Sustainable E-waste supply chain management with price/sustainability-sensitive demand and government intervention

Abstract

The impact of appropriate E-waste management practices on the environment, human health, and natural resources has made E-waste management an interesting research topic in recent decades. Research also shows that government intervention is an important factor in controlling the emission volume produced by waste management systems. This study considers a base E-waste supply chain in which a collection center is responsible for collecting E-waste and delivering it to a recycling center. The recycling center recovers valuable material and sells it to electronic device manufacturing companies using a price/sustainability-sensitive demand. E-waste material recovery generates emissions that are undesirable for manufacturing companies. Two extended cases regarding the base supply chain are studied, as well: (1) the recycling center is also active in E-waste collection. (2) There are two active recycling centers. Although the sustainability-sensitive demand is a controlling factor for material recovery emissions, government interferences through tariff and emission penalties make sure that sustainable issues are considered in the material recovery process. Each plant in this study makes a marginal profit by processing E-waste; therefore, it is important to know which plant is the primary decision-maker when it comes to price. Because of its capability in terms of solving interactive decision-making problems, game theory is used to model different scenarios in our problem. Equilibrium values are derived, and a numerical example with parameter sensitivity analysis is provided to show the applicability of the proposed models. The results show that the E-waste supply chain makes more profit and selects a higher level of material recovery sustainability if the plants work under a centralized decision-making framework. Moreover, it is more profitable for the entire E-waste supply chain if the recycling center undertakes a portion of the E-waste collection activity.

Appendices

Appendix A

Proof of Proposition 1

The value of $w$ maximizes the collection center’s profit function if it solves the first derivative of the profit function equal to zero:

$$ \begin{aligned} \frac{{\partial \Pi_{\text{CC}} }}{\partial w} &= D + c\alpha - (m + t + 2w)\alpha + \beta \theta = 0 \to w \\ &= \frac{D + c\alpha - (m + t)\alpha + \beta \theta }{2\alpha }. \end{aligned} $$

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Proof of Proposition 2

If we simultaneously solve the first derivative of ${{\Pi }}_{\text{RC}}$ with respect to $m$ and $\theta$, we find the following optimal values:

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 8i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

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$$ m = \frac{{\alpha ( { - 4i ( {c - ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {4i - be ( {be\alpha + \beta } )} )}}{{8i\alpha - ( {be\alpha + \beta } )^{2} }}. $$

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Proof of Proposition 3

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (4), we can find the value for $t$. We can substitute (15) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 8i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 8i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {8i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + a ( { - \alpha ( {8i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}}. $$

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Proof of Proposition 4

If we simultaneously solve the first derivative of ${{\Pi }}_{\text{CC - RC}}$ with respect to $m$ and $\theta$, we find the following optimal values:

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 4i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

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$$ m = \frac{{\alpha ( { - 2i ( {c - ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {2i - be ( {be\alpha + \beta } )} )}}{{4i\alpha - ( {be\alpha + \beta } )^{2} }}. $$

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Proof of Proposition 5

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (6), we can find the value for $t$. We can then substitute (18) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 4i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 4i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {4i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + a ( { - \alpha ( {4i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 4i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}} . $$

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Proof of Proposition 6

To find the optimal values for $m$, $w$, and $\theta$, we should simultaneously solve $\frac{{\partial \Pi_{\text{CC}} }}{\partial w} = 0$, $\frac{{\partial \Pi_{\text{RC}} }}{\partial \theta } = 0$, and $\frac{{\partial {{\Pi }}_{\text{RC}} }}{\partial m} = 0$. Therefore:

$$ w = \frac{{ - 2Di + 2i ( {ae + t} )\alpha + c ( { - 4i\alpha + ( {be\alpha + \beta } )^{2} } )}}{{ - 6i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

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$$ m = \frac{{\alpha ( { - 2i ( {c - 2ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {2i - be ( {be\alpha + \beta } )} )}}{{6i\alpha - ( {be\alpha + \beta } )^{2} }}, $$

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$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 6i\alpha + ( {be\alpha + \beta } )^{2} }}. $$

