## 3 Solution Algorithms
### Obtaining the Set of Feasible Routes  
The travel time of the rescue route should meet constraint (12); that is, the rescue route from the rescue unit to the location of accident is bounded. In addition, when determining the vehicle routing problem from the rescue center to the accident location, the travel time and reliability of the route should be considered simultaneously. Therefore, designing a multi-path algorithm with a bound constraint is necessary in order to obtain a set of feasible routes. 1. Determination of the boundary constraint The rescue route of each rescue center should meet the constraint of the maximum tolerable time constraint (12). Therefore, according to the definition of travel time reliability of rescue vehicle route \(k_{rs}\) (Equation (8)) and the probability density function for route travel time (Equation (7)), the reliability constraint (12) can be transformed into the following: \[E(t2_{k_{rs}})+D(t2_{k_{rs}})\cdot\inf\Bigl{\{}G\Bigl{|}G=\varphi^{-1}(\alpha) \Bigr{\}}\leq RT_{i}\] (18) where \(\varphi^{-1}(\cdot)\) is the inverse function of the distribution function of the standard normal distribution, and \(\inf\{G\bigl{|}G=\varphi^{-1}(\alpha)\Bigr{\}}\) represents the infimum of function \(\varphi^{-1}(\alpha)\). Thus, the boundary constraint of a rescue route can be determined by Equation (18). 2. Multi-path algorithm [PERSON]'s algorithm is an effective shortest path algorithm, which calculates many nodes during the process of solving the shortest path. If the label information of these nodes is retained, it is conducive to the calculation of multi-path. The rescue route studied in this paper is a simple path with a boundary constraint. Therefore, it is necessary to meet the boundary constraints when selecting temporary label nodes. Thus, this paper provides the definition of qualified nodes firstly. **Definition 1**. : _Let \(t2_{p_{rm}}\) represent the travel time of route \(p_{rm}\) from the rescue center \(\mathbf{r}\) to a node \(v_{m}\), and \(v_{n}\) is the adjacent node of \(v_{m}\). If \(v_{n}\) is not on route \(p_{rm}\) and the travel time from the rescue center \(\mathbf{r}\) to a node \(\mathbf{n}\) satisfies the following:_  
\[E(t2_{p_{rm}}+t_{mm})+D(t2_{p_{rm}}+t_{mm})\cdot\inf\Bigl{\{}K\Bigl{|}K= \varphi^{-1}(\alpha)\Bigr{\}}\leq RT_{i} \tag{19}\]  
_then \(v_{n}\) is called the qualified node for route \(p_{rm}\)._  
Definition 1 ensures that the generated route is a simple path and meets the bound constraint. Based on [PERSON]'s algorithm and the definition of qualified nodes, a multi-path algorithm can be designed by retaining the temporary information of nodes. For convenience, \(p_{k_{rs}}\) is used to represent the \(k_{rs}\)-th route from the rescue center \(r\) to the accident location \(s\), and \(T(\cdot)\) and \(P(\cdot)\) represent temporary label and permanent label, respectively. The specific steps of the multi-path algorithm are as follows:  
* Step 1: Let the number of routes \(k_{rs}=1\), the temporary label of other notes (\(v_{m}\in V\backslash r\))\(T(v_{m})=+\infty\), the departure time of the rescue center \(T(r)=t1_{ir}\), \(p_{k_{rs}}=r\), and let node \(r\) be a permanent label. * Step 2: Select the smallest permanent label \(v_{i}\) and check whether each adjacent node \(v_{j}\) of \(v_{i}\) is a qualified note according to Equation (18). If it is a qualified node, add the temporary label of \(v_{j}\) and \(T(v_{j})=T(v_{i})+d_{i}+t_{ij}\) and proceed to Step 3. If there is no qualified node, let \(P(v_{i})=+\infty\), and proceed to Step 3. * Step 3: Select the node \(v_{n}\) with the smallest expected value among all the temporary labels, and mark it as the permanent label; let \(P(v_{n})=\min_{j}\{E\big{[}T(v_{j})\big{]}\}\), \(p_{k_{rs}}=p_{k_{rs}}\cup n\). * Step 4: If \(v_{n}\) is the location of accident, add route \(p_{k_{rs}}\) to \(K_{k_{rs}}\) and mark the permanent label value of nodes on this route with \(+\infty\), \(k_{rs}=k_{rs}+1\); then, proceed to Step 3. Otherwise, proceed to Step 2.