## 3 Methodology
### Using Association Rule Mining to Create Itemsets  
ARM is a datamining approach used in this paper to examine the large transactional databases available at the distribution centers to create itemsets with items frequently found together in the order picking transactions.  
Let \(I=\{i_{1}\), \(i_{2}\),..., \(i_{K}\}\) be a set of 0-1 binary variables referred to items; also let \(X,Y\in I\) and \(X\cap Y=\emptyset\). An association rule is expressed as an implication, which is expressed as follows:  
\[X\Rightarrow Y \tag{5}\]  
The set \(X\) is called the antecedent and Y is called the consequent. Within the context of our application, if \(X=\{SKU_{1}\), \(SKU_{2}\}\) and \(Y=\{SKU_{3}\}\), the association rule states that a transaction with items \(\{SKU_{1}\), \(SKU_{2}\}\) implies that \(\{SKU_{3}\}\) is also present in the transaction.  
The Apriori algorithm, introduced in [18], is a datamining technique designed to extract, in an efficient manner, frequent itemsets and useful association rules from large transactional datasets, such as those available in distribution center applications. The extraction of itemsets and association rules must satisfy minimum thresholds for support and confidence, two of the item relationship measurements of association rules obtained with the Apriori algorithm:  
* Support, \(\xi_{\theta}\): A number between 0 and 1. Support is the probability of measures the importance of a rule. For \(\mid\theta\mid\) = 1, \(\xi_{i}\) is the proportion of order picking transactions that contain item \(i\). For \(\mid\theta\mid\) = 2, \(\xi_{ik}\) is the proportion of order picking transactions that contain items \(i\) and \(k\), concurrently. For \(\mid\theta\mid\) = 3, \(\xi_{ikm}\) is the proportion of order picking transactions that contain items \(i\),\(k\) and \(m\), concurrently, and so forth. The minimum support threshold is expressed as \(\varepsilon\).
* Confidence, \(conf(X\Rightarrow Y)\): A number between 0 and 1. Confidence of a rule \(X\Rightarrow Y\) is calculated as \(\text{support}(X\cap Y)/\text{support}(X)\). Confidence is the likelihood that the order contains itemset (i.e., \(Y\)), given itemset (i.e., \(X\)) is already picked up in the same order.
* Lift: the strength of the rules. It is calculated as \(P(X\cup Y)/\{P(X)P(Y)\}\) for an association rule \(X\Rightarrow Y\). If lift \(>\) 1, \(X\) and \(Y\) are positively correlated with each other, and vice versa.  
Apriori discards sets that do not meet minimum support or minimum confidence conditions at the same time. It is essential to specify a threshold value to determine at which level an itemset is deemed to be nonfrequent. R statistical software is used for the implementation of the Apriori algorithm. The pseudo-code for Apriori is presented in Algorithm 1. The Apriori algorithm result is an itemset \(L_{s}\), which consists of those itemsets of size less than or equal to \(s\) that meet the minimum support threshold, \(\varepsilon\).  
When relatively small values of support (i.e., the sample space includes itemsets that are not frequent) are involved, the computational expense can become extreme due to the plethora of frequent itemsets. In contrast, we have only few frequent itemsets if the minimum support threshold is too high. This paper used a calibration experiment to find support values that are small enough to detect enough rules to make a difference in the total performance of the order-picking tasks. Nonetheless, the support values should not be too small. Otherwise, the computational algorithm time may increase significantly.