A new methodological development for solving linear bilevel integer programming problems in hybrid fuzzy environment

Abstract

This paper deals with fuzzy goal programming approach to solve fuzzy linear bilevel integer programming problems with fuzzy probabilistic constraints following Pareto distribution and Frechet distribution. In the proposed approach a new chance constrained programming methodology is developed from the view point of managing those probabilistic constraints in a hybrid fuzzy environment. A method of defuzzification of fuzzy numbers using a−cut has been adopted to reduce the problem into a linear bilevel integer programming problem. The individual optimal value of the objective of each DMis found in isolation to construct the fuzzy membership goals. Finally, fuzzy goal programming approach is used to achieve maximum degree of each of the membership goals by minimizing under deviational variables in the decision making environment. To demonstrate the efficiency of the proposed approach, a numerical example is provided.

Keywords

• Bilevel programming,fuzzy goal programming,ranking function,stochastic programming

References

1. W. Candler, R. Townsley, A linear two-level programming problem, Computers and Operations Research, 9(1982), 59 – 76.
2. W.F. Bialas, M.H. Karwan, Two-level linear programming, Management and Science, 30(1984), 1004 – 1020.
3. A. Migdalas, Bilevel programming in traffic planning: Models, methods and challenge, Journal of Global Optimization, 7(1995), 381–405.
4. J.P. Cote, P. Marcotte, G. Savard, A bilevel modeling approach to pricing and fare optimization in the airline industry, Journal of Revenue and Pricing Management, 2(2003), 23–36.
5. B.Y. Kara, V. Verter, Designing a road network for hazardous materials transportation, Transportation Science, 38(2004), 188–196.
6. M.G. Nicholls, The applications of non-linear bi-level programming to the aluminum industry, Journal of Global Optimization, 8(1996), 245–261.
7. M.A. Amouzegar, K. Moshirvaziri, Determining optimal pollution control policies: an application of bilevel programming, European Journal of Operational Research, 119(1999), 100–120.
8. S. Dempe, J.F. Bard, Bundle trust-region algorithm for bilinear bilevel programming, Journal of Optimization Theory and Applications, 110(2001), 265–288.

9. M. Fampa, L.A. Barroso, D. Candal, L. Simonetti, Bilevel optimization applied to strategic pricing in competitive electricity markets, Computational Optimization and Applications, 39(2008), 121–142.
10. E. Roghanian, S.J. Sadjadi, M.B. Aryanezhad, A probabilistic bilevel linear multi-objective programming problem to supply chain planning, Applied Mathematics and Computation,188(2007), 786–800.
11. T. Uno, H. Katagiri, Single- and multi-objective defensive location problems on a network, European Journal of Operational Research, 188(2008), 76–84.
12. T. Uno, K. Katagiri, K. Kato, A multi-dimensionalization of competitive facility location problems, International Journal of Innovative Computing, Information and Control, 7(2011), 2593–2601.
13. G. B. Dantzig, Linear programming under uncertainty, Management Science,1(1955), 197-206.
14. A. Charnes, W.W. Cooper, Chance-constrained programming, Management Science, 6(1959), 73-79.
15. S. Kataoka, A stochastic programming model, Econometrica, 31(1963), 181–196.
16. A.M. Geoffrion, Stochastic programming with aspiration or fractile criterion, Management Science, 13(1967), 672–679.
17. L.A. Zadeh, Fuzzy sets, Information and Control,8(1965), 338–353.
18. H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1(1978), 45 – 55. M. Sakawa, H. Katagiri, T. Matsui, Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility, Fuzzy Sets and Systems,188(2012), 45–57.
19. H. Shih, E. S. Lee, Compensatory fuzzy multiple level decision making, Fuzzy Sets and Systems, 114(1) (2000), 71–87.
20. G. Zhang, J. Lu, Y. Gao, Fuzzy bilevel programming: multiobjective and multi-follower with shared variables, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,16 (2008),105–133.
21. B. B. Pal, B. N. Moitra, A fuzzy goal programming procedure for solving quadratic bilevel programming problems, International Journal of Intelligence Systems, 18(5) (2003),529-540.
22. M. A. Abo-Sinha, A bi-level non-linear multi-objective decision making under fuzziness, Operation Research Society of India (OPSEARCH), 38 (2001), 484–495.
23. M. S.Osman, M. A.Abo-Sinna, A. H.Amer,A multi-level non-linear multi-objective decision-making under fuzziness, Applied Mathematics and Computation, 153(1)(2004), 239-252.
24. I. A. Baky, Fuzzy goal programming algorithm for solving decentralized bi-level multi-objective programming problems, Fuzzy Sets Systems, 160(2009), 2701–2713.
25. S. R. Arora, R. Gupta, Interactive fuzzy goal programming approach for bilevel programming problem, European Journal of Operational Research, 194(2) (2009), 368-376.
26. S. Deng, L. Zhou, X. Wang, Solving the fuzzy bilevel linear programming with multiple followers through structured element method, Mathematical Problems in Engineering, 2014 (2014), 1-6.
27. N. Modak, A. Biswas, A Fuzzy Programming Approach for Bilevel Stochastic Programming, Advances in Intelligence Systems and Soft Computing, 236 (2014), 125 – 136.
28. M. Delgado, M.A. Vila,W. Voxman, On a canonical representation of fuzzy number, Fuzzy Sets and Systems, 93(1998), 125-135.