﻿ 供应中断风险下制造商应对策略研究
 上海理工大学学报  2021, Vol. 43 Issue (4): 409-420 PDF

Manufacturer’s coping strategies under supply interruption risk
KONG Jin, LI Fang
Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: In order to study the manufacturer’s coping strategy to supply interruption, the time-sensitive supply chain composed of two manufacturers and two suppliers was studied. Among them, supplier A had the risk of interruption, and manufacturer 1 could choose whether to assist supplier A, and manufacturer 2 can choose to adopt alternate supplier B and selected supplier A and chose whether to take assistance. On this basis, the decision-making model of the strategy c combination of 6 manufacturer 1 and 2 was established. The balanced solution is obtained by the game theory analysis and numerical study analysis is performed. The results show that the optimal coping strategy for the supply interruption of two manufacturers is affected by each other. While one or both of the two manufacturers take assistance measures at the same time, only when the aid strength is greater than the threshold, supply interruption can be effectively recovered. When both manufacturers choose the aid strategy, the recovery time for supply interruption is the shortest.
Key words: supply interruption     purchase strategy     manufacturer     expected profit equation

1 模型与基本假设

 图 1 结构示意图 Fig. 1 Structure diagram

a. 制造商、供应商、备用供应商均处于云环境之中，制造商1、制造商2、供应商A、供应商B信息共享。

b. 考虑制造商1和制造商2均为供应商A的重要客户，因此，供应商A会在完成所有订单后同一时间将货物发送给制造商1和制造商2。

c. 制造商生产的产品具有时间敏感性，即市场需求量同时受产品价格和制造商交货时间影响。制造商1和制造商2的需求量分别为 $D_1({p}_{1},T)= {\alpha }_{1}-$ $\mu {p}_{1}-\beta T$ $D_2({p}_{2},T)={\alpha }_{2}-\mu {p}_{2}-\beta T,\mu >0,\beta >0$ 。其中， $\,\mu$ 为价格敏感系数， $\,\beta$ 为时间敏感系数， ${p_1}$ 为制造商1零售价格， ${p_2}$ 为制造商2零售价格[23]T为产品实际进入市场的时间， $\alpha_1$ $\alpha_2$ 分别为制造商1、制造商2的潜在市场需求量。

d. 本文仅考虑制造商的采购成本，不考虑生产运营成本。

e. 供应商A具有产品价格低、可靠性低的特点，供应商B具有产品价格高、完全可靠的特点。

f. 供应商A发生中断时，实际交货时间延迟造成的损失由供应商A承担。

g. 供应商A与制造商1和制造商2存在长期合作关系，当发生供应中断时供应商A会主动采取措施恢复产能。

h. 考虑制造商2采用备用供应商B时，供应商B的交货时间与供应正常情况下供应商A的约定交货时间相同。

$\,\rho$ 表示主供应商A的供应中断的概率， $0 < \rho < 1$

$r$ 表示商品的残值， $r < {w_1} < {w_2}$

${c_1}$ 表示供应商A的生产成本；

${c_2}$ 表示供应商B的生产成本；

${w_1}$ 表示制造商向供应商A购买产品的批发价格；

${w_2}$ 表示制造商向供应商B购买产品的批发价格；

$c(t)$ 表示供应商A在无援助情况下付出的供应链恢　复成本， $c(t) = {t^2} - 2Ht + F$ H为供应商A允许　的最长交货时间，F为供应商A可能付出的最　大成本[23]

