A Production-Inventory System With Both Patient and Impatient Demand Classes

作者：Benjaafar Saif^*; Elhafsi Mohsen

来源：IEEE Transactions on Automation Science and Engineering, 2012, 9(1): 148-159.

DOI：10.1109/TASE.2011.2158645

• Inventory rationing
• Markov decision processes
• multiple demand classes
• optimal control
• production-inventory systems
• make-to-stock queues

We consider a production-inventory system with two customer classes, one patient and one impatient. Orders from the patient class can be backordered if needed, while orders from the impatient class must be rejected if they cannot be fulfilled from on-hand inventory. Orders backordered incur a backorder cost, while orders rejected incur a lost sales cost. The objective is to minimize the sum of inventory holding cost and the costs of backorders and lost sales. We formulate the problem as a Markov decision process and use this formulation to characterize the structure of the optimal policy. We show that the optimal policy can be described by two threshold functions that depend on the level of backorders from the patient class. These threshold functions specify: 1) when it is optimal to produce; 2) how to allocate units produced to either increase inventory or reduce backorder; and 3) when to fulfill orders from on-hand inventory and when to backorder (in the case of the patient class) and when to reject them (in the case of the impatient class). We show that the priority in inventory allocation among the two classes is not static and instead depends on the backorder level from the class of patient customers. In particular, it is possible to start out fulfilling orders from the impatient class and backordering orders from the patient class and then to switch to fulfilling orders from the patient class and rejecting orders from the impatient class. In addition to characterizing the structure of the optimal policy, we also describe an effective heuristic that retains the essential features of the optimal policy but is significantly simpler to implement. This heuristic performs nearly as well as the optimal policy and significantly outperforms other plausible heuristics. %26lt;br%26gt;Note to Practitioners-Most of the existing academic literature assumes that, in the case of inventory shortage, demand is always either backordered or rejected (or fulfilled through some other means at a higher cost). This implicitly assumes that either all customers are patient and willing to wait for their orders until they are fulfilled or all customers are impatient and walk away if their orders are not immediately fulfilled. However, in practice, there are many settings where both types of customers are present. The presence of both types of customers makes the management of such systems more difficult since it requires setting priorities on how inventory should be allocated and how this allocation should affect production decisions. In fact, very little is known about optimal control policies or even effective heuristics for such systems. In this paper, we provide insights into the optimal policy for production and inventory allocation. We show that the optimal policy can be complex and with counter-intuitive features. As an alternative to the optimal policy, we provide a simple and easy to implement heuristic that performs nearly as well.

• 出版日期2012-1