## 3 The Proposed Method
### Negotiate for Scout Task Assignment
#### 3.1.2 Collaborative Optimization of Reconnaissance Tasks Assignment  
When assigning the given reconnaissance and strike tasks, the algorithm should strive to maximize the proportion of task completion, minimize the total task completion time, and minimize the degree of task risk. Therefore, the objective function of reconnaissance task-allocation is defined as:  
\[\max_{\beta^{sc}}J(\beta^{sc}) \tag{8}\]  
where  
\[J(\beta^{sc}) =\sum_{\bigl{(}u_{i},t_{j},\tau_{j}^{s},g_{j}\bigr{)}\in\beta^{sc }}g_{j}(\beta^{sc}) \tag{9}\] \[g_{j}(\beta^{sc}) =r_{j}(\beta^{sc})\cdot e^{-\alpha_{\ell}\bigl{(}\tau_{j}^{s}- \tau\bigr{)}}\] (10) \[r_{j}(\beta^{sc}) =r_{\max}-\alpha_{t}\cdot\sum_{(u_{m},t_{m},t_{m},\beta_{m})\in \beta^{sc}}\left[\tau_{j}^{s}<\tau_{n}^{s}\right]\cdot f_{ssig}\big{(}d_{jn}/ \varphi\big{)} \tag{11}\]  
where the quaternion \(\left(u_{i},t_{j},\tau_{j}^{s},g_{j}\right)\) indicates that \(u_{i}\) will start reconnaissance tasks \(t_{j}\) at time \(\tau_{j}^{s}\), and the expected benefit is \(g_{j}\); \(\alpha_{r}\) is the coefficient of time for the discount of rewards; reconnaissance task plan \(\beta^{sc}\) is a collection of task assignment quaternions; \(\tau_{j}^{s}\) and \(\tau_{n}^{s}\) are the start execution times of tasks \(t_{j}\) and \(t_{n}\), respectively; \(r_{j}(\beta^{sc})\) is the expected reward for \(\beta^{sc}\) to complete \(t_{j}\), which is composed of the maximum reward \(r_{\max}\) and the estimated risk for completing the task, and \(\alpha_{t}\) is the weight of risk; \(\llbracket P\rrbracket=\left\{ \begin{array}{ll}1&\text{If $P$ is true}\\ 0&\text{Otherwise}\end{array} \right.\) is an Iverson bracket, and if the start time \(\tau_{n}^{s}\) is later than \(\tau_{j}^{s}\) in the scheme, \(t_{n}\) will pose a threat to \(t_{j}\). It can be inferred from Equation (9) that if adjacent tasks can be scouted at the same time, the threat to each other can be reduced and the reward can be increased. However, starting reconnaissance at the same time also means that some tasks need to be deliberately postponed, leading to a decline in overall reward. Therefore, it is necessary to optimize the time synergy of each plan. The time-collaborative optimization of a plan is to optimize the specific start time of each assigned reconnaissance task. For any two assignments \(\left(u_{i},t_{j},\tau_{j}^{s},g_{j}\right)\) and \(\left(u_{m},t_{n},\tau_{n}^{s},g_{n}\right)\) in plan \(\beta^{sc}\), if \(\tau_{j}^{s}<\tau_{n}^{s}\), \(u_{i}\) can postpone its task start time to the same as \(u_{m}\) to improve its task reward \(r_{j}(\beta^{sc})\). Express the updated plan as \(\beta^{sc}\), then there is \(\tau_{j}^{s\prime}=\tau_{n}^{s}\). Let \(\Delta\tau_{jn}^{s}=\tau_{n}^{s}-\tau_{j}^{s}\), then the gain of reward for time collaboration is:  
\[ \begin{split}\Delta g_{jn}&=g_{j}(\beta^{sc\prime} )-g_{j}(\beta^{sc})=r_{j}\big{(}\beta^{sc\prime}\big{)}\cdot e^{-\alpha_{r}( \tau_{n}^{s}-\tau)}-r_{j}(\beta^{sc})\cdot e^{-\alpha_{r}\left(\tau_{j}^{s}- \tau\right)}\\ &=e^{-\alpha_{r}\left(\tau_{j}^{s}-\tau\right)}\left[r_{j}\big{(} \beta^{sc}\big{)}e^{-\alpha_{r}\cdot\Delta\tau_{jn}^{s}}-r_{j}(\beta^{sc}) \right]\\ &=e^{-\alpha_{r}\left(\tau_{j}^{s}-\tau\right)}\left[\left(r_{j}( \beta^{sc})+\alpha_{t}\cdot f_{ssig}\big{(}d_{jn}/\varphi\big{)}\right)e^{- \alpha_{r}\cdot\Delta\tau_{jn}^{s}}-r_{j}(\beta^{sc})\right]\\ &=e^{-\alpha_{r}\left(\tau_{j}^{s}-\tau\right)}\left[r_{j}(\beta ^{sc})\left(e^{-\alpha_{r}\cdot\Delta\tau_{jn}^{s}}-1\right)+\alpha_{t}\cdot f _{ssig}\big{(}d_{jn}/\varphi\big{)}e^{-\alpha_{r}\cdot\Delta\tau_{jn}^{s}} \right]\end{split}  \tag{12}\]  
Since the time collaboration between any two assignments in a plan will depend on the recalculation of Equation (8), and the time collaborative optimization is an underlying algorithm that will be called repeatedly, we define Algorithm 1 based on Equation (12) to quickly optimize the time collaboration of the given plan. ```
0: The plan \(\beta^{sc}\) that needs to optimize its time collaboration
0: The updated plan \(\beta^{sc\prime}\).