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doi: 10.3934/jimo.2021033

## Pricing and energy efficiency decisions by manufacturer under channel coordination

 1 Coordinated Innovation Center for Computable Modelling in Management Science, Yango University, Fujian 350015, China 2 Coordinated Innovation Center for Computable Modelling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China 3 School of Mathematical Sciences, Sunway University, Malaysia

* Corresponding author: Shuhua Chang

Received  September 2020 Revised  November 2020 Published  February 2021

Fund Project: The first author is supported by the National Basic Research Program (2012CB955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), and the Major Program of Tianjin University of Finance and Economics (ZD1302). The third author is supported by the Humanities and Social Science Research Program of the Ministry of Education of China (19YJCZH174)

The profit level of the green supply chain under two decision modes is explored in cooperative and non-cooperative games, where the variable decision timing in direct channel and retail channel within the manufacturer management is studied based on the theory of observable delay game. This paper discusses how the total profit of the green supply chain can realize the profit of the inferior copy of the superior under the two decision-making modes. In the existing literature, it is usually assumed that the pricing decisions of the manufacturers and retailers are made simultaneously. In this study, our model takes into account not only the decision on the pricing, but also on the product innovation. Decision makers can choose the levels of the decision variables as well as the decision times. The observable delay game theory is applied to the study of the decision-making of price levels and energy efficiency levels in multi-channel supply chains. Early pricing decisions can be extended to the study of the product development process. This is more in line with the development of the market in reality. The results of our model show that: (a) an increase of the product efficiency innovation by the manufacturers will lead to

Citation: Shuhua Chang, Yameng Wang, Xinyu Wang, Kok Lay Teo. Pricing and energy efficiency decisions by manufacturer under channel coordination. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021033
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##### References:
Channels describe
The equilibrium matrix
Effect of $g$ on $\pi^{d}$ and $\pi^{c}$
Effect of $g$ on $q_{M}$ and $q_{R}$
Effects of $g$ and $\theta$ on $\pi^{d}$
Effects of $g$ and $\theta$ on $\pi^{c}$
Effects of $g$ and $\theta$ on $q_{M}$ and $q_{R}$
Effects of $g$ and $\theta$ on $\pi^{D}$
Notations
 $p_M$ Direct price $p_R$ Retail price $c$ Marginal cost $c_1$ Wholesale price $x$ Level of energy efficiency $g$ Sensitivity of energy efficiency levels to demand $\theta$ Cross-price sensitivity between the channels $a_M$ Intercept of the demand function for the direct channel $a_R$ Intercept of the demand function for the retail channel $b_M$ Sensitivity of demand in the direct channel to the direct channel price $b_R$ Sensitivity of demand in the retail channel to the retail channel price $q_M$ The demand at the direct end $q_R$ The demand at the retail end $t_{c1}$ Period when the manufacturer chooses the wholesale price $t_{p_M}$ Period when the manufacturer chooses the direct channel price $t_{p_R}$ Period when the retailer chooses the retail channel price $t_x$ Period when the manufacture chooses the level of energy efficiency $E$ Manufacturer precede retailer in setting direct price $S$ Manufacturer and retailer setting the order of direct price at the same time $L$ Manufacturer setting the order of direct price later than retailer $\pi_M$ Profit for the manufacture under decentralized decision-making $\pi_R$ Profit for the retailer under decentralized decision-making $\pi_V$ Profit of the whole supply chain under centralized decision-making
 $p_M$ Direct price $p_R$ Retail price $c$ Marginal cost $c_1$ Wholesale price $x$ Level of energy efficiency $g$ Sensitivity of energy efficiency levels to demand $\theta$ Cross-price sensitivity between the channels $a_M$ Intercept of the demand function for the direct channel $a_R$ Intercept of the demand function for the retail channel $b_M$ Sensitivity of demand in the direct channel to the direct channel price $b_R$ Sensitivity of demand in the retail channel to the retail channel price $q_M$ The demand at the direct end $q_R$ The demand at the retail end $t_{c1}$ Period when the manufacturer chooses the wholesale price $t_{p_M}$ Period when the manufacturer chooses the direct channel price $t_{p_R}$ Period when the retailer chooses the retail channel price $t_x$ Period when the manufacture chooses the level of energy efficiency $E$ Manufacturer precede retailer in setting direct price $S$ Manufacturer and retailer setting the order of direct price at the same time $L$ Manufacturer setting the order of direct price later than retailer $\pi_M$ Profit for the manufacture under decentralized decision-making $\pi_R$ Profit for the retailer under decentralized decision-making $\pi_V$ Profit of the whole supply chain under centralized decision-making
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