Jairo R. Montoya-Torres
Universidad de La Sabana, Colombia Angel A. Juan
Open University of Catalonia, Spain Luisa Huaccho Huatuco
University of Leeds, UK Javier Faulin
Public University of Navarre, Spain Gloria L. Rodriguez-Verjan
Ecole Nationale Supérieure des Mines de Saint-Étienne, France
Service, Computing and Manufacturing Systems:
Routing and Scheduling
Solutions
Hybrid algorithms for service, computing and manufacturing systems: routing and scheduling solutions / Jairo R. Montoya- Torres ... [et al.]
p. cm.
Includes bibliographical references and index.
Summary: “This book explores research developments and applications from an interdisciplinary perspective that combines approaches from operations research, computer science, artificial intelligence, and applied computational mathematics”-- Provided by publisher.
ISBN 978-1-61350-086-6 (hardcover) -- ISBN 978-1-61350-087-3 (ebook) -- ISBN 978-1-61350-088-0 (print & perpetual access) 1. Operations research. 2. Artificial intelligence. 3. Computer science--Mathematics. I. Montoya-Torres, Jairo R., 1977-
T57.6.H93 2012 658.5’3--dc23
2011023571
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Development Editor: Michael Killian Production Editor: Sean Woznicki
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Library of Congress Cataloging-in-Publication Data
Sana Belmokhtar, Research Center for Automatic Control of Nancy (CRAN), France Gaston Cedillo, COMIMSA, Mexico
Stéphane Dauzère-Pérès, École nationale supérieure des mines de Saint-Étienne, France Scott Grasman, Missouri University of Science and Technology, USA
M. Grazia Speranza, University of Brescia, Italy
Patrick Hirsch, University of Natural Resources and Applied Life Sciences, Austria Carlos A. Méndez, INTEC, Argentina
Helena Ramalhinho Lourenço, Universidad Pompeu Fabra, Spain Janet Smart, University of Oxford, UK
Andres Weintraub, Universidad de Chile, Chile
List of Reviewers
Edgar H. Alfonso, Ecole Nationale Supérieure des Mines de Saint-Étienne, France Burcin Bozkaya, Sabanci University, Turkey
Tom Burgess, University of Leeds, UK
Juan Pablo Caballero, Pontificia Universidad Javeriana, Colombia Ani Calinescu, University of Oxford, UK
Linda Castro, École Polytechnique de Montréal, Canada Gaston Cedillo, COMIMSA, Mexico
Julio Mario Daza, Universidad Autónoma del Caribe, Colombia Javier Faulin, Public University of Navarre, Spain
Andreas Fink, Helmut-Schmidt-University & UniBw Hamburg, Germany Jorge Freire de Sousa, Universidade do Porto, Portugal
Alvaro Garcia, Universidad Politécnica de Madrid, Spain Marco Gavanelli, Università di Ferrara, Italy
Alex Grasas, Universitat Pompeu Fabra, Spain
Scott Grasman, Missouri University of Science and Technology, USA Edgar Gutierrez, Universidad de La Sabana, Colombia
Patrick Hirsch, University of Natural Resources and Applied Life Sciences, Austria Luisa Huaccho Huatuco, University of Leeds, UK
Laura Lotero, Universidad Nacional de Colombia, Colombia Helena R. Lourenço, Universitat Pompeu Fabra, Spain Toni Mancini, Università di Roma La Sapienza, Italy Vanessa Manotas, Universidad de La Sabana, Colombia Alberto Márquez, University of Seville, Spain
Carlos A. Méndez, INTEC, Argentina
Jairo R. Montoya-Torres, Universidad de La Sabana, Colombia Miguel Ortega Mier, Universidad Politécnica de Madrid, Spain Pedro Palominos, University of Santiago of Chile, Chile Daniel Riera, Open University of Catalonia, Spain
Gloria L. Rodríguez-Verjan, Ecole Nationale Supérieure des Mines de Saint-Étienne, France Janet Smart, University of Oxford, UK
Elyn L. Solano-Charris, Universidad de La Sabana, Colombia M. Grazia Speranza, University of Brescia, Italy
Gülfem Tuzkaya, Yildiz Technical University, Turkey Eva Vallada, Universidad Politénica de Valencia, Spain Mario Vélez, EAFIT University, Colombia
Claude Yugma, Ecole Nationale Supérieure des Mines de Saint-Étienne, France
Foreword ...viii Preface ...xi Acknowledgment ...xvi
Section 1
Hybrid Algorithms for Routing Problems Chapter 1
Matheuristics for Inventory Routing Problems ... 1 Luca Bertazzi, University of Brescia, Italy
M. Grazia Speranza, University of Brescia, Italy Chapter 2
Vehicle Routing Models and Algorithms for Winter Road Spreading Operations ... 15 Nathalie Perrier, École Polytechnique de Montréal, Canada
James F. Campbell, University of Missouri-St. Louis, USA Michel Gendreau, École Polytechnique de Montréal, Canada André Langevin, École Polytechnique de Montréal, Canada Chapter 3
Routing Solutions for the Service Industry ... 46 Burcin Bozkaya, Sabanci University, Turkey
Buyang Cao, Tongji University, China
Kaan Aktolug, Aktolug Consultancy Ltd., Turkey Chapter 4
A Hybrid Genetic Algorithm-Simulated Annealing Approach for the Multi-Objective
Vehicle Routing Problem with Time Windows ... 79 Gülfem Tuzkaya, Yildiz Technical University, Turkey
Bahadır Gülsün, Yildiz Technical University, Turkey Ender Bildik, Yildiz Technical University, Turkey E. Gözde Çağlar, Yildiz Technical University, Turkey
Rita Ribeiro, Catholic University of Portugal (Porto), Portugal Chapter 6
A Hybrid Algorithm Based on Monte-Carlo Simulation for the Vehicle Routing Problem with
Route Length Restrictions ... 122 Angel A. Juan, Open University of Catalonia, Spain
Javier Faulin, Public University of Navarre, Spain Tolga Bektaş, University of Southampton, UK
Scott E. Grasman, Missouri University of Science and Technology, USA
Section 2
Hybrid Algorithms for Scheduling Problems Chapter 7
A Hybrid Particle Swarm Algorithm for Resource-Constrained Project Scheduling ... 137 Jens Czogalla, Helmut-Schmidt-University Hamburg, Germany
Andreas Fink, Helmut-Schmidt-University Hamburg, Germany Chapter 8
Marriage in Honeybee Optimization to Scheduling Problems ... 158 Pedro Palominos, University of Santiago of Chile, Chile
Victor Parada, University of Santiago of Chile, Chile Gustavo Gatica, University of Santiago of Chile, Chile Andrés Véjar, Université de Nancy, France
Chapter 9
Global Bacteria Optimization Meta-Heuristic: Performance Analysis and Application
to Shop Scheduling Problems ... 178 Elyn L. Solano-Charris, Universidad de La Sabana, Colombia
Libardo S. Gómez-Vizcaíno, Universidad Autónoma del Caribe, Colombia Jairo R. Montoya-Torres, Universidad de La Sabana, Colombia
Carlos D. Paternina-Arboleda, Universidad del Norte, Colombia Chapter 10
Hybrid Algorithms for Manufacturing Rescheduling: Customised vs. Commodity Production ... 195 Luisa Huaccho Huatuco, University of Leeds, UK
