# Unconstrained Estimation of Multitype Car Rental Demand
## 4 Case Study and Discussion
### Discussion  
As described previously, the demand data was estimated by four unconstrained estimation methods, and the three forecast models were used to predict the demand in the fourth presale lead time interval; then, 12 sets of demand data were obtained. In contrast to the real demand in the fourth presale lead time interval (see Table 12), the relative errors of the various results can be calculated (see Table 13). From the real demand and the prediction results (as shown in Tables 12 and 13), the following can be seen. For different car types, from the perspective of the predicted effect, comparing the Holt-Winters model with the BP neural network model, the former has a greater volatility, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Forecast Model**} & \multirow{2}{*}{**Car Type**} & \multicolumn{5}{c}{**Unconstrained Estimation Method**} \\ \cline{3-8}  & & **EM** & **PD** & **MSM** & **IMSM** \\ \hline \multirow{3}{*}{Holt–Winters} & 1 & 12.31\% & \(-\)3.62\% & \(-\)10.10\% & \(-\)12.10 \\  & 2 & 20.69\% & 17.24\% & 8.62\% & 7.47\% \\  & 3 & 6.00\% & 47.45\% & 32.50\% & \(-\)6.67\% \\ \hline \multirow{3}{*}{BP Neural Network} & 1 & 23.19\% & 21.74\% & 18.84\% & 7.27\% \\  & 2 & 29.31\% & 25.86\% & 17.24\% & 15.52\% \\  & 3 & 36.00\% & 35.00\% & 20.00\% & 10.00\% \\ \hline \multirow{3}{*}{Hybrid Model} & 1 & 18.84\% & 11.59\% & 7.25\% & 1.45\% \\  & 2 & 25.86\% & 22.41\% & 13.79\% & 8.66\% \\  & 3 & 24.00\% & 40.00\% & 25.00\% & 5.00\% \\ \hline Holt–Winters & \multirow{3}{*}{All types} & 14.96\% & 11.56\% & 3.06\% & 9.50\% \\ BP Neural Network & & 28.57\% & 25.17\% & 18.37\% & 10.89\% \\ Hybrid Model & & 23.12\% & 19.72\% & 12.24\% & 4.76\% \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Relative errors of forecast demand for each car type and all car types. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Car Type**} & \multicolumn{5}{c}{\(\Delta_{4}\)} & \multirow{2}{*}{**Subtotal**} & \multirow{2}{*}{**Total**} \\ \cline{3-8}  & & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \multirow{3}{*}{EM} & 1 & 10 & 13 & 10 & 11 & 9 & 14 & 15 & 82 & \\  & 2 & 10 & 11 & 9 & 8 & 8 & 15 & 12 & 73 & 181 \\  & 3 & 4 & 4 & 2 & 3 & 3 & 5 & 5 & 26 & \\ \hline \multirow{3}{*}{PD} & 1 & 12 & 9 & 10 & 10 & 8 & 13 & 15 & 77 & \\  & 2 & 13 & 8 & 6 & 10 & 9 & 12 & 13 & 71 & 176 \\  & 3 & 3 & 4 & 2 & 3 & 4 & 5 & 7 & 28 & \\ \hline \multirow{3}{*}{MSM} & 1 & 12 & 9 & 9 & 8 & 9 & 13 & 14 & 74 & \\  & 2 & 11 & 10 & 7 & 8 & 7 & 11 & 12 & 66 & 165 \\  & 3 & 5 & 3 & 3 & 2 & 3 & 4 & 5 & 25 & \\ \hline \multirow{3}{*}{MSM} & 1 & 11 & 10 & 8 & 9 & 8 & 12 & 12 & 2 & \\  & 2 & 10 & 11 & 7 & 8 & 7 & 10 & 10 & 63 & 154 \\ \cline{1-1}  & 3 & 4 & 3 & 2 & 1 & 3 & 4 & 4 & 21 & \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Hybrid forecasting results based on different estimation methods. while the latter is more stable. From the perspective of the adaptability of the unconstrained estimation model, when the unconstrained estimation data obtained by MSM is used, the prediction results of the three methods are relatively stable, and the hybrid forecasting model can obtain the best results. The relative errors of the three models are respectively: 1.45%, 8.66%, and 5.00% (see data in the box of Table 13). For the forecasted total, the Holt-Winters model is better than the BP neural network model in predicting the effect. From the perspective of the adaptability of the unconstrained estimation model, the Holt-Winters model has the best prediction effect when using MSM to repair the data, while the three methods can all obtain better prediction results when using IMSM, among which hybrid model has the smallest error. Considered comprehensively, when using MSM for unconstraining repair data, the relative error of the Holt-Winters model for predicting total demand is 3.06%. However, looking at the corresponding subtype prediction results, we find that the relative error of the third type is 32.5%.