Coordinating a Supply Chain with an EOQ Model

Abstract

In this paper, we consider a supply chain coordination scheme and issues in which a manufacturer supplies a product to a retailer. The retailer decides his optimal order quantity using an economic order quantity (EOQ) model which takes into consideration the shipment costs charged by the manufacturer. We show that under some circumstances, the manufacturer can offer a contract which includes a discount shipment fee per delivery and a shipment fee per unit to coordinate the supply chain and enhance the profits of both the manufacturer and the retailer. We also identify under which condition the manufacturer cannot coordinate the supply chain with shipment fees. This research highlights that the manufacturer needs to further investigate these conditions before offering and implementing a contract. Numerical examples are also included to illustrate the main results discussed in the paper.

Appendix

Proof of Proposition 1

For a given $C_{R}$, taking partial derivatives of (1) w.r.t. Q:

$$ \begin{gathered} {{\partial \prod \nolimits_{R} } \mathord{\left/ {\vphantom {{\partial \prod \nolimits_{R} } {\partial Q}}} \right. \kern-0pt} {\partial Q}} = {{(s_{R} + C_{R} )D} \mathord{\left/ {\vphantom {{(s_{R} + C_{R} )D} {Q^{2} }}} \right. \kern-0pt} {Q^{2} }} - {{h_{R} } \mathord{\left/ {\vphantom {{h_{R} } 2}} \right. \kern-0pt} 2}\quad {\text{and}} \end{gathered}$$

(A1)

$\begin{gathered}\hfill \\ {{\partial^{2} \prod \nolimits_{R} } \mathord{\left/ {\vphantom {{\partial^{2} \prod \nolimits_{R} } {\partial Q^{2} }}} \right. \kern-0pt} {\partial Q^{2} }} = - {{2(s_{R} + C_{R} )D} \mathord{\left/ {\vphantom {{2(s_{R} + C_{R} )D} {Q^{ 3} }}} \right. \kern-0pt} {Q^{ 3} }} < 0. \hfill \\ \end{gathered}$Therefore, there exists a unique optimal ordering quantity for the retailer, which is given by setting ${{\partial \prod \nolimits_{R} } \mathord{\left/ {\vphantom {{\partial \prod \nolimits_{R} } {\partial Q}}} \right. \kern-0pt} {\partial Q}} = 0$.

Proof of Proposition 2

With Q ^* in (3), taking partial derivatives of (2) w.r.t. $C_{R}$:

$$ \frac{{\partial \prod \nolimits_{M} }}{{\partial C_{R} }} = \frac{{\sqrt 2 D^{2} h_{R} \left( {(s_{M} + 2s_{R} + C_{R} )h_{R} p_{M} - h_{M} (s_{R} + C_{R} )(3p_{M} - 2D)} \right)}}{{4p_{M} (h_{R} D(s_{R} + C_{R} ))^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }} $$

(A2)

$$ \frac{{\partial^{2} \prod \nolimits_{M} }}{{\partial C_{R}^{2} }} = - \frac{{\sqrt 2 \left[ { - h_{M} (3p_{M} - 2D)(s_{R} + C_{R} ) + h_{R} p_{M} (3s_{M} + 4s_{R} + C_{R} )} \right]h_{R}^{2} D^{3} }}{{8p_{M} (h_{R} D(s_{R} + C_{R} ))^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0pt} 2}}} }}. $$

(A3)

${{\partial \prod \nolimits_{M} } \mathord{\left/ {\vphantom {{\partial \prod \nolimits_{M} } {\partial C_{R} }}} \right. \kern-0pt} {\partial C_{R} }} = 0$ gives

$$ C_{R}^{*} = \frac{{(h_{R} s_{M} - 3s_{R} h_{M} + 2h_{R} s_{R} )p_{M} + 2h_{M} s_{R} D}}{{(3h_{M} - h_{R} )p_{M} - 2h_{M} D}} . $$

(A4)

With (A4), we see that $\left. {{{\partial^{2} \prod \nolimits_{M} } \mathord{\left/ {\vphantom {{\partial^{2} \prod \nolimits_{M} } {\partial C_{R}^{2} }}} \right. \kern-0pt} {\partial C_{R}^{2} }}} \right|_{{C_{R} = C_{R}^{*} }} < 0$, suggesting that there exists a unique optimal ordering quantity for the retailer, which is given by (A3). Also, a positive $C_{R}$ requires $\frac{{h_{R} }}{{h_{M} }} < 3 - \frac{2D}{{p_{M} }} < \frac{{h_{R} }}{{h_{M} }}\frac{{(s_{M} + 2s_{R} )}}{{s_{R} }}$.

Proof of Proposition 3

Taking partial derivatives of (7) w.r.t. $Q_{I}$:

$$ {{\partial \prod } \mathord{\left/ {\vphantom {{\partial \prod } {\partial Q_{I} }}} \right. \kern-0pt} {\partial Q_{I} }} = {{(s_{R} + s_{M} )D} \mathord{\left/ {\vphantom {{(s_{R} + s_{M} )D} {Q^{2} }}} \right. \kern-0pt} {Q^{2} }} - {{h_{R} } \mathord{\left/ {\vphantom {{h_{R} } 2}} \right. \kern-0pt} 2} - {{3h_{M} } \mathord{\left/ {\vphantom {{3h_{M} } 2}} \right. \kern-0pt} 2} + {{h_{M} D} \mathord{\left/ {\vphantom {{h_{M} D} {p_{M} }}} \right. \kern-0pt} {p_{M} }}\quad {\text{and}} $$

(A5)

${{\partial^{2} \prod} \mathord{\left/ {\vphantom {{\partial^{2} \prod} {\partial Q_{I}^{2} }}} \right. \kern-0pt} {\partial Q_{I}^{2} }} = - {{2(s_{R} + s_{M} )D} \mathord{\left/ {\vphantom {{2(s_{R} + s_{M} )D} {Q_{I}^{ 3} }}} \right. \kern-0pt} {Q_{I}^{ 3} }} < 0.$

Therefore, there exists a unique optimal ordering quantity for the supply chain, which is given by setting ${{\partial \prod} \mathord{\left/ {\vphantom {{\partial \prod} {\partial Q_{I} }}} \right. \kern-0pt} {\partial Q_{I} }} = 0$.

Proof of Proposition 4

Comparing Eqs. (3) to (8), we have that the manufacturer can set a discount in $C_{R}$ so that $Q^{*} = Q_{I}^{*}$, where $C_{R}^{d} = \frac{{(h_{R} s_{M} - 3s_{R} h_{M} )p_{M} + 2h_{M} s_{R} D}}{{(3h_{M} + h_{R} )p_{M} - 2h_{M} D}}$. A nonnegative $C_{R}^{d}$ requires $\frac{{h_{R} }}{{h_{M} }}\frac{{s_{M} }}{{s_{R} }} \ge 3 - \frac{2D}{{p_{M} }}$. Comparing (9) to (5), it is obvious that $C_{R}^{d} < C_{R}^{*}$ for $h_{R} > 0$.