# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022119
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## Analysis of the short-term game and long-term evolutionary game behaviors of the waste battery remanufacturing market considering government subsidies

 1 School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China 2 School of Management, Hefei University of Technology, Hefei 230009, China

*Corresponding author: Lei Liu

Received  December 2021 Revised  March 2022 Early access July 2022

Fund Project: The authors acknowledge support from Key Project of Natural Science Research of Universities in Anhui Province (KJ2019A0662) and the Graduate Research and Innovation Fund of Anhui University of Finance and Economics (ACYC2020363)

The popularity of new energy vehicles is undoubtedly the best response to the current global call to protect the environment, and the subsequent waste battery recycling and remanufacturing industry is quietly emerging under government promotion. This paper considers the government provides production subsidy or consumption subsidy for a supply chain consisting of a manufacturer and a remanufacturer. The main theme of this research is to explore the impacts of these two types of subsidy on the short-term game and long-term evolutionary game behaviors of the manufacturer and the remanufacturer. To be specific, a Cournot model is built to exam short-term game behaviors in case of production subsidy and consumption subsidy, respectively. Furthermore, we investigate a nonlinear dynamic system to study the long-term evolutionary trend of the supply chain, including long-term operation strategies, the stability of the equilibrium, and how different subsidies impact the system. Results suggest that: (a) the production subsidy is better for higher output of remanufactured products when the eco-design cost of new products is relatively low. And this subsidy works better when consumers are more environmentally conscious. (b) The government is more likely to choose consumption subsidy when consumers have less acceptance of remanufactured products. (c) In the long-term market, the consumption subsidy is more beneficial for the market to remain stable. Nevertheless, excessive adjustments in output by the manufacturer and the remanufacturer can still lead the market to be uncontrolled.

