Chance Constrained Linear Plus Linear Fractional Bi-level Programming Problem

Abstract

We present fuzzy goal programming approach to solve chance constrained linear plus linear fractional bi-level programming problem. The chance constraints with right hand parameters as random variables of prescribed probability distribution functions are transformed into equivalent deterministic system constraints. We construct nonlinear membership functions based on deterministic system constraints. The nonlinear membership functions are transformed into linear membership functions by using first order Taylor's series approximation. In the bi-level decision making context, decision deadlock may arise due to the dissatisfaction of the lower level decision maker with the decision of upper level decision maker. To overcome this problem, decision maker of each level gives his preference bounds on decision variables under his/her control to provide some relaxation on their decisions. Fuzzy goal programming model is used to achieve highest membership goals by minimizing negative deviational variables. Euclidean distance function is used in order to find out the most satisfactory solution. We solve a chance constrained linear plus linear fractional bi-level programming problem to illustrate the proposed approach. General TermsBi-level programming, linear plus linear fractional programming.

A Taylor Series based Fuzzy Mathematical Approach for Multi Objective Linear Fractional Programming Problem with Fuzzy Parameters

“…Banerjee and Pramanik [39] presented chance constrained multi-objective linear plus linear fractional programming problem based on Taylor's series approximation. Pramanik et al [40] formulated chance constrained linear plus linear fractional bi-level programming problem. Several studies have been made using Taylor's polynomial series approximation to deal quadratic programming problem [41,42], quadratic bi-level programming problem [43], quadratic bi-level multi-objective programming problem [44] and Chance constrained quadratic bi-level programming problem [45].…”

Chance Constrained Multi-Level Linear Programming Problem

“…Pramanik and Banerjee [20] investigated fuzzy goal programming approach to chance constrained quadratic bilevel programming problem by extending the concept of Pramanik and Dey [19]. Pramanik et al [21] also developed FGP models to solve chance constrained linear plus linear fractional bi-level programming problem. In the present paper, Pramanik and Banerjee's concept [20] has been extended to chance constrained multilevel linear programming problem (CCMLPP).…”

A Decomposition Algorithm for Solving a Three Level Large Scale linear Programming Problem

“…To obtain the optimal solution of the TLDM problem; the TLDM solves his master problem by the decomposition method [13] as the FLDM and SLDM. …”

“…At the first phase of the solution algorithm, it begin by finding the convex hull of its original set of constraints then simplifying the equivalent problem by transforming it into a separate multi-objective decision-making problem and finally solving the resulted problem by using the ε-constraint method. Pramanik and Banerjee presented an approach to deal with fuzzy goal programming approach to solve chance constrained quadratic bi-level programming problem [13]. The presented approach convert the chance constraints into equivalent deterministic constraints with prescribed distribution functions and confidence levels.…”

“…Usually, this kind of problems can be solved by using different mathematical programming techniques ( [8], [11]). Most studies in multilevel field are focused on bi-level problem ( [10,13,14,15,16,17]). In [10], Emam proposed an algorithm for solving bi-level integer multi-objective fractional programming problem.…”