# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 3 Dual-Code Group Network Plan Diagram Model with Time Line
### Calculation of Time Parameters  
For the plan maker, it is relatively straightforward to empirically determine the duration of each mission and the cooperative relationship between missions. However, establishing these parameters alone is insufficient to generate an executable plan. It is also necessary to determine the start and end moments of each mission. This requires the calculation of the time parameters in the group network plan diagram. The primary objective is to generate an executable plan that provides a clear concept of time for the control of the planning process. The time parameters of the group network plan diagram can be calculated based on the sorted missions set obtained by Algorithm 1, the global start node \(S\) and global end node \(F\). In order to facilitate the representation of logical relationships among missions, we use dual-codes to represent missions. The dual-code form of a mission is represented as \((s,f)\). \(s\) and \(f\) represent the start node and end node numbers of the mission, respectively. The time parameters of the mission \((s,f)\) contain the earliest start time \(T_{ES}^{(s,f)}\), the earliest end time \(T_{EF}^{(s,f)}\), the latest start time \(T_{LS}^{(s,f)}\), the latest end time \(T_{LF}^{(s,f)}\), the total time difference \(R^{(s,f)}\), the single time difference \(R^{(s,f)}\), and so on. Let the duration of the mission \((s,f)\) be \(x^{(s,f)}\), then the equations for \(T_{ES}^{(s,f)}\) and \(T_{EF}^{(s,f)}\) are  
\[\left\{ \begin{array}{ll}T_{ES}^{(s,f)}&=0\\ T_{ES}^{(s,f)}&=\max_{(k,s)\in\mathcal{EM}_{(s,s)}}\Bigl{\{}T_{ES}^{(k,s)}+x^ {(k,s)}\Bigr{\}}\\ T_{EF}^{(s,f)}&=T_{ES}^{(s,f)}+x^{(s,f)}\end{array} \right. \tag{10}\]  
Equation (10) is a recursive equation, which can be computed according to Theorem 1 by following the sorted missions set \(\mathcal{M}_{s}\) recursively from the beginning to the end, one by one. That is, the earliest start time of all tasks \((S,j)\) from the global start node \(S\) is zero. Furthermore, the earliest start time of any task \((s,f)\) is determined by the earliest start times of all of its prior-missions \((k,f)\); and the earliest end time of a mission \((s,f)\) is equal to the sum of its earliest start time and its mission duration. The \(T_{LS}^{(s,f)}\) and \(T_{LF}^{(s,f)}\) are calculated as \[\left\{ \begin{array}{l}T_{LS}^{(s,F)}=T_{ES}^{(s,F)}\\ T_{LS}^{(s,f)}=\min_{(f,k)\in\mathcal{F}\mathcal{M}_{(s,f)}}\left\{T_{LS}^{(f,k) }-x^{(s,f)}\right\}\\ T_{LF}^{(s,f)}=T_{LS}^{(s,f)}+x^{(s,f)}\end{array} \right. \tag{11}\]  
Equation (11) is also a set of recursive equations that can be computed recursively from back to front one by one for the sorted mission set \(\mathcal{M}_{s}\). The latest start time of any mission \((s,F)\) that enters the global end node \(F\) must be equal to the earliest start time of the mission. The latest start time of any mission \((s,F)\) is determined by the latest start time of its post-mission mission; and the latest finish time of mission \((s,F)\) is obviously equal to the sum of its latest start time and the duration of the mission. The total time difference in a mission \(R^{(s,f)}\) refers to the maximum amount a mission can delay its start time without affecting the total duration of the mission and is expressed as  
\[R^{(s,f)}=T_{LF}^{(s,f)}-T_{EF}^{(s,f)} \tag{12}\]  
**Definition 1**. : _(Critical Mission). In the cooperative strike mission plan, a mission with a total time difference of zero is called a critical mission. It means that if mission \((s,f)\) is a critical mission, there is \(R^{(s,f)}=0\)._  
The set containing all critical missions is called the critical mission set \(\mathcal{M}^{*}\), and there is \(\mathcal{M}^{*}=\left\{(s,f)|R^{(s,f)}=0\right\}\).