On The Karush – Kuhn – Tucker Reformulation of Bi – Level Geometric Programming Problem with an Interval Coefficients as Multiple Parameters

Azza Hassan Amer

Azza Hassan Amer Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt

Keywords: Keywords, Bi- Level optimization problem, Geometric programming, KKT reformulation, Arithmetic mean, Geometric mean, Harmonic mean.

Abstract

This paper presents a new approach to solve a special class of bi – level nonlinear programming (NLP) problems with an interval coefficients as multiple parameters. Geometric programming (GP) is a powerful technique developed for solving nonlinear programming (NLP) problems and it is useful in the study of a variety of optimization problems. Many applications of GP in various fields of science and engineering are used to solve certain complex decision making problems. In this paper a new mathematical formulations for a new class of nonlinear optimization models called bi – level geometric programming (BLGP) problem is presented. This problems are not necessarily convex and thus not solvable by standard nonlinear programming techniques. This paper proposed a method to solve BLGP problem where coefficient of objective function as well as coeffiaent of constraints are multiple parameters. Especially the multiple parameters are considered in an interval which are the Arithmetic mean (A.M), Geometric mean (G.M) and Harmonic mean (H. M) of the end points of the interval. In this paper, the values of objective function in interval range of parameters for A. M., G. M. and H. M. are preserved the same relationship. Also, BLGP problem can be converted to a single objective by using the classical karush – kuhn – Tucker (KKT) reformulation and the ability of calculating the bounds of objective value in KKT is basically presented in this paper that may help researchers in constructing more realistic model in optimization field. Finally, numerical example is given to illustrate the efficiency of the method.

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