...are crucial to the operation of the customer's business and not having them when they are needed can cause major disruptions. As result, customers expect suppliers to be responsive in order fulfillment, typically allowing the suppliers less time than is required for production and delivery. Third, the products are slow-moving items, which makes the timing issue of inventory management as important as, if not more important than, the quantity issue. Due to the special products they are dealing with, the suppliers in these industries face the following dilemma in order fulfillment. On the one hand, if production is started too early when the demand forecast is poor, then they may incur a high holding cost and the risk is even greater if orders do not materialize. On the other hand, if production is started too late, the suppliers may not be able to achieve the high degree of responsiveness the customers expect. Cohen et al. (2003) discuss in great detail the order-fulfillment process in these industries with a focus on the semiconductor equipment industry. Our research provides a formal model to understand the dilemma and examines various comparative statics.

The basic purposes of inventory analysis in manufacturing and stockkeeping services is to specify: (i) when items should be produced (or ordered, delivered, etc.); and (ii) how large the quantity should be. The literature has disproportionately focused on the latter, and only a few studies deal with the timing issues directly (1). For example, in continuous-review models, which assume an exponential demand inter-arrival time, and all periodic-review models, the state of the system remains unchanged between demand arrivals; therefore, orders are placed only at the points when the demand arrives. As the demand arrivals determine exogenously when items should be ordered, the question of how much should be ordered is naturally the main focus. Schultz (1987), Katircioglu and Atkins (1996) and Moinzadeh (2001) study the classical single-stage, single-product, continuous-review inventory problem. They all recognize that the base-stock policy, which restricts ordering to the points when demand arrives, is not necessarily optimal when the demand inter-arrival time is more general than exponential. Katircioglu and Atkins (1996) show that a delayed (S-1, S) policy is optimal when the inter-arrival time distribution has an increasing failure rate. Schultz (1987) and Moinzadeh (2001) introduce policies that include a delay in ordering under different settings. These studies all show by numerical experiments that the improvement over the base-stock policy can be quite significant.

A recent paper by Cohen et al. (2003) models the order-fulfillment process of a supplier producing a customized capital good as a newsvendor problem with the time to start production as the decision variable. They estimate the corresponding imputed cost parameters based on empirical data. Their data reveals that the supplier is very conservative in commencing the order fulfillment, which leads the authors to conclude that the supplier perceives the holding cost and the cancellation cost to be about three and two times higher, respectively, than the delay cost.

We build our basic model based upon the elegant framework of Katricioglu and Atkins (1996) and apply it to the order-fulfillment process in capital goods industries. The optimal policy is first shown to be characterized by a critical time: if the order is canceled or confirmed before the critical time, then do nothing or start production immediately when it is confirmed, respectively; otherwise, start production at the critical time. We then investigate the comparative statics effects of lead time and lead time uncertainty on the process. We show that the supplier always responds to an increase in lead time by commencing production earlier, and that the adjustment in the time to produce as a result of a change in lead time may sometimes be greater than the change in lead time itself. Although the cost can always be reduced by shortening the lead time, both the expected inventory holding cost and the service level can go either way. The impact of lead time uncertainty, under some mild conditions, depends on the cost structure of the system in a very simple way. Then, the basic model is extended to include the supplier's risk aversion. We show that the structure of the optimal policy remains the same under risk aversion, but the critical time depends on the risk attitude. Although the relationship between risk aversion and commencement of production is context specific, it is clear that risk aversion plays a crucial role in this process.

Finally, we compare our model with the model considered by Cohen et al. (2003). Our model differs from their model in two important ways. First, the optimal policy of our model allows the supplier to choose different actions depending on the demand information received, whereas in the model by Cohen et al. (2003), the supplier must commit to start production at the critical time regardless of the information received. Second, they assume that the supplier is rational (cost minimizing) and risk neutral, but we consider the supplier's risk aversion. The empirical data collected by Cohen et al. (2003) reveals that the supplier is very conservative in commencing order fulfillment and they attribute this to high holding and cancellation costs relative to the delay cost. Our model, due to the two differences mentioned above, provides two alternative explanations. First, we show that in our model the supplier delays his or her production to a later critical time than in the model by Cohen et al. (2003). This suggests that the supplier's unwillingness to commit early may also be explained by his or her need to wait until the demand is more likely to arrive, other than costs. Second, risk aversion, alone or together with other factors, may have caused the supplier to be conservative.

