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An improved version of the augmented epsilon-constraint method (AUGMECON2) for finding the exact Pareto set in Multi-Objective Integer Programming problems

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  • Mavrotas, George
  • Florios, Kostas

Abstract

Generation (or a posteriori) methods in Multi-Objective Mathematical Programming (MOMP) is the most computationally demanding category among the MOMP approaches. Due to the dramatic increase in computational speed and the improvement of Mathematical Programming algorithms the generation methods become all the more attractive among today’s decision makers. In the current paper we present the generation method AUGMECON2 which is an improvement of our development, AUGMECON. Although AUGMECON2 is a general purpose method, we will demonstrate that AUGMECON2 is especially suitable for Multi-Objective Integer Programming (MOIP) problems. Specifically, AUGMECON2 is capable of producing the exact Pareto set in MOIP problems by appropriately tuning its running parameters. In this context, we compare the previous and the new version in a series of new and old benchmarks found in the literature. We also compare AUGMECON2’s performance in the generation of the exact Pareto sets with established methods and algorithms based on specific MOIP problems (knapsack, set packing) and on published results. Except from other Mathematical Programming methods, AUGMECON2 is found to be competitive also with Multi-Objective Meta-Heuristics (MOMH) in producing adequate approximations of the Pareto set in Multi-Objective Combinatorial Optimization (MOCO) problems.

Suggested Citation

  • Mavrotas, George & Florios, Kostas, 2013. "An improved version of the augmented epsilon-constraint method (AUGMECON2) for finding the exact Pareto set in Multi-Objective Integer Programming problems," MPRA Paper 105034, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:105034
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    1. Chalmet, L. G. & Lemonidis, L. & Elzinga, D. J., 1986. "An algorithm for the bi-criterion integer programming problem," European Journal of Operational Research, Elsevier, vol. 25(2), pages 292-300, May.
    2. Dessouky, M. I. & Ghiassi, M. & Davis, W. J., 1986. "Estimates of the minimum nondominated criterion values in multiple-criteria decision-making," Engineering Costs and Production Economics, Elsevier, vol. 10(2), pages 95-104, June.
    3. Thomas Erlebach & Hans Kellerer & Ulrich Pferschy, 2002. "Approximating Multiobjective Knapsack Problems," Management Science, INFORMS, vol. 48(12), pages 1603-1612, December.
    4. Murat Köksalan & Banu Lokman, 2009. "Approximating the nondominated frontiers of multi‐objective combinatorial optimization problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(2), pages 191-198, March.
    5. Shah, Ruchit & Reed, Patrick, 2011. "Comparative analysis of multiobjective evolutionary algorithms for random and correlated instances of multiobjective d-dimensional knapsack problems," European Journal of Operational Research, Elsevier, vol. 211(3), pages 466-479, June.
    6. Francis Sourd & Olivier Spanjaard, 2008. "A Multiobjective Branch-and-Bound Framework: Application to the Biobjective Spanning Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 20(3), pages 472-484, August.
    7. Jorge, Jesús M., 2009. "An algorithm for optimizing a linear function over an integer efficient set," European Journal of Operational Research, Elsevier, vol. 195(1), pages 98-103, May.
    8. Altannar Chinchuluun & Panos Pardalos, 2007. "A survey of recent developments in multiobjective optimization," Annals of Operations Research, Springer, vol. 154(1), pages 29-50, October.
    9. Rasmussen, L. M., 1986. "Zero--one programming with multiple criteria," European Journal of Operational Research, Elsevier, vol. 26(1), pages 83-95, July.
    10. Ted Ralphs & Matthew Saltzman & Margaret Wiecek, 2006. "An improved algorithm for solving biobjective integer programs," Annals of Operations Research, Springer, vol. 147(1), pages 43-70, October.
    11. Klein, Dieter & Hannan, Edward, 1982. "An algorithm for the multiple objective integer linear programming problem," European Journal of Operational Research, Elsevier, vol. 9(4), pages 378-385, April.
    12. Jaszkiewicz, Andrzej, 2004. "On the computational efficiency of multiple objective metaheuristics. The knapsack problem case study," European Journal of Operational Research, Elsevier, vol. 158(2), pages 418-433, October.
    13. Laumanns, Marco & Thiele, Lothar & Zitzler, Eckart, 2006. "An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method," European Journal of Operational Research, Elsevier, vol. 169(3), pages 932-942, March.
    14. Mavrotas, George & Diakoulaki, Danae & Kourentzis, Athanasios, 2008. "Selection among ranked projects under segmentation, policy and logical constraints," European Journal of Operational Research, Elsevier, vol. 187(1), pages 177-192, May.
    15. Kathrin Klamroth & Margaret M. Wiecek, 2000. "Dynamic programming approaches to the multiple criteria knapsack problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(1), pages 57-76, February.
    16. M. Ehrgott & S. Ruzika, 2008. "Improved ε-Constraint Method for Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 375-396, September.
    17. Florios, Kostas & Mavrotas, George & Diakoulaki, Danae, 2010. "Solving multiobjective, multiconstraint knapsack problems using mathematical programming and evolutionary algorithms," European Journal of Operational Research, Elsevier, vol. 203(1), pages 14-21, May.
    18. Sylva, John & Crema, Alejandro, 2007. "A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs," European Journal of Operational Research, Elsevier, vol. 180(3), pages 1011-1027, August.
    19. Alves, Maria Joao & Climaco, Joao, 2007. "A review of interactive methods for multiobjective integer and mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 180(1), pages 99-115, July.
    20. Özlen, Melih & Azizoglu, Meral, 2009. "Multi-objective integer programming: A general approach for generating all non-dominated solutions," European Journal of Operational Research, Elsevier, vol. 199(1), pages 25-35, November.
    21. Delorme, Xavier & Gandibleux, Xavier & Degoutin, Fabien, 2010. "Evolutionary, constructive and hybrid procedures for the bi-objective set packing problem," European Journal of Operational Research, Elsevier, vol. 204(2), pages 206-217, July.
    22. Alves, Maria João & Costa, João Paulo, 2009. "An exact method for computing the nadir values in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 198(2), pages 637-646, October.
    23. Sylva, John & Crema, Alejandro, 2004. "A method for finding the set of non-dominated vectors for multiple objective integer linear programs," European Journal of Operational Research, Elsevier, vol. 158(1), pages 46-55, October.
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    More about this item

    Keywords

    Multi-Objective Programming; ε-constraint method; exact Pareto set;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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