﻿ M-CVaR准则下具有风险偏好的双渠道供应链决策分析 Decision Analysis of Dual-Channel Supply Chain with Risk Preference under M-CVaR Criterion

Modeling and Simulation
Vol. 11  No. 03 ( 2022 ), Article ID: 51530 , 16 pages
10.12677/MOS.2022.113073

M-CVaR准则下具有风险偏好的双渠道供应链决策分析

M-CVaR准则，双渠道供应链，Nash博弈，风险偏好

Decision Analysis of Dual-Channel Supply Chain with Risk Preference under M-CVaR Criterion

Futian Wu, Yazheng Dang*, Niman Bai

School of Management, University of Shanghai for Science and Technology, Shanghai

Received: Apr. 17th, 2022; accepted: May 16th, 2022; published: May 23rd, 2022

ABSTRACT

This paper aims at the two-level supply chain system of dual-channel sales in which the supplier is risk-neutral and the retailer has risk preference. Based on the conditional value at risk (CVaR) criterion, this paper creatively puts forward the use of mean conditional value at risk (m-CVaR) to measure risk preference, integrates risk aversion and pessimism coefficient, and establishes a dual-channel sales supply chain model based on m-CVaR decision criterion. Finally, the influence of retailers’ different risk preference on the optimal decision-making of supply chain is discussed, and the feasibility of the model is verified by numerical analysis. The research shows that: 1) m-CVaR decision criterion can more completely measure the risk preference of decision-makers than conditional value at risk criterion; 2) The dual-channel supply chain with risk preference can find the optimal decision through the mean conditional value at risk decision criterion.

Keywords:M-CVaR Criterion, Dual-Channel Supply Chain, Nash Game, Risk Preference

1. 引言

2. 问题描述及假设

${D}_{r}=\left(1-\theta \right)a-{\beta }_{2}{p}_{r}+\gamma {p}_{m}+{\epsilon }_{r}$ (1)

${D}_{m}=\theta a-{\beta }_{1}{p}_{m}+\gamma {p}_{r}+{\epsilon }_{m}$ (2)

1)供应商和零售商都是各自最求自己最大的利润为目标进行决策；

2)不考虑零售商的销售成本和库存成本、供应商的销售成本和库存成本；

3) 供应商为风险中性，零售商具有风险偏好，风险偏好影响决策行为。

${\pi }_{r}={p}_{r}\mathrm{min}\left\{{y}_{r},{D}_{r}\right\}+s\cdot {\left[{y}_{r}-{D}_{r}\right]}^{+}-w\cdot {y}_{r}=\left({p}_{r}-w\right){y}_{r}-\left({p}_{r}-s\right){\left({y}_{r}-{D}_{r}\right)}^{+}$ (3)

$\begin{array}{c}{\pi }_{m}={p}_{m}\mathrm{min}\left\{{y}_{m},{D}_{m}\right\}+s\cdot {\left[{y}_{m}-{D}_{m}\right]}^{+}+w\cdot {y}_{r}-\left({y}_{r}+{y}_{m}\right)\cdot c\\ =\left(w-c\right){y}_{r}+\left({p}_{m}-c\right){y}_{m}-\left({p}_{m}-s\right){\left({y}_{m}-{D}_{m}\right)}^{+}\end{array}$ (4)

$E\left({\pi }_{m}\right)=\left(w-c\right){y}_{r}+{p}_{m}-c{y}_{m}-\left({p}_{m}-s\right){\int }_{L}^{{y}_{m}-{d}_{m}}F\left(x\right)\text{d}x$ (5)

CVaR度量了低于 $\alpha \left(0<\alpha <1\right)$ 分位数的收益平均值，而忽略了其收益会超出的那一部分，所以CVaR只能度量风险厌恶或者中性的情况。

${\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)=E\left[{\pi }_{r}\left({p}_{r},{y}_{r}\right)|{\pi }_{r}\left({p}_{r},{y}_{r}\right)\le {\beta }_{\alpha }\right]$ (6)

$\alpha \left(0<\alpha <1\right)$ 表示零售商的风险厌恶度， $\alpha$ 越小表示零售商对风险越厌恶，当 $\alpha =1$ 时零售商为风险中性。 ${\beta }_{\alpha }=\mathrm{sup}\left[v|p\left[{\pi }_{r}\left({p}_{r},{y}_{r}\right)\le v\right]\le \alpha \right]$ 为分位数，令

