Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation

Jun Huang; Ying Peng; Ruwen Tan; Chunxiang Guo

doi:10.3934/jimo.2020113

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doi: 10.3934/jimo.2020113

Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation

Jun Huang ^1, , Ying Peng ^1, , Ruwen Tan ^1,, and Chunxiang Guo ^2,

1.	College of Architecture and Environment, Sichuan University, Chengdu, 610065, China
2.	Business School, Sichuan University, Chengdu, 610065, China

^* Corresponding author: Ruwen Tan

Received January 2020 Revised March 2020 Published June 2020

Fund Project: This work was supported by the Ministry of Education in China Project of Humanities and Social Sciences [grant numbers 17YJA630078]; Sichuan Province Cyclic Economy Research Center Project [grant numbers XHJJ-1811]; the Sichuan Science and Technology Department Soft Science Project [grant numbers 2019JDR0148]; Chengdu Science and Technology Bureau Project [grant numbers 2017-RK00-00209-ZF]; and Sichuan University [grant numbers skqy201740]

Construction and demolition waste (CDW) recycling enterprises have an imperfect operation system, which leads to low recovery rate and low production profits. A feasible method to improve this situation involves seeking a high-efficiency enterprise alliance strategy in the CDW recycling system composed of manufacturers, retailers and recyclers and designing a reasonable and effective coordination mechanism to enhance their enthusiasm for participa-tion. First, we constructed a Stackelberg game model of CDW recycling under government regulation and analyzed the optimal alliance strategy of CDW recycling enterprises under punishment or subsidy by the government as a game leader. In order to ensure the stable cooperation of the alliance, we used the Shapley value method to coordinate the distribution of the optimal alliance profit and improved the fairness and effectiveness of the coordination mechanism through modification of the unequal rights factor. Finally, based on the survey data of Chongqing, we further verified the conclusion through numerical simulation and an-alyzed changes in various parameters at different product costs. The results show that the alliance strategy and coordination mechanism can improve the CDW recovery rate, improve the recycling market status, and increase the production profits of enterprises.

Keywords: Government regulation, construction and demolition waste, cooperative game, cooperative game, profit distribution.

Mathematics Subject Classification: Primary: 90B50; Secondary: 91A12.

Citation: Jun Huang, Ying Peng, Ruwen Tan, Chunxiang Guo. Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020113

References:

