Solving fuzzy linear fractional set covering problem by a goal programming based solution approach

Ali Mahmoodirad; Harish Garg; Sadegh Niroomand

doi:10.3934/jimo.2020162

Article Contents

2022, Volume 18, Issue 1: 439-456. Doi: 10.3934/jimo.2020162

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Solving fuzzy linear fractional set covering problem by a goal programming based solution approach

1.
Department of Mathematics, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran
2.
School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University), Patiala-147004, Punjab, India
3.
Department of Industrial Engineering, Firouzabad Institute of Higher Education, Firouzabad, Fars, Iran

^*Corresponding author: Harish Garg
^*Corresponding author: Harish Garg

Received: May 2020

Revised: August 2020

Early access: December 2020

Published: January 2022

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Abstract

In this paper, a fuzzy linear fractional set covering problem is solved. The non-linearity of the objective function of the problem as well as its fuzziness make it difficult and complex to be solved effectively. To overcome these difficulties, using the concepts of fuzzy theory and component-wise optimization, the problem is converted to a crisp multi-objective non-linear problem. In order to tackle the obtained multi-objective non-linear problem, a goal programming based solution approach is proposed for its Pareto-optimal solution. The non-linearity of the problem is linearized by applying some linearization techniques in the procedure of the goal programming approach. The obtained Pareto-optimal solution is also a solution of the initial fuzzy linear fractional set covering problem. As advantage, the proposed approach applies no ranking function of fuzzy numbers and its goal programming stage considers no preferences from decision maker. The computational experiments provided by some examples of the literature show the superiority of the proposed approach over the existing approaches of the literature.

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Mathematics Subject Classification: Primary: 68T35; 90B50; 62A86; 03E72.

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Figure 1. The fuzzy objective function obtained by the proposed approach and the approach of Gupta and Saxena [14] for Example 1

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Figure 2. The fuzzy objective function obtained by the proposed approach and the approach of Gupta and Saxena [14] for Example 2

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Table 1. The goals obtained for Example 1 from Step 3 of the proposed approach

				Objective
	$ X_1 $	$ X_2 $	$ X_3 $	function value
Model (30)	$ 1 $	$ 1 $	$ 0 $	$ Z^{1*}=0.622 $
Model (31)	$ 1 $	$ 1 $	$ 0 $	$ Z^{2*}=0.966 $
Model (32)	$ 1 $	$ 1 $	$ 0 $	$ Z^{3*}=1.034 $
Model (33)	$ 1 $	$ 1 $	$ 0 $	$ Z^{4*}=2.875 $

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