Ani Calinescu, University of Oxford, UK
Chapter 11
HMIP Model for a Territory Design Problem with Capacity and Contiguity Constraints ... 226 Fabian Lopez, Universidad Autónoma de Nuevo León (CEDEEM), México
Chapter 12
Hybrid Heuristics for the Territory Alignment Problem ... 258 Jorge Freire de Sousa, Universidade do Porto, Portugal
José Barros-Basto, Universidade do Porto, Portugal
Paulo Lima Júnior, Universidade do Vale do São Francisco, Brazil Chapter 13
A Hybrid Lagrangian Relaxation and Tabu Search Method for Interdependent-Choice
Network Design Problems ... 294 Chi Xie, The University of Texas at Austin, USA
Mark A. Turnquist, Cornell University, USA
S. Travis Waller, The University of Texas at Austin, USA
About the Contributors ... 325 Index ... 333
Foreword
Most fundamental problems in combinatorial optimization field have been proven to be computation- ally hard to solve to optimality and are known as NP-hard problems in the literature. Knowing that a problem of interest is NP-hard implies, on the one hand, that the problem is unlikely to be solved within a reasonable amount of computation time and, on the other, that one has to be satisfied with solv- ing the problem approximately or near-optimally.
An important class of algorithms that have shown their usefulness in solving many computationally hard optimization problems is that of meta-heuristics. This is by no chance –meta-heuristics methods possess many good features, among which we could distinguish: they are able to find high quality solu- tions in a reasonable amount of computation time, are robust, generic, flexible and easy to implement on sequential, parallel and networked computer systems. This, together with the fact that for most practical applications in industry and businesses high quality solution would suffice, have converted meta-heuristics into de facto approaches to cope in practice with the computationally hard optimization problems. In fact, even when a polynomial time algorithm is known for a certain problem, solving large-size/real-life instances (e.g. instances at enterprise scale) calls again for the application of meta-heuristics methods.
Not less importantly, meta-heuristic approaches can tackle with efficacy both single and multi-objective optimization problems.
Meta-heuristics methods have been applied for decades now. Besides using them as stand alone ap- proaches, during the last years, the attention of researchers has shifted to consider another type of high level algorithms, namely hybrid algorithms. These algorithms do not follow any concrete meta-heuristic, but rather combine meta-heuristics with meta-heuristics and/or other methods (e.g. divide-and-conquer, linear programming, dynamic programming, constraint programming or other AI techniques) yielding thus hybrid meta-heuristics. One fundamental question here is how can be achieved for hybrid approaches to outperform stand alone approaches? The hybridization aims at exploring the synergies among stand alone methods in order to achieve better results for the optimization problem under study. For instance, using hybrid approaches one can explore the synergies between exploration of solution space (through population based meta-heuristics, such as Genetic Algorithms—GAs) with the exploitation of the solu- tion space (through local search methods, such as Tabu Search –TS); the GA could them be used as a main search method while TS can improve locally the individuals of the population.
The rationale behind the hybridization resides in the ``no free lunch theorem” stating that ``... all algorithms that search for an extremum of a cost function perform exactly the same, when averaged over all possible cost functions. In particular, if algorithm A outperforms algorithm B on some cost functions, then loosely speaking there must exist exactly as many other functions where B outperforms A.” Based
on this theorem, existing algorithms can be used as components for designing new efficient search algorithms and expect improved performance of the newly obtained algorithm for some cost functions.
Naturally, there are major issues in designing hybrid meta-heuristics for a given optimization problem, such as: (a) how to choose heuristic and/or meta-heuristic methods to be combined (within the same family or from different families of existing algorithms), and, (b) how to combine the chosen methods into new hybrid approaches. Unfortunately, there are no theoretical foundations for these issues, yet there are interesting evidences, experiences and reports on the literature. For the former, different classes of search algorithms can be considered for the purposes of hybridization, such as exact methods, simple deterministic or random heuristic methods and meta-heuristics. Moreover, meta-heuristics themselves are classified into local search based methods, population based methods and other classes of nature inspired meta-heuristics. Therefore, in principle, one could combine any methods from the same class or methods from different classes. Regarding the later, there are some attempts for taxonomies of hybrid meta-heuristics; in fact, the common approach is to try out in smart ways, based on domain knowledge of problem at hand and characteristics of heuristics methods, different hybrid approaches and shed light on the performance of the resulting hybrid approach. The level of hybridization here plays an important role, namely the degree of coupling between the meta-heuristics (e.g. coercive vs. cooperative). It should as well be noted that frameworks that facilitate fast prototyping have been also provided in the hybrid meta-heuristics literature.
This book brings excellent contributions to the field of hybrid algorithms, their design, implemen- tation and experimental evaluation. The proposed hybrid approaches tackle fundamental problems in the domain of logistics, industry services, commercial distribution and manufacturing systems. The studied problems include routing, different forms of scheduling, such as permutation scheduling and shop scheduling problems, service allocation problems, etc. The proposed approaches include hybrid- ization of meta-heuristic methods with other meta-heuristic methods such as Genetic Algorithms (GA) and Simulated Annealing (SA) or the hybridization of meta-heuristics with the exact solution of one or several mathematical programming models.
