## 1 Introduction  
With the development of intelligent collaboration technology, multi-agent systems with the characteristics of low cost and replaceability are being increasingly used for complex task allocation [1]. The problem of task allocation is fundamental in the field of military operation research. Its primary purpose is to provide an optimal or quasi-optimal cooperative allocation scheme for multi-agents. This is achieved by fully understanding the intention of decision makers, so as to assist them in formulating action plans and maximizing the overall efficiency. Traditional multi-agent task allocation is mostly based on single-space domains such as land, sea, and air. The increasing adoption of cross-domain cooperation and the integration of new technologies to leverage their complementary advantages facilitate omnidirectional interactions. Thus, there is a need to address the challenges associated with large-scale cross-domain cooperative tasking, especially in dynamic and uncertain environments. This has become a key research direction in the field of task allocation [2, 3]. Cross-domain cooperative task allocation involves transferring the control of the load agent between multi-dimensional sea and air platforms through the handover and relay of the control of the surface agent and the air relay control agent in the cluster to the target beyond the line of sight, on the basis of sharing situation data. By comprehensively utilizing the internal resources of the cluster, this approach can compensate for the shortcomings of single-platform sensors in terms of detection, signal control distance, and accuracy. It maximizes the range capability and concealment of the load agent, thereby ensuring the safety of the delivery platform. In an actual task scenario, the number of targets is large, the density is large, the timeliness is strong, the load resources and the delivery platform resources are limited, and the command and control constraints are very complex; therefore, an assessment of the real-time performance of task allocation is greatly required [4; 5]. At present, the multi-agent system of decision making and control is developing towards network distribution. Various devices can serve as nodes within the information fusion center, which features a distributed network structure without a central node. This configuration enhances the system's decision-making capabilities and addresses the limitations of centralized task allocation algorithms, such as high communication demands, elevated computational complexity, poor fault tolerance, and limited system scalability. This decision mode requires that the decision control center receives support from each agent system to fully mobilize all the resources, comprehensively analyze the global situation, hierarchically develop the global task plan, and realize the maximization of global revenue. Simultaneously, the decision control center must inject the global initial optimal plan into each agent before it makes autonomous decisions. This process guides the agent in adjusting the task plan according to its own characteristics and completing autonomous actions [6]. In this research, we examined multi-agent task allocation methods under the background of distributed cross-domain cooperation, combined with the latest task allocation ideas. The existing research on task allocation methods has mainly focused on optimizing the mathematical model of task allocation and improving the solving algorithm of the model [7]. In order to ensure the achievement of goals, the traditional mathematical model of task allocation often considers task efficiency to be the primary index of goal allocation, and transforms the constraint function by constructing a penalty function. This makes the solution space of the model extremely complex, which can easily lead to allocation failure or oversaturation due to an unreasonable threshold setting for the efficiency index, leading to a waste of resources [8]. [PERSON] et al.