JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (7): 88-102, 110.doi: 10.6040/j.issn.1671-9352.0.2019.290

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Channel decisions and coordination of closed-loop supply chain under demand and cost disruptions

Dan-dan SUN(),Ke LI*(),Li-bin XIE,Zong-yu MU   

  1. School of Business, Qingdao University, Qingdao 266071, Shandong, China
  • Received:2019-05-15 Online:2020-07-20 Published:2020-07-08
  • Contact: Ke LI E-mail:17854239588@163.com;605495315@qq.com

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Abstract:

Considering demand and cost are disrupted by emergency events, there are three CLSC emergency models, i.e. retailer collecting, manufacturer collecting and third-party collecting. In the three decentralized collecting systems and centralized decision system, compared with equilibrium decisions in normal environment, when the disruptions of demand and cost are not large, sales quantity of products and collecting rate of used products have some robustness, but the wholesale price and selling price of products should be adjusted along the same direction with demand disruption; when the disruptions of demand and cost are large, the wholesale price, selling price, sales quantity of products and collecting rate of used products should be adjusted along the same direction with demand disruption; when the degree of demand decreasing and cost increasing are large, the three decentralized collecting systems and centralized decision system are all destroyed. In each case of demand and cost disruptions, manufacturer can obtain more profits by MRCRM system; when the degree of demand decreasing and cost increasing exceeds a certain threshold, the profits of centralized decision system decrease resulted from assuming more extra disposing costs, which results in its profits are less than MRCRM system. Furthermore, in other disruption cases, emergency revenue and expense sharing contract can solve "double marginalization" problems in MRCRM system, which improves system's operation efficiency and makes each member get Pareto improved profits.

Key words: closed-loop supply chain, demand, cost, channel choice, revenue and expense sharing contract

CLC Number: 

  • F272

Fig.1

Three types of recycling closed-loop supply chain systems"

Fig.2

Operation process of system under disturbance of emergencies"

Table 1

The parameters and notations"

符号 意义 符号 意义
cm 使用原材料生产新产品单位生产成本 β 市场上产品价格敏感系数
cr 使用废旧品生产再造品单位生产成本 q 产品的销售量(零售商的决策变量)
w 制造商给零售商产品单位批发价 A 回收商回收废旧品单位回收价
b 制造商从回收方处回收废旧品单位回收价(制造商的决策变量) τ 废旧品回收率, 0≤τ≤1
p 零售商处产品单位销售价(零售商的决策变量) CL 回收努力成本系数
ϕ 产品的最大市场需求规模 δ δ=cm-cr

Table 2

Comparison of three kinds of recovery system and centralized system in emergency decision and profit"

扰动区间 系统总利润 制造商利润
Δϕ>0, Δc＜0 $ 0 \le \Delta \phi - \beta \Delta c ＜ \frac{{4{C_{\rm{L}}}\left[ {\phi - \beta {c_{\rm{m}}} - \beta \left( {\delta - A} \right)} \right]\left( {\delta - A} \right)}}{{\left( {\delta - A} \right)}} + \beta {\lambda _2} $ $ {\mathit{\overline \Pi} ^{\rm{c}}} > \mathit{\overline \Pi} _{\rm{m}}^{\rm{R}} + \mathit{\overline \Pi} _{\rm{r}}^{\rm{R}} $ $ \mathit{\overline \Pi} _{\rm{m}}^{\rm{R}} > \mathit{\overline \Pi} _{\rm{m}}^{\rm{M}} > \mathit{\overline \Pi} _{\rm{m}}^{\rm{T}}$
Δϕ＜0, Δc>0 $ \sqrt {6\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)} - \left( {\phi - \beta {c_{\rm{m}}}} \right) - \beta {\lambda _1} ＜ \Delta \phi - \beta \Delta c \le 0 $ $ {\mathit{\overline \Pi} ^{\rm{c}}} > \mathit{\overline \Pi} _{\rm{m}}^{\rm{R}} + \mathit{\overline \Pi} _{\rm{r}}^{\rm{R}} $
$ - \left( {\phi - \beta {c_{\rm{m}}}} \right) - \beta {\lambda _1} \le \Delta \phi - \beta \Delta c ＜ \sqrt {6\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)} - \left( {\phi - \beta {c_{\rm{m}}}} \right) - \beta {\lambda _1} $ $ {\mathit{\overline \Pi} ^{\rm{c}}} \le \mathit{\overline \Pi} _{\rm{m}}^{\rm{R}} + \mathit{\overline \Pi} _{\rm{r}}^{\rm{R}} $

