**2. Model**

The model proposed in this research is a two-stage stochastic model. The first stage involves all the decisions taken before the disaster strikes, such as determining the inventories' locations and choosing the road sections to fortify. The problem assumes that a fortified road cannot be interdicted. The second stage takes place after the disaster strikes, which determines the number of people affected in the population centers and the roads that have been interdicted and cannot be used. In this stage, the emergency goods distribution takes place and is modeled as a flow problem with an unlimited supply. The objective is maximizing the number of affected people that receive relief. However, this objective presents multiple optima. Therefore, a second optimization step chooses among these optima the solution that minimizes the distribution time. Two different time measures are considered. Finally, the model also provides information regarding the desired inventory capacities: each inventory should be able to relieve as many persons as its maximum supply across all the scenarios. The optimization model is presented in the following.

## *2.1. Sets*

Let *<sup>G</sup>*(*<sup>V</sup>*, *A*) be a directed graph, being *V* the set of vertices, indexed by *i* and *j*, and *A* the set of arcs (*i*, *j*). The vertices represent population centers and road crossings. On the other hand, the arcs represent the road sections; specifically, each road segmen<sup>t</sup> is modeled by a pair of arcs, i.e., (*i*, *j*) and (*j*, *i*). Finally, Ω is the set of stochastic scenarios and is indexed by *ω*.

## *2.2. Parameters*

Two attributes, *P* and *Q*, specify the number of inventories to locate and the number of road sections to fortify, respectively. Vertices are characterized by a demand, *demandω i* , which represent the population affected by the disaster in a community and depends on the scenario. Arcs are characterized by a length, *lengthij*, which represents the traversal time. Also, each scenario specifies which arcs are not interdicted. This information is represented by the parameter *saf eωij* , that takes value 1 if the arc (*i*, *j*) can be traversed in scenario *ω*, and 0 if it is interdicted. Finally, the scenarios have an assigned probability distribution, *pω* ∀ *ω* ∈ Ω, which verifies ∑*ω*∈<sup>Ω</sup> *pω* = 1.

*Mathematics* **2020**, *8*, 529
