8:9Þ 2 2 2 whereais percent of storages which are performed using SC cycles. E(V) is the expected travel time from any corner of the rack to a randomly selected point or vice versa and can be obtained by dividing E(SC) by 2 (for E(SC) refer to Eq. 8.3), so 1 1 EðV Þ ¼ b2 þ 6 2

ð8:10Þ

Operations are equally storages and retrievals. K is the fixed travel time from the output to the input station. E(TB) is the expected travel time between to randomly selected points and it is given by the following expression, where b is the shape factor: EðTBÞ ¼

1 1 2 1 þ b  b3 3 6 30

ð8:11Þ

Second, considering the dwell-point strategy (2) (see Sect. 8.3.5), the following expression was obtained for the expected travel time model per operation E1(T) for this configuration, 

 1a 1 1 1 EðV Þða þ 2Þ þ EðTBÞð1  aÞ þ K E1 ð T Þ ¼ 2

2 2  2 a 3 1 þ EðV Þ þ EðTBÞ : ð8:12Þ 2 2 2 From the results it was observed that using the dwell-point strategy (2) generates a reduction in the expected travel time in comparison with the dwell-point strategy (1) for a SC cycle.

8.3.6.2 Input and Output at the Same End of the Aisle, but at Different Elevations For the second configuration, it was assumed to have the input station at the lower left-hand corner of the rack while the output station is located d time units above

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Fig. 8.10 Input and output at the same end of the aisle, but at different elevations (Modified After Bozer and White 1984)

the input station, where d \ b (Fig. 8.10). Furthermore, it was assumed that the vertical travel yields the value of b. As shown in Fig. 8.10, the rack can be visualized as being two separate racks (as indicated by the dashed line). However, on going from the random point to the output station, Eq. 8.10 is not appropriate. The output station may be considered to be located at the corner of racks A and B. Considering E0(V) as the expected travel time for returning to the output station, thus, 1 1 1 E0 ðV Þ ¼ b2 þ  dðb  dÞ: ð8:13Þ 6 2 2 Therefore, assuming the dwell-point strategy (1) (see Sect. 8.3.5), the expected travel time model per operation E2(T) for the this configuration was shown to be,

 a 1 1 E2 ðT Þ¼ aEðV Þ aEðTBÞþ ½EðV ÞEðTBÞþE0 ðV Þ 2 2 2

 1 1 1 þ ð1a=2Þ a½EðV ÞEðTBÞþE0 ðV Þþ ½EðV ÞþEðTBÞþE0 ðV Þþ d : 2 2 2 ð8:14Þ The values of E(V), E(TB) and E0(V) can be determined by using Eqs. 8.10, 8.11 and 8.13, respectively. From a travel time standpoint, it was found that the second configuration performs better than the first configuration. The reason for this is due to the fact that elevating the output station will save some travel time in the vertical direction.

8.3.6.3 Input and Output at the Same Elevation, but at a Midpoint in the Aisle The third configuration alternative considered was based on the I/O station being located at the center of the rack. Such a configuration can be visualized as having the delivery and take-away conveyors running halfway into the aisle, through a set of rack openings located at the midlevel on either side of the aisle. It was further

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assumed that vertical travel time is between 0 and b, and horizontal travel time is between 0 and 1. Hence, the I/O station is assumed to be located at (1/2, b/2) for the normalized rack where 0 B b B 1. Considering EM(V) as expected travel time from the center of rack to a randomly selected point, thus EM ð V Þ ¼

1 2 1 b þ : 12 4

ð8:15Þ

For this configuration, the input and output stations are coincident. Considering the dwell-point strategy (1) described earlier, the strategy is equivalent to the case where every trip originates and terminates at the I/O station. Hence, the expected travel time model per operation E3(T) for the this configuration was shown to be, E3 ðT Þ ¼ a½2EM ðV Þ þ ð1  aÞ½2EM ðV Þ þ EðTBÞ:

ð8:16Þ

The results indicated that this configuration provides a reduction in the expected travel time, in comparison with the second configuration. 8.3.6.4 Input and Output Elevated at the End of the Aisle The forth configuration alternative considers the situation where the I/O station has the location (0, d). As before, it is assumed that the maximum horizontal and vertical travel times are 1.0 and b, respectively. The analysis of the configuration is involving input and output stations at the end of aisle, but at different elevations. From previous discussions, it is straightforward to obtain the following expected travel times for SC and DC cycles: 1 EðSC Þ ¼ b2 þ 1  d ðb  d Þ; 3 EðDC Þ ¼

4 1 2 1 þ b  b3  d ðb  dÞ; 3 2 30

ð8:17Þ ð8:18Þ

Comparing Eqs. 8.17 and 8.18 with Eqs. 8.3 and 8.4, elevating the I/O station d time units introduces a correction factor of d(b - d) in the computation of cycle times. Randhawa et al. (1991) analyzed and compared the effect of the number of I/O stations on the mean waiting time and maximum waiting time, for three different unit-load AS/RSs operating under DC cycle. The AS/RS layouts differ in the number of I/O stations per aisle, and the relationship between the storage and retrieval sources. A simulation model was used to evaluate the systems on three performance criteria, including system throughput, mean waiting time and maximum waiting time. From the results it was observed that the efficiency of the AS/RS can be improved by the introduction of two I/O stations per aisle with the input/output pallets for each station being independent of each other, the input pallet storage based on closest open location policy, and output pallet withdrawal based on a nearest-neighbor policy (or its variant, with a maximum waiting time

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limit). Randhawa and Shroff (1995) extended the study of Randhawa et al. (1991) by means of an extensive simulation study. They evaluated and analyzed six different layouts with single I/O station using three different scheduling policies. The results were compared considering the system throughput as the primary criterion. Other performance measures investigated were storage and retrieval waiting times, and rejects due to the rack or input/output queues being fully utilized. Ashayeri et al. (2002) presented a geometrical-based algorithmic approach for determining the travel times and throughput for class-based storage assignment layouts in an AS/RS with single, double or multiple I/O stations. In the double I/O stations layout the AS/RS is equipped with two I/O ports at floor level on opposite ends of each aisle. The results indicated that, by applying the algorithm a significant reduction in the expected cycle time per transaction is produced. Vasili et al. (2008) proposed a new configuration for the I/O station in split-platform AS/RS (SP-AS/RS) in order to reduce average handling time of this system. In their proposed configuration the I/O station is located at the center of the rack. They developed a continuous travel time model for this new configuration. The travel time model was validated by using Monte Carlo simulation. The results and comparisons show that within a range of shape factors this new configuration is more preferable than those introduced by Chen et al. (2003) and Hu et al. (2005).

8.3.7 Storage Assignment Another topic that has received considerable attention in the literature is the assignment of incoming stock to the storage locations. A storage assignment policy serves to determine which products are assigned to which locations and establishes a framework for allocating the incoming products to the storage locations (Roodbergen and Vis 2009). A storage policy is considered optimal if it minimizes the average time required to store and retrieve a load while satisfying the various constraints placed upon the system (Goetschalckx and Ratliff 1990). A storage assignment policy based on the needs of manufacturing operations can increase not only the performance of the AS/RS but also the performance of the production system (Hsieh and Tsai 2001). Several storage assignment policies can be found in the literature for AS/RSs. The five often used policies are: randomized storage; closest open location storage assignment; class-based storage; full-turnover-based storage and dedicated storage (see e.g., Hausman et al. 1976; Graves et al. 1977; Schwarz et al. 1978; Goetschalckx and Ratliff 1990; Van den Berg 1999; Roodbergen and Vis 2009). Randomized storage policy allows the products to be stored anywhere in the storage area. Using this policy, all empty locations have an equal probability of having an incoming load assigned to them. If the closest open location storage is applied, the first empty location that is encountered will be used to store the

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products. This typically leads to an AS/RS where racks are full around the I/O stations and gradually more empty toward the back (if there is excess capacity). Class-based storage policy distributes the products based on their demand or movements frequency, among a number of classes, and for each class it reserves a region within the storage area. Accordingly, an incoming load is stored at an arbitrary open location within its class (randomized storage is applied within each class). Full-turnover storage policy determines storage locations for the products based on their demand or turnover frequency. Frequently requested products get the easiest accessible locations, usually near the I/O-points. Slow-moving products are located farther away from the I/O-point. An important assumption for this rule is that the turnover frequencies need to be known beforehand. Randomized and fullturnover storage policies are in fact extreme cases of the class-based storage policy. Randomized storage considers a single class and full-turnover storage considers one class for each product. The class-based storage policy and the fullturnover storage policy attempt to reduce the mean travel times for storage and retrieval operations by storing products with high demand at locations that are easily accessible. According to Van den Berg and Gademann (1999), the demand for a product may be estimated by the cube-per-order index (COI) which has been presented by Heskett (1963). Dedicated storage policy assigns each product type to a fixed location. These locations may be determined by activity and inventory levels or by stock number (Lee and Schaefer 1997). Replenishments of that product always occur at this same location. The main disadvantages of this policy are its high-space requirements and consequent low-space utilization. This is due to the fact that locations are reserved even for products that are out of stock. Furthermore, for each product type sufficient space must be reserved to accommodate the maximum inventory level that may occur. Most advantages of dedicated storage, such as locating heavy products at the bottom or matching the layout of stores, are related to non-automated orderpicking areas and are not as interesting for AS/RSs. For practical purposes it is easiest if a full-turnover policy is combined with dedicated storage. Hausman et al. (1976) investigated and compared the operating performance of the three storage assignment policies: randomized storage; class-based storage and full-turnover policy. It was observed that significant potential reductions in stacker crane travel times in automatic warehousing systems is possible based on classbased turnover assignment policies rather than closest open-location (essentially random) policies. However, in this study the interrelationship between storage assignment and requests sequencing rules was not investigated. Linn and Wysk (1987) presented a simulation study to consider the storage assignment rules similar to Hausman et al. (1976) but with other control decisions. Performance of different control algorithms for a unit-load AS/RS for various storage and retrieval rates under seasonal demand was analyzed. Furthermore, the effect of workload intensity on the control algorithms and the effect of product mix on the control algorithms were investigated. They used the following storage location assignment rules:

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1. Random assignment (RNDM). A location is randomly picked and assigned to the pallet to be stored if it is empty. Otherwise, another location will be picked. 2. Pattern search, lowest tier first (LTF). The storage location is selected by searching for the closest open location in the lowest tier first. If no empty one is found, the next lower tier will be searched. 3. Shortest processing time (SPT). The empty location with the minimum travel time from input station is assigned for next storage. 4. Turnover rate based zone assignment (ZONE). The storage rack is partitioned into number of zones, which is equal to the number of product types. The zone closest to the I/O station is assigned to store pallets of highest turnover rate. When searching for an empty location, if an empty location cannot be found in its own zone, the next lower turnover zone will be searched. If all the lower turnover zones are full, then the next higher turnover zone is searched. The results indicated that, the storage location assignment rules affect the system performance in the following manner: when the traffic intensity is low (below a critical value), the random location assignment is better; as the traffic becomes heavier, the pattern search (lowest tier first) becomes better; if the traffic intensity increases further, the shortest processing time rules and zone-based rules will be better rules. Rosenblatt and Eynan (1989) developed the optimal boundaries for a general n-class storage rack in AS/RS. A solution procedure was developed which required only a one-dimensional (1D) search procedure. It was shown that most of the potential improvement in the expected one-way-travel time can be obtained when the warehouse is divided into a relatively small number of regions (\10). Thus, there is no need to use the full-turnover approach, which is difficult to implement and administer. Goetschalckx and Ratliff (1990), with regard to storing unit-loads, classified the storage policies in two major classes: dedicated storage policies and shared storage policies. Dedicated storage policies require that a particular storage location be reserved for units of a single product during the entire planning horizon. Shared storage policies allow the successive storage of units of different products in the same location. Under these definitions, randomized and class-based storage policies are placed in the category of shared storage policies. They proposed an optimal storage policy based on duration-of-stay (DOS) with respect to travel time and storage space for the systems with balanced input and output. The DOS-based policy classifies the units of all items according to their expected DOS. Then the class of units having the shortest DOS is assigned to the closest AS/RS region. Based on the same principles, two heuristic policies were developed for more complex systems. Simulation results were presented to compare travel times for dedicated storage, random storage, turnover-based storage classes and DOS-based storage classes. It was shown that for SC storage and retrieval, shared storage policies based on duration-of-stay of individual unit-loads in the system have the potential to significantly decrease travel time. Kim and Seidmann (1990) presented a framework for obtaining analytic expressions of the expected throughput rate in AS/RSs. These expressions were developed based on generalized full-turnover

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item allocation policies and random storage and retrieval requests. Both SC and DC operations were considered and a general expression for the expected cycle time in a SC class-based system was also developed. The results demonstrated the potential for significant reductions in the expected cycle time in the case of fullturnover item allocation. Van den Berg (1996) investigated the class-based storage allocation problem. He presented a polynomial-time dynamic programing algorithm that distributes products and locations among classes such that the mean SC travel time is minimized. The algorithm outperforms previous algorithms (e.g., Graves et al. 1977; Hausman et al. 1976; Rosenblatt and Eynan 1989). He claimed that, this algorithm may be applied to a wide variety of warehousing systems, since it holds for any demand curve, any travel time metric, any warehouse layout and any positions of the input station and output station. Moreover, it allows that the inventory level varies and determines the storage space requirements per class by imposing a risklevel on stock overflow. Thonemann and Brandeau (1998) applied the turnover-based and class-based assignment policies of Hausman et al. (1976) to a stochastic environment. An expression for expected one-way travel time with given uniform and exponentially distributed demand was developed. It was observed that the turnover-based policy applied to the stochastic environment is optimal as it minimizes one-way travel time. Both the turnover-based and class-based assignment policies applied in the stochastic environment reduce the expected time of storage and retrieval in comparison with randomized assignment. These savings can be directly translated into increased throughput capacity for existing systems and can be used to improve the design of proposed systems. Based on the same approach as in Goetschalckx and Ratliff (1990), and using computer simulation, Kulturel et al. (1999) compared two shared storage assignment policies in an AS/RS. The AS/RS was assumed to operate under a continuous review, order quantity and reorder point inventory policy. The average travel time of the stacker crane for storing and retrieving products was used as the main performance measure. Sensitivity of the system to product variety, inventory replenishment lead time and demand rate were investigated, as well as the effects of the inventory policy and the product classification technique used. The results indicated that the turnover-based policy, in general, outperforms the duration-of-stay-based policy. However, the difference between the performances of the two policies becomes insignificant under certain conditions. A bill of materials (BOM) contains a listing of all of the assemblies, subassemblies, parts, and raw materials that are needed to produce one unit of finished product. Thus, each finished product has its own BOM (Stevenson 2005). Generally, the requirements of manufacturing operations are embedded in material attributes, and the BOM is the best source to link material attributes. By employing the BOM as the backbone structure of a production system, a computer integrated manufacturing (CIM) system can be thoroughly configured. In this regard, Hsieh and Tsai (2001) presented a BOM-oriented class-based storage assignment method for an AS/RS. The proposed method possesses not only the advantage of a

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class-based storage method, but also the feasibility to integrate an AS/RS into a CIM system. The effectiveness of the proposed method was illustrated through a case study. A random storage assignment method was also employed to obtain the solution for the illustrative example. From the results of comparative studies, the proposed BOM-oriented class-based AS/RS assignment method was shown to be efficient. Wen et al. (2001) presented compact forms of expected travel time under the class-based and full-turnover-based storage assignments considering various travel speeds with known acceleration and deceleration rates. Ashayeri et al. (2001) presented an exact, geometry-based analytical model for computing the expected cycle time for a stacker crane operating under SCs, DCs or both, in a rack structure that has been laid out in pre-specified storage zones for classes of goods. The rack may be either square-in-time or non-square-in-time. The approach is intuitively appealing, and it does not assume any certain layout shape, such as traditional ‘‘L-shaped’’ class layouts. The model can be used by designers as a tool for quickly evaluating alternative layout configurations with respect to expected S/R cycle time in an AS/RS, and thereby the throughput of an automated warehouse over time. In a later study, Ashayeri et al. (2002) presented the use and extension of the geometrical-based algorithmic approach proposed by Ashayeri et al. (2001), for determining the expected stacker crane cycle times, and therefore warehouse throughput, for class-based storage assignment layouts in an AS/RS. The algorithm may be used for the rack layouts with single double or multiple I/O stations. They derived the travel time expressions of the stacker crane for an AS/ RS having two I/O station, for SC and DC cycles as, XXX Pki  Pim ðEki þ Eim Þ; ð8:19Þ EðSC Þ ¼ i

EðDC Þ ¼

k

XXXX i

k

m

m

  Pki  Pij  Pjm Eki þ Eij þ Ejm :

ð8:20Þ

j

where Pk.i is the probability that a movement from input port k to zone i takes place. Eim is the probability that a movement from zone i to output port m takes place. Eki is the expected travel time between input port k and a random location in zone i. Eim is the expected travel time between a random location in a storage zone and an output port m. Eij is the expected travel time between a random point in zone i and a random point in zone j. Pij is the probability that a movement from zone i to zone j takes place. k and m represent I/O 1 and I/O 2, respectively. Note that the expected travel time in an AS/RS with a single I/O port located at one end of the aisle can be found by setting Pim ¼ 1 and Eki ¼ Eim in Eq. 8.19. In order to apply Eq. 8.19 in a more general form for an AS/RS with p input ports and q output ports (multiple I/O stations) allow the indices of k and m to range over all I/O ports, i.e., k ¼ I=O 1; 2; . . .; p; and m ¼ I=O 1; 2; . . .; q: Foley and Frazelle (1991) derived the distribution of the DC cycle time for a square-in-time rack under randomized storage, and used it to determine the throughput of miniload AS/RSs. Park et al. (2003) analyzed the travel time of

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minload AS/RSs for turnover-based storage systems and determined the mean and variance of DC travel times. Detailed numerical results for selected rack shape factors and ABC inventory profiles were presented and the effect of alternative rack configurations on travel time performance measures was investigated. They demonstrated how to determine the throughput of EOA miniload systems with turnover-based storage and exponentially distributed pick times. Petersen et al. (2004) compared the performance implications of class-based storage to both randomized and full-turnover-based storage for a manual order picking warehouse by means of simulation. In addition, the effect of the number of storage classes, the partition of storage classes and the storage implementation strategy applied in the warehouse were investigated. From the simulation results it was observed that class-based storage provides savings in picker travel over random storage and offers performance that approaches full-turnover-based storage. Park et al. (2006) investigated the performance of EOA miniload system with a square-in-time rack containing two storage zones (two-class storage): high turnover and low turnover. The distribution of the DC travel time and closed-form expressions for throughput for two important families of pick time distributions: deterministic and exponential were derived. In a later study, Park (2006) determined the same issues for systems with non-square-in-time racks. Yu and De Koster (2009a, b) extended the study of De Koster et al. (2006) on compact 3D AS/RS by investigating two different storage assignment policies. They derived the expected SC cycle time under the full-turnover-based storage policy and proposed a model to determine the optimal rack dimensions by minimizing this cycle time (Yu and De Koster 2009a). It was observed that, under the full-turnover-based storage policy, significant cycle time reduction can be obtained compared with the random storage policy. In the later study the optimal storage zone boundaries were determined for this system with two product classes: high- and low-turnover, by minimizing the expected stacker crane travel time (Yu and De Koster 2009b). They formulated a mixed-integer non-linear programing model to determine the zone boundaries. The results indicated that significant reductions of the machine travel time are obtainable by using class-based storage. Comparing all the mentioned storage policies in this section, randomized storage is the most commonly used method because it is simple to administer. However, the policies based on the demand frequency of products are generally most effective at improving performance, but they are information intensive and far more difficult to administer than a random storage policy (Bozer and White 1984; Petersen et al. 2004). White and Kinney (1982) stated that in comparison to class-based storage, random storage generally requires less storage space because the maximum aggregate storage requirement is generally less than the aggregate maximum storage requirements for each product in storage. In comparison to random storage, class-based storage results in reduced travel time if equal storage is assumed. However, since the class-based storage policy is based on turnover frequency (used to determine the classes) for each product, it is difficult to use, if the turnover frequencies of the products vary with time. Random storage policy is not affected by varying turnover frequencies.

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8.3.8 Request Sequencing Request sequencing rules determine the queuing discipline for the storage and retrieval request queues (Linn and Wysk 1987). Storage requests in distribution or production environments are usually stored according to the first-come-first-served (FCFS) principle. In sequencing retrievals usually due times of retrievals should be met. Hence, by sequencing the retrievals in a smart way, improvements in the overall throughput of the AS/RS can be obtained (Roodbergen and Vis 2009). When sequencing requests on an AS/RS, it is necessary to make a trade-off between efficiency and urgency (Van den Berg and Gademann 1999). The Request sequencing policies may also be used to improve the design of proposed systems to achieve a more desirable balance between throughput and storage capacity (Graves et al. 1977). The term ‘‘interleaving’’ refers to the pairing of storage and retrieval transactions on the same cycle to generate DC cycle cycles (Fukunari and Malmborg 2009). Hausman et al. (1976) investigated optimal storage assignment, without regard to interleaving. Graves et al. (1977) extended the study of Hausman et al. (1976) to include the following interleaving rules: (1) mandatory interleaving with FCFS queue discipline of retrieves and (2) mandatory interleaving with selection queue of K retrieves. The results indicated that significant reductions in crane travel time (and distance) are obtainable using the proposed interleaving rules. These reductions may be directly translated into increased throughput capacity for existing systems. Schwarz et al. (1978) by means of simulation, validated the analytical works presented by Hausman et al. (1976) and Graves et al. (1977), in a deterministic environment and extended the results to conditions of imperfect information. Later, Bozer and White (1984) developed analytical models for SC and DC cycles and FCFS sequencing of the storage and retrieval requests. In order to create a DC from the storage and retrieval requests, sequencing retrieval requests optimally is a complex problem. The list of retrievals continuously changes over time as old requests are filled and new requests appear. Han et al. (1987) suggested two alternatives to deal with this dynamic problem. The first alternative is to select a ‘‘block’’ of the most urgent storage and retrieval requests, sequence these requests in the block, and when the block of requests has been completed, select another block and so on. This is referred to as ‘‘block sequencing’’. The second alternative is to re-sequence the list every time a new request is added and employ due dates or priorities to ensure that a retrieval at the far end of the aisle is not excessively delayed (i.e., the whole retrieval queue is a block). This is called ‘‘dynamic sequencing’’. They showed that the throughput capacity can be increased by replacing the FIFO strategy with a new heuristics strategy, which is NN, when several retrieval requests are available and DC cycles are performed. The NN rule was studied for selecting storage locations and sequencing retrieval requests, so that the interleaving travel time between storage and retrieval locations in a DC cycle is reduced. Two simple greedy heuristics were developed to select a pair of S/R locations to minimize TBW (called the

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Fig. 8.11 An illustration of DC cycle for nearestneighbor (NN) and shortestleg (SL) policies

S

• T BW

TS

•R TR I/O

•

nearest-neighbor, NN) or TS ? TBW (called the shortest-leg, SL), where TBW is the interleaving time the crane travels empty to the retrieval location after depositing the storage load (Fig. 8.11). The results indicated that SL is better than NN in terms of dual cycle time in a few trials. However, over the long run SL is much worse than NN because SL attempts to occupy the locations near the I/O station (called cluster phenomenon) and drive the rest of the open locations farthest from the I/O station. The performance of both ‘‘block sequencing’’ and ‘‘dynamic sequencing’’ approaches differs per situation. Eben-Chaime (1992) showed that if the NN strategy is applied to the blocks of fixed size in a non-deterministic environment, it has destructive effects in terms of waiting times, queue length and system stability. They proposed instead to use dynamic nearest-neighbor (DNN) strategy where the whole retrieval queue is the block. It was observed that the performance level of DNN is surprisingly high in terms of average waiting time, average queue length and maximum queue length. Later, Lee and Schaefer (1996) proved that total travel time (TT) is superior to SL and NN strategies. TT selects a pair of S/R locations such that TS ? TBW ? TR (shortest total-travel, STT) is the minimum (See Fig. 8.11). Bozer et al. (1990) explained that the DC scheduling of AS/RSs can be formulated as a Chebyshev traveling salesman problem (CTSP), which has numerous applications in materials handling and information storage-retrieval. Several heuristic procedures based on geometric concepts have been developed for the CTSP. The study was concerned with evaluating the performance of geometric approaches as a function of the shape of the service region and the number of points to be sequenced. In the two aforementioned studies, Bozer et al. (1990) and Han et al. (1987) studied the sequencing problem of retrievals without reflecting the dynamic nature of an AS/RS, which is the realistic operating characteristic. These studies especially assumed that all storage and retrieval orders are known in advance. Taking the dynamic operating characteristics of an AS/RS into account, these studies either under-estimate or over-estimate the performance of the stacker crane. Thus, these studies could not provide feasible alternatives for the important design factors of an AS/RS, such as the buffer size, utilization of the stacker crane and so on. Linn and Wysk (1987) presented a simulation model to evaluate the following sequencing rules when the product demand shows seasonal trend:

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1. First-Come-First-Serve (FCFS). All the requests are served on FCFS basis. 2. Shortest Completion Time (SCT). The request which needs the shortest completion time is served first. 3. Shortest Completion Time with output priority (SCTop). This is a modified SCT rule, in which the retrieval requests have first priority, to clear the room for storage. 4. Shortest Completion Time with controlled output priority (SCTcop). This is another modified SCT rule in which the retrieval requests would have the first priority only when the retrieval queue is longer than the storage queue. The results indicated that, when arrival rate is such that the traffic intensity is low (below a critical value), the sequencing rules produce little improvement in system performance. When arrival rate increases until the traffic intensity goes beyond the critical value, job sequencing rules begin affecting the system performance. Considering the product mix, it was observed that SCTop sequencing rule was better than SCT rule in five product type system; however, SCT became better in ten product type system. Linn and Wysk (1990a, b) presented an expert system framework for the control of an AS/RS. Their expert system-based control uses a hierarchical control structure which partitions the AS/RS control decision process into strategic, tactical and process control levels, and employs a multi-pass simulation technique to tactically adapt control policies to system changes. The results demonstrated the ability of the system to include control flexibility, for adapting the system to fluctuations in demand and maintain quality performance. It was also observed that the system performed very well particularly at high demand levels. Linn and Xie (1993) presented a simulation study to investigate the effect of job sequencing rule on delivery performance of an ASRS, in an assembly environment with given due dates. The interaction of the sequencing rules with other control variables was also examined. Hwang and Song (1993) analyzed the order sequencing problem in a man-on-board storage and retrieval warehousing system which is suitable for storing items of small size and light weight. Considering the operating characteristics of the man-on-board system, a combined hull heuristic procedure was presented for the problem of sequencing a given set of retrieval requests. The procedure was validated through simulations and the results showed that the procedure performs satisfactorily. Lee and Schaefer (1996) presented an algorithm for the unit-load AS/RSs with non-dedicated storage. The algorithm combines the Hungarian method and the ranking algorithm (Murthy’s ranking algorithm) for the assignment problem with tour-checking and tour-breaking algorithms. They showed that their algorithm finds either a verified optimal or near-optimal solution quickly for moderate size problems. Lee (1997) presented an analytical stochastic approach for performance analysis of unit-load AS/RSs, using the queuing model. He established an M/M/1-type queuing model with two waiting spaces for storage requests and retrieval requests. The analysis assumed the exponential distribution of the travel times of a stacker crane. Experimental results showed that the proposed method was effective for both short-term and long-term planning of AS/RSs.

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Lee and Schaefer (1997) investigated the effect of sequencing storage and retrieval requests on the performance of AS/RS where a storage request is assigned a predetermined storage location. By exploiting this unique operating characteristic, several optimum and heuristic sequencing methods under static and dynamic approaches were presented. Applications of such sequencing methods include unitload AS/RS with dedicated storage, EOA miniload AS/RS and potentially unitload AS/RS with randomized storage. The results indicated that the sequencing methods can significantly reduce travel time by a storage and retrieval machine, thereby, increasing throughput, and that the dynamic heuristic method is simple and fast, yet considerably outperforms the others. Mahajan et al. (1998) developed a retrieval sequencing scheme for the purpose of improving the throughput of EOA miniload AS/RS in an order picking environment. It was assumed that an order comprised of retrieval requests is always available such that DC cycles are always performed. A NN retrieval sequencing heuristic was presented, an analytical model was developed to predict its performance, and this model was validated by means of simulation. The results showed that the heuristic improves the throughput of the system, over traditional FCFS retrieval sequencing. The heuristic achieves this improvement by properly sequencing the retrieval requests within an order and also optimizing the retrieval requests among successive orders. However, since the analysis assumes that the requests are always available, it represents an over-estimation of the system throughput. Van den Berg and Gademann (1999) studied the optimal sequencing of requests with dedicated storage using the block sequencing approach. It was assumed that a set of storage and retrieval requests are given beforehand and no new requests come in during operation. The objective for this static problem was to find a route of minimal total travel time in which all storage and retrieval requests may be performed. Considering the problem of retrievals sequencing equivalent to the traveling salesman problem (TSP), they showed that the special case of sequencing under the dedicated storage policy can be solved in polynomial time. Van den Berg and Gademann (2000) evaluated the performance of various control policies for the AS/RS by using computer simulation. For the sequencing of storage and retrieval requests they developed policies based on the heuristics presented in Van den Berg and Gademann (1999). By means of these policies, they analyzed the trade-off between efficient travel of the stacker cranes and response time performance. Hur et al. (2004) and Hur and Nam (2006) presented stochastic approaches for the performance estimation of a unit-load AS/RS by using an M/G/1 queuing model with a single server and two queues. They assumed that the storage and retrieval commands arrive at the system according to Poisson processes with different rates. Comparing the results with simulation results it was observed that the proposed approach gives satisfactory results with very high accuracy. Based on the same approach as in Linn and Wysk (1990a, b), Yin and Rau (2006) studied dynamic selection of sequencing rules for a class-based unit-load AS/RS. They developed a multi-pass and genetic algorithm (MPGA) simulation system which divides storage and retrieval requests or DCs into a series of blocks, and then

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conquers each block to find the most promising combination of sequencing rules. They considered the following sequencing rules: FCFS, shortest total-travel time (STT) and shortest due time (SDT), where these rules could be chosen dynamically in any decision points in the system. The results showed that the proposed approach with dynamic rules was much better than those approaches with any single rule used from the beginning to the end in the whole system. The results of this study provide a better way to control and manage the operation of AS/RS. Dooly and Lee (2008) presented a shift-based sequencing problem for twin-shuttle AS/RS, where replenishment and depletion (storage and retrieval operations) of items occur over different shifts. For instance, certain warehouses or distribution depots deplete their items in stock during morning shifts and replenish during later shifts. They showed that this problem can be transformed into the minimum-cost perfect matching problem and presented an efficient polynomial-time optimum method that can achieve a large throughput gain over other methods. Average-case and lower bound analyses for this problem were presented as well.

8.3.9 Order Batching Batching is a control policy which considers how one can combine different customer orders into a single tour of the crane (Roodbergen and Vis 2009). When orders are fairly large, each order can be picked individually (i.e., one order per picking tour). This way of picking is often referred as the single order picking policy (or discrete picking or pick-by-order). However, when orders are small, there is a potential for reducing travel times by picking a set of orders in a single picking tour. Order batching is the method of grouping a set of orders into a number of sub-sets, each of which can then be retrieved by a single picking tour (De Koster et al. 2007). Based on the previous literature, only a few papers have addressed the batching of orders in combination with the AS/RS, since this policy is mainly applicable to man-on-board AS/RS. In a man-on-board AS/RS, orders are combined into batches and each batch is processed in a tour of the stacker crane. Batching policy determines the way to combine orders to form batches. Since botching is an NP-hard (i.e., non-deterministic polynomial-time hard) problem, in order to obtain solutions for large problems in acceptable computation times, heuristic algorithms have been proposed (Pan and Liu 1995). Most heuristic algorithms for order batching basically follow the same three steps (Fig. 8.12): (1) a method of initiating batches by selecting a seed; (2) a method of allocating orders to batches (addition of orders to batches) and (3) a stopping rule to determine when a batch has been completed. An important assumption in all batching heuristics is the fact that a single order cannot be split over various batches, but needs to be picked as a whole (Pan and Liu 1995; Roodbergen and Vis 2009). De Koster et al. (2007) distinguished two types of order-batching heuristics: seed and savings algorithms. Seed algorithms construct batches in two phases: seed selection and order congruency. Seed selection rules

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Start

Select the seed based on a seed selection rule

Select an order to be included in the batch based on a order addition rule

Yes

Yes Decide if batch is complete by using a stopping rule

No Any more orders to be batched? No End

Fig. 8.12 Common procedure for order-batching Heuristics (Modified after Pan and Liu 1995; Roodbergen and Vis 2009)

define a seed order for each batch. Order congruency rules determine which unassigned order should be added next into the current batch. Usually, an order is selected, to be included in a batch, based on a measure of the ‘distance’ from the order to the seed order of the batch. In saving algorithms a saving on travel distance is obtained by combining a set of small tours into a smaller set of larger tours. Elsayed (1981) presented four heuristic algorithms for handling orders in single aisle man-on-board AS/RSs. He used the following heuristic algorithms: order with largest number of locations to be visited; order with smallest number of locations to be visited; order with largest volume; and order with smallest volume. The algorithms select the orders that will be handled in one tour in order to minimize the total distance travelled by the stacker crane within the warehouse system. The optimal tours for the four algorithms were found by using the traveling salesman algorithm. Elsayed and Stern (1983) used a cumulative rule for the seed selection in single aisle man-on-board AS/RSs. Contrary to a single seeding rule, a cumulative seeding rule uses all orders that are already in the batch as the seed. Hwang et al. (1988) and Hwang and Lee (1988) presented heuristic algorithms based on cluster analysis for order-batching problem in a single aisle man-on-board AS/RS. The algorithms process the orders by batching some of them according to the value of the similarity coefficient which is defined in terms of attribute vectors. In order to find the minimum travel time for each batch of orders, the traveling salesman algorithm was used. The performances of the algorithms were analyzed using simulations. The results and comparisons indicated that some algorithms developed represent satisfactory performance.

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Pan and Liu (1995) presented a comparative study of order-batching algorithms composed of four seed selection rules and four order addition rules for orderbatching problems based on average travel times. The performances of these algorithms were compared along the 3D of shape factor, capacity of the stacker crane and storage assignment policy. The results indicated that only the capacity of the stacker crane has effect in the selection of order-batching algorithms. It was concluded that the heuristic which was presented by Hwang and Lee (1988) generates the most efficient batches for a small capacity stacker crane as well as for large one. It was recommended to use this heuristic for order batching under any type of shape factor, capacity of the stacker crane, and storage assignment policy. The above-mentioned studies have not taken into account the time constraints on retrievals (e.g., order due time and the penalty of violating the due time). Elsayed et al. (1993) and Elsayed and Lee (1996) investigated the order-batching problem in a man-aboard system where a due date is specified for each retrieval order. The grouping of orders into batches (batching process) was performed based on a penalty function, which incorporates both the earliness and the tardiness of the orders. They developed efficient procedures for order sequencing and grouping the orders into batches such that the penalty function is minimized. All mentioned papers in this section assume that the arrival patterns of orders are known before the start of the operations. However, it is obvious that orderbatching problem can become more difficult when orders arrive on-line. It stands for reason that, for on-line arrivals there is a trade-off between reducing waiting times and reducing travel times.

8.3.10 Load Shuffling and Sorting Heuristics Although AS/RSs allow random access to any storage cells, often it is advantageous to shuffle (i.e., pre-sort, relocate or rearrange) the loads in order to minimize the retrieval time (Hu et al. 2010). Updating and shuffling of items and reconsidering storage assignment decisions can be vital in current dynamic environments to meet the fluctuating, short-term throughput requirements imposed on the AS/RSs. An AS/RS needs to store and retrieve loads in the shortest possible time period. Compared with storage, the quick response of retrievals is often more critical. This is because when a load is to be stored into an AS/RS rack, it can be put into any empty storage cell. While for retrieval, only the designated one is valid. In order to retrieve loads as quickly as possible, a solution is to shuffle the items to specified locations to minimize the response time of retrieval. In other words, the shuffling of the items with a high expectancy of retrieval closer to the I/O station of each storage rack during off-peak periods will reduce the expected travel time for the stacker cranes during future peak periods of the planning horizon (Hu et al. 2004; Jaikumar and Solomon 1990; Roodbergen and Vis 2009). Using the load-shuffling strategy to some extent can also speed up the storage operation. It stands for reason that during a storage operation, a load can be stored

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Fig. 8.13 (a, b) Matching for load shuffling in an AS/RS (Modified after Muralidharan et al. 1995)

into the most convenient storage location so as to reduce the storage travel time. Later during stacker crane idleness, the loads can be shuffled into the more suitable locations. Applying the shuffling scheme also imposes positive influences on the rack and stacker crane utilization (Hu et al. 2010). However, very little information about the load-shuffling strategy can be found in the literature. Jaikumar and Solomon (1990) presented an efficient algorithm that minimizes the number of load-shuffling operations in order to meet the expected throughput. It was assumed that there is sufficient time, so that travel time considerations were omitted from the model. Considering the fluctuation in order volume, they proposed two heuristic methods. The first method enlarges the order pick system by one new zone during peak periods; and the second one reduces the system by one zone during off-peak periods. The purpose of these heuristic methods is to maintain the regular workloads so that no picker is overloaded during peak periods and/or light-loaded during off-peak periods. Such adjustments balance the workloads among all pickers and keep the continuity of the pick lane. All the proposed methods were validated through simulation experiments. Muralidharan et al. (1995) proposed a shuffling heuristic-based approach that combines the random storage and class-based storage assignments for the storage location assignment in an AS/RS. They described the proposed shuffling algorithms and showed that the waiting time and service time reduced considerably for this storage policy. Their approach for shuffling the loads can be briefly described as follows. When the slacker crane is idle, a shuffling cycle is initialized. There may be a number of class A products (very important/high-turnover products) that have been stored farther away from the I/O station and class B and C products stored near the I/O station. As shown in Fig. 8.13a, all the class A products farther away from the I/O station are matched to the class B or C product or an empty location closest to the I/O location for shuffling. Each match is represented by a directed arc which corresponds to the planned movement of class A pallets.

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In order to move a class A pallet into the location of a class B or C pallet, the B or C pallet must he moved to a near-by empty location to make room for the class A pallet. So the class B or C product is matched to the nearest empty location away from the I/O station that is not matched with a class A product. These matchings are represented by another set of directed arcs shown in Fig. 8.13b. The direction of arrow corresponds to the planned movement of the class B or C pallets. For each pair of arrow and arc, the arrow must be traversed prior to traversing the arc. When forming a route through these arcs and arrows, the precedence constraints must be met. Based on the results they observed that the load shuffling is clearly an appropriate strategy to increase the AS/RS operating efficiency. By means of a simulation study Moon and Kim (2001) demonstrated that the load-shuffling strategy can be valuable if the quantities of products belonging to different classes vary over different time horizons. It helps to maintain stable throughputs with any storage assignment policies, and it can alleviate the waiting line for the AS/RS rack. Load-shuffling strategy does not cause a bottleneck in the stacker crane operations, since the time to shuffle the items in an AS/RS is too small to affect the stacker crane utilization. Hu et al. (2004, 2010) investigated the issue of shuffling loads in the SP-AS/RS which is suitable to handle extra heavy loads. The objective was to shuffle the loads into any specified locations in order to minimize the response time of retrievals. 1D, 2D and 3D AS/RS racks were designed in order to achieve the shuffling efficiently. They described the shuffling algorithms and derived expressions for calculating the response time of retrieval. Results of the analysis and numerical experiments showed that the proposed shuffling algorithms are quite efficient. Based on examination of the literature it can be observed that the load-shuffling strategies for AS/RS have not been adequately investigated in previous studies which implies a need for further studies in this area. Moreover, existing shuffling algorithms are applicable only during the slacker crane idleness. For AS/RSs with high system utilizations, as discussed earlier, it is not clear what opportunity exists in a practical sense to take advantage of existing load-shuffling strategies since the stacker crane will not be idle very often.

8.4 Conclusions and Further Research Issues From the literature survey and discussions in this chapter it can be observed that a considerable amount of research has been carried out over the years to evaluate, improve and optimize the physical structure, operational features and control policies of the AS/RSs. Most of the existing studies only discuss a fraction of these AS/RS issues. Therefore, development of comprehensive evaluating and improving procedures would seem to be necessary in order to simultaneously address all these issues. In addition, regardless of the actual improving and optimization procedures, a system of performance measurement is needed to evaluate the overall performance of the resulting system at every stage. In this regard, many

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publications have appeared on performance measurement. Most studies have analyzed the performance of AS/RSs under a balanced situation so that inbound work-flow is equal to the outbound work-flow. However, considering the dynamic nature and realistic operating characteristic of an AS/RS, during certain time slots, the system operates under an unbalanced situation. A perfectly balanced system is a very idealized situation which is unlikely to occur in real storage systems. Hence, the research in this field should move toward developing models, algorithms and heuristics that include the dynamic and stochastic aspects of current business. The performance of an AS/RS varies according to the definition of measure and the operating strategies. Performance measures for an AS/RS may include: system throughput, utilization of rack and stacker crane and expected travel time of stacker crane. Travel time estimates in different types of AS/RS configurations are appropriate analytical tools for evaluating and comparing the system performance and control policies. In the preceding sections the existing travel time models on different aspects of the AS/RS, especially its control policies were investigated. Considering different control policies, dwell-point policy of the stacker crane is the strategy that can affect and contribute to the system response time of AS/RS. Several dwell-point policies for AS/RS have been introduced in the literature. Meanwhile, development of expected travel time (i.e., average travel time) models for AS/RS based on different dwell-point policies has been the subject in much research over the past several years. Although many dwell-point strategies have been suggested, and an optimal strategy defined, however for AS/RSs with high system utilizations, the dwell-point strategies may have no significant effect on the system response time, since the stacker crane will not be idle very often. Other control policies for AS/RSs that have received considerable attention in the literature are storage assignment and request sequencing. Majority of the literature addresses single aisle AS/RSs with single I/O station. Hence, storage assignment and request sequencing policies for other types of configurations (e.g., multiple I/O stations) or nontraditional AS/RSs (e.g., multiple shuttle AS/RSs) deserve further study. Order batching policy is the method of grouping a set of orders into a number of sub-sets, each of which can then be retrieved by a single picking tour. Almost all research on the batching of orders has assumed that the arrival patterns of orders are known before the start of the operations. However, it is obvious that order-batching problem can become more difficult when orders arrive on-line. It stands for reason that, for on-line arrivals there is a trade-off between reducing waiting times and reducing travel times. Another strategy which can result in minimizing the AS/RS travel time and consequently increasing its throughput performance is to use the load-shuffling procedures. The objective is to shuffle (i.e., pre-sort, relocate or rearrange) the loads to specified locations to minimize the response time of retrieval. However, based on examination of the literature it can be observed that the load-shuffling strategies for AS/RS have not been adequately investigated in previous studies which implies a need for further studies in this area.

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Appendix A Figures 8.14, 8.15

Fig. 8.14 Generic Structure and principal constituents of an AS/RS

Fig. 8.15 Some common types of rack structures in AS/RSs. a Single wide aisle, single deep rack. b Single wide aisle, double deep rack. c Double wide aisle, double deep rack

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Chapter 9

Designing Unit Load Automated Storage and Retrieval Systems Tone Lerher and Matjazˇ Šraml

Abstract The successful performance of the unit load automated storage and retrieval systems is dependent upon the appropriate design and optimization process. In this chapter a model of designing unit load automated storage and retrieval system for the single- and multi-aisle systems is presented. Because of the required conditions that the unit load automated storage and retrieval systems should be technically highly efficient and that it should be designed on reasonable expenses, the objective function represents minimum total cost. The objective function combines elements of layout, time-dependant part, the initial investment and the operational costs. Due to the nonlinear, multi-variable and discrete shape of the objective function, the method of genetic algorithms has been used for the optimization process of decision variables. The presented model proves to be a useful and flexible tool for choosing a particular type of the single- or multi-aisle system in designing unit load automated storage and retrieval systems. Computational analysis of the design model indicates the model suitability for addressing industry-size problems.

Nomenclature AS/RS S/R machine SR I/O

Automated storage and retrieval system Storage and retrieval machine Storage rack Input/output location

T. Lerher (&) Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia e-mail: [email protected] M. Šraml Faculty of Civil Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_9,  Springer-Verlag London Limited 2012

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TUL SC DC 3D PP GA SIT TC R Y S Nx Ny Q (TUL) Pf (TUL/h) Min TC (EUR) Pz (m2) Dz (/) C1 (EUR/m2) C2 (EUR/m2) C3 (EUR/m2) C4 (EUR/m2) C5 (EUR/m) C6 (EUR/m) C7 (EUR/piece) C8 (EUR/PP) C9 (EUR/PP) C10 (EUR/m3) C11 (EUR/piece) C12 (EUR/m) C13 (EUR/piece) C14 (EUR/m) HWAR (m) Hmin (m) Hmax (m) LTZ (m) LRS (m) HRS (m) LWAR (m) Lmin (m)

T. Lerher and M. Šraml

Transport unit load Single command cycle Dual command cycle Three dimensional Palette position Genetics algorithms Square-In-Time Total cost Number of picking aisles (variable) Number of SR (variable); Y = 2R Number of S/R machines (variable) Number of storage compartments in horizontal direction (variable) Number of storage compartments in vertical direction (variable) Storage capacity Throughput capacity Minimum total cost Surface of the land for warehouse Share for the warehouse building Cost of buying the land Cost of laying the foundation of warehouse per square meter of foundation Cost of building the walls of warehouse per square meter of walls Cost of building the roof of warehouse per square meter of roof Cost of buying upright frames per meter Cost of buying rack beams per meter Cost of buying buffers per piece Cost of assembly per pallet position Cost of fire safety per pallet position Cost of air conditioning per cubic meter Cost of buying single-aisle S/R machine Cost of the picking aisle per meter Cost of buying multi-aisle S/R machine which includes aisle transferring machine Cost of the cross aisle per meter Height of the warehouse Minimum height of the warehouse Maximum height of the warehouse Length of the transport zone Length of the storage rack Height of the storage rack Length of the warehouse Minimum length of the warehouse

9 Designing Unit Load Automated Storage and Retrieval Systems

Lmax (m) WWAR (m) Wmin (m) Wmax (m) WRD (m) HRD (m) g (m) h (m) w (m) n (/) b1 (m) b2 (m) b3 (m) b4 (m) b5 (m) b6 (m) b7 (m) b8 (m) b9 (m) b10 (m) b20 (m) Lv (m) vx (m/s) vy (m/s) vi (m/s) ax (m/s2) ay (m/s2) ai (m/s2) T(SC) (s) nSC (/) T(DC) (s) nDC (/) Tshift (s) g (%) T01 (s) T02 (s)

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Maximum length of the warehouse Width of the warehouse Minimum width of the warehouse Maximum width of the warehouse Width of the S/R machine Lift height of the S/R machine Length of the palette/TUL Height of the palette/TUL Width of the palette/TUL Number of TUL in the storage compartment Safety addition to the width of the storage compartment Safety addition to the height of the storage compartment Width of the storage compartment Width of the upright frame Thickness of the upright frame Height of rack beams Elevation of the first level storage compartment from the floor Safety spacing between racks that are placed close to each other Safety addition to the height of the warehouse Addition to the width of the palette at input buffer Addition to the end of the warehouse Length of the rack beam Maximum velocity of the S/R machine in the horizontal direction Maximum velocity of the hoisted carriage in the vertical direction Maximum velocity of the transferring vehicle in the cross warehouse aisle Acceleration/deceleration of the S/R machine in the horizontal direction Acceleration/deceleration of the hoisted carriage in the vertical direction Acceleration/deceleration of the transferring vehicle in the cross warehouse aisle Expected single command travel time Number of single command cycles Expected dual command travel time Number of dual command cycles Time of one shift Efficiency of the S/R machine Pickup time Deposit time

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9.1 Introduction Warehouses are an absolute necessity for a continuous and optimum operation of the production and distribution processes. Warehouses are needed for various reasons, which can be grouped in the following way (Bartholdi 2010): (i) to facilitate the coordination between the production and customer demand by buffering products for a certain period of time, (ii) to accumulate and consolidate products from various producers for combined shipments, (iii) to provide the same-day delivery in production and to important customers, (iv) to support products customization activities, such as packaging, final assembly etc. In this chapter automated warehouses, also named automated storage and retrieval systems (AS/RS) are presented. In the past few decades, the share of AS/RS on the world scale, which in comparison with conventional warehouses enables a higher level of technological efficiency, has increased. In (Roodbergen and Iris 2008) an overview of the work from the past 30 years is presented. A comprehensive explanation of the current state of the art in AS/RS design is provided for a range of issues such as system configuration, travel time estimation, storage assignment, dwell-point location, and request sequencing. The application of the AS/RS received consideration decades ago, when in 1962 the company Demag created the first AS/AR. The aforementioned AS/RS was a high-bay warehouse measuring 20 m in height, which marked the beginning of a new era in the development of material handling equipment in Europe. The AS/RS consists of storage racks (SR), storage and retrieval machines (S/R machines), accumulating conveyors, input and output location (I/O location), and a computer system for managing and organizing activities in the warehouse. In comparison with conventional warehousing systems, the key advantages of the AS/RS are: (i) high throughput capacity Pf and storage capacity Q, (ii) high reliability and better control of the warehousing process, (iii) improved safety conditions, and (iv) decrease in damage and the loss of goods. Due to advanced technology and the complete automation of the system, the AS/RS demands extensive investment. Additionally, those AS/RS where the S/R machine operates only in the single picking aisle (single-aisle system) are rather inflexible as far as a possible change in the throughput capacity of the warehouse is concerned. Due to the well-known benefits of AS/RS a high initial investment is necessary for the success of such systems. In the total initial investment for automated warehouse, S/R machines alone represent approximately 40% or more of the costs (Rosenblatt and Roll 1993). A measure to reduce the initial investment cost is application of the multi-aisle AS/RS. In the abovementioned system, the S/R machine serves several picking aisles with the help of the aisle transferring vehicle, which ensures driving in the cross aisle. Many of the warehouse equipment producers such as Siemens Dematic, Stoecklin, and Dambach have begun to offer such systems served by automatic curve-going or automatic aisle-transferring S/R machines. Additional benefits of using those systems are: subsequent expansion of S/R machines is possible at any time; optimum use is made of space

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due to minimal overrun dimensions, and high throughputs resulting from pallet buffer positions on the transfer car (automatic aisle-transferring S/R machines). In order to evaluate the optimal number of S/R machines in multi-aisle AS/RS the average travel time for a storage and retrieval operation has to be determined. Travel time models of AS/RS have been the subject of many researchers over the past few decades. Hausman et al. (1976) and Graves et al. (1977) presented travel time models for unit-load AS/RS assuming that the SR is Square-In-Time (SIT). They analyzed different storage strategies, e.g. randomized, turnover-based, and class-based storage assignment rules. Gudehus (1973) presented basic principles for the determination of cycle times according to unit-load AS/RS. With regard to other cycle time expressions, he considered the impact of the acceleration and deceleration rate on travel times. Bozer and White (1984) presented analytical models for the calculation of single command (SC) cycle and dual command (DC) cycle for non SIT racks. Their models are based on randomized storage and retrieval with different I/O configurations of the input queue. Hwang et al. (1990) and Vössner (1994) presented travel time models considering the operating characteristics of the S/R machine for unit-load AS/RS and non SIT racks. In the design of AS/RS, the major objective is to satisfy the required warehousing needs, which are subject to operational and physical constraints, at minimal total costs. The total cost of the system is composed of the initial investment and of annual operating costs. The design of warehouses (not necessarily AS/RS) has been studied by several authors. One of the first publications in the subject of optimizing the warehouses is represented by the work of Basan et al. (1980), who analyzed optimum dimensions of the warehouse, considering the chosen warehouse volume of the warehouse in dependence on various storage strategies. Karasawa et al. (1980) presented a design model of the AS/RS. In their work, the objective function is defined as nonlinear and multi-variable, consisting of three main variables: (i) the number of S/R machines, (ii) the length of the SR, and (iii) the height of the SR; and also of constant values: cost of buying the land, cost of building the warehouse, cost of buying the SR construction, and cost of buying S/R machines. The main disadvantage of this model (Karasawa et al. 1980) is that it refers only to the single-aisle AS/RS and the warehousing operation of only the SC cycle. Further, Ashayeri et al. (1985) presented a design model of the AS/RS that enables the determination of the main influential parameters when designing warehouses. Unlike (Karasawa et al. 1980), they considered the warehousing operation of the DC cycle. Bafna and Reed (1972) and Perry et al. (1983) used a combination of the analytical model and the system of discrete event simulations when designing the warehouse. Perry et al. (1983) used a special search method to determine optimum solutions for the AS/RS, which they included in the simulation model of the AS/RS. As a measure of the efficiency of the system, they used the throughput capacity of the warehouse, in dependence on the number of S/R machines and the number of workplaces. The design of warehouses with regard to the influence of the storage policy was presented by Rosenblatt and Roll (1984). When describing total costs, the authors have taken into account: (i) cost of building the warehouse, (ii) cost of buying storage equipment, (iii) costs arising from overloading the warehousing system (temporary

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shortage of the storage space), and (iv) costs that depend on a particular storage policy. An in-depth overview of the area of designing and controlling warehouses has been presented by Rouwenhorst and Reuter (2000) in the form of the methodology of designing warehousing systems. The design process is presented with a structured approach, which takes into account the strategic, tactical, and operational level of decision making. De Koster et al. (2008) presented an optimal storage rack design for the 3D compact AS/RS. They introduced the combination of the S/R machine for transport unit load (TUL) movement in the horizontal and vertical directions and the system of inbound and outbound conveyors (powered or non-powered) for the depth movement. They analyzed the system performance and optimal dimension of the system for the random storage strategy in order to minimize the expected travel time of the S/R machine. In the continuation of their work, Yu and De Koster (2009a, b) derived the expected SC time under the full-turnover based storage policy. In order to implement the class-based policy, Yu and De Koster (2009a, b) introduced the model for determining the optimal storage zone boundaries for the 3D compact AS/RS. For the determination of the zone boundaries, they introduced the mixed-integer nonlinear model. The presented research is very useful and flexible tool when designing 3D compact AS/RS. The majority of described models refer only to the single-aisle AS/AR (Bafna and Reed 1972; Karasawa et al. 1980; Perry et al. 1983; Ashayeri and Gelders 1985). The difference between the approaches and models lies in the cost of elements included in the objective function, the decision on considered variables, and the use of optimization techniques. Less has been done for other types of warehouses, especially for systems where the number of S/R machines (S) is less than or equal to the number of picking aisles (R) (the condition S B R). Therefore our proposed analytical travel time models for the efficiency determination of multi-aisle AS/RS have been included in our design model (Lerher et al. 2005; Lerher et al. 2010). The main purpose of our contribution chapter is to present the design and optimization of the unit load AS/RS, with which an optimal solution to the minimization of the total cost of the system is developed. The adopted approach is to use optimization in order to create the most economical design of the unit load AS/RS. Due to nonlinear, discrete, and multi-variable objective function Minimal total cost (Min.TC), the heuristics method with genetic algorithms (GA) was used. The significant part of our research lies in the creation of a computer aided tool for designing and optimizing unit load AS/RS (Lerher and Potrcˇ 2006).

9.2 Designing Unit Load Automated Storage and Retrieval Systems Model for designing of the unit load AS/RS is based on the structured approach (Rouwenhorst and Reuter 2000; Ramu 1996; Park and Webster 1989; Lerher and Potrcˇ 2006) where all parameters influencing data analysis (storage capacity,

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throughput capacity), warehouse layout planning, warehouse equipment planning, the economic consideration, modelling of the objective function, and optimization are taken into account.

9.2.1 Data Analysis The activity data analysis is required for setting the storage capacity of the warehouse and preparing information which will be used in determining inventory levels, storage and handling equipment, and assignment of products to storage locations. It includes estimating and identifying the following (Ramu 1996): high and low demand products, trends and changes (variations) in demand pattern and mix, percentage of products, which could be ordered in full and partial TUL, percentage of demand for domestics and global markets, identification of seasonal products and the timing of their picks and lows, volume of the orders, etc.

9.2.2 Warehouse Layout Planning The warehouse layout can be subdivided into planning for the external layout and planning for the internal layout. In the external layout plan, size, shape, location, and capacity of the warehouse building are considered. The size, shape, and number of SR, I/O locations, transport and storage area, rack orientation, aisle width, number of aisles, etc. have to be considered while planning for the internal layout.

9.2.3 Warehouse Equipment Planning The type of material handling equipment to be chosen will have great impact on the aisle space, capital investment, flexibility of the warehouse, and manpower requirements. To manipulate the TUL in the warehouse the lift trucks, accumulated conveyors, different manipulators, and S/R machines are used. It is not always advisable to plan for highly automated material handling equipment but based on the situation and investment, the best possible equipment should be used to meet the efficient warehouse. It is thus important to consider the level of automation needed and the amount of equipment needed (Ramu 1996).

9.2.4 The Economic Consideration of the Warehouse Total costs for automated warehouse are composed on numerous elements. They can be categorized as (Park and Webster 1989):

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• Land cost—depending on the location of the warehouse, land cost may have a considerable effect on the final design. • Building cost—one place to look for improved profitability is the cost of building for selected handling equipment. Therefore the building cost is a function of the selected handling equipment and the height of the building. • Equipment cost—selection of the material handling equipment for use in the automated warehouse is extremely important, since the amount of required space, building, storage rack facility, and handling cost are directly dependant on the type of equipment used. • Storage rack facility cost—the total storage rack facility cost is also dependent upon the selected handling equipment type and is obtained by multiplying the storage rack cost per storage compartment by the total number of storage compartments in the system. • Labour cost—the annual labour cost is dependent on the selection of material handling equipment type used. • Maintenance cost—the annual maintenance costs are composed of the maintenance costs on the building and the material handling equipment used.

