Heuristic Algorithms for Solving a Class of Multiobjective Zero-One Programming Problems ZHANG CAIWEN (B.Eng.; M.Eng., Civil Eng.) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements I would like to express my profound gratitude to my supervisor, Associated Professor Ong Hoon Liong, for his invaluable advice, support, guidance and patience throughout my study and research My sincere thanks are conveyed to the National University of Singapore for offering me a Research Scholarship and the Department of Industrial and Systems Engineering for the use of its facilities, without any of which it would be impossible for me to complete this work I want to give special thanks to my research colleagues, Teng Suyan and Liu Shubin, who have helped me a lot in my research work I also want to take this opportunity to thank Ms Ow Lai Chun from the Department Office for her very nice and patient assistance Special thanks also go out to all my colleagues and friends at the Department of Industrial and Systems Engineering, the National University of Singapore, who made my stay at NUS an experience I will not forget Last but certainly not least, thanks to my family and my wife for their continuous concern, moral support and encouragement in this endeavor I Table of Contents Acknowledgements I Table of Contents II Nomenclature IV List of Illustrations .V Summary VIII Chapter Introduction Chapter A Literature Review 2.1 Approaches aiming to generate a good approximation of the set of nondominated solutions 10 2.2 Approaches aiming to generate the set of all nondominated solutions 12 2.3 Approaches aiming to generate one or several “best compromise” solutions for the decision maker 13 2.4 Real world applications of multiobjective optimization techniques 16 2.5 Methods for solving the two classes of problems under study 18 Chapter Basic Concepts and Methodologies 22 3.1 Some basic concepts and definitions 23 3.2 Problem formulations 27 3.3 A simple approach for solving biobjective integer programming problems 31 3.3.1 A simple procedure to generate all nondominated solutions .32 3.3.2 A simple approach to generate nondominated solutions of interest .35 3.4 Computational experiments and results 40 3.5 Summary and conclusions 49 Chapter Solving the Multiobjective 0-1 Knapsack Problem 51 II 4.1 Evaluation of an approximation to the nondominated set 52 4.2 A simple approach for solving biobjective integer programming problems 53 4.2.1 An LP-based heuristic for solving 0-1 integer linear programming problems 55 4.2.2 Evaluation of an ε-NS solution .63 4.3 Computational experiments and results for biobjective problems 65 4.4 Solving the 3-objective 0-1 knapsack problem 70 4.4.1 Property of nondominated solutions to a 3-objective integer programming problem .70 4.4.2 An interactive solution approach based on LPH 75 4.4.3 Computational experiments 78 4.5 Summary and conclusions 81 Chapter 5.1 Solving the Multiobjective Generalized Assignment Problem 82 An efficient approach for solving the biobjective generalized assignment problem 83 5.1.1 Property of the generalized assignment problem 86 5.1.2 An LP-based heuristic for solving the biobjective generalized assignment problem 89 5.1.3 Size of subproblem P(5.3) 92 5.1.4 Evaluation of an ε-NS solution .96 5.1.5 Solution strategies 96 5.2 Computational experiments and results 102 5.3 Summary and conclusions 110 Chapter Conclusions 111 References 117 III Nomenclature R The set of real numbers X Feasible region Z Feasible outcome region CKP Continuous knapsack problem DM Decision maker ENS Extreme nondominated solution LP Linear programming LPH LP-based heuristic CV Cardinality of the expanded FreeSet MT Maximum number of trials T Turning point ε-NS solution ε-Nondominated solution δ-NS solution δ-representative nondominated solution f i (x) i-th objective function ε, δ A small positive value q A positive value or equal to ∞ λ A positive vector with component λi cj Attribute value of item j wj Weight of item j W Capacity of the knapsack c kj The k-th attribute value of item j p ijk The k-th attribute value of item j if it is assigned to knapsack i wij Weight of item j if it is assigned to knapsack i ci Capacity of knapsack i zk The k-th objective Z LP Optimal solution to a linear programming problem FreeSet The set containing all free variables OneSet The set containing all the variables having value ZeroSet The set containing all the variables having value IV List of Illustrations • Figures Figure 3.1 An illustration of the two ENS’s to a biobjective integer programming problem Figure 3.2 Flowchart of the procedure in Section 3.3.1 Figure 3.3 Flowchart of the approach described in Section 3.3.2 Figure 3.4 Number of variables versus average number of nondominated solutions Figure 3.5 Number of variables versus time taken to generate all nondominated solutions Figure 3.6 Number of variables versus average time used to generate each nondominated solution Figure 3.7 A plot of nondominated solutions to a typical problem instance with 150 variables Figure 3.8 A plot of the 101 0.01-NS solutions (100 intervals) to a typical problem instance with 10000 variables Figure 3.9 A plot of the 51 0.01-NS solutions (50 intervals) to a typical problem instance with 50000 variables Figure 4.1 An illustration of the LP-based