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November  2021, 17(6): 3247-3268. doi: 10.3934/jimo.2020116

Channel leadership and recycling channel in closed-loop supply chain: The case of recycling price by the recycling party

1. 

College of Information Science and Engineering, Northeastern University, Fundamental Teaching Department of Computer and Mathematics, Shenyang Normal University, Shenyang, Liaoning, 110034, China

2. 

Research Institute of Business Analytics and Supply Chain Management, College of Management, Shenzhen University, Shenzhen, 518060, China

3. 

College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning, 110819, China

4. 

Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong, China

5. 

College of Computer Science and Engineering, Northeastern University, Shenyang, Liaoning, 110819, China

* Corresponding author: Min Huang

Received  June 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

Due to the fast growing of the waste electrical and electronic equipment (WEEE), the business values of closed-loop supply chains (CLSCs) have been well recognized. In this paper, we investigate the performance of the CLSCs under different combinations of the recycling channel and the channel leadership when the recycling price is determined by the recycling party. Specially, we consider a CLSC consisting of two channel members, i.e., a manufacturer and a retailer. Each member acting as the channel leader has three different channels to collect the used products, and they are (ⅰ) the manufacturer (M-channel), (ⅱ) the retailer (R-channel) and (ⅲ) the third-party (T-channel). Given the recycling party determines the recycling price, mathematical models are developed to investigate the performance of the CLSC under different combinations of the channel leadership and the recycling channel. Through a comparison analysis, we find that M-channel is the most effective recycling channel. Moreover, once the M-channel be adopted, the retailer-led structure is as good as manufacture-led structure. We find that the recycling channel structure could be more important than the channel leadership in the CLSC. Finally, we illustrate that the CLSC can be coordinated by a two-part tariff contract.

Citation: Zhidan Wu, Xiaohu Qian, Min Huang, Wai-Ki Ching, Hanbin Kuang, Xingwei Wang. Channel leadership and recycling channel in closed-loop supply chain: The case of recycling price by the recycling party. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3247-3268. doi: 10.3934/jimo.2020116
References:
[1]

E. Almehdawe and B. Matin, Vendor managed inventory with a capacitated manufacturer and multiple retailer: Retailer versus manufacturer leadership, Internat. J. Prod. Econ., 128 (2010), 292-302.  doi: 10.1016/j.ijpe.2010.07.029.

[2]

A. AtasuV. D. R. Guide and L. N. Van Wassenhove, Product reuse economics in closed-loop supply chain research, Prod. Oper. Manag., 17 (2010), 483-496.  doi: 10.3401/poms.1080.0051.

[3]

A. AtasuL. B. Toktay and L. N. Van Wassenhove, How collection cost structure drives a manufacturer's reverse channel choice, Prod. Oper. Manag., 22 (2013), 1089-1102.  doi: 10.1111/j.1937-5956.2012.01426.x.

[4]

M. Bhattacharyya and S. S. Sana, A mathematical model on eco-friendly manufacturing system under probabilistic demand, RAIRO Oper. Res., 53 (2019), 1899-1913.  doi: 10.1051/ro/2018120.

[5]

G. P. Cachon and A. G. Kök, Competing manufacturers in a retail supply chain: On contractual form and coordination, Manag. Sci., 56 (2010), 571-589.  doi: 10.1287/mnsc.1090.1122.

[6]

J.-M. Chen and C.-I. Chang, The co-operative strategy of a closed-loop supply chain with remanufacturing, Transpor. Res. Part E, 48 (2012), 387-400.  doi: 10.1016/j.tre.2011.10.001.

[7]

T.-M. ChoiY. Li and L. Xu, Channel leadership, performance and coordination in closed loop supply chains, Internat. J. Prod. Econ., 146 (2013), 371-380.  doi: 10.1016/j.ijpe.2013.08.002.

