Approximating the joint replenishment problem

The Joint Replenishment Problem is one of the fundamental problems in Inventory and Supply Chain Management. We are given a warehouse and the set of retailers {1,..., n}=[n]. We are given the discrete time horizon {1,..., T}=[T] and the set of demands D consisting of pairs (i, t) for retailer of type i∈[n] and time t∈[T]. Let dit be the amount of demand for retailer i that arrived at time t. The demand (i, t) must be satisfied by an order placed in the time interval [0, t]. If the demand for retailer i at time t is satisfied by an order placed at time s≤ t then it incurs per unit holding costs of hist. A retailer i can place an order at warehouse at any time τ and incur the retailer ordering cost Ki, at the same time the warehouse places an order and incurs the warehouse ordering cost of K0. Therefore, it makes sense to combine many retailer orders together. For a example if at time τ we place orders for retailers from the set S⊆[n] then we incur ordering cost of K0+∑ i∈ S Ki. The goal is to define the set of warehouse and retailer orders to satisfy all the demand and minimize the total ordering and holding costs.