## 4 A General Solving Methodology
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Demand**} & \multicolumn{3}{c}{**Remaining Capacity**} \\ \cline{2-4}  & **L** & **M** & **H** \\ \hline L & M & H & VH \\ M & L & M & H \\ H & VL & L & M \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The rules used in the Fuzzy system for the VRP. Figure 7: Fuzzy sets for the battery of each vehicle. of the next node and the remaining battery of the vehicle. This preference index has been defined between 0 and 1, i.e., \(0\leq p\leq 1\). When \(p=1\), the vehicle will definitely visit the next node in the route, since the vehicle will reach the node. On the contrary, when \(p=0\), we are sure that the vehicle will not reach the next node, and the vehicle will remain in the current node. In this case, the route will present a failure, because the vehicle fails to reach the final depot, and consequently, the total reward of the route has been onlyt partially collected. We have classified the preference as: very low (VL), low (L), medium (M), high (H), or very high (VH). Each of these categories is represented by a fuzzy set. Finally, we have established a set of reasoning rules (Table 2), which describe the knowledge needed to determine the preference to visit the next node. After a quick fine-tuning process, we have set the threshold value for visiting the next node to \(p\geq 0.45\). Note that this is a sensitive value, as a larger value could lead to generating overly conservative routes, while a value close to 0 could lead to risky decisions. In order to transform the input variables into a crisp value, the contribution of each membership function is combined on the inference, while a union operator is used to determine the output distribution. Subsequently, the center-of-gravity method is applied in order to obtain a crisp output value corresponding to the preference value. ## 6 Results and Discussion

Tables 3 and 4 display the results of selected instances with different characteristics for the VRP and the TOP problems, respectively. In the case of the VRP and the TOP, the results--with the exception of the gap column--are measured in terms of distance and reward units, respectively. The first column of the tables identifies the instances. We have divided the remaining columns into three different parts. First, our best-found deterministic solutions (OBD) are presented (these solutions do not consider stochastic or fuzzy variables, they refer to the deterministic version of the problem). We compare the gap of our solutions (column 2) with respect to the best-known solutions (column 1). In the second part of the table, we present the obtained solutions for the stochastic scenario. Column 3 displays the expected cost when the OBD is evaluated under a stochastic scenario, with the corresponding level of uncertainty. To compute the expected cost, a simulation process has been applied to the OBD solution. Similarly, the next column shows the expected cost obtained using our simheuristic approach for the stochastic version of the problem. The last part of the table reports the results obtained considering fuzzy scenarios. Thus, column 5 reports the best hybrid fuzzy-stochastic solutions. To compute these solutions, we assume that half of the nodes follow a log-normal distribution, and the remaining half are considered to be fuzzy. In the case of the TOP, where the uncertainty is related to the edges, we have considered the origin node to evaluate the type of uncertainty. Finally, the last column of the table reports the solutions obtained in a scenario with a high level of uncertainty, where all the uncertain variables are considered as fuzzy. Notice that, although the goal of this paper is not to solve the deterministic version of the problem, the results show that our approaches are highly competitive for the deterministic version of both problems. For the VRP problem, we obtain an average gap of 0.39%, with a maximum gap of 1.27%. Furthermore, the obtained gap is 0.0% for the TOP problem.