9.2.5 The Objective Function Min. TC The objective function Min.TC defines the combination of project variables, operational parameters, and total cost of automated warehouse. The optimization of the objective function is based on GA. The main objective of the optimisation of project variables in Min.TC is to define the optimal solution for the type of automated warehouse, considering conditions of technically high and economically optimal solution type (Lerher and Potrcˇ 2006).

9.3 Development of the Model for Designing Unit Load AS/RS When developing and creating the model, propositions and references from other authors were considered (Rouwenhorst and Reuter 2000; Ramu 1996; Park and Webster 1989). Figure 9.1 shows the algorithm of the design model of the unit load AS/RS, including the following main modules: • Design of the storage zone It includes the choice of the palette and the building of the basic TUL. The definition of the storage compartment, which represents a foundation for the storage system, is to come after. Next, the design of SR structure (upright frames and rack beams) depends on the weight of TUL and arrangement of TUL in the horizontal and vertical directions. Finally, with regard to the storage capacity Q, geometry and type of SR, the configuration of the storage zone can be determined.

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Fig. 9.1 The algorithm of the model for designing unit load AS/RS (Lerher and Potrcˇ 2006)

• Design of the transport zone and determination of the efficiency of the warehouse It considers selection of the material handling equipment, which depends, mainly, on the SR geometry, Pf and Q. Due to the Pf, two systems of handling equipments are possible: (i) single-aisle system and (ii) multi-aisle system. Lift trucks and conveyors are used for manipulating TUL to the storage rack zone. Depending on the combination of the material handling equipment Pf and Q, the dimensions of the transport zone can be determined. • Determination of the total cost

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Determination of the total cost is divided into: (i) costs of the static part of the warehouse, (ii) costs of the dynamic part of the warehouse, and (iii) operational costs of the warehouse defined with time. • Design of the objective function and optimization of parameters of the objective function Min. TC The objective function Min. TC consists of decision variables, operational parameters, and costs of building and operating the warehouse.

9.3.1 The Development of the Objective Function Min. TC When developing the objective function Min. TC, the following assumptions, notations, and constraints have been applied to the design model of the AS/RS. Assumptions • The warehouse is divided into picking aisles with SR on both sides; therefore there is double SR between the picking aisle and single SR along the warehouse walls. The I/O location of the warehouse (I/O warehouse) is located on the lower, extreme left side of the warehouse (Fig. 9.2). • The number of the S/R machines S is less than or equal to the number of picking aisles R (S B R). • The multi-aisle AS/RS travels in the cross aisle on the transferring vehicle, which enables access to adjacent picking aisle. • The SR has a rectangular shape, whereby the I/O location of the SR (I/O aisle i) is located on the lower left side of the SR. • The S/R machine enables the operation of SC and DC, to which a variable share of travel time for travelling in the cross aisle (S B R) must be added. • When performing the operation of the DC (S B R), two different cases have been used: (i) the storage and retrieval operation is performed in the same picking aisle i and (ii) the storage and retrieval operation is performed in two randomly chosen picking aisles i and j. • Real drive characteristics of the S/R machine (velocity v, acceleration and deceleration a) as well as the length and height of the SR are known. • The S/R machine travels in the picking aisle simultaneously in the horizontal and vertical directions. • The length and height of the SR are large enough for the S/R machine to reach its maximum velocity v in the horizontal and vertical directions. • The length of the cross aisle is large enough for the transferring vehicle with the S/R machine to reach its maximum velocity v in the cross direction. • Randomized storage is used, which means that any location in the storage compartment is equally selected for the storage or retrieval assignment. For the evidence of presented operational and physical parameters, the storage compartment and the storage rack are presented in Figs. 9.3 and 9.4.

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Fig. 9.2 The layout of the unit load AS/RS (Lerher and Potrcˇ 2006) Fig. 9.3 The storage compartment (Lerher and Potrcˇ 2006)

The Min. TC is represented with a mathematical model, which includes decision variables, all relevant operational and physical parameters, investment and operating costs (Lerher and Potrcˇ 2006). (a) Cost for the warehouse building • Cost of the land

CLand

  100 ¼ Pz   C1 Dz

ð9:1Þ

Equation 9.1 calculates cost of the land according to the anticipated surface of the land for warehouse building Pz = (LWAR  WWAR) and a share Dz for the warehouse building that is usually lower than 100 (the warehouse building needs for its

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Fig. 9.4 The storage rack (Lerher and Potrcˇ 2006)

operation parking spaces, roads to dockings, office building, etc.) and is determined according to experiences of warehouse designer. Cost of the land therefore encompasses cost for the warehouse building, parking spaces, roads to dockings, office building, etc. • Cost of the floor " CFloor ¼

ððw  n þ ðn þ 1Þ  b1 þ b4 Þ  Nx þ b4 þ b10 þ b20 Þ þ LTZ  ðR  WRD þ Y  g þ ðR  1Þb8 Þ

#  C2 ð9:2Þ

Equation 9.2 calculates cost per square meter of the floor according to various physical parameters and variables (see Fig. 9.2) which are multiplied with cost of laying the foundation of warehouse per square meter of foundation. • Cost of walls 2 6 CWalls ¼ 6 4

ððw  n þ ðn þ 1Þ  b1 þ b4 Þ  Nx þ b4 þ b10 þ b20 Þ þ LTZ þ ðR  WRD þ Y  g þ ðR  1Þb8 Þ   ðh þ b2 þ b6 Þ  Ny þ b7 þ b9

! 3 7 7  2C3 5 ð9:3Þ

Equation 9.3 calculates cost per square meter of walls according to various physical parameters and variables (see Figs. 9.2 and 9.4) which are multiplied with cost of building the walls of warehouse per square meter of walls.

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• Cost of the roof " CRoof ¼

# ððw  n þ ðn þ 1Þ  b1 þ b4 Þ  Nx þ b4 þ b10 þ b20 Þ þ LTZ   C4 ð9:4Þ ðR  WRD þ Y  g þ ðR  1Þb8 Þ

Equation 9.4 calculates cost per square meter of the roof according to various physical parameters and variables (see Fig. 9.2) which are multiplied with cost of building the roof of warehouse per square meter of roof. (b) Material handling equipment costs • Cost for the upright frames    CUp:frame ¼ ððNx þ 1Þ  Y Þ  b7 þ ðh þ b2 þ b6 Þ  Ny  b2  h  C5

ð9:5Þ

Equation 9.5 calculates cost for the upright frames. According to Fig. 9.3, number of upright frames in the storage rack equals number of storage compartments in horizontal direction and one single upright frame at the end of the storage rack. Total number of upright frames in the warehouse depends of the number of SR Y (Y = 2R). The whole number of upright frames is multiplied with height of the storage rack and finally with the cost of buying upright frames per meter. • Cost for the load-supporting beams CBeam ¼



  Nx  Ny  2  Y  Lv  C6

ð9:6Þ

Equation 9.6 calculates cost for the load-supporting beams. Number of loadsupporting beams depends on the number of storage compartments in the horizontal and vertical directions and number of SR. The number 2 in the equations deals with the fact that there are always two load-supporting beams in one storage compartment. The whole number of load-supporting beams is multiplied with the length of the rack beam and finally with the cost of buying rack beams per meter. • Cost for buffers and the assembly for the SR: CBuffer ¼ ½2  R  C7    Nx  Ny  n  Y  C8

CAssembly ¼

ð9:7Þ

Equation 9.7 calculates cost for buffers and the assembly for the SR. In case of single-aisle AS/RS, there are two buffers per picking aisle (this means that each storage rack has a buffer which is located in the lower left-hand corner of the rack). In case of multi-aisle AS/RS, the cost for buffers is set to zero, since this kind of system does not use buffers in the lower left-hand corner of the rack. The cost for the assembly is dependent on the number of total pallet positions (rack capacity Q) which is multiplied by cost of assembly per pallet position.

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(c) Fire protection cost CFireprot ¼



  N x  Ny  n  Y  C 9

ð9:8Þ

Equation 9.8 calculates fire protection cost, which is dependent on the number of pallet total positions (rack capacity Q) which is multiplied by cost of fire safety per pallet position. (d) Air ventilation cost CAirvent ¼ ½ðLWAR  HWAR  WWAR Þ  C10

ð9:9Þ

Equation 9.9 calculates air ventilation cost, which is dependent on the length, width, and height of the warehouse and multiplied by cost of air conditioning per cubic meters. (e) Investment for the single- multi-aisle S/R machines • Single-aisle S/R machine: CSR machine ¼ S  C11 þ ðLRS  C12 Þ  R

ð9:10Þ

Equation 9.10 calculates investment for the single-aisle S/R machine, which is dependent on the number of S/R machines, rails, switches, etc. • Multi-aisle S/R machine: CSR machine ¼ S  C13 þ ðLRS  C12 Þ  R þ ðWWAR  ð2g þ WRD ÞÞ  C14

ð9:11Þ

Equation 9.11 calculates investment for the multi-aisle S/R machine, which is dependent on the number of S/R machines, rails, switches, etc. The right part of the equation in bracket deals with the assumption that the length of the cross warehouse aisle is lower than the warehouse width (see Fig. 9.2). Given all the above definitions, an integer nonlinear, multi-variables discrete objective function Min. TC is developed, as follows: Min:TC ¼ CLand þ CFloor þ CWall þ CRoof þ CUpframe þ CBeam þ CBuffer þ CAssembly þ CFireprot: þ CAirvent: þ CSR machine

ð9:12Þ

The decision variables S, R, Y, Nx, Ny in the objective function Min. TC, are subject to the following constraints: (1) Geometrical constraints of the warehouse are imposed on the following:

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• Length of the warehouse LWAR Lmin  ðw  n þ ðn þ 1Þ  b1 þ b4 Þ  Nx þ b4 þ b10 þ b20 þ LTZ  Lmax

ð9:13Þ

• Width of the warehouse WWAR Wmin  R  WRD þ Y  g þ ðR  1Þb8  Wmax

ð9:14Þ

• Height of the warehouse HWAR Hmin  ðh þ b2 þ b6 Þ  Ny þ b7 þ b9  Hmax

ð9:15Þ

(2) Minimum required Q of the warehouse n  Nx  Ny  2R  Q

ð9:16Þ

(3) Number of S/R machines has to be lower than or equal to the number of picking aisles SR

ð9:17Þ

The determination of the efficiency and the number of S/R machines is based on the average travel time of SC and DC. In case of the single-aisle AS/RS (S = R), the analytical travel time models from Hwang and Lee (1990) have been applied to the design model. To determine the number of S/R machines in the case of the multiaisle AS/RS (S B R), the analytical travel time models by Lerher et al. (2005), Lerher et al. (2010) have been applied. With regard to the layout of the SR, the number of the required SC and DC and the average travel time of SC and DC, one can determine the necessary number of S/R machines S that should be able to manage the load activity. S¼

nSC  T ðSCÞ þ nDC  T ðDCÞ Tshift  g

ð9:18Þ

In the case of application of the single-aisle AS/RS, the algorithm in the design model assigns the S/R machine to each picking aisle, under the condition S = R. At the same time the algorithm examines whether the assigned S/R machine, which meets the condition S = R, is able to reach the required Pf of warehouse. If the calculated Pfi is lower than the required Pf, the algorithm chooses a new type of S/R machine which has a more efficient driving characteristic (higher vx and higher vy). When the multi-aisle AS/RS, which enables transferring in the cross aisle, is chosen, then a single S/R machine serves multiple picking aisles. If the

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calculated Pfi does not reach the required Pf, the algorithm determines the necessary number of S/R machines. Considering the discrete form of the objective function Min. TC, nonlinearity and proposed decision variables, the method of genetics algorithms to optimize decision variables in the Min TC has been applied. GA are heuristic search algorithms which are used to perform demanding analyses and to solve problems of optimization. The method of GA simulates evolutionary processes ‘‘the survival of the most flexible organism’’. In the design model, the GA randomly creates the required number of subjects in a generation—organism. The subject refers to the unit load AS/RS, whereas genes in the organism are demonstrated by decision variables Nx, Ny, and S. Variables Y and R are not directly considered in the organism, since they are directly dependent on variables Nx, Ny, and S. Based on the Min. TC and project constraints, the GA evaluates each subject in the generation and arranges them with regard to their evaluation—minimum total costs. The rest of the generations in the GA (e.g. 95%) are created by crossover, reproduction, and mutation. The optimization process of GA and the meaning of individual genetic and evolutionary operators are presented in detail in Holland (1975).

9.4 An Illustrative Example of Designing Unit Load AS/RS With the optimization of decision variables S, R, Y, Nx, Ny in the Min. TC, the optimal design of the AS/RS was searched for. The input data for this example are based mainly on information from practice and sales representatives of companies supplying the unit load AS/RS equipment. The analysis refers to the chosen model of the AS/RS which is determined by the following parameters: (i) the project constraints: the length of the unit load AS/RS: Lmin [0 and Lmax B 120 m, the width of the unit load AS/RS: Wmin[0 and Wmax B 50 m, the height of the unit load AS/RS: Hmin[0 and Hmax B 22 m, (ii) entry-level parameters: the storage volume of the warehouse Q = 15.000 TUL, throughput capacity of the warehouse Pf = 100 TUL/h, (iii) operational parameters of the warehouse: w = 0.800 m, g = 1.200 m, h = 0.800 m, b1 = 0.100 m, b2 = 0.200 m, n = 3, b3 = 1.100 m, b4 = 0.120 m, b5 = 0.065 m, b6 = 0.120 m, b7 = 0.200 m, b8 = 0.200 m, b9 = 1.000 m, b10 = 0 m, b20 = 0 m, LTZ = 10.000 m, T01 = 5 s, T02 = 5 s, n(SC) = 0, n(DC) = 50, (vi) material handling equipment: the multi-aisle S/R machine (Stoeklin AT RBG 0-Q): HRD = 22.000 m, WRD = 1.400 m, vx = 3 m/s, vy = 0,8 m/s, vi = 0,6 m/s, ax = 0,5 m/s2, ay = 0,8 m/s2, ai = 0,4 m/s2, the single-aisle S/R machine (Single MAN): HRD = 21.900 m, WRD = 1.400 m, vx = 3 m/s, vy = 1 m/s, ax = 0,5 m/s2, ay = 0,5 m/s2, (iv) costs: C1 = 150,00 EUR/m2, C2 = 160,00 EUR/m2, C3 = 50,00 EUR/m2, C4 = 50,00 EUR/m2, C5 = 30,00 EUR/m, C6 = 35,00 EUR/m, C7 = 200,00 EUR/piece, C8 = 5,00 EUR/PP, C9 = 5,00 EUR/PP, C10 = 5,00 EUR/m3, C11 = 150.000,00 EUR/ piece, C12 = 50,00 EUR/m, C13 = 240.000,00 EUR/piece, C14 = 50,00 EUR/m.

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9.4.1 The Optimization Process of Decision Variables in the Objective Function Min. TC Alternative proposals of the unit load AS/RS presented in Tables 9.1 and 9.2 show the results of the optimization of decision variables with the number of generations in the GA n = 100. The response to the optimization of decision variables S, R, Y, Nx, Ny in the Min. TC indicates total costs, in dependence on the number of storage compartments in the horizontal and vertical directions for the chosen single- or multi-aisle unit load AS/RS. The optimization of project variables was carried out according to the following evolutionary and genetics operators: the degree of crossover—0.8; the degree of mutation—0.05; the degree of elitism—0.05; the size of population—100; the number of generations—100. Values of crossover, mutation and elitism degrees are chosen in accordance with our extensive analyses and experience of researchers who have been engaged in the development and application of the GA method. The size of population depends greatly on the number of decision variables in the Min. TC, which indirectly influences the necessary number of generations. Due to the proposed decision variables S, R, Y, Nx, Ny in the Min. TC, the comprehensive analyses has indicated that in most cases the GA finds the most economical solution already with 100 generations. In the case of a higher number of generations, the GA would also arrive at a solution, but it would take more time to do so. Time spent for building each proposal of the unit load AS/RS was around 15 min and time spent for the optimization of each proposal was around 20 s. The GA forms a chosen number of random designs of the AS/RS for both types of the single- and multi-aisle unit load AS/RS. Warehouses that do not follow the required constraints, defined at the optimization of decision variables S, R, Y, Nx, Ny in the Min. TC, are deleted and not considered in next generations. The number of randomly chosen designs of the AS/RS is the same as the size of the population n or in most cases smaller than n. Because of the random selection of the number of unit load AS/RS, which present a further basis for optimization, values of overall costs in the Min. TC are the highest in the generation n = 1, which holds true for both types of the single- and multi-aisle unit load AS/RS. The majority of the most economical designs would lie in the area of a high number of storage compartments in the horizontal and vertical directions. Based on specified evolutionary and genetics operators, next generations n = (1–100) are carried out, whereby each generation is better or at least equally well. Alternative proposals of the unit load AS/RS presented in Tables 9.1 and 9.2 show the results of the optimization of decision variables with the generation n = 100. The most economical design of the unit load AS/RS refers to the AS/RS with Nx =28 and Ny = 18 (multi-aisle S/R machine) and with Nx = 35 and Ny = 18 (single-aisle S/R machine). It can be seen that overall costs are minimal (optimal) at relatively high and long SR (with regard to the given geometrical constraints of the warehouse) for both variants of the single- and multi-aisle unit load AS/RS. One can comment on the presented dependence that in the case of a large storage rack ([Nx and[Ny) we have a

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Table 9.1 Results—alternative proposals of the unit load AS/RS Unit load AS/RS Multi-aisle S/R machine

Unit load AS/RS Single-aisle S/R machine

Nx Ny LRO (m) HRO (m) GRS (m) LWAR (m) WWAR (m) HWAR (m) Y R S Q (TUL)

35 18 2.920 1.120 1.200 113.12 15.96 21.16 8 4 4 15.120

28 18 2.920 1.120 1.200 91.880 20.00 21.16 10 5 3 15.120

Table 9.2 Investment—alternative proposals of the unit load AS/RS Unit load AS/RS Unit load AS/RS Multi-aisle S/R machine (EUR) Single-aisle S/R machine (EUR) Land Warehouse building Storage construction Fire safety Air ventilation S/R machine Total cost

397.200,00 627.686,90 1.190.172,00 75.600,00 196.110,90 733.186,00 3.219.955,80

381.398,40 643.518,10 1.196.850,00 75.600,00 188.309,10 620.464,00 3.106.139,60

large storage volume Q, low number of SR, and consequently a small width of the warehouse\WWAR. The latter takes the consequence of a lower number of necessary numbers of S/R machines S (apparently obvious with the single-aisle AS/RS), which has a significant influence on the entire investment in the warehouse. In dependence on the number of the S/R machines S, when applying the singleaisle AS/RS we need 4 S/R machines, whereas when applying the multi-aisle AS/RS we need 3 S/R machines. Although the multi-aisle AS/RS requires the application of 3 S/R machines, the investment in the analyzed unit load AS/RS (Q = 15.000 TUL and Pf = 100 TUL/h) is approximately the same for both types of the single(3.106.139,60 EUR) and multi-aisle (3.219.955,80 EUR) unit load AS/RS. The cost of the S/R machine for the multi-aisle system is approximately 60% higher than the cost of the S/R machine for single-aisle system due to additional elements (aisle transferring vehicle for travelling in the cross aisle, additional controls and switches, extensive control system, etc.) which make it possible to carry out the warehouse operation. The decision on application of a particular single- or multi-aisle system depends mainly on the required Pf of the warehouse.

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9.5 Conclusion In this contribution chapter, an improved model of design and optimization of the AS/RS is presented. Due to the high complexity in designing and optimizing modern warehouses, the conventional design process rises at higher and more demanding levels, in the form of the computer aided design and optimization. The presented design model is based on the structured approach and refers to the single deep storage system with several picking aisles. The essential part in the design model is the application of two different systems: (i) the single-aisle unit load AS/RS and (ii) the multi-aisle unit load AS/RS. Unlike the single-aisle unit load AS/RS the multi-aisle unit load AS/RS has not been researched in detail yet. Therefore newly improved analytical travel time models for the single- and multi-aisle systems have been included in the presented design model. Due to requirements for the most economical design and at the same time a technically highly efficient warehouse, the objective function Min. TC. has been formed. The objective function is represented with a mathematical model, which includes decision variables (S, R, Y, Nx, Ny), all relevant operational and physical parameters, investment and operating costs. Due to the nonlinearity of the Min. TC, its discrete shape and proposed decision variables, the method of genetic algorithms has been applied, in order to optimize decision variables. On the basis of the results of the optimization of decision variables in the Min. TC and with regard to the single- and multi-aisle system, the following conclusions can be drawn. With regard to the total costs, in accordance with the number of storage compartments in the horizontal and vertical directions, it can be concluded that the most economical design (for both types of the unit load AS/RS) is achieved with a high and long storage rack. The abovementioned dependence refers to the analyzed unit load AS/RS and the prescribed project constraints ei. This finding can be explained with the fact that large SR enable the achievement of a high warehouse volume of the warehouse, which influences a small number of picking aisles and consequently a smaller width of warehouse. Therefore, the number of S/R machines is smaller (\S—particularly with the single-aisle unit load AS/RS), which proves in overall costs of the investment. In future research the main focus should be on implementation of existing models for designing and optimizing warehouse systems in practice, when making decisions in the early stages of design project of multi-aisle unit load AS/RS and when deciding which type of S/R machine will be most promising. Therefore, our main goals of further research will be on: order picking in AS/RS, introducing green logistics warehousing systems, centralized versus decentralized material handling systems, etc.

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References Siemens Dematic. http://www.siemens-dematic.com/. Accessed 8 Oct 2010 Stöcklin. http://www.stoecklin.com/. Accessed 8 Oct 2010 Ashayeri J, Gelders LF (1985) A microcomputer-based optimization model for the design of automated warehouses. Int J Prod Res 23(4):825–839. doi:10.1080/00207548508904750 Bafna KM, Reed R (1972) An analytical approach to design of high-rise stacker crane warehouse systems. J Ind Eng 4(10):8–14 Bartholdi JJ (2010) Warehouse and distribution science. http://www.warehouse-science.com. Accessed 8 Oct 2010 Bassan Y, Roll Y, Rosenblatt MJ (1980) Internal layout design of a warehouse. IIE Trans 12(4):317–322. doi:10.1080/05695558008974523 Bozer AY, White AJ (1984) Travel-time models for automated storage and retrieval systems. IIE Trans 16(4):329–338. doi:10.1080/07408178408975252 Dambach, http://www.dambach.de/. Accessed 8 Oct 2010 De Koster MBM, Le-Duc T, Yu Y (2008) Optimal storage rack design for a 3- dimensional compact AS/RS. Int J Prod Res 46(6):1495–1514. doi:10.1080/00207540600957795 Graves SC, Hausman WH, Schwarz LB (1977) Storage retrieval interleaving in automatic warehousing systems. Manag Sci 23(9):935–945. doi:10.1287/mnsc.23.9.935 Gudehus T (1973) Principles of order picking: operations in distribution and warehousing systems. W. Girardet Verlag, Essen Hausman HW, Schwarz BL, Graves CS (1976) Optimal storage assignment in automatic warehousing systems. Manag Sci 22(6):629–638. doi:10.1287/mnsc.22.6.629 Holland JH (1975) Adaption in natural and artificial systems, Technical Report, Michigan University Hwang H, Lee SB (1990) Travel time models considering the operating characteristics of the storage and retrieval machine. Int J Prod Res 28(10):1779–1789. doi:10.1080/00207549008942833 Karasawa Y, Nakayama H, Dohi S (1980) Trade-off analysis for optimal design of automated warehouses. Int J Syst Sci 11(5):567–576. doi:10.1080/00207728008967037 Lerher T, Potrcˇ I (2006) The design and optimization of automated storage and retrieval systems. J Mech Eng 52(5):268–291 ISSN: 0039-2480 Lerher T, Šraml M, Kramberger J, Borovinšek M, Zmazek B, Potrcˇ I (2005) Analytical travel time models for multi aisle automated storage and retrieval systems. Int J Adv Manuf Technol 30(3–4):340–356. doi:10.1007/s00170-005-0061-6 Lerher T, Šraml M, Potrcˇ I, Tollazzi T (2010) Travel time models for double-deep automated storage and retrieval systems. Int J Prod Res 48(11):3151–3172. doi:10.1080/00207540902796008 Park YH, Webster DB (1989) Modelling of three-dimensional warehouse systems. Int J Prod Res 27(6):985–1003. doi:10.1080/00207548908942603 Perry RF, Hoover SF, Freeman DR (1983) Design of automated storage and retrieval systems using simulation modeling. In: Proceedings of ICAW 1983, pp 57–63, Institute of Industrial Engineers, Atlanta, Georgia Ramu NV (1996) Design methodology for modelling warehouse internal layout integrated with operating policies. Dissertation, Clemson University Roodbergen KJ, Iris VFA (2008) A survey of literature on automated storage and retrieval systems. Eur J Oper Res 194(2):343–362. doi:10.1016/j.ejor.2008.01.038 Rosenblatt MJ, Roll J (1984) Warehouse design with storage policy considerations. Int J Prod Res 22(5):809–821. doi:10.1080/00207548408942501 Rosenblatt MJ, Roll J (1993) A combined optimization and simulation approach for designing automated storage and retrieval systems. IIE Trans 25(1):40–50. doi:10.1080/07408179308964264 Rouwenhorst B, Reuter B (2000) Warehouse design and control: framework and literature review. Eur J Oper Res 122(3):515–533. doi:10.1016/S0377-2217(99)00020-X Vössner S (1994) Spielzeit Berechnung von Regalförderzeugen. Dissertation, Graz University of Technology

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Yu Y, De Koster MBM (2009a) Designing an optimal turnover-based storage rack for a 3D compact AS/RS. Int J Prod Res 47(6):1551–1571. doi:10.1080/00207540701576346 Yu Y, De Koster MBM (2009b) Optimal zone boundaries for two class-based compact 3D AS/ RS. IIE Trans (2008) 41(3):194–208. doi:10.1080/07408170802375778

Chapter 10

Warehouse Management: Productivity Improvement in Automated Storage and Retrieval Systems Yaghoub Khojasteh-Ghamari

Abstract This chapter addresses productivity improvement in automated storage and retrieval systems (AS/RS) and order picking time reduction. A unit load multiaisle AS/RS with a single storage/retrieval (S/R) machine is considered. The S/R machine, which is dedicated to all aisles, can simultaneously move in vertical and horizontal directions to perform storage and retrieval operations. An order can be a request for one or more items, and each item can be found in several storage locations in the warehouse. The objective is to formulate the order picking problem mathematically, and to propose algorithms that minimize the total time traveled by the S/R machine to complete the retrieval process of an order. We first formulate the problem as a nonlinear programing problem. Then, we develop a genetic algorithm as a meta-heuristic method as well as a heuristic algorithm, and provide a performance comparison between them. Finally, some numerical results from a large set of problems are presented.

10.1 Introduction Warehouses and distribution centers are an absolute necessity for a continuous and optimum operation of the production and distribution processes. A successful performance of a warehouse depends on the appropriate design, layout and operation of the warehouse and material handling systems. Automated storage and retrieval systems (AS/RS) as basic components of automated warehouses are used in manufacturing, warehousing and distributions

Y. Khojasteh-Ghamari (&) Temple University, Japan Campus, 4-1-27 Mita, Minato-ku, Tokyo 108-0073, Japan e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_10,  Springer-Verlag London Limited 2012

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applications. Efficient utilization of the warehouse space, increased storage capacity to meet long-range plans, reduction of damage and loss of goods, improved inventory management and control, quick response to locate/store/ retrieve items and reduced labor cost due to automation are among the major advantages provided by AS/RS (Tompkins et al. 2003). AS/RS consist of one or more aisles; each aisle has storage racks on either side, a storage and retrieval (S/R) machine, input/output (I/O) stations and accumulating conveyors. The S/R machine in each aisle can access the storage rack on either side of the aisle. The number of S/R machines depends on the throughput capacity. When throughput capacity is high, one S/R machine should be employed in each storage aisle. However, when it is relatively low, multi-aisle AS/RS can be applied. In case of multi-aisle AS/RS, when the number of S/R machines is smaller than the number of pick aisles, a considerable saving of initial investment costs can be achieved. In order to evaluate the optimal number of S/R machines in multiaisle AS/RS, the mean travel time for a S/R operation has to be determined. One important operational aspect of the AS/RS is to minimize the total time traveled by the S/R machine to complete the retrieval process of customer orders. Order picking is a fundamental component of the retrieval function performed in warehouses. The main purpose of an order picking system is to fill customer orders by selecting the appropriate amount of material from a pre-designated storage medium known as the picking or forward area. Order picking represents only a subset of the material handling operations performed in warehousing. However, it is one of the most costly and time-consuming functions of warehousing. In many warehouses, the difference between profit and loss depends on how well the order picking operation is run (Bozer and White 1990). The remainder of this paper is organized as follows. A review of the related literature is presented in the next section. Description of the problem and its formulation are given in Sect. 10.3. Section 10.4 describes solution methods to solve the problem. A genetic algorithm is developed in Sect. 10.5. Numerical results are discussed in Sect. 10.6. Conclusions and future research are summarized in Sect. 10.7.

10.2 Literature Review There are many studies on order picking problems in automated warehousing systems and AS/RS. Ratliff and Rosenthal (1983) developed a graph-based algorithm to find the shortest path to visit a set of pick locations in a ladder layout. Roodbergen and de Koster (2001) extended the work of Ratliff and Rosenthal (1983). They considered the order picking problem in a parallel aisle warehouse in which order pickers can cross over the aisles at the end of aisles as well as at a middle cross aisle. They developed a dynamic programing algorithm to solve the problem. Van den Berg and Gademann (1999) developed a transportation problem (TP) model for a block sequencing in an AS/RS with dedicated storage and a

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single-load machine. They proved that the optimal solution of the TP problem is the optimal sequence of the machine to minimize the traveling time. Elsayed (1981) made a chain of studies on the problem of optimally batching several orders in a two-dimensional warehouse with ladder structure. Recognizing that the exact solutions of the problem are very difficult and time-consuming to obtain, Elsayed and Stern (1983) presented some heuristic algorithms, but reported that none of them produces consistently superior results through experimentations. Hwang et al. (1988) studied a similar order picking problem in a single-aisle AS/ RS and presented heuristic algorithms, which determine an efficient batching of orders for each tour of the S/R machine. The algorithms were based on cluster analysis with some similarity measures. Through simulation, they compared performances of the proposed algorithms with Elsayed and Sterns’ results in 1983. Bozer and White (1984), Han et al. (1987) and Lee and Schaefer (1996) proposed a procedure to optimize the sequencing of retrieval requests based on the solution of a linear assignment problem. Lee and Schaefer (1997) also presented several optimum and heuristic sequencing methods, where a storage request is assigned to a predetermined storage location. Mahajan et al. (1998) developed a retrieval sequencing scheme aimed at improving the throughput of miniload AS/RS. They proposed a nearest-neighbor retrieval sequencing heuristic and developed an analytical model to predict its performance. Amato et al. (2005) proposed an algorithm to optimally sequence the retrieval orders based on colored timed Petri nets framework. They also proposed two control algorithms to optimize the operations of the cranes and shuttle. Hsu et al. (2005) considered the order batching problem in a multi-aisle warehouse and proposed a genetic algorithm to minimize the total travel distance. Hwang and Cho (2006) presented a performance evaluation model for the order picking warehouse in a supply center. The objective of their study was to minimize the cost by minimizing the number of transporters and to calculate the performance and facility utilization rate. Khojasteh-Ghamari and Son (2008) addressed the order picking problem in a miniload AS/RS through a simulation study, where each item can be stocked at several storage locations within the warehouse. They also proposed a genetic algorithm to solve the problem. Their simulation results showed the efficiency of the genetic algorithm in finding a solution for the problem. To the best of our knowledge, it is the only study that has addressed the order picking problem in AS/ RS, where each item can be stocked at several storage locations. In fact, some manufacturers whose products have a large variety of types, shapes and sizes are faced with this property in their finished goods warehouses. Since classifying and zoning each individual type of product in the warehouse needs a warehouse with a large space, the storage of an item in several places is unavoidable. There are also some survey papers on warehouse operations. Van den Berg (1999) and Rouwenhorst et al. (2000) surveyed literature on warehouse planning and control. The planning includes the storage location assignment problem, and the control of a warehousing system includes routing, sequencing, scheduling, dwellpoint selection and order batching. Gu et al. (2007) presented a comprehensive

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review of research on warehouse operation. In a surveyed paper, Roodbergen and Vis (2009) presented a comprehensive explanation of the current state of the art in AS/RS. According to their paper, the majority of the reviewed models used a different approach for analyzing the system performance of warehouses. Goetschalckx and Wei (1994) presented a bibliography on order picking systems for 1985 through to 1992. De Koster et al. (2007) carried out a literature review on typical decision problems in design and control of manual order picking processes. They focused on optimal layout design, storage assignment methods, routing methods, order batching and zoning. As a study on travel-time modeling, Koh et al. (2002) proposed some models for travel times of the S/R machine in a warehouse with a tower crane. They derived the models for both single and dual command cycles based on the randomized storage assignment rule. They also calculated the travel-time under the turnover-based storage assignment rule through a numerical approach. In a recent study, Lerher et al. (2010) studied analytical travel time models for the computation of travel-time in multi-aisle AS/RS. Their models consider the operating characteristics of the S/R machine such as acceleration and deceleration and the maximum velocity. There are also some studies on design of AS/RS. Hwang et al. (2002) formulated a nonlinear mathematical model and developed an efficient heuristic solution procedure to design the AS/RS and determine the unit load size of the vehicle simultaneously. Koh et al. (2005) proposed an optimization model to find an optimal buffer size in an end-of-aisle order picking system, where a single S/R machine serves several aisles. De Koster et al. (2008) presented an optimal storage rack design for the 3D compact AS/RS. They have introduced the combination of the S/R machine for transport unit load movement in the horizontal and vertical directions and the system of inbound and outbound conveyors for the depth movement. They analyzed the system performance and optimal dimension of the system for the random storage strategy in order to minimize the mean travel time of the S/R machine. In the continuation of their work, Yu and De Koster (2009) derived the mean single command cycle time under the full-turnover-based storage policy. In this research, we address an order picking problem in a unit load multi-aisle AS/RS, in which a single S/R machine performs S/R operations, and each item can be found in several storage locations. Our objective is to formulate the problem mathematically, and propose algorithms that minimize the total time traveled by the S/R machine to complete the retrieval process of orders. We develop a genetic algorithm as well as a heuristic one, and provide a performance comparison between them. Numerical results from a large set of problems will be also presented.

10.3 Problem Description and Formulation In this research, we consider an end-of-aisle order picking system of unit load AS/ RS, where there are one or more aisles. Each aisle contains a storage rack on both sides of the aisle. There is an I/O station at the end of each aisle. The I/O stations

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are at the height of the first layer (row) of the storage racks. There is also a single S/R machine dedicated to all aisles of the system, which can simultaneously move in vertical and horizontal directions. Hence, the travel time between two points is equal to the maximum of the horizontal and vertical travels. The S/R machine can carry one unit load at a time and can either deposit an inbound load in an empty location in the rack or pick up a load from a location in the rack and deliver it to the output station of the aisle. If an item is retrieved from an aisle by the S/R machine, then it is delivered to the output station of the same aisle. In other words, as delivery, the item retrieved from an aisle is delivered to the output station of that aisle. As deposit, when a storage location of an aisle is allocated for an item, the item should be loaded through the input station of that aisle. Therefore, the machine has no load when changing aisles. The S/R machine is positioned at one of the aisles (at the respective I/O station) before the receipt of an order. This aisle is decided by the storage location (aisle) of the last item of the previous order. That is, after accomplishing the last retrieve of an order, the S/R machine stays at that aisle awaiting a new order. Figure 10.1 shows a schematic overhead view of an end-of-aisle order picking system with four aisles, 20 storage locations in a row of each rack and a single S/R machine, which is located at the end of aisle number 2. Both input and output stations are shown by shaded rectangle. The path between aisles ends at the dwell-point, where all the maintenance operations as well as long term stay of the S/R machine are done. In calculating the travel time of the S/R machine, constant velocities are used for horizontal and vertical travels. An order can be a request for more than one item. Also each item can be in several storage locations in the warehouse. When retrieval requests consist of multiple items and the items are in multiple stock locations, the S/R machine must travel to numerous storage locations to complete each order. Our objective is to propose algorithms to minimize the total time traveled by the S/R machine to complete the retrieval process of a customer order. In the following section, we analyze the travel time of the S/R machine and formulate the problem as a nonlinear mathematical model.

10.3.1 S/R Machine Travel-Time Analysis In this section, we first analyze the travel time of the S/R machine in multi-aisle AS/RS. Then, we formulate the problem and present the objective function with respective constraints. The following notation is introduced. the horizontal velocity of the S/R machine vx the vertical velocity of the S/R machine vy vw the cross velocity of the S/R machine L the rack length H the rack height

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2

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5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

8

4 7 6

3 5 4

2 3 2

1 1

Fig. 10.1 An end-of-aisle order picking system with four aisles and a single S/R machine

n m i j l h r w pij tij tx ty tw S sc f

number of rows of a storage rack (i.e., the number of layers in a rack) number of columns of a storage rack (i.e., the number of bins in each layer) the column count of a rack the layer count of a rack the length of a storage bin the height of a storage bin the length of the curve at the entrance of each aisle the distance between two consecutive racks a storage bin position of column i and row j total travel time of the machine to pick up an item at pij and deliver it to the I/O station the travel-time function of distance D on horizontal direction, tx ðDÞ ¼ D=vx  the travel-time function of distance D on vertical direction, ty ðDÞ ¼ D vy the travel-time function of distance D on the cross direction, tw ðDÞ ¼ D=vw number of storage aisles current-aisle number an index for rack face: = 1 for the rack on the right side of an aisle; 2 for the rack on the left side of the aisle

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Fig. 10.2 Front view of a storage rack with n rows and m columns

n … j

…

j×h

2 1 1

2

3

…

i

…

m

i×l

Cijfs Fijfs As ssml slrg I/OðsÞ N

=1 if the item in location (i,j,f,s) is selected for retrieval; 0 otherwise =1 if item in location (i,j,f,s) is selected to be retrieved first; 0 otherwise =1 if aisle s has an item selected by the S/R machine; 0 otherwise the smallest aisle number that is visited the largest aisle number that is visited input/output station of aisle s total number of items requested in a customer order

10.3.1.1 Travel Time Within a Single Aisle Assume that the S/R machine is located in aisle s (at the I/OðsÞ ) and the item at location (bin) pij (column i and row j) is selected for retrieval. Machine should travel the horizontal distance of i 9 l and the vertical distance of j 9 h, as depicted in Fig. 10.2. If we denote the horizontal and vertical travel distances by x and y, respectively, then we have x¼il

ð10:1Þ

y¼jh ð10:2Þ Let Tx and Ty be the horizontal and vertical travel times of the machine, respectively. Then, we have tx ðxÞ ¼

x ¼ Tx vx

ð10:3Þ

ty ðyÞ ¼

y ¼ Ty vy

ð10:4Þ

and,

where vx and vy are the horizontal and vertical velocities of the S/R machine, respectively.

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Since the machine moves simultaneously in vertical and horizontal directions, the time to travel from I/OðsÞ to the location pij will be the maximum value between Tx and Ty, which is also equal to the return travel time from pij to I/OðsÞ : Let tij be the total travel time of the S/R machine. Then, tij ¼ 2 maxðTx ; Ty Þ

ð10:5Þ

10.3.1.2 Travel Time Between Aisles Assume that the S/R machine is located in aisle s (at I/OðsÞ ) and the item in pij (column i and row j) of the aisle s0 is selected to retrieve. In this case, the machine needs (i) to exit the current aisle s (a horizontal distance of m 9 l), (ii) to travel the distance between the two aisles, which includes two curves (2r) and the distance between aisles s and s0 , and (iii) to reach to bin pij of aisle s0 . If these three horizontal distances are denoted by x1, x2 and x3, respectively, then we have the following. x1 ¼ m  l ¼ L

ð10:6Þ

x2 ¼ 2r þ js0  sj  w

ð10:7Þ

x3 ¼ ðm  iÞ  l:

ð10:8Þ

In Eq. 10.7, because we do not know whether s0 is larger than s, the absolute difference value is used. Since the horizontal velocity of the machine within an aisle (vx) is different from that in between aisles (vw), the horizontal travel time to reach the item in aisle s0 is given by Tx0 ¼ tx ðx1 þ x3 Þ þ tw ðx2 Þ ¼

x 1 þ x3 x 2 þ : vx vw

ð10:9Þ

Again, because of the simultaneous movement of the machine in horizontal and vertical directions, the travel time will be the maximum value between Tx0 and Ty. That is, Tg ¼ maxðTx0 ; Ty Þ:

ð10:10Þ

In Eq. 10.10, Tg is the travel time of the machine from I/OðsÞ to location pij in aisle s0 . To carry this item to I/Oðs0 Þ (the I/O station of the current aisle, s0 ), maxðTx ; Ty Þ time units is required as shown in Eq. 10.5. Thus, the total travel time of the S/R machine will be tij ¼ maxðTx0 ; Ty Þ þ maxðTx ; Ty Þ

ð10:11Þ

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10.3.2 Problem Formulation In this section, we formulate the order picking problem as a nonlinear programing problem. The objective function is to minimize the total time traveled by the S/R machine to retrieve and deliver all the items requested in an order. When the S/R machine must visit multiple aisles for retrievals, the total travel time is the sum of the travel time of the S/R machine within and between the aisles. The components of the total travel time of the S/R machine are given below. (i)

Travel time within the current aisle: If there is at least an item in the current aisle sc to be selected (i.e., 8i; j; f ; Cijfsc  1), the travel time of the machine to retrieve the item(s) is T1 ¼

m X n X 2 X

ðCijfsc  tij Þ

ð10:12Þ

i¼1 j¼1 f ¼1

(ii)

The time to exit an aisle and enter to another aisle/aisles:

T2 ¼ ½tx ðm  lÞ þ tw ð2rÞ 

S X

As

ð10:13Þ

s6¼sc

This includes traveling the length of the current aisle with the speed of vx ; and the two curves (one curve for the exit, and the other for entering to the new aisle) with speed of vw : (iii) Travel time within other aisle(s):

T3 ¼

m X n X 2 X S X i¼1 j¼1 f ¼1 s¼1

tij ðFijfs  ½tx ðm  iÞ  l þ  þ ð1  Fijfs Þ  Cijfs  tij Þ: 2

ð10:14Þ

If there is an item in aisle s (s = sc) for retrieval (i.e., 8i; j; f ; Fijfs ¼ 1), the t machine travels the distance (m-i) l within the aisle to reach the location, and 2ij to carry the item to I/OðsÞ : Note that, tij is the round trip travel time of the machine from an I/O to the location pij and back to the station of that aisle. The first part of Eq. 10.14 is to retrieve and deliver the first item in the aisle s. The second part  is the required  travel time to retrieve and deliver the rest of the items, if any 8i; j; f ; Fijfs ¼ 0 ; within the aisle s. (iv) Traveling time between the aisles: Based on the current location of the S/R machine, and in order to minimize the total travel time between aisles, we should find and select the optimal sequence of the aisles to be visited by the machine.

242

Y. Khojasteh-Ghamari

Let us define ssml and slrg as the smallest and largest aisle numbers to be visited by the S/R machine, respectively. That is, ssml ¼ minfsjAs ¼ 1g; and slrg ¼ maxfsjAs ¼ 1g:

ð10:15Þ

For example, consider Fig. 10.1, where the S/R machine is located in aisle 2 ðsc ¼ 2Þ: Assume that aisles 1 and 4 should be visited by the machine. So, ssml ¼ 1; and slrg ¼ 4: The shortest distance traveled between the aisles to retrieve all selected items, is given by dw ¼ w  minfslrg  ssml þ sc  ssml ; slrg  ssml þ slrg  sc g ¼ w  ½ðslrg  ssml Þ þ minfjsc  ssml j; jslrg  sc jg:

ð10:16Þ

In the example, dw ¼ w  ½ð4  1Þ þ minfj2  1j; j4  2jg ¼ w  ð3 þ 1Þ ¼ 4w; where, w is the distance between two consecutive aisles. This indicates that to minimize the total travel time between the aisles, the machine should visit aisle 1 first, then aisle 4. If the machine visits aisle 4 first, this distance will be 5w, which is not minimum. Thus, the optimal travel time of the above distance is given by T4 ¼ tw ðw  ½ðslrg  ssml Þ þ minfjsc  ssml j; jslrg  sc jgÞ;

ð10:17Þ

where, 8s 2 S; Assml ¼ 1

ð10:18Þ

8s 2 S; Aslrg ¼ 1

ð10:19Þ

8s 2 S; ssml  As  s

ð10:20Þ

8s 2 S; slrg  s  As :

ð10:21Þ

Let Ds be the smallest distance from either ssml or slrg to sc. That is, Ds ¼ minfjssml  sc j; jslrg  sc jg:

ð10:22Þ

In order to transform Eq. 10.17 and write it as a nonlinear statement in the objective function, we define the following. as ¼ 1 if aisle s is the smallest aisle number that is visited; 0 otherwise, bs ¼ 1 if aisle s is the largest aisle number that is visited; 0 otherwise, and c¼1 if Ds ¼ jssml  sc j; 0 otherwise. Now, Eq. 10.17 can be written as: S S S X X X T4 ¼ tx ðw  ½ ðbs  as Þ  s þ cð1  as  sÞ þ ð1  cÞð bs  s  1ÞÞ: s¼1

s¼1

s¼1

ð10:23Þ

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Therefore, the objective function is minimizing the sum of T1,T2,T3 and T4. That is, Minimize T ¼

m X n X 2 X

ðCijfsc  tij Þ þ ½tx ðm  lÞ þ tw ð2rÞ 

i¼1 j¼1 f ¼1

þ

S X

As

s6¼sc

m X n X 2 X S X i¼1 j¼1 f ¼1

tij ðFijfs  ½tx ðm  iÞ  l þ  þ ð1  Fijfs Þ  Cijfs  tij Þ 2 s¼1

S S S X X X þ tx ðw  ½ ðbs  as Þ  s þ cð1  as  sÞ þ ð1  cÞð bs  s  1ÞÞ s¼1

s¼1

s¼1

ð10:24Þ Subject to: m X n X 2 X S X

Cijfs ¼ Nði ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n; s ¼ 1; 2; . . .; S; f ¼ 1; 2Þ

i¼1 j¼1 f ¼1 s¼1

ð10:25Þ m X n X 2 X

Fijfs ¼ 1 8s 6¼ sc

ð10:26Þ

i¼1 j¼1 f ¼1

As 

m X n X 2 X

Cijfs ðs ¼ 1; 2; . . .; SÞ

ð10:27Þ

i¼1 j¼1 f ¼1

Fijfs  Cijfs  As ði ¼ 1; 2; . . .; m ; j ¼ 1; 2; . . .; n; s ¼ 1; 2; . . .; S; f ¼ 1; 2Þ ð10:28Þ S X

as ¼ 1

ð10:29Þ

bs ¼ 1

ð10:30Þ

s¼1 S X s¼1

Cijfs 2 f0; 1gði ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n; s ¼ 1; 2; . . .; S; f ¼ 1; 2Þ

ð10:31Þ

Fijfs 2 f0; 1g ði ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; n; s ¼ 1; 2; . . .; S; f ¼ 1; 2Þ

ð10:32Þ

As 2 f0; 1g ðs ¼ 1; 2; . . .; SÞ

ð10:33Þ

as ; bs 2 f0; 1g ðs ¼ 1; 2; . . .; SÞ

ð10:34Þ

c 2 f0; 1g

ð10:35Þ

244

Y. Khojasteh-Ghamari

Constraint (10.25) guarantees that all the items requested in the order are retrieved and delivered. Constraint (10.26) represents that if any aisle (other than the current one) has an item for retrieval, then there is only one item in that aisle which is selected first to retrieve. Constraint (10.27) represents that if an aisle has an item which is selected for retrieval, then that item must be retrieved. Constraint (10.28) represents if there is more than one item for retrieval in a non-current aisle, then only one item can be retrieved first. In other words, when two or more items of an aisle (other than the current one) have to be retrieved, then only one of those items which is placed in the closest location to the aisle entrance is first selected for retrieval. Constraint (10.29) represents that there is only one aisle with the smallest number to be visited. Constraint (10.30) represents that there is only one aisle with the largest number to be visited.

10.4 Solution Methods In order to solve the problem, an ordinary heuristic as well as a suitable genetic algorithm is presented. To show the superiority of the presented algorithms, it is necessary to compare them with other methods and/or the optimal solution. Therefore, we first present an algorithm to obtain the best solution for the problem, so called the enumeration algorithm. The results are used as benchmark solutions for performance comparison of the two proposed algorithms. In the enumeration algorithm, we identify all feasible solutions and compare them with each other to find the best solution. To do this, the method first finds all feasible ways to retrieve the items requested in an order. Then, it calculates the total time traveled by the S/R machine for each one, and finally selects the solution requiring the least amount of time to accomplish the order. This solution is considered as the optimum solution for the problem. Consider an order consisting of K distinct types of items, in which nK items of the kth type are requested. The total number of feasible solutions to pick the order is given by N!

 K  Y m k

k¼1

nk

¼ N!

K  Y k¼1

 mk ! ; nk !ðmk  nk Þ!

ð10:36Þ

P where, mk is the total number of item k exist in the warehouse, and N ¼ Kk¼1 nk : Having solved various problems by the enumeration algorithm and identifying the best solution that had the minimum amount of travel time, we observed that the existing items in the current aisle (i.e., the aisle in which the S/R machine is in at the beginning of the retrieving process) are usually selected and included in the final solution. A reason for this would be that those items have a high probability to be selected because of their lower retrieval times compared to the items located in a different aisle. This in fact acted as a hint to develop an algorithm that first looks for the items existing in the current aisle. We call this algorithm the current aisle heuristic (CAH) algorithm.

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In the CAH algorithm, the existing items in the current aisle are selected first for retrieval. Afterward, the remainder of the order (if any) is selected and all the various retrieval methods are studied. To describe this algorithm more precisely, let v be the number of the ordered items existing in the current aisle. If v = 0, the method then becomes same as the enumeration algorithm. If v = 1, then the method first calculates the time traveled by the S/R machine, t1, for only one existing item in the current aisle, and removes that item from the list of ordered items, and then for the remaining items (if any), it proceeds as same as the enumeration algorithm to obtain the minimum travel time, t2. The total travel time of the S/R machine will be sum of t1 and t2 as the final solution. If v [ 1, then the method first assigns the picking sequence to pick up all v items which exist in the current aisle. After calculating the travel time, t1, it removes the items from the list of ordered items. Then, for the remaining items (if any), it proceeds like the enumeration algorithm, i.e., finding all feasible ways followed by calculating the travel time for each one and finally selecting the minimum value among them, t2. The total traveled time of the S/R machine will be sum of t1 and t2 as the final solution. In Sect. 10.3.1.1, we explained how to calculate the travel time of the S/R machine to retrieve an item from the current aisle. If any ordered items exist in the current aisle, then the number of ways studied will be divided by the number of the items existing in the warehouse. Therefore, this task causes the total number of potential solutions to decrease dramatically, and hence, the CPU time (process time of the program) would be decreased as well. The proposed genetic algorithm is explained in the following section.

10.5 Genetic Algorithms Genetic algorithms are powerful and broadly applicable in statistic search and optimization techniques based on principles from evolution theory. A genetic algorithm is an optimization process that employs genotypes (individuals or chromosomes) in a population, and the genotypes are made of units called genes arranged in linear succession. Each genotype would represent a potential solution to a problem; an evaluation process run on a population of chromosomes corresponds to a search through a space of potential solutions. The basic genetic algorithms include three parts (Sivanandam and Deepa 2008): 1. Genetic representation genetic representation involves the possible parameters combinations of handling problem. In general, feasible solutions are encoded into strings called chromosomes. 2. Fitness function the fitness function evaluates the fitness value of each chromosome and the fitness value determines the survival chromosome for evolution. The fitness function is directly related to the objective function of the optimization problem.

246 Fig. 10.3 Representation of a potential solution

Y. Khojasteh-Ghamari E D B A C 2

7 4

1

3

3. Genetic operators genetic operators are the kernels of genetic algorithms. The operators’ main task is to generate new generation according to the fitness values of chromosomes. In general, it includes selection (reproduction), crossover and mutation operators. Selection operator is choosing a good basis of chromosome based on fitness function. Crossover operator is selecting two chromosomes randomly and exchange parts of chromosomes with a crossover probability. Mutation operator is randomly changing one or more genes of the chromosome. The mutation purpose is to prevent the population from converging to a local optimal and to introduce new possible solutions to the population. The mutation is carried out according to a mutation probability. In each generation, we evaluate each chromosome, select a new population with respect to the probability distribution based on fitness values, and recombine the chromosomes in the new population by mutation and crossover operators. After a number of generations, when no further improvement is observed, the best chromosome represents the best solution found by the algorithm. The algorithm is often terminated after a fixed number of iterations depending on speed and resource criteria (Michalewicz 1992).

10.5.1 Representation In the order picking problem, a chromosome represents a potential solution, where each one is viewed as a sequence of genes each with its own associated allele. By analogy, each gene in a chromosome represents the item type, and its associated allele represents the storage location. Therefore, each potential solution consists of a chromosome, in which the number of genes is equal to the number of items in the requested order. An example is given in Fig. 10.3. Figure 10.3 shows an example of a potential solution in which, items A, B, C, D and E have been requested for retrieval. In this solution, items E, D, B, A and C with respective location numbers 2, 7, 4, 1 and 3 have been selected for retrieval. It should be noted that the sequence of picking items has also been considered in the representation. In this example, item E with location number 2 will be retrieved first, followed by items D, B, A and C with location numbers 7, 4, 1 and 3, respectively.

10.5.2 Initialization The initial population is randomly generated. Each chromosome consists of a randomly generated sequence of the items requested in an order. In each

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chromosome, a number is assigned to each item. It should be noted that the condition of feasibility for each solution is necessary. Therefore, in the populationgenerating process, a suitable procedure is required to make each solution feasible. Thus, in each chromosome, the requested items are randomly distributed without repetition, and the location numbers for each item are randomly selected and assigned. The assigned number for item k is an integer number in range ½1; mk ; where mk is the total number of item k existing in the warehouse. Suppose that within the warehouse, there are totally 7, 6, 4, 9 and 5 number of items A, B, C, D and E, respectively. In order to form the solution depicted in Fig. 10.3, first the requested items are randomly selected (E, D, B, A and C). Then, for each item E, D, B, A and C, an integer number is randomly selected from the range [1, 5], [1, 9], [1, 6], [1, 7] and [1, 4], respectively.

10.5.3 Fitness Function and Evaluation At each generation, chromosomes are evaluated by a fitness function. The fitness function of a genetic algorithm is directly related to the objective function of the problem. A fitness value not only represents how good the solution is, but also corresponds to how close the chromosome is to the optimal one. In most optimization applications, fitness values are calculated based on the original objective function. In the order picking problem, the objective function is to minimize the travel time of the S/R machine. The total time traveled by the S/R machine is the criteria to select the chromosome for the next generation. The method of calculation the travel time of the S/R machine has been explained in Sect. 10.3.1. Because our problem is treated as a minimization problem, we must convert the objective function value for each chromosome into a fitness value, so that a fitter chromosome has a larger fitness value. This can simply be adopted by the inverse of its value as follows (Cheng et al. 1995). evalðkÞ ¼

1 ; k ¼ 1; 2; . . .; pop size f ðkÞ

ð10:37Þ

where, eval(k) is the fitness function for the k th chromosome and f(k) is the total time traveled by the S/R machine for the k th chromosome. Population size (pop_size) determines how many chromosomes should be in the population at any given time.

10.5.4 Selection Operator Selection (reproduction) operator is how to select the individuals from population as parents for crossover and mutation to produce the next generation. There are

248

Y. Khojasteh-Ghamari

some selection operators such as, roulette wheel, stochastic sampling and tournament. We use a roulette wheel as the basic selection method to reproduce the next generation based on the current population, in which a fitter chromosome has a large chance to be reproduced into the next generation. In this selection method, solutions with short travel times have higher probabilities of being chosen for the next generation. The roulette wheel is performed as follows 1. Calculate the total time traveled by the S/R machine f(k) for each chromosome k (k = 1, 2, …, pop_size). 2. Calculate the fitness value eval(k) for each chromosome k (k = 1, 2, …, pop_size) 3. Find the total fitness of the population, F. F¼

pop Xsize

evalðkÞ;

ð10:38Þ

k¼1

4. Calculate the probability of a selection pk for each chromosome k (k = 1, 2, …, pop_size). pk ¼ evalðkÞ=F;

ð10:39Þ

5. Calculate the cumulative probability qk for each chromosome k (k = 1, 2, …, pop_size). k X pi ; ð10:40Þ qk ¼ i¼1

The selection process is based on spinning the roulette wheel for pop_size times; each time a single chromosome is selected for the new population in the following way. (i) (ii)

Generate a real random number g in range [0, 1]; If g  q1 ; then select the first chromosome; otherwise select the kth chromosome ð2  k  pop sizeÞ such that qk1 \g  qk :

All chromosomes of the last population are then replaced by the newly generated chromosomes.

