Distribution channel strategies of suboptimal food supply chain under demand uncertainty

Abstract

The commercialization of edible suboptimal foods thrown away seems a promising lever for tackling food shortage and waste. This paper constructs a Stackelberg game model to analyze the optimal distribution channel decisions of the suboptimal food supply chain consisting of a manufacturer and a retailer under the wholesale and consignment mode, respectively. Moreover, this paper performs the sensitivity analysis of key parameters including suboptimal food demand potential, marketing service cost coefficient and retail price. The results show that the optimal distribution channel strategies of the manufacturer and retailer are obtained for those who are profit-driven under different conditions. The suboptimal food demand potential, the preference of consumers to the suboptimal food, marketing service cost coefficient, retail price and price discount are key factors. Interestingly, the Pareto improvement for the whole suboptimal food supply chain can be reached under consignment mode, which means that the manufacturer and retailer not only achieve higher expected profits but also reduce food waste. This study provides important support for increasing firms’ profits in the suboptimal food supply chain and promoting sustainable development from a game theoretic approach.

Appendices

Appendix: Proof of Lemma 1

The first partial derivative of $E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]$ with respect to $Q_{w}$, $s_{w}$:

$$\frac{{\partial E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial Q_{w} }} = - w\beta + \varphi p\left[ {\beta - \frac{{\beta \left( {\beta Q - \gamma s - \theta a + b\varphi p} \right)}}{n}} \right]$$

(A.1)

$$\frac{{\partial E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial s_{w} }} = - ks + \frac{{\gamma \varphi p\left( {\beta Q - \gamma s - \theta a + b\varphi p} \right)}}{n}$$

(A.2)

The second derivative of $E\left[ {\pi \left( {Q_{w} ,s_{w} } \right)} \right]$ with $Q_{w}$, $s_{w}$: $\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial Q_{w}^{2} }} = - \frac{{\beta^{2} \varphi p}}{n} < 0$, $\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial s_{w}^{2} }} = - k - \frac{{\gamma^{2} \varphi p}}{n} < 0$. And $\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial Q_{w} \partial s_{w} }} = \frac{p\beta \gamma \varphi }{n}$.

The Hessian matrix of $E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]$ is verified the joint concavity on $Q_{w}$, $s_{w}$:

$$\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial Q_{w}^{2} }} \times \frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial s_{w}^{2} }} - \left( {\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{w} ,s_{w} } \right)} \right]}}{{\partial Q_{w} \partial s_{w} }}} \right)^{2} = \frac{{k\beta^{2} \varphi {\text{p}}}}{n} > 0$$

(A.3)

The expected profit of the retailer is joint concavity on $Q_{w}$, $s_{w}$: $Q_{w}^{*} = \frac{{\left( {\varphi p - w} \right)\left( {kn + \gamma^{2} \varphi p} \right)}}{k\beta \varphi p} + \frac{\theta a - b\varphi p}{\beta }$, $s_{w}^{*} = \frac{{\gamma \left( {\varphi p - w} \right)}}{k}$. Moreover, only the potential market demand hold: $\theta a > b\varphi p - \frac{{\left( {\varphi p - w} \right)\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p}$, the retailer is willing to sell SFs under the wholesale mode.

Proof of Proposition 1

According to Lemma 1, we substitute $Q_{w}^{*}$, $s_{w}^{*}$ into $E\left[ {\pi_{M} \left( {Q_{w} ,s_{w} ,w} \right)} \right]$, then the first and second partial derivatives of $E\left[ {\pi_{M} \left( w \right)} \right]$ with $w$:

$$\frac{{\partial E\left[ {\pi_{M} \left( w \right)} \right]}}{\partial w} = \frac{{\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p + c - 2w} \right) + k\varphi p\left( {\theta a - bp\varphi } \right)}}{k\varphi p}$$

(A.4)

$$\frac{{\partial^{2} E\left[ {\pi_{M} \left( w \right)} \right]}}{{\partial w^{2} }} = - \frac{{2\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p} < 0$$

(A.5)

The manufacturer’s expected profit is concave in $w$. By solving ${{\partial E\left[ {\pi_{M} \left( w \right)} \right]} \mathord{\left/ {\vphantom {{\partial E\left[ {\pi_{M} \left( w \right)} \right]} {\partial w}}} \right. \kern-0pt} {\partial w}} = 0$, the optimal wholesale price is obtained: $w^{**} = \frac{1}{2}\left( {c + \frac{{p\varphi \left[ {\gamma^{2} \varphi p + kn + k\left( {\theta a - bp\varphi } \right)} \right]}}{{kn + \gamma^{2} \varphi p}}} \right)$.

