Operational Freight Carrier Planning: Basic Concepts, Optimization Models and Advanced Memetic Algorithms (GOR-Publications)

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Author: Jörn Schönberger

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GOR ■ Publications Managing Editor Kolisch, Rainer

Editors Burkard, Rainer E. Fleischmann, Bernhard Inderfurth, Karl Möhring, Rolf H. Voss, Stefan

Titles in the Series H.-O. Günther and P. v. Beek (Eds.) Advanced Planning and Scheduling Solutions in Process Industry VI, 426 pages. 2003. ISBN 3-540-00222-7

Jörn Schönberger

Operational Freight Carrier Planning Basic Concepts, Optimization Models and Advanced Memetic Algorithms

With 43 Figures and 24 Tables

123

Dr. Jörn Schönberger University of Bremen Lehrstuhl für Logistik Fachbereich 07 Wilhelm-Herbst-Straße 5 28359 Bremen Germany E-mail: [email protected]

Library of Congress Control Number: 2005922933

ISBN 3-540-25318-1 Springer Berlin Heidelberg New York This work is subject to copyright.All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag.Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner Production: Helmut Petri Printing: Strauss Offsetdruck SPIN 11407584

Printed on acid-free paper – 42/3153 – 5 4 3 2 1 0

Preface

This book represents the compilation of several research approaches on operational freight carrier planning carried out at the Chair of Logistics, University of Bremen. It took nearly three years from the first ideas to the final version, now in your hands. During this time, several persons helped me all the time to keep on going and to re-start when I got stuck in a dead end or when I could not see the wood for the trees. I am deeply indebted to them for their encouragement and comments. Prof. Dr. Herbert Kopfer, holder of the Chair of Logistics, introduced me into the field of operational transport planning. He motivated and supervised me. Furthermore, he supported me constantly and allowed me to be as free as possible in my research and encouraged me to be as creative as necessary. In addition, I have to thank Prof. Dr. Hans-Dietrich Haasis, Prof. Dr. Martin G. Mohrle and Prof. Dr. Thorsten Poddig. On behalf of all my colleagues, who supported me in numerous ways, I have to say thank you to Prof. Dr. Dirk C. Mattfeld, Prof. Dr. Christian Bierwirth, Henner Gratz, Prof. Dr. Elmar Erkens, Nadja Shigo and Katrin Dorow. They all helped me even with my most obscure and dubious problems. My family supported me all the time. They always showed me their trust and encouraged me continuously. Special thanks are dedicated to my parents Monika and Heinz-Jiirgen. However, there is somebody who helped and supported me much more than any other person. It's my beloved wife Ilka. She believes in me more often than I beheve in myself. But more importantly, she periodically rescues me from the jungle of science and guides my attention to other wonderful aspects of life. Thank you very much.

Bremen, January 2005

Jorn Schonberger

Contents

1

Transport in F'reight Carrier Networks . . . . . . . . . . . . . . . . . . . . . I 1.1 Recent Trends in Freight Transportation . . . . . . . . . . . . . . . . . . . 1 1.2 Carrier Tkansport Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Network Design, Configuration and Deployment . . . . . . . . . . . . . 9 1.4 Distribution and Collection Planning . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Aims of this Book and Used Methods . . . . . . . . . . . . . . . . . . . . . . 13

2

Operational Freight Transport Planning . . . . . . . . . . . . . . . . . . . 2.1 Decision Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Request Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Mode Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Freight Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hierarchical and Simultaneous Planning . . . . . . . . . . . . . . . . . . . . 2.2.1 Hierarchical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Simultaneous Routing and Freight Optimization . . . . . . . 2.3 Generic Models for Simultaneous Problems . . . . . . . . . . . . . . . . . 2.3.1 Maximal-Profit Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Bottleneck Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Selection with Compulsory Requests . . . . . . . . . . . . . . . . . 2.3.4 Selection with Postponement . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 16 17 19 20 22 22 23 24 25 25 26 27 29

3

Pickup and Delivery Selection Problems . . . . . . . . . . . . . . . . . . . 3.1 Problems with Pickup and Delivery Requests . . . . . . . . . . . . . . . 3.1.1 Problems with Depot-Connected Requests . . . . . . . . . . . . 3.1.2 Problems with Direct Delivery Requests . . . . . . . . . . . . . . 3.1.3 Simultaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pickup and Delivery Paths and Schedules . . . . . . . . . . . . . . . . . . . 3.3 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Problem Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 33 34 34 36 37

VIII

Contents

3.4.1 The PDSP with LSP Incorporation . . . . . . . . . . . . . . . . . . 3.4.2 The Capacitated PDSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The PDSP with Compulsory Requests . . . . . . . . . . . . . . . 3.4.4 The PDSP with Postponement . . . . . . . . . . . . . . . . . . . . . . 3.5 Test Case Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Generation of Pickup and Delivery Requests . . . . . . . . . . 3.5.2 Freight Tariff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Benchmark Suites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 39 39 40 42 42 45 46 48

4

Memetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Algorithmic Solving of Problems with PD-Requests . . . . . . . . . . 49 4.2 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.1 Terminus Technici . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.2 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.3 Applicability of Genetic Search . . . . . . . . . . . . . . . . . . . . . . 57 4.3.4 Limits of the Genetic Search . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Repairing and Improving the Genetic Code . . . . . . . . . . . . . . . . . 60 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5

Memetic Algorithm Vehicle Routing . . . . . . . . . . . . . . . . . . . . . . . 5.1 Genetic Sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Genetic Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Combined Genetic Sequencing and Clustering . . . . . . . . . . . . . . . 5.4 Advanced MA-Approaches: The State-of-the-Art . . . . . . . . . . . . 5.4.1 Multi-Chromosome Memetic Algorithms . . . . . . . . . . . . . 5.4.2 Co-Evolution with Specialization . . . . . . . . . . . . . . . . . . . . 5.4.3 Co-Evolution of Partial Solutions . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Memetic Search for Optimal PD-Schedules . . . . . . . . . . . . . . . . 77 6.1 Permutation-Controlled Schedule Construction . . . . . . . . . . . . . . 78 6.1.1 Construction of Routes for more than one Vehicle . . . . . 78 6.1.2 Parallel Time-Window-Based Routing . . . . . . . . . . . . . . . . 78 6.1.3 Algorithm Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.1.4 Determination of the Request Instantiation Order . . . . . 84 6.2 Representation of a PD-Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.3 Configuration of the Memetic Algorithm . . . . . . . . . . . . . . . . . . . 85 6.3.1 Initial Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3.3 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.3.4 Population Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.4 Computational Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4.1 Parameterization of the MA . . . . . . . . . . . . . . . . . . . . . . . . 94

65 65 68 71 71 72 74 75 76

Contents

IX

6.4.2 Impacts of Spatial Distribution and Time Window Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4.3 Identification of Profit-Maximum Request Selections . . . 100 6.4.4 Consideration of Capacity Limitations . . . . . . . . . . . . . . . 102 6.4.5 Identification of Deferrable Requests . . . . . . . . . . . . . . . . . 109 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7

Coping with Compulsory Requests . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1 Limits of Fitness Penalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.1.1 Static Penalties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.1.2 Dynamically Determined Penalties . . . . . . . . . . . . . . . . . . . 118 7.1.3 Adaptive Penalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.2 A Double-Ranking Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.3 Converging-Constraint Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3.1 Alternating and Converging Constraints . . . . . . . . . . . . . . 121 7.3.2 ACC-Algorithm Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.4 Assessing QC-MA and ACC-MA: Numerical Results . . . . . . . . . 125 7.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4.3 Impacts of Intermediate Cost Reductions: An Example . 130 . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133

8

Request Selection and Collaborative Planning . . . . . . . . . . . . . 135 8.1 The Portfolio Re-composition Problem . . . . . . . . . . . . . . . . . . . . .136 8.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.1.2 Formal Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 Configuration of the Groupage System . . . . . . . . . . . . . . . . . . . . . 139 8.2.1 Bundle Specification by the Carriers . . . . . . . . . . . . . . . . . 140 8.2.2 Bundle Assignment by the Mediator . . . . . . . . . . . . . . . . . 140 8.3 Computational Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.3.1 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.3.2 Collaborative Planning Approach . . . . . . . . . . . . . . . . . . . . 142 8.3.3 Reference Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.1 Understanding Freight Carrier Decision Problems . . . . . . . . . . . . 149 9.2 Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.3 Methodological Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Transport in Freight Carrier Networks

The division of labor among the continents, countries or regions over the world enables the production of goods in the most efficient manner. Goods are produced at different locations so that the overall costs are minimized. The manufacture of a certain product often concentrates on few places in a region, a country, a continent or even in the world. However, the demand for the products manufactured at certain locations in an economic zone is typically scattered over the complete zone. In order to satisfy this demand with the centrally produced goods, extensive transport is needed. Transport describes the spatial transformation of goods or persons with the goal of balancing supply and demand. The increase of goods transport is accompanied by a significant extension of passenger transport. The movement of manpower to the centralized production facilities becomes necessary and additionally, the enlarged incomes are used for private travel. In Sect. 1.1of this chapter, the economic importance of freight transport is explored. Some current trends, from which the demand for a reinforced planning arises, are shown by means of the examples European Union (EU) and United States (US). In Sect. 1.2 the structure of a freight carrier network and the transport processes in such a network are analyzed. Planning problems regarding the design, configuration and deployment of the transport system are discussed in Sect. 1.3. The distribution and collection of freight from providers or suppliers to a consolidation facility, and in the reverse direction, is identified as a very critical phase of the transport and the need for additional planning support is emphasized in Sect. 1.4. The goals and the organization of this thesis are given in Sect. 1.5.

1.1 Recent Trends in Freight Transportation The commonly used indicator for the performance of the goods transport sector is the amount of realized ton-kilometers (tkm) expressing the product of

2

1 Transport in Freight Carrier Networks

the quantities moved and the sum of traveled distances. For passenger transport the number of passenger-kilometers (pkm) gives an adequate measure for the quantity of passengers moved and the bridged distances. From 1990 to 2000, the performance of the transport sector has grown significantly in the US as well as in the EU. For the United States a growth of more than 20% (goods) and 24% (passengers) is reported (Fenn, 2004). The European rates show an increase of 29% for goods and 17% more passengers (Eurostat, 2002). At the same time, the Gross Domestic Product (GDP) of the US has expanded by 39% (Fenn, 2004) and the EU-GDP has improved by 21% in the observed decade(Eurostat, 2002). The absolute values of the performance indicators have increased from 5.76 billion tkm t o 6.93 billion tkm, from 6.23 billion pkm to 7.72 billion pkm (US), from 2.33 billion tkm to 3.08 billion tkm and from 4.041 billion pkm to 4.839 billion pkm (EU). Different studies forecast a further significant increase in required transport (Eurostat, 2002; Arendt and Achermann, 2002; ICF, 2002). An annual growth of around 3.4% (US) and 3.0% (EU) is expected in the transport of goods. Relative to 2000, a growth of 40% (US) and 34% (EU) in goods transport will be achieved by 2010. This thesis is about problems in planning the transport of goods, often called freight transport (Crainic and Laporte, 1997). Freight transport is performed by different means of transport: road transport by trucks and vans, rail transport by trains, waterway transport by inland navigation (barges) and vessels and pipeline transport of fluid products. The contributions of each mode (the modal split) have changed during the last decade. In both economies, the contribution of pipeline transport remains on a approximately unchanged level. Waterway transport's part has lost significantly in the US as well as in the EU. Different directions are observed for rail transport: in the US it has grown but in the EU it has declined. In both economic zones, the main share of the internal freight transport is performed on the roads. Trucks and vans make up 32% of the US-domestic freight transport (Moore, 2002) and 44% of the intra-EU transport of goods (Eurostat, 2002). The domain of road transport mostly takes place in the short or medium size quantity field with shipments of less than 45 tons and distances of typically not more than 500 km (Eurostat, 2002; Moore, 2002). Two modes of road transport are performed. In the own account (AC) mode, the owner of the vehicles and of the moved goods are identical. Typically, such a company requires transport in order to manage the flow of their goods along their supply chain from the suppliers, through production stages and warehouses to the customers. The contribution of AC transports to the performance of the transport sector in the US has decreased slightly from 28% to 24% (Moore, 2004) in the nineties. A significant performance loss for AC has been observed in the EU-zone. For example, in Germany it has fallen from 42% down to 30%, and in France the value has dropped from 28% down

1.1 Recent Trends in Freight Transportation

3

to 18% (Oberhausen, 2003). An increasing number of enterprises outsource their transport departments, basically to reduce their overhead costs. They hire independent logistics service providers to execute the necessary transport. Such a transport company is called a (freight) carrier: it operates in the so-called hire or reward (HR) mode. This sector's contribution to the overall performance varies from 50% (in Portugal) to over 76% (in the US) and nearly 90% (in Spain) with an average value of nearly 70%. In both modes, AC and HR, short-distance (max. 50km) and medium-distance transport operations (max. 150km) are the most often demanded services (Oberhausen, 2003) in the EU. In the United States the average haul distance is 730km, all other modes operate at longer average haul lengths (Moore, 2002). The environmental impacts of road transport are significant: 19% (US) and 24% (EU) of the annual carbon dioxide emission is produced by this mode of transport (Eurostat, 2002; Fenn, 2004). An increase of nearly 16% (US) and 20% (EU) during the nineties of the last century has been observed. Besides the emissions, several other negative impacts to the environment are observed: intensified traffic congestion lowers the performance of the road transport, large surfaces are sealed by roads, parking lots and transshipment areas and noise emissions lower the quality of residential areas. Regulations and laws have been announced in order to alleviate the pollution and to maintain several standards of life quality (European Commission, 2001). For the EU some additional current developments are observed which are expected to have significant impacts on road transport and the companies involved. The transport enterprises in the current EU member countries expect the complete integration of the accession countries, e.g. Poland or the Czech Republic. These new competitors can offer lower transport costs since their labor costs are low and the regulations are not as restrictive as the laws in the current full member countries of the EU. The low cost structure in the preaccession countries will lead to a new wave of competitors intruding onto the EU transport market. Therefore, the existing transport sector in the current EU now has to improve the efficiency of their business. Roads are free of charge in most EU countries. The construction and maintenance of the expressways, highways and other roads is under the responsibility of public authorities. For some special connections (typically including large bridges or tunnels) the payment of a toll is required. Recently, several member countries like Germany, Austria and the Netherlands announced their intention to establish a toll system for using their national roads, and in most of the pre-accession countries road pricing has been established for years. The consideration of this additional kind of costs cannot be avoided in the future. The public authorities are not able anymore to provide the necessary funding in inter-urban transport infrastructure (roads, bridges, tunnels). Private investors are sought for funding, constructing and maintaining these facilities. They are allowed to collect a toll from the users.

4


Several urban governments (e.g. such as the municipality of the City of London, UK) have recently either introduced or plan to introduce a toll for driving on the urban roads. The urban-toll has been introduced mainly for the reason of preventing traffic congestion caused by private road transport with the goal of improving the speed of business transport in the urban areas (Nash and Niskanen, 2003). As mentioned above, road transport is an important player in the US and European economy with a significant contribution to the overall performance of the economic zone. However, it is faced with several challenges in the near future, which make it unconditionally necessary to strive for an improvement in the efficiency of the height transport operations on the roads. Carrier companies, which only operate in HR-mode, will be most affected by the challenges due to the increasing importance of road transport in the United States and in the EU, and their unconditional need of external customers.

1.2 Carrier Transport Networks A customer request describes a single transport demand. The location of the pickup and the location of the delivery are specified as well as the quantity to be moved. Typically, additional requirements like time limits for the loading or unloading operation or special handling requirements are stipulated. A transport department or a carrier company derives internal processes from the requests in order to satisfy the customer demands using the available resources in an efficient manner. Several independent requests with coinciding or adjacent origins and coinciding or adjacent destinations are bundled to shipments or truckloads of larger quantity. A significant decrease of the relative price for the movement of one single request is achieved, because overhead costs are split into small amounts that are assigned to each request. The realization of maximum economies of scale requires consolidation processes, which both support as well as depend upon the operation mode of a transport company. Firstly, the AC-mode is analyzed. This mode is typically selected for the distribution of finished goods in industrial productions. From a small number of factories, large quantities of different goods are moved over relatively long distances to regional distribution centers (DCs) in order to replenish the DCs with goods. From a DC, the goods are distributed to customers, who are situated relatively near to the DC. Typically, the distance from a DC to a customer is less than 100 km, whereas the distance between a production facility and a DC is often more than several 100 km. The links between the production facilities, DCs and the customers defines a distribution network. In a distribution network, the flow of goods is uni-directional from few sources (the production facilities) to many sinks (the customers). Since the quantities moved for replenishment are large, this network topology supports the realization of economies of scale. In Fig. 1.1, a distribution network with two

1.2 Carrier Ti-ansport Networks

5

production facilities Fl and Fz is shown. These two factories produce goods that are used to replenish the three distribution centers DC1, DCz and DC3. Each customer is assigned to one of the DC from which he is supplied. The traveled distance and the moved quantities in the distribution are different from those observed in the replenishment. In distribution tasks, the distances to travel are significantly reduced and the quantities moved along a link are significantly smaller.

production facilities replenishment

distribution centers distribution customers ( Fig. 1.1. Structure of a distribution network

The strict uni-directional flow of goods from the factories to the DCs or customers leads to an inefficient use of the deployed vehicles. They drive fully loaded from the production facilities to the DCs, from a DC to customers or directly from the factory to a customer (if enough load is available). However, they have to travel back to the DC or production facilities. If no back-freight is available, which is carried on the way back to the DC or to the production facility, half of the traveled distance consists of empty miles. Often, the transport of finished goods cannot be combined with backfreights flowing in the reverse direction. National laws in several countries (e.g. Germany) do not allow vehicles operating in the AC-mode to transport goods that are owned by another company. In order to use the existing network and equipment also for moving goods of third parties, several manufacturing enterprises decided to outsource their distribution systems and let them operate (at own responsibility) as freight carriers in the HR-mode. A freight carrier transports goods for different customer companies. Particular batches of different customers are combined in one vehicle if the origins and the destinations match. A vehicle of a carrier picks up goods of one customer at a certain place and moves it either directly to the final destination or with intermediate loading, unloading or transshipment stops belonging to other requests. Afterwards, it continues to a new pickup location situated

6


near the destination of the former transport task and loads new goods that are then moved towards the specified destination. Typically, a carrier company operates for several customers. The flow of the goods is not limited to a small number of relations and it is not unidirectional, but rather bi-directional (Fleischmann, 1998). There are many sources and many sinks of flow spread over the whole operational area. The quantity of goods associated to a certain pair of origin and destination is small, and the particular flows are numerous. Long-distance transport demands and medium-distance demands must be satisfied as well as short-distance bridging. Exclusive origin-destination transport is typically not achievable (Trip and Bontekoning, 2002). Due to the small quantities of particular requests, an efficient consolidation strategy is necessary in order to reduce the part of the overhead costs that have to be assigned to each single request. The consolidation of small flows from a huge number of locations into large longdistance flows has to be supported as well as the deconsolidation into flows to the particular customer destinations. Therefore, the operations area of a carrier is hierarchically organized and the origin-to-destination transport process is partitioned in sub-processes according to the partition of the area. Few large regions form the operations area, and each region is divided into several small zones. A hierarchically organized network of transshipment facilities and connections between these facilities is maintained. The network structure permits a successive aggregation of flows from customer origins into high quantity flows and a subsequent resolution from bundled flows into deliveries to the several destinations. Whenever a single transport demand requires the crossing of an organizational border, the goods of a request are combined with goods of other requests or extracted from a large volume flow. Transport between different regions takes place only between hubs (H). A hub is a transshipment facility where all inbound flows from other regions are received and where all outbound flows are released. In each region, the hub receives the goods from different transshipment points (TP) situated in the zones and forwards incoming goods into the right destination zones. In each zone, the goods are distributed from the TP to the customer locations by vehicles within several tours. The same vehicles are used to collect goods from customer locations. These goods are delivered to the destinations in the same tour if the location is situated in the same zone. Otherwise, they are brought to the TP, where they are merged with other collected goods and forwarded to the regional hub. A typical layout of such a hierarchically structured freight transport network is given in Fig. 1.2. The overall operations area is partitioned into four regions. The thin continuous lines mark the borders. In each region one hub (HI,Hz, H3 and H4) is available. All extra-regional flows of goods out of a region are realized through this transshipment facility as well as all inbound flows. Each region is separated into several zones. Their borders are given by the thin dotted lines. In each zone one T P is maintained where the goods flowing out of the zone are bundled and forwarded and where the

1.2 Carrier Transport Networks

s

Hi= hub in region i

Ti= transshipment point

Fig. 1.2. Hierarchical network structure of a carrier network

flow of goods destined for this particular zone is received. The distance from the customer location to a T P in a zone is typically less than the distance between the T P and the corresponding regional hub. However, the distance between the hubs often causes the main part of the distance necessary for moving a packet from its origin to its destination. The hierarchically organization is typical for a freight transport network operated by a carrier. It permits the economically reasonable service of geographically scattered locations with averagely low flow between particular origins and destinations (Fleischmann, 1998). However, the described original structure is often modified adapted to meet the special requirements (Wlcek, 1998). A hub serves as the T P for a zone or a complete region, if the quantity of the flow of goods does not require the strong tree structure in a region. Direct origin-to-destination shipments among different regions or zones are offered in the event that the quantity of goods to be moved and the associated revenues are sufficiently large. In the following, the transport process for carrying the less-than-truckload (LTL) packet p, of the request r is analyzed in detail. Figure 1.3 shows the five phases of the process from an origin C3 to the destination D. Initially, the packet p, is collected. This phase is called collection. Typically, a vehicle of small or medium capacity is deployed for the fulfilment of


this task. This vehicle collects packets from several requests with origins in one selected zone and carries them to the transshipment point (TP) of the zone. A T P is a special facility, in which the packets of a zone, collected by several vehicles, are consolidated into larger quantities (truckloads). A truckload consists of all quantities belonging to requests originating out of a zone with destinations in zones embedded in different regions. Requests in which both the origin and the destination are included in the same zone are served without involving a TP. In the forward feeding phase, the truckload is carried from the T P of the origin zone to the regional hub. All truckloads of a region arrive synchronized at this large and high-performance transshipment facility (Fleischmann, 1998). In contrast to a TP, incoming and outgoing goods are merged while passing a hub. Bundled truckloads from different TPs are resolved before the packets are re-consolidated into shipments so that all packets in a shipment have to be carried to customer locations in the same destination region.

hub

\

I line haul

I origin O

TP

hub

hub

hub

TP destination 'l/

Fig. 1.3. Process-chain of an origin-destination carrier transport

The complete shipment containing p, is now carried to the hub of the destination region, which includes the final customer location. If the hubs are fully connected, then no intermediate stop at any other hub is necessary because there is a direct connection between each two hubs. Otherwise the shipment is moved to one or several intermediate hubs, reconsolidated if necessary, and then finally transferred to the hub of the destination region. In this line haul phase (Daganzo, 1999), large distances are traveled. The means of transport is often different to those in the previous phases. Since the flow of goods is continuous and of balanced quantity, the inter-hub connections are often served in a regular way following a fixed schedule (Crainic, 2000). For this reason, it is necessary that the feeder transport schedule be synchronized with the departures from the hub. All feeder truckloads should arrive in time so that they can be considered for the inter-hub transport departures. A synchronized arrival enables the most effective re-consolidation of the incoming truckloads from other hubs and from the TPs.

1.3 Network Design, Configuration and Deployment

9

At the destination hub, the shipment containing p, is resolved and merged with other incoming shipments into truckloads, so that each truckload comprises packets for different customers who are situated in the same distribution zone of the considered region. Each truckload is transported to the TP of the corresponding zone. This phase is denoted as backward feeding. At the T P of the zone, which contains the destination of r, the truckload is broken into the packets and the packets are distributed to the customers that are situated scattered over the destination zone. Delivering p, to the customer specified delivery location completes the request r . The first two phases of the transport process are subsumed under the name pickup and the last two phases are referred to as delivery. The pickup in an origin region is typically combined with the delivery of back-freight destined for this region. Thereby, the flows of goods in both directions are combined in an effective and efficient way. The correspondence of the five-phase transport process and the hierarchically organized network is shown by means of a transport of a packet from a customer situated at the small black point in the north-western zone to the location marked by the small black point in the south-eastern zone in Fig. 1.2. A vehicle following the route that visits all customer locations in the origin zone, shaded in grey, picks up the considered packet. At TI, the TP of the origin zone, the packet is consolidated with all other packets originating from this zone into a truckload and it is fed to the regional hub H I . With an intermediate stop a t H3,the packet is carried to H4, the hub in the destination region. There, it is transshipped and fed to T4, the T P in the destination zone. F'rom T4 it is delivered on the route visiting all grey shaded locations, including the particular delivery site.

1.3 Network Design, Configuration and Deployment The construction, the management and the usage of an effective and efficient freight carrier transport network require the solution of numerous often interdependent decision problems. Logistics System Design. The design of a freight transportation network affects several problems related to the location and the layout of the network components such as the TPs, hubs or traffic routes. Three main classes of design problems are distinguished (Crainic and Laporte, 1997): 0

0

How many hubs and TPs are needed? Where should they be installed? How large should they be? (location and layout) How should the hubs and TPs be linked? Which means of transport should be used for the connections? How should the flow of goods be distributed over the connections between the hubs? (network design) How can the network be protected against disadvantageous external influences and evolutions arising from infrastructure modifications, the evolu-

10


tion of demand and from new governmental or industrial policies? (regional multi modal planning) Network design problems are strategic. The necessary funding and the necessary time for the construction or modification of a n existing infrastructure do not allow short or medium-time changes. Design problems are solved using estimated data expressing the expected flows of goods. The decision for a certain network architecture is based upon the costs for installing, maintaining and using the facilities and the traffic links between them. Logistics System Configuration comprises three main mid-term planning categories (Crainic, 2000): service selection, traffic distribution and terminal policies. A service describes a repeated transport operation connecting hubs or hubs and TPs. In a service selection problem, the offered services in a network are determined. Typically, the repetition of a service follows a regular schedule. The departure and the arrival times a t the first, the last and the intermediate stops are defined and announced. The necessary work power and transport capacity to offer the intended services is procured. Several services are compiled into closed routes (itineraries). For each itinerary, a vehicle is allocated and the corresponding necessary terminal operations are fixed. Services are determined only for connections of hubs with T P s or other hubs. The derived schedule is valid for up to several months in the future, but exact long-term planning data are not available. The quantities of goods have to be estimated, e.g. based on observations from the past. However, reliable estimates need reliable input data, which is typically available only for the feeder or line haul connections. The consolidation of packets from the customers ensures a predictable and balanced flow of goods between the hubs. A terminal policy describes the offered activities a t a given hub or TP. The type of performed consolidation tasks a t a certain hub or T P is specified and defines the available throughput that can be handled. Additional resources have to be maintained in order to compensate for peaks in the demanded services. Logistics System Configuration aims at establishing services that allow efficient operations t o answer customer demands and t o ensure the profitability of the operations (Crainic, 2003). Efficiency is typically measured in terms of costs for fulfilling the customer demands a t a predicted quality that allows the customers t o maintain complex and reliable production systems (Rodrigue, 1999). Logistics System Deployment comprises short-term planning problems in a freight carrier network. In contrast to the design and configuration of the network, deployment decisions are mainly based on known problem data. These data are derived from the known or declared flows of goods extracted from the customer demands. The goal is to allocate labor and capacities in order to support the efficient fulfilment of known customer demands with respect to the policies and services determined in the configuration step. These

1.4 Distribution and Collection Planning

11

planning problems are solved following the rolling horizon planning paradigm in order to handle the continuously updated information about additional, cancelled or modified customer requests. The necessary operations for the next period are definitively determined together with a tentative determination of the operations planned for the subsequent planning periods. The following short-term planning problems occur (Crainic and Laporte, 1997):

0

0

Assignment of crews, reserve crews or maintenance teams t o vehicles or transshipment facilities in order t o support the planned operations (crew scheduling) Preparation of the operations for the next planning period (empty balancing) Scheduling of the services for the pickup and the delivery phases (vehicle routing and scheduling)

Logistics System Deployment mainly impacts the short-term planning of pickup and delivery operations. The operations during the long haul phase are determined by the valid regular schedules. For two reasons the determination of a long-term schedule for the pickup and the delivery operations is not achievable:

1. The locations that have to be visited are typically not known in advance. 2. There is no balanced flow of goods that permits the prediction of necessary services and/or necessary capacities in a zone. The costs for the operations in the first and in the last phase of the carrier transport process are, expressed in terms of money units per tkm, the most expensive part in the complete transport from the pickup location to the final delivery location. Herry (2001) states that the costs per tkm in shortdistance transport are a t least three times larger than the costs per tkm in the middle or long distance case. The reason for this extreme increase of the costs can be seen in the relatively small quantities that are delivered or picked up a t a customer site stop, and in the lack of consolidation options due t o the scattered locations of customer sites that requires a visit. Furthermore, the unconditional need for the consideration of tight time windows for the pickup and the delivery visits confines the realization of economies of scale that are otherwise achieved by the bundling of several requests (Punakivi et al., 2001). The consideration of time windows is necessary in order t o synchronize the transport processes performed by the carrier company with the internal processes of the customers.

1.4 Distribution and Collection Planning Each customer request is split into five internal requests according t o the subprocesses described in Sect. 1.2. A collection task is necessary t o carry the

12


demanded quantity from the customer specified origin t o the next TP. Within the forward feeding task, this quantity has to be carried t o the regional hub, typically together with quantities from other customer requests. The line haul tasks expresses the necessity to transport the quantity to the destination hub, the backward feeding task requires the movement of the quantity t o the TP in the destination region and the distribution task describes the final carriage to the destination. In the remainder of this thesis, the term request is used as a synonym for task or internal request. Each line haul task is assigned t o one of the regular services, which ensures the execution of the task. The remaining tasks cannot be assigned t o such a regular service, because there are no regular services for tasks within one region. It is necessary t o allocate resources for the remaining four tasks in each planning period. To fulfil the required tasks associated with different requests, company-owned vehicles can be used or, for the payment of a fee, other carriers can be instructed to complete selected tasks. Pickup and delivery planning problems comprise the allocation of the resources for fulfilling the tasks within a region for a given planning period. A solution of a pickup and delivery planning problem is the transportation plan (Crainic and Laporte, 1997) and the necessary costs are called fuZjilment costs. The transportation plan is determined only for the next planning period, which comprises just a day. or often even only several hours. For subsequent periods, the transportation plan is renewed considering the recently released information of additional, cancelled or modified requests and the currently available transport resources. Altogether, the following issues characterize pickup and delivery planning problems. The request portfolio cannot be modified. In some cases a postponement of some of these requests is allowed. The consideration of a large number of low-quantity packets instead of bundled shipments requires a large number of stops a t customer sites. The composition of several requests into routes is necessary in order t o achieve profitability (Trip and Bontekoning, 2002). Compared t o the line haul tasks, the costs for fulfilling a (regional) pickup task or delivery task are tripled (Herry, 2001). No regular services are available. For each planning period, a new transport plan has t o be determined (logistics system deployment problems). The flow in both directions t o and from the TP/hub is typically not synchronized. A time gap between freight and back freight has t o be managed. Several time constraints have to be taken into account: earliest departure times from the TP/hub (availability), latest arrival times a t the TP/hub and customer site time windows. The routes are determined following a rolling horizon planning, the period length depends upon the frequency of incoming feeder or long haul services

1.5 Aims of this Book and Used Methods

0

0

13

a t the TP/hub and upon the portfolio of released, but so far unconsidered, customer requests. The demanded transport capacity is not balanced: it varies from planning period to planning period. Other carrier companies are allowed to be ordered to fulfil tasks (subcontractor incorporating, externalization).

Pickup and delivery planning aims a t finding a reasonable trade-off between necessary costs (resulting in reasonable offered prices) and service quality. Mathematical optimization models are proposed in which the costs are expressed in a cost function that is minimized. In order to ensure customer satisfaction, several constraints have to be considered, especially time windows that restrict the delivery or collection time. Existing models for pickup and delivery problems typically refrain from involving external carriers. However, their consideration can be profitable if the charges are below the costs for using an own vehicle. Additionally, the occasional involvement of external carriers for a fixed charge allows a quick and temporary capacity expansion when the available resources do not suffice to serve all tasks.

1.5 Aims of this Book and Used Methods Systems for freight transport require an exact and efficient coordination of the different processes in order to offer a reliable service to customers and to keep the necessary costs within an acceptable budget. The setup of regular services in a particular zone or region is generally not targeted because the local flow of goods is unpredictable. Neither the demanded quantities nor the locations to be visited can be estimated in a sufficient manner. Since the costs for each performed tkm are significantly enlarged compared to the long-haul operations the determination of efficient long valid schedules is hardly possible. The determination of the necessary tasks to fulfill the demand for transport in a particular region is furthermore compromised by a significant reduction of available and maintained transport resources due to the current poor market conditions and due to the intrusion of low-cost carriers from the EU candidate countries into the market. The involvement of carrier companies to fulfill requests for a fixed charge becomes more and more attractive for freight carrier companies who offer and manage wide-area transport networks. Instead of maintaining own equipment in all regions (with often severe costs for employment, depreciation, maintenance and insurance), subcontractors are involved for a previously known charge whenever it is possible (due to lower costs) or necessary (due to a short-term capacity bottleneck). The planning of the regional collection and distribution traffic of a freight carrier company is a very sophisticated challenge, since it is impossible to

14


establish regular services between customer locations and transshipment facilities. Planning support for the incorporation of external carriers has received only minor attention so far, although it is extensively required in practice. Adequate planning models are rarely proposed but they are becoming more and more necessary t o support freight carrier planning. The intention of this thesis is to contribute new ideas to close this gap between real world requirements and available planning support. The first goal of this thesis is to derive general modeling approaches for incorporating external logistics service providers in the fulfillment planning of a freight carrier network. Chapter 2 provides a n introduction t o operational carrier planning problems. The corresponding scientific literature is surveyed and four main modeling approaches with special consideration of the external request fulfillment are derived. As the second goal, the extension of existing pure routing models by the incorporation of a carrier incorporation feature is performed. One of the most general routing problems which represents a variety of different planning situations, the pickup and delivery problem with time windows, is generalized. Therefore, the possibility of the usage of a n external carrier for unprofitable requests is added. Four different problem variants representing several special planning environments are set up. For each single variant, adequate test problems are generated. Chapter 3 comprises all these investigations of so-called pickup and delivery selection problems. A pickup and delivery selection problem can be modeled as a multi-constraint mathematical optimization problem. The solving of instances of such a problem requires massive computational effort even for small number of incorporated requests and vehicles. Optimality guaranteeing algorithms are not available. The third goal of this thesis is the derivation of adequate algorithms t o solve instances of pickup and delivery selection problems. The configuration of Memetic Algorithms is proposed. Memetic Algorithms combine the proven exploration capability of Genetic Algorithms for restricted combinatorial optimization problems with the exploitation capabilities of problem-specific algorithms. General concepts of the memetic search are presented in Chap. 4. This class of algorithms has been successively applied to routing problems for vehicles. The general configurations of a memetic search algorithm for routing problems are surveyed in Chap. 5 . The configuration of a Memetic Algorithm for solving pickup and delivery selection problem instances is presented in the Chapters 6 and 7. Its applicability is assessed within extensive computational experiments. It is shown, how it might possibly be incorporated into and exploited by a carrier service. The presented framework can be used for problems based on any of the four carrier incorporation approaches. The applicability of the derived models and algorithms is investigated for a cooperative planning scenario described in Chap. 8. This thesis concludes with a summary of the core findings of the investigations and formulates topics for future research in the field of planning models for freight transportation with the possibility of incorporating a paid carrier.

