## 3 Problem Description
\[OC=\sum_{j\in I}\sum_{l\in L}(f_{j}+o_{jl})y_{jl} \tag{1}\]  
\[RC=\sum_{e\in E}\sum_{k\in K}c_{e}w_{ek} \tag{2}\]  
\[FC=\min\{c_{reac},c_{prev}\} \tag{3}\]  
\(FC\) represents the cost incurred whenever the actual demand of a route is greater than the vehicle capacity, where \(c_{reac}\) and \(c_{prev}\) depend on the corrective action considered, namely:  
1. A reactive strategy with a cost \(c_{reac}\), in which a vehicle must perform a round-trip to its assigned facility for a replenishment if the actual current-customer demand is higher than the vehicle's current load. 2. A preventive strategy with a cost \(c_{prev}\), in which a vehicle must perform a detour to the facility before visiting the next customer. The decision about performing this detour depends on the type of demand of the next customer. If the demand is stochastic, the detour is carried out whenever the expected demand of the next customer is higher than the current capacity of the vehicle. Alternatively, if the demand is fuzzy, this decision depends on the comparison between the fuzzy values of both the demand of the next customer and the current capacity. Let \(\mathcal{O}\
eq S\subset V\) be a subset of nodes, \(\delta^{+}(S)\) the set of edges leaving \(S\), \(\delta^{-}(S)\) the set of edges entering \(S\), and \(A(S)\) the set of edges with both ends in \(S\). Hence, the location routing problem with facility sizing decisions and uncertain demands can be modeled as the following integer program:  
\[\textit{Minimize }\ TC \tag{4}\]  
_subject to:_  
\[\sum_{k\in K}\sum_{e\in\delta^{-}(i)}w_{ek}=1\quad\forall i\in I \tag{5}\]  
\[\sum_{i\in I}\sum_{e\in\delta^{-}(i)}D_{i}w_{ek}\leq(1-\%SS)q\quad\forall k\in K \tag{6}\]  
\[\sum_{e\in\delta^{+}(n)}w_{ek}=\sum_{e\in\delta^{-}(n)}w_{ek},\quad\forall k \in K,\forall n\in V \tag{7}\]  
\[\sum_{e\in\delta^{+}(I)}w_{ek}\leq 1\quad\forall k\in K \tag{8}\]  
\[\sum_{e\in A(S)}w_{ek}\leq|S|-1\quad\forall S\subseteq I,\forall k\in K \tag{9}\]\[OC=\sum_{j\in I}\sum_{l\in L}(f_{j}+o_{jl})y_{jl} \tag{1}\]0 \[OC=\sum_{j\in I}\sum_{l\in L}(f_{j}+o_{jl})y_{jl} \tag{1}\]1 (11) \[OC=\sum_{j\in I}\sum_{l\in L}(f_{j}+o_{jl})y_{jl} \tag{1}\]2 (12) \[OC=\sum_{j\in I}\sum_{l\in L}(f_{j}+o_{jl})y_{jl} \tag{1}\]3 (13) \[OC=\sum_{j\in I}\sum_{l\in L}(f_{j}+o_{jl})y_{jl} \tag{1}\]4  
The objective function (4) minimizes the total cost. Constraint (5) ensures that each customer is served by a single route and a single vehicle. Constraint (6) guarantees that the total demand served by a vehicle in a route does not exceed its capacity. This limit is reduced by a safety stock, which is a percentage of the vehicle capacity reserved to respond more effectively to the uncertain demand. Constraint (7) guarantees the continuity of each route. Constraint (8) ensures the return of each vehicle to its starting depot. Constraint (9) guarantees the subtour elimination. Constraint (10) ensures that a customer is served by a route departing from an open depot only if this customer is allocated to this depot. Constraint (11) guarantees that a customer is assigned to only one depot. Constraint (12) ensures that the total demand served from a depot does not exceed its assigned size. Constraint (13) guarantees that only one size is assigned to an open depot. Finally, Constraint (14) determines that all decision variables are binary.