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[21]	E. Kemahlioğlu-Ziya and J. J. Bartholdi, Centralizing inventory in supply chains by using shapley value to allocate the profits, Manufacturing & Service Operations Management, 13 (2011), 146-162. Google Scholar
[22]	A. Laruelle and F. Valenciano, Shapley-Shubik and Banzhaf indices revisited, Mathematics of Operations Research, 26 (2001), 89-104. doi: 10.1287/moor.26.1.89.10589. Google Scholar
[23]	B.-z. Li and X.-f. Luo, Study on profit allocation of enterprise' s original innovation with an industry-university-research cooperative mode based on the shapley value, Operations Research and Management Science, 22 (2013), 220-224. Google Scholar
[24]	Y. Li, X. Zhang, G. Ding and Z. Feng, Developing a quantitative construction waste estimation model for building construction projects, Resources, Conservation and Recycling, 106 (2016), 9-20. doi: 10.1016/j.resconrec.2015.11.001. Google Scholar
[25]	X. Li, Z. Mu and Z. Song, Research on manufacturers' collecting and remanufacturing the closed -loop supply chain model based on the game theory, Science Research Management, 34 (2013), 64-71. Google Scholar
[26]	J. Liu, J. Nie and H. Yuan, To expand or not to expand: A strategic analysis of the recycler's waste treatment capacity, Computers & Industrial Engineering, 130 (2019), 731-744. doi: 10.1016/j.cie.2019.03.016. Google Scholar
[27]	J. Liu, J. Nie and H. Yuan, To expand or not to expand: A strategic analysis of the recycler's waste treatment capacity, Computers & Industrial Engineering, 130 (2019), 731-744. doi: 10.1016/j.cie.2019.03.016. Google Scholar
[28]	Z. Lu and W. Zhu, Profit distribution for information production supply chain based on modified interval-valued fuzzy Shapley value: Profit distribution for information production supply chain based on modified interval-valued fuzzy Shapley value, Journal of Computer Applications, 33 (2013), 2960-2963. doi: 10.3724/SP.J.1087.2013.02960. Google Scholar
[29]	W. Lu, X. Chen, Y. Peng and L. Shen, Benchmarking construction waste management performance using big data, Resources, Conservation and Recycling, 105 (2015), 49-58. doi: 10.1016/j.resconrec.2015.10.013. Google Scholar
[30]	Y.-j. Lu, X.-f. Xu and X.-z. Ai, Collective decision-making of a closed-loop supply chain dual-channel model under the third-party economies of scale, Journal of Industrial Engineering and Engineering Management, 32 (2018), 207-217. Google Scholar
[31]	S. R. Madani and M. Rasti-Barzoki, Sustainable supply chain management with pricing, greening and governmental tariffs determining strategies: A game-theoretic approach, Computers & Industrial Engineering, 105 (2017), 287-298. doi: 10.1016/j.cie.2017.01.017. Google Scholar
[32]	N. G. Mankiw, Principles of economics / N. Gregory Mankiw, 5th edition, South-Western Cengage Learning, Mason, OH, 2009. Google Scholar
[33]	T. M. Mak, I. K. Yu, L. Wang, S.-C. Hsu, D. C. Tsang, C. Li, T. L. Yeung, R. Zhang and C. S. Poon, Extended theory of planned behaviour for promoting construction waste recycling in Hong Kong, Waste Management, 83 (2019), 161-170. doi: 10.1016/j.wasman.2018.11.016. Google Scholar
[34]	S. Mitra and S. Webster, Competition in remanufacturing and the effects of government subsidies, International Journal of Production Economics, 111 (2008), 287-298. doi: 10.1016/j.ijpe.2007.02.042. Google Scholar
[35]	T. H. Mills and E. Showalter, A cost-effective waste management plan, Cost Engineering, 41 (1999), 35. Google Scholar
[36]	S. Panda, Coordination of a socially responsible supply chain using revenue sharing contract, Transportation Research Part E: Logistics and Transportation Review, 67 (2014), 92-104. Google Scholar
[37]	E. S. Park, K. U. Kwon and K. S. An, A study on policy decision making and social conflict factors considering the stakeholder of public construction projects, KOREA SCIENCE & ART FORUM, 35 (2018), 187-201. Google Scholar
[38]	C. S. Poon, A. T. W. Yu, A. Wong and R. Yip, Quantifying the impact of construction waste charging scheme on construction waste management in Hong Kong, Journal of Construction Engineering and Management, 139 (2013), 466-479. doi: 10.1061/(ASCE)CO.1943-7862.0000631. Google Scholar
[39]	S. Panda, N. M. Modak and L. E. Cárdenas-Barrón, Coordinating a socially responsible closed-loop supply chain with product recycling, International Journal of Production Economics, 188 (2017), 11-21. Google Scholar
[40]	S. Saha, S. Sarmah and I. Moon, Dual channel closed-loop supply chain coordination with a reward-driven remanufacturing policy, International Journal of Production Research, 54 (2016), 1503-1517. doi: 10.1080/00207543.2015.1090031. Google Scholar
[41]	L. Sun and S.-r. Sun, Research on Cooperation Mechanism of Infrastructure Project Financing Alliance Based on Shapley Value, Industrial Engineering and Management, 22 (2017), 76-82. Google Scholar
[42]	S. Swami and J. Shah, Channel coordination in green supply chain management, Journal of the Operational Research Society, 64 (2013), 336-351. Google Scholar
[43]	Y. Teng, X. Li, P. Wu and X. Wang, Using cooperative game theory to determine profit distribution in IPD projects, International Journal of Construction Management, 19 (2019), 32-45. doi: 10.1080/15623599.2017.1358075. Google Scholar
[44]	M. M. M. Teo and M. Loosemore, A theory of waste behaviour in the construction industry, Construction Management and Economics, 19 (2001), 741-751. doi: 10.1080/01446190110067037. Google Scholar
[45]	G. Wang, X. Ai, C. Zheng and L. Zhong, Strategic inventory with competing suppliers, Journal of Industrial & Management Optimization, 2019. Google Scholar
[46]	Y. Wang, Q. Han, B. de Vries and J. Zuo, How the public reacts to social impacts in construction projects? A structural equation modeling study, International Journal of Project Management, 34 (2016), 1433-1448. doi: 10.1016/j.ijproman.2016.07.008. Google Scholar
[47]	J. Wu, M. Hu and S. Shi, Expanding Boundaries - Eco-Efficiency of Construction and Demolition Waste Recycling in Chongqing, China – J. Wu, M. Hu, S. Shi, T. Liu, C. Zhang, 1. Google Scholar
[48]	L.-l. Xie, B. Xia, Y. Hu, M. Shan, Y. Le and A. P. Chan, Public participation performance in public construction projects of South China: A case study of the Guangzhou Games venues construction, International Journal of Project Management, 35 (2017), 1391-1401. doi: 10.1016/j.ijproman.2017.04.003. Google Scholar
[49]	J. Yang and B. lai, Research on profit distribution strategy of supply chain cooperation with unequal power, Journal of Engineering Management, 33–37. Google Scholar
[50]	H. Yuan, L. Shen and J. Wang, Major obstacles to improving the performance of waste management in China's construction industry, Facilities, 29 (2011), 224-242. doi: 10.1108/02632771111120538. Google Scholar
[51]	J. Zhang and J. Chen, Information sharing in a make-to-stock supply chain, Journal of Industrial and Management Optimization, 10 (2014), 1169-1189. doi: 10.3934/jimo.2014.10.1169. Google Scholar
[52]	W. Zhao, H. Ren and V. Rotter, A system dynamics model for evaluating the alternative of type in construction and demolition waste recycling center – The case of Chongqing, China, Resources, Conservation and Recycling, 55 (2011), 933-944. doi: 10.1016/j.resconrec.2011.04.011. Google Scholar
[53]	W. Zhao, R. Leeftink and V. Rotter, Evaluation of the economic feasibility for the recycling of construction and demolition waste in China–The case of Chongqing, Resources, Conservation and Recycling, 54 (2010), 377-389. doi: 10.1016/j.resconrec.2009.09.003. Google Scholar
[54]	X.-X. Zheng, Z. Liu, K. W. Li, J. Huang and J. Chen, Cooperative game approaches to coordinating a three-echelon closed-loop supply chain with fairness concerns, International Journal of Production Economics, 212 (2019), 92-110. doi: 10.1016/j.ijpe.2019.01.011. Google Scholar

show all references

References:

