Coordination of supply chain with a dominant retailer under government price regulation by revenue sharing contracts

Abstract

As the demands of some important products such as oil, gas, and agricultural commodities are disrupted, the government often regulates the retail price that includes impositions of a price ceiling and a price floor. In this paper, we analyze the coordination of a supply chain with a dominate retailer under the government price regulation policy by a revenue sharing contract after demand disruption. First, we characterize the optimal decisions of the supply chain under normal circumstance by the revenue sharing contract as a benchmark. Then, when the demand is disrupted, we redesign the contract to coordinate the supply chain and obtain the corresponding revenue sharing contract in different scenarios. Finally, we give some numerical examples to illustrate our theoretical results and explore the impacts of government price regulations on the coordination mechanism.

Appendix

Proof of Theorem 1

In the decentralized supply chain, the dominant retailer decides the retail price to maximize his own profit. By solving the first order conditions of (3) with respect to p and s, we have

$$\begin{aligned} p_d^\Delta =\frac{2t\varphi _d \alpha +(2t\beta -\gamma \theta ^{2}\varphi _d )(w_d +c_1 )}{4t\varphi _d \beta -\gamma \theta ^{2}\varphi _d^{2}},\quad s_d^\Delta =\frac{\gamma ^{2}\theta ^{2}[\varphi _d \alpha -\beta (w_d +c_1 )]^{2}}{(4t\beta -\gamma \theta ^{2}\varphi _d )^{2}}. \end{aligned}$$

Let $p_d^\Delta =p^{{*}}$, $s_d^\Delta =s^{{*}}$ and $\varphi _{r}p^{*}=w_{r}+c_{1}$. We get (8).

Because $0<\varphi _d^{*} <1$ and $w_d^{*} >0$, the dominant retailer’s revenue share $\varphi _d^{*}$ must satisfy

$$\begin{aligned} \frac{\beta c_{1}}{\beta p^{{*}}-q_T^{*} }<\varphi _d^{*} <1 \end{aligned}$$

(19)

The second objective is in order to satisfy win–win condition for the chain partners by the contracts. Assume that $\pi _k^{*}$ is the optimal profit of the actor k ($k=m, d)$ in the supply chain with the revenue-sharing contracts and $\pi _m$ is the profit of the actor $k\,(k=m, d)$ without any contracts. Only when $\pi _k^{*} >\pi _{km}$, the chain partners prefer to design the revenue-sharing contracts. Based on the above inequalities and (19), we can obtain (9).

Proof of Corollary 1

For convenience, the objective function (10) can be differentiated into two cases. We can combine these two cases to give the optimal solutions for the centralized supply chain.

$$\begin{aligned}&\left\{ {{\begin{array}{l} {\mathop {\max }\limits _{\tilde{p},\tilde{s}} \tilde{T}^{1}=(\tilde{p}-c_1 -c_0 )(\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}})-\tilde{s}-c_u (\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}-q_T^{*} )} \\ {\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}\ge q_T^{*} } \\ \end{array} }} \right. \; \end{aligned}$$

(20)

$$\begin{aligned}&\left\{ {{\begin{array}{l} {\mathop {\max }\limits _{\tilde{p},\tilde{s}} \tilde{T}^{2}=(\tilde{p}-c_1 -c_0 )(\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}})-\tilde{s}-c_{\mathrm{s}} (q_T^{*} -\tilde{\alpha }+\tilde{\beta }\tilde{p}-\theta \sqrt{\tilde{s}})} \\ {\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}\le q_T^{*} } \\ \end{array} }} \right. \; \end{aligned}$$

(21)

Based on the above formulas, we can see that $\tilde{T}^{1}$ and $\tilde{T}^{2}$ are concave functions of the retail price $\tilde{p}$ and service investment $\tilde{s}$, thus the solutions that satisfy the first-order condition give the optimal retail prices and service investments. $\square$

The Kuhn–Tucker condition of Eq. (21)

$$\begin{aligned} \left\{ {{\begin{array}{l} {{\begin{array}{l} {\begin{array}{l} \frac{\partial \tilde{T}^{1}}{\partial \tilde{p}}-\lambda \frac{\partial (\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}-q_T^{*} )}{\partial \tilde{p}}=0 \\ \frac{\partial \tilde{T}^{1}}{\partial \tilde{s}}-\lambda \frac{\partial (\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}-q_T^{*} )}{\partial \tilde{s}}=0 \\ \end{array}} \\ {\lambda (\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}-q_T^{*} )=0} \\ \end{array} }} \\ {{\begin{array}{l} {\lambda \ge 0} \\ {\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}-q_T^{*} \ge 0} \\ \end{array} }} \\ \end{array} }} \right. \end{aligned}$$

