Numerical Analysis

We first present an example to illustrate the form of the optimal contract and the performance of the system and each party. The parameters of the example are as follows: D = 1000 units/year, Cp = $25/unit, Ss = $400/setup, Sb = $20/order,

r = 0.2, CM = $12/unit, and CN = $5/unit. The initial product quality level is

¯

p = 0.85. Investment may be made in improving the quality with i = 0.1, α = 12500, and β = −23714. Suppose that the prior distribution of the supplier’s quality level ¯p is uniform over the interval [0.75, 0.95] with a mean of 0.85 and a standard deviation

of 0.0577. Suppose that the prior distribution of the supplier’s investment sensitivity α is uniform over the interval [8750, 16250] with a mean of 12500 and a standard deviation of 2165.13. Thus, the buyer’s estimation of ¯p and α is unbiased in this example. For simplicity, we assume that the supplier’s highest cost for signing a contract with the buyer is Cmax

s = 6589.78, which is the same as his total cost with

no form of contract.

Table V presents five cases, including no form of contract, a contract with full information, a contract with asymmetric information of the quality level ¯p, a con- tract to improve the quality with full information, and a contract with asymmetric information of the investment sensitivity α. As Table V indicates, when each party has full information of the system, a contract based on the lot size can help reduce the total cost of the system by 22.22%, and the buyer extracts all of the savings from the supplier by initiating the contract. When the buyer has incomplete information of the supplier’s quality level ¯p, the total savings are almost the same as in the full information case since the buyer’s estimation is unbiased in this example. However, the buyer can only reduce her own cost by 13.78%, instead of 45.34% as in the full in- formation case, due to the information asymmetry. The supplier’s cost is reduced by 30.27% at the same time since he has full knowledge of his quality level. If a supplier development program is initiated by the buyer and both parties reach a contractual agreement, the total cost of the system can by further reduced by 28.20%, and the buyer extracts all of the savings from the supplier if she has full information of the system. If the buyer has incomplete information of the investment sensitivity to im- prove the quality level, her own cost can be reduced by 43% in contrast to 57.55% in the full information case. The total savings are almost the same as in the full information case since the buyer’s estimation of α is unbiased. The supplier’s cost is reduced by 13.36% at the same time. In this example, we observe that a contract

Table V Comprehensive Results under Supply Uncertainty

Variable No contract Contract with Contract with Contract to Contract to full infor. asym. ¯p improve (full) improve (asym. α)

Q 105 482 612 438 446 L - 3496.24 1718.29 3730.80 2905.25 ¯ p 0.85 0.85 0.85 0.936 0.919 Supplier’s cost 6589.78 6589.78 4595.01 6589.78 5709.34 % Savings - 0 30.27% 0 13.36% Buyer’s cost 6329.57 3459.60 5457.27 2686.67 3607.58 % Savings - 45.34% 13.78% 57.55% 43.00% Total costs 12919.35 10049.38 10052.28 9276.45 9316.92 % Savings - 22.22% 22.19% 28.20% 27.88%

would help reduce the system’s total cost, and a supplier development program would help achieve further savings. If the buyer has full information of the system, she is able to extract all of the savings from the supplier. However, information asymme- try does impair the buyer’s capability to extract savings. Next, we examine further how the buyer can design contracts under information asymmetry and the impact of information asymmetry on the system’s and both individuals’ costs.

Figure 6 illustrates the buyer’s contract pairs (Q, L(Q)) when she has incomplete information of the supplier’s quality level ¯p. We observe that a larger lot size Q will lead to a larger sharing rate L. From the supplier’s perspective, a larger lot size means a smaller setup cost, and he would receive less savings as well. The tradeoff for the supplier, which is designed by the buyer under the incentive-compatibility constraint, is that he balances his setup costs and sharing rate which guarantees that the supplier will announce his true product quality.

Figure 6 Buyer’s Contract Menu (Q, L(Q))

(Q, ¯p, L) when she has incomplete information of the supplier’s investment sensitiv- ity α. We observe that a higher quality expectation ¯p is associated with a smaller sharing rate L. Recall Proposition 14 which states that the realized lot size ¯pQ in the buyer’s contract menu is the same as the joint optimal realized lot size, and thus the joint setup cost and inventory holding cost are system-wide optimal. Thus, from the supplier’s perspective, a lower quality expectation means less investment expense, larger disposal costs, and less savings received. When the quality expectation is high, the buyer’s sharing rate drops significantly. This is because the total investment to improve the quality level increases dramatically, and thus the buyer needs to expend much more effort and investment to help the supplier improve product quality as well as to make sure that the supplier is still willing to reveal his true investment sensitivity under the contract menu.

Figure 7 ¯p and L in Buyer’s Contract Menu (Q, ¯p, L)

level ¯p or investment sensitivity α, she is unable to extract all of the savings from the supplier. Next, we examine explicitly the impact of information asymmetry on the system’s and both individuals’ costs.

Figure 8 illustrates the impact of the coefficient of variation of the buyer’s esti- mation of ¯p on the joint total cost T 1, the supplier’s cost S1 and the buyer’s cost B1. It indicates that the buyer’s estimation variation has little impact on the system’s cost. However, the buyer’s cost increases significantly as the coefficient of variation increases. That is, the buyer’s capability of extracting savings from the supplier is considerably impaired if she has information about ¯p that is not accurate. Of course, the supplier’s benefits from this and his cost are significantly reduced.

Figure 9 illustrates the impact of the coefficient of variation of the buyer’s esti- mation of α on the joint total cost T 2, the supplier’s cost S2 and the buyer’s cost B2. Again, we observe that the buyer’s capability of extracting savings from the supplier

Figure 8 The Impact of the Coefficient of Variation δp¯

µp¯

is considerably impaired if she has inaccurate information of α, while the supplier’s benefits from this and his cost is significantly reduced.

The findings reported here clearly show that ignoring incentive conflicts and qual- ity information issues can lead to undesirable behavior. The practical implications are significant. In designing a supply chain, it is tempting to take the perspective of a central planner and focus on improving overall efficiency. Few supply chains have any mechanism that can pass for a central planner, so this is rarely an option. This section proposes a framework for how lot size, quality level, and transactions should be structured to help reduce supply chain inefficiency due to individual incentives and private information.

In this section, to concentrate on the supply contract design, we assume that the supply is in statistical control. In the next section, we further develop the model when the supply is random.

Figure 9 The Impact of the Coefficient of Variation δα

µα