Reverse supply chain management with dual channel and collection disruptions: supply chain coordination and game theory approaches

Abstract

In reverse supply chain (RSC) systems, disruptions in the collection process of used items may negatively influence the efficiency of RSC participators. Inspired by a real case study, this paper contributes to the RSC systems coordination literature by analyzing the effect of collection disruptions on the efficiency of an RSC system with dual collection channels using a coordination contract approach. Moreover, this study explores the effect of collection competition between two collection channels (collector channel and remanufacturer channel) on the acquisition prices offered to consumers as incentive schemes for returning the used items. This study determines the equilibrium solutions for selling prices of remanufactured products, acquisition prices, and transfer price under decentralization, centralization, and coordination settings. Additionally, this study incorporates the impact of collection disruptions into the remanufacturer’s and collector’s problem considering four scenarios. Furthermore, this study proposes a disruption-based two-part tariff contract to accomplish channel coordination in the RSC system with dual-collection channel under collection disruptions. Our finding reveals that the suggested coordinated scheme efficiently coordinates the disrupted RSC with dual collection channels even when under high possibility of collection disruptions. Moreover, our coordination plan is of environmental and economic benefits, as it can boost the return quantities and can increase RSC participators’ profits.

Appendices

Appendix A. Proofs of propositions

Proof of Proposition 1 of Proposition 1

By taking the second-order derivative of $E\left({\varPi }_{c}\left({a}_{c}\right)\right)$ w.r.t. ${a}_{c}$ under the Remanufacturer–Stackelberg game, we have $\frac{{\partial }^{2}E\left({\varPi }_{c}\left({a}_{c}\right)\right)}{\partial {{a}_{c}}^{2}}=-2\beta \left(1-{r}_{c}\right)<0$. Thus, the TWEP of the collector is concave w.r.t.${a}_{c}$.$\square$

Proof of Proposition 2

The second-order derivative of $E\left({\varPi }_{rm}\left(T,{a}_{m}\right)\right)$ w.r.t. $T$ is $\frac{{\partial }^{2}E\left({\varPi }_{m}\left(T,{ a}_{m}\right)\right)}{\partial {T}^{2}} =-\left(\frac{{\tau }^{2}{\left(\beta -\gamma \right)}^{2}}{\mu {\left(1-{r}_{c}\right)}^{2}}+\frac{\beta }{1-{r}_{c}}\right)<0$. Therefore, the remanufacturer’ TWEP is concave w.r.t. $T$ for a given ${a}_{m}$ under the Remanufacturer–Stackelberg game.$\square$

Proof of Proposition 3

Taking the second-order derivative of $E\left({\varPi }_{SC}\left({a}_{m},{a}_{c}\right)\right)$ w.r.t. ${a}_{c}$, we have $\frac{{\partial }^{2}E\left({\varPi }_{SC}\left({a}_{m}, {a}_{c}\right)\right)}{\partial {{a}_{c}}^{2}} =-\left(2\beta \left(1-{r}_{c}\right)+\frac{2{\tau }^{2}}{\mu }{\left(\beta -\gamma \right)}^{2}\right)<0$. Thus, the TWEP of the whole RSC under the centralized structure is concave w.r.t. ${a}_{c}$ for a given ${a}_{m}$.$\square$

Proof of Proposition 4

The remanufacturer and the collector agree on the proposed reverse disruption-based two-part tariff contract if and only if their TWEPs are at least equal to those of the decentralized model. Therefore, we have:

$$ E\left( {\varPi_{c}^{Co} \left( {a_{c}^{Cen} } \right)} \right) \ge E\left( {\varPi_{c}^{dec} \left( {a_{c}^{dec} } \right)} \right) $$

(A1)

$$ E\left( {\varPi_{m}^{Co} \left( {T^{Co} ,a_{m}^{Cen} } \right)} \right) \ge E\left( {\varPi_{m}^{dec} \left( {T^{dec} ,a_{m}^{dec} } \right)} \right) $$

