Planning, Design, and Analysis of Cellular Manufacturing Systems

MANUFACTURING RESEARCHAND TECHNOLOGY24

PLANNING, DESIGN, AND ANALYSIS OF CELLULAR MANUFACTURING SYSTEMS

MANUFACTURING RESEARCH AND TECHNOLOGY

Volume4.

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Volume 6. Volume 7A. Volume 7B. Volume 8.

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Volume 10. Volume 11. Volume 12. Volume 13. Volume 14. Volume 15. Volume 16. Volume 17. Volume 18. Volume 19. Volume 20. Volume21. Volume 22. Volume 23. Volume 24.

Flexible Manufacturing: Integrating technological and social innovation. (P. T. Bolwijn, J. Boorsma, Q. H. van Breukelen, S. Brinkman and T. Kumpe) Proceedings of the Second ORSA/TIMS Conference on Flexible Manufacturing Systems: Operations research models ~nd applications (edited by K. E. Stecke and R. Suri) Recent Developments in Production Research (edited by A. Mital) Intelligent Manufacturing Systems I (edited by V. R. Mila~:i~) Intelligent Manufacturing Systems II (edited by V. R. Mila~i~) Proceedings of the Third ORSA/TIMS Conference on Flexible Manufacturing Systems: Operations research models and applications (edited by K. E. Stecke and R. Suri) Justification Methods for Computer Integrated Manufacturing Systems: Planning, design justification, and costing (edited by H. R. Parsaei, T. L. Ward and W. Karwowski) Manufacturing Planning and ControI-A Reference Model (F. R M. Biemans) Production Control- A Structural and Design Oriented Approach (J.W.M. Bertrand, J. C. Wortmann and J. Wijngaard) Just-in-Time Manufacturing Systems-Operational planning and control issues (edited by A. $atlr) Modelling Product Structures by Generic Bills-of-Materials (E. A. van Veen) Economic and Financial Justification of Advanced Manufacturing Technologies (edited by H.R. Parsaei, T.R. Hanley and W.G. Sullivan) Integrated Discrete Production Control: Analysis and SynthesisA View based on GRAI-Nets (L. Pun) Advances in Factories ofthe Future, CIM and Robotics (edited by M. Cotsaftis and F. Vernadat) Global Manufacturing Practices- A Worldwide Survey of Practices in Production Planning and Control (edited by D.C. Whybark and G. Vastag) Modern Tools for Manufacturing Systems (edited by R. Zurawski and T. S. Dillon) Solid Freeform Manufacturing-Advanced Rapid Prototyping (D. Kochan) Advances in Feature Based Manufacturing (edited by J. J. Shah, M. M~ntyl~ and D. S. Nau) Computer Integrated Manufacturing (CIM) in Japan (V. Sandoval) Advances in Manufacturing Systems: Design, Modeling and Analysis (edited by R. S. Sodhi) Flexible Manufacturing Systems: Recent Developments (edited by A. Raouf and M. Ben-Daya) Planning, Design, and Analysis of Cellular Manufacturing Systems (edited by A.K. Kamrani, H.R. Parsaei and D.H. Liles)

MANUFACTURING RESEARCH AND TECHNOLOGY 24

Planning, Design, and Analysis of Cellular Manufacturing

Systems

edited by

Ali K. Kamrani

University of Michigan Dearborn, MI, U.S.A.

Hamid R. Parsaei

University of Louisville Louisville, KY, U.S.A.

Donald H, Liles

University of Texas Arlington, TX, U.S.A.

ELSEVIER 1995 A m s t e r d a m - Lausanne- New York- Oxford-Shannon-Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0 444 81815 4

91995 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, Ma 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

TO OUR PARENTS Azizollah Khosravi-Kamrani and Fataemah(Mahin) Arjasebi Abolfazl Parsaei and Barat Atabaki Harold and Marilyn Liles

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vii

Contents Contributors Acknowledgements Preface

ix xiii xv

Part One DESIGN AND MODELING TECHNIQUES

Recent Advances in Mathematical Programming for Cell Formation Chao-Hsien Chu 0

Q

0

An Industrial Application of Network-Flow Models in Cellular Manufacturing Planning Alberto Garcia-Diaz and Hongchul Lee 47 Design Quality: The Untapped Potential of Group Technology Charles T. Mosier and Farzad Mahmoodi

63

Partitioning Techniques for Cellular Manufacturing Soha Eid Moussa and Mohamed S. Kamel

73

Manufacturing Cell Loading Rules and Algorithms for Connected Cells Gursel A. Siier, Miguel Saiz, Cihan Dagli, and William Gonzalez

97

Cellular Manufacturing Design: A Holistic Approach Lorace L. Massay, Colin O. Benjamin, and Yildirim (Bill) Omurtag

129

Part Two P E R F O R M A N C E MEASURE AND ANALYSIS

145

Measuring Cellular Manufacturing Performance David F. Rogers and Scott M. Sharer

147

viii Performance of Manufacturing Cells for Group Technology: A Parametric Analysis Atul Agarwal, Faizul Huq, and Joseph Sarkis 167

10. 11

Design of a Manufacturing Cell in Consideration of Multiple Objective Performance Measures Towhee Park and Hassock Lee

181

Machine Sharing in Cellular Manufacturing Systems Saifallah Benjaafar

203

Integration of Flow Analysis Results with a Cross Clustering Method Marc Barth and Roland De Guio

229

Part Three ARTIFICIAL INTELLIGENCE AND COMPUTER TOOLS

249

12.

Adaptive Clustering Algorithm for Group Technology: An Application of the Fuzzy ART Neural Network Soheyla Kamal 251

13.

Intelligent Cost Estimation of Die-Castings Through Application of Group Technology Raj Veeramani

283

14.

Production Flow Analysis Using STORM Sharokh A. Irani and R. Ramakrishnan

299

15.

A Simulation Approach for Cellular Manufacturing System Design and Analysis Ali K. Kamrani, Hamid R. Parsaei, and Herman R. Leep

351

Subject Index

383

ix

Contributors Atul Agarwal, Department of Information Systems and Management Sciences, University of

Texas at Arlington, Arlington, Texas 76019, USA. Marc Barth, Laboratoire de Recherche en Productique de Strasbourg, Ecole Nationale Superieure des Arts et Industries de Strasbourg, 24, bd. de la Victoire, F-67084 Strasbourg, FRANCE. Saifailah Benjaafar, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA. Colin O. Benjamin, Department of Engineering Management, University of Missouri-Rolla, Rolla, Missouri 65401, USA.

Chao-Hsien Chu, Department of Management, Iowa State University, Ames, Iowa 50011, USA. Cihan Dagli, Department of Engineering Management, University of Missouri-Rolla, Rolla, Missouri 65401, USA. Alberto Garcia-Diaz, Department of Industrial Engineering, Texas A&M University, College

Station, Texas 77843-3131, USA. William Gonzalez, Avon Lomalinda Incorporated, San Sebastian, Puerto Rico 00755, USA. Roland De Guio, Laboratoire de Recherche en Productique de Strasbourg, Ecole Nationale Superieure des Arts et Industries de Strasbourg, 24, bd. de la Victoire, F-67084 Strasbourg, FRANCE. Faizul Huq, Department of Information Systems and Management Sciences, University of Texas at Arlington, Arlington, Texas 76019, USA. Shahrokh A. Irani, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA. Soheyla Kamal, 5530 Heather Lane, Orefield, Pennsylvania 18069, USA.

Mohamed S. Kamel, Department of Systems Design Engineering, University of Waterloo,

Waterloo, Ontario, CANADA N2L 3G1. Ali K. Kamrani, Department of Industrial and Manufacturing Engineering, University of Michigan-Dearborn, Dearborn, Michigan 48128-1491, USA. Heeseok Lee, Department of Management Information Systems, Korea Advanced Institute of

Science and Technology, 207-43 Cheongryangridong, Dongdaemoongu, Seoul, Korea. Hongehul Lee, Department of Industrial Engineering, Korea University, Seoul, Korea. Herman R. Leep, Department of Industrial Engineering, University of Louisville, Louisville, Kentucky 40292, USA. Farzad Mahmoodi, Department of Management, Clarkson University, Potsdam, New York 13699-5790, USA. Loraee L. Massay, Department of Industrial Engineering, North Carolina A&T State

University, Greensboro, North Carolina 2741 l, USA. Charles T. Mosier, Department of Management, Clarkson University, Potsdam, New York 13699-5790, USA. Soha Eid Moussa, Department of Systems Design Engineering, University of Waterloo, Waterloo, Ontario, CANADA N2L 3G1. Yildirim (Bill) Omurtag, Department of Engineering Management, University of MissouriRolla, Rolla, Missouri 65401, USA. Taeho Park, Department of Organization and Management, San Jose State University, San Jose,

California 95192-0070, USA. Hamid R. Parsaei, Department of Industrial Engineering, University of Louisville, Louisville,

Kentucky 40292, USA. R. Ramakrishnan, Department of Mechanical Engineering, University of Minnesota,

Minneapolis, Minnesota 55455, USA. David F. Rogers, Department of Qualitative Analysis and Operations Management, University of Cincinnati, 531 Carl H. Linder Hall, Cincinnati, Ohio 45221-0130, USA. Miguei Saiz, Department of Industrial Engineering, University of Puerto Rico-Mayaguez, P.O. Box 5000, Mayaguez, Puerto Rico 00681-5000, USA.

xi Joseph Sarkis, Department of Information Systems and Management Sciences, University of Texas at Arlington, Arlington, Texas 76019, USA Scott M. Sharer, Department of Management, University of Miami, 414 Jenkins Building, Coral

Gables, Florida 33124-9145, USA. Gursel A. Suer, Department of Industrial Engineering, University of Puerto Rico-Mayaguez, P.O. Box 5000, Mayaguez, Puerto Rico 00681-5000, USA. Raj Veeramani, Department of Industrial Engineering, 1513 University Avenue, University of Wisconsin, Madison, Wisconsin 53706, USA.

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xiii

Acknowledgments We would like to take this opportunity to express our sincere gratitude to many of our colleagues who provided us with invaluable assistance during the course of this project. This volume would have not been possible without the outstanding contributions of our authors and those who assisted us in reviewing the contents. The following individuals made significant contributions to this work either by their submissions or assistance in the review process. Atul Agarwal, University of Texas at Arlington. Dongke An, University of Louisville. Marc Barth, Ecole Nationale Superieure des Arts et Industries de Strasbourg. David Ben-Arieh, Kansas State University. Saifallah Benjaafar, University of Minnesota. Colin O. Benjamin, University of Missouri-Rolla. Thomas O. Boucher, Rutgers University. Chao-Hsien Chu, Iowa State University. Cihan Dagli, University of Missouri-Rolla. Osama Ettouney, Miami University. Alberto Garcia-Diaz, Texas A&M University. Soha Eid Moussa, University of Waterloo. William Gonzalez, Avon Lomalinda Incorporated. Roland De Guio, Ecole Nationale Superieure des Arts et Industries de Strasbourg. Kevin Hubbard, University of Missouri-Rolla. Faizul Huq, University of Texas at Arlington. Shahrokh A. Irani, University of Minnesota. Sanjay Jagdale, University of Arizona. Soheyla Kamal, 5530 Heather Lane, Orefield, Pennsylvania. Mohamed S. Kamel, University of Waterloo. Dennis E. Kroll, Bradley University. Jerome P. Lavelle, Kansas State University. Heeseok Lee, University of Nebraska-Omaha. Yuan-Shin Lee, Kansas State University. Hongchul Lee, University of Iowa. Herman R. Leep, University of Louisville. Hampton Liggett, Northern Illinois University. Farzad Mahmoodi, Clarkson University.

xiv P. K. Mallik, University of Michigan-Dearborn. Lorace L. Massay, North Carolina A&T State University. Charles T. Mosier, Clarkson University. Yildirim (Bill) Omurtag, University of Missouri-Rolla. Taeho Park, San Jose State University. R. Ramakrishnan, University of Minnesota. Fahimah Rezayat, California State University-DH. David F. Rogers, University of Cincinnati. Miguel Saiz, University of Puerto Rico-Mayaguez. Joseph Sarkis, University of Texas at Arlington. Scott M. Shafer, University of Miami. Alice E. Smith, University of Pittsburgh. Gursel A. Suer, University of Puerto Rico-Mayaguez. Louis Tsui, University of Michigan-Dearborn. Raj Veeramani, University of Wisconsin-Madison. David H. H. Yoon, University of Michigan- Dearborn.

XV

Preface The introduction of computers into manufacturing in the late 1950's has brought new challenges to manufacturing companies in the United States and aborad. Computer-Aided Design, Computer-Aided Manufacturing, Cellular Manufacturing, Group Technology, Computer Integrated Manufacturing, and so forth, are considered as viable technologies and philosophies for increasing productivity, enhancing product quality, and reducing direct and indirect manufacturing costs. Cellular Manufacturing (CM) is one of the major concepts used in the design of flexible manufacturing systems. CM, also known as group production or family programming, can be described as a manufacturing technique that produces families of parts within a single line or cell of machines. The objective of Planning, Design, and Analysis of Cellular Manufacturing Systems is to report the latest developments and address the central issues in the design and implementation of cellular manufacturing systems. This book consists of 15 refereed chapters, written by leading researchers from academia and industry, that are organized in three parts. Part One, Design and Modeling Techniques, includes six chapters. In the first chapter, Chu presents a state-of-the-art review based on a systematic survey of the literature. In the second chapter, Garcia-Diaz and Lee, develop a network flow methodology for grouping machines into cells and forming part families in cellular manufacturing. The third chapter, by Mosier and Mahmoodi, expands the application domain of group technology oriented coding and retrieval systems to address the problem of design quality. Moussa and Kamel address the partitioning problem in cellular manufacturing systems in chapter 4. They review various partitioning techniques and demonstrate the effectiveness of these techniques by some sample results. In the fifth chapter, Suer et al. review and discuss several manufacturing cell loading rules and algorithms for connected cells. The last paper in this section, by Massay et al., presents a systematic method to the design of the cellular manufacturing systems. The methodology utilizes a holistic system design approach that facilitates the evaluation of the total system being developed. Part Two of the book is concemed with Performance Measure and Analysis. Five papers on this topic are included in this part. The first paper, by Rogers and Shafer, identifies several design objectives associated with cellular manufacturing. Then, based upon these design objectives, appropriate performance measures are discussed and compared. In the second

xvi article, Agarwal et al. use an analytical model to investigate the relative performance of a partitioned system compared to an unpartitioned system as a function of the ratio between setup time and processing time per unit, varying over a large range of domain vaiues. The third article in this part, by Park and Lee, presents a new approach to the design of a manufacturing cell with multiple performance objectives via the simulation-based design of experiments and compromise programming. The effects of the machine sharing on the performance of traditional cellular manufacturing systems is demonstrated by Benjaafar in the fourth article. The last article in this part, by Barth and De Guio, involves the integration of flow analysis results with a cross clustering method. Finally, Part Three presents the applications of artificial intelligence and computer tools in the design and analysis of cellular manufacturing systems. Four articles are included in this part. The first article, by Kamal, presents a new clustering algorithm based on the neural network techniques and fuzzy logic concepts. Veeramani, in the second article, describes ongoing work with the die-casting industry on the application of group technology in developing a computerintegrated system that will assist cost estimators in developing quotes for die-casting parts in a consistent, accurate, and timely manner. The third article, by Irani and Ramakrishnan, demonstrates a step-by-step implementation of the first three phases in production flow analysis including factory flow analysis, group analysis, and line analysis using standard algorithms available in the STORM package. Kamrani et al., in the last article in this section, present the application of linear programming to develop a methodology that uses design and manufacturing attributes to form machining cells. We are indebted to our authors and reviewers for their outstanding contribution and assistance in preparing this volume. We would also like to thank Dr. Herman R. Leep of the University of Louisville for his invaluable support and advice. Special word of thanks are due to Dongke An for providing exceptional help to make this endeavor possible. Finally, we would like to express our deepest gratitude to Drs. Amanda Shipperbottom and Eefke Smit of Elsevier Science Publishers for giving us the opportunity to initiate this project.

Ali K. Kamrani Hamid R. Parsaei Donald H. Liles December, 1994

PART ONE

DESIGN AND MODELING TECHNIQUES

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Planning, Design, and Analysis of Cellular Manufacturing Systems A.K. Kamrani, H.R. Parsaei and D.H. Liles (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

R e c e n t A d v a n c e s in M a t h e m a t i c a l P r o g r a m m i n g for Cell F o r m a t i o n Chao-Hsien Chu Department of Management, College of Business, Iowa State University of Science and Technology, 300 Carver Hall, Ames, Iowa 50011, USA In the past decade, cellular manufacturing has received considerable interest from practitioners and academicians. Cell formation, one major problem with cellular manufacturing, involves the process of grouping the parts with similar design features or processing requirements into part families and the corresponding machines into machine cells. Numerous analytical approaches to solving the problem have been introduced, among which mathematical programming models and heuristic procedures constitute the greatest part of the literature. But as yet no comprehensive study has synthesized the literature pertaining to the use of mathematical programming in cell formation. This chapter presents a state-ofthe-art review based on a systematic survey of the literature. Survey results should help answer or clarify many related questions for the cellular manufacturing community. Examples have been provided to help the interested reader use earlier studies to develop mathematical programming models.

1. I N T R O D U C T I O N

During the past decade, there has been a major shift in the design of manufacturing planning and control systems using such innovative concepts as just-intime (JIT) production, optimization production technology (OPT) (recently named the theory of constraints), flexible manufacturing systems (FMS), cellular manufacturing (CM), and group technology (GT). Cellular manufacturing in particular has received considerable interest from both practitioners and academicians because it allows small, batch-type, production to gain an economic advantage similar to that of mass production and still retaining the high degree of flexibility associated with job-shop production. The design of a CM system is quite challenging because so many strategic issues, e.g. the selection of part types suitable for manufacturing on a group of machines, the level of machine flexibility, the layout of cells, the types of material handling equipment, and the types and numbers of tools and fixtures, must be considered during design [74]. Furthermore, any meaningful cell design must be compatible with corporate tactical/operational goals such as high production rate, high product quality, high on-time delivery,

low work-in-process, low queue length at each work station, and high machine utilization. One of the first and most important problems, i.e., cell formation (CF), faced in CM practice involves the decisions surrounding the decomposition of manufacturing systems into cells. Part families and machine cells are identified such that (1) parts with similar design features, functions, materials, or processing requirements are produced in a cell sharing common resources such as machines, tools, and labor; (2) each part can be processed fully within a cell without the need for movement across cells; and (3) capital investment in resources is maintained at a level compatible with corporate strategy. Manufacturing cells can capture the inherent advantages of both mass and job-shop productions, such as reduced setup times, improved process planning, decreased lead time, reduced tool requirements, improved productivity, increased overall operational control, improved product quality, and reduced material handling costs. Common disadvantages such as lower machine and labor utilization rates and higher capital investment due to duplication of machines and tools exist, however [61]. Much effort has been directed at the cell formation problem. As a result, many procedures have been developed, among which mathematical programming models and heuristic procedures are most discussed in the literature. Notwithstanding, the cell formation problem has been proved nonpolynomial (NP); that is, finding an optimal solution becomes increasingly unlikely as problem size grows. Optimal solutions are worth pursuing for at least two reasons [71,72]: (1) they can serve as a benchmark against which to evaluate heuristics and (2) optimal algorithms and heuristics can work together. On one hand, the logic from an optimal algorithm can lead to an efficient heuristic; on the other, a heuristic solution can serve as a starting point from which to reduce computational time in the optimizing search. Although a number of studies [10,42,56] have attempted to synthesize the literature concerning the use of mathematical programming in cell formation, the scale of these studies has been rather small and the literature cited somewhat outdated. And although a number of studies [14,56,61,65,73] have provided state-of-the-art reviews of cell formation issues, problems, and techniques, the scopes of these reviews generally has seemed too broad, that is, has covered in insufficient detail the use of mathematical programming in cell formation. A comprehensive study of this topic is needed to answer many of the questions frequently asked by the CM community: 9 What kinds of mathematical programming models have been used for cell formation? 9 Which models are most popular? 9 What kinds of objective functions concerning cell formation can be modeled through mathematical programming? What have been the popular measures used in previous studies? Do these measures reflect CM practice? 9 What kinds of constraints related to cell formation can be represented by mathematical programming? What are the popular constraints used in prior research? Do they capture manufacturing reality?