[1]	F. H. S. a. H. Authority, Fuzhou Real Estate Joint Information Network - Innovative demolition of old construction waste disposal mechanism in our city, 2019, URL http://fgj.fuzhou.gov.cn/zz/xxgk/ggtz/zsgl/201804/t20180413_2180189.htm. Google Scholar
[2]	D. J. Bowersox, Strategic Marketing Channel Management, McGraw-Hill series in marketing, McGraw-Hill, New York, 1992. Google Scholar
[3]	Y. Chi, J. Dong, Y. Tang, Q. Huang and M. Ni, Life cycle assessment of municipal solid waste source-separated collection and integrated waste management systems in Hangzhou, China, Journal of Material Cycles and Waste Management, 17 (2015), 695-706. doi: 10.1007/s10163-014-0300-8. Google Scholar
[4]	C.-K. Chen and M. A. Ulya, Analyses of the reward-penalty mechanism in green closed-loop supply chains with product remanufacturing, International Journal of Production Economics, 210 (2019), 211-223. doi: 10.1016/j.ijpe.2019.01.006. Google Scholar
[5]	J. Chen, C. Hua and C. Liu, Considerations for better construction and demolition waste management: Identifying the decision behaviors of contractors and government departments through a game theory decision-making model, Journal of Cleaner Production, 212 (2019), 190-199. doi: 10.1016/j.jclepro.2018.11.262. Google Scholar
[6]	L. Devangan, R. Amit, P. Mehta, S. Swami and K. Shanker, Individually rational buyback contracts with inventory level dependent demand, International Journal of Production Economics, 142 (2013), 381-387. doi: 10.1016/j.ijpe.2012.12.014. Google Scholar
[7]	Z. Ding, G. Yi, V. W. Tam and T. Huang, A system dynamics-based environmental performance simulation of construction waste reduction management in China, Waste Management, 51 (2016), 130-141. doi: 10.1016/j.wasman.2016.03.001. Google Scholar
[8]	H. Ding, Y. Wang, S. Guo, X. Xu and C. Che, Game analysis and benefit allocation in international projects among owner, supervisor and contractor, International Journal of General Systems, 45 (2016), 253-270. doi: 10.1080/03081079.2015.1086575. Google Scholar
[9]	M. R. Esa, A. Halog and L. Rigamonti, Strategies for minimizing construction and demolition wastes in Malaysia, Resources, Conservation and Recycling, 120 (2017), 219-229. doi: 10.1016/j.resconrec.2016.12.014. Google Scholar
[10]	L. Feng, K. Govindan and C. Li, Strategic planning: Design and coordination for dual-recycling channel reverse supply chain considering consumer behavior, European Journal of Operational Research, 260 (2017), 601-612. doi: 10.1016/j.ejor.2016.12.050. Google Scholar
[11]	P. Fu, H. Li, X. Wang, J. Luo, S.-l. Zhan and C. Zuo, Multiobjective location model design based on government subsidy in the recycling of CDW, Mathematical Problems in Engineering, 2017 (2017), 1-9. doi: 10.1155/2017/9081628. Google Scholar
[12]	Wuxi Municipal Government, Implementation Opinions of the Municipal Government Office on the Management of Urban Construction Waste Recycling (Trial), 2019, URL http://js.wuxi.gov.cn/doc/2015/02/25/473368.shtml. Google Scholar
[13]	Jinan Municipal People's Government, Jinan Municipal People's Government Local Regulations Jinan City Construction Waste Management Regulations, 2019, URL http://www.jinan.gov.cn/art/2018/4/2/art_30123_331.html. Google Scholar
[14]	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. doi: 10.1016/j.ijpe.2019.03.013. Google Scholar
[15]	M. Hu and M. Zhou, Research on the Economic Benefits and Countermeasures of Construction Waste On-site Sorting during the Construction Process: A Case of Chongqing, Construction Economy, 39 (2018), 108-113. Google Scholar
[16]	B. Huang, X. Wang, H. Kua, Y. Geng, R. Bleischwitz and J. Ren, Construction and demolition waste management in China through the 3R principle, Resources, Conservation and Recycling, 129 (2018), 36-44. doi: 10.1016/j.resconrec.2017.09.029. Google Scholar
[17]	F. Huang, J. He and J. Wang, Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency, Journal of Industrial & Management Optimization, 15 (2019), 1753-1772. doi: 10.3934/jimo.2018121. Google Scholar
[18]	Y.-A. Hwang and Y.-H. Liao, Reduction and dynamic approach for the multi-choice Shapley value, Journal of Industrial and Management Optimization, 9 (2013), 885-892. doi: 10.3934/jimo.2013.9.885. Google Scholar
[19]	R. Jin, B. Li, T. Zhou, D. Wanatowski and P. Piroozfar, An empirical study of perceptions towards construction and demolition waste recycling and reuse in China, Resources, Conservation and Recycling, 126 (2017), 86-98. Google Scholar
[20]	C. Knoeri, C. R. Binder and H.-J. Althaus, Decisions on recycling: Construction stakeholders' decisions regarding recycled mineral construction materials, Resources, Conservation and Recycling, 55 (2011), 1039-1050. doi: 10.1016/j.resconrec.2011.05.018. Google Scholar
[21]	E. Kemahlioğlu-Ziya and J. J. Bartholdi, Centralizing inventory in supply chains by using shapley value to allocate the profits, Manufacturing & Service Operations Management, 13 (2011), 146-162. Google Scholar
[22]	A. Laruelle and F. Valenciano, Shapley-Shubik and Banzhaf indices revisited, Mathematics of Operations Research, 26 (2001), 89-104. doi: 10.1287/moor.26.1.89.10589. Google Scholar
[23]	B.-z. Li and X.-f. Luo, Study on profit allocation of enterprise' s original innovation with an industry-university-research cooperative mode based on the shapley value, Operations Research and Management Science, 22 (2013), 220-224. Google Scholar
[24]	Y. Li, X. Zhang, G. Ding and Z. Feng, Developing a quantitative construction waste estimation model for building construction projects, Resources, Conservation and Recycling, 106 (2016), 9-20. doi: 10.1016/j.resconrec.2015.11.001. Google Scholar
[25]	X. Li, Z. Mu and Z. Song, Research on manufacturers' collecting and remanufacturing the closed -loop supply chain model based on the game theory, Science Research Management, 34 (2013), 64-71. Google Scholar
[26]	J. Liu, J. Nie and H. Yuan, To expand or not to expand: A strategic analysis of the recycler's waste treatment capacity, Computers & Industrial Engineering, 130 (2019), 731-744. doi: 10.1016/j.cie.2019.03.016. Google Scholar
[27]	J. Liu, J. Nie and H. Yuan, To expand or not to expand: A strategic analysis of the recycler's waste treatment capacity, Computers & Industrial Engineering, 130 (2019), 731-744. doi: 10.1016/j.cie.2019.03.016. Google Scholar
