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The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm

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  • Serpil Say{i}n

    (College of Administrative Sciences and Economics, Koç University, Sar\iota yer, \.Istanbul, 34450 Turkey)

  • Panos Kouvelis

    (Olin School of Business, Washington University, 1 Brookings Drive, St. Louis, Missouri 63130)

Abstract

We study the multiple objective discrete optimization (MODO) problem and propose two-stage optimization problems as subproblems to be solved to obtain efficient solutions. The mathematical structure of the first level subproblem has similarities to both Tchebycheff type of approaches and a generalization of the lexicographic max-ordering problem that are applicable to multiple objective optimization. We present some results that enable us to develop an algorithm to solve the bicriteria discrete optimization problem for the entire efficient set. We also propose a modification of the algorithm that generates a sample of efficient solutions that satisfies a prespecified quality guarantee. We apply the algorithm to solve the bicriteria knapsack problem. Our computational results on this particular problem demonstrate that our algorithm performs significantly better than an equivalent Tchebycheff counterpart. Moreover, the computational behavior of the sampling version is quite promising.

Suggested Citation

  • Serpil Say{i}n & Panos Kouvelis, 2005. "The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm," Management Science, INFORMS, vol. 51(10), pages 1572-1581, October.
    • Handle: RePEc:inm:ormnsc:v:51:y:2005:i:10:p:1572-1581
    DOI: 10.1287/mnsc.1050.0413
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    References listed on IDEAS

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    1. Stephan Helfrich & Kathrin Prinz & Stefan Ruzika, 2024. "The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1187-1216, September.
    2. David Bergman & Merve Bodur & Carlos Cardonha & Andre A. Cire, 2022. "Network Models for Multiobjective Discrete Optimization," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 990-1005, March.
    3. Aytug, Haldun & Sayın, Serpil, 2012. "Exploring the trade-off between generalization and empirical errors in a one-norm SVM," European Journal of Operational Research, Elsevier, vol. 218(3), pages 667-675.
    4. Jonas Ide & Anita Schöbel, 2016. "Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(1), pages 235-271, January.
    5. Kerstin Dächert & Kathrin Klamroth, 2015. "A linear bound on the number of scalarizations needed to solve discrete tricriteria optimization problems," Journal of Global Optimization, Springer, vol. 61(4), pages 643-676, April.
    6. Kirlik, Gokhan & Sayın, Serpil, 2014. "A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems," European Journal of Operational Research, Elsevier, vol. 232(3), pages 479-488.
    7. Alexander Engau, 2017. "Proper Efficiency and Tradeoffs in Multiple Criteria and Stochastic Optimization," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 119-134, January.
    8. Bérubé, Jean-François & Gendreau, Michel & Potvin, Jean-Yves, 2009. "An exact [epsilon]-constraint method for bi-objective combinatorial optimization problems: Application to the Traveling Salesman Problem with Profits," European Journal of Operational Research, Elsevier, vol. 194(1), pages 39-50, April.
    9. Aytug, Haldun & SayIn, Serpil, 2009. "Using support vector machines to learn the efficient set in multiple objective discrete optimization," European Journal of Operational Research, Elsevier, vol. 193(2), pages 510-519, March.
    10. Angelo Aliano Filho & Antonio Carlos Moretti & Margarida Vaz Pato & Washington Alves Oliveira, 2021. "An exact scalarization method with multiple reference points for bi-objective integer linear optimization problems," Annals of Operations Research, Springer, vol. 296(1), pages 35-69, January.
    11. Hadi Charkhgard & Martin Savelsbergh & Masoud Talebian, 2018. "Nondominated Nash points: application of biobjective mixed integer programming," 4OR, Springer, vol. 16(2), pages 151-171, June.
    12. Elisabeth Köbis, 2015. "On Robust Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 969-984, December.
    13. C. Gutiérrez & L. Huerga & E. Köbis & C. Tammer, 2021. "A scalarization scheme for binary relations with applications to set-valued and robust optimization," Journal of Global Optimization, Springer, vol. 79(1), pages 233-256, January.
    14. Panos Kouvelis & Serpil Sayın, 2006. "Algorithm robust for the bicriteria discrete optimization problem," Annals of Operations Research, Springer, vol. 147(1), pages 71-85, October.
    15. Rafael Lazimy, 2013. "Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems," Annals of Operations Research, Springer, vol. 210(1), pages 73-99, November.
    16. Klamroth, Kathrin & Köbis, Elisabeth & Schöbel, Anita & Tammer, Christiane, 2017. "A unified approach to uncertain optimization," European Journal of Operational Research, Elsevier, vol. 260(2), pages 403-420.
    17. Botte, Marco & Schöbel, Anita, 2019. "Dominance for multi-objective robust optimization concepts," European Journal of Operational Research, Elsevier, vol. 273(2), pages 430-440.
    18. Stacey Faulkenberg & Margaret Wiecek, 2012. "Generating equidistant representations in biobjective programming," Computational Optimization and Applications, Springer, vol. 51(3), pages 1173-1210, April.

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