Besides advancing in the design of more sophisticated hybrid solution strategies for routing and scheduling problems, the contributions of the book have a practical focus for solving real life problems.
The aim is to support decision processes in companies and thus to enable achieving better business ob- jectives by solving the problems at company scale. Also, the use of benchmarks and software packages are good examples of best practices in the field.
The editors of this volume bring together experts and researchers from the field whose contributions explore new research findings, developments and future directions in the hybrid approaches for routing and scheduling problems arising in service, computing and manufacturing systems. Finally, although focused on the concrete field of the routing and scheduling for service, computing and manufacturing systems, most of the conclusions provided in the volume could be extended to routing and scheduling in other research fields.
Fatos Xhafa
Technical University of Catalonia, Spain
Fatos Xhafa holds a PhD in Computer Science from the Department of Languages and Informatics Systems (LSI) of the Techni- cal University of Catalonia (UPC), Barcelona, Spain. He was a Visiting Professor at the Department of Computer Science and Information Systems, Birkbeck, University of London, UK (academic year 2009-2010) and a Research Associate at College of Information Science and Technology, Drexel University, Philly, USA (Sept.2004-Feb.2005). Dr Xhafa holds a permanent position of Professor Titular at the Department of LSI, UPC (Spain). His research interests include parallel and distributed algorithms, combinatorial optimization, approximation and meta-heuristics, networking and distributed computing, Grid and P2P computing. Dr. Xhafa has widely published in peer reviewed international journals, conferences/workshops proceedings, book chapters and edited books in the field. Dr. Xhafa has an extensive editorial and reviewing service. He is Editor in Chief of the International Journal of Space-based and Situated Computing, Inderscience (http://www.inderscience.com/browse/index.
php?journalCODE=ijssc) and of International Journal of Grid and Utility Computing, Inderscience (http://www.inderscience.
com/browse/index.php?journalCODE=ijguc). He is also an associate/member of Editorial Board of several international peer- reviewed scientific journals and has guest co-edited several special issues of international journals.
Preface
PURPOSE OF THE BOOK
In a global and highly-competitive world, organizations face an increasingly difficult environment, with increasing economic pressure and customer demands for more complex products and services that are inexpensive and that can be provided at short notice. In relation to this, hybrid algorithms could play an important role in helping organizations achieve current imperative costs reduction and fast product development.
This book deals with the study of Hybrid Algorithms for Service, Computing and Manufacturing Systems. Solutions to current real-life problems, such as supply chain management require more than an individual algorithm. Hybrid algorithms take advantage of each individual algorithm and help find solu- tions which are stronger/faster or more efficient than those provided by individual algorithms. Recently, the use of hybrid algorithms has increased in popularity in different research areas and industries. One of the indicators of this situation is the number of sessions, workshops, and conferences dealing with the design and application of hybrid algorithms. In practice, hybrid algorithms have proved to be efficient in solving a wide range of complex real-life application problems in different domains, including: Lo- gistics, Bioinformatics and Computational Biology, Engineering Design, Networking, Environmental Management, Transportation, Finance and Business.
This book aims at exploring state-of-the-art research developments and applications in these research areas from an interdisciplinary perspective that combines approaches from Operations Research, Com- puter Science, Artificial Intelligence and Applied Computational Mathematics.
The interest in hybrid algorithms has risen considerably among academics in order to improve both the behavior and the performance of meta-heuristics. Meta-heuristics are a branch of optimization in Computer Science, Operations Research and Applied Computational Mathematics that are related to algorithms and computational complexity theory. The past few years have witnessed the development of numerous meta-heuristics in various communities that sit at the intersection of several fields, including Artificial Intelligence, Computational Intelligence, Soft Computing, and Mathematical Programming.
Most of the meta-heuristics mimic natural metaphors to solve complex optimization problems (e.g.
evolution of species, annealing process, behavior of ant colonies, particle swarm, immune system, bee colony, wasp swarm, or bacterial behavior).
Some results of many real-life application problems are presented in this book. As reported in the literature, hybrid algorithms could result from one of the following, by:
• Combining meta-heuristics with complementary meta-heuristics,
• Combining meta-heuristics with exact methods from mathematical programming,
• Combining meta-heuristics with constraint programming approaches developed in the artificial intelligence community,
• Combining meta-heuristics with machine learning and data mining techniques,
• Combining meta-heuristics or exact methods with simulation techniques (optimization through simulation).
TARGET AUDIENCE
This book is a valuable tool for Researchers, Practitioners and Managers, Master and PhD students, Consultants, Government officials and Policy makers, as explained below:
1. Researchers: Since this book presents cutting-edge research on hybrid algorithms for routing, scheduling and other real-life application of hybrid algorithms. As shown by the chapters included in this book, researchers in Operations Research, Management Science, and Computer Science are developing new hybrid algorithms to solve problems in Logistics (to a large extend), Supply Chain Management and Design, Production and Operations Management, Applied Operations Research and Applied Combinatorial Optimization.
2. Practitioners and Managers: Logistics and Operations Managers, Supply Chain Managers, Production Managers and Control Engineers are dealing with real-life problems on a daily basis. The hybrid algorithms presented in this book provide a valuable resource to draw upon to inform their daily practice.
3. Master and PhD students: The book can be of interest for lecturers and students in Operations Research, Computer Science, Applied Computational Mathematics, Discrete Mathematics, Logistics and Operations Management, Industrial Engineering and Systems Engineering. Especially use- ful for the teaching and learning of the following topics: vehicle routing, scheduling, applied and discrete mathematics and artificial intelligence.
4. Consultants: this book provides off-the-shelf hybrid algorithms that have been tried and tested which can be used by consultants and further applied to their clients.
5. Government officials and Policy makers: In general, decision-makers in both private and public sectors could benefit from this book by tailoring and assessing the hybrid algorithms according to their decision-making criteria.
OVERVIEW OF THE BOOK
The chapters published in this book are authored by a total of 38 researchers affiliated to Higher Education Institutions based in 13 countries, as follows (in alphabetical order): Brazil, Canada, Chile, Colombia, France, Germany, Italy, Mexico, Portugal, Spain, Turkey, United Kingdom and United States of America.
These manuscripts cover a range of hybrid algorithms theory and practice in three main areas: Routing, Scheduling, and Other real-life application problems. These areas correspond to the three sections of the book, as explained in detail below.
