# Scheduling Optimization of Mobile Emergency Vehicles Considering Dual Uncertainties
## 2 Model Construction
### Mobile Emergency Vehicle Scheduling Model Considering Dual Uncertainties
\[min(Z_{C})=min(C_{1}+C_{2}+C_{3}+C_{4}) \tag{4}\] \[=min(\sum_{i}f_{i}X_{i}+\sum_{jku}\left(z_{ij}^{k}\eta_{ij}^{k}c_{ ij}^{k}+z_{ij}^{u}\eta_{ij}^{u}c_{ij}^{u}\right)+\sum_{ju}j_{i}^{u}+\sum_{jku} \left(\theta_{j}^{k}\lambda_{j}^{k}+\theta_{j}^{u}\lambda_{j}^{u}\right))\] \[minZ_{T}=min(T_{1}+T_{2}+T_{3})\] (5) \[=min(\sum_{ijk}z_{ij}^{k}\eta_{ij}^{k}+\sum_{ju}z_{ij}^{u}\eta_{ij }^{u}+\sum_{ju}z_{ij}^{u}\eta_{j}^{u}+\sum_{jku}\left(\theta_{j}^{k}\lambda_{j }^{k}+\theta_{j}^{u}\lambda_{j}^{u}\right))\] \[s.t.\;n_{ij}^{k}+n_{ij}^{u}\leq h_{j}^{\max,u},\forall i\in I,j \in J,k\in K,u\in U\] (6) \[n_{ij}^{k}+n_{ij}^{u}\geq\gamma_{j}^{k,u}\delta_{j}^{k,u}+\overline {d_{j}^{k,u}},\forall i\in I,j\in J,k\in K,u\in U\] (7) \[y_{i}^{k,u}=\sum_{jku}\left(n_{ij}^{k}+n_{ij}^{u}\right),\forall i \in I,j\in J,k\in K,u\in U\] (8) \[\sum_{ijk}z_{ij}^{k}=1,\sum_{iju}z_{ij}^{u}=1,\forall i\in I,j\in J,k\in K,u\in U\] (9) \[\sum_{i}x_{i}\geq 1,\forall i\in I\] (10) \[\beta_{ij}>\beta_{\min},\forall i\in I,\forall j\in J\] (11) \[=min(\sum_{i}f_{i}X_{i}+\sum_{jku}\left(z_{ij}^{k}\eta_{ij}^{k}c_{ ij}^{k}+z_{ij}^{u}\eta_{ij}^{u}c_{ij}^{u}\right)+\sum_{ju}j_{i}^{u}+\sum_{jku} \left(\theta_{j}^{k}\lambda_{j}^{k}+\theta_{j}^{u}\lambda_{j}^{u}\right))\]0 (12) \[=min(\sum_{i}f_{i}X_{i}+\sum_{jku}\left(z_{ij}^{k}\eta_{ij}^{k}c_{ ij}^{k}+z_{ij}^{u}\eta_{ij}^{u}c_{ij}^{u}\right)+\sum_{ju}j_{i}^{u}+\sum_{jku} \left(\theta_{j}^{k}\lambda_{j}^{k}+\theta_{j}^{u}\lambda_{j}^{u}\right))\]1 (13) \[=min(\sum_{i}f_{i}X_{i}+\sum_{jku}\left(z_{ij}^{k}\eta_{ij}^{k}c_{ ij}^{k}+z_{ij}^{u}\eta_{ij}^{u}c_{ij}^{u}\right)+\sum_{ju}j_{i}^{u}+\sum_{jku} \left(\theta_{j}^{k}\lambda_{j}^{k}+\theta_{j}^{u}\lambda_{j}^{u}\right))\]2  
In the model, Formulas (4) and (5) are objective functions, and Formulas (6)-(14) are constraints. The objective function (4) minimizes the scheduling cost of mobile emergency vehicles, which is the sum of fixed costs, transportation costs, preparation costs, and penalty costs. The objective function (5) minimizes the scheduling time of mobile emergency vehicles, which includes transportation time, development time, and punishment time. Constraint (6) indicates that the sum \(n_{ij}^{k}+n_{ij}^{u}\) of all kinds of demands in the mobile emergency system should not exceed the upper limit of the system reserve \(h_{i}^{max,u}\). It represents a ceiling on the availability of emergency supplies and services, regardless of how mobile emergency vehicles are deployed between mobile centers. Constraint (7) describes that, under the premise of introducing the demand uncertainty measurement scheme, meeting the demand of each demand point should still be a prerequisite for the optimal overall objective of the model. Constraint (8) represents that when the mobile emergency center provides emergency materials or emergency services to the disaster area, the number of vehicles and materials stored \(y_{i}^{k,u}\) is equal to the amount of outflow \(\sum_{jku}\left(n_{ij}^{k}+n_{ij}^{u}\right)\). Constraint (9) indicates that a mobile emergency center only meets a certain kind of demand at each demand point. Constraint (10) represents that at least one mobile emergency center participates in this mobile emergency. The constraint (11) indicates that the lower limit of path integrity is constrained in the process of path uncertainty measurement. The constraints (12) to (14) are decision variable constraints.