# 生产规模不经济闭环供应链的应急决策研究Study on Disruption Decision-Making of Closed-Loop Supply Chain with Diseconomies of Production Scale

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The problems of disrupting decision and coordination are researched for closed-loop supply chain with one manufacture exhibiting diseconomies of production scale and one retailer collecting used products under demand disruption. The results show that compared with the equilibrium decision of normal operating environment, when the degree of disturbance is not large, the order quantity/sales volume and the amount of waste products recovered are robust. When the degree of disturbance is large, it should be adjusted according to the disturbance direction, the order/sales volume of the product and the amount of recycling of the waste product; when the degree of reduction is larger, the closed-loop supply chain system will be destroyed.

1. 引言

2. 基本模型

2.1. 模型描述

2.2. 模型假设及符号说明

${\Pi }_{m}\left(w,{q}_{r}\right)=w\left({q}_{n}+{q}_{r}\right)-{p}_{r}{q}_{r}-{c}_{n}{q}_{n}^{2}-{c}_{r}{q}_{r}^{2}$ (1)

${\Pi }_{r}\left({q}_{n}\right)=\left(\frac{\alpha -{q}_{n}-{q}_{r}}{\beta }-w\right)\left({q}_{n}+{q}_{r}\right)$ (2)

3. 正常运营环境下的均衡决策

${q}_{n}\left(w,{q}_{r}\right)$ 代入制造商的利润函数 ${\Pi }_{m}\left(w,{q}_{r}\right)$ 可得： ${\Pi }_{m}\left(w,{q}_{r}\right)=w\left(\frac{\alpha -{q}_{r}-\beta {q}_{r}-\beta w}{1+\beta }+{q}_{r}\right)-{p}_{r}{q}_{r}-{c}_{n}{\left(\frac{\alpha -{q}_{r}-\beta {q}_{r}-\beta w}{1+\beta }\right)}^{2}-{c}_{r}{q}_{r}^{2}$

$A=\frac{\partial {\Pi }_{m}^{2}\left(w,{q}_{r}\right)}{\partial {w}^{2}}=-\frac{2\beta \left(1+\beta +\beta {c}_{n}\right)}{{\left(1+\beta \right)}^{2}}<0$$B=\frac{\partial {\Pi }_{m}^{2}\left(w,{q}_{r}\right)}{\partial w\partial {q}_{r}}=\frac{\partial {\Pi }_{m}^{2}\left(w,{q}_{r}\right)}{\partial {q}_{r}\partial w}=-\frac{2{c}_{n}\beta }{\left(1+\beta \right)}$$C=\frac{\partial {\Pi }_{m}^{2}\left(w,{q}_{r}\right)}{\partial {q}_{r}^{2}}=-\frac{2\left(1+b{c}_{n}+b{c}_{r}\right)}{b}$ ，可知 $A<0$$AC-{B}^{2}=\frac{4\beta \left(1+\beta \right)\left[\left(1+{c}_{n}\right)\left(1+b{c}_{r}\right)+b{c}_{n}\right]}{b{\left(1+\beta \right)}^{2}}>0$ 。故函数 ${\Pi }_{m}\left(w,{q}_{r}\right)$ 的海塞矩阵 $\left[\begin{array}{cc}A& B\\ B& C\end{array}\right]$ 负定，其为严格凹函数，存在唯一最优产品批发价w和废旧品回收率 ${q}_{r}$ ，使制造商获得的利润最大。

$\frac{\partial {\Pi }_{m}\left(w,{q}_{r}\right)}{\partial w}=0$$\frac{\partial {\Pi }_{m}\left(w,{q}_{r}\right)}{\partial {q}_{r}}=0$ ，可得： ${w}^{*}=\frac{\alpha \left[\left(1+b{c}_{r}\right)\left(1+\beta {c}_{n}\right)+b{c}_{n}\right]-\beta a{c}_{n}}{\beta \left[\left(1+b{c}_{r}\right)\left(2+\beta {c}_{n}\right)+2b{c}_{n}\right]}$${q}_{r}^{*}=\frac{\alpha b{c}_{n}+a\left(2+\beta {c}_{n}\right)}{2\left[\left(1+b{c}_{r}\right)\left(2+\beta {c}_{n}\right)+2b{c}_{n}\right]}$

${w}^{*}$${q}_{r}^{*}$ 代入 ${q}_{n}\left(w,{q}_{r}\right)$ 可得： ${q}_{n}^{*}=\frac{\alpha \left(1+b{c}_{r}\right)-2a}{2\left[\left(1+b{c}_{r}\right)\left(2+\beta {c}_{n}\right)+2b{c}_{n}\right]}$

4. 突发事件干扰下的应急均衡决策

${\stackrel{¯}{\Pi }}_{m}\left(w,{q}_{r}\right)=w\left({q}_{n}+{q}_{r}\right)-{p}_{r}{q}_{r}-{c}_{n}{q}_{n}^{2}-{c}_{r}{q}_{r}^{2}-{\lambda }_{1}{\left(q-{q}^{d*}\right)}^{+}-{\lambda }_{2}{\left({q}^{d*}-{q}^{M}\right)}^{+}$ (3)

${\stackrel{¯}{\Pi }}_{r}\left({q}_{n}\right)=\left(\frac{\alpha +\Delta \alpha -{q}_{n}-{q}_{r}}{\beta }-w\right)\left({q}_{n}+{q}_{r}\right)$ (4)

