Order optimisation in multi-location models with hub-and-spoke structure

We investigate a system of M+1 locations. Location 0 is a warehouse with an ample amount of a given product, whereas the locations 1 to M have to meet a random demand for that product. To transship product from location 0 to a location i, i=1,…,M, one of a given number T of transportation units with given capacity Q can be used. After transhipping product to a location a transportation unit returns to location 0. Transshipment time is assumed to be negligible. The cost structure includes cost for holding inventory and backlogged demand in locations 1 to M, and cost for each transport. Under the restriction that order sizes must be integer multiples of Q the decision problem is to find order policies for locations 1 to M and an allocation of transportation units, which minimise the expected total cost.