[8]

C.-H. ChuangC. X. Wang and Y. Zhao, Closed-loop supply chain models for a high-tech product under alternative reverse channel and collection cost structures, Internat. J. Prod. Econ., 156 (2014), 108-123.  doi: 10.1016/j.ijpe.2014.05.008.

[9]

P. De Giovanni and G. Zaccour, A two-period game of a closed-loop supply chain, European J. Oper. Res., 232 (2014), 22-40.  doi: 10.1016/j.ejor.2013.06.032.

[10]

V. D. R. Guide and L. N. Van Wassenhove, The evolution of closed-loop supply chain research, Oper. Res., 57 (2009), 10-18. 

[11]

G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700.  doi: 10.1023/A:1014920510164.

[12]

L. FengK. Govindan and C. Li, Strategic planning: Design and coordination for dual-recycling channel reverse supply chain considering consumer behavior, European J. Oper. Res., 260 (2017), 601-612.  doi: 10.1016/j.ejor.2016.12.050.

[13]

H. Garg, Fuzzy inventory models for deteriorating items under different types of lead-time distributions, in Intelligent Techniques in Engineering Management, Intelligent Systems Reference Library, 87, Springer, Cham, 2015,247–274. doi: 10.1007/978-3-319-17906-3_11.

[14]

V. D. R. GuideR. H. Teunter and L. N. Van Wassenhove, Matching demand and supply to maximize profits from remanufacturing, Manufac. Service Oper. Manag., 5 (2003), 303-316.  doi: 10.1287/msom.5.4.303.24883.

[15]

X. HongZ. Wang and H. Zhang, Decision models of closed-loop supply chain with remanufacturing under hybrid dual-channel collection, Internat. J. Advanced Manufac. Tech., 68 (2013), 1851-1865.  doi: 10.1007/s00170-013-4982-1.

[16]

M. HuangM. SongL. H. Lee and W. K. Ching, Analysis for strategy of closed-loop supply chain with dual recycling channel, Internat. J. Prod. Econ., 144 (2013), 510-520.  doi: 10.1016/j.ijpe.2013.04.002.

[17]

A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272.  doi: 10.1287/mksc.2.3.239.

[18]

I. KarakayaliH. Emir-Farinas and E. Akcal, An analysis of decentralized collection and processing of end-of-life products, J. Oper. Manag., 25 (2007), 1161-1183.  doi: 10.1016/j.jom.2007.01.017.

[19]

Y. LiangS. Pokharel and G. H. Lim, Pricing used products for remanufacturing, European J. Oper. Res., 193 (2009), 390-395.  doi: 10.1016/j.ejor.2007.11.029.

[20]

H. LiuM. LeiH. DengG. K. Leong and T. Huang, A dual channel, quality-based price competition model for the WEEE recycling market with government subsidy, Omega, 59 (2016), 290-302.  doi: 10.1016/j.omega.2015.07.002.

[21]

P. Majumder and A. Srinivasan, Leadership and competition in network supply chains, Manag. Science, 54 (2008), 1189-1204.  doi: 10.1287/mnsc.1070.0752.

[22]

T. W. McGuire and R. Staelin, An industry equilibrium analysis of downstream vertical integration, Marketing Science, 2 (1983), 161-190.  doi: 10.1287/mksc.2.2.161.

[23]

P. R. Messinger and C. Narasimhan, Has power shifted in the grocery channel?, Marketing Science, 14 (1995), 189-223. 

[24]

S. Mitra, Revenue management for remanufactured products, Omega, 35 (2007), 553-562.  doi: 10.1016/j.omega.2005.10.003.

[25]

S. K. MukhopadhyayD.-Q. Yao and X. Yue, Information sharing of value-adding retailer in a mixed channel hi-tech supply chain, J. Business Res., 61 (2008), 950-958.  doi: 10.1016/j.jbusres.2006.10.027.

[26]

I. E. NielsenS. MajumderS. S. Sana and S. Saha, Comparative analysis of government incentives and game structures on single and two-period green supply chain, J. Cleaner Prod., 235 (2019), 1371-1398.  doi: 10.1016/j.jclepro.2019.06.168.

