Pricing rules of Green Supply Chain considering Big Data information inputs and cost-sharing model

Abstract

Big Data provide an opportunity for decision makers to implement green production and order plans with accurate and timely consumer demand information. Thus, many enterprise begun to invest in consumer preference information based on big data (CPIBD) and green production technology. These will add their extra costs, and cost-sharing model is an effective way to improve chain members’ benefits. However, in the new environment, how to price can improve their outcomes in different cost-sharing models? And which model is the best? Aims of this paper are to solve the proposed issues considering CPIBD input and the green technology R&D cost. We chose a green supply chain with one green manufacturer and one retailer as study subject. Then, based on game theory, we proposed three cost-sharing models, and their benefit functions were developed. Using the reverse induction, we analyzed and discussed the change rules of the retail price and the wholesale price with the product green degree and the unit CPIBD cost. Then, using Matlab2014, a numerical example based on actual data was implemented. Findings: (1) with the increase of the unit CPIBD cost, the optimal retail prices will grow in the proposed three models, and the change trend of the wholesale price has a relationship with the situation whether the retailer undertakes the unit CPIBD cost or the green technology R&D cost. (2) With the growth of the product green degree, the optimal retail price and the optimal wholesale price in the three models will decrease. (3) With the increase of the unit CPIBD cost and the product green degree, benefits of supply chain members in the proposed three models will reduce. If the retailer can undertake the unit CPIBD cost or the green technology R&D cost, benefits of supply chain will be higher.

Appendices

Appendix A

1.1 Analysis process of Property 1

Based on formulas (9) and (10), we can get ${{\partial p^{M*} } \mathord{\left/ {\vphantom {{\partial p^{M*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} > 0$ and ${{\partial w^{M*} } \mathord{\left/ {\vphantom {{\partial w^{M*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} > 0$. Similarly, ${{\partial p^{M*} } \mathord{\left/ {\vphantom {{\partial p^{M*} } {\partial {\text{g}}}}} \right. \kern-\nulldelimiterspace} {\partial {\text{g}}}} = {{ - 3\lambda } \mathord{\left/ {\vphantom {{ - 3\lambda } {4e}}} \right. \kern-\nulldelimiterspace} {4e}}$ and ${{\partial w^{M*} } \mathord{\left/ {\vphantom {{\partial w^{M*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda } \mathord{\left/ {\vphantom {{ - \lambda } {2e}}} \right. \kern-\nulldelimiterspace} {2e}}$. Because $\lambda > 0$ and $e > 0$, ${{\partial p^{M*} } \mathord{\left/ {\vphantom {{\partial p^{M*} } {\partial {\text{g}}}}} \right. \kern-\nulldelimiterspace} {\partial {\text{g}}}} < 0$ and ${{\partial w^{M*} } \mathord{\left/ {\vphantom {{\partial w^{M*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} < 0$. In addition, ${{\partial \pi_{r}^{M*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{M*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{ - N_{1} } \mathord{\left/ {\vphantom {{ - N_{1} } {8e}}} \right. \kern-\nulldelimiterspace} {8e}}$, ${{\partial \pi_{d}^{M*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{M*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{ - N_{1} } \mathord{\left/ {\vphantom {{ - N_{1} } {4e}}} \right. \kern-\nulldelimiterspace} {4e}}$, here, $N_{1} = a - e(c_{o} + c_{r} + c_{vr} ) \, - \, \lambda g - \, e\theta (c \, + \, c_{d} + c_{vd} ) > 0$, thus, ${{\partial \pi_{r}^{M*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{M*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} < 0$ and ${{\partial \pi_{d}^{M*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{M*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} < 0$. Similarly, ${{\partial \pi_{r}^{M*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{M*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda N_{1} } \mathord{\left/ {\vphantom {{ - \lambda N_{1} } {8e}}} \right. \kern-\nulldelimiterspace} {8e}} < 0$ and ${{\partial \pi_{d}^{M*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{M*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda N_{1} } \mathord{\left/ {\vphantom {{ - \lambda N_{1} } {4e}}} \right. \kern-\nulldelimiterspace} {4e}} - z_{1} \, g < 0$, In summary, Property 1 is proved.

