## 1 Introduction
On the other hand, because the initial and end positions of the inventory pod in the moving task are different, the moving task is considered to be an arc task. Because of the arc tasks, the cost between node tasks and arc tasks varies depending on the order in which they are completed. The cost between the tasks is asymmetric. After completing a series of tasks, each robot does not return to the starting position but stays at the last task point and waits for the following command. According to the above, the intelligent warehouse's multi-robot task assignment problem can be considered an open-path multi-depot asymmetric traveling salesman problem (OP-MATSP). We treat each task as a city node and each robot as a traveling salesman. Compared to earlier research, this work is the first to propose the OP-MATSP. Figure 1: Two-dimensional schematic of Amazon’s order fulfillment center [3]. The orange squares represent the robots. The green squares represent the inventory pods. The inventory stations are shown only on the left side and are usually located around the perimeter of warehouse. Some scholars have studied warehouse task assignment based on the traveling salesman problem (TSP) [18], but they mainly discuss the order picking problem [19; 20; 21]. When an order is received, the system sends a worker or a robot from the station to pick the items of the order from different inventory pods and finally return to the station. We treat each good in the order as a city node, each robot as a traveling salesman, and the station as the depot. In these studies, each robot moves to the inventory pods to pick up all the goods in the order and then returns to the start position. The above studies all belong to the \"worker-to-goods\" model. This paper's task assignment is based on the \"live to human\" model, which is more efficient in an intelligent warehouse. For the optimization objective of the problem, the studies in [19; 20; 21] only design the pickup route for each vehicle through TSP, shortening its own distance to fulfill the order. These studies do not consider the optimal total distance of all robots to complete the orders. In [10], the optimization objective is mainly to minimize the manufacturing time to ensure the system's operating efficiency without balancing other optimization objectives. In order to reduce the amount of charging and replacing, some scholars take the device's energy efficiency as an optimization objective. The scholars in [22] propose a distributed resource management mechanism to determine the optimal transmission power for each device, in which the interests, physical relationships, and energy availability among devices are all considered. In this paper, we assume that the power of each robot is infinite. This paper optimizes the total distance and total time for all robots to complete the task simultaneously. The total distance is also known as the sum-of-cost (SOC). The total time is known as the makespan (MS). OP-MATSP can be converted from the multi-depot traveling salesman problem (MTSP) [23]. As shown in Figure 3, each starting depot only has one traveling salesman who departs to visit the city node in the MTSP. When all city nodes have been visited, all traveling salesmen return to their depots. The cost between each city node is the same. The cost between each task node in this paper is asymmetric. By adding asymmetric restrictions, such as [24], we can modify the MTSP model to a multi-depot asymmetric traveling salesman problem (MATSP) model. In general, asymmetric models are more complex to solve than symmetric models [25]. In this paper, each robot stops at the last task node after completing a series of tasks and does not return to the starting depot, which is an open path problem. By eliminating the last edge of the traveling salesman returning to the starting depot, we can transfer the MATSP model to the OP-MATSP. Figure 2: The multi-robot task assignment in an intelligent warehouse. As a generalization of TSP, the OP-MATSP is also an NP-hard combinatorial optimization problem.