## 3 Proposed Model
### PrefixSpan Algorithm
It was found that the execution time depended on the minimum support setting. If the minimum support was smaller, the processing time was longer, and the resulting correlation rule was more common. Table 13 shows that the execution time of the PrefixSpan algorithm was better than those of the Apriori and FP-Growth algorithms. The four algorithms were executed in the same environment after setting the following hyperparameters: the PrefixSpan and MBHS settings were HMS = 25, HMCR = 0.9, and PAR = 0.1; the SA settings were initial temperature = 100, cooling rate = 0.99, and complete temperature = 0.01; and the GA settings were initial population size = 2000, pc = 0.2, pm = 0.08, and iteration = 200. Table 14 shows the experimentally finetuned parameters. Figure 5: Comparison of the execution times of the Apriori, FP-Growth, and PrefixSpan algorithms according to the percent of support. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithms** & **Execute Times (Second)** \\ \hline Apriori & 715.51 \\ FP-Growth & 92.18 \\ Prefixspan & 0.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Comparison of the execution time of Apriori, FP-Growth, and PrefixSpan. ### Discussion

To discuss the experiment, the PrefixSpan and MBHS, GA, and SA were applied to shelf layouts to evaluate profit and time consumption. Figure 6 shows a comparison of the profit volume and execution time as the number of iterations increased: Figure 6a profit volume of the algorithms and Figure 6b execution time of the algorithms. A comparison of the profit volume of the proposed method (PrefixSpan and MBHS), GA, and SA, respectively, were given the highest profit and convergence to the best answer when the number of iterations was higher. Comparison with the other algorithms shows convergence and stability within 100 iterations, which was less profit than the proposed model. In contrast, the aspect of the time consumption of the proposed and SA was similar within 50 iterations. The GA and NP-Hard dramatically increased until 200 iterations. The processing time of NP-Hard was longer when the number of iterations was higher. The product layout profitability, total closeness rating, and execution time of the three algorithms, proposed method as the PrefixSpan algorithm with mutation-based harmony search (PrefixSpan and MBHS), the GA, and the SA, respectively, were employed. Table 11 (where the y- and x-axes are the sum of the profit of the product layout and the cycle of the execution of each algorithm, respectively) shows a comparison of the profitability and reveals the best answer. Figure 6a shows the PrefixSpan and MBHS algorithms were better able to determine the suitability or profit of the product layout than the other two algorithms. A new multi-objective design was included to filter both the profit and closeness rating into unique functions in the search space. In addition, the PrefixSpan and MBHS algorithm showed the highest performance and enhance aspect in the other performance comparison. Therefore, the PrefixSpan and MBHS in Figure 6b show that among the algorithms, the GA and the SA were the most and least time-consuming, respectively. Regarding the number of NP-hard, the number of profits of the product layout is the least time-consuming, grew very rapidly even in the case of small-size products, and was unable to be solved when the iteration increased to the MBHS algorithm when it included the PrefixSpan algorithm to solve the initial set of the population in the heuristic search before reaching the search space. Moreover, the time consumption was reduced compared to others. SA was the worst-performing algorithm because the SA has the disadvantage of huge items, and the algorithm can be across the best answer because GA and SA are an initial set of random answers. Therefore, it can be concluded that the proposed multi-objective mutation-based harmony search algorithm was the most efficient one of the tested algorithms.