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AIMS Mathematics, 2019, 4(3): 763-778. doi: 10.3934/math.2019.3.763. Research article

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A multiple objective programming approach to linear bilevel multi-follower programming

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran

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In this paper, we investigate the relationship between a certain class of linear bilevel multifollower programming problems and multiple objective programming. We introduce two multiple objective linear programming problems with different objective functions and the same constraint region. We show that the extreme points of the set of efficient solutions for both problems are the same as those of the set of feasible solutions to the linear bilevel multi-follower programming problem. Based on this relationship, a new algorithm to find an optimal solution for the linear bilevel multifollower programming problem is developed. Some numerical examples are presented to show the feasibility of the proposed algorithm.

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Keywords: linear bilevel programming; multi-follower; multiple objective programming; efficient set

Citation: Habibe Sadeghi, Fatemeh Moslemi. A multiple objective programming approach to linear bilevel multi-follower programming. AIMS Mathematics, 2019, 4(3): 763-778. doi: 10.3934/math.2019.3.763

References:

1.J. F. Bard, Practical Bilevel Optimization, Kluwer Academic, Dordrecht, 1998.
2.M. S. Bazaraa, J. J. Jarvis, H. D. Sherali, Linear Programming and Network Flows, 2nd ed., Wiley, New York, 1977.
3.M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming, Theory and Algorithms, 3nd ed., Wiley, New York, 2006.
4.H. P. Benson, Optimization over the efficient set, J. Math. Anal. Appl., 98 (1984), 562-580.
5.S. Dempe, Foundations of Bilevel Programming, Kluwer Academic, Dordrecht 2003.
6.M. Ehrgott, Multicriteria Optimization, 2nd ed., Berlin, Germany: Springer-Verlag, 2005.
7.N. P. Faísca, P. M. Saraiva, B. Rustem, et al. A multi-parametric programming approach for multilevel hierarchical and decentralized optimisation problems, Computer Management Sciences, 6 (2009), 377-397.
8.J. Fülöp, On the equivalency between a linear bilevel programming problem and linear optimization over the efficient set, Technical Report WP 93-1, Laboratory of Operations Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences, 1993.
9.J. Glackin, J. G. Ecker, M. Kupferschmid, Solving bilevel linear programs using multiple objective linear programming, J. Optimiz. Theory App., 140 (2009), 197-212.
10.J. Han, G. Zhang, Y. Hu, et al. A solution to bi/tri-level programming problems using particle swarm optimization, Inform. Sciences, 370 (2016), 519-537.
11.R. Horst, N. V. Thoai, Y. Yamamoto, et al. On optimization over the efficient set in linear multicriteria programming, J. Optimiz. Theory App., 134 (2007), 433-443.
12.J. Jorge, A bilinear algorithm for optimizing a linear function over the efficient set of multiple objective linear programming problem, J. Global Optim., 31 (2005), 1-16.
13.J. Lu, C. Shi, G. Zhang, et al. Model and extended Kuhn-Tucker approach for bilevel multi-follower decision making in a referential-uncooperative situation, J. Global Optim., 38 (2007), 597-608.
14.W. Ma, M. Wang, X. Zhu, Improved particle swarm optimization based approach for bilevel programming problem-an application on supply chain model, Int. J. Mach. Learn. Cyb., 5 (2014), 281-292.
15.B. Metev, Multiobjective optimization methods help to minimize a function over the efficient set, Cybernetics and Information Technologies, 7 (2007), 22-28.
16.C. Miao, G. Du, R. J. Jiao, et al. Coordinated optimisation of platform-driven product line planning by bilevel programming, Int. J. Prod. Res., 55 (2017), 3808-3831.
17.C. O. Pieume, L. P. Fotso, P. Siarry, An approach for finding efficient points in multiobjective linear programming, Journal of Information and Optimization Sciences, 29 (2008), 203-216.
18.C. O. Pieume, L. P. Fotso, P. Siarry, Solving bilevel programming problems with multicriteria optimization techniques, Opsearch., 46 (2009), 169-183.
19.A. S. Safaei, S. Farsad, M. M. Paydar, Robust bi-level optimization of relief logistics operations, Appl. Math. Model., 56 (2018), 359-380.
20.S. Sayin, An algorithm based on facial decomposition for finding the efficient set in multiple objective linear programming, Oper. Res. Lett., 19 (1996), 87-94.
21.C. Shi, G. Zhang, J. Lu, A Kth-best Approach for Linear Bilevel Multi-follower Programming, J. Global Optim., 33 (2005), 563-578.
22.R. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986.
23.G. Zhang, J. Lu, Y. Gao, Multi-Level Decision Making: Models, Methods and Applications, Springer-Verlag Berlin AN, 2016.

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