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Top1. Introduction
There are numerous circumstances in real life where we need to find the shortest path from the source to destination. In tradition SPP, it is always determined that the parameter is fixed which can be effortlessly solved by fundamental graph theory; for example, Dijkstra's algorithm (Dijkstra, 1959) where the weighted graph is used to find the optimal path. Floyd–Warshall (Floyd, 1962) algorithm solves all pairs shortest paths, Viterbi Algorithm (Viterbi, 1967) is a based on a dynamic programming algorithm. By this algorithm, we can easily find the shortest path with an addition probabilistic weight on each connected node. Johnson’s Algorithm (Johnson, 1977) solved all pairs of the shortest path. This algorithm is faster than Floyd Warshall algorithm (Floyd, 1962) on sparse graph, but there are numerous circumstances in real life where we have to face with uncertain parameters such as flood, low visibility during winter, traffic congestion, uneven roads, strike, road accidents or bad human health arrives between the source to destination. In order to remove this uncertainty, we imply our problem into fuzzy problems. Zadeh (1965) was first to introduce the fuzzy numbers. The method of evaluating the Shortest distance with fuzzy parameters is known as the FSPP. This problem is an extension of fuzzy numbers, and it has many real-life applications in the field of scheduling (Nip, Wang, & Xing, 2016), transportation (Scutellà & Grazia, 1998), broadcast problems (Crescenzia, et al., 2016), air traffic management (Sherali & Hill, 2009), spanning tree problem (Dey & Pal, 2016), (Dey, Mondal, & Pal, 2017)and so on.
Dubois and Prade (1983) were first analyzed the FSPP by using the classical Floyd and FMB algorithms. However, the major drawback of their method is that the path derived by the algorithms may not actually exist in the network. FSP is one of the applications where we used the ranking index to find the shortest path. Okada & Gen (1994) proposed an algorithm on the basis of Dijkstra algorithm. They used fuzzy order relation to compare two path lengths, to find the FSP. Mahdavi et al. (2009) suggested a dynamic programming (DP) technique for the FSPP considering triangular and trapezoidal arc length. Tajdin et al. (2010) also proposed an algorithm based on DP technique using α-cut, Hassanzadeh et al. (2013) suggested genetic algorithm (GA), Zhang et al. (2013) suggested biological inspired optimization model and Yang et al. (2017) proposed optimal path selection approach for solving the FSP under mixed fuzzy arc length. Chuang & Kung (2005) proposed an algorithm which is combination of two steps, i.e. heuristic procedure for finding the FSP among all probable paths in the network and the similarity degree between FSPL. Moazeni (2006) algorithm is a combination of two steps, i.e. Hansen’s multiple labeling and Dijkstra’s SP algorithm. Dou et al. (2008) algorithm also contains two-steps, i.e. classical approach for finding the FSP and use of vague similarity measure to evaluate the similarity degree between vague path length. Dou et al. (2012) also suggest a model for finding the FSP using MCDM based on his previous approach, i.e., vague similarity measure. Effati et al. (2011) proposed fuzzy neural network(FNN) model, Haung & Ding (2012) proposed a model which combines with fuzzy simulation and genetic optimization, Mohiuddin et al. (2016) proposed fuzzy particle swarm optimization algorithm(FPSO) for the weight setting problem, Qu &Yi (2007) proposed algorithm based on Pulse-coupled neural network (PCNN), Dey et al. (2016) and Dey & Pal (2017) presented some model for solving the FSPP. So, there are different approaches to find the SPP under fuzzy environment.