## 4 A General Solving Methodology
``` Data: set of nodes \(V\), geometric distribution parameter \(\beta\), number of short simulations \(q_{short}\), number of long simulations \(q_{long}\), maximum number of iterations \(max_{iter}\) \(initSol\leftarrow\) BiasedRandomAlgorithm(\(V\), \(\beta\)) simulation(\(initSol\), \(q_{short}\)) bestSol\(\leftarrow\)initSol \(n_{iter}\gets 0\) while\(n_{iter}<max_{iter}\)do \(sol\leftarrow\) BiasedRandomAlgorithm(\(V\), \(\beta\)) if\(detCost(sol)<detCost(bestSol)\)then  simulation(\(sol\), \(q_{short}\)) if\(expCost(sol)<expCost(bestSol)\)then  bestSol\(\leftarrow\) sol \(Elite\leftarrow Elite\cup\{sol\}\)  end if  end if \(n_{iter}\gets n_{iter}+1\) end if foreach\(sol\in Elite\)do  simulation(\(sol\), \(q_{long}\)) end foreach\(Elite\leftarrow\) sort(\(Elite\)) be



## 5 Computational Experiments

The proposed fuzzy simheuristic has been implemented using Python 3.8 and tested on a common PC with a multi-core processor Intel \(i7\) and using 8 GB of RAM. The algorithm was executed five times with different seeds for a maximum time of 100 s for each instance. To the best of our knowledge, there are no instances in the literature for the stochastic-and-fuzzy problems described above. Accordingly, we have extended the well-known deterministic benchmarks proposed by [PERSON] et al. [9] and [PERSON] et al. [80] for the TOP and the VRP problems, respectively. The following subsections describe in detail the process used to transform these deterministic benchmarks into stochastic-fuzzy ones. ### A Fuzzy-Stochastic Approach for the VRP

In order to check the performance of our algorithm, we compare it with some benchmark instances that can be found in the literature. From [PERSON] et al. [80], we have chosen 29 of the classical instances that can be suitable for our study. The nomenclature of the instances is as follows: \(`L-nXX-kY`\) where \(L\in\{A,B,E\}\) is the set identification, \(XX\) denotes the number of customers and \(Y\) establishes the number of vehicles. For carrying out the experiments, we assumed that the demand \(d_{i}\) of each customer \(i\) is uncertain and, therefore, we have modeled it either as a stochastic or as a fuzzy variable. Regarding the stochastic scenario, the instances were extended by considering that the stochastic demand \(D_{i}\) follows a log-normal probability distribution. The parameters of this distribution were adjusted according to the mean \(E[D_{i}]=d_{i}\ \forall i\in N\), where \(d_{i}\) is the deterministic demand, and the variance \(Var[D_{i}]=c\cdot d_{i}\). The parameter \(c\) is a design parameter that allows us to set up the level of uncertainty. It is expected that, as \(c\) converges to zero, the results of the stochastic version will converge to those obtained in the deterministic scenario. In our experiments, we have utilized the value \(c=0.25\), which introduces a medium level of uncertainty. Concerning to the fuzzy scenario, we consider the demand \(D_{i}\), for each customer \(i\), as a fuzzy variable. This demand can be estimated as low, medium, or high \((L,M,H)\). Likewise, we assume that the vehicle remaining capacity, \(RC\), is an input variable of the fuzzy system. Besides, each of the aforementioned demand levels is defined by a triangular fuzzy number \(D_{i}=(d_{1i},d_{2i},d_{3i})\). Figure 4 shows the membership functions of these fuzzy sets. Similarly, the remaining vehicle capacity \(RC\) is represented by a triangular fuzzy number \(RC=(rc_{1},rc_{2},rc_{3})\), which takes the values low (L), medium (M) or high (H) capacity. Figure 5 displays the membership function of the capacity fuzzy sets. Note that both the demands and the remaining capacities are expressed as a percentage of the total vehicle capacity, i.e., \(0\leq D_{i}\leq 1\) and \(0\leq RC\leq 1\). Figure 4: Fuzzy sets for the customer \(i\) demand. For each node \(i\), we define a preference index, \(p_{i}\), as the output of the fuzzy system, such that \(0\leq p_{i}\leq 1\).