# 需求依赖库存水平且零售商为库存风险厌恶的供应链协调Supply Chain Coordination with Inventory Level Dependent Demand and Inventory Risk Averse Retailer

• 全文下载: PDF(1298KB)    PP.80-92   DOI: 10.12677/ORF.2019.91010
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The coordination issue of supply chain composed of a supplier and a retailer is discussed, in which the demand of the product is stochastic and affected by the inventory level, that is, when the in-ventory level is high, the demand increases, and when the inventory level is low, the demand de-creases. Taking the probability of surplus products exceeding the predetermined valve as the re-tailer’s inventory risk measurement, the retailer’s ordering strategy is analyzed and the way to achieve supply chain coordination by the rebate and penalty contract is explored. The results show that the rebate and penalty contract can coordinate the supply chain with certain conditions. Numerical analysis shows that with the increase of demand elasticity parameter, the optimal order quantity and the supply chain overall profit increase. However, with the increase of inventory risk averse, the optimal order quantity may decrease, then the supply chain overall profit decreases and the supply chain fails to coordinate.

1. 引言

2. 问题描述

$D\left(q,x\right)=x+\alpha \left(q-x-u\right)$(1)

$\mathrm{Pr}\left\{q-s\left(q,x\right)\ge v\right\}\le \xi$(2)

$\begin{array}{c}s\left(q,x\right)=\mathrm{min}\left(D\left(q,x\right),q\right)\\ =\left\{\begin{array}{ll}x+\alpha \left(q-x-u\right),\hfill & 0\le x\le q+\theta u;\hfill \\ q,\hfill & x>q+\theta u,\hfill \end{array}\end{array}$ (3)

$\begin{array}{c}S\left(q\right)={\int }_{0}^{q+\theta u}\left[x+\alpha \left(q-x-u\right)\right]f\left(x\right)\text{d}x+{\int }_{q+\theta u}^{+\infty }qf\left(x\right)\text{d}x\\ =q-\left(1-\alpha \right){\int }_{0}^{q+\theta u}F\left(x\right)\text{d}x.\end{array}$

$\begin{array}{c}I\left(q\right)=E{\left(q-s\left(x,q\right)\right)}^{+}\\ \text{=}{\int }_{0}^{q+\theta u}\left[q-x-\alpha \left(q-x-u\right)\right]f\left(x\right)\text{d}x\\ \text{=}\left(\text{1}-\alpha \right){\int }_{0}^{q+\theta u}F\left(x\right)\text{\hspace{0.17em}}\text{d}x.\end{array}$

${q}_{0}={F}^{-1}\left(\xi \right)-\left(\alpha u-v\right)/\left(1-\alpha \right)$(4)

$\begin{array}{c}\mathrm{Pr}\left\{q-s\left(q,x\right)\ge v\right\}=\mathrm{Pr}\left\{q-s\left(q,x\right)\ge v,x\le q+\theta u\right\}\\ +\mathrm{Pr}\left\{q-s\left(q,x\right)\ge v,x>q+\theta u\right\}\\ =\mathrm{Pr}\left\{x\le q+\left(\alpha u-v\right)/\left(1-\alpha \right)\right\}\\ =F\left(q+\left(\alpha u-v\right)/\left(1-\alpha \right)\right).\end{array}$

3. 集中式供应链

$\begin{array}{c}E{\text{π}}_{c}\left(q\right)=pS\left(q\right)+hI\left(q\right)-cq\\ =\left(p-c\right)q-\left(1-\alpha \right)\left(p-h\right){\int }_{0}^{q+\theta u}F\left(x\right)\text{d}x.\end{array}$ (5)

$\left(1-\alpha \right)F\left(\theta u\right)\le \frac{p-c}{p-h}<1-\alpha$(6)

${q}_{c}={F}^{-1}\left(\frac{p-c}{\left(1-\alpha \right)\left(p-h\right)}\right)-\theta u$(7)

$\frac{\text{d}E{\text{π}}_{c}\left(q\right)}{\text{d}q}=\left(p-c\right)-\left(1-\alpha \right)\left(p-h\right)F\left(q+\theta u\right)$

$\frac{{\text{d}}^{2}E{\pi }_{c}\left(q\right)}{\text{d}{q}^{2}}=-\left(1-\alpha \right)\left(p-h\right)f\left(q+\theta u\right)$

$\underset{q\to 0}{\mathrm{lim}}\frac{\text{d}E{\text{π}}_{c}\left(q\right)}{\text{d}q}=\left(p-c\right)-\left(1-\alpha \right)\left(p-h\right)F\left(\theta u\right)\ge 0$

