## 3 Modeling of Apron Support Vehicle Operation Scheduling Problem
### Construct Mathematical Model  
The scheduling of apron support vehicles is the RCPSP. The scheduling process of the security vehicles needs to be sorted by assigning tasks to the processor to optimize the objective function and complete certain tasks under certain constraints. If each task needs to work on each processor with an arbitrary order of jobs, it is called a free-order job. It is convenient to simplify the expression of the problem and use the triple (\(\alpha\mid\beta\mid\gamma\)) to describe the problem [36; 37]. A typical free-order problem expression is:  
\[O_{m}||C_{max}P=\left\{\begin{array}{cccc}p_{11}&p_{12}&\ldots&p_{1n}\\ p_{21}&p_{22}&\ldots&p_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ p_{m1}&p_{m2}&\ldots&p_{mm}\end{array}\right\} \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(O_{m}\) & \(m\) processors \\ \hline \(C_{max}\) & the time objective function \\ \hline \(C_{max}=\max\{C_{j}\}\) & equal to the end time of the last completed task \\ \hline \(P\) & job time matrix, representing the time when \(j\) task completes the corresponding job on the \(i\) processor. \\ \hline \(T_{min}\) & the minimum time for ground support operations \\ \hline \(M_{ij}\) & the j support operation of the \(i\) aircraft \\ \hline \(P_{ij}\) & the support vehicle for the support operation \(M_{ij}\) \\ \hline \(T_{ij}\) & the operation time of the guaranteed operation \(M_{ij}\) \\ \hline \(S_{ij}\) & the start operation time of the guarantee operation \(M_{ij}\) \\ \hline \(F_{ij}\) & the end operation time of the guarantee operation \(M_{ij}\) \\ \hline \(MaxF\) & the longest guarantee operation end time \\ \hline \(\Omega\) & the set of feasible solutions \\ \hline \(P\) & the parent solution which is selected from the population \\ \hline \(C\) & the child solution after crossover \\ \hline \(M\) & the population which has several initial solutions \\ \hline \(pi(i)\) & the probability which the chromosome is selected \\ \hline Re & refueling truck \\ \hline Ox & oxygen refilling truck \\ \hline Po & power van \\ \hline \(T_{o}\) & operation time, min \\ \hline \(T_{w}\) & waiting time, min \\ \hline \(T_{t}\) & total time, min \\ \hline \hline \end{tabular}
\end{table}
Table 2: Notation and terminology.  
where \(O_{m}\) refers to \(m\) processors; \(C_{max}\) represents the time objective function, \(C_{max}=max\{C_{j}\}\), which is equal to the end time of the last completed task; \(P\) stands for the job time matrix, representing the time when the \(j\) task completes the corresponding job on the \(i\) processor.  
Since the oxygenation, charging, and refueling in apron support preparations are not strictly sequenced, this problem belongs to free-sequence or open operations. The objective function of the mathematical model for the scheduling of apron support vehicles is expressed as follows [38]:  
\[T_{min}=max\sum_{i\in n}\sum_{j\in m}\{F_{ij}\} \tag{2}\]  
The constraints are as follows:  
\[MaxF-S_{ij}\geq T_{ij},1\leq i\leq n,1\leq j\leq m \tag{3}\]  
\[S_{qj}-S_{ij}\geq T_{ij},if\ x_{iqj}=1,1\leq i\
eq q\leq n,1\leq j\leq m \tag{4}\]  
\[S_{ih}-S_{ij}\geq T_{ij},if\ x_{ijh}=1,1\leq i\leq n,1\leq j\
eq h\leq m \tag{5}\]  
\[S_{ij}\geq 0,1\leq i\leq n,1\leq j\leq m \tag{6}\]  
\[(x,y)\in\Omega \tag{7}\]  
\[x_{iqj}=\begin{array}{c}\{&1\quad if\ M_{ij}\ is\ a\ pre-\ process\ of\ M_{qj},1\leq i\
eq q\leq m;1\leq j\leq n\\ &0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad  
When there is more than one support vehicle of the same type, the support vehicles are grouped first, and then, the operations are carried out in sequence.