American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2021047

Assembly system with omnichannel coordination

Zhisong Chen 1,2,, and  Shong-Iee Ivan Su 3,

1. 

Business School, Nanjing Normal University, Qixia District, Nanjing 210023, China
2. 

Stern School of Business, New York University, 44 West Fourth Street, New York, NY 10012, USA
3. 

Supply Chain and Logistics Management Research Lab, Department of Business Administration, School of Business, Soochow University, Taipei, Taiwan

* Corresponding author: Zhisong Chen

Received  May 2020 Revised  December 2020 Published  March 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (Grant No. 71603125), China Scholarship Council (Grant No. 201706865020), the National Key R & D Program of China (Grant No. 2017YFC0404600), China Postdoctoral Science Foundation (Grant No. 2019M651833), Social Science Foundation of Jiangsu Province in China (Grant No. 19GLC003), and Young Leading Talent Program of Nanjing Normal University

Assembly system with omnichannel is rarely studied in literature. This paper explores the equilibrium and coordination issues for an omnichannel assembly system. Four different game-theoretical model types applying four operational strategies - a total of sixteen analytical models - are developed and analyzed for both omnichannel and pure channel modes. The numerical analysis of an electronic product assembly system provides a clearer understanding of the solutions and their effects on the profits in the assembly system for different model types and operational strategies. A further sensitivity analysis with focus on an omnichannel with offline channel subsidy (OMS) creates better insights regarding how changes of key parameters affect the assembly system profits. It is found that the omnichannel mode with or without offline channel subsidy can deliver much better operational performance to the assembly system via mutual fusion effect than that of a pure online- or offline-channel mode. Furthermore, the offline channel subsidy can amplify to a very large extent the mutual fusion effect to increase the product demand dramatically and thus improving the operational performance of the assembly system in the omnichannel business scenario. The best operational strategy for the assembly system in the omnichannel business scenario is the coordination strategy with offline channel subsidy.

Keywords: Omnichannel, assembly system, game-theoretical model, coordination, offline channel subsidy.
Mathematics Subject Classification: Primary: 90B06, 91A10; Secondary: 91A12.
Citation: Zhisong Chen, Shong-Iee Ivan Su. Assembly system with omnichannel coordination. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021047
References:
[1]

G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 30-44.  doi: 10.1287/mnsc.1040.0215.  Google Scholar

[2]

G. Cai, Channel selection and coordination in dual-channel supply chains, Journal of Retailing, 86 (2010), 22-36.  doi: 10.1016/j.jretai.2009.11.002.  Google Scholar

[3]

S. M. Carr and U. S. Karmarkar, Competition in multiechelon assembly supply chains, Management Science, 51 (2005), 45-59.  doi: 10.1287/mnsc.1040.0216.  Google Scholar

[4]

X. Chen, X. 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.  doi: 10.1057/jors.2015.6.  Google Scholar

[5]

Z. Chen, L. Fang and S. I. Su, The value of offline channel subsidy in bricks and clicks: An O2O supply chain coordination perspective, Electronic Commerce Research, 273 (2019), 1-45.  doi: 10.1007/s10660-019-09386-z.  Google Scholar

[6]

Z. Chen and S. I. Su, Consignment supply chain cooperation for complementary products under online to offline business mode, Flexible Services and Manufacturing Journal, (2020). doi: 10.1007/s10696-020-09376-6.  Google Scholar

[7]

DT News Network, Dell EMC Believes in an Omni Channel Strategy, 2017. Available from: http://digitalterminal.in/news/dell-emc-believes-in-an-omni-channel-strategy/8853.html. Google Scholar

[8]

W. Duggan, What does O2O mean for the future of E-Commerce?, Yahoo Finance, (2015). Google Scholar

[9]

A. Dumrongsiri, M. Fan, A. Jain and K. Moinzadeh, 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.  Google Scholar

[10]

F. Gao and X. Su, Omnichannel service operations with online and offline self-order technologies, Management Science, 64 (2018), 3595-3608.  doi: 10.1287/mnsc.2017.2787.  Google Scholar

[11]

F. Gao and X. Su, Omnichannel retail operations with buy-online-and-pick-up-in-store, Management Science, 63 (2016), 2478-2492.  doi: 10.1287/mnsc.2016.2473.  Google Scholar

[12]

Investopedia, Online-to-Offline Commerce, 2018. Available from: https://www.investopedia.com/terms/o/onlinetooffline-commerce.asp#ixzz5GNPyJCIX. Google Scholar

[13]

J. Ji, Z. Zhang and L. Yang, Comparisons of initial carbon allowance allocation rules in an O2O retail supply chain with the cap-and-trade regulation, International Journal of Production Economics, 187 (2017), 68-84.  doi: 10.1016/j.ijpe.2017.02.011.  Google Scholar

[14]

Y. Jiang, L. Liu and A. Lim, Optimal pricing decisions for an omni-channel supply chain with retail service, International Transactions in Operational Research, 27 (2020), 2927-2948.  doi: 10.1111/itor.12784.  Google Scholar

[15]

L. Jiang and Y. Wang, Supplier competition in decentralized assembly systems with price-sensitive and uncertain demand, Manufacturing & Service Operations Management, 12 (2010), 93-101.  doi: 10.1287/msom.1090.0259.  Google Scholar

[16]

B. Kalkancl and F. Erhun, Pricing games and impact of private demand information in decentralized assembly systems, Operations Research, 60 (2012), 1142-1156.  doi: 10.1287/opre.1120.1084.  Google Scholar

[17]

Kjmx, Hardware cost disclosure of Huawei, apple, Samsung and Xiaomi, how much is the profit margin of each manufacturer?, Available from: http://finance.sina.com.cn/stock/relnews/us/2019-06-29/doc-ihytcerm0232231.shtml. Google Scholar

[18]

L. Kong, Z. Liu, Y. Pan, J. Xie and G. Yang, Pricing and service decision of dual-channel operations in an O2O closed-loop supply chain, Industrial Management & Data Systems, 117 (2017), 1567-1588.  doi: 10.1108/IMDS-12-2016-0544.  Google Scholar

[19]

M. A. Lariviere, A note on probability distributions with increasing generalized failure rates, Operations Research, 54 (2006), 602-604.  doi: 10.1287/opre.1060.0282.  Google Scholar

[20]

M. A. Lariviere and E. L. Porteus, Selling to the newsvendor: An analysis of price-only contracts, Manufacturing and Service Operations Management, 3 (2001), 293-305.  doi: 10.1287/msom.3.4.293.9971.  Google Scholar

[21]

N. M. Modak, Exploring Omni-channel supply chain under price and delivery time sensitive stochastic demand, Supply Chain Forum: An International Journal, 18 (2017), 218-230.  doi: 10.1080/16258312.2017.1380499.  Google Scholar

[22]

N. M. Modak and P. Kelle, Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research, 272 (2019), 147-161.  doi: 10.1016/j.ejor.2018.05.067.  Google Scholar

