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doi: 10.3934/jimo.2022115
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Optimal retail price and service level in a dual-channel supply chain with reference price effect

School of Business Administration, Hunan University, Changsha, Hunan Province, 410082, China

*Corresponding author: Jiawu Peng

Received  September 2021 Revised  March 2022 Early access July 2022

Fund Project: The authors would like to thank the editor and the anonymous referees for their helpful comments and suggestions that greatly improved the quality of this paper. The research is supported by National Natural Science Foundation of China under Grant No. 72071072, and Postgraduate Scientific Research Innovation Project of Hunan Province under Grant Nos.CX20200456

We study the optimal retail price and service level in a dual-channel supply chain in which the manufacturer may sell products through its online channel and offline retailer. Consumers with channel preference make their purchasing decision based on retail price, service level and channel valuation difference. We consider the reference price effect into a Hotelling utility function to formulate the competition of retail price and service level. We analytically derive the unique equilibrium separately in three power structures: (1) Manufacturer-led Stackelberg game (MS), (2) Retailer-led Stackelberg game (RS) and (3) Nash game (N). Particularly, we characterize the impacts of channel valuation difference and reference price effect intensity on the optimal decision and expected profit. In contrast to the previous literature on the dominance of the supply chain in that the dominance always can raise the dominant enterprise's profit, our findings indicate that supply chain members with channel advantages and channel disadvantages have different preferences for power structure. The member with channel disadvantages should give up dominance. Only when channel valuation difference is significant or reference price effect intensity is weak, the member with channel advantages will fight for dominance. Nash game is highly competitive in that the members only earn less profit. Numerical examples verified theoretical results.

Citation: Honglin Yang, Siqi Zhao, Jiawu Peng. Optimal retail price and service level in a dual-channel supply chain with reference price effect. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022115
References:
[1]

J. M. ChenW. Zhang and Z. Y. Liu, Joint pricing, services and quality decisions in a dual-channel supply chain, RAIRO-Operations Research, 54 (2020), 1041-1056.  doi: 10.1051/ro/2019024.

[2]

K. H. ChenL. C. Alwan and L. Zhang, Dynamic pricing in the presence of reference price effect and consumer strategic behavior, International Journal of Production Research, 58 (2020), 546-561. 

[3]

X. ChenP. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2017), 4389-4408. 

[4]

X. ChenX. Wang and X. Jiang, The impact of power structure on the retail service supply chain with an O2O mixed channel, Journal of the Operational Research Society, 67 (2016), 294-301. 

[5]

R. Chenavaz and W. Klibi, Dynamic pricing with reference price effects in integrated online and offline retailing, to appear, International Journal of Production Research. doi: 10.1016/j.ejor.2012.05.009.

[6]

W. Y. K. X. ChiangD. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20. 

[7]

B. DanG. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, International Journal of Production Economics, 139 (2012), 312-320. 

[8]

B. DanS. Zhang and M. Zhou, Strategies for warranty service in a dual-channel supply chain with value-added service competition, International Journal of Production Research, 56 (2018), 5677-5699.  doi: 10.1111/itor.12769.

[9]

A. DumrongsiriM. Fan and A. Jain, A supply chain model with direct and retail channels, European Journal of Operational Research, 187 (2008), 691-718.  doi: 10.1016/j.ejor.2006.05.044.

[10]

D. DzyaburaS. Jagabathula and E. Muller, Accounting for discrepancies between online and offline product evaluations, Marketing Science, 38 (2019), 88-106. 

[11]

F. Gao and X. Su, Online and offline information for omnichannel retailing, Manufacturing & Service Operations Management, 19 (2017), 84-98. 

[12]

S. M. Hosseini-MotlaghM. Johari and P. Pazari, Coordinating pricing, warranty replacement and sales service decisions in a competitive dual-channel retailing system, Computers & Industrial Engineering, 163 (2022), 107862. 

[13]

W. Hu and Y. Li, Retail service for mixed retail and E-tail channels, Annals of operations Researcht, 192 (2012), 151-171.  doi: 10.1007/s10479-010-0818-7.

[14]

Y. HuS. QuG. Li and S. P. Sethi, Power structure and channel integration strategy for online retailers, European Journal of Operational Research, 294 (2021), 951-964.  doi: 10.1016/j.ejor.2019.10.050.

