We study robotic mobile fulfilment systems for online retailers, where products are stored in movable shelves and robots transport shelves. While previous studies assume random assignment rule of workstations to robots, we propose an assignment rule based on handling speeds of workstations and design a neighbourhood search algorithm to find a near optimal assignment rule. We build semi-open queueing networks and use a two-phase approximate approach for performance estimation. We first replace workstation service processes by a composite service node and then solve the model by the matrix-geometric method. Simulations are used to validate the analytical models. Numerical experiments are conducted to compare random, handling-speeds-based, near optimal and optimal assignment rules, in terms of retrieval throughput time. The results show that the random assignment rule is not a good choice, the handling-speeds-based assignment rule significantly outperforms the random assignment rule when the workers have large handling time difference, and the neighbourhood search approach can provide an assignment rule that is very close to the optimal one, using a much shorter time. Moreover, we design the shelf blocks under the examined assignment rules, and find that the optimal width of shelf block decreases with the width to length ratio.
