In the application of multi-robot system for a pick-and-place task, the appropriate robot arms and their base positions should be rapidly selected to improve the productivity, reduce the purchase cost, and apply the system for a task quickly. However, in current factories, a large amount of computational time is required to select the appropriate robot arms and their base positions because they are selected by the experienced engineers through evaluating the performance index in several trials. In this paper, we call the selection of robot arms and their base positions as integrated design of multi-robot system and propose a method to rapidly realize the integrated design of multi-robot system. We use the multi-objective particle swarm optimization (MOPSO) to select an appropriate multi-robot system, use the particle swarm optimization (PSO) to search for the base positions of the robot arms, and use the M/M/1 queueing model with impatient customer to estimate the performance index. A simulation proves that the proposed method is effective and efficient in comparison to a comparative method that uses simulation-based statistical inference to estimate the performance index. The computational time for the proposed method is 0.48 hour, which is less than 1/20 of the computational time for the comparative method.