Managing the supply disruption risk: option contract or order commitment contract?

Abstract

Supply disruption is a common phenomenon in many industries. Motivated by the cases of “Nokia and Ericsson” and “HP”, our study investigates how a core firm utilizes the option contract and the order commitment contract to share and mitigate the supply disruption risk. In a decentralized supply chain, we establish the Stackelberg game models to explore the option and order commitment contracts in single and dual sourcing cases, respectively. We obtain the optimal production and procurement strategies under the two types of contracts, after which we investigate the value of the option contract and the optimal contract selection strategy of the core firm. The results demonstrate that the firm is insulated from the supply disruption risk, and that its profit is independent of the disruption risk and production investment under the option contract. In the single or dual sourcing case, the preference of the core firm for the two contracts depends on the disruption risk level and switches back and forth as the probability of disruption increases. A relatively low option price or a medium-risk operational environment heightens the value of the option contract; hence, the core firm is likely to choose the option contract correspondingly. When a reliable supplier is available, the existence of a reliable supplier is always beneficial to the core firm, however is a threat to an unreliable supplier.

Appendix

Proof of Proposition 1

Taking the first order and second order derivatives of $\pi_{S}^{o} (Z_{o} |Q_{o} )$ with respect to Z_o, we have

$$ \begin{aligned} \frac{{\partial \pi_{S}^{o} (Z_{o} { | }Q_{o} )}}{{\partial Z_{o} }} & = - c + (1 - \beta )c_{e} (1 - F(Z_{o} )), \\ \frac{{\partial^{2} \pi_{S}^{o} (Z_{o} |Q_{o} )}}{{\partial Z_{o}^{2} }} & = - (1 - \beta )c_{e} f(Z_{o} ) < 0. \\ \end{aligned} $$

Then, $\pi_{S}^{o} (Z_{o} |Q_{o} )$ is concave in Z_o. By the first order condition (FOC), we can obtain $F(Z_{o}^{*} ) = 1 - \frac{c}{{(1 - \beta )c_{e} }}$. Given that Z^*_o  ≤ Q_o, then the optimal reaction of the supplier satisfies $F(Z_{o}^{*} ) = \hbox{min} \{ F(Q_{o} ),1 - \frac{c}{{(1 - \beta )c_{e} }}\}$.

Taking the first order and second order derivatives of π^o_B (Q_o) with respect to Q_o, we have

$$ \begin{aligned} \frac{{\partial \pi_{B}^{o} (Q_{o} )}}{{\partial Q_{o} }} & = - c_{o} + (r - w)(1 - F(Q_{o} )), \\ \frac{{\partial^{2} \pi_{B}^{o} (Q_{o} )}}{{\partial Q_{o}^{2} }} & = - (r - w)f(Q_{o} ) < 0. \\ \end{aligned} $$

Then, π^o_B (Q_o) is concave in Q_o. By the FOC, $F(Q_{o}^{*} ) = 1 - \frac{{c_{o} }}{r - w}$. Then, the optimal production quantity of the supplier is $F(Z_{o}^{*} ) = \hbox{min} \left\{ {F(Q_{o}^{*} ),1 - \frac{c}{{(1 - \beta )c_{e} }}} \right\}$.

When $1 - \frac{{c_{o} }}{r - w} \le 1 - \frac{c}{{(1 - \beta )c_{e} }}$, it can be verified that $\beta \le 1 - \frac{(r - w)c}{{c_{e} c_{o} }}$. Then we can obtain $F(Q_{o}^{*} ) = F(Z_{o}^{*} ) = 1 - \frac{{c_{o} }}{r - w}$. When $1 - \frac{{c_{o} }}{r - w} > 1 - \frac{c}{{(1 - \beta )c_{e} }}$, that is, $\beta > 1 - \frac{(r - w)c}{{c_{e} c_{o} }}$, then we can achieve Q^*_o  > Z^*_o , $F(Q_{o}^{*} ) = 1 - \frac{{c_{o} }}{r - w}$ and $F(Z_{o}^{*} ) = 1 - \frac{c}{{(1 - \beta )c_{e} }}$. We use $\hat{\beta }_{o}$ to denote $1 - \frac{(r - w)c}{{c_{e} c_{o} }}$, then we can achieve Proposition 1. □

Proof of Corollary 1

The proof of Corollary 1 is omitted. □

Proof of Proposition 2

Taking the first order and second order derivatives of $\pi_{S}^{f} (Z_{f} |Q_{f} )$ with respect to Z_f, we have

$$ \begin{aligned} \frac{{\partial \pi_{S}^{f} (Z_{f} { | }Q_{f} )}}{{\partial Z_{f} }} & = - c + (1 - \beta )w_{e} (1 - F(Z_{f} )), \\ \frac{{\partial^{2} \pi_{S}^{f} (Z_{f} |Q_{f} )}}{{\partial Z_{f}^{2} }} & = - (1 - \beta )w_{e} F(Z_{f} ) < 0. \\ \end{aligned} $$

Then, $\pi_{S}^{f} (Z_{f} |Q_{f} )$ is concave inZ_f. By the FOC, we can derive $F(Z_{f}^{*} ) = 1 - \frac{c}{{(1 - \beta )w_{e} }}$. Because ofZ^*_f  ≥ Q_f, the optimal reaction of the supplier satisfies $F(Z_{f}^{*} ) = \hbox{max} \left\{ {F(Q_{f} ),1 - \frac{c}{{(1 - \beta )w_{e} }}} \right\}$.

Taking the second order derivative of $\pi_{B}^{f} (Z_{f} |Q_{f} )$ with respect to Q_f, we have

$$ \frac{{\partial^{2} \pi_{B}^{f} (Q_{f} )}}{{\partial Q_{f}^{2} }} = \left\{ {\begin{array}{*{20}l} { - (1 - \beta )rf(Q_{f} ) < 0} \hfill & {if} \hfill & {Z_{f}^{*} = Q_{f} } \hfill \\ { - (1 - \beta )w_{e} f(Q_{f} ) { < 0}} \hfill & {if} \hfill & {Z_{f}^{*} > Q_{f} } \hfill \\ \end{array} } \right.. $$

π^f_B (Q_f) is concave in Q_f. The first order derivative of π^f_B (Q_f) relies on the optimal reaction of the supplier. Thus, the first order optimality condition is

$$ \frac{{\partial \pi_{B}^{f} (Q_{f} )}}{{\partial Q_{f} }} = \left\{ {\begin{array}{*{20}l} {(1 - \beta )r(1 - F(Q_{f} )) - (1 - \beta )w} \hfill & {if} \hfill & {Z_{f}^{*} = Q_{f} } \hfill \\ { - (1 - \beta )w + (1 - \beta )w_{e} (1 - F(Q_{f} ) )} \hfill & {if} \hfill & {Z_{f}^{*} > Q_{f} } \hfill \\ \end{array} } \right.. $$

The optimal order quantity Q^*_f of the firm satisfies

$$ F(Q_{f}^{*} )) = \left\{ {\begin{array}{*{20}l} {1 - \frac{w}{r}} \hfill & {if} \hfill & {Z_{f}^{*} = Q_{f}^{*} } \hfill \\ {1 - \frac{w}{{w_{e} }}} \hfill & {if} \hfill & {Z_{f}^{*} > Q_{f}^{*} } \hfill \\ \end{array} } \right.. $$

Then, we can obtain the optimal production quantity Z^*_f of the supplier as follows

$$ F(Z_{f}^{*} ) = \hbox{max} \left\{ {F(Q_{f}^{*} ),1 - \frac{c}{{(1 - \beta )w_{e} }}} \right\}. $$

Next, we analyze the optimal decisions of the supplier and firm as shown below.

