MODIFIED SIMPLEX METHOD FOR SOLVING LINEAR FRACTIONAL/QUADRATIC BI-LEVEL PROGRAMMING PROBLEM

Abstract

The main objective of this study was to solve a Linear Fractional/Quadratic Bi-level Programming Problem in which the objective function of the first level (leader) is linear fractional and the lower level (follower) is convex quadratic. The variables associated with both level problems are related by linear constraints. Since the objective of the upper and lower level decision makers are potentially conflicting in nature, a possible transformation of an LFQBLPP into equivalent single level LFPP with complementary constraints are considered by providing KKT necessary and sufficient condition at the lower level. Lagrangian functions have been used to transform the constrained follower problem into unconstrained problem; thereafter modified simplex algorithm was applied to find the solution of an LFPP which satisfies the complementary constraints and hence the optimum solution of an LFPP with complementary constraints determines the optimum solution of the given LFQBLPP. We have stated that for the coefficient of all constraints are negative with the right hand side positive the initial point is infeasible so we can solve that type problem by adding an artificial variable. For LFPP with complementary constraint, optimality conditions were derived based on KKT conditions. The problem was solved analytically and numerical examples were solved to observe the efficiency of the employed method. The proposed method achieves efficient and feasible solution and it is evaluated by comparing with the references. Finally, this study was suggested a further study for searching an algorithm such as genetic algorithm for solving an LFQBLPP and showing the efficiency of the proposed algorithm for solving other kinds of BLPP such as quadratic bi-level programming problem and so on.