# Transportation and Reserve of Emergency Medical Supplies during Public Health Events
## 3 Materials and Methods
### Allocation and Storage Model of Emergency Medical Materials
#### 3.3.2 Allocation and Storage Model of Emergency Medical Materials
\begin{array}{rl}&1,\text{Vehicle $k$ transports material from $a$ to $b$}\\ &0,\text{Vehicle $k$ does not transport material from $a$ to $b$}\\ \end{array} \right.\)  
Under normal circumstances, the distribution center must deliver emergency medical materials within the time window \(ET_{a}\) to \(RT_{a}\) of the material demand point. In this paper, the emergency medical logistics path optimization model is constructed with the objective function of minimizing total transportation cost, and the particle swarm optimization (PSO) algorithm is designed to solve the model. In view of the high consumption of emergency medical materials needed during public health issues, the large scale of use, and the strong emergency demand, the waiting cost caused by the early arrival of vehicles is not taken into account. By establishing a mathematical model, this paper studies the best route selection of emergency medical materials in different situations. If the delivery vehicle arrives at the demand point beyond the latest time window during rescue, resulting in insufficient supply of emergency medical materials, thus affecting life and health and prevention and control of public health issues, the emergency logistics system should be punished. If the urgency of the demand for service is greater, the total delay time will be longer, and the corresponding punishment cost will also increase. In this paper, the reduction in delay loss is taken as the optimization objective to establish a model, and a particle swarm optimization algorithm is designed to solve it. The following is the \(B_{a}\) formula to describe the sum of delay times:  
\(B_{\text{a}}=\left\{ \begin{array}{lr}\sum\limits_{k=1}^{K}T_{ak}x_{ak}-RT_{ a}&\sum\limits_{k=1}^{K}T_{ak}x_{ak}\geq RT_{a}\\ 0&\sum\limits_{k=1}^{K}T_{ak}x_{ak}<RT_{a}\\ \end{array} \right.\) (14)  
The resulting penalty cost \(D_{a}\) formula is as follows:  
\(D_{a}=o\times Y_{a}\times\max[B_{a},0]\) (15)  
The total running time of the system refers to the total time required for all vehicles to complete the distribution of emergency medical materials according to the path generated by the calculation and solution at the beginning of the system, including the service time for the assembly and transport of the materials, and then returning to the distribution center, collectively. If we set the total system time to \(T\), then we can use the \(T\) formula to express it:  
\[T=\sum\limits_{k=1}^{K}\sum\limits_{a=1}^{h}\sum\limits_{b=1}^{h}t_{ab}^{k}u_{ abk}+\sum\limits_{a=1}^{h}\Delta_{a} \tag{16}\]  
Then the cost function \(Z_{2}\) for vehicle travel is as follows:  
\[Z_{2}=c_{1}+c_{0}v_{0}\sum\limits_{k=1}^{K}\sum\limits_{a=1}^{h}\sum\limits_{b= 1}^{h}t_{ab}^{k}u_{abk}=c_{1}+\sum\limits_{k=1}^{K}\sum\limits_{a=1}^{h}\sum \limits_{b=1}^{h}c_{0}d_{ab}u_{abk} \tag{17}\]  
Based on the above assumptions, this paper proposes a bi-objective path optimization function, which aims to minimize the penalty cost of each demand point and the overall running time of the system. To conveniently solve the problem, the bi-objective problem is transformed into a single-objective problem, and the weights \(p_{1}\) and \(p_{2}\) (\(p_{1}+p_{2}=1\)) are designed to obtain the optimal single-objective model as follows:  
\[\text{min}Z_{1}=\text{min}\sum\limits_{a=1}^{h}D_{a}=\text{min}\left(\sum \limits_{a=1}^{h}oy_{a}\text{max}[\sum\limits_{k=1}^{K}T_{ak}x_{ak}-RT_{a},0]\right) \tag{18}\]  
\[\text{min}Z_{2}=\text{min}\left(c_{1}+\sum\limits_{a=1}^{h}\sum\limits_{b=1}^ {h}\sum\limits_{k=1}^{K}c_{0}d_{ab}u_{abk}\right) \tag{19}\]  
\[\text{min}Z=p_{1}\text{min}Z_{1}+p_{2}\text{min}Z_{2} \tag{20}\]  
Among them, (18) is the penalty minimum objective function, (19) is the vehicle cost minimum function with the minimum total operating time of the system, and (20) is the single objective function with added weight. Create constraints as follows:  
\[s.t.\left\{