## 3 VRPs under Uncertainty  
Real-life VRPs involve demand, travel distance, or time uncertainty. This uncertainty could be of a stochastic or fuzzy nature. Thus, the traditional heuristics that solve deterministic VRPs cannot be used to solve the uncertain versions of these problems. Uncertainty requires other components in the solution approach, such as simulations. Simulations can be used to evaluate different systems, including those of vehicle routing under stochastic conditions, but they cannot be used to solve problems. To overcome this challenge, researchers have combined simulation and optimization methods [32]. For example, [PERSON] et al. [33] combined evolutionary particle swarm optimization and Monte Carlo simulation (MCS) to find a solution to a VRP. The optimization method could be metaheuristics, mathematical programming, or machine learning. Accordingly, several simulation-optimization approaches could be identified based on the utilized optimization method and the function of the simulation in the hybridized approach. For more details on these approaches, the reader is directed to [PERSON] and [PERSON] [34]. One simulation-optimization approach integrates a metaheuristic algorithm and a simulation. This integration is called simheuristics, and it has become a popular approach among researchers for solving stochastic problems [35]. According to [PERSON] and [PERSON] [34], the metaheuristic algorithm generates solutions that are evaluated by using a simulation in simheuristics. Thus, the simulation in this hybridization approach assigns values to each solution with respect to the objective function(s). This approach could be considered as a \"first resource\" methodology when handling NP-hard problems and large-scale instances under stochastic uncertainty. Figure 3 demonstrates the flow of a simheuristic approach. First, a deterministic version of the considered problem is defined. This definition might be achieved by replacing the stochastic random variables with their estimated values, e.g., means. Then, the metaheuristic algorithm generates solutions for the deterministic version of the problem. If the generated deterministic solution is promising, it is examined with a small number of simulations run under stochastic conditions. This process of generating and evaluating solutions continues until a pre-specified time has elapsed or a number of iterations has been reached. At this stage, an elite list of solutions is defined. These solutions are further examined with a more intense simulation process and a larger number of simulation runs in order to rank them. Researchers have utilized different simulation approaches that have integrated into simheuristics, such as MCS and Petri nets. [PERSON] et al. [36] used MCS integrated into a metaheuristic algorithm to solve a multi-depot VRP. Another example of simheuristic utilization is found in the work of [PERSON] et al. [37]. These authors considered two-dimensional VRPs. This type of problem appears in real-world cases when transported items cannot be stacked on top of each other. [PERSON] et al. [37] integrated MCS into an iterated local search framework. Their problem considered stochastic travel times, and it triggered the integration of a simulation into their solution approach. [38] proposed an intertwined protocol for integrating Petri nets into simheuristics to solve stochastic VRPs with the correlated demand. It was noticed that simulations integrated into metaheuristic algorithms have formed solution approaches for different versions of VRPs. In addition, electric vehicles have been considered in VRPs, and this combination constitutes a new class of problems. These problems raise additional constraints related to the limited driving range, making the resolution even more challenging. Hence, [PERSON] et al. [39] solved a version of this problem with stochastic travel times. They used  
Figure 3: Simheuristic approach. MCS and a multi-start approach.