# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021229
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## Low carbon joint strategy and coordination for a dyadic supply chain with Nash bargaining fairness

 1 School of Management, Guangdong University of Technology, 510520, Guangzhou, China 2 Faculty of Applied Mathematics, Guangdong University of Technology, 510520, Guangzhou, China 3 Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

*Corresponding author: Rui Hou

Received  December 2020 Revised  October 2021 Early access January 2022

Fund Project: The first author is supported by the Chinese National Funding of Social Science grant 19BGL094

In the paper, fairness concern criterion is utilized to explore the coordination of a dyadic supply chain with a fairness-concerned retailer (acting as a newsvendor), who is committed to low carbon efforts. Two models are developed for stochastic demand disturbances in the forms of multiplicative case and additive case, respectively. Firstly, the optimal joint decision of the retailer and the supply chain are proposed in two scenarios, i.e., decentralized decision and the centralized decision. Secondly, in order to realize channel coordination, the contract of revenue sharing combined with the mechanism of low-carbon cost sharing is designed. Moreover, the influences of the retailer's fairness concern and bargaining power on the joint decision and the contract parameters are also investigated. Finally, numerical examples are given to illustrate the theoretical results and some suggestions to supply chain management are also provided. The results show that the revenue sharing contract can make the supply chain achieved coordination with the cost sharing mechanism of low-carbon efforts. Furthermore, the optimal low-carbon effort level and ordering quantity decrease in terms of fairness-concerned parameter and Nash bargaining power parameter, which increases in unit cost. However, the optimal pricing makes the opposite change.

Citation: Jianxin Chen, Lin Sun, Tonghua Zhang, Rui Hou. Low carbon joint strategy and coordination for a dyadic supply chain with Nash bargaining fairness. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021229
##### References:

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##### References:
Relations between the retailer's expected utility and $\lambda, \gamma, w$ and $c$ in LCAD model
Relations between the retailer's expected utility and $\lambda, \gamma, w$ and $c$ in LCMD model
Relationship between objective and key parameters
Relations between $U(\pi^{r}_{\phi\varphi})$, $\pi^c$ and the retail price $p$, respectively
Notations
 $d(e, p, \xi)$ Stochastic demand $\xi$ Stochastic price-independent component $y(p)$ Deterministic demand $q_i$ Optimal ordering quantity $p_{i}$ Retail price $e_i$ Low-carbon effort level $w$ Wholesale price $c$ Unit cost $s$ Salvage value $\lambda$ Fairness-concerned parameter of the retailer $\gamma$ Retailer's bargaining power $\pi_i^r$ Retailer's profit $\pi_i^s$ Supplier's profit $E(\pi_i^r)$ Expected profit of retailer $E(\pi_i^s)$ Supplier's expected profit $U(\pi_i^r)$ Utility function of retailer $p_i^*$ Optimal ordering $i=M,A$ Multiplicative or additive demand
 $d(e, p, \xi)$ Stochastic demand $\xi$ Stochastic price-independent component $y(p)$ Deterministic demand $q_i$ Optimal ordering quantity $p_{i}$ Retail price $e_i$ Low-carbon effort level $w$ Wholesale price $c$ Unit cost $s$ Salvage value $\lambda$ Fairness-concerned parameter of the retailer $\gamma$ Retailer's bargaining power $\pi_i^r$ Retailer's profit $\pi_i^s$ Supplier's profit $E(\pi_i^r)$ Expected profit of retailer $E(\pi_i^s)$ Supplier's expected profit $U(\pi_i^r)$ Utility function of retailer $p_i^*$ Optimal ordering $i=M,A$ Multiplicative or additive demand
Influence of the important parameter on joint decision ($\nearrow$ and $\searrow$ denote the result is increasing and decreasing in corresponding parameters, respectively). * means that the impact needs certain conditions. Specifically, $p_M^*$ is decreasing in $s$ whereas $Q_M^*$ is increasing in $s$ if $p_M^*>(2w-s)+2\gamma\lambda(w-c)$
 $i=A, M$ $\lambda$ $\gamma$ $w$ $c$ $s$ $i=A$ $i=M$ $p_{i}^*$ $\nearrow$ $\nearrow$ $\nearrow$ $\searrow$ $\searrow$ $\ast$ $e_{i}^*$ $\searrow$ $\searrow$ $\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $Q_{i}^*$ $\searrow$ $\searrow$ $\searrow$ $\nearrow$ $\nearrow$ $\ast$
 $i=A, M$ $\lambda$ $\gamma$ $w$ $c$ $s$ $i=A$ $i=M$ $p_{i}^*$ $\nearrow$ $\nearrow$ $\nearrow$ $\searrow$ $\searrow$ $\ast$ $e_{i}^*$ $\searrow$ $\searrow$ $\searrow$ $\nearrow$ $\nearrow$ $\nearrow$ $Q_{i}^*$ $\searrow$ $\searrow$ $\searrow$ $\nearrow$ $\nearrow$ $\ast$
Influence of $\lambda$ under LCAD when $\xi \sim U(-50; 50)$
 $\lambda$ $w^*_M$ $p^*_M$ $e^*_M$ $Q^*_M$ $U(\pi_M^r)$ $\pi_M^s$ $U(\pi_M^r)+\pi_M^s$ 0 3.021 5.6186 0.6179 66.3602 99.9620 67.7538 167.7157 0.5 2.817 5.6185 0.6179 66.3683 119.9643 54.2030 174.1673 1.0 2.681 5.6184 0.6178 66.3716 133.2981 45.1692 178.4673 1.5 2.584 5.6182 0.6177 66.3748 142.8241 38.7164 181.5406
 $\lambda$ $w^*_M$ $p^*_M$ $e^*_M$ $Q^*_M$ $U(\pi_M^r)$ $\pi_M^s$ $U(\pi_M^r)+\pi_M^s$ 0 3.021 5.6186 0.6179 66.3602 99.9620 67.7538 167.7157 0.5 2.817 5.6185 0.6179 66.3683 119.9643 54.2030 174.1673 1.0 2.681 5.6184 0.6178 66.3716 133.2981 45.1692 178.4673 1.5 2.584 5.6182 0.6177 66.3748 142.8241 38.7164 181.5406
Influence of $\lambda$ under LCMD when $\xi\sim U(0.1; 1.9)$
 $\lambda$ $w^*_M$ $p^*_M$ $e^*_M$ $Q^*_M$ $U(\pi_M^r)$ $\pi_M^s$ $U(\pi_M^r)+\pi_M^s$ 0 2.675 5.7809 0.4652 53.5966 68.1482 36.1884 104.3366 0.5 2.540 5.7806 0.4652 53.6026 81.7826 28.9507 110.7334 1.0 2.450 5.7803 0.4652 53.6125 90.8785 24.1256 115.0041 1.5 2.386 5.7802 0.4651 53.6145 97.3717 20.6791 118.0508
 $\lambda$ $w^*_M$ $p^*_M$ $e^*_M$ $Q^*_M$ $U(\pi_M^r)$ $\pi_M^s$ $U(\pi_M^r)+\pi_M^s$ 0 2.675 5.7809 0.4652 53.5966 68.1482 36.1884 104.3366 0.5 2.540 5.7806 0.4652 53.6026 81.7826 28.9507 110.7334 1.0 2.450 5.7803 0.4652 53.6125 90.8785 24.1256 115.0041 1.5 2.386 5.7802 0.4651 53.6145 97.3717 20.6791 118.0508
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