Appendix B

2.1 Analysis process of Proposition 3

Proof Based on formula (14), by calculating the second-order partial derivative of $\pi_{d}^{MR} (w^{MR} ,g,c_{o} )$ with respect to $w^{MR}$, $c_{o}$, and ${\text{g}}$, we get Hessian matrix $H^{2}$. Based on $H^{2}$, we find $\frac{{\partial^{2} \pi_{{\text{d}}}^{MR} }}{{\partial (w^{MR} )^{2} }} = - e < 0$, $\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (g)^{2} }} = - z_{1} < 0$, and $\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (w^{MR} )^{2} }}\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (g)^{2} }} - (\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial w^{MR} \partial g}}\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial g\partial w^{MR} }})^{2} = ez_{1} > 0$, thus, $\pi_{d}^{MR} (w^{MR} ,g,c_{o} )$ is the union concave function of $w^{MR}$ and ${\text{g}}$. Because $\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (w^{MR} )^{2} }}\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (c_{o} )^{2} }} - (\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial w^{MR} \partial c_{o} }}\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial c_{o} \partial w^{MR} }})^{2} = [ - {{e\gamma^{2} + (1 - \gamma )({\text{e}}\gamma - 1)]} \mathord{\left/ {\vphantom {{e\gamma^{2} + (1 - \gamma )({\text{e}}\gamma - 1)]} 2}} \right. \kern-\nulldelimiterspace} 2}$, $\pi_{d}^{MR} (w^{MR} ,g,c_{o} )$ is not the union concave function of $w^{MR}$ and $c_{o}$. In summary, $\pi_{d}^{MR} (w^{MR} ,g,c_{o} )$ is not the union concave function of $w^{MR}$, $c_{o}$, and ${\text{g}}$.

$$ H^{2} = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (w^{MR} )^{2} }}} & {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial w^{MR} \partial g}}} & {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial w^{MR} \partial c_{o} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial g\partial w^{MR} }}} \\ {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial c_{o} \partial w^{MR} }}} \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (g)^{2} }}} \\ {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial c_{o} \partial g}}} \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial g\partial c_{o} }}} \\ {\frac{{\partial^{2} \pi_{d}^{MR} }}{{\partial (c_{o} )^{2} }}} \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - e} & {\begin{array}{*{20}c} 0 & {{{(e\gamma + 1 - \gamma )} \mathord{\left/ {\vphantom {{(e\gamma + 1 - \gamma )} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ {(e\gamma + 1 - \gamma )} \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - z_{1} } \\ 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ { - \gamma (1 - \gamma )} \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] $$

