Fuzzy Shortest Path Problem Research Articles

Fuzzy shortest path problem with finite fuzzy quantities

We discuss the shortest path problem from a specified node to every other nodes on a network in which a positive fuzzy quantity with finite support is assigned to each arc as its arc length. We define an order relation between fuzzy quantities with finite supports. Then by applying Hansen’s multiple labeling method and Dijkstra’s shortest path algorithm, we propose a new algorithm for finding the set of non-dominated paths with respect to the extension principle. Moreover, we show that the only existing approach for this problem, Klein’s algorithm, may lead to a dominated path in the sense of extension principle.

A new algorithm for the discrete fuzzy shortest path problem in a network

In a network, the arc lengths may represent time or cost. In practical situations, it is reasonable to assume that each arc length is a discrete fuzzy set. We called it the discrete fuzzy shortest path problem. There were several methods reported to solve this kind of problem in the open literature. In these methods, they can obtain either the fuzzy shortest length or the shortest path. In this paper, we propose a new algorithm that can obtain both of them. The discrete fuzzy shortest length method is proposed to find the fuzzy shortest length, and the fuzzy similarity measure is utilized to get the shortest path. An illustrative example is given to demonstrate our proposed algorithm.

The shortest path problem with discrete fuzzy arc lengths

In the past, the fuzzy shortest path problem in a network has attracted attention from many researchers for its importance to various applications. In this paper, we propose a new algorithm to deal with the fuzzy shortest path problem. It is composed of fuzzy shortest path length procedure and similarity measure. The former is presented to determine the fuzzy shortest path length from source node to the destination node in the network, and the latter is used to measure the similarity degree between fuzzy length sets. This algorithm not only can yield shortest length but also can offer the actual shortest path to decision makers. An illustrative example is also included to demonstrate our proposed algorithm.

The fuzzy shortest path length and the corresponding shortest path in a network

The fuzzy shortest path (SP) problem aims at providing decision makers with the fuzzy shortest path length (FSPL) and the SP in a network with fuzzy arc lengths. In this paper, each arc length is represented as a triangular fuzzy set and a new algorithm is proposed to deal with the fuzzy SP problem. First, we proposed a heuristic procedure to find the FSPL among all possible paths in a network. It is based on the idea that a crisp number is a minimum number if and only if any other number is larger than or equal to it. It owns a firm theoretic base in fuzzy sets theory and can be implemented effectively. Second, we propose a way to measure the similarity degree between the FSPL and each fuzzy path lengths. The path with the highest similarity degree is the SP. An illustrative example is given to demonstrate our proposed approach.

Fuzzy Shortest-Path Network Problems With Uncertain Edge Weights

This paper presents two new types of fuzzy shortest-path network problems. We consider the edge weight of the network as uncertain, which means that it is either imprecise or unknown. Thus, the first type of fuzzy shortest-path problem uses triangular fuzzy numbers for the imprecise problem. The second type uses level (1-β, 1-α) interval-valued fuzzy numbers, which are based on past statistical data corresponding to the confidence intervals of the edge weights for the unknown problem. The main results obtained from this study are: (1) using triangular fuzzy numbers and a signed distance ranking method to obtain Theorem 1, and (2) using level (1-β, 1-α) interval-valued fuzzy numbers, combining statistics with fuzzy sets and a signed distance ranking method to obtain Theorem 2. We conclude that the shortest paths in the fuzzy sense obtained from Theorems 1 and 2 correspond to the actual paths in the network, and the fuzzy shortest-path problem is an extension of the crisp case.

Fuzzy shortest path problem

Abstract We discuss the problem of finding the shortest paths from a fixed origin to a specified nodes in a network with arcs represented as intervals on real line. As a preliminary, we define the order relation between intervals by using two parameters. The decision maker's preference can be reflected on the order relation according to setting values of two parameters. An algorithm for this problem is proposed on the basis of Dijkstra's method. The some paths with minimal duration times can be obtained through the algorithm. As a large scale example, the United States driving map with 70 cities and 211 routes between cities is examined.

The fuzzy shortest path problem and its most vital arcs

The shortest path problem is to find the shortest distance between two specified nodes in a network. An arc is called a single most vital arc in the network, if its removal from the network results in the greatest increase in the shortest distance. The availability of arcs plays a key role in network problems. The most vital arcs problems provide a means by which the importance of arc's availability can be measured. In the traditional shortest path problem, the arc lengths are assumed to be crisp numbers. In this paper, we consider the case that the arc lengths are fuzzy numbers. In particular, we derive the membership function of the shortest distance by using a fuzzy linear programming approach. Based on this result, we then propose an algorithm for finding the single most vital arc in a network.