[15]

Y. Jeong and M. Maruyama, Positioning and pricing strategies in a market with switching costs and staying costs, Information Economics and Policy, 44 (2018), 47-57. 

[16]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, in Handbook of the Fundamentals of Financial Decision Making: Part I, Academic Press, 2013, 99–127. doi: 10.21236/ADA045771.

[17]

Y. Liu, J. Li, W. W. Ren and J. Y. L. Forrest, Differentiated products pricing with consumer network acceptance in a dual-channel supply chain, to appear, Electronic Commerce Research and Applications. doi: 10.1111/itor.12749.

[18]

Y. Liu and R. K. Tyagi, The benefits of competitive upward channel decentralization, Management Science, 57 (2011), 741-751. 

[19]

L. LuQ. GouW. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270. 

[20]

J. Ma and L. Xie, The comparison and complex analysis on dual-channel supply chain under different channel power structures and uncertain demand, Nonlinear Dynamics, 83 (2016), 1379-1393.  doi: 10.1007/s11071-015-2410-9.

[21]

T. Maiti and B. C. Giri, Two-period pricing and decision strategies in a two-echelon supply chain under price-dependent demand, Applied Mathematical Modelling, 42 (2017), 655-674.  doi: 10.1016/j.apm.2016.10.051.

[22]

Z. Pi, W. Fang and B. Zhang, Service and pricing strategies with competition and cooperation in a dual-channel supply chain with demand disruption, to appear, Computers & Industrial Engineering.

[23]

X. PuL. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Research and Applications, 22 (2017), 1-12. 

[24]

X. Pu, S. Zhang, B. Ji and G. Han, Online channel strategies under different offline channel power structures, to appear, Journal of Retailing and Consumer Services. doi: 10.1007/s10479-018-2994-9.

[25]

L. RenY. He and H. Song, Price and service competition of dual-channel supply chain with consumer returns, Discrete Dynamics in Nature and Society, 11 (2014), 1-10.  doi: 10.1155/2014/565603.

[26]

L. Sun, X. Jiao, X. Guo and Y. Yu, Pricing policies in dual distribution channels: The reference effect of official prices, to appear, European Journal of Operational Research. doi: 10.1016/j.ejor.2021.03.040.

[27]

A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing & Service Operations Management, 2 (2000), 372-391. 

[28]

A. A. Tsay and N. Agrawal, Channel conflict and coordination in the e-commerce age, Production and Operations Management, 13 (2004), 93-110. 

[29]

N. WangT. ZhangX. Zhu and P. Li, Online-offline competitive pricing with reference price effect, Journal of the Operational Research Society, 72 (2021), 642-653.  doi: 10.1007/s40305-018-0227-1.

[30]

R. Wang, When prospect theory meets consumer choice models: Assortment and pricing management with reference prices, Manufacturing & Service Operations Management, 20 (2018), 583-600. 

[31]

Y. YuL. Sun and X. Guo, Dual-channel decision in a shopping complex when considering consumer channel preference, Journal of the Operational Research Society, 71 (2020), 1638-1656. 

[32]

Y. ZhaL. ZhangC. Xu and T. Zhang, A two-period pricing model with intertemporal and horizontal reference price effects, International Transactions in Operational Research, 28 (2021), 1417-1440.  doi: 10.1111/itor.12613.

[33]

C. Zhang, Y. Liu and G. Han, Two-stage pricing strategies of a dual-channel supply chain considering public green preference, to appear, Computers & Industrial Engineering. doi: 10.1155/2021/6614692.

[34]

G. Zhang, G. Dai, H. Sun, G. Zhang and Z. Yang, Equilibrium in supply chain network with competition and service level between channels considering consumers' channel preferences, to appear, Journal of Retailing and Consumer Services.

[35]

J. Zhang and W. Y. K. Chiang, Durable goods pricing with reference price effects, to appear, Omega. doi: 10.1016/j.ejor.2018.04.029.

[36]

J. ZhangW. Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135. 

[37]

R. ZhangB. Liu and W. Wang, Pricing decisions in a dual channels system with different power structures, Economic Modelling, 29 (2012), 523-533. 