$t$ 表示供应商A实际交货时间；

t0表示供应中断发生时间；

${t_1}$ 表示正常情况下供应商A与制造商1、制造商2　约定的交货时间；

${t_2}$ 表示制造商1和制造商2生产时间，本文不考虑　制造商1和制造商2生产时间差异；

$\textit{π} _{\rm{s}}^{ij}(i = M,{{N}};j = O,M,N)$ 表示制造商1和制造商2不　同策略下，供应商A的期望收益；

$\textit{π} _{{\rm{m1}}}^{ij}(i = M,{{N}};j = O,M,N)$ 表示3种策略下，主要供应　商A的期望收益；

$\textit{π} _{{\rm{m2}}}^{ij}(i = M,{{N}};j = O,M,N)$ 表示3种策略下，备用供应　商B的期望收益；

2 模型的建立与求解

2.1 制造商1采用供应商A供货但不采取援助措施且制造商2采用备用供应商B供货情形（策略1）

 图 2 时间节点示意图1 Fig. 2 Schematic 1 for time node

T1表示无供应中断情形下产品投入市场的时间，T2表示策略1中制造商1生产产品实际投入市场的时间。

 $\begin{split}\pi _{\rm s}^{MO} =& (1 - \rho )\left\{ {({w_1} - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1})]} \right\} + \\ &\rho \{ ({w_1} - {c_1}){D_1}({p_1},{T_2}) + (r - {c_1})[{D_1}({p_1},{T_1})- \\ & {D_1}({p_1},{T_2})] - c(t) \}\\[-10pt]\end{split}$ (1)
 $\begin{split}\pi _{{\rm m}1}^{MO} =& (1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_2})]\end{split}$ (2)
 $\begin{split}\pi _{{\rm m}2}^{MO} =& (1- \rho )[({p_2} - {w_1}){D_2}({p_2},{T_1})] +\\ &\rho [({p_2} - {w_2}){D_2}({p_2},{T_1})]\end{split}$ (3)

 $\left\{ \begin{array}{l} \dfrac{{\partial \textit{π} _{\rm s}^{MO}}}{{\partial t}} = \rho (2H - 2t + r\beta - \beta {w_1}) \\ \dfrac{{\partial \textit{π} _{{\rm m}1}^{MO}}}{{\partial {p_1}}} = - t\beta \rho - 2\mu {p_1} + \beta ( - 1 + \rho ){t_1} - \beta {t_2} + \mu {w_1} + {\alpha _1} \\ \dfrac{{\partial \textit{π} _{{\rm m}2}^{MO}}}{{\partial {p_2}}} = - 2\mu {p_2} - \beta {t_1} - \beta {t_2} + \mu {w_1} - \mu \rho {w_1} + \mu \rho {w_2} + {\alpha _2} \end{array} ;\right.\quad \left\{ \begin{array}{l} \dfrac{{{\partial ^2}\textit{π} _{\rm s}^{MO}}}{{\partial {t^2}}} = - 2\rho < 0 \\ \dfrac{{{\partial ^2}\textit{π} _{{\rm m}1}^{MO}}}{{\partial {p_1}^2}} = - 2\mu < 0 \\ \dfrac{{{\partial ^2}\textit{π} _{{\rm m}2}^{MO}}}{{\partial {p_2}^2}} = - 2\mu < 0 \\ \end{array} \right.$ (4)

 $\dfrac{{\partial \textit{π} _{\rm s}^{MO}}}{{\partial t}} = 0;\dfrac{{\partial \textit{π} _{{\rm m}1}^{MO}}}{{\partial {p_1}}} = 0; \dfrac{{\partial \textit{π} _{{\rm m}2}^{MO}}}{{\partial {p_2}}} = 0$

 $\left\{ \begin{array}{l} {t^{MO*}} = \dfrac{{r\beta - \beta {w_1}}}{2} + H \\ {p_1}^{MO*} = \dfrac{{\beta ( - 1 + \rho ){t_1} - \beta {t_2} + \mu {w_1} - \dfrac{1}{2}\beta \rho (2H + r\beta - \beta w_1) + {\alpha _1}}}{{2\mu }} \\ {p_2}^{MO*} = \dfrac{{ - \beta {t_1} - \beta {t_2} + \mu {w_1} - \mu \rho {w_1} + \mu \rho {w_2} + {\alpha _2}}}{{2\mu }} \end{array} \right.$ (5)
2.2 制造商1采用供应商A供货但不采取援助措施且制造商2采用供应商A供货但不采取援助措施情形（策略2）

 图 3 时间节点示意图2 Fig. 3 Schematic 2 for time node

 $\begin{split}\pi _{\rm s}^{MM} =\, &(1 \!-\! \rho )\left\{ {({w_1} \!-\! {c_1})[{D_1}({p_1},{T_1}) \!+\! {D_2}({p_2},{T_1})]} \right\} \!+ \\ &\rho \left\{ ({w_1} - {c_1})[{D_1}({p_1},{T_2}) + {D_2}({p_2},{T_2})] + \right.\\ & (r - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1}) - \\ &\left.{D_1}({p_1},{T_2}) - {D_2}({p_2},{T_2})] - c(t) \right\}\\[-10pt]\end{split}$ (6)
 $\begin{split}\pi_{{\rm m}1}^{MM} =\,& (1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_2})]\end{split}$ (7)
 $\begin{split}\pi _{{\rm m}2}^{MM} =\,& (1 - \rho )[({p_2} - {w_1}){D_2}({p_2},{T_1})] +\\ &\rho [({p_2} - {w_1}){D_2}({p_2},{T_2})]\end{split}$ (8)

