Generation of the exact Pareto set in Multi-Objective Traveling Salesman and Set Covering Problems

Abstract The calculation of the exact set in Multi-Objective Combinatorial Optimization (MOCO) problems is one of the most computationally demanding tasks as most of the problems are NP-hard. In the present work we use AUGMECON2 a Multi-Objective Mathematical Programming (MOMP) method which is capable of generating the exact Pareto set in Multi-Objective Integer Programming (MOIP) problems for producing all the Pareto optimal solutions in two popular MOCO problems: The Multi-Objective Traveling Salesman Problem (MOTSP) and the Multi-Objective Set Covering Problem (MOSCP). The computational experiment is confined to two-objective problems that are found in the literature. The performance of the algorithm is slightly better to what is already found from previous works and it goes one step further generating the exact Pareto set to till now unsolved problems. The results are provided in a dedicated site and can be useful for benchmarking with other MOMP methods or even Multi-Objective Meta-Heuristics (MOMH) that can check the performance of their approximate solution against the exact solution in MOTSP and MOSCP problems.

[1]  Xavier Gandibleux,et al.  A survey and annotated bibliography of multiobjective combinatorial optimization , 2000, OR Spectr..

[2]  Anthony Przybylski,et al.  Two phase algorithms for the bi-objective assignment problem , 2008, Eur. J. Oper. Res..

[3]  Murat Köksalan,et al.  Multiobjective traveling salesperson problem on Halin graphs , 2009, Eur. J. Oper. Res..

[4]  Jacques Teghem,et al.  The Multiobjective Traveling Salesman Problem: A Survey and a New Approach , 2010, Advances in Multi-Objective Nature Inspired Computing.

[5]  William H. Press,et al.  Numerical Recipes in FORTRAN - The Art of Scientific Computing, 2nd Edition , 1987 .

[6]  Kalyanmoy Deb,et al.  Multi-objective optimization using evolutionary algorithms , 2001, Wiley-Interscience series in systems and optimization.

[7]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[8]  Thomas Stützle,et al.  A Two-Phase Local Search for the Biobjective Traveling Salesman Problem , 2003, EMO.

[9]  George Mavrotas,et al.  Multi-criteria branch and bound: A vector maximization algorithm for Mixed 0-1 Multiple Objective Linear Programming , 2005, Appl. Math. Comput..

[10]  Jacques Teghem,et al.  Two-phases Method and Branch and Bound Procedures to Solve the Bi–objective Knapsack Problem , 1998, J. Glob. Optim..

[11]  Anthony Przybylski,et al.  A two phase method for multi-objective integer programming and its application to the assignment problem with three objectives , 2010, Discret. Optim..

[12]  Gexiang Zhang,et al.  Multi-objective ant colony optimization based on decomposition for bi-objective traveling salesman problems , 2011, Soft Computing.

[13]  A. Volgenant.  Symmetric traveling salesman problems , 1990 .

[14]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[15]  Andrzej Jaszkiewicz,et al.  Speed-up techniques for solving large-scale biobjective TSP , 2010, Comput. Oper. Res..

[16]  Knut Richter,et al.  Solving a multiobjective traveling salesman problem by dynamic programming , 1982 .

[17]  Funda Samanlioglu,et al.  A memetic random-key genetic algorithm for a symmetric multi-objective traveling salesman problem , 2008, Comput. Ind. Eng..

[18]  Keld Helsgaun,et al.  An effective implementation of the Lin-Kernighan traveling salesman heuristic , 2000, Eur. J. Oper. Res..

[19]  Xin Yao,et al.  Performance Scaling of Multi-objective Evolutionary Algorithms , 2003, EMO.

[20]  Qingfu Zhang,et al.  Comparison between MOEA/D and NSGA-II on the Multi-Objective Travelling Salesman Problem , 2009 .

[21]  Kathrin Klamroth,et al.  An augmented weighted Tchebycheff method with adaptively chosen parameters for discrete bicriteria optimization problems , 2012, Comput. Oper. Res..

[22]  Gerhard Reinelt,et al.  TSPLIB - A Traveling Salesman Problem Library , 1991, INFORMS J. Comput..

[23]  Milan Stanojević,et al.  Computation Results of Finding All Efficient Points in Multiobjective Combinatorial Optimization , 2008, Int. J. Comput. Commun. Control.

[24]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[25]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[26]  R. S. Laundy,et al.  Multiple Criteria Optimisation: Theory, Computation and Application , 1989 .

