## 2 Cross-Domain Collaborative Task Allocation Model
### Problem Description  
The cross-domain cooperative task introduces the air cooperative control platform as the child node of task allocation. Consequently, it was necessary to consider not only the control and guidance function for the middle and late stages of the load, but also the relay control process and the control channel allocation problem. According to the functional process, this task was divided into two parts: target allocation and control allocation. The target allocation commences at the sensor boot time of the surface platform and is concluded in the data center of the surface platform, in accordance with the battlefield fusion situation. The control allocation commences at the moment that the surface platform launches the load agent in accordance with the target allocation result. The surface platform  
Figure 1: Tree diagram of existing algorithms for the WTA problem. is responsible for guiding the early and middle stages of the flight process, and then the control allocation is completed according to the detection and data link advantages of the air platform. In the final phase of the flight, the control is transferred to the air platform. It is assumed that the multi-agent cross-domain cooperative task allocation includes four basic types of agent units: \(a\) surface agents, forming the set \(W=\{W_{1},\ W_{2},\ W_{3}\dots\ W_{a}\}\); \(b\) air relay control agents, forming the set \(A=\{A_{1},\ A_{2},\ A_{3}\dots\ A_{b}\}\); \(c\) target agents, forming the set \(T=\{T_{1},\ T_{2},\ T_{3}\dots\ T_{c}\}\); and \(d\) load agents, forming the set \(M=\{M_{1},\ M_{2},\ M_{3}\dots\ M_{d}\}\). Among them, the \(n\)th surface agent \(W_{n}\) is responsible for the launch and pre-middle-stage guidance tasks of the \(i\)th load agent \(M_{i}\), and the \(k\)th air relay control agent \(A_{k}\) is responsible for the end-stage guidance task of the \(i\)th load agent \(M_{i}\). The two agents cooperate with each other and connect in an orderly manner. From the functional process, the cross-domain collaborative task can be divided into two stages: target allocation and control allocation. The former mainly solves the target allocation problem of the load agent \(M_{i}\) to the target platform \(T_{j}\), and the latter solves the control allocation problem of the relay control platform \(A_{k}\) to the load agent \(M_{i}\). In summary, in the multi-agent cross-domain collaborative task, the key is to obtain the optimal or quasi-optimal task allocation scheme of the target allocation matrix \(x_{i}=\{x_{i}(i,j)\}_{d\times c}\) and the control allocation matrix \(y_{i}=\{y_{i}(i,k)\}_{d\times b}\), where \(x_{i}(i,j)\) and \(y_{i}(i,k)\) are decision variables; a value of 1 means that the load agent \(M_{i}\) is assigned to the target agent \(T_{j}\) or the air relay control agent \(A_{k}\), and a value of 0 means that the load agent is not assigned. In order to reduce the time complexity of the task allocation solution and meet the actual task requirements, the following assumptions were made:  
1. There was no redistribution problem in the load agent, and the matching relationship between the load and the target was no longer changed by the control allocation after the target allocation was completed. It was assumed that the target allocation and the control allocation were independent of each other, and the optimal solution of the target allocation was the premise in the control allocation;
2. The number of load agents participating in the allocation exceeded the number of target agents or air relay control agents;
3. Each load agent was independent of the others, and the corresponding damage probability was calculated in advance without considering its type selection problem;
4. The air relay control agent had a unique type. In order to ensure the safety and reliability of the communication between it and the load agent, an attitude change was not allowed, and the flight was always level.