[23]

M. Nagarajan and Y. Bassok, A bargaining framework in supply chains: The assembly problem, Management Science, 54 (2008), 1482-1496.   Google Scholar

[24]

M. Nagarajan and G. Sošić, Coalition stability in assembly models, Operations Research, 57 (2009), 131-145.   Google Scholar

[25]

A. Orendorff, O2O Commerce: Conquering Online-to-Offline Retail's Trillion Dollar Opportunity, 2018. Available from: https://www.shopify.com/enterprise/o2o-online-to-offline-commerce. Google Scholar

[26]

S. Panda, N. M. Modak, S. S. Sana and M. Basu, Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease, Applied Mathematics and Computation, 256 (2015), 913-929.  doi: 10.1016/j.amc.2015.01.081.  Google Scholar

[27]

J. Paul, N. Agatz, R. Spliet and R. De Koster, Shared capacity routing problem? An omni-channel retail study, European Journal of Operational Research, 273 (2019), 731-739.  doi: 10.1016/j.ejor.2018.08.027.  Google Scholar

[28]

N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194.  doi: 10.1287/opre.47.2.183.  Google Scholar

[29]

S. Saha, N. M. Modak, S. Panda and S. S. Sana, Managing a retailer's dual-channel supply chain under price- and delivery time-sensitive demand, Journal of Modelling in Management, 13 (2018), 351-374.  doi: 10.1108/JM2-10-2016-0089.  Google Scholar

[30]

A. A. Tsay and N. Agrawal, Channel conflict and coordination in the E-Commerce age, Production and Operations Management, 13 (2004), 93-110.  doi: 10.1007/s10436-017-0298-8.  Google Scholar

[31]

USDOC, U. S., Department of Commerce Quarterly Retail Ecommerce Sales 4th Quarter 2016, Reported 2/17/17. Google Scholar

[32]

T. Wallace, The Complete Omni-Channel Retail Report: What Brands Need to Know About Modern Consumer Shopping Habits in 2018., Available from: https://www.bigcommerce.com/blog/omni-channel-retail/. Google Scholar

[33]

Y. Wang, L. Jiang and Z. J. Shen, Channel performance under consignment contract with revenue sharing, Management Science, 50 (2004), 34-47.  doi: 10.1287/mnsc.1030.0168.  Google Scholar

[34]

Y. Wang, Joint pricing-production decisions in supply chains of complementary products with uncertain demand, Operations Research, 54 (2006), 1110-1127.  doi: 10.1287/opre.1060.0326.  Google Scholar

[35]

E. Weise, In best buy deal, Amazon acknowledges you sometimes need to touch things to buy, USA TODAY. Google Scholar

[36]

G. Xu, B. Dan, X. Zhang and C. Liu, Coordinating a dual-channel supply chain with risk-averse under a two-way revenue sharing contract, International Journal of Production Economics, 147 (2014), 171-179.   Google Scholar

[37]

D. Yang and X. Zhang, Quick response and omnichannel retail operations with the ship-to-store program, International Transactions in Operational Research, 27 (2020), 3007-3030.  doi: 10.1111/itor.12781.  Google Scholar

[38]

S. Yin, Alliance formation among perfectly complementary suppliers in a price-sensitive assembly system, Manufacturing & Service Operations Management, 12 (2010), 527-544.  doi: 10.1287/msom.1090.0283.  Google Scholar

[39]

F. Zhang, Competition, cooperation, and information sharing in a two-echelon assembly system, Manufacturing & Service Operations Management, 8 (2006), 273-291.  doi: 10.1287/msom.1060.0108.  Google Scholar

[40]

J. Zhang, H. Chen and X. Wu, Operation models in O2O supply chain when existing competitive service level, International Journal of u- and e- Service, Science and Technology, 8 (2015), 279-290.   Google Scholar

[41]

J. Zhang, H. Chen, J. Ma and K. Tang, How to coordinate supply chain under O2O business model when demand deviation happens, Management Science and Engineering, 9 (2015), 24-28.   Google Scholar

show all references

References:
[1]

G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 30-44.  doi: 10.1287/mnsc.1040.0215.  Google Scholar

[2]

G. Cai, Channel selection and coordination in dual-channel supply chains, Journal of Retailing, 86 (2010), 22-36.  doi: 10.1016/j.jretai.2009.11.002.  Google Scholar

[3]

S. M. Carr and U. S. Karmarkar, Competition in multiechelon assembly supply chains, Management Science, 51 (2005), 45-59.  doi: 10.1287/mnsc.1040.0216.  Google Scholar

[4]

X. Chen, X. 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.  doi: 10.1057/jors.2015.6.  Google Scholar

[5]

Z. Chen, L. Fang and S. I. Su, The value of offline channel subsidy in bricks and clicks: An O2O supply chain coordination perspective, Electronic Commerce Research, 273 (2019), 1-45.  doi: 10.1007/s10660-019-09386-z.  Google Scholar

[6]

Z. Chen and S. I. Su, Consignment supply chain cooperation for complementary products under online to offline business mode, Flexible Services and Manufacturing Journal, (2020). doi: 10.1007/s10696-020-09376-6.  Google Scholar

[7]

DT News Network, Dell EMC Believes in an Omni Channel Strategy, 2017. Available from: http://digitalterminal.in/news/dell-emc-believes-in-an-omni-channel-strategy/8853.html. Google Scholar

[8]

W. Duggan, What does O2O mean for the future of E-Commerce?, Yahoo Finance, (2015). Google Scholar

[9]

A. Dumrongsiri, M. Fan, A. Jain and K. Moinzadeh, 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.  Google Scholar

[10]

F. Gao and X. Su, Omnichannel service operations with online and offline self-order technologies, Management Science, 64 (2018), 3595-3608.  doi: 10.1287/mnsc.2017.2787.  Google Scholar

[11]

F. Gao and X. Su, Omnichannel retail operations with buy-online-and-pick-up-in-store, Management Science, 63 (2016), 2478-2492.  doi: 10.1287/mnsc.2016.2473.  Google Scholar

[12]

Investopedia, Online-to-Offline Commerce, 2018. Available from: https://www.investopedia.com/terms/o/onlinetooffline-commerce.asp#ixzz5GNPyJCIX. Google Scholar

[13]

J. Ji, Z. Zhang and L. Yang, Comparisons of initial carbon allowance allocation rules in an O2O retail supply chain with the cap-and-trade regulation, International Journal of Production Economics, 187 (2017), 68-84.  doi: 10.1016/j.ijpe.2017.02.011.  Google Scholar

[14]

Y. Jiang, L. Liu and A. Lim, Optimal pricing decisions for an omni-channel supply chain with retail service, International Transactions in Operational Research, 27 (2020), 2927-2948.  doi: 10.1111/itor.12784.  Google Scholar

[15]

