Performance optimization of supply chain based on cooperative contract with disappointment-aversion strategic consumers

Abstract

Consumers’ strategic behavior and psychological perception have impact on supply chains. In this paper, we consider a supply chain with one supplier and one retailer to study the influence of disappointment-aversion strategic (DAS) consumers on supply chain members’ decisions and performance. The results of this study show that strategic consumers can reduce the profit levels of such a supply chain, whilst DAS consumers can alleviate this effect. We also study the effectiveness of a quantity commitment strategy for a centralized supply chain when faced with DAS consumers. Under certain conditions, the quantity commitment strategy can further reduce the loss of profit caused by strategic consumers. Then the maximum profit of the centralized supply chain under the quantity commitment strategy is used as a benchmark. We then analyze the coordination efficiency and implementability of a wholesale price contract and a price subsidy contract respectively in a decentralized supply chain. The results show that the wholesale price contract can achieve the optimal profit benchmark of the centralized supply chain, but it can only realize fixed profit allocation to supply chain members; the price subsidy contract can not only achieve the optimal profit of the centralized supply chain but also make arbitrary profit allocation among chain members.

Appendix

1.1 Proof for Proposition 1

According to equation (3), the profit function of the alliance is

$\Pi_{c} (Q,p) = pE(X \wedge Q) + sE(Q - X)^{ + } - cQ$. Substituting $E(Q - X)^{ + } = Q - E(Q \wedge X)$ into the above formula, we get.

$\Pi_{c} (Q,p) = (p - s)E(X \wedge Q) - (c - s)Q{ = }(p - s)[\int_{0}^{Q} {xf(x)dx} + \int_{Q}^{ + \infty } {Qf(x)dx} ] - (c - s)Q$.

Since $\frac{{\partial \Pi_{c} (Q,p)}}{\partial Q} = (p - s)\overline{F} (Q) - (c - s)$ and $\frac{{\partial^{2} \Pi_{c} (Q,p)}}{{\partial Q^{2} }} = - (p - s)f(Q) < 0$, $\Pi_{c} (Q,p)$ is a concave function of $Q$ that has a unique maximum value. Based on the first-order condition, let $\frac{{\partial \Pi_{c} (Q,p)}}{\partial Q} = 0$, and then $\overline{F} (Q) = \frac{c - s}{{p - s}}$. From Eq. (4), we can get

$$p_{c} = s + (v - s)\overline{F} (Q_{c} )[1 + kF(Q_{c} )]$$

(21)

Substituting (21) into the above equation, we can get formula (6). Substituting formula (6) into formula (21), we can get formula (7).

1.2 Proof for Proposition 4

We first prove that this contract can achieve the maximum profit benchmark of the centralized supply chain under quantity commitment. That is, the retailer's optimal ordering quantity $Q_{m} = Q_{q}^{*}$. When the supply chain achieves the maximum profit under the quantity commitment, for the wholesale price contract, the optimal ordering quantity $Q_{q}^{*}$ and the price $p^{*}$ are characterized by

$$\overline{F} (Q_{q}^{*} ) = \frac{{w^{*} - s}}{{p^{*} - s}}$$

(22)

$$p^{*} = s + (v - s)\overline{F} (Q_{q}^{*} )[1 + kF(Q_{q}^{*} )]$$

(23)

Let parameters $w_{m}$ and $m$ under the price subsidy contract satisfy

$$w_{m} - s - m = \chi (w^{*} - s)$$

(24)

and

$$p^{*} - s - m = \chi (p^{*} - s)$$

(25)

where $\chi > 0$. In this contract, the conditions of RE equilibrium are

$$\overline{F} (Q_{m} ) = \frac{{w_{m} - s - m}}{{p_{m} - s - m}}$$

(26)

$$p_{m} = s + (v - s)\overline{F} (Q_{m} )[1 + kF(Q_{m} )]$$

(27)

Therefore, under the price subsidy contract, the ordering quantity $Q_{m}$ and price $p_{m}$ based on the RE equilibrium satisfy

$$\overline{F} (Q_{m} ) = \frac{{\chi (w^{*} - s)}}{{(p_{m} - p^{*} ) + \chi (p^{*} - s)}}$$

(28)

$$p_{m} = s + (v - s)\overline{F} (Q_{m} )[1 + kF(Q_{m} )]$$

(29)

Comparing (22 and 28), if $p_{m} > p^{*}$, then $\overline{F} (Q_{m} ) < \overline{F} (Q_{q}^{*} )$; comparing (23 and 29), if $\overline{F} (Q_{m} ) < \overline{F} (Q_{q}^{*} )$, since when $0 < k < \overline{k}$, function $(1 - x)\sqrt {1 + kx}$ is a decreasing function of $x$($0 < x < 1$), then $p^{*} > p_{m}$, which contradicts with $p_{m} > p^{*}$. If $p_{m} < p^{*}$, it will also lead to a contradiction. Therefore, there must be $p^{*} = p_{m}$, so that $\overline{F} (Q_{m} ) = \overline{F} (Q_{q}^{*} )$, and thereby $Q_{m} = Q_{q}^{*}$. The price subsidy contract can achieve the maximum profit of the centralized supply chain under quantity commitment.

Then we prove that for any $\lambda \in [0,1]$, when ${{\chi = \lambda } \mathord{\left/ {\vphantom {{\chi = \lambda } {\lambda^{*} }}} \right. \kern-\nulldelimiterspace} {\lambda^{*} }}$, the retailer's profit is $\lambda \Pi_{q}^{*}$.

$$\begin{gathered} \Pi_{m}^{r} (Q_{m} ,p_{m} ) = (p_{m} - s - m)E(X \wedge Q_{m} ) - (w_{m} - s - m)Q_{m} \hfill \\ = (p^{*} - s - m)E(X \wedge Q_{q}^{*} ) - (w_{m} - s - m)Q_{q}^{*} \hfill \\ = \frac{\lambda }{{\lambda^{*} }}[(p^{*} - s)E(X \wedge Q_{q}^{*} ) - (w^{*} - s)Q_{q}^{*} ] = \frac{\lambda }{{\lambda^{*} }}\lambda^{*} \Pi_{q}^{*} = \lambda \Pi_{q}^{*} \hfill \\ \end{gathered}$$

Substituting ${{\chi = \lambda } \mathord{\left/ {\vphantom {{\chi = \lambda } {\lambda^{*} }}} \right. \kern-\nulldelimiterspace} {\lambda^{*} }}$ into (24 and 25), we can get $w_{m} = (1 - \frac{\lambda }{{\lambda^{*} }})p^{*} + \frac{\lambda }{{\lambda^{*} }}w^{*}$, $m = (1 - \frac{\lambda }{{\lambda^{*} }})(p^{*} - s)$.

Proofs for Proposition 2, 3 and 5 can be easily obtained from the context.