Impacts of retailer’s risk averse behaviors on quick response fashion supply chain systems

Abstract

Supply chain systems for fashion apparel products face a high level of risk as the market demand is very volatile and unpredictable. In order to cope with demand volatility, the quick response system which aims to shorten replenishment lead time has been well-established. With a shortened lead time, retailers can postpone the ordering decision and improve their demand forecast by gathering updated market information. However, there is a limit for quick response in which the market demand forecast is never fully accurate and uncertainty can never be fully eliminated. Thus, to keep the level of risk under control, retailers tend to possess a risk-averse behaviour in making their respective inventory decisions. In this paper, we explore the make-to-order quick response fashion supply chain with a risk averse retailer. We employ the mean-risk framework to incorporate the retailer’s risk averse behavior into the optimization model. With the focal point on uncovering the impacts brought by the retailer’s risk averse behavior to the quick response fashion supply chain system, we analytically derive important theoretical insights regarding the retailer’s optimal decisions, the implied inventory service levels, the values of quick response, and the contractual arrangements to attain Pareto improvement when the retailer is risk averse.

Appendix: All proofs

Proof of Lemma 5.1

First of all, from the literature (see, e.g, Chen and Federgruen 2000; Choi et al. 2008), we know that: (i) ${ EP}_i (q)|\mu _i$ is concave, increasing in the region between 0 and $q_{i,{ EP}^*} |\mu _i =\mu _i +\sigma _i \Phi ^{-1}\left( {\frac{r-c}{r-v}} \right)$, and its peak is attained at $q_{i,{ EP}^*} |\mu _i$, (ii) $\textit{VP}_i (q)|\mu _i$ is an increasing function. As a result, for Problem $(\mathrm{P}i)$, if $R_i (q_{i,{ EP}^*} )|\mu _i \le k\Leftrightarrow k\ge (r-v)^{2}\xi \left( {\Phi ^{-1}\left( {\frac{r-c}{r-v}} \right) } \right)$ holds, the constraint $R_i (q)|\mu _i \le k$ will become inactive. In this case, the optimal solution is $q_{i,{ EP}^*} |\mu _i =\mu _i +\sigma _i \Phi ^{-1}\left( {\frac{r-c}{r-v}} \right)$ and the retailer is just an expected profit maximizer (i.e., risk neutral). If $R_i (q_{i,{ EP}^*} )|\mu _i >k\Leftrightarrow k<(r-v)^{2}\xi \left( {\Phi ^{-1}\left( {\frac{r-c}{r-v}} \right) } \right)$ holds, the constraint $R_i (q)|\mu _i \le k$ will become active and binding. In this case, the optimal solution is less than $q_{i,{ EP}^*} |\mu _i$ and the retailer is risk averse. $\square$

Proof of Proposition 5.1

As we discussed in the proof of Lemma 5.1, since ${ EP}_i (q)|\mu _i$ is concave, increasing in the region between 0 and $q_{i,{ EP}^*} |\mu _i =\mu _i +\sigma _i \Phi ^{-1}\left( {\frac{r-c}{r-v}} \right)$, and $\textit{VP}_i (q)|\mu _i$ is an increasing function, when the retailer is risk averse with the level of risk tolerance of k:

(a) The optimal ordering quantity at Stage i is given by:

$$\begin{aligned} q_{i^*} |\mu _i = \mathop {\arg }\limits _q [R_i (q)|\mu _i =k] \Rightarrow \mu _i +\sigma _i \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) . \end{aligned}$$

Thus, we have: $q_{i^*} |\mu _i =\hat{{q}}_i (k)|\mu _i$.

(b) This part is proven by observing the necessary and sufficient condition for the risk averse behavior of the retailer as shown in Lemma 5.1. $\square$

Proof of Lemma 5.2

(a) By definition, the inventory service level with a quantity q is given by: $\Phi \left( {\frac{q-\mu _i }{\sigma _i }} \right)$.

