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Evaluating solutions and solution sets under multiple objectives

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  • Karakaya, G.
  • Köksalan, M.

Abstract

In this study we address evaluating solutions and solution sets that are defined by multiple objectives based on a function. Although any function can be used, we focus on mostly weighted Tchebycheff functions that can be used for a variety of purposes when multiple objectives are considered. One such use is to approximate a decision maker’s preferences with a Tchebycheff utility function. Different solutions can be evaluated in terms of expected utility conditional on weight values. Another possible use is to evaluate a set of solutions that approximate a Pareto set. It is not straightforward to find the Pareto set, especially for large-size multi-objective combinatorial optimization problems. To measure the representation quality of approximate Pareto sets and to compare such sets with each other, there are some performance indicators such as the hypervolume measure, the ε indicator, and the integrated preference functional (IPF) measure. A Tchebycheff function based IPF measure can be used to estimate how well a set of solutions represents the Pareto set. We develop the necessary theory to practically evaluate solutions and solution sets. We develop a general algorithm and demonstrate it for two, three, and four objectives.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:ejores:v:294:y:2021:i:1:p:16-28
    DOI: 10.1016/j.ejor.2021.01.021
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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. 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.
    3. Karakaya, G. & Köksalan, M. & Ahipaşaoğlu, S.D., 2018. "Interactive algorithms for a broad underlying family of preference functions," European Journal of Operational Research, Elsevier, vol. 265(1), pages 248-262.
    4. 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.
    5. Banu Lokman & Murat Köksalan & Pekka J. Korhonen & Jyrki Wallenius, 2016. "An interactive algorithm to find the most preferred solution of multi-objective integer programs," Annals of Operations Research, Springer, vol. 245(1), pages 67-95, October.
    6. Holzmann, Tim & Smith, J.C., 2018. "Solving discrete multi-objective optimization problems using modified augmented weighted Tchebychev scalarizations," European Journal of Operational Research, Elsevier, vol. 271(2), pages 436-449.
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    1. 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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