Operational Freight Transport Planning

This chapter is about operational (short-time) planning problems that must be resolved in order to determine the transportation plan of a carrier company for the next planning period. The focus of the investigations is on the coordination of the operations to realize the outer legs of the freight carrier transport process described in the first chapter. The aim is to determine the necessary operations to fulfill the demand for transport between customer locations and the next transshipment facility in a particular region. A carrier has to solve different decision problems in order to setup the transportation plan for a certain planning period. The first decision concerns the question as to whether the responsibility for a request fulfilment is taken on. Each accepted customer request is resolved into several internal tasks (internal requests) distinguishing between pickup, line-haul and delivery requests. Line-haul tasks are assigned to the designated regular and scheduled services. The second decision problem comprises the selection of the fulfilment mode of the pickup and the delivery requests. Mode selection means to decide whether a request is fulfilled with own vehicles (carrier controlled vehicles) or whether it is externalized and fulfilled by another carrier for a charge (external order processing). To utilize the own fleet in a most efficient manner it is necessary to solve a routing problem in which the requests are assigned to different vehicles and execution orders are setup (third decision problem). In order to reduce the overall sum of charges to be paid to other carriers, a freight charge minimization problem has to be solved (fourth decision problem). These decision problems are analyzed in Sect. 2.1. The four decision problems are interdependent. In a hierarchical solving approach, the problems are solved successively: Initially, the mode of execution is selected for each accepted request and afterwards the resulting routing and the resulting freight minimization problem are solved. In a simultaneous approach, the decisions upon the selected mode, the routing and the freight charge minimization are derived simultaneously within one closed optimiza-

16

2 Operational Freight Transport Planning

tion problem. Hierarchical and simultaneous approaches are described in Sect. 2.2. The simultaneous approach has received only minor attention so far although it is most promising and required in many real-world applications. Adequate decision models or problem representations are required. In Sect. 2.3, four generic frameworks for combined mode selection, routing and freight charge minimization problems are introduced. Each one represents a simultaneous problem for a special environment.

2.1 Decision Problems The determination of the transportation plan for one period requires the solving of different sophisticated decision problems. First of all, the carrier has to decide whether a certain request r is accepted for completion, then the mode of completion (with own equipment or by another carrier who receives a charge) is selected. Routes have to be established for the own vehicles and the sum of charges to be paid for all externalized requests has to be minimized. In both cases, the consolidation of several requests helps to reduce the costs for each moved capacity unit so that the solving of these problems requires the simultaneous treatment of more than one request.

2.1.1 Request Acceptance If a carrier accepts a customer request then it takes over the responsibility for the reliable completion of the customer specified transport demand. The carrier company receives a certain revenue for each satisfactorily completed request. If a request is not completed to the customer's satisfaction, the carrier is sanctioned. The agreed revenues and the agreed penalties are typically codified in a contract between the carrier and the customer. Solving the request acceptance problem for a request r means to decide whether r can be fulfilled in a profitable manner or not. Therefore, the request acceptance problem is a binary decision problem. A carrier company is faced with different request acceptance problems. Tactical request acceptance problems require a general decision about the future acceptance of different request types. Such a type comprises typically all the requests of a certain customer. This acceptance problem is a management problem, because the general acceptance of all requests of a customer requires medium- or long-term investments for additional transport or transshipment resources. The general acceptance is fixed in medium- or long-term contracts. In such a contract, the expected quality of the request fulfilment is codified. Furthermore, the revenues achieved for the request fulfilment are determined. The general acceptance is recommended if, and only if, the agreed revenues cover the sum of necessary investment and operation costs. A reasonable contribution to the profit has to be achieved.

2.1 Decision Problems

17

In a n o p e r a t i o n a l a c c e p t a n c e problem, the carrier company has to decide about the acceptance of particular requests whose fulfilment is not enforced by long-lasting contracts. Such a request is accepted if the expected revenues cover the expected additional costs caused by the consideration of this additional request. Negotiations with the customer are often necessary in order t o commit the customer to pay larger revenues. If the negotiations lead t o sufficiently large revenues then the request is accepted otherwise its completion is refused. In contrast to tactical acceptance problems, investment costs are not considered in the profitability calculations (Meier-Sieden, 1978). Two additional aspects compromise calculation based solution approaches for operational acceptance decisions. In the current market situation, several carriers compete for few customer requests. If a carrier refuses a customer demand then it can be expected that also all other requests of this customer are lost for this carrier. These lost revenues cannot be adequately incorporated into the calculation of the profitability of a request (Schmidt, 1989). A carrier company receives demands for transport successively during the whole working day. As soon as a new demand is expressed and submitted to the carrier company, a decision immediately becomes necessary, whether the responsibility for the request completion is overtaken or not. So far unknown requests cannot be considered while evaluating the worth of the current request (Riebel, 1986). Each accepted customer request is resolved into several internal tasks or internal requests according to the differentiation of a pickup, a delivery or a line-haul operation. These operations are coupled and have t o be synchronized. A line-haul task is assigned to one of the established regular and scheduled services. This is not possible for a pickup or a delivery task because there are no regular services to fulfill the operations within a region. In the remainder of this thesis, it is assumed that all available requests are internal requests derived from customer requests. It is aimed a t determining a most profitable transportation plan for fulfilling all requests within one region. In the following three subsections the necessary decision problems t o setup the corresponding transportation plan for a given set of internal requests are discussed.

2.1.2 M o d e Selection

A carrier company has two possibilities t o complete a request. It can use an own vehicle for the execution of the necessary tasks or it pays another carrier for fulfilling the corresponding tasks. Such a carrier is called a logistics service provider (LSP). In the following, the term 'carrier' is used for the company for which the vehicles have t o be routed. The term 'LSP' is dedicated t o each company that offers the fulfilment of requests for a previously known freight charge. Two reasons for the incorporation of an LSP are possible:

18


1. The considered carrier company does not maintain enough transport resources to fulfill all requests as contracted. 2. The external processing costs (freight charge paid to the LSP) lie below the additional costs for the utilization of carrier-owned equipment. The decision whether an LSP is incorporated or not is based upon the relevant variable costs. In the case of LSP incorporation these costs are the freight charge and in the case of the usage of own resources, the relevant costs are determined by the additional travel costs of the own used vehicle. If the available transport resources of the considered carrier are not scarce then a request r is assigned to an LSP if and only if its claimed freight charge is less than the additional (travel) costs caused by the incorporation of r . The situation is different in the case where the transport resources of the carrier company are scarce and some requests with travel costs below LSPcharges cannot be served with own equipment. The fleet of the considered carrier company serves those requests that utilize the bottleneck resource(s) in the most efficient manner. All remaining requests are assigned to an LSP. The set R+ comprises all requests served by own equipment and R- includes the requests served by an LSP. The ordered pair (R+,R-) is denoted as request selection. A request selection is feasible if 1. All requests contained in R+ can be served with carrier-owned and controlled vehicles without violating any desired restrictions and 2. If an LSP is available for each request assigned to R - . The goal of mode selection is to determine a most beneficial request selection. It is aimed at maximizing the profit contribution achieved from the selection ( R + , R - ) . This is the difference between gained revenues and expended costs. The selection is evaluated by the sum of paid LSP charges for the requests assigned to R- and costs that are caused by the own-equipmentservice of the requests in R+. This sum is denoted as fulfilment costs. The goal is to minimize this amount. It is assumed that the sum of achieved revenues does not depend on a particular selection so that the fulfilment cost minimization is equivalent to the maximization of the profit contribution. Schmidt (1989) proposes a calculation model for long-distance operations in which the costs for the execution of a request r with own equipment are compared to the LSP-charges for r . The requests with the highest difference between the costs for the LSP and the own travel expenses are assigned to the available own vehicles until their capacity is exhausted. All other requests should be assigned to an LSP. Initially, each request is considered separately. Cost savings that arise from the coupled fulfilment of two or more requests within one itinerary of a vehicle (coupling savings) are rarely considered. The large number of possible combinations of requests for an itinerary does not allow the identification of the most advantageous request compositions. Erkens (1998) proposes a refined calculation model. However, requests are calculated in isolation. The consideration of coupling savings is not incorporated.


19

There may appear situations in which special contracts between the carrier company and a customer prohibit the fulfilment of requests by an LSP. Furthermore, several requests cannot be served by LSPs because the LSPs are not equipped with the necessary vehicles like cooling tanks or special security containers. Mode selection problems are mainly treated with comparative calculation approaches in order to solve the question whether the LSP incorporation is cheaper than the usage of own equipment. However, in many situations the desideration of savings from request compositions and consolidations reduces the quality of the proposals derived from the solutions of the calculation models. An LSP can be incorporated whenever it becomes necessary, but the charges may increase if a request requires an urgent completion or if the specifications of the request do not match with the remaining request stock of the LSP.

2.1.3 Routing Routing describes the problem of determining the routes for the vehicles of a carrier company. These routes are used as the instructions for the drivers and describe the way in which the customer requests are served. A request is completed after the necessary customer locations have been visited and the specified quantities have been transferred between the right locations. Two concatenated decision problems have t o be solved. First, it has to be decided for each request which vehicle is used t o serve this request. Therefore, each request is assigned t o exactly one vehicle. The set of requests assigned t o a certain vehicle v is called the tour of vehicle v. In order t o determine the route for this vehicle, all customer locations requiring a visit of v, are arranged in a sequence that defines a visiting order, called the route. The set of the routes of all vehicles of the carrier company represents a solution of the routing problem. Typically, several solutions are available for one particular routing problem. In order t o maximize the contribution to the profit gained for the considered carrier company through the completion of the requests within R , the aim should be to minimize the travel costs connected with the implementation of a solution. Therefore, several requests are composed into the same route in order to realize maximum coupling savings. Routing problems are typically modeled as mathematical optimization problems consisting of a set of feasible solutions and a n objective function that represents the profit contribution of a particular solution. These models combine the assignment subproblem and the sequencing subproblem in one model. Since the routing problems usually appear in practical applications, several complicated additional conditions compromise the determination of the most advantageous routes. In a capacitated routing problem, the maximal allowed

20

2 Operational F'reight Transport Planning

load of the available vehicles is limited so that maximum coupling savings cannot be achieved. If the visit t o one or more customer locations is restricted to a certain time interval (time window) then profitable sequences are often not realizable. This type of routing problem is called time constrained. If several precedence relations in the routes have t o be kept then a precedenceconstrained routing problem has to be solved. Models for practical routing problems typically involve combinations of these generic problem variants. The most extensively studied vehicle routing problem is the Traveling Salesman Problem (TSP). The goal of the TSP is to determine the shortest length route for one vehicle in which all specified customer locations are visited exactly once (E.L. Lawler et al., 1990), hence solving a pure sequencing problem. Capacity, time restrictions or precedence constraints are not given. In the multiple TSP the request portfolio has to be partitioned into routes for two or more vehicles, an assignment problem and several sequencing problem have t o be solved in parallel (Domschke, 1985). In the Capacitated (time constrained) Vehicle Routing Problem (VRP), a knapsack constraint hinders the implementation of the most promising routes (Christofides, 1990). This constraint represents a maximum capacity of the used vehicles, a maximum route length or maximum trip duration. If visits to (customer) locations are restricted to well-defined time intervals (time windows) then the VRP is denoted as the Vehicle Routing Problem with Time Windows (VRPTW) (Solomon, 1987). Precedence constraints are typically imposed if the considered portfolio comprises both internal collection and internal distribution requests associated to the same customer request. This type of problems will be discussed in more details in the following chapters of this thesis. Routing models are solved by automatic solution approaches following different search paradigms. Optimal (exact) approaches guarantee the identification of a solution that is provable not to be dominated by other so far unconsidered solutions. However, their performance is often insufficient because the models a t hand are very complex and the algorithm's running times are excessively long. Heuristic algorithms (heuristics) try t o identify sufficiently good solutions, but they cannot guarantee that the optimal solution is found. Nevertheless, heuristics are often able t o find solutions whose quality is close t o the (unknown) optimum. Specialized heuristics exploit special problem structures or features. Their applicability is typically restricted t o a closely defined problem family. Meta-heuristic search algorithms follow generalized search paradigms that are problem independent. For this reason their applicability is not restricted to some special well-defined applications (Aarts and Lenstra, 1997; Reeves, 1993).

2.1.4 Freight Optimization The carrier company wants to minimize the costs for the fulfilment of the requests for which LSP incorporation has been selected. Each LSP uses a freight charge function in order to determine the price for the fulfilment of


21

a request. Typically, the charge depends upon the starting position and the destination of the requests, the kind of goods to be moved, their weight and their size as described in the request. The charge function is degressive so that the average cost of each transported weight or size unit decreases with each additional unit assigned to a particular relation between an origin and a destination (Kopfer, 1992). Every request r E R- is split up into a sequence of partial requests Fl, . . . ,Fny according to the subprocesses described for freight carriers in the previous chapter. The goods q, have to be move from o to d. f l describes the request to move q, from o to a location where q, is consolidated with goods from other requests. For the combined quantity the request F2 is specified in order to transport the quantities to another location where additional quantities are added. The last partial request fnTexpresses the demand for delivering q, from the final consolidation or resolving point to its final destination d.

Fig. 2.1. Assignment of partial requests t o relations

Figure 2.1 illustrates this approach. Four requests R1, R2, R3 and R4 have to be fulfilled by an LSP. Instead of instructing an LSP to carry the quantity of R1 directly from A to F, it is split into the concatenated requests A to B, B to Dl D to E and E to F. R3 is resolved into the two partial requests B to D and D to E. The request R4 is defined as the concatenation of C to B, B to D, D to E and E to F. Five requests are defined and assigned to LSPs: from A to B with the quantity of R1, from C to B with the quantity of R4. The request for the connection B to D comprises of the quantities from all four requests, whereas the request from D to E comprises the quantities of R1, R3 and R4. The quantities belonging to R1 and R4 are bundled into one request from E to F. The number of necessary requests is enlarged from four to five,

22


but the quantities for the relation B to D, D to E and E to F are large, so that the part of the costs assigned to each request is reduced. The search for a least cost partition of the outsourced requests and their assignment to different relations is called freight charge optimization. Mathematical optimization models and solution approaches for this type of carrier planning problem are investigated in Pankratz (2002) and Kopfer (1989,1984).

2.2 Hierarchical and Simultaneous Planning The four decision problems (request acceptance, mode selection, routing and freight optimization) have to be solved in order t o minimize the costs for the completion of all requests in the current portfolio. As discussed, the company's management typically decides about acceptance or refusals of requests. For this reason it is assumed that the completion of all requests in the portfolio R must be achieved. The decisions for the selected mode, the routing and the freight optimization remain the subject of pickup and delivery planning.

2.2.1 Hierarchical Approach In a hierarchical planning approach the mode selection decision is the first decision made. For each request r in the portfolio R it is determined whether r is completed by a vehicle of the considered carrier company or by a vehicle of an LSP. This decision is irreversible and determines the request selection (R+, R - ) . The marker x, is set to 1 if, and only if, x, is served by an own vehicle of the carrier company. If an LSP is incorporated for the completion of r then x, is set to 0. After the mode of completion has been determined for all requests in R , a routing problem is solved for all requests contained in R + . The vehicle v,- that serves a certain request f E R+ is chosen and the position p, in the visiting sequence of v, is set. Therefore, the ordered pair (v,,pF) is irrevocably fixed for each request f E R+ . If r is assigned to R- it has to be decided by which shipment sequence 1, this request is served. After r has passed the different planning stages, its attributes can be expressed in the quadruple (x,, v,, p,, 1,). If x, = 1then the information in the fourth component is ignored, otherwise the second and the third component remain unconsidered. In the hierarchical planning approach mentioned above, the set R of accepted customer requests is separated into two the disjoint sets R+ and R-. A set of routes has to be determined in order to serve all requests in R+.The requests in R- are distributed among different LSPs so that the overall sum of paid charge is minimized. If the mode of a request has been determined it cannot be modified anymore so that the assignment of the requests to the sub-portfolios R+ and R- is unalterable. This hierarchical planning approach is illustrated in Fig. 2.2.

2.2 Hierarchical and Simultaneous Planning

request

23

r

mode selection

routing

fulfilment of r

or freight optimization

fulfilment of r

Fig. 2.2. Hierarchical solution of the decision problems

2.2.2 Simultaneous Routing and Freight Optimization In general, it is not possible to identify the subset of the request portfolio that leads t o positive profit contributions if they are served by own equipment. It depends on the routes of the vehicles whether a certain request enlarges the profit contribution (joint-product production). For this reason the profitability of a request served by an own vehicle of a carrier company, can only be estimated roughly in the mode selection step. If, during the solving of the routing problem, it turns out that a certain request is unprofitable and compromises the generation of profitable routes it should be given to an LSP for completion. On the other hand, the change of the execution mode is advantageous, if the self-service of a tentatively externalized request with a carrier-controlled vehicle leads t o considerable additional profit contributions because the request fits appropriately into the routes. In simultaneous pickup and delivery planning problems, the changes of the execution modes of requests are allowed. For each request, a mode is determined tentatively. As soon as it turns out that a change of the execution mode of requests leads to additional profit contributions, the necessary changes are performed. In the model for a simultaneous pickup and delivery-planning problem, x, is decidable for each request r. As a consequence, the two other decisions (routing and freight minimization) have to be coded in such a model a t the same time in order to allow for the evaluation of the mode selection. A model for the planning of the pickup and delivery operations, in which all three decisions can be modified, is called simultaneous pickup and delivery selection model. Simultaneous planning approaches work as follows. Initially, a tentative separation of the request portfolio into R+ and R- is determined by selecting 0 or 1 for x, for each request r . For the requests in R+, tentative routes are

24


generated and the requests in R- are tentatively assigned to different LSPs. The execution modes of some requests are altered and this new separation is evaluated by solving a routing problem for the requests in R+ and a freight optimization problem for the requests in R - . If no additional profit contributions are detected the mode alternations are canceled and other requests are selected for a tentative mode alteration. The exchange is repeated until it can be guaranteed that the fulfilment costs (travel costs plus LSP charges) are minimized, which is equivalent to the maximization of the profit contribution. request r

mode selection

routing

C

+ freight optimization

fulfilment of

r

Fig. 2.3. Simultaneous solution of the decision problems

The information flow between the three decision problems mode selection, routing and freight optimization is shown in Fig. 2.3. Different modes are tentatively assigned to each request r until the optimal mode for each request with respect to the fulfilment costs is identified.

2.3 Generic Models for Simultaneous Problems In the following, different generic frameworks for simultaneous planning models are derived. It is assumed, that a request portfolio R is given. The request selection (R+, R - ) is evaluated by means of the corresponding revenues and costs. R ( R + ) represents the revenues obtained for the requests fulfilled with own equipment. In some applications, the LSP incorporation is not prohibited, but is sanctioned by a reduction of the paid revenues. However, in the remainder of this thesis it is assumed that the utilization of an LSP does not lead to revenue decreases. Therefore, the sum of achieved revenues remains unchanged as long as all requests are completed by own or LSP equipment. The cost evaluation of a request selection requires the determination of an explicit transportation plan (cf. Subsection 1.4) and therefore the solving of a

2.3 Generic Models for Simultaneous Problems

25

routing and a freight optimization problem. The transportation plan describes the determined routes of the own vehicles to serve the requests in R+ and the selected assignment of the request collected in R-. Let T(R+)be a least cost routing plan for serving the requests in R+. The costs for executing T(R+) are given by C(T(R+)). Let B(R-) be a cost minimal distribution of the requests in R- among the available LSPs and let F(B(R-))represent the summarized charges for the LSP incorporation.

2.3.1 Maximal-Profit Selection

A carrier company aims t o achieve maximum or a t least non-negative contributions t o the overall profit of the company. Therefore, it tries t o minimize the costs for fulfilling the given request portfolio. Since the sum of achievable revenues is fixed, the minimization of the fulfilment costs leads to a maximum difference between the gained revenues and the spent costs. In order to achieve maximum savings by LSP incorporation, it is unconditionally necessary that the most promising request selection can be generated. Therefore, all requests are allowed t o be served by an LSP. This allowance is typically available as long as the LSP can provide a n equivalent performance, reliability and service quality. The costs for a transportation plan are determined by the chosen request selection. This means, that a cost reduction can be achieved only by modifying the current request selection. Therefore, the fulfilment cost minimization of the carrier company can be represented as the following optimal selection problem.

+

min C(T(R+))F(B(R-)) s.th. (R+,R-) is a feasible request selection of R

(2.1)

(a.2]

Diaby and Ramesh (1995) and Pankratz (2002) investigate problems, in which costs are minimized by distributing the requests among the own vehicles and vehicles of LSPs.

2.3.2 Bottleneck Selection The fulfilment of a request consumes resources like fuel, load capacity or time. The transport costs typically comprise fuel because this is the only resource whose consumption can be determined exactly. Load capacity or time consumption cannot be measured in terms of monetary units in order to distribute these costs among the requests in a fair manner. However, there is only a fixed amount of such a resource available. The quantities to be transported within a zone are not balanced; they vary significantly from planning period to planning period. In order to operate a vehicle continuously in the most efficient manner, an averagely high usage level

26


of this vehicle is aimed at. The available capacity is therefore adjusted to the average flow (Erkens, 1998). Highest peaks in the flow cannot be served with this capacity and require the incorporation of an LSP to fulfill all requests. The collection and the pickup routes have to incorporate strict duration constraints. The collection has to be terminated in time, so that the picked up packets can be brought to the next consolidation center in time to perform a consolidation before the feeder operation departs from the consolidation facility. On the other hand, the frequency of deliveries from the T P s is not allowed to fall below a certain value. If the time between two adjacent deliveries from such a transshipment facility is too long then the on-site storage capacities are exceeded and the on-site handling efficiency decreases. Therefore, strict duration limits have to be considered and the execution of all required pickup and delivery tasks often becomes impossible. In general, assume that the utilization of own vehicles consumes a resource. The consumption is given by Res(T(R+)). The amount Resmax is available. This constraints the generation of a feasible request selection because several sets R+ require more than Resmax capacity units. In this situation, the set of feasible and implementable request selections is pruned by the bottleneck resource. The resource constraint has to be added to the selection model (2.1)(2.2).

+

min C ( T ( R + ) ) F(B(R-)) s.th. Res(T(R+)) Resmax

+ F(B(R-))

(R+, R-) is a feasible request selection of R

(2.6) (2.7) (2.8)

If it is not allowed for any request to be fulfilled by an LSP, which means that the set Rcompis equal to R then the problem is reduced to a pure routing problem.

2.3.4 Selection with Postponement The on-site storage capacity of a transshipment facility is typically constrained (Grunert and Sebastian, 2000). In order to prevent exceeding this capacity, it sometimes becomes necessary to desist from carrying the packets of all available requests to the terminal. If possible, requests should be postponed in order to release facility capacity to the most urgent customer demands. Long lasting contracts or customer satisfaction issues often forbid the acceptance refusal of unprofitable requests for which no adequate LSP can be found because the claimed charge is too expensive. In these cases it is often a remedy to postpone those requests into the future, expecting that currently unprofitable requests will be able to be combined with other so far unknown but expected requests so that the expected sum of revenues will lead to positive profit contributions. Rolling planning horizon approaches determine transportation plans for the next T planning periods P I , . . . , PT.Only the transportation plan for the period PI is mandatory. Transportation plans for subsequent periods can be modified until the relevant period is reached. After one planning period has been passed, a mandatory transportation plan is determined for the immediate next period. Modifications of the planning data are considered as known so far.

28


However, it is not necessary in pickup and delivery planning problems to determine a sequence of transportation plans for the periods P I , . . . , PT including all so far known demands. Requests are only scheduled if their fulfilment leads to positive profit contributions. It is allowed for the fulfilment time to be assigned to a later planning period so that the corresponding pickup and delivery time windows are met. In such a case, the determined fulfilment time can be updated in later planning runs. If the start of a request fulfilment cannot be postponed into periods later than PI (urgent request) then this request is scheduled in the recent transportation plan regardless of whether their fulfilment leads t o positive profit contributions or not. The execution mode is determined for all urgent requests. The remaining requests are left temporarily unconsidered. If their fulfilment becomes necessary in future planning periods, they are combined with other so far unreleased and unknown requests or they are assigned to a n LSP. It is expected, that the changes of the request portfolio during the sequence of planning periods allow the realization of additional positive contributions to the profit in later periods. Such a myopic planning approach is adequate only in the case where the problem data for the periods Pz, . . . , PT are highly unsure. In this case, the saving of the profits gain-able from the current request portfolio is preferable. Request selection with postponement means to solve a static profit contribution maximization problem before PI is opened. In contrast t o the so far considered situations, one of three different modes can be selected for each request: the completion with carrier equipment, the completion by a n LSP and the postponement. To take this trinity into account, a third mode is offered in the mode selection for each request. Each postponed request is inserted into the set RPP. A pd-request can only be postponed if the customer specified time windows allow a delay of execution, e.g. if both the pickup and the delivery task can be executed after P I . The execution of a n urgent request cannot be delayed, because either the pickup or the delivery task or both of them have to be completed in P I . Let Rurgentdenote the set of all urgent requests. To ensure that all urgent requests are routed in time, the urgent requests have to be routed or assigned to a n LSP preferentially. The objective function (2.9) guarantees the consideration of the most profitable remaining requests. Constraint (2.10) ensures that all urgent requests are included in the generated transportation plan. The profit contribution is targeted directly since the sum of revenues varies with respect to the requests selected for fulfilment next. Revenues for postponed requests remain unconsidered. max R ( R + U R-) - C ( T ( R + ) ) - F(B(R-)) s.th. R+ U R( R + , R-

> RUrgent ,R p p ) is a feasible request partition of R

(2.9) (2.10) (2.11)

2.4 Conclusions

29

Golden et al. (1984) investigates a problem of oil replenishment. Within this distribution problem, a fleet of gas tankers is available t o delivery fuel to consumers. Following a push-strategy, customers whose oil reserve is near empty should be visited within the next period. Since the period duration does not allow for visiting each customer, the most urgent customer deliveries have to be selected for fulfilment within the next period. Requests with the highest urgency are served immediately in the next period whereas customers with larger reserves are served only if time is still available within the considered period.

2.4 Conclusions The determination of a transportation plan requires the solving of several welldefined, but interdependent decision problems. These problems comprise the mode selection for each request: does a carrier-owned and controlled vehicle serve a certain request or is a n LSP ordered? Furthermore, the composition of maximum profit routes for the own vehicles is targeted. Finally, a n adequate assignment of externally fulfilled partial requests t o several LSP in order to minimize the costs for the LSP incorporation is determined. To realize the maximum success it is necessary to consider all three problems simultaneously. The resulting decision problems are combined selection (of modes), assignment (of requests to vehicles or LSPs) and sequencing (operations) problems. Several additional customer or managerial requirements and recommendations compromise the generation of optimal transportation plans. Four generic base models have been proposed t o model the simultaneous decision problems. The incorporation of the mode selection decision into the operational transport planning of a carrier company has received only minor scientific interest so far. However, it is of high practical relevance because carrier companies often need to incorporate LSPs if the usage of their own equipment is unprofitable or if additional transport capacity is required immediately.

Pickup and Delivery Selection Problems

This chapter is about mathematical optimization models for the generation of transportation plans. It is aimed at allocating resources for the fulfilment of the transport demands in a particular and restricted region. Decisions concerning the mode selection for the available requests, routing of the controlled vehicles and freight charge optimization for LSP-served requests are simultaneously considered in these models. A literature review about problems and models associated with the allocation of transport resources of freight carrier companies reveals that these kinds of transport planning problems have received only minor attention so far. The main results from this review are reported in Sect. 3.1. The General Pickup and Delivery Selection Problem is introduced in Sect. 3.2. It represents the problem of determining a transportation plan from a given portfolio of transportation requests between pairs of pickup and delivery locations. A mathematical model is proposed. Four special variants of the General Pickup and Delivery Selection Problem are described in Sect. 3.4. An explicit problem is described for each of the four generic simultaneous problems introduced in Sect. 2.3. A profit maximization problem (Subsect. 3.4.1), a bottleneck selection problem (Subsect. 3.4.2), a problem with compulsory requests (Subsect. 3.4.3) and a problem with postponement (Subsect. 3.4.4) are presented. All variants can be described by a modification of the mathematical model proposed for the General Pickup and Delivery Selection Problem. These models represent combinatorial optimization problems with several intricating constraints. The generation of artificial test instances for each of the proposed problems is described in Sect. 3.5.

3.1 Problems with Pickup and Delivery Requests A pickup and delivery request (pd-request) r expresses an indivisible transport demand between an origin location (pickup location) p$ and a destination

32

3 Pickup and Delivery Selection Problems

location (delivery location) p;. Goods of the quantity q, have to be moved. Loading the goods a t the pickup location is often allowed only during the pickup time window T,?, determined as the interval between an earliest and a latest allowed operations time. The unloading a t the destination location is allowed only during the delivery time window T,-. It is distinguished between three types of a pd-request r within a region. If r is a collection request then p- coincides with a transshipment facility in the considered region that receives the quantity q,. In the case of a distribution request the location p+ coincides with a transshipment facility that releases q,. A direct delivery request describes the transport demand between two customer locations p+ and p-, the quantity q, is not transshipped during request fulfilment. Figure 3.1 illustrates the different pd-request types in a region. The request from p: to p l is a distribution request because p; coincides with the depot (transshipment facility) of the region. The second request from p; to p; is a collection request, since the delivery location p z coincides with the depot. The third request from p$ to pq is a direct delivery request. The carriage of the corresponding goods to the depot is not necessary and typically not allowed.

outbound flclw

collection request

P3

direct delivery request

Fig. 3.1. Types of pd-requests in a region

3.1 Problems with Pickup and Delivery Requests

33

3.1.1 Problems with Depot-Connected Requests The research on planning problems with pd-requests has focused on two original optimization models, the Vehicle Routing Problem with Backhauls (VRPB) and the Pickup and Delivery Problem (PDP). The former one is a generalization of the well-known Vehicle Routing Problem (VRP). The goal of the VRP is to visit all customers using a fleet of vehicles at minimum cost. Either all customers receive packets from a single source or all customers provide some packets that have to be transported to a single sink. In the VRPB, both types of customers are considered simultaneously, but either the origin or the destination of each packet has to coincide with the single sink (depot). A survey of problems with backhauls is given by Casco et al. (1988). Toth and Vigo (1999) investigate the special version of the VRPB in which all receiving customers have to be visited before goods are allowed to be picked up. Angelelli and Mansini (2002) and Duhamel et al. (1997) deal with a VRPB in which customer specified time windows compromise the generation of least cost routes.

3.1.2 Problems with Direct Delivery Requests VRPB models do not cover problems with requests in which neither the pickup nor the delivery location coincides with the depot. They require additional constraints to ensure that the pickup location is visited before the corresponding delivery location is reached. Mathematical models for this kind of problem are called models for pickup a n d delivery problems for freight transportation or dial-a-ride problems if passenger transport is required (Desrosiers et al., 1998). Rego and Roucairol (1995), Frantzeskakis and Powell (1990) and Powell et al. (1988) investigate a pickup and delivery problem in which the consolidation of packets of different requests is prohibited (Full-Truckload or Vehicle Allocation Problem). A sequence of matching requests has to be determined for each vehicle so that the overall sum of traveled empty miles is minimized. Savelsbergh and Sol (1995) present a survey on Less-Than-Truckload (LTL) pickup and delivery problems. In an LTL problem it is allowed to consolidate packets of different customer requests in one vehicle. Nanry and Barnes (2000), MitroviC-Mini6 (1998) and Dumas et al. (1991) consider customer given time windows for both the pickup and the delivery visits. This problem is known as Pickup and Delivery Problem with Time Windows (PDPTW). The decisions in all these models include only the daily routing of the vehicles for realizing the collection and the delivery processes for the traffic to and from the next transshipment facility in a zone or a region. Synchronization with the long-distance operations is necessary (Rodrigue, 1999). This is typically achieved by deriving the earliest release dates and due dates for the collection or pickup operations from the departure or arrival times of the long haul services determined in the tactical context. In contrast, Stumpf (1998)

34


includes the assignment of the requests to the long-distance services, which follow a regular schedule, in the daily operations planning.

3.1.3 Simultaneous P r o b l e m s In the Full-Truckload problem investigated by Frantzeskakis and Powell (1990) the rejection of inappropriate transport demands is possible. Greb (1998) investigates a simultaneous LTL pickup and delivery problem with direct delivery requests. An LSP can be ordered to complete a particular demand. Requests that are served by carrier-owned vehicles are composed into cost minimal routes. Every out-sourced request requires the payment of a freight charge that is transferred to the serving LSP. The freight charge for a particular request is calculated by means of the freight tariff function and depends upon the demanded transport relation and the quantity of goods to be moved. A similar problem is the subject of the PhD-Thesis of by Pankratz (2002). He extends the problem of Greb (1998) and solves a freight charge optimization problem for the outsourced requests. These requests are assigned to different LSPs. The degression of the freight charge function is exploited in the way that requests are grouped into bundles so that the sum of charges to be paid can be reduced, cf. Subsection 2.1.4.

