Bi-Objective Bilevel Programming Problem: A Fuzzy Approach

Open access


In this paper, a likely situation of a set of decision maker’s with bi-objectives in case of fuzzy multi-choice goal programming is considered. The problem is then carefully formulated as a bi-objective bilevel programming problem (BOBPP) with multiple fuzzy aspiration goals, fuzzy cost coefficients and fuzzy decision variables. Using Ranking method the fuzzy bi-objective bilevel programming problem (FBOBPP) is converted into a crisp model. The transformed problem is further solved by adopting a two level Stackelberg game theory and fuzzy decision model of Sakawa. A numerical with hypothetical values is also used to illustrate the problem.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • T. Allahviranloo F. H. Lofti M. K. Kiasary N.A. Kiani and L. Alizadeh. Solving fully fuzzy linear programming problem by the ranking function. Applied Mathematical Sciences 2(1):19–32 2008.

  • G. Anandalingam. A mathematical programming model of decentralized multi-level systems. Journal of Operational Research Society 39:1021âĂŞ1033 1988.

  • J.F. Bard. A grid search algorithm for the linear bilevel programming problem. 14-th Annual Meeting of American Institude for Decision Science San Francisco CA.2:256–258 1982.

  • W. Bialas M.H. Karwan and J Shaw. A parametric complementary pivot approach for two-level linear programming. Technical Report State University of New York at Buffalo Operations Research 80-2 1980.

  • M. P. Biswal and S. Acharya. Transformation of a multi-choice linear programming problem. Applied Mathematics and Computation 210:182–188 2009.

  • M. P. Biswal and S. Acharya. Solving multi-choice linear programming problems by interpolating polynomials. Mathematical and Computer Modelling 54:1405–1412 2011.

  • C. T. Chang. Multi-choice goal programming. Omega 35(4):389–396 2007.

  • O.E. Emam. A fuzzy approach for bi-level integer non-linear programming problem. Applied Mathematics and Computation 172:62âĂŞ71 2006.

  • R. Ezzati E. Khorram and R. Enayati. A algorithm to solve fully fuzzy linear programming problems using the molp problem. Applied Mathematical Modelling In Press 2013.

  • K. Ganesan and P. Veeramani. Fuzzy linear programs with trapezoidal fuzzy numbers. Annals of Operations Research 143:305–315 2006.

  • W. C. Healy. Multiple choice programming. operations research. International journal of Mathematics and Applied Statistics 12:122–138 1964.

  • F. Hiller and G. Lieberman. Introduction to operations research. McGraw-Hill New York 1990.

  • Masahiro Inuiguchi and Jaroslav Ramık. Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy sets and systems 111(1):3–28 2000.

  • B. Liu. Stackelberg-nash equilibrium for multi-level programming with multi-follows using genetic algorithms. Computers math applications 377:79–89 1998.

  • J. Lu C. Shi and G. Zhang. An extended kth-best approach for linear bilevel programming. Applied Mathematics and Computation 164:843–855 2005.

  • MK Luhandjula. Multiple objective programming problems with possibilistic coefficients. Fuzzy Sets and Systems 21(2):135–145 1987.

  • R. Mathieu L. Pittard and G. Anandalingam. Genetic algorithm based approach to bilevel linear ptogramming. R. A. I. R. O Researche operationelle 28:1–21 1994.

  • Heinrich Rommelfanger. Fuzzy linear programming and applications. European journal of operational research 92(3):512–527 1996.

  • Heinrich Rommelfanger. The advantages of fuzzy optimization models in practical use. Fuzzy Optimization and Decision Making 3(4):295–309 2004.

  • M Sakawa. Interactive fuzzy goal programming for multiobjective nonlinear problems and its application to water quality management. Control and Cybernetics 13(2):217–228 1984.

  • M. Sakawa. Fuzzy sets and interactive multi-objective optimization. Plenum Press New York 1993a.

  • Masatoshi Sakawa. Fuzzy sets and interactive multiobjective optimization. Plenum New York 1993b.

  • Fumiko Seo and Masatoshi Sakawa. Multiple criteria decision analysis in regional planning: concepts methods and applications. Reidel Dordrecht The Netherlands 1988.

  • C. Shi J. Lu and G. Zhang. An extended kuhn-tucker approach for linear bilevel programming. Applied Mathematics and Computation 162:51–63 2005.

  • Roman Slowinski and Jacques Teghem. Stochastic vs. fuzzy approaches to multiobjective mathematical programming under uncertainty. Kluwer Academic Publishers 1990.

  • Behzad Bankian Tabrizi Kamran Shahanaghi and M. Saeed Jabalameli. Fuzzy multi-choice goal programming. Applied Mathematical Modelling 36:1415–1420 2012.

  • Jea Teghem Jr D Dufrane M Thauvoye and P Kunsch. Strange: an interactive method for multi-objective linear programming under uncertainty. European Journal of Operational Research 26(1):65–82 1986.

  • W.F. Bialas U.P. Wen. The hybrid algorithm for solving the three-level linear programming problem. Computers and Operations Research 13:367âĂŞ377 1986.

  • H-J Zimmermann. Fuzzy programming and linear programming with several objective functions. Fuzzy sets and systems 1(1):45–55 1978.

Journal information
Impact Factor
Mathematical Citation Quotient (MCQ) 2017: 0.06

Cited By
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 285 206 16
PDF Downloads 145 101 9