Objective Functions:

Due to the disruptions in the infrastructure network caused by the disaster, some of the demand vertices might be disconnected from all the inventories and, therefore, unreachable. Leaving an affected community to its own devices during an emergency leads to human suffering and might result in casualties. Thus, minimizing the population that is not reached by the distribution operations takes precedence over any other objective.

$$\min \sum\_{\omega \in \Omega} p\_{\omega} \cdot \sum\_{i \in V} demand\_{i}^{\omega} \cdot (1 - reach\_{i}^{\omega}) \tag{20}$$

Due to the stochastic nature of the model, objective function (20) minimizes the expected unsatisfied demand. This objective presents multiple optima, therefore, to discriminate among them, the solutions are evaluated in terms of the time required by the distribution operations. Two different time measures are proposed, presented in the following:

$$\min \sum\_{\omega \in \Omega} p\_{\omega} \cdot T^{\omega} \tag{21}$$

$$\min \sum\_{\omega \in \Omega} p\_{\omega} \cdot \sum\_{(i,j) \in A} length\_{ij} \cdot flow\_{ij}^{\omega} \tag{22}$$

The first objective (21) minimizes the expected maximum arrival time to a demand vertex across all the scenarios, while the second (22) is based on the classical min-cost flow objective function and minimizes the expected total distribution time across all scenarios.

Overall, the model includes |*A*| + |*V*| + |Ω| · (1 + 2 · |*A*| + 3 · |*V*|) variables. Of these, |*A*| + |*V*| + |Ω| · (|*A*| + |*V*|) are binary. The number of constraints in the model is: 2 + |*A*| + 3 · |Ω| · (|*A*| + |*V*|).