• (1) When $1 - \frac{w}{r} < 1 - \frac{c}{{(1 - \beta )w_{e} }}$, then $\beta < 1 - \frac{rc}{{ww_{e} }}$. Given that 0 ≤ β ≤ 1, then ww_e > rc. Therefore, $F(Z_{f}^{*} ) = 1 - \frac{c}{{(1 - \beta )w_{e} }} > 1 - \frac{w}{r} > 1 - \frac{w}{{w_{e} }}$. The optimal decisions satisfy Z^*_f  > Q^*_f , $Q_{f}^{*} = F^{ - 1} \left( {1 - \frac{w}{{w_{e} }}} \right)$ and $Z_{f}^{*} = F^{ - 1} \left( {1 - \frac{c}{{(1 - \beta )w_{e} }}} \right).$

• (2) When $1 - \frac{w}{{w_{e} }} < 1 - \frac{c}{{(1 - \beta )w_{e} }} \le 1 - \frac{w}{r}$ and $\beta < 1 - \frac{c}{w}$, then $1 - \frac{rc}{{ww_{e} }} \le \beta < 1 - \frac{c}{w}$. Given that β ≥ 0, β should satisfy $\hbox{max} \left\{ {1 - \frac{rc}{{ww_{e} }},0} \right\} \le \beta < 1 - \frac{c}{w}$. If the firm chooses to order $Q_{f}^{H} : = Q_{f}^{*} = F^{ - 1} \left( {1 - \frac{w}{r}} \right)$, then the supplier decides to produce $Z_{f}^{H} : = Z_{f}^{*} = F^{ - 1} \left( {1 - \frac{w}{r}} \right) = Q_{f}^{H}$. If the firm chooses to order $Q_{f}^{L} : = Q_{f}^{*} = F^{ - 1} \left( {1 - \frac{w}{{w_{e} }}} \right)$, then the suppler decides to produce $Z_{f}^{L} : = Z_{f}^{*} = F^{ - 1} \left( {1 - \frac{c}{{(1 - \beta )w_{e} }}} \right) > Q_{f}^{L}$. We assume that π^fL_B and π^fH_B represent the expected profits of the firm in the above two situations, respectively. Then,

  $$ \begin{aligned} \pi_{B}^{fH} & = (1 - \beta )r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) - (1 - \beta )wQ_{f}^{H} , \\ \pi_{B}^{fL} & = (1 - \beta )(r - w_{e} )\left( {Z_{f}^{L} - \int_{0}^{{Z_{f}^{L} }} {F(x)dx} } \right) - (1 - \beta )wQ_{f}^{L} \\ & \quad + (1 - \beta )w_{e} \left( {Q_{f}^{L} - \int_{0}^{{Q_{f}^{L} }} {F(x)dx} } \right), \\ \pi_{B}^{fH} - \pi_{B}^{fL} & = (1 - \beta )r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & - (1 - \beta )(r - w_{e} )\left( {Z_{f}^{L} - \int_{0}^{{Z_{f}^{L} }} {F(x)dx} } \right) - (1 - \beta )w_{e} \left( {Q_{f}^{L} - \int_{0}^{{Q_{f}^{L} }} {F(x)dx} } \right). \\ \end{aligned} $$

We assume that

$$ \begin{aligned} H(t) & = (1 - \beta )r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad - (1 - \beta )(r - w_{e} )\left( {t - \int_{0}^{t} {F(x)dx} } \right) - (1 - \beta )w_{e} \left( {Q_{f}^{L} - \int_{0}^{{Q_{f}^{L} }} {F(x)dx} } \right). \\ \end{aligned} $$

Then,

$$ \begin{aligned} H\left( {Q_{f}^{L} } \right) & = (1 - \beta )r\left( {\frac{w}{r}Q_{f}^{H} - \frac{w}{{w_{e} }}Q_{f}^{L} + \int_{{Q_{f}^{L} }}^{{Q_{f}^{H} }} {xf(x)dx} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad > (1 - \beta )r\left[ {\frac{w}{r}Q_{f}^{H} - \frac{w}{{w_{e} }}Q_{f}^{L} + Q_{f}^{L} \left( {F\left( {Q_{f}^{H} } \right) - F\left( {Q_{f}^{L} } \right)} \right)} \right] - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad = (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad = 0, \\ H\left( {Q_{f}^{H} } \right) & = (1 - \beta )w_{e} \left( {\frac{w}{r}Q_{f}^{H} - \frac{w}{{w_{e} }}Q_{f}^{L} + \int_{{Q_{f}^{L} }}^{{Q_{f}^{H} }} {xf(x)dx} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad < (1 - \beta )w_{e} \left[ {\frac{w}{r}Q_{f}^{H} - \frac{w}{{w_{e} }}Q_{f}^{L} + Q_{f}^{H} \left( {F\left( {Q_{f}^{H} } \right) - F\left( {Q_{f}^{L} } \right)} \right)} \right] - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad = (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{L} } \right) \\ & \quad = 0. \\ \end{aligned} $$

Given that $\frac{dH(t)}{dt} = - (1 - \beta )(r - w_{e} )(1 - F(t)) < 0$, H(t) is monotone decreasing in t when t ∊ [Q^L_f , Q^H_f ]. From the above analysis, we get H(Q^L_f ) > 0 and H(Q^H_f ) < 0. Therefore, there exists a unique t₀ ∊ [Q^L_f , Q^H_f ] which satisfies H(t₀) = 0. We know that $F(Z_{f}^{L} ) = 1 - \frac{c}{{(1 - \beta )w_{e} }}$. Hence, Z^L_f is decreasing inβ. When $\beta = 1 - \frac{rc}{{ww_{e} }}$, then $F(Z_{f}^{H} ) = 1 - \frac{w}{r} = F(Q_{f}^{H} )$. When $\beta = 1 - \frac{c}{w}$, then $F(Z_{f}^{L} ) = 1 - \frac{w}{{w_{e} }} = F(Q_{f}^{L} )$. Hence, there exists a unique $\hat{\beta }_{f}$ that satisfies $F(t_{0} ) = 1 - \frac{c}{{(1 - \hat{\beta }_{f} )w_{e} }}$ and H(t₀) = 0. In the following, we compare π^fH_B and π^fL_B in the following two situations.

• A. When $1 - \frac{rc}{{ww_{e} }} < \beta < \hat{\beta }_{f}$, then H(t) < 0 and π^fH_B  - π^fL_B  < 0. Hence, we can obtain $F(Q_{f}^{*} ) = 1 - \frac{w}{{w_{e} }}$ and $F(Z_{f}^{*} ) = 1 - \frac{c}{{(1 - \beta )w_{e} }}$.

