Multi-objective bilevel fuzzy probabilistic programming problem

On Various Approaches for Bi-level Optimization Problems

“…, , -, -≥ 0 (12) where -N , " = 1, … , ! + , are Lagrange multipliers, and note that complementary slackness (11) For " ∈ G r u , the variables -N or p N are assume any nonnegative value in the solution of model in Equation (8)(9)(10)(11)(12), without the complementary term 11 , so complementary slackness will not necessarily be satisfied. The proposed procedure is summarized in the following Bard and Moore in 1990 [11,17,27]:…”

“…Set -N = 0 for " ∈ G r S and p N = 0 for " ∈ G r t . Attempt to solve Equation (8)(9)(10)(11)(12), without (11) if the solution is infeasible, go to Step 6. Otherwise, put = + 1 and label the solution as r , r , -r , go to Step 3.…”

“…However, there have been attempts to model the ability of one planner to indirectly influence the decisions of others to his benefits. The BLOP, which we are interested in our work, are merely a special case of the multi-level decision making problems [6][7][8][9][10]. An important feature of hierarchy structures is that a planner at one level of the hierarchy may have his objective function and decision space determined, in a part, by the other level.…”

“…A modified TOPSIS method presented by Baky & Elsayed for BLOP with vague numbers [37]. Ranarahu et al presented method for solving a multiobjective BLOP [9]. Lachhwani in [38] tackled a solution for MLOP based on FGP approach.…”

“…The transformed problem, Equation(8)(9)(10)(11)(12), without the complementary slackness term,(11) is as follows: 2-+k +l −m = −1, ≥ 0, H ≥ 0, -N ≥ 0, " = 1,2, … ,5.and the complementary slackness product terms are: -− − H + 3 = 0 -−3 + 2H + 4 = 0 k −2 + H = 0 l 2 + H − 12 = 0 m H = 0At each iteration, a check is made to check the condition -N p N , H 0, the initial feasible solution for the transformed problem without the complementary slackness product terms using "LINGO programming" is , H 3,6 , -0,0.5,0,0,0 , with % =21. This point does not satisfy the condition -N p N , H 0, so branching variable is -, so G S 027, G t ∅, G u 01,3,4,57 and s 027.…”