An Equivalent Linear Programming Form of General Linear Fractional Programming: A Duality Approach

Mehdi Toloo

doi:10.3390/math9141586

Abstract

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.

Highlights

Fractional programming has been attracting the attention of a fair number of researchers all over the world
The author of [6] investigated the primal–dual relation in linear fractional programming when the constraints are in an equation form
Referencing the weak duality theorem for linear programming models (2) and (3) reveals that h(x0, λ) = f (x0) ≤ g(y0, z0) which completes the proof

Summary

Introduction

Fractional programming has been attracting the attention of a fair number of researchers all over the world (see [1]). The author of [4] studied the duality for a special class of linear fractional functionals programming problem where its dual is a linear programming problem. The authors of [8] extended the duality in linear fractional programming and developed a dual linear programming model for a general maximization linear fractional functionals programming problem. They proved duality theorems in linear fractional functionals programming. We utilize some interesting properties obtained in this study and the duality properties to prove (in an easy manner) that the linear programming and linear fractional functional programming models are equivalent. As a special case of the general maximization linear fractional functional programming problem, we consider the well-known data envelopment analysis (DEA) approach. The authors of [8] stated and proved the well-known weak duality and complementary slackness theorems for models (1) and (2)

Dual of the Dual in Linear Fractional Programming

Numerical Example

H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12