# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 7 Conclusions  
In this paper, we addressed the problem of searching the shortest path (or more efficient path, in a more general sense) between two nodes in a communication network, considering uncertainty in the measurements used in the cost function. The above referred uncertainty is inevitably caused by the network dynamics and the impossibility of making decisions with the instantaneous measurement of the state variables at each moment. Thus,  
Figure 8: Bar diagram of the GMDR for each strategy (NFSNET network). in general, over a given time interval, decisions on the routing of information are made using measurements obtained in the previous time intervals, which, naturally, can differ from the current ones. In order to incorporate our fuzzy cost functions and strategies into a communication network management system, we associate the network to a Type-V Fuzzy Graph. In this graph, nodes and links are clearly located (without uncertainty), but a triangular fuzzy number defines the cost for each connection. Moreover, a fuzzy version of [PERSON]'s routing algorithm was developed and exhaustively described. This is called Fuzzy Dijkstra Algorithm (FDA). Simulated experiments have shown its correct operation and competitiveness. It is essential to distinguish this approximation, in which the uncertainty lies on the value of the variables, from other approximations based on Fuzzy Inference, where these values are considered crisp (precise data) and the uncertainty lies on the decision rules used. For our experimental study, we implemented the most commonly used cost functions and strategies in the management of real networks based on crisp values (e.g., these using the _Instantaneous_ or _Mean Used Bandwidth_ or _Residual Bandwidth_ as cost function, Shortest-Widest (SW), Widest-Shortest (WS)) and we confronted them with their similar ones based on our definition of fuzzy costs. As an especially interesting contribution, we proposed a new fuzzy strategy (Strategy 8) that has no correspondence with any classic one, which adapts the definition of the fuzzy cost to the fact that the smaller the _Used Bandwidth_ at the \((n-1)\)-th time interval (where the measurements have been obtained) the more significant uncertainty in the _Used Bandwidth_ value considered in the \(n\)-th time interval. We made the experiments on two representative network topologies. Fuzzy Strategy 3 surpasses their analogous crisp ones (strategies 1,2) in a slightly, but statistically significant way. As for the fuzzy Strategies 5 and 7, we observed that they do not present statistically significant differences with their analogous crisp Strategies 4 and 6, respectively. However, our new fuzzy strategy (Strategy 8), with an effectiveness ratio very close to unity, clearly surpasses the rest of the analyzed strategies. Therefore, we conclude that our methodology to introduce the inherent uncertainty to the dynamic nature of the network in the routing algorithms has been successful. Conceptualization, L.V. and S.M.A. ; methodology, L.V. and A.A.; validation, L.V., A.A. and G.J. ; formal analysis, L.V. and S.M.A. ; software, A.A.; writing--original draft preparation, L.V., A.A. and A.T.; writing--review and editing, L.V., A.A., S.M.A., G.J. and A.T.; visualization, A.T.; supervision, G.J. ; project administration, G.J. ; funding acquisition, A.T. and G.J. All authors have read and agreed to the published version of the manuscript. This research was partially supported by the University of Malaga-Andalucia Tech through the Plan Propio de Investigacion y Transferencia and by the II Plan Propio Smart-Campus with the project \"Smart and Secure EV Urban Lab II\". Not applicable. The data used in our research are synthetic values generated by means simulation. All the data and experimental software routines are freely available at [[https://github.com/aarizaq/flowsimulator](https://github.com/aarizaq/flowsimulator)]([https://github.com/aarizaq/flowsimulator](https://github.com/aarizaq/flowsimulator)) (accessed on 31 July 2021).