(A2)

in which, $dec$ indicates Remanufacture–Stackelberg game structure, i.e., RMS. Eqs. (A1) and (A2), ensure the expectations of the collector and the remanufacturer for participating in the coordination model, are met, respectively. It is noteworthy that the lower and upper bounds of fixed fee, i.e., ${F}_{min}$ and ${F}^{max}$ are calculated by simplifying Eqs. (A1) and (A2), respectively.$\square$

Appendix B. Supplementary symbols

$$ \begin{aligned} \varphi_{1} & = \frac{{\tau \left( {\gamma - \beta } \right)}}{{2\mu \left( {1 - r_{c} } \right)}}\left( {\tau \chi + \frac{{\tau \left( {\beta - \gamma } \right)\left[ {\beta \left( {\left( {r_{c} - 1} \right)k_{c} - r_{c} \eta_{c} } \right) + \alpha \chi \left( {r_{c} - 1} \right)} \right]}}{{2\beta \left( {1 - r_{c} } \right)}}} \right) \\ & \quad + \frac{{\tau \left( {\beta - \gamma } \right)}}{{2\left( {1 - r_{c} } \right)}}\left( {\frac{{\lambda - \tau \chi - \tau \left( {\beta - \gamma } \right)\left[ {\frac{{\beta \left( {\left( {r_{c} - 1} \right)k_{c} - r_{c} \eta_{c} } \right) + \alpha \chi \left( {r_{c} - 1} \right)}}{{2\beta \left( {1 - r_{c} } \right)}}} \right]}}{\mu }} \right) \\ & \quad - \frac{{\tau \left( {\beta - \gamma } \right)}}{{2\left( {1 - r_{c} } \right)}}e_{m} - \alpha \chi - \frac{{\beta^{2} \left( {\left( {r_{c} - 1} \right)k_{c} - r_{c} \eta_{c} } \right) + \beta \alpha \chi \left( {r_{c} - 1} \right)}}{{2\beta \left( {1 - r_{c} } \right)}} \\ & \quad - \frac{\gamma }{{2\left( {1 - r_{c} } \right)}}\left[ {\left( {r_{m} - 1} \right)k_{m} - r_{m} \eta_{m} } \right] \\ \end{aligned} $$

(B1)

$$ \varphi_{2} = - \frac{{\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{{2\mu \left( {1 - r_{c} } \right)}} - \frac{\beta }{{1 - r_{c} }} $$

(B2)

$$ \varphi_{3} = - \left( {\frac{{\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{{\mu \left( {1 - r_{c} } \right)}}} \right)\left( {\frac{\gamma }{2\beta } + 1} \right) + \frac{{\gamma \left( {2 - r_{m} - r_{c} } \right)}}{{2\left( {1 - r_{c} } \right)}} $$

(B3)

$$ \varphi_{4} = \frac{{\lambda - \tau \chi - \tau \left( {\beta - \gamma } \right)\left[ {\frac{{\beta \left( {r_{c} - 1} \right)k_{c} - r_{c} \eta_{c} + \alpha \chi \left( {r_{c} - 1} \right)}}{{2\beta \left( {1 - r_{c} } \right)}}} \right]}}{\mu } $$

(B4)

$$ \varphi_{5} = \frac{\tau }{\mu }\left( {\beta - \gamma } \right)\left( {\frac{\gamma }{2\beta } + 1} \right) $$

(B5)

$$ \varphi_{6} = \frac{{\tau \left( {\beta - \gamma } \right)}}{{2\mu \left( {1 - r_{c} } \right)}} $$

(B6)

