One warehouse multiple retailer systems with vehicle routing costs

We consider distribution systems with a depot and many geographically dispersed retailers each of which faces external demands occurring at constant, deterministic but retailer specific rates. All stock enters the system through the depot from where it is distributed to the retailers by a fleet of capacitated vehicles combining deliveries into efficient routes. Inventories are kept at the retailers but not at the depot. We wish to determine feasible replenishment strategies i.e., inventory rules and routing patterns minimising infinite horizon long-run average transportation and inventory costs. We restrict ourselves to a class of strategies in which a collection of regions sets of retailers is specified which cover all outlets: if an outlet belongs to several regions, a specific fraction of its sales/operations is assigned to each of these regions. Each time one of the retailers in a given region receives a delivery, this delivery is made by a vehicle who visits all other outlets in the region as well in an efficient route. We describe a class of low complexity heuristics and show under mild probabilistic assumptions that the generated solutions are asymptotically optimal within the above class of strategies. We also show that lower and upper bounds on the system-wide costs may be computed and that these bounds are asymptotically tight under the same assumptions. A numerical study exhibits the performance of these heuristics and bounds for problems of moderate size.