## 3 Related Work
### Fuzzy Sets in the VRP and the TOP  
Fuzzy techniques are typically used to deal with non-probabilistic uncertainty. This approach is especially useful when insufficient historical data are available as to model uncertainty with probability distributions. Early work shows the potential of using fuzzy sets in the VRP by considering travel times between nodes [54] or customer demands [55] as fuzzy. On the one hand, [PERSON] and [PERSON] [54] propose the introduction of fuzzy arithmetic operations in the savings computation when solving the VRP. On the other hand, [PERSON] and [PERSON] [55] present a procedure to decide whether or not to visit the next customer on a route. To make this decision, these authors propose a preference index, which indicates the strength of the inclination to make this visit. The preference index is computed by comparing the fuzzy demand of the next customer and the fuzzy current load of the vehicle. Uncertainty in customer demand is a recurrent topic in the VRP literature using fuzzy techniques. For instance, [PERSON] and [PERSON] [56] propose a hybrid differential evolution algorithm to solve a VRP with uncertain demands, which is additionally formulated as a fuzzy chance constrained program. Different preference index thresholds are tested with the objective of minimizing the total distance traveled. A similar approach is employed by [PERSON] and [PERSON] [57] to solve an open vehicle routing problem with fuzzy demands. This parameter is also considered by [PERSON] et al. [58], who address a home healthcare open VRP with time windows. A fuzzy chance constraint model is proposed, as well as a hybrid genetic algorithm. A set of benchmark instances is used to test their approach. Fuzzy customer demands are considered as well by [PERSON] et al. [59] and [PERSON] and [PERSON] [60]. Time-windows constraints are also quite frequently considered uncertain in the fuzzy VRP literature. The idea is that the information about the earliest and latest times at which customers must be visited is imprecise or vague. For instance, [PERSON] et al. [61] address a realistic multi-objective dynamic VRP with time windows. In this case, the time windows are related to the level of customer satisfaction. This satisfaction is sought to be maximized. In turn, the aim is to minimize the number of vehicles used, the total distance traveled, and the waiting time of the vehicles. A solving approach based on a genetic algorithm is proposed. The relation between the level of customer satisfaction and the fuzzy time windows is also examined by [PERSON] et al. [62]. It is proposed a multi-objective model that seeks to both minimize the distance traveled and maximize the level of the customer service. Fuzzy time windows are also considered by [PERSON] et al. [63], [PERSON] and [PERSON] [64], and [PERSON] et al. [65]. The latter authors also consider the vehicle capacity as a fuzzy parameter. Finally, fuzzy sets are additionally used to model parameters such as service times [66; 67] and travel times [68]. Fuzzy techniques have hardly been used in the TOP. The TOP is similar to the VRP, but in the former a fixed fleet of vehicles needs to collect rewards by visiting customers, and since there is a maximum time or distance that each vehicle can cover, it is often the case that not all customers can be visited [9]. Hence, the main objective of the TOP is to maximize the collected reward without exceeding the route length threshold. The orienteering problem (OP) refers to the single-vehicle (and less challenging) version of the TOP. [PERSON] [69] and [PERSON] et al. [70] consider OPs in which both the collected rewards and the travel times are fuzzy. The former authors propose a parallel algorithm as a solving approach, whereas the latter employ a genetic algorithm. Regarding the TOP, [PERSON] et al. [71] propose a greedy randomized adaptive search procedure (GRASP) to solve this problem considering fuzzy rewards and fuzzy travel times.