2.2 Analysis process of Property 2

Based on formulas (15) and (16), we can get ${{\partial p^{MR*} } \mathord{\left/ {\vphantom {{\partial p^{MR*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} > 0$ and ${{\partial w^{MR*} } \mathord{\left/ {\vphantom {{\partial w^{MR*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{(2\gamma - 1)} \mathord{\left/ {\vphantom {{(2\gamma - 1)} 2}} \right. \kern-\nulldelimiterspace} 2} > 0$. Similarly, ${{\partial p^{MR*} } \mathord{\left/ {\vphantom {{\partial p^{MR*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = - {{3\lambda } \mathord{\left/ {\vphantom {{3\lambda } {4e}}} \right. \kern-\nulldelimiterspace} {4e}}$ and ${{\partial w^{MR*} } \mathord{\left/ {\vphantom {{\partial w^{MR*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = - {\lambda \mathord{\left/ {\vphantom {\lambda {2e}}} \right. \kern-\nulldelimiterspace} {2e}}$. Because $\lambda > 0$ and $e > 0$, ${{\partial p^{MR*} } \mathord{\left/ {\vphantom {{\partial p^{MR*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} < 0$ and ${{\partial w^{MR*} } \mathord{\left/ {\vphantom {{\partial w^{MR*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} < 0$. In addition, ${{\partial \pi_{r}^{MR*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{MR*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{ - N_{2} } \mathord{\left/ {\vphantom {{ - N_{2} } 8}} \right. \kern-\nulldelimiterspace} 8}$ and ${{\partial \pi_{d}^{MR*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{MR*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{ - N_{2} } \mathord{\left/ {\vphantom {{ - N_{2} } 4}} \right. \kern-\nulldelimiterspace} 4}$, here, $N_{2} = a - ec_{o} \, - \, \lambda g - \, e\theta (c_{r} + c_{vr} + c \, + \, c_{d} + c_{vd} ) > 0$, thus, ${{\partial \pi_{r}^{MR*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{MR*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} < 0$ and ${{\partial \pi_{d}^{MR*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{MR*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} < 0$. Similarly, ${{\partial \pi_{r}^{MR*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{MR*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda N_{2} } \mathord{\left/ {\vphantom {{ - \lambda N_{2} } {8e}}} \right. \kern-\nulldelimiterspace} {8e}} < 0$ and ${{\partial \pi_{d}^{MR*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{MR*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda N_{2} } \mathord{\left/ {\vphantom {{ - \lambda N_{2} } {4e}}} \right. \kern-\nulldelimiterspace} {4e}} - z_{1} \, g < 0$. In summary, Property 2 is proved.

Appendix C

3.1 Analysis process of Proposition 5

Proof Based on formula (20), by calculating the second-order partial derivative of $\pi_{d}^{MRT} (w^{MRT} ,g,c_{o} )$ with respect to $w^{MRT}$, $c_{o}$, and ${\text{g}}$, we get Hessian matrix $H^{2}$. Based on $H^{2}$, we find $\frac{{\partial^{2} \pi_{{\text{d}}}^{MRT} }}{{\partial (w^{MRT} )^{2} }} = - e < 0$, $\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (g)^{2} }} = - uz_{1} < 0$, and $\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (w^{MRT} )^{2} }}\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (g)^{2} }} - (\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial w^{MRT} \partial g}}\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial g\partial w^{MRT} }})^{2} = uez_{1} - \lambda^{2}$, thus, $\pi_{d}^{MRT} (w^{MRT} ,g,c_{o} )$ is not the union concave function of $w^{MRT}$ and ${\text{g}}$. Because $\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (w^{MRT} )^{2} }}\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (c_{o} )^{2} }} - (\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial w^{MRT} \partial c_{o} }}\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial c_{o} \partial w^{MRT} }})^{2} = {{[e(2\gamma - 1)]^{2} } \mathord{\left/ {\vphantom {{[e(2\gamma - 1)]^{2} } {16}}} \right. \kern-\nulldelimiterspace} {16}} - {{e^{2} \gamma (1 - \gamma )} \mathord{\left/ {\vphantom {{e^{2} \gamma (1 - \gamma )} 2}} \right. \kern-\nulldelimiterspace} 2}$, $\pi_{d}^{MRT} (w^{MRT} ,g,c_{o} )$ is not the union concave function of $w^{MRT}$ and $c_{o}$. In summary, $\pi_{d}^{MRT} (w^{MRT} ,g,c_{o} )$ is not the union concave function of $w^{MRT}$, $c_{o}$, and ${\text{g}}$.