3.2 Pickup and Delivery Paths and Schedules Assume a carrier company with m vehicles of homogeneous capacity CmaX. The request stock R consists of n pd-requests r l , . . . , r,. They have to be distributed among at most m trips. Each trip is assigned to one of the vehicles and each vehicle serves at most one trip. The sequence of visits for these trips is determined in the corresponding routes whereas the arrival and leaving times are fixed in the corresponding schedules for the vehicles. Every pd-request ri is specified by the quadruple (PUi, DLi, qi, REV,). The pickup activity PUi takes place at p+ whereas the delivery activity DLi is demanded at location p i . A time window T< := [tmi,(p+), tma,(p+)] is specified for each pickup activity and the time window T; := [tmi,(p,), tma,(pi)] has to be considered for the associated delivery activity. Load of volume qi is to be picked up at p+ and to be delivered to p i . REV,, gives the revenue associated to ri. An operation is a triple .rr := (p, a(p), e(p)), where p represents the location of a pickup, a delivery, an initial or a terminating activity. The second component a(p) carries the arrival time of the vehicle a t location p and e(p) represents the leaving time from p. If the vehicle arrives a t p before the associated time window has been opened then the execution of the operation is retarded until the earliest allowed operation time tmi,(.rr) for .rr is reached. Dwell times are not considered here. The vehicle leaves a location as soon as

3.2 Pickup and Delivery Paths and Schedules

35

the corresponding pickup or delivery operation has been executed. The leaving time of .rr has to precede the latest allowed operation time tma,(.rr) in order to ensure the implementability of this operation. A sequence of operations 17 = ( x f ,. . . ,.rr&) describes a route. In the remainder of this work the first component of .rri is denoted by pi and the start location is named po. The route 17 includes Nu requests requiring n u = 2Nu stops for pickup or delivery operations. The initial starting operation and the final terminating operation are not stored within the route for the sake of simplification. Let t,z,pi+, be the travel time between pi and pi+l. The arrival and the leaving times are calculated recursively:

e@i-1)

+

falls i = 0 tpi-,,pz,falls i 1.

>

The leaving time a t pi is achieved by

The vector Su = ( S f , .. . , SFn) describes the volumes that are collected along the route 17. For a pickup operation at pi it is Si 2 0 and for the associated delivery operation a t pj it is defined by S j := -Si. The capacity usage along I7 is determined recursively. Let w i denote the loaded quantity of a certain vehicle arriving at pi

W?

:= 0 and w"=

wEl

+ Si-l

(i > 1 ) .

(3.3)

The route 17 is called pd-path if it holds the following restrictions (Savelsbergh and Sol, 1995): 0

0

0

0

Either both operations of request r or none of them are contained in 17 (PAIRING), A pickup operation precedes its associated delivery operation ( PRECEDENCE), The maximal load is not exceeded for all i along the routes: "w Cmax (CAPACITY) and The leaving time for operation .rri falls into the specified time window: t m i n ( ~ il) e ( ~ il) tmax (pi) (TIME WINDOW).

In the following, it is assumed that the initial positions of the vehicles are known and that the vehicles terminate their routes in one of a set of specified termination points (for example depots). The set 17* contains the requests incorporated into the route 17. A pd-schedule 0 is a set of pd-paths U 1 , .. . ,Ilm so that each pd-request is assigned to a t most one of these paths.

36


Filling R+ with the requests contained in one of the pd-paths I l l , . . . , Dm and filling R- with the remaining requests, S := ( R + , 77-) is a request selection as defined in 2.1.2. The original request portfolio R has been partitioned into m 1 subsets I7*l, . . . ,17*m (forming R + ) and the set R- of non-self-fulfilled requests. Let 2 be a function that assigns a real value to each partition (IT1, . . . , Urn, R-) of the portfolio. This value can be interpreted as the value of the considered transportation plan.

+

3.3 Optimization Problem The pd-schedule represents the part of the transportation plan in which the operations of the carrier-controlled vehicles are determined. Pd-requests that are not included in the pd-schedule are fulfilled by adequate LSPs. A fixed charge has to be paid for each single pd-request. It is assumed that an LSP is available as soon as it becomes necessary and that the charge for each pd-request is known a priori. For reasons of simplification and better understanding it is furthermore assumed that the charge is additive, so that the tariffs of the LSPs do not provide a degression with increasing volume or weight on a relation. Therefore, it is not necessary to solve a freight charge minimization problem to determine the sum of charges. The overall sum of charges to be paid is the sum of the charges for the pd-requests not included in the pd-schedule. The complete transportation plan is determined by the pd-schedule and the set of unconsidered pd-requests. Generating a pd-schedule (transportation plan) with optimized 2 can now be formulated as a mathematical optimization problem. For each vehicle k the set Ij of possible pd-paths (including the empty paths from the vehicle's starting position to its allowed terminating positions) is finite so that a set partition formulation (Williams (1999) and Nemhauser and Wolsey (1988)) of the problem at hand is appropriate. From the set I := U k c 3 1 k U {*) of pd-paths the routes that form the schedule T are selected. Each request that is not served by any of the vehicles is assigned to the dummy *. Such a request is called 'externally served'. It is ensured that each vehicle can serve the received requests within one pd-path and that each request is assigned either to exactly one vehicle or to the set of externally served requests. The following binary decision variables are introduced. Let Xi= 1 if, and only if, route i E I is considered in the schedule S and Y , j = 1 + route i is served by vehicle k. It is assumed that the binary constant zij = 1 if, and only if, route i contains customer request j . The General Pickup and Delivery Selection Problem (GPDSP) is now defined as shown in (3.4)-(3.9).

3.4 Problem Variants

opt so that

Z

37

(3.4)

The objective (3.4) requires an optimization. Each vehicle k is assigned to exactly one pd-path (3.5) so that the maximal capacity of k is not exceeded (3.6), each pd-path is selected at most once (3.7) and no request is contained in more than one selected pd-path (3.8). In this model, (3.8) represents the partitioning constraint. The remaining three constraints ensure, that only executable request assignments are made. The GPDSP is an optimal path problem. A set of non-overlapping paths connecting given pairs of starting and terminating points is searched so that the chosen objective function is optimized (de Queir6s Vieira Martins et al. (1999)). However, the consideration of the four conditions PAIRING, PRECEDENCE, CAPACITY and TIME WINDOWS prevent the application or extension of results obtained for unconstrained shortest or longest path problems or optimal path problems with one additional constraint, which are surveyed by Ziegelmann (2001).

3.4 Problem Variants The GPDSP is a general model for the separation of a request portfolio into the subset of requests that are visited by own vehicles and into the subset of the remaining requests. The four conditions PRECEDENCE, PAIRING, TIME WINDOWS and CAPACITY are considered so that the proposed sequences of visits remain feasible and implementable. Four special variants of the GPDSP are introduced in this section. They cover the four general request selection approaches profit maximization, bottleneck selection, selection with compulsory requests and selection with postponement introduced in Sect. 2.3. The necessary definitions and assumptions are given and explained for each variant.

38


3.4.1 The PDSP with LSP Incorporation The Pickup and Delivery Selection Problem with Logistics Service Provider incorporation (PDSPLSP) describes a generalization of the PDPTW in which each request is allowed to be fulfilled by an LSP. The LSP is ordered to serve a particular request if the freight charge to be paid is less than the additional costs for its fulfilment by a vehicle of the considered carrier company. The PDSPLSP is a special variant of the GPDSP achieved after considering the following assumptions. The decidable costs are the travel costs and the costs of the LSP incorporation, thus the fulfilment costs. For each driven length unit of a vehicle one money unit is spent. The pd-schedule f2 is determined to fulfill the selected requests R+ within the pd-paths U1, . . . , Urn with the travel distances L(U1), . . . ,L(Um). The overall travel costs are calculated 8s

In order to express the travel costs in monetary units, the travel distance is multiplied with p, which modifies the measure unit from distance units to money units. The costs for serving a particular pd-request with an LSP depend upon the distance d(., .) between a pickup and the associated delivery location. For the sake of simplification, the consideration of the quantity to be moved remains unconsidered in the calculation of the freight charges. Let r be a pd-request with loading point p: and unloading point p;, the freight charge for the fulfilment of r by an LSP is calculated by the tariff function f:

The coefficient m, is used to incorporate additional travel distances to the pickup location and from the delivery location into the charge. In order to keep the computational effort on an acceptable level, it has been decided that the freight charge is calculated for each request in isolation. Bundling savings achieved due to a degressive tariff are not considered here. Therefore, the total sum of freight charges F ( R - ) to be paid to LSPs for fulfilling the set of requests R- is given as the sum of the freight charges calculated for each single request:

F(R-) :=

f(r).

The revenue achieved for a particular request does not vary if the corresponding request is served by an LSP. Since all requests have to be served, the


39

sum of the gained revenues is fixed for all partitions of the request portfolio. Therefore, the maximum profit contribution is achieved if, and only if, the fulfilment costs are minimized. The objective Z of the PDSPLSP is now defined as min Z ( 0 , R-)

=C(0)

+ F(R-)

(3.13)

The available transport resources (capacities) are not scarce. The maximum-profit selection approach (Subsection 2.3.1) is applied in the PDSPLSP for deciding which requests are fulfilled by carrier-owned equipment and which requests are served by an LSP. 3.4.2 The Capacitated PDSP

The Capacitated Pickup and Delivery Selection Problem (CPDSP) represents the simultaneous planning problem in which the generation of the most profitable request selection is compromised by scarce transport resources maintained by the considered freight carrier. A homogeneous fleet of vehicles is available to serve requests from R. Each vehicle has a finite capacity Cmax. This capacity is low and hinders the consolidation of a larger number of requests into its route. Time windows associated with the pickup and delivery operations of the requests prohibit the successive execution of requests. It is not possible to realize the most profitable routes. For this reason, some requests have to be served by an LSP, although the charge to be paid exceeds the additional costs for the consideration of these requests into the routes of the carrier. Requests, which use the available capacity in the most efficient manner, are selected to be served by the carrier's equipment. All other requests are selected to be fulfilled by an LSP. The consideration of the capacity constraint requires the incorporation of a bottleneck selection approach, introduced in Subsection 2.3.2. 3.4.3 The PDSP with Compulsory Requests The PDSPLSP allows an unconstrained outsourcing of requests in order to minimize the costs for the fulfilment of a request portfolio. In the CPDSP, the selection of the requests for a self-service is compromised by scarce transport capacities of the available vehicles of the carrier company. However, each single request is allowed to be served by an LSP. Requests, which are not allowed to be served by an LSP, but whose fulfilment with carrier-owned equipment is compulsory, require special handling. Let R be the set of pd-requests and RComp be the subset of R containing the compulsory requests. The currently valid request selection is denoted as


40

0= as

(urn k=l u * ~ R , - ) . The quote of accepted compulsory requests is defined

If no compulsory requests are contained in the current pd-schedule then the quote is zero. On the other hand, if all compulsory requests are included into 0 then the quote equals one. Thus, a necessary condition for a feasible pd-schedule, which means that all compulsory requests are served with carrierowned equipment, is given by the expression

or equivalently

Adding the constraint (3.16) to the G-PDSP given by (3.4)-(3.9) defines the Pickup and Delivery Selection Problem with Compulsory Requests (PDSPCR). If the set of compulsory requests coincides with the set of all available requests (RC= R) then the PDSP-CR is equivalent to the Pickup and Delivery Problem with Time Windows (PDPTW) investigated by (Nanry and Barnes, 2000) and (Dumas et al., 1991). If all pickup locations or all delivery locations of the requests coincide, then the PDSP-CR is equivalent to the Vehicle Routing Problem with Time Windows (VRPTW) (Solomon (1987)). The minimization of the costs counteracts with the incorporation of compulsory requests if these requests cannot be served for costs less than LSP charges. On the other hand, the incorporation of expensive compulsory requests often hinders the service of concurring requests with positive contributions to the overall profit of the considered freight carrier.

3.4.4 The PDSP with Postponement Requests become successively known. Typically, a transportation plan is not updated as soon as a new request is released by a customer. Instead, after a pre-specified time interval has been passed, the plan is updated and the additional requests and any cancellations or modifications are incorporated.


41

Let [To,TI] be the next planning period PI for which a new transportation plan has to be set up and let ]TI, T,,,] denote the remaining interval under consideration. All pd-requests known at To whose latest allowed pickup or delivery time are not later than TI, are urgent requests. They cannot be postponed and have to be considered in one of the generated pd-paths or consists of all they have to be given to an LSP for fulfilment. The set RUrgent urgent requests contained in the current request portfolio R . Additionally released requests or cancellations remain unconsidered until the time TI. At this time, the request portfolio is updated as well as the corresponding transportation plan. As mentioned in Subsection 2.3.4, not all requests contained in the current portfolio have to be served within the next transportation plan. The postponement of a particular request is possible and recommended if this request does not match with the other requests in R at To but it is expected that it can be combined with other requests in a profitable way in subsequent planning periods. Postponing such a request means to leave it unserved in the transportation plan under construction. If a request is postponed then the associated revenues are not yet realized. The profit contributions of the transportation plan, valid from To to TI, have to be maximized. It is not sufficient to minimize the costs since the sum of achieved revenues depends upon the postponed requests. Let (R+, R-, RPP) be a feasible request partition. The profit contribution achieved from this separation is given by

REV, TER+uR-

--'

revenues

-

p.

x ~ ( n * )x -p .

kE3

travel costs

TER-

m, . ~(JI:,~;). "

carrier costs

(3.17)

/

If the objective 2 is defined as (3.17) and if the 'urgency1-constraint (2.10) is added to the model (3.4)-(3.9) then the resulting variant of the GPDSP is called the Pickup and Delivery Selection Problem with Postponement, abbreviated by PDSP-PP. The PDSP-PP can be seen as a model for the planning problem that has to be solved for each planning period in a rolling horizon planning approach for online-planning problems as introduced by (Fiat and Woeginger, 1998), Psaraftis (1995) and Psaraftis (1988). An instance of the PDSP-PP consists of all pd-requests known at the beginning To of a planning period PI. Some of the requests have already been scheduled within a previously determined transportation plan. The remaining requests are those that have been postponed from former periods and have been inserted into the next period (for which the determination of a transportation plan is now required) or have recently become known. For the next period it has to be decided again which subset of the request portfolio is considered for a service with carrier equipment or by LSP and which subset will be postponed into a later planning period.

42


In order to keep announced arrival times, the time windows of requests scheduled in former periods, but not yet executed, are typically updated and now very tight.

3.5 Test Case Generation Several artificial benchmark instances are generated in order to allow the evaluation of automatic solution approaches for the PDSP-variants. Unlike for the VRPTW or PDPTW, established and commonly used test case instances are not available. 3.5.1 Generation of Pickup and Delivery Requests Different benchmark classes for the considered pickup and delivery selection problems are generated adapting an idea found in Nanry and Barnes (2000) and refined by Lau and Liang (2001). The main concept is to derive PDPTWinstances from the famous and generally accepted Solomon instances for the VRPTW (Solomon, 1987) and their optimal or high quality solutions. Therefore, the customer locations are paired randomly within the routes of the considered solution to obtain pickup and delivery requests. The first visited location according to the considered solution becomes the pickup location whereas the remaining one refers to the delivery place. The demand a t the selected pickup location becomes the volume to be moved between the pickup and the delivery location. Solomon (1987) introduces six sets of VRPTW benchmark instances. These are denoted as R1, R2, RC1, RC2, C1 and C2. Each set consists of 8 to twelve instances. Each comprises 100 customer locations with known demand, which have to be visited. A homogeneous fleet of vehicles is available. The customer locations in the instances of the classes R1 and R2 are randomly distributed over the operations area, whereas in the RC1 and RC2 instances half of the customer locations are clustered whereas the second half is spread randomly over the operations area. The instances taken from C1 or C2 provide geographically clustered customer locations. The time windows specified in the test instances taken from the sets R1, RC1 and C1 are tight. They hinder the consolidation of a larger number of requests into routes. On the other hand, the allowed service intervals predefine a rough order in which the requests should be served. The capacity of the available vehicles is small. Instances taken from R2, RC2 or C2 come along with significantly extended time windows so that a corresponding solution makes typically use of fewer vehicles because the requests can be adequately consolidated into few pd-paths. This is supported by a vehicle capacity that is fixed a t a level five times larger than in the other three benchmark sets.

3.5 Test Case Generation

43

Table 3.1. Instances of customer locations used for the generation of sets of pdrequests

class 1 instance I

origin of the used

(Lau and Liang, 2001) (Lau and Liang, 2001) (Gambardella et al., 1999) (Gambardella et al., 1999) (Lau and Liang, 2001) (Lau and Liang, 2001) (Lau and Liang, 2001) (Lau and Liang, 2001) (Lau and Liang, 2001) (Gambardella et al., 1999) (Gambardella et al., 1999) (Larsen, 2001) (Larsen, 2001) (Larsen, 2001) ~ambardellal Gambardella Gambardella

Table 3.1 shows the VRPTW instances from Solomon that have been used in order to derive sets of pd-requests together with a high quality reference routing plan. Let P be the set consisting of all used 18 Solomon VRPTW instances. In order to set up an instance with pickup and delivery requests, it proceeds = (PI,. . . ,p ~ describe ~ ) the route of the vehicle number i as follows. Let in the corresponding solution instance of the VRPTW-instance 4 E P.It is assumed that Ni is even otherwise one of the available nodes is duplicated. , paired randomly so that $) ordered The customer locations p l , . . . , p ~ are pairs (pil,piz), . . . , (piNi-l,piN,) with i k < il for k < I are obtained. From each of these pairs, a pickup and delivery operation is formed. Consider the pair of locations (pil, pi,). The pickup operation must be served at pi, within the time window associated with this location in the considered VRPTW instance 4 and the delivery action requested a t the location pi, within its corresponding time window. The demand associated with pi, describes the quantity that has to be moved between the pickup and the delivery location. Further pickup and delivery requests are derived from the remaining pairs in the same manner. For each vehicle used in the solution of the considered instance 4 , requests with pickup and delivery locations are formed. Only the vehicles considered

rt

The solutions of the C2-instances have been kindly provided by L.M. Gambardella, IDSIA, Lugano, Switzerland

44


Fig. 3.2. Generation of pickup and delivery requests (dashed lines) from the route of vehicle 6 (dotted line) taken from the best-known solution of the VRPTW instance 1-104

in the applied solution instance are added to the generated problem instance forming the available fleet of vehicles. The generation of pickup and delivery requests is shown in a n example in Fig. 3.2. The large black point represents the depot from which all vehicles start and where all routes end, the small black points mark the customer locations. The route of vehicle 6 is selected exemplary t o outline the derivation of pickup and delivery requests from a solution of a given Solomon instance. In the used solution, the vehicle follows the dotted path. Overall, 10 customers are visited. They are numbered from 1t o 10 describing the order in which they are served in the used solution. Five pickup and delivery requests are formed:


45

from location 1 to location 4, from customer 2 to customer 8, from customer 3 to customer 6, from customer 7 to customer 10 and finally from customer 5 to customer 9. These transport demands are plotted in dashed lines. The presented dotted route is a pd-path serving the five generated pickup and delivery requests. Generating pickup and delivery requests from the solution paths for all vehicles in the same manner as in the example determines a complete problem instance. The used solution instance describes a high quality or even near optimal solution of this instance. Pd-request sets (4,a ) are generated from all 18 VRPTW instances given in Table 3.1. For each instance, the pairing is seeded by a E G = {1,2,3). Overall, 1 %' I . 1 6 I= 18 . 3 = 54 instances are now available. Each instance contains between 50 and 60 pd-requests. 3.5.2 Freight Tariff

The freight charge f (r) of the request r is derived from the available solution of the considered Solomon instance. Let v, be the index of the vehicle serving r as proposed in the solution, D,, be the sum of distances between the pickup and the delivery location of the requests derived from the route of v,. The driven distance of v, is denoted as L,,. The freight determining coefficient m, is set to

for all requests r served by v,. This approach for the determination of m, corresponds to the derivation of the freight rate from data observed in the past. The tariff is 'fair'. It assumes approximately equal costs for the LSP and the carrier to fulfill a particular request. The coefficient m, allows the assignment of an adequate part of the socalled overhead distance to the request r. The overhead distance gives the sum of traveled distances which are not required in any customer request, but which must be traveled in order to move the considered vehicle to the customer locations. The meaning of m, is illustrated by means of two simple examples shown in Fig. 3.3. In the left figure, five requests rl, . . . ,rs have to be served. The sum of the demanded distances is 3 + 1.8 1 2.1 0.7 = 8.6 distance units (DU) and the traveled distance following the dotted route is 11.1 DU. The freight charge coefficient for all requests served by the currently considered = 1.29. The demanded distance D,* requires an vehicle is set to m, = additional overhead distance of 29%. This amount is distributed among the requests within the portfolio in the proportion to the demanded distance, leading to the tariff determined in (3.11). The situation in the right part of Fig. 3.3 is different. The demanded distance is 11.5 DU and the route length (equal to the traveled distance) is 10.8

+ + +

%

46


DU, leading t o m, = = 0.94 for all requests contained in the currently considered route. In this example, no overhead distances have t o be distributed among the requests. The demanded distances exceed the traveled distance. The positive difference between the demanded and the traveled distance is called shared distance. In this situation an LSP can offer the fulfilment of a request for one or less money unit for each demanded distance unit, because several requests starts from closely situated locations in the left upper corner and can be completed without significant extra distance while serving the request from p$ to p g . If the tariff function (3.11) is used, then the shared distance savings are distributed among the requests served within this route in proportion to their demanded distances.

Fig. 3.3. Determination of the freight charge coefficients. The route in the left figure leads to overhead costs whereas the route in the right figure is able to achieve shared distances

Greb (1998) proposes a similar approach for the simultaneous consideration of the demanded distance and the quantity of goods to be moved.

3.5.3 Benchmark Suites The sets of pd-requests and the tariff function are used t o define several benchmark suits for the four variants of the GPDSP.

Instances for the PDSPLSP In order to investigate the impacts of different freight charges on the LSP incorporation, the pd-request sets are modified, so that different freight tariffs have to be incorporated.


47

The parameter @ E [-I; 11 represents the level of the freight charge compared to the 'fair' level m,, which is replaced by m,.(l -@). A @-valuebetween 0 and 1 describes the discount in percent for the freight charge that has to be paid to the ordered LSP. The discounted charges make the LSP incorporation more attractive. If @ is smaller than 0 then the payable charge is increased. The variation of the freight charge for the requests is controlled by the parameter a E [0, 11. This value represents the part of the request portfolio for which the LSP charge is discounted or enlarged. A complete instance of pd-requests for the PDSPLSP is now described by the quadruple (4,a, a,@). Instances are generated for the values a E {0.25,0.5,0.75,1) and @ = (-0.75, -0.5, -0.25,0.25,0.5,0.75). The benchmark suit comprises 5 4 . 4 . 6 = 1296 instances. The available fleet of vehicles consists of the equipment defined in the corresponding Solomon-VRPTW-problem. The capacity of each vehicle is set to 200 capacity units, so that the available transport resource is surely not scarce.

Instances for the CPDSP The generated sets of pd-requests are combined with different fleets of vehicles. A particular fleet consists of the number of vehicles predetermined by the original Solomon instance. The fleets are distinguished by the capacity assigned to each vehicle. All vehicles in a fleet have the same capacity y. The analysis of the used solutions shows that a capacity of around 100-120 units is required to allow a least cost completion of the pd-requests within the generated instances without LSP-incorporation. In order to allow the investigation of impacts of reduced available capacity of the carriers' equipment, fleets with vehicle capacities y = 50,75,100,125,150,175 are generated. To study the impacts of scarce transport resources if discounted and overpriced LSP charges are applied, the fleets defined by y are combined with the PDSPLSP instances. The 5-tuple (4,a, a, @,y) determines a complete problem instance for the CPDSP. Overall, the benchmark suit for the CPDSP consists of 1296.6 = 7776 instances.

Instances for the PDSP-CR Compulsory requests hinder the realization of a least fulfilment cost distribution of requests between the carrier and the available LSPs. Given a collection of pd-requests, the possibility of assigning requests to LSPs decreases if the part of compulsory requests is enlarged. A subset of the pd-requests contained in the set (4, a) is labeled as compulsory. The parameter x determines the frequency of these requests. Each request is selected to be compulsory with the probability X . Benchmark instances are generated with low, average and high compulsory request frequencies x = 0.5,0.6,0.7,0.8,0.9,1.0.


48

The benchmark suite for the PDSP-CR consists of problem instances determined by the triple (4, a, x). The number of generated instances is 54.6 = 324. Again, the fleet of vehicles found in the Solomon instances is used as carrier's fleet.

Instances for the PDSP-PP The urgency of a particular request r in the set of pd-requests described by the ordered pair (4, a) depends upon the length of the next planning period. Let To denote the current time and let L determine the length of the currently considered planning period. If both the pickup and the delivery operation of r are allowed to be scheduled after the end of this planning period, hence after To L, then r is not urgent, otherwise it is. If L is decreased then the possibility that a certain request in the set described by (4,a) is urgent increases. In order t o investigate situations with different degrees of request urgencies, different planning period durations T are added to each pd-request set (4, a ) . The set (4, a) represents the pd-request portfolio for a particular planning period with the length L. This value is different for each Solomon instance 4. For R1 it is L = 230, for RC1 it is L = 240, for R2 the length is L = 1000 and for RC2 closes the latest time window at L = 960. Period lengths 6 . L are determined for E = 0.2,0.3, . . . ,1.0 so that overall 54 . 9 = 486 deterministic instances are generated. The carrier company and the LSPs use the same freight tariff function so that the revenue associated with a particular request r can be calculated by applying the function (3.10).

+

3.6 Conclusions The General Pickup and Delivery Selection Problem has been introduced. It represents the problem of determining an optimal transportation plan. The incorporation of LSPs is one important feature that has received only minor attention in previous contributions on carrier operations planning problems with pd-requests. Logical and customer demanded restrictions for the determination of the routes for the carrier-owned vehicles can be handled and are regarded. Four explicit problem variants are derived from the General Pickup and Delivery Selection Problem. Each problem utilizes exactly one of the request selection approaches introduced in the previous chapter. All these problems are complex combinatorial optimization problems. Artificial benchmark suits have been proposed in order to prepare and support the development of algorithms to solve instances of the new problems.

Memetic Algorithms

Sophisticated algorithms are required in order to solve instances of pickup and delivery planning problems. Section 4.1 provides a survey of algorithmic approaches that have been proposed for planning problems with pd-requests. Evolutionary Algorithms are globally acting optimization algorithms that sample a given search space for regions containing promising solutions. They have been successfully applied t o many complex combinatorial optimization problems. However, their usability for planning problems with pd-requests and LSP incorporation has seldom been investigated. An outline of the concepts of evolutionary algorithms is given in Sect. 4.2. Genetic Algorithms represent a special class of Evolutionary Algorithm implementations. Their usability is discussed in Sect. 4.3. Special attention is paid to concepts of Memetic Algorithms, which combines the exploration of genetic search with the exploitation of local search procedures (hill climbers). Ideas for the integration of both algorithm classes are discussed in Sect. 4.4.

4.1 Algorithmic Solving of Problems with PD-Requests The solving of the mathematical optimization models, which represent the problem of determining optimal transportation plans requires automatic search algorithms. Manual solving is not possible due to the very large number of possible solutions and the large number of complicated restrictions. Surveys on tested algorithms for planning problems with pd-requests are provided by Pankratz (2002), MitroviC-Mini6 (1998) and Dethloff (1994). Exact algorithms guarantee that a non-dominated solution s of the considered optimization model is identified. The combinatorial complexity of the models for pickup and delivery planning problems hinders the configuration and successful application of exact enumeration techniques such as branch&-bound algorithms (Williams, 1993) for instances with a large number of

50

4 Memetic Algorithms

requests. Analytical approaches like gradient methods fail because the PDSPtype problems do not come with the necessary requisites like linearity, continuity, convexity or separability. For this reason it is not surprising that only very few researchers have successfully applied exact optimization algorithms to pickup and delivery problems. Optimizing approaches for the case with one vehicle are investigated by Sexton and Choi (1986) and Psaraftis (1983). Sigurd et al. (2000), Domenjoud et al. (1998), Savelsbergh and Sol (1998) and Desrosiers et al. (1998) propose enumeration-based methods to identify optimal solutions for problems with more than one vehicle. Several heuristic approaches have been investigated for the generation of high quality transportation plans in the presence of pd-requests. Pankratz (2002) surveys heuristics, designed for a particular problem with pd-requests and classifies them into three categories. In successive approaches the requests are initially assigned t o a vehicle so that tours are set up. In the second step, routes are derived for the requests contained in a tour. Simultaneous construction approaches first order the request in a sequence. The requests are assigned to the available vehicles one after another according to the sequence. A request is immediately integrated into the so far set up existing route of the considered vehicle. Improvement approaches modify an existing solution of a problem instance so that its quality is improved. The setup and configuration of problem specific heuristics require a large amount of effort, but this heuristic is applicable only t o this particular problem. A re-use for other problem-types is typically not possible. Metaheuristics implement search ideas and concepts that are problem independent. They provide general ideas for the exploration of a search space that can be easily re-configured and applied to a huge variety of optimization problems. Blum and Roli (2001) classify the meta-strategies into trajectory methods (local search methods) and population-based approaches. A trajectory method generates a sequence of (feasible) solutions with increasing quality. The simplest form is called iterative improvement. An initial solution s is generated. Several copies of s are slightly modified and collected in the neighborhood N ( s ) of s. The solution s is now replaced by the member 5 E N(s)that provides the best improvement to the solution quality compared to s . Then the neighborhood N(s) of 5 is generated and again S is replaced by the most improved member taken from N ( 5 ) . The replacement is repeated until no improved solution can be found in the current neighborhood. When this is the case, the algorithm stops. The main problem of iterative improvement is its rapid convergence to a local optimum s*, whose neighborhood N ( s * ) does not contain a solution that has a better quality than s*. Different ideas have been implemented t o remedy this shortcoming and t o guide the trajectory out of a local optima. In the Simulated Annealing approach a decrease of the solution quality along the trajectory is partially accepted. An element 3 E N(s) is selected a t

4.1 Algorithmic Solving of Problems with PD-Requests

51

random. It becomes the next point of the trajectory if its quality is higher than the quality of s. If this is not the case then 3 is accepted as the next trajectory point only with a certain probability. This probability decreases with the number of visited solutions. The idea behind this approach is to allow an exploration of the search space in early stages of the search by tentatively visiting several local optima. In later stages, in order to save the achieved improvements, a decrease of the solution quality is seldom accepted. Comprehensive presentations of this idea are given by Aarts et al. (1997), Dowsland (1993) and van Laarhoven and Aarts (1987). Only Li and Lim (2001) and van der Bruggen et al. (1993) report about the application of the simulated annealing approach to a planning problem with pd-requests. Tabu Search follows another idea to prevent the trajectory from being stuck in local optima. A memory is maintained in which former modifications of the trajectory are saved. The information is used 1. To prohibit the selection of some elements of a neighborhood from becoming the next trajectory point and 2. To support the selection of promising next trajectory points from a neighborhood.

The Tabu Search idea was proposed initially by Hertz et al. (1997) and Glover and Laguna (1993). This variant of the iterative improvement has been applied successfully to problems with pd-requests by Cordeau and Laporte (2003), Lau and Liang (2001), Nanry and Barnes (2000), Greb (1998) and Rego and Roucairol (1995). Beside the lack of knowledge about adequate parameter settings, specialized local search approaches are often very hard to implement if additional constraints corrupt the set of feasible solutions. The necessary management of the maintained neighborhood and of the memory disadvantageously influences the overall runtime and performance of the overall algorithm. Population based algorithms develop two or more trajectories a t the same time. The decision for the next point within a trajectory is derived from all trajectories, after they have all been evaluated. The information exchange allows for a rapid exploration of the search space. Two main population based algorithm classes are distinguished: Ant Algorithms (Dorigo et al., 1999) and evolutionary algorithms. The idea behind the Ant Algorithm meta-strategy is to create a selfevolving central knowledge base (memory) whose information can be used for selecting the next point in the considered trajectory. A sequence of decisions determines a solution (trajectory point). For example, the shortest path between two locations i+, i- in a transport network is given by the sequence i+, il, . . . , in,, i-. Having chosen the intermediate location il, the necessary decision determines the next visited location denoted by il+1.The decision which of the locations should be visited next is derived with the support of knowledge contained in the memory. On the other hand, the value of a decision based upon the information taken from the memory is

52


evaluated by means of the quality of the completed solution. This results in a continuous update and evolution of the information in the memory during the construction of tentative solutions. In order to speed up the derivation of necessary knowledge, several solutions are constructed in parallel. Doerner et al. (2000) apply an Ant Algorithm to solve a pickup and delivery planning problem with full truckloads.

4.2 Evolutionary Algorithms Evolution means the continuous process of development and change. The most prominent example is given by the development of life in natural systems. A conglomerate (population) of entities (individuals) is subject of the evolution. Each individual is completely determined by a collection of attributes. The evolutionary process continuously changes the individuals of the population by varying their attributes and, as a result, modifies the population. The essential ingredients of evolution are (Fogel, 2000): reproduction, random variation, competition and selection. Reproduction describes the generation of new individuals that are formed from attributes of two or more individuals taken from the maintained population. It represents an exchange of attributes between members of the population that is not enriched with information from outside the population. The implementation of properties into a population that cannot be generated by reproduction is called mutation. It is typically performed establishing slight random variations of existing attributes. Each individual in the population is prepared by its attributes (properties) to cope with its environment. Some individuals are well equipped, whereas others are not able to deal with the other members of the population or the system properties that determine the environment. The well-equipped individuals tend to replace the other individuals. Low quality individuals have only a slight chance to mate and to transmit their properties into new individuals. Averagely equipped individuals compete to stay in the population. In several cases, they transmit their attributes into new individuals, whereas in other cases such an individual dies and does not contribute to the setup of subsequent populations. A selection mechanism chooses those individuals, which will be used to transfer their attributes into subsequent population stages. The evolution of natural (sub)systems has been successively proven to be able to protect a population faced with several challenges:

1. The population is able to cope with a modified environment shortly after the environment state has been changed (adaptation). 2. Evolution is "intelligent enough" to solve new problems by extrapolating solution strategies from previously solved problems. 3. Evolution produces individuals that act at high efficiency so that the resources of the environment are used in an optimal manner.