9 What kinds of solution procedures or strategies have been applied to solve the mathematical models? Are they efficient or powerful enough to deal with real-world problems? 9 What types of data are needed to model mathematical programming models? Can these data be obtained easily from the shop floor? 9 What unique features are considered in current cell formation studies? Have they addressed the issues and concerns raised by earlier researchers and practitioners? 9 What kinds of computer systems and software have been used in cell formation research? The purpose of this study is twofold: (1) to examine the state of the art of mathematical programming's use in cell formation (Results from this study would help answer many of the aforementioned questions.) and (2) to illustrate how a variety of cell formation problems can be formulated by means of mathematical programming. Five examples with different objectives, constraints, and structures are provided for illustration. These examples not only represent typical cell formation problems but also can be used to demonstrate how the same scenario can be modeled through either objectives or constraints. The chapter is organized as follows. In section 2, approaches to cell formation are summarized. Section 3 discusses the most recent results concerning the use of mathematical programming models in cell formation. The review proceeds according to the questions just outlined. In section 4, examples of typical cell formation scenarios are provided. Conclusions are given in section 5, which is followed by appendices and references.

2. APPROACHES TO CELL FORMATION Extensive work has been done in the area of cell formation, and numerous approaches have been developed [61]:

* Classification and c o d i n g systems. Under this approach, users first examine the design features or manufacturing attributes of parts from blueprints and use a coding system to assign symbols (or codes) to the parts. H u m a n eyes, statistical clustering algorithms [31], or mathematical programming models [29,31,32] are then used to scrutinize the codes for similarities and part families are formed. The process of assigning codes to parts is tedious and time consuming and sometimes subjective inasmuch as it depends on h u m a n experience and judgment. These methods can be used only to identify part families. * Array based clustering methods. This approach differs from the former in t h a t it is based upon a production flow analysis [7] which uses routing sheet or process plans. A common feature of this approach is t h a t it sequentially rearranges columns and rows of the machine/part matrix according to

9

9

9

9

9

an index until diagonal blocks are generated [15]. Methods of this type have received much attention because of their simplicity. Popular methods include rank order clustering [33,34,35], direct clustering [8] and bond energy algorithm [15,24]. Common criticisms of the methods are that (1) identification of exclusive groups in a block diagram sometimes requires subjective judgment; (2) most methods consider only binary routing information and neglect other important cost and operational factors; and (3) in most cases, bottleneck machines must be removed before any machine/part groups can be identified dearly [8,15,35]. Statistical clustering algorithms. Statistical cluster algorithms have been used quite often in the decomposition of manufacturing cells [14]. In particular, use of hierarchical clustering methods such as the single and the complete linkage methods has been studied extensively [12,45]. This approach requires a calculation of similarity coefficients between each pair of parts or machines. Parts or machines with close similarity coefficients then are arranged in the same group. One study also has used a nonhierarchical clustering scheme [9]. Several problems associated with this approach remain to be solved [12,14], for instance, the selection of clustering criteria, the selection of performance measure, and the determination of the number of part families. Graphic theoretical approaches. A number of papers based upon graph theory have been published [5,14,61]. The methods described represent vertices of graphs as machines or parts and weights of arcs as similarity coefficients. The major drawback inherent in this approach is that practical issues such as production volume and alternative plans are not addressed [61]. Mathematical programming and heuristic approaches. Numerous studies of cell formation have been conducted that employ mathematical programming and heuristics to improve clustering effectiveness. These approaches are flexible enough to incorporate most objective functions and constraints in a precise format; they suffer, however, in that they consider the problem only in a static sense for purely stable manufacturing environments [61]. Additionally, none of the methods considers uncertainty or vagueness, both of which normally are presented in the information required by the models. K n o w l e d g e based and pattern recognition methods. Emerging from artificial intelligence and pattern recognition techniques, expert systems offer many new opportunities for manufacturing systems analysis and design. Yet very few papers have applied these techniques to the cell formation problem [17,61]. Developing expert systems which can capture pattern recognition, optimization, and expert cognition processes to form manufacturing cells is a promising area for exploration [61]. F u z z y clustering and modeling approaches. Most early cell formation research assumes that the information used for cell formation, such as production cost, demand, and processing time, is certain and that the objectives

and constraints considered can be formulated precisely. This early research also assumes that each part can belong to only one part family, yet parts may exist whose membership is much less evident. Only a few researchers have addressed the issues of vagueness and uncertainty in the cell formation problem [16,18]. Fuzzy modeling and clustering approaches may provide a solution in such cases. For instance, a fuzzy c-mean clustering method was used in [16] to form part families (or machine cells) such that a part (or machine) could belong to more than one family (or cell), with different degrees of membership. Recently, a fuzzy mathematical programming approach [18] has been proposed to deal with the imprecise nature of objectives and constraints. N e u r a l network approaches. Neural network is an emerging algorithmic approach that has been the subject of intent study by mathematicians, statisticians, physicists, engineers, and computer scientists. The number of studies utilizing the rapid parallel processing capability of neural networks to solve the cell formation problem has been increasing significantly [19]. Networks such as backpropagation, self-organizing map (SOM), competitive learning, adaptive resonance theory (ART), interactive activation and competition learning, and fuzzy ART, have been applied successfully to the decomposition of manufacturing cells [19].

3. STATE-OF-THE-ART REVIEWS The cell formation problem can be formulated differently depending on the mathematical programming model used, the objective functions chosen, and the constraints considered. There also have been major variations on the solution procedures used, the formation logic applied, the special features considered, and the input data involved. In this section, the state of the art, as characterized by the literature review, is summarized. Detailed information regarding individual studies (models) appears in the appendices.

3.1. Types of mathematical programming models A variety of mathematical programming approaches (See Table 1) have been applied to model and to solve the cell formation problem. The complexities of these models are limited (linear programming and 0-1 integer programming), modest (mixed integer programming and 0-1 nonlinear programming) and very great (mixed integer nonlinear programming and 0-1 nonlinear fractional programming). About 39% of prior studies use the relatively simple 0-1 integer prograrnming models; only about 5% use very complicated models. Thus, even complicated manufacturing design problems such as cell formation can be modeled with mathematical models of limited complexity.

Table 1 Summary of mathematical programming models Rank

1

Model Type

2

0-1 integer programming Mixed integer programming 0-1 integer nonlinear programming

4

Frequency Percentage Used Used (%) 23 § 10

38.98 16.95 16.95

Linear programming

5

8.47

5

Assignment model

3

5.08

5

Network Model

3

5.08 3.39

2

10 ~

7

Mixed integer nonlinear programming

2

8

0-1 nonlinear fractional programming

1

1.69

8

Dynamic programming

1

1.69

8

Branch and bound

1

1.69

Total:

59

+ Including two goal programming models. # Including five goal programming models.

3.2. Objective functions chosen The success of a mathematical programming model depends heavily upon how accurately objectives and constraints can be expressed in precise mathematical relations. Because many objective functions and constraints are considered in the cell formation problem, the challenge is not limited to the construction of equations; a more important issue is the selection of appropriate objectives and constraints that capture and reflect CM reality. There are several ways of classifying objective functions in cell formation. For instance, in [56] objectives were divided into four major categories: (1) reducing the number of setups; (2) producing parts completely within the cell; (3) minimizing investment in new equipment; and (4) maintaining acceptable utilization levels. In [14], according to the nature of objectives, performance measures were classified as either cost or noncost based and subsequently classified as either individual or aggregate. In total, 34 objectives were considered in the prior cell formation studies. In Table 2, these objectives are classified roughly as coefficient based, cost based, or operation related. The purpose of cell formation based on coefficient criteria has been either to maximize total similarity or to minimize total dissimilarity of parts or machines. These coefficients can be computed with design features or with processing requirements. Some studies even have gone so far as to consider tooling requirements between machines and parts [25,26,27,53] and similarities between machines and operators [46]. Two problems may be encountered when these

approaches are used: (1) the coefficients can be involved only one at a time; thus, the model can consider only one objective at most and (2) coefficients are routing based primarily and do not take into account other important factors such as costs and operational issues. Several studies have taken the lead in addressing this deficiency -- for instance, by considering demand, processing time, and even processing sequence in computing the coefficient [12,53], but their impacts have not been testified formally. Recent studies have deviated more and more from this course by considering a variety of costs and operational factors during model formulation. Table 2, which reflects this shift, indicating that about 26% of prior studies focus on minimizing total costs of machine investment, followed by minimizing total costs of intercell movement (24%), minimizing total intercell movement (21%), maximizing machine utilization (12%), and minimizing total processing costs (10%). No coefficient based criterion is among the top five focuses. These objectives coincide with those often used in manufacturing practice [56,74]. Also, the majority of studies (54%) still consider only a single objective during cell formation. Although models with multiple objectives can reflect manufacturing practice with comparative accuracy, they also are more difficult to develop and require longer time to solve [23,53,56,72]. One trade-off is to consider multiple related criteria in an aggregated format. About 34% of prior research has used this approach. In-depth analysis of the data from Appendix B indicates that the following groups of objectives have been used most often by prior researchers: (1) total amount of interand intra- cell movement; (2) total costs of inter- and intra- cell movement, coupled with total machine investment; (3) total costs of intercell movement and machine duplication; (4) total setup cost and inventory holding cost; and (5) total costs of machine investment, tooling investment, and processing.

3.3. Manufacturing constraints Another key component involved in constructing mathematical programming models for cell formation is to define precisely the system constraints. To be practical, constraints should capture the actual restriction (limitation) of a system. Forty-five constraints have been considered in the cell formation literature. Each of these constraints can be placed in one of four categories: (1) logical, (2) cell size, (3) physical, and (4) modeling. Logical constraints prevent models from contradicting common sense, judgment, or theoretic logic. For example, each part, machine, operation, or operator can be assigned into only one cell. Cell size constraints normally are considered to restrict the number of parts, machines, or operators allowed in each cell from exceeding an upper bound because of concerns regarding span of control, and space and capacity limitations. It also makes sense to ensure that the number of parts or machines assigned into each cell exceed a minimum. In this way, the systems are prevented from over division which may result in excessive duplication and thus waste of resources. Physical constraints such as space, budget, capacity, and number of machines available for each machine type capture another type of system restriction. Finally, there is a

10 need for modeling constraints, which provide necessary connections among decision variables, parameters, and objective functions. Table 2 S u m m a r y of objective functions Rank

Obiective Function

Coefficient Based Measures"

Code+

Frequency Percenta~ze usect Used (%)~*

O1 5 8.62 Max. total similarity between parts 02 3 5.17 Max. total similarity between machines 03 3 5.17 Min. total dissimilarity between parts 04 3 5.17 Max. total compatibility between parts and machines O16 2 3.45 9 Min. total distance between parts O16 1 1.72 10 Max. total similarity between machines and operators O16 1 1.72 10 Min. total dissimilarity between machines 1 1.72 10 Min. total distance between classification codes O16 Cost Based Measures: 06 15 25.86 1 Min. total costs of intercell movement 05 14 24.14 2 Min. total costs of machine investment 08 6 10.34 5 .Min. t o t a l p r o c e s s i n g a n d m a c h i n e u t i l i z a tion costs 07 5 8.62 6 Min. total costs ofintracell movement Oll 5 8.62 6 Min. total machine duplication costs 09 3 5.17 8 Min. total costs of idle machine capacity O10 3 5.17 8 Min. tqtal setup costs due to sequence aepenaence 011 3 5.17 8 Min. total tooling and fixture costs 011 2 3.45 9 Min. total inventory or work-in-process (WIP) costs 011 1 1.72 l0 Min. total subcontracting costs 011 1 1.72 l0 Min. total space utilization costs 011 1 1.72 l0 Min. total penalty for late or early production 011 1 1.72 l0 Min. total costs to expand capacity 011 1 1.72 l0 Min. total labor costs Operation Related Measures: O13 12 20.7 3 M i n . t o t a l a m o u n t ( n u m b e r ) of i n t e r c e l l movements O15 7 12.07 4 Max. total machine utilization O14 4 6.9 7 Min. total amount (number) ofintracell movements O12 4 6.9 7 Min. n u m b e r of exceptional elements O16 3 5.17 8 Min. total setup times O16 2 3.45 9 Min. total machining hours O16 2 3.45 9 Min. total within cell load variation O16 1 1.72 l0 Max. total amount of parts produced O16 1 1.72 l0 Match each operator's skill O16 1 1.72 10 Max. average cell utilization O16 1 1.72 l0 Min. intracell load imbalance O16 1 1.72 l0 Min. intercell load imbalance + Corresponding to Appendix B. * Based upon 58 models. (One model [60] did not provide a detailed objective.) 6 8 8 8

ll Table 3 summarizes constraints actually considered in the models. The ten most used constraints are spread evenly over the first three categories and none appear in the fourth group. This is because most modeling constraints can be expressed directly in the objective function. For instance, the required number of intercell movements can be expressed in the objective function instead of as a constraint set. Examples 2, 3, 4, and 5 (of section 4) provide a contrast. Examples 2 and 3 include a constraint to determine the number of exceptional elements whereas models 4 and 5 have the relations built into the objective function. According to Table 3, some commonly used constraints are unique part assignment (52%), maximum number of machines allowed in each cell (46%), machine capacity (37%); unique machine assignment (26%); and minimum number of machines needed in each cell (24%). Clearly, all these constraints are quite realistic in terms of managing and operating manufacturing cells.

3.4. Solution procedures One major bottleneck that prevents mathematical programming models from wide use in the real world is that once problem size grows, it is inefficient to seek and almost impossible to find an optimal solution. This phenomenon is known as NP-complete or NP-hard. To overcome this deficiency, two strategies often have been used in cell formation studies. One strategy is to develop efficient heuristic procedures and the other is to decompose the model into submodels or to model the problem in multi-phases and solve each through an optimal or heuristic procedure. According to Table 4, about 29% of previous studies used the decomposition or multi-phase approach to find solutions and about 27% used heuristic procedures (including general search, simulated annealing, genetic algorithm, tabu search, and fuzzy-c clustering). At the same time, 44 % of the studies relied on optimal procedures to solve the single-phase model, and 10% used optimal procedures to solve multi-phase models. So, if we consider that almost all heuristic procedures were developed or extended from an optimal procedure, we conclude that optimal procedures seem to play an important role in cell formation research. Furthermore, interest in using search algorithm [10,64], especially simulated annealing [6,17, 66,69], genetic algorithm [28,68], or tabu search [66], to tackle the problem has been increasing. 3.5. Formation logics The cell formation literature can be divided into four categories, according to the formation logic used [74]: (1) grouping part families only; (2) forming part families and then machine cells or vice versa; (3) forming part families and machine cells simultaneously; and (4) grouping machine cells only. According to Table 5, the literature is spread very evenly over the first three formation logics, with somewhat less effort devoted to the last procedure. Several observations can be extracted from the details provided in the appendices: (1) Most coefficient based models can be used only to form part families or machine

12 Table 3 S u m m a r y of constraints Rank

Constraints

Code+

Frequency Percentage Used Used (%)*

Logical Constraints: 1

Unique part assignment- each part can be assigned to only one part family 4 Unique machine assignment - each machine can be assigned to only one machine cell 9 P a r t family formation logic - a part family m u s t b e f o r m e d b e f o r e p a r t s c a n b e ass i g n e d to t h a t f a m i l y 9 Linkage between machines & parts - to ensure that all machines needed by a part be assigned to the same cell 9 U n i q u e o p e r a t i o n s a s s i g n m e n t - e a c h ope r a t i o n c a n b e a s s i g n e d to o n l y o n e machine 12 Linkage between operations, parts, and machines 13 Unique routing selection - only one routing be selected 13 Linkage between routings and machines 14 Machine cell formation l o ~ c - a machine cell m u s t be formed before machines can be assigned to t h a t cell 14 All p a r t types assigned to p a r t families 14 Unique operator assignments - each operator can only be assigned to one machine cell 14 Layout constraints - to ensure t h a t the machines in each cell do not overlap 14 No. of cells must be less than no. of available operators 14 M i n i m u m level of machine similarity for grouping 14 M i n i m u m level of tool similarity for grouping 14 Minimum level of intercell movement for moving Cell Size Constraints: 2 M a x i m u m no. of m a c h i n e s a l l o w e d i n e a c h machine cell 5 Minimum no. of machines to be qualified as a cell 6 Maximum no. of parts allowed in each part family 10 Minimum no. of parts to be qualified as a part family 14 Exact no. of parts required for each part family 14 Maximum no. of operators allowed in each machine cell

C1

28

51.85

C2

14

25.93

C3

7

12.96

C4

7

12.96

C19

7

12.96

C 19 C 19 C 19 C 19

3 2 2 1

5.56 3.7 3.7 1.85

C 19 C19

1 1

1.85 1.85

C19

1

1.85

C 19

1

1.85

C 19

1

1.85

C 19 C 19

1 1

1.85 1.85

C5

25

46.3

C6 C7 C8

13 11 6

24.07 20.37 11.11

C 19 C 19

1 1

1.85 1.85

+ Corresponding to Appendix C. * Based on 54 models. (Five models have constraints embodied in the structure.)

13 Table 3 (Continued) S u m m a r y of c o n s t r a i n t s Rank

Constraints

Physical Constraints: 3 Mach_ine c a p a c i t y c o n s t r a i n t s - to e n s u r e t h a t t h e t o t a l o p e r a t i o n t i m e s a s s i g n e d to a cell won't exceed capacity 7 A c o n s t r a i n t to s p e c i f y t h e n u m b e r of required cells 8 C o n s t r a i n t s to c o n s i d e r t h e no. o f machine types available 9 B u d g e t c o n s t r a i n t s - to e n s u r e t h a t t h e t o t a l - c o s t o f b u y i n g m a c h i n e s , tools, a n d overhead won't exceed 11 12 14 14 14

Production r e q u i r e m e n t s constraints Space constraints - to ensure t h a t the total space of m a c h i n e s assigned to a cell can be accommodated Constraints to restrict the maximum no. of procurable machines Constraints to restrict the maximum no. of cells allowed Constraints to restrict a part from subcontracting

Code +

Frequency Percentage Used Used (%)*

CIO

19

35.19

C9

9

16.67

Cll

8

14.81

C13

7

12.96

C12 C14

4 3

7.41 5.56

C19

1

1.85

C19

1

1.85

C19

1

1.85

C15

4

7.41

C18

3

5.56

C19 C16 C17

3 2 2

5.56 3.7 3.7

C19

2

3.7

C19 C19 C19 C19 C19 C19 C19

2 1 1 1 1 1 1 212

3.7 1.85 1.85 1.85 1.85 1.85 1.85 3.93

Modeling Constraints: 11 12 12 13 13 13 13 14 14 14 14 14 14

Total:

C o n s t r a i n t s to compute the needed no. of machine types C o n s t r a i n t s to compute the total intercell movements Constraints to consider the sequence dependent setup C o n s t r a i n t s to identify the bottleneck p a r t s C o n s t r a i n t s to identify the exceptional elements C o n s t r a i n t s to compute the total skipping operations (intracell m o v e m e n t s ) Constraints to meet sequence requirements Constraints to compute the completion times Constraints to meet due date requirements Modeling constraints Constraints to estimate the amount of capacity change Constraints to restrict the undirected flows Constraims to link the stages of process

+ C o r r e s p o n d i n g to Appendix C. * Based on 54 models. (Five models have constraints e m b e d d e d in the structure.)

14 Table 4 Summary of solution procedures Rank Solution Procedure Single-Phase: 1 Optimal (O) procedure 2 Heuristic (H) procedure 4 Simulated annealing (H) 5 General search algorithm (H) 5 Branch and bound (O) 5 Network algorithm (O) 6 Genetic algorithm (H) 7 Tabu search (H) 7 Assignment algorithm (O) 7 Fuzzy-C clustering (H) Subtotal: Multi-Phase: 2 Optimal then optimal

2 Heuristic then heuristic * 3 Optimal then heuristic 7 Heuristic then optimal Subtotal:

Frequency Used

Percentage Used (%)

21 6 4 3 3 3 2 1 1 1 45

33.33 9.52 6.35 4.76 4.76 4.76 3.17 1.59 1.59 1.59 71.43

6 6 5 1 18

9.52 9.52 7.94 1.59 28.57

* One model [4] uses simulated annealing and then other heuristic.