The remainder of the paper is organized as follows. Section 2 describes the basic model. The impact of an increasing (deterministic) lead time on the optimal policy, cost, inventory holding and service level is discussed in Section 3. We also discuss the gap between the performance of our optimal policy and that of a commonly used policy in this section. The impact of adding risk to the lead time gap is addressed in Section 4. The basic model is extended to include the supplier's risk aversion in Section 5. We compare our model with the one presented in Cohen et al. (2003) and offer two alternative explanations to their empirical data in Section 6. Section 7 concludes the paper.

2. Basic model

At time 0, the supplier receives a soft (forecasted) order for one unit of product through online information systems and direct customer interaction from its sales and marketing department. A substantial difference exists between a soft order and a firm purchase order. There are two independent types of uncertainty in the soft order: (i) the time when the buyer places a firm order or cancels it; and (ii) the probability of cancellation. At some point in time [~.x], the uncertainty inherent in the soft order is resolved. That is, at time [~.x], the order is either confirmed or canceled. The random variable [~.x], which we call the demand arrival time for ease of exposition, has a known distribution function [PHI](dot), density function [phi](dot) and mean [mu]. The product has to be delivered d unit time after the order is confirmed, otherwise the supplier incurs a delay cost of p per unit time of late delivery. Having the product ready before it is needed, however, will cost the supplier h per unit time in holding. Furthermore, a cancellation cost c per unit time is incurred if the order is canceled after production has been started. The production lead time for the product is L. In our setting, if L [less than or equal to] d, then the optimal policy is trivial: the supplier should always wait until the order is confirmed. Therefore, throughout this paper we assume L > d. We denote the probability of the soft order eventually being confirmed as [alpha]. The supplier is assumed to have infinite capacity.

If the supplier decides to wait until some time t, two events may happen. First, the buyer may cancel the order or confirm the order before t, i.e., [~.x] t}, or

[[PHI].sub.t](y) = [[PHI](y + t) - [PHI](t)]/[bar.[PHI]](t),

where [bar.[PHI]](t) = 1 - [PHI](t). If we think of [PHI] as the life distribution of a product in reliability theory, then [[PHI].sub.t](y) is the conditional probability of failure during the next interval of duration y of the product of age t (see, for example, Barlow and Proschan (1975)).

The expected cost, V(t), of waiting to start production until time t or the demand signal [~.x], whichever is sooner, can be written by conditioning on the two possible events: {[~.x] [less than or equal to] t} and {[~.x] > t} as...

NOTE: All illustrations and photos have been removed from this article.



More articles from IIE Transactions
Benefits of considering inventory in service parts logistics network d..., February 01, 2007
Strategic network design for multi-zone truckload shipments.(Author ab..., February 01, 2007
A Lagrangean heuristic for integrated production and transportation pl..., February 01, 2007
Coordinating production and distribution of jobs with bundling operati..., February 01, 2007
An improved version of the NEH algorithm and its application to large-..., February 01, 2007

Looking for additional articles?
Search our database of over 3 million articles.

Looking for more in-depth information on this industry?
Search our complete database of Industry & Market reports by text, subject, publication name or publication date.

About Goliath
Whether you're looking for sales prospects, competitive information, company analysis or best practices in managing your organization, Goliath can help you meet your business needs.

Our extensive business information databases empower business professionals with both the breadth and depth of credible, authoritative information they need to support their business goals. Whether it be strategic planning, sales prospecting, company research or defining management best practices - Goliath is your leading source for accurate information.