Optimizing the sustainable decisions in a multi-echelon closed-loop supply chain of the manufacturing/remanufacturing products with a competitive environment

Abstract

Along with raised public awareness about environmental issues, some supply chains have begun to develop eco-friendly strategies aiming to manufacture and (or) remanufacture green products. This research presents a new preferable gaming structure in a multi-echelon closed-loop supply chain including a manufacturer, a retailer/remanufacturer, and an intermediate collection center (which is a governmental entity). The supply chain is structured to sell the products in a first and a secondary market. The significance of the research ahead is enhanced due to the use of game theory in sustainability in a new competitive supply chain structure. Toward addressing sustainability requirements, emission reductions within manufacturing, remanufacturing, and delivery functions are considered in a two-period game-theoretic-based model. Regarding the customer’s low-carbon preferences in production and delivery, the manufacturer wholesales new products to the retailer, and then, the retailer forwards them to the primary market via a full truckload policy in the first period. Even more striking, considering the environmental commitments in the second period, used products are gathered by the collection center in a reverse flow and sent to the retailer/remanufacturer for remanufacturing processes. In this regard, a Stackelberg, a Nash game, and also a novel bargaining structure are developed in the first and second periods. Then, the resulting decentralized approach is compared with a centralized model, illustrating a better performance of the decentralized scenario. Finally, through numerical analyses, it is observed that there are ascending relations between the triple dimensions of reduction rates (in manufacturing, remanufacturing, and delivering) and the corresponding profits, in which incorporating low-carbon considerations in remanufacturing has a greater efficacy.

Appendix

Proof 1

For a multivariable function to be concave, it is incumbent to show that the sign of its Hessian matrix is the same as $\left( { - 1} \right)^{\varepsilon }$ in which ε represents the order of the Hessian matrix. According to the defined profit function for $\pi_{\text{cen}}^{1}$, the Hessian matrix of $\pi_{\text{cen}}^{1}$ is calculated as follows:

$$H\left( {r_{1} , s_{1}^{\text{ma}} ,s_{1}^{\text{re}} } \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial r_{1}^{2} }}} & {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial r_{1} \partial s_{1}^{\text{ma}} }}} & {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial r_{1} \partial s_{1}^{\text{re}} }}} \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial s_{1}^{\text{ma}} \partial r_{1} }}} & {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial s_{1}^{\text{ma2}} }}} & {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial s_{1}^{\text{ma}} \partial s_{1}^{\text{re}} }}} \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial s_{1}^{\text{re}} \partial r_{1} }}} & {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial s_{1}^{\text{re}} \partial s_{1}^{\text{ma}} }}} & {\frac{{\partial^{2} \pi_{\text{cen}}^{1} }}{{\partial s_{1}^{\text{re2}} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & \theta & \beta \\ \theta & { - \lambda } & 0 \\ \beta & 0 & { - \sigma } \\ \end{array} } \right]$$

Regarding the Hessian matrix obtained above and knowing that $2 - \theta^{2} > 0$, if the condition $- 2\lambda \sigma + \theta^{2} \sigma + \beta^{2} \lambda < 0$ be satisfied, the joint concavity of the profit function on decision variables is proven.□

Proof 2

Similarly, the Hessian matrix of $\pi_{\text{cen}}^{2}$ is obtained to demonstrate that the corresponding profit function is concave and also a negative definite. Therefore,

$$\begin{aligned} H\left( {r_{2} , s_{2}^{\text{ma}} , s_{2}^{\text{re}} , r', s^{\text{co}} ,s^{\text{rm}} } \right) = & \left[ {\begin{array}{*{20}l} {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r_{2}^{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r_{2} \partial s_{2}^{\text{ma}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r_{2} \partial s_{2}^{\text{re}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r_{2} \partial r'}}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r_{2} \partial s^{\text{co}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r_{2} \partial s^{\text{rm}} }}} \hfill \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{ma}} \partial r_{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{ma2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{ma}} \partial s_{2}^{\text{re}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{ma}} \partial r'}}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{ma}} \partial s^{\text{co}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{ma}} \partial s^{\text{rm}} }}} \hfill \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{re}} \partial r_{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{re}} \partial s_{2}^{\text{ma}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{re2}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{re}} \partial r'}}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{re}} \partial s^{\text{co}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s_{2}^{\text{re}} \partial s^{\text{rm}} }}} \hfill \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r'\partial r_{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r'\partial s_{2}^{ma} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{\text{re}}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r'^{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r'\partial s^{\text{co}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial r'\partial s^{\text{rm}} }}} \hfill \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{co}} \partial r_{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{co}} \partial s_{2}^{\text{ma}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{co}} \partial s_{2}^{\text{re}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{co}} \partial r'}}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{co2}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{co}} \partial s^{\text{rm}} }}} \hfill \\ {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{rm}} \partial r_{2} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{rm}} \partial s_{2}^{\text{ma}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{rm}} \partial s_{2}^{\text{re}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{rm}} \partial r'}}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{rm}} \partial s^{\text{co}} }}} \hfill & {\frac{{\partial^{2} \pi_{\text{cen}}^{2} }}{{\partial s^{\text{rm2}} }}} \hfill \\ \end{array} } \right] \\ = & \left[ {\begin{array}{*{20}l} { - 2} \hfill & \theta \hfill & \beta \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \theta \hfill & { - \lambda } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \beta \hfill & 0 \hfill & { - \sigma } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & { - 2} \hfill & \alpha \hfill & \beta \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & \alpha \hfill & { - \lambda '} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & \beta \hfill & 0 \hfill & { - \sigma } \hfill \\ \end{array} } \right] \\ \end{aligned}$$

