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January  2022, 18(1): 541-560. doi: 10.3934/jimo.2020167

Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain

School of Mathematics and Physics, Anhui Jianzhu University, Hefei, Anhui 230601, China

* Corresponding author: Jie Min

Received  May 2020 Revised  September 2020 Published  January 2022 Early access  November 2020

Consider a supply chain consisting of one manufacturer and one retailer. The manufacturer may open direct channels through ex-ante or ex-post encroachment, and the retailer can provide consumers with ex-ante or ex-post service. We investigates the effects of encroachment and services on the optimal strategy for two members in three decision modes: MR mode (ex-ante encroachment), MRM mode (ex-post encroachment and ex-post service), and MRMR mode (ex-post encroachment and ex-ante service). The results show that in the MRM mode, both the wholesale and retail prices may become higher with encroachment. Improving the service efficiency may hurt the retailer, and increasing the operating cost for direct channels harms the retailer, while benefits the manufacturer. In addition, only in the MRM mode, the retailer maybe benefits from encroachment under certain conditions. We further study the equilibrium mode and the result shows as follows. The MR mode, widely adopted by the literature on manufacturer encroachment, always is worst for the manufacturer. Only when both the operating cost for direct channels and the service efficiency are low, the equilibrium decision mode is the MRMR mode, otherwise the MRM mode is the equilibrium decision mode.

Citation: Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial and Management Optimization, 2022, 18 (1) : 541-560. doi: 10.3934/jimo.2020167
References:
[1]

A. AryaB. Mittendorf and D. E. M. Sappington, The bright side of supplier encroachment, Management Science, 26 (2007), 589-730.  doi: 10.1287/mksc.1070.0280.

[2]

Q. BaiM. Chen and L. Xu, Revenue and promotional cost-sharing contract versus two-part tariff contract in coordinating sustainable supply chain systems with deteriorating items, International Journal of Production Economics, 187 (2017), 85-101.  doi: 10.1016/j.ijpe.2017.02.012.

[3]

S. Balasubramanian, A strategic analysis of competition between direct marketers and conventional retailers, Marketing Science, 17 (1998), 181-195.  doi: 10.1287/mksc.17.3.181.

[4]

Jordan brand launches the highly anticipated air jordan 2011, Business Wire, January 26, 2011.

[5]

G. Cai, Channel selection and corrdination in dual-channel supplys, Journal of Retailing, 86 (2010), 22-36. 

[6]

Z.-H. Cao, Y.-W. Zhou, J. Zhao and C.-W. Li, Entry mode selection and its impact on an incumbent supply chain coordination, Journal of Retailing & Consumer Services, 26 (2015), 1–13. doi: 10.1016/j.jretconser.2015.04.005.

[7]

K. Cattani, W. Gilland, H. Sebastian Heese and J. Swaminathan, Boiling frogs: Pricing strategies for a manufacturer adding a direct channel that competes with the traditional channel, Production & Operations Management, 15 (2006), 40–56. doi: 10.1111/j.1937-5956.2006.tb00002.x.

[8]

J. ChenL. LiangD.-Q. Yao and S. Sun, Price and quality decisions in dual-channel supply chains, European Journal of Operational Research, 259 (2017), 935-948.  doi: 10.1016/j.ejor.2016.11.016.

[9]

J. ChenH. Zhang and Y. Sun, Implementing coordination contracts in a manufacturer stackelberg dual-channel supply chain, Omega, 40 (2012), 571-583.  doi: 10.1016/j.omega.2011.11.005.

[10]

W.-Y. K. 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.  doi: 10.1287/mnsc.49.1.1.12749.

[11]

B. Dan, C. Liu, G. Xu and X. Zhang, Pareto improvement strategy for service-based free-riding in a dual-channel supply chain, Asia-Pacific Journal of Operational Research, 31 (2014), 1450050, 27 pp. doi: 10.1142/S021759591450050X.

[12]

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.  doi: 10.1016/j.ijpe.2012.05.014.

[13]

R. Farmer, Cinema advertising and the sea witch lost island film (1965), Historical J. Film. Radio Television, 36 (2016), 569-586.  doi: 10.1080/01439685.2015.1129709.

[14]

A. Ha, X. Long and J. Nasiry, Quality in supply chain encroanchment, Manufacturing & Service Operations Management, 18 (2016), 280–298. doi: 10.1287/msom.2015.0562.

