# Unconstrained Estimation of Multitype Car Rental Demand
## 2 Unconstrained Estimation
### Unconstrained Estimation Methods
#### 2.2.3 Multitype Spill Model  
The multitype spill model (MSM) takes into account the vertical recapture of different price grade requirements, where the buyer's behaviors of buyup and buydown indwell in different price levels. Therefore, the unconstrained estimation process based on MSM is closer to reality, which can effectively avoid the overestimation of real demand due to the repeated records of the revenue management system's actual observable demand between different car type prices. The spill model is mainly divided into two parts, the first part being the calculation of spillage, and the second part being the calculation of unconstrained demand [35]. The specific calculation process is as follows:  
1. Check if the rental demand is constrained in \(\Delta_{t}\). When \(I(i,j,t)=1\), the rental demand is not constrained and no repair is performed; otherwise, the repairing process is performed, go to next step. \[I(i,j,t)=\left\{\begin{array}{l}1\,,\,C(i,j,t)-B(i,j,t)<0\\ 0\,,\,C(i,j,t)-B(i,j,t)\geq 0\end{array}\right.\] (11)
2. Parameters initialization. The distribution parameters of the demand in \(\Delta_{t}\) are initialized by the observable \"unconstrained demand\", which is the same as the EM algorithm.
3. Calculation of spillage and cumulative spillage. According to the assumptions, the customer's willingness to pay is characterized from low to high car type, so the spillage should be calculated from the lowest car type.  
* For a single car type, cumulative spillage can be determined by the following formula: \[SU(1,j,t)=\left\{\begin{array}{l}0\,,\,I(1,j,t)=1\\ \int_{c}^{\infty}f_{i,t}(x)(x-c)\mathrm{d}x\quad,\,I(1,j,t)=0\end{array}\right.\] (12) where \(c=IB(i,j,t),f_{i,t}(x)\) is the probability density function of unconstrained demand for car type \(i\) in \(\Delta_{t}\). When \(I(1,j,t)=0\), rental demand was constrained,  
\[SU(1,j,t)=\int_{c}^{\infty}f_{i,t}(x)(x-c)\mathrm{d}x=\sigma\phi(b)+(\mu-c)[1 -\Phi(b)] \tag{13}\]  
where \(b=\frac{c-\mu}{\sigma}\), \(\phi(x)\) is the distribution function of standard normal distribution, and \(\Phi(x)\) is the distribution function of standard normal distribution [36].  
Figure 2: Estimation process for constrained observations using PD.  
For a multitype situation, when \(i=1\), considering cumulative spillage \(SU(I,j,t-1)\) at review point \(t-1\) from car type 1 to \(I\).  
\[SU(I,j,t)=\left\{\begin{array}{ll}SU(I,j,t-1)&,\ I(i,j,t)=1\\ \int_{c}^{\infty}f_{I,t}(x)[x-c+SU(I,j,t-1)]\mathrm{d}x&,\ I(i,j,t)=0\end{array}\right. \tag{14}\]  
When \(I(i,T_{j},1)=0\),  
\[SU(I,j,t)=\int_{c}^{\infty}f_{I,t}(x)[x-c+SU(I,j,t-1)]\mathrm{d}x=\sigma\phi(b )+[SU(I,j,t-1)-\sigma b][1-\Phi(b)] \tag{15}\]  
When \(i>1\), calculate \(SU(i,j,t)\) by considering the cumulative spillage \(SU(i-1,j,t)\) at review point \(t\) from car type 1 to \(i-1\).  
\[SU(i,j,t)=\left\{\begin{array}{ll}SU(i-1,j,t)&,\ I(i,j,t)=1\\ \int_{c}^{\infty}f_{I,t}(x)[x-c+SU(i-1,j,t)]\mathrm{d}x&,\ I(i,j,t)=0\end{array}\right. \tag{16}\]  
When \(I(i,T_{j},1)=0\),  
\[SU(i,j,t)=\int_{c}^{\infty}f_{I,t}(x)[x-c+SU(i-1,j,t)]\mathrm{d}x=\sigma\phi(b )+[SU(i-1,j,t)-\sigma b][1-\Phi(b)] \tag{17}\]  
Calculate spillage.  
\[U(i,j,t)=\left\{\begin{array}{ll}SU(1,j,t),&i=1\\ SU(i,j,t)-SU(i-1,j,t),&i>1\end{array}\right. \tag{18}\]  
Get the repaired result, namely unconstrained estimation.  
\[IU(i,j,t)=\left\{\begin{array}{ll}IB(i,j,t),&I(i,j,t)=1\\ IB(i,j,t)+U(i,j,t),&I(i,j,t)=0\end{array}\right. \tag{19}\]