[27]

A. ÖrsdemirE. Kemahlioğlu-Ziya and A. K. Parlaktürk, Competitive quality choice and remanufacturing, Prod. and Oper. Manag., 23 (2014), 48-64.  doi: 10.1111/poms.12040.

[28]

S. SahaN. M. ModakS. Panda and S. S. Sana, Managing a retailer's dual-channel supply chain under price- and delivery time-sensitive demand, J. Modelling Manag., 13 (2018), 351-374.  doi: 10.1108/JM2-10-2016-0089.

[29]

S. SahaN. M. ModakS. Panda and S. S. Sana, Promotional coordination mechanisms with demand dependent on price and sales efforts, J. Industrial Prod. Engrg., 36 (2019), 13-31.  doi: 10.1080/21681015.2019.1565451.

[30]

S. S. SanaJ. Ferro-CorreaA. Quintero and R. Amaya, A system dynamics model of financial flow in supply chains: A case study, RAIRO Oper. Res., 52 (2018), 187-204.  doi: 10.1051/ro/2017025.

[31]

R. C. SavaskanS. Bhattacharya and L. N. Van Wassenhove, Closed-loop supply chain models with product remanufacturing, Manag. Science, 50 (2004), 239-252.  doi: 10.1287/mnsc.1030.0186.

[32]

R. C. Savaskan and L. N. Van Wassenhove, Reverse channel design: The case of competing retailers, Manag. Science, 52 (2006), 1-14.  doi: 10.1287/mnsc.1050.0454.

[33]

J. ShiG. Zhang and J. Sha, Optimal production and pricing policy for a closed loop system, Resources Conservation Recycling, 55 (2011), 639-647.  doi: 10.1016/j.resconrec.2010.05.016.

[34]

S. TiwariC. K. Jaggi and S. S. Sana, Integrated supply chain of supplier and retailer for stochastic demand, Math. Model. Anal., 23 (2018), 582-595.  doi: 10.3846/mma.2018.035.

[35]

A. A. TaleizadehM. S. Moshtagh and I. Moon, Pricing, product quality, and collection optimization in a decentralized closed-loop supply chain with different channel structures: Game theoretical approach, J. Cleaner Prod., 189 (2018), 406-431.  doi: 10.1016/j.jclepro.2018.02.209.

[36]

R. H. Teunter, Determining optimal disassembly and recovery strategies, Omega, 34 (2006), 533-537.  doi: 10.1016/j.omega.2005.01.014.

[37]

M. ThierryM. Salomon and N. L. V. Wassenhove, Strategies issues in product recovery management, California Manag. Review, 37 (1995), 114-135.  doi: 10.2307/41165792.

[38]

A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing and Service Oper. Manag., 2 (2000), 93-110.  doi: 10.1287/msom.2.4.372.12342.

[39]

J. Vorasayan and S. M. Ryan, Optimal price and quantity of refurbished products, Prod. Oper. Manag., 15 (2006), 369-383.  doi: 10.1111/j.1937-5956.2006.tb00251.x.

[40]

Z. WangB. ZhangJ. Yin and X. Zhang, Willingness and behavior towards e-waste recycling for residents in Beijing city, China, J. Cleaner Prod., 19 (2011), 977-984.  doi: 10.1016/j.jclepro.2010.09.016.

[41]

W. WangY. ZhangK. ZhangT. Bai and J. Shang, Reward-penalty mechanism for closed-loop supply chains under responsibility-sharing and different power structures, Internat. J. Prod. Econ., 170 (2015), 178-190.  doi: 10.1016/j.ijpe.2015.09.003.

