Sizing of a heterogeneous fleet of robots in a logistics warehouse

The purpose is to determine the size of the robot fleet, that is, to determine the optimal types and number of robots for each type. We first consider the case where loads are all of the same type (homogeneous) before generalizing to several types of loads (heterogeneous). In both cases, we show that the problem can be formulated by an integer linear program. In the case of homogeneous loads, we consider a relaxation of the problem where the number of robots can be a real number. For this approximation, we show that it is optimal to use a single type of robot. It is also near-optimal when the number of robots must be an integer number.