[28]	Z. Lu and W. Zhu, Profit distribution for information production supply chain based on modified interval-valued fuzzy Shapley value: Profit distribution for information production supply chain based on modified interval-valued fuzzy Shapley value, Journal of Computer Applications, 33 (2013), 2960-2963. doi: 10.3724/SP.J.1087.2013.02960. Google Scholar
[29]	W. Lu, X. Chen, Y. Peng and L. Shen, Benchmarking construction waste management performance using big data, Resources, Conservation and Recycling, 105 (2015), 49-58. doi: 10.1016/j.resconrec.2015.10.013. Google Scholar
[30]	Y.-j. Lu, X.-f. Xu and X.-z. Ai, Collective decision-making of a closed-loop supply chain dual-channel model under the third-party economies of scale, Journal of Industrial Engineering and Engineering Management, 32 (2018), 207-217. Google Scholar
[31]	S. R. Madani and M. Rasti-Barzoki, Sustainable supply chain management with pricing, greening and governmental tariffs determining strategies: A game-theoretic approach, Computers & Industrial Engineering, 105 (2017), 287-298. doi: 10.1016/j.cie.2017.01.017. Google Scholar
[32]	N. G. Mankiw, Principles of economics / N. Gregory Mankiw, 5th edition, South-Western Cengage Learning, Mason, OH, 2009. Google Scholar
[33]	T. M. Mak, I. K. Yu, L. Wang, S.-C. Hsu, D. C. Tsang, C. Li, T. L. Yeung, R. Zhang and C. S. Poon, Extended theory of planned behaviour for promoting construction waste recycling in Hong Kong, Waste Management, 83 (2019), 161-170. doi: 10.1016/j.wasman.2018.11.016. Google Scholar
[34]	S. Mitra and S. Webster, Competition in remanufacturing and the effects of government subsidies, International Journal of Production Economics, 111 (2008), 287-298. doi: 10.1016/j.ijpe.2007.02.042. Google Scholar
[35]	T. H. Mills and E. Showalter, A cost-effective waste management plan, Cost Engineering, 41 (1999), 35. Google Scholar
[36]	S. Panda, Coordination of a socially responsible supply chain using revenue sharing contract, Transportation Research Part E: Logistics and Transportation Review, 67 (2014), 92-104. Google Scholar
[37]	E. S. Park, K. U. Kwon and K. S. An, A study on policy decision making and social conflict factors considering the stakeholder of public construction projects, KOREA SCIENCE & ART FORUM, 35 (2018), 187-201. Google Scholar
[38]	C. S. Poon, A. T. W. Yu, A. Wong and R. Yip, Quantifying the impact of construction waste charging scheme on construction waste management in Hong Kong, Journal of Construction Engineering and Management, 139 (2013), 466-479. doi: 10.1061/(ASCE)CO.1943-7862.0000631. Google Scholar
[39]	S. Panda, N. M. Modak and L. E. Cárdenas-Barrón, Coordinating a socially responsible closed-loop supply chain with product recycling, International Journal of Production Economics, 188 (2017), 11-21. Google Scholar
[40]	S. Saha, S. Sarmah and I. Moon, Dual channel closed-loop supply chain coordination with a reward-driven remanufacturing policy, International Journal of Production Research, 54 (2016), 1503-1517. doi: 10.1080/00207543.2015.1090031. Google Scholar
[41]	L. Sun and S.-r. Sun, Research on Cooperation Mechanism of Infrastructure Project Financing Alliance Based on Shapley Value, Industrial Engineering and Management, 22 (2017), 76-82. Google Scholar
[42]	S. Swami and J. Shah, Channel coordination in green supply chain management, Journal of the Operational Research Society, 64 (2013), 336-351. Google Scholar
[43]	Y. Teng, X. Li, P. Wu and X. Wang, Using cooperative game theory to determine profit distribution in IPD projects, International Journal of Construction Management, 19 (2019), 32-45. doi: 10.1080/15623599.2017.1358075. Google Scholar
[44]	M. M. M. Teo and M. Loosemore, A theory of waste behaviour in the construction industry, Construction Management and Economics, 19 (2001), 741-751. doi: 10.1080/01446190110067037. Google Scholar
[45]	G. Wang, X. Ai, C. Zheng and L. Zhong, Strategic inventory with competing suppliers, Journal of Industrial & Management Optimization, 2019. Google Scholar
[46]	Y. Wang, Q. Han, B. de Vries and J. Zuo, How the public reacts to social impacts in construction projects? A structural equation modeling study, International Journal of Project Management, 34 (2016), 1433-1448. doi: 10.1016/j.ijproman.2016.07.008. Google Scholar
[47]	J. Wu, M. Hu and S. Shi, Expanding Boundaries - Eco-Efficiency of Construction and Demolition Waste Recycling in Chongqing, China – J. Wu, M. Hu, S. Shi, T. Liu, C. Zhang, 1. Google Scholar
[48]	L.-l. Xie, B. Xia, Y. Hu, M. Shan, Y. Le and A. P. Chan, Public participation performance in public construction projects of South China: A case study of the Guangzhou Games venues construction, International Journal of Project Management, 35 (2017), 1391-1401. doi: 10.1016/j.ijproman.2017.04.003. Google Scholar
[49]	J. Yang and B. lai, Research on profit distribution strategy of supply chain cooperation with unequal power, Journal of Engineering Management, 33–37. Google Scholar
[50]	H. Yuan, L. Shen and J. Wang, Major obstacles to improving the performance of waste management in China's construction industry, Facilities, 29 (2011), 224-242. doi: 10.1108/02632771111120538. Google Scholar
[51]	J. Zhang and J. Chen, Information sharing in a make-to-stock supply chain, Journal of Industrial and Management Optimization, 10 (2014), 1169-1189. doi: 10.3934/jimo.2014.10.1169. Google Scholar
[52]	W. Zhao, H. Ren and V. Rotter, A system dynamics model for evaluating the alternative of type in construction and demolition waste recycling center – The case of Chongqing, China, Resources, Conservation and Recycling, 55 (2011), 933-944. doi: 10.1016/j.resconrec.2011.04.011. Google Scholar
[53]	W. Zhao, R. Leeftink and V. Rotter, Evaluation of the economic feasibility for the recycling of construction and demolition waste in China–The case of Chongqing, Resources, Conservation and Recycling, 54 (2010), 377-389. doi: 10.1016/j.resconrec.2009.09.003. Google Scholar
[54]	X.-X. Zheng, Z. Liu, K. W. Li, J. Huang and J. Chen, Cooperative game approaches to coordinating a three-echelon closed-loop supply chain with fairness concerns, International Journal of Production Economics, 212 (2019), 92-110. doi: 10.1016/j.ijpe.2019.01.011. Google Scholar