[42]

D.-Q. Yao and J. J. Liu, Competitive pricing of mixed retail and e-tail distribution channels, OMEGA, 33 (2005), 235-247.  doi: 10.1016/j.omega.2004.04.007.

show all references

References:
[1]

E. Almehdawe and B. Matin, Vendor managed inventory with a capacitated manufacturer and multiple retailer: Retailer versus manufacturer leadership, Internat. J. Prod. Econ., 128 (2010), 292-302.  doi: 10.1016/j.ijpe.2010.07.029.

[2]

A. AtasuV. D. R. Guide and L. N. Van Wassenhove, Product reuse economics in closed-loop supply chain research, Prod. Oper. Manag., 17 (2010), 483-496.  doi: 10.3401/poms.1080.0051.

[3]

A. AtasuL. B. Toktay and L. N. Van Wassenhove, How collection cost structure drives a manufacturer's reverse channel choice, Prod. Oper. Manag., 22 (2013), 1089-1102.  doi: 10.1111/j.1937-5956.2012.01426.x.

[4]

M. Bhattacharyya and S. S. Sana, A mathematical model on eco-friendly manufacturing system under probabilistic demand, RAIRO Oper. Res., 53 (2019), 1899-1913.  doi: 10.1051/ro/2018120.

[5]

G. P. Cachon and A. G. Kök, Competing manufacturers in a retail supply chain: On contractual form and coordination, Manag. Sci., 56 (2010), 571-589.  doi: 10.1287/mnsc.1090.1122.

[6]

J.-M. Chen and C.-I. Chang, The co-operative strategy of a closed-loop supply chain with remanufacturing, Transpor. Res. Part E, 48 (2012), 387-400.  doi: 10.1016/j.tre.2011.10.001.

[7]

T.-M. ChoiY. Li and L. Xu, Channel leadership, performance and coordination in closed loop supply chains, Internat. J. Prod. Econ., 146 (2013), 371-380.  doi: 10.1016/j.ijpe.2013.08.002.

[8]

C.-H. ChuangC. X. Wang and Y. Zhao, Closed-loop supply chain models for a high-tech product under alternative reverse channel and collection cost structures, Internat. J. Prod. Econ., 156 (2014), 108-123.  doi: 10.1016/j.ijpe.2014.05.008.

[9]

P. De Giovanni and G. Zaccour, A two-period game of a closed-loop supply chain, European J. Oper. Res., 232 (2014), 22-40.  doi: 10.1016/j.ejor.2013.06.032.

[10]

V. D. R. Guide and L. N. Van Wassenhove, The evolution of closed-loop supply chain research, Oper. Res., 57 (2009), 10-18. 

[11]

G. Ertek and P. M. Griffin, Supplier-and buyer-driven channels in a two-stage supply chain, IIE Transactions, 34 (2002), 691-700.  doi: 10.1023/A:1014920510164.

[12]

L. FengK. Govindan and C. Li, Strategic planning: Design and coordination for dual-recycling channel reverse supply chain considering consumer behavior, European J. Oper. Res., 260 (2017), 601-612.  doi: 10.1016/j.ejor.2016.12.050.

[13]

H. Garg, Fuzzy inventory models for deteriorating items under different types of lead-time distributions, in Intelligent Techniques in Engineering Management, Intelligent Systems Reference Library, 87, Springer, Cham, 2015,247–274. doi: 10.1007/978-3-319-17906-3_11.

[14]

V. D. R. GuideR. H. Teunter and L. N. Van Wassenhove, Matching demand and supply to maximize profits from remanufacturing, Manufac. Service Oper. Manag., 5 (2003), 303-316.  doi: 10.1287/msom.5.4.303.24883.

[15]

X. HongZ. Wang and H. Zhang, Decision models of closed-loop supply chain with remanufacturing under hybrid dual-channel collection, Internat. J. Advanced Manufac. Tech., 68 (2013), 1851-1865.  doi: 10.1007/s00170-013-4982-1.

