## 5 Computational Experiments
#### A Fuzzy Approach for the Demand and the Vehicle Available Capacity  
When considering customers with stochastic demands, the decision about visiting the next customer in a route is made simply by comparing its expected demand with the vehicle's current capacity. If this demand is greater, the vehicle will perform a detour to the depot for a replenishment. Nevertheless, when the next customer demand is fuzzy, the decision about serving it is made employing a preference index \(p_{i}\)([PERSON] et al., 2013). It indicates the strength of our inclination to visit the next node in a route. This index depends on both the estimated demand of the next node \(D_{i+1}\) and the vehicle capacity \(C_{i}\) that remains available after serving the customer \(i\in I\). \(C_{i}\) is expressed as a proportion of \(q\), i.e., \(0\leq C_{i}\leq 1\). It also can be treated as low (CL), medium (CM) or high (CH), and it is represented by a triangular fuzzy number \(C_{i}=(c_{1i},c_{2i},c_{3i})\). The membership function of the capacity fuzzy sets are displayed in Figure 3.  
The preference index is defined between 0 and 1, i.e., \(0\leq p_{i}\leq 1\). When \(p_{i}=1\), we will definitely visit the next node in a route since the vehicle available capacity can for sure meet its demand. When \(p_{i}=0\), we are sure that \(D_{i+1}\) exceeds \(C_{i}\) and the vehicle must return to  
Figure 3: Fuzzy sets for the vehicle available capacity after visiting the customer \(i\). Figure 2: Fuzzy sets for the demand of the customer \(i\). the depot for a replenishment. We consider that the preference can be very low (PVL), low (PL), medium (PM), high (PH) or very high (PVH). Each of these categories is represented by a fuzzy set, whose membership function is depicted in Figure 4. Additionally, we define a set of reasoning rules (Table 2) to determine the preference to visit the next node depending on the levels of both the demand and the vehicle available capacity. Figure 5 displays the procedure used to compute the preference index \(p_{i}\) after serving the customer \(i\in I\). This procedure is described as follows:  
1. Simulate the actual demand of each customer employing a fuzzy simulation approach. Based on the works by [PERSON] and [PERSON] (2018); [PERSON] et al. (2019) and [PERSON] (2019), we follow the steps described below: 1. Generate a random demand \(d_{i}\) between a lower bound and an upper bound. Since the objective is preserving the variability conditions similar to the stochastic demands, the lower and upper bounds are given by the expressions \(\frac{\phi_{i}-\sqrt{3\lambda\phi_{i}}}{q}\) and \(\frac{\phi_{i}+\sqrt{3\lambda\phi_{i}}}{q}\), respectively. 2. Calculate the membership degree \(\mu(d_{i})\) of this demand. Notice that \(\mu(d_{i})\in[0,1]\). 3. Generate a random number \(\rho\in[0,1]\). 4. Compare \(\rho\) and \(\mu(d_{i})\). If \(\rho\leq\mu(d_{i})\), then assume the actual demand of the customer \(i\) as \(d_{i}\); otherwise, repeat steps (a)-(d) until this condition is fulfilled. Figure 4: Fuzzy sets for the preference strength to visit the customer \(i\). Figure 5: Procedure used to compute the preference index \(p_{i}\). Calculate the vehicle available capacity subtracting from \(q\) the sum of the simulated demand of the first \(m\) customers visited in the current route, including the customer \(i\). Whenever the route fails and the vehicle must perform a trip to the depot for a replenishment, the counting of \(m\) starts again from \(1\). 3. Estimate the fuzzy demand and the fuzzy available capacity according to the categories previously defined: low, medium or high. 4. Determine the membership function of the preference index using the reasoning rules defined in Table 2.
5. Calculate a crisp preference index using the center of gravity as defuzzification method. Additional methods can be found in [PERSON] and [PERSON] (2017), and [PERSON] (2018). We define a known threshold \(p^{*}\), such that \(0\leq p^{*}\leq 1\).