4.2 Evolutionary Algorithms

53

These observations motivate researchers from different fields to implement computer-based simulations of evolution and to use the properties of evolution to solve as yet intractable problems in their research areas. Fogel (2000) identifies the main fields for the application of simulated evolution. Biologists need a simulation model for natural evolution in order t o get deeper insights and a deeper understanding of different biological systems. Software engineers are often faced with the problem of determining algorithm or software system configurations for different, yet similar, applications. They hope t o use simulated evolution for supporting and for promoting the automatic adaptation of software systems if the underlying real world problem requires such modifications. The most promising application field for simulated evolution seems t o be optimization (Yao, 2002). Optimization means t o identify solution instances, represented by individuals, which fulfill a given goal in the best manner. Applying evolution to find the best solution of a certain problem means to implement a feedback strategy that allows for answering the question whether a certain individual fulfills the goal a t a higher or lower level than the other individuals in the population. Optimization therefore describes a directed evolution process. If specialized and proven algorithms are not available to solve complex and highly constrained (optimization) problems, or if a problem cannot be transformed into a highly abstracted formulation that allows the application of these methods, simulated evolution approaches have proven their applicability for a large number of applications in single or multi-objective optimization problems. Evolutionary Algorithms have been applied to both continuous (numerical) and discrete (combinatorial) optimization problems. For a survey, refer to the article of Beasley (2000) and the book of Sarker et al. (2002). Algorithms that mimic evolution are subsumed under the term Evolutionary Algorithms (Yao, 2002). It unifies three main classes of algorithms, which use ideas found in evolution: Evolutionary Programming (Porto, 2000), Evolution Strategies (Rudolph, 2000) and Genetic Algorithms (Eshelman, 2000), (Michaelewicz, 1996), (Goldberg, 1989). These algorithms differ in two aspects from conventional mathematical or heuristic optimization approaches (Yao, 2002): First, they are populationbased and second, a n ongoing existence of communication and information exchange mechanisms among the individuals in the population is implemented. The communication is established by means of so-called search operators that recombine individuals to new individuals (offspring) by merging and interchanging attribute values and by means of the competition among the individuals in a population. These two properties distinguish evolutionary algorithms from other pseudo-parallel algorithms such as parallel tabu search (Voss, 1993) or local search restart-methods. In these methods, the communication is typically interrupted for longer periods so that a consolidation of the individually acquired material is performed only from time t o time.

54


The general framework of an Evolutionary Algorithm following Yao (2002) is presented in Fig. 4.1. Initially, the population counter i is initialized (1). Then the first population is generated (2). Therefore, individuals representing points in the search space are randomly generated. The fitness, that describes the quality of an individual, is determined for each member of the current population (4). Based on these values, individuals with high fitness are then selected in order to serve as donors for the material from which the subsequent population is built (5). Search operators (recombination or mutation) construct new individuals combining the selected parental individuals (6). The population counter is increased (7). If a specified termination criterion has not yet been fulfilled, a new population is generated from the recently set up one. Otherwise, the iteration stops. (1) set i := 0; (2) generate the initial population P(i) at random; (3) REPEAT

(4)

evaluate the fitness of each individual in P(i);

(5) select parents from P(i) based on their fitness; (6) apply search operators to the parents and produce population P(i (7) set i := i + 1; (8) UNTIL termination criterion is reached;

+ 1);

Fig. 4.1. Framework of an Evolutionary Algorithm

In Evolutionary Programming (Porto, 2000) an individual's internal representation is close to the corresponding problem. A vector of real numbers often represents a point in the search space. Mutation changes the values in one or more components of the vector at random. A probabilistic selection scheme is used to choose the parental individuals. Recombination is not performed. The ingredients of Evolution Strategies (Rudolph, 2000) are similar to those of Evolutionary Programming. Again, a real valued vector represents a solution instance. Mutation is the same as introduced above for Evolutionary Programming. Two differences are observed. First, the selection procedure must be deterministic. Secondly, mutation and also recombination are applied to generate new individuals. In a traditional Genetic Algorithm (GA), solution instances are coded into (binary) strings of fixed length. Mutation and recombination (crossover) are applied to produce new binary strings from one or more parental individuals. Both operators as well as the incorporated selection scheme are probabilistic. In the next section, GAS are presented in more detail.

4.3 Genetic Algorithms

55

4.3 Genetic Algorithms GAS are the most prominent implementations using evolutionary techniques. They have been successively applied to an extensive number of problems, often of the optimization type. The application field includes numerical and cornbinatorial optimization problems from chemistry and physics, control, economics, electronics and VLSI design, engineering, logistics, manufacturing and Operations Research. An indexed bibliography on GA implementations has been compiled by Alander (1999). The subject of this section is the presentation of the general framework of GAS and the compilation of its strengths and weaknesses.

4.3.1 Terminus Technici An individual is an entity in which a single solution for a problem instance is coded (cf. Fig. 4.2). The specifications of the coded instance are stored in one or more chromosome(s). A chromosome is an ordered sequence of genes. Each gene carries the instantiation value of a single parameter of the coded solution. The value is called allele and the position of the gene in the chromosome is denoted as locus. The Cartesian product of the domains of the evolvable parameters spans the genetic search space. A GA maintains a population of individuals. The population is iteratively replaced by a new population of individuals generated from individuals of the previous population. The i-th generated population is named i-th generation. internal representation

decoding

original problem

solution instance

I -

genotype

encoding

phenotype

Fig. 4.2. Internal and external problem representation

The representation of a problem describes the instructions how to encode a solution instance for the problem (the phenotype description) into a chromosome description (genotype) and how to decode a chromosome into a solution for the problem at hand. The genotype consists of the genetic information (genetic material) that determines an individual. To determine the competitiveness of an individual its genotype is decoded into the corresponding phenotype. The usability of the solution instance is assessed and expressed in a numerical value (the fitness value) that is assigned to the genotype. GAS originally use a binary representation. A solution instance is coded in a string consisting of a sequence of '0' and '1'. This representation allows

56


the derivation of some theoretical investigations which aim a t explaining why genetic algorithms work. The results are subsumed under the topic 'schema theorem' (Goldberg, 1989). However, it has been proven in several investigations that more problem-related representations like real-valued vectors or strings do not restrict the success of genetic searches and that they can be even an accelerating feature to improve the capability of GAS to find high quality solutions. For a survey of different successfully applied representations, refer to the chapters 14-21 in Back et al. (2000a). Michaelewicz (1996) even proposes giving up the strict string representation of a solution instance and instead using an arbitrary data structure that is capable of giving a complete description of a problem solution instance without additional coding. 4.3.2 General Framework

Figure 4.3 shows the framework of a GA. Let P ( t ) denote the population at iteration stage t (the t-th generation or population). The following steps are iteratively repeated until a termination criterion is satisfied. The initial population of a GA consists of genotypes that represent different points in the solution space of the considered problem. Typically, these points are completely or partially generated at random in order to provide a scattered sample over the available search space (2). (1) (2) (3) (4) (5) (6) (7)

t := 0; initialize P(t); evaluate individuals in P ( t ) ; while (termination condition not satisfied) do select mating pool M(t); recombine and mutate individuals in M(t) forming C ( t ) ; evaluate individuals in C(t); forming P ( t 1) from P(t) and C(t); (8) (9) destroy M ( t ) and C(t); (10) t := t 1; (11) end;

+

+

Fig. 4.3. Genetic Algorithm Framework

+

To generate the new population P ( t l),several individuals are selected for the mating-pool M ( t ) (5). The individuals in M ( t ) get the chance to transmit their genetic information into the next population. Selection is typically realized by roulette-wheel-selection. The probability of individual i of becomThe absolute fitness ing a member of the mating pool is proportional to of i is fi and the population fitness is f := CiGP(,) f i . Other, often problem specific selection realizations are surveyed in chapters 22-27 in Back et al. (2000a).

4.


57

A temporal population C ( t ) is opened and filled with offspring individuals (6). An offspring individual carries genetic information from a t least one individual of the mating pool. Typically, two mating individuals are randomly chosen and their genetic material is composed into one or two new genotypes. The combination of individuals t o offspring is called crossover. Crossover is one backbone feature of a GA, because it enables for the merging of the genetic material of a t least two individuals to derive one or more offspring. Crossover is applied in the hope that combinations of promising individuals are also promising or even of higher quality. New genetic material cannot be inserted into the population by the application of crossover. In general, the genetic material within a population does not allow for the composition of all possible genetic samples. Randomly selected alleles are varied replacing the current values with other values randomly taken from the corresponding domain(s) of the considered gene(s). These slight modifications are called mutations. They ensure that no genetic information is definitively lost if it does not appear in a population a t a certain stage. Since mutation counteracts crossover, it is executed only slightly. For a compilation of different genetic operators, refer to chapters 31-34 in Back et al. (2000a). As soon as C(t) is completed all individuals are evaluated in order t o prepare the selection step in the next iteration (7). Several individuals from P ( t ) are replaced by individuals from C(t). This new collection forms the next population P ( t 1). The instructions that describe which individuals are replaced are called the recombination scheme or population model (8). At the end of an iteration, the temporal population and the mating pool are destroyed (9). The iteration counter t is increased by one (10). Several parameters are necessary t o complete the configuration of a genetic algorithm. The population size determines the number of maintained individuals in each population. The crossover probability describes the frequency of the application of recombination operators whereas the mutation probability describes the frequency in which the mutation procedure is invoked. The iteration stops as soon as a given number of iterations have been passed, if a given time span has passed or if no improvements have been observed within the last iterations (4). A population has been converged if the frequency of the genetic information does not significantly change any more and no new genetic information is able t o survive. If the genetic material contained in a converged population represents high quality phenotypes then the genetic search was successful.

+

4.3.3 Applicability of Genetic Search The most important feature for a successful application of a GA seems to be the usage of a n adequate genotype and phenotype equivalence. Such equiv-

58


alence is canonically given for the whole class of combinatorial optimization problems and many other problems, which aim at determining an appropriate instantiation of different decision variables. Each gene then carries a value taken from the (discrete) domain of the corresponding decision variable. The fitness value is often given directly by using the objective function associated with the optimization problem at hand. Once a GA is implemented for a closely defined problem class it is often possible to use the same or slightly modified combinations of components to deal with a number of similar problem classes. For example, the implementation of a permutation-representation permits not only the application of a GA framework for TSP-type problems, but also allows for resource scheduling problems (Whitley, 2000). From the point of view of the application, GAS are very promising because they need only a little preparation and the necessary properties of a problem that allow for the application of a GA are a priori easy to fulfill. Furthermore, the concept of maintaining a population of solutions instead of a single solution instance offers two main promising features. First, the last population of the GA includes not only one solution instance of superior quality but also several different instances. If a proposed solution instance cannot be implemented or if it is not accepted, other proposals are immediately available. Secondly, a GA starts from different initial points so that excluding the unprofitable points in early selection or recombination operations can often compensate the selection of a bad initial point.

4.3.4 Limits of the Genetic Search Three deficiencies of GAS are observed: the population does not converge in any region of the search space, the population converges prematurely or the GA produces individuals which are infeasible with respect to constraints arising from the currently considered problem.

Missing Convergence If the number of genes within the chromosome is very large then the convergence velocity is jeopardized as it could be too slow. More concretely, promising genetic samples do not dominate the genetic information maintained in the population. Crossover, mutation and selection do not use the promising samples because their frequency of occurrence in the population is too small for a statistical dominance. This is a population related deficiency. The variation (reduction) of the number of maintained individuals sometimes remedies the observed lack of convergence. A second reason for missing convergence of a population is epistasis (Naudts et al., 1997; Mattfeld, 1996): the change of the allele a t a single locus cannot be realized without modifications of alleles at other locus. Such


59

genes are linked in a certain way. Epistasis contradicts the building block hypothesis (Goldberg, 1989), which assumes that highly-suitable individuals are composed of independent small, but highly-suitable, chromosome fragments (building blocks). Although the validation of the building block hypothesis remains an open question, it has been observed, that epistasis typically compromises the genetic search (Beasley et al., 1993).

Premature Convergence If a GA converges prematurely, the average fitness of the evolved individuals in the population converges a t an insufficient (or unexpected low) level. It is observed that in some instances even simple heuristics produce better results than the genetic search is able t o find. Premature convergence has two causes. The first one is caused in an inappropriate fitness measure. Individuals, whose genetic code differs only slightly from the code of other individuals, but whose phenotypes are similar, are overrated. They get an over-proportional chance of being selected for transmitting their genetic information into the next population. A small number of overrated individuals, or even one individuals, quickly flood the population. The genetic diversity is reduced. The second reason for premature convergence concerns the used representation. For many problems it is not possible to find a bijective string coding of solution instances. Let P be the set of phenotype representations of solutions of an optimization problem and let denote the set of possible representations for solution instances. The function D : 6 -+ P decodes each genotype solution into a phenotype instance. If D is injective, then different genotypes are decoded into different phenotypes. Otherwise different genotypes are assigned to the same phenotype. In the latter case, there is the danger that offspring are assigned t o the same phenotype as one of their parents. In this case, the application of crossover or mutation does not contribute t o the evolution of the population. Such a representation is called redundant (Rothlauf, 2003). If there is a large level of redundancy present i.e. if the number of genotypes assigned to the same phenotype is large then the application of crossover does not support the evolution towards optimal phenotype solution instances. In this case, the application of the genetic operators does not produce new variations of genetic code, which can prove their capability.

Feasibility Typically, a combinatorial optimization problem comes with a large family of intricating restrictions. These constraints separate the set of genotypes into two disjunctive subsets. The first one is the set of feasible genotypes that can be decoded into feasible phenotypes whereas the second one contains all genotypes whose corresponding phenotypes violates at least one of the problem restrictions (infeasible genotypes). In general, GAS are not able t o maintain a population of feasible individuals throughout the complete iteration process

60


if constraints shrink the search space spanned by the Cartesian product of the domains of the decision variables. Typically, the detection of a n infeasible individual is easy but there is no common mechanism for preventing the occurrence of such a population member. Several ideas for handling infeasibilities are proposed. A recent survey is given in Coello (2002). In a penalty approach the fitness value of a genotype is corrupted, so that its fitness is disadvantageously modified (Smith and Coit, 2000; Coit et al., 1996). If crossover and mutation preserve the feasibility of the decoded genotype, then the generation of a feasible initial population ensures that each generated offspring corresponds to a feasible problem solution instance (Michaelewicz, 2000). A decoder is a construction algorithm. It generates a solution instance according t o a given construction instruction. Instead of making the solutions subject of evolution, the instruction is optimized (Michalewicz, 2000a). To determine the fitness of a construction plan, a tentative solution instance is generated invoking the decoder instructed by the current construction plan. The decoder is often capable of avoiding generating infeasible solutions. Infeasible individuals can be repaired if the reasons for the infeasibilities are known and if it is possible to correct these deficiencies (Michalewicz, 2000b). The next section presents some ideas for incorporating mending approaches to repair infeasibilities. Such approaches are of great interest because most of the real world problems come along with complex and complicated restrictions that contradict the evolution of appropriate genetic information.

4.4 Repairing and Improving the Genetic Code A one-to-one relation between the set of feasible solution instances (phenotypes) and the genotypes is often not available. Genetic Operators produce offspring whose corresponding phenotypes are infeasible with respect t o the problem a t hand, even if all used parental individuals respect the whole set of problem-related constraints. Repairing a genotype p with a corresponding infeasible phenotype means transforming p into another genotype p' so that the corresponding phenotype of p' is a feasible one and complies with the problem's restrictions. Figure 4.4 shows the genetic search space G. Only the subset F maintains individuals whose corresponding solution instances are feasible. The individual p does not possess a phenotype that respects all the problem's restrictions, but its repaired version p' belongs to F. Figure 4.5 depicts a prototype of a repair procedure. Initially, the genotype p is decoded (2), the iteration step counter i is set t o zero (3) and p is checked for feasibility (4). If feasibility is detected p remains unchanged. Otherwise, p is iterated into p' ( 5 ) . Then p is updated by p' (6) and the feasibility check of the next iteration is prepared by decoding p (7). The counter is increased

4.4 Repairing and Improving the Genetic Code

61

Fig. 4.4. Local improvement towards the set of genotypes with a corresponding feasible phenotype

(8). The iteration is repeated until a feasible genotype p E F is generated or the given number of allowed modification steps is reached. In the latter case, repair() does not return a feasible modification of the original genotype passed to repair(). The genotype q cannot be repaired within the maximum allowed number of iterations. The returned genotype q' does not belong t o F. However, the partially repaired genotype is often more 'closer' to F in the sense, that the number of detected constraint violations is reduced. In Fig. 4.4 the individual q cannot be completely repaired but its modified version q' has been brought closer to F. Other techniques have been proposed in order to cope with this unrealizable solution (cf. Subsection 4.3.3) . (1) function repair (p) (2) D(p) :=decode(p); (3) i := 0; (4) while [(D(p) is infeasible) and (i < i,,,)] (5) pt:=iterate(p); (6) p (7) D(p):=decode(p); (8) i : = i + l ; (9) end while; (10) return(p); Fig. 4.5. Prototype of a repair algorithm The setup of the function iterate() depends upon the considered problem and upon the chosen problem representation. A similar algorithm can be used in order to improve the quality (fitness) of a feasible individual. Instead of following the goal of transforming an in-

62


dividual p into its 'nearest feasible neighbor' an attempt is made to improve the quality of p as much as possible. Therefore, p is modified so that the new individual remains feasible and has an increased fitness value in comparison to the original individual p. The framework of such an improvement algorithm is shown in Fig. 4.6. Initially, the phenotype D(p) is generated and the iteration counter i is set is derived from p applying the function to 0. Then the iterated individual is feasible and if iterate() in line (5). Its phenotype is generated (6). If its fitness-value f (D({)) dominates the fitness-value f (D(p)) associated with individual p then p' replaces p (8). Each new individual is altered in the next iteration (9). The iteration is repeated until a maximum number is reached (4). If an infeasible individual is detected or if the modified individual' fitness is not better than the fitness of the unaltered individual then the procedure is terminated prematurely (11).Therefore, the improve()-procedure can be understood as a hill-climber that exploits a particular subspace of the solution space. (1) (2) (3) (4) (5) (6)

function improve ( p ) D(p):=decode(p); i := 0; while ( i < imaz) pl:=iterate(p); D(i):=decode(p'); if [ ( D ( ~ is ' ) feasible) and (f (7) (8) p:=i; i := i 1; (9) (10) else i .- zmaz; . (11) (12) end while; (13) return(p);

( ~ dominates ( i ) )f ( D ( P ) ) ) ]

+

Fig. 4.6. Prototype o f an improvement algorithm

Different ideas are proposed in order to incorporate a repair or an improvement procedure into the GA-fr amework. 1. The population is evolved through the predefined number of iterations. The repair or improvement procedure is applied only to individuals in the final population. 2. The repair procedure is invoked as a directed mutation operator. It is aimed at moving infeasible individuals towards feasibility and t o modify or improve feasible individuals without violating constraints (Schonberger et al., 2004). 3. In order to evaluate the potential fitness of an individual p, this member of the population is tentatively repaired. The original individual is then eval-

4.4 Repairing and Improving the Genetic Code

63

uated by means of the fitness obtained from its related modified version p'. However, p' does not replace p in the population. Although the genetic information is not altered into the population this kind of evaluation often improves the genetic search. The observed effect is called Baldwin Effect (Withley et al., 1994). 4. If the replacement of p or the partial alternation is fed back into the maintained population, then the information, individually acquired in the repair procedure, is available for inheritance. This disturbed evolution process is called Lamarckian Evolution (Joines and Kay, 2002). In pure genetic search, the genetic information determines the instantiation of a solution instance. The incorporation of a repair or improvement function relaxes this direct equivalence. Genetic information that comes along within a particular individual can be interpreted as an alterable proposal or as the starting point for further improvements. Such a n alterable information unit is called a meme (Burke et al., 1996) and a genetic algorithm that incorporates a kind of repair procedure is called a Memetic Algorithm, (MA) (Moscato, 1989), hybrid genetic search (Joines and Kay, 2002) or genetic local search (Ibaraki, 1997). Figure 4.7 shows the general framework of a MA that implements Lamarckian Evolution by applying the modify ()-function that enables the repair or improvement of individuals. After a new population P ( t 1) has been completely set up, each individual p is repaired to p' that replaces p in P(t 1).

+

+

t := 0; initialize P ( t ) ; evaluate individuals in P(t); while (termination condition not satisfied) do select mating pool M(t); recombine and mutate individuals in M(t) forming C(t); (7) evaluate individuals in C(t); (8) forming P ( t + 1) from P(t) and C(t); for each p E P ( t + 1) do (9) (10) p':=modify(p); (11) p:=p ; (12) destroy M(t) and C(t); (13) t := t + 1; (14) end; (1) (2) (3) (4) (5) (6)

Fig. 4.7. Framework of a Memetic Algorithm implementing Lamarckian Evolution

Often the chromosome does not carry the genotype of a solution instance, but a construction plan that parameterizes a construction procedure as proposed in the decoder approach. In such a case, the repair feature is part of the decoder. A GA that makes use of a decoder in order to generate solution instances of the considered problem is also referred t o as an MA.

64


4.5 Conclusions MAS are hybrid approaches of GAS and hill-climbing or repair procedures that combine the exploration abilities of Genetic Search with exploiting hillclimbing procedures. They are promising approaches for severely constrained combinatorial optimization problems. The robust genetic search is supported by means of often powerful but simple hill-climbing or repair procedures that help to remedy the two main weaknesses of GAS: limited ability to exploit promising regions of the genetic search space and effective treatment of infeasibilities originating from restrictions belonging to (combinatorial) optimization problems.

Memetic Algorithm Vehicle Routing

This section surveys investigated MA approaches to solving sophisticated vehicle routing and scheduling problems. These approaches are classified into several categories. In Genetic Sequencing (Sect. 5.1) only promising precedence relations between customer requests are evolved by genetic search approaches. The assignment of requests to vehicles is left for so-called cluster building heuristics, which play the role of the local search (hill-climber) procedure in the presented MAS. Genetic Clustering (Sect. 5.2) aims at evolving promising partitions of requests, which means that only the assignment of requests to vehicles is subject of evolutionary improvement. The determination of the routes is left for route construction heuristics. Both decisions about sequencing and clustering of requests are subject of evolution in Combined Genetic Sequencing and Clustering (Sect. 5.3). Sequencing, Clustering and combined approaches use the components of genetic search that are proposed in the first development stages of genetic algorithm research. Recent investigations incorporate more general population systems. They refrain from the adaptation of one-string-chromosomes and often maintain several independently evolved populations to generate individuals with special properties. A survey of these so-called advanced-MA approaches is provided in Sect. 5.4.

5.1 Genetic Sequencing The goal of Genetic Sequencing is to determine promising visiting sequences for serving customer requests. Such a sequence is determined by a permutation of the locations to be visited. A permutation-based representation is appropriate. The allele of a gene at the certain locus i refers to a customer location to be visited after the locations in previous loci. The route of a vehicle

66

5 Memetic Algorithm Vehicle Routing

through a subset of the requests is determined by the permutation proposed in the individual. Genetic Sequencing is relevant for both single-vehicle and multi-vehicle problems. For one-vehicle problems the representation as a permutation determines a complete solution instance. Merz and F'reisleben (2001) investigate several MA approaches for the TSP whereas Tasgetiren and Smith (2000) investigate the Orienteering Problem, in which each location is allowed t o be left unvisited and the starting and the terminating location do not necessarily coincide. If time windows have to be considered, several permutations of the customer visits cannot be implemented because they do not respect the specified time window constraints. Repair procedures are recommended t o ensure the overall feasibility of the generated permutations. genotype

cluster builder --f

(permutation)

phenotype --f

(assignment rules)

(collection of routes)

Fig. 5.1. Genetic Sequencing: the cluster-builder divides the permutation of re-

quests into several parts, the i-th subsequence represents the route of the i-th vehicle Generally, a single vehicle cannot visit all locations. The visits have to be distributed among the different available vehicles, but the information about the assignment of requests t o vehicles cannot be derived from the permutation. Thus, a clear decoding of a chromosome into a phenotype solution is not possible. Rules that are not subject of evolution, so-called cluster builders, are necessary in order t o distribute the requests among the vehicles. The assignment of requests t o the available vehicles is left for a clustering procedure, which additionally ensures that no capacity constraints are violated and that maximum route lengths are not exceeded. The cluster builder is part of the problem representation (see Fig. 5.1). It applies the clustering rules in order to determine the tours of the vehicles. The stored permutations are appropriate for determining the visiting orders for subsets of customer locations. This information is used to generate the route of a single vehicle. In the following, this review focuses on the more sophisticated multi-vehicle problems. To encode a solution of a VRP or VRPTW instance in the chromosome, the depot is removed from all the routes. The remaining subroutes are subsequently stored in the chromosome. Blanton and Wainwright (1993) propose the following route first-cluster second strategy. Let a permutation of n customer requests stored in a chromosome. These requests have to be assigned to k vehicles. The first k requests are equally distributed among the vehicles and become their first visit. The remaining n - k requests are successively assigned to the vehicles. The next so far unconsidered request from the permutation is tentatively inserted as the

5.1 Genetic Sequencing

67

latest visited customer into each route. It is then definitively assigned t o the vehicle in which it leads t o least additional costs. This assignment procedure maintains the precedence relations between the customers in a route as proposed in the genotype. If the insertion of a request into a particular route leads to the violation of a given time or capacity constraint then the corresponding vehicle remains unconsidered for the current request and the next vehicle is tentatively selected. In Fig. 5.2, a n example for the application of this cluster builder is shown. The order in which a subset of requests has t o be visited is given by the request permutation stored in the chromosome (genotype). The available fleet consists of the four vehicles {1,2,3,4). Initially, the first four requests found in the four left genes in the chromosome are distributed among the vehicles. Each vehicle receives exactly one request t o serve. Request 3 is assigned to vehicle 1, request 7 to vehicle 2, request 2 to vehicle 3 and request 4 is served by vehicle 4. The remaining requests are assigned t o the vehicles in the sequence given in the chromosome: request 5 (to vehicle I ) , 6 (to vehicle 2), 8 (to vehicle 4) and 1 (to vehicle 1). Following this route construction scheme, all vehicles are considered and each serves a t least one request. phenotype

genotype 1317121415161 81 11

/ n

cluster builder

1 route of vehicle 1 17161 route of vehicle 2

14181

route of vehicle 3 route of vehicle 4

Fig. 5.2. Decoding of a permutation with the cluster builder of Blanton and Wainwright (1993)

The cluster builder of Blanton and Wainwright (1993) aims a t reducing the average length of the routes, so that the number of used vehicles is as large as possible. However, it is often recommended t o keep the number of routes as small as possible in order to prevent pendulum routes. Zhu (2000) and Kopfer et al. (1994) propose another idea t o cluster the sorted requests. Their cluster builder aims a t loading each used vehicle with as much goods as possible in order t o reduce the number of vehicles that have to travel. Following a n evolved permutation, the requests are successively assigned t o the first vehicle until a capacity, a travel time or a time windows constraint violation occurs. In such a case the violation-causing request is assigned to the next vehicle. All following requests in the permutation are also assigned to this vehicle until once again a constraint violation is detected that stipulates the incorporation of a third vehicle, and so on. An example is shown for a n eight-customer problem in Fig. 5.3. The visiting order 0 -, 3 -, 7 -+ 2 -+ 4 -+ 5 -+ 6 --+ 8 -. 1 --, 0 is coded in the

68

5 Memetic Algorithm Vehicle Routing phenotype

1 route of vehicle 1 genotype

'

1317/2141516/811

1411

route of vehicle 2 route of vehicle 3 route of vehicle 4

cluster builder Fig. 5.3. Decoding of a customer permutation with the cluster builder of Kopfer

et al. (1994) genotype. A '0' represents a visit a t the depot. After the cluster builder has been applied the following routes are determined: 0 -, 3 4 7 -+ 2 -+ 0, 044 5 4 0, 0 4 6 0 and 0 4 8 4 1 -+ 0. All these routes are generated according to the proposed visiting sequences, but the decisions about the grouping of the requests is left to the cluster builder that also ensures that capacity and travel length feasibility is kept. Ochi et al. (1998) propose a similar approach. A request is assigned to the vehicle with least remaining capacity that can serve the request. As a departure from the idea found in Kopfer et al. (1994), a vehicle for which a capacity or travel duration constraint violation has been detected in a previous step is re-considered in subsequent steps. Low-volume requests are successively used to fill the so far unused storage space. None of the presented cluster builders are able to guarantee that the decoded set of routes is the most advantageous solution that can be generated while respecting the evolved precedence relations. Obviously, a cluster builder runs the risk of disrupting the composition of promising request combinations in a route.

5.2 Genetic Clustering Genetic Clustering addresses the assignment of requests to the available vehicles. It is of interest only for multi-vehicle problems. The most intuitive and canonical assignment coding has been applied first by Baker and Ayechew (2003). Assume a VRP-type problem with n customer requests and m available vehicles. A solution instance is coded in a chromosome of n genes. The allele of the i-th gene is taken from the set { I , . . . ,m) and determines the vehicle to which request number i is assigned. The tours of the vehicles are taken from the individual. For each vehicle the route is determined by solving a TSP-instance (if required with additional constraints) consisting of the requests received according to the genotype proposal. An example of this decoding scheme is given in Fig. 5.4. Six requests {1,2,3,4,5,6} have t o be distributed among between vehicles {1,2}. According to the genotype, request 1 and request 3 are assigned t o vehicle 1, whereas

5.2 Genetic Clustering

69

the remaining requests are assigned to the second vehicle. The resulting tours are {1,3) and {2,4,5,6). For each tour a TSP-instance is solved leading to the routes 0 -+ 3 4 1 -+ 0 and 0 -+ 4 4 6 + 2 4 5 4 0 as shown on the bottom line in Fig. 5.4. More sophisticated procedures that partially reassign requests are needed in order to ensure the feasibility of the generated collection of routes.

genotype

111211(212121 I

genetic clustering tours routing routes

I

1

m] 1

ppTpqFq

Fig. 5.4. Decoding of a vehicle assignment genotype into a phenotype as proposed by Baker and Ayechew (2003)

Another cluster first-route second approach, called Genetic Sectoring, has been proposed by Thangiah (1995) and Thangiah et al. (1995). It exploits the geographical distribution of the available customer locations for the building of promising clusters (tours). Initially, the polar coordinates are calculated for the request locations. The requests are then sorted by decreasing angle component of their polar coordinates. Let aminbe the lowest polar angle and a,,, the highest polar angle. The relevant sector between aminand a,,, contains all customer requests. Genetic Sectoring determines partitions of the relevant sector into several smaller subsectors. The i-th vehicle serves all requests, whose corresponding locations are situated in the i-th subsector. The i-th sector is represented by the offset angle ai. Vehicle 1's tour contains all requests with polar angles a E [amin,amin a1[, the second tour comprises all requests with the polar angle a E [amin al, amin a1 a 2 [ and so on. An individual consists of one chromosome of length k (the number of available vehicles). The i-th gene in the chromosome carries the offset angle determining the i-th sector and therefore indirectly determines the set of requests assigned to the i-th vehicle. To decode a chromosome representation into a solution of the problem a t hand, two steps are performed. At first, the polar angles are taken from the chromosome and the corresponding subclusters are instantiated. Afterwards, the requests are assigned to the vehicles according to their subcluster affiliation as described above. To evaluate such a clustering, routes are constructed tentatively, applying a route construction heuristic to the tour of the currently considered vehicles. This procedure incorporates several features to ensure the capacity and the

+ +

+ +

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route length feasibility. It is hardly possible to describe such a procedure in general terms since it is problem dependent at a high level. For each route, the length is determined and the clustering chromosome is then evaluated with the sum of the lengths of the routes. A subsequent route improvement is often recommended in order to obtain capacity and time window feasibility and a reduced travel distance. An example for the application of Genetic Sectoring representation is shown in Fig. 5.5 by means of a four-vehicle problem. The customer locations are situated in the relevant sector Sobetween the polar angles a , f f and a,ff a1 . . . as. In the clustering step, So is partitioned into the subsectors S1,. . . , Sq, spanned by the polar angles a l , . . . ,ad. Next, the requests within a certain subcluster are sequenced by a route generation heuristic (see the right coordinate system).

+ +

+

genotype

route construction

.Fig. 5.5. Genetic Sectoring Representation Scheme

Pankratz (2002) applies a genetic clustering approach to a pickup and delivery problem with time windows. Its goal is to distribute the available requests among the vehicles. Furthermore, it permits the assignment of requests to LSPs. Let n be the number of vehicles and n be the number of available LSPs. Each chromosome has the length n n and the i-th allele refers to the set of requests that represents those requests that are assigned to vehicle/LSP i. In a subsequent step the phenotype is determined completely. Therefore, the routes for the own vehicles are determined using a sophisticated construction heuristic. Requests selected to be served by a certain LSP are bundled so that the necessary freight charge is minimized. The routing and the bundling tasks are not subject to evolution. This is the only known MA-approach for a simultaneous route, mode selection and freight charge optimization problem.

+

5.4 Advanced MA-Approaches: The State-of-the-Art

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5.3 Combined Genetic Sequencing and Clustering A string-based problem representation that merges both the clustering and the routing aspect into one chromosome is investigated by Machado et al. (2002) who proposes the introduction of separation genes for indicating the end of a route. These genes are treated like the other genes that refer to requests. Their positions are evolved simultaneously to the sequence of requests. In order to apply permutation-oriented crossover and mutation operators, the k - 1 separation genes refer to the dummy requests n 1,. . . , n (k - 1).The alleles within the genes of a chromosome form a permutation of the union set of dummy and customer requests.

+

+

Fig. 5.6. Permutation-based representation of a VRP solution with three route separator genes in the loci 4, 7 and 9

In Fig. 5.6, a chromosome of length 11 carrying 8 request genes and three separator genes is shown. The separator genes refer to the dummy requests 9, 10 and 11. The (tentative) route of the first vehicle is 0 4 3 4 7 4 2 --, 0, the one of the second vehicle is 0 -i 4 -t 5 -+ 0, the third vehicle follows the route 0 4 6 --, 0 and the last vehicle travels along 0 + 8 4 1 -i 0. The routes described by the positions of the separator genes are tentative because they generally exceed the allowed travel time or capacity constraints are ignored. A repair heuristic has to move each error-causing request into another route. The alleles of two or more separation genes are exchangeable without varying the phenotype of the considered individual. This redundancy can lead to performance problems. To prevent this obstacle, crossover and mutation operators have to cope with the existence of dummy requests. However, combined genetic sequencing and clustering has not yet received any significantly interest.