$\underset{q\to \infty }{\mathrm{lim}}\frac{\text{d}E{\text{π}}_{c}\left(q\right)}{\text{d}q}=\left(p-c\right)-\left(1-\alpha \right)\left(p-h\right)<0$

$\alpha =0$ 时，供应链系统的最优订购量为：

${q}_{c}={F}^{-1}\left(\frac{p-c}{p-h}\right)$

4. 分散式供应链

4.1. 批发价契约

$\begin{array}{c}E{\text{π}}_{r}\left(q\right)=pS\left(q\right)+hI\left(q\right)-wq\\ =\left(p-w\right)q-\left(1-\alpha \right)\left(p-h\right){\int }_{0}^{q+\theta u}F\left(x\right)\text{d}x.\end{array}$ (8)

$E{\text{π}}_{s}\left(q\right)=\left(w-c\right)q$(9)

$\begin{array}{ll}\mathrm{max}\hfill & E{\text{π}}_{r}\left(q\right)\hfill \\ s.t.\hfill & 0 (10)

$\left(1-\alpha \right)F\left(\theta u\right)\le \frac{p-w}{p-h}<1-\alpha$(11)

${q}_{r}={F}^{-1}\left(\frac{p-w}{\left(1-\alpha \right)\left(p-h\right)}\right)-\theta u$(12)

$\frac{\text{d}E{\text{π}}_{r}\left(q\right)}{\text{d}q}=\left(p-w\right)-\left(1-\alpha \right)\left(p-h\right)F\left(q+\theta u\right)$

$\frac{{\text{d}}^{2}E{\text{π}}_{r}\left(q\right)}{\text{d}{q}^{2}}=-\left(1-\alpha \right)\left(p-h\right)f\left(q+\theta u\right)$

$\underset{q\to 0}{\mathrm{lim}}\frac{\text{d}E{\text{π}}_{r}\left(q\right)}{\text{d}q}=\left(p-w\right)-\left(1-\alpha \right)\left(p-h\right)F\left(\theta u\right)\ge 0$

$\underset{q\to \infty }{\mathrm{lim}}\frac{\text{d}E{\text{π}}_{r}\left(q\right)}{\text{d}q}=\left(p-w\right)-\left(1-\alpha \right)\left(p-h\right)<0$

4.2. 回馈与惩罚契约

$\begin{array}{c}E{\text{π}}_{r1}\left(q\right)=pS\left(q\right)+hI\left(q\right)+\beta E\left(s\left(q\right)-T\right)-wq\\ \text{=}\left(p-w+\beta \right)q-\left(1-\alpha \right)\left(p-h+\beta \right){\int }_{0}^{q+\theta u}F\left(x\right)\text{d}x-\beta T.\end{array}$ (13)

$\begin{array}{c}E{\text{π}}_{s1}\left(q\right)=\left(w-c\right)q-\beta E\left(s\left(q\right)-T\right)\\ \text{=}\left(w-c-\beta \right)q+\beta \left(1-\alpha \right){\int }_{0}^{q+\theta u}F\left(x\right)\text{d}x+\beta T.\end{array}$ (14)

$\begin{array}{ll}\mathrm{max}\hfill & E{\text{π}}_{r1}\left(q\right)\hfill \\ s.t.\hfill & 0 (15)

$\left(1-\alpha \right)F\left(\theta u\right)\le \frac{p-w+\beta }{p-h+\beta }<1-\alpha$(16)

${q}_{r1}={F}^{-1}\left(\frac{p-w+\beta }{\left(1-\alpha \right)\left(p-h+\beta \right)}\right)-\theta u$(17)

$\frac{\text{d}E{\text{π}}_{r1}\left(q\right)}{\text{d}q}=p-w+\beta -\left(1-\alpha \right)\left(p-h+\beta \right)F\left(q+\theta u\right)$

$\frac{{\text{d}}^{2}E{\text{π}}_{r1}\left(q\right)}{\text{d}{q}^{2}}=-\left(p-h+\beta \right)f\left(q+\theta u\right)$

$\underset{q\to 0}{\mathrm{lim}}\frac{\text{d}E{\text{π}}_{r1}\left(q\right)}{\text{d}q}=p-w+\beta -\left(1-\alpha \right)\left(p-h+\beta \right)F\left(\theta u\right)\ge 0$

