## 4 A General Solving Methodology  
Both the VRP and TOP are NP-hard problems [72; 73]. Therefore, due to the combinatorial nature of these problems, the use of exact solving approaches is often limited by the size of the problem instances. When dealing which real-life instances, which are typically large-scale instances, the use of heuristics and metaheuristics has proven to be a good alternative [74]. Although metaheuristics are capable of finding near-optimal (or even optimal) solutions to many different combinatorial optimization problems in reasonable computing times, these approaches have been mainly designed to solve deterministic versions of these problems. Consequently, metaheuristics are not able to cope adequately with stochastic components, being their application constrained against uncertain scenarios as the ones proposed in this paper. In order to tackle uncertainty, metaheuristics have been combined with simulation methods in recent years. The resulting _simheuristic_ approaches, apart from finding cost-effective solutions for the deterministic problem--through the optimization component--are also able to provide efficient solutions for the stochastic scenario [6]. In many real-life situations, large-scale and complex optimization problems assume different degrees of uncertainty, not only of a stochastic but also of a fuzzy (non-stochastic) nature. The latter might occur, for instance, when the volume of observations is low or the available data are of insufficient quality. In this case, the accurate modeling of the uncertainty sources simply does not follow the natural pattern of modeling them only as stochastic variables following a probability distribution. Instead, fuzzy inference systems (FIS) are also considered to achieve this goal. In this paper, we propose a fuzzy simheuristic approach to solve both the VRP and the TOP under general uncertainty scenarios (i.e., those including both probabilistic as well as non-probabilistic uncertainty). This hybrid solution approach combines a multi-start (MS) metaheuristic with MCS and FIS to deal with stochastic and fuzzy variables, respectively. Specifically, this solution method is composed of three stages. The first stage refers to the construction of an initial feasible solution through a savings-based constructive heuristic, which is designed considering the characteristics of each problem. The second stage consists of enriching this heuristic with biased-randomized (BR) decisions [75], which are then incorporated into a MS framework-in order to generate multiple solutions. This stage, in addition to exploring different regions of the solution space, conducts a short number of simulations on a set of promising solutions in order to evaluate their efficiency under stochastic conditions. Finally, the third stage is a refinement one, in which a larger number of simulation runs are applied to a set of elite solutions. This procedure allows to obtain a more accurate estimation of the different solution properties. Since the number of solutions generated during the search can be large and the simulation process is time consuming, we have limited the number of short simulation runs to 100. Regarding the number of simulations in the refinement step, it has been set to 1000. Figure 3 depicts a high-level description of the proposed methodology. As explained, this process starts from solving the deterministic problem, whose corresponding solution is submitted to a short simulation procedure, i.e., the exploratory stage. Consequently, new solutionsare generated for both the stochastic and the fuzzy environment. These steps are repeated until a stopping criterion is met. Finally, the best-found solutions (or a set of elite solutions) are submitted to a large number of simulation runs--the intensive stage--in order to obtain a more accurate summary of output variables, such as total cost/reward and risk/reliability values. In order to facilitate reproducibility, the low-level details of each of the stages of the described methodology are provided below:  
1.