NEWSBOY TYPE INVENTORY MODEL
2. Optimal Strategies When the Vendor’s Available Funds Are Unlimited
In this section we introduce the general model and investigate the vendor’s optimal strategy. We assume that there are no limits on the funds the vendor has available for procurement and that either the cost of capital is low or the time horizon under consideration is short. (For an analysis of the vendor’s optimal purchase/consignment strategy when funding is limited see Pasternack [2000].) We note that because a vendor does not have to obtain the item on consignment, in order for any con- signment scheme to be attractive to a vendor, it must result in a higher expected profit than if the vendor obtains all units through outright purchase.
The notation we use is as follows:
= the vendor’s cost per unit if the vendor obtains the item from the manufacturer through outright purchase.
= the retail price per unit and therefore the vendor’s revenue per unit if the vendor obtains the item from the manufacturer though outright purchase.
= the vendor’s salvage value per unit if the vendor obtains the item from the manufacturer through outright purchase.
= the vendor’s goodwill cost per unit if the vendor is out of stock of the item.
= the manufacturer’s production cost per unit.
= the vendor’s cost per unit if the vendor obtains the item from the manufacturer on consignment.
= the vendor’s revenue per unit if the vendor obtains the item from the manufacturer on consignment. (Note that equals the revenue paid to the manufacturer if an item purchased on con- signment is sold by the vendor.)
= the number of units the vendor obtains from the manufac- turer through outright purchase.
= the number of units the vendor obtains from the manufac- turer on consignment.
Q = the number of units ordered if the supply channel is coordi- nated.
= the vendor’s expected profit if it purchases units and obtains units on consignment.
= the manufacturer’s expected profit if the vendor purchases units and obtains units on consignment.
EP(Q) = the total expected channel profit if the channel is coor- dinated and the order quantity is Q.
= the optimal number of units the vendor should obtain from the manufacturer through outright purchase if the vendor wishes to maximize its expected profit.
= the optimal number of units the vendor should obtain from the manufacturer on consignment if the vendor wishes to maximize its expected profit.
= the optimal number of units ordered for a coordinated supply channel.
= the probability density function of demand.
We will assume that due to competitive pressure, the values of and are fixed1, however the manufacturer has control over setting the values of and Given the cost structure set by the manufacturer, the vendor will then determine the order quantity that maximizes its expected profit.
Following this notation, we see that if the vendor purchases the item outright from the manufacturer, the vendor earns a gross profit of for each unit sold and the manufacturer earns a gross profit of
for each unit ordered by the vendor. If however, the vendor orders the item from the manufacturer on a consignment basis, the vendor would
1For example, videotapes are not only sold to stores that rent tapes, but also to stores which sell tapes outright to consumers. Even among stores that rent tapes, some may not wish to participate in a consignment scheme. For these reasons it is assumed that the vendor’s cost per unit, and the retail price per unit, are not affected by the introduction of the consignment scheme.
pay the manufacturer per unit plus an amount equal to for each unit sold. As a result, for each unit purchased on consignment and sold by the vendor, the vendor earns a gross profit of and the manufacturer earns a gross profit of
We further assume that if the vendor purchases the item outright from the manufacturer the salvage value, from each unsold unit accrues to the vendor, whereas if the item is obtained from the manufacturer on a consignment basis the salvage value accrues to the manufacturer. In all cases the goodwill cost per unit due to shortage, is assumed to accrue to the vendor.
The following conditions are assumed to hold regarding and
(The vendor’s revenue per unit is greater if it purchases the item than if it obtains the item on consignment.)
(The vendor’s cost per unit is greater if it purchases the item than if obtains the item on consignment.)
(The vendor’s cost per unit for purchasing the item is greater than its salvage value)
(For items purchased, the vendor’s revenue per unit is greater than the cost per unit.)
(For it to be worthwhile for the manufacturer to offer the item to the vendor on a consignment basis, the manufac- turer’s gross profit per unit from consignment should be greater than the gross profit from outright sale to the vendor. This also states that the vendor’s gross profit per unit from outright pur- chase is greater than if it obtains the item on consignment.) Note that we will not require That is, for items obtained on consignment, the vendor’s revenue per unit may be less than the cost per unit. While such a situation would rarely arise, it is conceivably possible that it may be worthwhile for the vendor to obtain goods on consignment even if they would be sold at a loss in order to avoid the potential of incurring extremely high goodwill costs.
Operationally, because a vendor earns a larger gross profit on the units it obtains through outright purchase than on consignment, we assume that the vendor will first sell the units it purchases outright and will only sell those it obtains on consignment after all purchased units have been sold. The basis for our analysis is to investigate the Karush-Kuhn-Tucker (KKT) conditions in order to determine optimal strategies. Based on
The partial derivatives of are as follows:
and
The first term in equation (6.2) arises due to the fact that any items ordered by the vendor on consignment and unsold are returned to the manufacturer who will then dispose of them for their salvage value.
The problem faced by the vendor is therefore:
Maximize and
the above notation, we have:
The following KKT conditions (see Hillier and Lieberman [2001]) are therefore required for optimality:
where and are the Lagrange multipliers (dual variables) for this problem.
We also see that:
and
By looking at the second-order partial derivatives for we
note that if then is concave.
Our interest lies in determining the structure of the optimal strategy for the vendor. This gives rise to the following three theorems.
Theorem 6.1 It is impossible for and
Proof: If then from KKT condition (2) Hence, from KKT condition (4) we have:
Substituting equation (6.8) into KKT condition (3) gives:
But if then and it would be impossible for equation (6.9) to be satisfied.
Theorem 6.1 states that it would never pay for the vendor to obtain the item from the manufacturer only on a consignment basis.
Theorem 6.2 If If then it is impossible for and
Proof: Suppose Then from KKT condition (1) we have If we also assume that from KKT condition (4) we have:
while from KKT condition (3) we have:
Since we know from Theorem 6.1 that obtaining the good only on consignment cannot be optimal, Theorem 6.2 gives conditions under which it is optimal to both purchase the item outright and obtain it by consignment. In essence, what Theorem 6.2 states is that if the ratio of revenue to cost for goods obtained on consignment is high enough, it will never be optimal for the vendor to only purchase the good. Such a condition will occur if the salvage value is low and the goodwill cost is high2. In such cases the optimal amount for the vendor to purchase and to obtain on consignment can be determined by recognizing that
and solving for KKT conditions (3) and (4). This gives the following relationships for and
and
Theorem 6.3 If then and
Proof: We show in this case that it is impossible for both and In particular, if and then KKT conditions (1)
2This observation can be seen by rewriting as
But, if
and (2) imply that Hence, equations (6.12) and (6.13) must be satisfied. But if then from equation (6.13) we would have which would be impossible. Also, we know from The- orem 6.1 that it is impossible for and Hence,
and The result follows from KKT condition (3).
Theorem 6.3 states that if the salvage value per unit is greater than then the vendor would obtain all items through outright purchase.
The purchase amount would be identical to the amount the vendor would purchase if there were no consignment option.
Given this as a background we now turn our attention to how the manufacturer can set the values of and to achieve channel coordi- nation. This is examined in the next section.