(22)

Here $\lambda$ is the Lagrangian multiplier. Solving (22), we get the flowing:

If there are no price regulations and $\frac{2\tilde{\alpha }\tilde{\beta }-2\tilde{\beta }^{2}(c_0 +c_1 )-(4\tilde{\beta }-\theta ^{2})q_T^*}{2\tilde{\beta }^{2}}\ge c_u$, the optimal decisions satisfy

$$\begin{aligned} \tilde{p}_1^{*} =\frac{2\tilde{\alpha }+(2\tilde{\beta }-\theta ^{2})(c_0 +c_1 +c_u )}{4\tilde{\beta }-\theta ^{2}},\quad \tilde{s}_1^{*} =\frac{\theta ^{2}[\tilde{\alpha }-\tilde{\beta }(c_0 +c_1 +c_u )]^{2}}{(4\tilde{\beta }-\theta ^{2})^{2}}. \end{aligned}$$

When the retail price is regulated, we need to discuss the magnitude of the optimal retail price without regulations.

If $\bar{{p}}<\frac{2\tilde{\alpha }+(2\tilde{\beta }-\theta ^{2})(c_0 +c_1 +c_u )}{4\tilde{\beta }-\theta ^{2}}$, the supply chain is optimal when the retail price is equal to the price ceiling. At this time, the optimal service investment satisfies $\tilde{s}_1^{*} =\frac{\theta ^{2}[\bar{{p}}-(c_0 +c_1 +c_u )]^{2}}{4}$.

If $\frac{2\tilde{\alpha }+(2\tilde{\beta }-\theta ^{2})(c_0 +c_1 +c_u )}{4\tilde{\beta }-\theta ^{2}}<\underline{p}<\frac{2\tilde{\alpha }+(2\tilde{\beta }-\theta ^{2})(c_0 +c_1 +c_u )}{4\tilde{\beta }-\theta ^{2}}+\;\frac{\sqrt{\Delta _1 }}{4\tilde{\beta }-\theta ^{2}}$, the supply chain is optimal when the retail price is equal to the price floor ($\Delta _1 =[2\tilde{\alpha }+(2\tilde{\beta }-\theta ^{2})(c_0 +c_1 +c_u )]^{2}+(4\tilde{\beta }-\theta ^{2})[(\theta ^{2}-4\tilde{\alpha })(c_0 +c_1 +c_u )+c_u q_T^{*} ])$. Then, $\tilde{s}_1^{*} =\frac{\theta ^{2}[\underline{p}-(c_0 +c_1 +c_u )]^{2}}{4}$.

And if the price floor is too large and is greater than zero point, that there is no solution for the supply chain coordination with demand disruptions.

If $-c_s<\frac{2\tilde{\alpha }\tilde{\beta }-2\tilde{\beta }^{2}(c_0 +c_1 )-(4\tilde{\beta }-\theta ^{2})q_T^*}{2\tilde{\beta }^{2}}<c_u$, the Lagrangian multiplier $\lambda >0$, which means that $\tilde{\alpha }-\tilde{\beta }\tilde{p}+\theta \sqrt{\tilde{s}}-q_T^{*} =0$. From Kuhn–Tucker condition, we obtain $\tilde{p}_2^{*} =\frac{\beta ^{2}(2\tilde{\beta }-\theta ^{2})}{\tilde{\beta }^{2}(2\beta -\theta ^{2})}(\beta p^{*}-\alpha )+\frac{\tilde{\alpha }}{\tilde{\beta }},\tilde{s}_2^{*} =\frac{\beta ^{2}}{\tilde{\beta }^{2}}s^{*}$.

Similarly, we discuss this issue in four conditions. However, when the optimal retail price exceeds the price regulations, we should discuss whether to change original production quantity nor not.

