###### Abstract  
In a multi-agent system, multi-job assignment is an optimization problem that seeks to minimize total cost. This can be generalized as a complex problem in which several variations of vehicle routing problems are combined, and as an NP-hard problem. The parameters considered include the number of agents and jobs, the loading capacity, the speed of the agents, and the sequence of consecutive positions of jobs. In this study, a deep neural network (DNN) model was developed to solve the job assignment problem in a constant time regardless of the state of the parameters. To generate a large training dataset for the DNN, the planning domain definition language (PDDL) was used to describe the problem, and the optimal solution that was obtained using the PDDL solver was preprocessed into a sample of the dataset. A DNN was constructed by concatenating the fully-connected layers. The assignment solution obtained via DNN inference increased the average traveling time by up to 13% compared with the ground cost. As compared with the ground cost, which required hundreds of seconds, the DNN execution time was constant at approximately 20 ms regardless of the number of agents and jobs. multi-agent system; assignment problem; vehicle routing problem; planning domain definition language; deep neural network +
Footnote †: journal: Applied sciences  
not have to return to a depot. As a result, the route terminates when the last job is completed. The MDVRP extends the CVRP by allowing depots to be placed in multiple locations. This study's allocation problem is a combination of all of the aforementioned conditions, which reflect a real-world scenario. Based on the problem definition, all jobs should be selected only once; one job only moves from the starting position to the target position; multiple agents can select and process any number of jobs; and there is no need to return to the agent's starting position when the jobs have been completed. Therefore, this can be regarded as a variation of the vehicle routing problem wherein an open VRP in which vehicles with different loading capacity limitations start from multiple depots pick up at a specific location in the job and deliver to another location, and the pickup of other jobs may occur between the pickup and delivery of a job. This is an NP-hard problem known as the multi-depot capacitated open vehicle routing problem with pickup and delivery (MDCOVRPPD). In previous studies that applied a metaheuristic algorithm to VRP variants such as MDVRP or VRPPD, the computation time increased by hundreds of seconds as the number of agents and jobs increased [8; 9; 10]. As such, the direct application of the algorithm to problem-solving in a real-world environment is difficult. In this study, we designed a deep neural network (DNN) that infers a result close to the optimal solution to solve the subproblem with a time complexity of \(O(1)\) for the MDCOVRPPD that considers several complex parameters. The results indicate that the proposed approach can be applied to real-world scenarios by solving the allocation problem within a specified time (e.g., tens of microseconds). The number of input agents and jobs for the inference of a DNN is limited owing to the limitations of the computing resources used in this study. Many agents and jobs in the real world are handled by subsets using divide-and-conquer methods. The following is a summary of the study's contributions:  
* This study entailed the development of a DNN model for solving the assignment problem between multiple agents and multiple jobs in a variant of the VRP with real-world parameters. As compared to the previously proposed methods, the proposed method executes quickly without sequential procedures. * We devised a method for generating a dataset for training the proposed deep neural network model by using the planning domain definition language (PDDL). Various parameter types were defined and relationships between parameters were expressed in the PDDL description of the domain.