[38]

J. ZhaoX. HouY. Guo and J. Wei, Pricing policies for complementary products in a dual-channel supply chain, Applied mathematical modelling, 49 (2017), 437-451.  doi: 10.1016/j.apm.2017.04.023.

[39]

B. ZhengC. YangJ. Yang and M. Zhang, Dual-channel closed loop supply chains: Forward channel competition, power structures and coordination, International Journal of Production Research, 55 (2017), 3510-3527. 

[40]

Y. W. ZhouJ. Guo and W. Zhou, Pricing/service strategies for a dual-channel supply chain with free riding and service-cost sharing, International Journal of Production Economics, 196 (2018), 198-210. 

show all references

References:
[1]

J. M. ChenW. Zhang and Z. Y. Liu, Joint pricing, services and quality decisions in a dual-channel supply chain, RAIRO-Operations Research, 54 (2020), 1041-1056.  doi: 10.1051/ro/2019024.

[2]

K. H. ChenL. C. Alwan and L. Zhang, Dynamic pricing in the presence of reference price effect and consumer strategic behavior, International Journal of Production Research, 58 (2020), 546-561. 

[3]

X. ChenP. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2017), 4389-4408. 

[4]

X. ChenX. Wang and X. Jiang, The impact of power structure on the retail service supply chain with an O2O mixed channel, Journal of the Operational Research Society, 67 (2016), 294-301. 

[5]

R. Chenavaz and W. Klibi, Dynamic pricing with reference price effects in integrated online and offline retailing, to appear, International Journal of Production Research. doi: 10.1016/j.ejor.2012.05.009.

[6]

W. Y. K. X. ChiangD. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20. 

[7]

B. DanG. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, International Journal of Production Economics, 139 (2012), 312-320. 

[8]

B. DanS. Zhang and M. Zhou, Strategies for warranty service in a dual-channel supply chain with value-added service competition, International Journal of Production Research, 56 (2018), 5677-5699.  doi: 10.1111/itor.12769.

[9]

A. DumrongsiriM. Fan and A. Jain, A supply chain model with direct and retail channels, European Journal of Operational Research, 187 (2008), 691-718.  doi: 10.1016/j.ejor.2006.05.044.

[10]

D. DzyaburaS. Jagabathula and E. Muller, Accounting for discrepancies between online and offline product evaluations, Marketing Science, 38 (2019), 88-106. 

[11]

F. Gao and X. Su, Online and offline information for omnichannel retailing, Manufacturing & Service Operations Management, 19 (2017), 84-98. 

[12]

S. M. Hosseini-MotlaghM. Johari and P. Pazari, Coordinating pricing, warranty replacement and sales service decisions in a competitive dual-channel retailing system, Computers & Industrial Engineering, 163 (2022), 107862. 

[13]

W. Hu and Y. Li, Retail service for mixed retail and E-tail channels, Annals of operations Researcht, 192 (2012), 151-171.  doi: 10.1007/s10479-010-0818-7.

[14]

Y. HuS. QuG. Li and S. P. Sethi, Power structure and channel integration strategy for online retailers, European Journal of Operational Research, 294 (2021), 951-964.  doi: 10.1016/j.ejor.2019.10.050.

[15]

Y. Jeong and M. Maruyama, Positioning and pricing strategies in a market with switching costs and staying costs, Information Economics and Policy, 44 (2018), 47-57. 

[16]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, in Handbook of the Fundamentals of Financial Decision Making: Part I, Academic Press, 2013, 99–127. doi: 10.21236/ADA045771.

[17]

Y. Liu, J. Li, W. W. Ren and J. Y. L. Forrest, Differentiated products pricing with consumer network acceptance in a dual-channel supply chain, to appear, Electronic Commerce Research and Applications. doi: 10.1111/itor.12749.

[18]

Y. Liu and R. K. Tyagi, The benefits of competitive upward channel decentralization, Management Science, 57 (2011), 741-751. 

[19]

L. LuQ. GouW. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270. 

[20]

J. Ma and L. Xie, The comparison and complex analysis on dual-channel supply chain under different channel power structures and uncertain demand, Nonlinear Dynamics, 83 (2016), 1379-1393.  doi: 10.1007/s11071-015-2410-9.