$$ \begin{aligned} \mu_{1} & = \left( {1 - r_{m} } \right)\left( {1 - r_{c} } \right)\left[ { - \alpha \chi - \beta k_{c} + \left( {\frac{{\tau \left( {\beta - \gamma } \right)\left( {\lambda - 2\tau \chi } \right)}}{\mu } - \tau \left( {\beta - \gamma } \right)e_{m} } \right)} \right. \\ & \quad \left. { + \gamma k_{m} } \right] + r_{m} \left( {1 - r_{c} } \right)\left[ { - \alpha \chi - \beta k_{c} + \left( {\frac{{ - 2\tau^{2} \chi \left( {\beta - \gamma } \right)}}{\mu } + \frac{{\tau \left( {\beta - \gamma } \right)\lambda }}{\mu }} \right.} \right. \\ & \quad \left. { - \tau \left( {\beta - \gamma } \right)e_{m} ) + \gamma \eta_{m} } \right] + r_{c} \left( {1 - r_{m} } \right)\left[ { - \beta \eta_{c} + \left( {\frac{{ - 2\tau^{2} \chi \left( {\beta - \gamma } \right)}}{\mu } + \frac{{\tau \left( {\beta - \gamma } \right)\lambda }}{\mu }} \right.} \right. \\ & \quad \left. { - \tau \left( {\beta - \gamma } \right)e_{m} ) + \gamma k_{m} } \right] + r_{c} r_{m} \left[ {\beta \eta_{c} + \left( {\frac{{ - 2\tau^{2} \chi \left( {\beta - \gamma } \right)}}{\mu } + \frac{{\tau \left( {\beta - \gamma } \right)\lambda }}{\mu }} \right.} \right. \\ & \quad \left. { - \tau \left( {\beta - \gamma } \right)e_{m} ) + \gamma \eta_{m} } \right] \\ \end{aligned} $$

(B7)

$$ \mu_{2} = \left( {1 - r_{c} } \right)\left[ { - 2\beta - \frac{{2\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{\mu }} \right] - r_{c} \frac{{2\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{\mu } $$

(B8)

$$ \begin{aligned} \mu_{3} & = \left( {1 - r_{m} } \right)\left( {1 - r_{c} } \right)\left[ {2\gamma - \frac{{2\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{\mu }} \right] + \left( {r_{m} + r_{c} - 2r_{m} r_{c} } \right) \\ & \quad \times \left[ {\gamma - \frac{{2\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{\mu }} \right] - r_{c} r_{m} \frac{{2\tau^{2} \left( {\beta - \gamma } \right)^{2} }}{\mu } \\ \end{aligned} $$

(B9)

$$ \begin{aligned} \varsigma & = - \tau \left( {\beta - \gamma } \right)e_{m} - \left( {\frac{\lambda }{\mu } - \frac{\tau }{\mu }\left( {\chi + \left( {\beta - \gamma } \right)a_{c}^{Cen} } \right)} \right)\left( {\frac{{\tau \left( {\beta - \gamma } \right)\left( {\mu - 1} \right)}}{\mu }} \right) \\ & \quad - \left( {r_{m} - 1} \right)\left( {\left( {1 - \alpha } \right)\chi - \gamma a_{c}^{Cen} } \right) - \beta \left( {\left( {r_{m} - 1} \right)k_{m} - r_{m} \eta_{m} } \right) \\ \end{aligned} $$

(B10)

$$ \begin{aligned} F_{min} & = \left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c}^{dec} + a_{m}^{dec} } \right)} \right) \\ & \quad \times \left( {\frac{\tau \lambda }{\mu } - \frac{{\tau^{2} }}{\mu }\left( {\chi + \left( {\beta - \gamma } \right)a_{c}^{dec} + a_{m}^{dec} } \right)} \right. \\ & \quad \left. { - \tau e_{m} } \right) - T^{dec} \left( {\alpha \chi + \beta a_{c}^{dec} - \gamma a_{m}^{dec} } \right) \\ & \quad + \left( {\left( {r_{m} - 1} \right)a_{m}^{dec} + \left( {r_{m} - 1} \right)k_{m} } \right. \\ & \quad \left. { - r_{m} \eta_{m} } \right) \times \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m}^{dec} - \gamma a_{c}^{dec} } \right) \\ & \quad - \left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c}^{Cen} + a_{m}^{Cen} } \right)} \right) \\ & \quad \times \left( {\frac{\tau \lambda }{\mu } - \frac{{\tau^{2} }}{\mu }\left( {\chi + \left( {\beta - \gamma } \right)a_{c}^{Cen} + a_{m}^{Cen} } \right) - \tau e_{m} } \right) \\ & \quad + T^{Co} \left( {\alpha \chi + \beta a_{c}^{Cen} - \gamma a_{m}^{Cen} } \right) - \left( {\left( {r_{m} - 1} \right)a_{m}^{Cen} } \right. \\ & \quad \left. { + \left( {r_{m} - 1} \right)k_{m} - r_{m} \eta_{m} } \right) \\ & \quad \times \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m}^{Cen} - \gamma a_{c}^{Cen} } \right) \\ \end{aligned} $$