Substituting $w^{**}$ into $Q_{w}^{*}$, $s_{w}^{*}$, the retailer’s optimal order quantity and marketing service level are obtained: $Q_{w}^{**} = \frac{{\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right) + k\varphi p\left( {\theta a - b\varphi p} \right)}}{{{\text{2k}}\beta \varphi p}}$, $s_{w}^{**} = \frac{{\gamma \left[ {\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right) - k\varphi p\left( {\theta a - b\varphi p} \right)} \right]}}{{2k\left( {kn + \gamma^{2} \varphi p} \right)}}$.

To ensure that SFSC is willing to provide and sell SFs, it must hold: $Q_{w}^{**} > 0$, $s_{w}^{**} > 0$, $w^{**} > 0$. Therefore, the potential market demand must hold: $b\varphi p - \frac{{\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p} < \theta a < b\varphi p + \frac{{\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p}$.

Proof of Corollary 1

(i) Solving the first partial derivative of $Q_{w}^{**}$ with ${\text{a}}$, $\lambda$, ${\text{k}}$, ${\text{p}}$, respectively:

$$\frac{{\partial Q_{w}^{**} }}{\partial a} = \frac{{\left( {1 - \varphi } \right)p}}{{2\beta \left( {1 - \lambda } \right)V}} > 0$$

(A.6)

$$\frac{{\partial Q_{w}^{**} }}{\partial \lambda } = \frac{{\left( {1 - \varphi } \right)pa}}{{2\beta \left( {1 - \lambda } \right)^{2} V}} > 0$$

(A.7)

$$\frac{{\partial Q_{w}^{**} }}{\partial k} = - \frac{{\gamma^{2} \left( {\varphi p - c} \right)}}{{2k^{2} \beta }} < 0$$

(A.8)

$$\frac{{\partial Q_{w}^{**} }}{\partial p} = \frac{{\frac{{a\left( {1 - \varphi } \right)p}}{{\left( {1 - \lambda } \right)V}} + \frac{cn}{{\varphi p}} - b\varphi p + \frac{{\gamma^{2} \varphi p}}{k}}}{2\beta p} > 0$$

(A.9)

(ii) Solving the first partial derivative of $s_{w}^{**}$ with ${\text{a}}$, $\lambda$, ${\text{k}}$, $\varphi$, respectively:

$$\frac{{\partial s_{w}^{**} }}{\partial a} = - \frac{{\gamma \varphi p\left( {1 - \varphi } \right)p}}{{2V\left( {1 - \lambda } \right)\left( {kn + \gamma^{2} \varphi p} \right)}} < 0$$

(A.10)

$$\frac{{\partial s_{w}^{**} }}{\partial \lambda } = - \frac{{\gamma \varphi p\left( {1 - \varphi } \right)pa}}{{2V\left( {1 - \lambda } \right)^{2} \left( {kn + \gamma^{2} \varphi p} \right)}} < 0$$

(A.11)

$$\frac{{\partial s_{w}^{**} }}{\partial k} = \frac{\gamma }{{2k^{2} }}\left( {\frac{{ - \left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right)^{2} + \varphi pk^{2} n\left( {\theta a - b\varphi p} \right)}}{{\left( {kn + \gamma^{2} \varphi p} \right)^{2} }}} \right)$$

(A.12)

Given $\theta a < b\varphi p + \frac{{\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p}$, Then $\frac{{\partial s_{w}^{**} }}{\partial k} < - \frac{{\gamma^{3} \varphi p\left( {\varphi p - c} \right)}}{{2k^{2} \left( {kn + \gamma^{2} \varphi p} \right)}} < 0$.

$$\begin{aligned} \frac{{\partial s_{w}^{{**}} }}{{\partial \varphi }} & = \frac{{\gamma p}}{{2k\left( {kn + \gamma ^{2} \varphi p} \right)}}\left[ {\frac{{ka\varphi p\left( {kn + \gamma ^{2} \varphi p} \right)}}{{V\left( {1 - \lambda } \right)}} + \left( {kn + \gamma ^{2} \varphi p} \right)^{2} } \right. \\ & \quad + \left. {kb\varphi p\left( {kn + \gamma ^{2} \varphi p} \right) - k^{2} n\left( {\theta a - b\varphi p} \right)} \right] \\ \end{aligned}$$

(A.13)

Given $\theta a < b\varphi p + \frac{{\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p}$, Then $\frac{{\partial s_{w}^{**} }}{\partial \varphi } > \frac{\gamma p}{{2k}}\left[ {\frac{ka\varphi p}{{V\left( {1 - \lambda } \right)}} + \gamma^{2} \varphi p + kb\varphi p + \frac{knc}{{\varphi p}}} \right] > 0$.

