## 3 Problem Description  
The location routing problem is a well-known problem in which three main decisions must be made: (i) locating one or more facilities; (ii) allocating customers to open facilities without exceeding their capacity; and (iii) designing a number of routes whose aggregated customers' demand does not exceed a vehicle capacity. Each route must start and finish at the same facility. Furthermore, we consider a location routing problem with facility sizing decisions, where the size of each open facility is also a variable to decide on. Furthermore, we also consider both stochastic and fuzzy demands. Hence, a percentage of the vehicles' capacity is reserved as a safety stock (\(SS\)), in case the demand is higher than expected. Therefore, the main decision variables in this problem are related to the number of facilities to open, the facilities' size and location, which customers must be allocated to each open facility, how many vehicles must be used and how to design the associated routes. This problem is _NP-hard_ since it contains, as special cases, the capacitated vehicle routing problem, the multidepot VRP and the facility location problem, all of them known to be computationally hard. Figure 1 provides an example of a complete LRP solution. Facilities are represented by diamonds and customers by circles. Black (solid) diamonds are the open facilities, while noncolored diamonds correspond to nonopen facilities. For each open depot, a set of routes starting and finishing at the corresponding depot location is designed to serve all customers' demands. Each route is assigned a single vehicle. Formally speaking, the LRP can be defined on a complete, weighted, and undirected graph \(G(V,E,C)\), in which \(V=J\cup I\) is the set of nodes (comprising the subset \(J\) of potential facility locations and subset \(I\) of customers), \(E\) is the set of edges, and \(C\) is the cost matrix of traversing each edge. Delivery routes are performed by a set \(K\) of unlimited homogeneous vehicles with limited capacity. This problem also assumes that all vehicles are shared by all facilities (i.e., no depot has a specific fleet) and each edge \(e\in E\) satisfies the triangle inequality. The customers' demands are uncertain and are modeled using stochastic values for a subset of customers \(I_{1}\), and fuzzy values for a subset of customers \(I_{2}\), such that \(I_{1}\cup I_{2}=I\). The variant of the LRP considered in this paper is the one in which a decision must be made about the size of the facilities to open. Hence, a set \(L\) of alternative sizes for each facility and associated fixed and variable opening costs are provided as inputs. Depots might have equal or different capacities. Each customer node must be served by exactly one vehicle that starts and finishes its route in the facility to which it has been allocated (i.e., split deliveries are not allowed). The following notation is used to describe our problem:  
**Parameters**  
\(s_{I}=\) Available size of type \(l\in L\)  
\(D_{i}=\) Uncertain demand of customer \(i\in I\)  
\(f_{j}=\) Fixed opening cost of depot \(j\in J\)  
\(o_{jl}=\) Variable opening cost of depot \(j\in J\) with size of type \(l\in L\)  
\(c_{e}=\) Cost of traversing arc \(e\in E\)  
Figure 1: Graphical representation of a location routing problem (LRP) solution. \(q\) = Capacity of each vehicle  
\(\%SS\) = Safety stock percentage  
**Decision variables**  
\(y_{jl}\) = Binary variable that indicates whether the depot \(j\in J\) is open with size \(l\in L\) or not. \(x_{ij}\) = Binary variable that indicates whether customer \(i\in I\) is assigned to the depot \(j\in J\) or not  
\(w_{ek}\) = Binary variable that indicates whether arc \(e\in E\) is used in the route performed by vehicle \(k\in K\) or not  
The objective is to minimize the total cost (\(TC\)), which includes opening facilities costs (OC), routing costs (\(RC\)), and failure costs (\(FC\)), i.e., \(TC=OC+RC+FC\). These parts are defined in Equations (1)-(3).