Table 5 Summary of formation logic used Rank

Formation Logic

Frequency Used

Percentage Used (%)

1

Group part families only

17

29.82

2

Form both part families and machine cells sequentially

16

28.07

2

Form both part families and machine cells simultaneously

16

28.07

3

Group machine cells only

8

14.04

Total:

57

15 cells because these procedures consider either machine similarity or part similarity, not both, in the formation stage. (2) Most models using an operation index in the formulation, i.e., considering processing sequences, can form part families and machine cells simultaneously. (3) Because of model complexity, it is virtually impossible to use optimal procedures to group part families and machine cells sequentially or simultaneously. In fact, most models rely on a heuristic procedure or use the decomposition strategy. (4) Some studies focus attention on grouping machine cells only, these studies often assume that part families already have formed in spite of its unreality.

3.6. Required input data Many different data are needed for cell formation. The minimum requirements are routing or design feature data. Because many new models have been developed to capture manufacturing reality more faithfully, additional cost and operational data also are needed. (See Table 6.) Most of these data are readily available from the shop floor although acquiring certain information such as number of cells, machine cells size, and cell overhead and budget requires additional effort. Both binary routing and cell number are highly demanded information (80%). Thus, (1) most mathematical programming approaches rely on routing information instead of design features to form manufacturing cells and (2) users normally (80%) need to specify required cell number on a prior basis if they wish to develop mathematical models. This requirement may be controversial because in practice it is very difficult for managers to know beforehand exactly how many cells are needed.

3.7. Special features considered Numerous practical issues concerning cell formation have been raised by early researchers [14,23,61,65,74]: (1) Many of the techniques developed to date fail to capture many of the realities of cell formation. Specifically, most consider only binary routing information in forming the cells and totally or partly neglect other important cost and operational information such as production demand, processing time, machine capacity, processing sequence, machine investment cost, materials handling, and cell overhead. (2) Very few studies have considered field's stochastic nature, sequence-dependent setups, machine tool process capability, alternative process plans, or layout. (3) Efficient approaches considering a number of objectives are needed. Table 7 demonstrates that the number of studies including such special features as alternative routings (process plans) or multiple functional machines, processing sequences, tooling requirements, and sequence-dependent setups has increased significantly. A few studies even have attempted to consider layout planning [4], scheduling [1], and labor allocation [46] during cell formation. One study [32] also considers both design features and processing information in cell formation.

16 Table 6 S u m m a r y of required input data Rank

Required Input Data

Frequency Percentage Used Used (%)*

Coefficients:

Similarity coefficients between parts

6

10.17

12

Similarity coefficients between machines

4

6.78

13

Dissimilarity coefficients between parts

3

5.08

13

Compatibility between machines and parts

3

5.08

14

Distance between parts

2

3.39

15

Dissimilarity coefficients between machines

1

1.69

15

Tool similarity

1

1.69

15

Similarity coefficients between machines and operators

1

1.69

15

Similarity coefficients between classification codes

1

1.69

5

Machine Investment (M)

19

32.2

6

Intercell material handling cost (H)

14

23.73

8

Cell overhead (0) - to setup and operate cell

9

15.25

9

Budget (B)

8

13.56

11

Intracell material handling cost (H)

5

8.47

12

Setup cost (S)

4

6.78

13

Tooling and fixture cost

3

5.08

13

Inventory (WIP) holding cost

3

5.08

14

Machine idle cost

2

3.39

15

Inspection cost (I)

1

1.69

15

Subcontracting cost

1

1.69

15

Wage rate

1

1.69

15

Cost for capacity expansion

1

1.69

15

Penalty for early finish or late

1

1.69

10

Costs:

* Based upon 59 models.

17 Table 6 (Continued) S u m m a r y of required input data Rank

Required Input Data

Frequency Percentage Used Used (%)*

Operation Factors: 1

Routing (binary)

47

79.66

1

Number of cells

47

79.66

2

Demand (production volume)

31

52.54

2

Processing time (P)

31

52.54

3

Size of machine cells (M)

30

50.85

4

Machine capacity

26

44.07

7

Routing (sequence)

11

18.64

7

Size of part families (P)

11

18.64

8

Number of machines available for each machine type

9

15.25

10

Alternative routings

6

10.17

11

Tooling requirements

5

8.47

12

Setup time (S)

4

6.78

13

Total available space

3

5.08

14

Batch size

2

3.39

14

Maximum machine utilization

2

3.39

14

Design features

2

3.39

15

Inspection time (I)

1

1.69

15

Maximum no. of operators allowed in each cell (O)

1

1.69

15

Space requirements

1

1.69

15

Distance

1

1.69

15

Due date

1

1.69

15

Arrival time

1

1.69

15

Required no. of parts for each part family

1

1.69

15

Maximum no. of parts each operator can handle

1

1.69

15

Skill matching factor

1

1.69

15

Minimum level of machine similarity for grouping

1

1.69

15

Minimum level of tool similarity for grouping

1

1.69

15

Minimum level of movement for moving

1

1.69

* Based upon 59 models.

18 Table 7 Summary of special features considered in the model Rank

Special Features

Frequency Percentage Used Used (%)*

1

Consider alternative routings

12

20.34

2

Can find an appropriate n u m b e r of cells

11

18.64

3

Use operations index in the formulation

8

13.56

4

Consider processing sequences

6

10.17

4

Consider tooling requirements

6

10.17

5

Consider setup dependent sequences

5

8.47

6

Consider design features

2

3.39

6

Deal with exceptional elements after CF

2

3.39

7

Consider layout planning with CF

1

1.69

7

Consider undirected flows of parts

1

1.69

7

Consider operator assignment in CF

1

7

Integrate with scheduling

1

1.69 1.69

* Based upon 59 models. 3.8. C o m p u t e r systems and software used If undertaken manually, optimal solution of even a small mathematical programming model demands a prohibitive amount of computational effort; all practical applications of mathematical programming therefore, require the use of computer and related software. Cell formation applications are no exception. In spite of the needs, about one-third of the previous studies nevertheless neglect to mention the type of computer and software used. Based upon the data released (Table 8), LINDO is the most popular software used by cell formation researchers. ZOOM is the next most popular, followed by SAS/OR. Surprisingly, MPSX, the once popular optimization software in the operations research field, has lost its shine because of an inefficient built-in 0-1 integer algorithm and high purchasing and maintenance cost (MPSX is available only in the IBM-compatible mainframe platform). Among the programming languages, PASCAL takes the lead, followed closely by FORTRAN and C. In terms of computer systems, personal computers have replaced mainframes as the most popular systems. This trend may be due to the facts that (1) the processing capability of PCs and of corresponding software packages have been enhanced and improved significantly over the past years and (2) the problems demonstrated by most studies are relatively small and uncomplicated.

19 Table 8 Summary of software and computer used Software or Computer System

Rank

Frequency Used

Percentage Used (%)

Software: 1

Not mentioned

20

33.33

2

LINDO

11

18.33

3

8

13.33

4 5

ZOOM SAS/OR PASCAL

6

C

6 5 3

10 8.33 5

6 7

FORTRAN BASIC

8 8

MPSX RELAXT III

3 2 1

5 3.33 1.67

1

1.67

Total: C o m p u t e r Systems: 1 P e r s o n a l c o m p u t e r (PC a n d Mac) 2 Not mentioned 3 Mainframe computer 4 Mini computer 5 Unix workstation 6 Super computer Total:

60

18 18 13 7 2 2 60

30 30 21.67 11.67 3.33 3.33

4. EXAMPLES OF MODEL D E V E L O P M E N T As the foregoing survey and discussion indicate, mathematical programming has played a seminal role in cell formation history. For instance, most newly proposed heuristic procedures such as simulated annealing, genetic algorithm, tabu search, fuzzy modeling, and neural networks are based upon mathematical models. Two common problems may be encountered in the development of mathematical programming models. The first problem relates to the selection of appropriate objective functions and constraints able to capture manufacturing reality. Survey results from this study can be utilized to support and to ease this selection decision. For instance, users can refer to Tables 2 and 3 and choose their own objectives and constraints from somewhere at the top of the list, without losing much generality. The second problem involves the actual formulation of selected objectives and constraints in a precise and simplistic format. An easier way of

20 doing this is to use a building block approach, i.e., to find and then adopt or modify similar formulations from prior studies instead of developing new models from scratch. In this section, five models with different objectives, constraints, and structures are provided for illustration. Model 1 is a traditional p-median formulation based upon similarity coefficients. A unique feature of this model is t h a t it represents the required n u m b e r of cells in a constraint. Model 2 minimizes the total opport u n i t y costs of producing bottleneck parts outside the cells. This model exemplifies how to use constraints to identify bottleneck parts. Model 3 provides an example of aggregating two compatible criteria into a single objective. It also depicts yet another way of identifying exceptional elements (both parts and machines) by means of constraints. Model 4 is a typical example of the multiobjective approach to cell formation. Other features of the model are t h a t it represents exceptional elements through the objective function and considers withincell workload between machines. Model 5 illustrates how processing sequences can be considered in cell formation. The model also shows users how to use constraints to determine the required number of machines for each type and how to solve a model with a nonlinear term in the objective function. Table 9 s u m m a rizes the characteristic of these models.

4.1. N o t a t i o n s u s e d Indices:

i j k l

= = = =

Decision

Bi = D~ = Lij k M~j k uij k v~jk

= = = =

Xik = X~kz = Yjk =

p a r t index; i = 1, ..., N. machine index;j = 1, ..., M. cell index; k = 1, ..., C. operation index of part i; l = 1, ..., JiVariables:

1, if p a r t i is a bottleneck part; 0, otherwise. 1, if part i needs to be produced outside, either due to machine capacity limits or due to B~ = 1; 0 otherwise. 1, if u~jk = 1 and part i will be processed at cell k; 0, otherwise. 1, if uij k = 1 and machine j will to be duplicated at cell k; 0, otherwise. 1, if a~j = 1 and X~k = 1, but Yjk = 0; 0, otherwise. 1, if a~j= 1 and Yjk =1, but Xik = 0; 0, otherwise. 1, if part i belongs to cell k; 0, otherwise. 1, if operation l of part i is performed in cell k; 0, otherwise. 1, if machine j belongs to cell k; 0, otherwise. n u m b e r of machines of type j required in cell k.

Zjk = Parameters:

B~kz = d~ = D = D~k~ = Fj

G

a d u m m y variable used in example 5 to eliminate a nonlinear term. d e m a n d (in batches) per period for part i. n u m b e r of elements in R; i.e., D = IR I 9 a d u m m y variable used in example 5 to eliminate a nonlinear term. = the set of parts processed by machinej. = an arbitrarily large number.

Table 9 Characteristics of examples Characteristics Obiectives:

Model 1 [36,371

Model 2 1711

Model 3 [I 71

Model 4 1681

Model 5 r661

- Min. total costs of intercell movement

- Min. total costs of machine investment

- Min. total intercell movement

- Max. total similarity between parts

- Min. total machine duplication costs - Min. number of exceptional elements - Min. total within cell load variation - Min. total opportunity cost Constraints:

- Unique part assignment

- Max. no. of machines allowed in each cell - Unique machine assignment - Min. no. of machines needed for each cell - A constraint to specify the required no. of

cells - Part formation logic - Linkage between machines and parts - Unique operations assignment - Constraints to compute the required no. of machines for each machine type - Constraints to identify the bottleneck parts - Constraints to identify the exceptional elements - Machine capacity constraints

Problem /Model Characteristics:

X

X

X

X

X

X

X X

X X X

X

X X 0-1 integer

0-1 integer

0-1 integer

- Number of decision variables

N~ N2+N+ I Part family only Modest Optimal procedure

(M+N)C+2N 2M+C+(2C+3)N P&M simultaneously Very complicated Optimal procedure

C(M+N+3 D) (N+M+C+DN) P&M simultaneously Complicated Simulated annealing

- Formation logic

- Model complexity - Solution procedure

X

X

- Type of mathematical programming model

- Number of constraints

X X

0-1 Integer Mixed integer Bi-criteria (N+M)C C(N L+M) (M+C) L(N+M+NC)+C P&M P&M simultaneously simultaneously Modest Very complicated Genetic Simulated annealing; algorithm Tabu search

22

Hi = cost to transport one batch of any part i between cells (bottleneck cost). = procurement cost per period of one machine of typej.

m o =

mk Pi Qi R S~k

= = = = =

wij

=

average cell load for machinej induced by part i; where

mi2 =

Z):= M ~w o

maximum number of machines allowed in cell k. the set of machines needed to produce part i. total number of machines in P~. set of pairs (i, j) such that a o = 1. similarity between part i and part k. to = processing time (setup plus run time) of part i on machinej. tik~ = processing time (setup plus run time) required to process one batch of part i through operation l on machine type j. Tj = available productive time (capacity) for each machine of type j per period. Uj = maximum acceptable utilization per machine of typej. workload on machinej induced by part i; where

wij -

tij x d i

Tj

9

4.2. M o d e l 1 [36,37,38] This model, called P-median, is one of the popular formulations used in early cell formation research. Its objective is to maximize total similarity between parts where the similarity between two parts can be defined in several different ways [12,58,59], of which Jaccard's similarity coefficient is used most frequent: Max

N

N

~

~,

(1)

S ik X i k

i=l k = l

Subject to: N

~, Xik

= 1

i = 1, . . . , N

(2)

k--1 N

EXkk = C

(3)

k=l

X~k -< Xkk

i,j

=

1,...,N

(4)

Constraint set (2) ensures that each part exactly belongs to one part family. Constraint (3) specifies the required number of part families. Constraint set (4) ensures that part i belongs to cell k only ff part family k has been formed. The P-median model is simple and easy to understand. The task of updating the formulation is straightforward, but the size of formulation is rather large. If N parts are to be produced, the model requires N 2 binary decision variables and N 2 + N + I constraints. Moreover, users must specify the number of part families by means of informal judgment, trial-and-error, or iteration [37]. Deciding which process to use is not an easy task and depends highly on experience, preference,

23 and judgment. The model, however, can be expanded easily to consider alternative routings or process plans [37,38]. According to [42], constraint (3) of the P-median model can be removed to limit the difficulty of determining the required number of cells. The model then will use embedded logic to find an appropriate number of part families that maximizes total similarity. And constraint set (4) can be aggregated into constraint set (5) to reduce the number of constraints and thus to improve computational efficiency [13,42]: N

Xik -< (N-1)Xkk

k = 1,...,N;i

~ k

(5)

i=l

The reduced model is more compact, consisting of only 2 N constraints. This model has been adapted and used successfully in [46] for grouping part families, machine cells, and operators simultaneously. An efficient heuristic also has been developed in [42] to improve computational efficiency further. 4.3. M o d e l 2 [71] Two major weaknesses are associated with model 1: (1) it can be used only to find part families and (2) it considers only binary routing information in the formation stage while neglecting many other important factors. Model 2 is an improvement, not only taking into consideration intercell movement cost but also forming part families and machine cells simultaneously. The purpose of the model is to minimize total cost of intercell movement, or what the authors call bottleneck costs: n

(6)

Min ]~ Hi Di i-1

Subject to: C

] = 1, . . . , M

(7)

~, Yjk 2 i=1

Constraint set (21) ensures that each machine is assigned to only one cell. Constraint set (22) ensures that each cell contains at least two machines. The model not only can form part families and machine cells simultaneously but also takes processing time, machine capacity and part demand into consideration. Though the formulation is very compact, consisting of (N+M)C binary decision variables and (M+C) constraints, solving a 0-1 integer bi-criteria programming model is

26 much more difficult than solving a single objective 0-1 integer programming model, because as yet there is no commercial optimization software available for such a purpose. The authors have proposed a genetic algorithm heuristic to demonstrate its applicability.

4.6. Model 5 [66] Thus far, none of the above examples has considered processing sequences in the cell formation stage; neither has any directly determined the number of machines needed for each machine type. Model 5 illustrates how a 0-1 nonlinear programming model can be developed to meet such requirements. The objective function is to consider the trade-off between the total costs of intercell movement and total machine investment: Min Z Z (Ij Zjk) ]=1 k=l

C

"k-

Hi

Xikt = 1

Z

Z IXik(l+l) - Zikll

i=l l=l k=l

i=

]

1 , . . . , N ; l = 1 , . . . , J~

(23) (24)

k=l Ji ~-~gl ~-~l Zikl tikl di

TjUj

M

i=1

< Zjk

] = 1, . . . , M ; l =

1, ..., Ji

k = 1, ..., C

Zjk i AIOIA2

Tam (1990) defines a similarity coefficient between two operations, d(x,y) as Definition 7:

d(x,y) is the smallest number of transformations required to derive y from x.

He then proceeds to assign weights to the transformations in order to generalize the similarity measure. This is done by assigning non-negative weights w s, w d, and w i to substitution, deletion and insertion respectively. The weighted similarity coefficient between two operation sequences x and y is defined by: dw(x,y) = min(wsn s + wdn d + wini) where n s is the number of substitution transformations n d is the number of deletion transformations n i is the number of insertion transformations. The similarity coefficient obtained using this equation represents the minimum number of transformations between two operation sequences. Furthermore, the similarity coefficient does not incorporate the number of common operations between two parts. Tam (1990) overcomes these problems by defining a new similarity coefficient as follows: Sc[i,j] =f(d~[i,j], c[i,j]) where dn[i,j] = dw[i,j]/max {dw[y,z] I 1 < y,z < number of parts } dn[i,j] is the normalized similarity coefficient between any two parts. eli,j] = I P i ~ Pj I / I P i u Pj I where Pi is part i and Pj is part j. c[i,j] is a coefficient representing the commonality of operations between two parts. and f i s a function that maps these two parameters onto a linear range. Kamel and Liu (1992) and Kamel, Ghenniwa and Liu (1994) use the following function to calculate the similarity coefficient: Sc[i J]

=

wndn[i,j ] + Wc(1-c[ij])

85 where 1 and w n, w c > 0. is the number of transformations between two parts w c is the commonality of two parts. w n + w c =

w n

The families are formed using a clustering algorithm which utilizes the similarity coefficients. After the part families are formed, the assignment algorithm can be applied to assign part families to machine cells. Figure 2 shows the part-family creation algorithm of Kamel, Ghenniwa and Liu (1994).

FIGURE 2.

Part-Families Algorithm

Read machine capabilities and part operations

While there are part pairs and the number of part families have not been exceeded find distinct part-pairs by choosing pair with smallest similarity coefficient.

Assign the part to be the seed of a part family

No

Yes

No

End Algorithm

Having operation sequences considered directly when solving the group technology problem at the machine assignment level is important because the operation sequences affect

86 set-up time which in turn affects processing time. As a result, Moussa and Kamel (1994) introduced an algorithm which takes into consideration the effect of operation sequences directly in the group technology machine assignment solution process. Figure 3 shows the flow-chart of their algorithm.

FIGURE 3.

Moussa and Kamers (1994) Algorithm Read machine capabilities and part operations

Calculate Similarity Coefficient for parts

Find all machine combinations for parts with minimum number of machines

For parts with more than one machine combination, search for largest similarity coefficient with another part I

Compare machine combinations of both parts; eliminate those which do not accommodate both parts I

Construct subgraphs of problem model and solve for all possible cells by linking all part nodes of same level to their appropriate combination nodes. I

Assign parts to cells to satisfy combination load balance I

Assign parts to machines to satisfy machine-load balance

End Algorithm

Moussa and Kamel (1994) favoured the use of the similarity coefficient introduced by Choobineh (1988) over Tam' s. Choobineh's similarity coefficient takes into consideration the operations sequences required to produce the various parts. His similarity coefficient calculates the number of operations having the same sequence in two parts. At the first level, the similarity coefficient calculates the number of single operations shared by two parts relative to the total number of

87 operations of the part requiring the least processing. At the second level, the similarity coefficient calculates the number of single and double operations in series shared by two parts, and so on for each level. N

Z qijqkj Sik(1 ) =

N

j=l

Z qq + q k j -

qijqkj

j=l

where

qij

1

= {0 ; qij = 1 if operation j is needed for part i, 0 otherwise.

