Interactive Fuzzy Decision Making Algorithm for Two Level Linear Fractional Programming Problems

Abstract

This paper offers an interactive fuzzy decision-making algorithm for solving two-level linear fractional programming (TLLFP) problem which contains a single decision maker at the upper level and multiple decision makers at the lower level. In the presented interactive mechanism, the fuzzy goals and associated weight of the objective at all levels are first determined and the satisfactory solution is attained by renewing the satisfactory degrees of decision makers including the overall satisfactory balance among all levels. Moreover, the value of distance function is used in order to verify the satisfaction grades. Finally, a numerical example is given to illustrate the performance of the presented algorithm.

Keywords

• Two level linear fractional programming problem,Fuzzy goals,Fuzzy programming,Fuzzy goals,Interactive methods

References

1. Bracken J & McGill JM. (1973). Mathematical programs with optimization problems in the constraints. Operational Research, 21, 37–44.
2. Bracken J & McGill JM. (1974). Defense applications of mathematical programs with optimization problem in the constraints. Operations Research, 22,1086–1096.
3. Aiyoshi E & Shimizu K (1981). Hierarchical decentralized systems and its new solution by a barrier method. IEEE Transactions on Systems, Man, and Cybernetics,11(6), 444–449.
4. Anandalingam G & Apprey V (1991). Multi-level programming and conflict resolution. European Journal of Operational Research, 51, 233–247.
5. Ben-Ayed O & Blair CE (1990). Computational difficulties of bilevel linear programming. Operations Research, 38, 556–560.
6. Candler, W & Towersley R (1982). A linear two-level programming problem. Computers and Operations Research, 9, 59–76.
7. Bialas WF & Karwan MH (1984). Two-level linear programming. Management Science, 30, 1004–1020.
8. Bard J& Falk J (1982). An explicit solution to the multi-level programming problem. Computers and Operations Research, 9, 77–100.

9. Savard G & Gauvin J (1994). The steepest descent direction for the nonlinear bilevel programming problem. Operations Research Letters, 15, 275–282.
10. Liu B (1998). Stackelberg–Nash equilibrium for multilevel programming with multiple followers using genetic algorithms. Computers & Mathematics with Applications, 36(7), 79–89.
11. Gao J & Liu B (2005). Fuzzy multilevel programming with a hybrid intelligent algorithm. Computers & Mathematics with Applications, 49(9-10), 1539–1548
12. Lai YJ (1996). Hierarchical optimization: a satisfactory solution. Fuzzy Sets and Systems, 77(3), 321-335.
13. Ahlatcioglu M & Tiryaki F (2007). Interactive fuzzy programming for decentralized two-level linear fractional programming (DTLLFP) problems. Omega, 35(4), 432-450.
14. Shih HS, Lai YJ & Lee ES (1996). Fuzzy approach for multi-level programming problems. Computer & Operations Research, 23, 73–91.
15. Shih HS & Lee ES (2000). Compensatory fuzzy multiple level decision making. Fuzzy Sets and Systems, 114(1), 71-87.
16. Sakawa M, Nishizaki I & Uemura Y (2000). Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters. Fuzzy Sets and Systems, 115(1), 93-103.
17. Sakawa M & Nishizaki I (2001). Interactive fuzzy programming for two-level linear fractional programming problems. Fuzzy Sets and Systems, 119(1), 31-40.
18. Sakawa M & Nishizaki I (2002). Interactive fuzzy programming for two-level nonconvex programming problems with fuzzy parameters through genetic algorithms. Fuzzy Sets and Systems, 127(2), 185-197.
19. Sakawa M, & Nishizaki I (2002). Interactive fuzzy programming for decentralized two-level linear programming problems. Fuzzy Sets and Systems, 125(3), 301-315.
20. Tiryaki F (2006). Interactive compensatory fuzzy programming for decentralized multi-level linear programming (DMLLP) problems. Fuzzy sets and systems, 157(23), 3072-3090.
21. Baky IA (2010). Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Applied Mathematical Modelling, 34(9), 2377-2387.
22. Arora SR & Gupta R (2009). Interactive fuzzy goal programming approach for bilevel programming problem. European Journal of Operational Research, 194(2), 368-376.
23. Wang G, Wang X & Wan Z (2009). A fuzzy interactive decision making algorithm for bilevel multi-followers programming with partial shared variables among followers. Expert Systems with Applications, 36(7), 10471-10474.
24. Yu PL (1973) A class of solutions for group decision problems. Management Science, 19(8), 936-946.
25. Pal BB & Moitra BN (2003). A fuzzy goal programming procedure for solving quadratic bilevel programming problems. International Journal of Intelligent Systems, 18(5), 529-540.
26. Baky IA & Abo-Sinna MA (2013). TOPSIS for bi-level MODM problems. Applied Mathematical Modelling, 37(3), 1004-1015.
27. Sakawa M & Nishizaki I (2012). Interactive fuzzy programming for multi-level programming problems: a review. International Journal of Multicriteria Decision Making, 2(3), 241-266.
28. Zheng Y, Liu J & Wan Z (2014). Interactive fuzzy decision making method for solving bilevel programming problem. Applied Mathematical Modelling, 38(13), 3136-3141.
29. Sakawa, M., & Matsui, T. (2013). Interactive fuzzy random two-level linear programming based on level sets and fractile criterion optimization. Information Sciences, 238, 163-175.
30. Emam, O. E. (2013). Interactive approach to bi-level integer multi-objective fractional programming problem. Applied Mathematics and Computation, 223, 17-24.
31. Toksarı, M. D., & Bilim, Y. (2015). Interactive Fuzzy Goal Programming Based on Jacobian Matrix to Solve Decentralized Bi-level Multi-objective Fractional Programming Problems. International Journal of Fuzzy Systems, 1-10.
32. Singh, S., & Haldar, N. (2015). A New Method To Solve Bi-Level Quadratic Linear Fractional Programming Problems. International Game Theory Review, 17(02), 1540017.
33. Kassem, M. A. E. H. (1995). Interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints. Fuzzy Sets and Systems, 73(2), 235-243.
34. Sakawa, M., & Nishizaki, I. (2009). Cooperative and noncooperative multi-level programming (Vol. 48). Springer Science & Business Media.
35. Pramanik, S., & Roy, T. K. (2007). Fuzzy goal programming approach to multilevel programming problems. European Journal of Operational Research, 176(2), 1151-1166.
36. Biswas A & Pal BB (2005). Application of fuzzy goal programming technique to land use planning in agricultural system. Omega, 33(5), 391-398.