• B. When $\hat{\beta }_{f} \le \beta < 1 - \frac{c}{w}$, then H(t) ≥ 0 and π^fH_B  - π^fL_B  ≥ 0. Hence, we can obtain $F(Z_{f}^{*} ) = F(Q_{f}^{*} ) = 1 - \frac{w}{r}$.

Based on (1) and (2) above, the Proposition 2 can be derived. □

Proof of Corollary 2

When the stochastic demand follows a uniform distribution, we assume that $f(\zeta ) = \frac{1}{D}$, ζ ∊ [0, D], then, $F(\zeta ) = \frac{\zeta }{D}$, F⁻¹(x) = Dx, $Q_{f}^{H} = D\left( {1 - \frac{w}{r}} \right)$, $Q_{f}^{L} = D\left( {1 - \frac{w}{{w_{e} }}} \right)$, $t_{0} = D\left( {1 - \frac{c}{{(1 - \hat{\beta }_{f} )w_{e} }}} \right)$ and

$$ H(t_{0} ) = (1 - \beta )D\left[ {\frac{r}{2}\left( {1 - \frac{{w^{2} }}{{r^{2} }}} \right) - w\left( {\frac{w}{{w_{e} }} - \frac{w}{r}} \right) - (r - w_{e} )\frac{{t_{0} }}{D}\left( {1 - \frac{{t_{0} }}{2D}} \right) - \frac{{w_{e} }}{2}\left( {1 - \frac{{w^{2} }}{{w_{e}^{2} }}} \right)} \right] = 0. $$

Substituting t₀ with $D\left( {1 - \frac{c}{{(1 - \hat{\beta }_{f} )w_{e} }}} \right)$, we can obtain

$$ \frac{r}{2}\left( {1 - \frac{{w^{2} }}{{r^{2} }}} \right) - w\left( {\frac{w}{{w_{e} }} - \frac{w}{r}} \right) - \frac{{(r - w_{e} )}}{2}\left( {1 - \frac{{c^{2} }}{{(1 - \hat{\beta }_{f} )^{2} w_{e}^{2} }}} \right) - \frac{{w_{e} }}{2}\left( {1 - \frac{{w^{2} }}{{w_{e}^{2} }}} \right) = 0. $$

By performing a simple algebraic operation, we can get $\hat{\beta }_{f} = 1 - \frac{c}{w}\sqrt {\frac{r}{{w_{e} }}}$. □

Proof of Proposition 3

(I) Under the option contract, (1) based on the expected profit function of the firm, it is easy to conclude that π^o*_B is independent of β. (2) For the supplier, (a) when $0 \le \beta < \hat{\beta }_{o}$, then $\frac{{\partial \pi_{S}^{o * } }}{\partial \beta } = - c_{e} (Z_{o}^{*} - \int_{0}^{{Z_{o}^{*} }} {F(x)dx} ) < 0$. Hence, π^o*_S is decreasing when $\beta \in [0,\hat{\beta }_{o} )$. (b) When $\hat{\beta }_{o} < \beta < 1 - \frac{c}{w}$, then

$$ \begin{aligned} \frac{{\partial \pi_{S}^{o * } }}{\partial \beta } & = - c\frac{{dZ_{o}^{*} }}{d\beta } - c_{e} \left( {Z_{o}^{*} - \int_{0}^{{Z_{o}^{*} }} {F(x)dx} } \right) + (1 - \beta )c_{e} (1 - F(Z_{o}^{*} ))\frac{{dZ_{o}^{*} }}{d\beta } \\ & \quad = - c_{e} \left( {Z_{o}^{*} - \int_{0}^{{Z_{o}^{*} }} {F(x)dx} } \right) \\ & \quad < 0. \\ \end{aligned} $$

Hence, π^o*_S is decreasing when $\beta \in [\hat{\beta }_{o} ,1 - \frac{c}{w})$. Now, we investigate the continuity of π^o*_S when $\beta = \hat{\beta }_{o}$. Given that

$$ \begin{aligned} \pi_{S}^{o * } \left( {Z_{o}^{*} } \right)|_{{\beta = \hat{\beta }_{o}^{ - } }} & = - cQ_{o}^{*} + c_{o} Q_{o}^{*} + (w - \beta c_{e} )\left( {Q_{o}^{*} - \int_{0}^{{Q_{o}^{*} }} {F(x)dx} } \right), \\ \pi_{S}^{o * } \left( {Z_{o}^{*} } \right)|_{{\beta = \hat{\beta }_{o}^{ + } }} & = - cQ_{o}^{*} + c_{o} Q_{o}^{*} + (w - \beta c_{e} )\left( {Q_{o}^{*} - \int_{0}^{{Q_{o}^{*} }} {F(x)dx} } \right), \\ \end{aligned} $$

then $\pi_{S}^{o * } (Z_{o}^{*} )|_{{\beta = \hat{\beta }_{o}^{ - } }} = \pi_{S}^{o * } (Z_{o}^{*} )|_{{\beta = \hat{\beta }_{o}^{ + } }}$. Hence, π^o*_S is continuous when $\beta = \hat{\beta }_{o}$. Therefore, π^o*_S is decreasing when $\beta \in [0,1 - \frac{c}{w})$. Given that π^o*_B is independent of β, the profit of the whole supply chain, π^o*_SC decreases when $\beta \in [0,1 - \frac{c}{w})$.

(II) Under the order commitment contract,

(1) for the supplier, (a) when $0 < \beta < \hat{\beta }_{f}$, then

$$ \begin{aligned} \frac{{d\pi_{S}^{f} }}{d\beta } & = - c\frac{{dZ_{f}^{*} }}{d\beta } - wQ_{f}^{*} + w_{e} \left( {Q_{f}^{*} - \int_{0}^{{Q_{f}^{*} }} {F(x)dx} } \right) - w_{e} \left( {Z_{f}^{*} - \int_{0}^{{Z_{f}^{*} }} {F(x)dx} } \right) \\ & \quad + (1 - \beta )w_{e} \left( {1 - F\left( {Z_{f}^{*} } \right)} \right)\frac{{dZ_{f}^{*} }}{d\beta } \\ &= - wQ_{f}^{*} - w_{e} \left( {Z_{f}^{*} - Q_{f}^{*} - \int_{{Q_{f}^{*} }}^{{Z_{f}^{*} }} {F(x)dx} } \right) \\ & \quad < 0. \\ \end{aligned} $$