[21]

T. Maiti and B. C. Giri, Two-period pricing and decision strategies in a two-echelon supply chain under price-dependent demand, Applied Mathematical Modelling, 42 (2017), 655-674.  doi: 10.1016/j.apm.2016.10.051.

[22]

Z. Pi, W. Fang and B. Zhang, Service and pricing strategies with competition and cooperation in a dual-channel supply chain with demand disruption, to appear, Computers & Industrial Engineering.

[23]

X. PuL. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Research and Applications, 22 (2017), 1-12. 

[24]

X. Pu, S. Zhang, B. Ji and G. Han, Online channel strategies under different offline channel power structures, to appear, Journal of Retailing and Consumer Services. doi: 10.1007/s10479-018-2994-9.

[25]

L. RenY. He and H. Song, Price and service competition of dual-channel supply chain with consumer returns, Discrete Dynamics in Nature and Society, 11 (2014), 1-10.  doi: 10.1155/2014/565603.

[26]

L. Sun, X. Jiao, X. Guo and Y. Yu, Pricing policies in dual distribution channels: The reference effect of official prices, to appear, European Journal of Operational Research. doi: 10.1016/j.ejor.2021.03.040.

[27]

A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing & Service Operations Management, 2 (2000), 372-391. 

[28]

A. A. Tsay and N. Agrawal, Channel conflict and coordination in the e-commerce age, Production and Operations Management, 13 (2004), 93-110. 

[29]

N. WangT. ZhangX. Zhu and P. Li, Online-offline competitive pricing with reference price effect, Journal of the Operational Research Society, 72 (2021), 642-653.  doi: 10.1007/s40305-018-0227-1.

[30]

R. Wang, When prospect theory meets consumer choice models: Assortment and pricing management with reference prices, Manufacturing & Service Operations Management, 20 (2018), 583-600. 

[31]

Y. YuL. Sun and X. Guo, Dual-channel decision in a shopping complex when considering consumer channel preference, Journal of the Operational Research Society, 71 (2020), 1638-1656. 

[32]

Y. ZhaL. ZhangC. Xu and T. Zhang, A two-period pricing model with intertemporal and horizontal reference price effects, International Transactions in Operational Research, 28 (2021), 1417-1440.  doi: 10.1111/itor.12613.

[33]

C. Zhang, Y. Liu and G. Han, Two-stage pricing strategies of a dual-channel supply chain considering public green preference, to appear, Computers & Industrial Engineering. doi: 10.1155/2021/6614692.

[34]

G. Zhang, G. Dai, H. Sun, G. Zhang and Z. Yang, Equilibrium in supply chain network with competition and service level between channels considering consumers' channel preferences, to appear, Journal of Retailing and Consumer Services.

[35]

J. Zhang and W. Y. K. Chiang, Durable goods pricing with reference price effects, to appear, Omega. doi: 10.1016/j.ejor.2018.04.029.

[36]

J. ZhangW. Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135. 

[37]

R. ZhangB. Liu and W. Wang, Pricing decisions in a dual channels system with different power structures, Economic Modelling, 29 (2012), 523-533. 

[38]

J. ZhaoX. HouY. Guo and J. Wei, Pricing policies for complementary products in a dual-channel supply chain, Applied mathematical modelling, 49 (2017), 437-451.  doi: 10.1016/j.apm.2017.04.023.

[39]

B. ZhengC. YangJ. Yang and M. Zhang, Dual-channel closed loop supply chains: Forward channel competition, power structures and coordination, International Journal of Production Research, 55 (2017), 3510-3527. 

[40]

Y. W. ZhouJ. Guo and W. Zhou, Pricing/service strategies for a dual-channel supply chain with free riding and service-cost sharing, International Journal of Production Economics, 196 (2018), 198-210. 

