A new algorithm for the discrete fuzzy shortest path problem in a network
Introduction
In the past decades, much attention has been paid to the shortest path problem in a network since it is important to a great deal of applications such as routing, communication, and transportation [1]. In a network, the shortest length is the minimum of all path lengths. In the crisp world, it is apparent to obtain the shortest length. For example, there are two path lengths in a network, l1 = 4 and l2 = 5. The shortest length is the minimum of l1 and l2. It is trivial to obtain the shortest length is 4. However, in real world, the arc length in the path of network may represent cost or time and it can be considered to be a discrete fuzzy set. If the arc lengths are fuzzy, it is not so obvious to obtain the shortest length. For example, in a network, there are two path lengths that are discrete fuzzy sets L1 and L2. Assume L1 = {1.0/5, 0.6/6} and L2 = {1.0/4, 0.7/7}, that is, L1 has lengths 5 and 6 with a membership grade of 1.0 and 0.6, respectively; L2 has lengths 4 and 7 with a membership grade of 1.0 and 0.7, separately. The shortest length is the minimum of L1 and L2. The length 4 in L2 is a possible shortest length since it is smaller than both of the possible lengths in L1. The length 7 in L2 is, however, not a possible shortest one since it is larger than both of the possible lengths in L1. On the other hand, both the lengths 5 and 6 in L1 are possible shortest length since they are smaller than the possible length 7 in L2. The lengths, 4, 5 and 6 are then possible shortest length while 7 is not. As mentioned above, L1 has two possible shortest lengths 5 and 6; L2 has one possible shortest length 4. Thus, neither L1 nor L2 can represent the final set of possible shortest length. In this paper, we propose a discrete fuzzy shortest length method to find the fuzzy shortest length in a network.
Lots of approaches were developed to deal with the fuzzy shortest path problem [3], [4], [5], [6], [7], [8], [9]. Dubois and Prade [3] presented a method based on Floyd’s algorithm and Ford’s algorithm [1], [2] to treat the fuzzy shortest path problem. Their method can obtain the shortest path length whereas the corresponding shortest path in the network perhaps does not exist. Later, Klein [7] proposed an improved algorithm that was based on dynamical programming recursion. Klein’s algorithm can get not only the shortest path length but also the corresponding shortest path in the network; nevertheless, the assumption that the possible arc lengths are 1 through a fixed integer seems to be impractical. In this paper, to overcome the drawback mentioned above, we develop a method to determine the fuzzy shortest length from source node to the destination node in the network. Then, by means of fuzzy similarity measure to evaluate similarity degree between fuzzy lengths, we decide a practical shortest path whose length is closest to the yielded fuzzy shortest length.
The rest of this paper is organized as follows. In Section 2, some related fuzzy set operations and fuzzy similarity measure are reviewed. In Section 3, a method to find the discrete fuzzy shortest length is presented. In Section 4, through combining the previous method with fuzzy similarity measure, a new algorithm is developed to get the shortest length as well as the shortest path. An illustrative example is also included to demonstrate our proposed algorithm. Finally, some conclusions are drawn.
Section snippets
A review of related fuzzy set operations and similarity measure
In this section, some basic fuzzy set operations and fuzzy similarity measure are reviewed. Both of them are beneficial to the treatment of the discrete fuzzy shortest problem.
The discrete fuzzy shortest length method
In real world, the arc length in a network can be considered to be a discrete fuzzy set and so can the path length. Suppose that there are m fuzzy path lengths Li for i = 1, 2, … , m, we expect to find the fuzzy shortest length. In the crisp world, a length is the shortest one if any other length is greater than or equal to it. In other words, if a length is the shortest one of a set of length, then this length does exist and the other length smaller than it does not exist. We extend this idea to
An algorithm for the discrete fuzzy shortest problem
Suppose that in a directed acyclic network the arc lengths are fuzzy. We aim at determining the fuzzy shortest length Lmin and the shortest path needed to traverse from source node s to destination node d. By combining the discrete fuzzy shortest length method with similarity measure, the new algorithm is proposed as follows:
- Step 1.
Form the possible paths from source node s to destination node d and compute the corresponding path lengths Li, i = 1, 2, … , m, for possible m paths.
- Step 2.
Find the fuzzy shortest
Conclusions
In the past, there were several methods reported to deal with the discrete fuzzy shortest path problem in the open literature. In these methods, they can obtain either the fuzzy shortest length or the shortest path. It is the purpose of this paper to propose a new algorithm that can obtain both of them. We present the discrete fuzzy shortest length method to find the fuzzy shortest length. This method is based on the idea that a crisp length is the shortest one if and only if any other length
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