## 5 Computational Experiments
### Mathematical Model
\begin{table}
\begin{tabular}{l} \hline Parameters \\ \hline \(n_{F}\): number of dairy plants. \\ (\(n_{F}\) + 1) or \(DC\): the distribution center. \\ \(n_{F}\): number of products. \\ \(TQ_{p}\): quarantine time of product \(p\). \\ \(TT_{f,f^{\prime}}\): transportation time between plant \(f\) and \(f^{\prime}\). \\ \(CT_{f,p}\): capacity of vehicle type \(t\) transporting product \(p\). \\ \(CF_{r,f^{\prime}}\): dairy plant \(f\) production capacity for product \(p\). \\ \(I0_{f,f}\): initial stock of product \(p\) at dairy plant \(f\). \\ \(DM_{p}\): product \(p\) demand. \\ \hline Decision Variables \\ \hline \(I_{f,p,d}\): product stock \(p\) at dairy plant \(f\) on the day \(d\). \\ \(Q_{f,p,d}\): product stock \(p\) at dairy plant \(f\) released from quarantine on the day \(d\). \\ \(y_{f,p,d}\): quantity of product \(p\) produced on the day \(d\) in the dairy plant \(f\). \\ \(\theta_{r^{\prime},p,d}\): quantity of the dependent product \(p^{\prime}\) produced on the day \(d\) in the dairy plant \(f\) from the product \(p\). \\ \(\tau_{r,p,d,f^{\prime},k}\): quantity of product \(p\) transported from the dairy plant \(f\) to \(f^{\prime}\) on the day \(d\) using the \(k\)-th vehicle of type. \\ \(\delta_{r,p,d,f^{\prime},k}\): quantity of the product \(p\) received at the dairy plant \(f^{\prime}\) from \(f\) on the day \(d\) using the \(k\)-th vehicle of type \(t\). \\ \(z_{r,p,d,f^{\prime},k}\): it assumes one (1) if the \(k\)-th type \(t\) is used to transport product \(p\) on the day \(d\) from dairy plant \(f\) to \(f^{\prime}\); more (0) otherwise. \\ \(w_{p,d}\): it assumes one (1) if the demand for the product \(p\) is met on the day \(d\); zero (0) otherwise. \\ \(Tmax\): minimum number of days required to satisfy total demand. \\ \hline \end{tabular}
\end{table} 
Table 4: \(Cont\). \[0\leq g_{f,f,d}\leq CF_{f,p} \forall f\in F,p\in P,p^{\prime}\in P_{r}\cap P_{r},d\in D \tag{16}\] \[x_{f,d,p^{\prime},d,h}\geq 0 \forall f\in F,p\in P_{f},d\in D,f\in\mathcal{F}_{f},t\in\mathcal{T }_{f},k\in K_{f,j}\] (17) \[\dot{x}_{f,d,p^{\prime},d,h}\geq 0 \forall f\in F,p\in\mathcal{P}_{f},d\in D,f\in\mathcal{F}_{f},t\in \mathcal{T}_{f},k\in K_{f,j}\] (18) \[z_{f,d,p^{\prime},d,h}\in\{0,1\} \forall f\in F,p\in P,d\in D,f\in\mathcal{F}_{f},t\in\mathcal{T}_{f},k \in K_{f,j}\] (19) \[w_{p,d}\in\{0,1\} \forall p\in P,d\in D\] (20) \[Tmax\geq 0 \forall d\in D \tag{21}\]  
The objective, represented in Equation (1), is to minimize the total execution time of a demand response plan at the distribution center. Equation (2) represents the stock conservation constraints, where the first part represents the sum of the previous day's production, the stock of products available after quarantine, and the products arriving from other factories. The second part of Equation (2) represents the decrease in stock, subtracting the products that are sent to other factories and also subtracting the stock benefited in dependent productions, reducing the amount of product input. Equations (3) and (4) define the quantity of products released for transport to the distribution center or benefiting in another dairy plant on the day after the end of quarantine. Equation (4) refers to the production of a product that depends on previous production. Equation (5) indicates that the quantity of product transported between factories must be equal to the quantity arriving at the factory destination after the transportation time. Equation (6) indicates that the load carried in the selected vehicle must not exceed its maximum capacity. Equation (7) imposes a constraint that ensures the \(k\)-th vehicle of type \(t\) is assigned to only one transit operation between an origin factory \(f\) and a destination factory \(f^{\prime}\) within the defined period. This restriction prevents a vehicle from being used in multiple transits concurrently. The formulation accounts for all product flows from factory \(f\), represented by the summation over \(p^{\prime}\in P_{f}\).