Skip to main content
SpringerLink
Log in

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

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bard JF (1998) Practical bilevel optimization: algorithms and applications. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Budnitzkia A (2015) The solution approach to linear fuzzy bilevel optimization problems. Optimization 64(5):1195–1209

    Article  MathSciNet  Google 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):320–327

    Article  MATH  Google 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)

  • Chanas S, Kuchta D (1996) Multiobjective programming in optimization of interval objective functions-a generalized approach. Eur J Oper Res 94(3):594–598

    Article  MATH  Google Scholar 

  • Colson B, Marcotte P, Savard G (2005) A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput Optim Appl 30:211–227

    Article  MathSciNet  MATH  Google Scholar 

  • Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S (2002) Foundations of bilevel programming. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3):333–359

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Kalashnikov V, Pérez-Valdés GA, Kalashnykova N (2015) Bilevel programming problems. Springer, Berlin

    Book  MATH  Google Scholar 

  • Dempe S, Dutta J (2012) Is bilevel programming a special case of a mathematical program with complementarity constraints? Math Program 131(1–2):37–48

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Franke S (2013) Bilevel programming: stationarity and stability. Pac J Optim 9(2):183–199

    MathSciNet  MATH  Google Scholar 

  • Dempe S, Zemkoho AB (2013) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Program 138:447–473

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24:279–300

    Article  MathSciNet  MATH  Google 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):1–13

    Article  Google Scholar 

  • Grzegorzewski P (2002) Nearest interval approximation of a fuzzy number. Fuzzy Sets Syst 130(3):321–330

    Article  MathSciNet  MATH  Google Scholar 

  • Gümüs ZH, Floudas CA (2001) Global optimization of nonlinear bilevel programming problems. J Glob Optim 20(1):1–31

    Article  MathSciNet  MATH  Google Scholar 

  • Hamidi F, Mishmast N (2013) Bilevel linear programming with fuzzy parameters. Iran J Fuzzy Syst 10(4):83–99

    MathSciNet  MATH  Google Scholar 

  • Henrion R, Surowiec T (2011) On calmness conditions in convex bilevel programming. Appl Anal 90(6):951–970

    Article  MathSciNet  MATH  Google 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 55:9–14

    Article  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):4332–4339

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Labbé M, Violin A (2013) Bilevel programming and price setting problems. 4OR-Q J. Oper Res 11:1–30

    Article  MATH  Google Scholar 

  • Liu B (2006) A survey of credibility theory. Fuzzy Optim Decis Ma 5:387–408

    Article  MathSciNet  MATH  Google Scholar 

  • Lodi A, Ralphs TK, Woeginger GJ (2014) Bilevel programming and the separation problem. Math Program 146:437–458

    Article  MathSciNet  MATH  Google 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):808–813

    MathSciNet  MATH  Google Scholar 

  • Meng Z, Dang C, Shen R, Jiang M (2012) An objective penalty function of bilevel programming. J Optim Theory Appl 153:377–387

    Article  MathSciNet  MATH  Google Scholar 

  • Mersha A, Dempe S (2011) Direct search algorithm for bilevel programming problems. Comput Optim Appl 49:1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Moore RE (1979) Method and applications of interval Analysis. SIAM, Philadelphia

    Book  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Glob Optim 5(3):291–306

    Article  MathSciNet  MATH  Google Scholar 

  • Wan Z, Mao L, Wang G (2014) Estimation of distribution algorithm for a class of nonlinear bilevel programming problems. Inform Sci 256:184–196

    Article  MathSciNet  MATH  Google 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 35:221–232

    Article  Google Scholar 

  • Wu HC (2007) The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res 176:46–59

    Article  MathSciNet  MATH  Google 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):1–12

    MathSciNet  Google 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 49:430–448

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353

    Article  MATH  Google Scholar 

  • 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

    Google Scholar 

  • Zimmermann HJ (1996) Fuzzy set theory and its applications. Kluwer, Norwell

    Book  MATH  Google Scholar 

Download references

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

  1. Department of Mathematics, Baoji University of Arts and Sciences, Baoji, 721013, Shaanxi, China

    Aihong Ren

  2. School of Computer Science and Technology, Xidian University, Xi’an, 710071, Shaanxi, China

    Yuping Wang

Authors
  1. Aihong Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aihong Ren.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-016-2144-8

Keywords

Search

Navigation