INCENTIVE MISALIGNMENT IN SUPPLY CHAIN MANAGEMENT

The last point that we stressed in the previous section is the potential difficulty due to the interaction of multiple noncooperative, if not competitive, decision makers. The example we consider is a generalization of the newsvendor model:⁶

1. Unlike the basic model, there are two decisions to be made. The ordering decision follows the same logic as the standard case, but there is another one, related to product quality, which influences the probability distribution of demand.
2. Since there are two decisions involved, we should distinguish two cases:
   • In the integrated supply chain, there is only one decision maker in charge of both decisions.
   • In the disintegrated supply chain, there are two decision makers; a producer, who is in charge of setting the level of her product quality, and a distributor, who is in charge of deciding his order quantity.

In order to be able to find analytical solutions, we depart from the usual assumption of normal demand, and we suppose a uniform distribution, say, between 500 and 1000. We will rely on the following notation; the lower bound of the distribution support is denoted by a and its width by w; hence, the expected value is a + w/2. In the numerical example, a = 500 and w = 1000 – 500 = 500. The item has a production cost of c = €0.20 and sells at a price p = €1.00. To keep things as simple as possible, we suppose that the unsold items are just scrapped, and there is no salvage value. With the numbers above, we see that service level should be⁷

where m = p – c is the profit margin and c_u = c is the cost of unsold items, when there is no salvage value. The optimal order quantity is

Since the probability distribution is uniform, we may easily compute the expected profit. This requires calculating fairly simple integrals involving the constant demand density

The expected profit depends on Q and it amounts to the expected revenue minus cost:

This expression of expected profit may be interpreted as profit related to the purchased quantity, minus the expected lost revenue due to unsold items.⁸ To get an intuitive feeling for Eq. (14.5), we may refer to Fig. 14.2, where lost revenue is plotted against demand, for a given order quantity Q. When demand is at its lower bound, D = a, lost revenue is p(Q – a); when demand is D = Q, lost revenue is 0. For intermediate values of demand, lost revenue is a decreasing function of demand and results in the triangle illustrated in the figure. To evaluate the expected value of lost revenue, we should integrate this function, multiplied by the probability density 1/w. But this is simply the area of the triangle in Fig. 14.2, P(Q – a)²/2, times 1/w, which in fact yields the second term in Eq. (14.5). With our numerical data, the optimal expected profit is

We may also express the expected profit as a function of the service level β, by plugging Eq. (14.4) into Eq. (14.5):

Fig. 14.2 A geometric illustration of Eq. (14.5).

So far, we have always assumed that a demand distribution cannot be influenced by the producer. However, she could improve the product or adopt marketing strategies to change the distribution a bit. The result depends on the effort she spends, which in turn has a cost. Let us measure the amount of effort by h, with unit cost 10. To model the effect of h on demand, we make the following assumptions:

1. For the sake of simplicity, we assume a pure shift in the uniform distribution of demand. Its lower bound a is shifted, but its support width w does not change.⁹
2. The shift in a should be a concave function of h, to represent the fact that effort is decreasingly effective for increasing levels (diminishing marginal return). One possible building block that we can use is the square-root function.

Hence, we represent the lower extreme of the distribution support by the following function of h:

For instance, with our numerical parameters, if h = 1 the lower bound a shifts from 500 to 600. Using Eq. (14.6), we see that, if we include the effort h, the expected profit becomes

Note that the optimal service level does not change, according to our model, since we do not change either the unit cost of the item or its selling price. To find the optimal effort, we should take the first-order derivative of expected profit (14.7) with respect to h and set it to 0:

We notice that this condition equates the marginal increase of expected profit contribution from production and sales with the marginal cost of the effort. With this effort, the new probability distribution is uniform between 900 and 1400, the optimal produced quantity increases to Q* = 1300, and by applying Eq. (14.7) the new total expected profit is

This is the optimal profit resulting from the joint optimal decisions concerning effort level h and purchased amount Q.