The first section contains six chapters and it is devoted to the application of hybrid algorithms for solving complex realistic Routing problems.
Chapter 1 is written by Bertazzi and Speranza. This invited chapter reviews the main heuristic ap- proaches for the solution of Inventory Routing Problems (IRP) by presenting the most recent and novel ideas for the design of a new class of heuristics called matheuristics. This class of heuristic algorithms embeds, in a heuristic or meta-heuristic scheme, the exact solution of one or several mathematical pro- gramming models.
Chapter 2 is authored by Perrier, Campbell, Gendreau, and Langevin. It provides a survey of recent optimization models and solution methodologies for the vehicle routing problem for spreading opera- tions. The chapter presents a detailed classification scheme of models developed over the past 40 years.
It highlights some factors that may be limiting the application of Operations Research models in practice and discuss promising future research trends.
The Vehicle Routing Problem with Time Windows (VRPTW) is dealt with in chapters 3 and 4, respec- tively. VRPTW consists on finding routes for the vehicles to serve all the customers at a minimal cost without violating the capacity and travel time constraints of the vehicles and the time window constraints set by the customers. Chapter 3 is written by Bozkaya, Cao and Aktolug. It provides the reader with the background, mathematical models and various solution approaches for the Vehicle Routing Problem with Time Windows (VRPTW). Three case studies are presented as “success stories” of implementation of decision-aid tools for solving the VRPTW at an enterprise scale. Chapter 4 is authored by Tuzkaya, Gülsün, Bildik, and Çağlar. The chapter considers a multiple objective Vehicle Routing Problem subject to Time Windows deliveries. A hybrid meta-heuristic algorithm based on Genetic Algorithm (GA) and Simulated Annealing (SA) is proposed.
Chapter 5 is written by Lourenço and Ribeiro. It studies the problem of designing market-efficient and cost-effective product distribution routes. This chapter explores three different distribution routing strategies: (i) the classical vehicle routing problem where total distance or cost is minimized, responding to the classical objective functions of a Logistics Department in a company, (ii) a master route strategy with daily adaptations with maximization of customer’s loyalty, as intended by Marketing Departments, and (iii) a strategy that takes into account the cross-functional planning between the logistics (cost-based objective) and the marketing (customer-oriented objective) objectives through a multi-objective model.
All strategies are analyzed in a multi-period scenario through a meta-heuristic algorithm based on the iterated local search scheme.
Chapter 6 is authored by Juan, Faulin, Bektaş and Grasman, discusses how simulation can be ef- ficiently integrated with a classical heuristic in order to solve Vehicle Routing Problems with route length constraints and customer service costs. The strategy behind the hybrid solution procedure is to combine Monte Carlo Simulation with the classical Clarke & Wright Savings (CWS) algorithm and a divide-and-conquer technique. Authors also discuss the advantages and disadvantages of the procedures in relation to other existing approaches from literature.
The second section of the book is composed of four chapters and it is devoted to the presentation of hybrid algorithms for the solution of various Scheduling problems.
Chapter 7 is authored by Czogalla and Fink, and proposes a Particle Swarm Optimization (PSO) ap- proach for the Resource-Constrained Project Scheduling Problem (RCPSP). It incorporates well-known procedures such as the serial Schedule Generation Scheme (SGS) and is hybridized with forward-backward improvement. The proposed procedure is compared against state-of-the-art methods from the literature through an extensive computational experiment using a benchmark instances.
Chapter 8 is written by Palominos, Parada, Gatica, and Vejar. It provides an exploratory analysis of the marriage of honey-bees optimization (MBO) algorithm to solve scheduling problems. The chapter also evaluates the potential utility and adaptability of this method. Two scheduling problems are consid- ered: minimization of earliness-tardiness penalties under single machine scheduling environment with a common due date constraint and the permutation flow shop problem.
Chapter 9 is written by Solano-Charris, Gómez-Vizcaíno, Montoya-Torres and Paternina-Arboleda.
It proposes the use of a novel bio-inspired algorithm to solve hard combinatorial optimization problems.
The procedure is called Global Bacteria Optimization (GBO) algorithm and emulates the movement of microscopic organisms (bacteria) in response to stimulus from light. Applications are considered to solve hard mono-objective and bi-objective jobshop scheduling problems, with makespan and due date-based objective functions.
Chapter 10 is authored by Huaccho Huatuco and Calinescu. In this chapter, manufacturing resched- uling of customized production versus commodity production is investigated. Five hybrid rescheduling algorithms are presented. These are obtained by combining two key rescheduling-related elements found in the literature: rescheduling criteria and level of disruption transmitted to the shop-floor due to resched- uling. The main advantage of the proposed hybrid rescheduling algorithm over individual rescheduling algorithms consists of their ability to combine the main features of two different algorithms in order to achieve enhanced performance, depending on the objective of the organization. The practical impact of the hybrid algorithms is analyzed in the context of three manufacturing companies.
The third section is composed of three chapters and it is devoted to the presentation of hybrid algo- rithms for solving Other real-life application problems.
The Territory Design and Alignment Problems are considered in chapters 11 and 12, respectively. The Territory Design Problem (TDP) consists on grouping small geographic Basic Units (BU) into larger geographic territories. Chapter 11 is written by Lopez. It considers a real-life case study from a large soft drinks distribution company that operates in the city of Monterrey, Mexico. The chapter proposes a new strategy based on a hybrid-mixed integer programming method (HMIP). By taking advantage from territory centers obtained through a relaxation of the p-median model, the procedure requires a small number of iterations to find connected solutions. The chapter highlights that the model is currently being used by the large beverage bottler in Latin America.
Chapter 12 is authored by Freire de Sousa, Barros-Basto and Lima Júnior. It briefly updates the review of the existing literature on the territory alignment problem, its applications and solution approaches.
The chapter also illustrates the most recent tendencies by proposing a hybrid resolution approach based on the Greedy Randomized Adaptive Search Procedure (GRASP) and Tabu Search meta-heuristics that is integrated to an interactive and user-friendly Geographic Information System (GIS) application.
