# Slotting Optimization Model for a Warehouse with Divisible First-Level Accommodation Locations
## 8 Results
### Results by Subproblems  
For subproblem \(P_{1,1}\), the minimum number of locations to be divided corresponds to 16.17% of the total locations, a value that includes 24.14% locations with double position and 75.86% with a triple position. Table 10 shows the number of locations for each family and subfamily, in which the latter is described by the set \(S\) with index \(\varepsilon\), where \(\varepsilon\) indicates the number of regrouping of a family. From Figure 13, it is possible to notice that the largest number of locations belongs to family 3, which has the highest percentage of simple locations. This fact reveals that this family is the most requested during this period. On the contrary, family 2 represents the highest percentage of triple locations, from which it can be concluded that this family contains mostly SKUs of type \"B\" and \"C\". \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Family**} & \multirow{2}{*}{**Total [\%]**} & \multirow{2}{*}{**Subfamily**} & **Simple** & **Double** & **Triple Locations [\%]** & **Total [\%]** \\  & & & **Locations [\%]** & **Locations [\%]** & **Triple Locations [\%]** & **Total [\%]** \\ \hline \multirow{4}{*}{Family 1} & & \(s_{1}\) & 0.00 & 0.19 & 0.56 & 0.74 \\  & & \(s_{2}\) & 2.97 & 0.00 & 1.12 & 4.09 \\  & & \(s_{3}\) & 0.00 & 0.00 & 0.19 & 0.19 \\  & & \(s_{4}\) & 0.00 & 0.00 & 0.19 & 0.19 \\  & & \(s_{5}\) & 0.28 & 0.00 & 0.19 & 0.46 \\  & & \(s_{6}\) & 0.37 & 0.00 & 0.19 & 0.56 \\ \hline \multirow{4}{*}{Family 2} & \multirow{4}{*}{2.79} & \(s_{1}\) & 0.09 & 0.19 & 0.37 & 0.65 \\  & & \(s_{2}\) & 0.46 & 0.37 & 0.56 & 1.39 \\  & & \(s_{3}\) & 0.19 & 0.00. & 0.56 & 0.74 \\ \hline \multirow{4}{*}{Fily 3} & \multirow{4}{*}{70.72} & \(s_{1}\) & 1.86 & 0.00 & 0.19 & 2.04 \\  & & \(s_{2}\) & 30.67 & 0.56 & 1.12 & 32.34 \\  & & \(s_{3}\) & 5.86 & 0.56 & 0.56 & 6.97 \\  & & \(s_{4}\) & 0.74 & 0.00 & 0.74 & 1.49 \\  & & \(s_{5}\) & 12.55 & 0.00 & 0.37 & 12.92 \\  & & \(s_{6}\) & 13.29 & 0.74 & 0.93 & 14.96 \\ \hline \multirow{4}{*}{Family 4} & \multirow{4}{*}{3.72} & \(s_{1}\) & 0.00 & 0.19 & 0.19 & 0.37 \\  & & \(s_{2}\) & 0.09 & 0.19 & 0.19 & 0.46 \\  & & \(s_{3}\) & 0.56 & 0.00 & 0.00 & 0.56 \\  & & \(s_{4}\) & 0.56 & 0.00 & 0.19 & 0.74 \\  & & \(s_{5}\) & 1.21 & 0.19 For the subproblem \(P_{1.2}\), the design of the distribution of families per aisle follows the order described in Figures 14 and 15, where they are assigned into the warehouse from left to right. Furthermore, the described configuration considers the results obtained in \(P_{1.1}\) and the relationship generated in Figure 9.  
The difference between both scenarios is highly observable in the distribution by subfamily, specifically in the right section of the warehouse, since it corresponds to the assignment of the largest family in SKUs covering the largest number of aisles, as can be seen in Figure 15. The gradient color (from light to dark) represents from lower to higher weight by subfamilies. On the other hand, the division of locations is represented in Figures 16a and 17a, where the execution of the model \(P_{1.3}\), reveals that the divided locations should be applied to the end of each aisle, while the simple ones should be considered at the beginning. Figure 14: Representation of the family layout in the study warehouse for \(E_{1}\). The gray variation corresponds in (**a**) for each family and in (**b**) for each subfamily. Figure 13: Percentage representation for the results obtained in \(P_{1.1}\). Figure 15: Representation of the family layout in the study warehouse for \(E_{2}\). The gray variation corresponds in (**a**) for each family and in (**b**) for each subfamily. However, the stackability priority motivates a reorder within each aisle, without changing the family layout (see Figures 16b and 17b). Simple locations correspond mainly to type \"A\" SKUs, representing 86.6% of the total of these locations, while double and triple locations mainly represent type \"C\" with SKUs 76.19% and 76.52%, respectively. The rest are assigned for SKUs of type \"B\".