Impact of power structures in a subcontracting assembly system

Abstract

We investigate the impact of power structures on the production and pricing strategies in a decentralized subcontracting assembly system consisting of two suppliers (key supplier and subcontractor) and one manufacturer (assembler). The key supplier, who is also the general contractor, negotiates with the manufacturer and assigns partial component production to the subcontractor. We first identify a single power regime (SPR), in which either the key supplier or the manufacturer determines the wholesale price or the order/production quantity. Under SPR, we consider three power structures, namely, KSA, KAS, and SKA. We find that the assembly system will substantially benefit under KAS. Results show that the subcontracting mechanism between the two suppliers can increase each firm’s profit and disperse the bargaining power. Such a decentralization of powers can weaken the horizontal decentralization between the suppliers and improve the system’s performance, thereby achieving a win–win situation. Furthermore, we extend our analysis to a dual power regime (DPR), in which the key supplier or the manufacturer decides on price and quantity. We show that the proposed assembly system performs optimally under DPR. Moreover, the system will benefit if the firm that is substantially near the end market makes the centralization decision. Compared with the classical pull and push contract model, the proposed assembly system provides the best performance under DPR.

Appendix

1.1 Proof for Lemma 1

Note that a critical condition $k = \frac{{c_{2} }}{w}$ guarantees the subcontractor’s profit positive. That is, the optimal subcontracting price ratio is $\frac{{c_{2} }}{w}$. We calculate the first and second derivatives of Eq. (5) on $q_{ksa}$ to obtain $\frac{{\partial \prod_{S1} }}{{\partial q_{ksa} }} = \left\{ {p[1 - g(q_{ksa} )] - \frac{{c_{1} + c_{2} }}{{\bar{F}(q_{ksa} )}}} \right\}\bar{F}(q_{ksa} )$ and $\frac{{\partial^{2} \prod_{S1} }}{{\partial q_{{_{ksa} }}^{2} }} = - 2\mathop F\limits^{\_\_} (q_{ksa} )g(q_{ksa} ) < 0$, where $g(x) = \frac{xf(x)}{{\mathop F\limits^{\_\_} (x)}}$. Thus, the supply chain’s equilibrium production quantity satisfies $p[1 - g(q_{ksa} )] = \frac{{c_{1} + c_{2} }}{{\mathop F\limits^{\_\_} (q_{ksa} )}}$. In addition, the optimal wholesale price and subcontracting price ratio are $w^{*} = \frac{{c_{1} + c_{2} }}{{1 - g(q_{ksa} )}}$ and $k^{*} = \frac{{c_{2} }}{{c_{1} + c_{2} }}[1 - g(q_{ksa} )]$, respectively.

1.2 Proof of Lemma 2

Note that the two suppliers’ optimal strategy has a unique Nash equilibrium as follows:

$$ {\prod}_{S1} (k,q_{1} ) = w(1 - k)E\hbox{min} (D,q_{1} ,q_{2} ) - c_{1} q_{1} $$

(A1)

$$ {\prod}_{S2} (q_{2} ) = kwE\hbox{min} (D,q_{1} ,q_{2} ) - c_{2} q_{2} . $$

(A2)

We combine Eqs. (A1) and (A2). If $q_{1} < q_{2}$, then we obtain the following equations:

$$ {\prod}_{S1} (k,q_{1} ) = w(1 - k)E\hbox{min} (D,q_{1} ) - c_{1} q_{1} $$

(A3)

and

$$ {\prod}_{S2} (q_{2} ) = kwE\hbox{min} (D,q_{1} ) - c_{2} q_{2} . $$

(A4)

We consider the first and second derivatives of Eqs. (A3) and (A4) to obtain $\frac{{\partial \prod_{S1} }}{{\partial q_{1} }} = w(1 - k)[1 - F(q_{1} )] - c_{1} ,\;and\;{\kern 1pt} \frac{{\partial^{2} \prod_{S1} }}{{\partial q_{1}^{2} }} = - w(1 - k)f(q_{1} ) < 0.$ In addition, $\frac{{\partial \prod_{S2} }}{{\partial q_{2} }} = - c_{2} \; < 0$. Thus, the two suppliers’ optimal production quantity is $q_{1} \wedge q_{2}$ and $q_{1}^{{}} = \bar{F}^{ - 1} \left( {\frac{{c_{1} }}{w(1 - k)}} \right)$.

If $q_{1} > q_{2}$, then we obtain the following equations:

$$ {\prod}_{S1} (k,q_{1} ) = w(1 - k)E\hbox{min} (D,q_{2} ) - c_{1} q_{1} $$

(A5)

and

$$ {\prod}_{S2} (q_{2} ) = kwE\hbox{min} (D,q_{2} ) - c_{2} q_{2} . $$

(A6)

When the first and second derivatives of Eqs. (A5) and (6) are calculated, we obtain $\frac{{\partial \prod_{S2} }}{{\partial q_{2} }} = kw[1 - F(q_{2} )] - c_{2} ,\;and\;\frac{{\partial^{2} \prod_{S2} }}{{\partial q_{2}^{2} }} = - kwf(q_{2} ) < 0$.

