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Top1. Introduction
To find the shortest distance between the source node and the destination node (known as shortest path problem) is a well-known topic of Operations Research (Ahuja et al., 1993). In general, to solve a shortest path problem, the distance between every two nodes is represented by a positive real number. However, it is not always possible to represent the distance between every two nodes by a positive real number e.g., the statement “distance between two nodes is approximately twenty kilometers” cannot be represented by a positive real number. In the literature (Zadeh, 1965; Nayeem and Pal, 2005; Kung et al., 2007; Hernandes et al., 2007; Li, 2008; Li et al., 2010; Li, 2010a; Li, 2010b; Li, 2010c; Li, 2011; Deng et al., 2012; Biswas et al., 2013; Li and Liu, 2014; Li and Nan, 2014; Nan et al. 2014; Wan and Li, 2015; Kumar et al., 2015; Okada and Soper, 2000; Niroomand et al., 2017; Enayattabar et al., 2019; Mahdavi et al., 2009; Sori et al., 2020; Li et al., 2020; Ye and Li, 2020; Deli, 2020; Deli 2021; Deli and Keles., 2021), fuzzy set and its various extensions are used to represent such statements.
Ebrahimnejad et al. (2020) claimed that till now no one has used an interval-valued triangular fuzzy number to represent such statements for the shortest path problems. To fill this gap, they proposed interval-valued triangular fuzzy shortest path problems and also proposed a method to solve interval-valued triangular fuzzy shortest path problems.
In Ebrahimnejad et al. (2020)’s method, firstly, an interval-valued triangular fuzzy linear programming problem is obtained corresponding to the considered interval-valued triangular fuzzy shortest path problem. Then, the obtained interval-valued triangular fuzzy linear programming problem is transformed into its equivalent crisp multi-objective linear programming problem. Finally, the transformed crisp multi-objective linear programming problem is solved using a lexicographic approach to obtain the shortest path and the corresponding shortest interval-valued triangular fuzzy distance.
In this paper, an alternative approach (named as Mehar approach) is proposed to solve interval-valued triangular fuzzy shortest path problems. Also, it is shown that less computational efforts are required to apply the proposed Mehar approach as compared to Ebrahimnejad et al. (2020)’s method.
This paper is organized as follows: In Section 2, some basic definitions are reviewed. In Section 3, the existing method for comparing interval-valued triangular fuzzy numbers, used by Ebrahimnejad et al. (2020), is discussed. In Section 4, Ebrahimnejad et al. (2020)’s method to solve interval-valued triangular fuzzy shortest path problems is discussed. In Section 5, an alternative approach (named as Mehar approach) is proposed to solve interval-valued triangular fuzzy shortest path problems. In Section 6, the interval-valued triangular fuzzy shortest path problem, considered by Ebrahimnejad et al. (2020) to illustrate their proposed method, is solved by their proposed method as well by the proposed Mehar approach. In Section 7, it is shown that less computational efforts are required to apply the proposed Mehar approach as compared to Ebrahimnejad et al. (2020)’s method. Section 8 concludes the paper.