10.5.5 Crossover Operator Crossover operator enables the algorithm to extract the best genes from different chromosomes and recombine them into potentially superior children. The idea behind crossover is that the new chromosome may be better than both of its parents if it takes the best characteristics from them. The crossover process utilized here generates a new offspring from two parent chromosomes. Among the described operators for permutation problems, the Partially Matched Crossover (PMX) is used for the order picking problem. PMX is viewed

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Fig. 10.4 An example of the PMX operator

Fig. 10.5 An example of the PMX operator with six items

p1 :

D M E G 6 2 3 4

p2 :

G 1

E M D 5 5 4

o1 :

D 6

E M G 5 5 4

o2 :

G M E D 1 2 3 4

p3 :

M S P R A H 2 3 8 7 5 6

p4 :

R P 5 2

o3 :

R A H 5 3 4

o4 :

M H P R A S 2 6 8 7 5 3

H M S A 4 6 1 3 M S 6 1

P 2

as a crossover of permutations, which guarantees that all items are found exactly once in each offspring. That is, both offspring receive a full complement of genes, followed by the corresponding filling in of alleles from their parents. Figures 10.4 and 10.5 illustrate the crossover operator used in this research. In Fig. 10.4, where four items D, M, E and G are requested, two parents are denoted by p1 and p2, and the crossover points are 1 and 2. According to the corresponding between items M and E, the repeated items are replaced. That is, M in p1 is replaced by E of p2, and E in p2 is replaced by M of p1. The generated offsprings are o1 and o2. Figure 10.5 shows another example of PMX operator. In this example, as depicted in the figure, the crossover points are 2 and 5. The genes of children o3 and o4 between crossover points are derived from the genes between those crossover points of the parents p4 and p3, respectively. After that, according to the corresponding between items P-R-A and H-M-S, the repeated items are replaced. It should be noted that, according to PMX operator in the order picking problem, the role of crossover operator is to change the sequence of the items in a chromosome, without changing the associated alleles.

10.5.6 Mutation Mutation is the process to alter one or more gene values of a chromosome from its initial state and the purpose is to add new gene values to the gene pool. This mutation provides genetic diversity and enables the genetic algorithm to search a

250 Fig. 10.6 The mutation operator

Y. Khojasteh-Ghamari o3 :

R A H M S P 5 3 4 6 1 2

o3 :

R A H M S P 5 3 4 3 1 2

boarder space. In fact, it is an important operator that prevents the population from stagnating at local optimal. Mutation occurs during evolution according to a mutation probability. Contrary to binary implementation that each gene is replaced with a complementary amount (0 with 1 and vice versa), in the order picking problem, the associated allele of each gene that has been selected by a mutation operator can be replaced with another allele in the range of total inventory of the item. This operator, on the other hand, does not have any role in changing the sequence of items, but can only select another number (storage location) for an item. Figure 10.6 illustrates the mutation operation used in this paper. Suppose that in o3, the fourth gene has been selected by mutation. Also, suppose that the total number of item M in various storage locations within the warehouse is seven. Therefore, the mutation operator generates an integer random number between 1 and 7 to replace the fourth gene. Of course, when the generated number is equal to the current number (namely 6), the operator repeats generating the random numbers until obtaining a number except 6. In this example, the number 3 has been randomly generated.

10.6 Numerical Experiments To compare the performance of the presented algorithms, we constructed a set of 36 different cases of physical specifications of a warehouse with five different types of orders. Each case was first solved by the enumeration algorithm to obtain the optimal travel time, which is the minimum travel time of the S/R machine, as well as the required CPU time. Then, each case was solved by the other two algorithms.

10.6.1 Simulation Models In designing the 36 different cases, three main warehouse parameters, warehouse capacity, warehouse density and shape factor were used. Warehouse capacity is proportional to the number of aisles in the warehouse. We considered four cases for warehouse capacity; a warehouse with one, two, three and four aisles. Each storage rack contains 780 storage bins. Since each aisle contains two racks, the

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capacity of each aisle is 1,560 storage bins. The main reason not to consider a warehouse with five or more aisles, is the assumption that there is only one S/R machine serving all aisles. Assigning a machine to a warehouse with a high number of aisles will decrease the practical efficiency of the system. As warehouse density, we considered some cases in which 60, 75 and 90% of total capacity of the warehouse is used. The configuration of rack or shape factor denoted by b, as described in Bozer and White (1984) is the time ratio of length and height of the rack, supposing that rack capacity and both horizontal and vertical velocity of the S/R machine are constant. Same as Han et al. (1987) we used three values of 0.6, 0.73 and 1 for the shape factor. In addition, in each aforementioned case, we considered five types of orders so that, in the orders type 1, 2, 3, 4 and 5, the number of requested items to retrieve is one, two, three, four and five items, respectively. In the proposed genetic algorithm, the crossover and mutation probabilities of 0.4 and 0.05 are used, respectively. Also, population size is determined based on the types of orders. In the samples, where the order includes one or two items, the population size is 20. However, in the other samples, where the number of requested items is three or more, the population size at each generation remains at 50. The genetic evolution stops after 200 generations.

10.6.2 Numerical Results and Discussion Numerical results are given in Tables 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, 10.8, and 10.9. Tables 10.1, 10.2, 10.3 and 10.4 show the travel time of the S/R machine, and the CPU time of the three algorithms, when the order includes two, three, four and five items, respectively. Each table shows the results for a set of 36 randomly sampled cases. In the all tables, the enumeration, the CAH, and the genetic algorithm are denoted by ‘Enumeration’, ‘CAH’ and ‘GA’, respectively. Also, all the times are in seconds unless specified otherwise. In Table 10.4, some travel times in the enumeration algorithm are missing. This is due to having a large number of feasible solutions that required a long CPU time that failed us to find those travel times. In the first case, where the order includes only one item, 75% of the solutions obtained by the genetic algorithm are optimal. In the other cases, where the order includes two, three, four and five items, the corresponding values are 67, 75, 69 and at least 65%, respectively. Most of the solutions generated by the genetic algorithm are optimal and in the remaining, the differences between the suboptimal and optimal solutions are very slight. However, in the CAH algorithm, optimal solutions are found in only very rare cases and the differences between the sub-optimal and optimal are relatively large. Increasing the number of items in an order affects the performance of these algorithms. In order to obtain the optimal solution, CPU time of the enumeration

252

Y. Khojasteh-Ghamari

Table 10.1 Performance of the three algorithms when the Sample Number of Shape Density Enumeration no aisles factor Travel- CPU time time

order includes two items CAH GA Travel- CPU time time

Travel- CPU time time

2 7 12 17 22 27 32 37 42 47 52 57

1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

43.6 29.9 41.4 75.7 28.1 27.3 40.2 28.1 27.2 40.4 29.0 92.7

0.09 0.08 0.03 0.08 0.08 0.07 0.14 0.14 0.13 0.19 0.20 0.24

76.8 29.9 45.2 75.7 45.9 64.7 40.2 45.9 63.6 53.0 53.4 104.0

0.03 0.03 0.08 0.13 0.09 0.08 0.07 0.15 0.14 0.19 0.14 0.19

43.6 29.9 41.4 75.7 33.7 34.3 40.2 33.7 27.2 44.3 29.0 92.7

1.72 1.66 1.69 1.66 1.70 1.70 1.65 1.69 1.64 1.64 1.77 1.71

62 67 72 77 82 87 92 97 102 107 112 117

1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

58.8 29.7 34.1 18.0 80.9 48.1 18.0 33.6 47.7 40.4 29.9 29.5

0.03 0.03 0.08 0.08 0.08 0.08 0.20 0.21 0.21 0.31 0.26 0.37

76.2 29.7 45.9 42.0 112.2 66.0 42.0 70.4 65.5 56.9 33.8 31.5

0.03 0.03 0.09 0.09 0.08 0.09 0.13 0.20 0.08 0.13 0.13 0.13

58.8 29.7 34.1 18.0 80.9 48.1 18.0 33.6 69.3 40.4 30.9 46.8

1.59 1.69 1.52 1.48 1.52 1.48 1.58 1.58 1.59 1.61 1.55 1.60

122 127 132 137 142 147 152 157 162 167 172 177

1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

44.9 29.5 42.7 32.5 26.0 105.7 33.8 25.2 36.7 29.7 34.6 32.5

0.08 0.03 0.08 0.08 0.13 0.14 0.25 0.25 0.30 0.48 0.42 0.42

75.7 29.5 43.7 92.0 41.6 113.6 96.3 40.2 63.2 31.7 47.7 36.2

0.08 0.09 0.08 0.13 0.08 0.09 0.13 0.14 0.13 0.20 0.14 0.21

44.9 29.5 42.7 36.2 26.0 105.7 33.8 32.7 36.7 29.7 38.4 32.5

1.55 1.49 1.52 1.57 1.61 1.61 1.67 1.70 1.66 1.75 1.76 1.72

algorithm is increased dramatically. In the CAH algorithm, the probability of existence of some or all the ordered items in the current aisle is greater. Therefore, the CPU time of this algorithm is decreased as the number of items in the order increases. However, compared to the other two algorithms, the CAH algorithm generated solutions with the largest travel times of the S/R machine. In the genetic

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Table 10.2 Performance of the three algorithms when the Sample Number of Shape Density Enumeration no aisles factor Travel- CPU time time

order includes three items CAH GA Travel- CPU time time

Travel- CPU time time

3 8 13 18 23 28 33 38 43 48 53 58

1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

55.7 50.3 27.3 46.8 43.8 58.7 67.0 42.9 58.7 51.6 47.4 101.0

0.09 0.03 0.03 0.37 0.38 0.36 1.64 1.55 1.64 3.56 3.68 4.98

96.1 102.7 91.8 46.8 51.1 58.7 70.7 50.8 58.7 91.0 84.5 112.50

0.08 0.07 0.08 0.08 0.08 0.08 0.14 0.14 0.14 0.20 0.14 0.20

55.7 50.3 27.3 46.8 43.2 58.7 76.4 46.9 58.7 51.6 60.4 101.4

2.17 2.26 2.24 2.33 2.33 2.30 2.45 2.44 2.45 2.43 2.50 2.60

63 68 73 78 83 88 93 98 103 108 113 118

1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

85.6 68.5 52.2 57.8 36.7 60.2 57.4 37.5 104.0 52.2 36.5 54.8

0.08 0.08 0.08 0.79 0.82 0.82 3.95 3.71 7.19 8.48 8.47 8.78

113.2 107.9 86.5 99.5 67.0 84.6 98.7 68.5 138.1 113.2 44.0 80.8

0.03 0.07 0.08 0.08 0.13 0.08 0.08 0.13 0.20 0.14 0.14 0.20

85.6 68.5 52.2 57.8 36.7 60.2 57.4 37.5 104.0 52.2 36.5 56.7

2.32 2.24 2.23 2.44 2.33 2.32 2.52 2.38 2.61 2.50 2.53 2.57

123 128 133 138 143 148 153 158 163 168 173 178

1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

27.1 45.6 33.1 40.2 57.1 49.5 40.8 54.3 100.9 35.5 50.1 43.9

0.13 0.14 0.13 1.32 1.32 1.55 7.58 7.57 10.70 18.12 18.13 19.70

106.8 84.6 66.5 51.5 70.0 53.4 52.2 66.5 124.9 64.6 71.4 61.6

0.03 0.08 0.09 0.08 0.08 0.09 0.14 0.08 0.13 0.13 0.15 0.14

27.1 45.6 33.1 40.2 57.1 49.5 40.8 54.3 114.3 35.5 52.0 43.9

2.21 2.28 2.26 2.36 2.33 2.36 2.56 2.56 2.59 2.50 2.61 2.65

algorithm, on average 70.2% of the solutions are optimal, and the differences between the sub-optimal and the optimal solutions are very slight. These results in terms of the number of aisles are given in Tables 10.5, 10.6, 10.7 and 10.8. These tables show the performance of the three algorithms in four different kinds of warehouses. Tables 10.5, 10.6, 10.7 and 10.8 show the average

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Table 10.3 Performance of the three algorithms when the Sample Number of Shape Density Enumeration no aisles factor Travel- CPU time time 4 9 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164 169 174 179

1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

79.7 72.2 49.8 94.6 61.7 82.4 78.2 65.6 46.5 75.8 66.5 105.0 58.2 63.2 59.9 76.1 58.1 56.4 78.3 58.0 49.5 75.1 60.6 47.4 60.8 63.6 71.7 68.0 67.5 63.7 56.2 70.9 75.5 55.7 70.4 83.3

0.08 0.03 0.09 2.01 1.37 1.38 14.82 12.66 15.05 86 80 141 0.20 0.21 0.19 7.31 7.99 7.36 62 63 97 306 278 294 0.81 0.70 0.85 23 22 28 164 174 260 973 960 1,187

order includes four items CAH GA Travel- CPU time time

Travel- CPU time time

100.2 113.2 68.5 141.4 88.8 118.0 125.6 134.3 126.6 121.8 136.4 146.2 140.3 100.6 87.1 120.8 113.6 92.0 97.2 130.2 120.7 101.0 136.4 115.2 105.8 117.8 81.1 81.4 99.9 86.1 68.3 113.0 110.4 67.8 110.1 103.0

79.7 72.2 49.8 96.4 61.7 82.4 78.4 65.6 46.5 81.5 66.5 112.3 58.2 63.1 59.9 78.9 58.1 56.4 78.3 58.0 49.8 81.3 60.6 47.4 60.8 63.6 71.7 75.7 67.5 63.7 59.9 73.9 75.5 60.4 75.3 83.3

0.08 0.07 0.03 0.08 0.08 0.08 0.13 0.15 0.04 0.14 0.14 0.26 0.03 0.08 0.07 0.09 0.14 0.08 0.14 0.14 0.14 0.14 0.20 0.13 0.08 0.08 0.08 0.08 0.08 0.13 0.13 0.14 0.13 0.14 0.13 0.14

3.02 3.04 3.08 3.22 3.25 3.05 3.22 3.31 3.35 3.68 3.52 3.72 3.09 3.20 3.00 3.26 3.32 3.22 3.50 3.57 3.60 3.85 3.75 3.67 3.05 3.11 3.17 3.49 3.39 3.37 3.63 3.70 3.61 3.98 3.83 4.07

travel time of the S/R machine as well as the average CPU time of the three algorithms when the warehouse has one, two, three and four aisles, respectively. In each table and for each kind of warehouse, 9 sample sets are considered, which are combination of the two assumed warehouse parameters, the warehouse density,

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Table 10.4 Performance of the three algorithms when the Sample Number of Shape Density Enumeration no aisles factor Travel- CPU time time 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180

1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4

1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60 1.00 0.73 0.60

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

100.8 96.6 74.7 128.5 78.4 98.1 132.7 78.4 102.7 107.2 89.5 – 75.8 79.9 81.7 86.8 73.4 75.7 89.5 75.1 74.4 – – – 81.4 82.0 98.0 82.9 74.5 78.3 – – – – – –

0.14 0.14 0.14 30.59 21.77 20.97 403 347 397 7,369 6,820 – 1.00 1.01 0.99 140 160 138 5,382 4,805 8,240 – – – 6.99 6.18 7.20 509 508 837 – – – – – –

order includes five items CAH GA Travel- CPU time time

Travel- CPU time time

128.4 138.2 97.8 195.5 121.7 137.8 197.0 121.7 144.4 168.2 162.7 135.6 167.6 123.6 118.0 131.2 125.4 111.2 135.4 128.4 110.8 100.8 129.1 117.3 130.4 144.0 117.7 117.7 125.6 101.0 83.5 99.2 132.5 89.8 103.1 102.4

100.8 96.6 74.7 130.5 78.4 98.1 136.5 83.3 102.7 108.2 104.4 93.9 75.8 79.9 81.7 86.8 73.4 75.7 92.4 75.1 91.1 89.8 90.7 56.7 81.4 82.0 100.0 82.9 84.0 78.3 85.0 93.1 104.9 78.7 99.2 78.0

0.03 0.08 0.08 0.14 0.08 0.08 0.13 0.15 0.13 0.21 0.14 0.16 0.09 0.08 0.08 0.08 0.09 0.08 0.13 0.20 0.14 0.11 0.11 0.16 0.03 0.08 0.03 0.08 0.08 0.08 0.10 0.10 0.37 0.10 0.17 0.10

3.50 3.53 3.39 3.78 3.78 3.71 3.94 4.09 4.00 4.44 4.31 4.21 3.55 3.60 3.51 3.81 3.81 3.77 4.16 4.31 4.40 4.45 4.44 4.39 3.59 3.56 3.69 3.89 4.07 3.95 4.17 4.23 4.28 4.71 4.72 4.71

and the shape factor. The values in each sample set represent the average values of the five order types. As it can be inferred from the Tables 10.5, 10.6, 10.7 and 10.8, in all kinds of warehouses the CAH algorithm attains solutions with the largest travel time of the S/R machine, but with a less CPU time than the other two algorithms. In other

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Table 10.5 Performance of the three algorithms in a warehouse with one aisle Samples set Enumeration CAH GA 1*5 6 * 10 11 * 15 61 * 65 66 * 70 71 * 75 121 * 125 126 * 130 131 * 135

Travel-time

CPU-time

Travel-time

CPU time

Travel-time

CPU time

61.5 51.9 43.8 62.9 50.4 48.5 47.1 46.2 54.0

0.10 0.07 0.07 0.27 0.27 0.27 1.62 1.42 1.67

88.5 78.9 65.8 107.9 74.5 72.8 91.7 77.2 66.9

0.06 0.06 0.07 0.05 0.06 0.08 0.05 0.07 0.08

61.5 51.9 43.8 62.9 50.4 48.5 47.1 46.2 54.4

2.27 2.26 2.25 2.27 2.31 2.22 2.25 2.25 2.30

Table 10.6 Performance of the three algorithms in a warehouse with two aisles Samples set Enumeration CAH GA 16 * 20 21 * 25 26 * 30 76 * 80 81 * 85 86 * 90 136 * 140 141 * 145 146 * 150

Travel-time

CPU time

Travel-time

CPU time

Travel-time

CPU time

75.0 46.5 56.3 49.8 64.3 53.2 47.5 47.9 77.2

6.62 4.74 4.56 30 34 29 107 106 173

97.7 68.2 83.5 84.2 98.2 78.1 77.8 73.4 88.6

0.10 0.09 0.08 0.10 0.12 0.09 0.09 0.08 0.11

75.7 47.7 57.7 53.8 64.3 54.8 53.1 51.3 77.2

2.37 2.39 2.32 2.37 2.37 2.34 2.43 2.46 2.44

Table 10.7 Performance of the three algorithms in a warehouse with three aisles Samples set Enumeration CAH GA 31 * 35 36 * 40 41 * 45 91 * 95 96 * 100 101 * 105 151 * 155 156 * 160 161 * 165

Travel-time

CPU time

Travel-time

CPU time

Travel-time

CPU time

69.5 47.2 50.0 50.7 46.0 60.3 36.3 41.2 56.8

84 72 83 1,090 974 1,669 43 45 68

92.6 77.2 86.3 80.0 85.8 94.3 67.7 62.0 83.9

0.11 0.14 0.12 0.12 0.16 0.14 0.13 0.12 0.13

72.1 50.1 50.0 54.6 47.1 69.5 37.2 43.8 60.2

2.43 2.48 2.46 2.53 2.54 2.62 2.19 2.21 2.17

words, the CAH algorithm requires a lower CPU time to provide a solution to the problem. However, the travel time of the S/R machine is much larger than those of the other two algorithms.

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Table 10.8 Performance of the three algorithms in a warehouse with four aisles Samples set Enumeration CAH GA 46 * 50 51 * 55 56 * 60 106 * 110 111 * 115 116 * 120 166 * 170 171 * 175 176 * 180

Travel-time

CPU time

Travel-time

CPU time

Travel-time

CPU time

59.5 50.1 97.1 46.1 37.3 38.3 34.7 43.9 44.3

1,492 1,381 36 79 72 76 248 245 302

92.1 91.0 113.0 76.0 60.1 62.8 45.5 63.9 54.6

0.19 0.14 0.22 0.13 0.15 0.15 0.15 0.14 0.15

61.6 55.7 98.7 47.7 37.6 43.1 35.8 46.8 45.3

2.61 2.61 2.22 2.21 2.19 2.18 2.28 2.27 2.30

Table 10.9 Performance of the three algorithms with respect to the shape factor, b b Number of aisles Enumeration CAH GA

0.6

1 2 3 4 0.73 1 2 3 4 1 1 2 3 4

Traveltime

CPU time Traveltime

CPU time Traveltime

CPU time

48.74 62.27 55.61 59.92 49.48 52.92 45.07 44.26 57.19 57.44 53.29 47.75

0.67 69 644 137 0.59 48 386 628 0.66 48 431 674

0.08 0.09 0.13 0.18 0.06 0.09 0.14 0.14 0.06 0.10 0.12 0.16

2.26 2.36 2.43 2.24 2.28 2.41 2.43 2.38 2.26 2.39 2.40 2.39

68.47 83.41 88.46 76.77 76.87 79.90 75.93 73.18 96.05 86.55 80.99 72.78

48.87 63.25 59.88 62.37 49.48 54.18 47.26 47.37 57.19 60.85 55.88 49.39

When a warehouse has only one aisle, approximately 89% of the solutions obtained by the genetic algorithm are optimal. In the remaining, the average difference between the sub-optimal and optimal solutions is only 0.74% (but requires larger CPU times). When the warehouse consists of two aisles, about 22% of the solutions found by the genetic algorithm are optimal. This value is about 11% for the warehouse with three aisles. In the remaining cases, however, the average difference between the obtained solution and the optimal one is slight and equivalent to 4.76, 6.52 and 4.65% in the warehouse with two, three and four aisles, respectively. Increasing the number of aisles in a warehouse affects the performance of these algorithms. Due to the increase of the total number of feasible solutions, the CPU time of the enumeration algorithm increases dramatically in order to obtain the optimal solution. The genetic algorithm, however, requires a less CPU time than

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that of the enumeration algorithm to achieve a solution with a reasonable travel time for the S/R machine, which is optimal in most cases or quasi-optimal. Also, the performance of the algorithms is influenced by the rack configuration, b, as well. The average travel time of the S/R machine and the average CPU time with respect to the number of aisles in three shape factors 0.6, 0.73 and 1 are given in Table 10.9. This table provides comparisons between performances of the algorithms in three different rack configurations as the warehouse capacity increases. In the case of a warehouse with only one aisle, the enumeration algorithm provides the best solution in the all levels of b, with a CPU time less than that of the genetic algorithm. However, if the warehouse has more aisles, the genetic algorithm requires less CPU time than the enumeration algorithm. As the performance of the three algorithms for different values of b are approximately the same, the rack configuration in the warehouse shows essentially no effect on the performance of the algorithms.

10.7 Conclusions In this research, we addressed an order picking problem in a multi-aisle automated warehouse served by a single S/R machine, in which each item can be found in several storage locations. We first formulated the problem mathematically. Then, to solve the problem, we developed an ordinary heuristic (CAH), and a suitable genetic algorithm. In order to show the efficiency of the proposed algorithms, we compared them with the enumeration algorithm, which attains the optimal solution, but does so with a long CPU time, hence making the method unsatisfactory. The CAH algorithm requires less CPU time, however, the achieved solutions are mostly sub-optimal with dramatically longer travel times for the S/R machine. With the genetic algorithm, most of the solutions are optimal or quasi-optimal solution (approximately 4.17% difference in average). Consequently, the proposed genetic algorithm is more efficient than the other two algorithms. In the future, meta-heuristic methods, as well as branch and bound algorithms would be evaluated against various storage methods for their utility and a dual command S/R machine cycle in generating optimal solutions for order picking problems in automated warehouses.

References Amato F, Basile F, Carbone C, Chiacchio P (2005) An approach to control automated warehouse systems. Control Eng Pract 13:1223–1241 Bozer YA, White JA (1984) Travel-time models for automated storage/retrieval systems. IIE Trans 16(4):329–338

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Bozer YA, White JA (1990) Design and performance models for end-of-aisle order picking systems. Manag Sci 36(7):852–866 Cheng R, Gen M, Sasaki M (1995) Film-copy deliverer problem using genetic algorithms. Comput Ind Eng 29:549–553 De Koster R, Le-Duc T, Roodbergen KJ (2007) Design and control of warehouse order picking: a literature review. Eur J Oper Res 182:481–501 De Koster MBM, Le-Duc T, Yu Y (2008) Optimal storage rack design for a 3-dimensional compact AS/RS. Int J Prod Res 46(6):1495–1514 Elsayed EA (1981) Algorithms for optimal material handling in automatic warehousing systems. Int J Prod Res 19:525–535 Elsayed EA, Stern RG (1983) Computerized algorithms for order processing in automated warehousing systems. Int J Prod Res 21:579–586 Goetschalckx M, Wei R (1994) Bibliography on order picking systems, vol 1. pp 1985–1992. http://wwwisyegatechedu/people/faculty/Marc_Goetschalckx/researchhtml Gu J, Goetschalckx M, McGinnis LF (2007) Research on warehouse operation: a comprehensive review. Eur J Oper Res 177:1–21 Han M-H, McGinnis LF, Shieh JS, White JA (1987) On sequencing retrievals in an automated storage/retrieval system. IIE Trans 19:56–66 Hsu CM, Chen KY, Chen MC (2005) Batching orders in warehouses by minimizing travel distance with genetic algorithms. Comput Ind 56:169–178 Hwang HS, Cho GS (2006) A performance evaluation model for order picking warehouse design. Comput Ind Eng 51:335–342 Hwang H, Baek W, Lee M-K (1988) Clustering algorithms for order picking in an automated storage and retrieval system. Int J Prod Res 26:189–201 Hwang H, Moon S, Gen M (2002) An integrated model for the design of end-of-aisle order picking system and the determination of unit load sizes of AGVs. Comput Ind Eng 42: 249–258 Khojasteh-Ghamari Y, Son JD (2008) Order picking problem in a multi-aisle automated warehouse served by a single storage/retrieval machine. Int J Inf Manag Sci 19(4):651–665 Koh SG, Kim BS, Kim BN (2002) Travel time model for the warehousing system with a tower crane S/R machine. Comput Ind Eng 43:495–507 Koh SG, Kwon HM, Kim YJ (2005) An analysis of the end-of-aisle order picking system: multiaisle served by a single order picker. Int J Prod Econ 98:162–171 Lee HF, Schaefer SK (1996) Retrieval sequencing for unit-load automated storage and retrieval systems with multiple openings. Int J Prod Res 34:2943–2962 Lee HF, Schaefer SK (1997) Sequencing methods for automated storage and retrieval systems with dedicated storage. Comput Ind Eng 32:351–362 Lerher T, Potrcˇ I, Šraml M, Tollazzi T (2010) Travel time models for automated warehouses with aisle transferring storage and retrieval machine. Eur J Oper Res 205:571–583 Mahajan S, Rao BV, Peters BA (1998) A retrieval sequencing heuristic for miniload end-of-aisle automated storage/retrieval systems. Int J Prod Res 36:1715–1731 Michalewicz Z (1992) Genetic algorithms ? data structures = evolution programs. Springer, Berlin Ratliff HD, Rosenthal AS (1983) Order-picking in a rectangular warehouse: a solvable case of the traveling salesman problem. Oper Res 31:507–521 Roodbergen KJ, de Koster R (2001) Routing order pickers in a warehouse with a middle aisle. Eur J Oper Res 133:32–43 Roodbergen KJ, Vis IFA (2009) A survey of literature on automated storage and retrieval systems. Eur J Oper Res 194(2):343–362 Rouwenhorst B, Reuter B, Stockeahm V, van Houtum GJ, Mantel RJ, Zijm WHM (2000) Warehouse design and control: framework and literature review. Eur J Oper Res 122:515–533 Sivanandam SN, Deepa SN (2008) Introduction to genetic algorithms. Springer, New York Tompkins JA, White JA, Bozer YA, Tanchoco JMA (2003) Facilities planning, 3rd edn. Wiley, New York

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Van den Berg JP (1999) A literature survey on planning and control of warehousing systems. IIE Trans 31:751–762 Van den Berg JP, Gademann AJRM (1999) Optimal routing in an automated storage/retrieval system with dedicated storage. IIE Trans 31:407–415 Yu Y, De Koster MBM (2009) Designing an optimal turnover-based storage rack for a 3D compact AS/RS. Int J Prod Res 47(6):1551–1571

Chapter 11

Analytical and Numerical Modeling of AS/RS Cycle Time in Class-Based Storage Warehousing Mauro Gamberi, Riccardo Manzini and Alberto Regattieri

Abstract This work presents an analytical model for the computation of travel time for automated storage and retrieval systems (AS/RS) with a three (ABC) class based storage in a rectangular in time warehouse. In particular, the authors provide a method for the analytical closed form evaluation of the expected mean travel time for the single/dual command cycles in the configuration with the input/output located in the bottom/left corner of the warehouse and varying the ABC curve. A numerical simulation analysis adopting a numerical modeling has been developed, in order to validate the proposed model accordingly with a multi scenario analysis. The performance of the system obtained by the adoption of the proposed analytical travel time models under different configurations of the warehousing system (shape and dimension of the classes, ABC curve), have been evaluated and discussed.

11.1 Introduction This chapter deals with the prediction of the travel time of Automated Storage and Retrieval Systems (AS/RS): the aim is to calculate the time spent for the pick up/drop off operations on unit loads (ULs). The considered AS/RS has aisle captive cranes: Fig. 11.1 depicts this system with the crane able to move horizontally and vertically from/to the ULs and the input/output zone is in front M. Gamberi Department of Management and Engineering—DTG, Padova University, Padova, Italy R. Manzini (&)  A. Regattieri Department of Industrial Mechanical Plants—DIEM, Bologna University, Bologna, Italy e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_11,  Springer-Verlag London Limited 2012

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Vertical dimension of racks Horizontal dimension of racks

Input/Output zone

ULs Racks Rails Crane moving UL

Fig. 11.1 Scheme of an aisle captive crane AS/RS with the input/output point in front of the aisle (source: Daifuku Co. Ltd)

of the aisle. This study wants to propose a model to quantify the cycle time spent for the storage/retrieval operations for each aisle. The operations outside the aisle are not considered in this discussion. This model is designed for ABC class-based rectangular in time AS/RS and provides an analytical solution for evaluating the throughput performance which is itself based on determining analytical expressions for the single and dual command travel times. An analytical closed form of the expected mean travel time for the single and dual command cycles is provided with the input/output point located in the bottom/left corner of the warehouse and varying the ABC curve. The AS/RS warehouse is managed with a (ABC) class-based storage allocation strategy. In the literature, several studies (see the next section for literature details) evaluate the single command (SC) cycle time under uniform distribution storage and class-based storage (CBS) for square in time (SIT) and rectangular in time (RIT) racks. In addition, the Dual Command (DC) cycle time has been investigated at great length, regarding uniform distribution of products within class-based storage area with SIT racks, while the problem is more complicated if a CBS warehouse with a RIT shape of racks is considered. In addition to the analytical predicting method of AS/RS cycle time, an innovative numerical simulation approach is developed to evaluate AS/RS. This approach is based on the same

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theoretical background developed for the analytical model, proposing an effective and robust method to numerically calculate the mean SC and DC travel time of any class-based warehouse configuration. This Chapter ends with a practical example of the presented methods, with both the analytical and the numerical analysis evaluating their convergence in different warehouse configurations (shape factor, dimension of the classes, and ABC curve).

11.2 Literature Review The issue of AS/RS has received a great deal of attention during the last two decades. Several researchers developed analytical models for evaluating the travel time for the pick up and drop off activities. Among the most promising policies for operating an automated warehouse is the CBS policy assuming ABC inventory profile and a basic Economic Order Quantity (EOQ) policy. Hausman et al. (1976) derived an expression for the expected SC travel time for uniform storage, turnover-based storage and class-based storage. Graves et al. (1977) developed an expression for the expected DC travel time for uniform storage, turnover-based storage and 2- and 3-CBS. Schwarz et al. (1978) investigated the effects of various assumptions on AS/RS behavior through simulation. For warehouses with uniform turnover, Bozer and White (1984) proposed an analytical model for the expected single and dual command travel time. Rosenblatt and Eynan (1989) developed an expression for the expected SC travel time in SIT racks and multi-class storage, and in 1994 the same authors extended their previous work to rectangular racks (Eynan and Rosenblatt (1994)). A framework for obtaining analytic expressions for the expected throughput rate in automated storage systems was presented by Kim and Seidmann (1990) that developed expressions for the expected SC travel time for turnover-based storage and multi-class storage. Following this, Kouvelis and Papanicolaou (1995) proposed an approach for 2-CBS in a square rack analyzing boundary formulas that minimize the expected SC or DC travel time and, a year later, Pan and Wang (1996) proposed a framework for the dual command cycle continuous travel time model under class-based assignment. Hwang and Lee (1990) and Chang et al. (1995) developed travel time models that considered acceleration/deceleration effects of the S/R machine. Chang and Wen (1997) investigated the impact of speed profiles on rack configuration and found that the expected travel times are insensitive to the slight deviation in the optimal rack configuration. Lee (1997) performed a stochastic analysis of an AS/RS unit load using a single-server queuing model, while Thonemann and Brandeau (1998) studied storage policies under stochastic demands. Assuming uniform turnover distribution over a SIT rack, Han et al. (1987) developed an approximate expression for the expected DC travel time under a nearest neighbor retrieval sequencing heuristic. After this, Eben-Chaime (1992);

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Eynan and Rosenblatt (1993); Lee and Schaefer (1996, 1997); Mahajan et al. (1998) studied the AS/RS sequencing problem and developed heuristic algorithms that solved the unit load and the order picking problem. Foley and Frazelle (1991) studied the DC travel time with uniformly distributed turnover and squared racks, while Park et al. (1999a, b) extended previous work to not squared racks with uniformly distributed and 2-CBS. Bozer and White (1990) approximated the DC travel time distribution by the uniform distribution. Van den Berg and Zijm (1999) discussed models for warehouse management presenting a classification of them and explained possible approaches to the main warehouse management problems. Van den Berg (2002) investigated and proposed an analytic expression for the optimal dwell point positioning in an AS/RS. Park et al. (2003) investigated the DC travel times computing the AS/RS mean cycle time, the variance and its total throughput in turnover-based storage; moreover, Park (2006) presented a analysis of RIT racks with two-class storage. Lee et al. (2005) discussed the optimal design of rack structure with modular cells in AS/RS. Manzini et al. (2006) also presented an analysis of the critical parameters involved in the design of AS/RS for order picking systems managed by CBS policy (Manzini et al. 2007). Another important point of view of the AS/RS analysis is the numerical simulation of this systems: Regattieri et al. (2008) present a numerical simulation of the AS/RS systems highlighting the versatility of the Monte Carlo approach to measure the SC and DC cycle time; Mohammadreza et al. (2008) presented an analytical statistical model for computing expected cycle time of AS/RS (SP-AS/ RS) split-platform being the accuracy of the proposed statistical model validated by Monte Carlo simulation.

11.3 System Definition and Assumptions This study presents a method to predict the cycle time of AS/RSs with aisle captive cranes: each aisle has an its own crane that is aimed to pick up/drop off unit loads from/to the input/output positioned at one end of the aisle (see Fig. 11.1). For brevity, in agreement with Eyan and Rosenblatt (1993); Kouvelis and Papanicolaou (1995); Park et al. (2003, 2006), from here onward the assumptions adopted in this approach are schematically summarized as follows: 1. RIT racks of normalized dimensions [1, b]; 2. Input/Output (I/O) point in the lower left corner of the aisle, origin of axis; 3. Products divided into three ABC classes according to their demand distribution, with a random allocation within each area; 4. Products with the same dimensions and continuously allocated into the rack (columns and rows are not considered); 5. ABC classes located in SIT contours (if possible);

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Fig. 11.2 Scheme definition of the AS/RS rack

265

z b

C(c 3 )

B(c 2 )

A(c 1) x

I/O R1

R2

b

1

6. Independent, identically distributed (i.i.d.) storage and retrieval locations and random access to the available locations; 7. Product inventory replenishment according to the EOQ model with demand curve assigned and constant in time; 8. Products ranked according to their own demand; 9. AS/RS cranes having the capacity of one unit load and operating in each aisle in SC and DC mode; 10. Constant times (pick up and drop off times); acceleration and deceleration during the cycles not considered. According to Fig. 11.2, let: th = one-way travel time from I/O to the highest vertical location; tL = one-way travel time from I/O to the highest horizontal location; b = shape factor of the rack, with:   tL t h b ¼ min ; ; b 2 ð0; 1 th tL

ð11:1Þ

R1, R2 = ABC class boundary limits with 0 \ R1 \ R2\ 1; t(x, y) = the Tchebychev distance between the I/O point and the random S/R location (x, y), with: tðx; yÞ ¼ maxðx; yÞ

ð11:2Þ

f(t) = mean access frequency function in terms of demand per unit time, per unit area of racks, whose distance from I/O is t unit time, with1:

1 In this work the three-classes (ABC) problem is considered but the proposed approach can be easily extended to the n-classes problem. It is sufficient to define Eq. 11.3 f(t)={c1, c2, c3,. . .cn} with boundary limits R1, R2, R3,. . . . . ., Rn-1.

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8 < c1 j0\t  R1 fðtÞ ¼ c2 jR1 \t  R2 : c3 jR2 \t  1

ð11:3Þ

where ci is the constant access frequency of class ith. g(t) = probability density function of t(x,y) that represents the probability density of having a generic random location (x, y) at t time unit distance from the I/O point (see Kouvelis and Papanicolaou 1995), where:  8 < 2tf ðtÞ 0\t  b A gðtÞ ¼ bf ðtÞ ð11:4Þ : b\t  1 A  A = normalizing factor of the probability density g(t) that represents the total demand of the whole rack, with: Zb

Z1

0

b

A ¼ 2 t  f ðtÞdt þ b f ðtÞdt

ð11:5Þ

11.4 Class-Based Allocation Strategy 11.4.1 Demand Distribution Theory The ABC analysis ranks all items in an inventory according to their contribution to the total demand. For a given application, it is always possible to estimate the distribution of pallet turnover and to numerically quantify it. For this application, a well-known approach suggested by Hausmann et al. (1976) will be adopted and, in the following, a complete summarization of his definitions and results are briefly reported (see Hausmann et al. (1976); Rosenblatt and Eynan (1989); Eynan and Rosenblatt (1993, 1994); Yu and De Koster (2008)): GðiÞ ¼ is

ð11:6Þ

with: G(i) = cumulative demand until the ith product in unit load per unit time; i = product item in the ith percentile, ranking the products in a decreasing demand sequence; s = skew parameter with s [ (0,1] Now, introducing a demand rate of product (i), D(i), the total annual demand for all product is: Z1

N ¼ Dð xÞdx 0

ð11:7Þ

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267

Then, a normalized demand rate D*(i), with D*(i) [ [0,1] is introduced: D  ði Þ ¼

DðiÞ N

ð11:8Þ

With these assumptions, the definition of the cumulative demand G(i) leads to: Zi

GðiÞ ¼ is ¼ D ð xÞdx

ð11:9Þ

0

Consequently, Eq. 11.7 leads to: D ðiÞ ¼

dGðiÞ ¼ s is1 di

ð11:10Þ

Thus, introducing the batch quantity, Q(i) in terms of unit loads, for assumption 7, using EOQ standard model the link between D(i) and Q*(i) can be quantified in Q(i) = (2KND*(i))0.5, where K is the ratio of order cost to holding cost, and assumes the same for all products. Given Q(i), the average inventory of item (i), in unit loads, is Q(i)/2 = (2KND*(i))0.5/2. The total inventory area of all products is: L¼

Z1

pffiffiffiffiffiffiffiffiffiffiffi 2KNs a  Qð xÞ dx ¼ a  sþ1

ð11:11Þ

x¼0

where a is the area occupied by each product into the rack. L represents the total rack area; consequently, it is also possible to explain: L¼b

ð11:11 bisÞ

Now, introducing the index j(i) representing the cumulative area required by all the products until the ith, it can be seen that: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Z1 Z1 2KNDð xÞ 2KNs sþ1 a  Qð xÞdx ¼ a ð11:12Þ dx ¼ a  i 2 jðiÞ ¼ 2 sþ1 x¼0

x¼0

that is, isolating i: i¼

ðs þ 1Þ2 j2 pffiffiffiffiffiffiffiffiffiffiffi a 2KNs

1 !sþ1

ð11:13Þ

for j [ [0,L]. In order to simplify the analysis and to generalize the results, it could be useful to rescale normalizing j from [0, L] to [0, 1]. So j is redefined as: jðs þ 1Þ j j ¼ j=L ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ a 2KNs b

ð11:14Þ

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% of cumulative product demand G

1 0,9 0,8

20/90

0,7

20/80

0,6

20/70

0,5

20/60

0,4 0,3 0,2 0,1 0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

% of storage area j* Fig. 11.3 Typical ABC curves: 20/60, s0 = 0.318; 20/70, s0 = 0.222; 20/80, s0 = 0.139; 20/90, s0 = 0.065

with j* [ [0,1] and substituting j with j* in Eq. 11.13 using Eq. 11.14, then: i¼

ðs þ 1Þ2 ðj  LÞ2 pffiffiffiffiffiffiffiffiffiffiffiffi a 2KNs

1 !sþ1

¼ jðsþ1Þ 2

ð11:15Þ

As a result, using Eqs. 11.15 in 11.6, the cumulative demand curve G becomes a function of the storage area j*: 2s 0 Gðj Þ ¼ jðsþ1Þ ¼ js

ð11:16Þ

Equation (11.16), depicted in Fig. 11.3 for some values of the s0 , links the cumulative demand to the first j*th percentile of storage area. The exponent s0 = 2s/(s ? 1) has been introduced to easily manage the equation.

11.4.2 ABC Class Demand Frequency Correlated to the Demand Distribution In the previous paragraph the turnover distribution has been introduced. Now it is necessary to examine the connection between the demand frequency of each ABC class and the turnover distribution. First of all, three cases have to be analyzed,

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Fig. 11.4 The three cases of ABC zones

269 (R 1, R 2)b

b c2

c3

c1

x

I/O b

R1

R2

1

being each case correlated to the relative position of the class boundary limits R1 and R2 compared to the shape factor b. Figure 11.4 depicts these three situations that can occur. For each case, the class demand frequency ci (average unit load demand per unit time of ith ABC class, see Eq. 11.3) can be computed by the ratio between the demand distribution of class i and its area:     G ji  G ji1 ci ¼ ð11:17Þ ji  ji1

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In Eq. 11.17, for i = 1, j*0 has to be considered equal to zero. So, by means of Eq. 11.17, the required ci access frequency can be correlated to the warehouse parameter b, R1 and R2: R21 R2 ; j2 ¼ 2 ; j3 ¼ 1 b b

 2s  2s 

 2s  2 sþ1 R22 sþ1 R21 sþ1 R2 b  b b 1 b b  2    c2 ¼ ¼ ; c 3 R2  R21 b  R22 j1 ¼

b

2s  2 sþ1

R1 b R21

c1 ¼

;

ð11:18Þ

with (R1, R2) \ b; R21 ; j2 ¼ R2 ; j3 ¼ 1 b

 2s  2 sþ1 2s R1 sþ1 b R2  b   c2 ¼ ; c3 ¼ bR2  R21 j1 ¼

b c1 ¼

2s  2 sþ1

R1 b R21

;

2s

1  Rsþ1 2 1  R2

! ð11:19Þ

with R1 \ b, R2 [ b; j1 ¼ R1 ; j2 ¼ R2 ; j3 ¼ 1 2s

Rsþ1 c1 ¼ 1 ; R1

c2 ¼

2s 2s ! R2sþ1  Rsþ1 1 ; R 2  R1

2s

c3 ¼

1  R2sþ1 1  R2

! ð11:20Þ

with (R1, R2) [ b. By means of Eqs. 11.18–11.20 it is possible to rearrange Eq. 11.3 in terms of R1 and R2 only. This is due to the relation between the area of each ABC class (it depends on the position of R1 and R2) and its access probability: Eq. 11.17 summarizes the connection of the ABC class area and the cumulative demand.

11.5 Single Command Cycle Time Using the Eq. 11.4, the expected single command cycle time, Esc, can be evaluated as: ESC ¼ 2  E½tðx; yÞ ¼ 2 

Z1

t  gðtÞdt;

ð11:21Þ

0

where the constant ‘‘2’’ is due to the two-way trip having a complete cycle and E[t(x, y)] represents the expected mean value of t(x, y).

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271

To compute Esc, Eq. 11.21 has to be combined with Eqs. 11.5 and 11.18–11.20. Omitting the mathematical details and considering the three cases introduced in Sect. 11.4.2, Esc can be computed: 0R 1 Z1 ZR1 Zb Z1 @ A A1 ¼ 2 t  c1 dt þ t  c2 dt þ t  c3 dt þ b c3 dt ð11:22Þ R1

0

R2

b

¼ c1 R21 þ c2 R22  c2 R21 þ bc3  c3 R22 RZ 1

ESC1 ¼ 2

0

2 RZ 2 2t 2  c 2t 2  c1 Zb 2t  c3 Z1 t  c3 2 dt þ dt þ dt þ b dt A1 A1 A1 A1 R2 R1 b

! ð11:23Þ

1 4ðc1  c2 Þ  R31 þ 4ðc2  c3 Þ  R32 þ c3  b3 þ 3b  c3 ¼  A1 3 with (R1, R2) \ b; A2 ¼ 2

RZ 1

t  c1 dt þ

¼ ESC2 ¼ 2

RZ 1 0

! t  c2 dt

þb

R1

0

c1 R21

Zb



c2 R21

RZ 2

c2 dt þ

Z1

! c3 dt

ð11:24Þ

R2

b

þ c2 b R 2 þ b c 3  b c 3 R 2

2 RZ 2 b  t  c 2t2  c1 Zb 2t  c2 Z1 b  t  c3 2 dt þ dt þ dt þ dt A2 A2 A2 A2 R1 R2 b

! ð11:25Þ

1 4ðc1  c2 Þ  R31 þ c2  b3 þ 3b  ðc2  c3 Þ  R22 þ 3b  c3 ¼  A2 3 with R1 \ b, R2 [ b; Zb

RZ 1

0

b

A3 ¼ 2 t  c1 dt þ b

c1 dt þ

RZ 2 R1

c2 dt þ

Z1

! c3 dt

R2

  ¼ b c1 R1 þ c2 ðR2  R1 Þþ c3  c3 R2 ESC3

! RZ 1 b  t  c RZ 1 b  t  c 2t2  c1 Z1 b  t  c3 1 2 dt þ dt þ dt þ dt ¼2 A3 A3 A3 A3 R2 R2 0 b   1 bðc1  b2 þ 3  c1  R21 þ 3  c2  R22  R21 þ 3c3 ð1  R22 ÞÞ ¼  3 A3

ð11:26Þ

Zb

ð11:27Þ

with (R1, R2) [ b. Applying Eqs. 11.23, 11.25 and 11.27 the exact Esc value can be easily determined for any AS/RS, operating with a 3-CBS product assignment. It can be highlighted that ESC1 and ESC2 become the same as the expressions proposed by Eynan and Rosenblatt (1994) and Kouvelis and Papanicolaou (1995)

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if R2 = R1 and c3 = c2, obtaining the SC cycle time in a 2-CBS for the case R \ b; Eq. 29 is equal to the models proposed by the same authors for the case R [ b when R2 = R1 and c3 = c2. Moreover, the previously computed ESC1, ESC2 and ESC3 in the case of s = 1 (equal access probability of all products or random allocation) are equal to the Bozer–White (1984) equation (1 ? b2/3).

11.6 Dual Command Cycle Time 11.6.1 Travel Between Computation The evaluation of the DC cycle time is significantly more complex than the SC cycle time because of the need for the so-called Travel Between (TB) time, defined as the time between the first drop off point and the next pick up point. Let (x1, y1) and (x2, y2) be two random points in the rack and t1 = t(x1, y1), and t2 = t(x2, y2) their distance from the I/O point. t1 and t2 are i.i.d. and g(t) is their probability density function (see Eq. 11.4). As initially suggested by Bozer and White (1984), the expected DC cycle time can be evaluated by: EDC ¼ ESC þ ETB ;

ð11:28Þ

where ETB is the expected TB time. ETB is evaluated by introducing the function E(t1, t2) that represents the mean distance between the locus of points with distance t1 from I/O and the locus of points with distance t2 from I/O. Unfortunately, E(t1, t2) has a different functional form depending on the values of t1 and t2. In particular, there are seven regions in the Euclidean space R2: = (t1, t2) in which E(t1, t2) is piecewise defined. These seven regions are depicted in Fig. 11.5, which shows where and when the functions Ei(t1, t2) have to be used. It can be seen that E7(t1, t2) disappears for racks with shape factor [1/2. In addition, it is also useful to note that in the SIT problem (b = 1), and E6(t1, t2) must not be considered because region 6 vanishes. Each Ei(t1, t2), i = 1,…,7, and mean travel distance between the locus of points t1 and the locus of points t2, can be computed by applying the Integral Mean Value Theorem in its own domains (the rack area is the weight). The values of Ei(t1, t2) obtained are now reported:

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Fig. 11.5 Graphical representation of the seven domains of the Ei(t1, t2) for the three cases b \ 1/2 and 1/2 \ bB2/3 2/3 \ bB1. The problem is symmetrical in (t1, t2)

273 t1

b < 1/2

1

t 1=t2 -b E6

t 1=t2 /2 E7

b

E5 E2

E3

E4

E1

t1

1

2b

b 1 2

t2

< b < 23

1

E6

b

t 1=t2 /2

E5 E2

t 1=t2 -b E4

E1

E3 1

b

t1

2 3

t2

> > > >   > > E1 ðt1 ; t2 Þ ¼ 12t11 t2  t13  9t12 t2 þ 12t1 t22 > > > > > > > > > > > > > > > > > > > > > >   > > E2 ðt1 ; t2 Þ ¼ 12t11 t2  17t13  33t12 t2 þ 24t1 t22  2t23 > > > > > > > > > > > > > > > > > > > > > > > > E3 ðt1 ; t2 Þ ¼  34 t1 þ t2 > > > > > > > > > > > > > > > > > > > > <  2 3 1 3 2  t  3 ð t þ 2b Þt þ 3 ð t þ b Þ t  ð t  b Þ E4 ðt1 ; t2 Þ ¼ 12bt 2 2 1 2 1 1 1 > > > > > > > > > > > > > > > >   ! > > > 17t13  3ð9t2 þ 2bÞt12 þ 3 5t22 þ 2bt2 þ b2 t1  3t23 > > 1 > > > E5 ðt1 ; t2 Þ ¼ 12bt1  > > þ3bt22  3b2 t2 þ b3 > > > > > > > > > > > > > > > > >  > > > E6 ðt1 ; t2 Þ ¼ 3b1 2  ðt1  t2 Þ3 þ3bðt1  t2 Þ2 þb3 > > > > > > > > > > > > > > > > > > > > > > > > E7 ðt1 ; t2 Þ ¼ t2  t1 > > > > > > :

0  t1  t2 =2; 0  t2 b t2 =2;  t1  t2 ; 0  t2 b 0  t1  b; t1 þ b  t2  1 t2  b  t1  t2 =2; b  t2 1 t2 =2  t1  b; b  t2 1 b  t1  t2  t1 þ b; t2 1 b  t1  t2  b; t2 1

ð11:29  11:35Þ Equations 11.29–11.35 agree both with Park et al. (2003) and with Kouvelis and Papanicolaou (1995), describing all possible situations. With the function E(t1, t2), now being known in all its domains, ETB can be evaluated by applying the Integral Mean Value Theorem by using Eq. 11.4 again, as probability density function in the domain R2: = (t1, t2) with t1, t2 [ [0,1], and:

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Analytical and Numerical Modeling

275 2 3
> > > > > > >
2 c3 ðA uc1 R1 þc2 R22 þc2 R21 þc3 R22 Þ > > > > c3 > > > > : ðA uc1 R21 c2 R22 þc2 R21 þc3 R22 Þ b c3

0t R1  t

ð11:39Þ

R2  t b\t

with R1, R2 \ b;

Ft1 ðuÞ ¼

8 > > > > > > >
> uA c1 R21 þc2 R21 > > > b c2 > > : uA c1 R21 þc2 R21 bc2 R2 þb2 c3 b c3

with R1 \ b, R2 [ b; qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 c1 A u > > c1 > > < uA 1 b c1 Ft ðuÞ ¼ uA b c1 R1 þbc2 R1 > > > bc > : uA b c1 R1 þbc2 R12b c2 R2 þR2 bc3 b c3

with R1, R2 [ b.

0  t and t  R1  t and t 

ð11:40Þ

b  t and t  R2 \t and t 

0  t and t  b b\t and t  R1 R1 \t and t  R2 R2 \t and t  1

ð11:41Þ

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Analytical and Numerical Modeling

281

Fig. 11.9 Errors measured between the numerical simulation and analytical model

11.8 Comparison Between Analytical and Numerical Results The analytical models presented in Sects. 11.5 and 11.6 have been compared to the numerical approach presented in Sect. 11.7. The case study presented here has the following constraints: 2/3 \ bB1, 1–b \ R1 B b/2, b \ R2 B 1. SC is represented by Eq. 11.25 and DC by Eq. 11.36. In this context, the target is to validate these equations performing a parametric analysis of their convergence to the numerical solution. The analysis was conducted investigating the behavior of an AS/RS in presence of different ABC demand curves, G(i) = is, assuming that each item is ordered using the EOQ model (Hausman et al. 1976). Moreover the other parameters b, R1 and R2, have been varied. In particular, the considered independent variables are the following: ABC curve s:

Shape factor b Class A boundary limit R1

– 0.065 curve 20/90 – 0.7 – 0.139 curve 20/80 – 0.75 – 0.222 curve 20/70 – 0.8

– (1 - b) ? (1 - b ? b/2)/4 – (1 - b) ? 2(1 - b ? b/2)/ 4 – (1 - b) ? 3(1 - b ? b/2)/ 4

Class B boundary limit R2 – b ? (1 - b)4 – b ? 2(1 - b)/4 – b ? 3(1 - b)/4

– 0.318 curve 20/60

These parameters identify 108 different scenarios. The other parameters c1, c2 and c3 have been determined as a function of R1, and R2 according to Eq. 11.18–11.20. A numerical simulation using the framework of Sect. 11.7 was carried out with 20,000 runs for each scenario. The results are presented and compared in Fig. 11.9.

282

M. Gamberi et al.

It can be seen that the model predictions effectively converge to the values obtained in the simulations, with all % errors being very small (from 0 to 2–3%).

11.9 Conclusions and Further Research This work explains a method to analytically model the expected SC and DC cycle times for an AS/RS with RIT racks and 3-CBS storage allocation of products. This is a general approach that can also be applied to n-CBS AS/RSs. The model makes it possible to predict the behavior of an S/R 3-CBS warehouse if its operating parameters are known (ABC curve, shape factor b, class boundary limits R1, R2) so that its throughput performance can be calculated. In addition, a numerical approach to simulate a CBS AS/RS is presented: this approach is an effective and versatile application of the Inverse Transform Sampling, based on the theoretical assumptions and basis of the analytical model. In this way, it is possible to simulate every kind of CBS automatic warehouses with a considerable low-computational effort. Both the analytical method and the numerical simulation have been applied to 108 different but related scenarios. Even though the model presented in this work can be applied to every CBS AS/RS, the results of this paper are to be viewed as a starting point and preliminary to the quantification of the relationship between the storage assignment and the DC cycle, finding, if it is possible, the optimal boundaries Ri for the n-CBS AS/RS that minimize the multi-command cycles.