Figure 1. Closed-loop supply chain of building materials products

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Figure 2. Government subsidies $ (\eta<0) $ comparison between different cases when $ c $ and $ \Delta $ changes

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Figure 5. Recycler's (Manufacturer's) profit $ \pi_{C}\left(\pi_{M}\right) $ comparison between different distribution cases

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Figure 3. Retail price $ p $ (Wholesale price $ \omega $) comparison between different alliance cases

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Figure 4. Recovery rate $ \tau $ (Government utility $ \pi_{g} $) comparison between different alliance cases

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Table100

Notoations	Definitions
$ \alpha $	Potential market demand for new building products; $ (\alpha>0) $
$ \beta $	Sensitivity factor of public construction project contractors (consumers) to new building products; $ (\beta>0) $
$ D(p) $	Demand function for new building products
$ D(p, \eta) $	The demand function of new building products under government regulation
$ \eta $	Government penalizes manufacturer's products ($ \eta>0 $) or subsidies ($ \eta<0 $)
$ c_{n} $	The unit cost of producing new building products from new raw materials purchased by manufacturers; $ \left(c_{n}>0\right) $
$ c_{r} $	The unit cost of the manufacturer's use of CDW to produce new building products; $ \left(c_{r}>0\right) $
$ \omega $	The wholesale price of the new building products that the manufacturer provides to the retailer; $ (\omega>0) $
$ p $	The retail price of the new building products that the retailer manufacturer provides to the contractor; $ (p>0) $
$ c $	Collector's unit collection cost; $ (c>0) $
$ b $	The repurchase price of the CDW paid by the manufacturer to the recycler; $ (b>0) $
$ \tau $	The proportion of recycled CDW to the market demand for new building products, referred to as recovery rate; $ \tau \in[0,1] $
$ \gamma $	Environmental benefit coefficient, environmental benefits brought by unit CDW recycling
$ m $	The difficulty degree for collecting CDW; (m > 0)
$ c_{z} $	The unit cost of new building products, as new construction products include new and remanufactured products, so $ c_{z} = \tau c_{r}+ $ $ (1-\tau) c_{n} = c_{n}+\tau\left(c_{r}-c_{n}\right) = c_{n}-\Delta \tau $