Hence, π^f_S is decreasing when $\beta \in [0,\hat{\beta }_{f} ]$. (b) When $\hat{\beta }_{f} < \beta < 1 - \frac{c}{w}$, then $\frac{{d\pi_{S}^{f} }}{d\beta } = - wQ_{f}^{*} { < }0$. Hence, π^f_S is decreasing when $\beta \in [\hat{\beta }_{f} ,1 - \frac{c}{w})$. Now, we investigate the continuity of π^f_S when $\beta = \hat{\beta }_{f}$. Given that

$$ \begin{aligned} \pi_{S}^{f} \left( {Z_{f}^{*} } \right))|_{{\beta = \hat{\beta }_{f}^{ - } }} & = - ct_{0} + (1 - \beta )wQ_{f}^{L} + (1 - \beta )w_{e} \left( {t_{0} - Q_{f}^{L} - \int_{{Q_{f}^{L} }}^{{t_{0} }} {F(x)dx} } \right), \\ \pi_{S}^{f} \left( {Z_{f}^{*} } \right)|_{{\beta = \hat{\beta }_{f}^{ + } }} & = - cQ_{f}^{H} + (1 - \beta )wQ_{f}^{H} , \\ \end{aligned} $$

then $\pi_{S}^{f} (Z_{f}^{*} )|_{{\beta = \hat{\beta }_{f}^{ - } }} \ne \pi_{S}^{f} (Z_{f}^{*} )|_{{\beta = \hat{\beta }_{f}^{ + } }}$. Hence, π^f_S is not continuous when $\beta = \hat{\beta }_{f}$. Therefore, π^f_S decreases when β is in the domain of $[0,\hat{\beta }_{f} ]$ and $[\hat{\beta }_{f} ,1 - \frac{c}{w})$.

(2) For the firm, (a) when $0 < \beta < \hat{\beta }_{f}$, then

$$ \frac{{d\pi_{B}^{f} }}{d\beta } = \left( {\frac{r}{{w_{e} }} - 1} \right)c\frac{{dZ_{f}^{*} }}{d\beta } - (r - w_{e} )\left( {Z_{f}^{*} - \int_{0}^{{Z_{f}^{*} }} {F(x)dx} } \right) + wQ_{f}^{*} - w_{e} \left( {Q_{f}^{*} - \int_{0}^{{Q_{f}^{*} }} {F(x)dx} } \right). $$

Given that

$$ \begin{aligned} H\left( {Z_{f}^{*} } \right) & = (1 - \beta )r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) - (1 - \beta )w\left( {Q_{f}^{H} - Q_{f}^{*} } \right) \\ & \quad - (1 - \beta )(r - w_{e} )\left( {Z_{f}^{*} - \int_{0}^{{Z_{f}^{*} }} {F(x)dx} } \right) - (1 - \beta )w_{e} \left( {Q_{f}^{*} - \int_{0}^{{Q_{f}^{*} }} {F(x)dx} } \right) < 0, \\ \end{aligned} $$

that is,

$$ \begin{aligned} & - (r - w_{e} )\left( {Z_{f}^{*} - \int_{0}^{{Z_{f}^{*} }} {F(x)dx} } \right) - w_{e} \left( {Q_{f}^{*} - \int_{0}^{{Q_{f}^{*} }} {F(x)dx} } \right) \\ & < - r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) + w\left( {Q_{f}^{H} - Q_{f}^{*} } \right), \\ \end{aligned} $$

then

$$ \begin{aligned} \frac{{d\pi_{B}^{f} }}{d\beta } & = \left( {\frac{r}{{w_{e} }} - 1} \right)c\frac{{dZ_{f}^{*} }}{d\beta } - (r - w_{e} )\left( {Z_{f}^{*} - \int_{0}^{{Z_{f}^{*} }} {F(x)dx} } \right) + wQ_{f}^{*} - w_{e} \left( {Q_{f}^{*} - \int_{0}^{{Q_{f}^{*} }} {F(x)dx} } \right) \\ & \quad < \left( {\frac{r}{{w_{e} }} - 1} \right)c\frac{{dZ_{f}^{*} }}{d\beta } + wQ_{f}^{*} - r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) + w\left( {Q_{f}^{H} - Q_{f}^{*} } \right) \\ & = \left( {\frac{r}{{w_{e} }} - 1} \right)c\frac{{dZ_{f}^{*} }}{d\beta } - r\int_{0}^{{Q_{f}^{H} }} {xf(x)dx} \\ & \quad < 0. \\ \end{aligned} $$

Hence, π^f_B decreases when $\beta \in [0,\hat{\beta }_{f} ]$.

(b) When $\hat{\beta }_{f} < \beta < 1 - \frac{c}{w}$,

$$ \begin{aligned} \frac{{d\pi_{B}^{f} }}{d\beta } & = - r\left( {Q_{f}^{*} - \int_{0}^{{Q_{f}^{*} }} {F(x)dx} } \right) + wQ_{f}^{*} \\ & = - r\left( {Q_{f}^{*} - Q_{f}^{*} F\left( {Q_{f}^{*} } \right) + \int_{0}^{{Q_{f}^{*} }} {xf(x)dx} } \right) + wQ_{f}^{*} \\ & = - r\left( {Q_{f}^{*} - Q_{f}^{*} \left( {1 - \frac{w}{r}} \right) + \int_{0}^{{Q_{f}^{*} }} {xf(x)dx} } \right) + wQ_{f}^{*} \\ & = - r\int_{0}^{{Q_{f}^{*} }} {xf(x)dx} \\ & \quad < 0. \\ \end{aligned} $$

Hence, π^f_B is decreasing when $\beta \in [\hat{\beta }_{f} ,1 - \frac{c}{w})$. Now, we investigate the continuity of π^f_S when $\beta = \hat{\beta }_{f}$. Given that

$$ \begin{aligned} \pi_{B}^{f} \left( {Q_{f}^{*} } \right)|_{{\beta = \hat{\beta }_{f}^{ - } }} & = (1 - \beta )r\left( {t_{0} - \int_{0}^{{t_{0} }} {F(x)dx} } \right) - (1 - \beta )wQ_{f}^{L} \\ & \quad - (1 - \beta )w_{e} \left( {t_{0} - Q_{f}^{L} - \int_{{Q_{f}^{L} }}^{{t_{0} }} {F(x)dx} } \right), \\ \pi_{B}^{f} \left( {Q_{f}^{*} } \right)|_{{\beta = \hat{\beta }_{f}^{ + } }} & = (1 - \beta )r\left( {Q_{f}^{H} - \int_{0}^{{Q_{f}^{H} }} {F(x)dx} } \right) - (1 - \beta )wQ_{f}^{H} , \\ \end{aligned} $$

then $\pi_{B}^{f} (Q_{f}^{*} )|_{{\beta = \hat{\beta }_{f}^{ + } }} - \pi_{B}^{f} (Q_{f}^{*} )|_{{\beta = \hat{\beta }_{f}^{ - } }} = \pi_{B}^{fH} - \pi_{B}^{fL} = H(t_{0} ) = 0$. Hence, π^f_B is continuous when $\beta = \hat{\beta }_{f}$. Therefore, π^f_B is decreasing when $\beta \in [0,1 - \frac{c}{w})$. For the profit of the whole supply chain, because of (1) and (2), π^f_SC is not continuous when $\beta = \hat{\beta }_{f}$, and π^f_S is decreasing when β is in the domain of $[0,\hat{\beta }_{f} ]$ and $[\hat{\beta }_{f} ,1 - \frac{c}{w})$. □

Proof of Proposition 4

Taking the first order and second order derivatives of $\pi_{S}^{od} (Z_{od} |Q_{od} ,R_{od} )$ with respect to Z_od, we have

$$ \begin{aligned} \frac{{\partial \pi_{S}^{od} (Z_{od} |Q_{od} ,R_{od} )}}{{\partial Z_{od} }} & = - c + (1 - \beta )c_{e} (1 - F(R_{od} + Z_{od} )), \\ \frac{{\partial^{2} \pi_{S}^{od} (Z_{od} |Q_{od} ,R_{od} )}}{{\partial Z_{od}^{2} }} & = - (1 - \beta )c_{e} f(R_{od} + Z_{od} ) < 0. \\ \end{aligned} $$

Then $\pi_{S}^{od} (Z_{od} |Q_{od} ,R_{od} )$ is concave in Z_od. By the FOC, we can derive $F(R_{od} + Z_{od}^{*} ) = 1 - \frac{c}{{(1 - \beta )c_{e} }}$. Because of Z^*_od  ≤ Q^*_od , it can also be written as $Z_{od}^{*} = \hbox{min} \left\{ {Q_{od} ,F^{ - 1} \left( {1 - \frac{c}{{(1 - \beta )c_{e} }}} \right) - R_{od} } \right\}$.