$g\left({p}_{r},{y}_{r},v\right)=v-\frac{1}{\alpha }E{\left[v-{\pi }_{r}\left({p}_{r},{y}_{r}\right)\right]}^{+}$ (7)

${\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)=\mathrm{max}{\pi }_{r}\left({p}_{r},{y}_{r},v\right)$ (8)

3. 模型建立及求解分析

3.1. 供应商的反应函数

$\frac{\partial E\left({\pi }_{m}\right)}{\partial {p}_{m}}={y}_{m}-{\int }_{L}^{{y}_{m}-{d}_{m}}F\left(x\right)\text{d}x-{\beta }_{1}\left({p}_{m}-s\right)F\left({y}_{m}-{d}_{m}\right)=0$ (9)

$\frac{\partial E\left({\pi }_{m}\right)}{\partial {y}_{m}}=\left({p}_{m}-c\right)-\left({p}_{m}-s\right)F\left({y}_{m}-{d}_{m}\right)=0$ (10)

$E\left[{\pi }_{m}\left({p}_{m},{y}_{m}\right)\right]$ 关于 $\left({p}_{m},{y}_{m}\right)$ 的海瑟矩阵为：

$H=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$

$A=\frac{{\partial }^{2}E\left({\pi }_{m}\right)}{\partial {p}_{m}^{2}}=-2{\beta }_{1}F\left({y}_{m}-{d}_{m}\right)-{\beta }_{1}^{2}\left({p}_{m}-s\right)f\left({y}_{m}-{d}_{m}\right)$

$B=\frac{{\partial }^{2}E\left({\pi }_{m}\right)}{\partial {p}_{m}\partial {y}_{m}}=1-F\left({y}_{m}-{d}_{m}\right)-{\beta }_{1}\left({p}_{m}-s\right)f\left({y}_{m}-{d}_{m}\right)$

$C=\frac{{\partial }^{2}E\left({\pi }_{m}\right)}{\partial {y}_{m}{p}_{m}}=1-F\left({y}_{m}-{d}_{m}\right)-{\beta }_{1}\left({p}_{m}-s\right)f\left({y}_{m}-{d}_{m}\right)$

$D=\frac{{\partial }^{2}E\left({\pi }_{m}\right)}{\partial {y}_{m}^{2}}=-\left({p}_{m}-s\right)f\left({y}_{m}-{d}_{m}\right)$

${F}^{-1}\left(\frac{{p}_{m}^{*}-c}{{p}_{m}^{*}-s}\right)+{d}_{m}\left({p}_{r},{p}_{m}^{*}\right)-{\beta }_{1}\left({p}_{m}^{*}-s\right)-{\int }_{L}^{{F}^{-1}\left(\frac{{p}_{m}^{*}-c}{{p}_{m}^{*}-s}\right)}F\left(x\right)\text{d}x=0$ (11)

${y}_{m}^{*}={F}^{-1}\left(\frac{{p}_{m}-c}{{p}_{m}-s}\right)+{d}_{m}\left({p}_{r},{p}_{m}^{*}\right)$ (12)

$\frac{\partial {p}_{m}^{*}}{\partial {p}_{r}}=\frac{\gamma }{2{\beta }_{1}-\frac{{\left(c-s\right)}^{2}}{{\left({p}_{m}-s\right)}^{3}f\left({F}^{-1}\left(\frac{{p}_{m}-c}{{p}_{m}-s}\right)\right)}}$ (13)

$2{\beta }_{1}-\frac{{\left(c-s\right)}^{2}}{{\left({p}_{m}-s\right)}^{3}f\left({F}^{-1}\left(\frac{{p}_{m}-c}{{p}_{m}-s}\right)\right)}>\frac{\left(c-s\right)\left[2{\left({p}_{m}-s\right)}^{2}-{\left(c-s\right)}^{2}\right]}{{\left({p}_{m}-s\right)}^{3}\left(c-s\right)f\left({F}^{-1}\left(\frac{{p}_{m}-c}{{p}_{m}-s}\right)\right)}>0$

