The Economic Lot Sizing Problem for Continuous Multi-grade Production with Stochastic demands

We study a variant of the Stochastic Economic Lot Scheduling Problem (SELSP) in which a single production facility must produce several grades 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 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 deterministic 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 we must decide whether to initiate a changeover to a neighboring grade or keep the setup of the production facility unchanged, based on the current state of the system, which is determined by the current setup of the facility and the FG inventory levels of all the grades. The goal is to minimize the infinite-horizon long-run average cost. For 2 and 3-grade problems we can numerically solve the resulting MDP problem using successive approximation. For problems with more than three grades, we develop a heuristic solution which is based on approximating the original multi-grade problem into many 3-grade subproblems and numerically solving each sub-problem using successive approximation. We present and discuss numerical results for incidences with 2, 4 and 5 grades, using both the exact numerical and the heuristic solution procedure.