[16]

M. HuangM. SongL. H. Lee and W. K. Ching, Analysis for strategy of closed-loop supply chain with dual recycling channel, Internat. J. Prod. Econ., 144 (2013), 510-520.  doi: 10.1016/j.ijpe.2013.04.002.

[17]

A. P. Jeuland and S. M. Shugan, Managing channel profits, Marketing Science, 2 (1983), 239-272.  doi: 10.1287/mksc.2.3.239.

[18]

I. KarakayaliH. Emir-Farinas and E. Akcal, An analysis of decentralized collection and processing of end-of-life products, J. Oper. Manag., 25 (2007), 1161-1183.  doi: 10.1016/j.jom.2007.01.017.

[19]

Y. LiangS. Pokharel and G. H. Lim, Pricing used products for remanufacturing, European J. Oper. Res., 193 (2009), 390-395.  doi: 10.1016/j.ejor.2007.11.029.

[20]

H. LiuM. LeiH. DengG. K. Leong and T. Huang, A dual channel, quality-based price competition model for the WEEE recycling market with government subsidy, Omega, 59 (2016), 290-302.  doi: 10.1016/j.omega.2015.07.002.

[21]

P. Majumder and A. Srinivasan, Leadership and competition in network supply chains, Manag. Science, 54 (2008), 1189-1204.  doi: 10.1287/mnsc.1070.0752.

[22]

T. W. McGuire and R. Staelin, An industry equilibrium analysis of downstream vertical integration, Marketing Science, 2 (1983), 161-190.  doi: 10.1287/mksc.2.2.161.

[23]

P. R. Messinger and C. Narasimhan, Has power shifted in the grocery channel?, Marketing Science, 14 (1995), 189-223. 

[24]

S. Mitra, Revenue management for remanufactured products, Omega, 35 (2007), 553-562.  doi: 10.1016/j.omega.2005.10.003.

[25]

S. K. MukhopadhyayD.-Q. Yao and X. Yue, Information sharing of value-adding retailer in a mixed channel hi-tech supply chain, J. Business Res., 61 (2008), 950-958.  doi: 10.1016/j.jbusres.2006.10.027.

[26]

I. E. NielsenS. MajumderS. S. Sana and S. Saha, Comparative analysis of government incentives and game structures on single and two-period green supply chain, J. Cleaner Prod., 235 (2019), 1371-1398.  doi: 10.1016/j.jclepro.2019.06.168.

[27]

A. ÖrsdemirE. Kemahlioğlu-Ziya and A. K. Parlaktürk, Competitive quality choice and remanufacturing, Prod. and Oper. Manag., 23 (2014), 48-64.  doi: 10.1111/poms.12040.

[28]

S. SahaN. M. ModakS. Panda and S. S. Sana, Managing a retailer's dual-channel supply chain under price- and delivery time-sensitive demand, J. Modelling Manag., 13 (2018), 351-374.  doi: 10.1108/JM2-10-2016-0089.

[29]

S. SahaN. M. ModakS. Panda and S. S. Sana, Promotional coordination mechanisms with demand dependent on price and sales efforts, J. Industrial Prod. Engrg., 36 (2019), 13-31.  doi: 10.1080/21681015.2019.1565451.

[30]

S. S. SanaJ. Ferro-CorreaA. Quintero and R. Amaya, A system dynamics model of financial flow in supply chains: A case study, RAIRO Oper. Res., 52 (2018), 187-204.  doi: 10.1051/ro/2017025.

[31]

R. C. SavaskanS. Bhattacharya and L. N. Van Wassenhove, Closed-loop supply chain models with product remanufacturing, Manag. Science, 50 (2004), 239-252.  doi: 10.1287/mnsc.1030.0186.

[32]

R. C. Savaskan and L. N. Van Wassenhove, Reverse channel design: The case of competing retailers, Manag. Science, 52 (2006), 1-14.  doi: 10.1287/mnsc.1050.0454.

