## 3 Proposed Model
### PrefixSpan Algorithm
From the prefix <A>, only the subsequence <_, B, D, C> was considered for sequential patterns of <A>. Note that <_,> represents the location of the prefix. <A> was found first in 2-length sequential patterns. From the DT database with <A> as the projected database, we found postfix sequences consisting of <_, B, C, D, F>, <_, D, C, E>, <_, B, D, C>, <_, F, C, B>. We found the following 2-length sequential patterns with <A> as the prefix: <A, B>: 3, <A, C>: 4, <A, D>: 3, <. A, F>: 2, and <A, E>: 1, where <pattern>: count represents the pattern followed by the frequency. The process of finding the 2-length sequential patterns of the < B>-projected database and <_n>_-projected database could be repeatedly recreated and searched, just like the <A>-projected database shown in Table 7. Then, sequential patterns from Table 7 were calculated to find support and confidence values. An example of the obtained results is shown in Table 8. \begin{table}
\begin{tabular}{c c} \hline
**Prefix** & **Projected Postfix Database** \\ \hline A & <_, B, C, D, F>, <_, D, C, E>, <_, B, D, C>, <_, F, C, B> \\ B & <A, C, D, F>, <_, D, C> \\ C & <_, D, F>, <_, E>, <_, B> \\ D & <_, F>, <_, C, E>, <_, C> \\ E & <_, F, A, B, D, C>, <_, A, F, C, B> \\ \hline \end{tabular}
\end{table} 
Table 6: Projected database. ### Adjacency Preferences

One of the most widely used approaches for measuring the effectiveness of the shelf layout problem is the REL chart [75; 76], which determines closeness ratings. A closeness rating reflects the correlation between shelves. Here, the proximity rating was defined as shown in Table 9, and an adjacency matrix (defined from the valuable rack pair) was created to determine the suitability of a shelf location. We used the PrefixSpan algorithm to match shelf correlations, support, and confidence, as shown in Table 8, where rack correlation pairs with a confidence value of greater than 60%, between 60 and 40%, between 40 and 20%, and between 20 and 10% are considered \"Absolutely necessary\", \"Especially important\", \"Important\", and \"Ordinary closeness\", respectively. The generated adjacency matrix is displayed in Table 10. Because shelf layout directly affects the sales of products, the adjacency matrix is used to measure the effectiveness of a shelf layout when in the process of determining proximity values. Proximity values alone should not be used to calculate the income or profit of a shelf; adjacency preferences should also be considered. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Department Pair** & **Support** & **Confidence** \\ \hline
1 & Meat Seafood–Produce & 16.19\% & 87\% \\
2 & International–Produce & 6.46\% & 87\% \\
3 & Dry goods and Pasta–Dairy eggs & 16.38\% & 83\% \\
4 & Breakfast–Dairy eggs & 13.87\% & 82\% \\
5 & Frozen–Produce & 31.49\% & 80\% \\
6 & Snacks–Produce & 33.78\% & 77\% \\
7 & Frozen–Dairy eggs & 30.05\% & 77\% \\
8 & Pets–Dairy eggs & 1.44\% & 69\% \\
9 & Household–Produce & 11.14\% & 57\% \\
10 & Breakfast–Snacks & 10.28\% & 61\% \\
11 & Pets–Beverages & 1.25\% & 60\% \\
12 & International–Partry & 4.20\% & 56\% \\
13 & Dry goods and Pasta–Frozen & 10.99\% & 55\% \\
14 & Bakery–Snacks & 15.25\% & 55\% \\
15 & Babies–Frozen & 2.57\% & 52\% \\
16 & Deli–Beverages & 12.93\% & 52\% \\
17 & Meat Seafood–Pantry & 8.97\% & 48\% \\
18 & International–Canned goods & 3.17\% & 42\% \\
19 & Deli–Bakery & 10.02\% & 40\% \\
20 & Dry goods and Pasta–Canned goods & 7.95\% & 40\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Example sequence pattern result. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Rating** & **Definition** & **Assigned Score** \\ \hline A & Absolutely necessary & 125 \\ E & Especially important & 25 \\ I & Important & 5 \\ O & Ordinary closeness & 0 \\ U & Unimportant & \(-25\) \\ X & Undesirable & \(-125\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: REL score chart.