(iii) Solving the first partial derivative of $w^{**}$ with ${\text{a}}$, $\lambda$, ${\text{k}}$, ${\text{p}}$, respectively:

$$\frac{{\partial w^{**} }}{\partial a} = \frac{{k\varphi p\left( {1 - \varphi } \right)p}}{{2V\left( {1 - \lambda } \right)\left( {kn + \gamma^{2} \varphi p} \right)}} > 0$$

(A.14)

$$\frac{{\partial w^{**} }}{\partial \lambda } = \frac{{ak\varphi p\left( {1 - \varphi } \right)p}}{{2V\left( {1 - \lambda } \right)^{2} \left( {kn + \gamma^{2} \varphi p} \right)}} > 0$$

(A.15)

$$\frac{{\partial w^{**} }}{\partial k} = \frac{{\gamma^{2} \varphi^{2} p^{2} \left[ {\theta a - b\varphi p} \right]}}{{2\left( {kn + \gamma^{2} \varphi p} \right)^{2} }} > 0$$

(A.16)

$$\frac{{\partial w^{**} }}{\partial p} = \frac{{\varphi \left[ {k\left( {2kn + \gamma^{2} \varphi p} \right)\left( {\theta a - b\varphi p} \right) + \left( {kn + \gamma^{2} \varphi p} \right)^{2} } \right]}}{{2\left( {kn + \gamma^{2} \varphi p} \right)^{2} }} > 0$$

(A.17)

Proof of Lemma 2

Taking the first derivative of $E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]$ with respect to $Q_{c}$, $s_{c}$:

$$\frac{{\partial E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial Q_{c} }} = \left( {1 - \omega } \right)\varphi p\left( {\beta - \frac{{\beta \left( {\beta Q - \gamma s - \theta a + b\varphi p} \right)}}{n}} \right)$$

(A.18)

$$\frac{{\partial E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial s_{c} }} = - ks + \frac{{\left( {1 - \omega } \right)\gamma \varphi p\left( {\beta Q - \gamma s - \theta a + b\varphi p} \right)}}{n}$$

(A.19)

Then solving the second partial derivative of $E\left[ {\pi \left( {Q_{c} ,s_{c} } \right)} \right]$ with $Q_{c}$, $s_{c}$: $\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial Q_{c}^{2} }} = \frac{{\beta^{2} \varphi p\left( {\omega - 1} \right)}}{n} < 0$, $\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial s_{c}^{2} }} = - k - \frac{{\gamma^{2} \varphi p\left( {1 - \omega } \right)}}{n} < 0$. And $\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial Q_{c} \partial s_{c} }} = \frac{{p\beta \gamma \varphi \left( {1 - \omega } \right)}}{n}$.

The Hessian matrix of $E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]$ is verified the joint concavity on $Q_{c}$, $s_{c}$:

$$\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial Q_{c}^{2} }} \times \frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial s_{c}^{2} }} - \left( {\frac{{\partial^{2} E\left[ {\pi_{R} \left( {Q_{c} ,s_{c} } \right)} \right]}}{{\partial Q_{c} \partial s_{c} }}} \right)^{2} = \frac{{k\beta^{2} \varphi {\text{p}}\left( {1 - \omega } \right)}}{n} > 0$$

(A.20)

The expected profit of the retailer is concave on $Q_{c}$, $s_{c}$. Then we get $Q_{c}^{*} = \frac{{kn + \left( {1 - \omega } \right)\gamma^{2} \varphi p}}{k\beta } + \frac{\theta a - b\varphi p}{\beta }$, $s_{c}^{*} = \frac{{\left( {1 - \omega } \right)\gamma \varphi p}}{k}$. Moreover, only the potential market demand hold: $\theta a > b\varphi p - \frac{{kn + \left( {1 - \omega } \right)\gamma^{2} \varphi p}}{k}$, the retailer is willing to sell SFs under the consignment mode.

Proof of Proposition 2

According to Lemma 2, we substitute $Q_{c}^{*}$, $s_{c}^{*}$ into $E\left[ {\pi_{M} \left( {Q_{c} ,s_{c} ,\omega } \right)} \right]$, then the first and second partial derivatives of $E\left[ {\pi_{M} \left( \omega \right)} \right]$ with $\omega$:

$$\frac{{\partial E\left[ {\pi_{M} \left( \omega \right)} \right]}}{\partial \omega } = \frac{{2c\gamma^{2} \varphi p + \varphi {\text{p}}\left[ {kn + 2k\left( {\theta a - b\varphi p} \right) + 2p\gamma^{2} \varphi \left( {1 - \omega } \right)} \right] - 2\omega \gamma^{2} \varphi^{2} p^{2} }}{2k}$$

(A.21)

$$\frac{{\partial^{2} E\left[ {\pi_{M} \left( \omega \right)} \right]}}{{\partial \omega^{2} }} = - \frac{{2\gamma^{2} \varphi^{2} p^{2} }}{k} < 0$$

(A.22)

So the manufacturer’s expected profit is concave in $\omega$. By solving $\frac{{\partial E\left[ {\pi_{M} \left( \omega \right)} \right]}}{\partial \omega } = 0$, the manufacturer’s optimal consignment revenue-sharing ratio is $\omega^{**} = \frac{{2\gamma^{2} \left( {\varphi p + c} \right) + kn + 2k\left( {\theta a - b\varphi p} \right)}}{{4\gamma^{2} \varphi p}}$.