1

Sik(L)

E

L

Cik (l)

~-~N-l+l

= Z Sik(1)+

j

/=2

where

Cik(1) is the number of common sequences of length I between parts i and k; L

is the level at which the similarity coefficient is calculated (i.e. when L=2, determine how many times a sequence of two operations is repeated). Sik(L) is the average similarity coefficient of order L between parts i and k,

0 < Sik(L ) 1 3

p

8

15 14

9

12 13

14 13 12

2

3

1

2

(I0>II

8

11

I0

10

9

8

4

5

4

6

7

8

6

7

4

9

7

6

S

(11)1 7 6 s 4 (lz) l 1 2 3 (13) 1 3 z (14)I 1 (lS) I

Indicating that column 13 is for eat interaction effect of factors ~ssigne4 to columns 5 and 8.

11

11

5

I

10

10 11

3

9

191 5. C O M P R O M I S E

PROGRAMMING

(CP)

The objective of the CP technique is to define the human behavior in seeking their goals under multiple objective situations. It should be noted that human decision making is not just to maximize or minimize a goal, but to search for stable patterns of harmony among the goals because of several conflicting goals Zeleny (1992). Each goal, in a decision making process, is expressed as a function with respect to the decision variables. Once target values of the goals which a decision maker attempts to achieve are set, CP is used to reach the best decision through an iterative target setting process which helps to reduce deviations of goal values from their target values. (For further details of the CP, readers are referred to Romero (1991), Shi and Yu (1989), Yu (1985), and Zeleny (1974).) The CP model is to minimize a regret function which combines all deviations of goals from their target values. For given n goals, suppose that a vector y = (Yl ..... Yr0 is a set of goal functions. Let y* be a target vector which is initially set by a decision maker. The regret of having y instead of achieving the target y* is represented by the distance between y and y*. Thus, the regret function is defined by r(y) = II y - y* II. It is thereby presented in the following form of Lp metric (p > 1) which represents a distance with p as a parameter defining the family of distance functions, 1

r(ylp) =

, where ki is a normalization value for i- th goal measure. i=l

ki

Since the goals in the decision-making process often have different degree of importance, n

importance weights (mi's) should be assigned to goals, yi's, where y~ mi = 1. 1

Thus,

i=l

r(ylp,w) = Ii____~lcoiP(lYi - YlI)PlP , where w = (031..... ton). ki

The estimation of the weight vector, w, is not a trivial task. First, each goal should be compared with the others in its importance. The results from all pairwise comparisons are recorded in a matrix /sk = [aij], where aij (i = 1..... n; j = 1..... n) indicates the relative importance of goal i compared to goal j. For instance, if goal i is twice as important as goal j, then aij = 2. All diagonal elements of the matrix/~ are set as 1, and its lower triangle is the inverse of the upper triangle. Then, weight vector, w, can be calculated by applying an eigenvalue method t o / ~ . (Refer to Saaty (1977) for a detailed explanation of the eigenvalue method.) The absolute value sign in the above regret function can be removed by introducing new +

variables of di's and di's as follows: for i = 1..... n,

192

Y i - Yi

ifYi > Yi

0

otherwise;

Yi - Yi

ifYi < Yi

+ di =

di =

f" 0

otherwise. ,

+

" ,

+

+

Then, ly i - yi I = d[ + d i , Yi " Yi = d[ - d i , and d[ x di = 0. Combined with the previous result, the regret function can be rewritten as: 1

r(ylp, w) =

)p 1 (:i)

co_~i (di + d~) p ill ~,, ki

9

It is noted that the minimization of the above regret function is equivalent to the minimization of r'(ylp, w) = ~ OOi i=l

(d;

+ d + ) p.

Therefore, design problems with multiple objectives can be presented in the following CP formulation: (For simplicity and without loss of generality, r' is replaced by r.)

Minimize Subject to

n {Oi r(ylp, w ) = ~ - - i=1 k~ ,

+ P (d i + di)

l) p +

Yi - f i ( X ) = d i - d i , V i = l ..... n IBX = ~ , where X is a decision vector, (x 1..... Xn), with decision variables of x i ( i = 1..... n), f i (X)is a i-th goal function, Yi, such that Yi = f i (X), ]!5 is a constraint coefficient matrix, and is a right-hand side vector of constraints.

It is not uncommon for system designers to check if the target values of goals are achieved after solving the above compromise programming problems with multiple goals. If target values of some goals are not obtained at the satisfactory level, the system designers may attempt to improve the goals by adjusting target values of other goals. Thus, this iterative compromising process will continue, as shown in Figure 4, until a certain satisfactory solution is reached.

193

Construct a design problem by x, f(x), y*, w, and p. _

Solve the problem using compromise progranuning.

No

Is the solution Yes ,

,

,,,,,i

Adopt the solution and terminate the iterative searching process Figure 4. An iterative Compromising Process in the Compromise Programming.

6.

A N U M E R I C A L E X A M P L E OF MANUFACTURING CELL DESIGN

THE

MULTIPLE

OBJECTIVE

6.1. Description of a Manufacturing Cell Problem A. Description of a Manufacturing Cell Configuration. A gear manufacturing cell with six workstations (or, machine centers) is, as illustrated in Figure 5, used to show a numerical example of the multi-objective cell design method presented in Section III. The six operations of the gear cell are: (1) turn side number 1 (of a forging blank), (2) turn side number 2 and bore, (3) hob teeth, (4) deburr/shave teeth, (5) broach keyway, and (6) inspection and package. The gear cell machines 98 different gears with processing and setup times varying from operation to operation. Buffer storage is located between two workstations to reduce imbalance of workloads among workstations. B. System modeling via simulation. The gear cell presented in Figure 5 is modeled using STARCELL which is a PC-based simulator to assist in the design and evaluation of flow line manufacturing cells. (Refer to Steudel and Park (1987) for details of the STARCELL.) Parts coming to the cell are processed on the first-come-first-served basis. Operator's travel time from a workstation to another are given and constant. To avoid the initial transient system performance, collection of statistics from the simulation results begins after 40 hours (i.e., a week), and the data collection ends at the time of 2,000 hours (i.e. a year with 50 weeks).

194

r D Turn/ Bore

Stock

ob Teeth~ ob Teeth] I ob Teeth~ C Bufler #3 ]

Finish Stock

i Inspect ~) ~ Packag~J

ir~ D

Figure 5. Layout of a Gear Manufacturing Cell Used for a Numerical Example. C. Design of Experiments via the Taguchi Method. The following eight decision variables are involved in the manufacturing cell simulation model: i) Machine setup policy ii) Number of operators iii) Operator assignment to workstations in a cell iv) Buffer storage # 1 between workstations 1 and 2 v) Buffer storage # 2 between workstations 2 and 3 vi) Buffer storage # 3 between workstations 3 and 4 vii) Buffer storage # 4 between workstations 4 and 5 viii) Buffer storage # 5 between workstations 5 and 6. The following four performance measures are to be optimized in this research: i) Minimize average tardiness per job (for simplicity, it will be called job tardiness throughout this paper), ii) Minimize in-process inventory, iii) Maximize production rate, iv) Maximize system utilization. The design of experiments for the manufacturing cell design problem includes main effects of eight factors and two levels in each factor for simplicity, which requires orthogonal array L12 of the TM. It should be noted that since in the L12 the interaction effects of factors are consistently spread across all columns, higher orthogonal arrays should be used for incorporating the interaction effects into a manufacturing cell design problem. The layout of the design is delineated in Table 3, and also the table shows the allocation of effects of interest to the columns.

195 Table 3 Manufacturing Cell Design Parameters and Their Levels. Design Parameters Sym.

Contents

A

Machine Setup Policy

B

No. of Operators

C

Assignment of operators to workstations

Levels Low 1 (All machines in a workstation should be setup to process a job.) 2 1 (Operators can work at any workstation.)

High 2 (Only one machine in a workstation should be setup to process a job.) 4 2 (Half of operators should work at workstations 1 and 3, and the other half should work at the rest of workstations.)

D

Size of Buffer #1

10

20

E

Size of Buffer #2

10

20

F

Size of Buffer #3

25

50

G

Size of Buffer #4

1

5

H

Size of Buffer #5

1

5

6.2. Statistical Analysis via the Taguchi Method Twelve simulation runs are conducted, parameter settings and system utilization of which are shown in Table 4. To investigate the significance of effects, the analysis of variance (ANOVA) table is constructed and presented in Table 5. Since some columns have very insignificant effects on a performance measure as compared with the others, they are compounded into an error term and indicated by "p" in the table. For example, effects of seven columns 5, 6, 7, 8, 9, 10, and 11 for the job tardiness are compounded into the error term due to very small mean sum of squares. Therefore, this compounding process results in 7 degrees of freedom of the pooled error. From the F-test with F6,1,0.05 = 5.99 at the significance level of 0.05, some main effects are found significant, and these significant effects are highlighted by underlines in Table 5. For instance, the significant factors for the job tardiness performance measure include machine setup policy (A), number of operators (B), operator assignment to machines (C), and buffer storage #1 (D). Since the sizes of buffer storages #2, #3, and #5 do not have significant effects on any performance measure, they are set at the lowest level.

196

Table 4 E x p e r i m e n t a l settings and response values of 12 test runs for four different performance measures. Settings of Factors Job Tardi- Prod. M/C Worker Rate Setup No. of Assign- Buffer Buffer Buffer Buffer Buffer ness Policy workers merit No. 1 No. 2 No. 3 No. 4 No. 5 Oars) (pcs/hr)

Test Run No. 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 1 2 2 2 2 2 2

2 2 2 4 4 4 2 2 2 4 4 4

1 1 2 1 2 2 2 2 1 2 1 1

10 10 20 20 10 20 20 10 20 10 20 10

10 10 20 20 20 10 10 20 20 10 10 20

25 50 25 25 50 50 25 50 50 25 50 25

1 5 1 5 1 5 5 5 1 1 1 5

1 5 1 5 5 1 5 1 5 5 1 1

3.74 3.94 6.88 2.17 2.03 2.21 16.40 14.31 13.77 9.67 10.52 8.10

4.97 4.98 4.41 8.17 8.00 8.05 3.55 3.28 2.62 5.85 5.39 6.31

System WIP Utiliza(pcs) tion (%) 2.50 2.57 16.88 20.83 17.20 19.89 26.29 15.15 4.48 27.07 35.96 31.27

36.37 36.49 31.89 60.21 58.63 58.89 26.28 26.81 34.44 47.27 41.57 53.26

Table 5 The A N O V A table from the results shown in Table 4. Col. no. in OA

1 2 3 4 5 6 7 8 9 10 II

Source

A B C D E F G H Dummy Dummy Dummy

Degrees . of Freedom Job Tardiness 1 1 1 1 1 1 1 1 1 1 I

582.30 582.30 18.61 22.40 p p p 1.07 p p p

F Values WlP

Production Rate

System Utilization

7.53 7.53 1.28 1.69 p 1.81 p p p p p

180.24 180.24 p 1.94 p 1.19 12.92 p p p p

14.75 14.75 0.83 p 1.79 p p 1.11 p p p

F r o m a regression analysis, the following equations corresponding to the four performance measures are established: (1) Job tardiness (Yl): Yl = -5.398 + 9.384 Xl - 1.829 x2 + 2.221 x3 + 0.106 x4, (2) W I P (Y2): Y2 = -16.58 + 10.05 Xl + 7.021 x2, (3) Production rate (Y3): Y3 = 2.429 + 0.571 xl + 0.286 x2 - 0.429 x s , (4) S y s t e m utilization (y4): Y4 = 29.87 - 12.257 Xl + 10.008 x2, w h e r e Xl ..... x5 are a machine setup policy, the n u m b e r of operators, an assignment policy of operators to workstations, size of buffer #1, and the size of buffer #4.

197

6.3. The Design of a Manufacturing Cell Using Compromise Programming With the regression equations for four performance measures presented in Section 6.2, the manufacturing cell design problem can be formulated as follows: Problem 1 (P1): (1) Objective functions: Min Yl = -5.398 + 9.384 X 1 - 1.829 x2 + 2.221 x3 i) Job tardiness (Yl): + 0.106 x4, Min Y2 = -16.58 + 10.05 Xl + 7.021 x2, ii) WlP (Y2): iii) Production rate (Y3): Max Y3 = 2.429 + 0.571 Xl + 0.286 x2 - 0.429 xs, iv) System utilization (y4): Max Y4 = 29.87 - 12.257 xl + 10.008 x2, (2) Constraints:

l<x1 0) when the previous batch on the same machine is of the same type, otherwise it incurs a major setup "g'major such that Zmajo r > "Cmino r. Minor setups are due to simple changeovers between batches (e.g., part placement and fixture positioning) while major setups may require changes in tooling, part programs, and fixtures as well as adjustment time for operators. The ratio of "Cmajor to "Crninor will depend on the degree of similarity between the different part types and the versatility of machines. In a system where part handling is automated and machines are highly flexible (e.g., machining centers), changeover times between different part types will be small. On the other hand, in a system handling a large variety of part types and relying on specialized manual labor for part fixturing, transportation and setup, changeover times could be significant. Setup could also be sequence dependent so that the value of ~major depends on the identity of the current and the previous batch.

208 Intuitively, it should be clear that machine sharing can be potentially beneficial to system performance due to the resulting increase in resource pooling. It should also be clear that these benefits can be eroded by the higher frequency of setups due to the increase in the product mix variety. The interaction between these two opposing effects is not however clear. The impact of system operating parameters such as batch sizes and batch sequencing rules is also not known. In the remainder of this chapter, we set out to examine these various tradeoffs that arise from machine sharing and investigate the effect of other system operating parameters on the realization of the benefits, if any, of this sharing. To allow for a better understanding of the various effects that are at play during machine sharing, we divide our discussion in three sections. In the first section, we study in the absence of setup times the impact of machine pooling on various measures of system performance. In the second section, we evaluate the validity of these results when setup times are included and consider the effect of different batch sizing strategies on the desirability of machine sharing. In the last section, we examine the sensitivity of our findings to alternative scheduling rules. In particular, we study the effect of adopting a setup based batch sequencing priority rule and its implications for various system performance measures. For the sake of brevity, we omit detailed proofs for many of the results. A full discussion of the various issues addressed in this chapter can be found in [21, [3] and [4]. 4. T H E P O O L I N G E F F E C T In order to isolate the effect of machine pooling, we first consider the case where setup times are negligible (i.e., ~'minor-" "Cmajor"- 0 ) . TO allow for a fair comparison between different pooling scenarios, we assume that part average demands are identical with D i = D, for all i = 1, 2 . . . . , m, and part processing requirements are homogeneous with a mean operation time 1/#. These assumptions ensure that machines are balanced with equal utilizations Pi = P = D/# for all i = 1, 2 . . . . . m. For ease of exposition, we also assume initially that batch inter-arrival and processing times are independent and exponentially distributed. Given the above assumptions, average batch flow time, which is also average part flow time, in a machine group of k machines with batches of size B can be obtained as that of a multi-server queuing system (i.e. a M/M/k queue) and is given by [3]

W(k,B)=

BZCk,B + B__, k(,u - D) la

(1)

where

Zrk,8 =

[

k!(1 -

Ok,8)

+ 1]

.

(2)

j=o j!(kpk,8) k-j From equality 1, it is easy to verify that machine sharing and production batching exert opposite effects on average flow time. The value of W(k, B) increases with B while it decreases with k. More specifically, we have W (k, B) = BW (k, 1). (3) where W(k, B) is the average flow time for a batch size of one. This result follows from the fact that in the absence of setups no benefits are obtained from increased batching. On the other hand, average flow time can be shown to be a strictly decreasing and convex function of k [3] [4]. In particular, average part waiting time in queue, Wq(k, B), is found to decrease by at least a factor of k when k machines are shared. That is, Wq(1, B) (4)

Wq(k, B) < ~ .

209 (a result that follows from the fact that 7rk,B < 7rl, B)" These results are illustrated in Figures 3 and 4 for various machine utilization levels. It is interesting to note that while batching has a linear effect on flow time, the effect of machine sharing is of the diminishing type. In fact, much of the reduction in flow time occurs at relatively low levels of sharing. A small increase in the number of shared machines has however a significant effect on performance. For instance, with the grouping of only two machines, waiting time can be cut by more than 50%. Thus, when setup times are negligible and lead time related performance is important, a strategy of machine sharing should be pursued whenever possible. Note that because of the diminishing impact of machine sharing, only limited sharing is of significant value. Hence, a strategy of partial machine pooling would yield comparable performance to that of total pooling. It can also be seen that the effect of machine sharing is particularly important at high utilization levels. In fact, the amount of reduction in flow time can be shown to strictly increase with increases in machine utilization [3]. In addition to its effect on average performance, machine pooling can be shown to have an equally beneficial effect on performance variance. This can be seen by considering the variance of waiting time in queue, Vq(k, 1), for a machine group of size k. The value of Vq(k, 1) is given by [3]: Vq(g, 1)= ~k, 1(2 - /~'k,1) . (5) [kp (1- Pk, 1 )]2 and is a decreasing function of k. Noting that zrk, I -~ z i , 1, we can easily show that waiting time variance for k shared machines is always smaller than that of k dedicated machines by at least a factor of k 2. That is,

Vq(k, 1)_< Vq(1, 1).

(6) k2 Thus with the grouping of two machines, for example, queueing time variance is reduced by more than 75%. This means that fluctuation in workloads among different machines is drastically reduced and the possibility of both bottleneck and starved machines is minimized. The effect of sharing on waiting time variance is depicted in Figure 5. Note that the degree of reduction in variance is particularly significant under conditions of high utilization. The fact that waiting time variance is reduced results in a reduction of overall flow time variance which in turn leads to greater consistency and predictability in lead time related performance. This is desirable in environments where being dependable in meeting due dates and having consistently short lead times is important. Furthermore, machine sharing can be shown to be an effective mechanism for dealing with system variability. In fact, the benefits of sharing can be shown to increase with increases in either demand or processing variability. This can be seen, for example, by considering the following approximation of average flow time for the same machine group described above when batch arrivals and processing times are generally distributed [3]: W (k, 1)

=

Zrk, 1(1 +

2kp

(1-

C2)(C2 + pk2,1 C2) Pk, 1)(1 + pk2,1 C2s)

1

+ --.

]2

(7)

where C2 and C 2 represent respectively the squared coefficients of variation (i.e. the ratio of variance over the squared mean) in customer inter-arrival and processing times. The value of C2 indicates the degree of variability in the part arrival process which can be due to either demand variability and/or to variability in time between part releases to the system. Similarly, the value of C2 indicates the degree of variability in the part processing times which can be due to either inherent variability in the process or to external interferences such as machine breakdowns, tool wear, and poor fixturing. The value of W(k, 1) can be easily shown to be decreasing in k. More importantly, the amount of performance improvement can

210 20

!

I

I

I

I

I

I

I

0.95 15-

10- p=(.9

5-p=o p- o

~

i

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

0 0

~

10

Figure 3. The effect of machine sharing on average flow time

(/g= 1)

I

200

I

I

I

!

I

I

I

I

150-

J

100-

p =0.9 /

50-

0

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

10

B

Figure 4. The effect of batch size on average flow time

(~= 1)

211 be shown to increase with increases in either Ca2 or Cs2 . This is illustrated in Figures 6 and 7. Machine sharing is thus particularly valuable for systems subject to high variability, with the provision of sharing almost eliminating the negative impact of this variability. Note also that in the absence of variability, that is when Ca2 = Cs2 = 0, machine pooling has no effect on performance as the dedicated and shared systems become equivalent. In other words, the value of machine sharing is contingent upon the existence of some degree of variability in the system.

I

100

I

I

I

I

I

I

I

I

I 3

I 4

I 5

I 6

I 7

I 8

I 9

0.9

8060 4o 0.8 20

0 0

I 1

I 2

10

k

Figure 5. Waiting time variance versus machine sharing (/~= 1)

5. T H E E F F E C T O F SETUPS In this section, we investigate the effect of setups on the validity of our previous results and the overall desirability of machine sharing. As mentioned earlier, we associate either a major or a minor setup with the processing of each batch. A major setup occurs when a machine switches between two different part types. On the other hand, only a minor setup is required when the succeeding batches are of the same type. Average batch processing time for parts of type i is thus given by t~, 8(i) = Pr(previous batch is of type

i)'gminor+ Pr(previous

batch is not of type

i)'~major+_B_B. //

Assuming steady state operation, random and independent part arrivals according to a renewal process, and first-come first served batch ordering, the probability that the previous batch "is" and "is not" of the same type in a machine group of size k are given respectively by Pr(previous batch is of type i) =

k

Di

2 De i=1

and

212 I

14

!

