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A cutting plane method for bilevel linear programming with interval coefficients

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  • Aihong Ren
  • Yuping Wang

Abstract

This article considers the bilevel linear programming problem with interval coefficients in both objective functions. We propose a cutting plane method to solve such a problem. In order to obtain the best and worst optimal solutions, two types of cutting plane methods are developed based on the fact that the best and worst optimal solutions of this kind of problem occur at extreme points of its constraint region. The main idea of the proposed methods is to solve a sequence of linear programming problems with cutting planes that are successively introduced until the best and worst optimal solutions are found. Finally, we extend the two algorithms proposed to compute the best and worst optimal solutions of the general bilevel linear programming problem with interval coefficients in the objective functions as well as in the constraints. Copyright Springer Science+Business Media New York 2014

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  • Aihong Ren & Yuping Wang, 2014. "A cutting plane method for bilevel linear programming with interval coefficients," Annals of Operations Research, Springer, vol. 223(1), pages 355-378, December.
    • Handle: RePEc:spr:annopr:v:223:y:2014:i:1:p:355-378:10.1007/s10479-014-1624-4
    DOI: 10.1007/s10479-014-1624-4
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    References listed on IDEAS

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    1. A. Bhurjee & G. Panda, 2012. "Efficient solution of interval optimization problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(3), pages 273-288, December.
    2. Yang, Hai & Zhang, Xiaoning & Meng, Qiang, 2007. "Stackelberg games and multiple equilibrium behaviors on networks," Transportation Research Part B: Methodological, Elsevier, vol. 41(8), pages 841-861, October.
    3. Omar Ben-Ayed & Charles E. Blair, 1990. "Computational Difficulties of Bilevel Linear Programming," Operations Research, INFORMS, vol. 38(3), pages 556-560, June.
    4. Clegg, Janet & Smith, Mike & Xiang, Yanling & Yarrow, Robert, 2001. "Bilevel programming applied to optimising urban transportation," Transportation Research Part B: Methodological, Elsevier, vol. 35(1), pages 41-70, January.
    5. Guangquan Zhang & Jie Lu, 2010. "Fuzzy bilevel programming with multiple objectives and cooperative multiple followers," Journal of Global Optimization, Springer, vol. 47(3), pages 403-419, July.
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    Cited by:

    1. Puchit Sariddichainunta & Masahiro Inuiguchi, 2017. "Global optimality test for maximin solution of bilevel linear programming with ambiguous lower-level objective function," Annals of Operations Research, Springer, vol. 256(2), pages 285-304, September.
    2. Masahiro Inuiguchi & Puchit Sariddichainunta, 2016. "Bilevel linear programming with ambiguous objective function of the follower," Fuzzy Optimization and Decision Making, Springer, vol. 15(4), pages 415-434, December.

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