Substituting $\omega^{**}$ into $Q_{c}^{*}$, $s_{c}^{*}$, the retailer’s optimal order quantity $Q_{c}^{**} = \frac{{2\gamma^{2} \left( {\varphi p - c} \right) + 3kn + 2k\left( {\theta a - b\varphi p} \right)}}{4k\beta }$ and service level $s_{c}^{**} = \frac{{2\gamma^{2} \left( {\varphi p - c} \right) - kn - 2k\left( {\theta a - b\varphi p} \right)}}{4k\gamma }$ are obtained.

In order to ensure that SFSC can be willing to provide and sell SFs, it must hold: $Q_{c}^{**} > 0$,$s_{c}^{**} > 0$, $\omega^{**} > 0$. Therefore, the potential market demand must hold: ${\text{max}}\left\{ {b\varphi p - \frac{{2\gamma^{2} \left( {\varphi p - c} \right) + 3kn}}{2k},b\varphi p - \frac{{2\gamma^{2} \left( {\varphi p + c} \right) + kn}}{2k}} \right\} < \theta a < b\varphi p + \frac{{2\gamma^{2} \left( {\varphi p - c} \right) - kn}}{2k}$.

Proof of Corollary 2

(i) Solving the first partial derivative of $Q_{c}^{**}$ with ${\text{a}}$, $\lambda$, ${\text{k}}$, ${\text{p}}$, respectively:

$$\frac{{\partial Q_{c}^{**} }}{\partial a} = \frac{{p\left( {1 - \varphi } \right)}}{{2V\beta \left( {1 - \lambda } \right)}} > 0$$

(A.23)

$$\frac{{\partial Q_{c}^{**} }}{\partial \lambda } = \frac{{\left( {1 - \varphi } \right)pa}}{{2\beta \left( {1 - \lambda } \right)^{2} V}} > 0$$

(A.24)

$$\frac{{\partial Q_{c}^{**} }}{\partial k} = - \frac{{\gamma^{2} \left( {\varphi p - c} \right)}}{{2k^{2} \beta }} < 0$$

(A.25)

$$\frac{{\partial Q_{c}^{**} }}{\partial p} = \frac{{\theta a - b\varphi p + \frac{{\gamma^{2} \varphi p}}{k}}}{2\beta p} > 0$$

(A.26)

(ii) Solving the first partial derivative of $s_{w}^{**}$ with ${\text{a}}$, $\lambda$, ${\text{k}}$, $\varphi$, respectively:

$$\frac{{\partial s_{c}^{**} }}{\partial a} = - \frac{{\left( {1 - \varphi } \right)p}}{{2\gamma \left( {1 - \lambda } \right)V}} < 0$$

(A.27)

$$\frac{{\partial s_{c}^{**} }}{\partial \lambda } = - \frac{{\left( {1 - \varphi } \right)pa}}{{2\gamma \left( {1 - \lambda } \right)^{2} V}} < 0$$

(A.28)

$$\frac{{\partial s_{c}^{**} }}{\partial k} = - \frac{{\gamma \left( {\varphi p - c} \right)}}{{2k^{2} }} < 0$$

(A.29)

$$\frac{{\partial s_{c}^{**} }}{\partial \varphi } = \frac{1}{2\gamma }\left( {bp + \frac{{\gamma^{2} p}}{k} + \frac{ap}{{\left( {1 - \lambda } \right)V}}} \right) > 0$$

(A.30)

(iii) Solving the first partial derivative of $\omega^{**}$ with ${\text{a}}$, $\lambda$, ${\text{k}}$, ${\text{p}}$, respectively:

$$\frac{{\partial \omega^{**} }}{\partial a} = \frac{{k\left( {1 - \varphi } \right)}}{{2V\gamma^{2} \left( {1 - \lambda } \right)\varphi }} > 0$$

(A.31)

$$\frac{{\partial \omega^{**} }}{\partial \lambda } = \frac{{ak\left( {1 - \varphi } \right)}}{{2V\gamma^{2} \left( {1 - \lambda } \right)^{2} \varphi }} > 0$$

(A.32)

$$\frac{{\partial \omega^{**} }}{\partial k} = \frac{{2\left( {\theta a - b\varphi p} \right) + n}}{{4pV\gamma^{2} \varphi }} > 0$$

(A.33)

$$\frac{{\partial \omega^{**} }}{\partial p} = - \frac{{kn + 2\gamma^{2} c}}{{4p^{2} \gamma^{2} \varphi }} < 0$$

(A.34)