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Proof of Proposition 7

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (4), we can find the value for $t$. We can then substitute (22) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 6i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 6i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {6i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + a ( { - \alpha ( {6i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 6i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}} . $$

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Proof of Proposition 9

If we simultaneously solve the first derivative of ${{\Pi }}_{\text{RC}}$ with respect to $m$ and $\theta$, we find the following optimal values:

$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 6i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

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$$ m = \frac{{\alpha ( { - 2i ( {c - 2ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {2i - be ( {be\alpha + \beta } )} )}}{{6i\alpha - ( {be\alpha + \beta } )^{2} }}. $$

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Proof of Proposition 11

To find the optimal values for $m$, $w$, and $\theta$, we should simultaneously solve $\frac{{\partial \Pi_{\text{CC}} }}{\partial w} = 0$, $\frac{{\partial \Pi_{\text{RC}} }}{\partial \theta } = 0$, and $\frac{{\partial {{\Pi }}_{\text{RC}} }}{\partial m} = 0$ from Eqs. (7) and (8). Therefore:

$$ w = \frac{{ - 2Di + 2i ( {ae + t} )\alpha + c ( { - 3i\alpha + ( {be\alpha + \beta } )^{2} } )}}{{ - 5i\alpha + ( {be\alpha + \beta } )^{2} }}, $$

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$$ m = \frac{{\alpha ( { - i ( {c - 4ae + t} ) + b^{2} e^{2} ( {c + t} )\alpha } ) + be ( {c - ae + t} )\alpha \beta - ae\beta^{2} + D ( {i - be ( {be\alpha + \beta } )} )}}{{5i\alpha - ( {be\alpha + \beta } )^{2} }}, $$

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$$ \theta = - \frac{{ ( {D - ( {c + ae + t} )\alpha } ) ( {be\alpha + \beta } )}}{{ - 5i\alpha + ( {be\alpha + \beta } )^{2} }}. $$

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Proof of Proposition 12

Solving $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{\partial t} = 0$ from Eq. (4), we can find the value for $t$. We can then substitute (28) into the other equations to find the optimal value for all decision variables:

$$ t = \frac{{\begin{array}{*{20}c} {D ( { - 5i\alpha - ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + \alpha ( - 2ae ( { - 5i\alpha + \beta ( {be\alpha + \beta } )} )} \\ { + c ( {5i\alpha + ( {be\alpha + \beta } ) ( { - \beta + b\alpha ( {e - 2\lambda } )} )} ) + + a ( { - \alpha ( {5i + b^{2} e^{2} \alpha } ) + \beta^{2} } )\lambda )} \\ \end{array} }}{{2\alpha ( { - 5i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}}. $$

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Proof of Proposition 16

To find the optimal values for $w_{z}$ and $m_{z}$, we should simultaneously solve $\frac{{\partial {{\Pi }}_{\text{CC}} }}{{\partial w_{z} }} = 0$, $\frac{{\partial {{\Pi }}_{{{\text{RC}}_{z} }} }}{{\partial m_{z} }} = 0$ from Eqs. (9) and (10). Therefore:

$$ w_{z} = \frac{{ - 12i ( {D - a\alpha \lambda } ) + ( { - 54i\alpha + 5 ( {\beta + b\alpha \lambda } )^{2} } )c_{z} + ( { - 6i\alpha + ( {\beta + b\alpha \lambda } )^{2} } )c_{3 - z} }}{{6 ( { - 12i\alpha + ( {\beta + b\alpha \lambda } )^{2} } )}}, $$

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$$ m_{z} = \frac{{\begin{array}{*{20}c} {6 ( {D ( { - 2i + be ( {\beta + b\alpha \lambda } )} ) + a ( {2i\alpha \lambda + e ( { - 12i\alpha + \beta^{2} + b\alpha \beta \lambda } )} )} )} \\ { + ( {18i\alpha - ( {\beta + b\alpha \lambda } ) ( {\beta + b\alpha ( {3e + \lambda } )} )} )c_{1} + ( { - 6i\alpha + ( {\beta + b\alpha \lambda } ) ( {\beta + b\alpha ( { - 3e + \lambda } )} )} )c_{2} } \\ \end{array} }}{{6 ( { - 12i\alpha + ( {\beta + b\alpha \lambda } )^{2} } )}}. $$