References Bozer YA, White JA (1984) Travel-time models for automated storage/retrieval systems. IIE Trans 16(4):329–338 Bozer YA, White JA (1990) Design and performance models for end-of-aisle order picking systems. Manag Sci 36(7):852–866 Chang DT, Wen UP (1997) The impact of rack configuration on the speed profile of the storage and retrieval machine. IIE Trans 29(7):525–531 Chang DT, Wen UP, Lin JT (1995) The impact of acceleration/deceleration on travel time models for automated storage/retrieval systems. IIE Trans 27(1):108–111 Eben-Chaime M (1992) Operations sequencing in automated warehousing systems. Int J Prod Res 30(9):2401–2409 Eynan A, Rosenblatt MJ (1993) An interleaving policy in automated storage/retrieval systems. Int J Prod Res 31(1):1–18 Eynan A, Rosenblatt MJ (1994) Establishing zones in single command class-based rectangular AS/RS. IIE Trans 26(1):38–46 Foley RD, Frazelle EH (1991) Analytical results for miniload throughput and the distribution of dual command travel time. IIE Trans 23(3):273–281 Graves SC, Hausman WH, Schwarz LB (1977) Storage-retrieval interleaving in automatic warehousing systems. Manag Sci 23(9):935–945

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Han M, McGinnis LF, Shieh JS, White JA (1987) On sequencing retrievals in an automated storage/retrieval system. IIE Trans 19(1):56–66 Hausman WH, Schwarz LB, Graves SC (1976) Optimal storage assignment in automatic warehousing systems. Manag Sci 22(6):629–638 Hwang H, Lee SB (1990) Travel-time models considering the operating characteristic of the storage/retrieval machine. Int J Prod Res 28(10):1779–1789 Kim J, Seidmann A (1990) A framework for the exact evaluation of expected cycle times in automated storage systems with full-turnover allocation and random service requests. Comput Ind Eng 18(4):601–612 Kouvelis P, Papanicolaaou V (1995) Expected travel time and optimal boundary formulas for a two-class-based automated storage/retrieval system. Int J Prod Res 33(10):2889–2905 Lee HF (1997) Performance analysis for automated storage/retrieval systems. IIE Trans 29(1):15–28 Lee HF, Schaefer SK (1996) Retrieval sequencing for unit load automated storage and retrieval systems with multiple openings. Int J Prod Res 34(10):2943–2962 Lee HF, Schaefer SK (1997) Sequencing methods for automated storage and retrieval systems with dedicated storage. Comput Ind Eng 32(2):351–362 Lee YH, Lee MH, Hur S (2005) Optimal design of rack structure with modular cell in AS/RS. Int J Prod Econ 98(2):172–178 Mahajan S, Rao BV, Peters BA (1998) A retrieval sequencing heuristic for miniload end-of-aisle automated storage/retrieval systems. Int J Prod Res 36(6):1715–1731 Manzini R, Gamberi M, Regattieri A (2006) Design and control of an AS/RS. Int J Adv Manuf Tech 28(7–8):766–774 Manzini R, Gamberi M, Persona A, Regattieri A (2007) Design of a class based storage picker to product order picking system. Int J Adv Manuf Tech 32(7–8):811–821 Mohammadreza V, Tang Sai H, Homayouni SM, Ismail N (2008) A statistical model for expected cycle time of SP-AS/RS: an application of Monte Carlo simulation. App Artif Intell 22(N 7–8):824–840 Pan C-H, Wang C-H (1996) A framework for the dual command cycle travel time model in automated warehousing systems. Int J Prod Res 34(8):2099–2117 Park BC (1991) Analytical, models and optimal strategies for automated storage/retrieval system operations. Unpublished Ph.D. thesis, Georgia Institute of Technology, Atlanta Park C (2006) Performance of automated storage/retrieval systems with non-square-in-time racks and two-class storage. Int J Prod Res 44(6):1107–1123 Park BC, Frazelle EH, White JA (1999a) Buffer sizing models for end-of-aisle order picking systems. IIE Trans 31(1):31–38 Park BC, Foley RD, Frazelle EH (1999b) Dual command travel time distribution and performance of miniload systems with 2-class storage. Working paper, Department of Industrial Engineering, Keimyung University, Taegu, Korea Park BC, Foley RD, White JA, Frazelle EH (2003) Dual command travel times and miniload system throughput with turnover-based storage. IIE Trans 35(4):343–355 Regattieri A, Gamberi M, Manzini R, Persona A (2008) Monte Carlo approach for performance evaluation in automatic storage and retrieval systems. J Ent Res Manag 26 (ISSN: 0897-8336) Rosenblatt MJ, Eynan A (1989) Deriving the optimal boundaries for class-based automatic storage/retrieval systems. Manag Sci 35(12):1519–1524 Schwarz LB, Graves SC, Hausman WH (1978) Scheduling policies for automatic warehousing systems: simulation results. IIE Trans 10(3):260–270 Thonemann UW, Brandeau ML (1998) Optimal storage assignment policies for automated storage and retrieval systems with stochastic demands. Manag Sci 44(1):142–148 Van den Berg JP (2002) Analytic expressions for the optimal dwell point in an automated storage/ retrieval system. Int J Prod Econ 76(1):13–25 Van den Berg JP, Zijm WHM (1999) Models for warehouse management: Classification and examples. Int J of Prod Econ 59(1):519–528 Yugang Yu, De Koster Renè BM (2008) Designing an optimal turnover based storage rack for a 3d compact AS/RS. Int J Prod Res 46(06):1551–1571

Chapter 12

A New Technology For Unit-Load Automated Storage System: Autonomous Vehicle Storage and Retrieval System Banu Yetkin Ekren and Sunderesh Sesharanga Heragu

Abstract Material handling is a growing, $80 billion industry that is central to the nation’s economic competitiveness and national defense. A major segment of this industry is unit load (UL) storage and retrieval (S/R) systems which form a critical link in global supply chains. The first step in UL S/R system design is the configuration from which subsequent design details are derived. Most UL S/R system life cycle costs and capabilities are established during the selection of S/R device technology, UL throughput rates, storage capacity and system configuration. The crane-based automation technologies were introduced in the 1970s. Automated storage and retrieval system (AS/RS) has been widely used in distribution and production environments to store and retrieve loads (Roodbergen KJ, Vis IFA (2009) A survey of literature on automated storage and retrieval systems. Eur J Oper Res 194:343–362). An AS/RS usually consists of conveyors, racks and automated S/R devices (cranes). The cranes are fully automated and can travel in aisles between the racks to pick up and drop off loads. Between 1994 and 2004, there has been a significant increase in the number of AS/RSs used in distribution environments in the United States (Roodbergen KJ, Vis IFA (2009) A survey of literature on automated storage and retrieval systems. Eur J Oper Res 194: 343–362). The most important advantages of using AS/RSs over non-automated systems are: savings in labour costs and floor space, increased reliability and reduced error rates. However, it has significant disadvantages including the high investment cost, less flexibility and higher investments in control systems (Zollinger H (1999) AS/RS application, benefits and justification in comparison to

B. Y. Ekren Deptartment of Industrial Engineering, Pamukkale University, Denizli, Turkey S. S. Heragu (&) Department of Industrial Engineering, University of Louisville, Louisville, LY 40292, USA e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_12,  Springer-Verlag London Limited 2012

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other storage methods: a white paper automated storage retrieval systems. Production Section of the Material Handling Industry of America).

12.1 Introduction Recent advances in ‘‘autonomous vehicle’’ (AV) hardware technologies have created the possibility of drastically reducing the scale of cost effective automation in moderate to low throughput UL storage systems, especially for UL S/R systems with high storage capacity to transaction demand ratios. Given the pervasiveness of UL S/R systems in global supply chains, there is high potential for AV technology to yield major cost reductions in a wide breadth of industries. Autonomous vehicle storage and retrieval system (AVS/RS) is introduced as a new technology for automated UL storage systems (Malmborg 2002). This new technology has been implemented at over fifty facilities in Europe. It utilizes a rail system running in two horizontal dimensions within a storage area. In an AS/RS, aisle captive cranes move ULs simultaneously in the horizontal and vertical dimensions while interfacing with endof aisle accumulation systems, (usually conveyors). However, AVS/RSs utilize sequential vehicle movement in two horizontal dimensions along a guide rail, (rectilinear travel), which allows vehicles to access any location on the same vertical level within the storage rack. Vertical movement is performed by lifts installed along the periphery of racks. Figure 12.1 shows an AVS/RS and its components. The system is composed of AVs, lifts, conveyors and storage racks. AVs function as S/R devices. They follow rectilinear movement in the horizontal travel and use lifts for the vertical movement. ULs are usually transferred by the conveyors from the production area. In this chapter, we investigate the performance of this system using two modelling approaches: we develop analytical and simulation models for a particular AVS/RS. We implement the study for a company that uses AVS/RS in France. Analytical modelling uses mathematical relationships between inputs and outputs. The most important advantage of an analytical method is that it is typically not time-consuming. It can evaluate the system’s performance in a reasonable time. However, to develop an analytical model for a complicated system is not a simple task. Also, changing an assumption in an analytical model may render the model invalid. These are some of the disadvantages of the analytical modelling approach. When properly designed, analytical models, however, are capable of providing reasonably accurate estimates of complex systems in a relatively short time. In this approach, we use the semi-open queuing network (SOQN) approach to model the system (Jia and Heragu 2009). The simulation modelling approach simulates the sequence of events that could occur over a period of time via a computer program. There are some advantages and disadvantages with the simulation methodology. The most important advantage is the ability to model complex systems in great detail, so that it provides more accurate results. For example, a verified and validated simulation model could provide estimates of key performance measures that are very close to those

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Fig. 12.1 Main components of an AVS/RS

seen in the actual system. However, this high accuracy comes at the expense of high modelling and computational effort. Developing a detailed, more accurate simulation model for a large system is time-consuming. Therefore, to be able to benefit from both the approaches, we model the system using the two methodologies and show the results. Section 12.2 reviews the related studies from the literature. In this section, we also explain the difference between our study and existing ones. In Sect. 12.3, we propose analytical models for AVS/RS. Two analytical models are implemented on a particular AVS/RS and they are also shown in this section. One is an approximate analytical model (AAM) and the other is the Matrix-geometric method (MGM). We also compare both analytical models’ results. In Sect. 12.4, the simulation modelling application of the AVS/RS is studied. First, we implement design of experiment (DOE) to identify factors affecting performance of the AVS/RS. Second, we implement a regression analysis for the AVS/RS to find out the best rack design under predefined scenarios. Figure 12.2 illustrates the flow of the chapter.

12.2 Background and Motivation There are various studies that evaluate the performance of AVS/RS (Malmborg 2002, 2003; Fukunari and Malmborg 2008, 2009; Kuo et al. 2007, 2008; Zhang et al. 2009; Ekren et al. 2010a, b, c; Ekren and Heragu 2010b).

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1 Introduction 2 Background and Motivation 3 Analytical Model for AVS/RS 3.1 SOQN 3.2 SOQN Modelling of the AVS/RS 3.2.1 Scenarios for movement kinematics 3.2.1.1 Six travel scenarios for storage transactions 3.2.1.2 Seven travel scenarios for retrieval transactions 3.2.2 Occurrence probability calculations for storage and retrieval scenarios 3.2.3 Customer combination 3.2.4 Service time calculations 3.2.4.1 Storage transactions service time calculations 3.2.4.2 Retrieval transactions service time calculations 3.2.5 SOQN of the AVS/RS 3.3 Approximate Analytical Solution of the AVS/RS 3.3.1 Approximate analysis of load dependent generally distributed queuing networks having low service time variability 3.3.1.1 Load dependent arrival rates 3.3.1.2 Load dependent service rates 3.3.1.3 Performance measures 3.3.2 AVS/RS implementation 3.4 Matrix-Geometric Solution of the AVS/RS

4 Simulation Models for AVS/RS 4.1 Design of Experiments 4.1.1 Simulation model of the AVS/RS 4.1.2 Design factors 4.1.2.1 Dwell point 4.1.2.2 Scheduling rule 4.1.2.3 I/O location 4.1.2.4 Interleaving rule 4.1.3 Experimental design and results 4.1.3.1 ANOVA results 4.1.4 Tukey’s test 4.2 Regression Analysis for the Rack Configuration of the AVS/RS 4.2.1 Problem description 4.2.1.1 Constraints 4.2.2 Regression modelling 4.2.3 Experiments 4.2.4 Regression function analysis 4.2.4.1 Stepwise Regression Results 4.2.4.2 Best Subset Regression Results 5 Conclusion 6 References

3.5 Comparison of Analytical Model Results

Fig. 12.2 Flow of the chapter

AVS/RS was first studied by Malmborg (2002). He proposed conceptualizing models for AVS/RSs to estimate their performance measures. The models are significant for developing the preliminary profiles and have been implemented on sample problems. Later, Malmborg (2003) proposed a state equation model to estimate the proportion of dual command cycles in AVS/RSs using interleaving. The model extends previous concepting tools to estimate this proportion to predict system utilization and throughput capacity. Although the model has some limitations, it provides a potentially useful addition to the tools available for AVS/RS concepting when opportunistic interleaving is used. Fukunari and Malmborg (2008) developed a cycle time model for AVS/RSs using random storage and opportunistic pairing of S/R transactions. The model can also compare AVS/RS and AS/RS designs. The results indicate that the technology selection decision has a significant impact on the initial costs. Later, Fukunari and Malmborg (2009) study a queuing network model to predict expected resource utilization in AVS/RSs. Their model overcomes the computational disadvantage of the state equation model and the inflexibility of the nested queuing model. However, their model lacks the capability for modelling the transaction queuing process directly. The results show that the model provides reasonably accurate estimates for the resource utilization performance measure. Kuo et al. (2007) develop a cycle time model for AVS/RS. Different from the existing studies, the model includes additional measures of system performance such as transaction waiting time except the vehicle utilization. They implement the

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model on an AVS/RS using random storage and point-of-service-completion dwell point rules. Although the model accurately estimates vehicle utilization, there are substantial errors in estimating the other performance measures. Kuo et al. (2008) also study a cycle time model based on a queuing network model for AVS/RS having class-based storage policy. The model aims to achieve sufficient accuracy and computational efficiency for system design conceptualization applications. The model is implemented on various real problems. A technique to estimate the transaction waiting times of AVS/RS by dynamically selecting among three alternative queuing approximations based on the variation of transaction inter-arrival times is proposed by Zhang et al. (2009). The technique can estimate the transaction waiting times well. The procedure is applied on series of realistically sized test problems. In Ekren et al. (2010a, b) two analytical models are proposed to model the AVS/RS. In the former study, they solve the AVS/RS using an AAM and in the latter study they solve the same model using the MGM. The existing analytical studies in the literature usually consider the utilization performance measure of AVs in AVS/RS using open queuing network (OQN) and/or closed-queuing network (CQN) algorithms. These analytical models have the ability to obtain average number of transactions waiting for AV and average waiting time in the AV queue. Also, it is the first time the system is modelled using the SOQN approach. AAM works faster than the MGM, however MGM gives better than the AAM when compared to those obtained via simulation. These studies are shown in Sect. 12.3. Recently, Ekren et al. (2010b, c) conducted simulation-based studies for performance evaluation of AVS/RS. They study DOE for an AVS/RS to identify factors affecting its performance (Ekren et al. 2010b). They also propose a regression analysis to find the near optimum rack configuration design under predefined scenarios of a number of vehicles and lifts in the system (Ekren et al. 2010c). By simulating the system, several performance measures such as the average cycle time, average utilization of the lifts, average waiting time in the lifts’ queue and average number of transactions waiting in the lifts’ queue (NL) can be obtaineed. It is the first time a regression analysis for the rack design of AVS/RS and DOE is applied by Ekren and Heragu (2010b), and Ekren et al. (2010c) respectively. In this chapter, we show our previous studies Ekren et al. (2010a, b, c) and Ekren and Heragu (2010b). The data for the modelled AVS/RS are obtained from a real company that utilizes AVS/RS.

12.3 Analytical Models for AVS/RS The analytical model of AVS/RS is modelled as SOQN. A queuing network is a collection of service stations organized in the sequence that customers visit in order to satisfy their service requirements. Queuing networks are usually classified into two types, open and closed queuing networks.

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An open queuing network (OQN) is characterized by one or more sources of customer arrivals from outside to one or more queues, receiving service at a finite number of servers in a specified sequence and departing the network from the last queue. The customers move from one station to another in the network in an order given by the routing probabilities. Assembly lines and traffic flowing through a series of intersections are examples of systems that can be modelled as OQNs. In a closed queuing network (CQN), customers neither enter nor depart from the network from a modeling perspective and recirculate within the network. A CQN seems unusual at first glance because it does not allow jobs to enter or leave the network. However, if we model the system assuming that an arriving customer is paired with a secondary resource (e.g., a kanban ticket or a pallet) and when the last operation on the customer is completed, the customer leaves the system but the secondary resource does not and instead immediately returns to the network with another customer, the CQN model could be used to estimate resource (and hence customer) waiting times and queue lengths accurately. SOQN is a variation of these two queuing network systems as explained below.

12.3.1 SOQN Like a CQN, an SOQN consists of customers, secondary resources and servers. Each customer is paired with a secondary resource and the two visit the set of servers required for processing the customer in the specified sequence. If a secondary resource (e.g., kanban card or pallet) is available, the customer enters the network immediately. Otherwise, it waits in an external queue until a resource becomes available (see Fig. 12.3). When the customer exits the system, the resource associated with the customer returns to a ‘pallet’ pool and waits to be paired with the next arriving customer. Buitenhek et al. (2000) and Dallery (1990) call this system an OQN with population constraint. However, we will use the SOQN terminology as Buzacott and Shanthikumar (1993). If we have an unlimited number of the secondary resource, then customers will never have to wait in the external queue. Thus, the system becomes an OQN. In contrast, if we always have a customer waiting outside the network so that the secondary resource never has to wait to be paired with a customer, the system becomes a CQN. SOQN is in between these two cases and is more realistic. Sometimes a customer may have to wait for a secondary resource or a secondary resource for a customer. There are many systems that can only be modelled using SOQN. For example, because of the limited number of bays in an automated car wash, only a corresponding limited number of customers may be allowed to enter the system. Multiprogramming computer systems (Avi-Itzhak and Heyman 1973) and communication networks with window flow control (Fdida et al. 1990) can also be modelled as SOQN. The term product-form, introduced by Jackson (1963) and Gordon and Newell (1967), states that all stations have equilibrium state probabilities which can be expressed as products of factors describing the state of each individual station

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Fig. 12.3 SOQN

within the network. So, the individual stations behave as if they are separate queuing systems. Jackson (1963) showed that for an OQN with Poisson arrivals, FCFS service disciplines, exponential service times and probabilistic routing, the steady-state joint probability has a product-form solution. Because of the external queue, an SOQN does not have an exact product-form solution. Therefore, we use approximate techniques to model the problem. We propose two analytical methods—an AAM and the MGM—for solving the SOQN model of the AVS/RS. The detailed procedures are explained below. First, we show how we model the particular AVS/RS via SOQN. Then, we solve the network using the two analytical approaches and obtain various performance measures.

12.3.2 SOQN Modelling of the AVS/RS The AVS/RS queuing system can be modelled as an SOQN (Ekren et al. 2010a, b). In the context of an AVS/RS, S/R transactions are jobs and the AVs are the secondary resources. The lifts and horizontal travel times to/from a storage space are the servers and the transactions are customers. Each transaction needs a vehicle before it can enter the network. Thus, the total number of transactions in service or waiting for service in the network cannot be larger than the number of vehicles. Figure 12.4 shows the SOQN model of an AVS/RS. If a vehicle is available, then an arriving transaction enters the network of servers immediately, along with this vehicle. Otherwise, it waits in the transaction queue until a vehicle becomes available. When a transaction exits the system, the vehicle returns to a ‘vehicle’ pool and waits to be paired with the next arriving transaction. The general view of SOQN of the AVS/RS is illustrated in Fig. 12.5. According to the figure there are two types of physical queues in the system, the vehicle queue and the lift queue. Transactions arriving to the system first enter the vehicle queue, then the lift queue. For a storage transaction, the vehicles travel to the I/O point to pick up the load. For a retrieval transaction, the vehicles travel to the load’s

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Fig. 12.4 SOQN of an AVS/RS

Fig. 12.5 General view of SOQN of the AVS/RS

location to retrieve it. For both transactions, the vehicles use a lift if it must travel to a tier other than its current tier. In Fig. 12.5, the server VT1 corresponds to the vehicle’s horizontal travel from its current position to the lift’s location. LT corresponds to the lift’s travel. Lifts are released after completion of vertical travel. There are as many lifts in the entire system as there are parallel servers in LT. VT2 corresponds to the vehicle’s horizontal travel to complete a storage or retrieval. In other words, it is the travel from the vehicle’s current position to the storage/retrieval position after it reaches the destination tier, including load/unload activities. The difference between VT1 and VT2 is that, VT2 considers the vehicle’s travel from the lift location to the destination plus the unload/load activities for the storage/retrieval process. VT1 is the vehicle’s travel to the lift not including load/ unload. In the real AVS/RS system, the vehicle is seized and not released until it completes the entire storage or retrieval. VT1 and VT2 are servers used to capture vehicle travel times and thus are not real servers. Therefore, dummy vehicle queues are shown before VT1 and VT2 in Fig. 12.5. It is assumed that there are as many vehicles in the entire system as there are parallel servers in VT1 and VT2. We also consider a synchronization station after the external queue where transactions and vehicles synchronize. After a vehicle completes a storage or a retrieval, it becomes free and returns to a virtual vehicle pool to be used by a waiting transaction.

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Table 12.1 Probabilities of storage and retrieval scenarios Storage scenarios

Retrieval scenarios

kS T1 Pr ðS1 Þ ¼ T1 T  kS þkR  T

kS T1 Pr ðR1 Þ ¼ T2 T  kS þkR  T

kS 1 Pr ðS2 Þ ¼ T1 T  kS þkR  T

R Pr ðR2 Þ ¼ kSkþk  T1 T R

S Pr ðS3 Þ ¼ T1  kS kþk  T1 T R

S Pr ðR3 Þ ¼ T1  kSkþk  T1 T R

R Pr ðS4 Þ ¼ kSkþk  T1 T R

kS 1 Pr ðR4 Þ ¼ T1 T  kS þkR  T

S Pr ðS5 Þ ¼ T1  kSkþk  T1 R

S pr ðR5 Þ ¼ T1  kS kþk  T1 R

R Pr ðS6 Þ ¼ kSkþk  T1 R

R Pr ðR6 Þ ¼ kSkþk  T1 R S Pr ðR7 Þ ¼ T1  kSkþk  T1 T R

In Fig. 12.5, s1 ; s2 and s3 are the mean service times of VT1, LT and VT2, respectively. c21 ; c22 and c23 are the squared coefficient of variations (scvs) of the s1 ; s2 and s3 ; respectively. Scv is the ratio of the variance to the square of its mean. In an AVS/RS, there are two types of transactions arriving into the system— storage and retrieval transactions. Storage transactions refer to the storage of a UL from the I/O point to an available location in the storage area. Retrieval transactions refer to the retrieval of a UL from its current location. All storage transactions are assumed to arrive at the I/O point and all retrieval transactions end at the I/O point. The assumptions and formulations of a typical configuration of an AVS/RS— the one seen in the warehouse in France—are given below: 1. The dwell point of a vehicle is the location where the last storage or retrieval transaction is completed. 2. The dwell point of the lift is where the last vertical movement is completed. 3. The system uses a pure random storage policy. 4. The actual distance between two aisles, the width of a bay and the height of one tier in an actual AVS/RS installation are used to derive the expected travel distances. 5. The number of aisles, columns (bays) and tiers of the actual installation are used to derive the occurrence probabilities of storage and retrieval scenarios (see Table 12.1) and service times of vehicles and lifts. 6. The storage area is divided into as many zones as there are lifts. 7. Each zone has one lift and three vehicles. Due to lift and vehicle speeds, the company has determined that it is best to pair a lift with three vehicles. This allows them to balance the utilization of the two resources so that neither is a bottleneck. 8. Lifts travel along the z-axis and are located at the middle of each individual zone (see Figs. 12.1 and 12.6). The reserve storage area of the warehouse has 42 aisles and 7 lifts. This area is divided into seven zones, each zone containing six aisles and one lift. Each lift is located at the front end of the third aisle in its corresponding zone.

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Fig. 12.6 The rack configuration of the AVS/RS

9. Each zone has I/O locations located near the lift (see Fig. 12.1). 10. The storage and retrieval transactions are modelled as independent Poisson arrival processes and their mean arrival rates are equal. 11. The transactions are served by the vehicles on a first-come, first-served (FCFS) rule. The vehicles requiring lifts for vertical movement are also served by FCFS order. Although the company sometimes uses a closest-request-first dispatching strategy, because our analytical model can only handle FCFS rule, we make this assumption. 12. The transfer times of a vehicle to and from a lift is assumed to be negligible. The notation used in the AVS/RS model is given below: A T W V Yx vL kS kR

Number of aisles B Number of tiers D The width of one storage bay H The number of vehicles L The distance from the firts bay to the TT cross aisle The velocity of the lift Vv The arrival rate of storage TL/U transactions per hour The arrival rate of retrieval transactions per hour

Number of bays (columns) per aisle The distance between two aisles The height of one tier The number of lifts Load or unload transfer time between the lift and the I/O point. The velocity of the vehicle The time to load/unload to or from the storage rack

The rack configuration of the AVS/RS under study is shown in Fig. 12.6.

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Because of the agreement with the company, we do not give the specific values of the parameters defined above, except the values summarized below: A T

=42 =7

L kS

=7 =225 Uls/h

B kR

=27 =225 ULs/h

At most, three unit-loads can be accommodated in each bay. Thus, the total storage capacity of the AVS/RS system is 27 9 3 9 2 9 42 9 7 = 47,628 UL positions. Racks are located on either side of an aisle.

12.3.2.1 Scenarios for Movement Kinematics To be able to derive the expected service time of a transaction, all probable scenarios are considered. In this section, we define all probable scenarios of transactions and their probabilities. Because of the service completion dwell point policy, an idle vehicle can be at the I/O point or at a storage rack. Also, because the expected travel distance depends on the position of the vehicle and the requested storage or retrieval location, the scenarios are described according to the idle vehicle’s position and the requested location. Consequently, the storage transaction can have six, and the retrieval transactions can have seven travel scenarios.

Six Travel Scenarios for Storage Transactions We define six travel scenarios for the storage transactions and their corresponding probabilities. Sf indicates that the requested storage location is on the first tier. Sn means the requested storage location is not on the first tier. Vo and Vf denote that the seized vehicle is at the I/O point, and at the first tier but not at the I/O point, respectively. Vn indicates that the seized vehicle is not on the first tier. Scenario 1 (VnSn). The seized vehicle is not on the first tier and the requested storage position is not on the first tier. Under this condition, the expected travel distance is calculated by considering five movements: (i) Vehicle travels from its current position to the lift, (ii) Lift travels from its current tier to the vehicle’s tier, (iii) Vehicle travels in the lift to the I/O tier, (iv) A load is deposited on the vehicle and, the lift and vehicle travel to the destination tier and (v) Vehicle travels to the destination storage location and deposits the load. Scenario 2 (VnSf). The seized vehicle is not on the first tier and the requested storage position is on the first tier. Under this condition: (i) The vehicle travels from its current position to the lift, (ii) Lift travels from the current tier to the vehicle’s tier, (iii) Vehicle in the lift travels to the I/O, (iv) The load is deposited on the vehicle and (v) The vehicle travels to the destination storage and deposits the load.

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Scenario 3 (VfSn). The seized vehicle is on the first tier and the requested storage position is not on the first tier. Under this condition: (i) The vehicle travels from its current position to the lift, (ii) Lift travels to the first tier, (iii) The load is deposited on the vehicle and, lift and vehicle travel to the destination tier, (iv) The vehicle travels to the storage rack and deposits the load. Scenario 4 (VoSn). The seized vehicle is at the I/O point and the requested storage position is not on the first tier. The expected travel distance is: (i) Lift travels from its current tier to the first tier, (ii) The load is deposited on the vehicle and, the lift and vehicle travel to the destination tier and (iii) The vehicle travels to the storage rack and deposits the load. Scenario 5 (VfSf). The seized vehicle is on the first tier and the requested storage position is on the first tier. The expected travel distance is: (i) The vehicle travels from its current position to the lift, (ii) Lift travels from its current tier to the first tier, (iii) The load is deposited on the vehicle and (iv) The vehicle travels to the destination storage and deposits the load. Scenario 6 (VoSf). The seized vehicle is at the I/O point and the requested storage position is on the first tier. (i) Lift travels from its current tier to the first tier, (ii) The load is deposited on the vehicle, (iii) The vehicle travels to the destination storage and deposits the load.

Seven Travel Scenarios for Retrieval Transactions The same calculations are also completed for the retrieval transactions. Because the travel sequences are different from the storage transactions, we describe one more scenario in retrieval scenarios. This is the first scenario, listed below. Rf and (Rn) denote that the retrieval location is (is not) on the first tier. Vo, Vf, Vn denote the same for storage transaction scenarios. The results are summarized below: Scenario 1 (Vn1Rn2). The seized vehicle and the requested retrieval locations are not on the same tier and the retrieval location is not on the first tier. Scenario 2 (VoRn). The seized vehicle is at the I/O point and the retrieval location is not on the first tier. Scenario 3 (VnRn). The seized vehicle and the retrieval locations are on the same tier other than the first. Scenario 4 (VnRf). The seized vehicle is not on the first tier and the retrieval location is on the first tier. Scenario 5 (VfRf). The seized vehicle and the retrieval locations are on the first tier. Scenario 6 (VoRf).The seized vehicle is at the I/O point and the retrieval location is on the first tier. Scenario 7 (VfRn). The seized vehicle is on the first tier and retrieval location is not on the first tier.

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12.3.2.2 Occurrence Probability Calculations for Storage and Retrieval Scenarios The occurence probabilities for the storage and retrieval scenarios are calculated in Table 12.1 (Ekren et al. 2010a, b). For example, Scenario 1 of the storage transactions is calculated by: Pr (VnSn) = Pr (the seized vehicle is not on the first tier) 9 Pr (the transaction type is storage) 9 Pr (requested storage position is not on the first tier).

12.3.2.3 Customer Combination Because each storage and retrieval transaction scenario has different routes and service times, each can be seen as a customer type in the queuing system. Thus, we have a multi-product SOQN which has 13 (6 ? 7) types of transactions (customers). We model the system’s network with generally distributed service times. When the network servers have general service time distributions, they are usually defined by two moments which are the mean and the scv of service times. We are interested in the global performance measures of the system such as, the average number of transactions in the external queue and in the system. We treat the multi-class queuing network as a single-class queuing network using Whitt’s formulations (Whitt 1983). We summarize Whitt’s (1983) formulations that are used in aggregating the first and second moments of the effective arrival and service times of each station (node). Notations that are used for this technique are given below: n Number of nodes k Number of classes of customers kr0 External arrival rate of class r

mj Number of servers at node j prij Routing probability of class r c2j The scv of the service-time distribution of node j

kri The arrival rate of class r to node i srj

The mean service time of class r at node j

First, we obtain the external arrival rates to each node as shown in (12.1): ( Xk b ¼ 1; if node j is the first node on route class r r ðb  k0 Þ k0j ¼ r¼1 0; otherwise ð12:1Þ 8j ¼ 1; 2; . . .; n The external arrival rate at node j, k0j ; is the sum of all arrival rates of classes whose first node is j. The flow rate from i to j is calculated by (12.2); the flow from i out of the network is calculated by (12.3).

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kij ¼

k X ðkri  prij Þ

8 i; j ¼ 1; 2; . . .; n

ð12:2Þ

8 i ¼ 1; 2; . . .; n

ð12:3Þ

r¼1

ki0 ¼

k X ðkri  pri0 Þ r¼1

From Eqs. 12.2 and 12.3 the overall routing matrix is obtained as shown in (12.4). kij P Pij ¼ 8i; j ¼ 1; 2; ::; n ð12:4Þ ki0 þ nl¼1 kil The combined mean and the scv of service-time of each node are obtained by (12.5, 12.6). Pk r r r¼1 ðsj  kj Þ ; 8j ¼ 1; 2; . . .; n ð12:5Þ sj ¼ kj Pk sj ðc2j

þ 1Þ ¼

r 2 r¼1 ððsj Þ

 krj  ðc2rj þ 1ÞÞ ; kj

8j ¼ 1; 2; . . .; n

ð12:6Þ

12.3.2.4 Service Time Calculations Each scenario defined in Six Travel Scenarios for Storage Transactions and Seven Travel Scenarios for Retrieval Transactions has has its own route and service times at each node. For instance, because the vehicles are at the I/O point in the S4 and S6 storage scenarios, the vehicles will not visit VT1. In addition, in the S3 and S5 retrieval scenarios, the transactions will go to VT2 directly after seizing the vehicle. In the following sections, the derivation of the service times of VT1, LT and VT2 of each scenario are shown. Because the service times have general distributions, we provide both means and the scv for all the service times. Storage Transactions Service Time Calculations For the service time calculations, we consider all probable scenarios. For example, VT1 is the vehicle travel time from its current position to the lift. Because the lift’s location is fixed, the vehicle’s travel from its current position to the lift depends only on the vehicle’s current position. The vehicle’s position is assumed to be uniformly distributed so that it can be at any of the 27 bays and 6 aisles. Hence, we consider 162 (27 9 6) scenarios for this travel time. The travel distance is calculated along two axes, x and y. The travel on the x-axis shows the travel between the aisles, whereas travel on the y-axis shows the travel between bays (see

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A New Technology For Unit-Load Automated Storage System

Table 12.2 Expected service times of six storage scenarios Scenario Arrival Rates (ULs/h) Scenario 1(VnSn): VT1 LT

VT2 Scenario VT1 LT VT2 Scenario VT1 LT VT2 Scenario VT1 LT VT2 Scenario VT1 LT VT2 Scenario VT1 LT VT2

  T 1 kS T 1    225 kS þ k R T T 18 ¼  225 49

299

Mean Service Time (min), and scv (0.5.11, 0.115) (0.631, 0.016)

(0.744, 0.054) 2 (VnSf): 3 49

 225

(0.511, 0.115) (0.493, 0.014) (0.744, 0.054)

3 49

 225

(0.511, 0.115) (0.429, 0.022) (0.744, 0.054)

3 (VfSn):

4 (VoSn): 3 7

 225

Null (0.429, 0.022) (0.744, 0.054)

5 (VfSf): 1 98

 225

(0.511, 0.115) (0.429, 0.022) (0.744, 0.054)

1 14

 225

Null (0.429, 0.022) (0.744, 0.054)

6 (VoSf):

Fig. 12.6). As a result, the total travel time is calculated by dividing the total distance by the vehicle’s velocity. In the service time calculations we also consider the vehicle’s and lift’s acceleration and deceleration times, and the vehicle’s turning delays when changing its current coordinate. All these are based on the company supplied data. Consequently, the expected arrival rates and service times of six storage scenarios are obtained as shown in Table 12.2. The arrival rates in Table 12.2 are calculated using the probabilistic routings in the network. For example, the first scenario’s arrival rate is calculated by: Pr (VnSn) 9 Arrival rate of storage transaction. From Table 12.2, it can be seen that all the servers are visited in each scenario except the scenarios 4 and 6. In Scenarios 4 and 6, the transactions do not visit VT1 because the vehicles are at the I/O point and they do not need to visit VT1 (see Six Travel Scenarios for Storage Transactions).

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Retrieval Transactions Service Time Calculations Similar calculations are also completed for retrieval transaction scenarios (see Table 12.3). Unlike a storage transaction, in the retrieval transaction queuing system, transactions may visit a node twice. For example, VT1 and LT are visited twice in Scenarios 1, 4, and 7. 12.3.2.5 SOQN Model of the AVS/RS Figure 12.7 shows the SOQN model of the AVS/RS. It should be noted that we aggregate storage and retrieval transactions’ service times (Tables 12.2 and 12.3 values), obtained in the previous section, using the formulations in Sect. 12.3.2.3. Hence, the system becomes a single class queuing network. According to this figure, there are two ways to leave the network, from LT or VT2. This is because the storage transactions leave the system after VT2 and the retrieval transactions leave the system after LT. P values show the probabilistic routings that are obtained by the arrival rates calculated in Tables 12.2 and 12.3. Consequently, P, s and c2j values are obtained as: P1 ¼ 0:464; P2 ¼ 0:5; P3 ¼ 0:036; P4 ¼ 0:659; P5 ¼ 0:341; P6 ¼ 0:5; P7 ¼ 0:5 s1 = 0.511 min. c21 = 0.115 (VT1) s2 = 0.413 min. c22 = 0.233 (LT) s3 = 0.756 min. c23 = 0.065 (VT2) We verify VT1, LT and VT2 values using simulation and then solve the Fig. 12.7 network using the AAM and MGM.

12.3.3 Approximate Analytical Solution of the AVS/RS In this section, we demonstrate an approximation solution procedure for the SOQN of the AVS/RS (Ekren et al. 2010a). Jia and Heragu (2009) applied the matrixgeometric method to analyzes SOQNs. Dallery also applies a closed queuing network (CQN) based solution procedure for SOQN (Dallery 1990). In this paper, we propose a method that uses ideas from the aggregation technique originally proposed by Avi-Itzhak and Heyman (1973) for single class exponential networks. It consists of four steps: 1. Transform the SOQN into a CQN in which the transactions are seen as customers. Identify the synchronization station as the first station where transactions may have to wait for a vehicle. 2. Consider all the stations except the synchronization station as a CQN, and obtain the load-dependent throughput rates. 3. Replace the synchronization station by a load-dependent exponential server with the values obtained in step 2.

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A New Technology For Unit-Load Automated Storage System

Table 12.3 Expected service times of seven retrieval scenarios Scenario Arrival rates (units/min) Scenario VT1 LT VT2 VT1 LT Scenario VT1 LT VT2 VT1 LT Scenario VT1 LT VT2 VT1 LT Scenario VT1 LT VT2 VT1 LT Scenario VT1 LT VT2 VT1 LT Scenario VT1 LT VT2 VT1 LT Scenario VT1 LT VT2 VT1 LT

301

Mean service Time (min), and scv

1 (Vn1Rn2):

15 49

 225

(0.511, (0.196, (0.744, (0.511, (0.493,

0.115) 0.132) 0.054) 0.115) 0.014)

Null (0.229, (0.744, (0.511, (0.493,

0.162) 0.054) 0.115) 0.014)

2 (VoRn):

3 7

 225

3 (VnRn):

3 49

 225

Null Null (1.088, 0.105) (0.511, 0.115) (0.493, 0.014)

 225

(0.511, (0.231, (0.744, (0.511, (0.492,

 225

Null Null (1.088, 0.105) (0.511, 0.115) (0.492, 0.002)

 225

Null (0.092, (0.744, (0.511, (0.492,

0.641) 0.054) 0.115) 0.022)

 225

(0.511, (0.229, (0.744, (0.511, (0.493,

0.115) 0.162) 0.054) 0.115) 0.014)

4 (VnRf):

3 49

0.115) 0.12) 0.054) 0.115) 0.022)

5 (VfRf):

1 98

6 (VoRf):

1 14

7 (VfRn):

3 49

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Fig. 12.7 SOQN model of the AVS/RS

4. Find the mean number of transactions in the external queue by considering the synchronization station in isolation and by solving the birth and death process (M/M/1) that describes its behaviour. We use Marie’s well-known approximation for generally distributed queuing network to calculate the load-dependent throughput rates in step 2. However, Marie (1979, 1980) has developed this technique for generally distributed networks having single server stations. In Fig. 12.7, the SOQN has load-dependent general servers (multiple servers at each node). Therefore, first we extend the Marie’s method to a load-dependent generally distributed queuing network (Ekren and Heragu 2010a). Then, we implement the new technique for our problem.

12.3.3.1 Approximate Analysis of Load-Dependent Generally Distributed Queuing Networks Having Low Service Time Variability Marie (1979, 1980) presents an efficient technique for the approximate analysis of a CQN with generally distributed service times and a single server. The basic idea is to approximate the performance of the network via a product-form CQN. It provides satisfactory results for a wide range of queuing networks. The proposed technique is based on an iterative procedure and the concept of the conditional throughputs, vj (n), of a station. In the method, each station is analysed under a Markovian process with load-dependent arrival rate, kj ðnÞ. Marie suggests using an Erlang type distribution for networks with low service time variability (Marie 1980). Therefore, by extending his approximation we analyse a CQN for kðnÞ=Ek =m and FCFS station types (Ekren and Heragu 2010a).

Load-Dependent Arrival Rates We assume that the sub-network has a product-form solution and has a local balance. In other words, the rate at which customers arrive at each node must be equal to the rate at which they leave. The kj ðnÞ values can be calculated using the convolution algorithm for MVA (Reiser 1981). We calculate the kj ðnÞ values, for each node j = 1, 2…, L using Marie (1980):

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A New Technology For Unit-Load Automated Storage System

kj ðn  1Þ  pj ðn  1Þ ¼ lj ðnÞ  pj ðnÞ

303

ð12:7Þ

lj ðnÞ is the service rate of station j when there are n jobs at the server. pj ðnÞ is the probability of having n jobs at station j when there are N jobs in the entire network. The MVA method gives the pj ðnÞ values automatically, when the iteration ends. Load-Dependent Service Rates Marie (1980) suggests an Erlang-k type distribution for a network with service time scv less than 0.5. For each separated sub-network, the number of phases kj is calculated by 1/c2j , where c2j ¼ k1j þ u. Here, u is a small number. The new load^j ðnÞ for Erlang-k is calculated by: dependent l ^j ðnÞ ¼ kj  lj ðnÞ l

ð12:8Þ

In an Erlang-k phase type distribution, the job starting from the first phase enters the next phase with probability of one until all phases are completed. The total time a job spends in a phase is exponentially distributed. The iterative algorithm is given in (Ekren and Heragu 2010a).

Performance Measures After the iterative procedure stops, we compute the performance measures using the formulas (12.9–12.12). Throughput of each station: kj ðVÞ ¼

V X

Pj ðnÞlj ðnÞ

ð12:9Þ

n¼1

Mean number of jobs at each station:  nj ðVÞ ¼

V X

nPj ðnÞ

ð12:10Þ

n¼1

Mean residence time: tj ðVÞ ¼

 nj ðVÞ kj ðVÞ

ð12:11Þ

Throughput of the entire network: L V 1X 1X Pj ðnÞvj ðnÞ L j¼1 ej n¼1

ð12:12Þ

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B. Y. Ekren and S. S. Heragu

Table 12.4 AAM, MGM and simulation results when A = 42, V = 21, L = 7 kS þ kR ¼ 425 Leq Ln Lp Weq

kS þ kR ¼ 450

AAM

MGM

Simulation

AAM

MGM

Simulation

0.607 15.147 5.853 0.600

0.647 15.127 5.873 0.640

0.8621:727:102 15.1092:146:102 5.8912:005:102 0.8521:523:102

1.085 16.138 4.862 1.013

1.125 16.238 4.762 1.050

1.2612:868:102 16.1363:706:102 4.8643:137:102 1.1772:443:102

kS þ kR ¼ 475 Leq Ln Lp Weq

kS þ kR ¼ 500

AAM

MGM

Simulation

AAM

MGM

Simulation

1.776 17.152 3.848 1.571

1.847 17.231 3.769 1.633

1.9105:300:102 17.168 7:182:102 3.832 5:831:102 1.6893:733:102

3.269 18.189 2.811 2.746

3.199 18.221 2.780 2.688

3.1158:317:102 18.215 9:133:102 2.785 8:705:102 2.6177:760:102

Here, ej is the mean number of visits that a job makes to station j and is calculated by (12.13) and pij is the routing probability. ej ¼

L X

ei pij

ð12:13Þ

j¼1

We implement our algorithm on ten different load-dependent CQNs and the performance measures are compared with the results from simulation. On average, the deviations from the simulation results are less than 5% (see Ekren and Heragu 2010a). Therefore, we implement it on a real AVS/RS in the following section. It should also be noted that the proposed extended algorithm in this section can be used for any load-dependent general network with low service time variability.

12.3.3.2 AVS/RS Implementation We apply the proposed SOQN approach to analyse a real-world implementation of the AVS/RS employed in a warehouse in northern France. All of the data were obtained from the AVS/RS manufacturer and system integrator and the assumptions were verified with them. The four-step approximate SOQN algorithm proposed in Sect. 12.3.3 is implemented for seven vehicle-lift (V-L) scenarios. Because the warehouse is divided into as many zones as there are lifts, and in the real-world application the manufacturer has determined that it is best to assign three or four vehicles per lift (due to the fact that the average travel time of the lift is approximately three times that of a vehicle), we can only investigate a limited number of V-L warehouse configurations. In all the experiments, we have chosen the number of aisles such that it is divisible by the number of zones or lifts—6, 7, 8 and 9. For instance, in Tables 12.4, 12.5, 12.6 and 12.7 the number of aisles

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305

Table 12.5 AAM, MGM and simulation results when A = 42, V = 28, L = 7 kS þ kR ¼ 475 Leq Ln Lp Weq

kS þ kR ¼ 500

AAM

MGM

Simulation

AAM

MGM

Simulation

0.2802 18.1610 9.8390 0.2477

0.3498 18.3610 9.6390 0.3094

0.54819:112:103 18.48617:556:103 9.51384:282:103 0.48471:132:104

0.5436 19.6070 8.3930 0.4566

0.6496 19.8917 8.1083 0.5458

0.78791:748:102 19.77402:617:102 8.22612:108:102 0.66191:399:102

kS þ kR ¼ 525 Leq Ln Lp Weq

kS þ kR ¼ 550

AAM

MGM

Simulation

AAM

MGM

Simulation

1.3565 21.2550 6.7450 1.0853

1.0973 21.2084 6.7916 0.8779

1.15232:487:102 21.10154:562:102 6.89853:914:102 0.92191:816:102

1.8597 23.0560 4.9440 1.4203

1.7956 22.3521 5.6479 1.3715

1.73794:281:102 22.45707:275:102 5.54306:894:102 1.32743:071:102

Table 12.6 AAM, MGM and simulation results when A = 42, V = 18, L = 6 kS þ kR ¼ 375 Leq Ln Lp Weq

kS þ kR ¼ 400

AAM

MGM

Simulation

AAM

MGM

Simulation

0.8764 13.4180 4.5820 0.8413

0.8976 13.4311 4.5689 0.8618

1.05572:076:103 13.42193:871:102 4.57815:494:103 1.01368:886:103

1.3728 14.4397 3.5603 1.2355

1.4284 14.4806 3.5194 1.2859

1.68453:055:102 14.45371:975:102 3.54637:003:102 1.51642:441:102

kS þ kR ¼ 425 Leq Ln Lp Weq

kS þ kR ¼ 450

AAM

MGM

Simulation

AAM

MGM

Simulation

3.038 15.5113 2.4887 2.5734

3.1120 15.5206 2.4794 2.6361

2.92114:780:103 15.50296:602:103 2.49719:011:103 2.47445:250:102

7.0026 16.6002 1.3998 5.6021

6.5877 16.6318 1.3682 5.2702

6.08902:007:102 16.56003:431:103 1.44009:198:103 4.87131:979:102

A = 42. We consider the following V-L combinations in Tables 12.4, 12.5, 12.6 and 12.7: V = 21 L = 7; V = 28, L = 7; V = 18, L = 6 and V = 24, L = 6. In Table 12.8, we consider A = 48 and V = 24, L = 6. In Table 12.9, A = 48 and V = 24, L = 8 and in Table 12.10, A = 45 and V = 27, L = 9. We also consider four different transaction arrival rates such that kS ¼ kR . The results are compared with those from simulation and the MGM. The simulation model is run for two years, with a warm-up period of three months. Ten independent replications are completed. The results are obtained with a 95% confidence interval. As a result, four performance measures for the AVS/RS system are observed both from the analytical and the simulation models. These are, the external queue length (Leq), average number of transactions (vehicles) in the network including those waiting for service (Ln), average number of vehicles in the vehicle pool (Lp) and average waiting time in the external queue (Weq). The results are illustrated in Tables 12.4, 12.5, 12.6, 12.7, 12.8, 12.9 and 12.10.

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B. Y. Ekren and S. S. Heragu

Table 12.7 AAM, MGM and simulation results when A = 42, V = 24, L = 6 kS þ kR ¼ 375 Leq Ln Lp Weq

kS þ kR ¼ 400

AAM

MGM

Simulation

AAM

MGM

Simulation

0.5098 16.6045 7.3955 0.4318

0.5567 16.6862 7.3138 0.4715

0.79024:012:102 16.79117:905:103 7.20893:262:102 0.66934:105:102

1.0076 18.0023 5.9977 0.8061

1.0989 18.0874 5.9126 0.8799

1.24853:700:102 18.11832:411:103 5.88176:061:103 0.99979:171:103

kS þ kR ¼ 425 Leq Ln Lp Weq

kS þ kR ¼ 450

AAM

MGM

Simulation

AAM

MGM

Simulation

2.2248 19.5112 4.4888 1.6862

2.1955 19.5282 4.4718 1.6640

2.07162:087:102 19.49907:876:102 4.50105:155:103 1.57018:132:103

4.3055 21.0628 2.9372 3.1000

4.2166 21.1002 2.8998 3.0359

3.79382:184:102 20.92546:791:102 3.07468:786:102 2.73159:794:103

Table 12.8 AAM, MGM and simulation results when A = 48, V = 24, L = 6 kS þ kR ¼ 425 Leq Ln Lp Weq

kS þ kR ¼ 450

AAM

MGM

Simulation

AAM

MGM

Simulation

0.5246 16.9891 7.0109 0.4444

0.5893 17.1074 6.8926 0.4991

0.79644:970:103 17.04466:266:103 6.95549:896:104 0.67458:776:103

1.0008 18.1033 5.8967 0.8006

1.0599 18.2101 5.7899 0.8479

1.24315:818:103 18.36537:702:103 5.63478:072:103 0.99458:782:104

kS þ kR ¼ 475 Leq Ln Lp Weq

kS þ kR ¼ 500

AAM

MGM

Simulation

AAM

MGM

Simulation

2.1876 19.7590 4.2410 1.6580

2.1694 19.7705 4.2295 1.6443

2.03913:282:103 19.73144:464:102 4.26867:891:102 1.54557:782:102

4.2114 21.1996 2.8004 3.0322

4.1979 21.2002 2.7998 3.0240

3.70926:834:103 21.13484:743:103 2.86522:191:102 2.67201:005:103

Table 12.9 AAM, MGM and simulation results when A = 48, V = 24, L = 8 kS þ kR ¼ 475 Leq Ln Lp Weq

kS þ kR ¼ 500

AAM

MGM

Simulation

AAM

MGM

Simulation

0.5002 16.8017 7.1983 0.5055

0.5797 16.8126 7.1874 0.5858

0.75253:282:103 16.83337:197:103 7.16678:851:103 0.76059:667:103

0. 8269 17.8359 6.1641 0.7938

0. 9012 17.8653 6.1347 0.8651

1.03922:330:103 17.85415:557:103 6.14598:509:104 0.99765:791:103

kS þ kR ¼ 525 Leq Ln Lp Weq

kS þ kR ¼ 550

AAM

MGM

Simulation

AAM

MGM

Simulation

1.5829 19.0013 4.9987 1.4472

1.5635 18.9134 5.0866 1.4296

1.46647:762:103 18.88219:044:103 5.11796:771:103 1.34085:682:103

2.3005 20.0022 3.9978 2.0077

2.2887 20.0153 3.9847 1.9974

2.14903:202:103 19.92434:418:103 4.07574:772:103 1.87555:850:103

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A New Technology For Unit-Load Automated Storage System

307

Table 12.10 AAM, MGM and simulation results when A = 45, V = 27, L = 9 kS þ kR ¼ 475 Leq Ln Lp Weq

kS þ kR ¼ 500

AAM

MGM

Simulation

AAM

MGM

Simulation

0.5016 18.9854 8.0146 0.4925

0.6003 19.1005 7.8995 0.5894

0.79582:015:102 19.08418:112:103 7.91597:007:103 0.78136:641:103

0.8392 20.0298 6.9702 0.7881

0.8699 20.1107 6.8893 0.8169

1.05933:131:103 20.09459:830:104 6.90553:981:103 0.99487:332:103

kS þ kR ¼ 525 Leq Ln Lp Weq

kS þ kR ¼ 550

AAM

MGM

Simulation

1.5602 21.1863 5.8137 1.4042

1.5233 21.1596 5.8404 1.3709

1.8498 1.43481:076:102 22.2009 21.11592:202:103 5.8841kS kR ¼ 200; 155; 209 4.7991 1.5982 1.29132:076:103

AAM

MGM

Simulation

1.8864 22.1936 4.8064 1.6298

1.98793:231:103 22.14567:463:103 4.85446:613:102 1.71755:275:103

Fig. 12.8 SOQN model of the AVS/RS with aggregated stations

12.3.4 Matrix-Geometric Solution of the AVS/RS In this section, we present an MGM solution of the network in Fig. 12.7. The MGM was developed by Marcel Neuts in the 1980s (Neuts 1980). It is a numerical approach to solve Markov processes with a special repetitive property called matrix-geometric property. For systems with large or possibly infinite number of states, exact solutions can only be obtained if one can utilize the structural properties of equations. The repetition allows us to determine a recursive solution for the stationary state probabilities since it implies that if the stationary probability for any state i is known, then stationary probability for state i ? 1 can be determined (Neuts 1980). The stationary state probability for the repeating portion of this process thus has a geometric form. If the states of Markov process can be grouped into vectors which possess a certain repetitive structure, then a recursive procedure can be used to determine the stationary state probabilities of the i ? 1’th vector in terms of the probabilities for the i’th vector. The form of the solution for the stationary state probabilities leads to Neuts’ matrix geometric form.

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The servers in Fig. 12.7 network have multi-services. Recall that the first and third stations have as many parallel servers as there are vehicles and, the second station has as many parallel servers as there are lifts in the system. When a network has more than two stations the state space increases drastically so, an MGM solution becomes difficult (Jia and Heragu 2009). Therefore, to be able to solve the problem first, we aggregate two stations of the network and reduce the network to a two-station network. The aggregation procedure is completed by a CQN algorithm. For the aggregation procedure, we use our extended procedure explained in Sect. 12.3.3.1 (see Ekren and Heragu 2010a). After aggregating the two stations, we assume that the aggregated stations are a single station with load-dependent, exponentially distributed service time. Then, we apply the MGM to solve the problem. In the MGM, the state of the system is denoted by (a, b), where a is the total number of transactions in external queue and the first stations, and b is the number of jobs at the second aggregated server. Although the two aggregated servers become a load-dependent exponentially distributed single server because the remaining station has generally distributed service time, we model that station by a phase-type distribution. Marie (1980) suggests Erlang-k type distribution for generally distributed networks with low service time variability (e.g., scv \ 0.5). The number of phases is defined by 1/scv (see Sect. 12.3.3.1). In an effort to reduce the number of phases in the generator matrix and, because the third station has the lowest scv, we prefer aggregating the last two stations of the network. It should be noted that in the aggregated network, the first station is VT1 and the second station is the aggregated stations (see Fig. 12.8). The generator matrix of Fig. 12.8 network is given by (12.14). According to this, the repetitive structure starts from the row including A, B and C matrixes. 2 3 B00 B01 ... ...7 6 B10 B11 C 6 A B C ...7 Q¼6 ð12:14Þ 7 4 A B ...5 .. .. . . . . . where, 3 2 2 3 0 B0 C0 .. C 7 6 A1 .B1 6 . 7; B ¼ ½ 0 . . . . . . A . . ; B ¼ B00 ¼ 4 5 .. 5 10 4 01 V1 ; B11 .. .. .. . AV2 BV2 C ¼ BV1 ð12:15Þ 2

ð0; 0Þ ð0; 0Þ 6 k ð0; 1Þ 6 0:5l 6 2 ð1Þ 6 B0 ¼ ð0; 2Þ 6 .. 6 6 . ð0; V  1Þ 4 ð0; VÞ

ð0; 1Þ

ð0; 2Þ

ðl2 ð1Þ þ kÞ 0:5l2 ð2Þ ðl2 ð2Þ þ kÞ .. .

...

..

ð0; V  1Þ

ð0; VÞ

. 0:5l2 ðV  1Þ ðl2 ðV  1Þ þ kÞ 0:5l2 ðVÞ ðl2 ðVÞ þ kÞ

3 7 7 7 7 7 7 7 5

ð12:16Þ

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2

ð1; 0Þ ð1; 1Þ kb ð0; 0Þ 6 6 ð0; 1Þ 6 kb 6 0:5l2 ð1Þb 6 0:5l2 ð2Þb C0 ¼ ð0; 2Þ 6 6 .. 6 . 6 ð0; V  1Þ 4 ð0; VÞ

ð1; 2Þ

309

ð1; V  1Þ

...

ð1; VÞ

7 7 7 7 7 7 7 7 7 5

kb ..

. 0:5l2 ðV  1Þb

kb 0:5l2 ðVÞb

3

k

ð12:17Þ 2

ð0; 0Þ

ð1; 0Þ 6 ð1; 1Þ 6 6 6 A1 ¼ ð1; 2Þ 6 .. 6 6 . ð1; V  1Þ 4 ð1; VÞ 2

ð1; 0Þ 6 S1  kI 6 6 0:5l2 ð1ÞI 6 6 B1 ¼ 6 6 6 6 ð1; V  1Þ 4 ð1; VÞ ð1; 0Þ ð1; 1Þ ð1; 2Þ .. .

ð0; 1Þ S01

ð1; 1Þ

ð0; 2Þ

ð0; 3Þ . . .

S01

S01

ð1; 2Þ

S1  ðk þ l2 ð1ÞÞI 0:5l2 ð2ÞI

..

S01

7 7 7 7 7 7 7 5

ð12:18Þ

ð1; V  1Þ

...

S1  ðk þ l2 ð2ÞÞI .. .

.

ð0; VÞ

3

.. . 0:5l2 ðV  1ÞI

S1  ðk þ l2 ðV  1ÞÞI 0:5l2 ðVÞb

ð1; VÞ

3 7 7 7 7 7 7 7 7 7 5

ðl2 ðVÞ þ kÞ

ð12:19Þ 2

ð2; 0Þ kI ð1; 0Þ 6 6 ð1; 1Þ 6 6 0:5l2 ð1ÞI 6 C1 ¼ ð1; 2Þ 6 6 .. 6 6 . 6 ð1; V  1Þ 4 ð1; VÞ

ð2; 1Þ

ð2; 2Þ

kI 0:5l2 ð2ÞI

kI

...

ð2; V  1Þ

ð2; VÞ

kI 0:5l2 ðVÞb

k

..

. 0:5l2 ðV  1ÞI

3 7 7 7 7 7 7 7 7 7 7 5

.. .

