# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 6 Experimentation and Results
### Results and Discussion  
As a previous question, we raise some considerations about the suitability of applying the Dijkstra Algorithm in our problem. It is known that it can present two major drawbacks. On one hand, it does not converge in graphs with negative costs, which may imply a negative cycle. That could result in an infinite cycle, which will block the algorithm to find a solution [35]. On the other hand, when there is more than one shortest path between origin and destination nodes, [PERSON] algorithm will always provide the first one it finds, which can generate unbalanced solutions. The first drawback does not apply to our problem since all the links in our graphs have positive weights (the three parameters of our fuzzy numbers correspond to positive quantities measured in the network as described in Section 5.3). Regarding the second drawback, the probability that several paths generate the same value of their fuzzy costs is very low because of the high variability of their fuzzy triplets. Under these considerations, we can conclude that our fuzzy version of the Dijkstra algorithm is a reliable tool for solving our problem. In any case, we have alternatively implemented a Dijkstra algorithm for k-Shortest paths with k = 10, which extracts the 10 best routes. If there are several best routes with the same cost, it randomly selects one of them. Although in practice this algorithm is not necessary, its implementation is available in GitHub with the rest of the source codes of the simulator. When tested for our problems, we derived similar results. To compare and rank the performance of the different strategies we calculate the Global Mean Delivery Rate (GMDR) such as it is defined above. This parameter informs about the ratio between the bits received and those sent during a complete simulation. Thus, comparison and ranking of the different strategies with different cost functions is possible. Tables 3 and 4 show these values together with the associated confidence intervals for NTT and NFSNET networks, respectively. Figures 7 and 8 show the same information in a bar diagram for NTT and NFSNET. We can see that the behavior is qualitatively similar for both networks in all the analyzed strategies. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Strategies**} & \multirow{2}{*}{**GMDR**} & \multirow{2}{*}{**Error**} & \multicolumn{2}{c}{**Conf. Interval \([\mathbf{CI}_{\min},\mathbf{CI}_{\max}]\)**} \\ \cline{3-5}  & & & \(\mathbf{CI}_{\min}\) & \(\mathbf{CI}_{\max}\) \\ \hline Strategy 1 & 0.811 & 0.002 & 0.809 & 0.813 \\ Strategy 2 & 0.809 & 0.0034 & 0.805 & 0.812 \\ Strategy 3 & 0.832 & 0.003 & 0.83 & 0.835 \\ Strategy 4 & 0.856 & 0.09 & 0.847 & 0.865 \\ Strategy 5 & 0.862 & 0.065 & 0.856 & 0.869 \\ Strategy 6 & 0.965 & 0.00045 & 0.964 & 0.9657 \\ Strategy 7 & 0.965 & 0.00046 & 0.965 & 0.966 \\ Strategy 8 & 0.9951 & 0.0001 & 0.9949 & 0.9952 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: For each Strategy, the GMDR and confidence interval associated. (NFSNET network). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Strategies**} & \multirow{2}{*}{**GMDR**} & \multirow{2}{*}{**Error**} & \multicolumn{2}{c}{**Conf. Interval \([\mathbf{CI}_{\min},\mathbf{CI}_{\max}]\)**} \\ \cline{3-5}  & & & \(\mathbf{CI}_{\min}\) & \(\mathbf{CI}_{\max}\) \\ \hline Strategy 1 & 0.860 & 0.004 & 0.856 & 0.864 \\ Strategy 2 & 0.861 & 0.0064 & 0.856 & 0.866 \\ Strategy 3 & 0.870 & 0.004 & 0.866 & 0.874 \\ Strategy 4 & 0.850 & 0.015 & 0.835 & 0.865 \\ Strategy 5 & 0.857 & 0.013 & 0.843 & 0.870 \\ Strategy 6 & 0.966 & 0.0025 & 0.963 & 0.968 \\ Strategy 7 & 0.966 & 0.0026 & 0.9632 & 0.9684 \\ Strategy 8 & 0.9982 & 0.0004 & 0.9978 & 0.9986 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: For each Strategy, the GMDR and confidence interval associated (NTT network). Figure 7: Bar diagram of the GMDR for each strategy (NTT network).