# 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
### Mission Relationship  
Based on the network diagram parameters, we can define several types of mission relationships. **Definition 3**. : _(Continuous Relationship). The continuous relationship means that two missions of the same executor are continuous in time. For two missions \((s_{1},f_{1})\) and \((s_{2},f_{2})\) belonging to the same executor, if \(f_{1}=s_{2}\), then \((s_{1},f_{1})\) and \((s_{2},f_{2})\) are mutually continuous missions, as shown in Figure 6. The set of \(n\) (\(n\geq 1\)) missions with mutually continuous relationships in a mission belonging to the same executor is called a continuous path \(\mathcal{L}\) with \(\mathcal{L}=\{(S,f_{1}),(f_{1},f_{2}),\ldots,(f_{n-2},f_{n-1}),(f_{n-1},F)\}\). The node with the smallest time scale in the continuous path \(\mathcal{L}\) is the start node \(S(\mathcal{L})\) of \(\mathcal{L}\) and the node with the largest time scale is the end node \(F(\mathcal{L})\) of \(\mathcal{L}\). Note that a mission can belong to several continuous paths._  
**Definition 4**. : _(Cooperative Relationship). Cooperative relationship means that two missions have an connection relationship in the network plan diagram. As shown in Figure 7, if mission \((s_{1},f_{1})\) is the immediate prior-mission of mission \((s_{2},f_{2})\), there is a cooperative relationship between \((s_{1},f_{1})\) and \((s_{2},f_{2})\). If mission \((s_{3},f_{3})\) is the prior-mission of mission \((s_{3},f_{3})\), there is a cooperative relationship between \((s_{3},f_{3})\) and \((s_{3})\)._  
**Definition 5**. : _(Parallel Relationship). There is a mission \((s_{A},f_{A})\) of executor \(A\) that belongs to the continuous path \(\mathcal{L}_{A}\), and a mission \((s_{B},f_{B})\) belongs to the continuous path \(\mathcal{L}_{B}\). Let the start and end nodes of \(\mathcal{L}_{A}\) be \(S_{A}\) and \(F_{A}\), respectively, and the start and end nodes of \(\mathcal{L}_{B}\) are \(S_{B}\) and \(F_{B}\), respectively. If there is a connection between \(S_{A}\) and \(S_{B}\), \(F_{A}\) and \(F_{B}\), then \((s_{A},f_{A})\) and \((s_{B},f_{B})\) are mutually parallel missions, as shown in Figure 8. It is worth noting that the two missions with parallel relationship must belong to different executors._  
**Definition 6**. : _(Constant Critical Mission). A mission \((s,f)\) is a constant critical mission if it is always a critical mission at any moment during the mission execution process. The decision theorem of constant critical mission is shown in Theorem 2. Specially for the cooperative strike mission scenario, the strike mission must comply with Theorem 2 due to the operational logic, so we define the strike mission as a constant critical mission._  
Figure 8: Diagram of mission parallel relationships. Figure 7: Diagram of mission cooperative relationships. **Theorem 2**. : _The mission \((s,f)\) is known to be a critical mission. If there are no parallel missions for mission \((s,f)\), \((s,f)\) will always be a critical mission and \((s,f)\) will always be on a particular critical path \(\mathcal{K}\) as long as the cooperative relationship between missions in the network plan diagram does not change. We define such a mission as a constant critical mission._  
Proof of Theorem 2.: Take any continuous path \(\mathcal{L}\) to which \((s,f)\) belongs, and \(\mathcal{L}\) starts and ends at \(S(\mathcal{L})\) and \(F(\mathcal{L})\), respectively. If there are no parallel missions for the mission \((s,f)\), then there is bound to be no feasible path that starts at \(S(\mathcal{L})\) and ends at \(F(\mathcal{L})\). Therefore \(\mathcal{L}\) is the only feasible path that starts at \(S(\mathcal{L})\) and ends at \(F(\mathcal{L})\). If \(\mathcal{L}\) belongs to the critical path, then \(\mathcal{L}\) will always belong to the critical path, and then the mission \((s,f)\) on the critical path will always be the critical mission. This completes the Proof of Theorem 2. A critical path can change if there are parallel missions for some of the missions on that critical path.