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Notoations	Definitions
$ \alpha $	Potential market demand for new building products; $ (\alpha>0) $
$ \beta $	Sensitivity factor of public construction project contractors (consumers) to new building products; $ (\beta>0) $
$ D(p) $	Demand function for new building products
$ D(p, \eta) $	The demand function of new building products under government regulation
$ \eta $	Government penalizes manufacturer's products ($ \eta>0 $) or subsidies ($ \eta<0 $)
$ c_{n} $	The unit cost of producing new building products from new raw materials purchased by manufacturers; $ \left(c_{n}>0\right) $
$ c_{r} $	The unit cost of the manufacturer's use of CDW to produce new building products; $ \left(c_{r}>0\right) $
$ \omega $	The wholesale price of the new building products that the manufacturer provides to the retailer; $ (\omega>0) $
$ p $	The retail price of the new building products that the retailer manufacturer provides to the contractor; $ (p>0) $
$ c $	Collector's unit collection cost; $ (c>0) $
$ b $	The repurchase price of the CDW paid by the manufacturer to the recycler; $ (b>0) $
$ \tau $	The proportion of recycled CDW to the market demand for new building products, referred to as recovery rate; $ \tau \in[0,1] $
$ \gamma $	Environmental benefit coefficient, environmental benefits brought by unit CDW recycling
$ m $	The difficulty degree for collecting CDW; (m > 0)
$ c_{z} $	The unit cost of new building products, as new construction products include new and remanufactured products, so $ c_{z} = \tau c_{r}+ $ $ (1-\tau) c_{n} = c_{n}+\tau\left(c_{r}-c_{n}\right) = c_{n}-\Delta \tau $

Table 1. Calculation of the Shapley value of the recycler and manufacturer

S	Recycler's calculation of Shapley value		Manufacturer's calculation of Shapley value
S	$ C $	$ C \cup M $	$ M $	$ C \cup M $
$ \mathrm{v}(\mathrm{s}) $	$ {\pi_\mathrm{C}}^{*} $	$ {\pi_\mathrm{CM}}^{*} $	$ {\pi_\mathrm{M}}^{*} $	$ {\pi_\mathrm{CM}}^{*} $
$ \mathrm{v}(\mathrm{s} / \mathrm{i}) $	0	$ {\pi_\mathrm{M}}^{*} $	0	$ {\pi_\mathrm{C}}^{*} $
$ \mathrm{v}(\mathrm{s})-\mathrm{v}(\mathrm{s} / \mathrm{i}) $	$ \pi_{\mathrm{C}}^{*} $	$ \pi_{\mathrm{CM}}^{}-\pi_{\mathrm{M}}^{} $	$ \pi_{\mathrm{M}}^{*} $	$ \pi_{\mathrm{CM}}^{}-\pi_{\mathrm{C}}^{} $
$ \|\mathrm{s}\| $	1	2	1	2
$ \omega(\|\mathrm{s}\|) $	$ \frac{1}{2} $	$ \frac{1}{2} $	$ \frac{1}{2} $	$ \frac{1}{2} $
$ \omega(\|\mathrm{s}\|)[\mathrm{v}(\mathrm{s})-\mathrm{v}(\mathrm{s} / \mathrm{i})] $	$ \frac{{\pi_\mathrm{C}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}}^{}}{2} $	$ \frac{{\pi_\mathrm{M}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{C}}^{}}{2} $
Recycler's Shapley value distribution	$ \frac{\pi_{\mathrm{CM}}^{}-\pi_{\mathrm{M}}^{}+\pi_{\mathrm{C}}^{*}}{2} $
Manufacturer's Shapley value distribution	$ \frac{\pi_{\mathrm{CM}}^{}-\pi_{\mathrm{C}}^{}+\pi_{\mathrm{M}}^{*}}{2} $

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S	Recycler's calculation of Shapley value		Manufacturer's calculation of Shapley value
S	$ C $	$ C \cup M $	$ M $	$ C \cup M $
$ \mathrm{v}(\mathrm{s}) $	$ {\pi_\mathrm{C}}^{*} $	$ {\pi_\mathrm{CM}}^{*} $	$ {\pi_\mathrm{M}}^{*} $	$ {\pi_\mathrm{CM}}^{*} $
$ \mathrm{v}(\mathrm{s} / \mathrm{i}) $	0	$ {\pi_\mathrm{M}}^{*} $	0	$ {\pi_\mathrm{C}}^{*} $
$ \mathrm{v}(\mathrm{s})-\mathrm{v}(\mathrm{s} / \mathrm{i}) $	$ \pi_{\mathrm{C}}^{*} $	$ \pi_{\mathrm{CM}}^{}-\pi_{\mathrm{M}}^{} $	$ \pi_{\mathrm{M}}^{*} $	$ \pi_{\mathrm{CM}}^{}-\pi_{\mathrm{C}}^{} $
$ \|\mathrm{s}\| $	1	2	1	2
$ \omega(\|\mathrm{s}\|) $	$ \frac{1}{2} $	$ \frac{1}{2} $	$ \frac{1}{2} $	$ \frac{1}{2} $
$ \omega(\|\mathrm{s}\|)[\mathrm{v}(\mathrm{s})-\mathrm{v}(\mathrm{s} / \mathrm{i})] $	$ \frac{{\pi_\mathrm{C}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}}^{}}{2} $	$ \frac{{\pi_\mathrm{M}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{C}}^{}}{2} $
Recycler's Shapley value distribution	$ \frac{\pi_{\mathrm{CM}}^{}-\pi_{\mathrm{M}}^{}+\pi_{\mathrm{C}}^{*}}{2} $
Manufacturer's Shapley value distribution	$ \frac{\pi_{\mathrm{CM}}^{}-\pi_{\mathrm{C}}^{}+\pi_{\mathrm{M}}^{*}}{2} $