(12)式两边对 ${p}_{r}$ 求偏导，可得：

$\frac{\partial {y}_{m}^{*}}{\partial {p}_{r}}=\left({\beta }_{2}+\frac{\left({p}_{m}-c\right)\left(c-s\right)}{{\left({p}_{m}-s\right)}^{3}f\left({F}^{-1}\left(\frac{{p}_{m}-c}{{p}_{m}-s}\right)\right)}\right)\frac{\partial {p}_{m}^{*}}{\partial {p}_{r}}>0$ (14)

3.2. 零售商的反应函数

$\begin{array}{c}g\left({p}_{r},{y}_{r},v\right)=v-\frac{1}{\alpha }E{\left[v-{\pi }_{r}\left({p}_{r},{y}_{r}\right)\right]}^{+}\\ =v-\frac{1}{\alpha }{\int }_{L}^{{y}_{r}-{d}_{r}}{\left[v-{t}_{1}-\left({p}_{r}-s\right)x\right]}^{+}f\left(x\right)\text{d}x-\frac{1}{\alpha }{\int }_{{y}_{r}-{d}_{r}}^{U}{\left(v-{t}_{2}\right)}^{+}f\left(x\right)\text{d}x\end{array}$

1) 当 $v\le {t}_{1}$ 时， $g\left({p}_{r},{y}_{r},v\right)=v$，故 $g\left({p}_{r},{y}_{r},v\right)$ 是关于v的单调递增函数；

2) ${t}_{1} 时，

$g\left({p}_{r},{y}_{r},v\right)=v-\frac{1}{\alpha }{\int }_{L}^{\frac{v-{t}_{1}}{{p}_{r}-s}}{\left[v-{t}_{1}-\left({p}_{r}-s\right)x\right]}^{+}f\left(x\right)\text{d}x$

$\frac{\partial g\left({p}_{r},{y}_{r},v\right)}{\partial v}=1-\frac{1}{\alpha }\left[F\left(\frac{v-{t}_{1}}{{p}_{r}-s}\right)-F\left(L\right)\right]-{\int }_{L}^{\frac{v-{t}_{1}}{{p}_{r}-s}}{\left[v-{t}_{1}-\left({p}_{r}-s\right)x\right]}^{+}f\left(x\right)\text{d}x$

$\frac{{\partial }^{2}g\left({p}_{r},{y}_{r},v\right)}{\partial {v}^{2}}=-\frac{1}{{p}_{r}-s}\frac{1}{\alpha }f\left(\frac{v-{t}_{1}}{{p}_{r}-s}\right)<0$，故 $g\left({p}_{r},{y}_{r},v\right)$ 是关于v的凹函数

3) 当 $v>{t}_{2}$ 时，

$g\left({p}_{r},{y}_{r},v\right)=v-\frac{1}{\alpha }{\int }_{L}^{{y}_{r}-{d}_{r}}\left[v-{t}_{1}-\left({p}_{r}-s\right)x\right]f\left(x\right)\text{d}x-\frac{1}{\alpha }{\int }_{{y}_{r}-{d}_{r}}^{U}\left(v-{t}_{2}\right)f\left(x\right)\text{d}x$

$\frac{\partial g\left({p}_{r},{y}_{r},v\right)}{\partial v}<0$，所以 $g\left({p}_{r},{y}_{r},v\right)$ 是关于v的单调递减函数。

${v}^{*}=\left\{\begin{array}{l}{t}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }v<{t}_{1}+\left({F}^{-1}\left(\alpha \right)+L\right)\left({p}_{r}-s\right).\\ {t}_{1}+\left({F}^{-1}\left(\alpha \right)+L\right)\left({p}_{r}-s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}v\ge {t}_{1}+\left({F}^{-1}\left(\alpha \right)+L\right)\left({p}_{r}-s\right).\end{array}$ (15)

${\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)={v}^{*}-\frac{1}{\alpha }{\int }_{L}^{\frac{{v}^{*}-{t}_{1}}{{p}_{r}-s}}\left[{v}^{*}-{t}_{1}-\left({p}_{r}-s\right)x\right]f\left(x\right)\text{d}x$