. . . . . I. . . . . . . . . . . . .1. .

.I

1210-

0,,

i

1

0

l-

J

2 3 4 Processing variability (Cs 2)

5

Figure 6. The effect of processing variability and machine sharing on flow time (# = 1, C~ = 1, p = O.S)

15

J"

k=l

10 k=2 5

0

u,

0

,

,

'~',

i

1

~

,"i

,

i

,

,

,

~'i

,

~'

~

,

i'~',

2 3 4 Demand variability (Ca 2)

,

~ ''

5

Figure 7. The effect of demand variability and machine sharing on flow time (/2 = 1, Cs2 = 1, p = 0.8)

213

Di

Pr(previous batch is not of type i) = 1

k

Z

.

Di

i=1

where D i is the average demand per period for part type i. The overall average batch processing time can then be obtained as k

kDi ( kDi "Cminor-t- (1- kDi )Tmajor).t_ ]11 n__, tk, B= Z i=l Z Di Z D i 2 Di i=1

i=1

i=1

which can also be rewritten as

k )2 Tminor+ (1 - ~_~

k ( kDi t~,8 = ~__~ i=1

Z

Di

i=1

i=1

B ) 2)'gmajor+ lZp

( k Di

Z

(8)

Di

i=1

We can see that average batch processing time is composed of two components: (1) a setup time component, determined by the degree of machine sharing and part mix composition, and (2) an operation time component, a function of operation time and batch size. The effect of these two components can be more clearly seen by considering the case where D i= D for all i = 1, 2 ..... k. Equality 1 then simplifies to tk, 8

= "Cmajor( ~ -) dr Tmikn~ q. --.B

(9)

#

From equality 9, we can clearly see that batch processing time increases with increases in either machine sharing or batch size. The increase due to machine sharing can be explained by the increase in the frequency of major setups, as k - 1 out of every k setups are of the major type. Note that the proportion of major setups, (k- 1)/k, grows rapidly with k so that the major setup term becomes dominant relatively quickly. In the more general case described by equality 1, setup time will also be determined by the part mix composition (i.e., demand distribution among part types). Batch processing time is maximum when D / = D2 . . . . . DI, = D, and is minimum when there exists a part type i such that D i > 0 a n d Dj = 0 for all j ;e i . More generally, average batch processing time increases as the difference in the production ratios of different part types decreases. This is a result of the increase in the likelihood of a major setup when the different part types are produced in equal proportions. This likelihood is reduced when only few part types dominate the part mix. Thus, batch processing time is not only affected by the number of shared machines but also by the relative variety in the associated part mix. Average batch processing time determines the maximum feasible throughput (production rate) and, consequently, system capacity. Assuming a uniformly distributed part mix (maximum part variety), the maximum throughput per machine, THma x, can be calculated as THmax =

B

q~major "Cmaj~ - "Cminor B k +l~

.

(11)

This maximum production rate decreases with machine sharing while it increases with batch size. In limit cases, we have limk _+ ~(THmax) =

B B' ~major +

(12)

and lim B ~ ~(THmax) = ~t.

(13)

214 Plots of THmax are given in Figures 8 and 9 for various values of k, B and Tmajor. With increased batch size, throughput becomes less affected by machine sharing as setup time is gradually eliminated. Thus, in environments where maintaining high production rates is important (e.g. make-to-stock environments), larger batch sizes should be used. For environments where lead times are more important (e.g., make-to-order environments), large batch sizes could be dangerous since they tend to increase batch processing times. This is however ironic since in a make-to-stock environment the number of different part types is typically limited and setups are not significant, while in a make-to-order environment the number of part types could be high and setups could be important. In addition to determining system capacity, batch processing time determines system utilization. Assuming a uniformly distributed product mix, average utilization per machine, P~ B, is given by

Dk, B = D( Tmajor(~__ ) + Tmin~ + -~B)

(14)

which can also be rewritten as

[Ok,B "- [9setup at- Doperation ,

(15)

Psetup = ~ ( T m a j o r ( ~ ) q- Tm~~~ )

(16)

where and is the proportion of time a machine is being set up (in reality, this is idle time), while

Doperation = D P

and is the proportion of time a machine is actually busy performing operations. This is also the machine utilization in the absence of setup times (see section 4). In order to ensure system stability, we need to have Pk, n < 1. This means that for a fixed level of machine sharing, a lower bound on the required batch size is given by

B( Tmajor(-~-~--) + Tmikn~ ) Brain =

max(l,

),

1 D

(17)

#

which can also be rewritten as Bmi n =

max(l, 1 -

DT

Poperation

)

(18)

where

T=

"Cmajor( k - 1 )

" k

+

Tminor k

and represents total setup time. The minimum required batch size increases linearly as a function of setup time and inversely as a function of actual machine utilization. In the absence of setups, the need for batching is eliminated and B m i n becomes 1. Since total setup time is in part determined by the degree of machine sharing, the value of Bmin will depend on the value of k. In fact, Bmin is a steep convex function of k with the following limit

limk ~ .o(Bmin) =

D'Cmajor 1 - Poperation

.

(19)

Similarly, for a fixed batch size, we can calculate the maximum feasible number of shared machines, kmax. The value of kmax is given by

215 I

1.2

I

I

I

I

I

I

I

I

1 -

E

0.8

--

0.6

--

0.4

--

0.2

--

0 0

B=5 B-=A__13--___3__ B=2 B=I

I 1

! 2

I 3

'1 4

I 5

I 6

I 7

I 8

I 9

10

k

Figure 8. The effect of machine sharing and batch sizes on system capacity (/.t = 1, ~najor = 5, "gminor = O)

I

1.2

I

I

I

!

I

I

I

I

1-

0.8E

0.6-

"t =1 maj~ 1; . = 2 mal__ x =3 maj

0.40.2-

1;

maj = 4&5

0 0

I

I

I

i

I

I

I

I

I

1

2

3

4

5

6

7

8

9

l0

k

Figure 9. The effect of machine sharing and setup times on system capacity (11 = l, B = 1, Zmino r = O)

216 kma x =

Tmaj~ - Tmino r Tmajor - n ( 1 -

1

(20)

--~)

where Vmajor - Tminor > 0 and Tmajor - B (1/D - l/p) > 0. The value of kma x decreases as a function of Tmajo r with a limitt

Lim~major -o .o (kmax) = 1. The value of kma x also decreases with operation machine utilization, Pope rati on , as higher utilization reduces the available capacity for setups. On the other hand, the number of allowable shared machines tend to increase as batch size increases. A result due to the reduction in the frequency of setups. In fact, for B > "Cmajor/(1/D - Ill.O, the value of kmax becomes unbounded (kmax = oo). The value of kmax also increases as ~'major approaches B(1/D - 1/#) with kmax being unbounded for Tmajor < _ B ( 1 / D - 1/p). In summary, increased machine sharing can have a negative impact on several system performance measures. In particular, machine sharing increases batch processing times by increasing the frequency of major setups. This in turn, limits the available capacity for actual operation and increases the proportion of machine idle time due to setups. Consequently, the maximum feasible production rate is also reduced. This also means an increase in the minimum feasible batch size. However, as we saw in the previous section, machine sharing can have a positive impact on a number of flow-related performance measures such as part flow time and flow time variance. In the remainder of this section, we examine the degree to which these benefits are undermined by the presence of setups and the extent to which the negative effect of setups can be mitigated by increased batch sizes. For ease of exposition, let us assume that batch inter-arrival times and processing times are exponentially distributed. Average part flow time in a machine group of k machines with batches of size B can then be obtained, as previously, as that of a multi-server queuing system and is given by [3]

W (k, B) =

~k, B

kp

_D

+ Tmajor(k_~) + ~m~,or + n

P"

(21)

k(p ('rmajor(k- 1) + "rminor) + kB B ) It can be seen that he presence of setups tend to shift the effect of sharing and batching in opposite directions. Larger setup times diminish the desirability of machine sharing while favoring larger batch sizes. For example, consider the case where k is set to 1 and B is allowed to vary. The expression for average flow time can then be rewritten as Tminor + n___

W(1, B ) =

P . (22) 1 D DTminor p B As B is initially increased from its lower bound Bmin (assuming Bmi n > 1), average flow time is dramatically reduced. This reduction eventually levels off and flow time starts to increase again with batch size. This behavior is graphically depicted in Figure 10. The initial decrease in flow time is due to the reduction in the frequency of setups. This is however gradually offset by the increase in processing times as batches become larger. Note that as B increases, average flow time becomes almost linear in B. Noting that W(1, B) is a convex function of B, the batch size value that minimizes flow time can be obtained by simple differentiation of equality 22 and is given by [29]

t We should note that, for a fixed batch size and machine sharing level, there is a limit on the maximum feasible major setup time. The value of this maximum setup time can be directly obtained from the stability condition.

217

P # B * - D'Cminor--. D ~-(1-~-)

(23)

This result illustrates the fact that some degree of batching can be desirable even in the absence of machine sharing, with the optimal batch size being an increasing function of setup time, Tmino r, and machine loading D. Further discussion of the effect of batch sizes in the single machine case can be found in [291, [30] and [31].

I

150

100-

Xmi = 0.75

50-

0

I

I

5

10

15

Figure 10. The effect of batch size and setup time on average flow time (p = 1, D = 0.8) Now let us consider the case where B = 1 but where k is variable. The expression of flow time becomes W (k, 1) = k(

7~k, 1 1

- D)

+ Tmajor(k_~) + "Cminor q_ 1 k

-p"

(24)

"Cmajor(~-~) q- "Cmikn~ -l- l-~P

It is useful to distinguish here between two scenarios. The first is where "Emajor - ( 1 / D - l/p) < 0 and the second is where Zmajo r - ( 1 / D - l/p) > 0. For the first scenario, we have shown earlier that k,nax is unbounded. In fact, in the limit case we have limk ~ oo(Wk, 1) = "Cmajor + 1__ P

and thus the optimal sharing level, k*, is also unbounded. On the other hand for

(25) "Cmajor - ( 1 / D

1/ p ) > O, kma x is finite and given by expression 20. As k approaches kmax, flow time grows without bound. The expression of flow can be shown to be monotonically increasing in k and therefore k* = 1. The behavior of flow time for both scenarios is depicted in Figures 11 and 12. Note that for the first scenario, flow time initially grows as k increases. However because of the diminishing increase in setup time, flow time eventually starts to decrease again with k. This decrease is itself of the diminishing kind with much of the reduction

-

218 ,I

100

!

I

!

I

I

' r / aj. = 0.301

80

60-

I

/

_

40/

20-

0 0

I 2

/

I 4

I 6

~

I 8

..... I 10

1: .=0 26 ma~ . . . . . I 12

I 14

16

Figure 11. The effect of machine sharing and setup time on average flow time (Tmajo r " relation. It is easy to verify t h a t ">" defines a partial order relation on the set of quasiseriation relations on M and I. A consensus is said to be a m a x i m a l consensus on a set of consensus' ~ if there is no consensus Zj such t h a t Zj > Z i. On figure 4, for example Z 3 et Z 4 are the maximal consensus' of the set ~={ Z 1, Z2,Z 3, Z 4 }.

/~Z=R

Z2 Z3 4b-- 4b

---

I

-

-

-

Rc_Z

!

!

I

"O-

I

I

!

9 Z4

6 Figure 4: R agreements with Zl, Z2, Z3, Z4 In this section we have shown t h a t a quasi-seriation relation could express the consensus of the working party. We have defined a relation denoted ">" allowing us to compare the proximity to relation R, defined by the working party, of two quasi-senation relations. As the relation ">" is a partial order r e l a t i o n , it generally does not allow us to define the quasi-senation relation "the closest to R", but only a set of consensus' "close to"R that are the maximal consensus.

234 2.2.

The

method

The proposed method gives two types of information to the working p a r t y . It is made up of two steps whose characteristics will be described below. Step 1: research of maximal consensus' about R - Step 2: research of maximal consensus' about relations generated at random -

Step

1:

Taking into account the tremendous number of consensus on the sets P and I, it is not possible, even with todays best computers, to list each consensus and check if it is a maximal consensus. So, we satisfy ourselves with the maximal consensus of an initial set of consensus, obtained with an heuristic method. The initial set of consensus is composed of solutions of the following optimisation problem t h a t we note PB1. PB 1: Find a quasi- seriation relation Z on sets M and I maximizing the criterion: F(Z, ~,R) = ( a ( Z , R ) - ~).card(Z) where: R is a n o n - e m p t y subset of M x I card(Z) is the cardinal number of Z e [0.. 1] is a real number

The solution of PB1 with 13=0 is obtained with the algorithm presented in [15]. Solutions of PB1 with 13e ]0..1] are obtained with the algorithm presented in [8], [19]. The relevance of PB1 solutions as eligible maximal consensus of the subsets of M x I is shown in section 2.3. The operating process is: -

-

collect the working party propositions in the form of a binary matrix A. The relation R will then be defined. solve PB1 for ~=0.05i with i=0,1,2,..20. The set ~ of consensus is hence obtained. compute the quantities a(Z,R) and g(Z,R) for each element Z of~g retain the maximal consensus of represent the maximal consensus on a plot similar to figure 5.

The choice of the value of 13 stems from the objective properties of the solutions to PB1 on one hand, and our experience on the other. The first of these properties (P1) is t h a t a solution Zl to PB1 verifies a(Z 1,R) > ~. Hence the chosen values of allow us to reach the solutions to PB1 which have high values of a(Zl,R). The second property (P2) is that the solutions to PB1 for ~ = 0 contain the maximal

235 consensus Z 0 which verifies g ( Z o , R ) = 1. F u r t h e r m o r e ,we note t h a t the n u m b e r of different solutions of the set ~g is finite. Experience shows t h a t in resolving PB1 for a g r e a t e r n u m b e r of values ~, t h e c a r d i n a l n u m b e r of ~g would not be increased.

Proof of P1

Let us first d e m o n s t r a t e theorem 1:

Theorem 1

A solution Z 1 to PB1 with ~ ~ [O,l[ verifies a ( Z l , R ) > 13 and card(Z 1) > 0 Proof of t h e o r e m 1: As R is a non-empty subset of M • I, there exists a consensus Z o such t h a t card (Z 0 ) = card (R (~ Z 0 ) = 1

(3)

So, a ( Z o , R ) = card(Z 0 (~ R) = 1 card (Z 0 )

(4)

and F(Zo,~,R) = 1-~ > 0

(5)

Let Z! be a quasi-seriation relation solution of PB1, we t h e n have: F ( Z I , ~ , R ) = ( a ( Z 1 , R ) - ~).card(Z 1) > F(Zo,~,R)

(6)

As card(Z 1) is a positive number, (6) and (5) lead to the conclusions t h a t a ( Z 1,R) > 13 and card(Z 1) > 0. (QED proof of theorem 1) Let us now show t h a t a solution Z 1 of PB1 with 13= 1, verifies a(Z 1,R) = 13= 1 Indeed, let Z 0 be defined as above (3). Then F(Zo, 1,R) = 0

(7)

Suppose t h a t Z 1 is a solution of PB1 for 13= 1 .Thus: F ( Z l , I , R ) = ( a ( Z l , R ) - 1).card(Zl) _>F(Z0,1,R)

(8)

E i t h e r card(Z l) = 0 and a(Z 1,R) = 1 by definition, or card(Z 1) > 0 . In the last case (7) and (8) imply t h a t a ( Z 1,R) _> 1. However by definition

(9)

236 a(Zl,R) < 1

(10)

T h u s we have a ( Z l , R ) = 13= 1 0 T h e o r e m 1 and the property we have established imply the validity of property P1.

Proof of P2 We consider the set S 1 of quasi-seriations such t h a t g(Z,R) = 1. L e t S1 = { Z / g ( Z , R ) = 1} and let us prove the the assertion: Z e S 1 r Z is a solution of PB1 for 13= 0. The consensus Z 0 = M x I belonging to S 1. So, S 1 is a non-empty set. The function F ( Z , ~ , R ) for an a r b i t r a r y Z and 15= 0 is given by: F(Z,0,R) = g(Z,R)card(R)

(11)

By definition g(Z,R) < 1 a n d card(R) is fixed, thus F ( Z , 0 , R ) is m a x i m i s e d by each quasi-seriation of S 1. Suppose t h a t Z is a solution of PB1 for 15= 0. Then, F ( Z , 0 , R ) = g ( Z , R ) c a r d ( R ) > F(Z0,0,R) = card(R)

(12)

T h u s g(Z,R) >_1

(13)

By definition g(Z,R) _ (a(Z, R ) - ~max )card(Z)

(15)

(g(Zmax, R ) - g(Z, R))card(R) > ~max (card(Zmax)- card(Z))

(16)

Z is a solution of PB1 for ]3 > ~max, thus: F(Zmax,~,R) < F(Z, ~, R)

(17)

Hence, from the definitions of a and g ( see (1) and (2)) (1-~)card(Zma x) < ( a ( Z , R ) - ~)card(Z)

(18)

(g(Z~,, R ) - g(Z, R))card(R) _card (Z)

(20)

From (20) and (18) we deduce that (1- ~)card(Z) < (a(Z,R)- ~)card(Z)

(21)

In the case where card(Z) = 0, (18) implies that card(Zma x) = 0. Thus, Zmax=Z. Meanwhile, we generally have card(Z) > 0 . This and (21) imply that a(Z,R) > 1. Since, by definition a(Z,R)< 1, we conclude that a(Z,R)= 1; which demonstrates the first part of theorem 2. Moreover, since R is non empty we can deduce from (20) and (16) that g(Zmax,R) > g(Z,R)

(22)

Since we also have a(Zmax,R) = a(Z,R) = 1, then Zmax _ Z, which demonstrates the second part of this theorem. 0 Theorem 3

I f ~max e ]0.. 1[/s the smallest value of [3for which Zmax is solution of PB1 where a(Zma x,R) = 1, then Zmax is a maximal consensus. Proof." Let Z be a quasi-seriation such that a(Z,R) = 1 and such that it is not solution of

239 a PB1. As Zmax is a solution of PB1 for ~max such that a(Zmax,R) = 1 we have (14), (15) and (16). Since a ( Z , R ) = 1 ; (15) becomes (1- ~max)card(Zma x ) >_(1-13max )card(Z)

(23)

Thus (20) is still verified. Moreover as R is non empty, we can deduce from (20) and (16) t h a t g(Zmax,R)> g(Z,R) which demonstrates that the maximal consensus of S 2 is solution of a PB1. This result, associated with that of theorem 2, implies that Zmax is a maximal consensus. Z o n e 3: a(Z,R)r 1 and g(Z,R)r 1 We now consider Z 1 solution of PB1 such that a(Zl,R) r 1 and g(Zl,R) r 1. This is equivalent to studying the solutions of PB1 for ~ e ]~min ,~max[, with ~max defined as in theorem 3. The next theorem demonstrates the relationship between the set of solutions of PB1 for ~ 9 ]~min,~max[ and the set Sma x of maximal consensus' of M x I. We consider whether given Z 1 solution of PB1 for[~ 9 ]~min ,~max[, there can exist another quasi-seriation Z 2 which is such t h a t Z 2 > Z 1. Theorem 4

A solution Zl of PB1 for ~ e ]~min, ~max [ is a maximal consensus Proof." Let Z 1 be a solution of PB1 for ~ e ]~min, ~max [ and let Z 2 be a quasi-seriation relation. i) Case where Z2 is not a solution of PB1 for ~: Thus F(ZI,[3,R) > F(Z2,~,R)

(24)

Then (1) and (2)imply: (a(Zl, R ) - [3).card(Z l) > (a(Z2, R) - ~). card(Z 2)

(25)

and g(Z l, R). c a r d ( R ) - ~. card(Z l) > g(Z2, R). c a r d ( R ) - ~. card(Z 2 ).

(26)

In order to have Z2 > Z 1 we must have in particular : a(Z2,R) > a(Z 1,R)

(27)

From theorem I we get a(Z 1,R) > ~.

