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An algorithm for optimizing a linear function over an integer efficient set

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  • Jorge, Jesús M.

Abstract

Optimizing a linear function over the efficient set of a multiobjective integer linear programming (MOILP) problem is a topic of unquestionable practical as well as mathematical interest within the field of multiple criteria decision making. As known, those problems are particularly difficult to deal with due to the discrete nature of the efficient set, which is not explicitly known, nor a suitable implicit description is available. In this work an exact algorithm is presented to optimize a linear function over the efficient set of a MOILP. The approach here proposed defines a sequence of progressively more constrained single-objective integer problems that successively eliminates undesirable points from further consideration. The algorithm has been coded in C Sharp, using CPLEX solver, and computational experiments have been undertaken in order to analyze performance properties of the algorithm over different problem instances randomly generated.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:195:y:2009:i:1:p:98-103
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    1. Odile Marcotte & Richard M. Soland, 1986. "An Interactive Branch-and-Bound Algorithm for Multiple Criteria Optimization," Management Science, INFORMS, vol. 32(1), pages 61-75, January.
    2. Jesús Jorge, 2005. "A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem," Journal of Global Optimization, Springer, vol. 31(1), pages 1-16, January.
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    4. 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.
    5. 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.
    6. 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.
    7. 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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    Cited by:

    1. Zerdani, Ouiza & Moulai, Mustapha, 2011. "Optimization over an integer efficient set of a Multiple Objective Linear Fractional Problem," MPRA Paper 35579, University Library of Munich, Germany.
    2. Jornada, Daniel & Leon, V. Jorge, 2016. "Biobjective robust optimization over the efficient set for Pareto set reduction," European Journal of Operational Research, Elsevier, vol. 252(2), pages 573-586.
    3. Melih Ozlen & Benjamin A. Burton & Cameron A. G. MacRae, 2014. "Multi-Objective Integer Programming: An Improved Recursive Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 470-482, February.
    4. Blanco, Víctor, 2011. "A mathematical programming approach to the computation of the omega invariant of a numerical semigroup," European Journal of Operational Research, Elsevier, vol. 215(3), pages 539-550, December.
    5. Melih Ozlen & Meral Azizoğlu & Benjamin Burton, 2013. "Optimising a nonlinear utility function in multi-objective integer programming," Journal of Global Optimization, Springer, vol. 56(1), pages 93-102, May.
    6. Sierra-Altamiranda, Alvaro & Charkhgard, Hadi & Eaton, Mitchell & Martin, Julien & Yurek, Simeon & Udell, Bradley J., 2020. "Spatial conservation planning under uncertainty using modern portfolio theory and Nash bargaining solution," Ecological Modelling, Elsevier, vol. 423(C).
    7. Murat Köksalan & Banu Lokman, 2015. "Finding nadir points in multi-objective integer programs," Journal of Global Optimization, Springer, vol. 62(1), pages 55-77, May.
    8. Vahid Mahmoodian & Iman Dayarian & Payman Ghasemi Saghand & Yu Zhang & Hadi Charkhgard, 2022. "A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1453-1470, May.
    9. Boland, Natashia & Charkhgard, Hadi & Savelsbergh, Martin, 2017. "A new method for optimizing a linear function over the efficient set of a multiobjective integer program," European Journal of Operational Research, Elsevier, vol. 260(3), pages 904-919.
    10. Alvaro Sierra Altamiranda & Hadi Charkhgard, 2019. "A New Exact Algorithm to Optimize a Linear Function over the Set of Efficient Solutions for Biobjective Mixed Integer Linear Programs," INFORMS Journal on Computing, INFORMS, vol. 31(4), pages 823-840, October.
    11. Hadjer Belkhiri & Mohamed El-Amine Chergui & Fatma Zohra Ouaïl, 2022. "Optimizing a linear function over an efficient set," Operational Research, Springer, vol. 22(4), pages 3183-3201, September.
    12. 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.
    13. Gokhan Kirlik & Serpil Sayın, 2015. "Computing the nadir point for multiobjective discrete optimization problems," Journal of Global Optimization, Springer, vol. 62(1), pages 79-99, May.
    14. Seyyed Amir Babak Rasmi & Ali Fattahi & Metin Türkay, 2021. "SASS: slicing with adaptive steps search method for finding the non-dominated points of tri-objective mixed-integer linear programming problems," Annals of Operations Research, Springer, vol. 296(1), pages 841-876, January.
    15. Wassila Drici & Fatma Zohra Ouail & Mustapha Moulaï, 2018. "Optimizing a linear fractional function over the integer efficient set," Annals of Operations Research, Springer, vol. 267(1), pages 135-151, August.
    16. Esmaeili, Somayeh & Bashiri, Mahdi & Amiri, Amirhossein, 2023. "An exact criterion space search algorithm for a bi-objective blood collection problem," European Journal of Operational Research, Elsevier, vol. 311(1), pages 210-232.
    17. Daniel Jornada & V. Jorge Leon, 2020. "Filtering Algorithms for Biobjective Mixed Binary Linear Optimization Problems with a Multiple-Choice Constraint," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 57-73, January.

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