 $\left\{ \begin{array}{l} {t^{MM*}} = H + r\beta - \beta {w_1} \\ {p_1}^{MM*} = \dfrac{{\beta ( - 1 + \rho ){t_1} - \beta {t_2} + \mu {w_1} - \beta \rho (2H + r\beta - \beta w_1) + {\alpha _1}}}{{2\mu }} \\ {p_2}^{MM*} = \dfrac{{\beta ( - 1 + \rho ){t_1} - \beta {t_2} + \mu {w_1} - \beta \rho (2H + r\beta - \beta w{}_1) + {\alpha _2}}}{{2\mu }} \end{array} \right.$ (9)
2.3 制造商1采用供应商A供货但不采取援助措施且制造商2采用供应商A供货并采取援助措施情形（策略3）

 图 4 时间节点示意图3 Fig. 4 Schematic 3 for time node

 $\begin{split}\pi _{\rm s}^{MN} =\, &(1 - \rho )\left\{ {({w_1} - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1})]} \right\}+\\ &\rho \left\{ ({w_1} - {c_1})[{D_1}({p_1},{T_3}) + {D_2}({p_2},{T_3})] +\right.\\ & (r - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1}) -\\ &\left.{D_1}({p_1},{T_3}) - {D_2}({p_2},{T_3})] - c(t) \right\}\\[-10pt]\end{split}$ (10)
 $\begin{split}\pi _{{\rm m}1}^{MM} =& (1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_3})]\end{split}$ (11)
 $\begin{split}\pi _{{\rm m}2}^{MM} = &(1 - \rho )[({p_2} - {w_1}){D_2}({p_2},{T_1})] +\\ &\rho [({p_2} - {w_1}){D_2}({p_2},{T_2}) - G]\end{split}$ (12)

 $\left\{ \begin{array}{l} \dfrac{{\partial \textit{π} _{\rm s}^{MN}}}{{\partial t}} = \rho (2H - 2t + r\beta - \beta {w_1}) \\ \dfrac{{\partial \textit{π} _{{\rm m}1}^{MN}}}{{\partial {p_1}}} = - t\beta \rho - 2\mu {p_1} + \beta ( - 1 + \rho ){t_1} - \beta {t_2} + \mu {w_1} + {\alpha _1} \\ \dfrac{{\partial \textit{π} _{{\rm m}2}^{MN}}}{{\partial {p_2}}} = - 2\mu {p_2} - \beta {t_1} - \beta {t_2} + \mu {w_1} - \mu \rho {w_1} + \mu \rho {w_2} + {\alpha _2} \end{array} \right.;\;\;\;\left\{ \begin{array}{l} \dfrac{{{\partial ^2}\textit{π} _{\rm s}^{MN}}}{{\partial {t^2}}} = - 2\rho < 0 \\ \dfrac{{{\partial ^2}\textit{π} _{{\rm m}1}^{MN}}}{{\partial {p_1}^2}} = - 2\mu < 0 \\ \dfrac{{{\partial ^2}\textit{π} _{{\rm m}2}^{MN}}}{{\partial {p_2}^2}} = - 2\mu < 0 \end{array} \right.$ (13)

 ${t^{MM*}} = (1 - \delta )H + (r - {w_1})\beta {(1 - \delta )^2}$

${t^{MM*}}$ 代入 ${p_1}^{MM*}$ ${p_2}^{MM*}$ 可得

 $\left\{\begin{array}{l}{t}^{MM*}=(1-\delta )H+(r-{w}_{1})\beta ({1}-\delta)^2\\ {p}_{1}{}^{MM*}=\dfrac{\beta (-1\!+\!\rho ){t}_{1}\!-\!\beta {t}_{2}\!+\!\beta (1\!-\!\delta )\delta \rho [H\!-\!\beta (-1\!+\!\delta )(r\!-\!{w}_{1})]\!+\!\beta (1\!-\!\delta )\rho [-H\!+\!\beta (-1\!+\!\delta )(r\!-\!{w}_{1})]\!+\!\mu {w}_{1}\!+\!{\alpha }_{1}}{2\mu }\\ {p}_{2}{}^{MM*}=\dfrac{\beta (-1\!+\!\rho ){t}_{1}\!-\!\beta {t}_{2}\!+\!\beta (1\!-\!\delta )\delta \rho [H\!-\!\beta (-1\!+\!\delta )(r\!-\!{w}_{1})]\!+\!\beta (1\!-\!\delta )\rho [-H\!+\!\beta (-1\!+\!\delta )(r\!-\!{w}_{1})]\!+\!\mu {w}_{1}\!+\!{\alpha }_{2}}{2\mu }\end{array} \right.$ (14)
2.4 制造商1采用供应商A供货但并采取援助措施且制造商2采用备用供应商B供货情形（策略4）

