1063-6706 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2018.2876690, IEEE Transactions on Fuzzy Systems 1 A variable neighborhood search algorithm for solving fuzzy number linear programming problems using modified Kerre’s method Reza Ghanbari 1,* , Khatere Ghorbani-Moghadam 2 , and Nezam Mahdavi-Amiri 2 Abstract—To solve a fuzzy linear program, we need to compare fuzzy numbers. Here, we make use of our recently proposed modified Kerre’s method for comparison of LR fuzzy numbers. We give some new results on LR fuzzy numbers and show that to compare two LR fuzzy umbers, we do not need to compute the fuzzy maximum of two numbers directly. Using the modified Kerre’s method, we propose a new variable neighborhood search (VNS) algorithm for solving fuzzy number linear programming problems. In our algorithm, the local search is defined based on descent directions, which are found by solving four crisp mathematical programming problems. In several methods, a fuzzy optimization problem is converted to a crisp problem but in our proposed method, using our modified Kerre’s method, the fuzzy optimization problem is solved directly, without changing it to a crisp program. We give some examples to compare the performance of our proposed algorithm with some available methods. We show the effectiveness of our proposed algorithm by using the non-parametric statistical sign test. Index Terms—Fuzzy linear programming problem; Modified Kerre’s method; Ranking function; VNS algorithm. I. I NTRODUCTION In many real-world situations, some parameters of a linear program are given by experts. However, experts and decision makers frequently are not aware of the precise values of the parameters. Since some optimization problems contain parameters with imprecise values (see [1]), fuzzy number linear programs (FNLPs) are very useful tools for modeling and solving real-world problems. One type of FNLP problem is defined as follows: min ˜ c T x s.t. (I.1) Ax ≤ b, x ≥ 0, where b ∈ R m , x ∈ R n , A ∈ R m×n and ˜ c ∈ F (R n ), with F (R) being the set of all fuzzy numbers. FNLPs arise from several applications in various areas such as location problems ([2]–[5]), mathematical modeling [6], network [7], [8] and graph theory [9]. There are a number of methods to solve a fuzzy linear programming problem. For example, Maleki et al. [10] used a 1 Faculty of Mathematical Sciences, Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran. 2 Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran. * Corresponding author, Tel:++98-51-38805658, Fax:++98-51-38828606, P. O. Box: 9177948953, rghanbari@um.ac.ir certain ranking function to solve FNLP problems making use of an auxiliary linear program. Mahdavi-Amiri and Nasseri [11] explored some duality properties in FNLP problems. They introduced the dual of fuzzy number linear programming primal problems by using a linear ranking function. Mishmast- Nehi et al. [12] proposed lexicographic ranking to order fuzzy numbers and solved FNLP problems by lexicographic ranking function. Ganesan and Veeramani [13] introduced a type of fuzzy arithmetic for symmetric trapezoidal fuzzy numbers and then proposed a primal simplex method for solving fuzzy linear programming problems in which the coefficients of the constraints were represented by real numbers and all the other parameters as well as the variables were represented by symmetric trapezoidal fuzzy numbers. Lin [14] proposed a genetic algorithm (GA) for solving a linear programming problem with the constraint coefficients not given precisely. They investigated the possibility of applying a GA to solve this kind of fuzzy linear programming problem without defining membership functions for fuzzy numbers, using the extension principle, interval arithmetic, and α-cut operations for fuzzy computations, and using a penalty method for constraint violations. Buckley and Feuring [15] applied an evolutionary algorithm to two classical fully fuzzified linear programming problems and showed that it could produce good approxi- mate solutions. Wang [16] considered a linear programming problem with fuzzy resources and crisp objective function. He proposed a GA taking the mutation along the weighted gradient direction as a genetic operator. This lead most points to the fuzzy optimal solution quickly. Several authors considered various types of fuzzy linear programming problems and proposed several approaches for solving them ([17]–[26]). In most proposed algorithms for solving FNLP, the fuzzy problem is converted to a crisp problem. Converting a fuzzy problem to a crisp problem using ranking functions may not be appropriate at times; for exam- ple, using ranking function in [10], [11], [21], two numbers ˜ A =(a, α, β) and ˜ A ′ =(a, α + γ,β + γ ), where γ →∞, will have the same rank and thus are considered to be equal, while using Kerre’s method they are not. From another point of view, converting a fuzzy problem to a crisp problem may not be adequately appropriate for real-world applications. For example, authors who use ranking functions for comparison of fuzzy numbers usually define a crisp model which is equivalent to an FNLP problem and then determine the optimal solution of the original problem as the optimal solution of the crisp problem (see [10], [11], [21]). Here, however, we intend to