In equilibrium, the two suppliers’ optimal production quantity is $q_{1} \wedge q_{2}$ and $q_{2}^{{}} = \bar{F}^{ - 1} \left( {\frac{{c_{2} }}{kw}} \right)$. Thus, we use Eqs. (A4) and (A5) to solve for $q_{1}$ and $q_{2}$ in terms of $w$ and $k$. Accordingly, $q_{1}^{{}} = \bar{F}^{ - 1} \left( {\frac{{c_{1} }}{w(1 - k)}} \right)$ and $q_{2}^{{}} = \bar{F}^{ - 1} \left( {\frac{{c_{2} }}{kw}} \right)$. Thereafter, we obtain $w = \frac{{c_{1} }}{{\mathop F\limits^{\_\_} (q_{1}^{{}} )}} + \frac{{c_{2} }}{{\mathop F\limits^{\_\_} (q_{2}^{{}} )}}$ and $k = \frac{{c_{2} }}{{w\mathop F\limits^{\_\_} (q_{2}^{{}} )}}$. Thus, the equilibrium production quantity of the KAS mode is defined by $q_{kas} = q_{1} \wedge q_{2}$. Moreover, the integrated system shares the same principle with this Lemma.

1.3 Proof of Lemma 3

When the manufacturer’s profit function in Eq. (10) is used, if $q_{1} \ge q_{2}$, then we obtain the following equation:

$$ {\prod}_{M} (q_{1} ,q_{2} ) = \left[ {p - \frac{{c_{1} }}{{\mathop F\limits^{\_\_} (q_{1} )}} - \frac{{c_{2} }}{{\mathop F\limits^{\_\_} (q_{2} )}}} \right]E\hbox{min} (D,q_{2} ), $$

(A7)

which is evidently decreasing in $q_{1}$. In addition, the optimal $q_{1}$ equals the threshold $q_{2}$. If $q_{1} \le q_{2}$, then we obtain the following equation:

$$ {\prod}_{M} (q_{1} ,q_{2} ) = \left[ {p - \frac{{c_{1} }}{{\mathop F\limits^{\_\_} (q_{1} )}} - \frac{{c_{2} }}{{\mathop F\limits^{\_\_} (q_{2} )}}} \right]E\hbox{min} (D,q_{1} ). $$

(A8)

When the first and second derivatives of Eq. (A8) are calculated, we can obtain $\frac{{\partial \prod_{M} }}{{\partial q_{1} }} = \left[ {p - \frac{{c_{2} }}{{\bar{F}(q_{2} )}}} \right]\bar{F}(q_{1} ) - c_{1} [1 + j(q_{1} )h(q_{1} )]$, and $\frac{{\partial^{2} \prod_{M} }}{{\partial q_{1}^{2} }} = - \left[ {p - \frac{{c_{2} }}{{\bar{F}(q_{2} )}}} \right]f(q_{1} ) - c_{1} [j(q_{1} )h(q_{1} )]^{'} < 0$, where $j(x) = \frac{S(x)}{{\mathop F\limits^{\_\_} (x)}}$ and $h(x) = \frac{f(x)}{{\mathop F\limits^{\_\_} (x)}}$.

Hence, Eq. (A8) is concave in $q_{1}$ and $q_{1}^{*} = \hbox{min} (q_{1} ,q_{2} )$. From $\frac{{\partial \prod_{M} }}{{\partial q_{1} }} = 0$, we can obtain $\frac{{\bar{F}(q_{1} )}}{{1 + j(q_{1} )h(q_{1} )}} = \frac{{c_{1} }}{{p - \frac{{c_{2} }}{{\bar{F}(q_{2} )}}}}$. Thus, the key supplier’s profit function can be written in terms of $q_{1}^{*}$ as $\prod_{S1} (q_{1}^{*} ) = \frac{{c_{1} }}{{\mathop F\limits^{\_\_} (q_{1}^{*} )}}E\hbox{min} (D,q_{1}^{*} ,q_{2} ) - c_{1} q_{1}^{*}$. If $q_{1} \ge q_{2}$, there is $\prod_{S1} (q_{2} ) = \frac{{c_{1} }}{{\mathop F\limits^{\_\_} (q_{2} )}}E\hbox{min} (D,q_{2} ) - c_{1} q_{2}$. Given that $\frac{{\partial \prod_{S1} }}{{\partial q_{2} }} = \frac{{f(q_{2} )S(q_{2} )}}{{\bar{F}^{2} (q_{2} )}}c_{1} > 0$, $\prod_{S1}$ is decreasing in $q_{2}$. Thus, the optimal $q_{2}$ equals $q_{1}$. If $q_{1} \le q_{2}$, there is $\prod_{S1} (q_{2} ) = \frac{{c_{1} }}{{\mathop F\limits^{\_\_} (q_{1} )}}E\hbox{min} (D,q_{1} ) - c_{1} q_{1}$. Given that $\frac{{\partial \prod_{S1} }}{{\partial q_{2} }} = \frac{{f(q_{1} )S(q_{1} )}}{{\bar{F}^{2} (q_{1} )}}\frac{{\partial q_{1} }}{{\partial q_{2} }}c_{1} < 0$, $\prod_{S1}$ is decreasing in $q_{2}$. Thus, the optimal $q_{2}$ equals $q_{1}$.