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Appendix B

Proof of Lemma 2

We should find the sign for the principal minors of a Hessian matrix:

$$ H = \left ( {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{{\partial m^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{\partial m\partial \theta }} \\ {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{\partial \theta \partial m}} & {\frac{{\partial^{2} {{\Pi }}_{\text{RC}} }}{{\partial \theta^{2} }}} \\ \end{array} } \right ) = \left ( {\begin{array}{*{20}c} { - \alpha } & {\frac{1}{2} ( { - be\alpha + \beta } )} \\ {\frac{1}{2} ( { - be\alpha + \beta } )} & { - 2i + be\beta } \\ \end{array} } \right ) . $$

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The first principal minor of $H$ is $H^{1} = - {{\alpha }} < 0$. The objective function is joint concave in $m$ and $\theta$ if $2i\alpha - \frac{1}{4} ( {be\alpha + \beta } )^{2} > 0$.

Proof of Lemma 3

If we substitute the values of $m$ and $\theta$ from proposition (2) into profit function (4), and take the second derivative with respect to $t$, we have:

$$ \frac{{\partial {{\Pi }}_{\text{Gov}} }}{{\partial t^{2} }} = \frac{{4i\alpha^{2} ( { - 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } )} )}}{{ ( { - 8i\alpha + ( {be\alpha + \beta } )^{2} } )^{2} }} . $$

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The government profit function is concave in $t$ if $\frac{{\partial {{\Pi }}_{\text{Gov}} }}{{\partial t^{2} }} < 0$. This is only true if $- 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } ) < 0$. If Lemma 2 holds true, $8i\alpha > ( {be\alpha + \beta } )^{2}$. Therefore, $- 8i\alpha + ( {be\alpha + \beta } ) ( {\beta + b\alpha \lambda } ) < 0$ holds true if and only if $e > \lambda$.

Proof of Lemma 4

We should find the sign for the principal minors of a Hessian matrix:

$$ H = \left ( {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{{\partial m^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{\partial m\partial \theta }} \\ {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{\partial \theta \partial m}} & {\frac{{\partial^{2} {{\Pi }}_{\text{CC - RC}} }}{{\partial \theta^{2} }}} \\ \end{array} } \right ) = \left ( {\begin{array}{*{20}c} { - 2\alpha } & { - be\alpha + \beta } \\ { - be\alpha + \beta } & { - 2i + 2be\beta } \\ \end{array} }\right ). $$

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The first principal minor of $H$ is $H^{1} = - 2{{\alpha }} < 0$. The objective function is joint concave in $m$ and $\theta$ if $4i\alpha - ( {be\alpha + \beta } )^{2} > 0$.

Proof of Lemma 7

We should find the sign for the principal minors of a Hessian matrix:

$$ H = \left ( {\begin{array}{*{20}c} {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{{\partial t^{2} }}} & {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{\partial t\partial \theta }} \\ {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{\partial \theta \partial t}} & {\frac{{\partial^{2} {{\Pi }}_{\text{Gov}} }}{{\partial \theta^{2} }}} \\ \end{array} } \right ) = \left( {\begin{array}{*{20}c} { - \frac{\alpha }{2}} & {\frac{1}{4} ( {\beta + b\alpha ( {2e - \lambda } )} )} \\ {\frac{1}{4} ( {\beta + b\alpha ( {2e - \lambda } )} )} & {\frac{1}{2} ( { - 4i + b ( {be\alpha + \beta } ) ( { - e + \lambda } )} )} \\ \end{array} } \right ) . $$

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The first principal minor of $H$ is $H^{1} = - \frac{\alpha }{2} < 0$. The objective function is joint concave in $t$ and $\theta$ if $\frac{1}{16} ( {16i\alpha - ( {\beta + b\alpha \lambda } )^{2} } ) > 0$.