[15]

S. KhalilpourazariA. MirzazadehG.-W. Weber and S. H. R. Pasandideh, A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process, Optimization, 69 (2020), 63-90.  doi: 10.1080/02331934.2019.1630625.

[16]

S. KhalilpourazariS. SoltanzadehG.-W. Weber and S. K. Roy, Designing an efficient blood supply chain network in crisis: neural learning, optimization and case study, Annals of Operations Research, 289 (2020), 123-152.  doi: 10.1007/s10479-019-03437-2.

[17]

G. LiL. Li and J. Sun, Pricing and service effort strategy in a dual-channel supply chain with showrooming effect, Transportation Research Part E, 126 (2019), 32-48.  doi: 10.1016/j.tre.2019.03.019.

[18]

Q. LiB. LiP. Chen and P. Hou, Dual-channel supply chain decisions under asymmetric information with a risk-averse retailer, Annals of Operations Research, 257 (2017), 423-447.  doi: 10.1007/s10479-015-1852-2.

[19]

Z. LiS. M. Gilbert and G. Lai, Supplier encroachment under asymmetric information, Management Science, 60 (2014), 449-462.  doi: 10.1287/mnsc.2013.1780.

[20]

Y. Liu and Z. J. Zhang, Research note-the benefits of personalized pricing in a channel, Marketing Science, 25 (2006), 97-105.  doi: 10.1287/mksc.1050.0120.

[21]

N. M. ModakS. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International Journal of Production Economics, 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.

[22]

C. Pan, Supplier encroachment and consumer welfare: Upstream firm's opportunism and multichannel distribution, ISER Discussion Paper No. 10520, 40pp. https://ssrn.com/ abstract=2840667. doi: 10.2139/ssrn.3124026.

[23]

A. RoyS. S. Sana and K. Chaudhuri, Joint decision on EOQ and pricing strategy of a dual channel of mixed retail and e-tail comprising of single manufacturer and retailer under stochastic demand, Computers & Industrial Engineering, 102 (2016), 423-434.  doi: 10.1016/j.cie.2016.05.002.

[24]

S. S. Sana, Price competition between green and non green products under corporate social responsible firm, Journal of Retailing and Consumer Services, 55 (2020). doi: 10.1016/j.jretconser.2020.102118.

[25]

J. Shin, How does free riding on customer service affect competition, Marketing Science, 26 (2007), 488-503.  doi: 10.1287/mksc.1060.0252.

[26]

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

[27]

L. WangH. Song and Y. Wang, Pricing and service decisions of complementary products in a dual-channel supply chain, Computers & Industrial Engineering, 105 (2017), 223-233.  doi: 10.1016/j.cie.2016.12.034.

[28]

L. Wang and J. Zhao, Pricing and service decisions in a dual-channel supply chain with manufacturer's direct online channel service and retail service, Wseas Transactions on Business and Economics, 11 (2014), 293–302.

[29]

R. Yan and Z. Pei, Retail services and firm profit in a dual-channel market, Journal of Retailing & Consumer Services, 16 (2009), 306–314. doi: 10.1016/j.jretconser.2009.02.006.

[30]

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.

[31]

D.-Q. YaoX. Yue and J. Liu, Vertical cost information sharing in a supply chain with value-adding retailers, Omega, 36 (2008), 838-851.  doi: 10.1016/j.omega.2006.04.003.

[32]

S. ZhangJ. Zhang and G. Zhu, Retail service investing: An anti-encroachment strategy in a retailer-led supply chain, Omega, 84 (2019), 212-231.  doi: 10.1016/j.omega.2018.05.005.

[33]

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.  doi: 10.1016/j.ijpe.2017.11.014.

show all references

References:
[1]

A. AryaB. Mittendorf and D. E. M. Sappington, The bright side of supplier encroachment, Management Science, 26 (2007), 589-730.  doi: 10.1287/mksc.1070.0280.

[2]

Q. BaiM. Chen and L. Xu, Revenue and promotional cost-sharing contract versus two-part tariff contract in coordinating sustainable supply chain systems with deteriorating items, International Journal of Production Economics, 187 (2017), 85-101.  doi: 10.1016/j.ijpe.2017.02.012.

[3]

S. Balasubramanian, A strategic analysis of competition between direct marketers and conventional retailers, Marketing Science, 17 (1998), 181-195.  doi: 10.1287/mksc.17.3.181.