L. Jiang and Y. Wang, Supplier competition in decentralized assembly systems with price-sensitive and uncertain demand, Manufacturing & Service Operations Management, 12 (2010), 93-101.  doi: 10.1287/msom.1090.0259.  Google Scholar

[16]

B. Kalkancl and F. Erhun, Pricing games and impact of private demand information in decentralized assembly systems, Operations Research, 60 (2012), 1142-1156.  doi: 10.1287/opre.1120.1084.  Google Scholar

[17]

Kjmx, Hardware cost disclosure of Huawei, apple, Samsung and Xiaomi, how much is the profit margin of each manufacturer?, Available from: http://finance.sina.com.cn/stock/relnews/us/2019-06-29/doc-ihytcerm0232231.shtml. Google Scholar

[18]

L. Kong, Z. Liu, Y. Pan, J. Xie and G. Yang, Pricing and service decision of dual-channel operations in an O2O closed-loop supply chain, Industrial Management & Data Systems, 117 (2017), 1567-1588.  doi: 10.1108/IMDS-12-2016-0544.  Google Scholar

[19]

M. A. Lariviere, A note on probability distributions with increasing generalized failure rates, Operations Research, 54 (2006), 602-604.  doi: 10.1287/opre.1060.0282.  Google Scholar

[20]

M. A. Lariviere and E. L. Porteus, Selling to the newsvendor: An analysis of price-only contracts, Manufacturing and Service Operations Management, 3 (2001), 293-305.  doi: 10.1287/msom.3.4.293.9971.  Google Scholar

[21]

N. M. Modak, Exploring Omni-channel supply chain under price and delivery time sensitive stochastic demand, Supply Chain Forum: An International Journal, 18 (2017), 218-230.  doi: 10.1080/16258312.2017.1380499.  Google Scholar

[22]

N. M. Modak and P. Kelle, Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research, 272 (2019), 147-161.  doi: 10.1016/j.ejor.2018.05.067.  Google Scholar

[23]

M. Nagarajan and Y. Bassok, A bargaining framework in supply chains: The assembly problem, Management Science, 54 (2008), 1482-1496.   Google Scholar

[24]

M. Nagarajan and G. Sošić, Coalition stability in assembly models, Operations Research, 57 (2009), 131-145.   Google Scholar

[25]

A. Orendorff, O2O Commerce: Conquering Online-to-Offline Retail's Trillion Dollar Opportunity, 2018. Available from: https://www.shopify.com/enterprise/o2o-online-to-offline-commerce. Google Scholar

[26]

S. Panda, N. M. Modak, S. S. Sana and M. Basu, Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease, Applied Mathematics and Computation, 256 (2015), 913-929.  doi: 10.1016/j.amc.2015.01.081.  Google Scholar

[27]

J. Paul, N. Agatz, R. Spliet and R. De Koster, Shared capacity routing problem? An omni-channel retail study, European Journal of Operational Research, 273 (2019), 731-739.  doi: 10.1016/j.ejor.2018.08.027.  Google Scholar

[28]

N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194.  doi: 10.1287/opre.47.2.183.  Google Scholar

[29]

S. Saha, N. M. Modak, S. Panda and S. S. Sana, Managing a retailer's dual-channel supply chain under price- and delivery time-sensitive demand, Journal of Modelling in Management, 13 (2018), 351-374.  doi: 10.1108/JM2-10-2016-0089.  Google Scholar

[30]

A. A. Tsay and N. Agrawal, Channel conflict and coordination in the E-Commerce age, Production and Operations Management, 13 (2004), 93-110.  doi: 10.1007/s10436-017-0298-8.  Google Scholar

[31]

USDOC, U. S., Department of Commerce Quarterly Retail Ecommerce Sales 4th Quarter 2016, Reported 2/17/17. Google Scholar

[32]

T. Wallace, The Complete Omni-Channel Retail Report: What Brands Need to Know About Modern Consumer Shopping Habits in 2018., Available from: https://www.bigcommerce.com/blog/omni-channel-retail/. Google Scholar

[33]

Y. Wang, L. Jiang and Z. J. Shen, Channel performance under consignment contract with revenue sharing, Management Science, 50 (2004), 34-47.  doi: 10.1287/mnsc.1030.0168.  Google Scholar

[34]

Y. Wang, Joint pricing-production decisions in supply chains of complementary products with uncertain demand, Operations Research, 54 (2006), 1110-1127.  doi: 10.1287/opre.1060.0326.  Google Scholar

[35]

E. Weise, In best buy deal, Amazon acknowledges you sometimes need to touch things to buy, USA TODAY. Google Scholar

[36]

G. Xu, B. Dan, X. Zhang and C. Liu, Coordinating a dual-channel supply chain with risk-averse under a two-way revenue sharing contract, International Journal of Production Economics, 147 (2014), 171-179.   Google Scholar

[37]

D. Yang and X. Zhang, Quick response and omnichannel retail operations with the ship-to-store program, International Transactions in Operational Research, 27 (2020), 3007-3030.  doi: 10.1111/itor.12781.  Google Scholar

[38]

S. Yin, Alliance formation among perfectly complementary suppliers in a price-sensitive assembly system, Manufacturing & Service Operations Management, 12 (2010), 527-544.  doi: 10.1287/msom.1090.0283.  Google Scholar

[39]

F. Zhang, Competition, cooperation, and information sharing in a two-echelon assembly system, Manufacturing & Service Operations Management, 8 (2006), 273-291.  doi: 10.1287/msom.1060.0108.  Google Scholar

[40]

J. Zhang, H. Chen and X. Wu, Operation models in O2O supply chain when existing competitive service level, International Journal of u- and e- Service, Science and Technology, 8 (2015), 279-290.   Google Scholar

[41]

J. Zhang, H. Chen, J. Ma and K. Tang, How to coordinate supply chain under O2O business model when demand deviation happens, Management Science and Engineering, 9 (2015), 24-28.   Google Scholar

