Two techniques to reduce the Pareto optimal solutions in multiobjective optimization problems

Articles in Press

Document Type : Original paper

Authors

Department of Mathematics‎, ‎Vali-e-Asr University of‎ ‎Rafsanjan, Rafsanjan‎, ‎Iran

Abstract

In this study, for a decomposed multi-objective optimization problem, we  propose the direct sum of the preference matrices of the  subproblems provided by the decision maker (DM). Then, using this matrix, we present a new generalization of the rational efficiency concept for solving the multi-objective optimization problem (MOP). A problem that sometimes occurs in multi-objective optimization is the existence of a large set of Pareto optimal solutions. Hence, decision making based on selecting a unique preferred solution becomes difficult. Considering models with the concept of generalized rational efficiency can relieve some of the burden from the DM by shrinking the solution set. This paper discusses both theoretical and practical aspects of rationally efficient solutions related to this concept. Moreover, we present two techniques to reduce the Pareto optimal solutions using. The first technique involves using the powers of the preference matrix, while the second technique involves creating a new preference matrix by modifying the decomposition of the MOP.

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References

[1] V.P. Berman and G.E. Naumov, Preference-relation with interval value tradeoffs in criterion space, Autom. Remote Control 50 (1989), no. 3, 398–410.
[2] S. Dempe, G. Eichfelder, and J. Fliege, On the effects of combining objectives in multi-objective optimization, Math. Methods Oper. Res. 82 (2015), 1–18.
[3] A. Engau and M.M. Wiecek, Cone characterizations of approximate solutions in real vector optimization, J. Optim. Theory Appl. 134 (2007), no. 3, 499–513.  https://doi.org/10.1007/s10957-007-9235-8
[4] M. Farina and P. Amato, On the optimal solution definition for many-criteria optimization problems, 2002 annual meeting of the North American fuzzy information processing society proceedings. NAFIPS-FLINT 2002 (Cat. No. 02TH8622), IEEE, 2002, pp. 233–238.  https://doi.org/10.1109/NAFIPS.2002.1018061
[5] B.J. Hunt, M.M. Wiecek, and C.S. Hughes, Relative importance of criteria in multiobjective programming: A cone-based approach, European J. Oper. Res. 207 (2010), no. 2, 936–945.  https://doi.org/10.1016/j.ejor.2010.06.008
[6] M.M. Kostreva and W. Ogryczak, Linear optimization with multiple equitable criteria, RAIRO Oper. Res. 33 (1999), no. 3, 275–297.  https://doi.org/10.1051/ro:1999112
[7] A. Mahmodinejad and D. Foroutannia, Generalized rationnal efficiency in multiobjective programming, UPB Sci. Bull. A: Appl. Math. Phys. 78 (2016), no. 1, 135–146.
[8] B. Malakooti, A decision support system and a heuristic interactive approach for solving discrete multiple criteria problems, IEEE Trans. Syst. Man. Cybern. 18 (1988), no. 2, 273–284. https://doi.org/10.1109/21.3466
[9] J. Molina, L.V. Santana, A.G. Hernández-DÍaz, C.A.C. Coello, and R. Caballero, g-dominance: Reference point based dominance for multiobjective metaheuristics, European J. Oper. Res. 197 (2009), no. 2, 685–692. https://doi.org/10.1016/j.ejor.2008.07.015
[10] V.D. Noghin, Relative importance of criteria: a quantitative approach, J. MultiCriteria Decis. Anal. 6 (1997), no. 6, 355–363.
[11] V.D. Noghin, What is the relative importance of criteria and how to use it in mcdm, Multiple Criteria Decision Making in the New Millennium: Proceedings of the Fifteenth International Conference on Multiple Criteria Decision Making (MCDM) Ankara, Turkey, July 10–14, 2000, Springer, 2001, pp. 59–68. https://doi.org/10.1007/978-3-642-56680-6_5
[12] S. Petchrompo, D.W. Coit, A. Brintrup, A. Wannakrairot, and A.K. Parlikad, A review of pareto pruning methods for multi-objective optimization, Comput. Ind. Eng. 167 (2022), Article ID: 108022. https://doi.org/10.1016/j.cie.2022.108022
[13] S. Petchrompo, A. Wannakrairot, and A.K. Parlikad, Pruning Pareto optimal solutions for multi-objective portfolio asset management, European J. Oper. Res. 297 (2022), no. 1, 203–220. https://doi.org/10.1016/j.ejor.2021.04.053
[14] V.V. Podinovskii, Quantitative importance of criteria, Autom. Remote Control 61 (2000), 817–828.
[15] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986.
[16] V. Venkat, S.H. Jacobson, and J.A. Stori, A post-optimality analysis algorithm for multi-objective optimization, Comput. Optim. Appl. 28 (2004), no. 3, 357–372. https://doi.org/10.1023/B:COAP.0000033968.55439.8b
[17] J.B. Yang, Multiple criteria decision analysis methods and applications, Hunan Publishing House, Changsha, PR China, 1996, 1996.
[18] J.B. Yang, Minimax reference point approach and its application for multiobjective optimisation, European J. Oper. Res. 126 (2000), no. 3, 541–556. https://doi.org/10.1016/S0377-2217(99)00309-4
[19] E. Zio, P. Baraldi, and N. Pedroni, Optimal power system generation scheduling by multi-objective genetic algorithms with preferences, Reliab. Eng. Syst. Saf. 94 (2009), no. 2, 432–444. https://doi.org/10.1016/j.ress.2008.04.004
[20] E. Zio and R. Bazzo, A clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems, European J. Oper. Res. 210 (2011), no. 3, 624–634. https://doi.org/10.1016/j.ejor.2010.10.021
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 07 June 2024

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