PARTIAL QUICK RESPONSE POLICIES IN A SUPPLY CHAIN
2. Partial Fulfillment Model
Consider a setting in which a manufacturer sells her product through a set of N independent (i.e. non-competing) retailers. The manufac- turer has two modes of production: one which is relatively inexpensive, but has a lead time sufficiently long that production quantities must be committed prior to the selling season; and the other which is more ex- pensive but allows production to be done during the selling season. We denote the per-unit production costs of these two modes by and respectively.
The retailers have two opportunities to order the product: before and after observing demand. However, the manufacturer does not guarantee that the reorder will be filled. In reality, retailers typically order once prior to the selling season in order to have the product available when customers want it, and then place reorders during the season if early season sales are strong. As discussed in Fisher and Raman, 1996, the information provided by these early season sales dramatically increases the accuracy of the demand forecast. To simplify the presentation of our analysis, we assume that the request for restocking occurs at the
end of the selling season, when the realization of demand has been fully observed. Although this eliminates the possibility that a retailer can both receive a second shipment and have excess stock at the end of the season, our model provides insight into the trade-off that the retailer faces between improved demand information versus higher costs and lower certainty of getting what he has ordered.
To analyze the effect of the manufacturer’s policy of filling reorders on supply chain performance, we assume that the manufacturer acts as a leader by announcing the fraction of reorder requests that she will fill. In practice, such an announcement could be made by establishing a reputation based on long term performance. We further assume that individual requests for restocking are either filled completely or not at all, such that from the perspective of an individual retailer, he will receive all of the units requested in a reorder with probability and none of the units with probability This assumption can be justified in terms of two practical considerations. First, if each retailer were allocated some fraction of the amount that he ordered, then there would be an incentive for retailers to inflate their orders. Second, this approach may reduce shipping costs relative to those associated with sending partially filled orders to all retailers.
In response to the manufacturer’s order refilling policy and the wholesale price each retailer places an initial order, denoted by The manufacturer then produces these quantities, at a cost of per unit, and delivers to the retailers.
After receiving his initial order quantity, each retailer experiences a single period of demand, earning revenue of per unit sold. If the realization of demand at retailer exceeds we assume that the excess demand can be backlogged at a cost of per unit, and the retailer places a reorder with the manufacturer for the number of units in the backlog.
The manufacturer then produces a fraction, of the total amount backlogged by all of the retailers. We assume that the manufacturer fills the fraction of all reorders in a manner that is perceived as random by the retailers. Note that this could result from either the manufacturer filling all requests for reorders on a randomly chosen set of the products that it produces, or by filling a portion of requests on all products. In other words, we assume that it is not necessary for the manufacturer to fill the fraction of requests for each realization of demand, so long as she fills the fraction of requests in expectation. This second production run incurs a cost of per unit and is sold to each of the retailers at per unit. The decision variables in the manufacturer’s optimization
problem are the fraction of reorders filled and the wholesale price for reorders.
If a retailer’s restocking request is not filled, then he experiences lost sales for the backlogged units. Alternatively, if a retailer receives his requested units, then he earns revenue of less a backlog cost of per unit. The last quantity captures the costs associated with special shipping to the customer or services necessary for special delivery. Thus, the backlog cost is not incurred if a retailer’s order is not filled by the manufacturer. Other than the backlog cost there is no other penalty incurred by the retailer for shortages, such as loss of goodwill cost, etc.
Each retailer faces independent identically distributed (i.i.d.) demand that has density and cumulative distribution function For simplicity, we assume that the manufacturer has the same information about the distribution of demand as do the retailers.
In order to analyze this model, let us first consider the problem faced by retailer in determining the appropriate amount to order at the first opportunity. Taking and as given, we can express the expected profits of retailer as follows:
where is the converse cumulative distribution evaluated at Q.
Assuming of course that it is easy to
confirm that (5.1) is concave with respect to and that the optimal order quantity for retailer can be expressed as:
Observe that the retailers’ order quantities are decreasing in α. Thus, as the manufacturer becomes more reliable in responding to restocking requests, the retailers decrease the amount that they order initially and become more apt to require restocking.
Let us now consider the perspective of the manufacturer whose ex- pected profits can be expressed in terms of the retailers’ optimal re- sponses to her announced restocking policy:
Since the manufacturer can induce each retailer to order quantity by setting and/or appropriately, we can alternatively express her profits as the following function of Q
where
is the expected amount of backlogged demand at a given retailer. Note that, since is a nonnegative random variable, the expected value
can be expressed as (Justification
is provided by Ross, 1998, among others.) Therefore,
2.1 Manufacturer controls only the Reorder Fulfillment Rate
Let us first assume that the manufacturer can control only the rate of fulfilling requests for reorders From (5.2), it can be shown that in order to induce an order quantity of Q, the manufacturer must fill reorders at rate:
Substituting (5.6) into (5.4) and rearranging, we obtain a new expression for the manufacturer’s expected profit as a function of the induced order quantity:
This expression allows us to make interesting interpretations of the in- dividual terms. Recall that is equal to the expected backlog at a given retailer. The term can be interpreted as the con- ditional expectation of the amount reordered by a retailer, given that his demand exceeds his initial order quantity. Unfortunately, M(Q) is in general neither concave nor convex, as indicated by the following Lemma.
Lemma 5.1 a) M(Q) is convex (concave) if and only if