A Linearization to the Sum of Linear Ratios Programming Problem

Mojtaba Borza,Azmin Sham Rambely

doi:10.3390/math9091004

Abstract

Optimizing the sum of linear fractional functions over a set of linear inequalities (S-LFP) has been considered by many researchers due to the fact that there are a number of real-world problems which are modelled mathematically as S-LFP problems. Solving the S-LFP is not easy in practice since the problem may have several local optimal solutions which makes the structure complex. To our knowledge, existing methods dealing with S-LFP are iterative algorithms that are based on branch and bound algorithms. Using these methods requires high computational cost and time. In this paper, we present a non-iterative and straightforward method with less computational expenses to deal with S-LFP. In the method, a new S-LFP is constructed based on the membership functions of the objectives multiplied by suitable weights. This new problem is then changed into a linear programming problem (LPP) using variable transformations. It was proven that the optimal solution of the LPP becomes the global optimal solution for the S-LFP. Numerical examples are given to illustrate the method.

Highlights

Ratios Programming Problem.Optimizing the sum of linear fractional functions over a set of linear inequalities (SLFP) is considered as a branch of a fractional programming problem with a wide variety of applications in different disciplines such as transportation, economics, investment, control, bond portfolio, and in cluster analysis, multi-stage shipping problems, queueing location problems, and hospital fee optimization [1,2,3,4,5,6,7,8,9,10].In optimization, if the objective function of a problem is strictly convex, its local minimizer is a unique global
Schaible demonstrated that the S-Linear fractional programming (LFP) is a global optimization problem [9]; this means that the problem has one or more local optimal solutions that cause some difficulties to find the global optimal solution
We show that the S-LFP can be changed into a weighted linear programming problem (LPP) where for some values of weights, the optimal solution of the LPP becomes a global optimal solution for the S-LFP

Summary

Introduction

In [15], Cambini et al introduced an iterative algorithm to deal with the sum of a linear ratio and a linear objective over a polyhedral They proved that an optimal solution exists on the boundary of the feasible region. Motivated by Kuno, Benson [24,25] presented branch and bound based algorithms to reach global optimal solutions for S-LFP. According to the theory of monotonic optimization introduced by Tuy [26], Phuong and Tuy [27] presented an iterative efficient unified method to address a wide category of generalized LFPPs. In [28], Benson presented and validated a simplicial branch and bound duality-bounds algorithm to find the global optimal solution for S-LFP.

Main Results

Numerical Example

Method of

Conclusions and Discussion