ð12:20Þ and where, A ¼ AV ; B ¼ BV ; C ¼ C1 ; b ¼ ½1 0. . .0; 3 0 0 lðaÞ lðaÞ 7 6 0 0 7 6 lðaÞ lðaÞ 7 0 6 Sa ¼ 6 7; Sa ¼ Sa e; a ¼ 1 ; 2 ; . . . ; V .. .. 7 6 5 4 . . 2

0

lðaÞ

0

lðaÞ

ð12:21Þ

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B. Y. Ekren and S. S. Heragu

Here, N is the number of vehicles in the system and k is the total arrival rate into the system ðks þ kR Þ. The size of b is equal to the number of phases of the current station, e is the column vector of ones, and be ¼ 1: Sa is the transition rate matrix among the phases and, lðaÞ0 ¼ ð1=SCVÞlðaÞ; a ¼ 1; 2; . . .; V: For example, if the state is (2, 3), then there are two transactions at the first station so, in 0 S2 ; lð2Þ ¼ ð1=scvÞlð2Þ We define pa ; a  0 as the stationary probability vector of level a which includes the probabilities of all micro-states in that level. From the matrix-geometric property, paþ1 ¼ pa R (Neuts, 1980). R is a constant matrix called the rate matrix and can be calculated iteratively by (12.22). R ¼ ðC þ R2 AÞB1

ð12:22Þ

After obtaining the R, we solve the boundary stationary probabilities p0 through pV by (12.23) which are obtained by the property of the generator and the normalization condition - p0 e þ p1 e þ    ¼ 1: ½p0 . . .pV pVþ1 . . .Q ¼ 0   B01 B00 ¼0 ½p0 . . .pV  B10 B11 þ RA

ð12:23Þ

We can then determine the solution for the boundary states by solving (12.24). Here, we delete the first columns of B00 and B10 and show them as B00 ; and B10 ; respectively.   0 B0;1 B0;0 e ½p0 . . .pV  ¼ ½1; 0 ð12:24Þ  ðI  RÞ1 e B1;0 B1;1 þ RA After obtaining the stationary probabilities, the expected external queue length—Leq—and, the queue lengths at each station including being serviced—q1, q2—are calculated by (12.25) and (12.26), respectively (Jia and Heragu 2009). 2 3 2 3 2 3 2 3 0 0 2 0 . . 6 .. 7 6 .. 7 6 3 7 617 6 7 6 7 7 6 7 Leq ¼ p1 6 4 0 5 þ p2 4 1 5 þ    þ pv 4 ... 5 þ    þ pNþ2 4 ... 5 þ    ð12:25Þ V 1 2 V þ2 q1 ¼

V1 X

pn ne þ pV F 2 Ne  Leq

n¼1

q2 ¼

V1 X

ð12:26Þ pn l þ pV Fl

n¼0

where F = (I – R) -1 and l = [0 1 … V]T. We solve the SOQN problem for seven V and L scenarios as in AAM. The results are compared with those from simulation and AAM.

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311

12.3.5 Comparison of Analytical Model Results We solve the SOQN of the AVS/RS using two analytical models—AAM and MGM—for seven vehicle-lift (V-L) combinations and four arrival rate scenarios, k ¼ 425; 450; 475; 500 and k ¼ 475; 500; 525; 550, as explained in Sect. 12.3.3.2. The experiments are considered for AV utilization values between 70– 90%. Utilizations outside of this range would not be of much use because high utilization significantly increases the transaction cycle times and low utilization would not be cost-effective because the capital cost of the equipment is rather high. We thus consider only the practical scenarios. The analytical results are compared with those from simulation. As a result, four performance measures for the AVS/ RS system are observed both from the analytical, MGM, and the simulation model. These are, the external queue length (Leq), average number of transactions (vehicles) in the network including waiting for service (Ln = q1 ? q2), average number of vehicles in the vehicle pool (Lp) and average waiting time in the external queue (Weq). The results are illustrated in Tables 12.4, 12.5, 12.6, 12.7, 12.8, 12.9 and 12.10. Several observations from Tables 12.4, 12.5, 12.6, 12.7, 12.8, 12.9 and 12.10 are also summarized below. • We are able to estimate the key performance measures Ln and Lp values within 5% of the mean simulation values for multiple values of arrival rates and number of vehicles and for both analytical models. • Our extended algorithm works reasonably well. It estimates the Ln and Lp values within 5% of the simulation values for multiple values of arrival rates and number of vehicles. More importantly, the absolute difference is never greater than one job (transaction) for any of the Leq, Ln and Lp values in the above referenced tables. • Although we aggregate the two stations in the network the MGM provides reasonable results for generally distributed SOQNs with medium to moderately heavy traffic. • It should be noted that when the utilization (Ln/(Ln ? Lp)) is between 80–90% the error between the AAM, MGM and the simulation models for the Leq and Weq performance measures gets smaller for all N–L scenarios. • The algorithm works better—provides better estimates of the transaction waiting time in the external queue (Weq)—under medium to moderately heavy traffic conditions than low traffic conditions. A similar result was also shown in Avi-Itzhak and Heyman (1973). The reason is probably due to the treatment of the synchronization station as an exponential single server and solving the model as M/M/1. • The performance measure estimates obtained via the MGM are usually better than the estimates obtained via the AAM. In particular, we can estimate the Leq and Weq performance measures better via the MGM than the AAM.

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12.4 Simulation Models for AVS/RS In this section, we perform a simulation-based experimental design to identify factors affecting the performance of AVS/RS (Ekren et al. 2010c). In addition, we study the rack configuration design of AVS/RS warehouse using a simulationbased regression analysis. In the DOE, we consider three performance measures, as well as four factors. After DOE, we implement Tukey’s test to find out the best experiment(s) that reflect the performance measure. In the regression analysis, we develop 30 regression functions based on various V-L combinations, and arrival rate scenarios. The regression functions are developed in terms of number of tiers, aisles and bays. We consider five performance measures. After obtaining the regression functions, we optimize them using the LINGO software.

12.4.1 Design of Experiments DOE is a design tool that makes changes to the independent (input) variables to determine their effect on the dependent (output) variables. It not only identifies the significant factors (independent variables) that affect the response (dependent variable), but also how these factors affect the response. Thus, the objective of our study is not only to investigate how the performance measures of an AVS/ RS are affected by the predefined factors but also to ascertain how the measures can be improved by adjusting these factors. The performance measures considered in DOE are the average cycle time per S/R transaction (P1) and the average utilizations of the vehicles (P2) and those of the lifts (P3). Cycle time is the time between when a request originates until it is fulfilled. Along with resource utilization, it is a vital performance measure and employed extensively in many industries. AVS/RS cycle times are determined by the storage area configuration, vehicle movement kinematics, storage policy and vehicle dwell point policy (DPP). The factors that are likely to have an effect on the performance measures are: DPP, scheduling rule (SR), I/O locations and interleaving rule (IR). Details of these design factors and their levels are explained in Sect. 12.4.1.2. First, we simulate the current AVS/RS. Second, we define various scenarios to select the best combination of lifts and vehicles in the system. Third, we conduct a DOE under these predefined number of lifts and vehicles, and four independent factors. We implement DOE for seven arrival rate scenarios which are ;5, ;10, ;15% of the initial arrival rate of the S/R transactions. Fourth, we analyse the results using ANOVA to determine the main and interaction effects of the factors. Finally, we complete a Tukey’s test to find out the best factor levels for each scenario.

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Create storage transaction arrivals at the I/O point (λ pallets/hour)

The transaction waits for an available vehicle

Specify an available place (tier, aisle, bay and side) for the arriving storage, randomly

The transaction requests vehicle based on FCFS at the I/O point of its zone

Is the seized vehicle on Tier > 1

Vehicle travels from its current location to the lift point on that tier

Yes

Vehicle requests lift (FCFS)

313

The vehicle transfers to the lift and travel to the first tier (I/O) in the lift

Vehicle waits for lift

No

Is the seized vehicle at the I/O

No vehicle movement

Yes

Vehicle charges load in the lift

Is the storage tier = 1

Yes

Vehicle releases the lift and travels from the I/O point to the storage address

Vehicle discharges the load at the storage address

Transaction releases the vehicle

No

No

Vehicle requests lift (FCFS)

Vehicle travels from its current location to the lift point on that tier

Vehicle waits for lift

Vehicle travels to the destination tier in the lift

Vehicle releases the lift and travels to the storage address

Fig. 12.9 Simulation flowchart for storage transaction

Create retrieval transaction arrivals at the I/O point (λ pallets/hour)

The transaction waits for the vehicle

Specify a loaded address in the rack (tier, aisle, bay and side), randomly

The transaction requests vehicle (FCFS)

Is the seized vehicle on a different tier than the retrieval address

Yes

Vehicle travels from its current location to the lift’s location on that tier

Vehicle requests lift (FCFS)

Vehicle transfers to the lift and travels to the destination tier in the lift

Vehicle waits for lift

Vehicle releases the lift and travels from the lift to the retrieval address

No Vehicle travels from its current location to the retrieval address

Vehicle charges the load at the retrieval address

Vehicle travels from the retrieval address to the lift point on that tier

Vehicle requests lift (FCFS)

Vehicle waits for lift

Is the retrieval tier = 1 Yes No Vehicle discharges the load in the lift

Transaction releases the lift and the vehicle

Vehicle transfers to lift and travels from the retrieval tier to the I/O point in the lift

Fig. 12.10 Simulation flowchart for retrieval transaction

12.4.1.1 Simulation Model of the AVS/RS Figures 12.9 and 12.10 illustrate the simulation flowcharts of S/R transactions, respectively. S/R transactions arrive into the system at the same rate. Each zone has its own queue. When a vehicle becomes available in a zone, the first transaction waiting in that zone’s queue seizes the free vehicle. If more than one vehicle is available, the transaction seizes the free vehicle randomly. For a storage transaction on a tier other than the first, the autonomous vehicle uses the lift in the corresponding zone. The other assumptions that are used in our DOE simulation model are summarized below (Ekren et al. 2010c): • Approximately 10% of the transactions consist of heavy unit-loads, and it is company policy to assign them to the first tier for safety and structural stability reasons. The remaining are assigned randomly to any of the tiers, including the first tier.

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Table 12.11 Six scenarios for lift and vehicle combinations Scenario Lifts (zones) Vehicles per zone P1 (min)

P2 (%)

P3 (%)

1 2 3 4 5 6

95.6 76.7 Blocked 86.7 85.5 59.3

76.5 75.7 Blocked 82.1 78.3 57.6

7 7 6 6 8 8

3 4 3 4 3 4

17.12 4.92 Blocked 6.43 6.22 3.45

• The system uses a pure random storage policy. • Arrivals to the system follow a Poisson process and the mean arrival rates for S/R are kS ¼ kR ¼ 182 pallets per hour, respectively. We also vary the arrival rates in the DOE which are kS ¼ kR ¼ 173; 191; 164; 200; 155; 209 pallets per hour. • The storage transactions originate at the input station (located at the 39th aisle), and the retrieval transactions end at the output station (located at the 3rd aisle). • The time for picking up and depositing the unit loads are assumed to be 14 s each. • The transfer time of a vehicle to and from a lift is assumed to be negligible. • The vehicles and lifts have an extra delay because of acceleration and deceleration times which are assumed to be 5 and 3 s, respectively. The simulation model is assumed to be a non-terminating system, allowing us to conduct a steady-state analysis. The warm-up is three months. The length of each simulation run is two years or 1,180,800 min. ((365*2*24*60)+ warm up period). The model is run for 10 independent replications. The current AVS/RS configuration has high P2 (see scenario 1 of Table 12.11). To identify the combination of lifts and vehicles that provides minimum P1, we examine those configurations which have P2 and P3 between 75–85%. Under the assumption that the warehouse is zoned, i.e., it is split into equally sized independent subsystems, six scenarios can be identified as shown in Table 12.11. Table 12.11 shows the average values of ten replications under the initial condition of considered factors (low levels of factors in Table 12.12). From Table 12.11, we see that the 2nd, 4 and 5th scenarios have utilizations in the desired range. However, because the fourth scenario has the smallest total number of vehicles (24) and lifts (6), we choose that scenario to conduct DOE. Simulation runs are conducted based on the four independent factors shown in Table 12.12. Each factor has two levels. So, we complete a 2k experimental design for each performance measure. A total of 160 (24*10) response values are used for ANOVA in MINITAB for each performance measure. In the simulation model, the common random numbers (CRN) variance reduction technique is used. CRN is a popular and useful variance reduction technique when we compare two or more alternative configurations. It requires synchronization of the random number streams, and uses the same random numbers to simulate all configurations. In CRN, a specific random number used for a specific purpose in one configuration is

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A New Technology For Unit-Load Automated Storage System

Table 12.12 Levels of Factors Factors 1

DP

2

SR

3

I/O

4

IR

1 2 1 2 1 2 1 2

315

Codes (level)

Levels

Low High Low High Low High Low High

POSC POLL FCFS SDT Initial Middle No rule Opportunistic int.

used for exactly the same purpose in all other configurations. Thus, variance reduction is ensured. 12.4.1.2 Design Factors The various design factors considered in our experiments are detailed in the following sections. Dwell Point The position where an idle vehicle waits after completing a transaction is dictated by the dwell point policy (DPP). A dwell-point is chosen to minimize the expected time to travel to the next request. In most AS/RSs, the S/R device resides at the pointof-service-completion (POSC). Under this type of policy, a vehicle waits at the destination of the last transaction, until it is seized by the next S/R transaction. Thus, the I/O location becomes the vehicle dwell point for cycles concluding with a retrieval transaction and the load storage address becomes the dwell point for cycles concluding with a storage transaction. Because the vehicles utilize lifts for vertical movement, it may be advantageous to position the vehicles close to the corresponding lift location. Thus, we also test the point-of-lift-location (POLL) DPP in our study. Scheduling Rule Storage requests in distribution or production environments are usually not timecritical. The exact point in time at which loads are stored is not of much importance for the performance of the system. Therefore, storage requests are usually made according to the FCFS principle. However, because retrieval requests are time-sensitive, sequencing becomes important. By sequencing the S/R efficiently, the overall cycle time can be improved. Besides, studies show that the S/R transactions have an impact on the AS/RSs (Randhawa and Shroff 1995; Bozer and White 1984; Roodbergen and Vis 2009). We consider FCFS and shortest destination time (SDT) rules in our study. It should be noted that because we do not

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know the idle vehicle’s position in advance (until an idle vehicle is seized) we calculate the destination time for one-way travel time. For example, for S/R transactions, the destination time is calculated by considering the travel time from the input point to the destination storage location and from the retrieval storage location to the output point, respectively.

I/O Location An output location is where retrieved loads are dropped off, and an input location is where incoming loads are picked up for storage. I/O locations can be co-located or they can be separated. Unit loads arrive at the I/O location of a warehouse via conveyors. The unit loads are stored in the reserve storage area by AVs and after a period of time they are retrieved again, for example, to be shipped to a customer. Because loads are picked up and dropped off at I/O location by the autonomous vehicles, the I/O location may impact the cycle times. We consider two levels for I/O location. In low level (1), the input location is near the first lift’s location, and the output point is near the last lift’s location. In other words, the I/O points are far away from each other. For the high level (2), the I/O locations are merged and near the middle of the x-axis of the warehouse (See Fig. 12.6).

Interleaving Rule A unit-load AVS/RS can operate in two ways, namely in a single command (SC) cycle or in a dual command (DC) cycle. In an SC cycle, the vehicle performs either a single storage or a single retrieval transaction. The storage cycle time then is equal to the sum of the times to pick-up a load at the input station, to travel to the storage location, and to place the load in the rack. The retrieval cycle time can be defined as the sum of the times to travel to the storage location, to pick-up a load from this location, and to return to the output station. If an AVS/RS performs both a storage and a retrieval transaction in a single cycle, we call this a DC cycle. In this case, the cycle time is defined as the sum of the times to pick-up the load, to travel to the storage location and store the load, the empty travel time (interleaving time) from the storage location to the retrieval location and the time to pick the UL and transport it to the output station. Clearly, the total time to perform all S/R transactions reduces if dual commands are performed. With the opportunistic IR, a storage or retrieval transaction is combined with a retrieval or a storage transaction at the start of the S/R cycle when one or more of both types are pending in the active queue. However, as more DC cycles are executed, queue sizes are reduced and SC cycles become more likely (Malmborg 2003). In the current simulation model, because the arrival rates are the same for S/R transactions, we expect more DC cycles, and hence decreased average cycle times in our model. Hence, we consider two levels for the IR factor which are the

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317

Table 12.13a Design matrix and responses for all factor combinations (kS ¼ kR ¼ 182; 173; 191; 164) Exp. DP SR I/O IR P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3 Loc.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2

1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

kS ¼ kR ¼ 182 pallets/h

kS ¼ kR ¼ 173 pallets/h

kS ¼ kR ¼ 191 pallets/h

kS ¼ kR ¼ 164 pallets/h

6.43 6.36 5.80 5.73 3.91 3.74 3.78 3.61 5.74 5.63 5.67 5.57 3.60 3.40 3.59 3.39

5.64 5.54 5.24 5.14 3.72 3.53 3.62 3.43 5.21 5.08 5.15 5.02 3.47 3.25 3.46 3.24

7.82 7.84 6.77 6.75 4.18 4.06 3.99 3.86 6.57 6.50 6.49 6.43 3.75 3.57 3.74 3.57

5.13 4.99 4.86 4.72 3.57 3.35 3.50 3.28 4.84 4.67 4.78 4.63 3.36 3.12 3.35 3.11

0.87 0.86 0.86 0.85 0.73 0.70 0.72 0.70 0.85 0.84 0.85 0.84 0.70 0.67 0.70 0.67

0.82 0.80 0.82 0.80 0.78 0.73 0.78 0.73 0.79 0.77 0.80 0.77 0.76 0.71 0.76 0.71

0.83 0.82 0.83 0.81 0.69 0.66 0.69 0.66 0.82 0.80 0.82 0.80 0.67 0.64 0.67 0.64

0.80 0.78 0.80 0.78 0.76 0.71 0.76 0.71 0.78 0.75 0.78 0.75 0.74 0.68 0.74 0.68

0.90 0.90 0.90 0.89 0.76 0.73 0.76 0.73 0.89 0.88 0.89 0.88 0.74 0.71 0.74 0.71

0.84 0.83 0.84 0.83 0.80 0.76 0.80 0.76 0.81 0.79 0.82 0.80 0.78 0.73 0.78 0.73

0.79 0.77 0.79 0.77 0.66 0.63 0.66 0.63 0.78 0.76 0.78 0.76 0.64 0.60 0.64 0.60

0.78 0.75 0.78 0.75 0.73 0.68 0.74 0.68 0.75 0.72 0.76 0.72 0.71 0.65 0.71 0.65

DC (opportunistic interleaving) and the ‘‘no rule’’ in which the waiting transactions are selected according to the ‘‘scheduling rule’’ (see Scheduling Rule).

12.4.1.3 Experimental Design and Results Because the AVS/RS under study is quite complex, it makes it difficult for a manager to identify not only the parameters that could affect the average cycle time and utilization in the system, but also the interaction effects of these factors. Hence, a carefully designed factorial experiment is undertaken to determine the relative importance of the factors and their interaction. The DOE table is shown in Tables 12.13a, b. The level assigned to each factor (1 or 2) corresponds to those in Table 12.12. The output performance measures, namely P1, P2 and P3 (based on ten replications) for seven arrival rate scenarios—173, 191, 164, 200, 155, 209 pallets/h—are also given in Table 12.13a, b. To run the simulation model for each experiment, four main models are developed based on different DP and I/O locations, because they have basic structural differences. Then, the SR and IR are integrated into the model via simple logical changes. To analyse the effects of the factors properly, they should be categorized into their interaction and main effects. The following sections explain the interaction and main effect results (Ekren et al. 2010c).

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Table 12.13b Design matrix and responses for all factor combinations ðkS ¼ kR ¼ 200; 155; 209Þ Exp. DP SR I/O Loc. IR P1 P2 P3 P1 P2 P3 P1 P2 P3 kS ¼ kR ¼ 200 pallets/h 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2

1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

10.89 10.95 8.69 8.75 4.57 4.52 4.29 4.21 7.98 7.97 7.93 7.89 3.93 3.78 3.93 3.78

0.94 0.93 0.94 0.93 0.79 0.77 0.79 0.76 0.92 0.91 0.92 0.92 0.77 0.74 0.77 0.74

kS ¼ kR ¼ 155 pallets/h 0.86 0.85 0.86 0.85 0.82 0.78 0.82 0.78 0.82 0.81 0.83 0.82 0.80 0.76 0.80 0.76

4.76 4.59 4.57 4.41 3.45 3.21 3.40 3.16 4.55 4.37 4.51 4.33 3.27 3.01 3.26 3.00

0.75 0.73 0.75 0.73 0.63 0.59 0.63 0.59 0.74 0.72 0.74 0.72 0.60 0.57 0.60 0.57

kS ¼ kR ¼ 209 pallets/h 0.76 0.72 0.76 0.72 0.71 0.65 0.71 0.65 0.73 0.69 0.73 0.69 0.69 0.62 0.69 0.62

22.18 22.67 15.01 15.21 5.27 5.30 4.77 4.77 11.08 11.14 11.11 11.21 4.16 4.04 4.16 4.04

0.97 0.97 0.97 0.97 0.82 0.80 0.82 0.80 0.95 0.95 0.95 0.95 0.80 0.77 0.80 0.77

0.87 0.87 0.88 0.87 0.83 0.80 0.84 0.80 0.83 0.82 0.84 0.83 0.81 0.78 0.82 0.78

ANOVA Results The factorial ANOVA is completed in MINITAB statistical software at the 95% confidence level. To be able to interpret the ANOVA results, the model adequacy should be met. The model adequacy not only requires that residuals be normally distributed, but also that they have a mean of zero and have constant variance. Even if one of these assumptions is not met, a suitable transformation such as, inverse, logarithm, natural logarithm, square root, inverse, inverse square root, etc. should be applied on the response to achieve model adequacy. In the current model, because the P1 response ANOVA residuals are not normally distributed, an inverse transformation is applied on this response. As a result, the model adequacy conditions are met. For example, Table 12.14 presents the ANOVA results for the inverse of P1 and Tables 12.15 and 12.16 illustrate the ANOVA results for P2 and P3 of the initial arrival rate scenario of the system ðkS ¼ kR ¼ 182 pallets/h), respectively. We interpret our results based on Tables 12.14, 12.15 and 12.16 in the following sub-sections. The F-test is used to evaluate the significance of the experimental factor effects. The bigger the F, the more likely it is that the factor is significant. The rows of all significant factors (P \ 0.05) are shown in bold in Tables 12.14, 12.15 and 12.16.

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Table 12.14 ANOVA results for inverse of P1 when kS ¼ kR ¼ 182 pallets/h Source DF Seq SS Adj SS Adj MS F DP SR I/O IR DP*SR DP*I/O SR*IR DP*IR SR*I/O I/O*IR DP*SR*I/O DP*SR*IR DP*I/O*IR SR*I/O*IR DP*SR*I/O*IR Error Total

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 144 159

0.002776 0.002118 0.444524 0.010594 0.000002 0.001342 0.001409 0.000098 0.000212 0.000710 0.000001 0.000002 0.000021 0.000095 0.000000 0.000032 0.463936

0.002776 0.002118 0.444524 0.010594 0.000002 0.001342 0.001409 0.000098 0.000212 0.000710 0.000001 0.000002 0.000021 0.000095 0.000000 0.000032

0.002776 0.002118 0.444524 0.010594 0.000002 0.001342 0.001409 0.000098 0.000212 0.000710 0.000001 0.000002 0.000021 0.000095 0.000000 0.000000

12425.18 9479.28 1989370.57 47412.21 7.75 6007.02 6304.05 437.97 948.21 3177.65 3.35 7.77 95.12 423.15 0.51

P 0.000 0.000 0.000 0.000 0.006 0.000 0.000 0.000 0.000 0.000 0.069 0.006 0.000 0.000 0.475

Four-Way Interactions Four-way interactions occur when three-way interactions differ as a function of the level of a fourth variable. In Tables 12.14, 12.15 and 12.16 the last column (labeled P) indicate whether or not the factor affects the performance measure. If the P value is smaller than 0.05 for any factor, it means that this factor is significant for that performance measure. Because all the P values are greater than 0.05 in Tables 12.14, 12.15 and 12.16, there is no four-way interaction effect on the responses. Three-Way Interactions Three-way interaction effect means that there is a two-way interaction that varies across levels of a third variable. Therefore, the presence of two-way effect might not necessarily be useful and indicative when a three-way interaction effect exists. In Table 12.14, except for the DP*SR*I/O three-way interaction effect, all the other three-way interaction effects are significant on the inverse P1 response. From Table 12.15, we see that only the DP*I/O*IR three-way interaction is significant on the P2 response. For P3, all the three-way interactions—except for the SR*I/O*IR three-way interaction effects—are significant.

Two-Way Interactions When two factors interact, the effect on the response variable of one explanatory variable depends on the specific value or level of the other explanatory variable.

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Table 12.15 ANOVA results for P2 when kS ¼ kR ¼ 182 pallets/h Source DF Seq SS Adj SS Adj MS DP SR I/O IR DP*SR DP*I/O DP*IR SR*I/O SR*IR I/O*IR DP*SR*I/O DP*SR*IR DP*I/O*IR SR*I/O*IR DP*SR*I/O*IR Error Total

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 144 159

0.017774 0.000026 0.941856 0.012569 0.000000 0.003083 0.000085 0.000002 0.000047 0.000964 0.000000 0.000000 0.000008 0.000000 0.000000 0.000017 0.976432

0.017774 0.000026 0.941856 0.012569 0.000000 0.003083 0.000085 0.000002 0.000047 0.000964 0.000000 0.000000 0.000008 0.000000 0.000000 0.000017

0.017774 0.000026 0.941856 0.012569 0.000000 0.003083 0.000085 0.000002 0.000047 0.000964 0.000000 0.000000 0.000008 0.000000 0.000000 0.000000

F

P

149831.59 220.86 7939845.66 105954.18 3.42 25990.37 713.70 18.15 400.25 8127.51 1.47 3.35 70.50 0.09 0.00

0.000 0.000 0.000 0.000 0.067 0.000 0.000 0.000 0.000 0.000 0.227 0.069 0.000 0.761 0.952

Because the lower order factor effects depend upon the presence of higher order factor effects and because there is significant three-way interaction effect, then there will also be significant two-way interaction effects. As seen in Tables 12.14, 12.15 and 12.16 all two-way interaction effects are significant (P \ 0.05) for each response, except the DP*SR interaction effect for the P2 (P [ 0.05).

Main Effects The main effect of a factor is the average change in the output due to the factor shifting from its low level to high level, while holding all other factors constant. Because each factor has a significant interaction effect in each response for at least one level, we assume that all factors have main effects on each response. We also see this result from Tables 12.14, 12.15 and 12.16. In those tables, the P values for all the main effects, I/O, DP, SR and IR are smaller than 0.05. It should be noted that the most significant factor that affects each response is the I/O, because it always has the largest F value in each table. We complete the same analyses for all arrival rate scenarios. The statistically significant and insignificant interaction and main effects based on the arrivals rates and performance measures that are obtained via the ANOVA are summarized in Table 12.17. In Table 12.17 ‘‘I’’ refers to ‘‘statistically insignificant’’ and ‘‘S’’ refers to ‘‘statistically significant’’. From the table we understand that for each arrival rate scenario, the ANOVA assumptions are met by the inverse of P1. For instance, when the arrival rate is 173 pallets per hour, only the DP*I/O*IR and SR*I/O*IR three-way interactions are significant for the inverse of P1. Besides,

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Table 12.16 ANOVA results for P3 when kS ¼ kR ¼ 182 pallets/h Source DF Seq SS Adj SS Adj MS DP SR I/O IR DP*SR DP*I/O DP*IR SR*I/O SR*IR I/O*IR DP*SR*I/O DP*SR*IR DP*I/O*IR SR*I/O*IR DP*SR*I/O*IR Error Total

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 144 159

0.050810 0.000147 0.116272 0.026455 0.000003 0.006504 0.000273 0.000019 0.000001 0.000271 0.000000 0.000001 0.000003 0.000000 0.000000 0.000011 0.200770

0.050810 0.000147 0.116272 0.026455 0.000003 0.006504 0.000273 0.000019 0.000001 0.000271 0.000000 0.000001 0.000003 0.000000 0.000000 0.000011

0.050810 0.000147 0.116272 0.026455 0.000003 0.006504 0.000273 0.000019 0.000001 0.000271 0.000000 0.000001 0.000003 0.000000 0.000000 0.000000

321

F

P

670827.90 1935.10 1535092.44 349271.09 33.29 85864.20 3609.21 252.42 14.62 3571.94 4.21 14.40 42.97 0.45 0.01

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.042 0.000 0.000 0.503 0.914

except DP*SR, all the other two-way interaction effects and main effects are significant for this response (Ekren et al. 2010c).

12.4.1.4 Tukey’s Test The Tukey’s test is employed to find out the statistically significant levels of the factors which provide the best values for the response variables. In our case, the response variables are P1, P2 and P3. The Tukey’s test is a statistical test generally used in conjunction with an ANOVA. It is used to identify those mean values that are significantly different from the others. It is an all pair-wise comparison test which compares the means of every treatment to the means of every other treatment, and identifies where the difference between two means is greater than the standard error. We show the Tukey’s test analysis with an example. After the ANOVA, for example we find that the highest order interaction effect in our case is the three-way interaction for each response of the initial arrival rate scenario. Therefore, for each combination of three factors that are significant, we conduct a Tukey’s test. From Table 12.17, we find that for the inverse P1 response, three combinations of three-way interactions are significant—SR*I/O*IR, DP*I/O*IR, DP*SR*IR. We analyse Minitab Tukey’s test output as follows. First, for a given combination of significant three-way interactions, we sort the experiments in decreasing order of the desired response variable. For example, consider the SR*I/O*IR three-way interactions and the inverse P1 response. Table 12.18 shows a sorted list for this set of interactions and response. Since the

Four-way DP*SR*I/O*IR Three-way DP*SR*I/O DP*SR*IR DP*I/O*IR SR*I/O*IR Two-way DP*SR DP*I/O SR*IR DP*IR SR*I/O I/O*IR Main DP SR I/O IR

I

I I S I

I S S S S S

S S S S

I S S S

S S S S S S

S S S S

P2

I

Inv P1

S S S S

S S S S S S

S S S I

I

P3

S S S S

I S S S S S

I I S S

I

Inv P1

S S S S

S S S S S S

I I S I

I

P2

S S S S

S S S S S S

I S S I

I

P3

S S S S

S S S S S S

I S S S

I

Inv P1

S S S S

S S S S S S

I I S I

I

P2

S S S S

S S S S S S

I S S I

I

P3

S S S S

S S S S S S

I S S S

I

Inv P1

S S S S

I S S S S S

I I S I

I

P2

S S S S

S S I S S S

I S S I

I

P3

S S S S

I S S S S S

I I S S

S

Inv P1

S S S S

I S S S S S

I S S I

I

P2

S S S S

S S S S S S

S S S S

S

P3

Table 12.17 Interaction and main effect results for each performance measure and arrival rate scenario Effects kS ¼ kR ¼ 182 kS ¼ kR ¼ 173 kS ¼ kR ¼ 191 kS ¼ kR ¼ 164 kS ¼ kR ¼ 200 pallets=h pallets=h pallets=h pallets=h pallets=h

S S S S

S S S S S S

I I S S

I

Inv P1

S S S S

I S S S S S

I I S I

I

P2

S S S S

S S I S S S

I I I I

I

P3

kS ¼ kR ¼ 155 pallets=h

S S S S

I S S S I S

I I S I

I

Inv P1

S S S S

I S S S S S

I I S S

I

P2

S S S S

S S S S S S

S S S S

I

P3

kS ¼ kR ¼ 209 pallets=h

322 B. Y. Ekren and S. S. Heragu

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Table 12.18 Inverse P1 for interaction SR*I/O*IR in descending order Exp. SR I/O loc. IR Inv. avg. cycle time (min) 1 2 3 4 5 6 7 8

2 1 2 1 2 1 2 1

2 2 2 2 1 1 1 1

2 2 1 1 2 2 1 1

0.287 0.286 0.271 0.261 0.178 0.176 0.173 0.156

combinations are sorted in decreasing order of inverse P1, we can simply identify the first sorted combination that is statistically different from the best combination. In our case, the first two experiments themselves are statistically different. Note from Table 12.19 that a comparison of the first experiment for which SR, I/O, IR are 2, 2, 2, respectively, with the second for which these values are 1, 2 and 2, indicates a P value of 0.004, which is obviously less than 0.05. Again, because the combinations are sorted from the most desirable (i.e., maximum inverse P1) to the least, once we find the first pair-wise that is significantly different, we know that all the remaining experiments will also be statistically different from those of the best combinations. We thus conclude that the best levels for factors SR, I/O and IR are two each. We repeat this comparison for the other significant, three-way combinations— DP*I/O, IR and DP*SR*IR—for the same response variable, namely inverse P1, one at a time. We find that the best levels of factors for DP*I/O*IR are two each. The same is true for DP*SR*IR. As a result, we conclude that the best response (minimum P1) is obtained at high levels of all the four factors SR, I/O, IR and DP. We conduct a similar analysis for the other response variables, namely P2 and P3. From Table 12.17, we see that only one three-way factor combination (DP*I/O*IR) is significant for the P2. Once again, Tukey’s test results (not shown) indicate that the best response for P2 is obtained at high levels of all the three factors DP, I/O and IR. To find the best level of the remaining factor SR, an examination of its significant two-way interactions SR*I/O and SR*IR, indicates that the P2 is lower at the high level of SR. As a result, we conclude that the best response (P2) is obtained at high levels of all the four factors DP, I/O, IR and SR. An analysis for the P3 response, leads to the same conclusion that the best response is obtained at high levels of all the four factors DP, I/O, IR, and SR. We complete the Tukey’s test analyses for each arrival rate scenario. The results are summarized in Table 12.20. Note that the best level combination of the factors for each arrival rate scenario is obtained at the same level of the four factors corresponding to the 16th experiment in Tables 12.13a, b.

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Table 12.19 All pair-wise comparisons among levels of SR*I/O*IR SR = 1 I/O = 1 IR = 1 subtracted from: SR I/O IR Difference of means SE of difference Adjusted 1 1 2 0.01953 0.000149 130.7 1 2 1 0.10504 0.000149 702.7 1 2 2 0.12993 0.000149 869.2 2 1 1 0.01705 0.000149 114.1 2 1 2 0.02164 0.000149 144.8 2 2 1 0.11442 0.000149 765.4 2 2 2 0.13051 0.000149 873.1 SR = 1 I/O = 1 IR = 2 subtracted from: SR I/O IR Difference of means SE of difference Adjusted 1 2 1 0.085511 0.000149 572.05 1 2 2 0.110396 0.000149 738.52 2 1 1 -0.002483 0.000149 -16.61 2 1 2 0.002107 0.000149 14.09 2 2 1 0.094884 0.000149 634.75 2 2 2 0.110974 0.000149 742.39 SR = 1 I/O = 2 IR = 1 subtracted from: SR I/O IR Difference of means SE of difference Adjusted 1 2 2 0.02488 0.000149 166.5 2 1 1 -0.08799 0.000149 -588.7 2 1 2 -0.08340 0.000149 -558.0 2 2 1 0.00937 0.000149 62.7 2 2 2 0.02546 0.000149 170.3 SR = 1 I/O = 2 IR = 2 subtracted from: SR I/O IR Difference of means SE of difference Adjusted 2 1 1 -0.1129 0.000149 -755.1 2 1 2 -0.1083 0.000149 -724.4 2 2 1 -0.0155 0.000149 -103.8 2 2 2 0.0006 0.000149 3.9 SR = 2 I/O = 1 IR = 1 subtracted from: SR I/O IR Difference of means SE of difference Adjusted 2 1 2 0.004589 0.000149 30.70 2 2 1 0.097366 0.000149 651.36 2 2 2 0.113457 0.000149 759.00

T-value

P-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

T-value

P-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

T-value

P-value 0.0000 0.0000 0.0000 0.0000 0.0000

T-value

P-value 0.0000 0.0000 0.0000 0.0040

T-value

P-value 0.0000 0.0000 0.0000 (continued)

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Table 12.19 (continued) SR = 2 I/O = 1 IR = 2 subtracted from: SR I/O IR Difference of means 2 2 1 0.09278 2 2 2 0.10887 SR = 2 I/O = 2 IR = 1 subtracted from: SR I/O IR Difference of means 2 2 2 0.01609

SE of difference 0.000149 0.000149

Adjusted T-value 620.7 728.3

P-value 0.0000 0.0000

SE of difference 0.000149

Adjusted T-value 107.6

P-value 0.0000

12.4.2 Regression Analysis for the Rack Configuration of the AVS/RS In this section we perform simulation based regression analysis for the rack configuration of the AVS/RS (Ekren and Heragu 2010b).

12.4.2.1 Problem Description As mentioned before, in an AVS/RS, there are two types of transactions arriving into the system—storage and retrieval. Storage transactions refer to the storage of a unit-load from the I/O point to an available location in the racks. Retrieval transactions refer to the retrieval of a unit-load from its current location. All storage transactions are assumed to arrive at the I/O point and all retrieval transactions end at the I/O point. The vehicles realize three-dimensional movements. We assume that travel between tiers takes place on the z-axis; travel between bays takes place on the y-axis; and travel between aisles take place on the x-axis (see Fig. 12.6). The total travel time completed by a vehicle is calculated by the total travel distance divided by the velocity of the vehicle. Similarly, the total travel time by the lift is calculated by the total travel distance divided by the velocity of the lift. Because the vehicle travels in the lift, the total vehicle travel time also includes the lift travel time. Clearly, the vehicle travel time depends on the number of tiers, aisles and bays. To find out the best rack configuration design based on the number of tiers, aisles and bays, we consider five performance measures. These are: the average cycle time of transactions for vehicles and lifts including waiting times (P1), average waiting times of transactions for vehicles (P2), average waiting times of vehicles for lifts (P3), average utilizations of vehicles (P4) and average utilizations of lifts (P5) (Ekren and Heragu 2010b).

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Constraints The AVS/RS warehouse has a limited area. Therefore, there are constraints on the maximum numbers of tiers (T), aisles (A) and bays (B). Besides, the warehouse storage capacity must be greater than 42,000 ULs. Because our aim is to minimize the performance measures, we can assume that the warehouse capacity is 42,000 unit-loads. This capacity is determined by considering the expected maximum storage demand that may occur, during a period of time. In other words, the plantmanager expects that in the AVS/RS, the possible maximum storage position that may be needed in a period is 42,000 unit-loads. Thus we seek the best rack configuration that provides the required number of storage positions with the number of tier, aisle and bay constraints. Because we define the rack configuration of the AVS/RS in terms of the three dimensions, the warehouse shape varies according to T, A and B. If T is small, then the warehouse footprint will be large to accommodate the required number of storage positions. If T is high, then the footprint will be small. Under the first type of design (few tiers) the vehicles may become the bottleneck, because they will need to realize long horizontal travels. However, in the second condition (large number of tiers), the lifts may become a bottleneck because this time long vertical movements will be required. Therefore, the performance of the system will also be affected by the number of vehicles (V) and lifts (L) in the system. We complete the regression analyses under three V-L combinations in the AVS/RS warehouse. They are defined as: 21 vehicles and 7 lifts, 24 vehicles and 8 lifts, and 27 vehicles and 9 lifts, respectively. We also consider 500 transactions per hour arrival rate in a second set of scenarios. We thus analyse the system under six scenarios and obtain 30 regression functions for five outputs (6*5). It should be noted that in each scenario, each lift still has three designated vehicles at its zone (V/L = 3). When we divide the capacity of the warehouse—T*A*B*2*3 = 42,000 unitloads—by six, then we obtain a value of 7,000 for T*A*B. This means that the combination of the three input variables, T, A and B, should be designed in such a way that their product should provide for 7,000 unit-loads. In addition, because there are limited areas on z-, x- and y-axes, it is assumed that the T, A and B can get the maximum values, 8, 65 and 25, respectively. The warehouse has a rectangular shape.

12.4.2.2 Regression Modelling Because the AVS/RS under study is quite complex, it makes it difficult for a manager to identify how the input parameters affect the outputs in the system. Regression analysis enables us to understand and evaluate the performance of the system in terms of the input variables. Regression analysis is a statistical tool for the investigation of relationships between the output and input variables. Usually, one seeks to ascertain the causal effect of one variable on another e.g., the effect of a price increase on demand, or

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Table 12.20 The best experimental combinations for each response of all arrival rate scenarios Response The best experiment Factor levels P1 P2 P3

16 16 16

DP

SR

I/O Loc.

IR

POLL POLL POLL

SDT SDT SDT

Middle Middle Middle

Opportunistic int. Opportunistic int. Opportunistic int.

the effect of changes in the money supply on the inflation rate, etc. To explore such issues, the investigator assembles data on the underlying variables of interest and employs regression to estimate the quantitative effect of the causal variables on the variable that they influence. Regression analysis with a single explanatory variable is termed simple regression; with multiple explanatory variables it is termed multiple regression. We develop a multiple regression analysis. One of the outputs of the system under investigation is the average cycle time of storage and retrieval transactions—P1—measured in minutes. The cycle time is the time between when a request originates until it is completed and so it includes waiting times in vehicle and lift queues. Along with resource utilization, it is a comprehensive performance measure and employed extensively in many industries. AVS/RS cycle times are determined by the storage rack configuration, vehicle movement kinematics, storage policy and vehicle dwell point policy. In this study, the vehicle movement kinematics, storage policy and vehicle dwell point policy are assumed to be fixed (see Sect. 12.3.2). However, the rack configuration is of interest. The input variables are the T, A and B. While T determines the height of the warehouse, A and B determine the footprint of the warehouse.

12.4.2.3 Experiments To develop the regression model, first the levels of the input variables are determined. Then, the simulation model is run for these levels under three different V and L combinations and two arrival rates. For the T input variable four levels—5, 6, 7 and 8; and for the A input variable seven levels—35, 40, 45, 50, 55, 60 and 65 are considered. Here, the level of the B input variable varies according to the T and A values. Each time it is calculated by dividing 7,000 by T*A. For example, if T = 7 and A = 40, then B is 25 (7,000/ (7*40)) and if T = 8 and A = 50, B is 18. It should be noted that due to the warehouse area constraint, the maximum values of A and B can be 65 and 25, respectively. Hence, A can get the minimum value of 35 which is calculated by dividing 7,000 by the product of the maximum values of T and B (7,000/ (8*25) = 35). With the same logic, the minimum value of B becomes 13. Where necessary, the division is rounded up to the closest integer. In addition, we also complete experiments at the boundary levels of B. For example, at B = 25, A is calculated as 7,000/(25*T). In some cases, for example at small values of T and A,

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Table 12.21 The input variable levels and completed experiments T A B T A B T A B 5 5 5

65 60 55

22 23 25

6 6 6 6 6

65 60 55 50 47

18 19 21 23 25

7 7 7 7 7 7

65 60 55 50 45 40

15 17 18 20 22 25

T

A

B

8 8 8 8 8 8 8

65 60 55 50 45 40 35

13 15 16 18 19 22 25

the value of B may be greater than the maximum constraint, 25. Under these conditions those experiments are not completed (e.g., T = 6 and A = 40, B = 29). The levels of the input variables and the completed experiments for each scenario are summarized in Table 12.21. As seen from Table 12.21 a total of 210 experiments are completed with ten replications for each performance measure.

12.4.2.4 Regression Function Analysis In the regression function analysis, we first trace the factors to determine whether or not a curvature exists. A curvature is not detected at any of the input levels. Thus, we model the system by a multiple linear regression. If a curvature existed, then a high order regression analysis would have to be completed. In a regression model, ANOVA analysis is used to decide which terms to include in the function. For example, if there are three input variables under study, then at most seven terms can be included in the regression function. These possible terms are, main effects—T, A, B, two-way interactions—T*A, T*B, A*B and threeway interactions—T*A*B. The ANOVA can suggest which terms to include by providing the statistically significant effect(s). However, in this experimental study, we are not able to complete a full factorial design because of the area constraint for each factor combination (see Table 12.21). Since, we cannot conduct full factorial design, we cannot apply ANOVA (Montgomery 1996). However, we may use four alternative methods for the regression analysis. These are: forward selection, backward elimination, stepwise regression and best subsets (Montgomery 1996). We employ the popular stepwise regression and best subsets methods to evaluate the possible regression functions and determine the best one. Stepwise regression is the combination of forward selection and backward elimination methods. The purpose of the stepwise regression method is to find a meaningful subset of independent input variables which predict the dependent variable correctly. At each iteration, the terms that must be included or excluded in the model are reassessed using their partial F statistics. The term excluded in the

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Table 12.22 The stepwise regression results by MINITAB for the average cycle time output (k ¼ 450) V = 21, L = 7 V = 24, L = 8 V = 27, L = 9 Step 8 Constant T T-value P-value T*A T-value P-value B T-value P-value T*B T-value P-value R-Sq R-Sq (adj)

Step 4 1.0045 0.283 6.81 0.000 0.00112 -2.4 0.018 0.070 6.29 0.000 -0.0069 -3.67 0.000 90.66 90.28

Step 5

A T-value P-value B T-value P-value T T-value P-value

0.4441 0.0088 4.28 0.000 0.0318 3.73 0.000 0.088 3.48 0.001

R-Sq R-Sq (adj)

86.18 85.62

T*B T-value P-value A*B T-value P-value T*A T-value P-value B T-value P-value R-Sq R-Sq (adj)

0.8733 0.00288 7.24 0.000 0.00024 2.94 0.004 0.00132 8.07 0.000 0.0176 4.35 0.000 94.97 94.72

model with the largest partial F statistic larger than FIN is added to the model. The term included in the model with the smallest partial F statistic, smaller than FOUT is removed from the model. Terms can enter the model and be removed from the model more than once. The best subset method finds the possible n best subsets of i terms (i = 1, 2, …, k) of the regression model. For each subset, it calculates the coefficient of determination—R2 and adjusted R2-R2adj values, so that we can choose a subset that has a good balance of high R2adj and small number of terms. R2 provides a measure of how well outputs are likely to be predicted by the regression model. The bigger the value, the better fit the model. However, only considering R2 is not adequate to evaluate a regression function because the R2 value always increases with the addition of a new input variable to the function, even if it is not significant. If the R2adj value is significantly lower than R2, it normally means that one or more explanatory variables are missing. Therefore, usually R2adj value is used for evaluating a regression function and it is preferable for R2adj to be large and close enough to the R2. Stepwise Regression Results The statistical analyses are completed in MINITAB at a 95% confidence level. The stepwise regression results of P1 of each V, L combinations scenario for arrival rate 450 obtained by MINITAB are shown in Table 12.22.

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Table 12.23 The best subset regression results of the average cycle time output for k ¼ 450 (V = 21, L = 7) T T T A * Terms R2 T A B * * * A R2adj A B B * B 1 1 2 2 3 3 4 4 5 5 6 6 7

86.0 85.6 89.4 87.2 90.1 89.7 90.7 90.4 90.8 90.7 90.8 90.8 92.3

85.9 85.5 89.2 86.9 89.8 89.4 90.3 90.0 90.3 90.2 90.3 90.2 91.7

X X X X X X X X X X X X X

X X X

X

X X

X X

X X X X X X X X

X X X X X X

X X X X X X X

X X X X

X X X

According to Table 12.22, the iterations in scenarios 1, 2 and 3 terminate at the 8, 4 and 5th steps, respectively. Terms having P-values smaller than 0.05 are statistically significant terms. MINITAB shows the significant terms, after completing the stepwise regression analysis. According to Table 12.2, for the first scenario, V = 21 and L = 7, the results show that four terms are significant on response at the 95% confidence level and should be in the regression function. These terms are, T, T*A, B and T*B (P-value \ 0.05). The R2 and R2adj values are also seen at the end of the table as 90.66 and 90.28%, respectively. For the second and third scenarios, the results show that three and four terms are significant on response at 95% confidence level, respectively and should be in the regression function. These terms are, A, B, T and T*B, A*B, T*A, B for the second and third scenarios, respectively. The R2 and R2adj values are 86.18 and 85.62%, for the second scenario, and 94.97 and 94.72% for the third scenario, respectively. Because all the R2 and R2adj values are relatively large and close enough to each other, we assume that the models are acceptable. However, we also complete the usual regression analysis for the model, using these significant terms. Namely, we form the regular regression function with those significant terms and evaluate the R2, R2adj and P-values. From the regression analyses, the first scenario’s regression function’s R2 and R2adj values are obtained as 90.7 and 90.3%, respectively. The second scenario’s R2 and R2adj values are obtained as 85.7 and 85.3%, respectively. And the third scenario’s R2 and R2adj values are obtained as 94.8 and 94.6%, respectively.

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Table 12.24 The subset regression results of the average cycle time output for k ¼ 450 (V = 24, L = 8) T T T A * Terms R2 T A B * * * A R2adj A S B * B 1 1 2 2 3 3 4 4 5 5 6 6 7

82.6 75.3 83.9 83.6 85.7 85.6 86.4 86.2 86.6 86.6 86.7 86.7 87.3

82.4 75.0 83.6 83.3 85.3 85.2 85.9 85.6 85.9 85.9 85.9 85.8 86.4

X X X X X

X X

X X

X

X X X X X

X X X

X

X X

X X X X X X X X

X X

X X

X

X X X X X

X X X X X

X X X X X

We assume that the regression models are good fits. The regression functions of the P1 for the first, second and third scenarios of V-L combinations and for k ¼ 450 transactions/hour obtained by MINITAB are given as in order: P1 ¼ 1:00 þ 0:283 T  0:00112 T  A þ 0:0696 B  0:00688 T  B P1 ¼ 0:285 þ 0:126 T þ 0:00766 A þ 0:0444 B

ð12:27Þ ð12:28Þ

P1 ¼ 0:829 þ 0:00301 T  B þ 0:000133 A B þ 0:00114 T  A þ 0:0204 B ð12:29Þ Best Subset Regression Results The second approach, best subset regression, is also implemented on the outputs. The average cycle time results for three scenarios are shown in Tables 12.23, 12.24, 12.25. In Tables 12.23, 12.24 and 12.25, the R2 and R2adj values are calculated for each alternative design. ‘‘X’’ in the table shows the terms in the regression function. The first column shows the total number of terms that will be in the regression function. From Tables 12.23, 12.24 and 12.25, we can choose the best regression subsets by considering the highest R2 and R2adj values. In each table, when the number of terms in the regression function decreases, the R2 and R2adj values also decrease. However, this decrease is not significant in every condition. For example in

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Table 12.25 The subset regression results of the average cycle time output for k ¼ 450 (V = 27, L = 9) A T T A * Terms R2 T A B * * * * R2adj A B B A B 1 1 2 2 3 3 4 4 5 5 6 6 7

85.4 79.6 91.9 90.6 94.5 94.4 94.8 94.8 95 94.9 95 95 95.1

85.3 79.4 91.8 90.4 94.3 94.2 94.6 94.6 94.7 94.7 94.7 94.7 94.7

X X X

X

X

X

X X X X X

X X X

X X X X X X X X

X X

X

X X X X X X X X

X X X X X X X

X X X X X X X

X X

Table 12.23, if we want to choose the function with five terms, the R2 and R2adj values are 90.8 and 90.3%, respectively. And, these values become 90.7 and 90.3% for the function with four terms. Because the R2 and R2adj values do not vary significantly with the four term regression model, we could choose the function with fewer terms rather than the others. This is because it is always easier and not much confusing to deal with fewer terms in a function. For instance, in Table 12.23 the last subset has seven terms (all the terms) in the regression function and has the largest R2 and R2adj values, 92.3 and 91.7%, respectively. However, usually because all the terms may not be significant on the response, we may want to seek an alternative design with fewer terms. When we go through the alternative regression designs with fewer terms, we observe almost the same R2 and R2adj values with the previous case. The R2 and R2adj values are usually around 90% until the four-term functions. If we also utilize our former method’s result (stepwise regression), then we choose the regression function with four terms. This is because the R2 and R2adj values are high enough and also it includes few terms in the model. Hence, the considered terms are: T, B, T*A and T*B and this function’s R2 and R2adj values are 90.7 and 90.3%, respectively. With the same logic, from Tables 12.24 and 12.25, we choose the best subsets same with the stepwise regression results. All the selected subsets are shown in bold in Tables 12.23, 12.24 and 12.25. According to this, the second and third scenarios have three and four terms respectively, in their regression functions. These terms are: T, A, B and B, T*A, T*B, B*A for the second and third scenarios, respectively.

= = = = =

1.00 ? 0.283*T–0.00112*T*A ? 0.0696*B–0.00688*T*B -0.485 ? 0.204*T ? 0.0481*B–0.000811*T*A–0.00471*T*B 0.0578 ? 0.0225*T ? 0.000237*A–0.000005*T*A*B ? 0.000016*A*B 0.469 ? 0.0213*T ? 0.00588*B–0.000281*T*B ? 0.000032*A*B 0.485 ? 0.0255*T–0.000014*A*B

= = = = =

0.285 ? 0.126*T ? 0.00766*A ? 0.0444*B -0.343 ? 0.000544*T*A ? 0.00316*T*B ? 0.000171*A*B 0.0732 ? 0.0168*T–0.000059*T*B 0.424 ? 0.0212*T ? 0.0035*B ? 0.000085*A*B–0.000016*T*A*B 0.424 ? 0.022*T–0.000012*A*B

= = = = =

P1 = 4.12–0.0886*A ? 0.608*T ? 0.00291*A*B–0.0174*T*B P2 = 1.95–0.0745*A ? 0.387*T ? 0.00353*A*B–0.111*B P3 = 0.0597 ? 0.0229*T ? 0.000312*A

77.7 78.3 99.8

0.829 ? 0.00301*T*B ? 0.000133*A*B ? 0.00114*T*A ? 0.0204*B -0.380 ? 0.000837*T*B ? 0.0144*B ? 0.000693*T*A 0.0674 ? 0.0144*T–0.000055*T*B 0.279 ? 0.0131*T ? 0.000903*A ? 0.0058*B 0.402 ? 0.0149*T–0.000779*A ? 0.000099*T*A

V = 21, L = 7 k ¼ 500 transactions/h

P1 P2 P3 P4 P5

V = 27, L = 9 k ¼ 450 transactions/h

P1 P2 P3 P4 P5

V = 24, L = 8 k ¼ 450 transactions/h

P1 P2 P3 P4 P5

Regression functions V = 21, L = 7 k ¼ 450 transactions/h

Table 12.26 Regression functions and optimum values for all the scenarios

94.8 86.9 99.8 96.3 99.7

85.7 82.3 99.8 92.7 100

90.7 89.6 99.9 97 100

R2 (%)

76.8 77.5 99.8

94.6 86.5 99.8 96.2 99.6

85.3 81.8 99.8 92.5 100

90.3 89.2 99.9 96.8 100

R2adj (%)

= = = = =

6, 6, 5, 7, 5,

A A A A A

= = = = =

65, 65, 55, 63, 65,

B B B B B

= = = = =

T = 6, A = 65, B = 18 T = 6, A = 65, B = 18 T = 5, A = 55, B = 25

T T T T T

18 18 25 16 22

18 18 25 15 22

18 18 25 18 22 = = = = =

= = = = =

B B B B B

B B B B B

65, 65, 55, 65, 65,

65, 65, 55, 65, 65,

= = = = =

= = = = =

A A A A A

A A A A A

6, 6, 5, 7, 5,

6, 6, 5, 6, 5,

= = = = =

= = = = =

T T T T T

T T T T T

Optimum points

(continued)

3.534 min. 1.562 min. 0.191 min.

2.121 min. 0.240 min. 0.133 min. 0.520 0.458

2.338 min. 0.411 min. 0.150 min. 0.599 0.517

2.771 min. 0.780 min. 0.171 min. 0.710 0.592

Optimum value

12 A New Technology For Unit-Load Automated Storage System 333

= = = = =

0.104 ? 0.0125*T*B ? 0.019*A -0.893 ? 0.131*T ? 0.0232*B ? 0.000282* A*B 0.0585 ? 0.0212*T–0.000079*T*B ? 0.00001*A*B 0.44 ? 0.00872*T ? 0.00391*B ? 0.000017*T*A*B 0.472 ? 0.0247*T–0.000014*A*B

P1 P2 P3 P4 P5

= = = = =

0.66 ? 0.036*B ? 0.00165*T*A ? 0.00301*T*B -0.613 ? 0.0207*B ? 0.00108*T*A ? 0.00174*T*B 0.0721 ? 0.0171*T–0.000075*T*B 0.314 ? 0.0156*T ? 0.00101*A ? 0.00637*B 0.360 ? 0.024*T ? 0.000004*T*A*B

V = 27, L = 9 k ¼ 500 transactions/hour

P1 P2 P3 P4 P5

V = 24, L = 8 k ¼ 500 transactions/hour

P4 = 0.567 ? 0.0193*T ? 0.004*B ? 0.000043*A*B P5 = 0.540 ? 0.0284*T–0.000017*A*B

Table 12.26 (continued) V = 21, L = 7 k ¼ 500 transactions/h

92.7 90.9 99.7 96.5 99.7

86.8 78.6 99.8 95.6 100

97.6 100

92.5 90.6 99.6 96.4 99.7

86.4 77.9 99.8 95.5 100

97.6 100

T T T T T

T T T T T

= = = = =

= = = = =

6, 6, 5, 7, 5,

6, 6, 5, 6, 5,

A A A A A

A A A A A

= = = = =

= = = = =

65, 65, 55, 65, 55,

65, 65, 55, 65, 65,

B B B B B

B B B B B

= = = = =

= = = = =

18 18 25 15 25

18 18 25 18 22

T = 6, A = 65, B = 18 T = 5, A = 55, B = 25

2.277 min. 0.369 min. 0.148 min. 0.584 0.508

2.689 min. 0.641 min. 0.168 min. 0.682 0.575

0.805 0.659

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The R2 and R2adj values are 85.7, 85.3, 94.8 and 94.6% for the second and third scenarios, respectively. It should be noted that the fitted functions in (12.27–12.29) have high enough R2 and R2adj values. They also meet the regression model adequacy. If the R2 and R2adj values were not as large as we would like and/or the model adequacy was not met, then we would apply a suitable transformation on the output(s) and/or try including higher order term(s) in the regression model. However, the current models in (12.27–12.29) meet all the requirements. Hence, we accept those regression functions as good fits. It should also be noted that the functions in (12.27–12.29) reflect the output for only the limited values of T, A and B (see Table 12.21). For example, for T = 5, A can be between 55 and 65, and B can only be between 22 and 25. For T = 6, A can only be between 47 and 65, and B can only be between 18 and 25. For T = 7, A can be between 40 and 65, and B can be between 15 and 25. For T = 8, A can be between 35 and 65, and B can be between 13 and 25. The other values will not be represented by the regression function. In addition, the chosen values for A, T and B should also provide T*A*B = 7,000 unit-loads. For example, for the values of T = 8, A = 58, and B = 15 from (12.27) we obtain the cycle time as 2.95734 min. From the function in (12.27) we see that whereas T*A and T*B have negative effects on the output, T and B have positive effects. From the functions (12.28 and 12.29) we see that all the significant terms have positive effects on the output. We also minimize all the regression functions. We optimize each function by considering the constraints explained in Constraints in LINGO. We complete the same procedure for all the V-L combinations for two arrival rates—450 and 500. The obtained regression functions and their optimum points are summarized in Table 12.26. From Table 12.26, we see that for instance, the optimum points of the performance measures, P1—P5, of the first scenario of V and L combinations— V = 21, L = 7—with k ¼ 450 transactions/hour are T = 6, A = 65, B = 18; T = 6, A = 65, B = 18; T = 5, A = 55, B = 25; T = 6, A = 65, B = 18 and T = 5, A = 65, B = 22, respectively. Their optimum values at those optimum points are given in the last column. Several interpretations from these results are summarized below: • The optimum points are typically obtained at the boundary conditions on the A in each scenario. This is probably because of the zoning policy of the AVS/ RS. Because each zone has its own I/O point at the middle of its location and the transactions arrive at those I/O points, traveling from I/O to the x-axis is shorter than traveling from I/O to a long y-axis. Therefore, increasing the A (x-axis) decreases the average cycle time. In other words, it is more efficient to have a long x-axis and a short y-axis. • The optimum number of tiers for the functions, P3 and P5 for each scenario is 5. This is because those performance measures represent the average waiting time of transactions for lifts and average utilizations of lifts and, it is obvious that the

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minimum waiting time and utilization for lifts can be obtained by short vertical travels (small T). • In the low traffic rate condition (k ¼ 450 transactions/h), the optimum average utilizations of vehicles are obtained with seven lifts. However, when the traffic rate is high ðk ¼ 500 transactions/h) the optimum average utilizations of vehicles are obtained at T = 6 except when L = 9. As a result, the manager may want to choose the rack design in which T = 6, A = 65 and B = 18 because it is the optimum rack design that is obtained in many of the scenarios.