(B11)

$$ \begin{aligned} F^{max} & = \left( {T^{co} + \left( {r_{c} - 1} \right)a_{c}^{Cen} + \left( {r_{c} - 1} \right)k_{c} - r_{c} \eta_{c} } \right)\left( {\alpha \chi + \beta a_{c}^{Cen} - \gamma a_{m}^{Cen} } \right) \\ & \quad - \left( {T^{dec} + \left( {r_{c} - 1} \right)a_{c}^{dec} + \left( {r_{c} - 1} \right)k_{c} - r_{c} \eta_{c} } \right) \\ & \quad \times \left( {\alpha \chi + \beta a_{c}^{dec} - \gamma a_{m}^{dec} } \right) \\ \end{aligned} $$

(B12)

Considering Eqs. (4) and (5), the TWEP of the remanufacturer can be formulated as follows:

$$ \begin{aligned} RMP:\max E\left( {\varPi_{m} \left( {p_{m} , T,a_{m} } \right)} \right) & = \left( {1 - r_{m} } \right)\varPi_{m}^{{s_{I} ,s_{III} }} \left( {p_{m} , T,a_{m} } \right) + r_{m} \varPi_{m}^{{s_{II} ,s_{IV} }} \left( {p_{m} , T,a_{m} } \right) \\ & = \left( {\lambda - \mu p_{m} } \right)\left( {p_{m} - e_{m} } \right) - \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right)T + \left( {\left( {r_{m} - 1} \right)a_{m} } \right. \\ & \quad \left. { + \left( {r_{m} - 1} \right)k_{m} - r_{m} \eta_{m} } \right)\left( {\left( {1 - \alpha } \right)\chi + \beta a_{m} - \gamma a_{c} } \right) \\ \end{aligned} $$

(B13)

$$ {\text{Subject\,to }}\quad D\left( {p_{m} } \right) = \tau \left( {q_{c} + q_{m} } \right) $$

(B13-1)

$$ a_{c} \in argmax E\left( {\varPi_{c} \left( {a_{c} } \right)} \right) $$

(B13-2)

From constraint (B13-1), ${p}_{m}$ can be determined as ${p}_{m}=\frac{\lambda }{\mu }-\frac{\tau }{\mu }\left(\chi +\left(\beta -\gamma \right)\left({a}_{c}+{a}_{m}\right)\right)$. Then, by substituting $p_{m}$ into Eq. (B13), Eq. (6) will be determined. Equation (11) can be equivalently written as follows:

$$ \begin{aligned} & RSC:maxE\left( {\varPi_{SC} \left( {p_{m} ,a_{m} ,a_{c} } \right)} \right) \\ & \quad = \left( {1 - r_{m} } \right)\left( {1 - r_{c} } \right)\varPi_{SC}^{{s_{I} }} \left( {p_{m} ,a_{m} ,a_{c} } \right) + r_{m} \left( {1 - r_{c} } \right)\varPi_{SC}^{{s_{II} }} \left( {p_{m} ,a_{m} ,a_{c} } \right) \\ & \quad \quad + \left( {1 - r_{m} } \right)r_{c} \varPi_{SC}^{{s_{III} }} \left( {p_{m} ,a_{m} ,a_{c} } \right) + r_{m} r_{c} \varPi_{SC}^{{s_{IV} }} \left( {p_{m} ,a_{m} ,a_{c} } \right) \\ & = \left( {1 - r_{m} } \right)\left( {1 - r_{c} } \right)\left[ { - \left( {a_{c} + k_{c} } \right) \times \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right) + { }\left( {\lambda - \mu p_{m} } \right)\left( {p_{m} - e_{m} } \right) - \left( {\left( {1 - \alpha } \right)\chi } \right.} \right. \\ & \quad \quad \left. {\left. { + \beta a_{m} - \gamma a_{c} } \right) \left( {a_{m} + k_{m} } \right)} \right] + r_{m} \left( {1 - r_{c} } \right)\left[ { - \left( {a_{c} + k_{c} } \right) \times \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right)} \right. \\ & \quad \quad \left. { + \left( {\lambda - \mu p_{m} } \right)\left( {p_{m} - e_{m} } \right) - \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m} - \gamma a_{c} } \right)\eta_{m} } \right] \\ & \quad \quad + \left( {1 - r_{m} } \right)r_{c} \left[ { - \eta_{c} \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right) + { }\left( {\lambda - \mu p_{m} } \right)\left( {p_{m} - e_{m} } \right) - \left( {\left( {1 - \alpha } \right)\chi } \right.} \right. \\ & \quad \quad \left. {\left. { + \beta a_{m} - \gamma a_{c} } \right)\left( {a_{m} + k_{m} } \right)} \right] + r_{m} r_{c} \left[ { - b_{c} \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right)} \right. \\ & \quad \quad \left. { + { }\left( {\lambda - bp_{m} } \right)\left( {p_{m} - e_{m} } \right) - ((1 - \alpha )\chi + \beta a_{m} - \gamma a_{c} )\eta_{m} } \right] \\ \end{aligned} $$