Figure 1.  The dual-channel supply chain structure
Figure 2.  The Hotelling model
Figure 3.  Analysis of consumer purchase behavior
Figure 4.  Impact of $ \alpha $ on $ p_m^j $
Figure 5.  Impact of $ \alpha $ on $ s_m^j $
Figure 6.  Impact of $ \alpha $ on $ \pi_m^j $
Figure 7.  Optimal retail prices under three power structures
Figure 8.  Optimal service levels under three power structures
Figure 9.  Demand under three power structures
Figure 10.  Expected profit under three power structures
Figure 11.  The equilibrium selection of power structures for members
Figure 12.  The impact of t on the equilibrium solutions
Table 1.  Comparison of closest literature
Literatures Decision variables Game model Demand function Reference price effect
Liu et al., (2020) $ (w, p_r, p_m, s_r, s_m) $ MS Utility No
Wang et al., (2021) $ (p_r, p_m) $ MS, RS, N Hotelling Yes
This paper $ (p_r, p_m, s_r, s_m) $ MS, RS, N Hotelling Yes
Literatures Decision variables Game model Demand function Reference price effect
Liu et al., (2020) $ (w, p_r, p_m, s_r, s_m) $ MS Utility No
Wang et al., (2021) $ (p_r, p_m) $ MS, RS, N Hotelling Yes
This paper $ (p_r, p_m, s_r, s_m) $ MS, RS, N Hotelling Yes
Table 2.  Summary of notations
Notation Definition
$ v $ Consumers' valuation of product
$ \theta $ The customers' acceptance degree of online direct channel
$ t $ Unit psychological cost caused by consumers' channel preference
$ w $ Wholesale price
$ x_0^j $ Consumer indifferent point of purchase under model $ j $,
where $ j=MS, RS, N $
$ \alpha $ Reference price effect intensity
$ U_i $ The utility of purchasing from different channel, where $ i=r, m $
$ R_i $ Reference price for different channel, where $ i=r, m $
$ p_i^j $ Retail price in different channel under model $ j $, where $ i=r, m $
and $ j=MS, RS, N $ (decision variables)
$ s_i^j $ Service level in different channel under model $ j $, where $ i=r, m $
and $ j=MS, RS, N $ (decision variables)
$ Q_i^j $ The demand in different channel under model $ j $, where $ i=r, m $
and $ j=MS, RS, N $
$ \pi_i^j $ The profits in different supply chain members under model $ j $,
where $ i=r, m $ and $ j=MS, RS, N $
Notation Definition
$ v $ Consumers' valuation of product
$ \theta $ The customers' acceptance degree of online direct channel
$ t $ Unit psychological cost caused by consumers' channel preference
$ w $ Wholesale price
$ x_0^j $ Consumer indifferent point of purchase under model $ j $,
where $ j=MS, RS, N $
$ \alpha $ Reference price effect intensity
$ U_i $ The utility of purchasing from different channel, where $ i=r, m $
$ R_i $ Reference price for different channel, where $ i=r, m $
$ p_i^j $ Retail price in different channel under model $ j $, where $ i=r, m $
and $ j=MS, RS, N $ (decision variables)
$ s_i^j $ Service level in different channel under model $ j $, where $ i=r, m $
and $ j=MS, RS, N $ (decision variables)
$ Q_i^j $ The demand in different channel under model $ j $, where $ i=r, m $
and $ j=MS, RS, N $
$ \pi_i^j $ The profits in different supply chain members under model $ j $,
where $ i=r, m $ and $ j=MS, RS, N $
Table 3.  The equilibrium solutions in MS game
Manufacturer (Leader) Retailer (Follower)
Retail price $ p_m^{MS*}=w+\frac{(8\alpha+3)(2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ p_r^{MS*}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $
Service level $ s_m^{MS*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)} $ $ s_r^{MS*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $
Demand $ Q_m^{MS*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5} $ $ Q_r^{MS*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5} $
Profit $ \pi_m^{MS*}=w+\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \pi_r^{MS*}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
Manufacturer (Leader) Retailer (Follower)
Retail price $ p_m^{MS*}=w+\frac{(8\alpha+3)(2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ p_r^{MS*}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $
Service level $ s_m^{MS*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)} $ $ s_r^{MS*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $
Demand $ Q_m^{MS*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5} $ $ Q_r^{MS*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5} $
Profit $ \pi_m^{MS*}=w+\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \pi_r^{MS*}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
Table 4.  The equilibrium solutions in RS game
Retailer(Leader) Manufacturer(Follower)
Retail price $ p_r^{RS*}=w+\frac{(8\alpha+3)(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ p_m^{MS*}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $
Service level $ s_r^{RS*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)} $ $ s_m^{RS*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $
Demand $ Q_r^{RS*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5} $ $ Q_m^{RS*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5} $