We can easily verify that the expected profit of the firm π^od_B (Q_od, R_od) is jointly concave in Q_od and R_od. R^*_od and Q^*_od can be given by the first-order optimality condition

$$ \left\{ \begin{array}{l} \frac{{\partial \pi_{B}^{od} (Q_{od} ,R_{od} )}}{{\partial Q_{od} }} = - c_{o} + (r - w)(1 - F(R_{od} + Q_{od} )) = 0 \hfill \\ \frac{{\partial \pi_{B}^{od} (Q_{od} ,R_{od} )}}{{\partial R_{od} }} = - w_{r} + r(1 - F(R_{od} + Q_{od} )) + w(F(R_{od} + Q_{od} ) - F(R_{od} )) = 0 \hfill \\ \end{array} \right.. $$

Then, we obtain

$$ \left\{ \begin{array}{l} F(R_{od}^{ * } ) = 1 - \frac{{w_{r} - c_{o} }}{w} \hfill \\ F(R_{od}^{ * } + Q_{od}^{ * } ) = 1 - \frac{{c_{o} }}{r - w} \hfill \\ \end{array} \right.. $$

We can also derive

$$ Z_{od}^{*} = \hbox{min} \left\{ {Q_{od}^{ * } ,F^{ - 1} \left( {1 - \frac{c}{{(1 - \beta )c_{e} }}} \right) - F^{ - 1} \left( {1 - \frac{{w_{r} - c_{o} }}{w}} \right)} \right\}. $$

Next, we analyze the optimal decisions of the unreliable supplier and firm below.

(I) If w_r ≥ c_o + w, then $F(R_{od}^{ * } ) = 1 - \frac{{w_{r} - c_{o} }}{w} < 0$. Because of R^*_od  > 0, thenR^*_od  = 0. When $1 - \frac{c}{{(1 - \beta )c_{e} }} > 1 - \frac{{c_{o} }}{r - w}$, that is, F(R^*_od  + Z^*_od ) > F(R^*_od  + Q^*_od ), then $Z_{od}^{*} = Q_{od}^{ * } = F^{ - 1} (1 - \frac{{c_{o} }}{r - w})$. When $1 - \frac{c}{{(1 - \beta )c_{e} }} \le 1 - \frac{{c_{o} }}{r - w}$, that is, $\beta \ge 1 - \frac{c(r - w)}{{c_{e} c_{o} }}$, then F(R^*_od  + Z^*_od ) ≤ F(R^*_od  + Q^*_od ), Z^*_od  < Q^*_od , $Q_{od}^{ * } = F^{ - 1} \left( {1 - \frac{{c_{o} }}{r - w}} \right)$ and $Z_{od}^{*} = F^{ - 1} \left( {1 - \frac{c}{{(1 - \beta )c_{e} }}} \right)$.

(II) If c_o < w_r < c_o + w, then $0 < F(R_{od}^{ * } ) = 1 - \frac{{w_{r} - c_{o} }}{w} < 1$. When $1 - \frac{{c_{o} }}{r - w} < 1 - \frac{{w_{r} - c_{o} }}{w}$, that is, $\frac{{c_{o} }}{r - w} > \frac{{w_{r} - c_{o} }}{w}$, then F(R^*_od  + Q^*_od ) < F(R^*_od ),Z^*_od  = Q^*_od  = 0, $R_{od}^{ * } = F^{ - 1} (1 - \frac{{w_{r} - c_{o} }}{w})$. When $1 - \frac{{c_{o} }}{r - w} \ge 1 - \frac{{w_{r} - c_{o} }}{w}$, that is, $\frac{{c_{o} }}{r - w} \le \frac{{w_{r} - c_{o} }}{w}$, then F(R^*_od  + Q^*_od ) ≥ F(R^*_od ), Q^*_od  ≠ 0. For $1 - \frac{{c_{o} }}{r - w} < 1 - \frac{c}{{(1 - \beta )c_{e} }}$, that is, $0 \le \beta < 1 - \frac{c(r - w)}{{c_{e} c_{o} }}$, then $Z_{od}^{*} = Q_{od}^{*} = F^{ - 1} \left( {1 - \frac{{c_{o} }}{r - w}} \right) - F^{ - 1} \left( {1 - \frac{{w_{r} - c_{o} }}{w}} \right)$, $R_{od}^{ * } = F^{ - 1} \left( {1 - \frac{{w_{r} - c_{o} }}{w}} \right)$. For $1 - \frac{{c_{o} }}{r - w} \ge 1 - \frac{c}{{(1 - \beta )c_{e} }}$, that is,$1 - \frac{c(r - w)}{{c_{e} c_{o} }} \le \beta < 1 - \frac{c}{w}$, thenZ^*_od  < Q^*_od . Because ofF(R^*_od  + Z^*_od ) ≥ F(R^*_od ), that is,$\beta \le 1 - \frac{cw}{{c_{e} (w_{r} - c_{o} )}}$ ,then, in the condition of $1 - \frac{c(r - w)}{{c_{e} c_{o} }} \le \beta < \hbox{min} \left\{ {1 - \frac{cw}{{c_{e} (w_{r} - c_{o} )}},1 - \frac{c}{w}} \right\}$, we can get $Q_{od}^{*} = F^{ - 1} \left( {1 - \frac{{c_{o} }}{r - w}} \right) - F^{ - 1} \left( {1 - \frac{{w_{r} - c_{o} }}{w}} \right)$, $Z_{od}^{*} = F^{ - 1} \left( {1 - \frac{c}{{(1 - \beta )c_{e} }}} \right) - F^{ - 1} \left( {1 - \frac{{w_{r} - c_{o} }}{w}} \right)$,$R_{od}^{ * } = F^{ - 1} \left( {1 - \frac{{w_{r} - c_{o} }}{w}} \right)$.