$v<{t}_{1}+\left({F}^{-1}\left(\alpha \right)+L\right)\left({p}_{r}-s\right)$ 时，

$\begin{array}{c}{\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)={t}_{2}-\frac{1}{\alpha }{\int }_{L}^{\frac{{t}_{2}-{t}_{1}}{{p}_{r}-s}}\left[{t}_{2}-{t}_{1}-\left({p}_{r}-s\right)x\right]f\left(x\right)\text{d}x\\ =\left({p}_{r}-w\right){y}_{r}-\frac{1}{\alpha }\left({p}_{r}-s\right){\int }_{L}^{{y}_{r}-{d}_{r}}F\left(x\right)\text{d}x\end{array}$ (16)

$H=\left(\begin{array}{cc}{A}^{\prime }& {B}^{\prime }\\ {C}^{\prime }& {D}^{\prime }\end{array}\right)$

${F}^{-1}\left(\frac{\alpha \left({p}_{r}^{*}-w\right)}{{p}_{r}^{*}-s}\right)+{d}_{r}\left({p}_{m},{p}_{r}^{*}\right)-{\beta }_{2}\left({p}_{r}^{*}-w\right)-\frac{1}{\alpha }{\int }_{L}^{{F}^{-1}\left(\frac{\alpha \left({p}_{r}^{*}-w\right)}{{p}_{r}^{*}-s}\right)}F\left(x\right)\text{d}x=0$ (17)

${y}_{r}^{*}={F}^{-1}\left(\frac{\alpha \left({p}_{r}^{*}-w\right)}{{p}_{r}^{*}-s}\right)+{d}_{r}\left({p}_{m},{p}_{r}^{*}\right)$ (18)

3.3. 模型最优解及分析

$\left\{\begin{array}{l}\mathrm{max}{\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right),\\ \mathrm{max}E\left[{\pi }_{m}\left({p}_{m},{y}_{m}\right)\right],\end{array}$ (19)

$X\frac{\partial {p}_{m}^{*}}{\partial \alpha }-\gamma \frac{\partial {p}_{r}^{*}}{\partial \alpha }=0$ (20)

$-\gamma \frac{\partial {p}_{m}^{*}}{\partial \alpha }+Y\frac{\partial {p}_{r}^{*}}{\partial \alpha }-Z=0$ (21)

$Z=\frac{1}{{\alpha }^{2}}{\int }_{L}^{{F}^{-1}\left(\frac{\alpha \left({p}_{r}^{*}-w\right)}{{p}_{r}^{*}-s}\right)}F\left(x\right)\text{d}x$

(12)式两边对 $\alpha$ 求偏导，结合(11)式可得 $\frac{\partial {y}_{m}^{**}}{\partial \alpha }>0$ ；(18)式两边对 $\alpha$ 求偏导，结合(17)式可得 $\frac{\partial {y}_{r}^{**}}{\partial \alpha }>0$。证毕。

(11)，(12)式和(17)，(18)式，分别对参数 ${\beta }_{1},{\beta }_{2}$$\gamma$ 求偏导，得到参数 ${\beta }_{1},{\beta }_{2}$$\gamma$ 对最优直销价，最优零售价，最优直销生产量，最优零售订购量的影响，可得性质4

$\frac{\partial {p}_{m}^{**}}{\partial \gamma }>0$$\frac{\partial {p}_{r}^{**}}{\partial \gamma }>0$$\frac{\partial {y}_{m}^{**}}{\partial \gamma }>0$$\frac{\partial {y}_{r}^{**}}{\partial \gamma }>0$

$\frac{\partial {p}_{m}^{**}}{\partial {\beta }_{1}}<0$$\frac{\partial {p}_{r}^{**}}{\partial {\beta }_{1}}<0$$\frac{\partial {y}_{m}^{**}}{\partial {\beta }_{1}}<0$$\frac{\partial {y}_{r}^{**}}{\partial {\beta }_{1}}<0$

$\frac{\partial {p}_{m}^{**}}{\partial {\beta }_{2}}<0$$\frac{\partial {p}_{r}^{**}}{\partial {\beta }_{2}}<0$$\frac{\partial {y}_{m}^{**}}{\partial {\beta }_{2}}<0$$\frac{\partial {y}_{r}^{**}}{\partial {\beta }_{2}}<0$