Citation: Daoming Dai, Lei Liu, Xuanyu Wu. Analysis of the short-term game and long-term evolutionary game behaviors of the waste battery remanufacturing market considering government subsidies. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022119
##### References:
 [1] M. Alfaro-Algaba and F. J. Ramirez, Techno-economic and environmental disassembly planning of lithium-ion electric vehicle battery packs for remanufacturing, Conservation and Recycling, 154 (2020), 104461. [2] J. D. Abbey, H. N. Geismar and G. C. Souza, Improving remanufacturing core recovery and profitability through seeding, Production and Operations Management, 28 (2019), 610-627. [3] S. C. Bulmus, S. Zhu and R. Teunter, Competition for cores in remanufacturing, European Journal of Operational Research, 233 (2014), 105-113.  doi: 10.1016/j.ejor.2013.08.025. [4] X. Chang, T. Pan, Y. Zhong, et al., Analysis of dual environmental responsible behavior in producing-remaufacturing competition system under EPR policy, Systems Engineering-Theory & Practice, 41 (2021), 905–918. [5] J. Chai, Z. Qian, F. Wang, et al., Process innovation for green product in a closed loop supply chain with remanufacturing, in Annals of Operations Research, Academic Press, 2021. doi: 10.1007/s10479-020-03888-y. [6] L. Cui, M. Lu, Q. Ou, et al., Analysis and circuit implementation of fractional order multi-wing hidden attractors, Chaos, Solitons & Fractals, 138 (2020), 1–14. doi: 10.1016/j.chaos.2020.109894. [7] J. Chen, S. Dimitrov and H. Pun, The impact of government subsidy on supply chains' sustainability innovation, Omega, 86 (2019), 42-58. [8] A. A. Elsadany and A. M. Awad, Dynamics and chaos control of a duopolistic Bertrand competitions under environmental taxes, Annals of Operations Research, 274 (2019), 211-240.  doi: 10.1007/s10479-018-2837-8. [9] D. Feng, C. Shen and Z. Pei, Production decisions of a closed-loop supply chain considering remanufacturing and refurbishing under government subsidy, Sustainable Production and Consumption, 27 (2021), 2058-2074. [10] X. Gu, L. Zhou, H. Huang et al., Electric vehicle battery secondary use under government subsidy: a closed-loop supply chain perspective, International Journal of Production Economics, 243 (2021), 108035. [11] S. Hao, Q. Dong and J. Li, Analysis and tendency on the recycling mode of used EV batteries based on cost accounting, China Environmental Science, 41 (2021), 4745-4755. [12] P. He, Y. He and H. Xu, Channel structure and pricing in a dual-channel closed-loop supply chain with government subsidy, International Journal of Production Economics, 213 (2019), 108-123. [13] S. M. Hosseini-Motlagh, M. Nouri-Harzvili, T. M. Choi, et al., Reverse supply chain systems optimization with dual channel and demand disruptions: Sustainability, CSR investment and pricing coordination, Information Sciences, 503 (2019), 606–634. doi: 10.1016/j.ins.2019.07.021. [14] S. M. Hosseini-Motlagh, T. M. Choi, M. Johari, et al., A profit surplus distribution mechanism for supply chain coordination: An evolutionary game-theoretic analysis, European Journal of Operational Research, 301 (2022), 561–575. doi: 10.1016/j.ejor.2021.10.059. [15] I. Hong, P. Chen and H. Yu, The effects of government subsidies on decentralised reverse supply chains, International Journal of Production Economics, 54 (2016), 3962-3977. [16] Z. Hong and X. Guo, Green product supply chain contracts considering environmental responsibilities, Omega, 83 (2019), 155-166. [17] M. Johari and S. M. Hosseini-Motlagh, Evolutionary behaviors regarding pricing and payment-convenience strategies with uncertain risk, European Journal of Operational Research, 297 (2022), 600-614.  doi: 10.1016/j.ejor.2021.05.012. [18] L. Li, F. Dababneh and J. Zhao, Cost-effective supply chain for electric vehicle battery remanufacturing, Applied Energy, 226 (2018), 277-286. [19] L. Meng, Q. Qiang, Z. Huang, et al., Optimal pricing strategy and government consumption subsidy policy in closed-loop supply chain with third-party remanufacturer, Sustainability, 12 (2020), 2411. [20] J. Ma, Y. Hou, W. Yang, et al., A time-based pricing game in a competitive vehicle market regarding the intervention of carbon emission reduction, Energy Policy, 142 (2020), 111440. [21] S. Mitra and S. Webster, Competition in remanufacturing and the effects of government subsidies, International Journal of Production Economics, 111 (2008), 287-298. [22] J. Nie, J. Liu, H. Yuan, et al., Economic and environmental impacts of competitive remanufacturing under government financial intervention, Computers & Industrial Engineering, 159 (2021), 107473. [23] H. Qiao and Q. Su, Impact of government subsidy on the remanufacturing industry, Waste Management, 120 (2021), 433-447. [24] E. Shekarian, A. Marandi and J. Majava, Dual-channel remanufacturing closed-loop supply chains under carbon footprint and collection competition, Sustainable Production and Consumption, 28 (2021), 1050-1075. [25] Z. Sabir and H. A. Wahab, Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind, Physica Scripta, 96 (2021), 125261. [26] Z. Sabir, M. A. Z. Raja, D. Baleanu, et al., Design of Gudermannian Neuroswarming to solve the singular Emden-Fowler nonlinear model numerically, Nonlinear Dynamics, 106 (2021), 3199–3214. [27] S. Tang, W. Wang and G. Zhou, Remanufacturing in a competitive market: A closed-loop supply chain in a Stackelberg game framework, Expert Systems with Applications, 161 (2020), 113655. [28] P. Toktas-Palut, An integrated contract for coordinating a three-stage green forward and reverse supply chain under fairness concerns, Journal of Cleaner Production, 279 (2021), 123735. [29] C. Wu, Price and service competition between new and remanufactured products in a two-echelon supply chain, International Journal of Production Economics, 140 (2012), 496-507. [30] S. Xiong, J. Ji and X. Ma, Environmental and economic evaluation of remanufacturing lithium-ion batteries from electric vehicles, Waste Management, 102 (2020), 579-586. [31] T. Xu and J. Ma, Feed-in tariff or tax-rebate regulation? Dynamic decision model for the solar photovoltaic supply chain, Applied Mathematical Modelling, 89 (2021), 1106-1123.  doi: 10.1016/j.apm.2020.08.007. [32] F. Yang, M. Wang and S. Ang, Optimal remanufacturing decisions in supply chains considering consumers' anticipated regret and power structures, Transportation Research Part E: Logistics and Transportation Review, 148 (2021), 102267. [33] Y. Zhang, Z. Hong, Z. Chen, et al., Tax or subsidy? Design and selection of regulatory policies for remanufacturing, European Journal of Operational Research, 287 (2020), 885–900. doi: 10.1016/j.ejor.2020.05.023. [34] X. Zhang, Q. Li, Z. Liu, et al., Optimal pricing and remanufacturing mode in a closed-loop supply chain of WEEE under government fund policy, Computers & Industrial Engineering, 151 (2021), 106951. [35] L. Zhao, J. Chang and J. Du, Dynamics analysis on competition between manufacturing and remanufacturing in context of government subsidies, Chaos, Solitons & Fractals, 121 (2019), 119-128.  doi: 10.1016/j.chaos.2019.01.034.

