## 2 The Reliability Based Rescue Resource Scheduling Model
### Analysis of Travel Time and Reliability
#### 2.2.2 Travel Time of a Rescue Route  
A rescue route shall be a simple route, and its travel time can be expressed by the travel time of each link and intersection included in the route. The travel time of a rescue route \(k\) can be formulated as follows.  
\[t2_{k_{rs}}=t1_{ir}+\sum_{e_{mn}\in E}t_{mn}\delta_{mn}^{k_{rs}}+\sum_{v\in V }d_{v}\delta_{v}^{k_{rs}} \tag{3}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Description** \\ \hline \(EM_{ir}\) & The supply quantity of rescue center \(r\) for resource type \(i\) \\ \(k_{rs}\) & Index of routes between rescue center \(r\) and accident location \(s\) \\ \(t2_{k_{rs}}\) & Travel time of route \(k_{rs}\) from the rescue center \(r\) to the accident location s \\ \(R_{k_{rs}}\) & Travel time reliability of route \(k_{rs}\) from the rescue center \(r\) to the accident location s \\ \(\delta_{mn}^{k_{rs}}\) & A binary variable. \(\delta_{mn}^{k_{rs}}\) = 1 means link \(e_{mn}\) on route \(k_{rs}\), otherwise \(\delta_{mn}^{k_{rs}}\) = 0 \\ \(\delta_{v}^{k_{rs}}\) & A binary variable. \(\delta_{v}^{k_{rs}}\) = 1 means intersection \(v\) on route \(k_{rs}\), otherwise \(\delta_{v}^{k_{rs}}\) = 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Decision variables.  
Assuming that the travel time of each link and intersection is independent of each other, the expected value and standard deviation of the travel time for route \(k\) can be expressed as follows:  
\[E(t2_{k_{rs}})=\sum_{e_{mn}\in E}t_{mn}^{0}\delta_{mn}^{k_{rs}}+\sum_{v\in V}d_{v }^{0}\delta_{mn}^{k_{rs}}+E(t1_{ir})+\sum_{e_{mn}\in E}E(\xi_{mn})\delta_{mn}^{ k_{rs}}+\sum_{v\in V}E(\psi_{v})\delta_{v}^{k_{rs}} \tag{4}\]  
\[D(t2_{k_{rs}})=D(t1_{ir})+\sum_{e_{mn}\in E}D(\xi_{mn})\delta_{mn}^{k_{rs}}+\sum _{v\in V}D(\psi_{v})\delta_{mn}^{k_{rs}} \tag{5}\]  
where \(E(\cdot)\) and \(D(\cdot)\) are the expected value and standard deviation of random variables, respectively.  
According to the central limit theorem, the travel time of a rescue route can be considered to obey the normal distribution with the expected value of \(E(t2_{k_{rs}})\) and standard deviation of \(D(t2_{k_{rs}})\)[36; 37].  
\[t2_{k_{rs}}\sim N\Big{(}E(t2_{k_{rs}}),D(t2_{k_{rs}})^{2}\Big{)} \tag{6}\]  
Meanwhile, the probability density function for the travel time of a rescue route is as follows.  
\[\phi(t)=\frac{1}{\sqrt{2\pi}D(t2_{k_{rs}})}\int_{-\infty}^{t}\exp\Bigg{(}- \frac{(t-E(t2_{k_{rs}}))^{2}}{2D(t2_{k_{rs}})^{2}}\Bigg{)}dt \tag{7}\]  
5. Route travel-time reliability  
According to the travel characteristics of dangerous goods accident rescue, the travel time reliability of a rescue route is the probability that the rescue vehicle can reach the accident location within the maximum tolerable time, which can be expressed as follows.  
\[R_{k_{rs}}=P\big{\{}t2_{k_{rs}}\leq RT_{i}\big{\}}=\phi(RT_{i}) \tag{8}\]  
Generally, the more routes that can be selected between the rescue center and the accident location, the greater the possibility that the rescue vehicle reaches the accident location within the maximum tolerable time, and the higher the reliability. If there are multiple routes between the rescue center and the accident location, it can be considered as a parallel system, and the multi-route travel time reliability can be formulated as follows [38].  
\[R_{rs}=1-\prod_{k_{rs}\in K_{rs}}(1-R_{k_{rs}}) \tag{9}\]