# Sales Forecasting for Fashion Products Considering Lost Sales
## 4 Numerical Experiments
### Results and Discussion
#### 4.5.3 Further Discussion  
In this subsection, further conclusion from the model for the fashion managers is provided. The conclusion consists of two parts: measure of product competitiveness and auxiliary index for inventory decision. Three important indicators are introduced first: average conversion rate \(\gamma_{a}\), marginal conversion rate \(\gamma_{m}\) and demand satisfied proportion \(\gamma_{d}\). The expression of each indicator is as follows:  
\[\gamma_{a}(I)=\frac{S(I)}{I}, \tag{10}\]  
\[\gamma_{m}(I)=S(I)-S(I-1), \tag{11}\]  
\[\gamma_{d}(I)=\frac{S(I)}{D}, \tag{12}\]  
where \(I\in N^{+}\), \(S(0)=0\), \(\gamma_{a}\) is the converting ratio of the total inventory to sales, and \(\gamma_{m}\) is the converting ratio of a unit of increased inventory to sales, also known as the marginal conversion rate. \(\gamma_{d}\) is defined as the proportion of satisfied demand. All ratios are related to \(q\) and \(\alpha\) in the proposed model. Figures 4 and 5 display the relationship between inventory and three ratios: the average conversion rate, marginal conversion rate and demand-satisfied proportion over the shoes and belts datasets. Figure 4a-c plots the curves of three clusters in the shoe dataset. The demand set uses the same method shown in Figure 3, where the three cluster in the shoe dataset have an average demand of 106, 89 and 90, respectively. The setting of Figure 5 is the same as Figure 4.  
In general, the graph shows a similarity based on the product category. In terms of \(\gamma_{m}\), major commodities, described in Figures 4c and 5b, have a relatively high \(\gamma_{m}\) and decline rapidly when the inventory level exceeds the demand. Common commodities and targeted commodities have smooth curves compared with major commodities, as shown in Figures 4a,b and 5a,c. Further, when the inventory level is the same, more demand is satisfied in major commodities than in the other two commodities. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline \multirow{2}{*}{**Category Method**} & \multicolumn{3}{c}{**Cluster 1**} & \multicolumn{3}{c}{**Cluster 2**} & \multicolumn{3}{c}{**Cluster 3**} & \multicolumn{3}{c}{**Total**} \\ \cline{3-10}  & & **MAPE** & **MAE** & **MAPE** & **MAE** & **MAPE** & **MAE** & **MAPE** & **MAE** \\ \hline \multirow{4}{*}{Shoses} & TLs & **0.1531** & **6.3392** & **0.1359** & **6.9151** & **0.1481** & **4.5365** & **0.1482** & **5.6539** \\  & LR & 0.2610 & 10.2327 & 0.2861 & 9.4790 & 0.2012 & 5.6482 & 0.2392 & 8.1336 \\  & GBDT & 0.1648 & 8.2280 & 0.1381 & 7.7411 & 0.1656 & 5.3156 & 0.1609 & 6.8932 \\  & SVR & 0.1631 & 9.2175 & 0.1482 & 10.0306 & 0.1530 & 5.8745 & 0.1564 & 7.9057 \\  & ANN & 0.2134 & 9.2214 & 0.2491 & 10.5849 & 0.1787 & 5.6153 & 0.2042 & 7.8846 \\ \hline \multirow{4}{*}{Belts} & TLs & 0.1468 & 8.1172 & **0.1488** & **6.3018** & **0.1365** & **3.8445** & **0.1466** & 5.9736 \\  & LR & 0.1110 & 4.8785 & 0.1559 & 6.4324 & 0.1537 & 4.2487 & 0.1530 & 5.9648 \\ \cline{1-1}  & GBDT & 0.1763 & 8.6786 & 0.1606 & 8.0955 & 0.1526 & 4.6342 & 0.1602 & 7.5236 \\ \cline{1-1}  & SVR & 0.2134 & 9.6330 & 0.1934 & 11.8215 & 0.1457 & 4.8524 & 0.1862 & 10.4831 \\ \cline{1-1}  & ANN & **0.1087** & **4.1958** & 0.1557 & 6.4796 & 0.1476 & 4.0534 & 0.1516 & **5.9291** \\ \hline \hline \end{tabular}  
* Notes; the column ‘Total’ is the forecasting result of all samples in clusters 1, 2 and 3 in the shoes and belts datasets.  
\end{table} 
Table 5: Forecasting performance of the five forecasters and one benchmark over two categories in the test data. (1) Measure of product competitiveness  
The average conversion rate when the inventory level is equal to demand is denoted by \(\gamma_{a(D)}\). \(\gamma_{a(D)}\) is defined as the inventory conversion level of this SKU, and also can be understood as the product competitiveness of this SKU. Through computing, \(\gamma_{a(D)}\) over each cluster in shoes is 0.6232, 0.6908 and 0.7678, respectively. For belts, the ratios are 0.6561, 0.7532 and 0.5771 for clusters 1 to 3.