[33]

J. ShiG. Zhang and J. Sha, Optimal production and pricing policy for a closed loop system, Resources Conservation Recycling, 55 (2011), 639-647.  doi: 10.1016/j.resconrec.2010.05.016.

[34]

S. TiwariC. K. Jaggi and S. S. Sana, Integrated supply chain of supplier and retailer for stochastic demand, Math. Model. Anal., 23 (2018), 582-595.  doi: 10.3846/mma.2018.035.

[35]

A. A. TaleizadehM. S. Moshtagh and I. Moon, Pricing, product quality, and collection optimization in a decentralized closed-loop supply chain with different channel structures: Game theoretical approach, J. Cleaner Prod., 189 (2018), 406-431.  doi: 10.1016/j.jclepro.2018.02.209.

[36]

R. H. Teunter, Determining optimal disassembly and recovery strategies, Omega, 34 (2006), 533-537.  doi: 10.1016/j.omega.2005.01.014.

[37]

M. ThierryM. Salomon and N. L. V. Wassenhove, Strategies issues in product recovery management, California Manag. Review, 37 (1995), 114-135.  doi: 10.2307/41165792.

[38]

A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing and Service Oper. Manag., 2 (2000), 93-110.  doi: 10.1287/msom.2.4.372.12342.

[39]

J. Vorasayan and S. M. Ryan, Optimal price and quantity of refurbished products, Prod. Oper. Manag., 15 (2006), 369-383.  doi: 10.1111/j.1937-5956.2006.tb00251.x.

[40]

Z. WangB. ZhangJ. Yin and X. Zhang, Willingness and behavior towards e-waste recycling for residents in Beijing city, China, J. Cleaner Prod., 19 (2011), 977-984.  doi: 10.1016/j.jclepro.2010.09.016.

[41]

W. WangY. ZhangK. ZhangT. Bai and J. Shang, Reward-penalty mechanism for closed-loop supply chains under responsibility-sharing and different power structures, Internat. J. Prod. Econ., 170 (2015), 178-190.  doi: 10.1016/j.ijpe.2015.09.003.

[42]

D.-Q. Yao and J. J. Liu, Competitive pricing of mixed retail and e-tail distribution channels, OMEGA, 33 (2005), 235-247.  doi: 10.1016/j.omega.2004.04.007.

