Optimistic Stackelberg solutions to bilevel linear programming with fuzzy random variable coefficients

In this paper, we consider a kind of bilevel linear programming problem where the coefficients of both objective functions are fuzzy random variables. The purpose of this paper is to develop a computational method for obtaining optimistic Stackelberg solutions to such a problem. Based on @a-level sets of fuzzy random variables, we first transform the fuzzy random bilevel programming problem into an @a-stochastic interval bilevel linear programming problem. To minimize the interval objective functions, the order relations which represent the decision maker's preference are defined by the right limit and the center of random interval simultaneously. Using the order relations and expectation optimization, the @a-stochastic interval bilevel linear programming problem can be converted into a deterministic multiobjective bilevel linear programming problem. According to optimistic anticipation from the upper level decision maker, the optimistic Stackelberg solution is introduced and a computational method is also presented. Finally, several numerical examples are provided to demonstrate the feasibility of the proposed approach.

[1]  Wayne F. Bialas,et al.  Two-Level Linear Programming , 1984 .

[2]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[3]  Masatoshi Sakawa,et al.  Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility , 2012, Fuzzy Sets Syst..

[4]  Masatoshi Sakawa,et al.  Stackelberg solutions for fuzzy random bilevel linear programming through level sets and probability maximization , 2012, Oper. Res..

[5]  Jie Lu,et al.  The Definition of Optimal Solution and an Extended Kuhn-Tucker Approach for Fuzzy Linear Bilevel Programming , 2005, IEEE Intell. Informatics Bull..

[6]  R. Kruse,et al.  Statistics with vague data , 1987 .

[7]  S. Chanas,et al.  Multiobjective programming in optimization of interval objective functions -- A generalized approach , 1996 .

[8]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[9]  Masatoshi Sakawa,et al.  Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization , 2012, Central Eur. J. Oper. Res..

[10]  Ichiro Nishizaki,et al.  Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters , 2000, Fuzzy Sets Syst..

[11]  Jie Lu,et al.  Fuzzy bilevel programming with multiple objectives and cooperative multiple followers , 2010, J. Glob. Optim..

[12]  Guangquan Zhang,et al.  A ${\bm \lambda}$-Cut and Goal-Programming-Based Algorithm for Fuzzy-Linear Multiple-Objective Bilevel Optimization , 2010, IEEE Transactions on Fuzzy Systems.

[13]  Guangquan Zhang,et al.  Fuzzy linear bilevel optimization: Solution concepts, approaches and applications , 2007 .

[14]  S. Dempe.  Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints , 2003 .

[15]  Andrzej P. Wierzbicki,et al.  The Use of Reference Objectives in Multiobjective Optimization , 1979 .

[16]  Abdel Lisser,et al.  On a stochastic bilevel programming problem , 2012, Networks.

[17]  Ichiro Nishizaki,et al.  Interactive fuzzy programming for multilevel linear programming problems with fuzzy parameters , 2000 .

[18]  Nguyen Van Hop,et al.  Fuzzy stochastic goal programming problems , 2007, Eur. J. Oper. Res..

[19]  Huibert Kwakernaak,et al.  Fuzzy random variables--II. Algorithms and examples for the discrete case , 1979, Inf. Sci..

[20]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[21]  H. P. Benson,et al.  Existence of efficient solutions for vector maximization problems , 1978 .

[22]  Masatoshi Sakawa,et al.  Interactive fuzzy programming for fuzzy random two-level linear programming problems through probability maximization with possibility , 2013, Expert Syst. Appl..

[23]  Gabriele Eichfelder,et al.  Multiobjective bilevel optimization , 2010, Math. Program..

[24]  Inés Couso,et al.  Random intervals as a model for imprecise information , 2005, Fuzzy Sets Syst..

[25]  Masatoshi Sakawa,et al.  Interactive fuzzy random two-level linear programming based on level sets and fractile criterion optimization , 2013, Inf. Sci..

[26]  Kalyanmoy Deb,et al.  Solving Bilevel Multi-Objective Optimization Problems Using Evolutionary Algorithms , 2009, EMO.

[27]  Masatoshi Sakawa,et al.  INTERACTIVE FUZZY PROGRAMMING BASED ON FRACTILE CRITERION OPTIMIZATION MODEL FOR TWO-LEVEL STOCHASTIC LINEAR PROGRAMMING PROBLEMS , 2010, Cybern. Syst..

[28]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[29]  Huibert Kwakernaak,et al.  Fuzzy random variables - I. definitions and theorems , 1978, Inf. Sci..

[30]  Wang Guangyuan,et al.  Linear programming with fuzzy random variable coefficients , 1993 .

[31]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications) , 2006 .

[32]  Guangquan Zhang,et al.  Model and approach of fuzzy bilevel decision making for logistics planning problem , 2007, J. Enterp. Inf. Manag..

[33]  M. K. Luhandjula.  Fuzziness and randomness in an optimization framework , 1996, Fuzzy Sets Syst..

[34]  Zhongping Wan,et al.  A fuzzy interactive method for a class of bilevel multiobjective programming problem , 2011, Expert Syst. Appl..

[35]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[36]  Baoding Liu,et al.  Fuzzy random chance-constrained programming , 2001, IEEE Trans. Fuzzy Syst..

[37]  M. Sakawa,et al.  Stackelberg Solutions to Multiobjective Two-Level Linear Programming Problems , 1999 .

[38]  Masatoshi Sakawa,et al.  Fuzzy random bilevel linear programming through expectation optimization using possibility and necessity , 2012, Int. J. Mach. Learn. Cybern..

[39]  Herminia I. Calvete,et al.  On linear bilevel problems with multiple objectives at the lower level , 2011 .

[40]  Masatoshi Sakawa,et al.  Interactive fuzzy random cooperative two-level linear programming through level sets based probability maximization , 2013, Expert Syst. Appl..

[41]  Heinrich J. Rommelfanger,et al.  A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values , 2007, Fuzzy Sets Syst..