 图 5 时间节点示意图4 Fig. 5 Schematic 4 for time node

 $\begin{split}\textit{π} _{\rm s}^{NO} = &(1 - \rho )\left\{ {({w_1} - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1})]} \right\} +\\ &\rho \{ ({w_1} - {c_1}){D_1}({p_1},{T_4}) + (r - {c_1})[{D_1}({p_1},{T_1}) -\\ &{D_1}({p_1},{T_4})] - c(t) \}\\[-10pt]\end{split}$ (15)
 $\begin{split}\textit{π} _{{\rm m}1}^{MO} =& (1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_4}) - {L_1}]\end{split}$ (16)
 $\begin{split}\textit{π} _{{\rm m}2}^{MO} =& (1 - \rho )[({p_2} - {w_1}){D_2}({p_2},{T_1})] + \\ &\rho [({p_2} - {w_2}){D_2}({p_2},{T_1})]\end{split}$ (17)

 $\left\{\!\!\!\!\begin{array}{l}{t}^{{{NO}}*}=(1-{\psi }_{1})H+\dfrac{(r-{w}_{1})\beta {(1-{\psi }_{1})}^{2}}{2}\\ {p}_{1}{}^{{{NO}}*}=\left\{\beta (-1+\rho ){t}_{1}-\beta {t}_{2}\!+\!\beta \rho {\psi }_{1}\left[H-H{\psi }_{1}+\dfrac{1}{2}\beta {(-1+{\psi }_{1})}^{2}(r-{w}_{1})\right]+\mu {w}_{1}-\right.\\ \qquad\quad\;\;\left.\dfrac{1}{2}\beta \rho (-1+{\psi }_{1})[-2H+r\beta (-1+{\psi }_{1})+(\beta -\beta {\psi }_{1}){w}_{1}]+{\alpha }_{1}\right\}\Biggr/{2\mu }\\ {p}_{2}{}^{{{NO}}*}=\dfrac{-\beta {t}_{1}-\beta {t}_{2}+\mu {w}_{1}-\mu \rho {w}_{1}+\mu \rho {w}_{2}+{\alpha }_{2}}{2\mu }\end{array} \right.$ (18)
2.5 制造商1采用供应商A供货并采取援助措施且制造商2采用供应商A供货但不采取援助措施情形（策略5）

 图 6 时间节点示意图5 Fig. 6 Schematic 5 for time node

 $\begin{split}\textit{π} _{\rm s}^{NM} =& (1 - \rho )\left\{ {({w_1} - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1})]} \right\}+\\ &\rho \left\{ ({w_1} - {c_1})[{D_1}({p_1},{T_5}) + {D_2}({p_2},{T_5})] +\right.\\ & (r - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1}) - \\ & \left.{D_1}({p_1},{T_5}) - {D_2}({p_2},{T_5})] - c(t) \right\}\\[-10pt] \end{split}$ (19)
 $\begin{split}\textit{π} _{{\rm m}1}^{NM} = &(1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_5}) - {L_2}]\end{split}$ (20)
 $\begin{split}\textit{π} _{{\rm m}2}^{NM} =& (1 - \rho )[({p_2} - {w_1}){D_2}({p_2},{T_1})] + \\ &\rho [({p_2} - {w_1}){D_2}({p_2},{T_5})]\end{split}$ (21)

