# Unconstrained Estimation of Multitype Car Rental Demand
## 2 Unconstrained Estimation
### Improved Multitype Spill Model
#### 2.3.3 Customer Choice Probability  
The multitype spill model is based on traditional revenue management. The assumption is that people in reality are irrational. When the product price is lower than the willingness to pay, the customer will buy it immediately. Obviously, this assumption ignores the initiative of the decision-making subject. In reality, individuals have various cognitive deviations. If we do not proceed from the customer's choice behavior, we cannot  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Object**} & \multicolumn{8}{c}{**Condition Attribute**} & \multirow{2}{*}{**Decision Attribute**} \\ \cline{2-2} \cline{4-9}  
find the customer's car rental rules and the changing trend of demand, which is very unfavorable to the car rental company's ability to set prices and inventory in the later stage. Therefore, in order to describe the customer's choice behavior, the Logit model is used to solve the above problems. The utility maximization theory is assumed, and the utility size is used to measure the probability that a customer selects a certain type of vehicle. The various attributes of different types constitute a comprehensive utility value. It is generally believed that the greater the utility of a customer to select a particular type of vehicle, the greater the probability that such a vehicle will be selected. Customers choose a variety of types and make a choice after evaluating the utility based on various attributes. The customer's choice behavior can be described by a utility function \(U_{i}=v_{i}+\xi_{i}\), where \(v_{i}\) and \(\xi_{i}\) are the average utility and the random utility error, respectively, corresponding to the customer's selecting. Assuming that the customer is rational, then each customer will choose the product that will maximize utility. The probability that the customer chooses car type \(i\) is  
\[\mathrm{P}_{i}=\mathrm{P}(v_{i}\geq v_{h},\forall h\in I,h\
eq i)=\mathrm{P} (v_{i}+\xi_{i}\geq v_{h}+\xi_{h},\forall h\in I,h\
eq i)=\mathrm{P}(v_{i}+ \xi_{i}\geq\max_{\forall h\in I,h\
eq i}(v_{h}+\xi_{h})) \tag{23}\]  
Assuming that \(\xi_{i}\) are mutually independent and obey Gumbel distribution, then the probability variable deviations of the two independent Gumbel distributions are subject to the Gumbel distribution, which yields the general form of the multinomial logit model:  
\[P_{i}=\frac{e^{v_{i}}}{\sum\limits_{i\in I}e^{v_{i}}} \tag{24}\]  
Since the RP survey is based on actual conditions, the survey respondents are customers who return cars to the rental sites. In the design of the questionnaire, the actual upgrade or downgrade has been fully reflected. Therefore, the corresponding utility when the customer's rental car type is upgraded or downgraded is  
\[U_{i,l}=0.292c_{3}+0.292c_{5}+0.083c_{6}+0.125c_{7}+0.083c_{8}+0.125c_{9} \tag{25}\]  
If the customer did not lease type \(i\) and then rented the type \(h\), the probability of upgrading or downgrading is  
\[\mathrm{P}_{i,h}=\frac{e^{v_{i,h}}}{\sum\limits_{ \begin{subarray}{c}I=0\\ \
eq h\end{subarray} }^{I}e^{v_{i,l}}} \tag{26}\]  
where \(\mathrm{P}_{i,h}\) is the preference probability of buyup or buydown with car type \(h\) when the customer's choice of car type \(i\) is rejected, and \(\mathrm{P}_{i,0}\) is the leaving probability when car type \(i\) is rejected. The improved multitype spill model (IMSM) needs to calculate the complete spillage of all constrained data first and then calculate the demand transferring from other car types which are constrained according to the customer preference probability. Excluding this part is the spillage; then, the sum of cumulative order quantity and spillage will be the unconstrained estimation. \[CP(i,j,t)=\left\{ \begin{array}{ll}0,&I(i,j,t)=1\\ \int_{c}^{\infty}f_{i,t}(x)(x-c)dx,&I(i,j,t)=0\end{array} \right.