Figure 1.  The manufacturer's tradeoff between the channel leadership and the recycling channel
Figure 2.  The retailer's tradeoff between the channel leadership and the recycling channel
Table 1.  Notations
Symbol Description
Parameters
$c_{m}$ Unit producing cost from original materials
$c_{0}$ Unit producing cost from returns
$\delta$ Unit saving cost by recovery, $\delta=c_{m}-c_{0}$
$A$ The size of the market
$\alpha$ Sensitivity of the consumers for the retail price, $\alpha>0$
$k$ The basic recovery quantity, which represents the level of
environmental awareness of consumers
$h$ Sensitivity of the customers for the recycling price, $h>0$
Decision variables
$p$ The unit retail price
$w$ The unit wholesale price
$b$ The unit recycling price in centralized decision system
$b_{j}$ The unit recycling price of the recycling party $j$, subscript
$j=t, r, m$ denotes the recycling by the third-party, the
retailer and the manufacturer, respectively
$b_{mj}$ The unit transfer price, $j=r, t$, denotes R-channel and
T-channel, respectively
Derived function
$D(p)$ The demand of the products
$R(b_{j})$ The amount of the recycling products
$\pi_{m}$ The profits of the manufacturer
$\pi_{r}$ The profits of the retailer
$\pi_{t}$ The profits of the third-party
$\Pi$ The profits of the system
Symbol Description
Parameters
$c_{m}$ Unit producing cost from original materials
$c_{0}$ Unit producing cost from returns
$\delta$ Unit saving cost by recovery, $\delta=c_{m}-c_{0}$
$A$ The size of the market
$\alpha$ Sensitivity of the consumers for the retail price, $\alpha>0$
$k$ The basic recovery quantity, which represents the level of
environmental awareness of consumers
$h$ Sensitivity of the customers for the recycling price, $h>0$
Decision variables
$p$ The unit retail price
$w$ The unit wholesale price
$b$ The unit recycling price in centralized decision system
$b_{j}$ The unit recycling price of the recycling party $j$, subscript
$j=t, r, m$ denotes the recycling by the third-party, the
retailer and the manufacturer, respectively
$b_{mj}$ The unit transfer price, $j=r, t$, denotes R-channel and
T-channel, respectively
Derived function
$D(p)$ The demand of the products
$R(b_{j})$ The amount of the recycling products
$\pi_{m}$ The profits of the manufacturer
$\pi_{r}$ The profits of the retailer
$\pi_{t}$ The profits of the third-party
$\Pi$ The profits of the system
Table 2.  Main results of the M-led models
Model MM Model MR Model MT
$p^*$ $p^{MM*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{MR*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{MT*}=\frac{3A+\alpha c_m}{4\alpha}$
$b^*_i$ $b^{MM*}_m=\frac{h\delta -k}{2h}$ $b^{MR*}_r=\frac{h\delta -3k}{4h}$ $b^{MT*}_t=\frac{h\delta -3k}{4h}$
$w^*$ $w^{MM*}=\frac{A+\alpha c_m}{2\alpha}$ $w^{MR*}=\frac{A+\alpha c_m}{2\alpha}$ $w^{MT*}=\frac{A+\alpha c_m}{2\alpha}$
$b^*_{mj}$ N/A $b^{MR*}_{mr}=\frac{h\delta -k}{2h}$ $b^{MT*}_{mt}=\frac{h\delta -k}{2h}$
$\pi^*_m$ $\pi^{MM*}_m=\frac{P_f}{2}+P_r$ $\pi^{MR*}_m=\frac{P_f+P_r}{2}$ $\pi^{MT*}_m=\frac{P_f}{2}+\frac{P_r}{2}$