Figure 1.  A Generic Assembly System with Omnichannel
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Figure 2.  Impact of Offline Channel Subsidy Factor ($ \delta $) Change on Profits
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Figure 3.  Impact of Price Elasticity Index of the Expected Demand ($ b $) Change on Profits
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Figure 4.  Impact of Mutual fusion Coefficient ($ \theta $) Change on Profits
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Table 1.  Modelling Notations and Explanations
Parameter/Variable Explanations
$ c $ Unit assembly cost of the final product
$ c_i $ Unit cost of the $ i^{\text {th }} $ module
$ w_i $ Wholesale price of the $ i^{\text {th }} $ module
$ c_e $ Operational cost of the online channel
$ c_s $ Operational cost of the offline channel
$ p $ Retail price of the final product in the online/offline channel
$ z $ Stock factor
$ a $ Positive constant number
$ b $ Price-elasticity index of the expected demand
$ \theta $ Mutual fusion coefficient between channels
$ \eta $ Clearance discount price factor, and $ 0<\eta<1 $
$ {\lambda _0} $ Market demand share of the online channel, and $ 0<\lambda_{0}<1 $
$ \delta $ The offline channel subsidy factor, and $ 0<\delta<1 $
$ \kappa $ The offline channel discount price factor, and $ \kappa=1-\delta $
$ h $ The reaction extent of $ \lambda(\kappa) $ w.r.t. the change of $ \kappa $
$ \phi $ Revenue keeping rate, and $ 0<\phi<1 $
$ x $ The random factor defined in the range $ [A, B] $ with $ B>A>0 $
$ \mu $ Mean value of random factor
$ \sigma $ Standard deviation of random factor
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Parameter/Variable Explanations
$ c $ Unit assembly cost of the final product
$ c_i $ Unit cost of the $ i^{\text {th }} $ module
$ w_i $ Wholesale price of the $ i^{\text {th }} $ module
$ c_e $ Operational cost of the online channel
$ c_s $ Operational cost of the offline channel
$ p $ Retail price of the final product in the online/offline channel
$ z $ Stock factor
$ a $ Positive constant number
$ b $ Price-elasticity index of the expected demand
$ \theta $ Mutual fusion coefficient between channels
$ \eta $ Clearance discount price factor, and $ 0<\eta<1 $
$ {\lambda _0} $ Market demand share of the online channel, and $ 0<\lambda_{0}<1 $
$ \delta $ The offline channel subsidy factor, and $ 0<\delta<1 $
$ \kappa $ The offline channel discount price factor, and $ \kappa=1-\delta $
$ h $ The reaction extent of $ \lambda(\kappa) $ w.r.t. the change of $ \kappa $
$ \phi $ Revenue keeping rate, and $ 0<\phi<1 $
$ x $ The random factor defined in the range $ [A, B] $ with $ B>A>0 $
$ \mu $ Mean value of random factor
$ \sigma $ Standard deviation of random factor
Table 2.  Framework of Game-Theoretical Decision Models
Section Channel Strategy Game-Theoretical Decision Models Theories Applied
4.1 Omnichannel mode without offline channel subsidy (OMO mode) 4.1.1 Centralized Decision Model OT & BC
4.1.2 Decentralized Decision Model SG & BC
4.1.2.1 Assembler's Decision SG & BC
4.1.2.2 Suppliers' Simultaneous Decision SG & BC
4.1.2.3 Suppliers' Sequential Decision SG & BC
4.1.3 Coordination Decision Model RSC & BC
4.2 Omnichannel OT+SG+RSC+BC
mode with offline Centralized/Decentralized/Coordination
channel subsidy Decision Models under OMS mode
(OMS mode)
4.3 Pure online/offline Centralized/Decentralized/Coordination Decision Models under POC/PFC mode OT+SG+RSC+BC
channel mode
(POC/PFC mode)
Notation: OT: Optimization Theory; BC: Bertrand Competition; SG: Stackelberg Game; RSC: Revenue Sharing Contract
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Section Channel Strategy Game-Theoretical Decision Models Theories Applied
4.1 Omnichannel mode without offline channel subsidy (OMO mode) 4.1.1 Centralized Decision Model OT & BC
4.1.2 Decentralized Decision Model SG & BC
4.1.2.1 Assembler's Decision SG & BC
4.1.2.2 Suppliers' Simultaneous Decision SG & BC
4.1.2.3 Suppliers' Sequential Decision SG & BC
4.1.3 Coordination Decision Model RSC & BC
4.2 Omnichannel OT+SG+RSC+BC
mode with offline Centralized/Decentralized/Coordination
channel subsidy Decision Models under OMS mode
(OMS mode)
4.3 Pure online/offline Centralized/Decentralized/Coordination Decision Models under POC/PFC mode OT+SG+RSC+BC
channel mode
(POC/PFC mode)
Notation: OT: Optimization Theory; BC: Bertrand Competition; SG: Stackelberg Game; RSC: Revenue Sharing Contract
Table 3.  Analytical Results under the OMO Mode
Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{d}\right)=F\left(z_{c}\right) $ $ F\left(z_{d^{\prime}}\right)=F\left(z_{c}\right) $ $ F\left(z_{c}\right)=\frac{1}{(1-\eta)(b-\theta)}+\frac{(b-\theta-1) \Lambda\left(z_{c}\right)}{(b-\theta) z_{c}} $
$ p_{*} $ $ p_{d}=\frac{b-\theta}{b-\theta-n} p_{c} $ $ p_{d^{\prime}}=\left(\frac{b-\theta}{b-\theta-1}\right)^{n} p_{c} $ $ p_{c}=\frac{b-\theta}{b-\theta-1} \frac{\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right] z_{c}}{z_{c}-(1-\eta) \Lambda\left(z_{c}\right)} $
$ q_{*} $ $ q_{d}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta} q_{c} $ $ q_{d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)} q_{c} $ $ q_{c}=y\left(p_{c}\right) z_{c} $
$ w_{i}^{*} $ $ w_{i}^{d}=\frac{1}{b-\theta-n}\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right]+c_{i} $ $ w_{i}^{d^{\prime}}=\frac{(b-\theta)^{i-1}}{(b-\theta-1)^{i}}\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right]+c_{i} $ $ w_{i}^{c}=\phi^{*} c_{i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{d}=\frac{b-\theta-1}{b-\theta}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{c} $ $ \Pi_{S_{i}}^{d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)-i+1} \Pi_{S C}^{c} $ $ \Pi_{S_{i}}^{c}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{d}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{c} $ $ \Pi_{A}^{d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)} \Pi_{S C}^{c} $ $ \Pi_{A}^{c}=\phi^{*} \Pi_{S C}^{c} $