Likewise, if $\frac{2\tilde{\alpha }\tilde{\beta }-2\tilde{\beta }^{2}(c_0 +c_1 )-(4\tilde{\beta }-\theta ^{2})q_T^*}{2\tilde{\beta }^{2}}\le -c_s$, the optimal solution of (22) is

$$\begin{aligned} \tilde{p}_3^{*} =\frac{2\tilde{\alpha }+(2\tilde{\beta }-\theta ^{2})(c_0 +c_1 -c_s )}{4\tilde{\beta }-\theta ^{2}},\quad \tilde{s}_3^{*} =\frac{\theta ^{2}[\tilde{\alpha }-\tilde{\beta }(c_0 +c_1 -c_s )]^{2}}{(4\tilde{\beta }-\theta ^{2})^{2}}. \end{aligned}$$

When the price is regulated, the optimal solutions are similar to previous proofs.

Combining these two cases, we can obtain the Corollary 1.

Proof of Theorem 2

Because $0<\tilde{\varphi }_d^{*} <1$ and $\tilde{w}_d^{*} >0$, $\tilde{\varphi }_d^{*}$ must satisfy:

$$\begin{aligned} \frac{\tilde{\beta }c_1 }{\tilde{\beta }\tilde{p}^{{*}}-\tilde{q}_T^{*} }<\tilde{\varphi }_d^{*} <1 \end{aligned}$$

(23)

We assume that $\tilde{\pi }_k^*$ and $\tilde{\pi }_{kr}$ is the optimal profit of the actor $k\, (k=d, m)$ in the supply chain with the new and the original revenue sharing contracts respectively. In order to assure the partners prefer the new contracts to the original ones, let $\tilde{\pi }_k^*>\tilde{\pi }_{kr}$. So we can obtain

$$\begin{aligned}&[\tilde{\varphi }_d^{*} \tilde{p}^{{*}}-(\tilde{w}_d^{*} +c_1 )]\tilde{q}_d^{*} -\tilde{t}^{*}\tilde{s}^{*}>[\varphi _d^{*} p^{{*}}-(w_d^{*} +c_1 )]\tilde{q}_{dr}^{*} -t^{*}s^{*}\nonumber \\&[\tilde{w}_d^{*} -c_0 +(1-\tilde{\varphi }_d^{*} )\tilde{p}^{{*}}]\tilde{q}_d^{*} +[\tilde{w}_r^{*} -c_0 +(1-\tilde{\varphi }_r^{*} )\tilde{p}^{{*}}]\tilde{q}_r^{*}\nonumber \\&\quad -(1-\tilde{t}^{{*}})\tilde{s}^{{*}}-c_u (\Delta \tilde{Q})^{+}-c_s (-\Delta \tilde{Q})^{+}>\tilde{\pi }_{mr} \end{aligned}$$

(24)

Substitute (16) to (24), we can obtain

$$\begin{aligned} \frac{2\tilde{\beta }(\tilde{T}^{{*}}-\tilde{\pi }_{mr} )}{\tilde{q}_T^{*} (2\tilde{q}_d^{*} -\gamma \theta \sqrt{\tilde{s}^{{*}}})}>\tilde{\varphi }_d^{*} >\frac{2\tilde{\beta }\{[\varphi _d^{*} p^{{*}}-(w_d^{*} +c_1 )]\tilde{q}_{dr}^{*} -t^{*}s^{*}\}}{\tilde{q}_T^{*} (2\tilde{q}_d^{*} -\gamma \theta \sqrt{\tilde{s}^{{*}}})}, \end{aligned}$$

(25)

From (23) and (25), we can get (17) (Koulamas 2006). $\square$

Similar to Assumption 1, the market share of the dominant retailer $\gamma$ should satisfy the following condition, so that the dominant retailer should provide the demand-stimulating service voluntarily, and the manufacturer would like to induce such service even when the supply chain is in the decentralized operation.

$$\begin{aligned}&\gamma [\tilde{p}^{{*}}-(c_0 +c_1 )](\tilde{\alpha }-\tilde{\beta }\tilde{p}^{{*}}+\theta \sqrt{\tilde{s}^{{*}}})-\tilde{s}^{{*}}-c_u (\gamma \Delta \tilde{Q})^{+}-c_s (-\gamma \Delta \tilde{Q})^{+}\nonumber \\&\quad >\gamma [\tilde{p}_w^*-(c_0 +c_1 )](\tilde{\alpha } -\tilde{\beta }\tilde{p}_w^*) -c_u (\gamma \Delta \tilde{Q}_w )^{+}-c_s (-\gamma \Delta \tilde{Q}_w )^{+} \end{aligned}$$

(26)

Therefore, we can obtain (18).