mathematics Article A Linearization to the Sum of Linear Ratios Programming Problem Mojtaba Borza and Azmin Sham Rambely *   Citation: Borza, M.; Rambely, A.S. A Linearization to the Sum of Linear Ratios Programming Problem. Mathematics 2021, 9, 1004. https:// doi.org/10.3390/math9091004 Academic Editor: Armin Fügenschuh Received: 26 February 2021 Accepted: 23 April 2021 Published: 29 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Mathematical Sciences, Faculty of Science & Technology, UKM Bangi, Selangor 43600, Malaysia; mborza@siswa.ukm.edu.my * Correspondence: asr@ukm.edu.my 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. Keywords: global optimization problem; local optimal solution; global optimal solution; membership function; linear programming; linear fractional programming 1. Introduction Optimizing the sum of linear fractional functions over a set of linear inequalities (S- LFP) 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 more specifically in cluster analysis, multi-stage shipping problems, queueing location problems, and hospital fee optimization [1–10]. In optimization, if the objective function of a problem is strictly convex, then its local minimizer is also a unique global. In the literature, it has been of interest to find conditions so that a local minimizer becomes also global. On this subject, we mention the studies of Mititelu [11], and Trată et al. [12]. Schaible demonstrated that the S-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. In addition, he proved that the sum of linear ratios is neither quasiconcave nor quasiconvex. In [13], Freund and Jarre showed that the problem is N-P hard. Thus, working on this kind of problem is important and beneficial. Linear fractional programming (LFP) is a specific class of S-FLP. The best method to deal with LFP was proposed by Charnes and Cooper [14]. They showed that an LFP can be changed into an equivalent linear programming (LP). 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. In [16], Almogy and Levin determined the sum-of-ratios to the sum-of-non-ratios by using the methodology introduced by Dinkelbach [17]. However, Falk and Palocsay [7] showed the proposed method by Almogy and Levin does not always come out with the global optimal solutions. In [7], an iterative method was also introduced to S-LFP in which linear programming is solved over the image of the feasible region in iterations. Mathematics 2021, 9, 1004. https://doi.org/10.3390/math9091004 https://www.mdpi.com/journal/mathematics