Chapter 13 is authored by Xie, Turnquist and Waller. It presents a hybrid Lagrangean relaxation and Tabu Search procedure for a class of discrete network design problems with complex interdependent- choice constraints. The algorithm is implemented on the solution of a network design problem with lane reversal and crossing elimination strategies, arising from urban evacuation planning.
In such a context, the reader will gain full exposure to some of the latest research models and frame- works with hybrid algorithms to solve practical real-life application problems. In addition, the literature is enriched by the discovery of current and future trends due to evolution of the research in the field.
Jairo R. Montoya-Torres
Universidad de La Sabana, Colombia Angel A. Juan
Open University of Catalonia, Spain Luisa Huaccho Huatuco
University of Leeds, UK Javier Faulin
Public University of Navarre, Spain Gloria L. Rodriguez-Verjan
Ecole Nationale Supérieure des Mines de Saint-Étienne, France
Acknowledgment
The completion of this book has involved the efforts by several people in addition to the Authors of the chapters. The Editors are grateful to the members of the Editorial Advisory Board, who are well-recog- nized academics in their fields, for supporting and overseeing the development of this Editorial project.
We also want to express our gratitude to Professor Fatos Xhafa for writing the foreword of this book.
The chapter submissions for this book were subjected to double-blind refereeing process which engaged up to three reviewers per chapter. The Editors thank the reviewers for reviewing, proposing improvements, and advising the status of the chapters considered for publication in this book, their names are provided in a list of reviewers elsewhere in this book. Without their efforts, this book could not have been completed according to schedule. We are also would like to extent our appreciation to those authors who considered publishing in this book, but whose chapters could not be included for a variety of reasons. We trust that their work will eventually appear elsewhere in the literature.
The Editors would like to express their recognition to their respective organizations and colleagues for the moral support and encouragement that have proved to be indispensable during the preparation and execution of this editorial project. This book is part of the CYTED IN3-HAROSA Network. Jairo R.
Montoya-Torres wishes to thank the financial support of Universidad de La Sabana under grants CEA- 24-2008, CEA-46-2008 and CEA-53-2010. Javier Faulin and Angel Juan wish also to thank the financial aid of “VERTEVALLÉE” research network (funded by the Working Community of the Pyrenees by means of the grant IIQ13172.RI1-CTP09-R2), the “Sustainable TransMET” research network (funded by the program Jerónimo de Ayanz from the Government of Navarre, Spain), and the project TRA2010- 21644-C03 (funded by the Spanish Ministry of Science and Technology). Luisa Huaccho Huatuco would like to acknowledge the financial support provided by the UK Engineering and Physical Sciences Re- search Council (EPSRC) “Complexity in the Supply Chain” project led by Principal Investigators: Dr Janet Efstathiou & Mr Gerry Frizelle (EPSRC Grant No. GR/M57842) and Leeds University Business School (LUBS) Seedcorn “Rescheduling and Complexity for Customised Products” project (Grant No.
54010803). Luisa would also like to thank the following colleagues for their help with this Editorial project: Dr Thomas F. Burgess (Leeds University Business School, University of Leeds), Dr Janet Smart (Saïd Business School, University of Oxford) and Dr Ani Calinescu (Computing Laboratory, University of Oxford). Gloria L. Rodríguez-Verjan wishes to acknowledge Dr. Claude Yugma for his collaboration, remarks and help during this editorial process, and Carlos Fernando Rodríguez for his moral support.
Last, but not least, the editors would like to acknowledge the relentless support of the IGI Global team, and in particular to Michael Killian, as their help and patience have been infinite and significant.
More importantly, the editors would like to extend their sincere appreciation to their respective families for their love, support, and patience.
Hybrid Algorithms for Routing
Problems
Chapter 1
DOI: 10.4018/978-1-61350-086-6.ch001
INTRODUCTION
The class of Inventory Routing Problems (IRPs) includes a variety of different optimization prob- lems that may have very different characteristics but all simultaneously consider a routing and an inventory management component of an optimiza- tion problem. Time may be discrete or continu- ous, demand may be deterministic or stochastic, specific application-dependent characteristics may be considered, inventory holding costs may be ac- counted for in the objective function or not. When in an IRP the holding costs are not included in the
objective function, a limited inventory capacity at the customers is available and cannot be exceeded.
IRPs have received little attention, if compared to plain vehicle routing problems (VRPs). However, the interest in this class of problems has been increasing from the beginning of the eighties. We refer for an overview of the available literature to the surveys Bertazzi et al (2008), Campbell et al (1998), Cordeau et al (2007), Federgruen and Simchi-Levi (1995), Moin and Salhi (2007). In this chapter we will focus on deterministic inven- tory routing problems.
IRPs model the simultaneous optimization of inventory management and routing that are tradi- tionally optimized separately and independently.
Luca Bertazzi University of Brescia, Italy
M. Grazia Speranza University of Brescia, Italy
Matheuristics for Inventory Routing Problems
ABSTRACT
In this chapter the authors review the main heuristic approaches for the solution of inventory routing problems and present the most recent and interesting ideas for the design of a new class of heuristics, that they call matheuristics. A matheuristic embeds, in a heuristic or metaheuristic scheme, the exact solution of one or several mathematical programming models and rely on the power of commercial software.
Clearly, the traditional separate optimization may lead to a global sub-optimum. The efforts invested in the optimization of the routing side of the problem may be vanished because the optimization ignores the relations between the routing and the inventory management sides of the problem. In VRPs usually the quantities to be delivered to the customers in a specific day by a fleet of capacitated vehicles are given. In a IRP, whereas the demands of the customers are known, the quantities to be delivered have to be decided.
An optimal solution of a IRP may suggest that customers located close to each other should be served the same days, that customers close to the factory should be served more frequently, while customers far from the factory should be served more rarely.
Obviously, IRPs are more complex to solve than VRPs. A time dimension is present in the IRPs that is rarely considered in the VRPs. Whereas the value of IRPs in supply chain optimization is widely accepted, the class of the studied IRPs is still quite limited and relatively little is known in terms of solution techniques. The plain VRPs are known to be computationally very hard to solve.
The IRPs are even harder. Few attempts to find the optimal solution to IRPs have been proposed in the literature. In most cases heuristic algorithms have been presented.
