## 4 Results and Sensitivity Analysis
### Results
Here, \(k=1\) indicates a family of products 1 and \(k=2\), a family of products 2. In Table 5, the comparison of both multi-stages developed models with respect to the here-and-now (RP) and wait-and-see approach (WS) is shown, as well as the expected value for perfect information (EVPI). Finally, CPU time and the number of iterations for solving each problem are reported. Here, it is clear that the second proposal model is more efficient and tractable for the solver. Appendix A shows the soluti \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Three Possible Realizations** & **Four Possible Realizations** \\ \hline \(D^{s}_{(t)1}\) & 330, 350, 370 & 330, 340, 360, 370 \\ \(\mathcal{C}P^{s}_{(t)1}\) & 8, 10, 12 & 8, 9, 11, 12 \\ \(D^{s}_{(t)2}\) & 230, 250, 270 & 230, 240, 260, 270 \\ \(\mathcal{C}P^{s}_{(t)2}\) & 7, 8, 9 & 7, 8, 9, 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input data for the stochastic programming model: demand and production capacity. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value Family of Products 1** & **Value Family of Products 2** \\ \hline \(C_{I(k)}\) & 70 & 65 \\ \(C_{X(k)}\) & 100 & 85 \\ \(C_{B(k)}\) & 200 & 175 \\ \(C_{W(k)}\) & 16,000 & 16,000 \\ \(LS_{k}\) & 0.9 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input data for the stochastic programming model: inventory, production, backlog, workers costs and service level. From Table 5, it is noted that the RP solution was the same for all the cases; however, the WS solution changes and, therefore, the EVPI. In Model-Ila and Model-IIb, the WS solutions were higher than Model-Ia and Model-IIb, respectively, which reduced the EVPI for the models with bounds, and then, the uncertainty was reduced. This has sense since we made a bound in the decision variables. On the other hand, the advantage when the bounds were used is noticed in the CPU time and iterations needed to solve the problems. For example, Model-Ib required more than half a million iterations and nearly 11 min for solving the deterministic equivalent, and Model-IIb only took less than 6 min and nearly 150 thousand iterations. In Figure 2, the percentage GAP between Model-Ia versus Model-Ila and Model-Ib versus Model-IIb is compared. The CPU time GAP was calculated, as Equation (26), and it is noted when the model increases in size, the use of the upper and lower bounds in the first-stage decision variables decreases to 52.38% of the CPU time when four possible realizations of random variables occur. \[CPU\;time\;GAP=\left|\frac{CPU\;time\;(Model-Ib)-CPU\;time\;(Model-Ia)}{CPU\;time\;( Model-Ia)}\right| \tag{26}\]  
Figure 2: Percentage GAP in CPU time and iterations for the developed models. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Criteria** & **Model-Ia** & **Model-Ila** & **Model-Ib** & **Model-IIb** \\ \hline RP solution & 1,037,470 & 1,037,470 & 1,037,470 & 1,037,470 \\ WS solution & 827,653.5 & 918,092.3 & 810,600.3 & 952,798.5 \\ EVPI & 209,816.5 & 119,377.7 & 226,869.7 & 84,671.51 \\ CPU time (s) & 17.02 & 15.96 & 690.67 & 328.87 \\ Iterations & 63,786 & 44,264 & 666,275 & 145,519 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the results of the models developed. Equation (27) calculated the iterations GAP, and it is noted when the model increases in size, the use of the upper and lower bounds in first-stage decision variables decreases in more than 78.16% of the number of iterations. \[Iterations\ GAP=\left|\frac{Iterations\ \ (Model-lb)-Iterations\ \ (Model-la)}{Iterations\ \ (Model-la)}\right| \tag{27}\]  
When the model only considers three possible outcomes of the random variables, the advantage is even noted because the iterations were reduced by around 31% and the CPU time by 6%. This indicates that the use of bounds in decision variables improves the speed for solving the stochastic programming models.