4. 零售商基于M-CVaR度量风险偏好的模型

4.1. M-CVaR基础模型

$\text{M-CVaR}=\mathrm{max}\left[\frac{\lambda -\alpha }{1-\alpha }\text{CVaR}\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)+\frac{1-\lambda }{1-\alpha }E\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)\right]$ (22)

4.2. M-CVaR度量风险偏好的模型

$E\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)={p}_{r}-w{y}_{r}-\left({p}_{r}-s\right){\int }_{L}^{{y}_{r}-{d}_{r}}F\left(x\right)\text{d}x$

$\begin{array}{c}{\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)=\frac{1}{\alpha }{\int }_{0}^{\alpha }{F}_{y}^{*}\left(u\right)\text{d}u\\ =\frac{1}{\alpha }{\int }_{L}^{{F}^{-1}\left(\alpha \right)}{F}_{y}^{-1}\left(F\left(x\right)\right)\text{d}F\left(x\right)\\ =\frac{1}{\alpha }{\int }_{L}^{{F}^{-1}\left(\alpha \right)}{\pi }_{r}\left({p}_{r},{y}_{r}\right)\text{d}F\left(x\right)\\ =\left(s-w\right){y}_{r}-\frac{1}{\alpha }\left(s-w\right){y}_{r}F\left(L\right)+\frac{1}{\alpha }\left({p}_{r}-s\right){\int }_{L}^{{F}^{-1}\left(\alpha \right)}{D}_{r}\text{d}F\left( x \right)\end{array}$

${y}_{r}-{d}_{r}<{F}^{-1}\left(\alpha \right)$ 时，

$\begin{array}{c}{\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)=\frac{1}{\alpha }{\int }_{L}^{{F}^{-1}\left(\alpha \right)}{\pi }_{r}\left({p}_{r},{y}_{r}\right)\text{d}F\left(x\right)\\ =\frac{1}{\alpha }\left[\left({p}_{r}-s\right){\int }_{L}^{{y}_{r}-{d}_{r}}{D}_{r}\text{d}F\left(x\right)+{y}_{r}\left[s\left[F\left({y}_{r}-{d}_{r}\right)-F\left(L\right)\right]\\ {}^{{}_{}{}^{}}+{p}_{r}\left[\alpha -F\left({y}_{r}-{d}_{r}\right)\right]+w\left[F\left(L\right)-\alpha \right]\right]\right]\end{array}$

$\begin{array}{l}{\text{CVaR}}_{\alpha }\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)\\ =\left\{\begin{array}{l}\left(s-w\right){y}_{r}-\frac{1}{\alpha }\left(s-w\right){y}_{r}F\left(L\right)+\frac{1}{\alpha }\left({p}_{r}-s\right){\int }_{L}^{{F}^{-1}\left(\alpha \right)}{D}_{r}\text{d}F\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{r}-{d}_{r}>{F}^{-1}\left(\alpha \right)\\ \frac{1}{\alpha }\left[\left({p}_{r}-s\right){\int }_{L}^{{y}_{r}-{d}_{r}}{D}_{r}\text{d}F\left(x\right)+\left(s-w\right){y}_{r}\left[F\left({y}_{r}-{d}_{r}\right)-F\left(L\right)\right]\\ {}^{{}^{}}+\left({p}_{r}-w\right){y}_{r}\left[\alpha -F\left({y}_{r}-{d}_{r}\right)\right]\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{r}-{d}_{r}<{F}^{-1}\left( \alpha \right)\end{array}\end{array}$

${y}_{r}-{d}_{r}<{F}^{-1}\left(\alpha \right)$，此时 $\frac{{\partial }^{2}{\text{CVaR}}_{\alpha }{\pi }_{r}}{\partial {y}_{r}^{2}}=sf\left({y}_{r}-{d}_{r}\right)>0$，不为凹函数，故最优解不在此区间。