show all references

##### References:
 [1] M. Alfaro-Algaba and F. J. Ramirez, Techno-economic and environmental disassembly planning of lithium-ion electric vehicle battery packs for remanufacturing, Conservation and Recycling, 154 (2020), 104461. [2] J. D. Abbey, H. N. Geismar and G. C. Souza, Improving remanufacturing core recovery and profitability through seeding, Production and Operations Management, 28 (2019), 610-627. [3] S. C. Bulmus, S. Zhu and R. Teunter, Competition for cores in remanufacturing, European Journal of Operational Research, 233 (2014), 105-113.  doi: 10.1016/j.ejor.2013.08.025. [4] X. Chang, T. Pan, Y. Zhong, et al., Analysis of dual environmental responsible behavior in producing-remaufacturing competition system under EPR policy, Systems Engineering-Theory & Practice, 41 (2021), 905–918. [5] J. Chai, Z. Qian, F. Wang, et al., Process innovation for green product in a closed loop supply chain with remanufacturing, in Annals of Operations Research, Academic Press, 2021. doi: 10.1007/s10479-020-03888-y. [6] L. Cui, M. Lu, Q. Ou, et al., Analysis and circuit implementation of fractional order multi-wing hidden attractors, Chaos, Solitons & Fractals, 138 (2020), 1–14. doi: 10.1016/j.chaos.2020.109894. [7] J. Chen, S. Dimitrov and H. Pun, The impact of government subsidy on supply chains' sustainability innovation, Omega, 86 (2019), 42-58. [8] A. A. Elsadany and A. M. Awad, Dynamics and chaos control of a duopolistic Bertrand competitions under environmental taxes, Annals of Operations Research, 274 (2019), 211-240.  doi: 10.1007/s10479-018-2837-8. [9] D. Feng, C. Shen and Z. Pei, Production decisions of a closed-loop supply chain considering remanufacturing and refurbishing under government subsidy, Sustainable Production and Consumption, 27 (2021), 2058-2074. [10] X. Gu, L. Zhou, H. Huang et al., Electric vehicle battery secondary use under government subsidy: a closed-loop supply chain perspective, International Journal of Production Economics, 243 (2021), 108035. [11] S. Hao, Q. Dong and J. Li, Analysis and tendency on the recycling mode of used EV batteries based on cost accounting, China Environmental Science, 41 (2021), 4745-4755. [12] P. He, Y. He and H. Xu, Channel structure and pricing in a dual-channel closed-loop supply chain with government subsidy, International Journal of Production Economics, 213 (2019), 108-123. [13] S. M. Hosseini-Motlagh, M. Nouri-Harzvili, T. M. Choi, et al., Reverse supply chain systems optimization with dual channel and demand disruptions: Sustainability, CSR investment and pricing coordination, Information Sciences, 503 (2019), 606–634. doi: 10.1016/j.ins.2019.07.021. [14] S. M. Hosseini-Motlagh, T. M. Choi, M. Johari, et al., A profit surplus distribution mechanism for supply chain coordination: An evolutionary game-theoretic analysis, European Journal of Operational Research, 301 (2022), 561–575. doi: 10.1016/j.ejor.2021.10.059. [15] I. Hong, P. Chen and H. Yu, The effects of government subsidies on decentralised reverse supply chains, International Journal of Production Economics, 54 (2016), 3962-3977. [16] Z. Hong and X. Guo, Green product supply chain contracts considering environmental responsibilities, Omega, 83 (2019), 155-166. [17] M. Johari and S. M. Hosseini-Motlagh, Evolutionary behaviors regarding pricing and payment-convenience strategies with uncertain risk, European Journal of Operational Research, 297 (2022), 600-614.  doi: 10.1016/j.ejor.2021.05.012. [18] L. Li, F. Dababneh and J. Zhao, Cost-effective supply chain for electric vehicle battery remanufacturing, Applied Energy, 226 (2018), 277-286. [19] L. Meng, Q. Qiang, Z. Huang, et al., Optimal pricing strategy and government consumption subsidy policy in closed-loop supply chain with third-party remanufacturer, Sustainability, 12 (2020), 2411. [20] J. Ma, Y. Hou, W. Yang, et al., A time-based pricing game in a competitive vehicle market regarding the intervention of carbon emission reduction, Energy Policy, 142 (2020), 111440. [21] S. Mitra and S. Webster, Competition in remanufacturing and the effects of government subsidies, International Journal of Production Economics, 111 (2008), 287-298. [22] J. Nie, J. Liu, H. Yuan, et al., Economic and environmental impacts of competitive remanufacturing under government financial intervention, Computers & Industrial Engineering, 159 (2021), 107473. [23] H. Qiao and Q. Su, Impact of government subsidy on the