We assume $\hat{\beta }_{od} = 1 - \frac{c(r - w)}{{c_{e} c_{o} }}$ and $c \le \frac{{c_{e} c_{o} }}{r - w} < w$ which is used to assure $0 \le \hat{\beta }_{od} < 1 - \frac{c}{w}$. In conclusion, Proposition 4 can be derived. □

Proof of Proposition 5

Taking the first order and second order derivatives of $\pi_{S}^{fd} (Z_{fd} |Q_{fd} ,R_{fd} )$ with respect toZ_fd, we have

$$ \begin{aligned} \frac{{\partial \pi_{S}^{fd} (Z_{fd} |Q_{fd} ,R_{fd} )}}{{\partial Z_{fd} }} & = - c + (1 - \beta )w_{e} (1 - F(R_{fd} + Z_{fd} )), \\ \frac{{\partial^{2} \pi_{S}^{fd} (Z_{fd} |Q_{fd} ,R_{fd} )}}{{\partial Z_{fd}^{2} }} & = - (1 - \beta )w_{e} f(R_{fd} + Z_{fd} ) < 0. \\ \end{aligned} $$

$\pi_{S}^{fd} (Z_{fd} |Q_{fd} ,R_{fd} )$ is concave in Z_fd. From the FOC and Z^*_fd  ≥ Q_fd, we can get

$$ F\left( {R_{fd} { + }Z_{fd}^{*} } \right) = \hbox{max} \left\{ {F(R_{fd} { + }Q_{fd} ),1 - \frac{c}{{(1 - \beta )w_{e} }}} \right\}. $$

To achieve the optimal ordering quantities of the firm, we consider the following two cases.

Case 1 when Z^*_fd  = Q_fd, then

$$ \begin{aligned} \pi_{B}^{fd} (R_{fd} ,Q_{fd} ) & = r\mu - w_{r} R_{fd} - (1 - \beta )wQ_{fd} - \beta w_{e} \left( {Q_{fd} - \int_{{R_{fd} }}^{{R_{fd} + Q_{fd} }} {F(x)dx} } \right) \\ & \quad - w_{e} \left( {\mu - R_{fd} - Q_{fd} + \int_{0}^{{R_{fd} + Q_{fd} }} {F(x)dx} } \right). \\ \end{aligned} $$

We can easily verify that π^fd_B (R_fd, Q_fd) is jointly concave in Q_fd and R_fd. From the first-order optimal condition

$$ \left\{ \begin{array}{l} \frac{{\partial \pi_{B}^{fd} (R_{fd} ,Q_{fd} )}}{{\partial Q_{fd} }} = (1 - \beta )(w_{e} - w) - (1 - \beta )w_{e} F(R_{fd} + Q_{fd} ) = 0 \hfill \\ \frac{{\partial \pi_{B}^{fd} (R_{fd} ,Q_{fd} )}}{{\partial R_{fd} }} = w_{e} - w_{r} - (1 - \beta )w_{e} F(R_{fd} + Q_{fd} ) - \beta w_{e} F(R_{fd} ) = 0, \hfill \\ \end{array} \right. $$

we can obtain

$$ \left\{ \begin{array}{l} F\left( {R_{fd}^{ * } + Q_{fd}^{ * } } \right) = 1 - \frac{w}{{w_{e} }} \hfill \\ F\left( {R_{fd}^{ * } } \right) = \hbox{max} \left\{ {\frac{{w_{e} - w_{r} - (1 - \beta )(w_{e} - w)}}{{\beta w_{e} }},0} \right\}. \hfill \\ \end{array} \right. $$

Note that $F\left( {R_{fd} { + }Z_{f}^{*} } \right) = \hbox{max} \left\{ {F(R_{fd} { + }Q_{fd} ),1 - \frac{c}{{(1 - \beta )w_{e} }}} \right\}$. If the condition of Z^*_fd  = Q_fd holds, then $1 - \frac{c}{{(1 - \beta )w_{e} }} \le 1 - \frac{w}{{w_{e} }}$ must be satisfied. However, it is contradictory with $0 \le \beta < 1 - \frac{c}{w}$. Then, the condition of Z^*_fd  = Q_fd cannot hold and Case 1 is nonexistent.

Case 2 when Z^*_fd  > Q_fd, then

$$ \begin{aligned} \pi_{B}^{fd} (R_{fd} ,Q_{fd} ) & = r\mu - w_{r} R_{fd} - (1 - \beta )wQ_{fd} - \beta w_{e} \left( {Z_{fd}^{ * } - \int_{{R_{fd} }}^{{R_{fd} + Z_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad - w_{e} \left( {\mu - R_{fd} - Z_{fd}^{ * } + \int_{0}^{{R_{fd} + Z_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad - (1 - \beta )w_{e} \left( {Z_{fd}^{ * } - Q_{fd} - \int_{{R_{fd} + Q_{fd} }}^{{R_{fd} + Z_{fd}^{ * } }} {F(x)dx} } \right). \\ \end{aligned} $$

We can easily verify that π^fd_B (R_fd, Q_fd)is jointly concave in Q_fd and R_fd. The first-order optimal condition is

$$ \left\{ \begin{array}{l} \frac{{\partial \pi_{B}^{fd} (R_{fd} ,Q_{fd} )}}{{\partial Q_{fd} }} = - (1 - \beta )w + (1 - \beta )w_{e} (1 - F(R_{fd} + Q_{fd} )) = 0 \hfill \\ \frac{{\partial \pi_{B}^{fd} (R_{fd} ,Q_{fd} )}}{{\partial R_{fd} }} = w_{e} - w_{r} - \beta w_{e} F(R_{fd} ) - (1 - \beta )w_{e} F(R_{fd} + Q_{fd} ) = 0. \hfill \\ \end{array} \right. $$

This implies that

$$ \left\{ \begin{array}{l} F\left( {R_{fd}^{ * } + Q_{fd}^{ * } } \right) = 1 - \frac{w}{{w_{e} }} \hfill \\ F(R_{fd}^{ * } ) = \hbox{max} \left\{ {\frac{{w_{e} - w_{r} - (1 - \beta )(w_{e} - w)}}{{\beta w_{e} }},0} \right\}. \hfill \\ \end{array} \right. $$

Given that $1 - \frac{w}{{w_{e} }} \le 1 - \frac{c}{{(1 - \beta )w_{e} }}$, the optimal decision of the unreliable supplier satisfies $F(R_{fd}^{ * } { + }Z_{fd}^{*} ) = 1 - \frac{c}{{(1 - \beta )w_{e} }}$.

