Abstract
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.
Similar content being viewed by others
References
Bard JF (1998) Practical bilevel optimization: algorithms and applications. Kluwer, Dordrecht
Budnitzkia A (2015) The solution approach to linear fuzzy bilevel optimization problems. Optimization 64(5):1195–1209
Calvete HI, Galé C, Oliveros M (2011) Bilevel model for production distribution planning solved by using ant colony optimization. Comput Oper Res 38(1):320–327
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)
Chanas S, Kuchta D (1996) Multiobjective programming in optimization of interval objective functions-a generalized approach. Eur J Oper Res 94(3):594–598
Colson B, Marcotte P, Savard G (2005) A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput Optim Appl 30:211–227
Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256
Dempe S (2002) Foundations of bilevel programming. Kluwer, Dordrecht
Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3):333–359
Dempe S, Kalashnikov V, Pérez-Valdés GA, Kalashnykova N (2015) Bilevel programming problems. Springer, Berlin
Dempe S, Dutta J (2012) Is bilevel programming a special case of a mathematical program with complementarity constraints? Math Program 131(1–2):37–48
Dempe S, Franke S (2013) Bilevel programming: stationarity and stability. Pac J Optim 9(2):183–199
Dempe S, Zemkoho AB (2013) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Program 138:447–473
Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626
Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24:279–300
Gao Y, Zhang GQ, Ma J, Lu J (2010) A \(\lambda \)-cut and goal-programming-based algorithm for fuzzy-linear multiple-objective bilevel optimization. IEEE Trans Fuzzy Syst 18(1):1–13
Grzegorzewski P (2002) Nearest interval approximation of a fuzzy number. Fuzzy Sets Syst 130(3):321–330
Gümüs ZH, Floudas CA (2001) Global optimization of nonlinear bilevel programming problems. J Glob Optim 20(1):1–31
Hamidi F, Mishmast N (2013) Bilevel linear programming with fuzzy parameters. Iran J Fuzzy Syst 10(4):83–99
Henrion R, Surowiec T (2011) On calmness conditions in convex bilevel programming. Appl Anal 90(6):951–970
Jiang Y, Li X, Huang C, Wu X (2014) An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programming problems. Knowl Based Syst 55:9–14
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):4332–4339
Kalashnikov VV, Dempe S, Pérez-Valdés GA (2015) Bilevel programming and applications. Math Probl Eng:16 (Article ID 310301)
Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317
Labbé M, Violin A (2013) Bilevel programming and price setting problems. 4OR-Q J. Oper Res 11:1–30
Liu B (2006) A survey of credibility theory. Fuzzy Optim Decis Ma 5:387–408
Lodi A, Ralphs TK, Woeginger GJ (2014) Bilevel programming and the separation problem. Math Program 146:437–458
Lv Y, Hu T, Wang G, Wan Z (2007) A penalty function method based on Kuhn-Tucker condition for solving linear bilevel programming. Appl Math Comput 188(1):808–813
Meng Z, Dang C, Shen R, Jiang M (2012) An objective penalty function of bilevel programming. J Optim Theory Appl 153:377–387
Mersha A, Dempe S (2011) Direct search algorithm for bilevel programming problems. Comput Optim Appl 49:1–15
Moore RE (1979) Method and applications of interval Analysis. SIAM, Philadelphia
Ruziyeva A (2013) Fuzzy Bilevel optimization, Ph.D. thesis. Freiberg: Technical University Bergakademie
Ruziyeva A, Dempe S (2013) Yager ranking index in fuzzy bilevel optimization. Artif Intell Res 2:55–68
Sakawa M, Nishizaki I, Uemura Y (2000) Interactive fuzzy programming for multilevel linear programming problems with fuzzy parameters. Fuzzy Set Syst 109(1):3–19
Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Glob Optim 5(3):291–306
Wan Z, Mao L, Wang G (2014) Estimation of distribution algorithm for a class of nonlinear bilevel programming problems. Inform Sci 256:184–196
Wang YP, Jiao YC, Li H (2005) An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handling scheme. IEEE Trans Syst Man Cybern 35:221–232
Wu HC (2007) The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res 176:46–59
Xu J, Wei P (2012) A bilevel model for location-allocation problem of construction and demolition waste management under fuzzy random environment. Int J of Civ Eng 10(1):1–12
Yu PL, Zeleny M (1975) The set of all non-dominated solutions in linear cases and a multicriteria simplex method. J Math Anal Appl 49:430–448
Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353
Zhang GQ, Lu J, Dillon T (2007) Fuzzy linear bilevel optimization: solution concepts, approaches and applications. Fuzzy Logic 215:351–379
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 53:13–19
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):1–7
Zimmermann HJ (1996) Fuzzy set theory and its applications. Kluwer, Norwell
Acknowledgments
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).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Additional information
Communicated by V. Loia.
Rights and permissions
About this article
Cite this article
Ren, A., Wang, Y. An approach for solving a fuzzy bilevel programming problem through nearest interval approximation approach and KKT optimality conditions. Soft Comput 21, 5515–5526 (2017). https://doi.org/10.1007/s00500-016-2144-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-016-2144-8