The above calculations are formally correct, but potentially flawed: They relyon the assumption that there is one decision maker maximizing overall expected profit and in control of both decisions. What if we have a supply chain with different stakeholders in charge of each decision, with possibly conflicting objectives? To see the impact of misaligned incentives, let us assume that there are two stakeholders:

1. The producer, who is in charge of determining the quality of the product and its potential for sales, through the effort h
2. The distributor, who is in charge of determining the order quantity Q, depending on the probability distribution of demand and the prices at which he can buy and sell

We cannot analyze such a system if we do not clarify not only who is in charge of deciding what, but also when and on the basis of what information. So, to be specific, let as assume the following:

• The producer is the leader and the distributor is the follower, in the sense that the producer decides her effort level h first, and then the distributor will select his order quantity Q. It must be stressed that, in deciding the effort level h, the producer should somehow anticipate how her choice will influence the choice of Q by the distributor.
• The producer is also the leader in the sense that she is the one in charge of pricing decisions, which we assume given. She keeps the selling price fixed at €1.00 and sells each item to the distributor for a price of €0.80, which is the purchase cost from the viewpoint of the distributor. Please note that the profit margin for the distributor is just €0.20; furthermore, he is the one facing all of the risk, as items have no salvage value and there is no buyback agreement in case of unsold items.
• Last but not least, everything is common knowledge, in the sense that the producer knows how the distributor is going to make his decision and both agree on the probability distribution of demand and how this is affected by the effort level h.

The last point implies that when the producer makes her decision, she can anticipate what the optimal decision of the distributor is going to be; hence, she can build a reaction function, also known as best response function, describing how the order quantity Q is influenced by her effort level h. From the point of view of the follower, i.e., the distributor, once the effort h is decided by the leader, the only thing he can do is to determine the order quantity as a function of h. Under our simple assumption of a pure shift in the demand distribution, we have

This expression look just the same as before but there is a fundamental difference. Under our assumptions, the production cost and the selling price do not change, but now, since the profit margin is shared, the optimal service level for the distributor is only

There is a remarkable reduction in the service level, from 80% to 20%, because of the high price at which the distributor buys from the producer. Indeed, the split of the profit is €0.60 to the producer and €0.20 to the distributor. Plugging numbers, the optimal order quantity is

This is how the producer can anticipate the effect of her choice of h on the decision of the distributor, which in turn influences her profit. The profit from the point of view of the producer is

Applying the first-order optimality condition, we obtain the optimal effort, and then the optimal order quantity:

The new optimal order quantity depends on the fact that the shifted probability distribution of demand, with that level of effort, is uniform between 800 and 1300, and optimal service level is just 20%. A first observation that we can make is that both effort and order quantity are reduced when supply chain management is decentralized. This implies that consumers will receive a worse product and a reduced service level. What is also relevant, though, is the change in profit for the two players. The profit of the producer is

This profit is considerably reduced with respect to expected profit for the producer when she also manages distribution, which is €720. Of course, this is not quite surprising, since the producer has given up a fraction of her profit margin, leaving it to the distributor. Unlike the producer, the distributor faces an uncertain profit. Its expected value can be obtained from Eq. (14.5), after adjusting the parameters to reflect the reduced profit margin

The expected profit for the distributor is less than the (certain) profit of the producer, but this is no surprise after all, considering how margins are split between the two parties. Last but not least, the total profit for the disintegrated supply chain is

which is €100 less than the total profit for the fully integrated supply chain. Again, this is reasonable, in light of the concepts of Section 14.2.

These observations raise a couple of points:

1. One might well wonder why the producer should delegate distribution to someone else. Actually, there is a twofold answer:
   • Her profit now is a certain amount and not an expected value, since the whole risk is transferred to the distributor.
   • We did not consider fixed costs related to distribution; they may not impact optimal decisions at the tactical or operational level, but they do have an impact on the bottom line and on strategic decisions.
   We may also add further considerations concerning the fact that a distributor is probably better equipped at making demand forecasts and has a better incentive to offer additional services to the customer.
2. In splitting profit margins and decentralizing decisions, something has been lost for everyone: the producer, the distributor, and, last but not least, the customers, who receive less quality and less service. In principle, we could try to find the “optimal split” of profit margins, i.e., the allocation of margins that maximizes the total profit. Of course, this is hardly feasible in practice, as it would require cooperation between the two stakeholders; furthermore, in general, the overall profit will remain suboptimal, anyway. There are practical ways to realign the incentives by shifting fixed amounts of money between the parties, as a lump payment does not affect the above decisions, since it is a constant amount and does not affect the calculation of the derivatives that are involved in the optimality conditions. An alternative is to redistribute risks by introducing buyback contracts;¹⁰ if risk is shared between the two stakeholders, the optimal service level for the distributor is increased. The best strategy depends on the specific problem setting, as well as on the relative strengths of the parties involved.

In closing this section, we should note that, although this supply chain problem involves uncertainty in demand, this is not the key point. Coordination issues arise in purely deterministic problems, like the ones we show in Sections 14.4 and 14.5.