5.4 Advanced MA- Approaches: The State-of-t he-Art Several modifications and extensions have been proposed to improve and accelerate the power, speed and applicability of MAS. These developments are subsumed under the term Advanced Memetic Algorithms. In this section, two special enhancements to MAS, which are relevant for an application to vehicle routing and scheduling problems, are analyzed: multi-chromosome representations and decomposition approaches.

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In a multi-chromosome representation of a VRP solution instance each single path of a vehicle is coded in a chromosome. A multi-chromosome individual includes all chromosomes of the available vehicles. Subsection 5.4.1 is dedicated to illustrating MA-approaches with multi-chromosome individuals. Special attention is paid to the discussion of adequate crossover and mutation operators that are needed to support the chosen multi-chromosome representation. All previously presented approaches construct a phenotype solution instance from a single individual. In an MA with decomposition, a solution instance is determined from material of two or more individuals. Typically, each used individual provides genetic material that has been evolved for a special purpose. The general problem with the application of MAS t o vehicle routing problems is the appearance of infeasible instances during the evolution process. A route length reduction typically leads to a n increase in capacity exceeding as well as t o the occurrence of time window related infeasibilities and vice versa. In Subsection 5.4.2, an MA is presented that maintains two uncoupled populations. The first population is evolved in order to generate new feasible individuals. These individuals are transferred into the second population, which is evolved towards minimal travel distances. A promising clustering is compromised by insufficient visit sequences and, vice versa, disadvantageous clusters corrupt promising routes if unnecessary route separations are implemented. A typical decomposition of a vehicle routing and scheduling problem into clustering and the sequencing decisions lead t o the parallel development of high quality tours and routes by a coevolutionary MA. Two independent evolution processes of independent populations are set up in order to develop potential components of a complete solution instance. Both components are only merged in the evaluation step. In contrast to genetic clustering and genetic sequencing, both the assignment of requests to vehicles and the determination of the routes are subject to evolution. Co-evolutionary MAS are presented in Subsection 5.4.3.

5.4.1 Multi-Chromosome Memetic Algorithms The sequential genetic representation of an instance often restricts the application of problem-specific operators that exploit the canonical problem representation in a set of paths. Michaelewicz (1996) states that "... there is really little point in arguing any further that the incorporation of problem-specific knowledge, by means of representation and specialized operators, may enhance the performance of a n evolutionary system in a significant way.". The support of problem-specific representations is emphasized in MAS. Instead of evolving a set of linear gene sequences, arbitrary objects are evolved. However, this requires more sophisticated and problem specific crossover and mutation operators t o exploit the special representation. In this subsection, the corre-


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sponding investigations performed so far in the field of vehicle routing and scheduling are reviewed. Potvin and Bengio (1996) initially refrain from using a strict string-based coding of a VRPTW instance within an MA framework. Instead, an individual consists of a collection of integer vectors. Each vector describes the route of a vehicle and for each vehicle exactly one vector is maintained. The vectors are numbered from 1 to m, representing the number of available vehicles. If only k < m vehicles are routed, then empty dummy routes are kept in the individual. The i-th component within vector j refers to the request that is served in the i-th position of the route of vehicle j. The vectors within an individual can be considered as chromosomes. However, in contrast to the previously presented approaches, the length of a chromosome is allowed to vary from individual to individual. Independently from Potvin and Bengio (1996), this direct coding of a VRPTW-instance is proposed by Pereira et al. (2002) and Tavares et al. (2003) under the name Genetic Vehicle Representation (GVR).

I

vehicle 2 chromosome vehicle 3 chromosome

individual

Fig. 5.7. Genetic Vehicle Representation of a solution instance with 17 requests and three vehicles

Figure 5.7 shows an example of a problem with 17 locations that have to be distributed among 3 vehicles. Vehicle number one follows the route 0 4 1 + 2 -+ 3 4 4 + 5 6 7 0, vehicle number two the route 0 4 8 4 9 + 10 + 11 0 and the visit sequence for vehicle number three is 0+12+13+14+15-+16+17-.0. GVR ensures the feasibility with respect to the condition, that each request must be served by exactly one vehicle. The condition is satisfied syntactically by means of the multi-chromosome problem coding. However, the appearance of capacity or time window violations cannot be prevented. For this reason, repair procedures have to be called. A remedy for handling capacity exceeding has been proposed by Potvin and Bengio (1996). As soon as a case of capacity exceeding is detected, say at visit 5 in the route of vehicle 1, the corresponding itinerary is divided into two segments. The route remains unchanged for all requests preceding the error-)

-)

--)

--)

74


causing visit. The remaining subsequence now forms a new itinerary, which means the incorporation of an additional vehicle. The route 0 4 1 4 2 4 3 4 4 -+ 5 4 6 --+ 7 -+ 0 is divided into the routes 0 -+ 1 -+ 2 -+ 3 -+ 4 4 0 and 0 4 5 4 6 4 7 -+ 0. An attempt is then made to reduce the number of incorporated vehicles down to the number of available vehicles before the minimization of the travel length is targeted. Also in the presence of request related time windows, in order to ensure the customized service of all requests, an additional route is opened if an arrival time is detected that is later than the closing time of the corresponding time window. Adequate problem specific crossover operators directly exploit the multichromosome representation. In the most simple crossover operator, two parental individuals pl and p2 are selected and duplicated. In each of these duplicates ql and q2, a route rl respectively 1-2 is selected a t random. These routes are now interchanged between ql and q2. It is likely, that some requests are served more than one time or not at all. These deficiencies are corrected in a subsequent repair step. In a slightly different crossover operator (Potvin and Bengio, 1996), rl and 7-2 are each divided into two subsequences rp, r!, r$ and r!. The sequence rl is then replaced by (rp, rk) and 7-2 is substituted by (r!, r$). Again, errors concerning missed or multiple visits are corrected in a subsequent repair step. Alvarenga et al. (2003) propose a crossover operator that tries to transfer as much information as possible from the parents without modification. Therefore, there is an initial attempt to integrate complete routes from the parents into the offspring. The remaining, so far unserved requests from the parental routes are integrated into the core routes proposed by both parents. Additional routes are established for requests that conflict with previously inserted requests. Mutation is oriented on the data structure used to store a single solution instance. Possible modifications are: move one or more requests from a route into another route, swap requests (within a route or between several routes) or apply a procedure that uses problem knowledge to modify one or more routes. To ensure feasibility of the generated offspring the re-call of a repair-procedure is necessary. 5.4.2 Co-Evolution with Specialization

Berger and Barkaoui (2002) propose the following MA-approach to generate feasible individuals and to reduce the average travel distance of the generated feasible individuals. Two populations PI and P2 are maintained. Each consists of genotypes that are de-codable to complete phenotype solutions. The population Pz is evolved towards the set of feasible solutions. The sum of detected constraint violations of an individual is minimized down to zero. Infeasibilities of individuals in PI are penalized. Both populations are evolved independently until


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a new feasible individual is detected in P2 • If the number of incorporated vehicles in the new feasible individual from P2 is lower than the number of vehicles in the best individual found so far in Pi, then the complete population Pi is replaced by P2 and the individuals are evolved with the goal of minimizing the traveled distance without violating a constraint. The second population is reinitialized and both populations are evolved independently until a new feasible solution with reduced vehicle number is found. Migration (transfer of individuals) from Pi into P2 is not performed. 5.4.3 Co-Evolution of P a r t i a l Solutions Machado et al. (2002) exploit the decomposition of VRP-type problems into a clustering and a sequencing decision. Two independent populations are managed and evolved simultaneously. The first population addresses the clustering of requests to vehicles. It is aimed at determining a partition of requests among the available vehicles. Remember that m describes the number of available vehicles and n gives the number of requests that have to be served. An individual of the first population is a chromosome of length m. The i-th gene carries the number of requests assigned to the z-th vehicle. On the other hand, an individual taken from the second population describes the order in which the requests should be served. It is fully described by a chromosome of length n. The j-th gene refers to the request that should be served at position j . Each individual within one of the population represents a partial solution for the currently considered problem. A cluster-individual can be merged with different sequence parts resulting in different solution instances. On the other hand, a sequence individual can be completed to different solutions instances if it is merged with different cluster proposals. cluster chromosome

| 3| 2| 1| 2ro1

sequence chromosome | 3| 7| 2| 4| o| 6| 8| l| cluster chromosome

| 31 71 21 ITTSl [6] fSTT] ^

Fig. 5.8. Decomposed representation of a VRP solution Figure 5.8 illustrates the construction of a complete phenotype from a fivevehicle cluster chromosome with an eight-request sequencing chromosome. At first, the number of requests assigned to the first vehicle is taken from the cluster chromosome (at the top in Fig. 5.8). This number is three. The first three requests found in the sequence chromosome describe the tentative route of vehicle one: 0—^3-^7—»2—»0. Next, the number of requests designated for the second vehicle is taken from the second gene in the cluster chromosome. This number is 2 and the next two requests from the sequence chromosome

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form the route of vehicle 2: 0 + 4 -+ 5 + 0 and so on. In the considered example, vehicle 5 remains unused. No requests are left for this vehicle. Again, it is likely that merging the customer sequences with the cluster information leads to infeasible routes because capacities are exceeded, maximum travel durations are transgressed or time windows cannot be satisfied. For this reason it remains necessary to call a repair procedure after the partial solution has been tentatively built up. In order to obtain a fair and averaged evaluation for a partial solution, each sequence part is combined with several clustering parts and each single cluster part is merged into solution instances with different sequence parts. The fitness value of a solution part is determined by averaging the obtained fitness values. An instantiated solution is valued by the sum of travel distances of the generated routes.

5.5 Conclusions Vehicle routing and scheduling problems are combined assignment and sequencing problems. Genetic Sequencing aims at finding promising precedence relations between customer locations in order to prepare the generation of an implementable solution instance. On the other hand, Genetic Clustering tries to determine promising partitions of the available request portfolio. Typically, only Genetic Sequencing or only Genetic Clustering is performed and the remaining decision is left to another heuristic. Recent developments try to combine Genetic Sequencing and Genetic Clustering in order to exclude the algorithmically expensive incorporation of heuristics. Limited capacities and customer specified time windows for visits compromise the generation of minimal length routes. In order to warrant the feasibility of the generated solutions, sophisticated heuristics have to be incorporated to support the genetic search. Tavares et al. (2003) perform several numerical experiments with the Solomon benchmark instances (Solomon, 1987). They observe remarkable improvements of GVR-based genetic search over the so far set up genetic algorithms in which a solution instance in treated as one large string. Furthermore, they conclude that the direct coding enables the consideration of problem specific knowledge in the definition of genetic search operators. Braysy (2001) compares the performance of several different genetic algorithms with other sophisticated meta-heuristic approaches. He reports, that the results of the GVR-based genetic search outperform the results obtained by cluster firstlroute second or route first/cluster second genetic search. Other elaborated metaheuristic approaches are reported to only slightly outperform genetic search approaches for the VRPTW.

Memetic Search for Optimal PD-Schedules

The subject of this chapter is the determination of high quality pd-schedules. An exact solution for the optimization models proposed in Chapter 3 cannot be expected. For this reason, it has been decided to configure a heuristic meta strategy. A memetic algorithm is configured. As discussed earlier, this search paradigm has rarely been applied to PDSP-related problems. The consideration of the intricating PAIRING, PRECEDENCE, TIME WINDOW and CAPACITY constraints is a very challenging issue. A new parallel construction heuristic for pd-schedules is described in Sect. 6.1. This procedure is controlled by a permutation of the requests in the available portfolio. The possibility of omitting requests in the routes of the own vehicles is extensively exploited. From the quality view, the generated pd-schedules are not convincing. However, the variation of the control permutation allows for the generation of a largely diversified set of pd-schedules for a given portfolio. For this reason, the construction heuristic is used to setup the initial population of the MA. Section 6.2 describes the used representation of a pd-schedule into the internal model modified by the memetic search. The selection, the clustering and the sequencing decisions are left for genetic search. A population of feasible pd-schedules is maintained throughout the evolution. The description of the MA configuration is given in Sect. 6.3 Problemspecific crossover and mutation operators support the evolutionary strive for high quality pd-schedules as well as a strong selection mechanism that allows only the best solutions to survive. The applicability of the proposed MA is assessed within a large number of computational experiments. They are described in Sect. 6.4 together with the presentation and discussion of the results observed from the experiments achieved for the benchmark instances for the PDSPLSP, the CPDSP and the PDSP-PP. This chapter terminates with a summary of the most important findings. The application of the MA to instances of the PDSP-CR requires modifications of the MA-framework, which are described in the subsequent Chapter 7.

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6 Memetic Search for Optimal PD-Schedules

For this reason, the investigations on the PDSP-CR are deferred until the next chapter.

6.1 Permutation-Controlled Schedule Construction A new construction heuristic for pd-schedules is presented in this section. It produces feasible pd-schedules for given PDSPLSP, CPDSP or PDSP-PP instances. The incorporation of compulsory requests cannot be guaranteed in every case.

6.1.1 Construction of Routes for more than one Vehicle Two types of construction approaches for multi-vehicle routing problems are distinguished. In a sequential approach the routes are generated one after another for the available vehicles. Initially, the vehicles are permuted. Then, the route for the first vehicle in the permutation is generated. If no request can be assigned to the current vehicle anymore, its route is completed, the route of the next vehicle from the permutation is initialized and filled with requests until no additional request can be routed in a feasible manner (Li and Lim, 2001; Nanry and Barnes, 2000) and so on. In the second method, the routes for the available vehicles are built up in a parallel manner. All routes are initialized. The requests are permuted. An appropriate vehicle is selected for the first unconsidered request in the permutation, the request is assigned definitively to this vehicle and the corresponding operations are inserted into its existing route (Lau and Liang, 2001; Greb, 1998).

6.1.2 Parallel Time-Window-Based Routing The heuristic introduced in this section sets up a collection of routes in a parallel manner. In contrast to the typically proposed construction procedures found in the scientific literature, it does not exclusively exploit the geographical properties of the customer locations. Here, special attention is paid to the consideration of time window constraints that often impose a natural sequencing of the operations along the time axis. It is aimed a t distributing the requests among the vehicles, so that the determined sequence defines the order in which a vehicle serves the pickup and the delivery operations of the obtained requests. Primary goal of the heuristic presented here is the capability to generate a collection of significantly different solutions in order to seed a memetic search algorithm. The generation of high quality near optimal solutions cannot be expected and is not aimed at.

6.1 Permutation-Controlled Schedule Construction

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6.1.3 Algorithm Steps The construction procedure consists of four phases. In the first phase, the operations are ordered along the time axis, in the second phase they are distributed among the vehicles and tentative routes are established. These routes satisfy the PAIRING and the PRECEDENCE constraints, but TIME WINDOW violations or CAPACITY exceeding cannot be avoided. Requests that cause constraint violations are successively removed from the tentative routes in a subsequent repair step (third phase). They remain unscheduled and have to be fulfilled by an LSP. Finally, the modified and repaired routes are improved, applying a 2-opt-procedure to each pd-path in order to reduce the travel length (fourth phase). The set of obtained pd-paths forms a pdschedule. denote the latest allowed operation time among the pickup and Let T,,, delivery operations associated with the requests contained in the portfolio is called the relevant part of the under consideration. The interval [0,T,,,] time axis. All used vehicles are assumed to leave their starting positions immediately at time O and the pickup and the delivery operations are completed not later than T,,, if the corresponding requests are selected to be served.

Phase 1 (Slot-Assignment) The relevant part of the time axis is partitioned into s > 1 equidistant slots. Each pickup or delivery operation is assigned to the slot in which its latest permissible service time falls. Computational experiments show that a . loglo(n. m)l is adequate (n=number of requests, slot number of s = m=number of vehicles). The slot of the pickup operation r+ (pickup slot) of request r is denoted as ps(r), whereas ds(r) describes the delivery slot in which the corresponding delivery operation r - falls. It is ensured that each pickup operation precedes its corresponding delivery operation, hence, the condition ps(r) < ds(r) is satisfied by all requests r.

[E

slot assignment for the operations

Fig. 6.1. Slot assignment for the operations

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Figure 6.1 shows an example for a problem with n = 6 pd-requests and m = 2 vehicles. The relevant part of the time axis, reaching from 0 up to the latest allowed operation time, is partitioned into s = 12 time slots.

Phase 2 (Route-Generation) Attempts are made to assign the customer requests successively to one of the available vehicles and integrate them into the existing tentative routes. Let R be a request permutation and V be a permutation of the available vehicles. In the example under consideration, the request sequence R is given by R = (2,5,4,1,6,3). In Fig. 6.1, the requests are sorted into the rows of the assignment scheme according to R. Request 2 is in the first row, followed by request 5 and so on. Each request from R is assigned to the first vehicle from V with a free pickup and free delivery slot. A request that cannot be assigned to any vehicle remains unserved in the schedule under construction and must be fulfilled by an LSP. The depot situated as near as possible to the last delivery operation of a vehicle is selected as its terminating point. The obtained routes n l , . . . ,17" satisfy the pairing and the precedence condition. They form the tentative schedule P*.This route generation procedure is referred to as CONSTRUCT(R, V). The iterative construction of a set of tentative routes for the example introduced above is illustrated in Fig. 6.2. The vehicle sequence is given by V = (1,2). Request 2 is considered first. It is assigned to vehicle 1 and blocks the slots 5 and 8 of vehicle 1 exclusively. Next, request 5 is assigned to vehicle 1 and allocates the slots 2 and 12. Request number 4 cannot be assigned to the first vehicle due to a conflict of its delivery operation with the operation 2-. The request 4 is assigned to vehicle 2. In the fourth iteration, request 1 is considered. Since its pickup slot is already in use in the route of vehicle 1 and its delivery slot is occupied by vehicle 2 it cannot be assigned to any vehicle and remains unscheduled. It will be given to an LSP. Next, request 6 is assigned to vehicle 2 and finally, request 3 is assigned to vehicle 1. The assignment of the operations to slots leads to two tentative routes. The route of vehicle 1 is S1 4 5+ 4 3+ 4 2+ 4 2- 4 3- 4 5- --+ TI (S1denotes the starting position of vehicle 1 and TI its final destination). The tentative route of vehicle 2 is S24 4+ 4 6+ -+ 4- -+ 6- -, T2. The load of the vehicles, the arrival and the leaving times of the operations are determined recursively as described in section 3 in the equations (3.1)(3.3).

Phase 3 (Repair Tentative Routes) In the next step, pd-paths are derived from the previously generated tentative routes by successively excluding all requests that violate time window or capacity constraints. Let R* be a permutation of the requests assigned to the


v

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slots

11 2 131 4

1 5 16/718 191101111

12

iteration 1 - request r z : assigned to vehicle 1 1111 2111

I 1 12+1112-11 I I ( 1 I I I ( 1 I I

iteration 2 - request 5: assigned to vehicle 1

I ~ + I 12-1 I I I I 11 1 / 1 1 11 I I

11115+11 2111

5-

iteration 3 - request 4: assigned to vehicle 2

I IP+I I 12-1 I I I I I ~ + I1114-11 I I

11115+1 2111

5-

iteration 4 - request 1 is rejected and given to an LSP

I~+III~-11 I I I I ~ + I1 1 1 4 - 1 1 I I

1~115+11 2111

5-

iteration 5 - request 6: assigned to vehicle 2 11115+1

11 I I

I IP+I I I

14+16+1

12-1 14-1

I I I I 16-1

5-

iteration 6 - reauest 3: assimed to vehicle 1

Fig. 6.2. Construction of tentative paths

vehicles in phase 2. The next so far unconsidered request from R * is checked for capacity and time window feasibility. If its associated pickup or its associated delivery operation causes such a conflict, then this request is removed and given to an LSP. Otherwise, it is confirmed and cannot be removed from its current route in subsequent iterations. The predetermined sequences remain unaltered. Then the next request from R * is checked and so on. In the following, this procedure is referred to as repair heuristic REPAIR(R*,P*), parameterized with a permutation R* of the tentatively routed requests and with the tentative schedule P* to be checked. It is assumed that R* is determined as (3,2,6,4,5) in the considered example. First, request 3 is checked to see if it can be fulfilled exclusively without violating the corresponding time windows and without exceeding the maximum capacity of vehicle number 1. The assumption is that no constraint violation occurs. Request number 3 is definitively assigned to vehicle number 1 and cannot be rejected anymore. As a consequence, the subsequently tested requests must take into account that request 3 is confirmed. Next, request 2 is checked taking into account that request number 3 must be served without

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violating a capacity or time window constraint and without violating the precedence relations determined in the tentative routes. It is assumed, that this is possible and, therefore, request 2 is also confirmed. The next request to be considered is request 6. It has been tentatively assigned to vehicle 2. Again, this request does not lead to a constraint violation and it can be definitively accepted. Request number 4 is assumed to cause a time window constraint violation if the precedence relations in the tentative route for vehicle 2 are respected. This request is removed from the route of vehicle 2 and remains unscheduled in the pd-schedule under generation. Finally, request number 5 is tested. It is assumed that no constraint violation is detected. Therefore, this request is also confirmed. The generated pd-paths are S1--, 5+ -+3+ -+ 2+ -+ 2- -+ 3- --+ 5 - -+ TI (for vehicle 1) and S2 -+ 6+ -, 6- -+ T2 (for vehicle 2). They form a pd-schedule. The requests 1 and 4 are assigned to an LSP.

Phase 4 (Improvement of Routes) As mentioned above, the sequencing of operations is performed according to their arrangement along the time axis. However, the customer specified time windows and the capacity limitations often allow rearrangements of subsequences of operations so that a travel distance reduction for a vehicle is obtained. Let p = (pl, . . . ,p,,) describe a tentative route consisting of the n,, locations of pickup and delivery operations referring to the corresponding customer locations. It is checked to see if a travel distance reduction can be achieved by swapping two adjacent operations. The savings si are calculated for i = 1 , . . . , np-1 as si := di-l,i+di,i+l+di+l,i+2-(di-l,i+l+di+l,i+di,i+2) (di,j describes the travel distance between pi and pj).

original route improved route

----+

-

Fig. 6.3. Swap of two adjacent operations in a route

Figure 6.3 contains a demonstration of the exchange of two adjacent operations pi and pi+l. The arcs (pi-1, pi) and (pi+l, pi+2) are replaced by the arcs (pi-1, pi+^) and (pi, pi+2). Finally, the arc (pi,pi+l) is inverted. This approach is a special variant of the 2-opt-heuristic (Golden and Stewart, 1985).


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The swap ( ~ ~ , p of~ two + ~ adjacent ) operations is feasible if a positive saving s(pi,pi+l) is achieved, if the PRECEDENCE feasibility is kept and if the modification does not lead to additional TIME WINDOW constraint or CAPACITY constraint violations. All feasible swaps are collected in the set

R.

If f i is empty, then a n improvement of the currently considered pdpath is not possible and the next vehicle is considered. Otherwise, the swap (pi., pi*+l) is selected. The exchange with its successor leads t o the maximum saving s(pi*,pi*+l) := max {s(pi, pi+l) I (pi, pi+l) E and p p + l are swapped and p is updated t o

R } . The

operations pi*

The re-calculation of the arrival times, the leaving times and the capacity utilization terminates the current iteration. The next iteration starts with the calculation of the possible savings in the modified route p (Fig. 6.4). This iterative improvement procedure is referred to as I M P R O V E ( p ) . It is applied to all pd-paths p E P * . IMPROVE(p) (1) calculate savings s l , . . . , sn,-1 (2) determine 2 (3) if 2 = 0 then goto (8) (4) select (pi*,pi*+~), s.th. s(pi*,pi*+~) = ma~{s(pi,pi+l)1 (pi,pi+l) E ( 5 ) update P := (PI, . . . Pi*-1 Pi*,Pi*+z,.. . ,Pn,) (6) update arrival times, leaving times and capacity utilization along p

6)

(7) goto (1)

(8) terminate Fig. 6.4. 2-opt-improvement procedure

In order to limit the computational effort, the number of iterations for each call of I M P R O V E ( . ) is limited t o 100 swaps in a route. The IMPROVE()-heuristic aims a t reducing the travel distance in a route. For this reason, earlier arrival times can be expected which lead t o a limited number of time window constraint violations. However, the IMPROVE()-heuristic is applied only after the REPAIR()-heuristic in order t o maintain a sufficiently diversification among the generated pdschedules.

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6.1.4 Determination of the Request Instantiation Order The determination of adequate instantiations orders R has been an open issue so far. Promising requests should obtain a larger probability to be routed and confirmed first in order to avoid slot conflicts resulting in time window or capacity constraint violations that make the rejection of such a promising request necessary. Therefore, a permutation of the requests is determined according to a Biased Random Drawing (BRD). A fitness value f, is assigned to each request r E R and the relative fitness __fT_ is calculated. A large relative fitness C r c fr ~ value indicates that the corresponding request is above-averagely promising (with respect to the pursued goal). The permutation R is determined successively component after component. A roulette wheel selection (Subsection 4.3.2) is performed to draw the first component rl of the permutation from R and rl is removed from R. The relative fitness values are re-calculated for the so far unconsidered requests, the next request r2 in the permutation is drawn and so on. At the end, a complete permutation of R is obtained in which requests with a larger fitness value are more likely to be contained in the first permutation positions. The fitness value of a request r is set to the revenues achievable from r. Filling the permutation in this way, a request associated with large revenue is more likely to appear in the first components of the request permutation R. As a consequence, it has a greater probability of being routed in its original time slot and to be considered in earlier stages of the construction or repair heuristic. The application of a BRD of the permutations ensures that a diversified population is generated.

6.2 Representation of a PD-Schedule An individual that represents a pd-schedule S has to carry four kinds of information. Firstly, the separation of the available set of requests into self-served and LSP-served request has to be considered. Secondly, the assignment of self-served requests to the available vehicles must be stored as well as the sequences in which the vehicles perform the necessary pickup and delivery operations. Finally, the termination points of the pd-paths are part of a complete solution instance and must be adequately coded. In contrast to the routing information, the arrival and leaving times are not stored in an individual. They are determined deterministically in a subsequent scheduling process in accordance to the given routes of the vehicles. Here, the arrival and departure times within a route are determined recursively following the order of visits given by the routes (cf. (3.1)-(3.3)). As mentioned in subsection 5.4.1, a direct representation of a solution is state-of-the-art and supports the memetic search. For this reason, it has been

6.3 Configuration of the Memetic Algorithm

,

85

individual (data structure hull)

route of vehicle 1 ( r l ) route of vehicle 2 ( r z ) route of vehicle 3 (r3)

11 I I I I k

route of vehicle 4 (re) termination point references T

Fig. 6.5. Data structure of an individual

decided to set up a representation in which an individual consists of a data structure that includes the paths determined for each vehicle. The paths pl, . . . ,p, of the vehicles are stored in strings. The i-th component of the j-th string refers to the i-th operation of vehicle j. The j-th component of the string T refers to the termination point determined for the j-th vehicle. The individual can be understood as a hull in which all data that describe a solution instance are contained, see Fig. 6.5.

6.3 Configuration of the Memetic Algorithm In this section, the procedure for the generation of the initial population, the genetic operators (recombination and mutation) and the population model are described. 6.3.1 Initial Population A diversified initial population is obtained by applying the construction heuristic with different request permutations. In the following, it is assumed that the available fleet of the carrier-controlled vehicles is homogeneous. In this case, the variation of the vehicle sequence leads to several pd-schedules that differ only in the assignment of the generated routes to the vehicles for a given BRD-sequence, but the set of generated routes (pd-paths) remains unchanged. The vehicle permutation V := (1,. . . , m) is used for the construction of all individuals. The generation of the initial population is as follows. At first, the iteration counter i is set to 0 and the relevant part of the time axis is divided into several equidistant time slots. Let N be the size of the initial population. The following iteration is repeated N times. A BRD-permutation Ri is generated and parameterizes the construction procedure CONSTRUCT(Ri, V). A tentative pd-schedule P: is determined.

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The permutation Ri of the routed requests is generated from R i . It is a duplicate of Ri except that so far rejected requests are removed from the permutation. According to Ri, the tentative pd-schedule is repaired, leading to the pd-schedule Pi. It is improved applying the IMPROVE()-procedure to each pd-path contained in Pi. The iteration counter is increased. If the initial population is completed, the procedure terminates, otherwise the iteration loop is re-called, see Fig. 6.6. The pd-schedules PI,. . . , PN are transformed into data structures described above. These structures form the initial population. Each one represents one individual. (1) V := ( 1 , .. . , m) (2) i := 0 (3) generate time slots ( 4 ) Ri := B R D ( R ) ( 5 ) P: := C O N S T R U C T ( R i ,V ) (6) ~i := Ri \{v E Ri I v 6 P,*) (7) pi := REPAIR@, P;) (8) for each p E Pi: p := I M P R O V E ( p ) (9) i :=i+l (10) if i < N then goto (4) (11) terminate Fig. 6.6. Generation of the initial population

6.3.2 Recombination Recombination aims at combining two data structures (parental individuals) into one new data structure (offspring individual) so that properties of each parental individual are transferred into the offspring individual. Here, four categories of properties are distinguished. I Separation of requests: if a customer request is rejected in both parental individuals then this should be the case in the offspring. Otherwise, if a request r is served in at least one of the parental individuals then r should be served in the offspring individual. I1 Assignment of requests to vehicles: if a vehicle v serves a request r in both parental individuals then this property should be retained in the offspring individual. Otherwise, r should be assigned to one of the vehicles v: or v: that serve r in the parental solution instances. If r is served only in one parental solution instance, maybe in the first parental individual by v:, but remains unscheduled in the second parental solution instance, then r should be served by v: in the offspring individual.


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I11 Precedence relations between operations:The routes in the parental structures define precedence relations between the operations. These relations should be preserved in the generated offspring t o as large as possible extend. A precedence relation that can be found in both parental individuals should be retained. In cases where the two parental individuals propose different precedence relations for a pair of operations, both of them have t o get a chance to be implemented in the offspring. IV Termination points of the paths: For each vehicle, the termination point in its offspring route has to be selected from the termination points in its parental routes. The following steps are proposed t o derive an offspring data structure from two given parental structures, so that the conditions I-IV are taken into account. Requests that are unserved in a solution instance are stored in a n additional dummy route that is maintained in the corresponding data structure. This route is treated as the visit sequence of a n artificial vehicle. It is numbered with m 1. Let pHff denote the offspring route of vehicle i, p! and p: denote the corresponding parental routes obtained from the selected parental data structures. The offspring routes are generated successively. The sequence in which the offspring routes are generated is determined as (il,. . . , i,, m 1). In order to support the exchange of type-I-information the dummy vehicle route is considered last. At first, a n offspring route for vehicle il is derived from the parental routes pt1 and p:l of vehicle i l , followed by the generation of a n offspring route associated with vehicle i z and so on. Since the dummy route is considered last, all requests that are served in a t least one parental pd-path are assigned to the vehicle that fulfills this request in the corresponding parental individual. If a request is rejected in both individuals, it remains unserved in the offspring individual, so that the demands of I are kept. After an offspring route has been established for vehicle j , all requests that are contained in this route and in the parental routes of the so far unconsidered vehicles j 1 , . . . ,m 1 are labeled as 'used', so that they cannot be routed for another vehicle. Thus, condition I1 is satisfied. If the same vehicle in both parental solution instances serves a request then this vehicle also serves it in the offspring solution instance. On the other hand, if different vehicles in the parental solution instances serve it, one of these vehicles (those that is considered first according to the order i l , . . . ,i,) serves this request in the offspring. The composition of an offspring route for a vehicle from two parental routes has so far not yet been addressed. The recombination operators proposed in the scientific publications, cf. Subsection 5.4.1, seem to be inadequate because they do not pay special attention to the problem characteristics expressed in

+

+

+

+

88


the PRECEDENCE and in the PAIRING condition. None of the proposed operators can ensure that both operations of a request are kept together in the offspring route and that the pickup operation precedes its corresponding delivery operation. Indeed, several recombination operators that preserve precedence properties are defined for merging two complete permutations. One example is the precedence preserving crossover operator PPX proposed by Bierwirth et al. (1996). In the following, a modified version, called modified P P X (mPPX), is described. The operator mPPX merges two parental subsequences of a permutation into a new offspring subsequence of the same permutation, so that precedence relations from both parents can be found in the offspring. The operator mPPX is defined as follows: The parental routes of the vehicle i include 1: and 1; locations. Let 6 t 2 be the number of stops that are included in both routes a t the same time. The offspring route pHff of length l y f f := 1; +1: -St2 is initialized. The procedure to fill this route is as follows. A binary string b = (bl, . . . , blzff) is generated at random. A '0' in the I-th position indicates, that the I-th stop in the offspring route is taken from p:, in case of bl = 1 the next stop included in the offspring route is taken from p:. The 1?-6?2/2 , that is probability to select '0' for an arbitrary position in b is set to \:jf the relative length of p: with regard to p H f f . Starting from 1 = 0 each position of pHf is successively filled, distinguishing three cases: 1. There are unconsidered stops in both parental routes; Set the I-th stop in p i f f to the first so far unconsidered stop in (bl = 0) or p: (bl = 1). These stops are labeled as used in both parental routes. It is continued with the next stop (1 := 1 1). 2. If p: contains no more unconsidered stops, the remaining stops in the offspring route are filled with the so far unconsidered stops from p: in the sequence predetermined by P:. 3. If p: contains no more unconsidered stops, the remaining stops in the offspring route are filled with the so far unconsidered stops from p: in the sequence predetermined by pf .

+

The relation a + b is satisfied if operation a precedes operation b in a subsequence of a permutation. If a + b is satisfied by both parental subsequences then this relation will re-appear in the offspring subsequence. If one parental subsequence satisfies a 4 b and the other one satisfies b + a , then either a + b or b + a is implemented in the offspring subsequence. In case that a and b are contained together in only one parental route, the given precedence relation is kept and re-appears in the offspring route. This crossover operator produces a new path that fulfills the pairing and the precedence constraint. Additionally, it does not destroy sequences appearing in both parental routes. A precedence relation of two locations that is


89

included only in one parental route, maybe pa, survives with the probability 1 12 li-6, 12 . cO ndition 111 is fulfilled.

