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A new method for solving multiobjective bilevel programs. (English) Zbl 1369.90160
Summary: We study a class of multiobjective bilevel programs with the weights of objectives being uncertain and assumed to belong to convex and compact set. To the best of our knowledge, there is no study about this class of problems. We use a worst-case weighted approach to solve this class of problems. Our “worst-case weighted multiobjective bilevel programs” model supposes that each player (leader or follower) has a set of weights to their objectives and wishes to minimize their maximum weighted sum objective where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto optimum concept, which we call “robust-weighted Pareto optimum”; for the worst-case weighted multiobjective optimization with the weight set of each player given as a polytope, we show that a robust-weighted Pareto optimum can be obtained by solving mathematical programing with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multiobjective optimization to a supply chain risk management under demand uncertainty. By the comparison with the existing weighted approach, we show that our method is more robust and can be more efficiently applied to real-world problems.
MSC:
90C29 Multi-objective and goal programming
90C30 Nonlinear programming
Software:
PATH Solver
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[1] Dempe, S., Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52, 3, 333-359, (2003) · Zbl 1140.90493
[2] Bonnel, H., Optimality conditions for the semivectorial bilevel optimization problem, Pacific Journal of Optimization, 2, 3, 447-467, (2006) · Zbl 1124.90028
[3] Bonnel, H.; Morgan, J., Semivectorial bilevel optimization problem: penalty approach, Journal of Optimization Theory and Applications, 131, 3, 365-382, (2006) · Zbl 1205.90258
[4] Liu, B.; Wan, Z.; Chen, J.; Wang, G., Optimality conditions for pessimistic semivectorial bilevel programming problems, Journal of Inequalities and Applications, 2014, article no. 41, (2014) · Zbl 1332.90212
[5] Zheng, Y.; Wan, Z. P., A solution method for semivectorial bilevel programming problem via penalty method, Journal of Applied Mathematics and Computing, 37, 1-2, 207-219, (2011) · Zbl 1297.90131
[6] Ben-Tal, A.; Goryashko, A.; Guslitzer, E.; Nemirovski, A., Adjustable robust solutions of uncertain linear programs, Mathematical Programming, 99, 2, 351-376, (2004) · Zbl 1089.90037
[7] Jeyakumar, V.; Lee, G. M.; Li, G., Characterizing robust solution sets of convex programs under data uncertainty, Journal of Optimization Theory and Applications, 164, 2, 407-435, (2015) · Zbl 1307.90136
[8] Ehrgott, M., Multicriteria Optimization, (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1132.90001
[9] Wang, S. Y., Existence of a Pareto equilibrium, Journal of Optimization Theory and Applications, 79, 2, 373-384, (1993) · Zbl 0797.90124
[10] Yu, J.; Yuan, G. X., The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods, Computers & Mathematics with Applications, 35, 9, 17-24, (1998) · Zbl 1005.91008
[11] Tsiporkova, E.; Boeva, V., Multi-step ranking of alternatives in a multi-criteria and multi-expert decision making environment, Information Sciences, 176, 18, 2673-2697, (2006) · Zbl 1102.68655
[12] Schoemaker, P. J. H.; Waid, C. C., An experimental comparison of different approaches to determining weights in additive utility models, Management Science, 28, 2, 182-196, (1982)
[13] Weber, M.; Borcherding, K., Behavioral influences on weight judgments in multiattribute decision making, European Journal of Operational Research, 67, 1, 1-12, (1993)
[14] Yin, Y., Multi-objective bilevel optimization for transportation planning and management problems, Journal of Advanced Transportation, 36, 1, 93-105, (2002)
[15] Deb, K.; Sinha, A., Solving Bilevel Multi-Objective Optimization Problems Using Evolutionary Algorithms, Evolutionary Multi-Criterion Optimization, (2009), Springer
[16] Pieume, C. O.; Marcotte, P.; Fotso, L. P.; Siarry, P., Solving bilevel linear multiobjective programming problems, American Journal of Operations Research, 1, 4, 214-219, (2011) · Zbl 1154.90543
[17] Eichfelder, G., Multiobjective bilevel optimization, Mathematical Programming. A Publication of the Mathematical Programming Society, 123, 2, 419-449, (2010) · Zbl 1198.90347
[18] Ji, Y.; Qu, S. J.; Yu, Z. S., Bi-level multi-objective optimization model for last mile delivery using a discrete approach, Journal of Difference Equations and Applications, (2016) · Zbl 1376.90057
[19] Calvete, H. I.; Galé, C., Linear bilevel programs with multiple objectives at the upper level, Journal of Computational and Applied Mathematics, 234, 4, 950-959, (2010) · Zbl 1190.90137
[20] Calvete, H. I.; Galé, C., On linear bilevel problems with multiple objectives at the lower level, Omega, 39, 1, 33-40, (2011)
[21] Hu, J.; Mehrotra, S., Robust and stochastically weighted multiobjective optimization models and reformulations, Operations Research, 60, 4, 936-953, (2012) · Zbl 1342.90074
[22] Jian, J.-B.; Li, J.-L.; Mo, X.-D., A strongly and superlinearly convergent SQP algorithm for optimization problems with linear complementarity constraints, Applied Mathematics and Optimization, 54, 1, 17-46, (2006) · Zbl 1136.90514
[23] Ferris, M. C.; Munson, T. S., Complementarity problems in GAMS and the PATH solver, Journal of Economic Dynamics & Control, 24, 2, 165-188, (2000) · Zbl 1002.90070
[24] Dierkse, S. P., Robust solution of mixed complementarity problems [Ph.D. thesis], (1994), Madison, Wis, USA: University of Wisconsin-Madison, Madison, Wis, USA
[25] Hu, X.; Ralph, D., Using EPECs to model bilevel games in restructured electricity markets with locational prices, Operations Research, 55, 5, 809-827, (2007) · Zbl 1167.91357
[26] Saaty, T. L., The modern science of multicriteria decision making and its practical applications: the AHP/ANP approach, Operations Research, 61, 5, 1101-1118, (2013) · Zbl 1291.90110
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