Whether R^*_fd is zero or not, there exists a threshold $\hat{\beta }_{fd} { = }1 - \frac{{w_{e} - w_{r} }}{{w_{e} - w}}$. Thus, Proposition 5 is proven. □

Proof of Proposition 6

(I) Under the option contract with a reliable supplier, (1) ifw_r ≥ c_o + w, when $0 \le \beta < \hat{\beta }_{od}$, then $\frac{{\partial \pi_{S}^{od} (Z_{od}^{ * } )}}{\partial \beta } = - c_{e} \left( {Q_{od}^{ * } - \int_{0}^{{Q_{od}^{ * } }} {F(x)dx} } \right) < 0$. Hence, π^od_S (Z^*_od ) is decreasing when $\beta \in [0,\hat{\beta }_{od} ]$. When $\hat{\beta }_{od} \le \beta < 1 - \frac{c}{w}$, then $\frac{{\partial Z_{od}^{ * } }}{\partial \beta } = - \frac{c}{{f(Z_{od}^{ * } )(1 - \beta )^{2} c_{e} }}$, $\frac{{\partial \pi_{S}^{od} (Z_{od}^{ * } )}}{\partial \beta } = - c_{e} \left( {Z_{od}^{ * } - \int_{0}^{{Z_{od}^{ * } }} {F(x)dx} } \right) < 0$. Hence, π^od_S (Z^*_od ) is decreasing when $\beta \in [\hat{\beta }_{od1} ,1 - \frac{c}{w})$. Now, we investigate the continuity of π^od_S (Z^*_od ) when $\beta = \hat{\beta }_{od}$. Given that

$$ \begin{aligned} \pi_{S}^{od} \left( {Z_{od}^{*} } \right)|_{{\beta = \hat{\beta }_{od}^{ - } }} & = - cZ_{od}^{ * } + c_{o} Q_{od}^{ * } + (w - \beta c_{e} )\left( {Q_{od}^{ * } - \int_{0}^{{Q_{od}^{ * } }} {F(x)dx} } \right), \\ \pi_{S}^{od} \left( {Z_{od}^{*} } \right)|_{{\beta = \hat{\beta }_{od}^{ + } }} & = - cZ_{od}^{ * } + c_{o} Q_{od}^{ * } + (w - \beta c_{e} )\left( {Q_{od}^{ * } - \int_{0}^{{Q_{od}^{ * } }} {F(x)dx} } \right), \\ \end{aligned} $$

then $\pi_{S}^{od} (Z_{od}^{*} )|_{{\beta = \hat{\beta }_{od}^{ - } }} = \pi_{S}^{od} (Z_{od}^{*} )|_{{\beta = \hat{\beta }_{od}^{ + } }}$. Hence, π^od_S (Z^*_o ) is continuous when $\beta = \hat{\beta }_{od}$. Therefore, π^od_S (Z^*_od ) is decreasing when $\beta \in [0,1 - \frac{c}{w})$.

(2) If c_o < w_r < c_o + w, (a) when $\frac{{c_{o} }}{r - w} > \frac{{w_{r} - c_{o} }}{w}$, then $\pi_{S}^{od} (Z_{od}^{ * } ) = \pi_{B}^{od} (Q_{od}^{ * } ,R_{od}^{ * } ){ = 0}$. (b) When $\frac{{c_{o} }}{r - w} \le \frac{{w_{r} - c_{o} }}{w}$, for $0 \le \beta < \hat{\beta }_{od}$, then $\frac{{\partial \pi_{S}^{od} (Z_{od}^{ * } )}}{\partial \beta } = - c_{e} (Q_{od}^{ * } - \int_{{R_{od}^{ * } }}^{{R_{od}^{ * } + Q_{od}^{ * } }} {F(x)dx} ) < 0$. Hence, π^od_S (Z^*_od ) is decreasing when $\beta \in [0,\hat{\beta }_{od} ]$. For $\hat{\beta }_{od} \le \beta < 1 - \frac{c}{w}$, Z^*_od  < Q^*_od , there exists

$$ \frac{{\partial \pi_{S}^{od} (Z_{od}^{ * } )}}{\partial \beta } = - c_{e} \left( {Z_{od}^{ * } - \int_{{R_{od}^{ * } }}^{{R_{od}^{ * } + Z_{od}^{ * } }} {F(x)dx} } \right) < 0. $$

Hence, π^od_S (Z^*_od ) is decreasing when $\beta \in [\hat{\beta }_{od} ,1 - \frac{c}{w})$. Now, we investigate the continuity of π^od_S (Z^*_od ) when $\beta = \hat{\beta }_{od}$. Given that

$$ \begin{aligned} \pi_{S}^{od} \left( {Z_{od}^{*} } \right)|_{{\beta = \hat{\beta }_{od}^{ - } }} & = - cZ_{od}^{ * } + c_{o} Q_{od}^{ * } + (w - \beta c_{e} )\left( {Q_{od}^{ * } - \int_{{R_{od}^{ * } }}^{{R_{od}^{ * } + Q_{od}^{ * } }} {F(x)dx} } \right), \\ \pi_{S}^{od} \left( {Z_{od}^{*} } \right)|_{{\beta = \hat{\beta }_{od}^{ + } }} & = - cZ_{od}^{ * } + c_{o} Q_{od}^{ * } + (w - \beta c_{e} )\left( {Q_{od}^{ * } - \int_{{R_{od}^{ * } }}^{{R_{od}^{ * } + Q_{od}^{ * } }} {F(x)dx} } \right), \\ \end{aligned} $$

then $\pi_{S}^{od} (Z_{od}^{*} )|_{{\beta = \hat{\beta }_{od}^{ - } }} = \pi_{S}^{od} (Z_{od}^{*} )|_{{\beta = \hat{\beta }_{od}^{ + } }}$. Hence, π^od_S (Z^*_o ) is continuous when $\beta = \hat{\beta }_{od}$. Therefore, π^od_S (Z^*_od ) is decreasing when $\beta \in [0,1 - \frac{c}{w})$.

From the expected profit function of the firm, we can easily conclude thatπ^od_B (Q^*_od , R^*_od ) is independent of β. As π^od_S (Z^*_od ) is decreasing when $\beta \in [0,1 - \frac{c}{w})$, the profit π^o*_SC of the whole supply chain is decreasing when $\beta \in [0,1 - \frac{c}{w})$.

(II) Under the order commitment contract with reliable supplier, (1) when0 < β < β_fd, then

$$ \frac{{\partial \pi_{S}^{fd} (Z_{fd}^{ * } )}}{\partial \beta } = - wQ_{fd}^{ * } - w_{e} (Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{Q_{fd}^{ * } }}^{{Z_{fd}^{ * } }} {F(x)dx} ) < 0. $$

Hence, π^fd_S (Z^*_fd ) is decreasing when β ∊ [0, β_fd]. For the firm, there exists

$$ \frac{{\partial \pi_{B}^{fd} (R_{fd}^{ * } ,Q_{fd}^{ * } )}}{\partial \beta } = wQ_{fd}^{ * } - w_{e} \left( {Q_{fd}^{ * } - \int_{0}^{{Q_{fd}^{ * } }} {F(x)dx} } \right). $$

If $wQ_{fd}^{ * } - w_{e} (Q_{fd}^{ * } - \int_{0}^{{Q_{fd}^{ * } }} {F(x)dx} ) < 0$ (> 0), then $\frac{{\partial \pi_{B}^{fd} (R_{fd}^{ * } ,Q_{fd}^{ * } )}}{\partial \beta } < 0$ (> 0). Hence, π^fd_B (R^*_fd , Q^*_fd ) is decreasing (increasing) when β ∊ [0, β_fd]. For the whole supply chain, because of $\frac{{\partial (\pi_{S}^{fd} + \pi_{B}^{fd} )}}{\partial \beta } = - w_{e} (Z_{fd}^{ * } - \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} ) < 0$, then π^fd_SC  = π^fd_S  + π^fd_B is decreasing when β ∊ [0, β_fd].

