## 1 Introduction
While these weighting approaches offer insight into an item assignment for a given layout, they failed to include item-item associations with item-specific information. It would be more realistic and desirable if an item assignment method could incorporate the existence of practical item-specific information, such as profit and sales volume. In [PERSON] et al., they took into consideration difference in selling possibility depending on allocated location, but we, ignoring such difference, rather focus on interaction among items depending on distance. The rationale is that we focus on practical settings with numerous sites and items as in retail and e-commerce. This study utilized association rule mining in the item assignment problem in a setting with a given layout by considering the weights of items according to profit and frequency counts. Specifically, this study aimed to improve the existing approaches of association rule mining for the item assignment problem using the following two criteria. First, an item assignment plan must consider not only the strength of correlation between items but also the profit incurred by cross-selling. The measure of association between items is called lift. Assuming that X and Y are the antecedent and consequent for item sets, respectively, lift (X \(\rightarrow\) Y) measures the possibility of cross-selling Y given that X was previously purchased. The measure lift (X \(\rightarrow\) Y) can be also written as \(P(X\cap Y)/(P(X)P(Y)\), which relates to point-wise mutual information. When lift (item A \(\rightarrow\) item B) and lift (item A \(\rightarrow\) item C) have the same association strength, item A can be located next to either item B or item C. However, if the profit gained by cross-selling item B with item A is higher than by cross-selling item C with item A, it is reasonable that item A should be located next to item B to maximize sales. Second, an assignment plan also needs to consider the purchasing frequency of items. Since the association among items is expressed as a ratio of probabilities, it is possible that lower frequency items have a stronger association with each other than higher frequency items. From an economic viewpoint, weakly correlated items with a high frequency can be more valuable than strongly associated items with a low frequency. In other words, items with low profit can still be beneficial to retailers if the items are sold quite frequently. Given layouts and item-purchase frequencies, an effective item arrangement is quite important as well. The task of finding an effective item assignment, which is computationally intensive, involves searching for the best assignment with the highest potential for cross-selling. Solvable by searching for all assignments in a factorial way, the item assignment problem can be viewed as a special case of the quadratic assignment problem (QAP) [22], which can be formulated as a mixed-integer linear programming (MILP), a well-known combinatorial optimization problem. QAP is NP-hard, so no exact algorithm can solve the problem in polynomial time. An exact algorithm for QAP such as branch-and-bound and cutting plane methods is possible for a small size \(n\). Recently, numerous researches have been made to develop well-performing heuristic and metaheuristic methods, which can find high-quality solutions in a reasonable time but with no optimality guaranteed. [PERSON] proposed robust tabu search (RTS) producing high-quality solutions with fast comparison of neighborhood in \(0(n^{2})\) instead of \(0(n^{3})\), in which \(n\) is the problem size [23]. [PERSON] et al. proposed breakout local search (BLS), which exploits local search and adaptive perturbation strategies to explore the search space [24]. Their subsequent research also presented a population-based memetic algorithm for QAP, which integrates BLS within the population-based evolutionary algorithm.