# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 2 Related Work  
In this Section, we will first review the strategies used to search the shortest (or more efficient) path in a communication network. Concerning classic strategies (based on crisp magnitudes), we consider some approaches to be among the most known and used at the moment. The _Shortest-Widest_ (SW) strategy presented in [10] searches for the path with the largest residual bandwidth path (that is, the more efficient path). When there are two paths with the same value for this metric, the criterion about the path length is used so that the one with the lower number of hops is selected. The _Widest-Shortest_ (WS) strategy, explained in [11], proceeds contrary to SW. WS searches for the path with the minimum number of hops, and if there are two paths with the same number of hops, the one with the maximum residual bandwidth is selected. The _Shortest Dist Path_ is describedin [12; 13]. This strategy uses a hyperbolic cost function based on the inverse of the residual bandwidth to identify the best path. In these works, the authors perform an exhaustive comparison between SW, WS and Shortest Dist Path algorithm. When there is traffic without a reserve of resources, the Shortest Dist Path algorithm is clearly advantageous, especially under overload conditions. In [14], the goal is to find the optimum path in order to reduce the _Bandwidth Rejection Ratio_ (BRR). A comparison of cost functions applied to software-defined networks is carried out. In fact, based on a cost function that uses a variant of the _Shortest Dist Path_ algorithm, the algorithm DORA 0.9 provides the best results in [15]. In [16], the authors carry out an extensive study of different strategies and different network topologies concluding that widest-shortest is the best strategy among the studied ones in high load conditions. Other approaches to the optimal path search problem in a communication network are based on the adaptation of control techniques based on fuzzy inference. In this case, the values of the cost functions on the links are crisp, but the uncertainty is applied to the inference process itself. Thus, in [17], the authors developed a fuzzy inference system using the Expected Transmission Time (ETX) and the number of hops as input variables to find the best server node in a P2P wireless mesh. The fuzzification of these variables and their introduction in the fuzzy decision rules allow us to outperform the efficiency of traditional methods, which select the server node based on a min-hop or random criteria. The work in [18] addresses the problem of file sharing. The authors proposed that the server node selected for this operation is the one with the highest degree of trustworthiness. In [19], the authors proposed a fuzzy inference system to find the path in a sensor network taking into account the remaining energy, the minimum number of hops and the node traffic load. In [20], the authors addressed the problem of selecting the best communication path to reach a gateway while performing a load balancing. Fuzzy-logic functions are evaluated for this purpose. Type-II fuzzy logic has also been applied in wireless communication networks to find the optimal paths. With this technique, the work in [21] focused on selecting the best path to reach the gateway in an Internet-connected network. An alternative fuzzy approach to the network modeling assumes that the uncertainty lies on the value of the cost functions. This uncertainty can be modeled by using fuzzy numbers. In these cases, the application of a classic path search algorithm such as Dijkstra Algorithm involves solving the problem of operating with fuzzy numbers (e.g., adding up the fuzzy costs of the links in a path) and comparing fuzzy numbers (e.g., the comparison between the total cost of two paths). In [22], the definitions of the Graded Mean Integration Representation and the expression for the sum and product of two (triangular or trapezoidal) fuzzy numbers were developed with particular mathematical rigor using an integral representation.