# Using Product Network Analysis to Optimize Product-to-Shelf Assignment Problems
## 4 Implementation and Experimental Results
### Sensitive Analysis
#### 4.3.3 Analysis of Attraction and Opportunity Items  
In this research, a product item is classified as attraction item, opportunity item, or trivial item based on its centrality measure and parameters of \(\alpha\) and \(\beta\). An item whose centrality value is higher than \(\alpha\) is called an attraction item. An item whose centrality value is between \(\alpha\) and \(\beta\) is called an opportunity item. Otherwise, an item is a trivial item. Figure 9 shows the relationship between the number of attraction items and threshold value \(\alpha\) in TBN using betweenness centrality. If \(\alpha\) is greater than 50, the number of attraction items will not exceed 12, which might be too few for location preference evaluation. Therefore, \(\alpha<50\) is suggested in this experiment when betweenness centrality is adopted. Table 7 shows the possible outcomes when changing parameters \(\alpha\) and \(\beta\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Minimum** \\ **Support Count** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ \end{tabular} & \begin{tabular}{c} **Isolated** \\ **Product** \\ \end{tabular} & \begin{tabular}{c} **Diameter** \\ \end{tabular} & \
\begin{tabular}{c} **Average Shortest** \\ **Distance** \\ \end{tabular} &
\begin{tabular}{c} **Clustering** \\ **Coefficient** \\ \end{tabular} \\ \hline
30 & 0.827 & 0 & 2 & 1.173 & 0.974 \\
40 & 0.707 & 3 & 3 & 1.257 & 0.951 \\
50 & 0.579 & 13 & 3 & 1.258 & 0.911 \\
60 & 0.374 & 20 & 3 & 1.445 & 0.829 \\
70 & 0.211 & 30 & 3 & 1.62 & 0.801 \\
80 & 0.118 & 47 & 5 & 1.886 & 0.767 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Five indexes under different minimum support count in CBN.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Minimum** \\ **Support Count** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ \end{tabular} & \begin{tabular}{c} **Isolated** \\ **Product** \\ \end{tabular} & \begin{tabular}{c} **Diameter** \\ \end{tabular} & \
\begin{tabular}{c} **Average Shortest** \\ **Distance** \\ \end{tabular} &
\begin{tabular}{c} **Clustering** \\ **Coefficient** \\ \end{tabular} \\ \hline
5 & 0.905 & 0 & 2 & 1.095 & 0.936 \\
10 & 0.621 & 0 & 3 & 1.389 & 0.856 \\
15 & 0.382 & 12 & 4 & 1.632 & 0.656 \\
20 & 0.124 & 29 & 4 & 1.936 & 0.491 \\
25 & 0.034 & 48 & 6 & 2.541 & 0.228 \\
30 & 0.007 & 80 & 7 Figure 10 shows the relationship between the number of attraction items and the threshold value \(\alpha\) in TBN using closeness centrality. If \(\alpha\) exceeds 170, the number of attraction items will not exceed 20, which might be too few for location preference evaluation. Therefore, \(\alpha<170\) is suggested in this experiment when betweenness centrality is adopted. Table 8 shows the possible outcomes when changing parameters \(\alpha\) and \(\beta\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\alpha\) & \(\beta\) & **Number of Attraction Items** & **Number of Opportunity Items** \\ \hline
30 & 10 & 28 & 37 \\
30 & 15 & 28 & 25 \\
30 & 20 & 28 & 18 \\
40 & 10 & 19 & 46 \\
40 & 15 & 19 & 34 \\
40 & 20 & 19 & 27 \\
50 & 10 & 12 & 53 \\
50 & 15 & 12 & 41 \\
50 & 20 & 12 & 34 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Number of attraction and opportunity items using betweenness centrality.  
Figure 10: Relationship between closeness centrality and \(\alpha\) value in TBN.  
Figure 9: Relationship between betweenness centrality and \(\alpha\) value in TBN.  
Attraction items bring customers into the store. Thus the number of attraction items should be maximized and their locations should remain consistent. By contrast, opportunity items should be moved to maximize cross-selling. Reassignment of a large number of opportunity items will entail considerable labor and expense, but limiting the number of opportunity items to be reassigned may limit the impact on purchasing.