In equilibrium, the optimal production quantity satisfies $q_{kas} = q_{1} = q_{2}$ and $\frac{{\bar{F}(q_{kas} )}}{{1 + j(q_{kas} )h(q_{kas} )}} = \frac{{c_{1} }}{{p - \frac{{c_{2} }}{{\bar{F}(q_{kas} )}}}}$. The optimal wholesale price and subcontracting price ratio are defined by $w^{*} = \frac{{c_{1} + c_{2} }}{{\mathop F\limits^{\_\_} (q_{kas} )}}$ and $k^{*} = \frac{{c_{2} }}{{c_{1} + c_{2} }}$, respectively, where $j(x) = \frac{S(x)}{{\mathop F\limits^{\_\_} (x)}}$ and $h(x) = \frac{f(x)}{{\mathop F\limits^{\_\_} (x)}}$.

1.4 Proof of Lemma 4

We use backward induction and start from the third stage, when the manufacturer decides the order quantity from the two suppliers. Given the manufacturer’s profit function in Eq. (8), solving the first and second orders condition of $\prod_{M}$ will yield $\frac{{\partial \prod_{M} }}{{\partial q_{m} }} = p\mathop F\limits^{\_\_} (q_{m} ) - w$ and $\frac{{\partial^{2} \prod_{M} }}{{\partial q_{m}^{2} }} = - pf(q_{m} ) < 0$. Thereafter, the optimal order quantity satisfies $q_{ska} = \bar{F}^{ - 1} \left( {\frac{w}{p}} \right)$. That is,

$$ w(q_{ska} ) = p\mathop F\limits^{\_\_} (q_{ska} ). $$

(A9)

Thus, Eq. (9) is used to write the key supplier’s profit in terms of $q_{ska}$. Hence,

$$ {\prod}_{S1} (q_{ska} ) = [(1 - k)p\mathop F\limits^{\_\_} (q_{ska} ) - c_{1} ]q_{ska} . $$

(A10)

When the first and second derivatives of Eq. (A10) are calculated, we can obtain that $\frac{{\partial \prod_{S1} }}{{\partial q_{ska} }} = p(1 - k)[\bar{F}(q_{ska} ) - q_{ska} f(q_{ska} )] - c_{1}$ and $\frac{{\partial^{2} \prod_{S1} }}{{\partial q_{ska}^{2} }} = - 2p(1 - k)f(q_{ska} ) < 0$. Thereafter, Eq. (A10) is concave in $q_{ska}$ and the optimal order quantity $q_{ska}$ satisfies $p(1 - k)\bar{F}(q_{ska} )[1 - g(q_{ska} )] = c_{1}$, where $g(x) = \frac{xf(x)}{{\mathop F\limits^{\_\_} (x)}}$. Thereafter, substituting Eq. (A9) into the subcontractor’s profit function will yield as follows:

$$ {\prod}_{S2} (q_{ska} ) = \left[ {p\mathop F\limits^{\_\_} (q_{ska} ) - \frac{{c_{1} }}{{1 - g(q_{ska} )}} - c_{2} } \right]q_{ska} . $$

(A11)

Similarly, the derivative of Eq. (A11) yields $\frac{{\partial \prod_{S2} }}{{\partial q_{ska} }} = p\bar{F}(q_{ska} )[1 - g(q_{ska} )] - \frac{{c_{1} }}{{1 - g(q_{ska} )}} - c_{2} - q_{ska} \frac{{c_{1} g^{'} (q_{ska} )}}{{[1 - g(q_{ska} )]^{2} }}$. Given that $\frac{{\partial \prod_{S2} }}{{\partial q_{ska} }}\left| {{}_{{q_{ska} = 0}}} \right. = p - c_{1} - c_{2} > 0$, $\frac{{\partial \prod_{S2} }}{{\partial q_{ska} }}\left| {{}_{{q_{ska} = \infty }}} \right. < 0$, and $\prod_{S2}$ is continuous, there exists a maximum $q_{ska}$ at least to maximize $\prod_{S2}$. Moreover, the maximum $q_{ska}$ satisfies $p\mathop F\limits^{\_\_} (q_{ska} )[1 - g(q_{ska} )] - q_{ska} \frac{{c_{1} g^{'} (q_{ska} )}}{{[1 - g(q_{ska} )]^{2} }} = \frac{{c_{1} }}{{1 - g(q_{ska} )}} + c_{2}$. Subsequently, the optimal wholesale price and subcontracting price ratio are defined by $w^{*} = p\mathop F\limits^{\_\_} (q_{ska} )$ and $k^{*} = 1 - \frac{{c_{1} }}{{p\mathop F\limits^{\_\_} (q_{ska} )[1 - g(q_{ska} )]}}$, where $g(x) = \frac{xf(x)}{{\mathop F\limits^{\_\_} (x)}}$ and $h(x) = \frac{f(x)}{{\mathop F\limits^{\_\_} (x)}}$.