## 3 Item Assignment Based on Weighted Associations
### Effects of Profit and Frequency  
In Section 2, the final assignment of items for cross-selling was obtained based only on the correlations between the items. However, these associations between the items do not consider the profit increase due to cross-selling. Since all retailers are profit-seeking, increased profit is a more important factor than the strength of correlations among items for item assignment in a retail environment.  
Given BASE_ASSIGN, assume that the expected profits increased due to cross-selling are as shown in Table 2a. Then, the numbers considering both lifts and the profit is shown in Table 2b. Allocating item C next to item D promotes more profit than placing item C next to other items according to Table 2b. In the same sense, item B should be located next to item F.  
As shown in Figure 2a, the item assignment, denoted as PROFIT_ASSIGN, is an updated version from BASE_ASSIGN, considering profit as well as lifts. Logically, PROFIT_ASSIGN is a better assignment than BASE_ASSIGN from an economic viewpoint.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**(a)** & **A** & **B** & **C** & **D** & **E** & **F** \\ \hline A & - & 10 & 10 & 20 & 10 & 10 \\ B & 10 & - & 10 & 10 & 10 & 30 \\ C & 10 & 10 & - & 20 & 10 & 10 \\ D & 20 & 10 & 20 & - & 10 & 10 \\ E & 10 & 10 & 10 & 10 & - & 10 \\ F & 10 & 30 & 10 & 10 & 10 & - \\ \hline
**(b)** & **A** & **B** & **C** & **D** & **E** & **F** \\ \hline A & - & 16 & 0 & 26 & 0 & 12 \\ B & 16 & - & 11 & 14 & 0 & 39 \\ C & 0 & 11 & - & 26 & 15 & 0 \\ D & 26 & 14 & 26 & - & 0 & 13 \\ E & 0 & 0 & 15 & 0 & - & 13 \\ F & 12 & 39 & 0 & 13 & 13 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: (**a**) Expected profit increase due to cross-selling items. (**b**) Expected profit considering lifts.  
Figure 2: (**a**) Item assignment, denoted by PROFIT_ASSIGN. (**b**) Item assignment, denoted by PROF_FRE_ASSIGN.  
The PROFIT_ASSIGN in Figure 2a considers not only the associations between the items but also the profit increase due to cross-selling. Note that the increased profit applied to the item assignment problem is a unit profit for one occurrence of the purchased items. Considering the frequency of purchasing items, in addition to profit, is a realistic approach for the item assignment problem.  
Suppose that the number of transactions containing items cross-sold in a database is provided as shown in Table 3a. The numbers in Table 3 indicate how frequently the paired items are sold. Then, the values in Table 3b correspond to expected profits by taking into account the frequencies of occurrence as well as the lifts. The PROFIT_ASSIGN in Figure 3 needs to be changed when considering the frequencies of occurrence.  
Figure 3: Flowchart of the proposed hybrid genetic algorithm (HGA).  
For example, according to Table 3b, assigning item B next to item D can promote higher sales than other assignments. In the same way, item C should be located next to item E. The assignment denoted as PROF_FRE_ASSIGN in Figure 2b is an updated version from PROFIT_ASSIGN, exchanging B and E with C and F, respectively. Then, PROF_FRE_ASSIGN is a better assignment than PROFIT_ASSIGN when considering increased profit as well as frequency.