(2) When $\beta_{fd} \le \beta < 1 - \frac{c}{w}$, there exists

$$ \frac{{\partial \pi_{S}^{fd} (Z_{fd}^{ * } )}}{\partial \beta } = - wQ_{fd}^{ * } - ((1 - \beta )w - c)\frac{{\partial R_{fd}^{ * } }}{\partial \beta } - w_{e} \left( {Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{R_{fd}^{ * } + Q_{fd}^{ * } }}^{{R_{fd}^{ * } + Z_{fd}^{ * } }} {F(x)dx} } \right). $$

Given that $\frac{{\partial R_{fd}^{ * } }}{\partial \beta } = \frac{{w_{r} - w}}{{\beta^{2} w_{e} f(R_{fd}^{ * } )}} > 0$, then $\frac{{\partial \pi_{S}^{fd} (Z_{fd}^{ * } )}}{\partial \beta } < 0$. Hence, π^fd_S (Z^*_f ) is decreasing when $\beta \in [\beta_{fd} ,1 - \frac{c}{w})$. For the firm, there exists

$$ \frac{{\partial \pi_{B}^{fd} (R_{fd}^{ * } ,Q_{fd}^{ * } )}}{\partial \beta } = wQ_{fd}^{ * } - w_{e} \left( {Q_{fd}^{ * } - \int_{{R_{fd}^{ * } }}^{{R_{fd}^{ * } + Q_{fd}^{ * } }} {F(x)dx} } \right). $$

If $wQ_{fd}^{ * } - w_{e} (Q_{fd}^{ * } - \int_{{R_{fd}^{ * } }}^{{R_{fd}^{ * } + Q_{fd}^{ * } }} {F(x)dx} ) < 0$ (> 0), then $\frac{{\partial \pi_{B}^{fd} (R_{fd}^{ * } ,Q_{fd}^{ * } )}}{\partial \beta } < 0$ (> 0). Hence, π^fd_B (R^*_fd , Q^*_fd ) is decreasing (increasing) when $\beta \in [\beta_{fd} ,1 - \frac{c}{w})$. For the whole supply chain, because of

$$ \begin{aligned} \frac{{\partial (\pi_{S}^{fd} + \pi_{B}^{fd} )}}{\partial \beta } & = - ((1 - \beta )w - c)\frac{{\partial R_{fd}^{ * } }}{\partial \beta } - w_{e} \left( {Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{R_{fd}^{ * } + Q_{fd}^{ * } }}^{{R_{fd}^{ * } + Z_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad - w_{e} \left( {Q_{fd}^{ * } - \int_{{R_{fd}^{ * } }}^{{R_{fd}^{ * } + Q_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad < 0, \\ \end{aligned} $$

then π^fd_SC  = π^fd_S  + π^fd_B is decreasing when $\beta \in [\beta_{fd} ,1 - \frac{c}{w})$. Now, we investigate the continuity of π^fd_S (Z^*_fd ), π^fd_B (R^*_fd , Q^*_fd ) and π^fd_SC when $\beta = \hat{\beta }_{od}$. Given that

$$ \begin{aligned} \pi_{S}^{fd} \left( {Z_{fd}^{ * } } \right)|_{{\beta = \hat{\beta }_{fd}^{ - } }} & = - cZ_{fd}^{ * } + (1 - \beta )wQ_{fd}^{ * } + (1 - \beta )w_{e} \left( {Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{Q_{fd}^{ * } }}^{{Z_{fd}^{ * } }} {F(x)dx} } \right), \\ \pi_{S}^{fd} \left( {Z_{fd}^{ * } } \right)|_{{\beta = \hat{\beta }_{fd}^{ + } }} & = - cZ_{fd}^{ * } + (1 - \beta )wQ_{fd}^{ * } + (1 - \beta )w_{e} \left( {Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{Q_{fd}^{ * } }}^{{Z_{fd}^{ * } }} {F(x)dx} } \right), \\ \pi_{B}^{fd} \left( {R_{fd}^{ * } ,Q_{fd}^{ * } } \right)|_{{\beta = \hat{\beta }_{fd}^{ - } }} & = r\mu - (1 - \beta )wQ_{fd}^{ * } - \beta w_{e} \left( {Z_{fd}^{ * } - \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad - w_{e} \left( {\mu - Z_{fd}^{ * } + \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right) - (1 - \beta )w_{e} \left( {Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{Q_{fd}^{ * } }}^{{Z_{fd}^{ * } }} {F(x)dx} } \right), \\ \pi_{B}^{fd} \left( {R_{fd}^{ * } ,Q_{fd}^{ * } } \right)|_{{\beta = \hat{\beta }_{fd}^{ + } }} & = r\mu - (1 - \beta )wQ_{fd}^{ * } - \beta w_{e} \left( {Z_{fd}^{ * } - \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad - w_{e} \left( {\mu - Z_{fd}^{ * } + \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right) \\ & \quad - (1 - \beta )w_{e} \left( {Z_{fd}^{ * } - Q_{fd}^{ * } - \int_{{Q_{fd}^{ * } }}^{{Z_{fd}^{ * } }} {F(x)dx} } \right), \\ \pi_{SC}^{fd} |_{{\beta = \hat{\beta }_{fd}^{ - } }} & = r\mu - cZ_{fd}^{ * } - \beta w_{e} \left( {Z_{fd}^{ * } - \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right) - w_{e} \left( {\mu - Z_{fd}^{ * } + \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right), \\ \pi_{SC}^{fd} |_{{\beta = \hat{\beta }_{fd}^{ + } }} & = r\mu - cZ_{fd}^{ * } - \beta w_{e} \left( {Z_{fd}^{ * } - \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right) - w_{e} \left( {\mu - Z_{fd}^{ * } + \int_{0}^{{Z_{fd}^{ * } }} {F(x)dx} } \right). \\ \end{aligned} $$

Then $\pi_{S}^{fd} (Z_{fd}^{ * } )|_{{\beta = \hat{\beta }_{fd}^{ - } }} = \pi_{S}^{fd} (Z_{fd}^{ * } )|_{{\beta = \hat{\beta }_{fd}^{ + } }}$, $\pi_{B}^{fd} (R_{fd}^{ * } ,Q_{fd}^{ * } )|_{{\beta = \hat{\beta }_{fd}^{ - } }}~=~\pi_{B}^{fd} (R_{fd}^{ * } ,Q_{fd}^{ * } )|_{{\beta = \hat{\beta }_{fd}^{ + } }}$, and $\pi_{SC}^{fd} |_{{\beta = \hat{\beta }_{fd}^{ - } }} = \pi_{SC}^{fd} |_{{\beta = \hat{\beta }_{fd}^{ + } }}$. Hence, π^fd_S (Z^*_fd ), π^fd_B (R^*_fd , Q^*_fd ), and π^fd_SC are all continuous when β = β_fd. Therefore, π^od*_S and π^fd*_SC are decreasing in β, and when $wQ_{fd}^{ * } - w_{e} (Q_{fd}^{ * } - \int_{{R_{fd}^{ * } }}^{{R_{fd}^{ * } + Q_{fd}^{ * } }} {F(x)dx} ) < 0$ (> 0), π^od*_B is decreasing (increasing) in β. □