[4]

Jordan brand launches the highly anticipated air jordan 2011, Business Wire, January 26, 2011.

[5]

G. Cai, Channel selection and corrdination in dual-channel supplys, Journal of Retailing, 86 (2010), 22-36. 

[6]

Z.-H. Cao, Y.-W. Zhou, J. Zhao and C.-W. Li, Entry mode selection and its impact on an incumbent supply chain coordination, Journal of Retailing & Consumer Services, 26 (2015), 1–13. doi: 10.1016/j.jretconser.2015.04.005.

[7]

K. Cattani, W. Gilland, H. Sebastian Heese and J. Swaminathan, Boiling frogs: Pricing strategies for a manufacturer adding a direct channel that competes with the traditional channel, Production & Operations Management, 15 (2006), 40–56. doi: 10.1111/j.1937-5956.2006.tb00002.x.

[8]

J. ChenL. LiangD.-Q. Yao and S. Sun, Price and quality decisions in dual-channel supply chains, European Journal of Operational Research, 259 (2017), 935-948.  doi: 10.1016/j.ejor.2016.11.016.

[9]

J. ChenH. Zhang and Y. Sun, Implementing coordination contracts in a manufacturer stackelberg dual-channel supply chain, Omega, 40 (2012), 571-583.  doi: 10.1016/j.omega.2011.11.005.

[10]

W.-Y. K. 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.  doi: 10.1287/mnsc.49.1.1.12749.

[11]

B. Dan, C. Liu, G. Xu and X. Zhang, Pareto improvement strategy for service-based free-riding in a dual-channel supply chain, Asia-Pacific Journal of Operational Research, 31 (2014), 1450050, 27 pp. doi: 10.1142/S021759591450050X.

[12]

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.  doi: 10.1016/j.ijpe.2012.05.014.

[13]

R. Farmer, Cinema advertising and the sea witch lost island film (1965), Historical J. Film. Radio Television, 36 (2016), 569-586.  doi: 10.1080/01439685.2015.1129709.

[14]

A. Ha, X. Long and J. Nasiry, Quality in supply chain encroanchment, Manufacturing & Service Operations Management, 18 (2016), 280–298. doi: 10.1287/msom.2015.0562.

[15]

S. KhalilpourazariA. MirzazadehG.-W. Weber and S. H. R. Pasandideh, A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process, Optimization, 69 (2020), 63-90.  doi: 10.1080/02331934.2019.1630625.

[16]

S. KhalilpourazariS. SoltanzadehG.-W. Weber and S. K. Roy, Designing an efficient blood supply chain network in crisis: neural learning, optimization and case study, Annals of Operations Research, 289 (2020), 123-152.  doi: 10.1007/s10479-019-03437-2.

[17]

G. LiL. Li and J. Sun, Pricing and service effort strategy in a dual-channel supply chain with showrooming effect, Transportation Research Part E, 126 (2019), 32-48.  doi: 10.1016/j.tre.2019.03.019.

[18]

Q. LiB. LiP. Chen and P. Hou, Dual-channel supply chain decisions under asymmetric information with a risk-averse retailer, Annals of Operations Research, 257 (2017), 423-447.  doi: 10.1007/s10479-015-1852-2.

[19]

Z. LiS. M. Gilbert and G. Lai, Supplier encroachment under asymmetric information, Management Science, 60 (2014), 449-462.  doi: 10.1287/mnsc.2013.1780.

[20]

Y. Liu and Z. J. Zhang, Research note-the benefits of personalized pricing in a channel, Marketing Science, 25 (2006), 97-105.  doi: 10.1287/mksc.1050.0120.

[21]

N. M. ModakS. Panda and S. S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International Journal of Production Economics, 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.

[22]

C. Pan, Supplier encroachment and consumer welfare: Upstream firm's opportunism and multichannel distribution, ISER Discussion Paper No. 10520, 40pp. https://ssrn.com/ abstract=2840667. doi: 10.2139/ssrn.3124026.

[23]

A. RoyS. S. Sana and K. Chaudhuri, Joint decision on EOQ and pricing strategy of a dual channel of mixed retail and e-tail comprising of single manufacturer and retailer under stochastic demand, Computers & Industrial Engineering, 102 (2016), 423-434.  doi: 10.1016/j.cie.2016.05.002.

[24]

S. S. Sana, Price competition between green and non green products under corporate social responsible firm, Journal of Retailing and Consumer Services, 55 (2020). doi: 10.1016/j.jretconser.2020.102118.