Profit $ \pi_r^{RS*}=\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \pi_m^{RS*}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
Retailer(Leader) Manufacturer(Follower)
Retail price $ p_r^{RS*}=w+\frac{(8\alpha+3)(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ p_m^{MS*}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $
Service level $ s_r^{RS*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)} $ $ s_m^{RS*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $
Demand $ Q_r^{RS*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5} $ $ Q_m^{RS*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5} $
Profit $ \pi_r^{RS*}=\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \pi_m^{RS*}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
Table 5.  The equilibrium solutions in Nash game
Retailer Manufacturer
Retail price $ p_r^{N*}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)(6\alpha+2)} $ $ p_m^{N*}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $
Service level $ s_r^{N*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)} $ $ s_m^{N*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $
Demand $ Q_r^{N*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)} $ $ Q_m^{N*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)} $
Profit $ \pi_r^{N*}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $ $ \pi_m^{RS*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $
Retailer Manufacturer
Retail price $ p_r^{N*}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)(6\alpha+2)} $ $ p_m^{N*}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $
Service level $ s_r^{N*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)} $ $ s_m^{N*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $
Demand $ Q_r^{N*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)} $ $ Q_m^{N*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)} $
Profit $ \pi_r^{N*}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $ $ \pi_m^{RS*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $
Table 6.  The equilibrium solutions in Section 7.1
Retailer Manufacturer
Retail price $ \tilde{p}_r^{MS^*}=w+\frac{2t((2\alpha+1)(5t+(1-\theta)v-2))}{(2\alpha+1)(8t(2\alpha+1)-3)} $ $ \tilde{p}_m^{MS^*}=w+\frac{(4t(2\alpha+1)-1)(2\alpha+1)(3t-(1-\theta)v-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)} $
$ \tilde{p}_r^{RS^*}=w+\frac{(4t(2\alpha+1)-1)((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)} $ $ \tilde{p}_m^{RS^*}=w+\frac{2t((2\alpha+1)(5t-(1-\theta)v)-2)}{(2\alpha+1)(8t(2\alpha+1)-3)} $
$ \tilde{p}_r^{N^*}=w+\frac{2t((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)} $ $ \tilde{p}_m^{N^*}=w+\frac{2t((2\alpha+1)(3t-(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)} $
Service level $ \tilde{s}_r^{MS^*}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)} $ $ \tilde{s}_m^{MS^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)} $
$ \tilde{s}_r^{RS^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)} $ $ \tilde{s}_m^{RS^*}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)} $
$ \tilde{s}_r^{N^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)} $ $ \tilde{s}_m^{N^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)} $
Demand $ \tilde{Q}_r^{MS^*}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{8t(2\alpha+1)-3} $ $ \tilde{Q}_m^{MS^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{8t(2\alpha+1)-3} $
$ \tilde{Q}_r^{RS^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{8t(2\alpha+1)-3} $ $ \tilde{Q}_m^{RS^*}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{8t(2\alpha+1)-3} $
$ \tilde{Q}_r^{N^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{6t(2\alpha+1)-2} $ $ \tilde{Q}_m^{N^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{6t(2\alpha+1)-2} $
Profit $ \tilde{\pi}_r^{MS^*}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t+(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2} $ $ \tilde{\pi}_m^{MS^*}=w+\frac{((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)} $
$ \tilde{\pi}_r^{RS^*}=\frac{((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)} $ $ \tilde{\pi}_m^{RS^*}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t-(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2} $
$ \tilde{\pi}_r^{N^*}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2} $ $ \tilde{\pi}_m^{N^*}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2} $
Retailer Manufacturer
Retail price $ \tilde{p}_r^{MS^*}=w+\frac{2t((2\alpha+1)(5t+(1-\theta)v-2))}{(2\alpha+1)(8t(2\alpha+1)-3)} $ $ \tilde{p}_m^{MS^*}=w+\frac{(4t(2\alpha+1)-1)(2\alpha+1)(3t-(1-\theta)v-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)} $
$ \tilde{p}_r^{RS^*}=w+\frac{(4t(2\alpha+1)-1)((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)} $ $ \tilde{p}_m^{RS^*}=w+\frac{2t((2\alpha+1)(5t-(1-\theta)v)-2)}{(2\alpha+1)(8t(2\alpha+1)-3)} $
$ \tilde{p}_r^{N^*}=w+\frac{2t((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)} $ $ \tilde{p}_m^{N^*}=w+\frac{2t((2\alpha+1)(3t-(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)} $