In this chapter we will illustrate traditional and new ideas for the design of effective heuristics by making use of a simple instance of a basic IRP. We will also review the use of heuristics for IRPs in the literature in a separate section. A set of customers have to be served by a factory (or a warehouse) over a discrete time horizon, measured for example in days, by using a fleet of capacitated vehicles. The factory and the custom- ers have an initial inventory and limited inventory capacity. The production rate of the factory and the consumption rate of the customers are known and are time-dependent. The problem consists in deciding which customers to serve each day of the time horizon and which routes the vehicles
travel to serve the customers of each day in such a way that a cost function is minimized. The cost function includes the routing costs and possibly the inventory costs at the factory and/or at the customers. We call this problem the basic IRP.
The scope of this chapter is to present and motivate the study of a new class of heuristics, the so-called matheuristics, that make use of math- ematical programming models, typically Mixed Integer Linear Programming Problems (MILPs), inside a heuristic scheme. The computational ef- fectiveness of commercial optimization software makes it interesting and promising the design of heuristic solution approaches that make use of the optimal solution of MILPs. We will present different ways to embed MILPs in a heuristic scheme. Some of these are straightforward evo- lutions of traditional heuristics, others are based on more recent ideas that have recently appeared in the literature. An optimal solution approach has been proposed for the basic IRP in Archetti et al (2007) for the case of one vehicle and the optimal solution of benchmark instances with up to 3 days and 50 customers and 6 days and 30 customers is known. This will allow us to test the different heuristic solutions against the optimum and calculate the exact errors. Recent ideas for a matheuristic have been proposed for the basic IRP in Archetti et al (2010), where a tabu search heuristic includes, as an improvement step, the solution of two different mixed integer linear programming (MILP) models. The MILP models are run whenever a new best solution is obtained in the tabu search. The computational results show that this hybrid approach is more effective than the plain tabu search. Although the MILP models are shown to be NP-hard, CPLEX can solve the models to optimality in a short computational time for instances with up to 200 customers.
We will complete the chapter with a review of the literature on heuristic approaches for IRPs.
The chapter is organized as follows. In Section 1 we describe and formally define the basic IRP we consider. In Section 2 we discuss the evolu-
tion of heuristics used to solve VRPs and IRPs.
In Section 3 we discuss several matheuristics for the IRP, most of them inspired by traditional heuristics, and show how they behave on a simple instance of the basic IRP. Finally, in Section 4 we review the literature.
1. PROBLEM DESCRIPTION
We define here the basic IRP we consider in this chapter. A graph G=(V,E) is considered, where V is the set of vertices and E is the set of edges. The set of vertices V includes a depot, denoted by 0, and a set of customers M={1,2,…,n}. An edge (i,j), where i,j∈V, represents the possibility to travel directly between vertices i and j. A length or traveling cost cij is associated to each edge i,j∈E. We assume cij=cji, i,j∈V. A time horizon of length H is considered. At each discrete time t∈T={1,…,H} a quantity r0t is made available in 0 and a quantity rit is consumed at customer i∈M.
A starting inventory level B0 in 0 is given. Each customer i has a maximum capacity Ui and a given starting inventory Ii0. Obviously, Ii0≤Ui. We denote by h0 the unit inventory cost of the depot 0 and by hi the unit inventory cost of customer i∈M. Shipments from the depot to the customers can be performed at each time t∈T by a vehicle of capacity C. We denote by T′ the set T ∪{H +1}.
We will make use of the following variables:
• xit is the quantity shipped from 0 to i, i∈M, at time t∈T;
• Bt is the inventory level at the depot 0 at time t T∈ ′;
• Iit is the inventory level at the customer i∈M at time t T∈ ′;
• zit is a binary variable equal to 1 if i∈V is visited at time t and 0 otherwise. Note that z0t=1 means that the depot is “visited” at time t, that is a route is traveled at time t;
• yijt is a binary variable equal to 1 if j∈M immediately follows i∈M, j<i, in the route traveled at time t∈T and 0 otherwise;
• yit0 is an integer variable not greater than 2 for each customer i∈M at time t∈T. It is equal to 0 if the customer i is not the last customer visited in the route traveled at time t, equal to 1 if it is the last one and equal to 2 if it is the only customer visited in this route.
The objective of the problem is to find a fea- sible solution that minimizes the overall cost. A feasible solution is a solution that does not cause stock-out at the depot (i.e. Bt ≥0 ∀t ) and at the customers (i.e. Iit ≥0, ∀i t, ), such that the level of the inventory of any customer i is never greater than Ui and the total quantity delivered at any given time does not exceed the vehicle capac- ity C. The overall cost is the sum of the routing cost and of the inventory cost, at the depot and at the customers, over the time horizon. The time H+1 is included in the computation of the inven- tory cost in order to take into account the conse- quences of the operations performed at time H.
We now recall the mixed-integer linear pro- gramming formulation of this problem introduced in Archetti et al (2007). The objective function expresses the minimization of the inventory cost at the depot, the inventory cost at the customers and the routing cost:
min
t T t
i M t T i it
i V j V j i t T ij ij
h B h I c yt
∈ ′ ∈ ∈ ′ ∈ ∈ ∈
∑
0 +∑∑
+∑ ∑ ∑
, <
. (1) The constraints of the problem are the fol- lowing:
1. Inventory definition at the depot: The inventory level of the depot at time t is given by the level at time t−1, plus the product quantity r0t-1 made available at time
t−1, minus the total quantity shipped to the customers at time t−1, that is
Bt Bt rt x t T
i M it
= −1 0 1− 1 ,
∈ −
+ −
∑
∈ ′(2) where r00=0 and xi0=0, i∈M.
2. Stock-out constraints at the depot: These constraints guarantee that for each time t∈T the inventory level at the depot is sufficient to ship the total quantity delivered to the customers:
Bt x t T
i M it
≥ ∈
∑
∈ . (3)3. Inventory definition at the customers:
The inventory level at time t is given by the level at time t−1, plus the quantity xit-1 shipped from the depot to the customer i at time t−1, minus the quantity rit-1 consumed at time t−1, that is
Iit =Iit−1+xit−1−rit−1 i∈M t T∈ ′, (4) where xi0=ri0=0, i∈M.
