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Approximation Methods for Multiobjective Optimization Problems: A Survey

Author

Listed:
  • Arne Herzel

    (Department of Mathematics, University of Kaiserslautern, D-67663 Kaiserslautern, Germany)

  • Stefan Ruzika

    (Department of Mathematics, University of Kaiserslautern, D-67663 Kaiserslautern, Germany)

  • Clemens Thielen

    (TUM Campus Straubing, Technical University of Munich, D-94315 Straubing, Germany)

Abstract

Algorithms for approximating the nondominated set of multiobjective optimization problems are reviewed. The approaches are categorized into general methods that are applicable under mild assumptions and, thus, to a wide range of problems, and into algorithms that are specifically tailored to structured problems. All in all, this survey covers 52 articles published within the last 41 years, that is, between 1979 and 2020. Summary of Contribution: In many problems in operations research, several conflicting objective functions have to be optimized simultaneously, and one is interested in finding Pareto optimal solutions. Because of the high complexity of finding Pareto optimal solutions and their usually very large number, however, the exact solution of such multiobjective problems is often very difficult, which motivates the study of approximation algorithms for multiobjective optimization problems. This research area uses techniques and methods from algorithmics and computing in order to efficiently determine approximate solutions to many well-known multiobjective problems from operations research. Even though approximation algorithms for multiobjective optimization problems have been investigated for more than 40 years and more than 50 research articles have been published on this topic, this paper provides the first survey of this important area at the intersection of computing and operations research.

Suggested Citation

  • Arne Herzel & Stefan Ruzika & Clemens Thielen, 2021. "Approximation Methods for Multiobjective Optimization Problems: A Survey," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1284-1299, October.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:4:p:1284-1299
    DOI: 10.1287/ijoc.2020.1028
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    References listed on IDEAS

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    1. Thomas Erlebach & Hans Kellerer & Ulrich Pferschy, 2002. "Approximating Multiobjective Knapsack Problems," Management Science, INFORMS, vol. 48(12), pages 1603-1612, December.
    2. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
    3. Arthur Warburton, 1987. "Approximation of Pareto Optima in Multiple-Objective, Shortest-Path Problems," Operations Research, INFORMS, vol. 35(1), pages 70-79, February.
    4. Matthias Ehrgott & Xavier Gandibleux, 2004. "Approximative solution methods for multiobjective combinatorial optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(1), pages 1-63, June.
    5. Refael Hassin, 1992. "Approximation Schemes for the Restricted Shortest Path Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 36-42, February.
    6. Safer, Hershel M. & Orlin, James B., 1953-, 1995. "Fast approximation schemes for multi-criteria combinatorial optimization," Working papers 3756-95., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    7. Bazgan, Cristina & Jamain, Florian & Vanderpooten, Daniel, 2017. "Discrete representation of the non-dominated set for multi-objective optimization problems using kernels," European Journal of Operational Research, Elsevier, vol. 260(3), pages 814-827.
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    Cited by:

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    2. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2024. "Using scalarizations for the approximation of multiobjective optimization problems: towards a general theory," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 27-63, August.
    3. Yijie Li, 2023. "Bicriteria fabrication scheduling of two-component jobs on a single machine," Operational Research, Springer, vol. 23(4), pages 1-13, December.

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