Fig. 6.7. Example of the mPPX-operator

An example is illustrated in Fig. 6.7. The first parental route of vehicle 1 consists of 6 operations (1: = 6) and the second one of 10 operations (1: = 10). Four operations are contained in both parental routes ( ~ 5 : ~= 4). The offspring route consists of 10+6-4=12 operations. In a typical crossover vector of the entries are '0' and = $ of the components are bl, = '1'. The probability that a precedence relation is taken from pi is and a precedence relation from p? is preserved with the probability of $. According to the vector bl specified in the example, the first offspring operation is 5+ taken from p:. The second offspring operation 6+ is also taken from p:. This operation appears also in p:. It is labeled as used in order to avoid a multiple incorporation of this operation. The third offspring operation 9+ is taken from the first parental route and the next offspring operation is also obtained from pi. This is 9-, because the operation 9+ has already been inherited from p:. It proceeds in this way until the complete offspring route is filled. After all m 1 offspring routes have been determined, the terminating points of the routes are merged applying a uniform crossover operator, so that IV is adequately implemented. All requests served either in the first or in the second parental route are combined in one route if they are not contained in a route of a previously considered vehicle. None of these requests is moved to another vehicle. The application of mPPX to two empty parental routes produces an empty route and the combination of a non-empty route with an empty route by means of mPPX produces the non-empty route again. The mPPX-operator is part of an operator that combines two parental data structures into one new structure so that distinguishing properties of at least one parental structure are preserved and precedence relation that are part of both donating structures are kept unchanged. This operator is called Property Preserving Structure Crossover (PPSX). The complete definition is presented in Fig. 6.8. The offspring routes are determined successively. For a certain vehicle i the route is initialized first (2), the offspring route is

9

+

90


( 1 ) k := 1 (2) p Y f := init() (3) p",f := m P P X ( p i , p E ) ( 4 ) T , " := ~~ random draw({^^,^^)) ( 5 ) R k := { r E R I r served in p i f (6) j := k + 1 (7) label r E Ri as 'used' in Pb, (8) j := j 1 ( 9 ) if ( j 5 m 1 ) then goto (7) (10) k := k 1 (11) if ( k 5 m 1 ) then goto (2) (12) terminate f ,

+

+

+

+

Fig. 6.8. Property Preserving Structure Crossover (PPSX)

derived (3)-(4) and the requests contained in the recently generate route are labeled as 'used' in the parental routes of the so far unconsidered vehicles ( 5 ) (9). Each generated offspring route is checked for time window and capacity constraint violations. If such a violation is detected then the causing requests are re-assigned to an LSP and the corresponding pickup and the corresponding delivery operation are inserted into the dummy-route by applying the mPPXoperator. The dummy route is used, if the considered individual is selected as a parental individual in subsequent iterations. To demonstrate the PPSX-operator, it is applied exemplarily to the parental pd-schedules shown on the top of Fig. 6.9. The termination point of the route of vehicle i in parent j is denoted as T i . The generation of the offspring route for vehicle 1 has been shown in Fig. 6.7. Additionally, in this example the selection of the termination point is considered. The right value in the bi-row indicates whether parent 1's or parent 2's termination location is drawn. 6.3.3 Mutation

Mutation modifies a single individual. The inserted data does not originate from two specified parents, but arbitrary changes are performed. Mutation works complementarily with respect to the presented recombination operator PPSX. Three slight changes are performed.

1. The termination point of an arbitrarily selected route is replaced a t random. 2. Within a randomly selected route (including the dummy route) an arbitrarily selected location is re-positioned at random. The precedence feasibility is not violated.


91

parent 1 vehicle 1 9++ 6++ 9--+ 6--+ 8'4 82 4++ 2++ 4-+ 2 - 4 5++ 53 lo++ 7++ lo--+ 3+-+ 7--+ 3dummy 4 1'4 1-

parent 2 vehicle 1 5 + - +6 + - +9 + - +4+-+ 6-+ 9 - 4 1+-+ 4-+l 2 3++ lo+-+ lo--+ 33 2+-+ 2--+ 8+-+ 8--+ 7++ 7dummy 4

Offspring route for vehicle 1 (cf. fig. 6.7) p: P?

bl

pOf f 1

g++ 6++ 9--+ 6--+ 8+-+85++ 6++ 9++ 4++ 6-+ 9-+ I++ 4 - 4 11-1-0-0-1-1-0-1-1-0-1-1 5+-+ 6+-+ g++ 9-+ 4++ 6-7- 8++ I++ 4--+ 8--+ 1

Offspring route for vehicle 2 2+-+ 2- (request 4, 5 already used) 3++ lo+-+ 1 0 - 4 3bZ 0-1-0-1-1-0

pi p$

P2" f f

2++3++2-+

lo++ 10-+3-

Offspring route for vehicle 3 7++ 7- (requests 10, 3 already used) p; 7++7- (requests 2, 8 already used) p; b3 p;ff

0-1 7++7-

Offspring route for vehicle 4 (dummy vehicle) no customer operations (request 1 already used) no customer operations

pi p;

b3

pif

-

no customer operations Fig. 6.9. Example for the PPSX-operator

3. A request from the dummy route is re-assigned to the route of a n arbitrarily selected vehicle. Therefore, the operations belonging t o an arbitrarily selected request in the dummy route are deleted from this route and inserted randomly into the route of the determined vehicle, so that the precedence feasibility is preserved. The three modifications are performed successively to one data structure. If the mutation frequency is large, then the frequency of requests that are tentatively inserted into the route of the available vehicles is enlarged. The composition of advantageous routes is supported.

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6.3.4 Population Model In the population model it is determined how the individuals in a population are replaced. (1) (2) (3) (4) (5) (6) (7)

i := 1; generate initial population Pi; T := 0; j := 0; p := random([O,11); select parent1 and parent2 from Pi at random; i f ( p< p x o ) then indivj := PPSX(parentl,parent2); else indivj := parentl; ( 8 ) p := random([O,11); ( 9 ) i f ( p< P M U T ) then indivj := mutate(indivj); ( l o ) indivj := I M P R O V E ( i n d i v j ) ; (11) R := B R D ( i n d i v j ) ; (12) indivj := R E P A I R ( R ,indivj); (13) indivj := I M P R O V E ( i n d i v j ) ; (14) evaluate(indivj); (15) T := T U {indivj); (16) j := j + 1; (17) i f ( j 5 N ) then goto ( 5 ) ; (18) sort Pi U T b y decreasing fitness: s := (indivi,, . . . ,indiv;,,); (19) Pi+l := {indivi,, . . . ,indiviN); (20) i := i 1; (21) i f (i 5 M A X P O P ) then goto ( 3 ) ; (22) terminate;

+

Fig. 6.10. Memetic Algorithm Framework

The complete MA is outlined in Fig. 6.10. Initially, the population counter is set to 1 and the first population is generated (1)-(2). The population is evolved iteratively (3)-(21). Each iteration consists of four phases. In the first phase, offspring individuals (data structures) are determined (5)-(9). A new tentative solution instance is obtained by recombining two parental instances or by duplicating an existing instance (7). The crossover probability p x o describes the frequency of crossover. With a certain proba, offspring is undergoing mutation (9). bility p n / r u ~ an The second phase (10)-(13) is dedicated to improve the generated offspring and to obtain its overall feasibility. The IMPROVE-procedure is invoked in order to reduce the number of time window constraint violations

6.4 Computational Experiments

93

(10) before the REPAIR-procedure is applied in order to exclude constraintviolating requests from the paths. Computational experiments have shown that this does not limit the genetic diversity in the longer run, as long as the initial population is diversified enough. A BRD-permutation is determined for the scheduled requests and parameterizes the REPAIR-procedure whose application modifies the offspring, so that it fulfills the four constraints PAIRING, PRECEDENCE, TIME WINDOW and CAPACITY (11)-(12). Then, the IMPROVE-procedure is called again in order to reduce the travel distances (13). Next, in the third phase the fitness of the offspring is evaluated (14). Therefore the sum of traveled distances, the collected revenues and the necessary LSP charges are calculated. The fitness value is determined as described by the objective function in the considered problem variant. Now, the offspring is inserted into the temporal population T (15). These three phases are repeated until the desired number of N offspring individuals is obtained. In the fourth phase (18)-(20) of the iteration, the existing population is replaced. Therefore, the 2N individuals in the union set of the temporal population T and the existing population Pi are sorted by decreasing fitness (18). The best N individuals form the new population Pi+l(19). This replacement scheme is called p X population model (Back et al., 2000a). It ensures that the best observed individual can only be replaced by an individual with higher fitness. Computational experiments show that it is necessary for a successful evolution to ensure the transmission of the best individuals in the new population and to give the worst individuals no possibility of transmitting their specifications into a subsequent population. The iteration is repeated until the specified number of M A X P O P populations has been evolved (21). The selection of requests has crucial impacts on the obtained solution quality. The REPAIR-procedure tends to reject requests because they lead to constraint violations. However, recombination and mutation tentatively insert additional requests in the routes. The interaction of these two algorithm components (local search and genetic operators) allows an extensive and intensive sampling of different request selections and pd-paths. The IMPROVEprocedure and the genetic operators support the search for profitable request selections combining the selected requests into efficient routes.

+

6.4 Computational Experiments The recently configured MA is applied to the prepared benchmark instances for the PDSPLSP, CPDSP and the PDSP-PP. By means of these experiments, the capability of the MA to identify efficient pd-schedules is assessed. As already mentioned in to introductory paragraphs of this chapter, the proposed MA-framework requires extensions in order to deal with PDSP-CR-type problems and to ensure that all compulsory requests are finally contained in

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a t least one generated pd-path. These modifications are extensively discussed in Chapter 7 and the corresponding experiments are discussed together with the observed results a t the end of that chapter. The remainder of this chapter describes several performed numerical experiments. Initially, several MA-runs are performed in order to find an adequate parameter setting for the MA (Subsection 6.4.1). Next, the MA is applied to problem instances in which only fair freight charges (cf. 3.18) have to be paid for the LSP incorporation (Subsection 6.4.2). The impacts of different frequencies of discounted or overpriced tariffs for LSPs are studied in Subsection 6.4.3. In the previously listed experiments the available capacity of the carrier-controlled vehicles are not scarce, so that the competition between the considered carrier company and the LSPs is only based upon the price for the fulfillment of the requests. In Subsection 6.4.4, impacts of scarce capacities for the controlled fleet of vehicles are studied. In the final experiments, the postponement of requests into subsequent planning periods is assessed within several computational experiments (Subsection 6.4.5).

6.4.1 Parameterization of the MA The parameterization of the previously described MA requires the determination of four values. The population size is set to N = 100 individuals and the evolution is limited to a sequence of M A X P O P = 200 populations. A computational experiment is set up in order t o identify appropriate frequencies for the application of the crossover and the mutation operator. The crossover operator PPSX tries to insert so far unconsidered requests into the route of a vehicle. If the crossover frequency is large, then the number of tentative request insertions is also enlarged and the probability for a successful insertion of a request increases. The feature 3 of the proposed mutation operator also enforces the insertion of so far unconsidered requests in the route of an arbitrarily selected vehicle. However, the remaining two features of the mutation operator destroy evolved operation sequences by re-arranging the operations randomly. For this reason, a n over extensive usage of the mutation operator jeopardizes the achievement of promising solution properties. In order t o identify the averagely best suited pair (p,,, pmut) of crossover frequency p,, and mutation frequency pm,t, the MA is configured with different frequencies and applied t o a representative subset P of PDSPLSP benchmark instances with randomly scattered, semi-scattered or clustered customer locations. The subset P consists of the instances (r103,1,0,0), (r202,1,0,0), (rc104,1,0,0), (rc202,1,0,0), (c101,1,0,0) and (c202,1,0,0). Each instance is solved by the MA NexP times for each configuration V := (p,,,pmUt) E {0,0.1,. . . , I ) ~ In . the following, NexP is set to 3. Each unrouted request is given to a n LSP. The fair freight charge has to be paid for such a request. Let Qi E P be one of the problem instances considered in the experiment. The average of the observed best objective values (fulfillment costs) for the configuration (p,,,pmUt) E V associated with the instance cP is denoted by


95

&(pxo,pmut). The reference objective value f F f is defined as the fulfillment costs taken from the solution of the used Solomon instance that comes along with all properties of a pd-path for the associated problem instance. E V is evaluated by its relative average Each configuration (p,,,p,,t) objective value

If f (pxo,pm,t) > 1 then the average costs of a solution achieved for the configuration (p,,, pm,t) exceeds the costs associated with the reference solution, otherwise the MA has identified solutions that are averagely equal or even cheaper than the reference solutions.

Fig. 6.11. Relative average objective value f@,,,pmUt) settings ( p x o ,pmut)

for different parameter

The results obtained are shown in Fig. 6.11. If the crossover frequency or the mutation frequency (or both of them) are smaller then 0.1 then the relative objective value is significantly larger than 1. The set of these configurations is labeled by I. For all these configurations, the observed costs f (pX,,pmut) exceed the costs f y f of the reference solutions. If at least one of the frequencies is further enlarged up to 0.2, then the MA is able to achieve solutions that are averagely better then the reference solutions. However, the savings vary only between 0 and 2.5% (area 11). If p,, is further enlarged up to 0.7 then the averagely achieved objective value falls below the reference value and cost improvements up to 5% are observed (area 111). A further increase of the frequencies leads to average costs that are up to 7.5% below the costs in the reference solution (area IV). The MA finds the least cost solutions with

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a crossover frequency of 100% and a mutation frequency between 50% and 60%.

Fig. 6.12. Percentages a(pxo,p,,t) of requests served by carrier-owned vehicles for different parameter settings (p,, ,p,,t)

In order t o assess the MA's ability to generate reasonable pd-paths, the number of considered requests in the proposed plans is analyzed in more detail. denote the average percentage of requests routed for Let Ti@(p,,, p,,,) carrier-owned vehicles observed in the experiments described above. The averagely carrier-served part of the request portfolio is given by

The averagely observed percentages of requests that can be incorporated in a route of the considered carrier are shown in the isolines-chart in Fig. 6.12. Overall, the percentages of carrier-served requests a(pxO,pmut)in the optimal solutions vary between 35% and 70%. The highest percentages of routed requests are observed for configurations with a large crossover frequency of more than 90%. However, a weak mutation frequency hinders the identification of profitable request routings. The configuration (1.0,0.5) seems t o provide the best trade-off between the conservation of identified request compositions and the tentative insertion of additional requests into the routes of the considered carrier company. An average incorporation rate a(1.0,0.5) 2 0.70 is observed for this configuration.


97

Only a minor fraction of the set of requests is served by a n LSP. However, the applied freight tariff provides charges comparable with the travel costs of the carrier-owned vehicles. Therefore, the generated pd-paths must be promising, otherwise the part of LSP-served requests would be larger. For the following experiments, the memetic search is configured with p,, = 1 and pmUt = 0.5. This configuration permits the generation of a sequence of converging populations with a high fitness value level. The averagely achieved best solutions are comparable to the reference values. The execution of a single MA lasts between 1.5 and 3 minutes on a Personal Computer with an AMD K5 processor operating a t 1 GHz.

6.4.2 Impacts of Spatial Distribution and Time Window Tightness In this subsection the performance of the MA is investigated separately for problems with randomly, semi-randomly scattered and with clustered pickup and delivery locations. Furthermore, a distinction is made between problem instances with tight and with large time windows. The least observed objective value (fitness value) within the i-th population obtained in the j-th run is denoted by f itj(i) and the average fitness-value observed in the same population of the j-th run is labeled as f;tj(i). The relative improvement of the best and the average fitness in population i compared t o the initial population observed for the j-th run is defined as

Fj(i):=

f itj(i) F,(i) fit, (0) ' -

fit, (i)

:= -

fit, (0) '

The average values from all performed Nexp MA-runs is calculated as

These two fitness measures map the observed results in the interval [O,1] and allow for a comparison of the results obtained from experiments for different problem classes. An experiment has been performed in order to identify the differences in the evolution processes for problems with randomly scattered and clustered customer locations and tight or relaxed time windows. Figure 6.13 shows the evolution of the average fitness of the individuals of the population during the evolution and the development of the best so far observed fitness value during the performed iterations. Again, the representative set P of problem instances is used. It contains exactly one instance with tight or relaxed time windows and scattered, semiclustered or clustered customer locations. The MA in the above configuration


98

treats each instance @ E P within three independent runs. The observed values for F and F are shown in the graphs in Fig. 6.13. The grey line represents the graph of F and the black line represents the graph of F .

2'0 io 1

-

6'0

io

160 i i o

140

rb +A

160 180 zdo od

io

loo

iio iio

180 l i o zdo

1

-0 8 -0 6 0 4 -

(rc104,1,0,0)

-0 4

0 2 -

Oo

(rc201,1,0,0)

-

-0 2 -

zb rb &I s o 160 i i o ido 160 180 zoo

Oo

2'0

40

6'0 &I

160 120

do 180 160 zoo

1

-

0 8

0

0 2 -

-0 2 -

Oo

zb 4b sb sb

zoo i i o

id0

Fig. 6.13. Evolution of

F

160 l i o zoo

8

Oo

(c202,1,0,0) sb

4'0

&I sb loo l i o

140

160 i i o

200

(grey dotted graphs) and F (black graphs)

Two developments are striking. If the customer locations are scattered randomly over the operational area then the maximum observed improvements do not exceed 35% percent. For (r104,1,0,0), the asymptotes of F ( i ) and F(i) seem to be parallel lines to the x-axis with an asymptotic function value FZ 65%. In case of problem (r204,1,0,0),the asymptotical function value is around 75%. For semi-clustered customer locations, the observed improvements are significantly larger. The asymptotical values are around 60% which means an improvement of 40% compared to the initial values F ( 0 ) and F(0). However, if the customer locations are clustered and geographically grouped then the achieved improvements are more than 70%. The asymptotical values are around 30%. These values are independently observed for problems with tight time windows as well as for problems with large time windows. The second remarkable result concerns the variety of the genetic material in the population. For problems with tight time windows, the difference


99

between F(i) and F ( i ) remains significant only for the first 80 populations. In later evolution stages, the fitness of the best observed individual F ( i ) does not differ significantly any more from the average fitness F ( i ) . As a result, the subsequently improvements are only small. However, if the available time windows are relaxed and enlarged, then this phenomenon is not observed before iteration 120. To conclude the observations, two main statements are remarkable. At first, the improvement rate of the initial individuals increases if the spatial variety of the customer locations decreases. Secondly, if the length of the time windows is enlarged, then the iteration number from which the best observed and the averagely observed fitness values do not differ significantly anymore increases. No further significant improvements of F or F are achieved in subsequent iterations. Similar observations are made for other instances but the results are not presented here for reasons of clarity. The results obtained are also evaluated with special attention to the mode selection feature. It is of interest to understand how the modes of the requests are changed during the evolution of a population. The averagely observed fraction of requests associated with the fittest individual in population i during run j is denoted as aj(i) and the averagely observed fraction of routed requests during run j in this population is stored in a j (i). The relative values of these rates are defined as

The average rates observed over the NexP runs of the MA are defined by

A different behavior of A(i) and A(i) during the evolutionary iteration is observed for problem instances with randomly or semi-randomly scattered locations and problems with clustered locations. Figure 6.14 shows the development of the rates A(i) (gray dotted line) and A(i) (black line). In the former instances (top and middle row), both values A(i) and A(i) increase significantly. The number of incorporated requests increases. In the latter cases (bottom row), where the visiting locations are clustered, the first improvements of F ( i ) and F ( i ) are accompanied by a decrease of A(i) and A(i). In subsequent iterations, the average rates of carrier-served requests re-increase again. In the problem with tight time windows (left plot in the third row), the initial rate is not restored and in the problem with large time windows, only slight improvements compared to the initial rates are observed. For problems with tight time windows (left column plots) the improvements of A(i) and A(i) are significantly smaller than the improvements obtained for problems with relaxed time windows (right column plots). However,

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if the spatial distribution of the locations to be visited increases, then also the achieved improvements of the rates A ( i ) and A(i) increase. The number of requests incorporated in the pd-paths increases only slightly after iteration 80 if the time windows are tight. In the case of larger time windows, the pd-paths are satisfied approximately in iteration 120. A similar result has been observed for the average costs.

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zb 4'0 &I s o 160 o l i i i o i i o i i o zoo Oo

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Fig. 6.14. Evolution of the number of requests incorporated into the routes of the carrier-owned vehicles, the black lines show the A(i) curves and the grey dotted lines represent the A(i) graphs

6.4.3 Identification of Profit-Maximum Request Selections

The following investigations are dedicated to the PDSPLSP. Special attention is paid to the competition for requests between the considered carrier company and an LSP that applies the fair tariff defined in (3.18). Additionally, the impacts of enlarged or discounted charges are targeted.


101

Experimental Setup

MA runs have been executed for each PDSPLSP instance of the benchmark field introduced in 3.5.3. Since the MA is a randomized procedure average results are taken from three independent runs applied to each of the 1296 instances. Therefore. overall 3788 instances are evaluated. Numerical Results Table 6.1 shows the average results obtained for the six problem classes if the freight charge is neither discounted nor enlarged ( a = 0). The first line represents the results compared to the reference objective values. Only for problems in the R2 class, do the average observed results lie above the reference values. For all other problem classes, the MA is able to identify reduced cost solutions by incorporating an LSP. The largest improvements are observed for C1 problems in which a cost reduction of 16% is realized. The second line represents the percentages of requests that are fulfilled by LSPs. This percentage increases significantly if the spreading of the pickup and delivery locations decreases. The third row shows the percentage of customer locations a t which the serving vehicle can execute the corresponding pickup or delivery operation without waiting time for the opening of the corresponding time window. This value tends to decrease if the spreading of the customer locations is reduced. For problems with completely scattered customer locations (R1 and R2) and for problems with clustered locations, the number of no-wait-operations reduces if the time windows are relaxed. This phenomenon is not observed for problems with semi-scattered locations. Table 6.1. Results obtained for different problem classes

group R1 R2 RC1 RC2 C1 C2 relative costs 0.98 1.03 0.98 0.91 0.84 1.00 LSP-served 0.19 0.15 0.21 0.27 0.33 0.39 no-wait-service 0.70 0.68 0.65 0.73 0.63 0.50 Table 6.2 shows the results obtained for experiments with diversified LSPcharges. In all tables, the divergences from the case without discounted or surcharged values (P = 0) are shown and calculated for each problem class separately. The top set of rows shows the variation of the sum of costs for a slight diversification ( a = 0.25), followed by the set of rows with results from the a = 0.5 experiment. The third set of rows represents the results from the o = 0.75 experiment whereas the last set of rows contains the results observed for the experiments with a complete enlargement/discount ( a = 1.0). Two main observations can be stated. If the surcharge is increased from 25% up to 75% above the fair tariff then the overall costs also increase and if

102


the LSP-charge discount is enlarged then the savings also increase. Secondly, if the frequency of surcharge-requests is increased (increase of a ) then the additional costs (compared to the /3 = 0 costs) also increase. The savings achieved increase significantly if the frequency of discounted LSP-charges is increased. The largest savings are realized for problems with relaxed time windows (R2, RC2 and C2). Their savings are significantly larger than the savings observed for the problems with tight time windows. This is mainly caused by a reduced number of self-served requests and therefore by an intensified exploitation of the discounted LSP-charges (cf. Table 6.3). If the frequency of expensive LSP-charges is enlarged, then additional costs are observed. If the extra charge is too large, then the MA recognizes that the corresponding requests can be served by own equipment in a cheaper fashion. However, the savings achieved by not using the LSP are so great that it is not necessary to insert these requests in the most profitable way in the existing routes. For this reason, significant additional costs are observed for very large surcharges.

6.4.4 Consideration of Capacity Limitations Two different experiments are performed in order to study the impacts of scarce carrier-controlled transport resources as represented by the generated CPDSP benchmark instances. In the first experiment, request portfolios (4,o) have to be fulfilled. The considered freight carrier is provided with different fleets that can be used to serve the acquired requests. In each generated problem instance, a homogeneous fleet of vehicles with capacity y is available. Additionally, a subcontractor can be ordered if necessary. In contrast to the previously performed experiments, the capacities of the carrier-controlled vehicles are scarce. At the end, this available fleet is not able to serve all requests and the most attractive requests have to be selected for carrier fulfillment, whereas the remaining requests are selected for sub-contractor's fulfillment. The MA is applied three times to each CPDSP instance introduced in Subsection 3.5.3. Table 6.4 shows the average costs determined for the problem classes achieved for different capacities y. All results are given in relation to the observed results for y = 175 capacity units. A reduction of the vehicles' capacity down to y = 100 does on average lead to significantly increased fulfillment costs. An increase of only 1% is observed (cf. last row of Table 6.4). This value is further increased up to 6% additional costs compared to the reference value as soon as y is further decreased. The moderate exceeding of the reference value is mainly caused by an intensified usage of LSP resources that are available for the fair and comparable charge described in equation (3.18). If the carrier-controlled vehicles are completely blocked then the remaining requests are given to LSPs without additional costs. Table 6.5 shows the changes of the


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Table 6.2. Overall costs for the request fulfillment

a

surcharge (-/3)

discount (p)

0.75 0.50 0.25 0.00 0.25 0.50 0.75

number of requests served by carrier-controlled vehicles, again compared to the reference value with y = 175. This number decreases if the capacity of the carrier-controlled vehicles is reduced. The observed decrease of carrier-served requests is significantly larger for problems with larger time windows (R2, RC2, C2). It can be expected that the MA is incapable of composing those requests into high quality routes. For this reason, it is cheaper t o source them out. The second experiment aims a t investigating the algorithm's capability to identify the most valuable requests in scarce resource situations with discounted and overpriced LSP-charges. Therefore, the memetic algorithm is applied t o the CPDSP instances (4,a, y , a, P ) with the vehicle capacities y = 50,75,100,125,150 and portfolio diversifications (a,P ) E {0.25,0.5,0.75,1.0) x{-0.75, -0.5, -0.25,0,0.25,0.5,0.75).

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Table 6.3. Self-served requests CY

surcharge (-P) discount (p) 0.75 0.50 0.25 0.00 0.25 0.50 0.75

Table 6.4. Minimized costs in case of decreasing average vehicle capacities (relative t o y = 175) 50 R1 0.02 R2 0.07 RC1 0.08 RC2 0.06 C1 0.04 C2 0.08 avg. 0.06

vehicle capacity y 75 100 125 150 0.00 -0.01 0.00 0.00 0.03 0.01 0.02 0.01 0.03 0.01 0.02 0.01 0.04 0.01 -0.00 -0.01 0.04 -0.02 0.02 -0.00 0.08 0.03 0.02 -0.01 0.04 0.01 .OO 0.00

175 0.00 0.00 0.00 0.00 0.00 0.00 0.00


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Table 6.5. Number of self-served requests in case of decreasing average vehicle capacities (relative t o y = 175) vehicle capacity r R1 R2 RC1 RC2 C1 C2

-0.07 -0.47 -0.12 -0.64 -0.17 -0.72

-0.01 -0.27 -0.02 -0.30 -0.04 -0.54

-0.02 -0.13 0.01 -0.07 -0.01 -0.19

0.00 -0.16 0.03 -0.10 -0.01 -0.12

0.00 -0.07 0.02 0.12 0.02 0.05

0.00 0.00 0.00 0.00 0.00 0.00

avg. -0.37 -0.20 -0.07 -0.06 2.33 0.00

Table 6.6. Average costs for the fulfillment of the complete request portfolio (relative t o the average costs observed for instances with y = 150, a = 0, ,B = 0) surcharge (-P) discount (p) 0.75 0.50 0.25 0.00 0.25 0.50 0.75 0.25 50.000.15 0.13 0.10 0.06 0.01-0.06-0.14 75.00 0.12 0.09 0.07 0.04-0.01 -0.07-0.15 100.00 0.09 0.07 0.05 0.01 -0.02 -0.06 -0.14 125.00 0.08 0.06 0.04 0.01 -0.02 -0.07 -0.14 150.00 0.06 0.04 0.03 -0.00 -0.03 -0.07 -0.14 0.5 50.00 0.24 0.18 0.12 0.06 -0.03 -0.16 -0.31 75.00 0.17 0.13 0.09 0.04 -0.04 -0.15 -0.29 100.00 0.14 0.10 0.06 0.01 -0.04 -0.15 -0.29 125.00 0.11 0.08 0.06 0.01 -0.05 -0.16 -0.29 150.00 0.11 0.09 0.04 -0.00 -0.05 -0.15 -0.28 a

y

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The reduction of the carrier costs for the external request fulfillment leads to a decrease of the overall fulfillment costs for a given request portfolio. Table 6.6 shows the average observed costs for different capacities y and different frequencies a. The presented results show the relative costs compared t o the reference situation with y = 150, a = 0 and P = 0. An extension of the capacities y of the available vehicles influences the observed costs. In the a = 0.75-example, the enlargement of y from 50 up to 150 results in a cost reduction of between 29% above the reference costs down to 13% above the reference costs in the -P = 0.75-situation (extreme surcharges). On the other hand, if the charges are discounted (P = 0.75) then only slight savings are obtained. This is mainly caused by the fact that most of the unprofitable requests have already been given t o a sub-contractor. Similar results are observed for charges scaled between the two extreme values independent of the frequency of discounted or surcharged requests. If the frequency of discounted or surcharged carrier charges is increased from a = 0.25 up t o a = 1 then the differences between the lowest and the largest observed costs also increases. The differences in the observed costs mainly depend upon the different numbers of self-served requests. These numbers are shown in Table 6.7 as relative values compared to the reference instances as mentioned above. At first, it is observed that the number of externally fulfilled requests increase if the freight charges for the externalized requests decrease. If the capacity y is enlarged then the difference between the maximum number of self-fulfilled requests (P = -0.75) and the observed minimum number (P = 0.75) increases. As an example, the a = 0.75-case is considered. If the capacity of the available controllable vehicles is small (y = 50) then a reduction of the number of routed requests from -12% down to -70% below the reference value is observed. In case of large capacities y = 150 the maximum number of self-served requests is observed as 19% above the reference value (maximum surcharge) and this observed request number is reduced down t o 62% below the reference value (maximum discounted charge cases). Secondly, it can be observed that in case of surcharges the number of selfserved requests is increased more distinctly than in the case of a capacity extension in a situation with discounted charges. In order to analyze the capability of the proposed MA t o separate surcharge-requests from those with discounted freight carrier charges, additional experiments are performed. Let m+,,,,,p,, be the number of requests for which a discounted freight charge is available and let h4,,,,,p,, be the number of sub-contracted requests with an available discounted freight charge. The expected number S,,p,, of erroneous requests for a given situation ( a , p , y) is defined and calculated by


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Table 6.7. Averagely observed numbers of routed requests (relative to the average observed numbers for instances with y = 150,a = 0, ,B = 0) a

y

surcharge (-/3)

discount

(p)

0.75 0.50 0.25 0.00 0.25 0.50 0.75

where n,,p,, represents the number of experiments for the given parameters a,0, y. This value represents a frequency measure for the event that a request with a surcharge is accidentally sub-contracted. The values achieved are presented in Table 6.8. They are shown relatively t o the reference results for ,6' = 0 and y = 50. The error probability decreases if the surcharge amount increases from = 0.25 up t o ,6' = 0.75. For larger capacities y,the error probability reduction is not as distinctive as for smaller capacities. Let f+,,,,,p,, denote the corresponding number of those requests included into the path of a vehicle of the considered carrier company. Such a request is called an erroneous request because its fulfillment by a sub-contractor is cheaper so that its fulfillment by a carrier-controlled vehicle represents an error of the separation of the portfolio. Again, n,,p,, represents the number of experiments performed for the situation ( a ,p, 7). The expectance ecu,p,, of such erroneous requests for a given situation (a, 0,y) is defined and calculated by

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Table 6.8. Probability &,p,, of the occurrence of an erroneous request that is out-

sourced by a mistake (relative to the average observed values with y = 150,P = 0 for each a-value) a

y

surcharge (-P) 0.75 0.5 0.25

0

The value ~,,p,, represents the probability that a request is integrated into the route of one carrier-controlled vehicle, although a cheaper sub-contractorservice is available. Table 6.9 contains the observed probabilities relative to the probabilities observed for the reference situations with ,8 = 0 and y = 50 for the different a-values. It is the first observation that the error probability decreases if the freight charges decrease. As soon as a slight discount is offered (P = 0.25), the MA is able t o detect those requests and transfers them t o the sub-contracted carriers. The largest savings are observed for the largest discount class (P = 0.75). The variation of the capacity y does not come along with a parallel change of the observed discounts. Quite the contrary, the error probability seems to be


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Table 6.9. Probability ~ , , p , , of the occurrence of an erroneous request (relative to the average observed values with y = 150, /3 = 0 for each a-value) a

7

discount ( p ) 0.00 0.25 0.50 0.75

independent from the chosen capacity y in situations with discounted freight carrier charges. 6.4.5 Identification of Deferrable Requests

The postponement of requests is of special interest in stochastic planning problems with a continuously proceeding time measure. The customer requests are typically released subsequently a t unpredictable times. Additionally, the customer specified time windows prevent the collection of the complete planning data before the plan is setup. Since several requests require a quick response, the time axis is typically partitioned into several consecutive time intervals. Each interval represents a planning period. The transportation plan (pd-schedule) that determines the operations to be performed in the current period is fixed just before the period starts and it remains valid for the complete period. Requests scheduled out of this planning period are allowed t o be re-scheduled. Typically, it is sufficient t o fix the length of the planning periods in advance, and t o update the necessary planning data periodically before the next planning period starts. The planning routine is executed once for each period. The length of the planning period determines the time gap between two adjacent calls of the planning period. In some situations the modifications of the planning data are so important that it is not beneficial t o wait until the end

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of a period but to re-plan immediately. A new run of the planning procedures is invoked as soon as the additional data require a re-planning. In both situations a sequence of static and deterministic optimization problems have to be solved. It is often not necessary to schedule all requests available at the beginning of a certain planning period. In some cases, a postponement is performed with the expectation that forthcoming requests can be coupled with the postponed request in a profit-enlarging way. A postponed request remains unconsidered in the current transportation plan. The following experiments are setup in order to assess the general applicability of the proposed model and the developed MA to solve the mentioned static and deterministic instances of dynamic and stochastic variants of pickup and delivery selection problems. Ghiani et al. (2003), Psaraftis (1995) or Psaraftis (1988) discuss special features of this kind of decision problem. Fleischmann et al. (2003) and Slater (2002) describe the setup of systems for dynamic vehicle routing and scheduling. Special problems of this kind are investigated, among others, by Pankratz (2002), Powell et al. (2000) or Rego and Roucairol (1995). Experimental Setup

The MA is used to determine a pd-schedule with maximum profit contributions. This value is given by the sum of the revenues associated with the requests, whose fulfillment is announced in the recently generated transportation plan, minus the fulfillment costs for the non-deferred request. Requests that are not included into the routes of the available vehicles of the considered carrier company are postponed and are not considered for this planning period. However, a request that cannot be postponed and that is not considered in the determined pd-schedule is sub-contracted. To decide if a request can be postponed and executed into a later planning period it has to be checked to see whether there is a departure time tdept for leaving the pickup location that lies outside the current planning period [T, T AT] after T AT. Additionally, it has to be checked to see whether an arrival a t the corresponding delivery location within the customer specified time window is possible. Each request is classified as self-served (SS), outsourced ( 0 s ) or postponed (PP).A request that is contained in the generated pd-schedule is classified as SS. Let t,l,,,(r) denote the latest allowed delivery time of the request r and t(r) denotes the minimum travel time between the pickup and the delivery location of request r . This request is classified as PP, if and only if it fulfills the following two properties:

+

+

1. There exists t in the time window associated with the pickup operation, so that t T A T and 2. The earliest possible arrival time fulfills the condition Tarriw:= T A T '('1 5 tdose (r).

