تعیین اندازه دسته تولید و تصمیم گیری در زمان بین سفارش و تحویل کالا در سیستم های تولید / موجودی

Traditionally, lot sizing decisions in inventory management trade-off the cost of placing orders against the cost of holding inventory. However, when these lot sizes are to be produced in a finite capacity production/inventory system, the lot size has an important impact on the lead times, which in turn determine inventory levels (and costs). In this paper we study the lot sizing decision in a production/inventory setting, where lead times are determined by a queueing model that is linked endogenously to the orders placed by the inventory model. Assuming a continuous review (s, S) inventory policy, we develop a procedure to obtain the distribution of lead times and the distribution of inventory levels, when lead times are endogenously determined by the inventory model. This procedure allows to determine the optimal inventory parameters within the class of (s, S) policies that minimize the expected ordering and inventory related costs over time. We numerically show that ignoring the endogeneity of lead times may lead to inappropriate lot sizing decisions and significantly higher costs. This cost discrepancy is very outspoken if the lot size based on the economic order quantity deviates significantly from desirable production lot sizes. In these cases, the endogenous treatment of lead times is of particular importance.

A century ago, Ford Whitman Harris presented the Economic Order Quantity (EOQ) model as a simple, yet powerful model to determine how many parts to make (or order) at once, so as to balance the fixed costs per lot against the inventory carrying costs (Harris, 1913). Although various assumptions are underlying the model, the EOQ model proves to be a robust solution to many lot sizing decisions in practice. To apply the EOQ model, it is common practice to additionally define a reorder point based on the distribution of demand during lead time, so that a fixed order quantity Q (equal to the EOQ) is ordered as soon as the inventory position reaches the reorder point r. Arrow et al. (1951) introduced a slightly modified version of this model, i.e. the (s , S ) inventory policy, in which a reorder point s and an order-up-to level S are established: no order is placed until inventories fall to s or below, whereupon an order is placed to restore the inventory position to the level S . In other words, orders are placed with a lot size which is always larger than or equal to the value of S−sS−s (in many heuristics, the batching parameter S−sS−s is set equal to the EOQ), but in this case the order size is stochastic: the more the inventory position falls below s (which happens, for instance, in case of a large demand size), the more the order quantity will exceed S−sS−s; we call this the random overshoot. Several authors showed, assuming constant lead times, that an (s, S) policy is optimal when a fixed order cost is present ( Scarf, 1960, Iglehart, 1963, Veinott, 1966 and Porteus, 1971). If orders do not cross, the (s, S) optimality result under constant lead times carries over to stochastic, non-crossing lead times ( Muharremoglu and Tsitsiklis, 2008). To our knowledge, there is no analytical work that shows the optimality of the (s, S) policy in finite capacity production/inventory systems. The (s, S) inventory policy is still of main importance today to inventory theory and ordering policies and is incorporated in business software of many companies all over the world ( Caplin and Leahy, 2010). The traditional (s, S) inventory literature treats lead times exogenously with respect to the inventory policy. This means that the lot sizing decision is made in a local inventory environment, where production lead times are assumed to be exogenous and independent with respect to the lot size. Treating lead times as exogenous to the inventory model is justified when both production and inventory are decoupled through a large inventory at the production; if the owner of the production system guarantees a fixed delivery date; or if transportation lead times are much longer than production lead times ( Benjaafar et al., 2005). In these environments, the inventory policy does not have a (significant) impact on lead times. For some recent examples of inventory systems with exogenous lead times, we refer to Glock (2012a) and Hoque (2013). However, these assumptions do not hold in integrated production/inventory systems. In a production environment, there is a relationship between lot sizes and the lead times. Multiple small batches may cause an increase in traffic intensity at the production if there is a setup time per batch, resulting in lengthy queues and long waiting times (Karmarkar, 1987). At the other extreme, if lot sizes are very large, lead times approach an increasing function of the lot size. In such a system we need to take the dependency between lot sizes and lead times into account to determine the optimal lot size and reorder point parameters. In this paper, we examine the lot sizing decision in a production/inventory environment, in which the order quantities generated by the inventory model determine the production lot sizes, and thus the (production) lead times. These lead times in turn affect the parameters of the inventory policy. We show that the inclusion of endogenous lead times (as opposed to assuming lead times are exogenous to the inventory model) leads to different lot sizing decisions. Ignoring the endogeneity in lead times may lead to incorrect lot sizing decisions and, as a result, to higher costs.

In this paper, we study the lot sizing decision in a finite capacity production/inventory system, where lead times are endogenously determined by the orders placed in the inventory model. In the inventory management literature, lot sizes are often set equal to the economic order quantity to trade-off fixed ordering costs with inventory carrying costs. However, in a finite capacity production environment, the lot size determines the lead time distribution in production, which in turn impacts inventory costs. Therefore, when minimizing total (inventory and ordering) costs, setting the lot size equal to the economic order quantity can be far from optimal. We restrict our analysis to the class of (s, S) policies and provide a Markov-based procedure to compute the distribution of the time an order spends in the production system (i.e., in queue, in setup and in production) and the distribution of inventory levels at a random point in time for a given set of (s, S) parameters. In a numerical analysis, we show that ignoring the endogeneity of lead times in a production/inventory system may lead to inappropriate lot sizing decisions and higher expected total costs. This cost discrepancy is very outspoken if the EOQ value deviates significantly from desirable production lot sizes: in these cases, the endogenous treatment of lead times is of particular importance to set the right parameters of the (s, S) policy that minimize total costs.