# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 1 Introduction
However, it allows us to define them using statistical methods in such a way that we can keep the consistency with the physics of the problem and incorporating its fuzzy nature. Thus, in our case, the proposed fuzzy numbers for the time interval \(t\) are defined as follows: the central value corresponds to the mean value of the costs over the entire \(t-1\) time interval, which will generate a minor error, and the extreme values correspond to the maximum and minimum values of cost measurements in the above mentioned \(t-1\) time interval. In our proposal, we model the network as a non-directed Type-V fuzzy Graph. In this type of fuzzy graph, the vertexes and the edges are physically well established, but the costs of the links are defined by a fuzzy number. In our particular case, the costs are triangular fuzzy numbers to incorporate the uncertainty [4]. Finally, the application of a fuzzy version of the Dijkstra algorithm, like the shortest path search algorithm, should give us the path between any pair of source-destination nodes with the minimum cost. This fuzzy version of [PERSON]'s algorithm can be considered as classical since it does not incorporate any consideration of uncertainty in its execution, but it incorporates uncertainty in the value of the variables (here considered as fuzzy) and in the definition of the mathematical operation between them. A very basic application of this work can be found in [5], where a pair of shortest-disjoint paths is identified with a simpler version of the Modified Dijkstra Algorithm. However, the complete description of the algorithm, the operation of the fuzzy-based costs and their comparison with other classic approaches are firstly described in a scientific journal. In particular, we now include the mathematical framework to operate with the fuzzy costs in a modified Dijkstra algorithm. In addition, the present paper defines how classical cost functions can be incorporated into the algorithm and compare them with some fuzzy-based cost functions for two illustrative network topologies. The algorithm is completely described, including the complementary procedures of the initialization phase and the relaxation of the vertexes. However, the modified Dijkstra algorithm and the preliminary operations can be found as a technical report in the repository of the University of Malaga [6]. We are also interested in highlighting the differences between this methodology, which uses fuzzy values of variables (triangular fuzzy numbers) and the so-called Fuzzy Inference (or Control) (see [7; 8]), in which the measurements have precise values, but the uncertainty lies on the decision process itself. Summarizing, the main contribution of this work is the presentation of a methodology to optimize the search for the shortest path between two nodes during the global management of a communication network. This methodology is characterized by modeling the network as a Non-directed Type-V Fuzzy Graph and applying a fuzzy Dijkstra algorithm that is adapted to operate with fuzzy numbers. To show the goodness of our approach, we have two objectives:  
1. To show the competitiveness of the methodology based on the fuzzy model versus that based on the traditional model using crisp magnitudes. To do this, we reconstruct the currently used classical cost functions and strategies for searching the optimal path in the fuzzy model. Then, we compare both visions (classical and fuzzy) in the same experimental environment for two representative network topologies. The results show that the fuzzy model is more efficient (or equal in the worst case), being this conclusion validated by a statistical parameter. 2. To illustrate that the fuzzy model allows us to define new metrics that take advantage from the experimental uncertainty above mentioned. Thus, we provide a new strategy, and its cost function (referred to as Strategy 8), which has no equivalent in the classic model and by far exceeds all the other strategies analyzed in effectiveness.