Impacts of Vendor-Managed Strategic Partnership on Fashion Supply Chains with Markdown Money Policy

Abstract

A vendor-managed inventory (VMI) partnership is a popular approach to promote channel performance. This paper is motivated by real-industrial practices in the fashion industry and explores the issue of how a VMI partnership with markdown money policy (MMP) operates in the fashion supply chain. MMP is also known as a vendor agreement whereby a vendor supports the profitability of their particular brand with a specific retailer. A vendor allowance (VA) in the form of markdown money is issued to the said retailer in a certain period, e.g., every quarter or six-month season. We propose a model in the context of a two-echelon supply chain with one single supplier and one single retailer trading via a VMI partnership with MMP in both decentralized and centralized supply chains. We find that under the VMI mode with an MMP, the supply chain is able to achieve coordination, and the retailer’s profit is better off but the supplier suffers. We then conduct a numerical study to further explore the impact of VMI within the supply chain with the MMP. Important insights into industry practices are discussed.

Appendix: Proofs

Proof of Proposition 1

From (6.1), we have

$${\text{EP}}_{\text{S}} (q_{\text{S}} (b)) = (w - c)q_{\text{S}} (b) - b\int\limits_{0}^{{q_{\text{S}} }} {F(x){\text{d}}x}$$

$$\begin{aligned} \frac{{{\text{dEP}}_{\text{S}} (b,q_{\text{S}} (b))}}{{{\text{d}}b}} & = (w - c)\frac{{{\text{d}}q_{\text{S}} (b)}}{{{\text{d}}b}} - \int\limits_{0}^{{q_{S} }} {F(x){\text{d}}x - bF(q_{\text{S}} (b))\frac{{{\text{d}}q_{\text{S}} (b)}}{{{\text{d}}b}}} \\ & = (w - c - bF(q_{\text{S}} (b)))\frac{{{\text{d}}q_{\text{S}} (b)}}{{{\text{d}}b}} - \int\limits_{0}^{{q_{\text{S}} }} {F(x){\text{d}}x.} \\ \end{aligned}$$

Substitute (6.2) in the above equation, and we know $\int_{0}^{{q_{S} }} {F(x){\text{d}}x > 0}$, thus ${\text{dEP}}_{\text{S}} (q_{\text{S}} (b))/{\text{d}}b < 0$, that indicates ${\text{EP}}_{\text{S}} (q_{{{\text{S}}^{*} }} (b))$ is decreasing in b, $\forall 0 \le b \le w - v$. When q is optimal quantity $q_{{\text{S}^{*} }} (b)$, we have

$$\begin{aligned} \frac{{{\text{dEP}}_{\text{S}} \left( {b,q_{{\text{S}^{*} }} (b)} \right)}}{{{\text{d}}q_{{\text{S}^{*} }} (b)}} & = (w - c) - b^{*} F\left( {q_{{\text{S}^{*} }} (b)} \right) = 0, \\ \frac{{{\text{d}}q_{{{\text{S}}^{*} }} (b,w)}}{{{\text{d}}b}} & = - \frac{{F(q_{{{\text{S}}^{*} }} (b,w))}}{{bf(q_{{{\text{S}}^{*} }} (b,w))}} < 0. \\ \end{aligned}$$

The result shows that $q_{{{\text{S}}^{*} }} (b)$ is decreasing in b.

Proof of Proposition 2

For retailer, the expected profit is

$${\text{EP}}_{\text{R}} (q_{{{\text{S}}^{*} }} (b)) = (r - w)q_{{{\text{S}}^{*} }} (b) - (r - b - v)\int\limits_{0}^{{q_{{{\text{S}}^{*} }} }} {F(x){\text{d}}x}$$

In order to determine the optimal b, we have

$$\begin{aligned} \frac{{{\text{dEP}}_{\text{R}} (b,q_{{{\text{S}}^{*} }} (b))}}{{{\text{d}}b}} & = (r - w)\frac{{{\text{d}}q_{{{\text{S}}^{*} }} (b)}}{{{\text{d}}b}} + \int\limits_{0}^{{q_{{{\text{S}}^{*} }} }} {F(x){\text{d}}x - (r - b - v)F(q_{{{\text{S}}^{*} }} (b))\frac{{{\text{d}}q_{{{\text{S}}^{*} }} (b)}}{{{\text{d}}b}}} \\ & = \left[ {(r - w) - (r - b - v)F(q_{{{\text{S}}^{*} }} (b))} \right]\frac{{{\text{d}}q_{{{\text{S}}^{*} }} (b)}}{{{\text{d}}b}} + \int\limits_{0}^{{q_{{{\text{S}}^{*} }} }} {F(x){\text{d}}x} . \\ \end{aligned}$$