12.5 Conclusion In this chapter we develop analytical and simulation models to investigate the performance of an AVS/RS. In the analytical section, we model the AVS/RS via an SOQN. The study on SOQNs has been going on for more than two decades. According to that, we assume that the AVs are pallets and transactions are customers in the system. Because of the limited number of vehicles in the system the AVS/RS can be effectively modelled as an SOQN. Moreover, we consider general service time stations in the system having small scv. Because the obtained network has load-dependent stations and the existing techniques are for load-independent generally distributed network, first we extend Marie’s (1980) approximation to load-dependent generally distributed networks (see Sect. 12.3.3.1). Second, we apply that model in SOQN modelling of the AVS/RS and obtain its performance measures via the AAM given in Sect. 12.3.3. We also solve the system using the MGM as shown in Sect. 12.3.4, analytically. Four different performance measures for the AVS/RS system are observed both from the analytical and simulation models. These are, the external queue length (Leq), average number of transactions (vehicles) in the network including waiting for service (Ln), average number of vehicles in the vehicle pool (Lp) and average waiting time in the external queue (Weq). We compare the two techniques‘performances based on the simulation results. The results show that the algorithm works better under heavy traffic conditions than low traffic conditions in the AAM. We can estimate the key performance measures—the Ln and Lp values—within 5% the simulation estimates under any condition. The MGM solution can estimate the performance measures better than the AAM. In the simulation modelling in Sect. 12.4.1, we model the AVS/RS using simulation-based DOE. Our aim here is to identify the predefined factors affecting the performance of AVS/RS. First, we find out the best V-L combination among the predefined scenarios of the system. Then, we apply DOE on this AVS/RS. We consider four factors which are DP, SR, I/O locations and IR; and, three different responses which are the storage and retrieval transactions’ average cycle time,

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average utilizations of vehicles and lifts. However, because the ANOVA assumptions are not met for the average cycle time response, an inverse transformation method is applied on this response. The statistical results are analyzed in MINITAB. They show that all factors have significant effects on the responses at a 95% confidence level. After determining the main and the interaction effects, a Tukey’s test analysis is completed on the responses to determine the best levels of the factors. The 48th experiment which is the high level combinations of the factors is found to be the best combination for all three performance measures. In Sect. 12.4.2, we present regression analysis for rack configuration of the AVS/RS under three V—L and two arrival rate scenarios. By developing a regression function, we gain a better understanding of the relationship between the output and the input variables. In the regression analysis, we consider five performance measures (the average cycle time of the storage and retrieval transactions, average waiting times of transactions for vehicles, average waiting times of vehicles for lifts, average utilizations of vehicles and average utilizations of lifts), and three input variables (T, A and B) that pertain to the warehouse configuration. The simulation model of the system is developed using ARENA 12.0. The statistical analyses are completed in MINITAB statistical software. We obtain the possible optimum points of the regression functions. We use the LINGO software to minimize the functions. The optimum points and the functions’ values at those points are also provided. In many cases, the results suggest that the warehouse design be as long as is practically possible in the x-axis. This study can be extended in many directions by considering an optimization procedure using OptQuest in ARENA. For instance, the rack design and the number of AVs and lifts can be considered as decision variables at the same time in OptQuest. Hence, numerous design options can be evaluated simultaneously. In addition, other simulation based optimization procedures can also be integrated in the study. Finally, it is noted that the performance comparison of AVS/RS and its alternative systems (e.g., AS/RS) should also be considered. Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. CMMI 0522798 and CMMI 0946706. We are grateful for the support. Parts of this paper have been reproduced from journal papers written by the authors included in the reference section, namely Ekren and Heragu (2010a), Ekren and Heragu (2010b), Ekren et al. (2010b) and Ekren et al. (2010c).

References Avi-Itzhak B, Heyman DP (1973) Approximate queueing models for multiprogramming computer systems. Oper Res 21:1212–1230 Bozer YA, White JA (1984) Travel time models for automated storage and retrieval systems. IIE Transact 16:329–338 Buitenhek R, van-Houtum G-J, Zijm H (2000) AMVA-based solution procedures for open queuing networks with population constraints. Ann Oper Res 93:15–40

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Buzacott JA, Shanthikumar JG (1993) Stochastic models of manufacturing systems. New Jersey, Prentice-Hall Dallery Y (1990) Approximate analysis of general queuing networks with restricted capacity. Perform Eval 11:209–222 Ekren BY, Heragu SS (2010a) Approximate analysis of load dependent generally distributed queuing networks with low service time variability. Eur J Oper Res 205:381–389 Ekren BY, Heragu SS (2010b) Simulation-based regression analysis for the rack configuration of an autonomous vehicle storage and retrieval system. Int J Prod Res 48:6257–6274 Ekren BY, Heragu SS, Krishnamurthy A, Malmborg CJ (2010a) An approximate solution for semi-open queuing network model of autonomous vehicle storage and retrieval system. Working paper Ekren BY, Heragu SS, Krishnamurthy A, Malmborg CJ (2010b) Matrix-geometric solution for semi-open queuing network model of autonomous vehicle storage and retrieval system. Working paper Ekren BY, Heragu SS, Krishnamurthy A, Malmborg CJ (2010c) Simulation based experimental design to identify factors affecting performance of AVS/RS. Comput Ind Eng 58:175–185 Fdida S, Perros HG, Wilk A (1990) Semaphore queues: modeling multilayered window flow controlmechanisms. IEEE Trans Commun 38:309–317 Fukunari M, Malmborg CJ (2008) An efficient cycle time model for autonomous vehicle storage and retrieval systems. Int J Prod Res 46(12):3167–3184 Fukunari M, Malmborg CJ (2009) A network queuing approach for evaluation of performance measures in autonomous vehicle storage and retrieval systems. Eur J Oper Res 193(1): 152–167 Gordon WJ, Newell GF (1967) Closed queueing networks with exponential servers. Oper Res 15:254–265 Jackson JR (1963) Jobshop-like queueing systems. Manag Sci 10:131–142 Jia J, Heragu SS (2009) Analysis of semi-open queuing networks via analytical matrix geometric methods. Oper Res 57:391–401 Kuo PH, Krishnamurthy A, Malmborg CJ (2007) Design models for unit load storage and retrieval systems using autonomous vehicle technology and resource conserving storage and dwell point policies. Appl Math Model 31(10):2332–2346 Kuo PH, Krishnamurthy A, Malmborg CJ (2008) Performance modelling of autonomous vehicle storage and retrieval systems using class-based storage policies. Int J Comput Appl Tech 31(3–4):238–248 Malmborg CJ (2002) Conceptualizing tools for autonomous vehicle storage and retrieval systems. Int J Prod Res 40:1807–1822 Malmborg CJ (2003) Interleaving rule dynamics in autonomous vehicle storage and retrieval systems. Int J Prod Res 41:1057–1069 Marie R (1979) An approximate analytical method for general queueing networks. IEEE Trans Softw Eng SE- 5:530–538 Marie R (1980) Calculating equilibrium probabilities for k(n)/Ck/1 N queues. ACM Sigmetrics Perform Evaln Rev 9:117–125 Montgomery DC (1996) Design and analysis of experiments, 4th edn. Wiley, New York Neuts M (1980) Matrix-geometric solutions in stochastic modeling. The Johns Hopkins University Press, Baltimore Randhawa SU, Shroff R (1995) Simulation-based design evaluation of unit load automated storage/retrieval systems. Comput Ind Eng 28:71–79 Reiser M (1981) Mean value analysis and convolution method for queueing dependent servers in closed queueing networks. Perfor Eval 1:7–18 Roodbergen KJ, Vis IFA (2009) A survey of literature on automated storage and retrieval systems. Eur J Oper Res 194:343–362 Whitt W (1983) Performance of the queueing network analyzer. Bell Sys Tech J 62: 2779–2815

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Zhang L, Krishnamurthy A, Malmborg CJ, Heragu SS (2009) Variance-based approximations of transaction waiting times in autonomous vehicle storage and retrieval systems. Eur J Ind Eng 3(2):146–168 Zollinger H (1999) AS/RS application, benefits and justification in comparison to other storage methods: a white paper automated storage retrieval systems. Production Section of the Material Handling Industry of America. http://www.mhia.org/industrygroups/as-rs/technicalpapers

Chapter 13

Intelligent Optimization Methods for Industrial Storage Systems Mirko Ficko, Simon Klancnik, Simon Brezovnik, Joze Balic, Miran Brezocnik and Tone Lerher

Abstract The presented chapter introduces intelligent methods, which can be used for designing and managing of modern warehouses. Because of the ever-increasing complexity of such systems, the traditional methods cannot assure optimal or nearoptimal solutions in design and operation. Demands for high utilization, flexibility, and the capacity to work reliably, even in changeable environments, can be met by adding intelligence to artificial system. The most promising intelligent methods are evolutionary computation and swarm intelligence which are unique methods of nondeterministic solving and optimizing. They proved to be effective and robust for planning and management of real systems. Evolutionary computation and swarm intelligence are methods, which were obtained from the observation of nature. Nature has some of the best answers to the problem of design and management. Therefore, this chapter tries to present intelligent methods to wider audience, and especially to experts and students of warehousing design and management.

13.1 Introduction 13.1.1 Difficulties When Planning and Managing Complex Systems Most of the more and more complex and highly demanding artificial systems are still created by humans when designing, an increasing number of requirements have to be considered, leading to systems consisting of great numbers of elements M. Ficko (&)  S. Klancnik  S. Brezovnik  J. Balic  M. Brezocnik  T. Lerher Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_13,  Springer-Verlag London Limited 2012

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and links. Therefore, the design process poses increasing demands, since the solution selected must best suit the user’s requirements. Moreover, due to economical and ecological pressures, not just mere operating but near-optimal operating is expected. Thus, high utilization, flexibility, and the capacity to work reliably, even in changeable environments, are expected from artificial systems. Since the early 1950s, when the conventional theory of system planning and managing was first developed, engineers have developed quite a few distinct methods, based on deterministic rules (Jamshidi 1996), for the analyzing and managing of systems. In today’s complex systems these methods are often of insufficient help. Indeed, a conclusion can be made that no mathematically demonstrable theory can ensure solutions (Rodd et al. 1992). Analytical solutions cannot be reached primarily because of: • • • • • •

Process instability Combinations of continuous and serial operations Incomplete and/or too extensive information Unidentified processes and events Changing processes Disturbances.

In complex industrial systems there are too many unknown parameter sizes and too many approximations. Therefore, each parameter cannot be controlled and thus appropriate actions for each value or combination of the parameters is unavailable. Many parameters, however, are mutually dependent and these dependencies are, in most cases, unclear. Furthermore it is impossible to determine a mathematical description of the problem. Without an exact description a definite solution cannot be reached, which would ensure near-optimal operation. Even in cases with exact mathematical descriptions it is often hard or even impossible to obtain a solution.

13.1.2 Intelligent Optimization Methods in Warehousing Design and Management In general, we can distinguish between applications of intelligent optimization methods in planning and design stage, and operating of existing warehouse. Planning a modern warehouse with all necessary equipment requires significant investment. Therefore, proper early warehouse planning and design stage is a key to successful operating, from technical and economical viewpoint. Poor planning and layout will decrease warehouse utility and performance while increasing your operational costs. Nowadays most used method for planning and predicting the future warehouse operating is simulation modeling. Simulating a warehouse is based on a computer model and testing it by executing computer-based experiments with different combinations of parameters based. However, simulations serve only as a benchmark and do not include solutions or even optimization. Therefore, we can expect in future intensive development in the field of planning and design of warehouses.

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While operating the existing warehouse we can use intelligent optimization methods to loosen bottlenecks, shorten order cycle times, reduce inventory, increase worker productivity, improve workplace safety, etc. The very common problem is order picking which is the process of collecting and retrieving items from the warehouse in response to a specific order picking list. This problem can be already solved in very good way by artificial intelligence. Order picking is identified as the most labor intensive and costly activity for almost every warehouse in the practice. Consequently, warehousing professionals consider order picking as the most prominent area of productivity improvements with the objective of reducing the warehouse cost. The design of an order picking system is often complicated and complex due to a large spectrum of external and internal factors, which influence the design choices. The most common objective of order picking systems during the design phase is to maximize the efficiency of the order pickers (the definition of the minimal travel distance) and to minimize the cost. The efficiency of the order picker is influenced greatly by using a proper routing policy. Using the intelligent order picking policy for routing order pickers, the reduced meantime for making one batch and in this way reducing the cost of order picking, would be applied. Because of the complexity of order picking systems (non-linearity, discrete and multi-variable fitness function), the methods of artificial intelligence are suitable for development of the new order picking policy for routing order pickers. Methods of artificial intelligence can take into account also other objectives, such as minimization of the throughput time of processing an order, minimization of the labour costs, and maximization of the use of warehouse space, maximization of resource and equipment utilization. Both cases presented in this chapter comprise the problem, which is found at planning and managing of warehouses.

13.1.3 Classification of Systems Each system works within a particular environment and is usually part of a larger system. The core of a system relates to the transformation of objects (system elements) coming into the system from the environment and returning back to it after complete transformation. The division of those systems of interest in relation to simulation is as follows ˇ eric´ 1993): (Banks et al. 1984; C • Static and dynamic systems. The system is defined as dynamic, when the state of the system changes depending on time, while the state of the static system remains unchanged. • Deterministic and stochastic systems. The deterministic system is a system whose state is known in time (t ? 1) with respect to the system state in time (t). In contrast to the deterministic system, the stochastic system is a system, where the system state in time (t ? 1) is unknown with respect to the system state in time (t).

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• Finite and infinite systems. Finite systems are systems reaching their state at a particular moment and remain unchanged. Infinite systems change continuously, depending on time. • Discrete and continuous systems. In discrete systems, the system-state changes only over certain time intervals called ‘‘events’’. When examining the operation of a supermarket, the variables are analyzed in regard to the number of buyers in a waiting queue. The value of the variable can change over a particular timeinterval when a new buyer joins the queue. In continuous systems the systemstate changes, in dependence of time. An example of continuous simulations is the rocket flight, where the position and the speed of the rocket changes continuously independent of time. The systems studied in logistic processes are always dynamic, they may be deterministic or stochastic, finite or infinite, but always discrete (Lerher 2005).

13.1.4 Optimization Process The optimization is a process of searching for the best solution within the space of possible solutions from the model (cost-function) describing the problem. So far, conventional optimization methods (e.g., the gradient-method, hill-climbing method) have proved satisfactory, but have several disadvantages. As they use deterministic operations, they can quickly slide into the local optimum. The costfunction (target function) must also be continuous and derivable. In addition, the complex systems cannot be efficiently optimized when using them. It has been established that deterministic description is only possible in very simple systems or if an exacting system is studied within a very limited range. Basically, there are two fundamental methods when searching for the optimal problem solution: • Analytical problem-solving computation • Solving by tests. In order to solve a system analytically, the entire system including the relationships inside individual subsystems must be described analytically. This method is sometimes impossible due to the complexity and uncertainty of a system (Lerher 2005). Analytical problem solving and/or searching for optimal states is based on a transformation of the technical problem into an analytically written descriptive problem. The next step comprises the search for a solution by analytical methods and then re-transferring the solution to a real problem. Transformation of analytical solutions for this problem is often unreliable, particularly due to the simplification and modification of the analytical model. Therefore, an excellent analytical solution can, however, be incapable of producing a good real-problem solution. Experimental solving may comprise of experimentation on a real system which, however, is often economically unjustifiable due to system complexity and the

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costs incurred during the experimental operation of the system. Nevertheless, this is frequently the only applicable method. Of course, the basic prerequisite is to have a system which exists and is available. Within the system design stages, of course, the system is unavailable. Therefore, another type of experimentation is used, i.e., experimenting with a system model, the principle difference being that the tests are not preformed directly on the system, but on a virtual model. Usually, those systems are not yet existing and therefore allowing study only by simulation or analytical methods. The principle prerequisite is to know the target or costfunction of the system. These two principles of searching for solutions can be adequately illustrated by solving the square equation (where x represents a variable, and a, b, and c, constants): ax2 þ bx þ c ¼ 0

ð13:1Þ

Though this example is trivial, it can be solved by two methods. The first method is analytical treatment giving the solution: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b  b2  4ac x1;2 ¼ ð13:2Þ 2a Using the second method, all possible values of the unknown x would be selected and tested until the solution is obtained. The set of values x is called the solution space. The number of possible solutions in the solution space is so large as to forbid an exhaustive search for the optimal solution. Of course, it would not make sense to search for the roots of the square equation in this way but, in reality, there are many problems that cannot be solved by application of deterministic principles. The selection of a possible solution, and its verification afterwards is one of more universal techniques for solving problems where the sequence of steps required to reach the solution cannot be determined in advance. It is of use almost everywhere, where the target function is known. Searching can be affected by blind strategies or by heuristic strategies. The blind-search strategies do not use information about the problem, while the heuristic search strategies use additional information for determining the best search directions. Therefore, when applying search strategies, it is necessary to distinguish: • utilization of the best solution and • searching in the search space. The up-hill climbing strategy is an example of strategy utilizing the best solution when searching for possible improvement, and neglecting the remaining search space. Random-searching is an example of solution searching which searches for the solution within the entire search space, and neglects the promising space in the search space.

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13.1.5 Artificial Intelligence For the most part, especially regarding simple systems, a suitable solution (scenario) on the basis of known events can be prepared, but the number of solutions is in most cases uncontrollable, therefore, all possibly needed solutions cannot be prepared in advance. The disturbances coming from the external world and from the system itself require new scenarios. If the disturbance is of unknown form and source, an unpredictable event and/or conflict-ridden situation is in question. As in many cases, an answer to this problem can be reached by observing nature; as in nature the structurally complex systems need added intelligence (Brezocˇnik 2001; Brezocˇnik and Balicˇ 1997) to deal with new situations. Over recent years, researchers have managed to develop different methods of artificial intelligence which deals with such problems and were successfully included in manmade systems. For the time being, it would be difficult to speak about intelligent systems; however, their behavior can be called intelligent since they do not use the pre-defined solutions. Needed solutions are made with the use of non-deterministic methods. The use of intelligent methods in industrial system allows for an efficient answer to changes within the economic-technical world, since its aim is: • efficient use of natural resources, and lower pollution • more economical and user-friendly production • greater independence of the production system. Although an industrial system requires high efficiency and reliability simultaneously which, in the same way, is in disagreement with the traditional perception of non-deterministic rules, nevertheless artificial intelligence methods have priority during incorporation into industry, since the inaccuracy is inseparably associated with the thinking model of the rule’s creator––human (Rodd et al. 1992). Within the industrial environment, information for model formation is more often unclear and/or cannot be correctly measured. Often, human reduce the quality and quantity of information (Singh 1996). All these are reasons why the potential of artificial intelligence methods in the planning and managing process of artificial systems must be re-considered. Since the beginning of artificial intelligence research in the 1950s, certain more or less usable applications of artificial intelligence have operated within production practice, efficiently solving industrial problems. Today, most attention is attracted by solutions based on evolutionary computation, neural networks, fuzzy-logic, and swarm intelligence. Fuzzy-logic is used for monitoring, recognition, quantitative analyses (statistics), and consequence rule (export systems for diagnostic, planning prediction, and acquisition of information). Neural networks are used for the approximation of functions, prediction, classification and pattern recognition in data processing. From among artificial intelligence methods, the evolutionary compotation and swarm intelligence are more frequently used for optimization. Swarm intelligence algorithm functioning is very similar to other evolutionary computation methods. The principal common property is the searching for

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solutions (optimum) using algorithmic iterations. Thus, some author’s groups swarm intelligence into evolutionary algorithms (Engelbrecht 2005). This context uses the evolutionary algorithm which states that an evolutionary algorithm is an optimization process imitating biological evolution, where the solution can be described within three basic operations: mutation, selection, and recombination. Since these operations are absent in swarm intelligence they are classified into separate groups featuring not only the biological, but also the social-biological behavior of individuals.

13.2 Evolutionary Computation 13.2.1 General Evolutionary Computation For many real problems, the simulation of natural evolution has proved to be a better approach than conventional methods. Evolutionary computation is the term used to describe computer-based problem-solving systems which use computational models of evolutionary processes as key elements in their design and implementation (Spears et al 1993). Over recent decades, a considerable number of new methods, based on Darwinistic principles, have been introduced for the studying, planning, optimizing, and predicting complex systems. The idea of evolutionary computation was introduced by Reichenberg (1974) in his work ‘‘Evolutionary strategies’’. His work was then continued by other researchers. Thus in (1975) John Holland developed genetic algorithms and 15 years later John Koza (1992) introduced genetic programing. To date, a group of evolutionary algorithms have been researched, among them the most popular being: • • • •

Genetic algorithms Genetic programing Evolutionary programing Evolution strategies.

All of them are based on the Darwinistic principle, and are commonly called methods of evolutionary computation. They are based on simulating the evolution of individual structures via processes of selection, mutation, and reproduction. The evolution of these structures is based on the following facts: • Individuals in populations compete for resources and partners. • Individuals, more successful in mutual competition, will have more offspring than the less successful ones. • Genes of successful individuals propagate in the population so that two successful parents can create descendants who are better than their parents. If the basic principles of evolution and genetics are moved to artificial, virtual environments, they can be beneficially used to improve artificial structures.

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Table 13.1 Explanation of terms used by evolutionary computing Genetic algorithms

Explanation

Chromosome or organism Gene Locus Alleles Phenotype Genotype

Coded solution Part of solution Location of gene Values of the gene Decoded solution Encoded solution

Evolutionary computational terminology is usually used in the evolution of biological systems. Table 13.1 indicates the terminology used in evolutionary computation. The structure improved by artificial evolution is the problem solution, while the problem or target function is the environment in which the organism is evaluated. Evolutionary computation maintains a population (group) of structures, which evolve according to the rules of selection and other operations, such as recombination and mutation. Each individual in the population receives a measure of its fitness in the environment. Reproduction focuses attention on the fitness values of individuals. Although simplistic from a biologist’s viewpoint, these algorithms are sufficiently complex to provide robust and powerful adaptive search mechanisms. Figure 13.1 shows the general algorithm or pseudo-code of evolutionary computation. The known successful examples of using evolutionary computation are in • • • • •

Planning and time scheduling Vehicle and robot control Preparation of floor layout plans Transport management Creation of functional dependence among variables.

13.2.2 Genetic Algorithms Hereinafter, attention will be focused on genetic algorithms (Goldberg 1989), which are the most outstanding and more frequently used approaches of evolutionary computation. They were developed by John Holland (1975) in the 1960s and early 1970s. These are adaptive heuristic search algorithms used for solving demanding search and optimization problems. The principal steps of the method out of which steps 2–4 are repeated in each generation are: 1. Creation of an initial generation of organisms. 2. Evaluation of organisms by means of a fitness function. 3. Selection of organisms which solve the problem above average in population.

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Fig. 13.1 Pseudo-code of evolutionary algorithms

4. Creation of a new generation by crossover, reproduction, and mutation. New generations are created until enough high-quality solutions have been found. Genetic algorithms differ from conventional optimization methods and search procedures (Mitchell 1997): • They work with coded solutions and not with solutions themselves. • They search for solutions from both population of solutions and individual solutions. • For evaluating the quality of a solution they use cost-functions and not approximations. • Genetic algorithms use probability and not deterministic rules. The greatest advantage of genetic algorithms is that, within a very large search space, they find the solution very quickly by efficiently avoiding the local optimum. Genetic algorithms search for solutions very efficiently, since they use a combination of directed and general search. Some other advantages are: • For optimization they do not need special mathematical preparatory steps. Because of their evolutionary nature they search for the solution irrespective of the problem’s contents. They can work at searching for the optimum of any function, even the non-continuous ones. • Genetic algorithms are very efficient at searching for global solutions. • Genetic algorithms are very flexible at searching for solutions and are applicable to a very wide area. • Genetic algorithms can search for solutions to problems consisting of complex interdependent parts, and would normally be very hard to model (Mitchell 1997). • Genetic algorithms are applicable for time-variant problems. They progress to the optimum, but do not interrupt the search when it has been reached (Drstvenšek 1998). Thus, they can deal with optimizing problems within a changing environment.

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Therefore, genetic algorithms are totally applicable for the optimization of multi-parameter functions, and the solving of combinatorial problems. Both types of problems frequently appear within the industry. Figure 13.2 shows the general processes of genetic algorithms. Genetic algorithms treat one set of possible solutions at a time. Solving an optimization problem (usually) starts with a set of randomly created possible solutions. Each randomly generated organism represents a more or less accurate solution of the optimization problem. Then, the organisms are evaluated. The greater probability of cooperating in the selection and variation operations is assigned to those organisms that represent more accurate solution of problem. The selection operation assures the survival of fitter individuals of population and their advance into the next iteration, called generation. The variation operation has an effect on one or more parental organisms and their offspring are created from them. After completion of selection and variation, a new generation is obtained that is also evaluated. The process is repeated until the termination criterion of the process is fulfilled. This can be a prescribed number of generations or a sufficient quality of solutions.

13.2.2.1 Coding of Solutions When working with genetic algorithms, the solution must be suitably coded into organisms. Figure 13.3 shows an example of solutions coding into an organism. The organisms consist of genes representing part of the solution. Several different kinds of coding are available depending on the type of problem solved. The principal coding methods are: • • • •

Binary coding––the organism is a set of binary values Permutation coding––the organism is a set of natural numbers (Fig. 13.3). Coding with values––organism is a set of values Tree coding––the organism consists of genes interconnected into hierarchical relationships (used particularly on genetic programing).

13.2.2.2 Evolutionary and Genetic Operators The operations, which are executed on organisms, are called ‘evolutionary’ and ‘genetic’ operations. The evolutionary operator of selection decides which organisms (solutions) will progress into the next generation. The two genetic operators, ‘crossover’ and ‘mutation’ execute the process of inheriting the genes, i.e., they create the offspring in each generation. Evolutionary and genetic operators can perform in very different ways, depending at most on the problem’s nature and the coding used. In general, there are no rules on how to use these operators.

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Fig. 13.2 General process of genetic algorithms

Selection is the motive force in the development in genetic algorithms. It leads to searching into those areas of the search space promising the best solutions. It works in a similar way to that already presented by Darwin. Reproduction assigns greater probability to more successful (more fit) organisms which are transferred into the next generation either changed by genetic operations or unchanged (reproduction). The best organisms are selected by different methods but most often by the methods of: • Roulette wheel • Ranking • Tournament.

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7

4

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6 10 1 8

3

B

11

5

coding

A

coding of possible solution of travel from A to B

10 9 11 gene organism (coded solution)

Fig. 13.3 An example of permutation coding for a possible solution of the combinatorial problem when searching for a path from A to B

The crossover operation takes place on two organisms, out of population, at a time by combining their genes. The crossover ensures the exchange of genes among organisms. Two descendants are created from two parental organisms. The two descendants produced become the members of the next generation. Even better descendants may be produced by combining certain parts of good organisms. There are several different manners regarding crossover. The simplest crossover manner is affected by cutting the genotype of both parental organisms at a random but same point of crossover, and then by combining the parts (Fig. 13.4). Mutation, on the other hand, introduces new genetic material at the level of genes into the organisms (Fig. 13.5). The aim of the mutation is the conservation of variety inside the population and avoiding premature convergence (to a suboptimal solution). In general, the mutation (Gen 1997): • replaces the genes in organisms by lost genes omitted from evolution but able to contribute to better results in other organisms • introduces into the population those genes which were absent in the initial generation. Use of sole selection leads to filling the entire population by copies of the best individual. Selection and crossover lead to a good but suboptimal solution. Mutation itself implies random motion in the search space, but together with selection and without crossover, it causes too slow an approach to the solution. In most cases a combination of all three operations are the most appropriate.

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Fig. 13.4 Crossover in Gas

Fig. 13.5 Mutation in genetic algorithms

13.2.3 Case of Search for Optimal Row Based Layout The problem of creating an optimal or at least near-optimal layout for manufacturing system is designated as the floor layout problem. The general definition of floor layout problem is ‘‘the determination of relative locations for, and the allocation of, available space among a number of workstations’’ (Azadivar and Wang 2000). The problem of finding an optimal or near-optimal layout for is one of the NP-hard combinatorial optimization problems (Heragu and Kusiak 1988). Applicable mathematical solutions for such a type of problem do not exist (Garey and Johnson 1979). Therefore, an analytical approach is unsuitable. The complexities of such problems increase exponentially with the number of devices. Most often, it is derivable with such layouts to find layouts or objects in rows, so that the total path with required frequencies of travel between objects will be the shortest. The fitness function is in this case the sum of variable costs over a time period. For determination of fitness function value, it is necessary to know the transport quantities between the individual devices N within time period. Also the variable transport costs, depending on the transport means used, must be known. Thus, different transport costs per unit length also result. In order to find, by means of these data, the optimum layout of the devices N, it is necessary to find the minimum of the following fitness function: f ¼

N X N X i¼1 j¼1

fij  cij  Lij

ð13:3Þ

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fij is the frequency of trips between the locations i and j, cij is the variable transport costs for the quantity unit, and Lij is the length of path between the locations i and j. The number of all locations is equal to N. Fitness function heavily depends on the distances Lij between the locations. The value of the cost-function is, thus, the sum of all values obtained for all the pairs of devices. The complete procedure of optimization using genetic algorithms is divided into these main steps (Ficko et al. 2004): • Acquisition of the technological and physical conditions needed. • Determination of layout by genetic algorithm (determining the sequences of devices and rows). • Calculating the coordinates of devices and mutual operation points. • Determinating of the distances between every possible pair of devices. • Calculating of the value of cost-function. For such a manner of solving of the problem, it is necessary to know the dimensions of devices and the minimum allowable distances between all the pairs of devices. Furthermore it is necessary to know the transport quantities between the individual devices during a certain time period. In addition, the variable transport costs used must be known, depending on the transport means. It is also necessary to know the width of transport paths and the greatest length of the row. Figure 13.6 shows the general procedure of optimization with genetic algorithm. In the first step of solving using genetic algorithms, the initial population is created at random. Only correct organisms are created, representing a feasible solution. This initial generation enters the evolutionary-loop of the genetic algorithm. The evaluation itself is not linked to genetic algorithm. Firstly, the actual layout is formed on the basis of the organism genotype and afterward evaluated in the ordinary way. The selection of organisms using the roulette wheel method is made after evaluation of the population. In the next step the operations of reproduction and crossover with probability pr and pc, follow respectively. The operation of mutation is executed with probability pm. Thus a new population is obtained. When the genetic algorithms cannot improve, the layout evolution is stopped, and a best layout, according to fitness function, is taken as a solution of the problem. The layout is then evaluated by the expert with respect to the criteria not included in the cost function, and in the technological database. Each organism represents one possible solution of the problem regarding arranging, and each gene represents one device. The most natural coding for such a solution is permutation coding (Gen 1997). Such a type of coding, together with wisely chosen genetic operators, ensures the legality and feasibility of all organisms. In this way, the efficiency of the overall optimization procedure is increased. The sequence of genes in the organism is equal to the sequence of working devices in the layout of rows, where the gene represents the device i and its position in the organism represents the position of the row. However, such a gene would represent the arrangement in one row only. Therefore, the arrangement into rows is determined based on the parameter of length of row and technological limitations. The number of devices in one row is limited to the maximum length of row a.

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Fig. 13.6 The main steps of genetic algorithm procedure for solving of floor layout plan

Fig. 13.7 Decoding of organisms into valid solution

When the length of the row is greater than a, the next device is placed into a new row. The procedure is repeated until all devices have been arranged into rows. Such a manner of coding guarantees that all organisms are correct, even after completion of genetic operations (Gen 1997). The whole coding procedure is shown in Fig. 13.7. The method of selection for reproduction and crossover is roulette wheel selection. In this type of selection the organisms which represent better solution have a better possibility of taking part in the next generation.

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Fig. 13.8 Partial mapped crossover

Many genetic operators for crossover exist for the type of coding implemented in this research. The partial mapped crossover was selected for the crossover, which was proposed by Goldberg and Lingle (1985) and is shown in Fig. 13.8. Partial mapped crossover can be viewed as an extension of two-point crossover. In addition, it uses a special repairing procedure to resolve the illegitimacy caused by a simple two-point crossover. This type of crossover has been widely used in the field of combinatory problems (Tavakkoli-Moghaddain and Shayan 1998). Organisms reached by the operation of reproduction and crossover are further modified using the mutation operation. Reciprocal mutation was selected as the

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Fig. 13.9 Value of cost function during several runs of evolution

mutation operator. The two randomly selected genes in the original organism change places. Therefore, the offspring organism represents the feasible solution. No procedure is needed for correction of the organism. The fitness of each layout is evaluated using the above-mentioned fitness function. The best organism represents the solution with the lowest value for the fitness function. Until the evaluation, no information is available about the solution other than the sequence of devices. The arrangement into rows is determined for evaluation of the individual organisms. Conversion takes place in the organism representing the sequence itself, as shown in Fig. 13.7. The coordinates of the points of operation are determined on the basis of the layout. When calculating coordinates the conditions and limitations from the technological database are also taken into account. The test example with 14 locations was used for testing the system. Problem of such size (N = 14) yields at least 14! possible solutions, which is impossible to solve by trying all possible solutions. In the first step the technological database was filled with all necessary data. The evolution was run several times. After three evolutions, as presented on the Fig. 13.9, we have four different near-optimal solutions. If the solutions reached are evaluated, it can be seen that the model prepared similarly good layouts, especially if it is known that the worst solutions had up to five times higher cost values. The genetic algorithm itself does not ensure optimum solutions, but may yield near-optimum solutions. Our model proved to be successful in the search for optimal layout in rows. The resulting layout can be either layouts in single-rows or multiple-rows. The model does not limit itself to one solution only, but can propose several equally good solutions, which can differ greatly.

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13.3 Swarm Intelligence 13.3.1 General Information About Swarm Intelligence Swarm intelligence is the discipline that deals with natural and artificial systems composed of many individuals that coordinate using decentralized control and self-organization. In particular, the discipline focuses on the collective behavior that results from the local interactions of individuals with each other and with their environments. Examples of systems studied by swarm intelligence, and used in optimization methods are: • • • • •

Colonies of ants and termites Bee colony Shoals of fish Flocks of birds Herds of land animals.

13.3.2 Particle Swarm Optimization Particle swarm optimization is a population-based stochastic optimization technique developed by Eberhart and Kennedy in 1995, inspired by the social behavior of a flock of birds (Engelbrecht 2005). Particle swarm optimization shares many similarities with evolutionary computation techniques such as genetic algorithms (Hassan and Weck 2005). The potential solutions, called particles, fly through the problem space. Each particle represents a candidate solution to the optimization problem. The position of a particle is influenced by the best position visited by itself (i.e., its own experience), and the position of the best particle in its neighborhood (i.e., the experience of neighboring particles). When the neighborhood of a particle is the entire swarm, the best position in the neighborhood is referred to as the best global particle, and the resulting algorithm is referred to as the gbest algorithm. When smaller neighborhoods are used, the algorithm is generally referred to as the lbest algorithm. The performance of each particle (i.e., how close the particle is to the global optimum) is measured using a fitness function that varies depending on the optimization problem. Over several years past, particle swarm optimization has been successfully applied in many research and application areas. It has been demonstrated that particle swarm optimization gets better results in a faster, cheaper way when compared to other methods. Another reason that method is attractive is that there are few parameters to adjust. One version, with slight variations, works well within a wide variety of applications. Particle swarm optimization has been used for approaches that can be used across a wide range of applications, as well as for specific applications focused on a specific requirement.

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13.3.2.1 Particle Swarm Algorithm Supposing the following scenario: a group of birds are randomly searching for food over an area. There is only one piece of food in the area being searched for and not all the birds know where the food is. However, they know how far the food is in each iteration. The effective strategy for finding the food is to follow the bird which is nearest to the food. In algorithm, each single solution is a ‘‘bird’’ or particle within a search space. All the particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles. Particle swarm optimization is initialized with a group of random particles (solutions) and then searches for optima by updating generations. Within the iteration, each particle is updated by following the two better values. The first one is the best solution (fitness) achieved by the particle so far. This fitness value is also stored and is called pbest. The another ‘‘better’’ value tracked by the particle swarm optimizer is the best value obtained so far by any particle in the population. This better value is globally the best and called gbest. After finding the two better values, the particle updates its positions using the equation: xi ¼ xi þ vi

ð13:4Þ

and velocity by Eq. 13.5: vi ¼ vi þ c1 randð Þðpi  xi Þ þ c2 Randð

Þðpg  xi Þ

ð13:5Þ

In Eqs. 13.4 and 13.5, the designations have the following meanings: • • • • • •

c1 and c2 are learning factors rand () and Rand() are independent random numbers between (0,1) xi = (xi1, xi2, …, xiD) is a ith particle (D is the dimension of the particle) pi is the best position for the ith particle in the history (pbest) pg is globally the best position among all particles (gbest) vi is the particle velocity.

Equation 13.5 consists of three parts. The first part of the equation is inertia, the second part represents personal influence, and the last part represents social influence. The pseudo-code of the procedure is shown in Fig. 13.10. The algorithm performs the update Eqs. 13.4 and 13.5, repeatedly, until a specified number of iterations have been exceeded, or velocity updates are close to zero (Fig. 13.11). The quality of particles is measured using a fitness function, which reflects the optimality of a particular solution.

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Fig. 13.10 The Pseudo-code of particle swarm optimization algorithm

Fig. 13.11 Flow chart for the particle swarm optimization algorithm

13.3.3 Ant-Colony Optimization For a better understanding of an artificial ant-colony intelligence, it is necessary to state some of the rules of the life cycle of natural ants with which it is desirable to draw a parallel between the natural, and artificial intelligence. The following describes relevant facts from the nature of ant colonies, which is fruitfully exploited in the case of artificial ants. Most ants are ‘‘social’’ insects that live in family communities or colonies, and usually build large nests. With all ‘‘social’’ insects a group problem-solving ability has been observed, based on indirect communication––insects communicate by changing their local environment. Example: When foraging, some species of ants are capable of choosing the shortest path from the anthill to food. During the trip, they leave behind a chemical trail (a pheromone) that attracts other ants, so they often choose the path that is richer in pheromone. Since the ants that chose the shortest route return first, the shorter routes contain more pheromone than the longer. This indirectly stimulates

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other ants from the anthill to choose the shortest path. Such communication was studied by the French entomologist Pierre-Paul Grasse (1895–1985) during 1950s and he named it stigmergy as a neologism of Greek words stigma (send), and ergos (work).

13.3.3.1 Ant-Colony Algorithm In nature, ants are able to find the shortest route from food source to the nest without using visual information. In addition, they are able to adapt to changes in the environment, for example, to find a new shorter path when the current one is interrupted by obstacles. Ants move along this path connecting the food source to their nest. They use pheromones for communication. The more popular routes contain a large quantity of pheromone. When a barrier appears, the ants cannot continue following the path. In this case, it can be expected that half of the ants turn right and the other half left. A similar situation arises on the other side constraints. Interestingly, those ants, which incidentally choose a shorter route around the obstacle, reconstruct the interrupted path more quickly than those that chose the longer route. Therefore, the shorter path will gather more pheromone per unit of time and over time, the number of ants choosing the shorter route will increase. Ants tend to give priority to paths with a higher content of pheromone, which is why the pheromone gathers more quickly on shorter paths (Fig. 13.12). Optimization using ant-colony optimization is part of a larger area, which researches algorithms based on swarm intelligence, and deals with the algorithmic approaches that have been derived from the behavior of ant colonies, and other social insects. Of particular relevance are the collective activities, such as gathering food, caring for the brood, and building nests, which are the mechanisms of self-organization, communication, and task distribution. Ant-colony optimization is modeled on the behavior of ants searching for food. Ants use special pheromone traces to mark the path to a source of food. Optimization using an ant-colony is used to solve difficult combinatory optimization processes, such as a square allocation problem, the problem of classification, traveling salesman problem, and dynamic steering problems. Unfortunately, it is difficult to theoretically analyze the ant-colony algorithms; the main reason for this is that they are based on sequences of random decisionmaking, which is not usually independent and where the probability distributions vary from iteration to iteration.

13.3.3.2 Ant-System The ant-system is the ancestor of all research on ant-algorithms. The ant-system uses graph representation, together with costs d(r, s) and the desirability of s(r, s), called pheromone. Ants renew the pheromone during execution. If the ant-system used in a symmetric travelling salesman problem (TSP) type is s(r, s) = s(r, s), the

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Fig. 13.12 The ants choose the shorter route from the anthill to food

asymmetric species, it is possible that s(r, s) = s(r, s). The ant-system works in such a way that each ant generates the entire path by selecting sites based on the rule of transition positions (state transition rule)––ants prefer to move to locations associated with shorter connections (lower costs), and with larger amounts of pheromone. When all the ants made their way, the rule of global pheromone update (global pheromone updating rule) is used––part of the pheromone evaporates on all connections. Then each ant places the pheromone in proportional amounts on the connections belonging to its path, relative to its path’s length (connection, which belongs to many shorter paths, receives a larger amount of pheromone). The process is then repeated. The standard transition state is called a random proportions rule (random-proportional rule) and is shown in Eq. 13.6. This rule gives the states of the transition probability with which ant k in city r chooses the place in which to move (Di Caro and Dorigo 1998; Dorigo and Gambardella 1997; Dorigo et al. 1996; Gambardella et al. 1999): 8 b > < ½sðr; sÞ  ½gðr; sÞ ð13:6Þ pk ðr; sÞ ¼ ½sðr; uÞ  ½gðr; uÞb ; if is s ¼2 Jk ðrÞ > : 0

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Fig. 13.13 Pseudo-code of ant colony-system

where s pheromone, g = 1/d is the inverse cost value, Jk(r) is the set of points that remain to be visited by ant k and are associated with r (so that the solution is possible), and b is a parameter that determines the relative importance of pheromone compared to the cost (b [ 0). In Eq. 13.6, if the amount of pheromone on the connection (r, s) is multiplied with the corresponding heuristic value (r, s), priority is given to the selection of connections that are shorter and have a greater amount of pheromone. The global update rule is implemented as follows: when the ants build their way, the pheromone is updated on all the connections based on Eq. 13.7: ð1  aÞ  sðr; sÞ þ

sðr; sÞ

m X

Dsk ðr; sÞ

ð13:7Þ

k¼1

where  Dsk ðr; sÞ ¼

1 Lk

;

if is ðr; sÞ 2 path from ant k 0

ð13:8Þ

0 \ a \ 1 is the pheromone decay parameter, Lk is the length of the path created from ant k, and m is the number of ants. The pheromone updating is intended for the allocation of large quantities of pheromone to shorter paths. Pheromone updating formula is designed to imitate the changing quantities of the pheromone due to the addition of new routes, which were visited by ants and pheromone evaporation (Pesl et al. 2006). Pheromone placed on the connections plays the role of long-term distributed memory; this memory is not stored locally within the individual ants, but is distributed across the connections of the graph, which allows for an indirect way of communication. The time required to solve major problems is inadequate although the system is useful for finding good solutions for small traveling salesman problem (to 30 cities). Three major changes have, therefore, been proposed to improve the algorithm, which led to the definition of ant-colony system (Fig. 13.13).

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Fig. 13.14 Communication of food location through ‘‘threshold’’ dance

13.3.4 Bee-Colony Optimization Colonies can cover an area of more than 10 km where they have access to large amounts of food. A colony successfully develops a method for successfully gathering food over a large area. Flowers, which contain large amounts of nectar and pollen, are visited by more bees than these flowers containing less nectar or pollen. When the food gathering process is initiated, the bee-colony sends out scouts to find an area with a lot of food (many flowers). Scout-bees inspect random flowers. When the scout-bees return to the hive, they drop the pollen or nectar and do some kind of ‘‘threshold’’ dance, by which they communicate three important pieces of information about the flowers: directions to the field, precise distance from the hive to the food, and an assessment of the food location’s quality. This dance is essential for communication within the colony. This type of communication is important for a precise organization of the bees because they do not need a guide or map to find the site with a lot of food. Each individual piece of information communicated by scout-bees to the hive is gathered exclusively through dance. Dancing allows the colony to evaluate the quantities of the flowers, the quality of the food that will be gathered, and the amount of energy needed for gathering. After the dance, the scout-bee returns to the location together with those bees she was able to convince with her dancing. The more convincingly the scout-bee dances, the more bees she convinces to follow her. These things allow the colony an effective and rapid gathering of food. The gathering period depends on the amount of food. If there is enough food at a certain location, the bees will point it out at the next dance (Fig. 13.14), and thus attracts even more bees, which will help them gather food (Smart 2002; Jones et al. 1999; M and D 2007).

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Fig. 13.15 The general pseudo-code for the bee-colony algorithm

13.3.4.1 Bee-Colony Algorithm The bee-colony algorithm is an optimization algorithm, which is taken from wild bees and ensures the optimal solutions. The pseudo-code in Fig. 13.15 presents the bee-colony algorithm in a simple form. The following parameters are needed for the operation of this algorithm. The number of scout-bees (n), the number of locations outside the selected (n) locations (m), the number of good locations outside (m), selected locations (e), the number of bees (good gatherers) (e) locations (nep), the number of bees (other gatherers) (m-e) selected locations (nsp), initial number of flowers (ngh), including the near-by locations, and determining the program-stopping criteria. The algorithm starts with n number of scout-bees, which are randomly distributed across the search area. Evaluation of sites that are visited by scout-bees takes place in the second step. In the fourth step, bees with the highest estimated value are selected, called ‘selected bees’. Selected-bees visit selected locations in the surrounding area. In the fifth and sixth steps in the search algorithm, more bees search the best e selected locations. Bees can be selected directly based on the estimated value and the selected locations, which the bees visit. The estimated value is a criterion by which the bees choose. The bees gathering food at good e locations have more chances or probability of being be joined by other bees than the bees gathering food in bad locations. Information about good and bad locations is shared by dancing in front of the hive. In addition to scouting, differential recruitment is the key operation of a bee-colony algorithm (Fig. 13.16).

13.3.5 Case of Search for an Optimal Unconstrained Layout of Objects With Particle Swarm Optimization In the previous section, the case of searching for optimal row-based layout with genetic algorithm was presented. Here; an identical problem will be solved, the optimal layout of objects will be searched for without being limited to row-layout, but the system can propose a random layout (position and rotation) of objects inside the available space (Ficko et al. 2010). Our goals were limited to one of the

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Fig. 13.16 Demonstration of the bee-colony algorithm in pseudo-code

more important criteria; minimal cumulative length for all the transport parts or, in other words, minimal transport costs. The evaluation function given in Eq. 13.3 is used for evaluation of the solution. Because an analytical solution for this type of problem does not exist (Wiendahl and Fu 1992; Garey and Johnson 1979), it was decided to use a method which does not create an optimal solution, but searches for it. After the evaluation of different optimization methods, it was decided to use particle swarm optimization. This decision was made because the algorithm is fairly simple and fast (Curkovic and Jerbic 2007). Particle swarm optimization is more computationally efficient (uses a lesser number of function evaluations) than the genetic algorithm (Hassan et al. 2005). On the other hand, it is different from other evolutionary methods, in such a way that it has no evolutionary operators, such as crossover and mutation. Another advantage is its ease of use with fewer parameters to adjust. In general, this system is composed of two subsystems. One of them serves for creating layout (subsystem for creating a layout), and the second serves for evaluating a layout corresponding to the lengths of those paths between devices (subsystem for evaluation). During the first step, a swarm of layouts is created by

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Fig. 13.17 Schematic representation of the system working

Fig. 13.18 Schematic representation of particle

the subsystem on a random basis, whereas in all following steps the layouts are made following arithmetic operators. The forming of the layout with the proposed system is shown in Fig. 13.17. Steps 2 and 3 are repeated until a good solution is found. The coding of solutions into layouts/particles is carried out on the following principle: a particle is composed of central positions and orientations for all devices in the system, as presented in Fig. 13.18. In the particles, the values of the devices’ positions are presented as real numbers, so special coding is unnecessary. For the evaluation of layout, it is necessary to find the shortest path between two points in a 2D space (Fig. 13.19), which represent the serving points of the device. This method works on the following principle: in the first step the starting node A is examined then all the neighbors of A are examined, followed by all the neighbors of all the neighbours of A, and so on, until the desired target node has been reached or until there are no nodes left to be examined (in this case no path exists). We need to keep a track of all the nodes to ensure that no node is processed more than once. This is accomplished by linking the field ‘‘Status’’ with all the nodes. The outline of the algorithm is as follows: 1. Initialize all nodes to the ready state (Status = Ready) 2. Put the starting node A in a queue and change its status to the waiting state (Status = Waiting) 3. Repeat steps a and b until the queue is non-existent: 4. Remove the first node N of the queue. Process N, and change the status of N to the processed state (Status = Processed) 5. Add to the rear of the queue all those neighbors of N in the ready state (Status = Ready) and change their status to the waiting state (Status = Waiting) 6. Exit

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Fig. 13.19 Path finder result

Fig. 13.20 Convergence of the solution

13.3.5.1 Results and Discussion It was decided to use a larger space than necessary to avoid any unfeasible solutions in the first phase of the optimization process. In the preliminary runs of the system, a combination of parameters was searched for with an acceptable probability of success. The parameters depend largely on the characteristics of each particular problem. Initially, the first runs showed good convergence ability but after fine tuning of the subsystem for creation, a better one was obtained. Figure 13.20 shows the process of creating good layout for the test case.

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The results show the ability of the proposed system to obtain very good solutions for loosely constrained layout. As such, the proposed system can be used as a decision-support tool for the human expert.

13.4 Conclusion Evolutionary computation and swarm intelligence are subjects, which describe two unique methods of non-deterministic solving and optimizing. Cases of searching of optimal layout in multiple-rows and non-constrained the optimal layout according to the cost or time function solved by genetic algorithms and particle swarm intelligence obviously show advantages of this approach. This problem is widely present also in warehousing design and management. Traditional methods do not assure effective solution. Solutions are usually formed from expert ‘‘by feeling’’. In today’s modern world, we cannot be satisfied by this process. The so called ‘‘intelligent methods’’ offer solution for this problem. Evolutionary computation and particle swarm intelligence proved to be very effective and robust for planning and management of real systems, which are dynamic and include some uncertainty. Although these methods have advantages because of non-deterministic manner users try to avoid them. This is mainly without reason because of ignorance; proper inclusion may produce better results. Therefore, this chapter tries to present intelligent methods to wider audience, and especially to experts and students of warehousing design and management. In future, we can expect many successful implementations on this field.

References Azadivar F, Wang J (2000) Facility layout optimization using simulation and genetic algorithms. Int J Prod Res 38(17):4369–4383 Banks J, Carson J, Nelson BL, Nicol D (1984) Discrete-event system simulations. Georgia Institute of Technology, Atlanta Brezocˇnik M (2001) A genetic-based approach to simulation of self-organizing assembly. Robotics Comp Integrated Manuf 17(1–2):113–120 Brezocˇnik M, Balic J (1997) Comparison of genetic programming with genetic algorithm. Proceedings of 3rd international conference design to manufacture in modern industry. Portorozˇ, Slovenia Cˇeric´ V (1993) Simulacijsko modeliranje. University of Zagreb, Zagreb Curkovic P, Jerbic B (2007) Honey-bees optimization algorithm applied to path planning problem. Int J Simul Model 6(3):154–164 Di Caro G, Dorigo M (1998) AntNet: distributed stigmergetic control for communications networks. J Artif Intell Res 9:317–365 Dorigo M, Gambardella LM (1997) Ant colonies for the traveling salesman problem. BioSystems 43:73–81 Dorigo M, Maniezzo V, Colorni A (1996) The ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern 6:29–41 Drstvenšek I (1998) Model tehnološke baze obdelovalnih operacij v postopkih optimiranja rezalnih pogojev z uporabo genetskih algoritmov. University of Maribor, Maribor

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Engelbrecht A (2005) Fundamentals of computational swarm intelligence. Wiley, New York Ficko M, Brezocnik M, Balic J (2004) Designing the layout of single- and multiplerows flexible manufacturing system by genetic algorithms. J Mat Process Technol 157–158:150–158 Ficko M, Brezovnik S, Klancnik S, Balic J, Brezocnik M, Pahole I (2010) Intelligent design of an unconstrained layout for a flexible manufacturing system. Neurocomputing 73(4–6):639–647 Gambardella LM, Taillard ED, Dorigo M (1999) Ant colonies for the quadratic assignment problem. J Opl Res Soc 50:167–176 Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, New York Gen MC (1997) Genetic algorithms and engineering design. Wiley, New York Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. AddisonWesley, Reading Goldberg D, Lingle R (1985) Alleles, loci and the traveling salesman problem. 1st Conference on genetic algorithms, pp. 154–159 Hassan R, Weck C (2005) Comparison of particle swarm optimization and the genetic algorithm. 46th AIAA/ASME/ASCE/AHS/ASC structures: structural dynamics and materials conference. Austin, Texas Hassan R, Cohanim B, Weck O (2005) Comparison of particle swarm optimization and the genetic algorithm. 46th AIAA/ASME/ASCE/AHS/ASC structures: structural dynamics and materials conference. Austin, Texas Heragu S, Kusiak A (1988) Machine layout problem in flexible manufacturing systems. Oper Res 36(2):258–268 Holland J (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor Jamshidi M (1996) Large scale, systems: modeling, control, and fuzzy logic. Prentice-Hall, New Jersey Jones JL, Flynn A, Seiger BA (1999) Mobile robots. Natick: A. K. Peters cop, Massachusetts Koza J (1992) Genetic programming: on the programming of computers by means of natural selection. MIT Press Cambridge, Massachusetts Lerher T (2005) Simulacijski model visokoregalnega skladišcˇnega sistema (Doctor thesis izd.). University of Maribor, Maribor Mitchell TM (1997) Machine learning. The McGraw-Hill Companies, New York Pesl I, Zumer V, Brest J (2006) Optimization by ant colonies. J Electrotech Rev 73(2–3):93–98 Rechenberg I (1974) Evolutionsstrategie: optimierung technischer systeme nach Prinzipien der biologischen evolution. Frommann-Holzboog Verlag, Stuttgart Rodd M, Verbruggen H, Krijgsman A (1992) Artificial intelligence in real-time control. Eng Appl Artif Intell 5(5):385–399 Singh N (1996) Systems approach to computer-integrated design and manufacturing. Wiley, New York Smart WD (2002) Making reinforcement learning on real robots. Dissertation. Department of Computer Science, Brown University Spears WM, Jong KA, Bäck T, Fogel DB, Garis HD (1993) An overview of evolutionary computation. Proceedings of the European conference on machine learning, vol. 667, 442–459. Springer, London Tavakkoli-Moghaddain R, Shayan E (1998) Facilities layout design by genetic algorithms. Comp Ind Eng 35(3–4):527–530 Wiendahl EHP, Fu Z (1992) Computer-aided analysis and planning of set-up process. Ann CIRP 41(1):497–500

Part III

Applications and Case Studies

Chapter 14

Correlated Storage Assignment and Iso-Time Mapping Adopting Tri-Later Stackers. A Case Study from Tile Industry Riccardo Manzini, Filippo Bindi, Emilio Ferrari and Arrigo Pareschi

Abstract The process of picking products from storage locations to fill customer orders is a very critical and labour-intensive logistic activity, especially in the presence of many orders made of a few orderliness. The storage strategy is one of the main factors affecting the order picking system performance. Object of this chapter is the development and application of an original and systematic procedure for the minimisation of the variable travelling cost in a picker-to-part high-level order picking system with tri-lateral stackers. This procedure takes inspiration from a correlated storage assignment strategy developed and applied by the authors, and is based on the introduction of similarity indices, as measures of correlation between products, the application of hierarchical clustering algorithms and positioning rules. The authors propose different storage allocation/positioning rules supported by an iso-time mapping of the storage area. A case study from tile industry is illustrated and the results of a what-if analysis are discussed.