 $\left\{\!\!\!\begin{array}{l}{t}^{NM*}\!=\!(1-{\psi }_{2})H\!+\!(r\!-\!{w}_{1})\beta ({1}\!-\!\psi)^2\\ {p}_{1}{}^{NM*}\!=\!\dfrac{\beta (-1\!+\!\rho ){t}_{1}\!-\!\beta {t}_{2}\!+\!\beta (1\!-\!{\psi }_{2}){\psi }_{2}\rho [{{H}}\!-\!\beta (-1\!+\!{\psi }_{2})(r\!-\!{w}_{1})]\!+\!\beta (1\!-\!{\psi }_{2})\rho [-{{H}}\!+\!\beta (-1\!+\!{\psi }_{2})(r\!-\!{w}_{1})]\!+\!\mu {w}_{1}\!+\!{\alpha }_{1}}{2\mu }\\ {p}_{2}{}^{NM*}\!=\!\dfrac{\beta (-1\!+\!\rho ){t}_{1}\!-\!\beta {t}_{2}\!+\!\beta (1\!-\!{\psi }_{2}){\psi }_{2}\rho [{{H}}\!-\!\beta (-1\!+\!{\psi }_{2})(r\!-\!{w}_{1})]\!+\!\beta (1\!-\!{\psi }_{2})\rho [-{{H}}\!+\!\beta (-1\!+\!{\psi }_{2})(r\!-\!{w}_{1})]\!+\!\mu {w}_{1}\!+\!{\alpha }_{2}}{2\mu }\end{array} \!\!\!\right.$ (22)
2.6 制造商1采用供应商A供货并采取援助措施且制造商2采用供应商A并采取援助措施情形（策略6）

 图 7 时间节点示意图6 Fig. 7 Schematic 6 for time node

 $\begin{split}\text{π} _s^{NN} =& (1 - \rho )\left\{ {({w_1} - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1})]} \right\}+\\ &\rho \left\{ ({w_1} - {c_1})[{D_1}({p_1},{T_6}) + {D_2}({p_2},{T_6})] +\right.\\ &(r - {c_1})[{D_1}({p_1},{T_1}) + {D_2}({p_2},{T_1}) - \\ &\left.{D_1}({p_1},{T_6}) - {D_2}({p_2},{T_6})] - c(t) \right\}\\[-10pt] \end{split}$ (23)
 $\begin{split}\text{π} _{m1}^{NN} =& (1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_6}) - {\rm{L}}]\end{split}$ (24)
 $\begin{split}\text{π} _{m1}^{NN} =& (1 - \rho )[({p_1} - {w_1}){D_1}({p_1},{T_1})] +\\ &\rho [({p_1} - {w_1}){D_1}({p_1},{T_6}) - {\rm{L}}]\end{split}$ (25)

 $\left\{ \begin{array}{l}{t}^{NN*}=(1-\xi )H+(r-{w}_{1})\beta ({1}-\xi)^2\\ {p}_{1}{}^{NN*}=\dfrac{\beta (-1\!+\!\rho ){t}_{1}\!-\!\beta {t}_{2}\!+\!\beta (1\!-\!\xi )\xi \rho [{{H}}\!-\!\beta (-1\!+\!\xi )(r\!-\!{w}_{1})]\!+\!\beta (1\!-\!\xi )\rho [-{{H}}\!+\!\beta (-1\!+\!\xi )(r\!-\!{w}_{1})]\!+\!\mu {w}_{1}\!+\!{\alpha }_{1}}{2\mu }\\ {p}_{2}{}^{NN*}=\dfrac{\beta (-1\!+\!\rho ){t}_{1}\!-\!\beta {t}_{2}\!+\!\beta (1\!-\!\xi )\xi \rho [{{H}}\!-\!\beta (-1\!+\!\xi )(r\!-\!{w}_{1})]\!+\!\beta (1\!-\!\xi )\rho [-{{H}}\!+\!\beta (-1\!+\!\xi )(r\!-\!{w}_{1})]\!+\!\mu {w}_{1}\!+\!{\alpha }_{2}}{2\mu }\end{array} \right.$ (26)
2.7 模型分析

a. ${t^{MM*}} < {t^{MO*}}$ 恒成立；

b. 当 $\delta > \dfrac{H}{{(r - {w_1})\beta }} + 2$ 时， ${t^{MN*}} < {t^{MM*}} < {t^{MO*}}$

c. 当 $\delta < \dfrac{H}{{(r - {w_1})\beta }} + 2$ $- H\delta + \beta (r - {w_1}){( - 1 + \delta )^2} +$ $\!\dfrac{{({w_1} \!-\! r)\beta }}{2} \! < 0$ 时， ${t^{MM*}} < {t^{MN*}} < {t^{MO*}}$

d. 当 $\delta < \dfrac{H}{{(r - {w_1})\beta }} + 2$ $- H\delta + \beta (r - {w_1}){( - 1 + \delta )^2} +$ $\dfrac{{({w_1} - r)\beta }}{2} > 0$ ${t^{MM*}} < {t^{MO*}} < {t^{MN*}}$