$ \Pi_{S C}^{*} $ $ \Pi_{S C}^{d}=\left[(n+1)-\frac{n}{b-\theta}\right]\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{c} $ $ \Pi_{S C}^{d^{\prime}}=\left[(b-\theta)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}-(b-\theta-1)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)}\right] \Pi_{S C}^{c} $ $ \Pi_{S C}^{c}=\frac{1}{b-\theta-1}\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right] q_{c} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline{\phi}=\max \left\{\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1}, \left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}\right\} $, $ \bar{\phi}=\min _{i \in N}\left\{1-\frac{\sum_{i=1}^{n} c_{i}}{c_{i}} \max \left\{\frac{b-\theta-1}{b-\theta-n}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta}, \left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)-i+1}\right\}\right\} $
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Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{d}\right)=F\left(z_{c}\right) $ $ F\left(z_{d^{\prime}}\right)=F\left(z_{c}\right) $ $ F\left(z_{c}\right)=\frac{1}{(1-\eta)(b-\theta)}+\frac{(b-\theta-1) \Lambda\left(z_{c}\right)}{(b-\theta) z_{c}} $
$ p_{*} $ $ p_{d}=\frac{b-\theta}{b-\theta-n} p_{c} $ $ p_{d^{\prime}}=\left(\frac{b-\theta}{b-\theta-1}\right)^{n} p_{c} $ $ p_{c}=\frac{b-\theta}{b-\theta-1} \frac{\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right] z_{c}}{z_{c}-(1-\eta) \Lambda\left(z_{c}\right)} $
$ q_{*} $ $ q_{d}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta} q_{c} $ $ q_{d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)} q_{c} $ $ q_{c}=y\left(p_{c}\right) z_{c} $
$ w_{i}^{*} $ $ w_{i}^{d}=\frac{1}{b-\theta-n}\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right]+c_{i} $ $ w_{i}^{d^{\prime}}=\frac{(b-\theta)^{i-1}}{(b-\theta-1)^{i}}\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right]+c_{i} $ $ w_{i}^{c}=\phi^{*} c_{i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{d}=\frac{b-\theta-1}{b-\theta}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{c} $ $ \Pi_{S_{i}}^{d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)-i+1} \Pi_{S C}^{c} $ $ \Pi_{S_{i}}^{c}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{d}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{c} $ $ \Pi_{A}^{d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)} \Pi_{S C}^{c} $ $ \Pi_{A}^{c}=\phi^{*} \Pi_{S C}^{c} $
$ \Pi_{S C}^{*} $ $ \Pi_{S C}^{d}=\left[(n+1)-\frac{n}{b-\theta}\right]\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{c} $ $ \Pi_{S C}^{d^{\prime}}=\left[(b-\theta)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}-(b-\theta-1)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)}\right] \Pi_{S C}^{c} $ $ \Pi_{S C}^{c}=\frac{1}{b-\theta-1}\left[c+\lambda_{0} c_{e}+\left(1-\lambda_{0}\right) c_{s}+\sum_{i=1}^{n} c_{i}\right] q_{c} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline{\phi}=\max \left\{\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1}, \left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}\right\} $, $ \bar{\phi}=\min _{i \in N}\left\{1-\frac{\sum_{i=1}^{n} c_{i}}{c_{i}} \max \left\{\frac{b-\theta-1}{b-\theta-n}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta}, \left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)-i+1}\right\}\right\} $
Table 4.  Analytical Results under the OMS Mode
Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{sd}\right)=F\left(z_{sc}\right) $ $ F\left(z_{sd^{\prime}}\right)=F\left(z_{sc}\right) $ $ F\left(z_{s c}\right)=\frac{1}{(1-\eta)(b-\theta)}+\frac{(b-\theta-1) \Lambda\left(z_{s c}\right)}{(b-\theta) z_{s c}} $
$ p_{*} $ $ p_{sd}=\frac{b-\theta}{b-\theta-n} p_{sc} $ $ p_{sd^{\prime}}=\left(\frac{b-\theta}{b-\theta-1}\right)^{n} p_{sc} $ $ p_{s c}=\frac{b-\theta}{b-\theta-1} \frac{\left\{c+\lambda(\kappa) c_{e}+[1-\lambda(\kappa)] c_{s}+\sum_{i=1}^{n} c_{i}\right\} z_{s c}}{z_{s c}-(1-\eta) \Lambda\left(z_{s c}\right)} $
$ q_{*} $ $ q_{sd}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta} q_{sc} $ $ q_{sd^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)} q_{sc} $ $ q_{s c}=\left\{\lambda(\kappa) \kappa^{\theta}+[1-\lambda(\kappa)] \kappa^{-b}\right\} y\left(p_{s c}\right) z_{x} $
$ w_{i}^{*} $ $ w_i^{sd} = \frac{1}{{b - \theta - n}}\left\{ {c + \lambda (\kappa ){c_e} + [1 - \lambda (\kappa )]{c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right\} + {c_i} $ $ w_{i}^{s d^{\prime}}=\frac{(b-\theta)^{i-1}}{(b-\theta-1)^{i}}\left\{c+\lambda(\kappa) c_{e}+[1-\lambda(\kappa)] c_{s}+\sum\nolimits_{i=1}^{n} c_{i}\right\}+c_{i} $ $ w_{i}^{sc}=\phi^{*} c_{i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{s d}=\frac{b-\theta-1}{b-\theta}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{s c} $ $ \Pi_{S_{i}}^{s d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n{(b-\theta)-i+1}} \Pi_{S C}^{s c} $ $ \Pi_{S_{i}}^{s c}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{s c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{s d}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{s c} $ $ \Pi_{A}^{s d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)} \Pi_{S C}^{s c} $ $ \Pi_{A}^{s c}=\phi^{*} \Pi_{S C}^{s c} $