Again, if the determinant of the above matrix be more than zero—i.e., if the condition $4\lambda \sigma \left( {\lambda^{\prime}\sigma - 0} \right) + 2\sigma \theta^{2} \left( {\lambda^{\prime}\sigma - 0} \right) - 2\lambda \beta \left( {\lambda^{\prime}\sigma - 0} \right) = 4\lambda \lambda^{\prime}\sigma^{2} + 2\theta^{2} \lambda^{\prime}\sigma^{2} - 2\lambda \lambda^{\prime}\beta \sigma > 0$ be satisfied, the Hessian matrix of $\pi_{\text{cen}}^{2}$ is a negative definite and also a joint concave function with respect to related decision variables.□

Proof 3

The concavity of the Hessian matrix of $\pi_{{{\text{ma}}_{\left( 1 \right)} }}^{n}$ must be shown. Therefore,

$$H\left( {\varphi_{1} , s_{1}^{\text{ma}} } \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 1 \right)} }}^{n} }}{{\partial \varphi_{1}^{2} }}} & {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 1 \right)} }}^{n} }}{{\partial \varphi_{1} \partial s_{1}^{\text{ma}} }}} \\ {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 1 \right)} }}^{n} }}{{\partial s_{1}^{\text{ma}} \partial \varphi_{1} }}} & {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 1 \right)} }}^{n} }}{{\partial s_{1}^{{{\text{ma}}2}} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2\sigma } & {\theta \sigma } \\ {\theta \sigma } & {\beta^{2} \lambda - 2\sigma \lambda } \\ \end{array} } \right]$$

The Hessian matrix of obtained above is a negative definite and also concave with respect to $\varphi_{1}$ and $s_{1}^{\text{ma}}$ if the corresponding condition $2\sigma^{2} \lambda - \lambda \sigma^{2} \beta^{2} - \theta^{2} \sigma^{2} > 0$ be satisfied.□

Proof 4

The second-order derivatives of $\pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n}$ with respect to $\varphi_{2}$ and $s_{2}^{\text{ma}}$ are, respectively, $\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n} }}{{\partial \varphi_{2}^{2} }} = - \frac{1}{2}$ and $\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n} }}{{\partial s_{2}^{{{\text{ma}}2}} }} = - \lambda$. Therefore, $\pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n}$ is jointly concave if the condition $\frac{\lambda }{2} - \theta^{2} > 0$ be satisfied (as shown below).

$$H\left( {\varphi_{2} , s_{2}^{\text{ma}} } \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n} }}{{\partial \varphi_{2}^{2} }}} & {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n} }}{{\partial \varphi_{2} \partial s_{2}^{\text{ma}} }}} \\ {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n} }}{{\partial s_{2}^{\text{ma}} \partial \varphi_{2} }}} & {\frac{{\partial^{2} \pi_{{{\text{ma}}_{\left( 2 \right)} }}^{n} }}{{\partial s_{2}^{{{\text{ma}}2}} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \frac{1}{2}} & \theta \\ \theta & { - \lambda } \\ \end{array} } \right]$$

□

Proof 5

In order to prove that the profit function $\pi_{{{\text{re}}_{\left( 2 \right)} }}^{n}$ is concave and also a negative definite, the corresponding Hessian matrix is gained as follows:

$$H\left( {r_{2} , s_{2}^{\text{re}} } \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{{{\text{re}}_{\left( 2 \right)} }}^{n} }}{{\partial r_{2}^{2} }}} & {\frac{{\partial^{2} \pi_{{{\text{re}}_{\left( 2 \right)} }}^{n} }}{{\partial r_{2} \partial s_{2}^{\text{re}} }}} \\ {\frac{{\partial^{2} \pi_{{{\text{re}}_{\left( 2 \right)} }}^{n} }}{{\partial s_{2}^{\text{re}} \partial r_{2} }}} & {\frac{{\partial^{2} \pi_{{{\text{re}}_{\left( 2 \right)} }}^{n} }}{{\partial s_{2}^{{{\text{re}}2}} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & \beta \\ \beta & { - \sigma } \\ \end{array} } \right]$$

Thereupon, if the determinant of $H\left( {r_{2} , s_{2}^{\text{re}} } \right)$ be more than zero ($2\sigma - \beta^{2} > 0)$, the function $\pi_{{{\text{re}}_{\left( 2 \right)} }}^{n}$ is strictly concave and also a negative definite.□

Proof 6

It can be shown that $\pi_{\text{rm}}^{c}$ is a concave function with respect to $r'$ and $s^{\text{rm}}$ as (33):

$$H\left( {r', s^{\text{rm}} } \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{\text{rm}}^{c} }}{{\partial r^{'2} }}} & {\frac{{\partial^{2} \pi_{\text{rm}}^{c} }}{{\partial r'\partial s^{\text{rm}} }}} \\ {\frac{{\partial^{2} \pi_{\text{rm}}^{c} }}{{\partial s^{\text{rm}} \partial r'}}} & {\frac{{\partial^{2} \pi_{\text{rm}}^{c} }}{{\partial s^{\text{rm2}} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & \beta \\ \beta & { - \sigma } \\ \end{array} } \right]$$

It is obvious that $\pi_{\text{rm}}^{c}$ is strictly concave if the condition $2\sigma - \beta^{2} > 0$ be satisfied.□