Stability of Solutions for Parametric Inverse Nonlinear Cost Transportation Problem

Abstract

This paper investigates the solution for an inverse of a parametric nonlinear transportation problem, in which, for a certain values of the parameters, the cost of the unit transportation in the basic problem are adapted as little as possible so that the specific feasible alternative become an optimal solution. In addition, a solution stability set of these parameters was investigated to keep the new optimal solution (feasible one) is unchanged. The idea of this study based on using a tuning parameters λ∈Rm in the function of the objective and input parameters υ∈Rl in the set of constraint. The inverse parametric nonlinear cost transportation problem P(λ,υ), where the tuning parameters λ∈Rm in the objective function are tuned (adapted) as less as possible so that the specific feasible solution x∘ has been became the optimal ones for a certain values of υ∈Rl, then, a solution stability set of the parameters was investigated to keep the new optimal solution x∘ unchanged. The proposed method consists of three phases. Firstly, based on the optimality conditions, the parameter λ∈Rm are tuned as less as possible so that the initial feasible solution x∘ has been became new optimal solution. Secondly, using input parameters υ∈Rl resulting problem is reformulated in parametric form P(υ). Finally, based on the stability notions, the availability domain of the input parameters was detected to keep its optimal solution unchanged. Finally, to clarify the effectiveness of the proposed algorithm not only for the inverse transportation problems but also, for the nonlinear programming problems; numerical examples treating the inverse nonlinear programming problem and the inverse transportation problem of minimizing the nonlinear cost functions are presented.