remanufacturing industry, Waste Management, 120 (2021), 433-447. [24] E. Shekarian, A. Marandi and J. Majava, Dual-channel remanufacturing closed-loop supply chains under carbon footprint and collection competition, Sustainable Production and Consumption, 28 (2021), 1050-1075. [25] Z. Sabir and H. A. Wahab, Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind, Physica Scripta, 96 (2021), 125261. [26] Z. Sabir, M. A. Z. Raja, D. Baleanu, et al., Design of Gudermannian Neuroswarming to solve the singular Emden-Fowler nonlinear model numerically, Nonlinear Dynamics, 106 (2021), 3199–3214. [27] S. Tang, W. Wang and G. Zhou, Remanufacturing in a competitive market: A closed-loop supply chain in a Stackelberg game framework, Expert Systems with Applications, 161 (2020), 113655. [28] P. Toktas-Palut, An integrated contract for coordinating a three-stage green forward and reverse supply chain under fairness concerns, Journal of Cleaner Production, 279 (2021), 123735. [29] C. Wu, Price and service competition between new and remanufactured products in a two-echelon supply chain, International Journal of Production Economics, 140 (2012), 496-507. [30] S. Xiong, J. Ji and X. Ma, Environmental and economic evaluation of remanufacturing lithium-ion batteries from electric vehicles, Waste Management, 102 (2020), 579-586. [31] T. Xu and J. Ma, Feed-in tariff or tax-rebate regulation? Dynamic decision model for the solar photovoltaic supply chain, Applied Mathematical Modelling, 89 (2021), 1106-1123.  doi: 10.1016/j.apm.2020.08.007. [32] F. Yang, M. Wang and S. Ang, Optimal remanufacturing decisions in supply chains considering consumers' anticipated regret and power structures, Transportation Research Part E: Logistics and Transportation Review, 148 (2021), 102267. [33] Y. Zhang, Z. Hong, Z. Chen, et al., Tax or subsidy? Design and selection of regulatory policies for remanufacturing, European Journal of Operational Research, 287 (2020), 885–900. doi: 10.1016/j.ejor.2020.05.023. [34] X. Zhang, Q. Li, Z. Liu, et al., Optimal pricing and remanufacturing mode in a closed-loop supply chain of WEEE under government fund policy, Computers & Industrial Engineering, 151 (2021), 106951. [35] L. Zhao, J. Chang and J. Du, Dynamics analysis on competition between manufacturing and remanufacturing in context of government subsidies, Chaos, Solitons & Fractals, 121 (2019), 119-128.  doi: 10.1016/j.chaos.2019.01.034.
Remanufacturer profit comparisons between different subsidy policies
The change of $q_n^{NS}$, ${e^{NS}}$ and $q_r^{NS}$ along with $s$ and $\delta$
The change of $q_n^{RS}$, ${e^{RS}}$ and $q_r^{RS}$ along with $s$ and $\delta$
The profit comparisons between different subsidies
The feasible domains of government subsidies
Bifurcation diagram of $k_1^j$
Bifurcation diagram of $k_3^j$
Lyapunov exponent and Entropy with respect to $k_1^{NS}$
Lyapunov exponent and Entropy with respect to $k_1^{RS}$
The average profits with respect to $k_1^j$
The average profits with respect to $k_3^j$
3D-bifurcation with respect to $k_1^j$ and $s$
3D-bifurcation with respect to $k_3^j$ and $s$
Notation and description
 Notation Description ${p_i}$ The price of new or manufactured product ${c_i}$ The unit production cost of new or manufactured product ${q_i}$ The output of new or manufactured product $e$ Eco-design level of new product $\theta$ Consumers' acceptance of remanufactured product $\delta$ Consumers' environmental awareness $s$ Government subsidy $v$ Consumers' willingness to pay for new products $\eta$ The manufacturer' marginal cost in product eco-design $CS$ The total consumer surplus $GS$ Subsidy expenditure ${\pi _m}$ The profit of the manufacturer ${\pi _r}$ The profit of the remanufacturer ${\pi _g}$ The profit of the government
 Notation Description ${p_i}$ The price of new or manufactured product ${c_i}$ The unit production cost of new or manufactured product ${q_i}$ The output of new or manufactured product $e$ Eco-design level of new product $\theta$ Consumers' acceptance of remanufactured product $\delta$ Consumers' environmental awareness $s$ Government subsidy $v$ Consumers' willingness to pay for new products $\eta$ The manufacturer' marginal cost in product eco-design $CS$ The total consumer surplus $GS$ Subsidy expenditure ${\pi _m}$ The profit of the manufacturer ${\pi _r}$ The profit of the remanufacturer ${\pi _g}$ The profit of the government
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