# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021155
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## Joint emission reduction dynamic optimization and coordination in the supply chain considering fairness concern and reference low-carbon effect

 School of Management, Shanghai University, Shanghai, China

* Corresponding author: Liying Yu

Received  April 2021 Revised  May 2021 Early access September 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China (NSFC) under Grants 12071280, 11671250

In the context of low-carbon economy, in order to explore the impact of the fairness concern and reference low-carbon effect on supply chain members' balanced emission reduction decisions and profits, supply chain joint emission reduction dynamic optimization models under four different scenarios are built, in which the manufacturer's optimal emission reduction strategy, the retailer's optimal low-carbon promotion strategy and other equilibrium solutions are solved by differential game theory. On the basis of analysis, a contract is designed to achieve the coordination of the supply chain when members are fairness concern. Some findings are as follows. First, when consumers' purchasing behavior is significantly affected by the reference low-carbon effect, and they have higher expectations for the product's emission reduction level, consumers' reference low-carbon effect will discourage the manufacturer's enthusiasm to reduce emissions, and do harm to the profits of the manufacturer and the retailer. Second, the fairness concern behavior of both parties will aggravate the adverse effects of reference low-carbon effect, bring a detrimental effect on the performance of the supply chain, aggravate the double marginal effect of the supply chain, and cause continuous negative social influence. Third, the bilateral cost-sharing contract can encourage the manufacturer to increase emission reduction investment, the retailer to increase low-carbon promotion investment, and can achieve a Pareto improvement of both parties' profits and utilities. In addition, the two cost-sharing ratios are only proportional to the marginal revenue and fairness concern intensity of both parties. Finally, when the two cost-sharing ratios and the revenue-sharing coefficient meet a certain relationship and are within a reasonable range, the bilateral cost sharing-revenue sharing hybrid contract can reduce the double marginal effect and achieve supply chain coordination.