[27]  Andrzej Jaszkiewicz,et al.  Pareto memetic algorithm with path relinking for bi-objective traveling salesperson problem , 2009, Eur. J. Oper. Res..

[28]  Murat Köksalan,et al.  An Exact Algorithm for Finding Extreme Supported Nondominated Points of Multiobjective Mixed Integer Programs , 2010, Manag. Sci..

[29]  El-Ghazali Talbi,et al.  On dominance-based multiobjective local search: design, implementation and experimental analysis on scheduling and traveling salesman problems , 2012, J. Heuristics.

[30]  Thomas Stützle,et al.  An Analysis of Algorithmic Components for Multiobjective Ant Colony Optimization: A Case Study on the Biobjective TSP , 2009, Artificial Evolution.

[31]  Andrzej Jaszkiewicz,et al.  A Comparative Study of Multiple-Objective Metaheuristics on the Bi-Objective Set Covering Problem and the Pareto Memetic Algorithm , 2004, Ann. Oper. Res..

[32]  Thomas Stützle,et al.  Design and analysis of stochastic local search for the multiobjective traveling salesman problem , 2009, Comput. Oper. Res..

[33]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems , 2002, Genetic Algorithms and Evolutionary Computation.

[34]  G. Laporte.  The traveling salesman problem: An overview of exact and approximate algorithms , 1992 .

[35]  Ghasem Tohidi,et al.  A method for generating all efficient solutions of 0-1 multi-objective linear programming problem , 2005, Appl. Math. Comput..

[36]  Murat Köksalan,et al.  Pyramidal tours and multiple objectives , 2010, J. Glob. Optim..

[37]  Bodo Manthey,et al.  Approximation Algorithms for Multi-Criteria Traveling Salesman Problems , 2006, Algorithmica.

[38]  Benjamin A. Burton,et al.  Multi-Objective Integer Programming: An Improved Recursive Algorithm , 2011, Journal of Optimization Theory and Applications.

[39]  Fred W. Glover,et al.  Multi-objective Meta-heuristics for the Traveling Salesman Problem with Profits , 2008, J. Math. Model. Algorithms.

[40]  Jacques Teghem,et al.  Two-phase Pareto local search for the biobjective traveling salesman problem , 2010, J. Heuristics.

[41]  Christian Prins,et al.  Two-phase method and Lagrangian relaxation to solve the Bi-Objective Set Covering Problem , 2006, Ann. Oper. Res..

[42]  Panos M. Pardalos,et al.  A survey of recent developments in multiobjective optimization , 2007, Ann. Oper. Res..

[43]  Marco Laumanns,et al.  An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method , 2006, Eur. J. Oper. Res..

[44]  Andrzej Jaszkiewicz,et al.  Genetic local search for multi-objective combinatorial optimization , 2022 .

[45]  Michael Pilegaard Hansen.  Use of Substitute Scalarizing Functions to Guide a Local Search Based Heuristic: The Case of moTSP , 2000, J. Heuristics.

[46]  Madjid Tavana,et al.  An integrated multi-objective framework for solving multi-period project selection problems , 2012, Appl. Math. Comput..

[47]  Francisco Herrera,et al.  A taxonomy and an empirical analysis of multiple objective ant colony optimization algorithms for the bi-criteria TSP , 2007, Eur. J. Oper. Res..

[48]  Clarisse Dhaenens,et al.  Parallel partitioning method (PPM): A new exact method to solve bi-objective problems , 2007, Comput. Oper. Res..

[49]  Thomas Stützle,et al.  Stochastic Local Search Algorithms for Multiobjective Combinatorial Optimization , 2006, Handbook of Approximation Algorithms and Metaheuristics.

[50]  Michel Gendreau,et al.  An exact epsilon-constraint method for bi-objective combinatorial optimization problems: Application to the Traveling Salesman Problem with Profits , 2009, Eur. J. Oper. Res..

[51]  George Mavrotas,et al.  An improved version of the augmented ε-constraint method (AUGMECON2) for finding the exact pareto set in multi-objective integer programming problems , 2013, Appl. Math. Comput..

[52]  George Mavrotas,et al.  Effective implementation of the epsilon-constraint method in Multi-Objective Mathematical Programming problems , 2009, Appl. Math. Comput..

[53]  Jacques Teghem,et al.  Very Large-Scale Neighborhood Search for Solving Multiobjective Combinatorial Optimization Problems , 2011, EMO.

[54]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.