$ \Pi_{S C}^{*} $ $ \Pi_{S C}^{s d}=\left[(n+1)-\frac{n}{b-\theta}\right]\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{s c} $ $ \Pi_{S C}^{s d^{\prime}}=\left[(b-\theta)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}-(b-\theta-1)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)}\right] \Pi_{S C}^{s c} $ $ \Pi_{S C}^{s c}=\frac{1}{b-\theta-1}\left\{c+\lambda(\kappa) c_{e}+[1-\lambda(\kappa)] c_{s}+\sum_{i=1}^{n} c_{i}\right\} q_{s c} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline{\phi}=\max \left\{\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1},\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}\right\} $, $ \bar{\phi}=\min _{i=N}\left\{1-\frac{\sum_{i=1}^{n} c_{i}}{c_{i}} \max \left\{\frac{b-\theta-1}{b-\theta-n}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta},\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)-i+1}\right\}\right\} $
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Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{sd}\right)=F\left(z_{sc}\right) $ $ F\left(z_{sd^{\prime}}\right)=F\left(z_{sc}\right) $ $ F\left(z_{s c}\right)=\frac{1}{(1-\eta)(b-\theta)}+\frac{(b-\theta-1) \Lambda\left(z_{s c}\right)}{(b-\theta) z_{s c}} $
$ p_{*} $ $ p_{sd}=\frac{b-\theta}{b-\theta-n} p_{sc} $ $ p_{sd^{\prime}}=\left(\frac{b-\theta}{b-\theta-1}\right)^{n} p_{sc} $ $ p_{s c}=\frac{b-\theta}{b-\theta-1} \frac{\left\{c+\lambda(\kappa) c_{e}+[1-\lambda(\kappa)] c_{s}+\sum_{i=1}^{n} c_{i}\right\} z_{s c}}{z_{s c}-(1-\eta) \Lambda\left(z_{s c}\right)} $
$ q_{*} $ $ q_{sd}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta} q_{sc} $ $ q_{sd^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)} q_{sc} $ $ q_{s c}=\left\{\lambda(\kappa) \kappa^{\theta}+[1-\lambda(\kappa)] \kappa^{-b}\right\} y\left(p_{s c}\right) z_{x} $
$ w_{i}^{*} $ $ w_i^{sd} = \frac{1}{{b - \theta - n}}\left\{ {c + \lambda (\kappa ){c_e} + [1 - \lambda (\kappa )]{c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right\} + {c_i} $ $ w_{i}^{s d^{\prime}}=\frac{(b-\theta)^{i-1}}{(b-\theta-1)^{i}}\left\{c+\lambda(\kappa) c_{e}+[1-\lambda(\kappa)] c_{s}+\sum\nolimits_{i=1}^{n} c_{i}\right\}+c_{i} $ $ w_{i}^{sc}=\phi^{*} c_{i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{s d}=\frac{b-\theta-1}{b-\theta}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{s c} $ $ \Pi_{S_{i}}^{s d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n{(b-\theta)-i+1}} \Pi_{S C}^{s c} $ $ \Pi_{S_{i}}^{s c}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{s c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{s d}=\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{s c} $ $ \Pi_{A}^{s d^{\prime}}=\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)} \Pi_{S C}^{s c} $ $ \Pi_{A}^{s c}=\phi^{*} \Pi_{S C}^{s c} $
$ \Pi_{S C}^{*} $ $ \Pi_{S C}^{s d}=\left[(n+1)-\frac{n}{b-\theta}\right]\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1} \Pi_{S C}^{s c} $ $ \Pi_{S C}^{s d^{\prime}}=\left[(b-\theta)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}-(b-\theta-1)\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)}\right] \Pi_{S C}^{s c} $ $ \Pi_{S C}^{s c}=\frac{1}{b-\theta-1}\left\{c+\lambda(\kappa) c_{e}+[1-\lambda(\kappa)] c_{s}+\sum_{i=1}^{n} c_{i}\right\} q_{s c} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline{\phi}=\max \left\{\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta-1},\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta-1)}\right\} $, $ \bar{\phi}=\min _{i=N}\left\{1-\frac{\sum_{i=1}^{n} c_{i}}{c_{i}} \max \left\{\frac{b-\theta-1}{b-\theta-n}\left(\frac{b-\theta-n}{b-\theta}\right)^{b-\theta},\left(\frac{b-\theta-1}{b-\theta}\right)^{n(b-\theta)-i+1}\right\}\right\} $
Table 5.  Analytical Results under the POC Mode
Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{o d}\right)=F\left(z_{o c}\right) $ $ F\left(z_{o d^{\prime}}\right)=F\left(z_{o c}\right) $ $ F\left(z_{o c}\right)=\frac{1}{(1-\eta) b}+\frac{(b-1) \Lambda\left(z_{o c}\right)}{b z_{o c}} $
$ p_{*} $ $ p_{o d}=\frac{b}{b-n} p_{oc} $ $ p_{o d^{\prime}}=\left(\frac{b}{b-1}\right)^{n} p_{o c} $ $ p_{o c}=\frac{b}{b-1} \frac{\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right) z_{o c}}{z_{o c}-(1-\eta) \Lambda\left(z_{o c}\right)} $
$ q_{*} $ $ q_{o d}=\left(\frac{b-n}{b}\right)^{b} q_{o c} $ $ q_{o d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-\theta)} q_{o c} $ $ q_{o c}=y\left(p_{o c}\right) z_{o c} $
$ w_{i}^{*} $ $ w_{i}^{o d}=\frac{1}{b-n}\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right)+c_{i} $ $ w_{i}^{o d^{\prime}}=\frac{b^{i-1}}{(b-1)^{i}}\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right)+c_{i} $ $ w_{i}^{o c}=\phi^{*} c_{i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{o d}=\frac{b-1}{b}\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{o c} $ $ \Pi_{S_{i}}^{o d^{\prime}}=\left(\frac{b-1}{b}\right)^{n b-i+1} \Pi_{S C}^{o c} $ $ \Pi_{S_{i}}^{oc}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{o c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{o d}=\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{o c} $ $ \Pi_{A}^{o d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-1)} \Pi_{S C}^{o c} $ $ \Pi_{A}^{o c}=\phi^{*} \Pi_{S C}^{o c} $
$ \Pi_{S C}^{*} $ $ \Pi_{S C}^{o d}=\left[(n+1)-\frac{n}{b}\right]\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{o c} $ $ \Pi_{S C}^{o d^{\prime}}=\left[b\left(\frac{b-1}{b}\right)^{n(b-1)}-(b-1)\left(\frac{b-1}{b}\right)^{n b}\right] \Pi_{S C}^{o c} $ $ \Pi_{S C}^{o c}=\frac{1}{b-1}\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right) q_{o c} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline \phi = \max \left\{ {{{\left( {\frac{{b - n}}{b}} \right)}^{b - 1}},{{\left( {\frac{{b - 1}}{b}} \right)}^{n(b - 1)}}} \right\},\bar \phi = {\min _{i \in N}}\left\{ {1 - \frac{{\sum\nolimits_{i = 1}^n {{c_i}} }}{{{c_i}}}\max \left\{ {\frac{{b - 1}}{{b - n}}{{\left( {\frac{{b - n}}{b}} \right)}^b},{{\left( {\frac{{b - 1}}{b}} \right)}^{nb - i + 1}}} \right\}} \right\} $
