MANAGING DEMAND UNCERTAINTY FOR SHORT LIFE CYCLE PRODUCTS
3. The Analysis Framework
3.2 The Advance Booking Discount Model
Notice from (4.1) that the term
corresponds to the sum of the expected overstock and understock costs associated with the optimal order quantity Q*. Thus, one can reduce the impact of these costs by reducing the demand variance We now discuss how the ABD program can enable a retailer to achieve variance reduction. Under this program, the retailer offers a discount price per unit of Brand A (i.e., the discount factor is equal to prior to
the beginning of the season, where If customers accept this offer, then they place an order by pre-paying per unit prior to
the beginning of the season and pick up this order during the season. If customers decline this offer, they can always purchase the product during the season by paying regular price per unit; however, the availability of the product will not be guaranteed.
The ABD program affects the two segments of customers as follows.
First, among those customers who plan to buy Brand A during the selling season (i.e., will commit their orders at a lower price prior to the selling season and will purchase the product at regular price during the selling season.4 Second, for those customers who plan to buy Brand B during the selling season (i.e.,
will switch from buying Brand B to Brand A at a lower price prior to the selling season and the remaining will buy Brand B during the selling season as planned.5 We assume that the
It is conceivable that the consumption may increase as a result of price discount. However, since the product is perishable and since the customers can pick up the product only during the selling season, the customers may not be able to consume more during the selling season as a result of the ABD program. Thus, it seems reasonable to assume that the consumption remains the same.
5Since this segment plans to buy Brand B during the selling season, they would not buy Brand A during the selling season at the regular price However, some of them may switch
where and and are the distribution and the
4
functions and are bounded between 0 and 1 and decreasing in so that more customers will buy Brand A prior to the selling season as decreases.
Let be the pre-committed orders occurred prior to the season and let be the demand that occurs during the season, where:
Notice that because the ABD
program generates additional demand for Brand A due to customers who switch from buying Brand B at its regular price to Brand A at the discount price
Suppose that the joint distribution of is a bivariate normal distribution with means and standard deviations and
and correlation coefficient where
where Then it is well known (Bickel and Doksum (1977)) that
the distribution of given (i.e., is also
to buy Brand A prior to the selling season because of the discount price associated with the ABD program.
normally distributed with mean and standard deviation where6:
We offer two observations. First, notice from (4.8) that Thus, the ABD program will reduce the variance of the demand that occurs during the selling season. Second, it can be shown from (4.8 ) that is concave in and that is decreasing in
for Thus, positively correlated demands further re- duce demand variance due to the additional information associated with pre-committed orders generated from those customers switching from Brand B to Brand A. These two observations illustrate the basic mechanism by which the ABD program enables the retailer to increase sales, obtain an improved forecast, and place a more accurate order so as to achieve higher expected profits.
In the remainder of this section, we shall evaluate the optimal ex- pected profits associated with the ABD program. To obtain tractable analytical results, we shall consider the case in which the correlation coefficient between and is equal to 0; (i.e.,
However, the same analysis can be extended numerically to include the case for which
When it is easy to check from (4.4), (4.6) and (4.8) that and can be expressed as follows:
6The bivariate normal distribution allows us to obtain simple expressions for and and to simplify our analysis. To elaborate, if one uses the conjugate prior distributions to determine the posterior distribution of the updated demand, then the mean and the standard deviation of the posterior distribution is quite complex and would complicate the analysis significantly. Also, for the case when the retailer offers no discount; i.e., when we have
and has a degenerate bivariate normal distribution that has In such case, we have and
3.2.1 No Demand Forecast Updating. Consider the case when the retailer offers the ABD program with the discount factor To isolate the benefits of variance reduction and improved forecast due to updating in the ABD program, we first assume that the retailer is unable to utilize the pre-committed orders to update the distri- bution of This scenario is plausible when the retailer lacks the infrastructure to capture or analyze sales data.
Since the order is placed at the start of selling season, the retailer can order the exact amount to fulfill the pre-committed orders observed prior to the selling season. Hence, the profit generated from those pre-committed orders will equal Although the retailer does not use to update the distribution of
is still normally distributed with mean and standard deviation given by (4.3) and (4.5), respectively. In this case, the retailer orders an additional quantity so as to cover the demand during the selling season. Thus, the profit generated from the demand is equal to The total expected profit associated with the ABD program without demand forecast updating, denoted by can be expressed as follows:8
By using the standard newsvendor result, the expected profit can be expressed as:
We now compare the expected profit for any discount factor given in (4.12) with the optimal profit associated with the base case
given in (4.1). When and Thus, the
optimal expected profit must be at least equal to In this case, we have proved the following Lemma:
Lemma 4.1
The above lemma implies that the ABD program, even without updated demand forecasts, can increase the retailer’s expected profit due to two
7We shall consider the case in which the retailer would utilize the early sales information to update the demand forecast in section 3.2.2
8Without loss of generality, we omit the fixed promotion cost associated with the ABD program. Clearly, when the promotion cost is prevalent, the decision maker can always check to see if the expected benefit of the ABD program outweighs this promotion cost.
reasons. First, this program generates additional demand because it offers an economic incentive for customers to switch from Brand B to Brand A by paying a reduced price Second, this program reduces the demand uncertainty because a portion of the demand has been made certain (i.e., the pre-committed orders) prior to the sales season.
3.2.2 Demand Forecast Updating. Consider the case in which the retailer offers the ABD program and utilizes the pre-committed orders to update the distribution of In this case, the profit
generated from those pre-committed orders equals to
Since the updated distribution of is the retailer would order additional quantity Q' so as to cover the demand during the season.9 Thus, the profit generated during the season is equal to The optimal total ex- pected profit associated with the ABD program with demand forecast updating, denoted by can be expressed as:
Since is normally distributed with mean and standard deviation we can utilize (4.7), (4.11), and the newsvendor result to express as:
We now compare the expected profit given in (4.13) with the expected profit given in (4.12). For any given discount factor it can be shown that
Therefore, we have proved the following Lemma:
Lemma 4.2
The above lemma implies that, when implementing the ABD program, the retailer can realize higher expected profits if the retailer utilizes the pre-committed orders to update the demand distribution
This is because the variance of the demand will be further reduced due to updating. Overall, by increasing sales and reducing demand variance, the ABD program enables the retailer to increase the expected profit.
9The total order quantity is now equal to