> +

+

+


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delivery operation pickup operation

Fig. 6.15. Postponement of requests into a later planning period

If r violates a t least one of the two conditions, and if r is not classified as SS then r is classified as 0 s . Such a request cannot be postponed and is immediately fulfilled during the next planning period by an LSP. In Fig. 6.15, three different situations are shown. In each situation it has to be decided whether the corresponding request r l , r 2 or r3 can be postponed from the current planning period [T,T AT] into a later planning period. Request rl is deferrable. If the departure time T A T is selected then the arrival time T T i "lies in the time window (the upper rectangle in the rl-section) of the delivery operation. This arrival time is approximately calculated by adding the necessary travel time to the departure time. The remaining two requests cannot be postponed. In case of 7-2, the earlilies after est possible arrival time at the delivery location, denoted by T,a,TrZ" the closing time of the corresponding time window. The third request r3 has to be scheduled in the current planning period because the latest possible departure time at the pickup location lies within this period. Since both operations of a request have to be scheduled simultaneously, this request has to be incorporated into the recent transportation plan. The length of the current planning period is controlled by the parameter E E {0.2,0.3,. . . ,1.0), and the period [O; L] denotes the overall considered relevant part of the time axis as introduced in 3.5. The length of the considered planning period is determined by E . L. The MA is applied to the instances ( a ,/3, E ) in three independent runs. The fair freight carrier charge (cf. (3.18)) has to be paid for each sub-contracted request.

+

+

Presentation and Discussion of Numerical Results The results of the experiments are presented in the tables 6.10-6.12. Firstly, the number of postponed requests is shown in Table 6.10. The quantity of

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requests that are not contained in the determined pd-schedule decreases if the length of the considered planning period is enlarged from 0.2 . L up to L time units. This general observation is valid for all six problem classes. For the R1 class the largest part of deferred requests is 16% and it decreases down to 0%. Finally, all request have to be considered in the determined pd-schedule if the planning period covers the complete relevant part [O; L] of the time axis. Instances with relaxed time windows come along with a significantly enlarged initial postponement percentage. For example, 36% of the portfolio within an R2-instance is postponed in the shortest period case with E = 0.2. Table 6.10. Percentages of postponed requests

Not all deferrable requests are postponed as shown in Table 6.11. The values presented represent the average observed part of the subset of deferrable requests for which a postponement is realized. In contrast to the previously discussed table, a monotone development of the percentage within one problem class cannot be observed. For instances with tight time windows (Rl, RC1 and C l ) , a trend towards a convergence down to zero is recognizable. However, for problems with relaxed time windows, the averagely observed exploitation percentage for E 5 0.9 is significantly larger than 20%. This is mainly caused by the long duration of the pickup and delivery time windows. These intervals allow for a more extensive shifting of the execution times along the time axis. In a problem with tight time windows, such flexibility is not given. Postponement is established in order to allow the combination of currently unprofitable requests with other currently still unknown, but highly expected, requests. Following this way, the aim is the realization of additional positive profit contributions from those requests that are currently unprofitable. If the possibility of the deferment of unprofitable requests is constrained, then the negative profit contributions are realized in the current planning period. The number of available requests is limited in the used benchmark instances. If the planning period length is extended, then the number of deferrable requests decreases due to customer specified time windows. Additional negative profit contributions are realized in the planning period under consideration as can be seen in Table 6.12, in which the achieved profit contributions are shown

6.5 Conclusions

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Table 6.11. Exploitation percentages of the postponement capability

for different combinations of problem classes and planning period lengths. All results are presented relatively to the results obtained from the E = 0.2 experiments. As expected, a decrease of the profit contribution can be observed as soon as the planning period length is extended. Table 6.12. Averagely achieved profit contribution (relative to the

E

= 0.20-cases)

6.5 Conclusions A simple time-window oriented and permutation controlled construction heuristic for pd-schedules has been developed. The generation of a large diversity of pd-schedules for a given set of pd-requests is possible by varying the control permutation. For this reason, the procedure is applied in order t o generate a diversified initial population of a memetic search algorithm. The MA is configured for improving the initial pd-schedules of low quality up t o individuals representing transportation plans of above average quality. The used representation of an individual is non-standard. Routes are stored into several strings of variable length instead of into one single string of a fixed length. Only requests that are served by carrier-controlled vehicles are contained in the routes. The precedence and the pairing constraints are kept syntactically.

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The genetic search is supported by a local search in two ways. Firstly, a locally working procedure is applied that repairs time window and capacity constraint violations and therefore ensures that each individual represents a feasible solution. Secondly, a simple route improvement procedure, based on the Zopt-idea, is applied to each pd-path generated in the offspring. The core operator of the MA is the new crossover operator PPSX that generates a feasible offspring from two feasible parental individuals. It is based on a generalization of a crossover operator designed for and successfully applied to merging complete permutations while keeping precedence constraints alive. A problem-specific mutation operator supports and completes the PPSX operator. The applicability of the developed MA-framework has been tested successfully within several computational experiments in which the MA is applied t o the benchmark instances introduced in Chapter 3. All observed results are promising and consistent.

Coping with Compulsory Requests

The MA proposed in the previous Chapter 6 is able to generate pd-paths that cope appropriately with the goals of cost minimization or profit maximization. Locally acting heuristics support the genetic search in the determination of high quality visiting sequences although time windows and maximum capacities make the composition of promising routes a very challenging task. In this chapter, the route generation is additionally hindered by the quoteconstraint Q(R+)= I that requires the consideration of all compulsory requests in the generated routes in addition to the preservation of the time window and capacity feasibility. So far, each request that does not match with the other requests contained in a generated route is removed temporarily or finally from this route. For a compulsory request, it has to be ensured that it is re-inserted into exactly one of the pd-paths belonging to the transportation plans generated in the later evolution stages. The quote-constraint requires an extension of the MA so that undesired request sub-contractions are prevented. Unfortunately, the additional condition contradicts the goal of minimizing the travel distances within the generated routes. For this reason, one cannot expect the genetic search for the least cost transportation plans to account for the quote-constraint without any additional effort. Furthermore, the re-insertion of a temporarily unconsidered compulsory request into an existing route is already a very hard problem. It has to be ensured that the time windows' requirements are met and that the capacity limitations are kept after the insertion. Additionally, the REPAIR()-procedure, that ensures the time window and the capacity feasibility, also counteracts the insertion of compulsory requests into the generated paths because it excludes constraint-violating requests from the pd-paths. All these reasons lead to the necessity for extending and modifying the MA-framework in order to generate promising pd-paths that serve all compulsory requests in a feasible way a t least fulfillment costs.

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7 Coping with Compulsory Requests

In this chapter, several extensions of the previously proposed MA-framework are investigated and their capabilities for composing the compulsory request into the least cost pd-paths are tested. The application of penalties for a degradation of the quality of transportation plans with unrouted compulsory requests is investigated in Sect. 7.1. Next, a two-phase genetic search approach is proposed in Sect. 7.2. It first tries to identify pd-schedules in which all compulsory requests are contained in the pd-paths. Afterwards, it aims at reducing the fulfillment costs by evolving these solutions towards the goal of least fulfillment costs. The feasibility of the generated solutions is preserved. Finally, an alternating and converging constraint approach is discussed in Sect. 7.3. Initially, the quote constraint is strongly relaxed. Only an arbitrarily selected fraction of the subset of compulsory requests has to be served. If the evolved individuals satisfy this reduced quote, the travel costs are reduced. Then, the required quote is slightly enlarged and the population is evolved again towards feasibility and so on. Both approaches are tested within several computational experiments. The obtained results are presented and discussed in Sect. 7.4.

7.1 Limits of Fitness Penalization Penalization during the fitness evaluation process is a typical approach to direct the evolution towards feasible solution instances of a constrained optimization problem (Smith and Coit (2000)). Several experiments have been performed in order to assess the impacts of penalizing unrouted compulsory requests within the previous experiments for the PDSPLSP with tariff diversifications. The penalization of sub-contracted compulsory requests is realized by enlarging their freight charges for externalized requests over the costs for a self-fulfillment, hence over the fair freight tariff.

7.1.1 Static Penalties In an approach with static penalty values, constraint violations are accepted, but the fitness of such an individual is unadvantageously modified by adding (minimization) or subtracting (maximization) the static and previously known penalty value. Let T denote a benchmark instance from the suite introduced in 3.5.3 in which all requests are compulsory (PDPTW-instances). The least fulfillment costs lj(T) achieved in run j are compared with the reference costs 1(T)associated with the donating solution. The deviation Alj (T) := - 1 is smaller l(?) than zero if the fulfillment costs of the generated transportation plan are less than l(T), otherwise the fulfillment costs of the generated transportation plan exceed the comparison value.

7.1 Limits of Fitness Penalization

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The averagely observed deviation AL(Y) from the reference values for Y is defined now as

A similar measure AQ(Y)is introduced for the quote of served compulsory requests: qj(Y) denotes the quote of compulsory requests served in the pdpaths for the solution of Y achieved in run j . The associated obtained average percentages Aq(T) of self-fulfilled compulsory requests in Y are calculated by

The instances of each problem class C E {Rl, R2, RC1, RC2, C1, C2) are averaged leading to the measures

Table 7.1 shows the average observed relative travel distances and quotes, averaged for PDPTW instances within each class. PDPTW-instances are the hardest ones because no request is allowed to be rejected. The surcharges are considered as penalties that have to be paid for each sub-contracted request. Table 7.1. Solutions obtained from the static penalty approach with surcharges for sub-contracted requests class AL(C) AQ(C) AL(C) AQ(C) AL(C) AQ(C)

The observed results, already presented in a slightly different fashion in Subsection 6.4.3, show that even for a severe penalization (P = -0.75) the

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incorporation of all requests is not achieved. For problems with tight time windows ( R l , RC1 and C1) some requests remain excluded from the pd-paths and furthermore their fulfillment costs exceed the fulfillments costs associated with the reference solutions that serves as the donator of the request portfolio. If the penalization is alleviated then the observed travel distances fall below those from the reference solutions in every problem class, but the part of requests served in a t least one generated pd-path further recedes. The reason for these bad results can be found mainly in the determination of the penalty parameters. If the value that penalizes rejected requests is too large then the MA tends to decide only on routing or externalization (LSP-incorporation) of requests and inserts additional requests in the pd-paths without considering the travel distance needed. This is because the costs for serving the additional requests are negligible compared to the costs of an LSP-incorporation. Thus, the MA does not differentiate between individuals with better or worse fulfillment costs but only between individuals, in which a n increased or a decreased number of requests is contained in pd-paths. The improvement of the route length (travel costs) does not significantly enlarge the chance for an individual t o transfer its genetic information into the subsequent population(s). On the other hand, if an unrouted request is penalized with a too small value, then its externalization leads t o travel cost savings that compensate or dominate the penalty costs. In this case, the incorporation of all compulsory requests is not achieved. Computational experiments show that it is necessary t o re-determine the penalty values a t adequate levels for each problem instance anew. This makes the application of global and static penalty values unattractive. An over severe penalization of infeasibilities in early iteration stages guides the population towards feasible regions of the search space, but often prevents the search from identifying feasible optimal or suboptimal solutions. The population converges prematurely a t a poor average solution quality level. Otherwise, an over lenient penalization in latter iteration stages leads to dominating infeasible solutions, especially if the consideration of associated constraints contradicts the improvement of the followed objective. These infeasibilities cannot be solved in the remaining evolution steps.

7.1.2 Dynamically Determined Penalties Smith and Coit (2000) propose a dynamic determination of the penalty values. Instead of adding (or subtracting) a static value from the individual's fitness value for each detected error in each population, the applied penalty value is determined subject to the progress of the evolution, typically measured by the number of already performed iteration steps. A monotonic function assigns each iteration a penalty that is used to depreciate the fitness of each infeasible population member. Therefore, the penalty value is varied from population to population.

7.1 Limits of Fitness Penalization

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The dynamically determined fitness values support the search for feasible solutions with high-quality fitness values. A lenient penalization of infeasibilities in early evolution steps enables the population to evolve towards high quality solutions and prevents premature convergence. In latter iterations, a severe devaluation of infeasibilities is unconditionally recommended in order to guarantee the feasibility of the finally generated solutions. However, the setup of the dynamic penalty function is a non-trivial task. Again, it turns out that each problem instance requires a calibration of its own used penalty function. The convergence levels of the fitness values of optimal or suboptimal solutions differ among the PDSP-CR instances, so that a rescaling of the penalty function is necessary before the MA can be applied successfully to a particular instance. This is not recommended and unacceptable so that these approaches also remain unfollowed. 7.1.3 Adaptive Penalization

In order to avoid expensive parameterizations, the determination of the penalty parameters is often coupled to the development of the maintained population (Coit et al. (1996)). If the frequency of a certain infeasibility in the current population is large, then the transmission of the mistakes into the next population cannot be prevented. It is more worthwhile to differentiate the individuals with respect to their original fitness (e.g. fulfillment costs). If only few infeasibilities are detected in the currently maintained population, a severe penalization of the errors prevent the transmission of the defective genetic material into the next population. Consequently, the penalty-values are coupled to the number of detected errors in the current population. They are enlarged as soon as the number of detected infeasibilities is reduced and decreased as soon as the frequency of observed errors is increased. A successful implementation of such a selfadjusting penalty approach can be found in Schonberger et al. (2004) for a multi-constraint combinatorial optimization problem. However, the implementation of such an adaptive penalty strategy for the PDPTW, which represent the most challenging PDSP-CR instances, has failed. Again, there is some evidence that its malfunction is also based on the different levels of convergence of the fitness values belonging to each problem instance. Thus, a manual re-parameterization is still required. The application of penalty-based approaches does not lead to satisfying results. The quote-constraint cannot be kept. For this reason, implementing and using penalization of unconsidered compulsory requests for PDSP-CR instances is not used.

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7.2 A Double-Ranking Approach In this section, a n MA approach is proposed that simulates an evolution process recognizing both important performance measures: customer satisfaction and fulfillment costs. Customer satisfaction is measured by the quote of routed compulsory requests and it is a t a maximum if the quote equals 1. Customer satisfaction is given priority for the minimization of the associated costs in the sense that each selection S1 = (R:, 72:) with ~(72:) < 1 is not as valuable as another selection S2 = (R:,R;) with Q ( R i ) = 1. In every case, a selection with the latter property is preferred t o one with the first property, independent of the corresponding fulfillment costs. In order to preferentially generate individuals with large quotes and in order t o enlarge their appearance frequency in subsequent populations, the ( p A) reproduction model introduced in Chapter 6 is modified. Again, a temporal offspring population T is derived from the current population Pk. The quote of served compulsory requests within the pd-paths associated with individual indivi is denoted as qi. A permutation of the individuals in the set union T U Pk is determined. In the permutation, an individual indivi with quote qi precedes another individual indivj if the latter one has a quote of less than qi. If the quotes of both individuals are equal (qi = qj), then the individual indivi precedes indivj if the necessary fulfillment costs li associated with indivi are less than the costs lj associated with indivj. This order relation is denoted as D (indivi D indivj "indivi precedes indivj" ) and formally defined as a t least one of the conditions indivi D indivj

+

*

is satisfied. The relation D induces a double ranking within the considered mating pool T U Pk.In the outer ranking all individuals are sorted by decreasing quote q of self-served (routed) compulsory requests. Different quote classes are established. All individuals within such a class serve the same quote of compulsory requests. Within a quote-class, the corresponding individuals are resorted by increasing fulfillment costs (inner ranking). The population is evolved towards pd-schedules in which all compulsory requests are considered and the fitness of the individuals is measured by their quote of incorporated compulsory requests. Individuals with the largest quotes survive and transfer their genetic material t o the next population because the ( p A) selection saves only those individuals, which are located in the first half of the permutation of T U Pk that has been determined according t o D. Since the maximization of customer satisfaction is preferentially targeted, its fulfillment is the first goal of the memetic search. Thus, the profit maximization (or equivalently: cost minimization) objective proposed in the MA-

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7.3 Converging-Constraint Approach

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framework in Chapter 6 is replaced by the objective of maximizing the quote of incorporated compulsory requests. It is aimed at evolving a population so that at least all compulsory requests are served in a feasible way, respecting the pairing, the precedence, the time window and the capacity constraints. The inner ranking ensures the preferred selection of individuals with the highest quotes but least fulfillment costs in this quote class. As soon as quote feasible individuals have been generated, the inner ranking supports the pursuit of a reduction of the fulfillment cost minimization by placing these individuals in leading permutation positions. The domination of the quote maximization hinders a significant cost reduction before the population contains a sufficient large number of feasible individuals. For this reason, the double-ranking approach is denoted as Quote First - Cost Second (QC).

7.3 Converging-Constraint Approach The weakness of the QC-approach is its downstream consideration of the fulfillment costs associated with the generated transportation plans in the iterations before the maximum quote 1 is achieved. This means, the costs of the so far generated infeasible transportation plans are not sufficiently considered during the evolution. As a consequence, the initially determined feasible solutions are of very low quality and come along with very high fulfillment costs. To remedy this deficiency, the MA framework is extended in a way that establishes a selection procedure, in which the fulfillment costs are already considered in a more sufficient manner before the first feasible transportation plan is found.

7.3.1 Alternating and Converging Constraints The problem of solving a PDSP-CR instance aims at minimizing the fulfillment costs for serving a given request portfolio containing compulsory requests. Therefore, the generated pd-schedules have to satisfy the constraint

which expresses the requirement that all compulsory requests are contained in the routes of the carrier-owned vehicles. The previous investigations have shown that the generation of such a pdschedule is already a very hard problem if the frequency x of compulsory requests is large. In the QC-approach, an attempt is first made to evolve the population into a region with feasible pd-schedules and afterwards an attempt is made to improve the identified feasible solutions without re-loosing their feasibility.

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The focus on reaching the first goal quote-feasibility (7.5) leads to a complete negligence of minimizing the fulfillment costs. After the population contains quote-feasible individuals, the diversification of the genetic material contained in the so far evolved population is reduced. A specialization with regard to the incorporation of all compulsory requests has been taken place. The derivation of genetic information from the maintained population in order to improve the so far generated pd-paths is not possible anymore. Consequently no significant travel distance reductions are observed in the length reduction phase of the QC-approach. As a remedy, the following concept is proposed. Instead of trying to reach the quote 1 directly, this constraint is temporarily relaxed. Let 0 < t l < 1 denote the temporarily required quote. The initial population Po is evolved towards the temporary goal

To reach this goal, the (p+X) selection strategy is applied after the permutation of the generated population induced by D has been established. This can be seen as the pursuit for enlarging the quote of routed compulsory requests in the maintained individuals. A sequence PI, . . . , Pi, of populations is generated. In Pi,, an individual that fulfills the temporary quote tl is detected. Now the selection strategy is modified. Again, a (/.+A) selection strategy is applied, but the population is sorted according to the order relation 7, defined as at least one of the conditions indivi 7indivj

*

is satisfied, where l(indivi) denotes the fulfillment costs associated with the transportation plan coded in indivi. This relation realizes another double ranking. In the outer sorting, all individuals are sorted by increasing fulfillment costs. Individuals that come along with the same costs in their associated transportation plans form a length class. During an inner sorting step, all requests within the same class are re-sorted and re-arranged so that the individuals within such a length class are strung by decreasing quotes. The temporary population consisting of the original population Pi, and the generated offspring individuals is sorted according to 7 . The first half of the individuals in this list is selected to form the next population Pi,+l so that the population is now evolved towards individuals with reduced costs. A sequence of populations Pi,+l, . . . , Pj, is generated so that the average fulfillment costs associated with the individuals decrease. Now the quote-constraint is reinforced. Therefore, a new temporary quote t2 with tl < t2 < 1 is selected and the constraint (7.6) is updated to

7.3 Converging-Constraint Approach

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For the next iterations, a sequence of populations Pj,+l, . . . , Pi,is derived from Pj,. The selection is prepared by applying an D-based permutation until at least one individual that fulfills the reinforced quote constraint (7.7) is detected in population Pi,. This population is now evolved along a trail with decreasing average fulfillment costs by applying the order relation 7 in the selection procedure until a sufficient low level has reached. Then the quote constraint (7.7) is again reinforced and updated to

so that t2 < t3 < 1 and so on. This swapping of the objectives is repeated

NQ times. In each repetition, the quote constraint is reinforced so that at the end feasible individuals are identified for the constraint

The sequence tl, t 2 , . . . , tNQ is monotonically increasing and bounded by 1. For this reason the quote bound becomes tighter and tighter in each repetition and motivates the name Alternating and Converging Constraint (ACC) approach for this genetic search strategy. The framework for an ACC-based memetic algorithm is outlined in Fig. 7.1. The procedure starts with the generation and evaluation of the initial population (1).A marker CURRENTGOAL E {maxquote, mincosts) is introduced. It carries the information about the currently used selection scheme. If CURRENTGOAL is set to maxquote then D is applied in the ( p A) selection model. On the other hand, if CURRENTGOAL equals mincosts then 3 is applied. Initially, D is applied (2). Then, the iteration is started (3)-(8). At first, it is checked to see if the currently followed goal has to be swapped (4). In case that the swap-criterion is satisfied, CURRENTGOAL is modified, otherwise is it left unchanged. Depending on the value of the marker, the next population is generated (5)-(6). The iteration terminates with the increase of the iteration counter i. The motivation for the ACC development arises from the observation that the QC-approach generates feasible solutions Q(.) = 1 that are far away from the cost minimum. The periodical interruption in the strive for quote-feasible solutions provided by the ACC-MA is used to inject and evolve genetic material in the population that can be used for improving the generated pd-paths in the so far evolved transportation plans.

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(1) generate and evaluate the initial population; (2) CURRENTGOAL := maxquote; (3) 2 = 0;

(4) if(swap-criterion is satisfied) then CURRENTGOAL:=swap(CURRENTGOAL); (5) if(CURRENTGOAL==maxquote) then generatenext- pop(^) (6) else generate-next-pop(3); (7) 2 +; (8) if(iiMAXP0P) goto (4);

+

Fig. 7.1. ACC-MA framework

7.3.2 ACC-Algorithm Control The determination of the control parameter of an ACC-algorithm has remained an open issue so far. It has to be decided, when the selection scheme has to be swapped from the D-based selection to the 7-based selection and vice versa. More concretely it has to be decided

1. How the sequence t l , t 2 , . . . ,t ~ is , determined and 2. When the cost minimization is terminated. The following static rule is implemented. Instead of predetermining the temporary goal quotes t l , t2, . . . , t ~ , ,the number of iterations in which the goal quote maximization is followed, is specified. Let LE be a positive number larger than zero. For the first LE instances, the population is evolved using the D-based selection. After the LE-th population has been generated and evaluated, the 3-based selection is applied for the next L D populations. After the LE Lo-th population has been completely generated and evaluated the next LE populations are evolved following the D-based selection and so on. There are two advantages for applying this static control strategy. At first, if an explicit value ti is specified, it cannot be guaranteed that the genetic search is able to generate individuals that satisfy the first quote-constraint. The search process cannot use the further ACC-advantages. Secondly, it must be ensured that the length reduction phase is terminated after reasonable improvements. However, no dynamic or adaptively approach that ensures the satisfaction of an externally given goal is known. The ordered pair (LE, LD) is called the control of the ACC-algorithm. To accelerate the ACC-approach, an elitism-based modification is installed (Sarma and Jong, 2000). In order to prevent the destruction of quote-feasible solutions during a length reduction phase, the best individual according to D is saved at the end of each quote maximization phase. After the subsequent length-reduction phase has been completed, the saved individual is re-inserted into the population.

+

7.4 Assessing QC-MA and ACC-MA: Numerical Results

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7.4 Assessing QC-MA and ACC-MA: Numerical Results The extension of the basic QC-MA framework to the ACC-MA framework has significant impacts on the quality of the produced transportation plans. In the following, the results, which have been achieved within several computational experiments performed in order to assess and to compare the recently introduced MA frameworks are reported and discussed.

7.4.1 Experimental Setup Both the QC-MA and the ACC-MA are applied to the benchmark instances (4, a, X ) introduced in Subsection 3.5.3. Each of the 324 instances is treated in N ( @ ,x) := 3 independent runs, @ = (4,a ) . Overall 972 individual experiments are performed. An initial analysis of the numerical results shows that the so far used parameter setting is inappropriate for the QC-MA as well as for the ACCMA. The maximum quote 1 is not achieved in most of the cases. For this reason, two parameters are re-adjusted. It has been observed that the evolved population converges prematurely at a quote level significantly below 1. To overcome this shortcoming the population size has been doubled up to 200 maintained individuals in order to prevent premature convergence. On the other hand, the mutation frequency is increased from 0.5 up to 0.75. Enlarging the frequency of the inserting operators and procedures intensifies the insertion of temporarily unrouted requests into the pd-paths. The control of the ACC-MA is set to (15,20). For these values, the best average results have been observed in a preparatory experiment. In this experiment the ACC-MA has been parameterized with different control settings and it has been applied to the instances of the representative set P introduced in Subsection 6.4.1. The average running time for a problem instance within the 200-individual population MA on a Pentium 4 PC operating at 1 GHz is now 4.5 minutes. 7.4.2 Numerical Results In contrast to the experiments reported in Chapter 4, the genetic search performed by QC-MA and ACC-MA consists of two phases. In the first phase, an attempt is made to generate feasible individuals for the PDSP-CR and in the second phase, an attempt is made to improve the established feasible individuals. However, it cannot be guaranteed that feasible individuals are actually found. For this reason, the first experiment is dedicated to investigate the capability of QC-MA and ACC-MA to find feasible solutions of a given PDSP-CR instance (@,X ) .

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Let "N $;, (@,X) denote the number of instances for which QC-MA has successfully identified a t least one feasible solution. The corresponding value for the ACC-MA is N?~A(@,x). The feasibility quotes for the problem instance (@,X) are defined as

for QC-MA and

for ACC-MA. These quotes are averaged for each class C. Each class consists of nine instances so that the averaged feasibility quotes of a class for a given X-value are given by

and

The observed quotes are shown in Table 7.2. The upper table contains the quotes observed from the QC-MA runs and the lower table contains the quotes achieved in the ACC-MA experiment. For instances taken from the class R2, QC-MA seems to be more appropriate for identify feasible solutions. The average feasibility quotes for the QC-MA are significantly larger than the ACC-MA quotes. Both MAS are reliable for the identification of feasible solutions for instances taken from RC1. In all executed runs, they find feasible solutions. ACC-MA outperforms QC-MA for the challenging instances taken from R1 with very large frequencies of compulsory requests ( x = 0.90,l). In these cases ACC-MA is more reliable. Especially for the PDPTW-instances (x = I), ACC-MA finds feasible solutions in 81% of the performed runs whereas QCMA is successful in only 67% of the experiments. Neither QC-MA nor ACC-MA is able to generate feasible solutions for instances from C1 or C2. For this reason, further results achieved from applying the algorithms to problems of these two classes are no longer presented. Since the generated solutions are infeasible, the results cannot be compared


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Table 7.2. Feasibility quotes

in a meaningful way with the feasible solutions obtained from instances of the remaining four classes. So far, it has been seen that the QC-MA is more reliable and identifies feasible solutions with a higher frequency. However, the quality of the found feasible solutions was left unconsidered in the previous experiment. In order to compare the quality of the best feasible solutions, the results obtained for QC-MA and ACC-MA are evaluated with respect to the costs of the represented transportation plans. To allow the comparison of the results achieved for different instances, the relative costs of the solutions are calculated. Let Lref (@,1) the reference fulfillment costs that are derived from the x ) respectively I?:&(@ X) be , ($ ;,,I donating Solomon instance and let@ the set of indices of the runs, in which feasible solutions have been found. The fulfillment costs of the generated feasible solution obtained in run i E Qc ear (@,X) and i E $ z~ ,:@ (, X) are stored in L&(@, X) (for QC-MA) and LAcc (@,x ) (for ACC-MA). The relative fulfillment costs

are calculated. They map all observed fulfillment costs into the interval [O, 11. These values are averaged for each problem instance @ and each frequency x of compulsory requests. This average is given by


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for results obtained from the QC-MA application and

for the evaluation of solutions obtained in ACC-MA runs. For each problem class C and each frequency X, the averaged relative fulfillment costs

OEC

and

are calculated. The observed values for L?&(C, X) and LT&(C, presented in Table 7.3.

x)

are

Table 7.3. Relative costs

It is the first observation that ACC-MA significantly outperforms QC-MA in all cases (C, x). In order to explain these results, the fraction VQc(C, X) respectively VACC(C,x) of non-compulsory requests within the set on routed requests (voluntarily routed requests) is analyzed (cf. Table 7.4). The reported percentages of routed (non-externalized) requests EQc(C, X) and EACC(C,X) in Table 7.5 show that ACC-MA incorporates more requests into the routes than QC-MA is able to do.


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Table 7.4. Percentages of non-compulsory routed request

Table 7.5. Percentages of routed requests (compulsory and non-compulsory)

The presented results show that the ACC-MA is able t o produce transportation plans with significantly reduced fulfillment costs compared to the solutions obtained from the QC-MA. Additionally, ACC-MA generates transportation plans in which significantly more customer locations are contained in the associated pd-paths. This permits the conclusion that the quality of the routes produced by ACC-MA runs is significant higher than the quality of the pd-paths that are part of the solutions proposed by QC-MA. The recurring intermediate cost reduction during the ACC-MA evolution leads t o significant improvements of the produced pd-paths.

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7.4.3 Impacts of Intermediate Cost Reductions: An Example

In order to analyze the impacts of intermediate cost reduction during the evolution of a population of transportation plans, two key measures are observed during the iterations. The fulfillment costs associated with individual indivj in population Pi is denoted as L;. This value is compared to a given reference value Lref leading Lz.

to the relative travel distance 1;. := &. The quote of routed compulsory requests associated with the individual j taken from population Piis given by q;. The average relative costs i g within population Piand the average quote qi are defined as

where N is the size of the maintained population. &) describes the state of the i-th population The ordered pair Si := is called the trail S of the population and the sequence So, $, . . . , SMAXPoP through the search space. Let a ( i ) be the index of the first element within the permutation of the population forming the i-th population and induced by D. The trail of the and best individual in each population is defined as So, S1, . . . , SMAXPoP si = ( ~ u ( i )L(i)). , Initially, the state of the population obtained from the construction heuristic is So= (4'0,lo). Typically, @ is significantly smaller than 1. Figure 7.2 shows the trail observed for the evolution of the population to solve the instance (R104,0,1). The initial population state is 30= (0.3,1.54), thus the mean percentage of routed compulsory requests is 30 percent and the average relative costs are 1.54, which means costs that exceed the reference value by 54%. The population is further evolved towards a mean acceptance rate of 92 percent and the corresponding relative costs are 1.42 (SS0= (0.92,1.42)). Additional requests are incorporated without significant cost changes so that quote-feasible individuals are generated at the mean relative costs of 1.40. The population is further evolved visiting the intermediate state Sloe = (l.0,1.24). This value is reduced down to around 21% above the known comparison value leading to Slso = S 2 0 0 = (1.0,1.21). The trail of the best individual is different from the trail of the population. Initially, around 37 percent of the compulsory requests are served and relative costs of 1.68 are observed: So = (0.37,1.68). The quote is increased up to 98% whereas the corresponding relative costs are reduced down to 1.44%.

(a,


costs li and

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Ti

quote qi and qi Fig. 7.2. QC-approach: Trail of the population (continuous line) and trail of the best solutions observed (dotted line)

It is SS0= (0.98,1.44). Up to iteration 50, the trail forms a zigzag-course. For a reached quote, the costs are reduced, so that additional requests can be inserted. After the maximum quote has been reached, the costs of the generated pd-schedules are successively improved. It is Sloo= (1.0,1.24) and SlsO= SzoO = (1.0,1.21) . The trail 2 produced in the ACC-approach applied t o the instance := ( ~ iff ), ) . The MA is controlled (R104,0,1) is denoted as Z1,. . . , Zzoo(Zi by (15,20). Figure 7.3 show both trails. The dashed line represents the trail S achieved from the QC-approach and the dotted line show the trail 2 generated by the ACC-MA. Both algorithms produce feasible results. However, significant differences between S and 2 are observed. At first, the final quality of the best observed feasible solution produced by the ACC is essentially better than the quality of the best solutions found by the QC-algorithm. The ACC approach even produces a final feasible solution whose costs lie below the reference value. It is ifoo = 1.21 and i:oo = 0.99. In contrast t o the trail of the QC-algorithm, the ACC-approach includes temporary decreases of the quote of routed compulsory requests. In order to get a n understanding of the functionality of the ACC-MA, the trail 2 is analyzed in a deeper way. The first phase comprises the stages Z1,. . . , Z15. In these iterations the MA enlarges the number of routed compulsory quotes. Figure 7.4 shows that the ACC-MA succeeds in the first three quote-maximizing phases.