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Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{o d}\right)=F\left(z_{o c}\right) $ $ F\left(z_{o d^{\prime}}\right)=F\left(z_{o c}\right) $ $ F\left(z_{o c}\right)=\frac{1}{(1-\eta) b}+\frac{(b-1) \Lambda\left(z_{o c}\right)}{b z_{o c}} $
$ p_{*} $ $ p_{o d}=\frac{b}{b-n} p_{oc} $ $ p_{o d^{\prime}}=\left(\frac{b}{b-1}\right)^{n} p_{o c} $ $ p_{o c}=\frac{b}{b-1} \frac{\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right) z_{o c}}{z_{o c}-(1-\eta) \Lambda\left(z_{o c}\right)} $
$ q_{*} $ $ q_{o d}=\left(\frac{b-n}{b}\right)^{b} q_{o c} $ $ q_{o d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-\theta)} q_{o c} $ $ q_{o c}=y\left(p_{o c}\right) z_{o c} $
$ w_{i}^{*} $ $ w_{i}^{o d}=\frac{1}{b-n}\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right)+c_{i} $ $ w_{i}^{o d^{\prime}}=\frac{b^{i-1}}{(b-1)^{i}}\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right)+c_{i} $ $ w_{i}^{o c}=\phi^{*} c_{i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{o d}=\frac{b-1}{b}\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{o c} $ $ \Pi_{S_{i}}^{o d^{\prime}}=\left(\frac{b-1}{b}\right)^{n b-i+1} \Pi_{S C}^{o c} $ $ \Pi_{S_{i}}^{oc}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{o c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{o d}=\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{o c} $ $ \Pi_{A}^{o d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-1)} \Pi_{S C}^{o c} $ $ \Pi_{A}^{o c}=\phi^{*} \Pi_{S C}^{o c} $
$ \Pi_{S C}^{*} $ $ \Pi_{S C}^{o d}=\left[(n+1)-\frac{n}{b}\right]\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{o c} $ $ \Pi_{S C}^{o d^{\prime}}=\left[b\left(\frac{b-1}{b}\right)^{n(b-1)}-(b-1)\left(\frac{b-1}{b}\right)^{n b}\right] \Pi_{S C}^{o c} $ $ \Pi_{S C}^{o c}=\frac{1}{b-1}\left(c+c_{e}+\sum_{i=1}^{n} c_{i}\right) q_{o c} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline \phi = \max \left\{ {{{\left( {\frac{{b - n}}{b}} \right)}^{b - 1}},{{\left( {\frac{{b - 1}}{b}} \right)}^{n(b - 1)}}} \right\},\bar \phi = {\min _{i \in N}}\left\{ {1 - \frac{{\sum\nolimits_{i = 1}^n {{c_i}} }}{{{c_i}}}\max \left\{ {\frac{{b - 1}}{{b - n}}{{\left( {\frac{{b - n}}{b}} \right)}^b},{{\left( {\frac{{b - 1}}{b}} \right)}^{nb - i + 1}}} \right\}} \right\} $
Table 6.  Analytical Results under the PFC Mode
Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{f d}\right)=F\left(z_{f c}\right) $ $ F\left(z_{f d^{\prime}}\right)=F\left(z_{f c}\right) $ $ F\left(z_{f c}\right)=\frac{1}{(1-\eta) b}+\frac{(b-1) \Lambda\left(z_{f c}\right)}{b z_{fc}} $
$ p_{*} $ $ p_{f d}=\frac{b}{b-n} p_{f c} $ $ p_{f d^{\prime}}=\left(\frac{b}{b-1}\right)^{n} p_{f c} $ $ p_{f c}=\frac{b}{b-1} \frac{\left(c+c_{s}+\sum_{i=1}^{n} c_{i}\right) z_{f c}}{z_{f c}-(1-\eta) \Lambda\left(z_{f c}\right)} $
$ q_{*} $ $ q_{f d}=\left(\frac{b-n}{b}\right)^{b} q_{f c} $ $ q_{f d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-\theta)} q_{f c} $ $ q_{f c}=y\left(p_{f c}\right) z_{f c} $
$ w_{i}^{*} $ $ w_i^{fd} = \frac{1}{{b - n}}\left( {c + {c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right) + {c_i} $ $ w_i^{f{d^\prime }} = \frac{{{b^{i - 1}}}}{{{{(b - 1)}^i}}}\left( {c + {c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right) + {c_i} $ $ w_i^{fc} = {\phi ^*}{c_i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{f d}=\frac{b-1}{b}\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{f{c}} $ $ \Pi_{S_{i}}^{f d^{\prime}}=\left(\frac{b-1}{b}\right)^{n b-i+1} \Pi_{S C}^{f_{c}} $ $ \Pi_{S_{i}}^{f c}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{f c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{f d}=\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{f c} $ $ \Pi_{A}^{f d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-1)} \Pi_{S C}^{f c} $ $ \Pi_{A}^{f_{c}}=\phi^{*} \Pi_{S C}^{f c} $
$ \Pi_{S C}^{*} $ $ \Pi _{SC}^{fd} = \left[ {(n + 1) - \frac{n}{b}} \right]{\left( {\frac{{b - n}}{b}} \right)^{b - 1}}\Pi _{SC}^{{fc}} $ $ \Pi _{SC}^{f{d^{\prime}}} = \left[ {b{{\left( {\frac{{b - 1}}{b}} \right)}^{n(b - 1)}} - (b - 1){{\left( {\frac{{b - 1}}{b}} \right)}^{nb}}} \right]\Pi _{SC}^{fc} $ $ \Pi _{SC}^{{fc}} = \frac{1}{{b - 1}}\left( {c + {c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right){q_{fc}} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline{\phi}=\max \left\{\left(\frac{b-n}{b}\right)^{b-1},\left(\frac{b-1}{b}\right)^{n(b-1)}\right\}, \bar{\phi}=\min _{i \in N}\left\{1-\frac{\sum_{i=1}^{n} c_{i}}{c_{i}} \max \left\{\frac{b-1}{b-n}\left(\frac{b-n}{b}\right)^{b},\left(\frac{b-1}{b}\right)^{n b-i+1}\right\}\right\} $
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Decentralized (Equilibrium) strategy Coordination strategy
Suppliers' Simultaneous Actions Suppliers' Sequential Actions
$ F\left(z_{*}\right) $ $ F\left(z_{f d}\right)=F\left(z_{f c}\right) $ $ F\left(z_{f d^{\prime}}\right)=F\left(z_{f c}\right) $ $ F\left(z_{f c}\right)=\frac{1}{(1-\eta) b}+\frac{(b-1) \Lambda\left(z_{f c}\right)}{b z_{fc}} $
$ p_{*} $ $ p_{f d}=\frac{b}{b-n} p_{f c} $ $ p_{f d^{\prime}}=\left(\frac{b}{b-1}\right)^{n} p_{f c} $ $ p_{f c}=\frac{b}{b-1} \frac{\left(c+c_{s}+\sum_{i=1}^{n} c_{i}\right) z_{f c}}{z_{f c}-(1-\eta) \Lambda\left(z_{f c}\right)} $
$ q_{*} $ $ q_{f d}=\left(\frac{b-n}{b}\right)^{b} q_{f c} $ $ q_{f d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-\theta)} q_{f c} $ $ q_{f c}=y\left(p_{f c}\right) z_{f c} $
