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Quantitative Comparison of Approximate Solution Sets for Multicriteria Optimization Problems with Weighted Tchebycheff Preference Function

Author

Listed:
  • Bilge Bozkurt

    (Department of Industrial Engineering, Middle East Technical University, 06531 Ankara, Turkey)

  • John W. Fowler

    (School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, Arizona 85287)

  • Esma S. Gel

    (School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, Arizona 85287)

  • Bosun Kim

    (School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, Arizona 85287)

  • Murat Köksalan

    (Department of Industrial Engineering, Middle East Technical University, 06531 Ankara, Turkey)

  • Jyrki Wallenius

    (Department of Business Technology, Helsinki School of Economics, FIN-00100 Helsinki, Finland)

Abstract

We consider the problem of evaluating the quality of solution sets generated by heuristics for multiple-objective combinatorial optimization problems. We extend previous research on the integrated preference functional (IPF), which assigns a scalar value to a given discrete set of nondominated points so that the weighted Tchebycheff function can be used as the underlying implicit value function. This extension is useful because modeling the decision maker's value function with the weighted Tchebycheff function reflects the impact of unsupported points when evaluating sets of nondominated points. We present an exact calculation method for the IPF measure in this case for an arbitrary number of criteria. We show that every nondominated point has its optimal weight interval for the weighted Tchebycheff function. Accordingly, all nondominated points, and not only the supported points in a set, contribute to the value of the IPF measure when using the weighted Tchebycheff function. Two- and three-criteria numerical examples illustrate the desirable properties of the weighted Tchebycheff function, providing a richer measure than the original IPF based on a convex combination of objectives.

Suggested Citation

  • Bilge Bozkurt & John W. Fowler & Esma S. Gel & Bosun Kim & Murat Köksalan & Jyrki Wallenius, 2010. "Quantitative Comparison of Approximate Solution Sets for Multicriteria Optimization Problems with Weighted Tchebycheff Preference Function," Operations Research, INFORMS, vol. 58(3), pages 650-659, June.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:3:p:650-659
    DOI: 10.1287/opre.1090.0766
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    References listed on IDEAS

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    1. Kim, B. & Gel, E.S. & Fowler, J.W. & Carlyle, W.M. & Wallenius, J., 2006. "Evaluation of nondominated solution sets for k-objective optimization problems: An exact method and approximations," European Journal of Operational Research, Elsevier, vol. 173(2), pages 565-582, September.
    2. Robert F. Dell & Mark H. Karwan, 1990. "An interactive MCDM weight space reduction method utilizing a tchebycheff utility function," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(2), pages 263-277, April.
    3. Ralph E. Steuer & Joe Silverman & Alan W. Whisman, 1993. "A Combined Tchebycheff/Aspiration Criterion Vector Interactive Multiobjective Programming Procedure," Management Science, INFORMS, vol. 39(10), pages 1255-1260, October.
    4. Pekka Korhonen & Jyrki Wallenius, 1988. "A pareto race," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(6), pages 615-623, December.
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    2. Karakaya, G. & Köksalan, M., 2021. "Evaluating solutions and solution sets under multiple objectives," European Journal of Operational Research, Elsevier, vol. 294(1), pages 16-28.
    3. Markus Leitner & Ivana Ljubić & Markus Sinnl, 2015. "A Computational Study of Exact Approaches for the Bi-Objective Prize-Collecting Steiner Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 118-134, February.
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    5. 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.

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