## 1 Introduction  
When designing and managing supply chains, one of the most relevant problems is the simultaneous location of distribution facilities and the routing of vehicles to deliver products to a set of geographically dispersed customers. The former is considered a strategic decision, while the latter is operational. This problem is known in the scientific literature as the location routing problem (LRP). The LRP addresses these two types of decisions in an integrated manner. From the formal view of the operational research community, the LRP is known to be _NP-hard_, since it can be reduced to either the facility location problem (FLP), the vehicle routing problem (VRP) or the multidepot VRP, which are all known to be _NP-hard_. This computational complexity means that optimal solutions are really difficult to obtain in a reasonable computational time. Thus, heuristic approaches are required to solve medium- and large-sized instances. Due to its complexity, some of the first studies tackled the problem by splitting it into the corresponding subproblems [1, 2]. Nevertheless, this approach might lead to suboptimal solutions. Due to the increase in computational power and the development of fast heuristic approaches, the LRP has been studied in an integrated way, which clearly has improved the obtained results [3]. One of the most studied versions of the LRP is the capacitated LRP, in which both depot and vehicle capacity constraints must be satisfied (the acronym LRP will henceforth refer to this version). However, all previous works consider the depot capacity as a fixed value for each location. This could not be a suitable approach when dealing with realistic problems, since it is usual that decision-makers can select the size of a facility from a discrete set of known available sizes, or even freely. For real-world problems, this set is usually associated with investment activities, such as building facilities [4], purchasingequipment [5] or qualifying workforce [6]. From an academic point of view, despite the increasing number of published works on the LRP, the consideration of flexible sizes for facilities has been rarely addressed in the literature. Nevertheless, real-life examples from both LRP [4; 7; 8] and non-LRP [5; 6] contexts show the relevance of considering a variety of facility sizes to select from. Traditional LRP approaches consider that parameters are deterministic or crisp, i.e., they assume that inputs are known in advance. This assumption is far from reality in many applications, such as waste collection, humanitarian logistics and urban freight distribution, where uncertainty is a key factor to consider. Despite this, the literature on the LRP addressing uncertain parameters is still scarce. In order to overcome this problem, articles employing stochastic approaches can be found in the literature. Customers' demand is one of the most addressed stochastic parameters [9; 10; 11; 12; 13]. Other parameters might also be considered as stochastic, such as transportation costs and travel speeds [14] or logistic costs and travel distance [15]. In general, many articles addressing stochasticity in routing problems hybridize simulation models with heuristic or metaheuristic algorithms to tackle efficiently both uncertainty and _NP-hardness_. In many real-life situations, however, it might not be possible to accurately model all uncertainty sources as stochastic variables following a probability distribution. This might be the case, for instance, when the volume of observations is low or the available data does not have enough quality [16]. Hence, uncertainty in the LRP has also been tackled through the use of fuzzy sets. Parameters such as customers' demands [17; 18; 19; 20], travel times [21; 22] or time windows [23] have been modeled as fuzzy in several studies. Notice that, whenever possible, modeling uncertainty as stochastic variables might allow a deeper statistical analysis of the results.