## 4 Numerical Study
### Comparison between the Association Rule Mining-Based and the Class-Based Methods  
We further designed and analyzed a statistical experiment to compare the ARM algorithm to the traditional class-based layout (CBL) method. In these experiments, the class-based method was applied separately to each of the weight categories (heavy, medium, light) to make a direct comparison to the ARM-based method. Two partitions were obtained from the dataset to test for repeatability of the proposed algorithms. SKUs were selected at random to form the partitions. The first partition (referred to as Scenario 1) included 39 orders with a total number of 8141 SKUs. The number of unique SKUs in this test scenario was 2638. The second partition (referred to as Scenario 2) included 39 orders with 8122 SKUs with 2721 unique SKUs. For the implementation of the ARM-based heuristic's step 1, Scenarios 1 and 2 were divided into three weight classes: heavy, medium and light. Tables 2 and 3 show the weight distributions per class. For Scenario 1, Class 1-1 accounts for 30.40% of all the SKUs in the sample dataset containing 802 SKUs with weight greater than 9 pounds; Class 1-2 includes 756 SKUs with weight between 5.28 and 9 pounds, which is 28.66% of all the sample SKUs; Class 1-3 has 1080 SKUs with weight less than 5.28 pounds, which is 40.94% of all the SKUs in the selected sample. For Scenario 2, Class 2-1 accounts for 29.58% of all the SKUs in  
Figure 1: The distribution center under study follows an S-shape route. the sample dataset containing 805 SKUs with weight greater than 8.85 pounds; Class 2-2 includes 902 SKUs with weight between 5 and 8.85 pounds, which is 33.15% of all the sample SKUs; Class 2-3 has 1014 SKUs with weight less than 5 pounds, which is 37.27% of all the SKUs in selected sample. The application of the ARM-based heuristic found a total of 74 association rules in Scenario 1 (i.e., 10 rules in Class 1-1, 48 rules in Class 1-2 and 18 rules in Class 1-3), and a total of 59 association rules in Scenario 2 (i.e., 14 rules in Class 2-1, 32 rules in Class 2-2, and 13 rules in Class 2-3). Figures 2-4 show the plots of association rules in each weight class for Scenario 1. The numbers in these plots are the SKUs, while the circles represent the associations between SKUs. The size of circles is proportional to the level of support of the rules (i.e., a larger circle size represents a higher level of support). Thus, we can see that SKU 83,626 is frequently ordered together with SKU 103,427 in Figure 2. Additionally, the color of the circles is indicative of the lift of the rule (i.e., a darker circle color represents a higher lift). The arrow starts from the antecedent of the rule and ends at the consequent of the rule. It is important that the family of SKUs can be discovered from the results of ARM analysis. For instance, SKUs 10,829, 83,626, 103,427, 241,880 and 249,707 in the upper right corner of Figure 2 can be regarded as the members of a family, since SKU 103,427 is frequently ordered together with the others. Moreover, such SKUs are assigned to the slots in proximity by the heuristic method proposed in this paper. Similar trends are observed in the plots we obtained for Scenario 2, but we omitted these figures from this paper due to space limitations. Figures 5 and 6 show the average order picking times for Scenarios 1 and 2, as generated by CBL and ARM. Table 4 shows the mean order picking completion times and standard deviation for Scenarios 1 and 2. As the results show, the mean order picking time decreased from 3345.4 to 3006.9 s in Scenario 1, which represents a 10.1% improvement by applying the ARM-based method over the class-based layout method. Similar trends were observed in the second scenario. The mean order picking time decreased from 3489.0 to 3010.8 s in Scenario 2, which represents a 13.7% improvement by applying the ARM-based method over the class-based layout method.