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1.4

costs

if and if

1.3 1.2

1.1 1.0 0.3 0.4 0.5

0.6 quote

0.7 0.8 0.9

0.9 1.0

qf and qf

Fig. 7.3. Comparison of the QC-MA (dashed line) and the ACC-MA (dotted line) trails

In the first one, cjf is enlarged from 0.3 up to more than 0.65 from i = 1,. . . ,15. During the second one, i = 36, . . . ,50, the quote is enlarged from around 0.75 up to almost 0.95 and In the third quote-targeting phase from i = 71 to i = 85 qf is re-enlarged up to 0.97. The success of the ACC-approach is mainly based on its behavior during the cost reduction phases. Within such an iteration subsequence, the most unadvantageously routed requests are temporarily removed from the pd-paths. They are re-inserted in a more profitable manner or substituted by other requests. In the initial iterations of the first cost reduction phase, i = 16, . . . ,20, savings are achieved by reducing the quote of routed compulsory requests. During the remaining iteration in the cost reduction leg, i = 21,. . . ,35, additional requests are incorporated into the pd-paths in a way that reduces the fulfillment costs. The savings are achieved by reducing the number of subcontracted requests and by composing profitable routes without detours. At the end of the cost reduction phase, the fulfillment costs have been significantly reduced and the quote of routed compulsory requests is also preserved or even enlarged. It can be expected that the unprofitable requests are temporarily removed from the routes and then re-inserted in a more appropriate and cheaper fashion.

7.5 Conclusions

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In the first cost reduction phase, qf is enlarged from around 0.65 up to 0.75 and a t the same time the fulfillment costs are reduced from around 1.7 down to less than 1.3. Similar observations are made during the second cost reduction phase i = 50,. . . ,70. Again, in the first iterations of the cost reduction phase the quote is reduced, but at the end the quote is slightly enlarged (compared to the start of the phase) and the costs have been significantly reduced from around 1.3 down to less than 1.1.

I?

Fig. 7.4. Trail of an ACC-MA run for the instance (r104,0,1)

7.5 Conclusions The consideration of the quote-constraint belonging to PDSP-CR instances requires the extension of the previously proposed population model. Identifying feasible solutions is already a very challenging task. A new kind of population model has been proposed. The quote constraint is initially relaxed at a level that permits for the generation of feasible solutions (with regard to the relaxed constrained). As soon as new feasible solutions are found, the so far identified solutions are improved with special attention to the fulfillment costs. Afterwards, the quote constraint is reinforced and so on. The application of the MA-framework proposed in the previous chapter together with the new population models allows for the generation of transportation plans that satisfy the customer's requirements (by preventing the

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externalization of compulsory requests) and the carrier's wishes (least fulfillment costs). The generated solutions in the performed computational experiments are of convincing quality.

Request Selection and Collaborative Planning

Vehicle routing problems have their origins in distribution and/or collection problems. Typically, it is assumed that a sufficiently large fleet of vehicles is exclusively available for fulfilling the demanded transport tasks (requests). This assumption reflected with the real world for several years. In recent years, a large number of companies outsourced their fleets to independent carrier companies and allowed them to operate on their own. For such a carrier, the operational short-term route generation problem differs considerably from the standard models problems, especially the vehicle routing problem with time windows (VRPTW) and the pickup and delivery problem with time windows (PDPTW). Since the daily request portfolio cannot be anticipated exactly, and since the available transport resources cannot be immediately adapted t o the needed demand, bottleneck-situations appear. In these situations the demanded transport resources exceed the available capacities. The solution of such a bottleneck situation has received only minor attention so far in the vehicle routing context. However, such situations often occur in practical applications. A typical remedy is to order a logistics service provider to serve these requests. Typically, such a service is available, but its charge often exceeds the revenues associated with the corresponding requests. Since the ad-hoc incorporation of a logistics service provider is not convenient, cooperations between several independent carriers, called GroupageSystems (GS), are setup. Their purpose is to find an equilibrium between the demanded and available transport resources within several carriers by interchanging customer requests in order to prevent the necessity of incorporating expensive additional resources Kopfer and Pankratz (1998). Instead, requests are negotiated between participants of the GS for fair charges. Each participant should benefit from this cooperation by enlarging its efficiency. This chapter proposes a simple optimization model for the determination of a maximum profit interchange of requests between carriers forming a GS. To maximize the success of the participating carrier companies, each of them is allowed to specify and to propose bundles of requests, that they can fulfill with non-negative profit contributions (earned revenues minus spent costs).

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8 Request Selection and Collaborative Planning

If several carriers compete for the same requests then an independent coordinator determines the final assignment of the bundles to the carriers. Thereby, the minimization of costs for external request fulfillments is targeted. The organization of this chapter is as follows. In Sect. 8.1, the optimization model is derived. In Sect. 8.2, its application in a GS is described and a simulation environment is set up. Numerical results from experiments that simulate the configured GS are reported in Sect. 8.3. This chapter ends with summarizing remarks and some ideas for further research topics.

8.1 The Portfolio Re-composition Problem A GS aims at interchanging requests between carriers, so that each carrier achieves an updated request portfolio that can be served leading to a maximized profit contribution. Requests that do not lead to a positive contribution to the profit of a certain carrier are allowed to be fulfilled by another carrier. The corresponding revenues are simultaneously transmitted from the emitting to the serving carrier. Other charges are not paid. The success of a GS depends upon its ability to determine appropriate exchanges of requests among the carriers forming the GS. A model for optimizing the request interchange in a GS is presented. It is aimed at recomposing the request portfolios of the carriers in order to reduce the sum of travel costs and service provider costs. Attempts are made to transfer unprofitable requests to the portfolio of other GS participants. On the other hand each carrier tries to acquire additional profitable requests from other GS participants. The presented model can be configured for several GS environments.

8.1.1 Literature Review

A planning domain describes a part of a value-added chain that is under the control of one planning organization. The incorporation of two or more planning domains into one single planning domain leads to collaborative planning Kilger and Reuter (2002). A cooperation is a framework in which the rules for the domain integration are described Seuring (2001). It permits the coordination of the planning processes in order to achieve &%&ncy improvements. The exchange of data that is relevant for the planning process is a core aspect of the cooperation. A cooperation is instantiated if each single co-operator cannot achieve a given goal because the competencies or resources are insufficient or if the cooperation leads to additional benefits for each co-operator. The simplest form of a cooperation of more than one carrier is established for only one request or one bundle of requests. A carrier pays another carrier for serving one or several request(s), cf. Diaby and Ramesh (1995). This special form of cooperation is called carrier service. The corresponding collaborative

8.1 The Portfolio Re-composition Problem

137

planning problem is a combined route generation and freight optimization problem Kopfer (1992). A cooperation of several independent carriers constitutes a GS if each participating carrier is allowed to offer and to receive requests from one or more other participating carriers Kopfer and Pankratz (1998). Instead of paying charges to the executing carriers, the emitting carriers let the receiving carriers have the associated revenues. To establish a GS, rules that define the possible interactions of the participants are defined. The request assignment must take into account some organizational restrictions that represent the demand of the partners in the cooperation. Additionally, each single participant must be protected against unfair request distribution. The goal of a GS is to find an assignment of requests to the available carriers that maximizes the overall global profit or, equivalently, minimizes the total costs. However, the participants themselves who want to maximize their own profit and not necessarily the profit of the GS contradict this goal. Instead of global optimization approaches, agent-based bilateral negotiating systems are proposed ((Kopfer and Pankratz, 1998), (Gomber et al., 1997)). 8.1.2 Formal Problem Statement

Assume several independent carrier companies C1, . . . , C M .Each carrier Ci comes along with the request portfolio R i . This set is partitioned into the two sets 72' and R c . The former set contains all requests served by the fleet of Ci following the routing plan IT(R'). All remaining requests, collected in the latter set, are transferred to an external logistics service provider for execution. The execution costs for carrier Ci are denoted by c(IT(R+)) and the freight charge paid to the incorporated logistics service provider is FC(R,). The portfolio composition

causes overall costs of the amount

A request r that is unprofitable for a carrier Ci could be inserted into the portfolio of another carrier Ci. The latter carrier receives the revenues REV, associated with r . A request interchange between two carriers is profitable for both carriers if, and only if,

138

0


The additional costs for Cj caused by the incorporation of r are less than REV,, and The costs for serving r with Ci's own equipment or for the incorporation of a logistics service provider (from outside the GS) to serve r are larger than REV,.

A profitable request interchange between two carriers enlarges the profit contributions of both carriers. The emitting carrier looses the revenues, but it saves the service provider or travel costs, which exceed the earned revenues. On the other hand, the receiving carrier enlarges its profit contribution, because the additional revenues exceed its additional (travel) costs. Let Rij denote the set of requests transferred from the portfolio of carrier Ci to the portfolio of Cj. The updated request portfolio Ri is given by

All unfavorable requests are given to another participant of the GS, or they have been inserted into the pool 7?M+1 of requests that the GS assigns to external service providers. Therefore, U2: = 0 can be assumed. The updated portfolio composition is given by (21,. .., R M , ~ M + I ) .

(8.4) The Portfolio Re-Composition Problem (PRP) is aimed at identifying pairwise disjoint request transfers Rij leading to an updated portfolio composition with minimized costs. The P R P can be stated more formally in the optimization model

The goal is to modify the given portfolio composition. Therefore, the necessary request transfers between the carriers are determined. Constraint (8.6) ensures that only available requests are transferred and (8.7) allows only the unique transfer of a certain request. The determination of a minimum cost partition of the request portfolio Ri of a certain carrier Ci can be formulated in terms of the Pickup and Delivery Selection Problem with Postponement. In this context, all externalized or postponed requests are completely ignored by the considered carrier Ci in the hope that another member of the GS fulfills these requests. If no other member is interested in fulfilling such a request, it is served by an LSP paid by the GS.

8.2 Configuration of the Groupage System

139

8.2 Configuration of the Groupage System A pure centralized assignment of requests to carriers is often hindered by the participants themselves who worry about disadvantages resulting from the disclosure of internal data or unbalanced request allocations. However, a complete absence of a central coordination which oversees the overall success of the GS and prevents tactical or strategic acting to exclude certain weak customers from further participation within the cooperation, is often observed. The aim of a PRP is a global reassignment of requests among the independent carriers to reduce the sum of costs. However, this is not realizable in real world applications. As mentioned already above, each single customer has to protect its own economic success so that no carrier would allow handing out the responsibility of its complete request portfolio. Otherwise, a carrier runs the danger of being forced to emit its most profitable requests and to serve other less profitable requests. In a global optimal solution of a PRP instance, the preservation of the interests of certain carriers cannot be guaranteed. Therefore, a heuristic two-step approach for the determination of the transfer sets Rij is proposed. Initially, each carrier selects the requests leading to maximum profit contributions. Afterwards, an independent mediator decides about the assignment of requests to the carriers with the goal of minimizing the sum of carrier service costs. In the first step, each carrier is allowed to select the requests that it wants to fulfill. Besides the requests from its own portfolio, it is allowed to select requests acquired by other members of the cooperation. This feature permits each cooperator to select the most profitable requests and enables a request interchange between the carriers. Additionally, no carrier has to serve any request that it does not want to fulfill. Typically, a single request cannot be served in a profitable way. For this reason, the carrier composes several requests into routes in order to achieve positive profit contributions. The carriers do not only specify single requests, but bundles of requests that they can serve in a profitable way. Such a bundle consists of the requests served within one route. Each carrier is allowed to specify several bundles. In the second step, the proposed bundles are assigned to carriers. Generally, several carriers compete for the most profitable requests. A conflict occurs as soon as two or more carriers declare at least one request as profitable and claim to serve it. It is not possible to partition one of the bundles in order to solve the conflict. Only one of the conflicting bundles can be assigned to the corresponding carrier. The other bundles are completely ignored. An independent mediator is called. He decides with the aim of minimizing additional costs arising from the incorporation of a non-cooperating logistics service provider. Attempts are made to keep these costs as small as possible. However, necessary service provider costs are distributed among the cooperators so that each one has to pay the same part of the carrier costs.

140


8.2.1 Bundle Specification by the Carriers Let R := R1 U . . . U RM be the union set of the request portfolios provided by the cooperation carriers C1, . . . , CM, n :=I R I. Each carrier specifies the request bundles in complete blindness about the selection of the other carriers. It has to decide only knowing the specifications of the requests and of its own equipment. The bundles assigned to carrier Ci are determined solving a PDSP instance in which R gives the set of available requests and the available fleet is set to the fleet of Ci. Let l7i contain the routes determined for serving the requests selected by carrier Ci and let Ni denote the number of routes in Di. Carrier The bundle B,?'contains all requests Ci specifies the bundles B!, . . . , contained in the j-th route in Ili. Each bundle B{ is expected to contribute a non-negative amount to the overall profit of the carrier company. Otherwise, the route would not be in a solution of the PDSP instance of carrier Ci. Its absence would increase the overall profit contribution of this carrier. For this reason, BS describes a collection of requests that can be served by Ci in an advantageous way.

BY.

8.2.2 Bundle Assignment by the Mediator After each carrier has bundled its selected requests, the bundles are assigned to the carriers. Since the bundles of a certain carrier only consist of the most promising requests (from the point of view of the carriers), it cannot be ensured that all requests contained in R are served by cooperating carriers. For requests that cannot be assigned to a vehicle of any participating carrier, an external carrier service has to be incorporated. The carriers forming the GS must pay the carrier service providers. Each participant has to pay a part of the overall amount of carrier service costs. To support the maximization of the profit contribution of each single carrier company, the bundles are assigned to the carriers in a way that minimizes the costs incurred when of using external carriers. The carrier companies have specified the bundles M

'R* := U { B ~ ., . .,BY}, i=l that they want to serve completely or not at all. Let N be the number of bundles in R*. For each bundle B E R* the specifying carrier is denoted as dB). The binary decision variable XB is equal to one if, and only if, bundle B E R* is selected, that means carrier c ( B ) is selected to serve the requests in B. The composites of the bundles are stored in the binary matrix (YrB) E (0, l ) n x N The . entry YrB is set to 1 if, and only if, request r is contained


141

in bundle B. If request r is contained in bundle B and B is selected to be assigned completely to c(B), then the expression Y,g . x g is equal to one, otherwise it is zero. The goal is to minimize the remaining carrier costs for requests that are not assigned to any cooperating carrier company from the GS. Equivalently, the goal can be re-expressed as to minimize the negative sum of prevented carrier costs cl . . c, where cr denotes the costs for incorporating a carrier service to fulfill request r . For each request the resulting carrier costs are determined by

+ .+

If request r is contained in a bundle assigned to a carrier then c, gives the corresponding carrier service costs that are saved, otherwise it is 0. The factor f, describes the additional charge for the incorporation of the carrier service. If f, > 1 then the costs c, exceed the obtained revenues REV,. The objective function can be defined as in (8.10). It is ensured that each request is assigned to a t most one participating carrier (8.11). The problem of finding a minimal freight charge request assignment can now be formulated in terms of the optimization problem (8.10)-(8.12). It is of a special type, called combinatorial auction Vohra and s. de Vries (2003).

BER*

x g E { O , l ) YB E R*.

(8.12)

The combinatorial auction (8.10)-(8.12) is an integer linear program with

N binary decision variables and n constraints. Special algorithms have been developed to deal with very large instances of combinatorial auctions (see, e.g. (Vohra and s. de Vries, 2003), (Sandholm, 2002)). However, the problem instances a t hand are so small that an exact solution can be obtained engaging ILOG1sMixed-Integer-Linear-Program solver in the CPLEX-package (Version 7.1).

8.3 Computat ional Experiments Several numerical experiments have been performed to show the general applicability of the collaborative two-step assignment procedure for determining the portfolio modifications Rij.

142


8.3.1 Test Cases Some benchmark instances are generated in order to perform numerical experiments that show the general applicability of the presented approach. They describe the special situation in which the carriers are strongly spatially distributed among the operational area. A cooperation is intended because each single carrier cannot cover the whole area: the fulfillment of a large number of requests is hindered by latest fulfillment times associated with each pickupand each delivery-operation of the requests. The operations area O is defined as O := [O; 10012. Four Carriers are considered. Each provides one vehicle. The depot of the first carrier is situated in the northwestern corner (0,100) of 0;the remaining carriers are positioned in the northeastern (100, loo), the southeastern (0,100) and the southwestern corner (0,O). The vehicles start from these positions and must return there after having finished their operations. Requests are generated at random for each carrier. The pickup and the delivery locations are arbitrarily determined, so that each request is completely included into exactly one of the clusters O1 := [O, 5012, O2 := [50,100]x [O, 501, O3 := [50,100] x [50,100] or O4 := [50, 10012. In each cluster R requests are derived for each carrier, so that each carrier provides a request portfolio of 4 . R requests. The demanded transport volume is set to one capacity unit for each request and the maximum allowed capacity of each vehicle is set to co. The capacities of the available vehicles do not compromise the composition of requests into routes. However, each request must be completely served within the customer specified time window [0,TI. The latest allowed execution time T is determined so small that it is not possible for any carrier to serve all available requests. It is intended that, in the cooperation, each carrier serve all the requests that are situated near its depot and that can be executed within [0,TI. Each benchmark instance is determined by the triple ( I , R, T), where I = 0 , . . . , 4 denotes the seeding of the request generator, R = 6 , . . . , l o specifies the number of requests for each carrier in each cluster and T = 200,300,400,500 gives the latest allowed finishing time of each request. Overall 5 . 5 . 4 = 100 instances are derived.

8.3.2 Collaborative Planning Approach Requests can be shifted between the cooperating carriers. If a carrier has to serve a request acquired by another carrier within the cooperation, then the complete revenues are transferred from the original to the serving carrier. For requests that remain unserved after the bundles have been assigned to the carriers, an external carrier service is engaged. It must be paid for each request that it fulfills. The charge is 20% larger than the revenues associated with the currently considered request. Hence, the carrier service incorporation


143

is unprofitable. The sum of the carrier costs is distributed among the participants of the cooperation so that each company has to pay the same part of the sum. Every independent carrier provides one vehicle to participate in the cooperation. 8.3.3 Reference Approach The setup of the non-collaborative planning approach that serves as a referential system is as follows. Each carrier company only knows about its own requests that are spread across the complete operational area 0. Requests of other carriers are not allowed to be fulfilled. Another cooperating carrier cannot serve a request acquired by a certain carrier. For each excluded request a carrier service must be incorporated. The costs for the carrier service incorporation exceed the associated revenues by 20%, i.e. 1.2.REV, monetary units must be paid for an externally served request r. It is assumed that such a carrier is always available. 8.3.4 Results Each benchmark instance is evaluated once by the collaborative approach and once by the reference approach. Since the MA used for the request selection is a randomized procedure, average results taken from five independent runs are presented. First, it can be observed, that the cooperative request assignment is able to incorporate significantly more requests. In Table 8.1 the expected percentages of self-served requests are presented and grouped for different problem classes (., R, T ) .The upper table shows the results obtained from the cooperative approach whereas the lower one includes the results of the reference approach. It can be stated, that the expected number of requests served by the cooperation increases if the latest allowed completion time is extended, if the set of available requests is enlarged or if both occur. Comparing both approaches, the expected number is drastically enlarged in the cooperative approach. It can be seen, as expected, that the overall percentage of served requests in the collaboratively generated routes dominates the percentage of served requests in the routes generated by the reference approach. Table 8.2 shows the average improvements of the quotes of requests served without incorporating expensive external carrier services for the different problem instances (., R, T ) .For several instances with large time windows the quote is more than doubled, whereas for shorter allowed serving intervals, it is enlarged by at least twenty percent. Table 8.3 shows the percentage of assigned bundles. It can be seen that the corresponding quotes for R = 6, T = 500 and R = 7, T = 500 are significantly smaller than the remaining quotes. In the other cases, the quotes of assigned bundles vary between 75 and 100 percent.

144


Table 8.1. Expected percentage of self served requests, obtained for different prob-

lem instances (., R, T) using the collaborative (upper table) and the reference assignment procedure (lower table)

Table 8.2. Expected improvement of the quote of requests served without incorporation of external service providers, achieved by applying the collaborative planning approach

In the collaborative assignment approach, requests are shifted and exchanged between the participating carriers. Table 8.4 shows the composition of the request stock of each single carrier for different time windows. For each single carrier 0,1,2 or 3 it is broken down from which carrier the served requests are acquired. The quotes of shifted requests vary between 20 and 31 percent. For example in the case of T = 300, 26 percent of the requests assigned to carrier 0 are self-acquired, 26 percent have been provided by carrier 1, respectively 23 percent of the transport demands are taken from carrier 2 and finally 25 percent originates from carrier 3. The observed results match the expectation that each carrier provides around a quarter of the requests to fulfill due t o their spatial distribution. The percentages change only slightly if the latest allowed finishing time is varied.


145

Table 8.3. Average ratios of bundles accepted in the second step in the collaborative approach

Table 8.4. Interchanges of requests between the cooperating carriers: composition of the schedules with requests acquired by different carriers

Table 8.5 shows the average profit contribution achieved by a single bundle for different instances (., R, T). It increases if the time windows are expanded and if the carrier companies do not compete for the requests. If R is enlarged, the profitability is also improved. In both cases the number of available requests in raised up, which supports the consolidation of profitable routes. Since in the collaborative request assignment approach the set of available requests is expanded again, the profitability of the routes in the cooperation dramatically outperforms the routes obtained in the reference request assignment scheme as shown in Table 8.5. The profit is multiplied by an average factor around 4. The same factor describes the enlargement of the request backlog available to consolidate appropriate bundles.

146


Table 8.5. Average profit received from the operations of a single vehicle for the problem instances (R, T, I) using the collaborative assignment (upper table) and the reference assignment procedure

Table 8.6 shows, that the costs for incorporating external carrier services from outside the cooperation (assigned to each carrier company) can be reduced due to the capability of generating more profitable routes. Unprofitable requests are often not necessarily served by expensive carrier services but by carrier companies belonging to the cooperation. Table 8.6. Average carrier service costs obtained for the problem instances (R, T, I) using the collaborative assignment (upper table) and the reference assignment procedure

8.4 Conclusions

147

A consequence of the enlargement of the number of incorporated and cooperation-served requests is the increasing of the obtained overall average profit. The amounts achieved for the considered benchmark field are presented in Table 8.7. For instances whose results are marked by * the collaborative request assignment does not work. This seems to be caused by conflicting bundles resulting from a competition between the carriers to obtain requests that are profitable to different carriers. For instances that are labeled by a 0 , tight time windows and/or the small number of available requests seem t o hinder the consolidation of requests into appropriate bundles. For the unmarked instances in the upper table representation in Table 8.7, the collaborative request assignment leads to an improved overall profit. Requests that cannot be served by the acquiring carriers are fulfilled by other cooperation members at reasonable quotes. Table 8.7. Average profit obtained for the problem instances ( R , T ,I) using the collaborative assignment (left) and the reference assignment procedure

8.4 Conclusions This investigation addresses the determination of profitable interchanges of requests among several independent, but cooperating, carriers. Requests that cannot appropriately be assigned to any carrier are fulfilled by external carriers for extra charges. An important feature is the blind selection of profitable requests that enforce the necessity of each carrier to select the requests that are profitable without knowing the selections of other carriers. Tactical, but unprofitable, bargaining is excluded.

148


The obtained results are promising. Future research should be dedicated to the investigation of situations in which the regional specialization is varied and in which the competition for requests is stronger. The generation of bundles that lead t o non-negative profit contributions seems to be a crucial aspect of the cooperative request assignment. Additionally, there should be investigation into if and how the application of different planning approaches (routing algorithms) by some carriers influences the success of single carriers in a cooperation.

Conclusions

To conclude this thesis, the main answers found for the three research topics mentioned in the introduction of this book, are summarized. In 9.1, the main results on the analysis of the short-term freight carrier planning problems are presented. In 9.2, the ideas for modeling simultaneous routing and freight optimization problems are summarized and in 9.3 the presented extensions of the memetic search method are listed. For each topic further research requirements are pointed out.

9.1 Understanding F'reight Carrier Decision Problems The geographical distribution of customer locations and the unpredictable demand for transport means that the operational short-term planning of a freight carrier company is very important. The local collection and distribution activities from the customers to local transshipment and consolidation facilities and vice versa cannot be pre-determined in advance within long-lasting schedules. The small quantities on the initial and last leg of a transport served by the freight carrier do not justify the installation of repetitive itineraries. Unbalanced demand over the long run in the local areas lead to bottleneck situations in which the carrier-owned fleet cannot serve all requests in a reliable manner. Subcontractors (LSPs) are ordered to fulfill those requests. The fulfillment mode is determined for each request: it is decided whether a request is given to an LSP or not. All requests have to be considered simultaneously, a sequential treatment of the requests leads to inappropriate mode selections. The derivation of the fulfillment mode requires the solving of a simultaneous model in which the benefits of both modes are compared. To evaluate the sense of using own equipment, a routing and scheduling problem has to be solved and for evaluating the costs of subcontractor incorporation, a freight optimization problem requires solving. Thus, the operational freight carrier planning problem is a composed vehicle routing and freight optimization problem. The two previously studied problems are coupled by the (bi-

150

9 Conclusions

nary) mode decisions for each request. Freight carrier planning problems bring two so far separately considered problem classes together. As soon as a sequence of consecutive planning periods is considered, additional benefits can be realized by selecting the most promising period for the fulfillment of a request. The implications and benefits or problems associated with the postponement or acceleration of the execution have only been initially studied in this thesis. Further research effort should be spent on this topic in order to investigate the symbiosis of request sub-contraction and postponement or acceleration of request completions.

9.2 Model Building The setup of decision models for a freight carrier planning problem requires the merging of models for vehicle routing and scheduling problems, and for the freight optimization. Besides the representation of the decisions within these submodels, additional decisions that couple both submodels have to be coded. For each single request an additional coupling decision variable is required. It is set true if, and only if, the corresponding request is served by carrier-owned equipment, and it is set false if, and only if, an LSP is ordered to serve this request. Three models with binary coupling decisions variables have been presented. In the Pickup and Delivery Selection Problem with Logistics Service Provider Incorporation (PDSPLSP), the costs of both modes are calculated for the requests. If the self-fulfillment is cheaper than the LSP incorporation, then the corresponding requests are inserted into the routes of the own equipment. Otherwise, LSPs are ordered to serve the mentioned requests. A knapsack-type constraint hinders the determination of the cheapest mode for each request in the Capacitated Pickup and Delivery Selection Problem (CPSDP). Since the capacity of the own fleet is scarce, some requests have to be given to an LSP. In the Pickup and Delivery Problem with Compulsory Requests (PDSPCR), the mode for the compulsory requests cannot be modified. These requests cannot be given to an LSP. Each of the three models represents a generic modeling approach. In the PDSPLSP, the determination of the coupling variables is unconstrained and their instantiation is performed subject to the evaluation of the corresponding modes. The knapsack constrained in the CPDSP prevents the selection of the true values for all coupling variables. In the PDSP-CR the predetermination of the values for the decision variables associated with the compulsory requests forbids the other extreme solution that all binary coupling variables are set to false. The main problem in the Pickup and Delivery Selection Problem with Postponement (PDSP-PP) is to determine the monetary value of a postponement or acceleration of a request execution. If the postponement opportunity

9.3 Methodological Enhancements

151

should be applied in a dynamic scenario more effort should be spent on the valuation of this third decision possibility. The second interesting topic is the investigation of impacts of different and more realistic freight tariffs for the LSP incorporation. The first proposals of Pankratz (2002) should be incorporated into the four derived basic models in order to bring the so far academic models closer to real world applications.

9.3 Methodological Enhancements The proposed Memetic Algorithms are able to solve the instances of the pickup and delivery selection problems. The solution quality is convincing. To cope with the intricate constraints, some extensions of the memetic search paradigm have been successfully implemented. So far, these additional features have not received special attention in the scientific literature. The abandonment of the string-based representation and the usage of the problem-specific structure-base representation have proven their applicability. This motivates the application of the memetic approach to problems for which a string-based representation is not available. However, the definition of the required problem specific search operators remains a very challenging task. The introduction and successful application of the alternating and converging constraint memetic algorithm (ACC-MA) represents a new idea for handling constraints that are not accessible for the other feasibilityachieving and -preserving techniques, such as penalization or repairing. This method should be applied to other combinatorial optimization problems with complicated constraints. However, the main problem of the memetic search paradigm is that it is missing scalability. The computational effort for the determination of the initial population, for the calls of the repair and the improvements procedures leads to unattractive running times. The solved benchmark problems contains only between 50 and 60 requests. Real-world applications have to cope with significantly larger instances. Considering the running time observed so far, it can be expected that the memetic search paradigm is inappropriate for larger instances and the configuration of another meta-strategy should be taken into account. Nevertheless the developments of the memetic search presented in this thesis are promising for a variety of similar applications. If the computational effort in the hill-climber calls can be reduced, then Memetic Algorithms are very promising and will remain comparable to other metaheuristics.

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Index

adaptation, 52 adaptive penalties, 119 advanced memetic algorithm, 71 aggregation of flows, 6 alternating and converging constraint, 123 ant algorithms, 51 auction combinatorial, 141 back-freight, 5 backward feeding, 9 baldwin effect, 63 benchmark instances, 42 biased random drawing, 84 bottleneck resource, 18 bottleneck selection, 25 capacitated vehicle routing problem, 20 carrier, 17 charge function, 21 chromosome, 55 cluster builder, 66 cluster first-route second, 69 co-evolution of partial solutions, 75 with specialization, 74 co-evolutionary memetic algorithm, 72 collaborative planning, 136 collaborative planning approach, 142 collection, 7 combinatorial auction, 141 compulsory request, 27 consolidation strategy, 6

construction approach parallel, 78 sequential, 78 control alternating and converging constraints, 124 cooperation, 136 coupling savings, 18 crew scheduling, 11 crossover, 57 modified precedence preserving crossover, 88 mPPX, 88 PPX, 88 precedence preserving crossover, 88 data hull, 85 decision problem, 16 decomposition approach, 71 deferrable request, 109 delivery, 9 destination, 6 dial-a-ride-problem, 33 direct representation, 84 distribution center, 4 distribution network, 4 domain integration, 136 double ranking, 120 dynamic penalties, 118 empty balancing, 11 empty miles, 5 enonomies of scale, 4 environmental impacts, 3

162

Index

evolutionary algorithms, 52 evolutionary programming, 53 evolutionary strategies, 53 five-phase transport-process, 9 flow of goods aggregation, 6 bi-directional, 6 uni-directional, 4 forward feeding, 8 freight carrier, 5 freight charge, 17 freight charge optimization, 22 freight charge optimization problem, 22, 34 freight tariff function, 34 freight transport network, 7 fulfillment costs, 12 fulfilment mode, 17 full truckload problem, 33 gene, 55 genetic algorithm, 53 genetic clustering, 68 genetic local search, 63 genetic sectoring, 69 genetic sequencing, 65 genetic vehicle representation, 73 genotype, 55 groupage system, 135 heuristic, 50 heuristic algorithms, 20 heuristics meta-, 20 hierarchical planning approach, 22 hill climber, 62 hub, 6 hybrid genetic search, 63 improvement approach, 50 individual, 52 inner ranking, 120 inst ant iation order, 84 itinerary, 10 lamarckian evolution, 63 less-than-truckload, 7 line haul, 8

local search methods, 50 location in networks, 9 locus, 55 logistics service provider, 17 incorporation, 19 logistics system configuration, 10 logistics system deployment, 10 logistics system design, 9 mating pool, 56 means of transport, 2 meme, 63 memetic algorithm, 63 advanced, 71 co-evolutionary, 72 meta-heuristics, 20 modal split, 2 mode fulfilment, 17 selection, 18 mode selection problem, 19 modes of road transport hire or reward, 3 own account, 2 multi-chromosome representation, 71 multiple traveling salesman problem, 20 mutation, 52 myopic planning, 28 network design, 9 network layout, 9 online planning problems, 41 operation, 34 optimal approaches, 20 origin, 6 overhead cost, 4 passenger-kilometer , 2 pd-path, 35 pd-schedule, 35 penalty adaptive, 119 dynamic, 118 static, 116 penalty value, 118 performance of transport, 2 phenotype, 55 pickup, 9 pickup and delivery selection problem

Index capacitated, 39 general, 36 pickup and delivery planning problems, 12 pickup and delivery problem with time windows, 33 pickup and delivery path, 35 pickup and delivery planning problem simultaneous, 23 pickup and delivery problem, 33 pickup and delivery request, 31 pickup and delivery schedule, 35 pickup and delivery selection problem with compulsory requests, 40 pickup and delivery selection model simultaneous, 23 pickup and delivery selection problem with logistics service provider incorporation, 38 with postponement, 41 planning collaborative, 136 population, 52 model, 92 population based approaches, 50 portfolio sub-, 22 portfolio re-composition problem, 138 postponement, 27 problem portfolio re-composition, 138 profit contribution maximization, 18 quote, 117 quote first-cost second, 121 quote-class, 120 rail transport, 2 ranking double, 120 inner, 120 regional multimodal planning, 10 rejection of transport demands, 34 replenishment, 4 representation, 55 direct, 84 multi-chromosome, 71 of a pd-schedule, 84

163

permutation based, 65 problem specific, 72 reproduction, 52 reproduction model (CL A), 120 request, 31 compulsory, 27 deferrable, 109 urgent, 28 request acceptance, 16 operational, 17 problem, 16 tactical, 16 request selection, 18 requests internal, 17 revenue, 16 road transport, 2 rolling planning horizon, 27 roulette-wheel-selection, 56 route, 19 improvement, 82 route construction heuristic, 69 route first-cluster second, 66 routes, 19 tentative, 80 routing, 19 routing problem, 19

+

search algorithms, 49 selection, 52 bottleneck, 25 maximal-profit , 25 with compulsory requests, 26 with postponement, 27 separation genes, 71 sequencing genetic, 65 shared distances, 46 shipment, 4 simulated annealing, 50 simultaneous approach, 23 simultaneous construction approach, 50 simultaneous planning models generic, 24 static penalties, 116 sub-portfolio, 22 successive approaches, 50

164

Index

tabu search, 51 third party, 5 ton-kilometer , 1 tour, 19 trail of a population, 130 trajectory methods, 50 transport process, 7 transportation plan, 12 transshipment, 6 traveling salesman problem, 20 truckload, 4

vehicle allocation problem, 33 vehicle routing and scheduling, 11 vehicle routing problem capacitated, 20 vehicle routing problem with backhauls, 33 vehicle routing problem with time windows, 20 waterway transport, 2

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