A time variant multi-objective particle swarm optimization algorithm for solving fuzzy number linear programming problem
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A time variant multi‑objective particle swarm optimization algorithm for solving fuzzy number linear programming problems using modified Kerre’s method Reza Ghanbari1 · Khatere Ghorbani‑Moghadam1 · Nezam Mahdavi‑Amiri2 Accepted: 27 September 2020 © Operational Research Society of India 2020
Abstract Recently, Ghanbari et al. (IEEE Transactions on Fuzzy Systems 27:1286–1294, 2019) have proposed modified Kerre’s method for comparison of LR fuzzy numbers. Here, we make use of the modified Kerre’s method to solve fuzzy linear programming problems with LR coefficients. In an approach to solve a fuzzy linear program with fuzzy LR coefficients, a bi-objective optimization problem is formulated. For the associated bi-objective optimization problem, we present a time variant multiobjective particle swarm optimization (TV-MOPSO) algorithm to compute the Pareto front, a set containing a large number of solutions. Contrary to methods that change the fuzzy optimization problem to a crisp problem by use of a ranking function, using modified Kerre’s method, the fuzzy optimization problem is solved directly, with no need for changing it to a crisp program. A comparative investigation using illustrative examples with triangular fuzzy coefficients show the effectiveness of the proposed algorithm. Keywords Fuzzy linear programming · Modified Kerre’s method · LR fuzzy numbers · Particle swarm optimization
* Nezam Mahdavi‑Amiri [email protected] Reza Ghanbari [email protected] Khatere Ghorbani‑Moghadam [email protected] 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
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OPSEARCH
1 Introduction A linear programming model may represent a real world situation involving a number of parameters whose values are assigned by experts. However, experts and decision makers frequently don’t know the precise values of the parameters. In most optimization problems, there are parameters with imprecise or fuzzy values (see [8]). Fuzzy set-based optimization was introduced by Bellman and Zadeh [3], using some concepts of fuzzy constraint, fuzzy objective and fuzzy decision. Afterwards, these concepts were abundantly used, applied and extended by many investigators. In the past two decades, fuzzy linear programming problems (specially fuzzy LR linear programs) with imprecise decision variables or parameters have been developed from several applications in various areas such as mathematical modeling [58, 65], management science [51], game theory [2, 59], engineering and economics [6], graph theory [12], transportation [19–21] and manufacturing [1, 43]. Fuzzy linear programming problems (FLPPs) are of extensive interests in operations research [32]. FLPPs are classified into three groups [32]: (1) fuzzy linear programming problems with fuzzy decision variables, (2) fuzzy linear programming problems with fuzzy parameters and (3) fully fuzzy linear programming pro
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