Proof of Proposition 3

To make sure that the retailer is willing to operate under two modes, i.e., $Q_{w}^{*} > 0$, $Q_{c}^{*} > 0$, We set: $\theta a_{1} = b\varphi p - \frac{{\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p - w} \right)}}{k\varphi p}$, $\theta a_{2} = b\varphi p - \frac{{kn + \gamma^{2} \varphi p\left( {1 - \omega } \right)}}{k}$. If $a > a_{1}$, $Q_{w}^{*} > 0$; If $a > a_{2}$, $Q_{c}^{*} > 0$. Comparing $a_{1}$ and $a_{2}$, only $a > max\left\{ {a_{1} ,a_{2} } \right\}$, $Q_{w}^{*} ,Q_{c}^{*} > 0$ and vice versa. $\theta a_{1} - \theta a_{2} = \frac{nw}{{\varphi p}} + \frac{{\gamma^{2} \left( {w - \omega \varphi p} \right)}}{k}$. When ${\text{w}} \ge \frac{{\omega \left( {\gamma \varphi {\text{p}}} \right)^{2} }}{{kn + \gamma^{2} \varphi {\text{p}}}}$, $a_{1} \ge a_{2}$; and when ${\text{w}} < \frac{{\omega \left( {\gamma \varphi {\text{p}}} \right)^{2} }}{{kn + \gamma^{2} \varphi {\text{p}}}}$, $a_{1} < a_{2}$.

Therefore, when ${\text{w}} \ge \frac{{\omega \left( {\gamma \varphi {\text{p}}} \right)^{2} }}{{kn + \gamma^{2} \varphi {\text{p}}}}$: (i) $Q_{w}^{*} > 0$, $Q_{c}^{*} > 0$ if $a > a_{1}$; (ii) $Q_{w}^{*} \ge 0$, $Q_{c}^{*} = 0$ if $a_{2} \le a \le a_{1}$; (iii) $Q_{w}^{*} = 0$, $Q_{c}^{*} = 0$ if $a \le a_{2}$; when ${\text{w}} < \frac{{\omega \left( {\gamma \varphi {\text{p}}} \right)^{2} }}{{kn + \gamma^{2} \varphi {\text{p}}}}$: (i) $Q_{w}^{*} > 0$, $Q_{c}^{*} > 0$ if $a > a_{2}$; (ii) $Q_{w}^{*} = 0$, $Q_{c}^{*} \ge 0$ if $a_{1} \le a \le a_{2}$; (iii) $Q_{w}^{*} = 0$, $Q_{c}^{*} = 0$ if $a \le a_{1}$.

Proof of Proposition 4

The comparison of the optimal order quantity and marketing service level are as follows:

$$Q_{w}^{*} - Q_{c}^{*} = \frac{{ - knw + \gamma^{2} \varphi p\left( {\omega \varphi p - w} \right)}}{kp\beta \varphi }$$

(A.35)

Solving the first partial derivative of $Q_{w}^{*} - Q_{c}^{*}$ with ${\text{w}}$:

$$\frac{{\partial \left( {Q_{w}^{*} - Q_{c}^{*} } \right)}}{\partial w} = \frac{{ - kn - \gamma^{2} \varphi p}}{k\beta \varphi p} < 0$$

(A.36)

Let $w_{1} = {{\omega \left( {\gamma \varphi p} \right)^{2} } \mathord{\left/ {\vphantom {{\omega \left( {\gamma \varphi p} \right)^{2} } {\left( {kn + \gamma^{2} \varphi p} \right)}}} \right. \kern-0pt} {\left( {kn + \gamma^{2} \varphi p} \right)}}$, we can get: if $w < w_{1}$, $Q_{w}^{*} > Q_{c}^{*}$.

$$s_{w}^{*} - s_{c}^{*} = \frac{{\gamma \left( {\omega \varphi p - w} \right)}}{k}$$

(A.37)

Solving the first partial derivative of $s_{w}^{*} - s_{c}^{*}$ with ${{w}}$:

$$\frac{{\partial \left( {s_{w}^{*} - s_{c}^{*} } \right)}}{\partial w} = - \frac{\gamma }{k} < 0$$

(A.38)

Let $w_{2} = \omega \varphi p$, we can get: if $w < w_{2}$, $s_{w}^{*} > s_{c}^{*}$.

Obviously, $w_{1} - w_{2} = \left( {\frac{{\gamma^{2} }}{{kn + \gamma^{2} \varphi p}} - 1} \right)\omega \varphi p < 0$, so $max\left\{ {w_{1} ,w_{2} } \right\} = w_{2}$. Therefore, (i) $Q_{w}^{*} > Q_{c}^{*}$, $s_{w}^{*} > s_{c}^{*}$ if $w < w_{1}$; (ii) $Q_{w}^{*} \le Q_{c}^{*}$, $s_{w}^{*} \ge s_{c}^{*}$ if $w_{1} \le w \le w_{2}$; (iii) $Q_{w}^{*} < Q_{c}^{*}$, $s_{w}^{*} < s_{c}^{*}$ if $w > w_{2}$.