Citation: Ziyuan Zhang, Liying Yu. Joint emission reduction dynamic optimization and coordination in the supply chain considering fairness concern and reference low-carbon effect. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021155
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##### References:
The impact of the bilateral cost sharing-revenue sharing hybrid contract.(a)the effect of $\tau_m$ and $\tau_r$ on $\theta$.(b)the effect of $\theta$ on both parties' profits
(a) The evolution path of low-carbon goodwill $G(t)$.(b)The evolution path of reference low-carbon level $R(t)$
(a) The evolution path of the manufacturer's profit $\pi_m(t)$.(b)The evolution path of the retailer's profit $\pi_r(t)$.(c) The evolution path of overall supply chain profit $\pi_{sc}(t)$.(d) The evolution path of the manufacturer's utility $U_m(t)$.(e) The evolution path of the retailer's utility $U_r(t)$.(f) The evolution path of the utility difference $\Delta U(t)$
(a) The impact of $\tau_m$ and $\tau_r$ on the low-carbon promotion cost-sharing ratio.(b)The impact of $\tau_m$ and $\tau_r$ on the emission reduction cost-sharing ratio.(c) The impact of $\tau_m$ and $\tau_r$ on the manufacturer's emission reduction investment.(d) The impact of $\tau_m$ and $\tau_r$ on the retailer's low-carbon promotion investment.(e) The impact of $\tau_m$ and $\tau_r$ on the manufacturer's profit $\pi_m^f$. (f) The impact of $\tau_m$ and $\tau_r$ on the retailer's profit $\pi_r^f$
(a) The impact of $\epsilon$ on the manufacturer's emission reduction investment.(b)The impact of $\alpha$ and $\epsilon$ on the manufacturer's emission reduction investment.(c) The impact of $\tau_m$ and $\epsilon$ on the manufacturer's emission reduction investment.(d) The impact of $\tau_m$ and $\alpha$ on the manufacturer's emission reduction investment.(e) The impact of $\alpha$ and $\epsilon$ on the manufacturer's profit $\pi_m^f$. (f) The impact of $\alpha$ and $\epsilon$ on the retailer's profit $\pi_r^f$.(g) The impact of initial reference low-carbon level $R_0$ on the manufacturer's profit.(h) The impact of initial reference low-carbon level $R_0$ on the retailer's profit
Some literature most relevant to this paper
 Author Reference Low Carbon on Effect Fairness Concern Low Carbon Goodwill Supply Chain Coordination Wang et al. [28] $\surd$ Liu et al. [18] $\surd$ $\surd$ Yu et al. [32] $\surd$ Zhang et al. [41] $\surd$ Zou et al. [45] $\surd$ This paper $\surd$ $\surd$ $\surd$ $\surd$
 Author Reference Low Carbon on Effect Fairness Concern Low Carbon Goodwill Supply Chain Coordination Wang et al. [28] $\surd$ Liu et al. [18] $\surd$ $\surd$ Yu et al. [32] $\surd$ Zhang et al. [41] $\surd$ Zou et al. [45] $\surd$ This paper $\surd$ $\surd$ $\surd$ $\surd$
Related parameters and decision variables
 Decision variables Definition $E(t)$ Manufacturer's emission reduction investment at time $t$ $A(t)$ Retailer's low-carbon promotion investment at time $t$ Parameters Definition $\alpha$ Influence coefficient of reference low-carbon effect on the demand $\beta$ Influence coefficient of manufacturer's emission reduction on the demand $k$ Influence coefficient of retailer's low-carbon promotion on the demand $\rho$ Discount rate $\mu$ Influence coefficient of low-carbon goodwill on the demand $\pi_1$ Manufacturer's marginal revenue $\pi_2$ Retailer's marginal revenue $G(t)$ Low-carbon goodwill level at time $t$ $R(t)$ Consumer's reference low-carbon level at time $t$ $\lambda$ Influence coefficient of the low-carbon promotion on the low-carbon goodwill $\sigma$ Natural decay rate of low-carbon goodwill $\epsilon$ Memory parameter $\phi$ Low-carbon promotion Cost-sharing ratio $\gamma$ Emission reduction cost-sharing ratio $\theta$ Revenue-sharing coefficient under the supply chain coordination $\tau_m$ Manufacturer's fairness concern intensity $\tau_r$ Retailer's fairness concern intensity $\eta_m$, $\eta_r$ Emission reduction cost coefficient and low-carbon promotion cost coefficient.
 Decision variables Definition $E(t)$ Manufacturer's emission reduction investment at time $t$ $A(t)$ Retailer's low-carbon promotion investment at time $t$ Parameters Definition $\alpha$ Influence coefficient of reference low-carbon effect on the demand $\beta$ Influence coefficient of manufacturer's emission reduction on the demand $k$ Influence coefficient of retailer's low-carbon promotion on the demand $\rho$ Discount rate $\mu$ Influence coefficient of low-carbon goodwill on the demand $\pi_1$ Manufacturer's marginal revenue $\pi_2$ Retailer's marginal revenue $G(t)$ Low-carbon goodwill level at time $t$ $R(t)$ Consumer's reference low-carbon level at time $t$ $\lambda$ Influence coefficient of the low-carbon promotion on the low-carbon goodwill $\sigma$ Natural decay rate of low-carbon goodwill $\epsilon$ Memory parameter $\phi$ Low-carbon promotion Cost-sharing ratio $\gamma$ Emission reduction cost-sharing ratio $\theta$ Revenue-sharing coefficient under the supply chain coordination $\tau_m$ Manufacturer's fairness concern intensity $\tau_r$ Retailer's fairness concern intensity $\eta_m$, $\eta_r$ Emission reduction cost coefficient and low-carbon promotion cost coefficient.
The relationship between related parameters
 $\pi_1$ $\pi_2$ $\tau_m$ $\tau_r$ $\eta_m$ $\eta_r$ $\alpha$ $\beta$ $k$ $\epsilon$ $\sigma$ $\lambda$ $\mu$ $\rho$ $E$ $+$ $-$ $-$ $\times$ $-$ $\times$ $+$ $+$ $\times$ $-$ $\times$ $\times$ $\times$ $+$ $A$ $-$ $+$ $\times$ $-$ $\times$ $-$ $\times$ $\times$ $+$ $\times$ $-$ $+$ $+$ $-$ $G_{\infty}$ $-$ $+$ $\times$ $-$ $\times$ $-$ $\times$ $\times$ $+$ $\times$ $-$ $+$ $+$ $-$ $R_{\infty}$ $+$ $-$ $-$ $\times$ $-$ $\times$ $+$ $+$ $\times$ $-$ $\times$ $\times$ $\times$ $+$
 $\pi_1$ $\pi_2$ $\tau_m$ $\tau_r$ $\eta_m$ $\eta_r$ $\alpha$ $\beta$ $k$ $\epsilon$ $\sigma$ $\lambda$ $\mu$ $\rho$ $E$ $+$ $-$ $-$ $\times$ $-$ $\times$ $+$ $+$ $\times$ $-$ $\times$ $\times$ $\times$ $+$ $A$ $-$ $+$ $\times$ $-$ $\times$ $-$ $\times$ $\times$ $+$ $\times$ $-$ $+$ $+$ $-$ $G_{\infty}$ $-$ $+$ $\times$ $-$ $\times$ $-$ $\times$ $\times$ $+$ $\times$ $-$ $+$ $+$ $-$ $R_{\infty}$ $+$ $-$ $-$ $\times$ $-$ $\times$ $+$ $+$ $\times$ $-$ $\times$ $\times$ $\times$ $+$
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