The Stochastic Economic Lot Sizing Problem for Continuous Multi-Grade Production

We study a variant of the stochastic economic lot scheduling problem (SELSP) in which a single production facility must produce several different grades of a family of products to meet random stationary demand for each grade from a common finished-goods (FG) inventory buffer with limited storage capacity. Demand that can not be satisfied directly from inventory is lost. Raw material is always available, and the production facility continuously produces at a constant rate. When the facility is set up to produce a particular grade, the only allowable changeovers are from that grade to next lower or higher grade. All changeover times are constant and equal to each other. There is a changeover cost per changeover occasion, a spill-over cost per unit of product in excess whenever there is not enough space in the FG buffer to store the produced grade, and a lost-sales cost per unit short whenever there is not enough FG inventory to satisfy demand. We model the SELSP as a discrete-time Markov decision process (MDP), where in each time period the decision is whether to initiate a changeover to a neighboring grade or keep the set up of the production facility unchanged, based on the current state of the system which is defined by the current set up of the facility and the FG inventory levels of all the grades. The goal is to minimize the (long-run) expected average cost per period. For 2-and 3-grade problems, we numerically solve the exact MDP problem using the value iteration method. For problems with more than three grades, we develop a heuristic solution procedure which is based on approximating the original multi-grade problem by several 3-grade sub-problems and numerically solving each sub-problem using value iteration. We present numerical results for problem examples with 2-5 grades. For the 2-and 3-grade examples, we use the exact solution procedure to obtain insights into the structure of the optimal changeover policy. For the 4-and 5-grade examples, we compare the performance of the heuristic solution procedure against that of the exact procedure.