(B14)

$$ {\text{Subject\,to }}\quad D\left( {p_{m} } \right) = \tau \left( {q_{c} + q_{m} } \right) $$

(B15)

Based upon constraint (B15), ${p}_{m}$ can be determined as ${p}_{m}=\frac{\lambda }{\mu }-\frac{\tau }{\mu }\left(\chi +\left(\beta -\gamma \right)\left({a}_{c}+{a}_{m}\right)\right)$, and then by substituting ${p}_{m}$ into $E\left({\varPi }_{SC}\left({p}_{m},{a}_{m},{a}_{c}\right)\right)$, the TWEP of the whole RSC is converted to:

$$ \begin{aligned} & RSC:maxE\left( {\varPi_{SC} \left( {a_{m} ,a_{c} } \right)} \right) \\ & \quad = \left( {1 - r_{m} } \right)\left( {1 - r_{c} } \right)\left[ { - \left( {a_{c} + k_{c} } \right)\left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right)} \right. \\ & \quad \quad + \left( {\tau \left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right)} \right) \times \left( {\frac{\lambda }{\mu } - \frac{\tau }{\mu }\left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right) - e_{m} } \right) \\ & \quad \quad \left. { - \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m} - \gamma a_{c} } \right) \times \left( {a_{m} + k_{m} } \right)} \right] + r_{m} \left( {1 - r_{c} } \right) \\ & \quad \quad \times \left[ { - \left( {a_{c} + k_{c} } \right)\left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right) + \left( {\tau \left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right)} \right) \times \left( {\frac{\lambda }{\mu }} \right.} \right. \\ & \quad \quad \left. {\left. {\left. { - \frac{\tau }{\mu }(\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right) - e_{m} } \right) - \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m} - \gamma a_{c} } \right)\eta_{m} } \right] \\ & \quad \quad + \left( {1 - r_{m} } \right)r_{c} \times \left[ { - \eta_{c} \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right) + \left( {\tau \left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right)} \right) \times \left( {\frac{\lambda }{\mu }} \right.} \right. \\ & \quad \quad \left. {\left. { - \frac{\tau }{\mu }\left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right) - e_{m} } \right) - \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m} - \gamma a_{c} } \right) \left( {a_{m} + k_{m} } \right)} \right] \\ & \quad \quad + r_{m} r_{c} \left[ { - \eta_{c} \left( {\alpha \chi + \beta a_{c} - \gamma a_{m} } \right) + \left( {\tau \left( {\chi + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right)} \right) \times \left( {\frac{\lambda }{\mu } - \frac{\tau }{\mu }\left( \chi \right.} \right.} \right. \\ & \quad \quad \left. {\left. {\left. { + \left( {\beta - \gamma } \right)\left( {a_{c} + a_{m} } \right)} \right) - e_{m} } \right) - \left( {\left( {1 - \alpha } \right)\chi + \beta a_{m} - \gamma a_{c} } \right)\eta_{m} } \right] \\ \end{aligned} $$

(B16)