Table 3

Emergency decision and profit comparison of three kinds of recovery system and centralized decision system"

扰动范围 (-(ϕ-βcm)-βλ1, -βλ1] (-βλ1, βλ2) $ \left[ {\beta {\lambda _2}, \frac{{4{C_{\rm{L}}}\left[ {\phi - \beta {c_{\rm{m}}} + \beta \left( {\delta - A} \right)} \right]\left( {\delta - A} \right)}}{{\left( {\delta - A} \right)}} + \beta {\lambda _2}} \right] $
价格 MRCRM $ {p^{{{\rm{R}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ {p^{{{\rm{R}}^*}}} + \frac{{\Delta \phi }}{\beta } $ $ {p^{{{\rm{R}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
MRCM $ {p^{{{\rm{M}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ {p^{{{\rm{M}}^*}}} + \frac{{\Delta \phi }}{\beta } $ $ {p^{{{\rm{M}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
MRCTM $ {p^{{{\rm{T}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{4{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ {p^{{{\rm{T}}^*}}} + \frac{{\Delta \phi }}{\beta } $ $ {p^{{{\rm{T}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{4{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
集中式 $ {p^{{{\rm{c}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ {p^{{{\rm{c}}^*}}} + \frac{{\Delta \phi }}{\beta } $ $ {p^{{{\rm{c}}^*}}} + \frac{{\Delta \phi }}{\beta } - \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
产量 MRCRM $ {q^{{{\rm{R}}^*}}} + \frac{{{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ qR* $ {q^{{{\rm{R}}^*}}} + \frac{{{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
MRCM $ {q^{{{\rm{M}}^*}}} + \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ qM* $ {q^{{{\rm{M}}^*}}} + \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
MRCTM $ {q^{{{\rm{T}}^*}}} + \frac{{4{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ qT* $ {q^{{{\rm{T}}^*}}} + \frac{{4{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
集中式 $ {q^{{{\rm{c}}^*}}} + \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ qc* $ {q^{{{\rm{c}}^*}}} + \frac{{2{C_{\rm{L}}}\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
回收率 MRCRM $ {\tau ^{{{\rm{R}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{2\left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ τR* $ {\tau ^{{{\rm{R}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{2\left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
MRCM $ {\tau ^{{{\rm{M}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ τM* $ {\tau ^{{{\rm{M}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
MRCTM $ {\tau ^{{{\rm{T}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ τT* $ {\tau ^{{{\rm{T}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
集中式 $ {\tau ^{{{\rm{c}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}}{{4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $ τc* $ {\tau ^{{{\rm{c}}^*}}} + \frac{{\left( {\delta - A} \right)\left( {\Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}}{{4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} $
利润 MRCRM M $ \frac{{{C_{\rm{L}}}\left[ {{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2} - 2\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{2\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left( {\phi - \beta {c_{\rm{m}}}} \right)\left[ {\phi - \beta {c_{\rm{m}}} + 2\left( {\Delta \phi - \beta \Delta c} \right)} \right]}}{{2\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left[ {{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2} - 2\beta {\lambda _2}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{2\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
R $ \frac{{{C_{\rm{L}}}{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2}}}{{4\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}{{\left( {\phi - \beta {c_{\rm{m}}}} \right)}^2}}}{{4\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2}}}{{4\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
S $ \frac{{{C_{\rm{L}}}\left[ {3{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2} - 4\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{4\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left( {\phi - \beta {c_{\rm{m}}}} \right)\left[ {3\left( {\phi - \beta {c_{\rm{m}}}} \right) + 4\left( {\Delta \phi - \beta \Delta c} \right)} \right]}}{{4\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left[ {3{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2} - 4\beta {\lambda _2}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{4\beta \left[ {4{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
MRCM M $ \frac{{{C_{\rm{L}}}\left[ {{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2} - 2\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left( {\phi - \beta {c_{\rm{m}}}} \right)\left[ {\phi - \beta {c_{\rm{m}}} + 2\left( {\Delta \phi - \beta \Delta c} \right)} \right]}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left[ {{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2} - 2\beta {\lambda _2}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