[25]

J. Shin, How does free riding on customer service affect competition, Marketing Science, 26 (2007), 488-503.  doi: 10.1287/mksc.1060.0252.

[26]

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

[27]

L. WangH. Song and Y. Wang, Pricing and service decisions of complementary products in a dual-channel supply chain, Computers & Industrial Engineering, 105 (2017), 223-233.  doi: 10.1016/j.cie.2016.12.034.

[28]

L. Wang and J. Zhao, Pricing and service decisions in a dual-channel supply chain with manufacturer's direct online channel service and retail service, Wseas Transactions on Business and Economics, 11 (2014), 293–302.

[29]

R. Yan and Z. Pei, Retail services and firm profit in a dual-channel market, Journal of Retailing & Consumer Services, 16 (2009), 306–314. doi: 10.1016/j.jretconser.2009.02.006.

[30]

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.

[31]

D.-Q. YaoX. Yue and J. Liu, Vertical cost information sharing in a supply chain with value-adding retailers, Omega, 36 (2008), 838-851.  doi: 10.1016/j.omega.2006.04.003.

[32]

S. ZhangJ. Zhang and G. Zhu, Retail service investing: An anti-encroachment strategy in a retailer-led supply chain, Omega, 84 (2019), 212-231.  doi: 10.1016/j.omega.2018.05.005.

[33]

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.  doi: 10.1016/j.ijpe.2017.11.014.

