## 4 A General Solving Methodology
When this index takes the maximum value (\(p_{i}=1\)) then the next node of a route will be visited for sure as the remaining capacity \(RC\) of the vehicle can meet the demand \(D_{i+1}\). Moreover, if \(p_{i}=0\), then we are sure that \(D_{i+1}>RC\) and, consequently, the vehicle needs a replenishment at the depot. The preference index is classified into very low (VL), low (L), medium (M), high (H) and very high (VH) levels. The membership function related to each of these categories can be seen in Figure 6. The reasoning rules that determine the preference to travel to the next node-depending on the levels of the demand and the remaining capacity-are featured in Table 1. After performing a set of fine-tuning experiments, we established the threshold value to visit the next node to \(p=0.25\). This means that whenever the calculated \(p_{i}\) is greater than 0.25, the next node will be visited; otherwise, the vehicle will return to the depot for a replenishment. The calculation of a specific value for \(p_{i}\) requires converting the input variables into a crisp value. Hence, the estimated crisp values of the demand and the remaining capacity, the membership functions and the reasoning rules are employed in a fuzzification-defuzzification process to obtain the preference index. In our case, the defuzzification method applied was the well-known center-of-gravity method to obtain the output crisp value. Figure 5: Fuzzy sets for the remaining capacity after visiting customer \(i\). Figure 6: Fuzzy sets for the preference strength to travel to customer \(i\). ### A Fuzzy-Stochastic Approach for the TOP

The deterministic benchmark used contains a total of 320 instances that are distributed in 7 subsets. The instances are identified following the nomenclature '_pa.b.c_', where \(a\) represents the subset, \(b\) defines the number of available vehicles, and \(c\) identifies the specific instance under study. For experimentation purposes, we have considered that the uncertainty is located in the travel times between two pairs of nodes. To extend the instances to be employed in the stochastic scenarios, we have assumed that travel times, \(T_{ij}\), follow a log-normal probability distribution. In setting up the stochastic instances, we assume that \(E[T_{ij}]=t_{ij}\,\forall(i,j)\in N\), where \(t_{ij}\) is the travel time for the corresponding deterministic instance. We set the variability in the travel times with reference to the deterministic travel time such that \(Var[T_{ij}]=c\cdot t_{ij}\), and \(c\geq 0\). As in the VRP problem, we have utilized the value \(c=0.25\) to induce a medium level of uncertainty in the travel times. With the aim of extending the instances to be used also in fuzzy scenarios, we have considered the travel times for each pair of nodes, \(t_{ij}\), as a fuzzy variable. This variable has been modeled using a fuzzy inference system. We have assumed the case of electric vehicles and used their battery levels, as well as the reward of each node, as the input variables of the fuzzy system. The battery level (\(Q\)) of each vehicle can be estimated as low (L), medium (M), or high (H). The low and high levels are represented by a triangular fuzzy number \(Q=(q_{1},q_{2},q_{3})\), while the medium level follows a trapezoidal fuzzy number \(Q=(q_{1},q_{2},q_{3},q_{4})\). All battery values are expressed as a proportion of the total battery level, i.e., \(0\leq Q\leq 1\). The membership function of this fuzzy set is displayed in Figure 7. Similarly to the battery level, the reward of each node has been categorized using three fuzzy sets: low (L), medium (M), or high (H), where each of them follows a triangular distribution. The reward values have been represented as a proportion of the maximum reward that can be collected at any node of all the possible nodes to be visited. Finally, the output of the fuzzy system gives a preference index, \(p\), which indicates the inclination to visit the next node in the route. This index depends on both the reward