heuristic Figure 4.2 Comparison of the performances of the LPH and CPLEX Figure 4.3 Performances of the 101 0.01-NS solutions generated by the LPH and CPLEX for a typical problem instance of size 100 Figure 4.4 Performances of the 101 0.01-NS solutions generated by the LPH and CPLEX for a typical problem instance of size 500 Figure 4.5 Performances of the 101 0.01-NS solutions generated by the LPH and CPLEX for a typical problem instance of size 50000 Figure 4.6 An illustration of the first typical case of the distribution of the ENS’s Figure 4.7 An illustration of the second typical case of the distribution of the ENS’s V Figure 4.8 An illustration of the first category nondominated solutions for the first typical case Figure 4.9 An illustration of the first category nondominated solutions for the second typical case Figure 4.10 An illustration of the small squares in the interactive approach Figure 4.11a Generated approximate nondominated frontier to a 3-objective problem instance of size 1000 Figure 4.11b Generated approximate nondominated frontier to a 3-objective problem instance of size 1000 Figure 4.12a Generated approximate nondominated frontier to the LP relaxation of a 3-objective problem instance of size 1000 Figure 4.12b Generated approximate nondominated frontier to the LP relaxation of a 3-objective problem instance of size 1000 Figure 5.1 ε-NS solutions generated by LPH with the first strategy Figure 5.2 An illustration of the searching process of the first strategy Figure 5.3 An illustration of the searching process of the second strategy Figure 5.4 ε-NS solutions generated by LPH with the second strategy Figure 5.5 An illustration of the searching process of the third strategy Figure 5.6 ε-NS solutions generated by LPH with the third strategy Figure 5.7 Relationship between problem size and ε value Figure 5.8 Relationship between problem size and time use Figure 5.9 Relationship between the ratio of n to m and the ε value Figure 5.10 Generated approximate nondominated frontier for a typical problem instance of size (200×50) Figure 5.11 Generated approximate nondominated frontier for a typical problem instance of size (400×50) Figure 5.12 Generated approximate nondominated frontier for a typical problem instance of size (600×50) VI Figure 5.13 Generated approximate nondominated frontier for a typical problem instance of size (800×50) Figure 5.14 Generated approximate nondominated frontier for a typical problem instance of size (1000ì50) ã Tables Table 3.1 Computational results of generating exact nondominated solutions Table 3.2 Computational results of generating ε-NS solutions Table 3.3 Time taken to generate 101 0.01-NS solutions (100 intervals) Table 4.1 Time taken by the LPH and CPLEX to generate 101 0.01-NS solutions (100 intervals) Table 5.1a Relationship between the number of split items n2 and problem size (for problem instances of data set 1) Table 5.1b Relationship between the number of split items and problem size (for problem instances of data set 2) Table 5.1c Relationship between the number of split items and problem size (for problem instances of data set 3) Table 5.2 Summary of the computational results VII Summary This thesis is devoted to solving multiobjective zero-one integer linear programming problems Although this class of problems has been studied for many years, relative few effective solution methods have been developed in this field This study is particularly concerned with the design and development of heuristics for solving this class of problems We present some useful concepts and propose some heuristic methods for finding the ε-nondominated solutions The proposed solution method is quite simple and useful This method decomposes a multiobjective zero-one programming problem into a series of single objective problems Efficient LP-based heuristics, which capitalize on the similarity between the optimal solution of a zeroone integer linear programming problem and that of its corresponding LP problem, are employed to solve these single objective problems In particular, the proposed methods are applied to two classes of multiobjective zero-one programming problems, i.e the multiobjective zero-one knapsack problem and the multiobjective generalized assignment problem, in this study Extensive computational experiments have demonstrated that the methods we proposed are very effective for solving these classes of problems These methods can also be extended to other multiobjective zero-one programming problems, especially the other members from the family of the knapsack problems VIII Chapter Introduction ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Chapter Introduction ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Chapter Solving the Multiobjective Generalized Assignment Problem ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– GAP400x50_LP 39000 GAP400x50_MIP Objective2 34000 29000 24000 19000 180000 230000 280000 330000 380000 Objective1 Figure 5.11 Generated approximate nondominated frontier for a typical problem instance of size (400×50) 59000 GAP600x50_LP GAP600x50_MIP 54000 Objective2 49000 44000 39000 34000 29000 295000 345000 395000 445000 495000 545000 595000 Objective1 Figure 5.12 Generated approximate nondominated frontier for a typical problem instance of size (600×50) ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 108 Chapter Solving the Multiobjective Generalized Assignment Problem ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– GAP800x50_LP GAP800x50_MIP 76000 71000 Objective2 66000 61000 56000 51000 46000 41000 400000 450000 500000 550000 600000 650000 700000 750000 Objective1 Figure 5.13 Generated approximate nondominated frontier for a typical problem instance of size (800×50) GAP1000x50_LP 95000 GAP1000x50_MIP 90000 85000 Objective2 80000 75000 70000 65000 60000 55000 50000 490000 540000 590000 640000 690000 740000 790000 840000 890000 940000 990000 Objective1 Figure 5.14 Generated approximate nondominated frontier for a typical problem instance of size (1000×50) ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 109 Chapter Solving the Multiobjective Generalized Assignment Problem ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 5.3 Summary and conclusions In this chapter, we propose an efficient approach based on an LPH algorithm to solve the biobjective generalized assignment problem It is shown that this approach is able to generate a good approximation of the nondominated frontier to a large biobjective generalized assignment problem in a short time The successful computational results demonstrate the high efficiency of the proposed approach and the LPH algorithm The method looks simple, but it is able to provide very efficient and helpful results for the DM ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 110 Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 111 Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– In this thesis, we attempt to tackle the multiobjective zero-one programming problems, which imply linear constraints, linear objective functions, and binary decision variables Although the study of this class of problems occurred in the 1970s, not too many effective techniques or methods have been developed to solve them In view of this, we aim to contribute some new ideas and techniques to this field in this study Firstly, we introduced three concepts of ε-Nondominated Solution, Extreme Nondominated Solution and δ-representative Nondominated Solution for solving multiobjective integer programming problems These concepts have proved to be very helpful for the solution of these problems The concept of ε-Nondominated Solution is a natural extension of the ε-optimality concept within the single objective optimization context The concept of Extreme Nondominated Solution helps to effectively restrict the searching region and facilitate the solution of multiobjective integer programming problems The concept of δ-representative Nondominated Solution reflects the characteristic of the generated nondominated solutions We also proposed a simple and useful approach for generating a good approximation to the nondominated set of multiobjective integer programming problems The basic idea underlying this approach is to divide the feasible outcome region into some smaller sub-regions and focus on one of them at a time In this way, a multiobjective integer programming problem can be converted into a series of single objective problems, which makes the solution process much easier Therefore, we don’t have to ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 112 Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– tackle a multiobjective integer programming problem directly, but rather we can solve it through solving a series of single objective integer programming problems This research is particularly concerned with the two multiobjective 0-1 programming problems: the multiobjective 0-1 knapsack problem and the multiobjective generalized assignment problem They are the multiobjective versions of the two classical NP-hard problems: the 0-1 knapsack problem and the generalized assignment problem, respectively We proposed an efficient LP-based heuristic to solve each of them These LPH algorithms are based on the particular properties exhibited by these two classes of problems, i.e the strong similarity between the optimal solution of a 0-1 integer linear programming problem and that of its LP relaxation problem Although we did not prove this similarity from a theoretical point of view in this study, it does seem to imply some underlying theoretical necessity As a result, one of the further extensions of this work is to verify this empirically observed property For the multiobjective 0-1 knapsack problem, we have tackled both the biobjective case and 3-objective case But for the multiobjective generalized assignment problem, only the biobjective case has been addressed However, the proposed approach is easily extended to the 3-objective case in the same manner as the 0-1 knapsack problem Although we have only addressed two classes of problems with the approaches developed in this study, according to our experience they are applicable or extendable to the other members of the family of knapsack problems, including the bounded knapsack problem, the subset-sum problem, the change-making problem, the 0-1 multiple knapsack problem, and the bin-packing problem For more detailed ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 113 Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– information of the family of knapsack problem, readers are referred