$\pi^*_r$ $\pi^{MM*}_r=\frac{P_f}{4}$ $\pi^{MR*}_r=\frac{P_f+P_r}{4}$ $\pi^{MT*}_r=\frac{P_f}{4}$
$\pi^*_t$ N/A N/A $\pi^{MT*}_t=\frac{P_r}{4}$
$\Pi^*$ $\Pi^{MM*}=\frac{3P_f}{4}+P_r$ $\Pi^{MR*}=\frac{3(P_f+P_r)}{4}$ $\Pi^{MT*}=\frac{3(P_f+P_r)}{4}$
Model MM Model MR Model MT
$p^*$ $p^{MM*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{MR*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{MT*}=\frac{3A+\alpha c_m}{4\alpha}$
$b^*_i$ $b^{MM*}_m=\frac{h\delta -k}{2h}$ $b^{MR*}_r=\frac{h\delta -3k}{4h}$ $b^{MT*}_t=\frac{h\delta -3k}{4h}$
$w^*$ $w^{MM*}=\frac{A+\alpha c_m}{2\alpha}$ $w^{MR*}=\frac{A+\alpha c_m}{2\alpha}$ $w^{MT*}=\frac{A+\alpha c_m}{2\alpha}$
$b^*_{mj}$ N/A $b^{MR*}_{mr}=\frac{h\delta -k}{2h}$ $b^{MT*}_{mt}=\frac{h\delta -k}{2h}$
$\pi^*_m$ $\pi^{MM*}_m=\frac{P_f}{2}+P_r$ $\pi^{MR*}_m=\frac{P_f+P_r}{2}$ $\pi^{MT*}_m=\frac{P_f}{2}+\frac{P_r}{2}$
$\pi^*_r$ $\pi^{MM*}_r=\frac{P_f}{4}$ $\pi^{MR*}_r=\frac{P_f+P_r}{4}$ $\pi^{MT*}_r=\frac{P_f}{4}$
$\pi^*_t$ N/A N/A $\pi^{MT*}_t=\frac{P_r}{4}$
$\Pi^*$ $\Pi^{MM*}=\frac{3P_f}{4}+P_r$ $\Pi^{MR*}=\frac{3(P_f+P_r)}{4}$ $\Pi^{MT*}=\frac{3(P_f+P_r)}{4}$
Table 3.  Main results of the R-led models
Model RM Model RR Model RT
$p^*$ $p^{RM*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{RR*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{RT*}=\frac{3A+\alpha c_m}{4\alpha}$
$b^*_i$ $b^{RM*}_m=\frac{h\delta -k}{2h}$ $b^{RR*}_r=\frac{h\delta -3k}{4h}$ $b^{RT*}_t=\frac{h\delta -3k}{4h}$
$w^*$ $w^{RM*}=\frac{A+3\alpha c_m}{4\alpha}$ $w^{RR*}=\frac{A+3\alpha c_m}{4\alpha}$ $w^{RT*}=\frac{A+3\alpha c_m}{4\alpha}$
$b^*_{mj}$ N/A $b^{RR*}_{mr}=\frac{3h\delta -k}{4h}$ $b^{RT*}_{mt}=\frac{h\delta -k}{2h}$
$\pi^*_m$ $\pi^{RM*}_m=\frac{P_f}{4}+P_r$ $\pi^{RR*}_m=\frac{P_f+P_r}{4}$ $\pi^{RT*}_m=\frac{P_f}{4}+\frac{P_r}{2}$
$\pi^*_r$ $\pi^{RM*}_r=\frac{P_f}{2}$ $\pi^{RR*}_r=\frac{P_f+P_r}{2}$ $\pi^{RT*}_r=\frac{P_f}{2}$
$\pi^*_t$ N/A N/A $\pi^{RT*}_t=\frac{P_r}{4}$
$\Pi^*$ $\Pi^{RM*}=\frac{3P_f}{4}+P_r$ $\Pi^{RR*}=\frac{3(P_f+P_r)}{4}$ $\Pi^{RT*}=\frac{3(P_f+P_r)}{4}$
Model RM Model RR Model RT
$p^*$ $p^{RM*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{RR*}=\frac{3A+\alpha c_m}{4\alpha}$ $p^{RT*}=\frac{3A+\alpha c_m}{4\alpha}$
$b^*_i$ $b^{RM*}_m=\frac{h\delta -k}{2h}$ $b^{RR*}_r=\frac{h\delta -3k}{4h}$ $b^{RT*}_t=\frac{h\delta -3k}{4h}$
$w^*$ $w^{RM*}=\frac{A+3\alpha c_m}{4\alpha}$ $w^{RR*}=\frac{A+3\alpha c_m}{4\alpha}$ $w^{RT*}=\frac{A+3\alpha c_m}{4\alpha}$
$b^*_{mj}$ N/A $b^{RR*}_{mr}=\frac{3h\delta -k}{4h}$ $b^{RT*}_{mt}=\frac{h\delta -k}{2h}$
$\pi^*_m$ $\pi^{RM*}_m=\frac{P_f}{4}+P_r$ $\pi^{RR*}_m=\frac{P_f+P_r}{4}$ $\pi^{RT*}_m=\frac{P_f}{4}+\frac{P_r}{2}$
$\pi^*_r$ $\pi^{RM*}_r=\frac{P_f}{2}$ $\pi^{RR*}_r=\frac{P_f+P_r}{2}$ $\pi^{RT*}_r=\frac{P_f}{2}$
$\pi^*_t$ N/A N/A $\pi^{RT*}_t=\frac{P_r}{4}$
$\Pi^*$ $\Pi^{RM*}=\frac{3P_f}{4}+P_r$ $\Pi^{RR*}=\frac{3(P_f+P_r)}{4}$ $\Pi^{RT*}=\frac{3(P_f+P_r)}{4}$
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