${t^{MM*}} \!-\! {t^{MN*}} \!=\! \delta [ - H \!+\! r\beta ( - 2 \!+\! \delta ) \!-\! \beta ( - 2 \!+\! \delta ){w_1}] \!> \! 0$ ，则 $\delta \! >\! \dfrac{H}{{(r \!-\! {w_1})\beta }} \!+\! 2$ ，此时 ${t^{MN*}} \!< \! {t^{MM*}}$ ，则当 $\delta > \dfrac{H}{{(r - {w_1})\beta }} + 2$ 时， ${t^{MN*}} < {t^{MM*}} < {t^{MO*}}$

$\delta < \dfrac{H}{{(r - {w_1})\beta }} + 2$ 时，同时 ${t^{MN*}} - {t^{MO*}} = - H\delta + \beta ( - 1 + \delta )^2(r - {w_1}) + \dfrac{({w_1} - r)\beta }{2} < 0$ 时， ${t^{MM*}} < {t^{MN*}} < {t^{MO*}}$

$\delta < \dfrac{H}{{(r - {w_1})\beta }} + 2$ 时，同时 ${t^{MN*}} - {t^{MO*}} = - H\delta + \beta {( - 1 + \delta )^2}(r - {w_1}) + \dfrac{{({w_1} - r)\beta }}{2} > 0$ 时， ${t^{MM*}} < {t^{MO*}} < {t^{MN*}}$

a. 当 $\xi + {\psi _2} \geqslant 2 + \dfrac{H}{{(r - {w_1})\beta }}$ $H({\psi _1} - {\psi _2}) + \beta (r - {w_1})\Biggr[( - 1 + {\psi _2})^2 - \dfrac{{{{( - 1 + {\psi _1})}^2}}}{2}\Biggr] < 0$ 时， ${t^{NN*}} < {t^{NM*}} < {t^{NO*}}$

b. 当 $\xi + {\psi _2} \geqslant 2 + \dfrac{H}{{(r - {w_1})\beta }}$ $H({\psi _1} - {\psi _2}) + \beta (r - {w_1})\Biggr[{( - 1 + {\psi _2})^2} - \dfrac{{{{( - 1 +{\psi _1})}^2}}}{2}\Biggr] \geqslant 0$ $H({\psi _1} - \xi ) + \beta (r - {w_1})\Biggr[( - 1 + \xi )^2 -$ $\dfrac{1}{2}{( - 1 + {\psi _1})^2}\Biggr] \geqslant 0$ 时， ${t^{NO*}} < {t^{NN*}} < {t^{NM*}}$

c. 当 $\xi + {\psi _2} \geqslant 2 + \dfrac{H}{{(r - {w_1})\beta }}$ $H({\psi _1} - {\psi _2}) + \beta (r - {w_1})\Biggr[{( - 1 + {\psi _2})^2} - \dfrac{{{{( - 1 + {\psi _1})}^2}}}{2}\Biggr] \geqslant 0$ $H({\psi _1} - \xi ) + \beta (r - {w_1})\Biggr[( - 1 + \xi )^2 -$ $\dfrac{1}{2}{( - 1 + {\psi _1})^2}\Biggr] < 0$ 时， ${t^{NN*}} < {t^{NO*}} < {t^{NM*}}$

d. $\xi + {\psi _2} < 2 + \dfrac{H}{{(r - {w_1})\beta }}$ $H({\psi _1} - {\psi _2}) + \beta (r - {w_1})\Biggr[( - 1 + {\psi _2})^2 - \dfrac{{{{( - 1 + {\psi _1})}^2}}}{2}\Biggr] \geqslant 0$ 时， ${t^{NO*}} < {t^{NM*}} < {t^{NN*}}$

e. $\xi + {\psi _2} < 2 + \dfrac{H}{{(r - {w_1})\beta }}$ $H({\psi _1} - {\psi _2}) + \beta (r - {w_1})\Biggr[( - 1 + {\psi _2})^2 - \dfrac{{{{( - 1 + {\psi _1})}^2}}}{2}\Biggr] < 0$ $H({\psi _1} - \xi ) + \beta (r - {w_1})\Biggr[{( - 1 + \xi )^2} -$ $\dfrac{1}{2}{( - 1 + {\psi _1})^2}\Biggr] \geqslant 0$ 时， ${t^{NM*}} < {t^{NO*}} < {t^{NN*}}$