R $ \frac{{4C_{\rm{L}}^2{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2}}}{{\beta {{\left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $ $ \frac{{4C_{\rm{L}}^2{{\left( {\phi - \beta {c_{\rm{m}}}} \right)}^2}}}{{\beta {{\left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $ $ \frac{{4C_{\rm{L}}^2{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c - \beta {\lambda _2}} \right)}^2}}}{{\beta {{\left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $
S $ {C_{\rm{L}}}\left\{ \begin{array}{l} \frac{{{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2}\left[ {12{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}}{{\beta {{\left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}}\\ \;\;\;\;\;\;\;\frac{{2{\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)}}{{8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} \end{array} \right\} $ $ {C_{\rm{L}}}\left( {\phi - \beta {c_{\rm{m}}}} \right)\left\{ \begin{array}{l} \frac{{\left( {\phi - \beta {c_{\rm{m}}}} \right)\left[ {12{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}}{{\beta {{\left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}}\\ \;\;\;\;\; + \frac{{2\left( {\Delta \phi - \beta \Delta c} \right)}}{{\beta \left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} \end{array} \right\} $ $ {C_{\rm{L}}}\left\{ \begin{array}{l} \frac{{{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2}\left[ {12{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}}{{\beta {{\left[ {8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}}\\ \;\;\;\; + \frac{{2{\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)}}{{8{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}} \end{array} \right\} $
MRCTM M $ \frac{{2{C_{\rm{L}}}\left[ {{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2} - 2\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{2{C_{\rm{L}}}\left( {\phi - \beta {c_{\rm{m}}}} \right)\left[ {\phi - \beta {c_{\rm{m}}} + 2\left( {\Delta \phi - \beta \Delta c} \right)} \right]}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{2{C_{\rm{L}}}\left[ {{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2} - 2\beta {\lambda _2}\left( {\phi - \beta {c_{\rm{m}}}} \right)} \right]}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
R $ \frac{{16C_{\rm{L}}^2{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2}}}{{\beta {{\left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $ $ \frac{{16{C_{\rm{L}}}^2{{\left( {\phi - \beta {c_{\rm{m}}}} \right)}^2}}}{{\beta {{\left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $ $ \frac{{16C_{\rm{L}}^2{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c - \beta {\lambda _2}} \right)}^2}}}{{\beta {{\left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $
T $ \frac{{{C_{\rm{L}}}{{\left( {\delta - A} \right)}^2}{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2}}}{{{{\left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $ $ \frac{{{C_{\rm{L}}}{{\left( {\delta - A} \right)}^2}{{\left( {\phi - \beta {c_{\rm{m}}}} \right)}^2}}}{{{{\left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $ $ \frac{{{C_{\rm{L}}}{{\left( {\delta - A} \right)}^2}{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c - \beta {\lambda _2}} \right)}^2}}}{{{{\left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}^2}}} $
$ \frac{{{C_{\rm{L}}}}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}\left( {\phi - \beta {c_{\rm{m}}}} \right)}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_{\rm{L}}}}}{{\beta \left[ {16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $
S $ \left\{ \begin{array}{l} \frac{{\left[ {48{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2}}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}}\\ \;\;\;\;\; - 4\beta {\lambda _1}\left( {\phi - \beta {c_{\rm{m}}}} \right) \end{array} \right\} $ $ \left\{ \begin{array}{l} \frac{{\left[ {48{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]\left( {\phi - \beta {c_{\rm{m}}}} \right)}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}}\\ \;\;\;\;\; + 4\left( {\Delta \phi - \beta \Delta c} \right) \end{array} \right\} $ $ \left\{ \begin{array}{l} \frac{{\left[ {48{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}} \right]{{\left( {\phi - \beta {c_{\rm{m}}} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2}}}{{16{C_{\rm{L}}} - \beta {{\left( {\delta - A} \right)}^2}}}\\ \;\;\;\;\; + 4\beta {\lambda _2}\left( {\phi - \beta {c_{\rm{m}}}} \right) \end{array} \right\} $
集中式 $ \frac{{{C_L}\left[ {{{\left( {\phi - \beta {c_m} + \Delta \phi - \beta \Delta c + \beta {\lambda _1}} \right)}^2} - 2\beta {\lambda _1}\left( {\phi - \beta {c_m}} \right)} \right]}}{{\beta \left[ {4{C_L} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_L}\left( {\phi - \beta {c_m}} \right)\left[ {\phi - \beta {c_m} + 2\left( {\Delta \phi - \beta \Delta c} \right)} \right]}}{{\beta \left[ {4{C_L} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $ $ \frac{{{C_L}\left[ {{{\left( {\phi - \beta {c_m} + \Delta \phi - \beta \Delta c + \beta {\lambda _2}} \right)}^2} - 2\beta {\lambda _2}\left( {\phi - \beta {c_m}} \right)} \right]}}{{\beta \left[ {4{C_L} - \beta {{\left( {\delta - A} \right)}^2}} \right]}} $

