An approach for solving a fuzzy bilevel programming problem through nearest interval approximation approach and KKT optimality conditions
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In this paper, we consider a kind of bilevel linear programming problem where the coefficients of both objective functions are fuzzy numbers. In order to deal with such a problem, the original problem can be approximated by an interval bilevel programming problem in terms of the nearest interval approximation of a fuzzy number. Based on the Karush–Kuhn–Tucker (KKT) optimality conditions for the optimization problem with an interval-valued objective function, the interval bilevel programming problem can be converted into a single-level programming problem with an interval-value objective function. To minimize the interval objective function, the order relations of interval numbers are used to transform the uncertain single-objective optimization into a multi-objective optimization solved by global criteria method (GCM). Finally, illustrative numerical examples are provided to demonstrate the feasibility of the proposed approach.
KeywordsBilevel programming problem Fuzzy number Interval programming Nearest interval approximation KKT conditions
This work was supported by National Natural Science Foundation of China (No. 61272119), Science Foundation of the Education Department of Shaanxi Province of China (No. 15JK1036) and Science Foundation of Baoji University of Arts and Sciences (No. ZK16049).
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Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
- Camacho-Vallejo JF, Cordero-Franco AE, González-Ramírez RG (2014) Solving the bilevel facility location problem under preferences by a Stackelberg-evolutionary algorithm. Math Probl Eng:14 (article ID 430243)Google Scholar
- Jiang Y, Li XY (2013) Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bilevel programming problem. Appl Math Comput 219(9):4332–4339Google Scholar
- Kalashnikov VV, Dempe S, Pérez-Valdés GA (2015) Bilevel programming and applications. Math Probl Eng:16 (Article ID 310301)Google Scholar
- Ruziyeva A (2013) Fuzzy Bilevel optimization, Ph.D. thesis. Freiberg: Technical University BergakademieGoogle Scholar
- Ruziyeva A, Dempe S (2013) Yager ranking index in fuzzy bilevel optimization. Artif Intell Res 2:55–68Google Scholar
- Zhang GQ, Lu J, Dillon T (2007) Fuzzy linear bilevel optimization: solution concepts, approaches and applications. Fuzzy Logic 215:351–379Google Scholar
- Zhang T, Hu T, Guo X, Chen Z, Zheng Y (2013) Solving high dimensional bilevel multiobjective programming problem using a hybrid particle swarm optimization algorithm with crossover operator. Knowl Based Syst 53:13–19Google Scholar
- Zhang GQ, Lu J (2005) The definition of optimal solution and an extended Kuhn-Tucker approach for fuzzy linear bilevel programming. IEEE Intell Informatics Bull 6(2):1–7Google Scholar