f. $\xi + {\psi _2} < 2 + \dfrac{H}{{(r - {w_1})\beta }}$ $H({\psi _1} - {\psi _2}) + \beta (r - {w_1})\Biggr[( - 1 + {\psi _2})^2 - \dfrac{{{{( - 1 + {\psi _1})}^2}}}{2}\Biggr] < 0$ $H({\psi _1} - \xi ) + \beta (r - {w_1})\Biggr[{( - 1 + \xi )^2} -$ $\dfrac{1}{2}{( - 1 + {\psi _1})^2}\Biggr] < 0$ 时， ${t^{NM*}} < {t^{NN*}} < {t^{NO*}}$

a. ${\psi _1} < 2 + \dfrac{{2H}}{{(r - {w_1})\beta }}$ 时， ${t^{MO*}} < {t^{NO*}}$ ${\psi _1} > 2 + \dfrac{{2H}}{{(r - {w_1})\beta }}$ 时， ${t^{MO*}} > {t^{NO*}}$

b. ${\psi _2} < 2 + \dfrac{H}{{(r - {w_1})\beta }}$ 时， ${t^{MM*}} < {t^{NM*}}$ ${\psi _2} > 2 + \dfrac{H}{{(r - {w_1})\beta }}$ 时， ${t^{MM*}} > {t^{NM*}}$

c. $\xi + \delta < 2 + \dfrac{H}{{(r - {w_1})\beta }}$ 时， ${t^{MN*}} < {t^{NN*}}$ $\delta + \psi > 2 + \dfrac{H}{{(r - {w_1})\beta }}$ 时， ${t^{MN*}} > {t^{NN*}}$

3 算例分析

3.1 策略3～6中期望收益与恢复系数关系的数值算例分析

 图 8 策略3中援助系数与制造商2期望收益关系 Fig. 8 Relationship between recovering coefficient and manufacturer 2 expected revenue in strategy 3

 图 9 策略4中援助系数与制造商1期望收益的关系 Fig. 9 Relationship between recovering coefficient and manufacturer 1 expected revenue in strategy 4

 图 10 策略5中援助系数与制造商1期望收益关系 Fig. 10 Relationship between recovering coefficient and manufacturer 1 expected revenue in strategy 5

 图 11 策略6中援助系数与总体期望收益的关系 Fig. 11 Relationship between recovering coefficient and overall expected revenue in strategy 6

3.2 关于制造商生产完成时间数值算例分析

3.3 关于制造商2的3种策略数值算例分析 3.3.1 制造商1不援助时制造商2的3种策略数值算例分析

3.3.2 制造商1援助时制造商2的3种策略数值算例分析

3.4 关于制造商1的2种策略数值算例分析 3.4.1 制造商2采用备用供应商B时制造商1的2种策略数值算例分析

 图 12 策略1和策略4中制造商1的期望收益对比 Fig. 12 Comparison of expected revenue of manufacturer 1 in strategy 1 and strategy 4
3.4.2 制造商2采用供应商A但不援助时制造商1的2种策略数值算例分析

 图 13 策略2和策略5中制造商1的期望收益对比 Fig. 13 Comparison of expected revenue of manufacturer 1 in strategy 2 and strategy 5
3.4.3 制造商2采用供应商A并援助时制造商1的2种策略数值算例分析

 图 14 策略3和策略6中制造商1的期望收益对比 Fig. 14 Comparison of expected revenue of manufacturer 1 in strategy 3 and strategy 6
4 结论与展望

a. 当供应商A发生中断时，制造商1（或制造商2）单独采取援助措施时，合适的援助成本能够使制造商1（或制造商2）获得更高的收益。

b. 当供应商A发生中断时，制造商1和制造商2共同援助时，供应商A实际交货时间最短。当制造商1的市场需求量相较于制造商2更大时，制造商1（单独援助时）的最优援助措施相对制造商2（单独援助时）对供应中断恢复更为有效。

c. 当制造商1采用供应商A供货但不采取援助措施时，制造商2采用供应商A供货并采取援助措施时的期望收益相对其他策略更高；当制造商1采用供应商A供货并采取援助措施时，制造商2采用供应商A供货但不采取援助措施时的期望收益相对其他策略更高。

d. 当制造商2采用备用供应商B供货或采用供应商A供货但并不采取援助措施时，制造商1采用供应商A供货并采取援助措施时的期望收益均高于其他策略；当制造商2采用供应商A供货并采取援助措施时，制造商1策略选择的优劣与中断概率大小有关，当中断概率较小时，制造商1选择供应商A供货并采取援助措施相对其他策略能够获得更高的收益，当中断概率较大时，制造商选择供应商A但不采取援助措施时相对其他策略能够获得更高的收益。

