## 3 Methodology  
### Formulating and Solving the Storage Location Assignment Problem  
Let us consider the Storage Location Assignment Problem in Liu [7]. This problem is represented by a Quadratic Generalized Assignment Model. This model seeks to determine the optimal storage location of SKUs, herein referred to as \"items\", for inputs, such as the similarity coefficients between items, the throughput-to-storage ratios, the distance between pairs of storage slots and the distance from the input/output point to the storage slots. Unlike Liu [8], this paper computes the similarity coefficients by using ARM, and thereby the resulting mathematical programming model is as follows:  
\[\text{Min}\ Z=\frac{1}{2}\sum_{i=1}^{K}\sum_{j=1}^{P}\sum_{k=1}^{K}\sum_{l=1}^ {P}t_{i}c_{ikl}d_{jl}x_{ij}x_{kl}+\ \sum_{i=1}^{K}\sum_{j=1}^{P}t_{i}c_{i}r_{j}x_{ij}\]  
\(d_{jl}=\) the picker's travel distance between slots \(j\) and \(l\). \(r_{j}=\) the relative distance of slot \(j\) to the input/output point. \(\varsigma_{i}=\) the support of item \(i\) (obtained from association rule mining). \(\varsigma_{ik}=\) the support of itemset \(\{i,k\}\) (obtained from association rule mining). The objective function in Equation (1) minimizes the total expected travel distance to complete all order picking operations in a given amount of time. The first term in Equation (1) represents the expected distance to travel from slot \(j\) to slot \(l\) provided that item \(i\) is assigned to slot \(j\) and item \(k\) is assigned to slot \(l\). The parameter \(t_{i}\) represents the rate at \(u\) which an item is ordered. The parameter \(\varsigma_{ik}\) is the probability, or support, of having items \(i\) and \(k\) together in an order as measured by ARM. A high value of the parameter \(\varsigma_{ik}\) indicates that the item \(i\) is ordered together with item \(k\) with higher frequency. In contrast, a low value indicates that item \(i\) is ordered together with item \(k\) with lower frequency. The parameter \(d_{jl}\) is the distance traveled from slot \(j\) to pick up item \(i\) to slot \(l\) to pick up item \(k\). Therefore, \(t_{i}\varsigma_{ik}d_{jl}\) estimates the expected distance traveled from slot \(j\) to slot \(l\). The second term in Equation (1) represents the expected distance to travel from the input/output point to slot \(j\) provided that item \(i\) is assigned to slot \(j\). The parameter \(\varsigma_{i}\) represents the support value of each item \(i\) as determined by the ARM algorithm. A high value of the parameter \(\varsigma_{i}\) indicates that item \(i\) is ordered with a higher frequency. In contrast, a low value of the parameter \(\varsigma_{i}\) suggests that item \(i\) is ordered with a lower frequency. We refer the reader to Section 3.2. for more details on how the parameters \(\varsigma_{i}\) and \(\varsigma_{ik}\) are obtained from ARM. Equations (2)-(4) are the constraints for this mathematical model. The constraint in (2) ensures that one item is assigned to precisely one slot. The constraint in (3) ensures that one slot contains exactly one item. The constraint in (4) restricts the decision variable to assume zero or one values. The SLAP model formulated in (1)-(4) is NP-hard; thus, finding the optimal solution for the most common applications is exceptionally challenging. [PERSON] [8] proposed a novel heuristic to achieve a quick practical layout. [PERSON]'s heuristic uses three phases: ranking, clustering and interchanging. A modified version of the heuristic is presented below, which incorporates parameters obtained with ARM. [PERSON] [9] explains that weight is a critical item attribute that must be considered to maintain the integrity of products while building an order. Order pickers built their pallets by placing the heaviest items at the bottom of the pallet to give a solid foundation to the pallet; lighter items are placed on heavier items, with the lightest items placed at the top of the pallet. Order pickers try to follow this sequence to reduce the number of damaged products.