Table 2. Calculation of the Shapley value of the recycler and manufacturer

Player i	Recycler	Manufacturer
$ \lambda_{i} $	$ \frac{\sqrt{t_{1}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}} $	$ \frac{\sqrt{t_{2}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}} $
$ \Delta \lambda_{i} $	$ \frac{\sqrt{t_{1}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}}-\frac{1}{2} $	$ \frac{\sqrt{t_{2}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}}-\frac{1}{2} $
$ \Delta \varphi_{i}(v) $	$ \left(\frac{t_{1}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{C M}}^{*} $	$ \left(\frac{t_{2}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{C M}^{*}} $
$ \varphi_{i}(v) $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}}^{}+{\pi_\mathrm{C}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{C}}^{}+{\pi_\mathrm{M}}^{*}}{2} $
$ \varphi_{i}^{*}(v) $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}^{}}+{\pi_\mathrm{C}}^{}}{2}-\left(\frac{t_{1}}{t_{1}+t_{2}}-\frac{1}{2}\right)\cdot \epsilon \cdot {\pi_{C M}}^{} $	$ \frac{\pi_{\mathrm{CM}^{}}-{\pi_\mathrm{C}^{}}+{\pi_\mathrm{M}}^{}}{2}-\left(\frac{t_{2}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{CM}}^{} $

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Player i	Recycler	Manufacturer
$ \lambda_{i} $	$ \frac{\sqrt{t_{1}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}} $	$ \frac{\sqrt{t_{2}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}} $
$ \Delta \lambda_{i} $	$ \frac{\sqrt{t_{1}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}}-\frac{1}{2} $	$ \frac{\sqrt{t_{2}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}}-\frac{1}{2} $
$ \Delta \varphi_{i}(v) $	$ \left(\frac{t_{1}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{C M}}^{*} $	$ \left(\frac{t_{2}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{C M}^{*}} $
$ \varphi_{i}(v) $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}}^{}+{\pi_\mathrm{C}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{C}}^{}+{\pi_\mathrm{M}}^{*}}{2} $
$ \varphi_{i}^{*}(v) $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}^{}}+{\pi_\mathrm{C}}^{}}{2}-\left(\frac{t_{1}}{t_{1}+t_{2}}-\frac{1}{2}\right)\cdot \epsilon \cdot {\pi_{C M}}^{} $	$ \frac{\pi_{\mathrm{CM}^{}}-{\pi_\mathrm{C}^{}}+{\pi_\mathrm{M}}^{}}{2}-\left(\frac{t_{2}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{CM}}^{} $

Table 3. The maximum profit for each alliance

	$R U C$	$R \cup M$
$\eta^{*}$	$\frac{(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(2+\beta \mu))\left(\alpha-\beta c_{n}\right)}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	$\frac{((c-\Delta)(c-4 \gamma-\Delta)+4 m \mu)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$
$\omega^{*}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-4 m\right)+\beta^{2}\left(c^{2}+4 \gamma \Delta+\Delta^{2}-2 c(2 \gamma+\Delta)+4 m \mu\right) c_{n}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	---
$b^{*}$	$\frac{c+\Delta}{2}$	$\frac{c+\Delta}{2}$
$p^{*}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-6 m\right)+\beta(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(1+\beta \mu)) c_{n}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-2 m\right)+\beta(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(-1+\beta \mu)) c_{n}}{\beta(\zeta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$
$\tau^{*}$	$\frac{2(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$	$\frac{2(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$
	$C \cup M$	$R \cup C \cup M$
$\eta^{*}$	$\frac{(2 \beta \gamma(-c+\Delta)+m(2+\beta \mu))\left(\alpha-\beta c_{n}\right)}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	$\frac{(2 c \gamma-2 \gamma \Delta-m \mu)\left(-\alpha+\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$
$\omega^{*}$	$\frac{\alpha\left(-2 m+\beta(c-\Delta)^{2}\right)+\beta^{2}(-2 c \gamma+2 \gamma \Delta+m \mu) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	---
$b^{*}$	$\frac{c+\Delta}{2}$	$\frac{c+\Delta}{2}$
$p^{*}$	$\frac{\alpha\left(-3 m+\beta(c-\Delta)^{2}\right)+\beta(m+2 \beta \gamma(-c+\Delta)+m \beta \mu) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	$\frac{\alpha\left(-m+\beta(c-\Delta)^{2}\right)+\beta(2 \beta \gamma(-c+\Delta)+m(-1+\beta \mu)) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$
$\tau^{*}$	$\frac{(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$	$\frac{(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$
$\pi_{\mathrm{rc}}^{*}$	$\frac{2 m\left(8 m+\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{rm}}^{*}$	$\frac{4 m\left(4 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{cm}}^{*}$	$\frac{m\left(4 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{2 \beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{rcm}}$	$\frac{m\left(2 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{2 \beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}}$