to Martello and Toth (1990) The high efficiency of the proposed approach and the LP-based heuristic algorithms is demonstrated through extensive computational experiments The results of our computational experiments are superior to those reported in the open literature, either in terms of the time consumption or the quality of the generated solutions or even both Furthermore, this research is the first attempt to extend the classical generalized assignment problem to the multiobjective context and propose efficient and effective solution to solve it, to our knowledge This research, among other things, also proposes the three criteria proximity, uniformity and coverage to evaluate the quality of the generated approximation to the nondominated frontier to a multiobjective integer programming problem They reflect the quality information in three different respects Consequently, the integration of them is able to reflect the comprehensive information regarding the quality of the approximation For a method to be used it should be not only easy to use, but meaningful and provide helpful results (Zionts, 1992) Our methods live up to these requirements They are characterized by the ease to use and understand, requiring little input from the DM, and providing very efficient and helpful results These properties are greatly desired in practice As a result, they provide the practitioners with a good way to tackle the multiobjective integer programming problems ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 114 Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Certainly, every approach has its limitations The approaches proposed in this study are not exceptional One of the limitations of the LPH algorithm for solving multiobjective 0-1 knapsack problem is that the determination of some parameters such as CV has to be based on experience Likewise, one of the limitations of proposed approaches lies in the fact that the position of the turning point T and the positions of the overlap T1 and T2 have to be determined empirically as well As a result, further studies on these issues could be worthwhile In addition, three further extensions to this work are possible The first one is to verify the empirically observed property of close similarity between the optimal solution to the 0-1 knapsack problem, generalized assignment problem and other knapsack problems and the optimal solution to their respective LP relaxation problem from a theoretical point of view The second one is to extend the proposed methods to some other multiobjective integer programming and combinatorial optimization problems The third one is to develop some other heuristic or exact approaches to solve the subproblem P(5.3), in place of relying on CPLEX, and compare the performances The contributions of this study are briefly summarized as follows: We have proposed three concepts of ε-Nondominated Solution, Extreme Nondominated Solution and δ-representative Nondominated Solution that have proven very useful for solving multiobjective integer programming problems ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 115 Chapter Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– We have developed a methodological framework to solve multiobjective integer programming and combinatorial optimization problems, which converts a multiobjective problem into a series of single-objective problems according to the necessity of the decision maker We have proposed an efficient and effective method for solving multiobjective 0-1 knapsack problems We have proposed an efficient and effective method for solving biobjective generalized assignment problems We also have proposed the three criteria proximity, uniformity and coverage to evaluate the quality of the generated approximation to the nondominated frontier to a multiobjective integer programming problem ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 116 References ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– References Aksoy Y (1990) An Interactive 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an efficient LP-based heuristic, to appear in European Journal of Operational Research 67 Zhang C.W., Ong H.L (2003) Solving the biobjective generalized assignment problem by an efficient LP-based heuristic, to appear in Naval Research Logistics 68 Zionts Stanley (1979) A survey of multiple criteria integer programming methods Annals of Discrete Mathematics 5, 389-398 69 Zionts Stanley (1992) Multiple Criteria Decision Making: The challenge that lies ahead, Proceedings of the 10th International Conference on Multiple Criteria Decision Making, 1992: Taipei, Taiwan ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 122 ... usual Lagrangean and surrogate relaxations The Lagrangean/surrogate relaxation relaxes first a set of constraints in a surrogate way; and then the Lagrangean relaxation of the surrogate constraint... competitive and thus can not be attained optimally at the same time As a result, the concept of Nondominated Solution is adopted in the area of multiobjective mathematically programming Usually a multiobjective. .. the research in the area of multiobjective combinatorial optimization and presented an extensive annotated bibliography of it Jones, Mirrazavi and Tamiz (2002) gave an overview of the articles