Improved Randomized Approximation Algorithms for Lot-Sizing Problems

We consider in this paper multi-product, lot-sizing problems that arise in manufacturing and inventory systems. We describe the problem in a manufactruring setting. There is a set N of products. For each product j E N there is a set ~-j (called predecessors of product j) of products consumed in producing product j. We define the product network G to be a directed network with node set N and arc set A = {(i,j) : i E ~rj). In other words, the network G corresponds to the flow of materials in the system and contains no circuit. External demand di for product i is assumed to be constant in time. Clearly in order to satisfy the demand orders should be placed for the products dynamically in time. If an order is placed for product i, an ordering cost Ki is incurred. Moreover, an incremental echelon holding cost hi is incurred per unit time the item spends in inventory. The production rate is assumed to be infinite. The objective is to schedule orders for each of the products over an infinite horizon so as to minimize long-run average cost. As the optimal dynamic policy can be very complicated, the research community (see for instance Roundy [18, 19], Jackson, Maxwell and Muckstadt [10], Muckstadt and Roundy [14]) has focused on stationary and nested policies defined as follows: Orders are placed periodically in time at equal intervals, for each of the products in the system (stationary policies). If product j precedes product i, then an order is placed for product j only when an order is placed for product i at the same time (nested policies). Therefore, under a stationary and nested policy the objective is to decide the period Ti that an order is placed. The reason stationary and nested policies are attractive is that they are easy to implement. Muckstadt and Roundy [14] discuss in detail the rationale of using order intervals ~ as variables. The problem of designing an optimal stationary and nested policy can then be formulated (see [18]) as the following nonlinear integer programming problem.