4. Stock-out constraints at the customers:
These constraints guarantee that for each customer i∈M the inventory level Iit at each time t T∈ ' is non-negative:
Iit ≥0 i∈M t T∈ '. (5) 5. Capacity constraints at the customers: These
constraints guarantee that for each customer
i∈M the maximum quantity Ui is never exceeded:
xit ≤Ui−Iit i∈M t T∈ . (6) 6. Routing constraints: These constraints
guarantee that, for each time t∈T, a feasible route is determined to visit all customers served at time t. They can be formulated as follows:
a. The total quantity loaded on the vehicle at each time t∈T does not exceed the transportation capacity:
i Mxit Czt t T
∑
∈ ≤ 0 ∈ . (7)b. If a positive quantity is delivered to customer i at time t, the customer i has to be visited:
xit ≤Czit i ∈M t T∈ . (8) c. If deliveries are made at time t (i.e. zit is equal to 1 for some i∈V), then the route traveled at time t has to contain one arc entering every vertex i of the route and one arc leaving every i:
j V j i ij t
j V j i ji t
y y zit i V t T
∈
∑
+ ∈∑
∈ ∈, < , >
= 2 .
(9) d. Subtours elimination constraints (see
Fischetti et al, 1998, and Gendreau et al, 1998):
i S j S j i ij t
i S it kt
y z z S M t T
∈ ∈ ∈
∑ ∑
≤∑
− ⊆ ∈, <
(10)
for some k∈S.
7. Non-negativity and integrality constraints:
xit ≥0 t T∈ i ∈M (11)
Bt ≥0 t T∈ ′ (12)
zit ∈{0,1} i V∈ t T∈ (13) yijt ∈{0,1} i∈M j ∈M j, <i t T∈
(14) yit0 ∈{0,1, 2} i ∈M t T∈ . (15)
We refer to this problem as Problem P.
2. EVOLUTION OF HEURISTICS Over the last decades the increasing computa- tional power of computers and the advancement in algorithms design has made it possible to solve exactly instances of hard problems of increasing size. Commercial software for linear and mixed integer linear mathematical programming models has become extremely powerful, incorporating the most recent algorithmic advancement. At the same time, more and more research efforts have been devoted to the design of heuristics for problems of increasing complexity.
Although it is out of the scope of this chapter to overview the evolution of heuristics, if we restrict the attention to the area of routing problems, we may summarize such evolution as follows:
• greedy heuristics;
• local search heuristics;
• metaheuristics;
• hybrid heuristics and matheuristics.
The nearest neighbor is a classical example of greedy heuristic for the Traveling Salesman Problem (TSP). The tour is built starting from the origin. The next visited vertex is the vertex closest to an extreme of the partial tour. The cheapest insertion heuristic for the TSP is another classical example of greedy heuristic. At each iteration, given a partial tour (at the beginning the tour contains the depot only), for any vertex not yet included in the partial tour the cheapest way to insert it into the partial tour is identified.
The vertex with the minimum cheapest insertion cost is chosen and inserted.
The 2-opt heuristic is a well known example of a local search heuristic for the Vehicle Routing Problem (VRP). Given any solution, however built for the VRP, a local search is performed by evaluating all the solutions where two vertices, belonging to two different tours, are exchanged.
If the best of the evaluated solutions improves the previous solution, a new solution is obtained and a new local search is performed. A sequence of better and better solutions is built until no more improvement is achieved.
Local search heuristics can be very effective but in many cases they end up in a local minimum where they remain trapped. Different so called metaheuristics have been designed to escape from local minima. Simulated annealing, tabu search, genetic and evolutionary are the most successful metaheuristics proposed.
The most recent heuristics for routing prob- lems explore new directions and take new names, hybrid heuristics, optimization-based heuristics, matheuristics. We believe that an extremely in- teresting direction for the design of new classes of heuristics is related to the use of the power of commercial software. As a commercial software can be used to solve instances of complex problems up to a certain size, a commercial software can be used to solve sub-problems, but also to solve different problems that are embedded in a heu- ristic scheme and are used to effectively explore promising parts of the solution space.
At present, the word hybrid is used to indi- cate a solution approach that combines ideas or techniques that are typically used separately. A matheuristic is a heuristic that embeds a math- ematical programming model. An optimization- based heuristic is a heuristic that is based on the solution of an optimization problem. In this chapter we will use the term matheuristic to identify any heuristic that embeds a step where a mathematical programming model is solved. A hybrid heuristic may be seen as more general than a matheuristic.
There exist several ways to embed a mathemati- cal programming model in a heuristic scheme:
1. To solve sub-problems: The IRP jointly considers a routing problem and an inventory management problem. A natural decom- position approach consists in treating the two problems separately. In the inventory management sub-problem we will determine the days of service and the quantities to be delivered to the customers, whereas in the routing sub-problem we will determine the routes of the vehicles, given the days of service and the quantities. We will solve the two sub-problems of the basic IRP optimally to evaluate the impact of the decomposition on the quality of the solution.
2. To solve parts of an instance: A direct way to reduce the complexity of an instance to be solved is to decompose the instance and solve the same problem on sub-instances of smaller size. In the case of the IRP, when the customers are geographically distributed in clusters, it may be possible to decompose the instance and solve the IRP on different sub-instances of smaller size, one for each cluster.
3. To restrict the search space: Restricting the search space is another way to reduce the complexity. While what discussed in the previous two points may be seen as special case of restricting the solution space, there are also different ways to do it. In IRPs the
space can be restricted by focusing the search of specific policies that are, for different reasons, attractive.
4. To explore neighborhoods: Some of the most innovative and recently proposed heuristics belong to this class. Promising neighborhoods are explored in depth by means of a mathematical programming model. The challenge consists in identifying the neighborhoods that contain a high den- sity of high quality solutions and to design an appropriate mathematical programming model that finds the best solution of each neighborhood.
3. OLD AND NEW IDEAS FOR MATHEURISTICS TO SOLVE THE BASIC IRP
In this section we discuss several matheuristics.
Most of them are inspired by traditional heuris- tics. The difference here is that some operations, usually executed heuristically, are now executed more accurately by means of MILPs that are solved with a commercial software. Some new ideas for matheuristics are also discussed.