14.1 Introduction Warehousing and the relative order picking process are an essential component of any supply chain (SC). Advanced approaches to improve order picking efficiency can reduce customer response time in a SC, decrease overall costs and improve related customer service level. Gu et al. (2007) present a classification of main problems and topics dealing with the design of the systems and the management of the operations in a R. Manzini (&)  F. Bindi  E. Ferrari  A. Pareschi Department of Industrial Mechanical Plants, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_14,  Springer-Verlag London Limited 2012

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warehousing system. In the warehouse design the problems classified by Gu et al. (2007) and de Koster (2007) deal with decisions such as setting the overall structure (sizing and dimensioning), designing the department layout, selecting the equipment and identifying the operations strategy. Typical warehouse operations are: receiving and shipping, storage (including stock keeping unit—SKU department assignment, zoning and storage location assignment), order picking including batching, routing and sequencing and sorting. Each problem involves a set of critical decisions. For example the storage location assignment problem deals with the determination of the SKU positioning and the determination of the best location of products within the storage area. Similarly the order picking problem deals with batching (batch size and orderbatching assignment), routing and sequencing (order picking tours definition, dwell point selection in automated storage and retrieval systems—AS/RS) and sorting (order-lane assignment). In particular, the order picking (OP) is the process of retrieving products from a storage area in response to a specific customer request: incoming items are usually received and stored in large-volume unit loads while customers order small volumes of different products. These are the so-called less than unit loads systems (Manzini et al. 2007). de Koster (2004) distinguishes two main classes of order picking systems (OPS). 1. Manual systems. There are three different subcategories of systems employing humans: • picker-to-parts systems: high-level picking systems with man-aboard and low-level picking systems (labour-intensive and manual pick) • put system • parts-to-picker system (AS/RS miniloads, carousels, etc.). 2. Mechanized/automated systems. There are two main subcategories of systems employing machines: • automated picking (e.g. dispensers, fully automated AS/RSs, etc.) • robot picking. A framework for the classification of OPSs is illustrated in Fig. 14.1. This chapter focuses on the development and application of original models and tools for the correlated storage assignment in OPS. In particular this study takes inspiration from the approach proposed by Bindi et al. (2009) illustrating an application of the correlated storage assignment strategy by the adoption of a similarity-based systematic procedure in a low-level picker-to-parts OPS (see Fig. 14.1). Literature does not present significant case studies yet, models and methods to support the storage allocation in a high-level picker-to-parts OPS, where the problem of locating products within the storage area is three-dimensional (3D), and in presence of a correlation between products.

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Fig. 14.1 Classification of the OPSs (de Koster 2004, 2007)

The study examined in this chapter has been developed in presence of a tri-later stacker and modelling the distance between two generic locations, P1 and P2, adopting the Chebyshev metric defined as: dcheby ðP1 ; P2 Þ ¼ maxfjx1  x2 j; jy1  y2 jg

ð14:1Þ

where (x1, y1) and (x2, y2) are the coordinates of P1 and P2 respectively. The aim of this chapter is to demonstrate and quantify the expected benefits from the application of some original storage allocation and positioning rules suited for 3D picker-to-part order picking systems. A case study is provided to demonstrate how the proposed rules can be used in an actual production system to reduce the overall order picking time and improve performance. The Chebyshev metric is also the basis for the modelling of not fully (and fully) automated parts-to-picker systems, e.g. AS/RS (Manzini et al. 2005, 2006). The remainder of the chapter is organised as follows. Section 14.2 introduces the storage assignment problem and illustrates main existing strategies as proposed by the literature. Section 14.3 presents the proposed storage assignment rules and storage positioning rules as patterns for the correlated storage assignment. Section 14.4 presents a case study and discusses the obtained results from an experimental what-if analysis. Finally Sect. 14.5 presents conclusions and further research.

14.2 Storage Assignment Problem The storage assignment problem deals with the assignment of products to storage locations in order to identify which is the best location for the generic item (Cahn 1948), called SKU. The assignment problem has been formalised by Frazelle and Sharp (1989) and classified as a non-polynomial (NP) hard problem.

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A list of typical indices adopted to rank the SKUs for the assignment of storage locations follows. • Popularity (P). This is defined as the number of times an item belongs to an order in a given set of picking orders which refer to a period of time T: Pi; T ¼

X order j in period T

xij

ð14:2Þ

where 

1 if item i occures in order j xij ¼ 0 otherwise   xij product  order incidence matrix • Cube per order index (COI). This can be defined as the ratio of volume storage to inventory for the generic SKU to the average number of occurrences of SKU in the order picking list for a given period of time (Haskett 1963). Given an SKU i, COI is defined as the ratio of the volume of the stocks to the value of its popularity in the period of interest T. Formally: COIi; T ¼ P

vi; T order j in period T

xij

ð14:3Þ

where vi,T average storage level of SKU i in time period T. • Order closing index (OC). Order Completion (OC) assignment is based on the OC principle introduced by Bartholdi and Hackman (2003). Bindi (2010) introduces the Order Completion rule based on an index called OC index that evaluates the probability of a generic item being part of the completion of an order, composed of multiple orderlines of different products (items). The OC index is the sum of the fractions of orders the generic item performs. For a generic SKU and a time period T, OC is defined as follows: X f ð14:4Þ OCi; T ¼ j ij;T where fij;T

mð jÞ

xij mðjÞ order j in P period T xzj z¼1 ¼

number of order lines for the picking order j:

ð14:5Þ

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Table 14.1 Incidence matrix (numerical example) Order 1 Order 2 Order 3

Order 4

Order 5

Item Item Item Item Item Item

1 0 0 0 0 0

0 0 0 0 0 0

1 2 3 4 5 6

1 0 1 0 1 1

1 1 0 0 0 0

0 1 0 1 0 1

Table 14.2 Order closing OC evaluation (numerical example) Order 1 Order 2 Order 3 Order 4

Order 5

Sum

Item Item Item Item Item Item

0 0 0 0 0 0

7/4 5/6 1/4 1/3 1/4 7/12

1 2 3 4 5 6

1/4 0 1/4 0 1/4 1/4

1/2 1/2 0 0 0 0

0 1/3 0 1/3 0 1/3

1 0 0 0 0 0

Table 14.1 presents an exemplifying item-order incidence matrix in presence of six different items and five customer orders. Item 1 belongs to Order 1 and at the same time three different items belong to the same order, so the Item 1 has a 1/4 of fraction of Order 1. The OC index for Item 1 is the sum of all fractions related to all the orders, which is equal to 7/4 in the example (see Table 14.2, column ‘‘Sum’’). According to the previous hypotheses the OC index for a certain item can assume the following special values: – Minimum value = 1/Total Number of Items, when the item belongs to all the customer orders. – Maximum value = number of orders, when the item belongs to all customers orders and there are no other items. • Turn Index (T). Given an SKU i, it is defined as the ratio of the picked volume during a specific period of time T to the average stock stored in T. The index can be written as: P pij j Ti; T ¼ ð14:6Þ order j in vi;T period T

where pij Picked volume of product i in the order j. The unit of measurement is the same as vi, T

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14.2.1 Storage Assignment Policies The literature presents several storage assignment policies that can be classified in one of the following main categories (Van der Berg and Zijm 1999; Manzini et al. 2006, 2007). • Randomised storage. This policy provides for SKUs randomly assigned to the first available space in the warehouse. The random storage policy is widely adopted in the warehousing industry because it is easy to use, often requires less space than other storage methods and uses all the picking aisles intensively. • Dedicated storage. This policy reserves specific locations for each SKU within the warehouse. It requires more space in the pick area for storage but allows the pickers to memorise fixed locations of SKUs producing time labour saving. The choice of dedicated location to assign a generic item follows one of the following rules. – Class-based storage rule. This rule defines several classes as groups of SKUs located in storage areas more or less favourable to satisfy particular criteria. Frazelle (2002) punctually states the two most frequently used criteria used to assign a class of products to storage locations are popularity and the COI as defined above. – Ranked index-based rules. They are based on the ascending or descending values of one of the previously introduced indices e.g. P, COI, OC, or T defined for each SKU. The P-based assignment rule considers a list of items sorted by decreasing value of popularity and assigns the highest of them to the nearest location from the depot area (I/O point) i.e. the most favourable and free (available to receive items) locations. The COI-based assignment rule considers a list of items sorted by increasing value of COI and assigns the lowest of them to the most favourable locations. The OC-based assignment rule arranges items in a similar way to the P-based rule: it considers a list of items sorted by decreasing value of OC index and assigns the highest of them to the most favourable locations. The turn-based assignment rule assigns items in the same way as for the previous OC rule but uses Turn index instead of OC. – Correlated storage policy. This is illustrated in detail in the following section. Literature presents several studies on storage location policies but many of the existing contributions provide performance evaluation comparing only at least a couple of rules and the proposed methods are very often compared to the randomised storage rule, ignoring that popularity is the criterion most widely adopted as class-based storage in warehousing practice (Frazelle 2002). Furthermore, literature is mainly focused on low-level forward-reserve OPS thus lacking in high-level systems where the picking area is spread over several higher levels. This is the case of typical storage systems for niche industrial sector managing medium to high volume items, with high unit value, low turn and few pick lines per order (e.g. ceramic, glass and wood decoration sector).

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The purpose of this study is firstly to present original storage assignment rules based on correlated storage policy. Secondly, this study tests and compares the proposed correlated-based rules with the standard and not correlated-based ones (popularity, COI, turn, etc.). There are several criteria for the performance evaluation of OPS, but literature on man-on-board OPSs considers the travel distance minimisation as a primary objective in warehouse design and optimisation. This is because the travel time for manual system is an increasing function of the travel distance (Petersen et al. 2004). In addition, the authors extend the performance evaluation to other significant indices: the total travel time and the travelled aisles. An evaluation of main logistic costs is also presented. Focusing on a case study based on an OPS served by man-on-board tri-lateral stackers, we also introduce and take into account the acceleration and deceleration effects on the average operation time for a picking tour. In the following sections this concept is widely argued. In particular, the total number of travelled aisles is introduced as a performance index measure of congestions: given a set of retrievals, i.e. orderliness, a lower value of that index might indicate more congestion in the system.

14.3 Correlated Storage Assignment The correlated storage assignment policy locates items with a high degree of correlation close to each other. The measure of correlation between two generic items is usually based on the frequency of being in different picking orders. The allocation of products within a storage area can be based on different types of correlation existing between products. Once the correlation has been calculated for all pairs of products, the couples with the highest value of correlation are stored together. For example, customers may usually order a certain item together with another. These products might reasonably have high correlation and it may be useful to locate them close together within the system to reduce the travelling distance and time during the picking activity. In order to group products, the statistical correlation between them should be known or at least be predictable, as described by Frazelle and Sharp (1989) and Brynzér and Johansson (1996). The approach adopted in this paper has been proposed and illustrated by Bindi et al. (2009) and Manzini et al. (2010) who respectively discuss its application to the OP problem and to cellular manufacturing (CM). It is composed of the following three main steps (see Fig. 14.2): i.

Correlation analysis

This evaluates the degree of correlation (also called similarity) between the items on the picking list for a significant period of time. The level of similarity is usually measured by introducing an index of similarity. The literature presents several indices of similarity. One of them, the Jaccard index (McAuley 1972),

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Fig. 14.2 Systematic procedure for the correlated storage assignment [Bindi et al. 2009]

which is frequently used in statistics and in CM, can be adopted to assess the relationship between SKUs. This step directly influences the shape of the clusters, as some products may be close to one another according to a measure of similarity and further away according to another measure. Given a set of K customer orders in a period of time T, and the two items i and j, a problem oriented similarity index, called Proposed Index, and inspired to the index proposed by Bindi et al. (2009) for the best storage allocation of products in a picker-to-parts low-level OPS, is now defined as:   PK min Turni ðTÞ; Turnj ðTÞ k¼1 aijk   ð14:7Þ  Si; j ¼ PK  max Turni ðTÞ; Turnj ðTÞ k¼1 aijk þ Dðbijk þ cijk Þ where k = 1, …, K historical customer orders in a period of time T

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 aijk ¼

0 1 ¼ 0  1 ¼ 0 

bijk cijk

1

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if both i and j belong to the same order k else if both i belongs to the order k else if both j belongs to the orderk else

Turni(T) Turn value for the product i Turnj(T) Turn value for the product j D corrective parameter D 2 ½0; 1: The value of D depends upon the problem density as the number of zero within the incidence matrix orders/items. In particular, given the products i and j, in case of a lot of orders visiting i and not j, D assumes values close to zero. In the case study illustrated in Sect. 14.5, the value of D is assumed to be 0.25. ii.

Clustering

This second step concerns the clustering techniques used to form groups of similar products (SKUs) in order to ensure that the items within the same group are highly correlated with each other and poorly correlated with those in other clusters. Two of the most popular clustering algorithms are applied in this paper, that is, the farthest neighbour (fn) and the nearest neighbour (nn) as proposed by Aldenderfer and Blashfield (1984). The dendrogram is the graphical representation of the process of degradation of the similarity level as a result of the grouping process executed by the clustering algorithm. Figure 14.3 shows a dendrogram generated by the execution of the clustering process. The results of the clustering analysis depend on the minimum admissible level of correlation adopted for the generic group of clustered items. Consequently, the choice of a threshold group correlation measurement strongly influences the number and configuration of the groups of products, that is, the partitioning of the whole set of items (product mix). In this paper a percentile-based threshold value of similarity, Tvalue%, is adopted as proposed by the authors for cellular manufacturing (Manzini et al. 2010). This value corresponds to a range of group similarity measurements and cuts the dendrogram at the percentile number of the aggregations (nodes of the dendrogram) identified by the adopted clustering rule, as follows: 

  Tvalue% ð%p Þ 2 Lf %p  N g; Lf %p  N ð14:8Þ where N is the number of aggregations necessary to obtain one cluster starting from a number of clusters equal to the number of products, each cluster being composed of a single product. %p is the percentile of aggregations, expressed as a percentage on the whole set of aggregations as generated by the process of

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Fig. 14.3 Dendrogram example

grouping, and is graphically visible in the dendrogram. The value of percentile %p = 100 percentile corresponds to the last admissible grouping which generates a single cluster with the worst, i.e. the lowest, value of similarity. L{A} is the similarity value which corresponds to the A grouping/clustering i.e. the A node given N nodes. Different values of %p generate a different number and configuration of clusters. iii.

Priority list

Given the configuration of clusters, their locations within the fast-pick area need to be determined by following appropriate criteria. For this purpose Bindi (2009) adopts previously introduced indices (see Sect. 14.2) defined for a SKU, and quantifies them for each cluster of similar products. As a consequence, the authors introduce the following assignment rules. • Cluster-based rule (CB). A weight parameter of the cluster given by the number of items it contains is defined. The assignment cluster-based (CB) rules aim to sort clusters by increasing weight, and assign the lowest of them to the most favourable locations. • Cluster-based & popularity (C&P). Popularity for a cluster is estimated as the average of the popularity of items belonging to it. The assignment CB&P rule arranges clusters by decreasing popularity and assigns the most visited to the most favourable locations. • Cluster-based & COI (C&C). COI for each cluster is calculated as the average value of COI measured for all the items belonging to the selected cluster. The assignment C&C rule plans to sort clusters by increasing value of COI and assigns the lowest of them to the most favourable locations.

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• Cluster-based and & order closing (C&OC). This rule arranges clusters using the same method adopted for C&P rule but uses the index OC instead of P. • Cluster-based & turn (C&T). This rule arranges clusters following the same method adopted for C&OC rule but using the Turn index instead of OC. iv.

Storage positioning

After all the products have been ordered in the priority list, they can be located in the most suitable way within the storage areas. Different rules are illustrated in next section.

14.3.1 Storage Positioning Rules Two general classes of positioning rules are argued in the literature: the former considers the exact distances between the available locations and the I/O input output area, the latter refers to the so-called storage-allocation patterns according to which items types are assigned to locations following different frequency areas, e.g. three areas (high, medium and low frequency), (Jarvis and McDowell 1991; Petersen 1999; Wäscher 2004). Literature defines positioning rules purely for low-level forward-reserve OPS thus lacking in high-level systems where the forward area is spread over several higher levels. Yu and de Koster (2009a, b) argue some attempts to develop positioning rules for a three-dimensional storage system but the approach is oriented to the AS/RS automated storage. This section introduces some original positioning rules to overcome this constraint and a useful set of recommendations for supporting the design and control activities of managers and practitioners in high-level forward-reserve pickerto-part OPS adopting the Chebyshev metric. Consider a storage system made of multiple levels whose first ones, e.g. the first and the second, and the highest, e.g. the last and the last before, are reserve storage areas. The picking activity involves all the other available levels: this is the forward fast-pick area. Figure 14.4 exemplifies the side view of a rack in a forward-reserve OPS made of 9 levels, whose level 1 and level 9 are not picking areas. The three positioning rules introduced in this section are an extension of the storage allocation patterns: • Stripes positioning rule. This rule divides the storage system in equal width stripes (e.g. three and a half spans in Fig. 14.5) following an across-aisle storage policy and the direction generated by the composition of the horizontal velocity and the vertical velocity of the picking vehicle. As shown in Fig. 14.5, products according to the priority list are allocated side by side in the slots of shelf 1 and shelf 2 following the path identified by the red arrow (see the sequence of numbers printed on the SKUs). Once all the slots from the first stripe (e.g. three

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and a half spans) are completely filled, the same procedure skips to the next aisle. Then the procedure is repeated to the end. • Iso-time mapping. The second and the third positioning rules, which have been developed for the case study illustrated in next section, are based on the so-called iso-time mapping. Warehouse locations are divided into iso-time areas according to the necessary total travel time considering the speed profile of a storage/retrieval (S/R) vehicle (see an example in Fig. 14.6, whose values refer to the case study illustrated and discussed in next section) to reach the locations belonging to an iso area from the I/O point. The so-called Iso-time 1 presents three iso-time areas, while Iso-time 2 is made of six areas according to the rule introduced by Sturges (1926) for the identification of the optimal number of classes. As shown in Fig. 14.7, A locations have a travel time lower than tA (e.g. 10.9 s in the case study), locations in area B have a travel time between tA and tB (e.g. 22.3 s in the case study), while locations from area C have a travel time between tB and tC (e.g. 33.6 s in the case study). Similarly, Fig. 14.8 illustrates the iso-time areas for the Iso-time 2 positioning rule. In particular, the reported values of time refer to the case study illustrated in next section. The adopted procedure to establish the definitive position of each product within each class of location is the same used for the Stripes rule.

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Isotime A (Total travel time to reach a random location in Isotime A area from shipping dock is below 5.3 sec) Isotime B (Total travel time to reach a random location in Isotime B area from shipping dock is over 5.3 sec and below 10.9 sec) Isotime C (Total travel time to reach a random location in Isotime C area from shipping dock is over 10.9 sec and below 16.6 sec ) Isotime D (Total travel time to reach a random location in Isotime D area from shipping dock is over 16.6 sec and below 22.3 sec) Isotime E (Total travel time to reach a random location in Isotime E area from shipping dock is over 22.3 sec and below 27.9 sec) Isotime F (Total travel time to reach a random location in Isotime F area from shipping dock is over 27.9 sec and below 33.6 sec ) Reserve Area

Fig. 14.8 Iso-time 2 Positioning rule—side view of the shelf

14.4 Case Study The proposed approach has been tested on SKU reprofiling for an Italian leading manufacturing company operating in the ceramic sector for flooring and tiles at the luxury high end of the market. The case study is focused on the storage system for the business unit of ceramic decorations made of mosaics, listellos and inserts.

14.4.1 Current Operation The total footprint of the current picking zone is 630 m2: 90 m long and 7 m wide. The width of the picking aisle is 1.8 m. Each picking aisle has 1,116 stock keeping unit locations. The layout of the ceramic broken-case picking zone is shown in Fig. 14.9. The layout configuration is set up according to a length-wise shape of the racks. The overall ceramic decoration picking zone is organised with two aisles and the I/O point located in the middle of the front of the system. The order picker travels down the picking aisle from I/O area till the pick location adopting a return routing policy, so it can enter and exit at the same side of an aisle (Manzini et al. 2006). The current storage system is a simple pallet rack with 9 levels. The company has established that the reserve area includes the level 1 and level 9 of the rack, while the forward pick area comprises the levels from 2 to 8. Once the forward area has been ‘‘emptied’’, a restocker refills it by dropping/arising the products from the two reserve levels during a shift completely dedicated to the refilling process, known as ‘‘restocking’’. A schematic side view of the rack is depicted in Fig. 14.4. The current picking method is single order picking, i.e. retrieval of one customer order per trip. Tri-lateral stackers are used as retrieval equipment. The capacity of the picking stackers, set according to the fragility of the products and limitations on stacking, only allows the picker-to-pick 0.8 m3 per picking trip. The speed profile of the tri-lateral stackers depicted in Fig. 14.6 has two speed segment

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respectively of 2.5 m/s and an acceleration/deceleration of 0.55 m/s2 for travelling, and 0.45 m/s and an acceleration/deceleration of 0.08 m/s2 for raising/ lowering the forks of the stacker. It is noticeable that the raising/lowering speed has a significant impact on the total travel time for the order picking. As a consequence the vertical drive has been taken into account for the computation of the total travel time. The horizon time for the analysis embraces 18-month order profile data. The initial data have been divided into two sets, the training set (15 months) and the test set (3 months). The training set has been used to train the approach and define the storage allocation of products within the system, while the test set is used to simulate the retrieval of customer orders from the obtained forward storage area and to evaluate the results in terms of performance. The splitting of initial data is a necessary step in order to avoid the so-called overfitting data problem and outcome bias. The average number of orders dispatched per day was 103. The standard deviation of the number of orders processed per day is 44. It can be concluded that the picking operation is small and demand variability is high, therefore a flexible order picking system is preferred. The distribution of the number of lines per order (see Fig. 14.10) suggests that almost all the orders have more than 8 lines; the most frequent order size falls in the range between 8 and 15. Mulcahy (1994) defines large orders as those that have more than 10 lines and recommends that single order picking may yield an efficient picking tour for large orders. He also suggests that batch picking is especially

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effective for small orders that have 1 up to 5 lines. Therefore, considering the distribution of order lines, batch picking may not be appropriate for the operation but single order picking may be suitable.

14.4.2 Experimental Analysis An experimental analysis has been conducted in order to identify the most critical factors affecting the system performance and to set the system for minimising picking costs. In particular Fig. 14.11 describes the most important factors and related values, called attributes or levels, combined in the study. In detail, the routing factor refers to the use (level ‘‘on’’) or nonuse (level ‘‘off’’), of a sequencing rule of the visits given an allocation of products and a generic customer order. In a single OPS a customer order made of several orderliness is assigned to a picker who can visit the slots of the system in a number of possible tours equal to the permutation n!, where n is the number of orderliness. The level ‘‘on’’ corresponds to the application of a travelling salesman problem (TSP) heuristic procedure for the determination of the minimum cost of sequencing visits (i.e. stops) in a multi-stop picking route that corresponds to a single customer order. The experimental analysis is the result of the combination of the following sets of values for each factor: • nine different storage assignment rules (i.e. all the previously introduced storage allocation rules—C, T, P, OC, CB, C&T, C&P, C&C, C&OC); • two clustering algorithms (fn and nn);

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Table 14.3 Experimental analysis, extract of the results (case study) Assignment Similarity Total travel Total travel Travelled Total travel raising/ rule index time [s] distance [m] aisles [visits] lowering distance [m] C&C C&OC C&P C&T CB

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404,975 403,208 348,098 345,247 310,268 287,981 403,372 402,856 409,874 398,990

218,090 240,285 219,329 176,124 153,503 131,316 263,609 230,191 197,373 257,350

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• two types of similarity indices (general purpose Jaccard coefficient and the product oriented Proposed similarity index); • three percentile threshold cut values (40, 60, and 70) related to the level of similarity/correlation by Eq. 14.8; • two modes of routing (‘‘off’’ identify a not ordered picking list, ‘‘on’’ an arranged picking list with an optimal routing); • three positioning rules (‘‘stripes’’, ‘‘iso-time 1’’ and ‘‘iso-time 2’’). These factors are combined in a what-if analysis, and the best system configuration in terms of travel distance and travel time is identified. For each scenario simulated, the total travel distance (vertical drive ? horizontal drive), the travel time, and the total number of visited aisles associated with retrieving products from the storage area in response to the customer requests is quantified.

14.4.3 Results Table 14.3 is an extract of the results obtained in terms of total travel time [s/month], total travel distance [m/month], travelled aisles [visits/month], raising/ lowering distance [m/month] for different system configurations, ordered by the adopted storage allocation rules. In particular, the so-called raising/lowering distance is travelled distance measured on the vertical direction.

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Considering the minimum value obtained on 384 simulations for each storage allocation level, the C&P allocation rule performs better than the others, especially when the proposed similarity index is used to estimate the correlation between products. Table 14.3 reports in bold the best obtained performance and in italic format the second best performance demonstrating that cluster-based strategy performs better than non-cluster-based rules. Even if P performs satisfactory in terms of total travelled distance, C&P performs significantly better. In particular the C&P rule compared to the P rule obtains a total travel time reduced by 28.3%; it guarantees a global saving of about 21.2% on total travel distance and about 26.5% on the raising/lowering distance. By this clustering strategy, it is clear that the storage allocation rules are significantly best performing when the proposed similarity index, instead of the Jaccard coefficient, is adopted. The correlated storage assignment rules perform better than the others in terms of the number of travelled aisles. This result could be reasonably explained referring to the influence of the product correlation. Products with high correlation are indeed stored together resulting in less visited aisles. As a consequence, congestion may occur and hence it is a natural extension to consider the waiting times between two pickers for future study. Figures 14.12 and 14.13 show the trend of the mean values of the total travel time and the total travel distance for different storage assignment rules and different positioning rules. The C&P rule performs better than P rule in terms of travel time and similarly to the P rule in terms of travel distance. Table 14.4 compares the results obtained for different storage assignment rules including (and not including) the acceleration/deceleration effect, and adopting a

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tri-lateral stacker. Firstly a reduction in travel time occurs for all the rules in the results not considering the acceleration/deceleration modelling. But, while the storage assignment rules considering the product correlation have a moderate increase, the other rules (e.g. C, OC, P and T) generate significant variations of the performance. As a consequence, a simulation for a 3D high level storage system not considering the acceleration/deceleration effect could produce an incorrect performance evaluation, undervaluing the performance gap between the storage allocation rules adopting product correlation and the others. Figure 14.14 illustrates 3D ball-graphs of frequency of visits of the pickers within the system. It is a representation of the popularity distribution within the storage area. Different settings of the proposed procedure generate different distributions of calls for the picker. In particular the C&P correlated-based storage

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Fig. 14.14 Ball/popularity analyses adopting cluster-based storage allocation

assignment generates the 3D distribution of popularity illustrated in Fig. 14.14. It demonstrates that most visited products and locations are not in the front of the system because of the importance of correlation between products within the same group named cluster.

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Fig. 14.15 Main effects plot for the total travel distance (correlated storage assignment), Minitab Statistical Software

In order to get a complete comprehension of the main factors affecting the system performance and identify the optimal configuration for the case study a statistical analysis has been conducted as reported in Figs. 14.15 and 14.16. These analyses refer to the set of results obtained adopting the correlated storage assignment rules. From the main effect plot for the total travel distance and the interaction plot for the total travel time it can be seen that: • Iso-time 2 rule performs better referring to the other positioning rules. • The use of an ordered picking list and an optimal routing obviously may reduce the travel distance. • fn clustering algorithm obtains on average a reduced travel distance for the storage assignment policy based on the clustering process. • The simulations conducted adopting the proposed similarity index result in a lower total travel distance referring to the simulations adopting Jaccard coefficient as the correlation measurement. • 40 percentile threshold cut value is preferred. • C&P performs better than the other storage assignment rules. These results are not general, i.e. of general use, but they demonstrate the importance of best setting the picking system. A brief illustration of the economic impact of the best performing correlationbased storage assignment rule is finally presented. The so-called C&P rule guarantees a saving of about 21.5% when compared to the P assignment rule as a result

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Fig. 14.16 Interaction plot for the total travel time, Minitab Statistical Software

of the following hypotheses in coherence with the case study coming from the tile industry: • cost of a tri-lateral stacker: 65,000 €; • a 5 year cost amortisation model; • number of hours of use of the stacker in a day: 12 h/die including 4 h/die for restocks; • global industrial cost of workload: 24 €/hour. By the assumptions reported above the global unit cost in an hour of a stacker is about 4.299 €/hour, without including maintenance and energy costs. The expected total cost due to the travelling activity of the stacker is about 3232 €/month adopting the popularity-based assignment rule and 2535 €/month adopting the C&P rule. All costs include the contribution of maintenance, the variable cost of travelling due to energy consumption, and the mean availability of the generic stacker (A = 0.92).

14.5 Conclusions This chapter provides a basic approach for the storage allocation process in OPSs for reducing the total travel time and distance assuming some essential system features such as product correlation, three-dimensional allocations and considering

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stackers speed profiles. Introducing some original storage allocation and positioning rules, the effectiveness of the proposed approach and an optimal combination of factors have been identified by the execution of a test, as an experimental analysis, on a real world case study. In particular, similarity-based storage allocation performs better than traditional allocation strategies and rules: travel time, distance and costs can be reduced of more than 20% renouncing to the adoption of the widely used popularity-based allocation of products to the storage system. Bindi et al. (2009) quantifie savings of about 10% in travelling distance adopting a correlated-based storage rule and renouncing to the random assignment policy. This significant result confirms the good performance of the proposed correlated strategy, systematic approach and rules both in a lower-level picker-to-part 2-dimensional OPS (Bindi et al. 2009) and in high-level 3D OPS. Further research are expected on storage allocation of products including the determination of the best storage level for each product and the measurement of the number of restocks in presence of both a reserve/bulk area and the forward picking area. Which are the effects of a correlated storage assignment strategy and policies in presence of the optimal storage levels of products quantified in accordance with the analytical models introduced by Bartholdi and Hackman (2003)? Further research are also expected on warehousing system design and management including inventory management issues (e.g. fulfilment/replenishment system development and safety stock determination) and distribution issues (e.g. facility location, location and allocation problem—LAP, packaging, vehicle loading, routing and grouping). New applications to industrial case studies are also expected in order to identify best practices useful for practitioners and managers of industry to face a design/optimisation problem in warehousing science and in presence of different categories of problem as a result of the number of products, orders typology, packaging system, customers’ behaviour, level of automation, presence of workload, etc. Acknowledgments The authors would like to thank John J. Bartholdi III professor at the Georgia Institute of Technology for the useful suggestions to the research.

References Aldenderfer MS, Blashfield RK (1984) Cluster analysis. Sage University paper series on quantitative applications in the social sciences No. 07-044, Beverly Hills, Sage Bartholdi J, Hackman ST (2003) Warehouse and distribution science, http://www2.isye.gatech.edu/ people/faculty/John_Bartholdi/wh/book/editions/history.html. Accessed Jan 2010 Bindi F (2010) advanced models and tools for inbound and outbound logistics in supply chain. PhD Thesis, University of Padua Bindi F, Manzini R, Pareschi A, Regattieri A (2009) Similarity-based storage allocation rules in an order picking system. An application to the food service industry. Int J Logist Res Appl 12(4):233–247 August 2009 Brynzér H, Johansson MI (1996) Storage location assignment: using the product structure to reduce order picking times. Int J Prod Econ 46:595–603

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Cahn AS (1948) The warehouse problem. Bull Am Math Soc 54:1073 de Koster R (2004) How to assess a warehouse operation in a single tour report. RSM Erasmus University, The Netherlands de Koster R, Le-Duc T, Roodbergen KJ (2007) Design and control of warehouse order picking: a literature review. Eur J Oper Res 182(2):481–501 Frazelle EH (2002) World-class warehousing and material handling. McGraw Hill, New York Frazelle EH, Sharp GP (1989) Correlated assignment strategy can improve any order picking operation. Indus Eng 21:33–37 Gu J, Goetschalckx M, McGinnis LF (2007) Research on warehouse operation: a comprehensive review. Eur J Oper Res 177:1–21 Haskett JL (1963) Cube per order index–a key to warehouse stock location. Trans Distrib Manag 3:27–31 Jarvis JM, McDowell ED (1991) Optimal product layout in an order picking warehouse. IEE Trans 23:93–102 Manzini R, Gamberi M, Regattieri A (2005) Design and control of flexible order picking systems. Int J Oper Prod Manag 16(1):18–35 Manzini R, Gamberi M, Regattieri A (2006) Design and control of an AS/RS. Int J Adv Manuf Technol 28:766–774 Manzini R, Gamberi M, Persona A, Regattieri A (2007) Design of a class based storage picker to product order picking system. Int J Adv Manuf Technol 32:811–821 Manzini R, Bindi F, Pareschi A (2010) The threshold value of group similarity in the formation of cellular manufacturing systems. Int J Prod Res 48(10):3029–3060 McAuley J (1972) Machine grouping for efficient production. Prod Eng 51:53–57 Mulcahy DE (1994) Warehouse distribution and operations handbook. McGraw-Hill, New York Petersen CG (1999) The impact of routing policies on warehousing efficiency. Int J Oper Prod Manag 17:1098–1111 Petersen CG, Aase G, Heiser DR (2004) Improving order-picking performance through the implementation of class-based storage. Int J Phys Distrib Logist Manag 34(7):534–544 Sturges HA (1926) The choice of a class interval. J Am Stat Assoc 21(153):65–66 Van den Berg JP, Zijm WHM (1999) Models for warehouse management: classification and examples. Int J Prod Econ 59:519–528 Wäscher G (2004) Order picking: a survey of planning problems and methods. In: Dyckhoff H, Lackes R, Reese J (eds) Supply chain management and reverse logistics. Springer-Verlag, Berlin, pp 323–347 Yu Y, De Koster R (2009a) Optimal zone for two-class-based compact three-dimensional automated storage and retrieval systems. IIE Trans 41(3):194–208 Yu Y, De Koster R (2009b) Designing an optimal turnover-based storage rack for a 3D compact automated storage and retrieval system. Int J Prod Res 47(6):1551–1571

Chapter 15

Design and Optimization of Picking in the Case of Multi-Item Multi-Location Multi-Pallet Customer Orders R. Gamberini, B. Rimini, M. Dell’Amico, F. Lolli and M. Bianchi

Abstract Order picking-related costs may account for up to 65% of the total expense of warehouse management. Hence, the implementation of robust design and optimization procedures for planning picking is addressed by researchers and practitioners. In this chapter the case of warehouses served by humans, in pickerto-parts systems, with a discrete picking organization is studied. Specifically, the case of orders including multiple different items, located in different aisles and requiring more than one forklift load to completely satisfy customer requests is analyzed, with the aim of minimizing the time for retrieving an order. Specifically, two aspects are studied: (1) the grouping of orders into a finite number of forklift missions, by assuring that each required item is picked in the required amount (2) the optimization of the routing to be followed by handling facilities in accordance with the objective of minimizing the total traveled distance and the computation of the number of handling facilities necessary for serving the warehouse aisles.

15.1 Introduction Order picking, the activity aimed at inserting or retrieving products from storage (or buffer areas) in accordance with pre-defined customer requests, has been identified as a labor-intensive and costly process in a wide variety of warehouses. Related costs (mainly due to annual depreciation of investments, i.e. handling vehicles such as forklifts, operator loading and unloading products from pallets along with driving

R. Gamberini (&)  B. Rimini  M. Dell’Amico  F. Lolli  M. Bianchi University of Modena and Reggio Emilia, Via Amendola, 2, Padiglione Morselli, 42100 Reggio Emilia, Italy e-mail: [email protected]

R. Manzini (ed.), Warehousing in the Global Supply Chain, DOI: 10.1007/978-1-4471-2274-6_15,  Springer-Verlag London Limited 2012

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handling vehicles, fuel and/or electric power required by forklifts, operators, materials, equipment and spare parts necessary for maintenance) may account for up to 65% of the total expense of warehouse management (Coyle et al. 1996). Hence, the implementation of robust design and optimization procedures for planning picking is addressed by researchers and practitioners (Manzini and Gamberini 2008). Picking systems are classified according to whether humans or automated facilities are used. Furthermore, picker-to-parts (where the pickers move along the warehouse aisles and pick items) or parts-to-picker systems (where automated cranes moving along the aisle retrieve stocking units, bring them to a pick station, wait for picking and insert the remaining portion of the stocking units again in the appropriate warehouse locations) are identified. Alternatively, in accordance with the picking organization implemented, picking systems are classified as picking by article (or batch picking), picking by zone and picking by order (or discrete picking). In the former two cases, multiple orders are served simultaneously. Hence a sorting operation follows picking (de Koster et al. 2007). In this chapter the case of warehouses served by humans, in picker-to-parts systems, with a discrete picking organization is studied. Specifically, the case of orders including multiple different items, located in different aisles and requiring more than one forklift load to completely satisfy customer requests is analyzed. Optimal order picking system management focuses on the maximization of the service level given constraints related to available resources, such as operators, handling facilities, capital for investments and operative costs (Goetschalckx and Ashayeri 1989). Specifically, the service level includes a variety of factors: • • • • •

minimizing the throughput time of an order minimizing the overall throughput time (e.g. to complete a batch of orders) maximizing the use of space maximizing the use of equipment maximizing the use of labor.

Hence a crucial aspect to be pursued is related to the minimization of time for retrieving an order, whose main component is represented by the time spent for traveling and handling in the storage area (Dekker et al. 2004). Numerous contributions in the literature focus on the problem of sequencing items on the pick list to ensure a good route through the warehouse. A critical review is reported in Sect. 15.3.2. Nevertheless, a problem that occurs in practice is related to the need to update the fleet of machines serving the warehouse area, due to the necessity of: • replacing old handling facilities • introducing new models offered on the market • introducing alternatives that are better fitting with changeable needs arising during warehouse management. Hence, the process of order picking requires analysis and re-definition, by studying two different aspects:

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• the grouping of orders into a finite number of forklift missions, by assuring that each required item is picked in the required amount • the optimization of the routing to be followed by handling facilities in accordance with the objective of minimizing the total traveled distance (or the time required for missions). As a consequence, forklift routing is only one part of the problem. Furthermore, the number of handling facilities required for serving the warehouse aisles is unknown data to be discovered and optimized. The aspects concerning forklift load and number optimization are even more important as the analyzed system involves the handling of orders made up of numerous items, which are small (hence a wide variety of combinations for grouping them into forklifts loads are available). Thus, as mentioned above, in this chapter warehouses served by humans, in picker-to-parts systems, with a discrete picking organization are studied. Specifically, the problem of defining the number of machines belonging to the warehouse serving fleet is tackled in the case of orders including multiple items, located in different aisles and requiring more than one forklift load in order to fully satisfy customer requests. Finally, a case study is discussed. This chapter is organized as follows. The problem statement is described in detail in Sect. 15.2. After a review of the related research (Sect. 15.3), the notation that is adopted in the subsequent sections is presented (Sect. 15.4). Section 15.5 describes the proposed solving framework, while in Sect. 15.6 a case study is discussed. Finally, some conclusions are provided in Sect. 15.7.

15.2 The Problem Statement As described in the introduction, in this chapter the problem of defining the number of machines necessary for satisfying picking requests in a warehouse area organized with human pickers implementing discrete picking of orders including a large number of multiple different items is tackled. Specifically, the initial data of the problem are represented by: • the warehouse layout, including the dimensions and characteristics of the areas available for receiving, storing and shipping products, the position, size and typology of the shelves, the dimensions and characteristics of the aisles, the allowable directions for forklift flows and constraints limiting forklift movements • the location of stored items • the features of stored items (in terms of weight and volume) and quantity • the shape of stored items • the composition of the picking orders in a predefined time horizon. Specifically, each order is made up of the typology, quantity and potentially the position in the warehouse

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• the characteristics of forklifts available for serving customer requests (specifically in terms of average moving speed, maximum weight allowed and maximum volume allowed) • the time required for serving each location and for moving among the warehouse aisles (potentially including an estimate of initial acceleration and final deceleration along with the time required for loading or unloading items) • the maximum weight allowed for the forklift load (evaluated by considering the aforementioned forklift characteristics along with additional constraints due to shelf limits, item design and resistance and load stability) • the maximum volume allowed for the forklift load (evaluated by considering the aforementioned forklift characteristics along with additional constraints due to shelf limits, item design and resistance and load stability) • constraints related to items overlapping (i.e. due to their maximum weight allowed, their shape and aesthetic features along with materials adopted). These data constitute the assumptions of the problem and are constraints for the available alternative solutions. The aim of the problem is to determine the following output: • for each order, the composition of items in each forklift load. Specifically, when long orders occur, more than one forklift load is required • the routing of each forklift • the number of forklifts needed to satisfy customer orders in a predefined time horizon. As occurs in Duhamel et al. (2011), Table 15.1 is set in order to trace a comparison between the problem statement managed in this chapter and problems frequently tackled in the literature. Columns contain input and output listed above. Rows enumerate similar problems faced in the literature (specifically, the binpacking problem, the routing of forklifts among aisles in a warehouse and the capacitated vehicle routing problem), the features of the proposed problem statement, described in depth in the case study, and characteristics of the presented solving framework. A detailed literature review is subsequently carried out in Sect. 15.3.

15.3 Literature Review As anticipated in Sect. 15.2, related research in the literature consists of: • solving approaches for the bin-packing problem • solving approaches for the routing of forklifts among aisles in a warehouse • solving approaches for the capacitated vehicle routing problemwhose published contributions are respectively analyzed in Sects. 15.3.1, 15.3.2 and 15.3.3.

Bin packing solving approaches Forklift routing solving approaches Capacitated vehicle routing solving approaches coupled with loading aspects Case study Proposed framework

Problem

x

x

x x

x

x

x x

x x

x

x

x x

x

x

x

x x

x

x

x x

x

x

x

Stored Composition Warehouse Stored Stored Stored item item of layout item item location weight quantity shape picking orders and volume

Input

Table 15.1 Published related research

x x

x

x

x x

x

x

x x

x

x

Characteristics Maximum Time weight required of forklifts allowed for on the serving forklift each location

x x

x

x

x x

x

x

Maximum Item overlapping volume constraints allowed on the forklift

x

Compo sition of forklift load

Output

x x

x

x

Compo sition of forklift load

x x

x

x

x x

x

x

Forklift Number routing of forklifts required

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15.3.1 Published Approaches for the Bin-Packing Problem The bin-packing problem aims to pack a given set of items into the minimum number of containers. Specifically, both a two-dimensional and a three-dimensional version of the problem are available. Given the input and output list characterizing the problem statement described in Sect. 15.2, the bin-packing solving approaches available in the literature could be used to solve only a portion of the problem. Specifically, given: • the features of stored items, in terms of weight, volume, quantity, shape, overlapping limits • the composition of the picking orders in a predefined time horizon • the characteristics of available forklifts, including the maximum weight and volume allowed for loading the existing bin packing solving approaches could be used for evaluating, for each order, the composition of items in each forklift load and consequently, for the whole of the orders, the number of forklifts required. Nevertheless, as underlined in Table 15.1, the routing of forklifts among the warehouse aisles is outside the aims of the problem. For further details concerning related literature, readers should refer, for example, to Iori and Martello (2010), Wu et al. (2010), Caprara and Monaci (2009), Crainic et al. (2008), Alves and de Carvalho (2007), Fekete et al. (2007), Martello et al. (2000, 2007), Pisinger and Sigurd (2007), Boschetti and Mingozzi (2003), Faroe et al. (2003) and Martello and Vigo (1998).

15.3.2 Published Approaches for the Routing of Forklifts in a Warehouse The problem of routing in a warehouse area aims to sequence items on the pick list in order to ensure minimum occupation of forklifts and operators (and consequently space and capital) (Gu et al. 2007). Usually, it is tackled as a special case of the traveling salesman problem—TSP (see Lawler et al. 1985; Reinelt 1994; Gutin and Punnen 2002; Applegate et al. 2007; D’Ambrosio et al. 2010; Letchford and Lodi 2010). Hence, given a set of vertexes of a graph representing the aisles where items are stored and given a predefined subset of the aforementioned vertexes representing locations of items included in a pick list, the solving approaches for the TSP could be used for defining the route to visit each required location only once, by assuring the optimal use of available resources in terms of a prespecified objective function (i.e. traveled distance). In particular, a warehouse configuration with aisles grouped into a single block is studied in Ratliff and Rosenthal (1983), Hall (1993), Petersen (1997), de Koster

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and Van der Poort (1998) and Makris and Giakoumakis (2003). The multiple block version of the problem is studied in Vaughan and Petersen (1999), Roodbergen and de Koster (2001a, b) and Theys et al. (2010). Nevertheless, by referring to the input and output list used for describing the problem statement in Sect. 15.2, it emerges that the aforementioned contributions disregard some input data: • the characteristics of the stored items (in terms of weight and volume, as well as shape) • the characteristics of forklifts available for serving customer requests (specifically in terms of maximum weight allowed and maximum volume allowed) • maximum weight allowed on the forklift (including analysis of the forklift characteristics and constraints due to shelf limits, item design and resistance and load stability, etc.) • maximum volume allowed on the forklift (again including analysis of the forklift characteristics and constraints due to shelves and item design and resistance and load stability, etc.) • constraints existing for loading forklifts and overlapping items. In particular, the composition of the forklift loads requiring a route definition in a predefined time horizon is assumed as given and satisfying the aforementioned constraints (see Table 15.1). Otherwise, in the problem statement that is the object of the paper the composition of forklifts load is a required output.

15.3.3 Published Approaches for the Capacitated Vehicle Routing Problem The capacitated vehicle routing problem (CVRP) assumes that a central depot is located at vertex 0, where a fleet of m identical vehicles are available, while l customers are located at vertices 1; 2; . . .; l. All vehicles have the same capacity D, and each customer has a demand dz (with 0  dz  D for z ¼ 1; 2; . . .; l). The CVRP is to find a set of maximum m circuits (called routes), each visiting the depot and a subset of customers, such that: (1) each customer is visited by exactly one vehicle; (2) each vehicle performs one route at most; (3) the sum of the demands on each route is no greater than D; (4) the sum of the costs of the traveled edges/arcs is a minimum. Interesting reviews are reported in Laporte (1992), Cordeau et al. (1999, 2005), Toth and Vigo (2002), Golden et al. (2008) and Baldacci et al. (2007, 2010). By focusing on the combination of CVRP with loading aspects, the survey contribution by Iori and Martello (2010) is crucial. Interesting recent solving approaches combining CVRP with loading needs are cited: Iori et al. (2007), Gendreau et al. (2006, 2007a, b, 2008), Fuellerer et al. (2009, 2010), Tarantilis et al. (2009), Zachariadis et al. (2009) and Duhamel et al. (2011).

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Nevertheless, even if as described in Table 15.1, the published solving approaches for the CVRP with loading constraints require the same input and assure the same output characterizing the problem statement described in Sect. 15.2, the aforementioned contributions disregard some aspects: • the grouping of product fragility in only two subsets is a limit for some market fields (i.e. in the case of bathroom furnishings where different classes of item overlapping capability can be identified. Tiles, boxes containing taps, boxes containing lamps and packets containing decorations can be stacked by following such an order. Alternatives are not usually feasible and product integrity may not be assured. Obviously, individual cases may represent exceptions to be evaluated one by one). • the capacity D (both in terms of weight and volume) of a forklift load is sometimes not a strict constraint. Some variability ranges may exist and induce interesting advantages (see Gamberini et al. 2005, 2008). • the time required for picking execution in two warehouse locations may change in accordance with the forklift loading composition: when fragile items are transported or when the cargo slightly exceeds limits initially imposed, particular care may be requested by the operators. Alternatively, when loads are moved that are still far from their capability saturation, a higher speed may be considered. Hence, a solving framework for complex scenarios occurring in practice is required.

15.4 Notation n m l I

wðiÞ qðiÞ W Q aðiÞ chk

total number of the items maximum number of required vehicle loads total number of positions in the storage area number of iterations of the solving framework consequently assuring the same final number of required forklifts for serving the warehouse in the studied time horizon weight of item i, with i ¼ 1; . . .; n volume of item i, with i ¼ 1; . . .; n maximum weight allowed for each vehicle maximum volume allowed for each vehicle number of items i in a specific order, with i ¼ 1; . . .; n cost associated with the arc ðh; kÞ that is the cost to go from h to k, with h; k ¼ 1; . . .; l

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Variables: ( 1 if item i is loaded into vehicle j; with j ¼ 1; . . .; m and i ¼ 1; . . .; n xji ¼ 0 otherwise ( 1 if the arcðh; kÞ belongs to the tour, with h; k ¼ 1; . . .; l nhk ¼ 0 otherwise ( 1 if vehicle j is used,with j ¼ 1; . . .; l yj ¼ 0 otherwise

15.5 The Solving Framework Given the analysis of related published research, the proposed problem can be seen as a modified version of the picking problem, where additional aims are expected: the definition of the forklift load when the fulfillment of each order requires more than one handling machine and the number of forklifts for satisfying customer demands in a predefined time horizon. Only in the case of simple warehouse configurations the problem of defining picking routing is optimally solved (de Koster 2007). Otherwise, heuristics are preferred. Hence, a heuristic solving framework is also proposed for the modified version of the picking problem referred to by this chapter, whose main steps are described as follows and depicted in Fig. 15.1: Step 1: Pareto analysis of stored items, in order to determine items that better influence picking activities. The problem statement focuses on the definition of forklifts required in a predefined time horizon rather than on the strict definition of picking routes. A middle term standpoint is assumed, justifying investments in forklifts along with personnel insertion. As a consequence, the items that mainly influence warehouse flows are analyzed, rather than the routing of orders per day. Step 2: Pareto analysis of time buckets belonging to the time horizon during which forklifts should operate and selection of a robust time horizon for data collection. Step 3: Pareto analysis of orders during the time horizon defined in step 2, with the aim of identifying a sample of orders that represent warehouse activities well. As a consequence of the considerations reported in step 1, a selection of orders that mainly influence warehouse flows is also carried out. Step 4: Given the forklifts available on the market for satisfying customer requests, a definition of the minimum number of machine loads required per order, by adopting a solving algorithm for the bin-packing problem. Step 5: Optimization of forklift routing per order, by adopting a solving algorithm for the traveling salesman problem.

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Fig. 15.1 The solving framework proposed

Step 6: Adjustments of the solution proposed in the previous step, by considering additional constraints typical of practice (i.e. load stability, item fragility, limits on overlapping and traffic congestion) or relaxing existing ones (i.e. those concerning maximum weight and volume allowed). The adopted solving approach for the bin-packing problem may not include the complete set of constraints required by the case study. Hence, modifications to the proposed solutions may be introduced. Alternatively, some constraints may be relaxed in order to evaluate new solutions potentially near to the selected ones in the solution space. Step 7: Grouping of routing per available handling machine in order to determine the minimum number of forklifts required for orders defined in step 3. Step 8: if the last iterations of the solving approach have assured solutions with the same number of required forklifts, then go to step 10, otherwise go to step 9.

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Step 9: If alternative optimal solutions for the bi-dimensional bin-packing problem are available, then go to step 4, otherwise go to step 10. Step 10: Propose the solution with the minimum number of required forklifts in the studied time horizon, by specifying: per order, the composition of items in each forklift load and the routing of each forklift along with, obviously, the number of forklifts required.

15.6 A Case Study The analysis of a warehouse of an Italian company operating in the field of bathroom furnishings is carried out. The study focuses on the work in progress and final product storage area, depicted in Fig. 15.2. The solving framework described in Sect. 15.5 and depicted in Fig. 15.1 is applied in the following, in order to define the number of forklifts required to satisfy customer orders. As a consequence, the forklift load and routing are also expected as output. Step 1: Pareto analysis of stored items Initially, the Pareto analysis of stored items is carried out. Specifically, the study of past customer demand has underlined that products that mainly influence warehouse flows are stored in the portion of shelves depicted in Fig. 15.3, on which the study will focus below. Step 2: Pareto analysis of time buckets belonging to the time horizon during which forklifts should operate and selection of a robust time horizon for data collection. Through analysis of customer demand profiles, a peak of orders in the month of March has emerged, since most companies operating in the building field are interested in buying bathroom furnishings for existing sites or ones that are due to open. In subsequent months, a similar or lower demand has been recorded. Hence a time horizon for the order data collection has been set and selected coinciding with the month of March, representing the most critical situation pickers have to face. Step 3: Pareto analysis of orders with the aim of identifying a sample of orders that represent warehouse activities well. By collecting orders referring to products contained on the shelves depicted in Fig. 15.3 and arising during the month of March, a pool of 405 orders is recorded. An example order is reported in Table 15.2, where the columns contain the data required during the following steps of the solving framework: typology and quantity of products requested, weight and volume per item along with their location in the warehouse (identified by a number for the rack, bay and level, respectively). An order is uniquely defined by its progressive number, the issue date and the customer requiring it. In Table 15.2, data are reported for order number 174, issued by customer 20 on 13th of March.

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Fig. 15.2 The warehouse analyzed in the case study

Fig. 15.3 Portion of shelves containing products mainly influencing warehouse flows

20 20 20 20 20 :

174 174 174 174 174 …

13/03 13/03 13/03 13/03 13/03 :

.MET0161SP .MET0366S .MET0416S .MET1942 .PLA0330 …

1 2 3 4 5 …

1.00 2.00 2.00 1.00 1.00 …

0.0013 0.01 0.008 0.0008 0.0003 …

Table 15.2 Data collected for order 174 emitted by customer 20 on 13th of March Customer Order Date Item Id item Quantity Volume (m3) 0.0013 0.02 0.016 0.0008 0.0003 …

Total volume (m3) 0.4 0.95 0.5 0.25 0.25 …

Weight (Kg)

0.4 1.9 1 0.25 0.25 …

Total weight (Kg)

R10503C R10502B R10503D 1904A2 2302A1 …

Position

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Step 4: Definition of the minimum number of machine loads required per order, by adopting a solving algorithm for the vector bin packing, where the analyzed dimensions refer to weight and volume. Specifically, in the case study the mathematical model reported in Eqs. 15.1– 15.6 is solved. The problem of loading n items in m loads (possible load layouts), without considering the constraints posed by their shape, but only those imposed by their volume and weight, is tackled. Obviously, an a priori defined condition is that the weight and volume of each item do not exceed the limit imposed by each vehicle load (hence at least one solution exists, that loading one item per vehicle). m X

yj

ð15:1Þ

xji ai wi  Wyj

j ¼ 1; . . .; m

ð15:2Þ

xji ai qi  Qyj

j ¼ 1; . . .; m

ð15:3Þ

min

j¼1

subject to: n X i¼1 n X i¼1 m X

xji ¼ 1

i ¼ 1; . . .; n

ð15:4Þ

j¼1

xji 2 f1; 0g i ¼ 1; . . .; n j ¼ 1; . . .; m

ð15:5Þ

yj 2 f1; 0g j ¼ 1; . . .; m

ð15:6Þ

The objective function (Eq. 15.1) aims to minimize the total number of adopted vehicle loads. Equations 15.2 and 15.3 bind the load in order not to exceed the weight and the volume admissible on the vehicle, respectively W and Q. Furthermore, Eqs. 15.2 and 15.3 impose the following logic relations between the binary decision variables xji and yj : • if yj ¼ 0, then xji ¼ 0 (with i ¼ 1; . . .; n). Thus, the vehicle load j is not used; • if xji ¼ 1 for at least one i (with i ¼ 1; . . .; n), then yj ¼ 1. Thus, the vehicle load j is inserted in the solution. Equation 15.4 requires each item i to be assigned exactly to one vehicle j. Moreover, Eqs. 15.5 and 15.6 guarantee that the decision variables xji and yj are binary. Even if solving approaches are available in the literature for more complex versions of the problem, this pioneering one is selected in order to define and evaluate a wide variety of solutions laying in the solution space, subsequently handled and modified in order to tackle a wide variety of problem features and constraints, some of which avoid the implementation of existing published solving methodologies. A more detailed explanation is inserted when step 6 is described and commented.

