## 4 Results and Sensitivity Analysis
### Results  
For the solution of the deterministic equivalent of the multi-stage stochastic programming model, a Dell Inspiron 5570 computer with an Intel1 CoreTM i5-8250U processor, 1.6 GHz and 4 Gb RAM memory was used. The mathematical modeling language LINGO 19.0 was used with the Branch-and-Bound solver. For the relaxation of the problem four cores of the processors, simplex and dual-simplex algorithms were used. For the pre-integer solver, the maximum number of heuristics allowed in LINGO 19.0 was used (100). Constraint cuts were applied using all the ability options and both directions (up and down) were considered in the branching process. Other settings in the solver were in default; an example of report solution is in Appendix A. In Table 2, the comparison between the size of the deterministic equivalent is shown. It is evident that when the number of scenarios increases, the size of the model also increases, which is because of the deterministic equivalent increase proportionality with respect to the increments in the number of stages and/or realizations of the random variable. Because the Branch-and-Bound solver is used, the number of branches also increases when the number of scenarios does, even if the maximum number of heuristics is used for pre-solving the problem. For the solutions, we defined the here-and-now solution of the recourse problem (RP) as Equation (23), which indicates the optimal solution before the realization of the random vector \(\xi\). This approach seeks to find an optimal policy over all the possible scenarios in the stochastic programming model [26]. \[RP=min\ E_{\xi}z(x,\xi) \tag{23}\]  
The wait-and-see solution, expressed as Equation (24), solves the expected value of the stochastic programming model. This solution considers that all the possible outcomes \begin{table}
\begin{tabular}{p{113.8 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Number of Scenarios for Each Random Variable** & **Number of Scenarios in the Deterministic Equivalent** & **Variables** & **Integer Variables** & **Constraints** \\ \hline
3 & 6561 & 91,854 & 65,610 & 130,733 \\ \hline
4 & 65,536 & 917,504 & 655,360 & 1,309,183 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the size of the models developed. Figure 1: Matrix structure of the developed models, developed in LINGO 19.0.  
of the random vector \(\xi\) are known, hence the non anticipativity constraints are dropped, in other words, the relaxation of the here-and-know solution [44]. \[WS=E_{\xi}[min\;z(x,\xi)] \tag{24}\]  
For a minimized problem, it is clear that the wait-and-see solution always will be less or equal to the here-and-now solution, and in a maximized problem, the wait-and-see solution will be greater or equal to the here-and-now solution. These valid inequalities are important because the deterministic equivalent of the stochastic programming is complicated to solve under the here-and-now solution. A good alternative is to use the wait-and-see solution as an approach that only solves a deterministic problem by dropping the non anticipativity constraints. Finally, the absolute difference between the RP solution and the WS solution is called the expected value for perfect information, denoted in Equation (25), which indicates the maximum amount that the decision maker is willing to pay to know the uncertainty in advance [26]. \[EVPI=|RP-WS| \tag{25}\]  
Table 3 indicates the input for the deterministic parameters in the multi-product AAP under uncertainty. The input values were used for solving the stochastic programming model with LINGO 19.0, and the model considers two families of products, each family having a demand and production capacity. Table 4 indicates the input for the stochastic parameters in the multi-product AAP under uncertainty. The values were obtained with Gaussian quadrature, proposed by [PERSON] [PERSON] [45]. The probabilities are the same for three discrete points and are 0.167, 0.66 and 0.167; meanwhile, in the case of four discrete points, they are 0.05, 0.45, 0.45 and 0.05.