We find that if $b = (r - v)(w - c)/(r - c)$, then

$$\left[ {(r - w) - (r - b - v)F(q_{{{\text{S}}^{*} }} (b))} \right] = 0 \Rightarrow {\text{dEP}}_{\text{R}} (b,q_{{{\text{S}}^{*} }} (b))/{\text{d}}b > 0 .$$

Thus, $b^{*} > (r - v)(w - c)/(r - c) > (w - c)$ and $b^{*}$ are in the region $(w - c,w - v]$.

Regarding the first derivation of ${\text{EP}}_{\text{R}}$ when $b = w - v$, if $\left. {{\text{dEP}}_{\text{R}} (b,q_{{{\text{S}}^{*} }} (b))/{\text{d}}b} \right|_{b = w - v} \ge 0$, then optimal $b^{*}$ over the upper bound value $w - v$ and therefore $b^{*} = w - v$; if $\left. {{\text{dEP}}_{\text{R}} (b,q_{{{\text{S}}^{*} }} (b))/{\text{d}}b} \right|_{b = w - v} < 0$, then optimal $b^{*}$ satisfied ${\text{dEP}}_{\text{R}} (b,q_{{{\text{S}}^{*} }} (b))/{\text{d}}b = 0$.

Proof of Proposition 3

First, we proof Eq. (6.4). Consider the supply chain’s expected profit

$${\text{EP}}_{\text{SC}} (q_{\text{SC}} ) = (r - c)q_{\text{SC}} - (r - v)\int\limits_{0}^{{q_{\text{SC}} }} {F(x){\text{d}}x.}$$

Take the derivative of ${\text{EP}}_{\text{SC}} (q_{\text{SC}} )$ with the respect to $q_{\text{SC}}$, we get

$$q_{{{\text{SC}}^{*} }} = F^{ - 1} \left( {\frac{r - c}{r - v}} \right).$$

Since the supply chain is coordinated, i.e., $q_{{{\text{S}}^{*} }} = q_{{{\text{SC}}^{*} }}$, we can determine the optimal b

$$b_{\text{SC}}^{*} = \frac{(w - c)(r - v)}{(r - c)}.$$

Substituting $b_{\text{SC}}^{*}$ and $q_{{\text{SC}^{*} }}$ into the expected profit of supply chain ${\text{EP}}_{\text{SC}} (q_{\text{SC}} )$, we can find that $b_{\text{SC}}^{*}$ is able to allocate supply chain profit between the retailer and the supplier.

Proof of Proposition 4

Following the proof of Proposition 2, first we consider the case when $b^{*} = w - v$. In this case, since $b^{*} - b_{\text{SC}}^{*} = w - c -$ $[(r - v)(w - c)]/(r - c) = (r - w)(c - v) > 0$ and ${\text{EP}}_{R} (b,q_{{\text{S}^{*} }} (b))$ is increasing in b in the range of $(w - c,w - v]$. Therefore, the maximum ${\text{EP}}_{\text{R}} (b,q_{{\text{S}^{*} }} (b))$ is always larger than ${\text{EP}}_{\text{R}} (b_{\text{SC}}^{*} ,q_{{\text{SC}^{*} }} )$. When $b^{*}$ is smaller than $w - v$, we have proved that $b^{*} > b_{\text{SC}}^{*}$ in the proof of Proposition 2. Thus, the maximum ${\text{EP}}_{\text{R}} (b,q_{{\text{S}^{*} }} (b))$ is greater than ${\text{EP}}_{\text{R}} (b_{\text{SC}}^{*} ,q_{{\text{SC}^{*} }} )$.

For the retailer, since that the optimal integrated markdown money is lower than the markdown money decided in the decentralized supply chain. We find that  ${\text{EP}}_{\text{S}}$ is decreasing in b, when achieves channel coordination.