$\begin{array}{c}{\stackrel{¯}{\Pi }}_{m}\left(w,{q}_{r}\right)=w\left(\frac{\alpha +\Delta \alpha -\beta w}{2}+{q}_{r}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{q}_{r}^{2}}{b}+\frac{a}{b}{q}_{r}-{c}_{n}{\left(\frac{\alpha +\Delta \alpha -2{q}_{r}-\beta w}{2}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{c}_{r}{q}_{r}^{2}-{\lambda }_{1}\left(\frac{\alpha +\Delta \alpha -\beta w}{2}+{q}_{r}-{q}^{*}\right)\end{array}$ (5)

$\left\{\begin{array}{l}\frac{\alpha +\Delta \alpha -2{\stackrel{¯}{q}}_{r}-2\beta \stackrel{¯}{w}}{2}+{\stackrel{¯}{q}}_{r}+\beta {c}_{n}\left(\frac{\alpha +\Delta \alpha -\beta \stackrel{¯}{w}-2{\stackrel{¯}{q}}_{r}}{2}\right)+\frac{\beta {\lambda }_{1}}{2}-\frac{\beta {\gamma }_{1}}{2}=0\\ -\frac{2{\stackrel{¯}{q}}_{r}}{b}+\frac{a}{b}+{c}_{n}\left(\alpha +\Delta \alpha -2{\stackrel{¯}{q}}_{r}-\beta \stackrel{¯}{w}\right)-2{c}_{r}{\stackrel{¯}{q}}_{r}+{\gamma }_{2}=0\\ {\gamma }_{1}\left(\frac{\alpha +\Delta \alpha -\beta \stackrel{¯}{w}}{2}-{q}^{d*}\right)=0,{\gamma }_{2}\left({\stackrel{¯}{q}}_{r}-{q}_{r}^{*}\right)=0\\ {\gamma }_{1},{\gamma }_{2}\ge 0,\stackrel{¯}{w}\ge 0,\stackrel{¯}{q}\ge {q}^{*}>0,{\stackrel{¯}{q}}_{r}\ge {q}_{r}^{*}>0\end{array}$ (6)

$\begin{array}{c}{\stackrel{¯}{\Pi }}_{m}\left(w,{q}_{r}\right)=w\left(\frac{\alpha +\Delta \alpha -\beta w}{2}+{q}_{r}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{q}_{r}^{2}}{b}+\frac{a}{b}{q}_{r}-{c}_{n}{\left(\frac{\alpha +\Delta \alpha -2{q}_{r}-\beta w}{2}\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{c}_{r}{q}_{r}^{2}+{\lambda }_{2}\left(\frac{\alpha +\Delta \alpha -\beta w}{2}+{q}_{r}-{q}^{*}\right)\end{array}$ (7)

$\left\{\begin{array}{l}\frac{\alpha +\Delta \alpha -2{\stackrel{¯}{q}}_{r}-2\beta \stackrel{¯}{w}}{2}+{\stackrel{¯}{q}}_{r}+\beta {c}_{n}\left(\frac{\alpha +\Delta \alpha -\beta \stackrel{¯}{w}-2{\stackrel{¯}{q}}_{r}}{2}\right)-\frac{\beta {\lambda }_{4}}{2}+\frac{\beta {\gamma }_{3}}{2}=0\\ -\frac{2{\stackrel{¯}{q}}_{r}}{b}+\frac{a}{b}+{c}_{n}\left(\alpha +\Delta \alpha -2{\stackrel{¯}{q}}_{r}-\beta \stackrel{¯}{w}\right)-2{c}_{r}{\stackrel{¯}{q}}_{r}-{\gamma }_{4}=0\\ {\gamma }_{3}\left({q}^{*}-\frac{\alpha +\Delta \alpha -\beta \stackrel{¯}{w}}{2}\right)=0,{\gamma }_{4}\left({q}_{r}^{*}-{\stackrel{¯}{q}}_{r}\right)=0\\ {\gamma }_{3}、{\gamma }_{4}\ge 0,\stackrel{¯}{w}\ge 0,{q}^{*}\ge \stackrel{¯}{q}>0,{q}_{r}^{*}\ge {\stackrel{¯}{q}}_{r}>0\end{array}$ (8)

1) 当最大市场需求规模的正扰动程度不大，即 $0<\Delta \alpha <\beta {\lambda }_{1}$ 时，制造商应保持废旧品的回收量不变，而应提高产品的单位批发价格，以获取更多的利润；同时，零售商应保持产品的订购/销售量不变。

2) 当最大市场需求规模的负扰动程度不大，即 $-\beta {\lambda }_{2}<\Delta \alpha <0$ 时，制造商应保持废旧品的回收量不变，而应降低产品的单位批发价格，以此来激励零售商保持产品的订购/销售量不变。

3) 当最大市场需求规模的扰动程度较大，即 $\Delta \alpha \ge \beta {\lambda }_{1}$$\Delta \alpha \le -\beta {\lambda }_{2}$ 时，制造商应按扰动的方向调整废旧品的回收量和产品的单位批发价格，零售商应按扰动的方向调整产品的订购/销售量。即在产品的最大市场需求规模变的较大时，提高其价格和销量，以获取更多的利润；在产品的最大市场需求规模变的较小时，降低其价格和销量，以此来拉动需求，避免利润的过多损失。

4) 当最大市场需求规模减少的程度很大，即 $\Delta \alpha <\frac{2a}{\left(1+b{c}_{r}\right)}-\alpha -\beta {\lambda }_{2}$ 时，因产品的订购/销售量小于零，闭环供应链系统会停产。

5. 结论

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