(28)

We also have card(Z 2 ) r 0

(29)

240

Indeed, by definition a ( Z 2 , R ) =

card (R n Z 2 ) card(Z 2 )

(30)

thus card(Z 2) = 0 ~ a(Z2,R) = 0.

(31)

Now (27) and (28) imply that a(Z2,R) > ~ > 0.

(32)

Thus (25) and (27) give us card(Z1) a(Z2,R)- ~ > >1 card(Z2) a(Z~,R)- ~

(33)

Hence card(Z~ ) > card(Z 2). Then (26)implies that:

(34)

(g(Z 1,R) - g(Z 2,R)). card(R) > ~(card(Z1)-card(Z 2)) > 0

(35)

Thus g(Z~,R) > g(Z2,R)

(36)

In this case, we deduce from (27) and (36) that Z, and Z 2 cannot be compared. Hence there does not exist a quasi-senation relation Z 2 non solution of PB1 for ~, such that Z 2 > Z,. ii) Case where Z, and Z 2 are two solutions of PB1 for ~ e ]~lmm,~m,x[such that the co-ordinates of Z, and Z 2 differ in the plane (a,g): Then F(Z 1,~,R) = F(Z2,~,R)

(37)

So we have: (a(Z 1, R ) - ~). card(Zl) = (a(Z2, R) - ~). card(Z 2)

(38)

g(Z~, R). card(R) - ~. card(Z~) = g(Z2, R).card(R ) - ~. card(Z2)

(39)

In order to have Z 2 > Z~, we must have (27) If a(Z2,R) = a(Z~,R)

(40)

then (38) implies that card(Z 1) = card(Z:)

(41)

Since R is a non empty set, from (39) we have g(Zl,R) = g(Z2,R)

(42)

Thus (40) and (42) show that the co-ordinates of Z 1 and Z 2 are the same, which contradicts the initial hypothesis.

241 (43)

If a(Z2,R) > a(Z~,R) as card(Z2) > 0 (see theorem 1), (38) and (43) imply card(Z 1) = a(Z2,R)-~ > 1 card(Z 2) a(Z 1,R)-~

(44)

Thus card (Zl) > card (Z 2)

(45)

from (39) and (45) we have (g(Z 1,R)-g(Z2, R)). card(R)= ~. (card(Z 1)-card(Z 2)) > 0

(46)

Hence g(Zl,R) > g(Z 2,R) Hence (43) and (47) show that Z1 and Z2 are not comparable and there exists no solution Z2 of PB1 for ~e ]~n,~m=[ which maximises Zl. Thus any consensus solution of PB1 that belongs to zone 3 is a maximal consensus. 0 We have proved that the maximal consensus' of the solutions of PB1 are maximal consensus' on the whole set of quasi-seriations on M x I.

3.

APPLICATIONS

3.1 A p p l i c a t i o n to t h e s c h o l a r case:

Let us return to the example in the introduction. The solutions proposed by the working party are: 1. 2. 3. 4. 5. 6. 7. 8.

Cell Cell Cell Cell Cell Cell Cell Cell

I1,1: I1,2: I1,3: I2,1: I2,2: I3,1: I3,2: I4,1:

M01, M02, M07, M01, M02, M01, M02, M07,

M05, M03, M09, M05, M03, M03, M04, M09

M06, M10, M04, M08, M l l M06, M04, M05, M07,

M10 M07, M08, M09, M l l M06, M10 M08, M09, M l l

Let us apply step l of the method presented in point 2.2. The binary matrix figure 6 represents workcenters to cells assignment relation R. The matrix includes eleven rows corresponding to the eleven workcenters M01 to M l l and eight columns representing the four solutions in 8 cells. We have then resolved PB1 for 21 values of ~ and calculated the consensus couples (a,g). Three consensus only are different; furthermore, they are maximal. We call them Zl, Z2 and Z3. Figures 7,8,9 specify the consensus Zl,

242 Z2 a n d Z3 as well as the value of the couples (a, g).. I1,1 11,2 11,3 12,1 I2,2 I3,1 I3,2 14,1 1 1 . MO1 1 M02 9 i : . i . i M03 9 1 . . 1 1 M04 i 1 9 i 1 i i 9 M05 M06 1 . . 1 . 1 . . M07 1 1 . 1 1 1 . 1 M08 . i i " 1 1 i M09 " i . i MIO i Mll ,

,

.

,

,

I1,1 12,1 I3,1 11,2 12,2 13,2 11,3 I4,1 M03 MO1 M05 M06 MIO M02 M04! M08 Mll M07 M09

i

1

1 1 1

1 1 1

1 1 1

9

9

9

9

9

,

9

,

9

1

9

.

.

.

9

.

9

9

~

.

.

~

9

~

~

.

9

9

~

9

28+0

9

9

9

1 1 1 1

1 1 1 1

1 1 1 1

. . .

. . .

9

1 1

1 1

1 1

1 1

. 28

Figure 6: the relation R

1

i

28 g=~=0.80 28+7

=1.00

Figure 7" relation Z l I1,1 I2,1 I3,1 11,2 12,2 13,2 I1,3 I4,1 MO1 M05 M06 MIO M02 M03 M04 :M08 Mll M07 M09

1 1 1 1

1 1 1 1

1 1 1 1

9

.

1

9

i

i

1

1

1

i

1 1 . 9

1 1 1 1

1 1 1 1

1 ,

9

30 a=~=0.97 30 + 1

9

g-

9

i

I1,1 12,1 I3,1 I1,2 12,2 13,2 11,3 14,1 1 1 MO1 1 1 1 M05 1 1 1 1 MIO~o 1 1 1 i i i : : M02 1 1 M03 M04 1 1 1 " " 1 1 i i M07 i l l M08 11ii M09 i 1 1 . . Mll 9

9

1 1

30 -0.86 30 + 5

Figure 8: relation Z2

1 1

9

.

34 a=~=0.72 34 + 13

.

.

.

.

.

.

.

9

.

.

.

.

.

.

9

.

1

34 g=~=0.97 34 + 1

Figure 9: relation Z3

We can now proceed with step 2. F i g u r e 10 p r e s e n t s the couples ( a , g ) obtained from the resolution of PB1 for 100 randomly generated matrices a n d 21 v a l u e s of [3. E a c h m a t r i x possesses 11 rows, 8 columns a n d 35 one v a l u e s (cardR=35). A statistical study [20] has shown t h a t the borders of the 2100 points of the r a n d o m l y generated matrices contain 99% of the points of the m a x i m a l consensus of the m a t r i c e s with n rows, n columns and a n u m b e r of one values equal to cardR. These borders are drawn on figure 10.

243 g

J

1.0-

9 l" l ~ _

"'" ":[;. ~

0.8" 0.6-

~ID

Z3

4,~zli,Z2

bord

0.4" 0.2" 0

I

I

I

I

0.2 0.4 0.6 0.8 "" randomized matrices 9 relation Z ]

I"--

1.0

Figure 10" graph of consensus' We can complete the former results with a few remarks. The positioning of Zl,Z2,Z3 on figure 10 relative to the borders of the randomly generated matrices points allows us to state t h a t the probability of the consensus being a m a t t e r of chance is less t h a n 1%. Each of these consensus is candidate to a detailed analysis. The analysis of Z3 figure 9 reveals t h a t the workcenters within the group GI={M1,M05,M06,M10} are systematically in the same cell. On this p o i n t , there is indeed a consensus between all members of the working party. The same consensus shows that workcenters within G1 and G2={M02,M03,M04,M07,M08,M09,Mll} are not, with the exception of M03 included in a solution of the working party. Hence, there is a consensus within the working party on the separation of the group of workcenters G1 and G2. The co-ordinates of Z3 on the graph of maximal consensus figure 10 foretells this result. Indeed, g(Z3,R)=0.97 indicates t h a t the workcenters in a group of workcenters of Z3 are almost never associated with those of a n o t h e r group. Solutions Z2 and Z l are close to one another on the graph of consensus. The analysis of matrices figures 7 and 8 shows t h a t the consensus Z l and Z2 are nearly identical. These findings are generalised in effect. Two close consensus on the graph of consensus are generally similar. We shall limit our analysis to t h a t of Zl. As a ( Z l , R ) = I we know that the Zl groups of workcenters are very often gathered within the same cell by the working party members. This solution will probably reveal consensus of the working party. The detailed analysis of solution Z l confirms this presumption. Indeed, the workcenters of group G'I={M01,M05, M06,M10},G'2={M02,M04,M08,M11}, G'3={M07,M09} are systematically gatherd within the solutions elaborated by the working party. There is very much a consensus of the working party on this point. Furthermore, the workcenters of groups G'I and G'2 are never clustered in a working party solution. Likewise for the workcenters of groups G'I and G'3. We can guess from the value g(Zl,R)=0.80 this last kind of results for consensus Zl.

244 3.2

Industrial application In this study, we were asked to evaluate the advisability of reorganising workcenters in m a n u f a c t u r i n g cells. The workshop studied includes 244 workcenters producing 200 items. The feasibility study was carried out using the method of project management [4]. The material flow analysis was performed with the help of the specialised software of process planning clustering, SAFIR [23]. The n u m b e r of propositions of the reorganisation committee was about twenty. The relation R matrix possesses 244 rows and 1046 columns. The cardinal number of relation R is 4880. The graph C1 is the maximal consensus obtained with the propositions of the working party in stepl. The borders of 99% of the points of the randomly generated matrices is given by C2.

g 1.0-

.

.

.

0.8

.

.

9.

9.

.

zl z2

5 J z7

I

0.6 0.4

I

C2 L

0.2 0.2

0.4

0.6

0.8

1

Figure 11" Consensus for W2 Step 1 should provide us with 21 maximal consensus. However, we note that on figure 11 seven couples (a,g) only (Zl to Z7) are obtained from this process. In fact different values of an~n for PB1 can give rise to the same maximal consensus. Practical trials have shown that an increase in the number of values of a ~ , used does not, in general, increase the number of maximal consensus obtained. The fact that the number of maximal consensus is limited is an interesting property in practice, since it means that the working party can quickly and efficiently analyse and compare all the possible solutions. Each proposed consensus is obviously useful to elaborate the final decision. If solutions Zl to Z7 are far away from the borders; it is unlikely due to chance. The quantity g(Zi,R) for i~7 is greater than 0.8. This presupposes the fact that the solutions Z l to Z6 show up groups of workcenters generally dissociated in the working party solutions, and thus consensus. Consensus of a different nature are revealed by solutions Z7 for which a(Z7,R)=I. Analysis of consensus Z3 and Z7 allows us to quickly and definitely classify 80% of workcenters in a family with the working p a r t y agreement. Thus, the former was able to focus on the remaining 20%.

245 4.

CONCLUSION

For m a n y years, several authors have proposed methods to recognise manufacturing cells from the analysis of the machine-part incidence matrix. In fact, the problem of reorganisation of a workshop into cells is very complex, because a lot of technical constraints and social and economic points of view must be considered for industrial applications. So depending on the assumptions used in the methods, the results are various and it is difficult to deal with them. Here, we propose a method which permits us to point out the similarities of the different points of view. The method is based on a cross clustering technique. Using the a parameter and the overlapping rate g, the working party can easily identify the best grouping of workcenter into cells reflecting its various practical aspects. This study raises, however, number of points t h a t have not been resolved. From a mathematical point of view, we have shown that the set of maximal consensus is included in the solutions of the quasi-seriation problem (PB1). It remains for us to precisely define the relations between these two groups in order to be able to evaluate, for example, the exact number of maximal consensus. Statistical trials t h a t we have carried out for the randomly generated curves, show that 2100 matrices are sufficient to define the boundaries of these curves at a level of 99%. These binary matrices are subject to the number of rows, columns and percentage of I values being fixed. We are in the process of formulating and putting into practice a rigorous method which will allow us to minimise the number of random matrices used. From a practical point of view, problems concerning the robustness of solutions may occur if the distance from the random curve to the curve of consensus formulated by the working party is great. Therefore, we find it necessary to concentrate on this point and create a measure of the robustness of the reorganisation comitte solutions. Another practical aspect which could be improved is the analysis of the consensus considered as being interesting from the maximal consensus curve. We are working on the systematisation of these matrices analysis and on the automation of these simple and repetitive tasks. Not any firm is ready to embark on a relocation scheme, which is costly in study hours, without strong presumptions on the results. The method we have presented enables us to answer the following question: Is it reasonable to embark on a cell relocation study? Indeed, if a firm has got at its disposal production m a n a g e m e n t software, it is inexpensive to extract manufacturing routings and to classify them according to various points of view, with the help of a software's package, and then analyse the solutions obtained with the method presented in this paper.

246 REFERENCES

1. BALLAKUR A., STEUDEL H. J. " A within-cell utilisation based heuristic for designing cellular manufacturing systems"- Int. J. Res. 1987, Vol. 25, n ~ 5, 639-665. 2. BARTH M. and MUTEL B., "Data for management layout reorganisation A systemic approach", 8th international Conference on CAD/CAM, Robotics and Factories of the Future, Metz, France, August, 17-19, 1992, pp. 670-680. 3. BARTH M., DE GUIO R., MUTEL B., "An Help for Solving Dilemmas Encountered in Flow Analysis"- IEEE Computer Soc. Press, ISBN 0-8186-40308193,1993, 46-51. 4. BARTH M., "Methodologic contribution to workshop reorganisation" - PhD dissertation University of Nancy 1. F-54000 Nancy France- december1991. 5. BEDECARRAX C., "Quadri-d~composition en analyse relationnelle et applications ~ la s~riation" - IBM Paris Scientific Center Technical Report Fl17 1987. 6. BURBIDGE J. L., "Production flow analysis" - Production Ingeneering, 1975, 742-752 7. DE GUIO R., BARTH M., OSTROSI E., MUTEL B., "Workshop reorganisation and computer aided support for PFA" - Proc. 4 th International Congress of Industrial Engineering, IUSPIM]University of Aix-Marseille 3 FRANCE, December 1993, tome 2,. 269-275. 8. DE GUIO R., "Contribution to workshop b r e a k u p " - Phd dissertation University Louis Pasteur F-67084 Strasbourg France 1991 9. DE GUIO R., MUTEL B., "A general approach of the part workcenter grouping problem for the design of cellullar manufacturing problem", - In advances in factory of the future, CIM and Robotics, Elsevier ISBN 0444 898 565, 1992. 10. DE WITTE J., "The use of similarity in production flow analysis" - Int. J. Prod. Res.,Vol. 18, NO. 4, 1980, 503-514. 11. HALL R. W., "Attaining manufacturing excellence", Dow Jones-Irwin, 1987, p. 128, ISBN 0-87094-925-X 12. KAPARTHI S., SURESH N. C., "Workcenter-component cell formation in group technology : a neural network approach"- International Journal of Production Research 1992, Vol. 30, N ~ 6, 1353-1367. 13. KUSIAK A., CHUNG Y., "GT/ART : using neural networks to form workcenter cells"- Manufacturing review, Vol. 4, n~ December 1991. 14. KUSIAK A., "EXGT-S : A knowledge base system for group technology" Int. J. Prod. Res., Vol. 26 NO.5, 1988, 887-904. 15. KUSIAK A.; CHOW W.S, Efficient Solving of the Group Technology Problem, Journal of Manufacturing Systems Volume 6/NO.2,117-124. 16. MARCOTORCHINO F., "A unified approach of the block seriation problem" - Journal of applied Stochastic Models and Data Analysis,Vol 3, NO. 2 J. Wiley 1987. 17. Mc AULEY, "Workcenter grouping for efficient production" - The production Ingeneer 1972, 53-57. 18. MOON Y.B., "Forming part-workcenter families for cellular manufacturing : a neural-network approach"- International Journal of Advanced Manufacturing Technology 1990 5 : 278-291.

247 19. MUTEL B., BOUZID L., and DE GUIO R.,"Application of conceptual learning techniques to generalized group technology" Applied Artificiel InteUigence:6:, 1992,443-458 20. O'REGAN N.,"Seriation and quasi-seriation", Master's dissertation in mathematics, University of Strasbourg I, France, 1994. 21. RAJAGOPALAN R., "An ideal seed non hierarchical clustering algorithm for cellular manufacturing" - Int. J. Prod. Res., 1986, Vol. 24, n~ 451-464. 22. RAJAGOPALAN R., "An ideal seed non hierarchical clustering algorithm for cellular manufacturing" - Int. J. Prod. Res., Vol. 24, NO.2, 1986, 451-464. 23. RAJAGOPALAN R., "Design of cellular production systems. A graph theoretic approach"- Int. J. Prod. Res, Vol. 13, NO.6, 1975, 567-579. 24. SAFIR is a product of the Laboratoire de Recherche en Productique de S t r a s b o u r g - E N S A I S , 24 bld. de la Victoire F-67084 S t r a s b o u r g - Cedex FRANCE. 25. VANELLI A. and KUMAR K.R., "A method for finding minimal bottle neck cells for grouping part workcenters families", Int. Prod. Res., 1986, Vol. 24, n~ 387-401.

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PART THREE

ARTIFICIAL INTELLIGENCE AND C O M P U T E R TOOLS

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Planning, Design, and Analysis of Cellular Manufacturing Systems A.K. Kamrani, H.R. Parsaei and D.H. Liles (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

251

Adaptive Clustering Algorithm for Group Technology: An Application of the Fuzzy ART Neural Network Soheyla K a m a l 5530 H e a t h e r Ln Orefield, PA 18069

FACT (Fuzzy art with Add Clustering Technique) is a new clustering algorithm based on the neural network techniques and fuzzy logic concept. In this paper, structure and application of this algorithm with respect to the group technology (GT) problem is presented. Characteristics and abilities of the algorithm are shown through several examples. To evaluate the quality of the clustering results, two sets of performance measures are considered. The first set evaluates performance of the clustering results independent of the group technology application. The second set includes three group technology dependent measures. These measures are used for evaluating performance of the FACT algorithm with respect to several GT clustering algorithms. A comparison of the results of the FACT algorithm with several other GT clustering algorithms published in the literature has shown that FACT's results dominate other algorithms.

1.0 Introduction Clustering based algorithms are the most common methods for solving the group technology problem in the cellular manufacturing environment. Production flow analysis is the technique that provides required information for these clustering algorithms [1]. Ideally, each part family will map to a unique machine cell, and the entire family need not ever leave the cell in order to complete all necessary processing. Practically, this may be either impossible or computationally infeasible to

252 achieve. In most cases, the actual goal changes to satisfying objectives such as minimizing the number of inter-cell moves, minimizing the number of cells, maximizing utilization of machines, minimizing duplication of machines in different cells, maximizing the percentage of operations of a part processed within a single cell, etc. Cluster analysis approaches group objects (parts or machines) into homogeneous dusters (groups) based on object features. The existing clustering approaches to the group technology problem can be classified as (1) matrix based methods [2, 3, 4, 5, 6, 7, 8]; (2) mathematical programming algorithms [9, 10, 11, 12, 13]; (3) graph theory based methods [14, 15, 16]; (4) pattern recognition techniques [17, 18, 19, 20, 21]; (5) fuzzy logic approaches22, 23]; (6) expert system based methods [1]; (7) neural network based methods [24, 25, 26, 27, 28, 29]. For a complete review of literature interested readers are referred to Kamal [30]. All the above methods, with the exception of neural network based methods, are serial algorithms requiring significant time for processing. Moreover, these methods also require storage and manipulation of large matrices and virtually always focus on binary attributes. Even though neural networks are, at present, usually simulated via serial algorithms, they still require significantly less storage and processing time than conventional approaches. The next section presents information required by a GT clustering algorithm. In section 3, the fuzzy ART neural network which is used in the present algorithm is introduced. The structure of the FACT algorithm is presented in section 4. The performance measures used to evaluate goodness of the clustering results are discussed in section 5. An example from the literature illustrates the technique in section 6. In section 7, performance of the FACT algorithm is compared with several GT clustering algorithms. Finally, conclusions are presented.

2.0 Clustering Based Approaches to Group Technology The information required by a GT clustering algorithm is usually provided in the form of a part-machine incidence matrix A. If the information is represented in binary form, element [Aij] indicates whether part i requires machine j (Aij=l)or not (Aij=0). Alternatively, the transpose of the matrix, the

253 machine-part matrix A, gives element [A~] which is I if machine j is required by part i; 0 otherwise. Table 1 presents an incidence matrix based on the parts process routes.