(b) The inventory service level (achieved with the retailer’s optimal ordering quantity) under SR =$\Phi \left( {\frac{q_{0^*} |\mu _0 -\mu _0 }{\sigma _0 }} \right) =\Phi \left[ {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right]$. The inventory service level (achieved with the retailer’s optimal ordering quantity under QR) =$\Phi \left( {\frac{q_{1^*} |\mu _1 -\mu _1 }{\sigma _1 }} \right) =\Phi \left[ {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right]$. Since from Proposition 5.1, we have: $q_{0^*} |\mu _0 <q_{0,{ EP}^*} |\mu _0$ and $q_{1^*} |\mu _1 <q_{1,{ EP}^*} |\mu _1$, thus: $q_{i,{ EP}^*} |\mu _i =\mu _i +\sigma _i \Phi ^{-1}\left( {\frac{r-c}{r-v}} \right)$ and $q_{i^*} |\mu _i =\mu _i +\sigma _i \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right)$ yields: $\Phi \left[ {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right] <\left( {\frac{r-c}{r-v}} \right)$.

(c) The inventory service level is $\Phi \left[ {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right]$. As $\Phi (\cdot )$ and $\xi ^{-1}(\cdot )$ are both increasing functions, we know $\Phi \left[ {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right]$ is also increasing in k. $\square$

Proof of Proposition 6.1

From (6.2), we have: ${ EVQR}^{(R)}=(\sigma _0 -\sigma _1 )\left[ (r-v)\right. \left. \Psi \left( {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) +(c-v)_1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) \right]$.

As we consider the case when the inventory service level is larger than 0.5, we know that $\Psi \left( {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right)$ and $\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right)$ must be positive. Thus, ${ EVQR}^{(R)}$ is positive.

Denote $\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right)$ by $\beta$. Taking partial derivative with respect to $\beta$, we have:

$$\begin{aligned} \frac{\partial { EVQR}^{(R)}}{\partial \beta }= (c-v)(\sigma _0 -\sigma _1 )+(r-v)(\sigma _0 -\sigma _1 )\frac{\partial \Psi (\beta )}{\partial \beta }. \end{aligned}$$

As $\frac{\partial \Psi (\beta )}{\partial \beta }=-(1-\Phi (\beta ))$, we can simplify $\frac{\partial { EVQR}^{(R)}}{\partial \beta }$ to be the following:

$$\begin{aligned} \frac{\partial { EVQR}^{(R)}}{\partial \beta }= (\sigma _0 -\sigma _1 )[c-v-(r-v)+(r-v)\Phi (\beta )]. \end{aligned}$$

Since from Lemma 5.2(b), we have $\Phi \left[ {\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right] <\left( {\frac{r-c}{r-v}} \right)$ which implies $\beta =\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) <\Phi ^{-1}\left( {\frac{r-c}{r-v}} \right)$. Thus, we have:

$$\begin{aligned}&\frac{\partial { EVQR}^{(R)}}{\partial \beta }<(\sigma _0 -\sigma _1 )\left[ c-v-(r-v)+(r-v)\Phi \left( {\Phi ^{-1}\left( {\frac{r-c}{r-v}} \right) } \right) \right] \\&\Rightarrow \frac{\partial { EVQR}^{(R)}}{\partial \beta }<0. \end{aligned}$$

As $\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right)$ is increasing in k, we have: $\frac{\partial { EVQR}^{(R)}}{\partial k}=\left( {\frac{\partial { EVQR}^{(R)}}{\partial \beta }} \right) \left( {\frac{\partial \beta }{\partial k}} \right) <0$.

Thus, ${ EVQR}^{(R)}$ is a decreasing function of k. Plus, from the analytical expression of ${ EVQR}^{(R)}$, it is crystal clear that it is an increasing function of $(\sigma _0 -\sigma _1 )$. $\square$

Proof of Proposition 6.2

(a)(b) Similar to the proof of Proposition 6.1, and by direct observation.

(c) Checking the expressions of ${ EVQR}^{(R)}$ and $\left| {{ EVQR}^{(M)}} \right|$ reveals that ${ EVQR}^{(R)}-\left| {{ EVQR}^{(M)}} \right|$ is always positive. $\square$