 [1] IVANOV D, SOKOLOV B, SOLOVYEVA I, et al. Dynamic recovery policies for time-critical supply chains under conditions of ripple effect[J]. International Journal of Production Research, 2016, 54(23): 7245-7258. DOI:10.1080/00207543.2016.1161253 [2] PALAKA K, ERLEBACHER S, KROPP D H. Lead-time setting, capacity utilization, and pricing decisions under lead-time dependent demand[J]. IIE Transactions, 1998, 30(2): 151-163. [3] ZHANG X M, SONG M L, LIU G D. Service product pricing strategies based on time-sensitive customer choice behavior[J]. Journal of Industrial and Management Optimization, 2017, 13(1): 297-312. DOI:10.3934/jimo.2016018 [4] 李焕之. 云平台环境下制造供应链资源共享博弈研究[D]. 大连: 大连理工大学, 2019: 31–43. [5] 张雪梅, 李盈盈. 基于时间和渠道敏感顾客选择偏好的双渠道定价策略[J]. 运筹与管理, 2019, 28(7): 179-186. [6] 谢祥添, 张毕西. 需求不确定下承诺交货时间和产能决策[J]. 中国管理科学, 2016, 24(11): 73-80. [7] 朱玉炜, 徐琪. 考虑消费者时间敏感的双渠道供应链竞争策略[J]. 计算机集成制造系统, 2013, 19(6): 1363-1368. [8] IYER A V, DESHPANDE V, WU Z P. Contingency management under asymmetric information[J]. Operations Research Letters, 2005, 33(6): 572-580. DOI:10.1016/j.orl.2004.11.007 [9] KUMAR M, BASU P, AVITTATHUR B. Pricing and sourcing strategies for competing retailers in supply chains under disruption risk[J]. European Journal of Operational Research, 2018, 265(2): 533-543. DOI:10.1016/j.ejor.2017.08.019 [10] LI G, KANG Y C, LIU M Q. Dual-source procurement strategies for manufacturers with supply disruption risks[J]. Journal of Intelligent & Fuzzy Systems, 2017, 33(5): 2637-2645. [11] BERGER P D, GERSTENFELD A, ZENG A Z. How many suppliers are best? A decision-analysis approach[J]. Omega, 2004, 32(1): 9-15. DOI:10.1016/j.omega.2003.09.001 [12] BURKE G J, CARRILLO J E, VAKHARIA A J. Single versus multiple supplier sourcing strategies[J]. European Journal of Operational Research, 2007, 182(1): 95-112. DOI:10.1016/j.ejor.2006.07.007 [13] GUPTA V, CHUTANI A. Supply chain financing with advance selling under disruption[J]. International Transactions in Operational Research, 2020, 27(5): 2449-2468. DOI:10.1111/itor.12663 [14] 徐蓉. 制造商应对供应链中断风险的采购策略研究[D]. 泉州: 华侨大学, 2020: 19–64. [15] 王静, 陈希. 考虑供应链中断风险的制造商风险应对方案研究[J]. 工业工程与管理, 2019, 24(3): 27-34. [16] 李姗姗, 何勇. 供应中断情况下动态混合应急策略研究[J]. 工业工程与管理, 2020, 25(2): 17-23, 16. [17] 张广胜, 刘伟, 高志军. 考虑供给风险的物流服务供应链能力应急采购设计[J]. 统计与决策, 2018, 34(8): 43-47. [18] 何青, 伍星华, 李志鹏. 供应中断风险下制造商的后备采购与改善努力策略比较研究[J]. 数学的实践与认识, 2020, 50(7): 54-63. [19] 颜荣芳, 王倩, 胡文丰. 需求中断下具有提前期的双渠道供应链的风险规避决策[J]. 经济数学, 2020, 37(1): 48-52. DOI:10.3969/j.issn.1007-1660.2020.01.007 [20] 何璐, 郭健全. 碳税下考虑供应中断的制造/再制造企业最优策略[J]. 上海理工大学学报, 2017, 39(6): 556-562. [21] 李新军, 王建军, 达庆利. 供应中断情况下基于备份供应商的应急决策分析[J]. 中国管理科学, 2016, 24(7): 63-71. [22] 陈崇萍, 陈志祥. 供应商产出随机与供应中断下的双源采购决策[J]. 中国管理科学, 2019, 27(6): 113-122. [23] 桂华明, 马士华. 供应链下游订单处置能力扩张协调策略研究[J]. 管理科学, 2007, 20(3): 2-8. DOI:10.3969/j.issn.1672-0334.2007.03.001