BILEVEL LINEAR PROGRAMMING WITH FUZZY PARAMETERS

Bilevel linear programming is a decision making problem with a two-level decentralized organization. The "leader" is in the upper level and the "follower", in the lower. Making a decision at one level affects that at the other one. In this paper, bilevel linear programming with inexact parameters has been studied and a method is proposed to solve a fuzzy bilevel linear programming using interval bilevel linear programming. Bilevel linear programming (BLP), first introduced by Stackelberg (22), is a nested optimization problem including a leader and a follower problem. Making a decision at one level, affects the objective function and the decision space of the other and the order is from top to bottom. "Kth-best" (6), "Branch and bound" (4) and "Complementary pivot" (13) are among the most important methods presented for the solution of this problem. In real cases, the parameters of an optimization problem may not be exact, i.e. they may be interval or fuzzy; same is the case with BLP problems. It is worth mentioning that when the parameters of a Fuzzy Bilevel Linear Programming (FBLP) problem are fuzzy quantities, the objective functions are fuzzy quantities too. Recently, Calvete et al. (9) have presented two algorithms, similar to the Kth-best, for the calculation of the best and the worst optimal values of the Interval Bilevel Linear Programming (IBLP) objective function. They are based on the ordering of the extreme points. In the IBLP problem, investigated by the above researchers, only the coefficients of the leader and the follower objective functions are interval; the other coefficients are ordinary. These algorithms can be extended to the solution of the IBLP problems wherein all parameters are interval. In this paper, using " -cut", an IBLP is obtained from a fuzzy one. The best and the worst optimal values of the objective function will be found first, and then a linear piecewise trapezoidal approximate fuzzy number will be presented for the optimal value of the leader objective function related to the FBLP problem.