Table 1- Clustering parts based on their process routes

Machines --)

.

.

.

.

.

.

.

.

Parts $ 1 2 3 .

.

.

.

.

.

.

.

.

.

.

.

.

.

1

.

.

.

.

.

.

.

1 1 0 .

.

2

.

.

.

.

.

.

.

3

1 1 1

.

.

.

.

.

.

.

.

.

.

4

0 0 1

.

.

.

.

.

.

.

.

.

.

5

1 1 0 .

.

.

.

.

.

.

.

.

.

0 1 1 .

.

.

.

.

.

.

.

.

1 1 1

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Information related to production volume, demand for a part on each machine, and processing time of a part on each machine can be considered in clustering parts and machines. To consider these information entries to the incidence matrix should be continuous values. Table 2 presents a continuous incidence matrix. Entries in this matrix are processing time of each part on each machine.

Table 2- Clustering parts based on their process routes and process times | , , ,

Machines --} .......

.

.

.

.

.

.

.

.

Parts $ 1 2 3 .

.

.

~

~

. . . . . . . . . .

1 0.5 0.2 0

_ .

.

2

.

.

.

3

7 0.3 7 .

.

.

.

4

0 0 0.3 ,

~

.

.

.

5

0.2 9 0 .

.

.

.

.

0 4 0.1 ~.

6 8 0.1 6

.

__~._~

3.0 N e u r a l Networks and the GT Problem Neural networks are excellent at some of the things that biological systems do, such as pattern classification [31]. Unsupervised neural networks are applied to pattern classification and clustering problems. Group technology problem can be viewed as a clustering problem. One of the well-known families of the unsupervised neural networks is the ART (adaptive resonance theory) family [32, 33, 34]. ART neural networks have been used for several clustering problems

254 [35]. The architecture of our clustering algorithm is based on fuzzy ART which is the latest development in this family of neural networks.

3.1 Fuzzy ART Fuzzy Art is an unsupervised category learning and pattern recognition network [36]. It incorporates computations from fuzzy set theory [37] into the Adaptive Resonance Theory (ART) based neural network. Fuzzy Art is capable of rapid, stable clustering of analog or binary input patterns. This network consists of two layers, the input (F1) and the output (F2) layers. The number of possible categories (output nodes) can be chosen arbitrarily large. At first each category is said to be uncommitted. A category becomes committed after being selected to code an input pattern. Each input I is presented by an M-dimensional vector I=(II,I2,...,IM), where each component I i is in the interval [0,1]. One set of weight v e c t o r s Wj'-(Wjl,Wj2,...,WjM) are used to represent each output category j. Initially Wjl=Wj2. . . . . WjM=I, for all j. To categorize input patterns, the output nodes receive net input in the form of a choice function, Tj. The following choice function is used

I I^ Wjl Tj=a+ IWj I

(1)

where ^ is the fuzzy MIN operator (Zadeh 1965), defined as

(X^Y)i-min (xi,Yi)

(2)

and the norm I ~ I is defined by M

IXI- ~xi i=1

(3)

A Category (output node) with the highest value of Tj becomes nominated to claim the incoming pattern where

255 (4)

Tj= max{Tj 9j=I...N} To accept the nomination of the category the match function should exceed the vigilance parameter; i.e.,

IIAWjl III >p

(5)

In the fast learning mode, if the first nominated category does not pass the similarity test, an uncommitted node should be committed to the input pattern. The weight vector of the winner category is updated as follow: Wj(new) =

13(IAWj(~

(~

(6)

Fuzzy ART has three independent parameters: 1) The choice parameter 0~ > 0, which Carpenter et al. suggested to be close to zero, affects the search procedure. The choice parameter controls the choosing of a category whose weight vector Wj is the largest coded subset of the input vector I (if such a category exists). The following example shows this property. Consider a two dimensional input pattern I=(.8, 1). Assume there are two categories W1=(.1, .2) and W2=(.4, .8) whose weight vectors are subsets of the input pattern. If we consider c~=0, the values of the choice function for two categories will be equal as follows: T1=(.1 + .2)/(.3)=1 T2=(.4 + .8) / (1.2)=1. Here, there is a tie between categories 1 and 2. Since T1 is visited first, it will be chosen even though it is not the best choice. Considering a small value of (z such as c~=.01, will change this condition so that: T1=(.1 + .2)/(.01 + .3)=.9677 T2=(.4 + .8)/(.01 + 1.2)=.9917.

256 Here, the second category which is more similar to the input will be chosen.

2) the learning parameter ~e[0,1], which defines the degree to which the weight vector Wj is updated (recoded) with respect to an input vector claimed by node J; and 3) the vigilance parameter pc[0,1], which defines the required level of similarity of patterns within clusters.In the fast learning mode, Carpenter et al (1991) suggest 13=1.Fuzzy A R T learns in fast commit-slow recode mode, which is 13=I for the first time committing an uncommitted node (fast learning/commitment), and 132 (#2)- > 0 (#2)- >I (#1)- >4 (#1)- > 0 (#1)- >I (#2)- >7 (#2)- >9 (#2)- > 0 (#2)

part 6 I (#2) -> 6 (#2) -> 10 (#2) -> 7 (#2) -> 0 (#2) -> I (#1) -> 8 (#1) -> 0 (#1) -> I (#2) -> 9 (#2) -> 0 (#2)

Figure 15 Cell and Shop Layouts for 2 Cell Partition

MACHINE

MACHINE

MACHINE

Figure 16 Travel Chart for Intercell Flows in 3-cell Partition

338

INTERCELL FLOWS

W W 00

339 CELL 1

CELL2

CELL 3

16]

I7 !

1,11

/

L

I ,o I

I i, I

~

.'0

!7 I

I'

'1

171

11 [

7r~

|

Fig 17(a)

Flow Path for Part # 10 in the 3 Cell Partition

I (#1)- >4 (#1)- >O (#1)- >I ( # 2)- >7 (# 2)- >O (#2)- >I (#1)- >4 (#1)- >8 (#1)- >O (#1)

|

340 CELL 1

CELL 2

CELL 3

[11[

171

i

161 121141 151

141

131

181

Ill |

191 |

F i g 17(b)

171

11~

6I

I ~~ i

Flow Path for Part #15 in 3 Cell partition

I (#2). >1 (#2) - > 0 (#2) -> I (#3) - >7 (#3)- >11 (#3) - > 0 (#3) - >I (#2)- >10 (#2) - > 0 (#2)- >I (#3) - >11 (#3) - >12 (#3) - > 0 (i

341

CELL 1

CELL 2

161

CELL 3

lTf

121 I /

!111

Ilol lloj

I111

!1ii

171

ilol

/~

112j

@ \

|

/

Fig 17(c) F l o w p a t h for P a r t #2 in 3 Cell p a r t i t o n (#1)- >1 (#1) - > 4 (#1) -> 0 (#1) -> I (#2) -> 7 (# 2)- >O (#2)- >I (#1)- >4 (#1)- >8 (#1)- >0 (#1) -> I (# 2)- >10 (#2) -> O (#2)

342 CELL 1

CELL 2

CELL 3

16i

17 I

I"1

-/14 t

I, ~1,o i

!" I

I,, I

i 7I

1,01

171

1,21

@

F i g 17(d)

|

F l o w P a t h f o r P a r t #3 in 3 Cell p a r t i t i o n

I (#1)- >1 (#1) - >2 (#1) - >4 (#1)- >O (#1) - >I (#2)- >7 (# 2)- >O (#2)- >I (#1)- >8 (#1)- >9 (# 1) - >o(#1)

343 CELL 1

CELL 2

lol 121 I5 I

CELL 3

171 141

.~

171xll01

,6 i lO,

i"i I I'l

I"l

171

I,ol

171

1,21

@

|

Fig 17(e) Flow Path for P a r t #4 in 3 Cell partition I (#2)- >1 (#2)- >O (#2)- >I (#1)- >4 (#1)- >O (#1)- >I (#2)- >7 (#2)- >9 (#2)- >O (#2)

344

CELL 2

CELL 1

CELL 3

l lll

161

121

141

151

141

13t

I,I

171 lloi

D

i,ll

I I, I

171

l,ol

171

1121

|

|

F i g 17(t") F l o w p a t h f o r P a r t # 6 i n 3 C e l l p a r t i t i o n I (#2)- >6 (#2) - >10 (#2) - >7 (#2) - > 0 (# 2) - >I (#1)- >8 (#1)- > 0 (#1) - >I (#2) - >9 (#2) - > 0 (#2)

345

I-o

1

2

3

4

5

6

FROM 1

-

-

-

1 2 3

2 3 4

568 483

.

-

172 .

-

-

2 0

.

.

.

6

576

.

.

-

-

12

54 -

.

-

-

-

380

-

.

-

-

F i g u r e 18(a) A s s y m m e t r i c T r a v e l C h a r t f o r I n t e r c e l l F l o w s

To FROM 1 2 3

1

2

3

4

5

6

-

568 -

483 -

123 172 20

12 -

630 -

.

5

.

.

.

.

. .

.

.

. .

380 .

.

F i g u r e 18(b) S y m m e t r i c T r a v e l C h a r t f o r I n t e r c e l l F l o w s

346

, ,,,,

(~)i .,~~~,Q ,,,oo 1

'ok [ 1

12

Fig 18(c) Maximum Spanning Tree generated from the Symmetric Travel Chart

AISLE

/

/

J

Figl8(d) Shop Layout generated by manipulation of the Maximum Spanning Tree

APPENDIX

348

S T O R M EDITOR : F A C I L I T Y L A Y O U T M O D U L E Title: Permutation of Machines Number of departments 912 Number of departments down "1 Department height 91 Successful evaluations : 66

R15 :C6 FIXED LOC USER SOLN FLOW 1 FLOW 2 FLOW 3 FLOW 4 FLOW 5 FLOW 6 FLOW 7 FLOW 8 FLOW 9 FLOW10 FLOWl 1

M1

M2

Distance(EUCL/RECT) :RECT Department width 91 Symmetric Matrices 9BOTH

M3

M4

33.

36. 25. 25.

13.

M5

M6

33. 100. 25.

9. 13. 29. 25. 29.

349 FACILITY LAYOUT :PROCESS 1. Edit the current data set 2. Save the current data set 3. Print the current data set 4. Execute the module with the current data set Select Option 4

I

FACILITY LAYOUT :ITERATION 1 1) Go to final solution 2) Go to next solution 3) Draw layout 4) Department interaction values 5) Objective function by departments I

Select Option 1

I

FACILITY LAYOUT : POST SOLUTION ANALYSIS 1) Draw Layout 2) Department interaction values 3) Objective function by departments 4) Generate a random initial solution and resolve 5) Perform user defined exchanges 6) Restart solution process from current solution 7) Save best solution found as user solution Select Option 1

I

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Planning, Design, and Analysis of Cellular Manufacturing Systems A.K. Kamrani, H.R. Parsaei and D.H. Liles (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

351

A Simulation A p p r o a c h for Cellular Manufacturing S y s t e m D e s i g n and A n a l y s i s Ali K. Kamrani a, Hamid R. Parsaei b, and Herman R. Leep b aDepartment of Industrial & Manufacturing Systems Engineering, The University of MichiganDearborn, Dearborn, Michigan, 48128-1491, USA bDepartment of Industrial Engineering, University of Louisville, Louisville, Kentucky 40292, USA

1. ABSTRACT Often, real world decisions require some degree of human judgment. These decisions may require a set of tools that can assist the decision maker. Simulation modeling, MRP, MRPII, decision trees, and linear programming are some examples of the types of tools used for Decision Support Systems. This chapter presents the application of linear programming to develop a methodology that uses design and manufacturing attributes to form machining cells. The methodology is implemented in four phases. In Phase I, parts are coded based on the proposed coding system. In Phase II, parts are grouped into families based on their design and manufacturing dissimilarities. In Phase III, the optimum number of resources (e.g., machines, tools, and fixtures) are determined and grouped into manufacturing cells based on relevant operational costs and the various cells are assigned part families. Finally, in Phase IV, a simulation model of the proposed system is built and analyzed. This model is executed so that data from the proposed system may be gathered and evaluated to justify the feasibility of the system by introducing real-world scenarios such as breakdown, maintenance, and on-off shifts. The developed mathematical and simulation models are used to solve a sample production problem. The results from these models are compared, and are used to justify the final design of the cell. By using these modeling techniques and tools, cellular manufacturing systems can be designed, analyzed, and finally optimized. 2. GROUP TECHNOLOGY The philosophy of group technology (GT) is an important concept in the design of flexible manufacturing systems and manufacturing cells. Group technology is a manufacturing philosophy that identifies similar parts and groups them into families. In addition to assigning unique codes to these parts, the group technology developers use the part similarities during the design and manufacturing phases. GT is not the answer to all manufacturing problems, but it is a good management technique with which to standardize efforts and eliminate duplication.

352 Group technology classifies parts by assigning them to different families based on their similarities in the following areas: design attributes (physical shape and size) and/or manufacturing attributes (processing sequence). Methods that are available for solving the GT problems in manufacturing can be classified into the following categories: classification, production-flow analysis, and cluster analysis. Classification and cluster analysis methods are the two most widely used.

2.1. Classification Classification is defined as a process of grouping parts into families based on some set of principles. This approach can be further categorized into the visual method (ocular) and coding procedure. Grouping based on the ocular method is a process of identifying part families by visually inspecting parts and assigning them to families and the production cells to which they belong. This approach is limited to parts with large physical geometries and it is not an optimal approach because it lacks accuracy and sophistication. The coding method of grouping is considered to be the most powerful and reliable method. In this method, each part is inspected individually by means of its design and processing features. Coding can be defined as a process of tagging parts with a set of symbols that reflect the part's characteristics. A part's code can consist of a numerical, alphabetical, or alpha-numerical string. Three types of coding structures exist. These structures are described below. 9 Hierarehial (Monocode) Structure: In this structure, each digit is a further expansion of the previous digit. This method requires that the meaning of a digit is dependent on the meaning of the previous digit in the code's string. The advantage of this method is due to the amount of information that the code can represent in a relatively small number of digits. However, a coding system based on this structure is complicated and very difficult to implement. 9 Chain (Attribute or Polycode) Structure: In this structure, the meaning of each digit is independent of any other digit within the code string. In this approach, each attribute of a part is tagged with a specific position in the code. This structure is simple to implement, but a large number of digits may be required to represent the characteristics of a part. 9 Hybrid Structure: Most of the coding systems available are implemented using the hybrid structure. A hybrid coding system is a combination of both the monocode and polycode structures, taking advantage of the best characteristics of the two previously described structures.

2.2. Clustering The clustering process involves the grouping of similar objects. This approach has been practiced for many years. This method requires the calculation of a clustering factor known as the similarity or dissimilarity coefficient by using a clustering criterion as an objective in order to optimize the system performance. Similarity and dissimilarity coefficients are calculated values that represent the relationships between parts. Most research in this area has been based n the fact that these coefficients range from 0 to 1. This condition indicates that the dissimilarity

353

coefficient = 1.0 - similarity coefficient or vice versa. Clustering methods for the grouping of parts and the design of manufacturing cells have gained the attention of both researchers and practitioners. Both hierarchical clustering and nonhierarchical clustering methods are available. The hierarchial method results in a graph known as dendrogram, which illustrates the data grouped into smaller clusters based on their similarity or dissimilarity measures. The hierarchial method is accomplished in two forms, agglomerative and divisive. In the agglomerative hierarchial approach, the procedure begins with m objects that are to be classified. At each step, the two most similar objects are merged into one single cluster. After m-1 steps, all objects belong to one large cluster. Many such methods, differing in criteria, are used to decide which individual elements or clusters should be merged together and how the similarity between a newly obtained cluster and other clusters or objects is defined. In the hierarchial method, the structure of the set of objects can also be obtained by dividing the set into two or more subsets and continuing the division until all objects have been completely separated. This hierarchial clustering method is known as the divisive method. The divisive method has been studied and used much less than the agglomerative procedures. The nonhierarchial method uses partitioning clustering algorithms to search for a division of a set of objects into a number of clusters, k, in such a way that the elements of the same cluster are close to each other and the different clusters are well separated. Because the k clusters are generated simultaneously, the resulting classification is nonhierarchial. The number of clusters can be either given or determined by an optimization algorithm. When k is unknown, several iterations of the algorithm can provide several values of k. In this way, it is possible to analyze the performance of the developed clusters and select the optimal one. 3. CELLULAR MANUFACTURING The main objective of designing manufacturing cells is to develop a production environment of machining centers, either as a line or in cells, operated manually or automatically for the production of part families that are grouped according to a number of similarities in their design and manufacturing features. This type of manufacturing is known as cellular manufacturing and is used for manufacturing a product in batches. Cellular manufacturing or group production will foster an environment where the cost effectiveness of mass production and the flexibility of job-shop production can be achieved for a batch production environment. This approach will require the analysis of parts and the selection of part families where the part's attributes are similar enough to allow processing with a minimum number of or no change overs. When families are created and the lot sizes are acceptable, the layout of the cell can be establish and the production of part families can start. The advantages derived from using cellular manufacturing include the reduction of work in process (WIP), improved quality, better utilization of high-investment machines, and reduction of scrapped parts. A number of methodologies have been developed and are available for grouping parts and developing machines cells. The next section will introduce a new design methodology using a customized coding system for the design of flexible manufacturing cells.

354 4. A N E W D E S I G N M E T H O D O L O G Y

FOR MACHINE CELLS FORMATION

A new methodology is proposed and implemented. The design is described below: 4.1. C o d i n g System and its Structure A new coding system is proposed for part code assignment. The required information for this coding system can be easily retrieved from the firm's design and manufacturing databases. This system consists of 18 digits and is based on the hybrid structure. The attributes and components used for this coding structure are as follows:

Attribute 1: General Shape of the Part 9Rotational 9 (CAI-1) 9 (CA1-2) 9 (CA1-3) 9Nonrotational 9 (CA1-4) 9 (CA1-5) 9 (CA1-6) 9 (CA1-7)

R-Bar R-Tube R-Hexagonal Bar NR-Plate NR-Square Bar NR-Sheet Plate NR-Rectangular Bar

Attribute 2: Material 9 (CA2-1) 9 (CA2-2) 9 (CA2-3) 9 (CA2-4) 9 (CA2-5)

Aluminum Alloys Copper-Zinc Alloys Steels Cast Irons Plastics

Attribute 3: Maximum Diameter 9 (CA3-1) 9 (CA3-2) 9 (CA3-3) 9 (CA3-4) 9 (CA3-5)

D ~ 0.75 in. 0.75 < D ~ 1.50 in. 1.50 < D ~ 4.00 in. D > 4.00 in. N/A (Nonrotational Part)

Attribute 4." Overall Length 9 (CA4-1) 9 (CA4-2) 9 (CA4-3) 9 (CA4-4)

L ~ 6 in. 6 < L ~ 18 in. 18 < L ~ 60 in. L > 60 in.

Attribute 5: Diameter of Inside Hole 9 (CA5-1)d ~ 0.5 in. 9 (CA5-2) 0.5 < d ~ 1.0 in. 9 (CA5-3) 1.0 < d ~ 5.0 in. 9 (CA5-4) d >5.0 in.

