Scope and Definition of B2B E-Commerce
Remark 3.3 We should note that we could relax the last assumption related to from being constant (of value to just being non-
For all K the penalty is linear in This is similar to the ap- proaches by Li and Kouvelis (1999) as well as by Barnes-Schuster et al. (1998).
The cost of goods is always increasing with i.e., it costs more to buy more. As demand is random, the manufacturer and supplier commit to a risk-sharing agreement in the following sense:
the penalty in total supply cost for “under-ordering” with respect to K becomes smaller as the order quantity approaches K.
is independent of K. This assumption is for pedagogic reasons only. However, it can be interpreted as the result of the negotiation between manufacturer and supplier. The first wants to have non-increasing in K, whereas the latter prefers to see
being non-decreasing in K, which may lead to
independent of K. (See Araman et al. (2000b) for a more detailed justification and a model without this assumption).
Based on the previous assumptions, a linear function that fulfills all of these conditions has to be of the form:
Thus the penalty is given by: See Figure 3.4 below for an illustration of the long-term pricing schedule
Remark 3.3 We should note that we could relax the last assumption
that describes the behavior of these markets precisely and since indus- try experts predict that different market mechanisms will co-exist on each exchange, we have exercised great caution in making assumptions about the spot market and define it as a random price-quantity func- tion. Thus the unit price is a random variable that depends on the number of units ordered, such that for a fixed is defined by its density function is in addition dependent on and most likely positively correlated with the total demand received by the buyer.
Since we consider the optimal mix with respect to the expected value of the procurement costs, it is sufficient to consider that the spot market
price is given by where for every order and
a total end market demand is the expected value of the price. In addition, as we did for the long-term supply pricing scheme, we assume that the function is increasing in i.e., the total cost of goods increases with the quantity of goods bought from the spot market. We let On the other hand, as we stressed in the first sections of this chapter, one of the main issues related to spot markets is their liquidity, as manifested by the current low transaction volume of most spot markets. That means that companies may not be able to fill all of their demands using only the spot market. We therefore consider in the general case a spot market with a capacity constraint, i.e., C is the maximum amount available at the time when the buyer places her order. We assume that C is random, possibly dependent on the total demand and bounded by a constant value
For clarity of exposition we make the assumption that is indepen- dent of and that C is constant (possibly infinite). Again the main results hold true in the general case (see Araman et al. (2000a), and Araman et al. (2000b) for details). Therefore, unless stated otherwise,
we consider We finally assume that:
Problem Formulation. Our first approach is to consider a risk- neutral buyer interested in minimizing her expected total procurement cost G(K). As we will see below, the model is defined such that higher moments of the demand distribution will come into play. Thus, despite the assumption that the decision-maker is risk-neutral, which makes the problem tractable, the model still captures the volatilities of both the demand and the spot market. If we denote the total random cost associated with a reserved capacity level K, the optimization problem of the manufacturer can be written as follows: (We will use the follow- ing notation: E[X;A] that should be read E[X . I(A)], where I is the indicator function. It means the expected value of the random variable X, under the event A, which is not to be confused with a conditional expectation.)
When the demand of the manufacturer is less than K, she will only use the long-term contract, and therefore:
However when the demand is higher than K, exactly K units will be supplied by the long-term contract and an additional units will have to be purchased on the spot market. Again, since most spot markets are currently struggling with low transaction volumes, the manufacturer may not be able to purchase as many units on the spot market as needed and the amount available to the manufacturer is given by max
where C is the spot market capacity. We assume that the manufacturer incurs a fixed penalty cost per unit, on those units that she could not
buy on the spot market.
We will divide the total expected cost into two parts, slightly differently than just computed. The first term, represents the expected cost that is spent via the long-term contract, whereas the second, is the expected cost of purchases on the spot market, including the penalty cost for a possible undersupply. We rearrange the terms of the previous equations to obtain:
and the total expected cost is again the sum of the two terms:
Proposition 3.4 is a positive, increasing, and concave func- tion of K such that and H converges asymptotically to a finite constant
Proposition 3.5 is a positive decreasing function of K such that and converges asymptotically to 0 as
These two last propositions are very intuitive. As K increases the (expected) amount paid to the long-term supplier increases and, simul- taneously, the amount spent on the spot market decreases in expectation.
See Figure 3.5 for an illustration of and Notice that in the graph below, is greater than or in other words, it is more expensive in expectation to only purchase on the spot market than to go exclusively with the long-term supplier.
Main Result. We now describe a set of results that indicate that the spot market is valuable under fairly general conditions, and also characterize the values of the (optimal) contracted capacity K.
Theorem 3.6 Let K* be the optimal value of K that minimizes (3.2)