# Unconstrained Estimation of Multitype Car Rental Demand
## 4 Case Study and Discussion
### Applying IMSM  
According to the rental price, the car type is classified as 1 under $40 per day, 2 within $40-70 per day, and 3 above $70 per day. Due to cost and time constraints, a random sampling method was used to conduct a questionnaire survey at some rental sites of China Auto Rental Inc. in Chongqing (see Figure 5). The sample size of this RP survey was 400, and 363 questionnaires were collected, of which 320 were valid and 220 of which included buyup or buydown. Discretize the survey data, and let \"very satisfied = 10\", \"satisfied = 8\", \"general = 5\", \"not very satisfied = 3\", and  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{8}{c}{\(\Delta_{1}\)} & \multicolumn{8}{c}{\(\Delta_{2}\)} & \multicolumn{8}{c}{\(\Delta_{3}\)} \\ \cline{  
\"not satisfied = 0\". The influencing factor score, utilities, and possibilities corresponding to various choice behaviors can be obtained (Table 5). Based on a week's data from the International Convention & Exhibition Center, the biggest site of China Auto Rental Inc. in the Nan'an district, EM, PD, MSM, and IMSM were used to repair the car rental demand, and the unconstrained repair results were compared with the real needs. The convergence conditions were \(\theta\)= 0.1 and \(\tau\)= 0.6, set in EM and PD (see Table 6). According to the results listed in the above table, when demand repair is not carried out, the average gap between constrained demand and actual demand is large, which is 10.1%. After various demand estimation methods are adopted, the average error of the obtained demand data is small, among which the average error of IMSM is the smallest, 3.06%. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{
\begin{tabular}{c} **Estimation** \\ **Method** \\ \end{tabular} } & \multirow{2}{*}{**1**} & \multirow{2}{*}{**2**} & \multirow{2}{*}{**3**} & \multirow{2}{*}{**4**} & \multirow{2}{*}{**5**} & \multirow{2}{*}{**6**} & \multirow{2}{*}{**7**} \\ \cline{3-3} \cline{5-8}  & & & & & & & & & \\ \hline \(IB(1,j,t)\) & & 45 & 46 & 53 & 51 & [43] & 49 & [47] \\ \(TU(1,j,t)\) & & 49 & 46 & 53 & 51 & [47] & 49 & [49] \\ \(I(1,j,t)\) & & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\  & EM & 51.76 (5.63\%) & & & & 51.59 (9.77\%) & & [52.16 (6.45\%)] \\  & PD & 50.11 (2.27\%) & & & & 49.84 (6.04\%) & & [50.7 (3.47\%)] \\  & MSM & 49.82 (1.67\%) & & & & 49.76 (5.87\%) & & [50.04 (2.12\%)] \\  & IMSM & 49.56 (1.14\%) & & & & 47.68 (1.45\%) & & [50.04 (2.12\%)] \\ \hline \(IB(2,j,t)\) & & 24 & 24 & 24 & 23 & [20] & 20 & 26 \\ \(TU(2,j,t)\) & & 24 & 24 & 24 & 23 & [22] & 20 & 26 \\ \(I(2,j,t)\) & & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\  & EM & & & & & 24.38 (10.82\%) & & & \\  & PD & & & & & 23.5 (6.82\%) & & & \\  & MSM & & & & & 23.27 (5.77\%) & & & \\  & IMSM & & & & & & 20.5 (6.82\%) & & \\ \hline \(IB(3,j,t)\) & & 12 & 14 & 12 & 12 & [10] & 15 & 15 \\ \(TU(3,j,t)\) & & 13 & 14 & 12 & 12 & [13] & 15 & 15 \\ \(I(3,j,t)\) & & 0 & 1 & 1 & 1 & 0 & 1 & 1 \\  & EM & 14.88 (14.46\%) & & & & 14.64 (12.62\%) & & \\  & PD & 14 (10.82\%) & & & & 13.63 (14.45\%) & & \\  & MSM & 12.85 (1.15\%) & & & & 13.56 (4.31\%) & & \\  & IMSM & 13.12 (0.92\%) & & & & 12.23 (5.92\%) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Repair results and errors analysis of 1 week.