Proof of Proposition 5

For the retailer and manufacturer, their optimal expected profits under the wholesale and consignment modes are as follows:

$$E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] = \frac{{\left( {\varphi p - w} \right)\left[ {\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p - w} \right) + 2k\varphi p\left( {\theta a - b\varphi p} \right)} \right]}}{2k\varphi p}$$

(A.39)

$$E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right] = \frac{{\varphi {\text{p}}\left( {1 - \omega } \right)\left[ {kn + \gamma^{2} \varphi p\left( {1 - \omega } \right) + 2k\left( {\theta a - b\varphi p} \right)} \right]}}{2k}$$

(A.40)

$$E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] = \frac{{\left( {w - c} \right)\left[ {\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p - w} \right) + k\varphi p\left( {\theta a - b\varphi p} \right)} \right]}}{k\varphi p}$$

(A.41)

$$E\left[ {\pi_{M}^{c} \left( {\omega^{*} } \right)} \right] = \frac{{\omega \varphi p\left[ {kn + 2\gamma^{2} \varphi p\left( {1 - \omega } \right) + 2k\left( {\theta a - b\varphi p} \right)} \right] - 2c\left[ {kn + \gamma^{2} \varphi p\left( {1 - \omega } \right) + k\left( {\theta a - b\varphi p} \right)} \right]}}{2k}$$

(A.42)

Comparing the optimal expected profits of the retailer:

$$\begin{gathered} E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right] = \frac{1}{2}\left( {\frac{1}{k}\left( {w\gamma^{2} - 2k\theta a + \varphi p\left( {2bk + \gamma^{2} \left( { - 2 + \omega } \right)} \right)} \right)\left( {w - p\varphi \omega } \right) + n\left( { - 2w + \frac{{w^{2} }}{\varphi p} + \omega \varphi p} \right)} \right) \hfill \\ \end{gathered}$$

(A.43)

Taking the first and second partial derivative of $E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right]$ with ${{w}}$:

$$\frac{{\partial \left( {E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right]} \right)}}{\partial w} = - \frac{{n\left( {\varphi p - w} \right)}}{\varphi p} - \frac{{\gamma^{2} \left( {\varphi p - w} \right) - k\left( {\theta a - b\varphi p} \right)}}{k}$$

(A.44)

$$\frac{{\partial^{2} \left( {E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right]} \right)}}{{\partial w^{2} }} = \frac{{kn + \gamma^{2} \varphi p}}{k\varphi p} > 0$$

(A.45)

Let $w_{3} = \frac{{A_{2} - \sqrt {A_{2}^{2} - A_{1} A_{3} } }}{{A_{1} }}$, $w_{4} = \frac{{A_{2} + \sqrt {A_{2}^{2} - A_{1} A_{3} } }}{{A_{1} }}$. We can obtain: when $w < w_{3}$ or $w > w_{4}$, $E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] > E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right]$; when $w_{3} \le w \le w_{4}$, $E\left[ {\pi_{R}^{w} \left( {Q_{w}^{*} ,s_{w}^{*} } \right)} \right] \le E\left[ {\pi_{R}^{c} \left( {Q_{c}^{*} ,s_{c}^{*} } \right)} \right]$.

Comparing the optimal expected profits of the manufacturer:

$$\begin{aligned} E\left[ {\pi _{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] - E\left[ {\pi _{M}^{c} \left( {\omega ^{*} } \right)} \right] = & \frac{{2c\left( {knw + \gamma ^{2} \varphi p\left( {w - \omega \varphi p} \right)} \right) + 2\gamma ^{2} \varphi p\left( { - w^{2} + w\varphi p + \left( { - 1 + \omega } \right)\omega \varphi ^{2} p^{2} } \right)}}{{2k\varphi p}} \\ & - \frac{{k\left[ {2\varphi p\left( {\theta a - b\varphi p} \right)\left( {\omega \varphi p - w} \right) + n\left( {2w^{2} - 2pw\varphi + p^{2} \varphi ^{2} \omega } \right)} \right]}}{{2k\varphi p}} \\ \end{aligned}$$

(A.46)

Taking the first and second partial derivative of $E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] - E\left[ {\pi_{M}^{c} \left( {\omega^{*} } \right)} \right]$ with ${\text{w}}$:

$$\frac{{\partial \left( {E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] - E\left[ {\pi_{M}^{c} \left( {\omega^{*} } \right)} \right]} \right)}}{\partial w} = \frac{{\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p - 2w + c} \right) + k\varphi p\left( {\theta a - b\varphi p} \right)}}{k\varphi p}$$

(A.47)

$$\frac{{\partial^{2} \left( {E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] - E\left[ {\pi_{M}^{c} \left( {\omega^{*} } \right)} \right]} \right)}}{{\partial w^{2} }} = \frac{{ - 2\left( {kn + \gamma^{2} \varphi p} \right)}}{k\varphi p} < 0$$

(A.48)

Let $w_{5} = \frac{{A_{4} - \sqrt {A_{4}^{2} - A_{1} A_{5} } }}{{2A_{1} }}$, $w_{6} = \frac{{A_{4} + \sqrt {A_{4}^{2} - A_{1} A_{5} } }}{{2A_{1} }}$. We can obtain: when $w_{5} \le w \le w_{6}$, $E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] \ge E\left[ {\pi_{M}^{c} \left( {\omega^{*} } \right)} \right]$; when $w < w_{5}$ or $w > w_{6}$, $E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{*} } \right)} \right] < E\left[ {\pi_{M}^{c} \left( {\omega^{*} } \right)} \right]$.