Proof of Corollary 7.1

(a) Pareto improvement for QR implementation can be attained if the retailer offers a credit transfer L as low as $\hat{{L}}=(c-m)(\sigma _0 -\sigma _1 )\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right)$ because this value is equal to the expected loss suffered by the manufacturer after implementing QR, i.e. $\left| {{ EVQR}^{(M)}} \right|$, and it is lower than ${ EVQR}^{(R)}$. On the other hand, in order to ensure the retailer is not worse off after adopting QR, the credit transfer is bounded above by $\overline{L} =(\sigma _0 -\sigma _1 )J(k)$ which is equal to the ${ EVQR}^{(R)}$. Therefore, Pareto improvement for QR implementation can be attained if the retailer offers a credit transfer L which is bounded in the range of $[\hat{{L}},\;\overline{L} ]$, i.e. $\hat{{L}}\le L\le \overline{L}$, under the two-part tariff contract to the manufacturer. (b) From $\hat{{L}}=(c-m)(\sigma _0 -\sigma _1 )\xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right)$, it is obvious that the minimum required credit transfer $\hat{{L}}$ is an increasing function of k. $\square$

Proof of Corollary 7.2

(a) Under the wholesale pricing contract, Pareto improvement for QR implementation can be attained if the retailer offers a wholesale price $\hat{{c}}$ under QR such that the manufacturer’s expected loss is fully compensated:

$$\begin{aligned}&(\hat{{c}}-m)\left( {\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) =(c-m)\left( {\mu _0 +\sigma _0 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) \\&\quad \Leftrightarrow \hat{{c}}=m+(c-m)G(k). \end{aligned}$$

Notice that $\hat{{c}}=m+(c-m)G(k)$ is the minimum required wholesale price to achieve Pareto improvement. On the other hand, in order to ensure the retailer does not get worse off after exercising QR, we denote the maximum extra unit wholesale price that the retailer is willing to grant to the manufacturer under QR to be $\bar{{w}}$, and hence we have:

$$\begin{aligned}&\bar{{w}}\left( {\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) = { EVQR}^{(R)}\\&\Leftrightarrow \bar{{w}}= \frac{(\sigma _0 -\sigma _1 )J(k)}{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }. \end{aligned}$$

Thus, the maximum unit wholesale price that the retailer can grant to the manufacturer is $\bar{{c}}=c+\bar{{w}}$ which is: $\bar{{c}}=c+\frac{(\sigma _0 -\sigma _1 )J(k)}{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }$.

(b) Checking the first order partial derivative reveals that $\hat{{c}}$ is an increasing function of k.

$\square$

Proof of Corollary 7.3

(a) Under the manufacturing cost sponsorship scheme, suppose that the minimum new manufacturing cost under the sponsorship scheme (with QR) is $\hat{{m}}$. Pareto improvement for QR implementation can be attained if the retailer offers $\hat{{m}}$ under QR such that:

$$\begin{aligned}&(c-\hat{{m}})\left( {\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) =(c-m)\left( {\mu _0 +\sigma _0 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) \nonumber \\&\Leftrightarrow c-\hat{{m}}=\left( {\frac{\mu _0 +\sigma _0 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }} \right) (c-m). \end{aligned}$$

(8.1)

Since $\left( {\frac{\mu _0 +\sigma _0 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }} \right) >1$, we define:

$$\begin{aligned} \left( {\frac{\mu _0 +\sigma _0 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }} \right) =1+\varepsilon ,\hbox { where } \varepsilon >0. \end{aligned}$$

(8.2)

Substituting (8.2) into (8.1) gives:

$$\begin{aligned}&(c-\hat{{m}})\left( {\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) =(c-m)\left( {\mu _0 +\sigma _0 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) \\&\Leftrightarrow c-\hat{{m}}=(1+\varepsilon )(c-m)\\&\Leftrightarrow \hat{{m}}=m-\varepsilon (c-m). \end{aligned}$$

Thus, $\hat{{m}}<m$. In addition, $m-\hat{{m}}=\varepsilon (c-m)$, which implies that the minimum amount of needed sponsor is equal to $\varepsilon (c-m)$, i.e. $\hat{{\Omega }}=\frac{\hat{{L}}}{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }$.

In order to ensure the retailer is not worse off after granting the manufacturing cost sponsorship, we denote the maximum manufacturing cost sponsorship as $\bar{{\Omega }}$, and we have:

$$\begin{aligned}&\bar{{\Omega }}\left( {\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) } \right) = { EVQR}^{(R)}\\&\Leftrightarrow \bar{{\Omega }}=\frac{(\sigma _0 -\sigma _1 )J(k)}{\mu _0 +\sigma _1 \xi ^{-1}\left( {\frac{k}{(r-v)^{2}}} \right) }. \end{aligned}$$

(b) Checking the first order partial derivative reveals that $\hat{{\Omega }}$ is an increasing function of k.

$\square$