$\begin{array}{l}{H}_{r}=\frac{\lambda -\alpha }{1-\alpha }\text{CVaR}\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)+\frac{1-\lambda }{1-\alpha }E\left({\pi }_{r}\left({p}_{r},{y}_{r}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1-\lambda }{1-\alpha }\left[{p}_{r}-w{y}_{r}-\left({p}_{r}-s\right){\int }_{L}^{{y}_{r}-{d}_{r}}F\left(x\right)\text{d}x\right]+\frac{\lambda -\alpha }{1-\alpha }\left[\left(s-w\right){y}_{r}-\frac{1}{\alpha }\left(s-w\right){y}_{r}F\left(L\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{1}{\alpha }\left({p}_{r}-s\right){\int }_{L}^{{F}^{-1}\left(\alpha \right)}{D}_{r}\text{d}F\left(x\right)\right],\text{\hspace{0.17em}}{y}_{r}-{d}_{r}>{F}^{-1}\left(\alpha \right)\end{array}$ (23)

$H=\left(\begin{array}{cc}J& K\\ I& M\end{array}\right)$

$h\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}>\frac{1}{{\beta }_{2}\left(c-s\right)}$ 时，存在均衡零售价 ${p}_{r}^{*}$ 和零售商订购量 ${y}_{r}^{*}$ 满足以

${y}_{r}^{*}={F}^{-1}\left(\frac{{p}_{r}^{*}-w}{{p}_{r}^{*}-s}+\frac{\left(\alpha -\lambda \right)\left(w-s\right)\left[\alpha -F\left(L\right)\right]}{\alpha \left(1-\lambda \right)\left({p}_{r}^{*}-s\right)}\right)+{d}_{r}\left({p}_{m},{p}_{r}^{*}\right)$ (24)

$\begin{array}{l}\frac{1-\lambda }{1-\alpha }\left[{F}^{-1}\left(\frac{{p}_{r}^{*}-w}{{p}_{r}^{*}-s}+\frac{\left(\alpha -\lambda \right)\left(w-s\right)\left[\alpha -F\left(L\right)\right]}{\alpha \left(1-\lambda \right)\left({p}_{r}^{*}-s\right)}\right)+{d}_{r}\left({p}_{m},{p}_{r}^{*}\right)\\ -{\beta }_{2}\left({p}_{r}^{*}-w+\frac{\left(\alpha -\lambda \right)\left(w-s\right)\left[\alpha -F\left(L\right)\right]}{\alpha \left(1-\lambda \right)}\right)-{\int }_{L}^{{F}^{-1}\left(\frac{{p}_{r}^{*}-w}{{p}_{r}^{*}-s}+\frac{\left(\alpha -\lambda \right)\left(w-s\right)\left[\alpha -F\left(L\right)\right]}{\alpha \left(1-\lambda \right)\left({p}_{r}^{*}-s\right)}\right)}F\left(x\right)\text{d}x\right]\\ +\frac{\lambda -\alpha }{1-\alpha }\frac{1}{\alpha }\left[{\int }_{L}^{{F}^{-1}\left(\alpha \right)}{D}_{r}f\left(x\right)\text{d}x-{\beta }_{2}\left({p}_{r}^{*}-s\right)\left(\alpha -F\left(L\right)\right)\right]=0\end{array}$ (25)

5. 数值分析

5.1. CVaR模型分析

Figure 1. The impact of risk preference on optimal pricing

Figure 2. The effect of risk preference on order quantity

5.2. M-CVaR模型分析

Figure 3. The effect of risk preference on supplier expected profit and retailer CVaR

Figure 4. The impact of risk preference and pessimism coefficient on direct selling prices

Figure 5. The impact of risk preference and pessimism coefficient on retail prices

Figure 6. The impact of risk preference and pessimism coefficient on direct selling production

Figure 7. The effect of risk preference and pessimism coefficient on retailer’s order quantity

5.3. CVaR和M-CVaR模型仿真对比

Figure 8. The effect of risk preference and pessimism coefficient on retailers’ M-CVaR value

Figure 9. Impact of risk preference and pessimism coefficient on supplier expected profit

6. 结论及展望

Figure 10. The impact of risk preference on suppliers’ expected profits

Figure 11. The effect of risk preference on retailers’ CVaR and M-CVaR

Decision Analysis of Dual-Channel Supply Chain with Risk Preference under M-CVaR Criterion[J]. 建模与仿真, 2022, 11(03): 781-796. https://doi.org/10.12677/MOS.2022.113073

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18. NOTES

*通讯作者。