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Table 15.3 Example of a bin-packing solution for order 174, issued by customer 20 on 13th of March

411

Order 174 ITEM—Description

ITEM—Sequential Bin-packing solution number (forklift loads required)

MET0161SP MET0366S MET0416S MET1942 PLA0330 MET2560 A369204SP A369209SP A369211Z A369213Z ACOM0531RW ACOM0627Z ACOM0634Z AMAC0659A AMOB02712SRWSP AOLC0815S AOLC0818S AOLC0829S AOLC0833S AOLC0835S AOLC0844S APOR0560 ARUB0918S ARUB0968SP ASPE012P ASPE014RW

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

1 2 2 2 2 3 1 2 2 2 1 2 3 3 3 2 1 1 1 2 2 3 3 3 2 2

Furthermore, in the case study the model is solved in a few seconds by the commercial optimizer Xpress, selected as a robust solving tool also in Bard and Jarrah (2009) and available in a wide variety of companies. An example of a solution obtained is reported in Table 15.3, where a solution for order 174, issued by customer 20 on 13th of March (see Table 15.1) is reported. Specifically, the identification number of the required forklift load is reported in the last column on the right. For example, item 1 is loaded on vehicle 1, item 2 on vehicle 2 and so on. Therefore, in order to satisfy order 174, three forklifts loads are necessary and thus three optimum routes have to be found in the following steps. Step 5: Optimization of forklift routing per order The definition of forklift routing identified in the previous step requires input data concerning the layout of the storage area in terms of item positions and distances and the performances of the handling equipment in terms of velocity

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(along with, potentially, the initial and final time required for acceleration and deceleration). In the analyzed case study, all the items are stored in 110 positions, served by electric porters. Table 15.4 presents a diagonal matrix containing all the distances between the 110 positions in the storage area (111 with the input–output point) expressed in meters. De Koster et al. (2007) underlines that the problem of defining routing for forklifts among the aisles is solvable as a modified version of the TSP. The TSP aims to find the shortest path that passes just once through all the points requiring a loading/unloading mission and then returns to the starting point. If a graphical representation is applied, then TSP is equivalent to finding the path of a single vehicle, from and to a starting node, which minimizes the total cost of the routing passing through all the nodes of the network just once. Thus, each retrieval point is a node on the graph, while each route from one node to another is an arc with a cost depending on its length. In particular, when the graph is oriented, if the cost to go from a node z to a node k is different from the opposite from k to z, then the problem is called asymmetric traveling salesman problem (ATSP) and it is briefly described below. Given a graph G ¼ ðV; AÞ, where V ¼ f1; . . .; lg is the set of the nodes and A ¼ fðh; kÞ : h; k 2 V g is the set of the arcs, a non-negative cost chk is associated with each arc ðh; kÞ. The cost czz is set equal to 1, with z ¼ 1; . . .; l, in order to prevent loops in the network. The ATSP asks for a selection of l arcs that form a tour visiting each vertex exactly once and having a minimum total cost. For more details, see Eqs. 15.7–15.11: min

l X l X

chk nhk

ð15:7Þ

h¼1 k¼1

subject to: l X

nhk ¼ 1

k ¼ 1; . . .; l

ð15:8Þ

nhk ¼ 1

h ¼ 1; . . .; l

ð15:9Þ

h¼1 l X k¼1

XX

nhk  jSj  1

h ¼ 1; . . .; l k ¼ 1; . . .; l 8S  V; S 6¼ 0

ð15:10Þ

h2S k2S

nhk 2 f1; 0g

h; k ¼ 1; . . .; l

ð15:11Þ

The objective function (Eq. 15.7) aims to find the tour of the graph that implies minimum cost. The decision variable nhk is binary (Eq. 15.11) with the following mean: if nhk ¼ 1, then the arc ðh; kÞ belongs to the tour, otherwise not. Note that just one arc could enter each node (Eq. 15.8) or exit each node (Eq. 15.9).

in/out 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18    110

0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0201 0202 0203 0204 0301 0302 0303 0304    2502

Rack/ Bay

x

14

15

16

17

18

   110

46.6 49.4 52.2 x 2.8 5.6 x 2.8 x

55 8.4 5.6 2.8 x

57.8 11.2 8.4 5.6 2.8 x

60.6 14 11.2 8.4 5.6 2.8 x

63.4 16.8 14 11.2 8.4 5.6 2.8 x

66.2 19.6 16.8 14 11.2 8.4 5.6 2.8 x

69 22.4 19.6 16.8 14 11.2 8.4 5.6 2.8 x

71.8 25.2 22.4 19.6 16.8 14 11.2 8.4 5.6 2.8 x

46.6 0 2.8 5.4 8.4 11.2 14 16.8 19.6 22.4 25.2 x

49.4 2.8 0 2.8 5.6 8.4 11.2 14 16.8 19.6 22.4 2.8 x

52.2 5.6 2.8 0 2.8 5.6 8.4 11.2 14 16.8 19.6 5.6 2.8 x

55 8.4 5.6 2.8 0 2.8 5.6 8.4 11.2 14 16.8 8.4 5.6 2.8 x

41.4 8.8 15.2 22 19.2 16 19.2 22 24.8 27.6 30.4 12.4 15.2 22 19.2 x

44.2 11.6 18 19.2 16.4 13.2 16.4 19.2 22 24.8 27.6 15.2 18 19.2 16.4 2.8 x 47 14.4 20.8 16.6 13.6 10.4 13.6 16.6 19.2 22 24.8 18 20.8 16.4 13.6 5.6 2.8 x

49.8 17.2 23.6 13.6 10.8 7.6 10.8 13.6 16.4 19.2 22 20.8 23.6 13.6 10.8 8.4 5.6 2.8 x

                   x

                    x

                     x

36.6 56 56.8 51.2 48.4 45.6 42.8 40 37.2 34.4 31.6 56.8 56.8 51.2 48.4 51.6 48.8 46 43.2    x

0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0201 0202 0203 0204 0301 0302 0303 0304    2502

Table 15.4 Matrix of distances between the 110 positions of the storage area, expressed in meters in/ 1 2 3 4 5 6 7 8 9 10 11 12 13 out

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Moreover, Eq. 15.10 guarantees that each node is visited, avoiding both sub-tours and loops. Interesting published solving approaches for the picking problem are reviewed in Sect. 15.3.2. and are available for predefined configurations of shelves. Nevertheless, given the complexity of the warehouse layout analyzed in the case study, a general algorithm for the asymmetric version of TSP is preferred, as suggested in Theys et al. (2010). Specifically, the contribution of Carpaneto et al. (1995) is selected, since Climer and Zhang (2006) demonstrate that it could reach optimal results in low- computational time in small problems (such as those encountered in the case study). The solution obtained in a few seconds for order 174 issued by customer 20 on 13th of March (see Table 15.1) is reported in Table 15.5, where the third column describes the order in which item locations (listed in the second column) are visited. Finally, the fourth column evaluates the route length (expressed in meters). An alternative cost factor could be introduced, i.e. economical evaluations could be coupled with technical analysis similar to that carried out in this chapter. Nevertheless, in this work available operators represent the resource the company desires to optimize. Hence, their time savings represent the main studied aspects. Each obtained route starts and ends with the point 0, which corresponds to the input/output point of Table 15.4. Despite order 174 being made up of 26 items (see Table 15.3), the final solution contains only 23 positions because some items are stored in the same rack and in the same bay. To summarize, in order to satisfy order 174, three vehicle loads are necessary and thus three routes are established. The same procedure has to be applied to all 405 orders needing to be fulfilled in the time horizon considered. The same computational time is registered. Step 6: Adjustments of the solution proposed in the previous step, by considering additional constraints typical of practice: i.e. load stability, item dimensions and volume, maximum allowed load dimensions, forklift speed as a function of the typology of loaded items and traffic congestion. The solution of the adopted vector bin-packing problem indicates the composition of forklift loads in order to address the use of the minimum number of forklifts. Nevertheless, some constraints are disregarded, concerning: • • • • •

product shape load stability (for some guidelines see Bischoff et al. 1995) item fragility overlapping limits list of visited storage locations (in order to ensure that pallet loading is practically implementable while ensuring the picking order designed in step 5) • traffic congestion in some zones of the layout.

Furthermore, by relaxing constraints related to maximum volume and weight allowed, alternative solutions may be evaluated (in accordance with guidelines outlined in Gamberini et al. (2005, 2008), where the possibility of considering item loading characteristics, i.e. in terms of volume or weight, as weak rather than strict

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Table 15.5 The solution of the forklift routing problem for order 174 issued by customer 20 on 13th of March

415

Order 174 Forklift

ITEM—warehouse location

1 R10503 A41303 0903 1401 1402 2 1904 1403 1501 0110 A31304 1503 0704 3 R10502 R10503 1904 A31302 A41302 1403 1401 1402 0101 0105 2302

Routing

Distance (m)

0 1401 1402 A41303 0903 R10503 0 0 1501 0704 1403 A31304 0110 1503 1904 0 0 A31302 R10503 R10502 0101 0105 1401 1402 A41302 1403 2302 1904 0

128.8

160

184

constraints, is a sometimes implementable option, ensuring improvements in logistics performance). Analogously, additional positive or negative time may be added to the solution obtained by the ATSP solving approach in order to consider the effects on forklift movements induced by loading defined in step 4. Hence, the analysis of the applicability of solutions obtained by the subsequent implementation of steps 4 and 5 is carried out. Then, adjustments are implemented. Step 7: Grouping of routing for each available truck in order to determine the minimum number of forklifts required. By considering the available operating time for each existing forklift typology or to be inserted in the warehouse, the number of forklifts required is computed. Specifically, an availability of 7.5 h per shift is considered for porters and

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Table 15.6 Results obtained in the NEW-1 scenario Date Order Distance Mean Distance Items (m) velocity covered in the (m/ time order min) (min) 13/03 17/03 24/03 --Average

174 199 312 ---

472.8 251.2 451.2 --124.65

116.67 116.67 116.67 ---

4.05 2.15 3.87 ---

52 17 105 ---

Go up ? Go down (min)

Load ? Unload (min)

Fixed time (min)

Total Time (min)

0.05 0.05 0.05 ---

0.30 0.30 0.30 ---

18.20 5.95 36.75 ---

22.25 8.10 40.62 --5.57

electrical transpallets, while an availability of 7.8 h per shift is registered in the field for manual transpallets, requiring fewer maintenance activities. Now steps 4, 5, 6 and 7 can be repeated until alternative solutions for the binpacking problem ensure optimizations in the number of handling machines required. If consistent improvements are not recorded for I successive iterations, then the procedure stops and the final optimal solution found is printed by specifying: per order, the composition of items in each forklift load, the routing of each forklift and, obviously, the number of forklifts required. By repeating the solving framework for the 405 orders in the case study, the solutions described below are obtained. Specifically, existing forklift loading and routing procedures are compared with the proposed one. Furthermore two initial situations are analyzed: AS-IS-1 where manual transpallets are adopted for material handling AS-IS-2 where electric transpallets are adopted for material handling. In the new configuration (referred to below as NEW) the electric porter (scenario named NEW-1), manual porter (scenario named NEW-2) and electric transpallets (scenario named NEW-3) are alternatively inserted. Hence, AS-IS-1 and AS-IS-2 are characterized by the same old loading and routing procedures, but different material handling systems, while NEW scenarios are characterized by both improved loading and routing approaches along with a new model of forklift, available for insertion in the warehouse and depicted in Fig. 15.4. Its possibility of transporting both a pallet and an operator improves required traveling time. The results obtained are reported in Tables 15.6, 15.7 and 15.8 (respectively for NEW-1, NEW-2 and NEW-3 scenarios). As already underlined, the mean velocities of the vehicles change. Moreover, manual and electric transpallets do not present go-down and go-up times. Total time is calculated as the sum of fixed time, which does not depend on the number of items in the order, with the distance covered time. The last rows in Tables 15.6, 15.7 and 15.8 contain the average distance covered and the average total time on the total 405 orders after application of the optimizing procedure explained.

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Fig. 15.4 An alternative electric porter available for use in the case study (www.fantozzi.com—accessed May 2011)

Table 15.7 Results obtained in the NEW-2 scenario DATE Order Distance Mean Distance Items (m) velocity covered in the (m/min) time (min) order 13/03 174 17/03 199 24/03 312 --Average

472.8 251.2 451.2 --124.65

50.00 50.00 50.00 ---

9.46 5.02 9.02 ---

52 17 105 ---

Table 15.8 Results obtained in the NEW-3 scenario Date Order Distance Mean Distance Items (m) velocity covered in the (m/min) time (min) order 13/03 17/03 24/03 --Average

174 199 312 ---

472.8 251.2 451.2 --124.65

60.00 60.00 60.00 ---

7.88 4.19 7.52 ---

52 17 105 ---

Load ? Unload (min)

Fixed time (min)

Total time (min)

0.30 0.30 0.30 ---

15.60 5.10 31.50 ---

25.06 10.12 40.52 --6.35

Load ? Unload (min)

Fixed time (min)

Total time (min)

0.30 0.30 0.30 ---

15.60 5.10 31.50 ---

23.48 9.29 39.02 --5.94

The AS-IS situation represents the benchmark term that has to be compared with the achieved solution for the same handling equipment, in terms of either the average handling time or the average distance covered during the time horizon considered. In order to compare the current situation with that obtainable through the application of the optimizing procedure, the real handling time for each order has to be evaluated. In Table 15.9 the obtained results are reported on a sample of three orders in the event of using the current handling equipment, which is made up of manual transpallets and electric transpallets, supposing that the orders are fully satisfied by the same type of vehicle. Note that electric porters are not adopted yet in the current situation, thus they are not involved with the AS-IS situation. On the sample analyzed, only a single machine is required, nevertheless, the application of the optimizing procedure permits an average time saving of 10%

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Table 15.9 Comparison between optimized and as-is situations by varying the handling equipment After the application of the optimizing procedure AS-IS situation Order

174 199 312 … Average on sample orders

Covered distance (m) 472.8 251.2 451.2 … 124.65

Total time (min) NEW-1 Electric porter

NEW-2 NEW-3 Manual Electric transpallets transpallets

22.25 8.10 40.62 … 5.57

25.06 10.12 40.52 … 6.35

23.48 9.29 39.02 … 5.94

Covered distance (m)

Total time (min)

694.07 289.63 887.58 … 161.02

29.48 10.89 49.05 … 7.08

AS-IS-1 AS-IS-2 Manual Electric transpallets transpallets 27.17 9.93 46.13 … 6.55

and 9%, respectively for the manual and electric transpallets. The use of an electric porter induces savings of up to 21%. The results are summarized in Tables 15.10, 15.11 and 15.12 where the NEW scenarios are respectively compared with the AS-IS ones.

15.7 Conclusion Order picking, the activity that aims to insert or retrieve products from storage (or buffer areas) in accordance with predefined customer requests, has been identified as a process which is responsible for most of the costs relating to warehouses. Hence, the implementation of robust design and optimization procedures for planning picking are addressed by researchers and practitioners. In this chapter the case of warehouses served by humans, in picker-to-parts systems, with a discrete picking organization is studied. Specifically, the case of orders including multiple items, located in different aisles and requiring more than one forklift load to completely satisfy customer requests is analyzed with the aim of determining the optimal fleet of handling machines needed to satisfy customer orders. Specifically, two aspects are studied: • the grouping of orders in a finite number of forklift missions, by ensuring that each required item is picked in the required amount • the optimization of the routing handling facilities to be followed in accordance with the objective of minimizing the total traveled distance (or the time required for missions).

Covered distance (m)

472.8 251.2 451.2 … 124.65

Order

174 199 312 … Average on sample orders

22.25 8.10 40.62 … 5.57

NEW-1 Electric porter

Total time (min)

694.07 289.63 887.58 … 161.02

Covered distance (m) -32% -13% -49% … -23%

29.48 10.89 49.05 … 7.08

AS-IS-1 Manual transpallets

Improvement Total time (min)

Table 15.10 Comparison between optimized and as-is situations with scenario NEW-1 After the application of the optimizing procedure AS-IS situation

-25% -26% -17% … -21%

27.17 9.93 46.13 … 6.55

Improvement AS-IS-2 Electric transpallets

-18% -18% -12% … -15%

Improvement

15 Design and Optimization of Picking 419

Covered distance (m)

472.8 251.2 451.2 … 124.65

Order

174 199 312 … Average on sample orders

25.06 10.12 40.52 … 6.35

NEW-2 Manual transpallets

Total time (min)

694.07 289.63 887.58 … 161.02

Covered distance (m) -32% -13% -49% … -23%

29.48 10.89 49.05 … 7.08

AS-IS-1 Manual transpallets

Improvement Total time (min)

Table 15.11 Comparison between optimized and as-is situations with scenario NEW-2 After the application of the optimizing procedure AS-IS situation

-15% -7% -17% … -10%

27.17 9.93 46.13 … 6.55

Improvement AS-IS-2 Electric transpallets

-8% +2% -12% … -3%

Improvement

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Covered distance (m)

472.8 251.2 451.2 … 124.65

Order

174 199 312 … Average on sample orders

23.48 9.29 39.02 … 5.94

NEW-3 Electric transpallets

Total time [min]

694.07 289.63 887.58 … 161.02

Covered distance (m) -32% -13% -49% … -23%

29.48 10.89 49.05 … 7.08

AS-IS-1 Manual transpallets

Improvement Total time (min)

Table 15.12 Comparison between optimized and as-is situations with scenario NEW-3 After the application of the optimizing procedure AS-IS situation

-20% -15% -20% … -16%

27.17 9.93 46.13 … 6.55

Improvement AS-IS-2 Electric transpallets

-14% -6% -15% … -9%

Improvement

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Even if algorithms for solving the capacitated vehicle routing with loading constraints are available for solving similar problems, aspects emerging in complex practical scenarios such as those described in the case study are disregarded: • the grouping of product fragility in only two subsets is a limit for some market fields (i.e. in the case of bathroom furnishings where different classes of item overlapping capability can be identified. Tiles, boxes containing taps, boxes containing lamps and packets containing decorations can be stacked by following such an order. Alternatives are not usually feasible and product integrity may not be ensured. Obviously, individual cases may represent exceptions to be evaluated one by one) • the capacity (both in terms of weight and volume) of a forklift load is sometimes not a strict constraint. Some variability ranges may exist and induce interesting advantages. • the time required for picking execution in two warehouse locations may change in accordance with the forklift loading composition: when fragile items are transported or when the cargo slightly exceeds limits initially imposed, particular care may be requested by the operators. Alternatively, when loads are moved that are still far from their capability saturation, a higher speed may be considered. Hence, a solving framework is proposed and applied in a case study where the warehouse of a company operating in the field of bathroom furnishings is analyzed. Initially product characteristics are identified and a Pareto analysis of both time buckets during which orders occur along with the pool of past orders is carried out. Subsequently, alternative solutions for forklift loads and routing are studied. The results obtained are compared with those occurring in the existing situation. It emerges that mean savings of up to 9% are ensured if traditional manual and electric transpallets are adopted. Otherwise, local savings of up to 26% are achieved if new electric porters are selected.

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Chapter 16

The Logistics Reengineering Process in a Warehouse/Order Fulfillment System: A Case Study Alberto Regattieri, Riccardo Manzini and Mauro Gamberi

Abstract The logistics reengineering process (LRP) is a useful industrial engineering and management technique for achieving significant improvements in operational efficiencies for products quality services in a warehouse/order fulfillment system. In warehousing systems the picking process usually has a significant impact on logistic performance, customer service levels and costs, hence improvement activities are attractive and important. This chapter presents the application of an LRP process in an Italian distribution company, which is a distributor of home furnishings and health care products. In particular, the proposed optimization process is focused on the Order Fulfillment Process (OFP). The main aim of this chapter is to present a methodology to make an effective analysis of an OFP system and, mainly, to present the results, opportunities and criticalities arising from its application. The benefits are significant both in terms of traveled distance savings and manpower usage reduction. These results demonstrate that ‘‘soft’’ reengineering improvements can significantly affect processes, procedures, rules and strategies, can reduce logistics costs and improve customer service levels without introducing ‘‘hard’’ improvements and system modifications, e.g. new equipment, personnel, and machinery.

A. Regattieri (&)  R. Manzini DIEM—Department of Industrial and Mechanical Plants, Bologna University, v.le Risorgimento 2, 40136 Bologna, Italy e-mail: [email protected] M. Gamberi Department of Management and Engineering, DTG University, Padova, Italy

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16.1 Introduction During recent decades the Logistics Reengineering Process (LRP) has been used by many companies in the pursuit of efficiency through technology and organizational developments. In manufacturing warehouses the picking systems usually have a significant impact in terms of logistic process, customer service levels and costs; hence improvement activities are attractive and important. Ozcelik (2010) and Chien-wen (2007) examine whether the implementation of logistics process reengineering projects improves company performance by analyzing a comprehensive set of data on firms in the United States and Taiwan, respectively. Moreover, a series of studies in the early 1990s found that a significant number of LRP initiatives failed or delivered less than they promised (Chien-wen 2007). LRP process is potentially very attractive but is not always easy. This chapter presents the application of an LRP process in an Italian distribution company, named Alpha (a disguised name), which sells home furnishings and health care products. The materials received from the suppliers are checked, accepted and then stored in a warehouse system. The customer orders generate intensive split case picking activities and finally the materials are shipped out. To satisfy and respond quickly to customer demand, Alpha is now strongly focusing on warehouse management and on the order fulfillment process with the aim of strengthening its competitiveness. Alpha is engaged in activities ranging from the supply of input materials to the production and delivery of products, in an attempt to obtain the best added value while reducing the entire supply chain cost (Croom et al. 2000; Piramuthu 2005). In general there is a large amount of complete literature dealing with Business Process Reengineering (BPR), but there are very few sources that explicitly discuss this process applied to the Order Fulfillment System. In general the Order Fulfillment Process (OFP) has been recognized as one of the core business processes in any organization (Kritchanchai and MacCarthy 1999; Turner et al. 2002; Manzini et al. 2005, 2007; Park and Lee 2007). An OFP starts with receiving customer orders and ends with delivering products. It includes activities such as purchase orders (PO) planning, customer order processing, stock checking, order picking activities, final shipping assembly and delivery. While in different companies, OFPs are executed differently in accordance with their unique business characteristics (e.g. product characteristics and lead times), they fulfill two common objectives (De Koster et al. 2007; Ozcelik, 2010): 1. The delivery of products to satisfy customer expectations at the right time, right place, and right quantity. 2. Effective management of the uncertainties from internal and external environments (delay in supply, quality problems, etc.). In any OFP, the wide range of activities involved is carried out by operators from different functional units (e.g. sales personnel for processing customer orders and logistics personnel to manage picking and packing activities). As a result, an

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OFP is complicated and requires activity integration and coordination. In this condition, it is necessary for companies to reengineer their OFPs so that they can be better integrated into the corresponding supply chains (Kritchanchai and MacCarthy 1999; Turner et al. 2002; Waller et al. 1995). Furthermore, logistics systems are dynamic, which leads to the importance for companies to reengineer their OFPs systematically to exploit saving and efficiency. The studies in the literature focusing on Order Fulfillment Systems usually pay attention to success factors such as impetus, opportunities and strategies, but do not address the operational issues (Grover Kettinger 2000; Kleiner 2000). Others studies deal with several particular problems such as order batching, picking area design and product customization (Bartholdi and Hackman 2010; Yu and de Koster 2009; de Koster et al. 2007; Zhang et al. 2010). In view of these limitations this chapter seeks to present a case study dealing with a practical investigation of an OFP system. The final aim is to propose a methodology to make an effective analysis of an OFP system and, mainly, to present the results, opportunities and criticalities arising from its application. This chapter, by means of a case study, discusses original and innovative approaches for reengineering Order Fulfillment Systems based on new material allocation strategies and product clustering. The remainder of this chapter is organized as follows: Sect. 16.2 presents the logistic chain of the company (subject of interest as a case study). Section 16.3 presents the results of the AS-IS analysis; the main collected criticalities are discussed in Sect. 16.4. Finally, several reengineering activities are presented in Sect. 16.5.

16.2 Alpha Company Logistic Chain Alpha sells home furnishings and health care products. Its supply chain involves a great number of suppliers and customers (i.e. hundreds). Figure 16.1 shows the activities related to the order fulfillment system, normally placed along the supply chain. The gray box lists the logistics activities directly connected with material handling. Reengineering allows breakthrough improvements by discarding existing malfunctioning processes. This effort may not be rewarded without a systematic analysis of the AS-IS condition. The improvement activities must be developed considering the results of this assessment. At Alpha the AS-IS analysis concentrates on the material handling process, allowing several steps, in particular: • Layout analysis The aim is to study the layout to locate the warehouse area, the other areas (e.g. receiving area, order setup area, etc.) and their characteristics; • Analysis of material receiving procedures and activities This part focuses on the investigation of procedures connected with the

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acceptance and inspection of material from suppliers waiting in the checking area. The main aspects considered are workloads, (i.e. personnel and shifting), the equipment used and the activities performed (in term of targets, durations and information technology support needed); The approach mentioned in the last bullet point is repeated to study the other activities, in particular in the following steps: • Analysis of the storage process This process deals with the handling activities performed on the accepted materials from the check-in area to the storage area; • Analysis of the picking and refilling processes The picking and refilling processes are the core logistics tasks executed. In these phases the goods are collected considering customer orders and the picking areas are replenished. The relationship between the materials in the stock area and in the picking area plays a crucial role in the performance of the entire order fulfillment system. • Analysis of order setting and shipment processes. The materials and related orders are checked and then the checked goods are packed and forwarded to the shipment process and finally delivered to customers. The AS-IS analysis identifies a set of main criticalities and of further development areas that are the starting point for the following improvement activities. The following section presents the entire AS-IS analysis and some examples of improvement activities on the order picking process.

16.2.1 Definition and Notation of Symbols ASj CLi fi IASj ICj LRP OP OPi OPTi PDA Rj RTc RTf SKU Vi

available space in cluster j products cluster i rate of material flow through the warehouse for the item i available space for an item of cluster j number of items in cluster j Logistics reengineering process Order picking operator i optimal fraction of available space devoted to item i Personal digital assistant residuals for day j, between forecasted and collected discharging time discharging time collected for day j discharging time forecasted for day j Stock keeping units volume of product i

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Wi WMS

429

weight of product i Warehouse Management System

16.3 AS-IS Analysis 16.3.1 Layout Figure 16.2 shows the layout of Alpha. Excluding the office building, the total area is about 10,000 m2, of which roughly 5,000 m2 are devoted to storage. The shelves are equipped with pallet racks that are organized into 13 aisles. The racks have six levels: the ground level is dedicated to picking activities; levels 1–5 contain the Stock Keeping Units (SKUs). The forward-reserve picker to part policy (Caron et. al. 2000) is adopted. Lower rack levels are used for manual Order picking (OP) (the forward area), while higher levels contain bulk storage (the reserve area). The SKU storage capacity is about 9,200. The SKU used as a reference is the pallet EUR-EPAL 800 9 1200 mm. A secondary storage area is available, about 480 m2, where out of shape materials are located, which have larger external dimensions than the racks. The building presents 12 truck docks, seven used for shipping material and five used for receiving material. Between the truck docks and shelves two depot areas, 450 and 700 m2 respectively, allow temporary storage of goods. Materials arriving (from suppliers) are discharged and placed in a temporary receiving area. The check-in procedure requires a physical check and data entry in the Enterprise Resource Planning System (ERP). When this procedure is complete the storage activities are executed. The shipment area collects the material waiting for delivery to customers. In this area packing activities are carried out, when necessary. Considering the retrieval activities, picking is a Less than unit load with multiple stops per trip. The pickers retrieve sets of items or multiple handling units of the same item on a single Order Picking cycle (Manzini et al. 2007). Each picker usually picks a complete customer order during a mission. All the activities are supported by two software packages: an ERP that manages the entire supply chain from customer orders to customer invoicing (including supplier orders) and a Warehouse Management System (WMS), which focuses on supporting the warehouse activities (e.g. material acceptance, labeling, picking and order pack listing). In the following paragraphs all the logistics activities represented in the gray box in Fig. 16.1 are discussed with the same approach, such as the analysis of the workload involved and the equipment used, the description of activities by means of a flowchart and finally the analysis of statistics and observations.

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Fig. 16.1 Alpha company supply chain (order fulfillment system detail)

Alpha activities C orders

orders generation

forecast

Selection of suppliers Negotiation with suppliers Issuing & purchasing Order paying

Reception of materials

CUSTOMERS

SUPPLIERS

S orders

Inspecting quantity & quality

Storaging materials Material picking & refilling Material Shipping

Invoicing

16.3.2 Acceptance of Materials Four operators are normally employed in the receiving area (named Op1, …, Op4). Their shifts during the week are organized and shown in Table 16.1. Their fleet of vehicles consists of five pallet trucks, 1,200 kg load capacity, 2,390 mm Duplex mast legs and a forklift truck with load capacity of 1,600 kg, Triple mast legs. The workload in the receiving area is concentrated in the morning. This is a normal condition due to the use of industrial vehicles for transporting the materials both from suppliers and to customers.

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Table 16.1 Workload in receiving area 6 6 6 6

6 AM–1.45 PM 6 AM–1.45 PM 12 AM–7.45 PM 6 AM–1.45 PM

AM–1.45 AM–1.45 AM–1.45 AM–1.45

Wednesday PM PM PM PM

6 6 6 6

AM–1.45 AM–1.45 AM–1.45 AM–1.45

Thursday PM PM PM PM

6 6 6 6

AM–1.45 AM–1.45 AM–1.45 AM–1.45

6 AM–1.45 PM 12 AM–7.45 PM 6 AM–1.45 PM 6 AM–1.45 PM

U. S.

Hm=7.75

Friday PM PM PM PM

1.30 2.15

Tuesday

Op1 Op2 Op3 Op4

2,50 4,20

Operators Monday

2.50 3.50

1

2.50 3.50

2 3 4

order setup area & shipment area

5

2.50 3.50

2.50 3.50

2.50 3.50

2.50 3.50

6

docks

2.50 3.50

2.50 3.50

7

2.50 3.50

pallet racking area

8 2.50 3.50

9 2.50 3.50

10 11

receiving area

2.50 3.50

2.50 3.50

12 13

out of shape storage area

recharging forklift area

office

Fig. 16.2 Layout of Alpha company

The operators involved in receiving are responsible for the activities of vehicle hauling in a dock to the material depot in the receiving area. These material handling activities are supported by several operations performed in the ERP system and in the WMS system. Figure 16.4 shows a flowchart describing all these tasks. Different kinds of industrial vehicles arrive at Alpha. Figure 16.3 shows the average percentage collected over a 3 month assessment.

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Fig. 16.3 Percentage of different kinds of industrial vehicles arriving to docks

van 17%

lorry trailer 19%

truck 23% tractor trailer 41%

Figure 16.5 presents the average durations, collected during the assessment, for carrying out the entire procedure depicted in Fig. 16.4, from vehicle docking to vehicle departure and item labeling. Different kinds of vehicles have different load capacities hence the time taken is quite different. Another interesting observation emerges from the analysis of the number of packages unloaded during the assessment period (i.e. in an SKU there are usually several packages), which vary according to the different days of the week. Monday and Friday are definitely more critical compared to the other days, in terms of the amount of material to be unloaded. This typical behavior asks for the introduction of variable workloads but there is currently a fixed number of operators for this task. Figure 16.6 shows the average number of packages unloaded during the week and Fig. 16.7 presents the corresponding analysis of variance. In terms of analysis of variance the average number of packages processed per day is 9,781, but the spectrum is wide as shown in the Figure. Over recent years, Alpha has developed a model based on the concept of standard unloading times for each different transport system, in an attempt to schedule the arrivals of the supplier vehicles with the aim of balancing the workload in the goods receipt process. This model appears to be insufficiently accurate. Figure 16.8 shows the analysis of residuals Rj, (minutes) for day j, as the result of comparison between the forecasted and the collected daily discharging time values. Rj ¼ RTf  RTc with: RTf discharging time forecasted for day j (min); RTc discharging time collected for day j (min);

ð16:1Þ

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Fig. 16.4 Receiving activities flowchart

433

Truck docking Packing list delivering (from truck driver)

SKU unload no

Equal to previous SKU? yes

SKU bar code reading Packagesbar code reading Single itemsbar code reading Comparison packing list/order in WMS system

picking and reserve area SKU handling in receiving area

Another SKU to unload?

yes

no

Empty SKUs loading on truck (if needed) Document ruling (to truck driver)

Truck leaving

Packing list data entry in WMS system Exporting data in ERP system Labels printing and items labeling stop

Fig. 16.5 Receiving average times and average value of SKUs unloaded by kind of vehicle

receiving time (min)

SKUs unloaded (average)

87.4 72.6 53.2 36.7

43.7 30.7 10.1

lorry trailer

tractor -trailer

truck

2.5 van

434 Fig. 16.6 Average number of packages processed per day in receiving area

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8961

Monday

5914

6247

Tuesday

Wednesday

Thursday

Friday

Fig. 16.7 Number of average packages received per day: analysis of variance

Fig. 16.8 Discharging time model accuracy: analysis of variance of residuals Rj

In conclusion, the scheduling of vehicles arriving is not accurate and the effect of the beginning and the end of week due to the commercial requirements of suppliers and freight forwarders is significant. This results in serious inefficiencies and extra costs (i.e. work overtime, traffic congestion, equipment damage and industrial accidents).

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Table 16.2 Workload in storage area Operators

Monday

Tuesday

Wednesday

Thursday

Friday

Op5 Op6 Op7 Op8 Op9 Op10

12 AM–7.45 PM 6 AM–1.45 PM 12 AM–7.45 PM 6 AM–1.45 PM 12 AM–7.45 PM 6 AM–1.45 PM

6 AM–1.45 PM 12 AM–7.45 PM 12 AM–7.45 PM 12 AM–7.45 PM 6 AM–1.45 PM 6 AM–1.45 PM

12 AM–7.45 PM 6 AM–1.45 PM 6 AM–1.45 PM 12 AM–7.45 PM 12 AM–7.45 PM 6 AM–1.45 PM

6 AM–1.45 PM 12 AM–7.45 PM 6 AM–1.45 PM 6 AM–1.45 PM 6 AM–1.45 PM 12 AM–7.45 PM

6 AM–1.45 PM 6 AM–1.45 PM 6 AM–1.45 PM 6 AM–1.45 PM 6 AM–1.45 PM 12 AM–7.45 PM

Fig. 16.9 Storage activities flowchart

Receiving area empty ?

yes

stop

no

SKU bar code reading by PDA Handling to storage location suggested by WMS

Storage location eligible ?

yes

Location bar code reading by PDA

Searching of an eligible storage location

New location data entering in WMS

SKU stocking

16.3.3 Storage After the acceptance procedure (para 3.2) the materials wait in the receiving area until operators carry them into the storage area (i.e. reserve area). This operation is usually performed by six operators (named Op5, …, Op10). Their shifts during the week are organized and shown in Table 16.2. Their fleet of vehicles consists of 6 reach trucks, 1,600 kg load capacity and Triple mast legs. In all the storage activities operators are supported by the WMS system by means of personal digital assistant (PDA) devices. PDA operators know the assigned storage location, read bar codes and, if needed, can edit incorrect information on the WMS. Figure 16.9 presents a flowchart dealing with storage activities.

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Fig. 16.10 Time sampling of storage activities

Class of times Table 16.3 Product clusters assignment in picking area

Aisle

Assigned clusters of products

1 2 3 4 5 6 7 8 9 10 11 12 13 Out of shape

CL1, CL2 CL3 CL4 CL5, CL6 CL7 CL8 CL9, CL10, CL11, CL12 CL13, CL14, CL15 CL16, CL17, CL18, CL19 CL20 CL21 CL22, CL23 CL24 CL25

During the assessment much of the sampling is focused on the duration of the storage activities revealing an average time of about 6.59 min from the SKU pick up operation in the receiving area to the SKU drop off operation in the storage location. Figure 16.10 goes into more depth on the analysis of variance of the storage time. The storage locations of products (reserve area) are related to their picking locations (forward area). Both positions are managed by the WMS system by means of product clustering and dedicated aisles. Alpha manages about 7,150 different products. They are grouped into 25 clusters (named CL1, …, CL25) stratified by volume, dimensions and weight. In the picking area a single cluster or group of clusters is assigned to storage slots pertaining to racks looking towards the same aisle. The reserves of products in the storage area are located in higher levels as near as possible to the corresponding

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IASj

Fig. 16.11 Available space for an item of cluster j (IASj) in reserve and in forward area

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

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IASj reserve area (SKUs locations) IASj forward area (picking locations)

1

2

3

4

5

6

7

8

9

10 11 12 13

Aisle

picking location on the lower level. This procedure is entirely supported by the WMS using PDA devices. Table 16.3 shows the product clusters assigned to different aisles. The allocation of clusters/aisles appears to be slightly off balance in terms of the space required and the allocated space, both in the picking area and the reserve area. Consider IASj as the available space for an item of cluster j in terms of SKU storage locations in the reserve area and in terms of allocated picking space in the forward area: IASj ¼

ASj ICj

ð16:2Þ

with: ASj available space in cluster j (SKUs locations in the reserve area or picking locations in the forward area); number of items in cluster j ICj IASj has a wide range and is dispersed as clearly shown in Fig. 16.11 Alpha applies different supply policies to different products. Furthermore, the products have very different characteristics in terms of shape and dimensions. In conclusion clusters need a different amount of space both in the reserve area and the forward area. But these differences do not explain the wide range of variation in the IASj parameter. Furthermore several values of IASj are less then the unit. This behavior results in the continuous use of the manual location of materials by operators in an aisle not suggested by the WMS. In these conditions picking traveling time and distances clearly increase, along with the number of refilling missions (restocks).

16.3.4 Restocking Process The order fulfillment activities collect the products in the picking locations, then refilling, i.e. restocking, is necessary. This restocking task is done by the same

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A. Regattieri et al. Reading on PDA of 1° item in restockslist Travelling in storage location and SKU picking Scanning of item’s bar code and of storage location’s bar code

Travelling in Picking location and forwardarea refilling

Scanning of picking location’s bar code and dropingoff confirmation no

no

Restocks list empty ?

On hand materials ?

yes

yes

stop

Original stock location is ok? (by operator)

no

Selection of a new location in storage area (by operator)

yes

Travelling in storage location and SKU droping off Scanning of storagelocation’s bar code and droping off confirmation

Fig. 16.12 Refilling of picking area activities

personnel who stocks the materials (i.e. Table 16.2). WMS, by means of PDA, guides the operators to get restocks following a list that ranks product in hand order and products which have the empty locations in the picking area. The sampling activities carried out during the assessment have highlighted two main characteristics regarding the restock process: the average duration of a restock mission is about 8’55’’ min and the average distance travelled is about 34.0 m. Figure 16.12 presents a flowchart describing the refilling of picking area activities.

16.3.5 Order Picking Orders fulfillment is the core activity in Alpha. Its importance emerges due to strongly differentiated products with shorter life cycles, low volume and low customer delivery time accepted that Alpha normally manage.

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Table 16.4 Workload in storage area Operators

Monday

Op11 Op12 Op13 …. Op25

7 7 6 6 6

AM–2.45 AM–2.45 AM–1.45 AM–1.45 AM–1.45

Tuesday PM PM PM PM PM

7 7 6 6 6

AM–2.45 AM–2.45 AM–1.45 AM–1.45 AM–1.45

Wednesday PM PM PM PM PM

7 7 6 6 6

AM–2.45 AM–2.45 AM–2.45 AM–2.45 AM–2.45

Thursday PM PM PM PM PM

7 7 6 6 6

AM–2.45 AM–2.45 AM–1.45 AM–1.45 AM–1.45

Friday PM PM PM PM PM

7 7 6 6 6

AM–2.45 AM–2.45 AM–1.45 AM–1.45 AM–1.45

PM PM PM PM PM

Retrieval activities can be defined as less than unit load systems with multiple stops per trip. Order pickers retrieve sets of items or multiple SKUs of the same item on a single order picking cycle. They visit different slots in the forward area before going to the shipment area. Each picker is responsible for picking a complete single customer order during a mission. The picking operation is usually performed by 6 operators (named Op11, …, Op25). Their shifts during the week are organized and shown in Table 16.4. Their fleet of vehicles consists of 15 order picker trucks, 2,000 kg load capacity, 2,350 mm legs. Figure 16.13 shows the flow of activities to perform a customer order picking cycle. The starting point is the delivery of the customer order list to the picker by the dispatch department. The pickers are supported by WMS through a PDA system including voice technology and speech recognition to communicate. In particular, the pickers use headsets and microphones to receive instructions by voice, and verbally confirm their actions back to the system. As mentioned before the adopted storage location assignment policy is based on cluster allocation in defined aisles. The products are grouped into clusters according to their characteristics (i.e. volume, weight, dimensions, etc.), and the clusters are assigned to different aisles as reported in Table 16.3. Both the cluster assignment to aisles and the space allocation of the picking area to different products are suggested by the WMS, exclusively dictated by empirical evaluation and the working experience of the warehouse managers. During the assessment the characteristics of customer orders are analyzed, in particular the number of different lines per order and the corresponding volume of material. Each order line implies a trip to reach the appropriate slot in the forward area and several retrieval activities on single or multiple packages of product. The items marketed by Alpha are very different in terms of weight and size, hence the orders have very different volumes in terms of space used. The capacity of the order picker truck can almost always contain the volume of material corresponding to a customer order. The average number of lines per order is 83.7 and the average order volume is 1424 dm3, but the values are very dispersed. Figures 16.14 and 16.15 show the statistical analysis of these two parameters for orders collected during the period December 2008–December 2009.

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Target aisle and location communicating (voice)

Target aisle and location travelling

Target reached confirming (voice)

Product quantity (to pick) communicating (voice)

Forward product quantity sufficient ?

no

yes

No pickup communicatingand confirming (voice)

Packages picking up

Picking up confirming (voice)

End of order?

no

yes

Travelling to shipping area

Order packing

Fig. 16.13 Picking activities flowchart

During the assessment the productivity of the pickers is evaluated both by means of tests carried out in the field and statistical tracing of WMS data. The average order picking mission takes 39.2 min and includes about 83.1 order lines corresponding to 118.2 packages. The high number of lines per order in comparison to the number of aisles results in a complete crossing of the warehouse area for each customer order fulfillment. In other words during an order mission a picker visits all the aisles typically with a traversal policy (Manzini et al. 2005; Bindi et al. 2009). The unbalanced product

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Fig. 16.14 Lines per customer order (Dec 2008–Dec 2009)

Fig. 16.15 Volume per customer order (Dec 2008–Dec 2009)

clustering and aisle assignment, as revealed by IASj analysis (Fig. 16.11), cause an unbalanced use of slots and aisles. The analysis of popularity of products (i.e. the number of requests for a given products, so the number of times a picker must travel to a storage location for a given product) related to the spatial position in the layout of warehouse reveals a critical situation typically resulting in the over-crowding of trucks and a high number of restocks. Figure 16.16 shows the popularity of products in the forward area along different spatial coordinates of warehouse. The diameter of the circle is proportioned to the popularity value.

16.3.6 Orders Shipment Setting The pickers leave the material collected in the shipping area and several activities to prepare this material for the final loading into the industrial vehicle must be

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Fig. 16.16 Analysis of popularity of product versus spatial coordinates

performed. The load capacity of a vehicle includes material corresponding to several customer orders, typically from two to ten orders. These operations are usually performed by four operators (named Op26, …, Op29). Their shifts during the week are organized and shown in Table 16.5. Their fleet of vehicles consists of one reach truck, 1,600 kg load capacity, triple mast legs and three robopacks to apply the plastic film to the pallets. Normally before picking operations, the orders are clustered by the dispatch department into different groups considering the customer locations, the volume of materials and the kind of available vehicles for the considered delivery date. Considering a single cluster of orders, the analysis of customer material acceptance times (often there are significant constraints) and the best vehicle routing (also taking into consideration traffic congestion) give the pallet priority in vehicle loading activities.

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Table 16.5 Workload in shipping area Operators Monday Op26 Op27 Op28 Op29

6 AM–13.45 12 AM–7.45 12 AM–7.45 12 AM–7.45

Tuesday PM PM PM PM

Wednesday

6 AM–13.45 12 AM–7.45 12 AM–7.45 12 AM–7.45

Fig. 16.17 Orders shipment setting flowchart

PM PM PM PM

6 AM–13.45 12 AM–7.45 12 AM–7.45 12 AM–7.45

Thursday PM PM PM PM

6 AM–13.45 12 AM–7.45 12 AM–7.45 12 AM–7.45

Friday PM PM PM PM

6 AM–13.45 12 AM–7.45 12 AM–7.45 12 AM–7.45

PM PM PM PM

Gathering of arrival time and route plan of vehicle

Analysis of assignment Vehicle –Group of orders (by dispatching office)

Design of pallet loading order on vehicle (based on vehicle routing)

Pallet packing (plastic film) and labeling

Pallet loading on vehicle

no

Group of orders completed? yes

Packing list delivering (todispatching office)

Vehicledeparture

A customer order is usually split into different pallets (EUR-EPAL 1,200 9 800 mm). At the end, the operators hand over the final packing list to the dispatch department, which performs the final check and allows the truck to depart. Figure 16.17 shows the flowchart of typical order shipment setting activities. During the assessment several sampling activities on the time duration of activities were carried out. The reference must be the pallet because the orders are split into a variable number. The average times are: pallet setting before applying the plastic film 2.3 min, film application 1.5 min and pallet handling to vehicle 2.5 min.

16.4 Main Criticalities The AS-IS analysis is proposed to identify the critical processes. Process-flow diagrams, process analysis worksheets and data summary charts are prepared as effective tools to get a thorough understanding of the existing process with a view

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to focus on the possible areas of improvement. In particular procedures, workflows, and resource consumption in terms of workforce, equipment and information technology support are investigated and discussed. In the acceptance process the number of packages processed per day is different on different days of the week. Typically Monday and Friday are overloaded considering that the workforce is constant throughout the week. The final result is significant mental stress for operators and need to work overtime and hence extra costs. To provide the time consumption in the SKU unloading process, Alpha uses a model based on standard time for each kind of industrial vehicle. Experimental evidence shows that the accuracy is not sufficient and the model does not represent valid support for the vehicle arrival planning process. Another important criticality deals with the space allocation policy used both for product clusters and for single products in a single cluster. Alpha adopted a fixed assignment aisle-cluster of products in the picking area. One or more clusters are dedicated to racks corresponding to a single aisle in the picking area. The products in the reserve area are allocated to higher levels of racks as near as possible to the position of the same products in the picking area (lower level). When picking area is empty product in reserve area are pulled down usually by forklift. This assignment aisle-product clusters is managed by WMS but only considering the experience of the managers that force materials in positions which are different in comparison to WMS suggestions. Their experience also guides the allocation of the space of racks for products into a single cluster. Considering the constraint of the total space for the cluster or the group of clusters (i.e. the assigned aisle of racks) managers divide the racks into four kinds of slots which have different capacities, ranging from one to four single units of slots, and assign different products to these different slots. This empirical double assignment (rack aisles-product clusters and slotsproducts) and the following allocation of material in the reserve area lead to several problems. The products have very different sales levels in term of quantity but similar space is allocated in the picking area, so order pickers often do not find sufficient product quantity to fulfill the order (and temporarily skip the item) or pick up all the product stock in the picking area. In both cases a high number of refilling missions (restocks) are needed. This unbalanced space assignment, also revealed by the popularity analysis, generates significant traffic congestion in aisles 6–11. To limit this problem, pickers often force manual allocation of materials in different slots from the positions suggested by the WMS. This bad procedure leads to long distances between the same material in the picking area (lower level) and the reserve area (higher levels) increasing the costs of the refilling missions. These kinds of missions require long distances in terms of space because the allocated space in the reserve area is not sufficient to contain all the material due to the empirical distribution between rack aisles and product clusters.

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The next section presents several significant improvement activities developed in the Alpha case. Refilling policies along with reserve and picking strategies are interesting areas for improvements in the fulfillment system. Different proposals are developed both in terms of receiving goods and customer order fulfillment. To reduce the risk of LRP, several simulative approaches are set for evaluating and analyzing the reengineering solutions. The logistics reengineering process has proved to be a useful industrial engineering and management technique for achieving significant improvements in operational efficiencies for good quality services in the warehouse/order fulfillment system.

16.5 Reengineering Activities The reengineering process of the picking system starts from the following considerations/assumptions: i. ii.

iii.

iv.

v.

This is a low-level picker to part OPS. In the fast pick area the storage location assignment is fixed, that is, the location for a generic SKU is fixed. This is a single order picking system. The company rejected the idea of organizing the picking and retrieval activities from the fast pick area adopting a batch strategy, e.g. assigning more than one order (or different parts of different orders) to a single picker. The average number of orderlines is very high and does not justify the adoption of new routing procedures and rules in presence of a single order picking strategy: for each customer order the picker visits a large number of locations, i.e. a large number of aisles. The storage capacity of the fast pick area and the storage capacity of the reserve area are constant from the AS-IS to the reengineered system configurations (called TO-BE). The configuration of families/clusters of items and the assignment of families to aisles are the same in AS-IS and TO-BE configurations.

The last two hypotheses can be considered not useful in a reengineering process, but they are necessary because the company did not accept changing the AS-IS configuration of racks which are very variable from one aisle to another. In particular the level of products stored in the forward area significantly changes from one item to the next. The subject of this section is the illustration of the procedure adopted for redesigning the OPS in terms of storage allocation, that is, the determination of the best storage level for each item, and assignment of storage allocation within the fast pick (forward) area and the bulk (reserve) area. The adopted procedure is made up of the following decisional steps as illustrated in Fig. 16.18:

446 Fig. 16.18 Adopted procedure for OPS reengineering

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• Forward area (AS-IS analysis) capacity • Reserve area (AS-IS analysis) capacity Storage capacity • SKUs or packages' locations mapping on AS-IS and SKUs' locations determination

Storage capacity determination for itemsand families of items

• OPT rule (Bartholdi and Hackmann, 2003)

• rank-based rule for assignment Aisles and locations • SKUs or packages' barycentric coordinates determination assignment in forward area

• distance from SKUs or packages' locations in forward and reserve configurations • number of restocks, • cost of travelling for restocking, KPI measurement • cost of travelling doe picking/retrieving.

1. Determination of the available storage capacity and the percentage of capacity used in the forward and reserve areas. By the previous (iv) assumption the capacity in AS-IS and TO-BE configuration is the same (Storage capacity of an SKU and usage) 2. Mapping of SKUs or packages locations, i.e. determination of the barycentric coordinates, in the AS-IS configuration and for both forward and reserve areas. 3. Determination of the storage capacity, i.e. the maximum admissible level of storage, for a generic item in the forward area adopting a dedicated storage assignment strategy and the single location hypothesis, i.e. at least one location for each item. (Storage allocation of products in the forward area and restocking). 4. Determination of the storage capacity assigned to each family of products; 5. Assignment of aisles and locations to products in the forward area adopting a rank-based rule. (Storage location assignment). 6. Determination of the barycentric coordinates of SKU in TO-BE configuration and for both forward and reserve areas. 7. KPI measurements and comparison (AS-IS vs. TO-BE configurations): distances from packages or SKU locations in forward and reserve configurations; number of restocks; cost of traveling for restocking; cost of traveling for retrieving items. (KPI measurement and comparison (AS-IS vs. TO-BE)). An in-depth analysis of the AS-IS configuration of the system shows that: the historical restock number is very high and depends on the maximum storage level defined for a generic SKU; 99.30% of SKUs, which corresponds to about 7,910 items, have a single fixed location in the forward area. The number of locations for each SKU in the reserve area is very variable, but most items have 1, 2 or 3 different locations. Figure 16.19 illustrates the statistical distribution of the

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The Logistics Reengineering Process in a Warehouse/Order Fulfillment System

Fig. 16.19 Distribution of the number of locations in the fast pick area

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number of locations in the reserve area. This is a typical circumstance in the presence of seasonal products and when a randomized storage assignment is adopted. The historical period of interest is 1 year. The statistical distribution of popularity distinguishes ‘‘very high’’ moving items from ‘‘very slow’’ ones. In particular, in the forward area the number of visits (the total popularity) is 13,078,000 corresponding to 7,966 items moved and 32,731,363 dm3/year (dm3 stays for 1 L). The value of popularity is very variable: from one to more than 5,000 accesses in a year. In the forward (i.e. ‘‘stock’’) area the number of items restocked is 2,760, while the popularity, i.e. the number of visits to the reserve area, is 32,705. Now a brief description of the adopted procedure and the results obtained for some of the steps cited in Fig. 16.18 follows.

16.5.1 Step 1: Storage Capacity of an SKU and Usage The forward (i.e. the ‘‘pick’’) capacity is 3,885,036 dm3 corresponding to 18% of the whole system capacity: 1,237,163 dm3 of this capacity (corresponding to about 32%) is used assuming a ‘‘fluid’’, i.e. continuous, hypothesis of using the storage volume. The reserve capacity is 18,644,922 dm3 corresponding to 82% of the whole system capacity: 5,777,498 dm3 of this capacity (corresponding to about 31%) is used assuming a fluid hypothesis of using the storage volume.

16.5.2 Step 3: Storage Allocation of Products in the Forward Area and Restocking The adopted model for the determination of the fraction of storage volume to be allocated to a single item is the following as proposed by Bartholdi and Hackman (2003):

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pffiffiffiffi f OPTi ¼ P pi ffiffiffiffi i fi With: fi OPTi

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rate of material flow through the warehouse for the item i; optimal fraction of available space devoted to item i

Bartholdi and Hackman (2003) demonstrate that this strategy minimizes the number of restocks from the reserve area. Then the minimum number of restocks found is the same for each item i. This could represent a useful guideline for evaluating whether or not the adopted storage allocation strategy is out of balance and produces as many restocks as required. In particular, for this case study, the large variability of restocks for each product in a period of time demonstrates that the OPS is not operating under optimal conditions. For example consider the code 00879006, whose AS-IS capacity is 2,268 dm3 corresponding to 1.48 pallets. The popularity is 2,484 pieces, and the corresponding rate of material flow is about 39,529 dm3/year. Equation (16.3) quantifies a fraction of storage volume equal to 0.069266%, that is about 2,691 dm3 and 1.76 pallets. As a consequence, the allocated capacity for code 00879006 increases by about 18.65% from the AS-IS to TO-BE configuration. Figure 16.20 shows the effect of the reallocation of storage capacity to the whole set of items in the forward area. In particular, the statistical distribution of the ratio of allocated capacity in AS-IS and the allocated capacity in TO-BE is reported. More than 50% of items have an assigned capacity over the optimal

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value for restock minimization. The large variability in the rate of materials generates a large variability in the assigned storage capacity. The adopted hypothesis for the reserve area is that the storage capacity of an SKU in TO-BE is the same as in AS-IS. The expected reduction of restocks in one year is about 23.5% corresponding to about 13,396 restocks/year.

16.5.3 Step 5: Storage Location Assignment This section deals with the determination of the storage level assigned to each item in accordance with the assignment of families of products to the aisles, that is, in agreement with the pick list sequence. The pick list is a sequence of retrieval items and orderlines, arranged based on the volume and weight of each bin (see Sect. 16.3.3). For example, during the retrieval process items with high levels of weight and/or low volume (dm3 values) have to be stored at the first levels of picked unit of load. As a consequence most critical items are assigned to the most favorable locations given the pre-assignment of families and items to the aisles (see previous hypothesis (v)). By the reallocation of storage capacity in the forward area the storage level assigned to a generic family of products can significantly change from AS-IS to TO-BE configurations. Given the list of items to be assigned to a specific aisle, a ranking procedure is adopted to define the exact location of each item in the forward area and in accordance with the reallocated storage capacity (see Step 3). The adopted procedure is based on the value of popularity (considering descending values) and not on the value of the so-called index of assignment (AI) defined as numbers of bins moved in a historical period T and adopted by the company in an AS-IS configuration. As a consequence, in a TO-BE system configuration the most visited items are located in accordance with the number of accesses in T, i.e. the number of visits requested of pickers, and not with the flow of bins moved in T (as in AS-IS system configuration). Items with high popularity are located in available and most favorable locations, i.e. near the I/O depot area. According to the aforementioned (v) hypothesis/constraint the features of storage racks and aisles are given and it is possible to modify the assignment of locations to products given the updated values of storage level area and the preassignment of families of items to the most favorable aisles. Following the re-assignment procedure the storage locations refer to a single aisle corresponding to two racks and made of two areas: the first is the low-level area and corresponds to the forward storage volume; the second is the high-level area and corresponds to the reserve area. The basic idea of the adopted greedy rule for storage assignment is to cut the fast pick area and the reserve area of the generic aisle obtaining slides of different widths.

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Fig. 16.21 Virtual level and stock and pick areas (a). Slice construction at pick and stock (b)

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Fig. 16.22 Storage location assignment procedure

Figure 16.21 illustrates this filling process. The warehousing system is made up of several aisles, two racks each, and each aisle has a rack level used for the picked products (see Fig. 16.21a). In a single aisle this level is supposed to be fixed and is called ‘‘virtual level’’. For a given aisle the whole AS-IS area spent for picking is known and is also supposed to be constant in a generic TO-BE. As a consequence the AS-IS configuration defines a virtual level for each aisle, as a couple of racks. The generic rack is cut into slices respecting its virtual level and separating ‘‘picked’’ from ‘‘bulk’’ storage quantities. Different slices represent the assigned storage capacity to different items. The assignment of items is constructed by assigning a single item with another starting from the first available locations close to the I/O depot area (see Fig. 16.21b). Obviously, with two items assigned to the same rack, the corresponding slices have different widths as illustrated in Fig. 16.21b. Literature demonstrates that in the presence of class-based storage allocation of products the right number of classes is three (A, B and C). So in order to reduce the number of classes from the AS-IS configuration made up of 25 different classes, they have been grouped in accordance with the combined values of volume—V, high (H) and low (L), and weight—W, high (H) and low (L). Then the groups of

The Logistics Reengineering Process in a Warehouse/Order Fulfillment System

Fig. 16.23 Storage location assignment and slices configuration

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items of each class of families are assigned to the most suitable aisles in order to pick items with high weight and low volume at the beginning of the picking tour. In fact, all picking missions start in a corner of the warehousing system first visiting aisles that are closer to the depot area. Figure 16.22a and b exemplify the process of assignment families and products to an aisle. The generic family is a tower of bricks, each corresponding to a specific product and made up of two contributions: pick (for the forward storage area) and stock (for the reserve storage area). The assignment is parallel in order to minimize the distance between the picked quantity and the reserve quantity. The priority adopted for the assignment is the popularity: given two or more products to be assigned to an available and favorable location the most ‘‘popular’’ is selected and related slices are identified for both the forward and the reserve areas. The result of the definition of slices at pick and stock is exemplified in Fig. 16.23.

16.5.4 Step 7: KPI Measurement and Comparison (AS-IS vs. TO-BE) Figure 16.24 shows the distribution of distance, in meters, between the pick and stock locations in a specific aisle and for a specific family of products. The mean expected saving on the traveled distance for a single restock is about 6%. But there is another important saving due to the reduction in the number of restocks as illustrated in Sect. 16.5.2. The average cost of a restock is 34.0 m, while the duration, including the variable traveling time and fixed times, is about 8 min and 55 s. The saving distance for a single restock is about 2.2 m. The reduction in restocks in a year generates a saving of 456 km/year. The saving due to the reduction of 1.6 m generates a saving of 95 km/year. The saving time is 2,040 h/year when a picker works about 1,760 h/year. As a consequence the saving corresponds to at least one picker and related annual costs which are about 33 k€/year.

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Fig. 16.24 Distribution of distance between pick and stock locations

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