Service level $ \tilde{s}_r^{MS^*}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)} $ $ \tilde{s}_m^{MS^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)} $
$ \tilde{s}_r^{RS^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)} $ $ \tilde{s}_m^{RS^*}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)} $
$ \tilde{s}_r^{N^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)} $ $ \tilde{s}_m^{N^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)} $
Demand $ \tilde{Q}_r^{MS^*}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{8t(2\alpha+1)-3} $ $ \tilde{Q}_m^{MS^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{8t(2\alpha+1)-3} $
$ \tilde{Q}_r^{RS^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{8t(2\alpha+1)-3} $ $ \tilde{Q}_m^{RS^*}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{8t(2\alpha+1)-3} $
$ \tilde{Q}_r^{N^*}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{6t(2\alpha+1)-2} $ $ \tilde{Q}_m^{N^*}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{6t(2\alpha+1)-2} $
Profit $ \tilde{\pi}_r^{MS^*}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t+(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2} $ $ \tilde{\pi}_m^{MS^*}=w+\frac{((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)} $
$ \tilde{\pi}_r^{RS^*}=\frac{((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)} $ $ \tilde{\pi}_m^{RS^*}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t-(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2} $
$ \tilde{\pi}_r^{N^*}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2} $ $ \tilde{\pi}_m^{N^*}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2} $
Table 7.  The equilibrium solutions in Section 7.2
Retailer Manufacturer
Retail price $ \check{p}_r^{MS^*}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $ $ \check{p}_m^{MS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2 (16\alpha+5)} $
$ \check{p}_r^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ \check{p}_m^{RS^*}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $
$ \check{p}_r^{N^*}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $ $ \check{p}_m^{N^*}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $
Service level $ \check{s}_r^{MS^*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $ $ \check{s}_m^{MS^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)} $
$ \check{s}_r^{RS^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)} $ $ \check{s}_m^{RS^*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $
$ \check{s}_r^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)} $ $ \check{s}_m^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)} $
Demand $ \check{Q}_r^{MS^*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5} $ $ \check{Q}_m^{MS^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5} $
$ \check{Q}_r^{RS^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5} $ $ \check{Q}_m^{RS^*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5} $
$ \check{Q}_r^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)} $ $ \check{Q}_m^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)} $
Profit $ \check{\pi}_r^{MS^*}=\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $ $ \check{\pi}_m^{MS^*}=w+\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $
$ \check{\pi}_r^{RS^*}=\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \check{\pi}_m^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
$ \check{\pi}_r^{N^*}=\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $ $ \check{\pi}_m^{N^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $
Retailer Manufacturer
Retail price $ \check{p}_r^{MS^*}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $ $ \check{p}_m^{MS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2 (16\alpha+5)} $
$ \check{p}_r^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ \check{p}_m^{RS^*}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)} $
$ \check{p}_r^{N^*}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $ $ \check{p}_m^{N^*}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)} $
Service level $ \check{s}_r^{MS^*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $ $ \check{s}_m^{MS^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)} $
$ \check{s}_r^{RS^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)} $ $ \check{s}_m^{RS^*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)} $
$ \check{s}_r^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)} $ $ \check{s}_m^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)} $
Demand $ \check{Q}_r^{MS^*}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5} $ $ \check{Q}_m^{MS^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5} $
$ \check{Q}_r^{RS^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5} $ $ \check{Q}_m^{RS^*}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5} $
$ \check{Q}_r^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)} $ $ \check{Q}_m^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)} $