Proof of Proposition 6

Setting: $\theta a_{3} = b\varphi p - \frac{{3kn + 2\gamma^{2} \left( {\varphi p - c} \right)}}{2k}$, $\theta a_{4} = b\varphi p - \frac{{\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p - c} \right)}}{k\varphi p}$, $\theta a_{5} = b\varphi p + \frac{{2\gamma^{2} \left( {\varphi p - c} \right) - kn}}{2k}$, $\theta a_{6} = b\varphi p + \frac{{\left( {kn + \gamma^{2} \varphi p} \right)\left( {\varphi p - c} \right)}}{k\varphi p}$. It can be obtained: $Q_{w}^{**} > 0$ if $a > a_{4}$; $Q_{c}^{**} > 0$ if $a > a_{3}$; $s_{w}^{**} > 0$ if $a < a_{6}$; $s_{c}^{**} > 0$ if $a < a_{5}$.

Comparing the relative sizes of $a_{3} ,a_{4} ,a_{5} ,a_{6}$:

$$\theta a_{4} - \theta a_{3} = n\left( {\frac{1}{2} + \frac{c}{p\varphi }} \right) > 0$$

(A.49)

$$\theta a_{5} - \theta a_{6} = n\left( {\frac{3}{2} - \frac{c}{p\varphi }} \right) > 0$$

(A.50)

$$\theta a_{5} - \theta a_{4} = n\left( {\frac{1}{2} - \frac{c}{p\varphi }} \right) + \frac{{2\gamma^{2} \left( {\varphi p - c} \right)}}{k}$$

(A.51)

Hence, we can obtain: when $c \le \frac{{\varphi p\left( {kn + 4\gamma^{2} \varphi p} \right)}}{{2\left( {kn + 2\gamma^{2} \varphi p} \right)}}$, $a_{3} < a_{4} < a_{5} < a_{6}$; when $c > \frac{{\varphi p\left( {kn + 4\gamma^{2} \varphi p} \right)}}{{2\left( {kn + 2\gamma^{2} \varphi p} \right)}}$, $a_{3} < a_{5} < a_{4} < a_{6}$ Therefore, (i) $Q_{w}^{**} = 0$, $Q_{c}^{**} = 0$, $s_{w}^{**} > 0$, $s_{c}^{**} > 0$ if $a < a_{3}$; (ii) $Q_{w}^{**} > 0$, $Q_{c}^{**} > 0$, $s_{w}^{**} = 0$, $s_{c}^{**} = 0$ if $a > a_{6}$; (iii) When $c \le \frac{{\varphi p\left( {kn + 4\gamma^{2} \varphi p} \right)}}{{2\left( {kn + 2\gamma^{2} \varphi p} \right)}}$, $Q_{w}^{**} \ge 0$, $Q_{c}^{**} > 0$, $s_{w}^{**} > 0$, $s_{c}^{**} \ge 0$ if $a_{4} \le a \le a_{5}$; (iv) When $c > \frac{{\varphi p\left( {kn + 4\gamma^{2} \varphi p} \right)}}{{2\left( {kn + 2\gamma^{2} \varphi p} \right)}}$, $Q_{w}^{**} \ge 0$, $Q_{c}^{**} > 0$, $s_{w}^{**} \ge 0$, $s_{c}^{**} = 0$ if $a_{4} \le a \le a_{6}$.

Proof of Proposition 7

For the manufacturer and retailer, their optimal expected profits are as follows:

$$E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{**} } \right)} \right] = \frac{{\left[ {\left( {\varphi p - c} \right)\left( {kn + \gamma^{2} \varphi p} \right) + k\varphi p\left( {\theta a - b\varphi p} \right)} \right]^{2} }}{{4k\varphi p\left( {kn + \gamma^{2} \varphi p} \right)}}$$

(A.52)

$$E\left[ {\pi_{M}^{c} \left( {\omega^{**} } \right)} \right] = \frac{1}{16}\left[ {\frac{{4\gamma^{2} \left( {\varphi p - c} \right)^{2} }}{k} + 4\varphi p\left( {n + 2\theta a - 2b\varphi p} \right) + \frac{{k\left( {n + 2\theta a - 2b\varphi p} \right)^{2} }}{{\gamma^{2} }} - 4c\left( {3n + 2\theta a - 2b\varphi p} \right)} \right]$$

