Summary: Due to exorbitant product variety, very limited space, and other factors, organizing efficient and timely deliveries of parts and subassemblies to final assembly within the factory is one of the most pressing problems of modern mixed-model assembly production. Many automobile producers have implemented the so-called “supermarket” concept to transfer material to the assembly line frequently and in small lots. Supermarkets are decentralized logistics areas on the shop floor where parts are intermediately stored for nearby assembly cells, to be ferried there by small transport vehicles (called tow trains or tuggers). This paper tackles the operational problem of drawing up schedules for these tow trains, such that the assembly line never starves for parts while also minimizing in-process inventory, thus satisfying just-in-time goals. We prove strong NP-completeness of the problem and present exact and heuristic solution methods. In a computational study, the procedures are shown to perform very well, solving realistic instances to (near-)optimality in a matter of minutes, clearly outperforming the simple cyclic schedules commonly used in industrial practice. We also provide some managerial insight into the right degree of automation for such a part feeding system.