$$ H^{3} = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (w^{MRT} )^{2} }}} & {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial w^{MRT} \partial g}}} & {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial w^{MRT} \partial c_{o} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial g\partial w^{MRT} }}} \\ {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial c_{o} \partial w^{MRT} }}} \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (g)^{2} }}} \\ {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial c_{o} \partial g}}} \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial g\partial c_{o} }}} \\ {\frac{{\partial^{2} \pi_{d}^{MRT} }}{{\partial (c_{o} )^{2} }}} \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - e} & {\begin{array}{*{20}c} \lambda & {{{e(2\gamma - 1)} \mathord{\left/ {\vphantom {{e(2\gamma - 1)} 4}} \right. \kern-\nulldelimiterspace} 4}} \\ \end{array} } \\ {\begin{array}{*{20}c} \lambda \\ {{{e(2\gamma - 1)} \mathord{\left/ {\vphantom {{e(2\gamma - 1)} 4}} \right. \kern-\nulldelimiterspace} 4}} \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - uz_{1} } \\ {{{ - \lambda \gamma } \mathord{\left/ {\vphantom {{ - \lambda \gamma } 4}} \right. \kern-\nulldelimiterspace} 4}} \\ \end{array} } & {\begin{array}{*{20}c} {{{ - \lambda \gamma } \mathord{\left/ {\vphantom {{ - \lambda \gamma } 4}} \right. \kern-\nulldelimiterspace} 4}} \\ {{{ - e\gamma (1 - \gamma )} \mathord{\left/ {\vphantom {{ - e\gamma (1 - \gamma )} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] $$

3.2 Analysis process of Property 3

Based on formula (18), we can get ${{\partial p^{MRT*} } \mathord{\left/ {\vphantom {{\partial p^{MRT*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} > 0$ and ${{\partial w^{MRT*} } \mathord{\left/ {\vphantom {{\partial w^{MRT*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{(2\gamma - 1)} \mathord{\left/ {\vphantom {{(2\gamma - 1)} 2}} \right. \kern-\nulldelimiterspace} 2} > 0$. Similarly, ${{\partial p^{MRT*} } \mathord{\left/ {\vphantom {{\partial p^{MRT*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = - {{3\lambda } \mathord{\left/ {\vphantom {{3\lambda } {4e}}} \right. \kern-\nulldelimiterspace} {4e}}$ and ${{\partial w^{MRT*} } \mathord{\left/ {\vphantom {{\partial w^{MRT*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = - {\lambda \mathord{\left/ {\vphantom {\lambda {2e}}} \right. \kern-\nulldelimiterspace} {2e}}$. Because $\lambda > 0$ and $e > 0$, ${{\partial p^{MRT*} } \mathord{\left/ {\vphantom {{\partial p^{MRT*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} < 0$ and ${{\partial w^{MRT*} } \mathord{\left/ {\vphantom {{\partial w^{MRT*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} < 0$. In addition, ${{\partial \pi_{r}^{MRT*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{MRT*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{ - N_{2} } \mathord{\left/ {\vphantom {{ - N_{2} } 8}} \right. \kern-\nulldelimiterspace} 8} < 0$, ${{\partial \pi_{d}^{MRT*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{MRT*} } {\partial c_{o} }}} \right. \kern-\nulldelimiterspace} {\partial c_{o} }} = {{ - N_{2} } \mathord{\left/ {\vphantom {{ - N_{2} } 4}} \right. \kern-\nulldelimiterspace} 4} < 0$, ${{\partial \pi_{r}^{MRT*} } \mathord{\left/ {\vphantom {{\partial \pi_{r}^{MRT*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda N_{2} } \mathord{\left/ {\vphantom {{ - \lambda N_{2} } {8e}}} \right. \kern-\nulldelimiterspace} {8e}} < 0$, and ${{\partial \pi_{d}^{MRT*} } \mathord{\left/ {\vphantom {{\partial \pi_{d}^{MRT*} } {\partial g}}} \right. \kern-\nulldelimiterspace} {\partial g}} = {{ - \lambda N_{2} } \mathord{\left/ {\vphantom {{ - \lambda N_{2} } {4e}}} \right. \kern-\nulldelimiterspace} {4e}} - 4uz_{1} \, g < 0$.