$\underset{q\to \infty }{\mathrm{lim}}\frac{\text{d}E{\text{π}}_{r1}\left(q\right)}{\text{d}q}=p-w+\beta -\left(1-\alpha \right)\left(p-h+\beta \right)<0$

$E{\text{π}}_{r1}^{*}=E{\text{π}}_{r1}\left({q}_{c}\right)$$E{\text{π}}_{s1}^{*}=E{\text{π}}_{s1}\left({q}_{c}\right)$$E{\text{π}}_{c}^{*}=E{\text{π}}_{c}\left({q}_{c}\right)$$E{\text{π}}_{r}^{*}=E{\text{π}}_{r}\left({q}_{r}\right)$$E{\text{π}}_{s}^{*}=E{\text{π}}_{s}\left({q}_{r}\right)$。接下来，我们探讨用回馈与惩罚契约来实现供应链协调的途径。

$\left(1-\alpha \right)F\left(\theta u\right)\le \frac{p-c}{p-h}\le \left(1-\alpha \right)F\left({F}^{-1}\left(\zeta \right)+\frac{v}{1-\alpha }\right)$(18)

$\beta =\frac{p-h}{c-h}w-\frac{p-h}{c-h}c$(19)

${T}_{1}\le T\le {T}_{2}$(20)

${T}_{1}=\frac{\left(c-h\right)E{\text{π}}_{s}^{*}+\left(w-c\right)E{\text{π}}_{c}^{*}}{\left(p-h\right)\left(w-c\right)}$

${T}_{2}=\frac{\left(w-h\right)E{\text{π}}_{c}^{*}-\left(c-h\right)E{\text{π}}_{r}^{*}}{\left(p-h\right)\left(w-c\right)}$

$\frac{p-w+\beta }{p-h+\beta }=\frac{p-c}{p-h}$(21)

$\left(1-\alpha \right)F\left(u\theta \right)\le \frac{p-c}{p-h}\le \left(1-\alpha \right)F\left({F}^{-1}\left(\zeta \right)+\frac{v}{1-\alpha }\right)<1-\alpha$(22)

$\left\{\begin{array}{c}E{\text{π}}_{r1}^{*}\ge E{\text{π}}_{r}^{*}\\ E{\text{π}}_{s1}^{*}\ge E{\text{π}}_{s}^{*}\end{array}$

$E{\text{π}}_{r1}^{*}=\frac{w-h}{c-h}E{\text{π}}_{c}^{*}-\frac{\left(p-h\right)\left(w-c\right)}{c-h}T$(23)

$E{\text{π}}_{s1}^{*}=\frac{c-w}{c-h}E{\text{π}}_{c}^{*}+\frac{\left(p-h\right)\left(w-c\right)}{c-h}T$(24)

${T}_{1}\le T\le {T}_{2}$ 时， $E{\text{π}}_{s1}^{*}\ge E{\text{π}}_{s}^{*}$$E{\text{π}}_{r1}^{*}\ge E{\text{π}}_{r}^{*}$。此时，供应链的协调定义的三个条件均得到满足，则供应链实现了协调。

5. 数值例子

5.1. 协调分析

Figure 1. The expected profit curves of the retailer and the supply chain system

Table 1. The impacts of T on E π r 1 * and E π s 1 *

Table 2. The impacts of α on q r 1 * and T

5.2. 参数对供应链的影响

5.2.1. 参数w和β

Figure 2. The impacts of w on the optimal order quantity

Figure 3. The impacts of $w$ on $E{\text{π}}_{r1}$ and $E{\text{π}}_{s1}$

$\xi =0.6$$v=50$$w=6.5$$T=90$。将 $\beta$ 从0到6取值，绘制出最优订货量和供应链成员利润关于 $\beta$ 的曲线，如图4图5所示。

Figure 4. The impacts of β on the optimal order quantity

Figure 5. The impacts of $\beta$ on $E{\text{π}}_{r1}$ and $E{\text{π}}_{s1}$

5.2.2. 参数α

$w=6.5$$\beta =5$$u=60$$v=50$$\xi =0.6$$T=90$。当 $\alpha$ 变化时，结果如表3所示。

Table 3. The impacts of α on the optimal order quantity and the supply chain profit

5.2.3. 参数ξ

$w=6.5$$\beta =5$$\alpha =0.2$$u=60$$T=90$，则 ${q}_{r1}={q}_{c}=160$。当 $\xi$ 变化时，结果如表4所示。

Table 4. The impacts of ξ on the optimal order quantity and the supply chain profit

6. 结语

NOTES

*通讯作者。

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