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	$R U C$	$R \cup M$
$\eta^{*}$	$\frac{(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(2+\beta \mu))\left(\alpha-\beta c_{n}\right)}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	$\frac{((c-\Delta)(c-4 \gamma-\Delta)+4 m \mu)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$
$\omega^{*}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-4 m\right)+\beta^{2}\left(c^{2}+4 \gamma \Delta+\Delta^{2}-2 c(2 \gamma+\Delta)+4 m \mu\right) c_{n}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	---
$b^{*}$	$\frac{c+\Delta}{2}$	$\frac{c+\Delta}{2}$
$p^{*}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-6 m\right)+\beta(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(1+\beta \mu)) c_{n}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-2 m\right)+\beta(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(-1+\beta \mu)) c_{n}}{\beta(\zeta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$
$\tau^{*}$	$\frac{2(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$	$\frac{2(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$
	$C \cup M$	$R \cup C \cup M$
$\eta^{*}$	$\frac{(2 \beta \gamma(-c+\Delta)+m(2+\beta \mu))\left(\alpha-\beta c_{n}\right)}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	$\frac{(2 c \gamma-2 \gamma \Delta-m \mu)\left(-\alpha+\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$
$\omega^{*}$	$\frac{\alpha\left(-2 m+\beta(c-\Delta)^{2}\right)+\beta^{2}(-2 c \gamma+2 \gamma \Delta+m \mu) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	---
$b^{*}$	$\frac{c+\Delta}{2}$	$\frac{c+\Delta}{2}$
$p^{*}$	$\frac{\alpha\left(-3 m+\beta(c-\Delta)^{2}\right)+\beta(m+2 \beta \gamma(-c+\Delta)+m \beta \mu) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	$\frac{\alpha\left(-m+\beta(c-\Delta)^{2}\right)+\beta(2 \beta \gamma(-c+\Delta)+m(-1+\beta \mu)) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$
$\tau^{*}$	$\frac{(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$	$\frac{(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$
$\pi_{\mathrm{rc}}^{*}$	$\frac{2 m\left(8 m+\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{rm}}^{*}$	$\frac{4 m\left(4 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{cm}}^{*}$	$\frac{m\left(4 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{2 \beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{rcm}}$	$\frac{m\left(2 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{2 \beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}}$

Table 4. Calculation of the Shapley value of the retailer

s	$ R $	$ R \cup C $	$ R \cup M $	$ R \cup C \cup M $
$ \mathbf{v}(s) $	$ \pi_{r 3}^{*} $	$ \pi_{r 3}^{}+\pi_{c 3}^{} $	$ \pi_{\mathrm{rm} 4}^{*} $	$ \pi_{\mathrm{rcm} 7}^{*} $
$ \mathbf{v}(s/i) $	0	$ \pi_{\mathrm{c} 3}^{*} $	$ \pi_{\mathrm{m} 3}^{*} $	$ \pi_{\mathrm{cm} 5}^{*} $
$ \mathbf{v}(s)-\mathbf{v}(s/i) $	$ \pi_{\mathrm{r} 3}^{*} $	$ \pi_{\mathrm{r} 3}^{*} $	$ \pi_{\mathrm{rm} 4}^{}-\pi_{\mathrm{m} 3}^{} $	$ \pi_{\mathrm{rcm} 7}^{}-\pi_{\mathrm{cm} 5}^{} $
$ \|s\| $	1	2	2	3
$ \omega (\|s\|) $	$ \frac{1}{3} $	$ \frac{1}{6} $	$ \frac{1}{6} $	$ \frac{1}{3} $
$ \omega(\|s\|)[v(s)-v(s)i)] $	$ \frac{\pi_{\mathrm{r} 3}^{*}}{3} $	$ \frac{\pi_{\mathrm{r} 3}^{*}}{6} $	$ \frac{\pi_{\mathrm{rm} 4}^{}-\pi_{\mathrm{m} 3}^{}}{6} $	$ \frac{\pi_{\mathrm{rcm} 7}^{}-\pi_{\mathrm{cm} 5}^{}}{3} $
Retailer 's Shapley value distribution	$ \frac{m^{2}(c-\Delta)(c-4 \gamma-\Delta)(\beta(7 c-12 \gamma-7 \Delta)(c-\Delta)+8 m(-2+\beta \mu))\left(\alpha-\beta c_{n}\right)^{2}}{3(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}} $

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s	$ R $	$ R \cup C $	$ R \cup M $	$ R \cup C \cup M $
$ \mathbf{v}(s) $	$ \pi_{r 3}^{*} $	$ \pi_{r 3}^{}+\pi_{c 3}^{} $	$ \pi_{\mathrm{rm} 4}^{*} $	$ \pi_{\mathrm{rcm} 7}^{*} $
$ \mathbf{v}(s/i) $	0	$ \pi_{\mathrm{c} 3}^{*} $	$ \pi_{\mathrm{m} 3}^{*} $	$ \pi_{\mathrm{cm} 5}^{*} $
$ \mathbf{v}(s)-\mathbf{v}(s/i) $	$ \pi_{\mathrm{r} 3}^{*} $	$ \pi_{\mathrm{r} 3}^{*} $	$ \pi_{\mathrm{rm} 4}^{}-\pi_{\mathrm{m} 3}^{} $	$ \pi_{\mathrm{rcm} 7}^{}-\pi_{\mathrm{cm} 5}^{} $
$ \|s\| $	1	2	2	3
$ \omega (\|s\|) $	$ \frac{1}{3} $	$ \frac{1}{6} $	$ \frac{1}{6} $	$ \frac{1}{3} $
$ \omega(\|s\|)[v(s)-v(s)i)] $	$ \frac{\pi_{\mathrm{r} 3}^{*}}{3} $	$ \frac{\pi_{\mathrm{r} 3}^{*}}{6} $	$ \frac{\pi_{\mathrm{rm} 4}^{}-\pi_{\mathrm{m} 3}^{}}{6} $	$ \frac{\pi_{\mathrm{rcm} 7}^{}-\pi_{\mathrm{cm} 5}^{}}{3} $
Retailer 's Shapley value distribution	$ \frac{m^{2}(c-\Delta)(c-4 \gamma-\Delta)(\beta(7 c-12 \gamma-7 \Delta)(c-\Delta)+8 m(-2+\beta \mu))\left(\alpha-\beta c_{n}\right)^{2}}{3(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}} $

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American Institute of Mathematical Sciences

Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation

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American Institute of Mathematical Sciences

Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation

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