# An approach for solving a fuzzy bilevel programming problem through nearest interval approximation approach and KKT optimality conditions

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## 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.

### Keywords

Bilevel programming problem Fuzzy number Interval programming Nearest interval approximation KKT conditions## Notes

### 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).

### Compliance with ethical standards

### Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

### References

- Bard JF (1998) Practical bilevel optimization: algorithms and applications. Kluwer, DordrechtCrossRefMATHGoogle Scholar
- Budnitzkia A (2015) The solution approach to linear fuzzy bilevel optimization problems. Optimization 64(5):1195–1209MathSciNetCrossRefGoogle Scholar
- Calvete HI, Galé C, Oliveros M (2011) Bilevel model for production distribution planning solved by using ant colony optimization. Comput Oper Res 38(1):320–327CrossRefMATHGoogle Scholar
- 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
- Chanas S, Kuchta D (1996) Multiobjective programming in optimization of interval objective functions-a generalized approach. Eur J Oper Res 94(3):594–598CrossRefMATHGoogle Scholar
- Colson B, Marcotte P, Savard G (2005) A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput Optim Appl 30:211–227MathSciNetCrossRefMATHGoogle Scholar
- Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256MathSciNetCrossRefMATHGoogle Scholar
- Dempe S (2002) Foundations of bilevel programming. Kluwer, DordrechtMATHGoogle Scholar
- Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3):333–359MathSciNetCrossRefMATHGoogle Scholar
- Dempe S, Kalashnikov V, Pérez-Valdés GA, Kalashnykova N (2015) Bilevel programming problems. Springer, BerlinCrossRefMATHGoogle Scholar
- Dempe S, Dutta J (2012) Is bilevel programming a special case of a mathematical program with complementarity constraints? Math Program 131(1–2):37–48MathSciNetCrossRefMATHGoogle Scholar
- Dempe S, Franke S (2013) Bilevel programming: stationarity and stability. Pac J Optim 9(2):183–199MathSciNetMATHGoogle Scholar
- Dempe S, Zemkoho AB (2013) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Program 138:447–473MathSciNetCrossRefMATHGoogle Scholar
- Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626MathSciNetCrossRefMATHGoogle Scholar
- Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24:279–300MathSciNetCrossRefMATHGoogle Scholar
- 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):1–13CrossRefGoogle Scholar
- Grzegorzewski P (2002) Nearest interval approximation of a fuzzy number. Fuzzy Sets Syst 130(3):321–330MathSciNetCrossRefMATHGoogle Scholar
- Gümüs ZH, Floudas CA (2001) Global optimization of nonlinear bilevel programming problems. J Glob Optim 20(1):1–31MathSciNetCrossRefMATHGoogle Scholar
- Hamidi F, Mishmast N (2013) Bilevel linear programming with fuzzy parameters. Iran J Fuzzy Syst 10(4):83–99MathSciNetMATHGoogle Scholar
- Henrion R, Surowiec T (2011) On calmness conditions in convex bilevel programming. Appl Anal 90(6):951–970MathSciNetCrossRefMATHGoogle Scholar
- 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 55:9–14CrossRefGoogle 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
- Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317MathSciNetCrossRefMATHGoogle Scholar
- Labbé M, Violin A (2013) Bilevel programming and price setting problems. 4OR-Q J. Oper Res 11:1–30CrossRefMATHGoogle Scholar
- Liu B (2006) A survey of credibility theory. Fuzzy Optim Decis Ma 5:387–408MathSciNetCrossRefMATHGoogle Scholar
- Lodi A, Ralphs TK, Woeginger GJ (2014) Bilevel programming and the separation problem. Math Program 146:437–458MathSciNetCrossRefMATHGoogle Scholar
- 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):808–813MathSciNetMATHGoogle Scholar
- Meng Z, Dang C, Shen R, Jiang M (2012) An objective penalty function of bilevel programming. J Optim Theory Appl 153:377–387MathSciNetCrossRefMATHGoogle Scholar
- Mersha A, Dempe S (2011) Direct search algorithm for bilevel programming problems. Comput Optim Appl 49:1–15MathSciNetCrossRefMATHGoogle Scholar
- Moore RE (1979) Method and applications of interval Analysis. SIAM, PhiladelphiaCrossRefGoogle 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
- Sakawa M, Nishizaki I, Uemura Y (2000) Interactive fuzzy programming for multilevel linear programming problems with fuzzy parameters. Fuzzy Set Syst 109(1):3–19CrossRefMATHGoogle Scholar
- Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Glob Optim 5(3):291–306MathSciNetCrossRefMATHGoogle Scholar
- Wan Z, Mao L, Wang G (2014) Estimation of distribution algorithm for a class of nonlinear bilevel programming problems. Inform Sci 256:184–196MathSciNetCrossRefMATHGoogle Scholar
- 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 35:221–232CrossRefGoogle Scholar
- Wu HC (2007) The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res 176:46–59MathSciNetCrossRefMATHGoogle Scholar
- 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):1–12MathSciNetGoogle Scholar
- Yu PL, Zeleny M (1975) The set of all non-dominated solutions in linear cases and a multicriteria simplex method. J Math Anal Appl 49:430–448MathSciNetCrossRefMATHGoogle Scholar
- Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353CrossRefMATHGoogle 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
- Zimmermann HJ (1996) Fuzzy set theory and its applications. Kluwer, NorwellCrossRefMATHGoogle Scholar