(A.53)

$$\begin{aligned} E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] = & \frac{{c^{2} \left( {kn + \gamma^{2} \varphi p} \right)}}{8k\varphi p} - \frac{{2c\left[ {\gamma^{2} \varphi p + kn + k\left( {\theta a - b\varphi p} \right)} \right]}}{8k} \\ & + \frac{{\varphi p\left[ {kn + \gamma^{2} \varphi p - k\left( {\theta a - b\varphi p} \right)} \right]\left[ {kn + \gamma^{2} \varphi p + 3k\left( {\theta a - b\varphi p} \right)} \right]}}{{8k\left( {kn + \gamma^{2} \varphi p} \right)}} \\ \end{aligned}$$

(A.54)

$$E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right] = - \frac{{\left[ {2\gamma^{2} \left( {c - \varphi p} \right) + kn + 2k\left( {\theta a - 2b\varphi p} \right)} \right]\left[ {2\gamma^{2} \left( {\varphi p - c} \right) + 3kn + 6k\left( {\theta a - 2b\varphi p} \right)} \right]}}{{32k\gamma^{2} }}$$

(A.55)

Comparing the optimal expected profits of the manufacturer:

$$E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{**} } \right)} \right] - E\left[ {\pi_{M}^{c} \left( {\omega^{**} } \right)} \right] = \frac{{nc\left( {\varphi p + c} \right)}}{4\varphi p} - \frac{{kn\left( {\gamma^{2} \varphi p\left( {n + 4\theta a - 4b\varphi p} \right) + k\left( {n + 2\theta a - 2b\varphi p} \right)^{2} } \right)}}{{16\gamma^{2} \left( {kn + \gamma^{2} \varphi p} \right)}}$$

(A.56)

Taking the first and second partial derivative of $E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{**} } \right)} \right] - E\left[ {\pi_{M}^{c} \left( {\omega^{**} } \right)} \right]$ with ${{c}}$:

$$\frac{{\partial \left( {E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{**} } \right)} \right] - E\left[ {\pi_{M}^{c} \left( {\omega^{**} } \right)} \right]} \right)}}{\partial c} = \frac{{n\left( {\varphi p + 2c} \right)}}{4\varphi p} > 0$$

(A.57)

Let $c_{1} = \frac{{ - \gamma^{2} \varphi p\left( {kn + \gamma^{2} \varphi p} \right) + \sqrt {\gamma^{2} \varphi p\left( {kn + \gamma^{2} \varphi p} \right)\left[ {kn + \gamma^{2} \varphi p + 2k\left( {\theta {\text{a}} - b\varphi p} \right)} \right]^{2} } }}{{2\gamma^{2} \left( {kn + \gamma^{2} \varphi p} \right)}}$. We can obtain: when $c \ge c_{1}$, $E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{**} } \right)} \right] \ge E\left[ {\pi_{M}^{c} \left( {\omega^{**} } \right)} \right]$; when $0 < c < c_{1}$, $E\left[ {\pi_{M}^{w} \left( {{\text{w}}^{**} } \right)} \right] < E\left[ {\pi_{M}^{c} \left( {\omega^{**} } \right)} \right]$.

Comparing the optimal expected profits of the retailer:

$$E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right] = \frac{{3kn\left[ {\gamma^{2} \varphi p\left( {n + 4\theta a - 4b\varphi p} \right) + k\left( {n + 2\theta a - 2b\varphi p} \right)^{2} } \right]}}{{32\gamma^{2} \left( {kn + \gamma^{2} \varphi p} \right)}} - \frac{{nc\left( {\varphi p - c} \right)}}{8\varphi p}$$

(A.58)

Taking the first and second derivative of $E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right]$ with ${{c}}$:

$$\frac{{\partial \left( {E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right]} \right)}}{\partial c} = \frac{{n\left( {2c - \varphi p} \right)}}{8\varphi p}$$

(A.59)

$$\frac{{\partial^{2} \left( {E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] - E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right]} \right)}}{{\partial c^{2} }} = \frac{n}{4\varphi p} > 0$$

(A.60)

Let $c_{2} = \frac{{B_{2} - \sqrt {B_{2}^{2} - B_{2} B_{3} } }}{{2B_{1} }}$, $c_{3} = \frac{{B_{2} + \sqrt {B_{2}^{2} - B_{2} B_{3} } }}{{2B_{1} }}$. We can obtain: when $c < c_{2}$ or $c > c_{3}$, $E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] > E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right]$; when $c_{2} \le c \le c_{3}$, $E\left[ {\pi_{R}^{w} \left( {Q_{w}^{**} ,s_{w}^{**} } \right)} \right] \le E\left[ {\pi_{R}^{c} \left( {Q_{c}^{**} ,s_{c}^{**} } \right)} \right]$.