The matheuristics for IRPs that we will illus- trate through an example for the basic IRP are:
• minimizing the routing cost (Routing- based matheuristic);
• first solving the inventory part of the problem without routing and then find- ing the best routes to serve the custom- ers (Inventory-first Routing-second matheuristic);
• first cluster the customers into sets and then solve the inventory routing problem for each set separately (Cluster-first Inventory Routing-second matheuristic);
• by iteratively inserting in the partial solu- tion the optimal solution of the subprob-
lem of a single customer (Customer-based matheuristic);
• restricting the search space to simplifying policies (Policy-based matheuristic);
• MILPs in a tabu search scheme (Intensified tabu search matheuristic).
The Routing-based and the Inventory-first Routing-second matheuristics belong to the class 1 of the previous section, whereas the Cluster-first Inventory Routing-second and the Customer- based matheuristics belong to the class 2. The Policy-based matheuristic belongs to class 3 and the last one to the class 4.
We now describe an instance of the basic IRP that we will use as an example to illustrate the matheuristics and also to show that the design of a good matheuristic is not trivial, as the most natural or traditional ideas may generate poor or infeasible solutions.
A set M={1,2,3,4} of customers is served by the depot 0 over a time horizon H=3. The depot is located at (50,50), while the customers are located at (0,0), (0,100), (100,100) and (100,0), respectively. The traveling costs cij, i,j∈V, are:
cij 0 1 2 3 4
0 0 71 71 71 71
1 71 0 100 141 100 2 71 100 0 100 141 3 71 141 100 0 100 4 771 100 141 100 0
and correspond to the rounded off Euclidean distances. The quantity r0t made available at the depot 0 at each time t∈T is equal to 100, while the quantity rit consumed at each customer i∈M at each time t∈T is 25. The starting inventory level B0 at the depot is 100, while the starting inven- tory level Ii0 at each customer i∈M is 25. The unit inventory cost h0 at the depot is 0.01, while the inventory cost at the customers are h1=h2=1
and h3=h4=10. The maximum capacity Ui of each customer i∈M is 50 and the transportation capacity C is 70. Note that, since for each customer i∈M the initial inventory level Ii0 is equal to 25 and the total demand is 75, then in any feasible solu- tion at least 50 units have to be delivered to each customer during the time horizon. Moreover, due to the initial inventory and the maximum capacity Ui=50, at most 25 units can be delivered to each customer i at time 1.
Let us first show the optimal solution obtained by solving exactly Problem P in which the total cost is minimized. The optimal solution has been obtained with CPLEX. The following table shows the quantity xit sent to each customer i∈M at each time t∈T and the route traveled by the vehicle at each time t∈T:
t x1t x2t x3t x4t Route Routing Cost
1 25 25 10 0 0 1 2 3 0 342
2 25 0 15 30 0
→ → → →
→
→ → → →
→ → → →
1 4 3 0 342
3 0 25 25 20 0 2 3 4 0 342
50 50 50 50
Total Quantity Total RRouting Cost 1026
A total quantity of exactly 50 units is delivered to each customer. The vehicles used at times 2 and 3 have a full load. On the basis of the values of these decision variables, the corresponding values of the inventory levels Bt and Iit and the inventory cost can be easily computed as follows:
t B I I I I
Tot
t 1t 2t 3t 4t
1 100 25 25 25 25
2 140 25 25 10 0
3 170 25 0 0 5
4 200 0 0 0 0
aal Inventory Inventory Cost
610 75 50 35 30 6.1 75 50 350 300
Therefore, the total cost is 1807.1, given by the sum of the routing cost (1026), the inventory cost at the depot (6.1) and the inventory cost at the customers (75+50+350+300=775).
Routing-Based Matheuristic
We first consider a matheuristic based on the minimization of the routing cost. The rationale of this type of matheuristics is two-fold. The first is based on the fact that, since the inventory costs are not explicitly paid, it is common from the operational point of view to solve the problem by simply focusing on the routing cost. The second is that several heuristics are known for the solution of routing problems. We show that, if the rout- ing cost only is taken into account, bad solutions may be obtained, even if the routing part of the problem is optimally solved.
Consider our instance. We solve with CPLEX the model (1)-(15) where in (1) we only consider the third term, that is the routing cost only.
The following table shows the quantity xit sent to each customer i∈M at each time t∈T and the route traveled by the vehicle at each time t∈T, when the objective function of Problem P is replaced by the routing cost only:
t x1t x2t x3t x4t Route Routing Cost
1 25 25 0 20 0 2 1 4 0 342
2 0 0 50 10 0
→ → → →
→33 4 0 242
3 25 25 0 20 0 2 1 4 0 342
50 50 50 50
→ →
→ → → →
Total Quantity Total Routting Cost 926
A total quantity of 50 is delivered to each customer. The vehicles used at times 1 and 3 have a full load. On the basis of the values of these decision variables, the corresponding values of the inventory levels Bt and Iit and the inventory cost can be easily computed as follows:
t B I I I I
Tot
t 1t 2t 3t 4t
1 100 25 25 25 25
2 130 25 25 0 20
3 170 0 0 25 5
4 200 0 0 0 0
aal Inventory Inventory Cost
600 50 50 50 50 6 50 50 500 500
Therefore, the total cost is 2032, given by the sum of the routing cost (926), the inventory cost at the depot (6) and the inventory cost at the customers (50+50+500+500=1100). Although the routing cost has been significantly reduced, the percent increase of the total cost is about 12.5%
with respect to the total cost of the optimal solu- tion of Problem P.
Inventory-First Route- Second Matheuristic
The Inventory-first Route-second matheuristic is a two phase algorithm. In the first phase the inven- tory cost is minimized, while in the second phase the optimal routes are found, given the quantities to be delivered at each time.
In the first phase the following Problem Inv is solved.
Problem Inv:
min
t T t
i M t T i it
h B h I
∈ ′ ∈ ∈ ′
∑
0 +∑∑
(16)Bt Bt rt x t T
i M it
= −1 0 1− 1
∈ −
+ −
∑
∈ ′(17)
Bt x t T
i M it
≥ ∈
∑
∈ (18)Iit =Iit−1+xit−1−rit−1 i∈M t T∈ ′ (19) Iit ≥0 i ∈M t T∈ ' (20) xit ≤Ui−Iit i∈M t T∈ (21)
i Mxit C t T
∑
∈ ≤ ∈ (22)xit ≥0 t T∈ i ∈M (23)
Bt ≥0 t T∈ ′. (24)