Table 4

Emergency equilibrium decision results of three kinds of recovery channel system and centralized decision system"

Δϕ-βΔc 回收模式 分散式决策 集中式决策
零售商 制造商 第三方 分散式总利润
${\bar p}$ ${\bar q}$ τ Πr ${\bar w}$ ${\bar b}$ Πm Πt Πd ${\bar p}$c ${\bar q}$c τc Πc
-20 MRCRM 6.748 5.036 0.072 5.036 5.000 1 1.259 — 6.295 6.496 10.072 0.144 2.518
MRCM 6.749 5.018 0.072 5.036 4.896 — 1.254 — 6.290
MRCTM 6.750 5.009 0.036 5.018 4.996 0.90 1.252 0.01 6.279
-7 MRCRM 7.639 8.309 0.119 13.710 5.900 1 18.608 — 32.318 6.078 16.619 0.237 37.216
MRCM 7.644 8.280 0.118 13.710 5.691 — 18.541 — 32.251
MRCTM 7.647 8.265 0.059 13.661 5.696 0.90 18.508 0.02 32.194
-5 MRCRM 7.837 8.813 0.126 15.423 6.194 1 22.032 — 37.455 6.075 17.626 0.252 44.065
MRCM 7.844 8.781 0.125 15.422 6.087 — 21.953 — 37.376
MRCTM 7.847 8.766 0.063 15.367 6.094 0.90 21.914 0.03 37.309
-2 MRCRM 8.037 8.813 0.126 15.423 6.300 1 27.320 — 42.743 6.275 17.626 0.252 54.640
MRCM 8.044 8.781 0.125 15.422 6.287 — 27.222 — 42.645
MRCTM 8.047 8.766 0.063 15.367 6.294 0.90 27.174 0.03 42.568
0 MRCRM 8.237 8.813 0.126 15.423 6.500 1 30.845 — 46.268 6.475 17.626 0.252 61.691
MRCM 8.244 8.781 0.125 15.422 6.487 — 30.735 — 46.157
MRCTM 8.247 8.766 0.063 15.367 6.494 0.90 30.680 0.03 46.075
2 MRCRM 8.437 8.813 0.126 15.423 6.700 1 34.371 — 49.793 6.675 17.626 0.252 68.741
MRCM 8.444 8.781 0.125 15.422 6.687 — 34.247 — 49.670
MRCTM 8.447 8.766 0.063 15, 367 6.694 0.90 34.186 0.03 49.581
5 MRCRM 8.837 8.813 0.126 15.423 7.100 1 39.658 — 55.081 7.075 17.626 0.252 79.317
MRCM 8.844 8.781 0.125 15.422 7.087 — 39.516 — 54.939
MRCTM 8.847 8.766 0.063 15, 367 7.094 0.90 39.445 0.03 54.840
20 MRCRM 9.727 12.590 0.180 31.475 8.000 1 71.763 — 103.240 6.453 25.180 0.360 143.520
MRCM 9.738 12.545 0.179 31.474 7.982 — 71.505 — 102.980
MRCTM 9.744 12.522 0.089 31.362 7.991 0.90 71.377 0.03 102.770
100 MRCRM 15.670 32.734 0.468 212.770 14.000 1 434.350 — 647.120 4.338 65.468 0.935 868.710
MRCM 15.710 32.616 0.466 212.760 13.950 — 432.790 645.560
MRCTM 15.730 32.558 0.233 212.980 13.980 0.90 432.020 0.04 644.410

Table 5

Results of a closed-loop supply chain coordinated by emergency revenue sharing contract"

Δϕ-βΔc θ范围 θ取值 w(θ) b(θ) $ \mathit{\overline \Pi} _{\rm{r}}^{\rm{d}} $ $ \mathit{\overline \Pi} _{\rm{m}}^{\rm{d}} $ $ \mathit{\overline \Pi} _{\rm{m}}^{\rm{d}} + \mathit{\overline \Pi} _{\rm{r}}^{\rm{d}} $
-7 (0.15, 0.5) 0.435 2.437 0.435 16.198 21.018 37.216
-5 (0.19, 0.5) 0.443 2.483 0.443 19.538 24.527 44.065
-2 (0.22, 0.5) 0.474 2.465 0.474 25.901 28.739 54.640
0 (0.25, 0.5) 0.282 1.409 0.282 17.381 44.310 61.691
2 (0.28, 0.5) 0.480 2.302 0.480 32.960 35.781 68.741
5 (0.35, 0.5) 0.369 1.698 0.369 29.272 50.045 79.317
20 (0.22, 0.5) 0.247 0.987 0.247 35.405 108.121 143.525
100 (0.25, 0.5) 0.491 0.982 0.491 426.572 442.133 868.705
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