Profit $ \check{\pi}_r^{MS^*}=\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $ $ \check{\pi}_m^{MS^*}=w+\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $
$ \check{\pi}_r^{RS^*}=\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \check{\pi}_m^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
$ \check{\pi}_r^{N^*}=\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $ $ \check{\pi}_m^{N^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $
Table 8.  The equilibrium solutions in Section 7.3
Retailer Manufacturer
Retail price $ \hat{p}_r^{MS^*}=w+\frac{2(2\alpha+1)(5+(1-\theta)v+h)-4}{(2\alpha+1)(16\alpha+5)} $ $ \hat{p}_m^{MS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)}{(2\alpha+1)^2 (16\alpha+5)} $
$ \hat{p}_r^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ \hat{p}_m^{RS^*}=w+\frac{2(2\alpha+1)(5-(1-\theta)v-h)-4}{(2\alpha+1)(16\alpha+5)} $
$ \hat{p}_r^{N^*}=w+\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(6\alpha+2)} $ $ \hat{p}_m^{N^*}=w+\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(6\alpha+2)} $
Service level $ \hat{s}_r^{MS^*}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{(2\alpha+1)(16\alpha+5)} $ $ \hat{s}_m^{MS^*}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(16\alpha+5)} $
$ \hat{s}_r^{RS^*}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(16\alpha+5)} $ $ \hat{s}_m^{RS^*}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{(2\alpha+1)(16\alpha+5)} $
$ \hat{s}_r^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{2(2\alpha+1)(6\alpha+2)} $ $ \hat{s}_m^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{2(2\alpha+1)(6\alpha+2)} $
Demand $ \hat{Q}_r^{MS^*}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{16\alpha+5} $ $ \hat{Q}_m^{MS^*}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{16\alpha+5} $
$ \hat{Q}_r^{RS^*}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{16\alpha+5} $ $ \hat{Q}_m^{RS^*}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{16\alpha+5} $
$ \hat{Q}_r^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1+h}{2(6\alpha+2)} $ $ \hat{Q}_m^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1-h}{2(6\alpha+2)} $
Profit $ \hat{\pi}_r^{MS^*}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v+h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $ $ \hat{\pi}_m^{MS^*}=w+\frac{((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $
$ \hat{\pi}_r^{RS^*}=\frac{((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \hat{\pi}_m^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
$ \hat{\pi}_r^{N^*}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $ $ \hat{\pi}_m^{N^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $
Retailer Manufacturer
Retail price $ \hat{p}_r^{MS^*}=w+\frac{2(2\alpha+1)(5+(1-\theta)v+h)-4}{(2\alpha+1)(16\alpha+5)} $ $ \hat{p}_m^{MS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)}{(2\alpha+1)^2 (16\alpha+5)} $
$ \hat{p}_r^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)}{(2\alpha+1)^2(16\alpha+5)} $ $ \hat{p}_m^{RS^*}=w+\frac{2(2\alpha+1)(5-(1-\theta)v-h)-4}{(2\alpha+1)(16\alpha+5)} $
$ \hat{p}_r^{N^*}=w+\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(6\alpha+2)} $ $ \hat{p}_m^{N^*}=w+\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(6\alpha+2)} $
Service level $ \hat{s}_r^{MS^*}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{(2\alpha+1)(16\alpha+5)} $ $ \hat{s}_m^{MS^*}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(16\alpha+5)} $
$ \hat{s}_r^{RS^*}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(16\alpha+5)} $ $ \hat{s}_m^{RS^*}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{(2\alpha+1)(16\alpha+5)} $
$ \hat{s}_r^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{2(2\alpha+1)(6\alpha+2)} $ $ \hat{s}_m^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{2(2\alpha+1)(6\alpha+2)} $
Demand $ \hat{Q}_r^{MS^*}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{16\alpha+5} $ $ \hat{Q}_m^{MS^*}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{16\alpha+5} $
$ \hat{Q}_r^{RS^*}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{16\alpha+5} $ $ \hat{Q}_m^{RS^*}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{16\alpha+5} $
$ \hat{Q}_r^{N^*}=\frac{(2\alpha+1)(3+(1-\theta)v)-1+h}{2(6\alpha+2)} $ $ \hat{Q}_m^{N^*}=\frac{(2\alpha+1)(3-(1-\theta)v)-1-h}{2(6\alpha+2)} $
Profit $ \hat{\pi}_r^{MS^*}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v+h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $ $ \hat{\pi}_m^{MS^*}=w+\frac{((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $
$ \hat{\pi}_r^{RS^*}=\frac{((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)} $ $ \hat{\pi}_m^{RS^*}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2} $
$ \hat{\pi}_r^{N^*}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $ $ \hat{\pi}_m^{N^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2} $
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