Figure 1.  The sequence of the events under three decision modes
Figure 2.  The effects of $ c $ on the optimal outcomes in the MRM mode
Figure 3.  Impacts of $ \frac{c}{a} $ and $ S $ on the profits in the MRM mode
Figure 4.  The effects of $ c $ on the optimal outcomes in the MRMR mode
Figure 5.  The effects of $ S $ on the profits in three decision modes
Figure 6.  The impact of $ \frac{c}{a} $ and $ S $ on the equilibrium decision mode
Table 1.  List of notations
$ a $The maximum potential market size
$ c $The unit operating cost for the direct channel
$ \lambda $The efficacy of the service level
$ \eta $The coefficient for the service cost
$ S $The service efficiency, where $ S = \frac{\lambda^2}{\eta} $
$ p_r $The retail price for the retail channel, where $ p_n = u_n+w_n $
$ p_d $The retail price for the direct channel
$ \pi_M $/$ \pi_R/\pi_C $Manufacturer's /Retailer's/Whole chain's profit
Decision variables
$ w $The wholesale price
$ Q_d $The sale quantity for the direct channel
$ Q_r $The sale quantity for the retail channel
$ s $The service level for the retail channel
Superscript
$ MR/RM/RMR $The optimal outcome in the MR/MRM/MRMR mode
$ B $The optimal outcome in the benchmark without encroachment
$ a $The maximum potential market size
$ c $The unit operating cost for the direct channel
$ \lambda $The efficacy of the service level
$ \eta $The coefficient for the service cost
$ S $The service efficiency, where $ S = \frac{\lambda^2}{\eta} $
$ p_r $The retail price for the retail channel, where $ p_n = u_n+w_n $
$ p_d $The retail price for the direct channel
$ \pi_M $/$ \pi_R/\pi_C $Manufacturer's /Retailer's/Whole chain's profit
Decision variables
$ w $The wholesale price
$ Q_d $The sale quantity for the direct channel
$ Q_r $The sale quantity for the retail channel
$ s $The service level for the retail channel
Superscript
$ MR/RM/RMR $The optimal outcome in the MR/MRM/MRMR mode
$ B $The optimal outcome in the benchmark without encroachment
Table 2.  The outcomes in the benchmark without encroachment
$ w^B $ $ Q_r^B $$ s^B $$ p_r^B $ $ \pi_M^B $ $ \pi_R^B $ $ \pi_C^B $
$ \frac{a}{2} $$ \frac{a}{4-S} $$ \frac{\lambda a}{2\eta(4-S)} $$ \frac{(6-S)a}{2(4-S)} $$ \frac{a^2}{2(4-S)} $$ \frac{a^2}{4(4-S)} $$ \frac{3a^2}{4(4-S)} $
$ w^B $ $ Q_r^B $$ s^B $$ p_r^B $ $ \pi_M^B $ $ \pi_R^B $ $ \pi_C^B $
$ \frac{a}{2} $$ \frac{a}{4-S} $$ \frac{\lambda a}{2\eta(4-S)} $$ \frac{(6-S)a}{2(4-S)} $$ \frac{a^2}{2(4-S)} $$ \frac{a^2}{4(4-S)} $$ \frac{3a^2}{4(4-S)} $
Table 3.  The optimal outcomes in the MR mode
$ w^{MR} $ $ Q_r^{MR} $ $ s^{MR} $$ p_r^{MR} $ $ \pi_M^{MR} $ $ \pi_R^{MR} $
$ c<\frac{(2-S)a}{4-S} $$ \frac{a}{2} $$ \frac{c}{2-S} $ $ \frac{\lambda c}{2\eta(2-S)} $$ \frac{a}{2}+\frac{c}{2-S} $ $ \frac{(a-c)^2}{4}+\frac{c^2}{4-2S} $$ \frac{(4-S)c^2}{4(2-S)^2} $
$ c\geq\frac{(2-S)a}{4-S} $$ \frac{a}{2} $$ \frac{a}{4-S} $ $ \frac{\lambda a}{2\eta(4-S)} $$ \frac{(6-S)a}{2(4-S)} $ $ \frac{a^2}{2(4-S)} $$ \frac{a^2}{4(4-S)} $
$ w^{MR} $ $ Q_r^{MR} $ $ s^{MR} $$ p_r^{MR} $ $ \pi_M^{MR} $ $ \pi_R^{MR} $
$ c<\frac{(2-S)a}{4-S} $$ \frac{a}{2} $$ \frac{c}{2-S} $ $ \frac{\lambda c}{2\eta(2-S)} $$ \frac{a}{2}+\frac{c}{2-S} $ $ \frac{(a-c)^2}{4}+\frac{c^2}{4-2S} $$ \frac{(4-S)c^2}{4(2-S)^2} $
$ c\geq\frac{(2-S)a}{4-S} $$ \frac{a}{2} $$ \frac{a}{4-S} $ $ \frac{\lambda a}{2\eta(4-S)} $$ \frac{(6-S)a}{2(4-S)} $ $ \frac{a^2}{2(4-S)} $$ \frac{a^2}{4(4-S)} $
