Abstract
The shortest path problem (SPP) is one of the most important combinatorial optimization problems in graph theory due to its various applications. The uncertainty existing in the real world problems makes it difficult to determine the arc lengths exactly. The fuzzy set is one of the popular tools to represent and handle uncertainty in information due to incompleteness or inexactness. In most cases, the SPP in fuzzy graph, called the fuzzy shortest path problem (FSPP) uses type-1 fuzzy set (T1FS) as arc length. Uncertainty in the evaluation of membership degrees due to inexactness of human perception is not considered in T1FS. An interval type-2 fuzzy set (IT2FS) is able to tackle this uncertainty. In this paper, we use IT2FSs to represent the arc lengths of a fuzzy graph for FSPP. We call this problem an interval type-2 fuzzy shortest path problem (IT2FSPP). We describe the utility of IT2FSs as arc lengths and its application in different real world shortest path problems. Here, we propose an algorithm for IT2FSPP. In the proposed algorithm, we incorporate the uncertainty in Dijkstra’s algorithm for SPP using IT2FS as arc length. The path algebra corresponding to the proposed algorithm and the generalized algorithm based on the path algebra are also presented here. Numerical examples are used to illustrate the effectiveness of the proposed approach.
Highlights
The problem of finding the shortest path from a specified source node to a destination node is a fundamental and well known combinatorial optimization problem in graph theory
We propose an algorithm to find a shortest path based on Dijkstra’s algorithm in a fuzzy graph, where the arc lengths are trapezoidal interval type-2 fuzzy set (IT2FS)
The shaded eU, region is the FOU. It is bounded by an upper membership function (UMF), denoted by μ Ãi or A
Summary
The problem of finding the shortest path from a specified source node to a destination node is a fundamental and well known combinatorial optimization problem in graph theory. All the possible paths between source node and destination node and their corresponding path lengths are computed in their proposed algorithm For this purpose, the authors have added type-2 fuzzy numbers corresponding to the arcs present in the path. In [56], the authors assigned a discrete type 2 fuzzy number to each arc of the network and computed the path lengths of all possible paths between source node and destination node using the same method as in [55]. For comparison purpose they have used a function to find the minimum of two discrete type 2 fuzzy numbers representing two paths based on which a metric called similarity measure is computed.
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