# Scheduling Optimization of Mobile Emergency Vehicles Considering Dual Uncertainties
## 2 Model Construction
### Uncertainty Risk Measurement  
The randomness and destructiveness of sudden disasters determine that mobile emergency vehicle scheduling has many uncertain characteristics. Compared with the daily vehicle scheduling, its uncertainties are generally manifested as emergency path uncertainty, data uncertainty, personnel uncertainty, and demand uncertainty, etc. The path uncertainty and demand uncertainty are closely related to mobile emergency vehicles, which determine the mechanism and goal of scheduling. Thus, this paper proposed an uncertainty-based mobile emergency vehicle scheduling model that focuses on path and demand uncertainty. The objective of the model is to minimize the cost and time of mobile emergency service. In order to effectively characterize these two types of uncertainty, we adopted the following metric method. 1. Path uncertainty measurement during the decision-making period of mobile emergency vehicle scheduling  
Disaster inevitably leads to a certain degree of damage to the road, and traffic congestion caused by disasters will further limit the speed of vehicles. Considering that mobile emergency vehicles are equipped with GPS navigation devices, it is easy to obtain the location of the vehicle and the distance from the destination at any time. Therefore, the main impact on the rescue routing is the road conditions. We can use path integrity to concretely represent the path damage. The stronger the path integrity, the shorter the passage time, the faster the average speed, and the longer the generalized path distance. In addition, path complexity refers to road intersections, overpasses, or areas prone to daily traffic accidents. Because transportation cost and time are strongly related to road conditions in an emergency logistics system, Formula (1) usually is adopted to define the unit logistics cost and single transportation time of mobile emergency vehicles on the premise of introducing path integrity and complexity. Here, the symbol \"*\" can be \(k\) or \(u\), representing the process of emergency material distribution or the process of emergency rescue. That is, \(c_{ij}^{k}\) and \(c_{ij}^{u}\) are the unit transportation costs of two different processes under path uncertainty conditions. \(t_{ij}^{k}\) and \(t_{ij}^{u}\) are the single transportation times correspondingly. In Formula (1), \(\alpha_{ij}\) represents the path complexity from point \(i\) to point \(j\). It is defined as the number of loops from the departure point \(i\) to the destination \(j\) in a given area and can be measured by McCabe metric. \(\beta_{ij}\) represents the path integrity from point \(i\) to point \(j\). \(\alpha_{ij}/\beta_{ij}\) denotes the path influence coefficient. Moreover, \(\beta_{min}\) is the lower limit of the path integrity of this segment. When the path integrity is lower than this value, the path will not be considered an alternative route. Thus, when path uncertainty is merged into the scheduling optimization model, the cost of emergency service \(c_{ij}^{*}\) and the corresponding time consuming \(t_{ij}^{*}\) are obviously affected by the path influence coefficient. The higher the path complexity or the lower the path integrity, the higher the emergency service cost and time consumption. In addition, if he path integrity \(\beta_{ij}\) is lower than \(\beta_{min}\), it means that the demand point cannot be reached, so \(c_{ij}^{*}\) and \(t_{ij}^{*}\) are infinite. \[c_{ij}^{*}= \begin{cases}\frac{c_{ij}^{*}\alpha_{ij}}{\beta_{ij}},\beta_{ij}> \beta_{min},i\in I,j\in I^{*}_{ij}=\begin{cases}\frac{t_{ij}^{*}\alpha_{ij}}{ \beta_{ij}},\beta_{ij}>\beta_{min},i\in I,j\in J\\ +\infty,\beta_{ij}\leq\beta_{min}\end{cases},i\in I,j\in J\end{cases}  \tag{1}\]Demand uncertainty measurement  
In the initial stage of emergency service following the occurrence of a sudden disaster, the change in the demand of emergency supply is untraceable. Many factors make it difficult to determine the need for emergency supplies in disaster areas.