355 9 (CA5-5) No Hole Attribute 6. Product Type 9 (CA6-1) Commercial 9 (CA6-2) Electrical 9 (CA6-3) Industrial 9 (CA6-4) Military 9 (CA6-5) Special 9 (CA6-6) Other Attribute 7. Number of Processing Steps Attribute 8. Processing Type and Sequence 9 ( C A 8 - 1 ) Turning 9 (CA8-2) Drilling 9 (CA8-3) Reaming 9 (CA8-4) Boring 9 (CA8-5) Tapping 9 (CA8-6) Milling 9 (CA8-7) Grinding 9 (CA8-8) Broaching 9 (CA8-9) Sawing Attribute 9. Minimum Number of Machines Required for Processing Attribute 10. Processing Machine Type 9 (CA10-1) CNC Turning 9 (CA10-2) CNC Drilling/Tapping 9 (CA10-3) Vertical/Horizontal CNC Milling 9 (CA10-4) External/Internal Grinding 9 (CA10-5) Broaching 9 (CA10-6) Band/Circular Sawing Attribute 11: Number of Tool Types Attribute 12. Tool Type and Sequence 9 ( C A 1 2 - 1 ) Insert 9 (CA12-2) Twist Drill 9 (CA12-3) Adjustable Reamer 9 (CA12-4) Adjustable Boring Bar 9 (CA12-5) Tap 9 (CA12-6) Milling Cutter 9 (CA12-7) Grinding Wheel 9 (CA12-8) Broach 9 (CA12-9) Band/Circular Saw Blade

356

Attribute 13."Number of Fixture~Jig Types Attribute 14: Fixture~Jig Type 9 (CA14-1) Special Fixture 9 (CA14-2) Multipurpose Fixture 9 (CA14-3) Adjustable Fixture 9 (CA14-4) Rotational Adjustable Fixture 9 (CA14-5) Nonrotational Adjustable Fixture 9 (CA14-6) Special Jig 9 (CA14-7) Multipurpose Jig 9 (CA14-8) Adjustable Jig Attribute 15: Number of End Operations Attribute 16: End Operation Type and Sequence 9 (CA16-1) Clean 9 (CA16-2) Polish 9 (CA16-3) Buff 9 (CA16-4) Coat 9 (CA16-5) Paint 9 (CA16-6) Assemble 9 (CA16-7) Pack 9 (CA16-8) Inspect Attribute 17: Minimum Number of Devices Requiredfor End Operations Attribute 18: Devices Usedfor End Operations 9 (CA18-1) Dip Tank 9 (CA18-2) Disk Grinder 9 (CA18-3) Process Robot 9 (CA18-4) Material Handling Robot 9 (CA18-5) Painting Robot 9 (CA18-6) Assembly Robot 9 (CA18-7) Packaging Machine 9 (CA18-8) Vision System 4.2. Part-Family Formation By examining the structure of the proposed coding system, four types of variables (binary, nominal, ordinal, and continuous) are identified. The linear disagreement index between parts i and j for attribute k, which is either a binary or nominal variable type, is measured by the following: 1, if Rik * Rjk dijk =

{

(1) 0, otherwise

357 where

disagreement index between parts i and j for attribute k Rik- rank of part i for attribute k Rjk= rank of part j for attribute k. dijk =

The linear disagreement index for an ordinal variable is determined by the following equation: dijk = [Rik--Rjk where

m m-1

I/(m-I)

(2)

= number of classes for attribute k = maximum rank difference between parts i and j.

The linear disagreement index for a continuous variable is determined by the following equation: dijk = I Rik-Rjk

I/x

(3)

where Xk is the range of values for the variable. The linear disagreement index for Attributes 1,2, and 6 is calculated using Eq. (1) because the general shape of part is considered to be a binary variable and the material and product types are considered to be nominal variables. The linear disagreement index for Attributes 3, 4, and 5 is calculated using Eq. (2) because there is class associated with these variables and, therefore, they are considered to be ordinal variables. The linear disagreement index for the processing and end-operation sequences is calculated using McAuly's equation as follows: dij = 1- ~ (qio * qjo)/oX (qio + qjo- qio * qjo)

(4)

where 1, Part i requires operation o

qio = { 0, otherwise. The linear disagreement index for tools and fixtures can be calculated by the following: dij = (NTi + NTj- 2NTij) / (NT~ + NTj)

(5)

where NTj = number of tools required by part i NT~j= number of tools common to both parts i and j. The linear disagreement index for processing machines and end-operation devices is calculated using the Hamming Metric as follows:

358 dij = Z ~(Xim, m

Xjm)

(6)

1, if part i is made on machine m where

Xi m = {

0, otherwise 1, ifXim ~ Xjm and

~(Xim , Xjm )

-- {

0, otherwise. After the evaluation of these parameters, the analyst can assign weights to represent his or her subjective evaluation of the variables and group parts based on their assigned priority. These weights can be categorized as critical (1.00), very important (0.75), important (0.50), and not important (0.25). Finally, the weighted dissimilarity measure (DISij) between parts i and j can be determined by the following: DlSij = ]~(Wk * dijk)/E Wk k

k

(7)

where DISij Wk

dijk

= weighted dissimilarity coefficient between parts i and j = weight assigned to attribute k = disagreement index between parts i and j for attribute k.

In order to select the part families, the K-Medium formulation for nonhierarchical clustering is used to minimize the total sum of dissimilarities as follows: Minimize ~I~I DISij * xij for all i,j = 1,2 ..... p ij

subject to x 0 = 1 for all i = 1,2,...,p j =1.....p

~

xjj = K

j=l .....p

x~j ~ xjj for all i,j = 1,2,...,p where P k DISij

= number of parts = required number of part families = dissimilarity measure between parts i and j, DISij = DISji

(8)

359 1, if part i is a member of group j and

Xij --- { 0, otherwise.

4.3. Machine Cell Formation In this section of the methodology, the main objectives are to determine the number of machines, tools, and cells for the assignment of the proper families to the proper cells. This phase is more of an economic issue rather than one of design. The methodology takes into account several relevant operating costs and intends to minimize the total sum of these costs. This study uses several costs, including the following:

9 Machine investment cost 9 Tool investment cost 9 Fixture investment cost 9 Machine production cost 9 Inspection cost 9 Setup cost 9 Intracell material handling cost A mixed-integer mathematical model for the overall operating cost was developed and used in this part of the methodology. A number of assumptions were made in order to develop this mathematical model. These assumptions include the following: 9 Operating time for each part is known. 9 Demand for each part is known. 9 Machine types can be placed in any selected cell. 9 Each part has a fixed routing. 9 No intercellular material handling is allowed. 9 Inspection time for each part is known, and each part is inspected after being processed on a machine type. The mathematical model developed for this stage is as follows: Index sets

o: Operation p: Part m: Machine type t: Tool type f: Fixture type g: Part Group c: Cell, C ~ G

o = 1,2 .... ,O p - 1,2,...,P m = 1,2,...,M t - 1,2 .... ,T f - 1,2,...,F g = 1,2,...,G c - 1,2,...,C

360

Machine investment cost MIC = Illl CM m * NMmc

(9)

me

where C M m = annual investment and maintenance costs for machine type m NMme = number of machine types m assigned to cell c.

Tool investment cost TIC = II II II C T t * NTtme* O.'tm

(10)

tmc

where CTt = annual investment and maintenance costs for tool type t NTtmc = number of tool types t assigned to machine type m in cell c 1, if tool type t is required by machine type m

,

Otm

--

{ O, otherwise.

Fixture investment cost FIC = II II ~ C F f * NFfme* Ofm

(11)

free

where C F f = annual investment and maintenance costs for fixture type f NFfmc = number of fixture types f assigned to machine type m in cell c

1, if fixture type f is required by machine type m Ofm = { 0, otherwise.

Machine production cost In order to calculate the machine production cost, the processing time required for each part type should be determined on all machine types required for its production. This time is dependent on the type of operation and its processing time, and the machine performance rates for that particular operation, because a process can be performed by more than one machine type. Total demand of a particular part, the total machining time required to meet the annual demand of the part is calculated as:

361 TMDpm = Y, PRFom * TOop * dp * ~omp

for all m,p

(12)

o

where TMDpm = total time required to meet annual demand of part p on machine type m PRFom = machine performance rate for operation o on machine type m TOop = time to complete operation o on part p dp = total number of parts p demanded annually

~omp

-{

1, if operation o of part p requires machine type m

O, otherwise Because it is assumed that each part is inspected after each operation, the cost of rework and machine reliability should be included in the equation. This machine production cost (MPC) is calculated by the following expression: MPC =~m~ [CPpm * Rm + C R W m * (1- Rm)] * Bpm * TMDpm * Y~

(13)

where CPpm = cost of processing part p on machine type m

Rm = machine type m reliability C R W m = rework cost on machine type m

1, if part p is processed on machine type m Bpm = {

O, otherwise 1, if part p is assigned to cell c Yp~= { 0, otherwise.

Inspection cost IC =II II II Clpm * Tlpm * dp * ~omp * NMmr pmo

(14)

where CIpm = inspection cost of part p after being processed on machine type m TIpm= inspection time required to inspect part p after being processed on machine type m dp -- total number of parts p demanded annually

362 1, if operation o of parts p requires machine type m ~l,omp

"-- {

0, otherwise NMmr = number of machine types m is assigned to cell c. Setup cost SC=Yl, ~ ~ TSmgc * CSmg c * iI'mg * Xg c rngc

(15)

where TSmg e = average setup time for machine type m for group g in cell c CSmgc = average setup cost for machine type m for group g in cell c

rng

{

1, if machine type m is used for group g production

O, otherwise 1, if group g is assigned to cell c Xg c

--- {

O, otherwise. Intracell material handlin~ cost

IMHC = Y,~ ( N M V p - 1 ) * dp * CH~ * Yp~

(16)

pc

where

N M V p = number of moves for part p dp = number of parts p demanded annually CHp~ = average material handling cost of part p in cell c

1, if part p is assigned to cell c Yp~={ O, otherwise. The objective of this mathematical model is to minimize the total sum of the costs described above. The constraints for this mathematical model are formulated as follows. Budget v

The following constraints will restrict the amount of capital expenditure to the annual budgets for machines, tools, inspection, and material handling.

363 X] X] C M m * N M m c < B M

(17)

YI.X] C T t * NTtm c * Otm ' ~' B T tmc

(18)

me

Xl~Z CFf * fmc

NFfmc *

(19)

Otm < B F

(20)

X]]~Z CIpm * YIpm * dp * ~tomp * NMme ~ BI pmo X]X ( N M V p - 1 ) * dp * CHpe * pc

Ypr ~ B M H

(21)

where BM BT BI BMH

= = -

budget available to purchase and maintain machines of all types budget available to purchase and maintain tools of all types budget available for inspection of all parts budget available for material handling of all parts in all cells.

Machine Capacity Equations for this constraint will ensure that the capacity of each machine type in each cell is not exceeded. If it is exceeded, the number of duplicate machines available will be calculated and proposed to meet the annual demand for parts. ~] gpm * TMDpm * Ypc ~ TPm * NMmc for all p

m,c

(22)

1, if part p is processed on machine type m where

Bpm

-- {

0, otherwise TMDpm

=

total time required to meet annual demand of part p on machine type m

Ype

-{

1, if part p is assigned to cell c

O, otherwise TP m = total annual processing time available on machine type m NMme = number of machine types m assigned to cell c.

Tool life Equations developed in this section will ensure that the tool life of each tool type on each machine assigned in each cell is not violated. If a cutting time exceeds the life of the tool in question, the number of duplicate tools available will be calculated.

364 P Bpm * B'tp * O'tm * TMDpm * Yp~ ~ TL t * NTtm c for all m,c,t

(23)

where 1, if part p is processed on machine type m Bpm -- { 0, otherwise TMDpm = total time required to meet annual demand of part p on machine type m 1, if tool type t is required by part p ~'tp: { 0, otherwise 1, if tool type t is required by machine type m

O'tm'-{ O, otherwise 1, if part p is assigned to cell c

0, otherwise. TL t = total life of tool type t NTtm~ = number of tool types t assigned to machine type m in cell c

Machine-Fixture Balance Since each process machine will require at least one fixture for its production this constraint will assure the minimum number of required fixture in case of machine duplications, NFfmc * o fm> NMmr

(24)

Cell Capacity In order to have a high degree of flexibility in each cell, a limit must be set for the total number of parts assigned to each cell. This constraint is formulated as follows: p

dp * Yp~ ~ ICe

for all c

(25)

365 where dp

=

total number of parts p demanded annually. 1, if part p is assigned to cell c

Ypc= { O, otherwise ICc= maximum number of parts allowed in cell c

Part-Group Assignment In order to reassure the assignment of each part family to only one cell and the assignment of the parts within these families to the same cell, the following constraints are proposed: l~ Xgc = 1

c

for all g

Igpg * Ypr Xg c for all p,g,c

(26)

(27)

where 1, if group g is assigned to cell c

Xg c -- { O, otherwise

1, if part p is a member of group g

IXpg= { O, otherwise.

1, if part p is assigned to cell c Yr~={ O, otherwise

Binary and integerality conditions of decision variables Xgc Ypr NMmc NTtmc

e (0,1) for all g,c e (0,1) for all p,c > 0 and integer for all m,c > 0 and integer for all t,m,c.

(28) (29) (30) 01)

366 NFfm~ 2 0 and integer for all f,m,c.

(32)

5. NUMERICAL EXAMPLE Production of 15 parts is required. These parts require 15 operations (nine process operations and six end operations). There are eight process machines and five end-operation devices available. A rating factor of 1 is assumed for all machines performing the process operations. This assumption indicates that the machine selected is the most suitable one for performing the process operations. Nine types of tools are available for the process operations. Using the proposed formulation in Stage II, and by setting K to be 4, parts and their associated families are as follows: G1(1,3,5), G2(2,4,6,8,11,14), G3(7,9,10,15), and G4(12,13). Table 1 lists the annual investment and maintenance costs associated with each machine and tool. It also contains the annual available machining time on each machine and the tool life associated with each tool. The annual demand for each part is given in Table 2. Machine reliability is illustrated in Table 3. The inspection time and cost are assumed to be similar for all parts, and all machines have the same rework cost. Setup cost and time are assumed to be similar for all machines. The intracellular material handling cost associated with parts is similar in all cells. Table 4 illustrates these values. The annual budgets for machines, tools, inspection, and material handling, and the upper limits for the number of parts in each cell, in order to maintain cell flexibility, are also illustrated in Table 4. The model is solved by using LINDO software and the results are listed in Tables 5, 6, 7, and 8. 6. THE SIMULATION MODEL The output from the mathematical model forms the numerical basis for the simulation model. Further assumptions for development of the simulation model are required. The incorporated assumptions in this model are as follows: Operational times (for both processing and end operations) are represented by exponentially distributed random variables. The cell produces part types in random sequence, but each part type is produced in proportion to its share of overall demand. The machine cell is operated for 20 hours out of every 24-hour period, with the remaining four hours being devoted to preventive maintenance, tool

367 Table 1 CMm, TPm, CMt, and TLt Values Machine Type

6 7 8 9 10 11 12 13

CMm

TPm (min)

$ 30,000 $ 42,000 $ 24,000 $ 35,000 $ 41,000 $ 22,000 $ 20,75O $ 30,00O $ 22,460 $ 27,000 $ 32,000 $ 26,000 $ 30,490

102,000 139,000 111,000 114,000 140,000 100,000 162,000 156,000 130,000 155,000 90,000 99,000 89,000

Tool

CMt

TLt (min)

$ 2538 $ 2576 $ 2526 $ 2436 $ 2562 $ 2334 $ 2454 $ 2154 $ 2244

490 444 430 427 488 413 412 414 442

Type

Table 2 Annual demands for various parts (dp) Table 3 Machine reliability (Rm)

Part Type 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15

1728 2000 2145 1729 1948 2263 2226 2236 2160 1777 1758 1824 2089 1929 2308

Machine Type 1

2 3 4 5 6 7 8 9 10 11 12 13

Rm 93 % 85% 94 % 89% 75% 85% 95 % 94 % 97 % 85% 82 % 85% 91%

368

TIpm = 0.5

min/part

CIpm = $ 1 . 5 0 / m i n / p a r t

CHp~ = $ 2.00 C R W m = $ 6.00 TSmgc = 15.0 m i n CSmg c -- $ 5.00

Table 6 Number of machine types and their assignments

BM = $1,500,000 BT = $15,000,000 BI = $ 750,000 BMH = $ 250,000 IC 1,2 = 12,000 parts I C 3 , 4 -- 10,000 parts

Machine Type

Table 4. Data for the Sample Problem Table 5 Cell Configuration Part Number

Family Number

1 3 5 12 13

1,4

2 4 6 8 11 14 7 9 10 15

Cell Number

1 2 3 8 9 10 11 12

5 8 9 10 11 12 13 1 3 4 8 9 10 11 12 13

Cell Number

Number of Duplicates

369 Table 8 Number of fixture types and their assignments

Table 7 Number of tool types and their assignments Tool

Machine Type

Cell Type

No. of Duplicate 176 194 404 91 263 137 83 49 196 189 136 763 51 71 78 318 121 117 73 227

Fixture Type

Machine Type

Cell No.

No. of Duplicate

370 reconditioning, setup, and other activities of this nature. The cell is operated six days per week. A transient period is incorporated into every simulation run. A duration of 400 simulation hours is estimated. This value was determined by observing the behavior of several random varieties and choosing the time at which these values reached steady state. The preprocess buffer capacities for processing machines are set at 100 parts. Queue capacities for inspection stations and end-operation machines are set at 50 parts. Parts are removed from queues for processing according to the first-in-first-out rule. Each processing operation has a corresponding inspection operation. These inspection operations will have durations that are exponentially distributed, each with a mean of 0.5 minute. If more than one operation is performed on a part as it is processed by a particular machine, then a corresponding number of inspections will be performed sequentially on that part at the inspection station. In this case, the inspection time will be distributed as the sum of the appropriate number of exponential functions, each with a mean of 0.5 minute. In other words, duration = Erlang(0.5 minute, n) where n = cumulative number of exponential distributions 0.5 min = mean of each exponential distribution. There are no inspections after end operations. The nature of the end operations being performed is such that they are completed successfully virtually 100% of the time. The probability of any part failing inspection is given by the reliability of the machine on which the part was processed. The probability of a part failing inspection after undergoing multiple processes on a single machine is given by 1 - R n, where R = machine reliability n = number of processing operations performed on a single processing machine. Parts failing inspection are returned to the proper processing machine for rework. Parts may be reworked twice before being scrapped. Rework operations are assumed to take the same amount of time as the original processing operation. As a result, the cost of the rework operation, in terms of time, tool wear, and machine operating cost, is exactly the same as the cost of producing a "raw" part. When parts are returned to the queues serving processing machines for rework, they will not be given priority, but will instead be placed at the end of the queue according to the first-in-first-out scheduling rule.

371 The simulation model does not attempt to account for fixture wear, replacement, or duplication. Both processing and end-operation machines are assumed to periodically break down as the result of machine failures, part jams, broken tools, and so forth. The only exception is the dip tank, which (due to its simplicity) is assumed never to malfunction. Machine breakdowns are th___~emost significant source of randomness in discrete manufacturing processes. Parts undergoing process operations at the time that a machine breakdown occurs are not damaged. Parts being processed at the time of a breakdown are completed before the machine is shut down. These parts will have the same probability of meeting specifications as parts that were processed when no breakdown occurred. The time intervals between breakdowns are based on "calendar time" (elapsed simulation time), and not on machine busy time (time that the machine is actually in operation). Assuming the Central Limit Theorem applies, the time intervals between breakdowns are modeled with normal distributions. The practical effect of the use for the normal distribution is that the value of the duration for the time interval between breakdowns can be allowed to vary significantly from one breakdown to the next. The mean time between breakdowns for processing machines is assumed to be 325 hours. The standard deviation of the normal distribution used to model this time interval is assumed to be one tenth of the mean, or 32.5 hours. The mean time between breakdowns for end-operation machines is assumed to be twice that of the interval between breakdowns for processing machines, or 650 hours. The standard deviation of the distribution used for these machines is likewise taken to be twice the value of the standard deviation used for processing machines, or 65 hours. Repair of machines is assumed to occur according to the following phases: diagnosis, disassembly, and reassembly. The time required to perform each of these tasks will be assumed to be exponentially distributed, and each phase will be assumed to last approximately two hours. The entire repair procedure will be modeled as an activity with a duration following the Erlang distribution. The exponential distributions contributing to these Erlang distributions will each have a mean value of two hours, and the number of contributing distributions will be 3.25. Machine capacity is known for each machine. When a machine has reached a time-inoperation which exceeds its life, it is duplicated. Tool life is assumed to be known for each type of tool. Tools are replaced when their time-in-operation exceeds the known life of the tool.

372 7. ANALYSIS OF SIMULATION OUTPUT SLAMSYS software, due to its ease of use and capabilities, was selected to perform this analysis. Figures 1, 2, 3, 4, 5, and 6 illustrate some sample SLAMSYS networks developed for modeling the machining cell.

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