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An improved algorithm for solving biobjective integer programs

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  • Ted Ralphs
  • Matthew Saltzman
  • Margaret Wiecek

Abstract

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented. Copyright Springer Science + Business Media, LLC 2006

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  • 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.
    • Handle: RePEc:spr:annopr:v:147:y:2006:i:1:p:43-70:10.1007/s10479-006-0058-z
    DOI: 10.1007/s10479-006-0058-z
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    Cited by:

    1. Nathan Adelgren & Pietro Belotti & Akshay Gupte, 2018. "Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 324-338, May.
    2. Natashia Boland & Hadi Charkhgard & Martin Savelsbergh, 2015. "A Criterion Space Search Algorithm for Biobjective Integer Programming: The Balanced Box Method," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 735-754, November.
    3. Nathan Adelgren & Akshay Gupte, 2022. "Branch-and-Bound for Biobjective Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 909-933, March.
    4. 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.
    5. Hartillo-Hermoso, María Isabel & Jiménez-Tafur, Haydee & Ucha-Enríquez, José María, 2020. "An exact algebraic ϵ-constraint method for bi-objective linear integer programming based on test sets," European Journal of Operational Research, Elsevier, vol. 282(2), pages 453-463.
    6. Jesús Sáez-Aguado & Paula Camelia Trandafir, 2018. "Variants of the $$ \varepsilon $$ ε -constraint method for biobjective integer programming problems: application to p-median-cover problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(2), pages 251-283, April.
    7. Kerstin Dächert & Tino Fleuren & Kathrin Klamroth, 2024. "A simple, efficient and versatile objective space algorithm for multiobjective integer programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 351-384, August.
    8. Jie Wu & Liang Liang, 2012. "A multiple criteria ranking method based on game cross-evaluation approach," Annals of Operations Research, Springer, vol. 197(1), pages 191-200, August.
    9. De Santis, Marianna & Grani, Giorgio & Palagi, Laura, 2020. "Branching with hyperplanes in the criterion space: The frontier partitioner algorithm for biobjective integer programming," European Journal of Operational Research, Elsevier, vol. 283(1), pages 57-69.
    10. 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.
    11. Kyle Cooper & Susan R. Hunter & Kalyani Nagaraj, 2020. "Biobjective Simulation Optimization on Integer Lattices Using the Epsilon-Constraint Method in a Retrospective Approximation Framework," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1080-1100, October.
    12. 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.
    13. Özlem Karsu & Hale Erkan, 2020. "Balance in resource allocation problems: a changing reference approach," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 42(1), pages 297-326, March.
    14. Samira Fallah & Ted K. Ralphs & Natashia L. Boland, 2024. "On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 175-220, August.
    15. Belhoul, Lyes, 2014. "Résolution de problèmes d'optimisation combinatoire mono et multi-objectifs par énumération ordonnée," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/14672 edited by Vanderpooten, Daniel.
    16. 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.
    17. Ralf Borndörfer & Guillaume Sagnol & Thomas Schlechte & Elmar Swarat, 2017. "Optimal duty rostering for toll enforcement inspectors," Annals of Operations Research, Springer, vol. 252(2), pages 383-406, May.
    18. 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.
    19. 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.
    20. Stephan Helfrich & Tyler Perini & Pascal Halffmann & Natashia Boland & Stefan Ruzika, 2023. "Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems," Journal of Global Optimization, Springer, vol. 86(2), pages 417-440, June.
    21. Gutjahr, Walter J. & Katzensteiner, Stefan & Reiter, Peter & Stummer, Christian & Denk, Michaela, 2010. "Multi-objective decision analysis for competence-oriented project portfolio selection," European Journal of Operational Research, Elsevier, vol. 205(3), pages 670-679, September.
    22. Dächert, Kerstin & Klamroth, Kathrin & Lacour, Renaud & Vanderpooten, Daniel, 2017. "Efficient computation of the search region in multi-objective optimization," European Journal of Operational Research, Elsevier, vol. 260(3), pages 841-855.
    23. Sune Lauth Gadegaard & Andreas Klose & Lars Relund Nielsen, 2018. "A bi-objective approach to discrete cost-bottleneck location problems," Annals of Operations Research, Springer, vol. 267(1), pages 179-201, August.

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