## 3 Proposed Model
### PrefixSpan Algorithm
Figure 4a shows the original product layout, and Table 12 shows the proposed product layout with shelf improvements. This layout was developed with the experimental results by using the profit volume calculated from Equation (3) and finding the suitability of the closeness rating. Table 11 shows the profit volume values following shelf adjustment, which were calculated by determining the solution of a rack from 19 departments. The solution comprised a 1 \(\times\) 19 matrix, with the best answer determined by the highest profit sum and the TCR determined by the original product layout (as illustrated in Figure 4a). The proposed product layout is shown in Figure 4b. According to the results of the experiment, the improvements of the proposed and conventional shelf layouts had profit volume values of 228,139.50 (THB) and 441,477.71 (THB), respectively, as shown in Table 12.

Figure 3: Mutation operation: (**a**) swap operator, (**b**) insert operator and (**c**) reverse operator. Table 12 presents the sum of the total closeness ratings. The results show that the original product layout had a TCR of 205, but the proposed product layout had a TCR of 350. In other words, the proposed method provided a higher profit volume and a better TCR value. This new shelf layout was additionally determined to be able to lead to an increase in customer satisfaction levels. In summary, the proposed method can effectively improve the placement of store shelves and increase future sales by using correlations between shopping lists and shelf layouts. ### Comparison of Algorithms

In this experiment, we evaluated the execution time of the Apriori algorithm, the FP-Growth algorithm, and the proposed PrefixSpan algorithm on the same large dataset, as shown in Table 13. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Iteration**} & \multicolumn{2}{c}{**PrefixSpan and MBHS(Proposed Algorithm)**} & \multicolumn{2}{c}{**GA**} & \multicolumn{2}{c}{**SA**} \\ \cline{2-10}  & **Profit** & **TCR** & **Time(s)** & **Profit** & **TCR** & **Time(s)** & **Profit** & **TCR** & **Time(s)** \\ \hline
1 & 311,235.91 & 105 & 0.03 & 255,865.51 & \(-\)5 & 3.82 & 249,138.03 & 225 & 0.008 \\
25 & 390,576.34 & 205 & 0.46 & 367,919.9 & 10 & 9.62 & 307,112.12 & 285 & 0.026 \\
50 & 396,787.27 & 180 & 0.84 & 372,192.96 & 35 & 13.38 & 324,907.14 & 160 & 0.031 \\
75 & 416,880.68 & 205 & 1.32 & 390,894.29 & 100 & 17.23 & 321,803.74 & 305 & 0.036 \\
100 & 433,363.4 & 210 & 1.69 & 390,894.29 & 100 & 20.37 & 341,573.87 & 275 & 0.042 \\
150 & 441,331.74 & 345 & 2.80 & 392,172.53 & 150 & 26.68 & 341,573.87 & 275 & 0.052 \\
180 & 441,477.71 & 350 & 3.12 & 392,172.53 & 150 & 30.05 & 341,573.87 & 275 & 0.058 \\
190 & 441,477.71 & 350 & 3.21 & 392,172.53 & 150 & 31.17 & 341,573.87 & 275 & 0.060 \\
200 & 441,477.71 & 350 & 3.55 & 392,172.53 & 150 & 32.29 & 341,573.87 & 275 & 0.062 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Profit volumes, total closeness rating, and execution time for the three algorithms. Figure 4: Product layouts: (**a**) the original product layout; (**b**) the proposed product layout. Table 13 displays a comparison of the time needed by the Apriori, FP-Growth, and PrefixSpan algorithms to correlate all occurring rules using the same dataset. We set a minimum support value of 1%. The PrefixSpan algorithm performed better than the other algorithms. The average execution times of the Apriori, FP-Growth, and PrefixSpan algorithms were 1411.02, 279.37, and 30.27 s, respectively. Figure 5 shows a comparison of the execution times of the Apriori, FP-Growth, and PrefixSpan algorithms with minimum supports of 20%, 40%, 60%, and 80%. According to the time processing results shown in Figure 5, at the minimum support of 20%, the average execution times of the Apriori, FP-Growth, and PrefixSpan algorithms were 715.51, 92.18, and 1.67 s, respectively. At the minimum support of 80%, the average execution times of the Apriori, FP-Growth, and PrefixSpan algorithms were 73.40, 11.15, and 0.29 s, respectively.