Table 4.  The optimal outcomes in the MRM mode
$ c $$ [0, \frac{(3-2S)a}{5-2S}) $$ [\frac{(3-2S)a}{5-2S}, \frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}}) $$ [\frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}}, a) $
$ w^{RM} $$ \frac{a}{2}-\frac{c}{2(3-2S)} $$ \frac{(3-S)c-(1-S)a}{2} $$ \frac{a}{2} $
$ Q_r^{RM} $$ \frac{2c}{3-2S} $$ a-c $$ \frac{a}{4-S} $
$ s^{RM} $$ \frac{\lambda c}{\eta(3-2S)} $$ \frac{\lambda (a-c)}{2\eta} $$ \frac{\lambda a}{2\eta(4-S)} $
$ Q_d^{RM} $$ \frac{a}{2}-\frac{(5-2S)c}{2(3-2S)} $$ 0 $$ 0 $
$ p_r^{RM} $$ \frac{a}{2}+\frac{c}{2(3-2S)} $$ \frac{(2-S)c+Sa}{2} $$ \frac{(6-S)a}{2(4-S)} $
$ p_d^{RM} $$ \frac{a}{2}+\frac{(1-2S)c}{2(3-2S)} $$ - $$ - $
$ \pi_M^{RM} $$ \frac{(a-c)^2}{4}+\frac{c^2}{3-2S} $$ \frac{(a-c)[(3-S)c-(1-S)a]}{2} $$ \frac{a^2}{2(4-S)} $
$ \pi_R^{RM} $$ \frac{(2-S)c^2}{(3-2S)^2} $$ \frac{(2-S)(a-c)^2}{4} $$ \frac{a^2}{4(4-S)} $
$ c $$ [0, \frac{(3-2S)a}{5-2S}) $$ [\frac{(3-2S)a}{5-2S}, \frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}}) $$ [\frac{(3-S+\sqrt{4-S})a}{4-S+\sqrt{4-S}}, a) $
$ w^{RM} $$ \frac{a}{2}-\frac{c}{2(3-2S)} $$ \frac{(3-S)c-(1-S)a}{2} $$ \frac{a}{2} $
$ Q_r^{RM} $$ \frac{2c}{3-2S} $$ a-c $$ \frac{a}{4-S} $
$ s^{RM} $$ \frac{\lambda c}{\eta(3-2S)} $$ \frac{\lambda (a-c)}{2\eta} $$ \frac{\lambda a}{2\eta(4-S)} $
$ Q_d^{RM} $$ \frac{a}{2}-\frac{(5-2S)c}{2(3-2S)} $$ 0 $$ 0 $
$ p_r^{RM} $$ \frac{a}{2}+\frac{c}{2(3-2S)} $$ \frac{(2-S)c+Sa}{2} $$ \frac{(6-S)a}{2(4-S)} $
$ p_d^{RM} $$ \frac{a}{2}+\frac{(1-2S)c}{2(3-2S)} $$ - $$ - $
$ \pi_M^{RM} $$ \frac{(a-c)^2}{4}+\frac{c^2}{3-2S} $$ \frac{(a-c)[(3-S)c-(1-S)a]}{2} $$ \frac{a^2}{2(4-S)} $
$ \pi_R^{RM} $$ \frac{(2-S)c^2}{(3-2S)^2} $$ \frac{(2-S)(a-c)^2}{4} $$ \frac{a^2}{4(4-S)} $
Table 5.  The optimal outcomes in the MRMR mode
$ c $$ [0, \frac{(128-96S+9S^2)a}{256-144S+9S^2}) $$ [\frac{(128-96S+9S^2)a}{256-144S+9S^2}, a) $
$ w^{RMR} $$ \frac{a}{2}-\frac{9S^2c}{2(128-96S+9S^2)} $$ \frac{a}{2} $
$ s^{RMR} $$ \frac{3\lambda (16-3S)c}{\eta(128-96S+9S^2)} $$ \frac{\lambda a}{2\eta(4-S)} $
$ Q_d^{RMR} $$ \frac{(128-96S+9S^2)a-(256-144S+9S^2)c}{2(128-96S+9S^2)} $$ 0 $
$ Q_r^{RMR} $$ \frac{4(16-3S)c}{128-96S+9S^2} $$ \frac{a}{4-S} $
$ p_r^{RMR} $$ \frac{a}{2}+\frac{(128-24S-9S^2)c}{2(128-96S+9S^2)} $$ \frac{(6-S)a}{2(4-S)} $
$ p_d^{RMR} $$ \frac{a}{2}+\frac{(128-120S+9S^2)c}{2(128-96S+9S^2)} $$ - $
$ \pi_M^{RMR} $$ \frac{(128-96S+9S^2)(a-c)^2+128c^2 }{4(128-96S+9S^2)} $$ \frac{a^2}{2(4-S)} $
$ \pi_R^{RMR} $$ \frac{(16-9S)(16-3S)^2c^2}{(128-96S+9S^2)^2} $$ \frac{a^2}{4(4-S)} $
$ c $$ [0, \frac{(128-96S+9S^2)a}{256-144S+9S^2}) $$ [\frac{(128-96S+9S^2)a}{256-144S+9S^2}, a) $
$ w^{RMR} $$ \frac{a}{2}-\frac{9S^2c}{2(128-96S+9S^2)} $$ \frac{a}{2} $
$ s^{RMR} $$ \frac{3\lambda (16-3S)c}{\eta(128-96S+9S^2)} $$ \frac{\lambda a}{2\eta(4-S)} $
$ Q_d^{RMR} $$ \frac{(128-96S+9S^2)a-(256-144S+9S^2)c}{2(128-96S+9S^2)} $$ 0 $
$ Q_r^{RMR} $$ \frac{4(16-3S)c}{128-96S+9S^2} $$ \frac{a}{4-S} $
$ p_r^{RMR} $$ \frac{a}{2}+\frac{(128-24S-9S^2)c}{2(128-96S+9S^2)} $$ \frac{(6-S)a}{2(4-S)} $
$ p_d^{RMR} $$ \frac{a}{2}+\frac{(128-120S+9S^2)c}{2(128-96S+9S^2)} $$ - $
$ \pi_M^{RMR} $$ \frac{(128-96S+9S^2)(a-c)^2+128c^2 }{4(128-96S+9S^2)} $$ \frac{a^2}{2(4-S)} $
$ \pi_R^{RMR} $$ \frac{(16-9S)(16-3S)^2c^2}{(128-96S+9S^2)^2} $$ \frac{a^2}{4(4-S)} $
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