A Lower-side Attainment Degree Approach for Bilevel Optimization under Uncertainty

In this study, a new solution approach based on the lower-side attainment degree is developed for bilevel linear programming problems with fuzzy coefficients in both objective functions and constraint functions. In order to handle fuzzy uncertainties, we adopt the lower-side attainment degree to defuzzify fuzzy terms, and convert the fuzzy bilevel programming problem into the equivalent deterministic bilevel one. Compared with some traditional defuzzifying techniques, this kind of transformation does not produce complicated intermediate models and complex computation process, and provides a simple deterministic bilevel linear model. The resulting bilevel linear model is coped with by the extended Kth-best approach. Furthermore, we extend the developed approach to deal with the fuzzy random bilevel programming problem with the aid of expectation. Finally, we provide several numerical examples to demonstrate the feasibility and efficiency of the proposed method.

[1]  Stephan Dempe,et al.  The bilevel programming problem: reformulations, constraint qualifications and optimality conditions , 2013, Math. Program..

[2]  Tapabrata Ray,et al.  An Enhanced Memetic Algorithm for Single-Objective Bilevel Optimization Problems , 2016, Evolutionary Computation.

[3]  Yuping Wang,et al.  An approach based on reliability-based possibility degree of interval for solving general interval bilevel linear programming problem , 2019, Soft Comput..

[4]  M. Sakawa,et al.  Bilevel linear programming with fuzzy random variables through absolute deviation minimisation , 2015 .

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

[6]  Amelia Bilbao-Terol,et al.  Linear programming with fuzzy parameters: An interactive method resolution , 2007, Eur. J. Oper. Res..

[7]  Xingsi Xue,et al.  An interval programming approach for the bilevel linear programming problem under fuzzy random environments , 2014, Soft Comput..

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

[9]  Masatoshi Sakawa,et al.  Fuzzy random non-cooperative two-level linear programming through fractile models with possibility and necessity , 2013 .

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

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

[12]  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..

[13]  Kalyanmoy Deb,et al.  A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications , 2017, IEEE Transactions on Evolutionary Computation.

[14]  Yuping Wang,et al.  Optimistic Stackelberg solutions to bilevel linear programming with fuzzy random variable coefficients , 2014, Knowl. Based Syst..

[15]  Nguyen Van Hop.  Solving linear programming problems under fuzziness and randomness environment using attainment values , 2007, Inf. Sci..

[16]  Nataliya I. Kalashnykova,et al.  Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks , 2015 .

[17]  Jie Lu,et al.  Fuzzy Bi-level Decision-Making Techniques: A Survey , 2016, Int. J. Comput. Intell. Syst..

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

[19]  Kalyanmoy Deb,et al.  Transportation policy formulation as a multi-objective bilevel optimization problem , 2015, 2015 IEEE Congress on Evolutionary Computation (CEC).

[20]  Guangmin Wang,et al.  A bilevel improved fruit fly optimization algorithm for the nonlinear bilevel programming problem , 2017, Knowl. Based Syst..

[21]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[22]  Debjani Chakraborty,et al.  Solving bi-level programming problem with fuzzy random variable coefficients , 2017, J. Intell. Fuzzy Syst..

[23]  Xinwang Liu,et al.  Measuring the satisfaction of constraints in fuzzy linear programming , 2001, Fuzzy Sets Syst..

[24]  Hideki Katagiri,et al.  Possibilistic Stackelberg solutions to bilevel linear programming problems with fuzzy parameters , 2013, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS).

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

[26]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[27]  Nataliya I. Kalashnykova,et al.  Bilevel Programming and Applications , 2015 .

[28]  H. Zimmermann,et al.  Fuzzy Set Theory and Its Applications , 1993 .

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

[30]  Jie Lu,et al.  An extended Kth-best approach for linear bilevel programming , 2005, Appl. Math. Comput..

[31]  H. Farhad,et al.  BILEVEL LINEAR PROGRAMMING WITH FUZZY PARAMETERS , 2013 .

[32]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[33]  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..