## 1 Introduction  
Sequential pattern mining (SPM) is a sequential pattern query technique used to find data correlations during transactions; it was first presented by [PERSON] and [PERSON] [1; 2; 3]. Currently, due to competition in the retail industry, supermarkets are trying to improve their revenue by becoming more efficient and, therefore, profitable. Usually, supermarkets keep records of their customers' purchases. This shopping information can be used to find the most common purchasing patterns. Therefore, sequential pattern mining has been applied to determine purchasing patterns with common purchasing frequencies. Previous studies [4; 5; 6; 7; 8; 9] have presented models for determining the groupings of product series on shelves in order to maximize profits. Product layout can not only catch the eyes and draw the attention of consumers but also increase customer purchases and satisfaction. Previous studies [10; 11; 12; 13] have also implemented sequential pattern mining to explore frequent sequential patterns, which are considered necessary for determining the frequency of product purchases [14]. There are two key issues that must be considered in shelf management: shelf management strategies and purchases frequencies. Product shelf management strategies, which are under the umbrella of shelf layout, directly influence customers' purchasing decisions and affect store profits. It is also challenging to manage shelf displays to increase sales without first considering the product layout. The challenges of shelf layouts and patterns have caught the attention of market researchers. The authors of [15; 16; 17; 18] found that factors that may significantly influence customers' purchasing decisions include shelf space allocation and layout display. Personal or social factors may also influence the urge to purchase another product, a phenomenon that has been studied via Market Basket Analysis [19; 20; 21; 22; 23; 24; 25], which analyzes the correlation of products that customers are most likely to buy at the same time [26; 27; 28]. The management of shelf space was found to be a problem affecting consumers' purchasing behaviors; the effectiveness of shelf layouts between nearby supermarkets was compared to investigate how to increase profits and improve customer satisfaction [29; 30; 31]. To deal with this critical problem, a PrefixSpan algorithm [32; 33; 34] has been implemented. Prior to this algorithm, authors [35; 36; 37; 38] paid attention to the management of the location of each specific shelf in a supermarket. Although this method is able to increase sales, it does not consider possible correlations between products on different shelves by analyzing customers' shopping lists, a gap that could lead to crucial sales losses. Sequential patterns can be determined with various algorithms, such as the Apriori, GSP, Freespan [32; 33; 34], and projection-based PrefixSpan algorithms. The PrefixSpan algorithms can be used to mine sequence patterns and count the frequency of item sets in search spaces without the need for candidate generation. It can also be used for small-scaled problem solving to better increase consumer efficiency compared to other algorithms. In contrast to the GSP algorithm, the PrefixSpan algorithm can even be used to search for local minima and scan sequences patterns. To summarize, the PrefixSpan algorithm can effectively improve consumer time efficiency and filter redundant patterns, and the metaheuristic method can be used to aid the algorithm's pattern optimization. Aspects of the metaheuristic method have been employed to optimize answer or fitness values when solving complex problems. Well-known metaheuristic algorithms include the genetic algorithm (GA) [39], genetic programming (GP), evolutionary programming (EP), particle swarm optimization (PSO) [40; 41], simulated annealing (SA) [42], artificial bee colony (ABC) [43; 44], and harmony search (HS) [45; 46; 47; 48].