Primal-dual algorithms for deterministic inventory problems

We consider several classical models in deterministic inventory theory: the single–item lot–sizing problem, the joint replenishment problem, and the multistage assembly problem. These inventory models have been studied extensively, and play a fundamental role in broader planning issues, such as the management of supply chains. For each of these problems, we wish to balance the cost of maintaining surplus inventory for future demand against the cost of replenishing inventory more frequently. For example, in the joint replenishment problem, demand for several commodities is specified over a discrete finite planning horizon, the cost of maintaining inventory is linear in the number of units held, but the cost incurred for ordering a commodity is independent of the size of the order; furthermore, there is an additional fixed cost incurred each time a nonempty subset of commodities is ordered. The goal is to find a policy that satisfies all demands on time and minimizes the overall holding and ordering cost.