## 6 Discussion of Results  
Table 7 shows the dates when each product reaches its demand in the DC and \(Tmax\) and the computational effort spent by each method. When analyzing Table 7, it is observed that the Exact Method met the demand of most products before the other approaches in the shortest computational time. This demonstrates that for a MILP, good mathematical modeling, together with a good solver may have superior performance to other methods, especially in smaller dimensions, such as this case study. APS required more time to meet the products' demand requirements. New production cycles for a given product start only after the quarantine period has expired. This behavior resulted in a production plan that extrapolated the predicted time horizon. The product \(p_{1}\), which has the longest quarantine time and is also used as input for the production of \(p_{2}\), determines the \(Tmax\). In the Exact Method and in the GA, it was sought to meet the demand of \(p_{2}\) while the production of \(p_{1}\) remains in quarantine. However,  
Figure 31: \(p_{2}\) stock behavior at the distribution center—Exact Method. *: corresponds to the moment of meeting the demand. in the APS, the decision was to prioritize meeting the demand of \(p_{1}\) before supplying the demand of \(p_{2}\), leaving its solution unfavorable. Among the APS limitations, the computational cost due to solving complex temporal constraints and sequential decisions is worth mentioning, which significantly increases the time required to find a solution compared to the Exact Method and the GA. Since APS models problems using constraint programming, its multiple temporal and conditional dependencies contribute to expanding the search space. These factors impact both the solution time and the quality of the results, making classical approaches more competitive for this type of problem (see Table 7). In addition, APS has accumulated a large stock of products in the factories. The GA algorithm and the Exact Method do not present residual stocks. Figures 32-34 graphically illustrate the arrival of products in the DC to meet demand. Figure 32 shows that the decision of the APS to supply the demand of \(p_{1}\) before \(p_{2}\) results in a gap of 21 days where no arrival of products occurs in the DC. Figures 33 and 34 show the increase of products in the \(DC\) stock for both the GA and the Exact Method, so one can see that the demands are met within the time horizon at \(d_{26}\). The GA is more flexible in terms of modeling than the APS, although it still depends on adjustments to fit changes. The Exact Method stands out for its flexibility when adaptations are necessary. It demonstrates a more direct and efficient modeling process independent of the size of the problem. We can easily add new products and factories. APS showed the worst performance among the three methods and did not obtain a solution nor come close to the optimal, evidencing limitations in computational efficiency and the quality of solutions found. These difficulties in obtaining the solution by APS demonstrate its ineffectiveness in treating this combinatorial optimization problem. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(P\) & APS & GA & Exact Method \\ \hline \(p_{1}\) & 44 & **26** & **26** \\ \(p_{2}\) & 65 & **8** & 12 \\ \(p_{3}\) & **10** & 26 & 25 \\ \(p_{4}\) & **16** & 22 & 25 \\ \(p_{5}\) & 39 & 26 & **22** \\ \(p_{6}\) & 41 & 24 & **22** \\ \hline \(Tmax\) & 65 & **26** & **26** \\ Time (s) & 444.64 s & 24.92 s & **4.42 s** \\ \hline \hline \end{tabular} Bold highlights the best performance.  
\end{table} 
Table 7: Meeting the demand for each product in days and computational effort in seconds. Figure 32: Stock evolution in the \(DC\) to meet demand—APS. Although the GA is an evolutionary algorithm, it achieved the optimal result, demonstrating its capacity to explore the solutions space. The Exact Method performs best, achieving the optimal result in the shortest execution time of the compared approaches.