$ w_{i}^{*} $ $ w_i^{fd} = \frac{1}{{b - n}}\left( {c + {c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right) + {c_i} $ $ w_i^{f{d^\prime }} = \frac{{{b^{i - 1}}}}{{{{(b - 1)}^i}}}\left( {c + {c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right) + {c_i} $ $ w_i^{fc} = {\phi ^*}{c_i} $
$ \Pi_{S_{i}}^{*} $ $ \Pi_{S_{i}}^{f d}=\frac{b-1}{b}\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{f{c}} $ $ \Pi_{S_{i}}^{f d^{\prime}}=\left(\frac{b-1}{b}\right)^{n b-i+1} \Pi_{S C}^{f_{c}} $ $ \Pi_{S_{i}}^{f c}=\frac{c_{i}}{\sum_{i=1}^{n} c_{i}}\left(1-\phi^{*}\right) \Pi_{S C}^{f c} $
$ \Pi_{A}^{*} $ $ \Pi_{A}^{f d}=\left(\frac{b-n}{b}\right)^{b-1} \Pi_{S C}^{f c} $ $ \Pi_{A}^{f d^{\prime}}=\left(\frac{b-1}{b}\right)^{n(b-1)} \Pi_{S C}^{f c} $ $ \Pi_{A}^{f_{c}}=\phi^{*} \Pi_{S C}^{f c} $
$ \Pi_{S C}^{*} $ $ \Pi _{SC}^{fd} = \left[ {(n + 1) - \frac{n}{b}} \right]{\left( {\frac{{b - n}}{b}} \right)^{b - 1}}\Pi _{SC}^{{fc}} $ $ \Pi _{SC}^{f{d^{\prime}}} = \left[ {b{{\left( {\frac{{b - 1}}{b}} \right)}^{n(b - 1)}} - (b - 1){{\left( {\frac{{b - 1}}{b}} \right)}^{nb}}} \right]\Pi _{SC}^{fc} $ $ \Pi _{SC}^{{fc}} = \frac{1}{{b - 1}}\left( {c + {c_s} + \sum\nolimits_{i = 1}^n {{c_i}} } \right){q_{fc}} $
$ \phi^{*} $ - - $ {\phi ^*} \in [\underline \phi ,\bar \phi ] $
note: $ \underline{\phi}=\max \left\{\left(\frac{b-n}{b}\right)^{b-1},\left(\frac{b-1}{b}\right)^{n(b-1)}\right\}, \bar{\phi}=\min _{i \in N}\left\{1-\frac{\sum_{i=1}^{n} c_{i}}{c_{i}} \max \left\{\frac{b-1}{b-n}\left(\frac{b-n}{b}\right)^{b},\left(\frac{b-1}{b}\right)^{n b-i+1}\right\}\right\} $
Table 7.  Parameter Values for Numerical Analysis
Parameters Value
$ c $ Assembly cost (USD/unit) 50
$ c_1 $ $ 1^{\text {st }} $ module cost (USD/unit) 149
$ c_2 $ $ 2^{\text {nd }} $ module cost (USD/unit) 53
$ c_3 $ $ 3^{\text {rd }} $ module cost (USD/unit) 80
$ c_4 $ $ 4^{\text {th }} $ module cost (USD/unit) 60
$ c_e $ Operational cost of the online channel (USD/unit) 26
$ c_s $ Operational cost of offline channel (USD/unit) 39
$ a $ Positive constant number 1E+18
$ b $ Price-elasticity index of the expected demand 5.0
$ \theta $ Mutual fusion coefficient between channels 0.5
$ \eta $ Clearance discount price factor 50%
$ {\lambda _0} $ Market demand share of the online channel 0.6
$ \delta $ The offline channel subsidy factor, and $ 0< \delta< 1 $ 0.1
$ \kappa $ The offline channel discount price factor, and $ \kappa = 1 - \delta $ 0.9
$ h $ the reaction extent of $ \lambda \left( \kappa \right) $ w.r.t. the change of $ \kappa $ 0.5
$ \phi $ Revenue keeping rate 0.7
$ \mu $ Mean value of random factor 100
$ \sigma $ Standard deviation of random factor 10
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Parameters Value
$ c $ Assembly cost (USD/unit) 50
$ c_1 $ $ 1^{\text {st }} $ module cost (USD/unit) 149
$ c_2 $ $ 2^{\text {nd }} $ module cost (USD/unit) 53
$ c_3 $ $ 3^{\text {rd }} $ module cost (USD/unit) 80
$ c_4 $ $ 4^{\text {th }} $ module cost (USD/unit) 60
$ c_e $ Operational cost of the online channel (USD/unit) 26
$ c_s $ Operational cost of offline channel (USD/unit) 39
$ a $ Positive constant number 1E+18
$ b $ Price-elasticity index of the expected demand 5.0
$ \theta $ Mutual fusion coefficient between channels 0.5
$ \eta $ Clearance discount price factor 50%
$ {\lambda _0} $ Market demand share of the online channel 0.6
$ \delta $ The offline channel subsidy factor, and $ 0< \delta< 1 $ 0.1
$ \kappa $ The offline channel discount price factor, and $ \kappa = 1 - \delta $ 0.9
$ h $ the reaction extent of $ \lambda \left( \kappa \right) $ w.r.t. the change of $ \kappa $ 0.5
$ \phi $ Revenue keeping rate 0.7
$ \mu $ Mean value of random factor 100
$ \sigma $ Standard deviation of random factor 10
Table 8.  Numerical Analysis Results for the Omnichannel mode
OMO Mode OMS Mode
Decentralized strategy Coordination strategy Decentralized strategy Coordination strategy
Simultaneous actions Sequential actions Simultaneous actions Sequential actions
$ {z_ * } $ 99 99 99 99 99 99
$ {p_ * } $ 6, 418 1, 949 713 6, 604 2, 005 734
$ {q_ * } $ 731 156, 071 14, 385, 462 825 176, 168 16, 237, 802
$ w_1^ * $ 1, 238 305 104 1, 269 309 104
$ w_2^ * $ 1, 142 253 37 1, 173 259 37
$ w_3^ * $ 1, 169 337 56 1, 200 345 56
$ w_4^ * $ 1, 149 391 42 1, 180 400 42
$ \Pi_{A}^{*} $ 1, 023, 117 66, 336, 309 1, 566, 289, 055 1, 188, 375 77, 051, 259 1, 819, 283, 383
$ \Pi_{S_{1}}^{*} $ 795, 757 24, 275, 793 292, 452, 468 924, 292 28, 196, 932 339, 690, 757
$ \Pi_{S_{2}}^{*} $ 795, 757 31, 211, 734 104, 026, 717 924, 292 36, 253, 199 120, 829, 598
$ \Pi_{S_{3}}^{*} $ 795, 757 40, 129, 372 157, 021, 459 924, 292 46, 611, 256 182, 384, 299
$ \Pi_{S_{4}}^{*} $ 795, 757 51, 594, 907 117, 766, 094 924, 292 59, 928, 757 136, 788, 224
$ \Pi_{S C}^{*} $ 4, 206, 146 213, 548, 114 2, 237, 555, 793 4, 885, 542 248, 041, 403 2, 598, 976, 261
Range of $ \phi^{*} $ [0.029646773, 0.868565973]; set at 0.7 [0.029646773, 0.868565973]; set at 0.7
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OMO Mode OMS Mode
Decentralized strategy Coordination strategy Decentralized strategy Coordination strategy
Simultaneous actions Sequential actions Simultaneous actions Sequential actions
$ {z_ * } $ 99 99 99 99 99 99
$ {p_ * } $ 6, 418 1, 949 713 6, 604 2, 005 734
$ {q_ * } $ 731 156, 071 14, 385, 462 825 176, 168 16, 237, 802
$ w_1^ * $ 1, 238 305 104 1, 269 309 104
$ w_2^ * $ 1, 142 253 37 1, 173 259 37
$ w_3^ * $ 1, 169 337 56 1, 200 345 56
$ w_4^ * $ 1, 149 391 42 1, 180 400 42
$ \Pi_{A}^{*} $ 1, 023, 117 66, 336, 309 1, 566, 289, 055 1, 188, 375 77, 051, 259 1, 819, 283, 383
$ \Pi_{S_{1}}^{*} $ 795, 757 24, 275, 793 292, 452, 468 924, 292 28, 196, 932 339, 690, 757
$ \Pi_{S_{2}}^{*} $ 795, 757 31, 211, 734 104, 026, 717 924, 292 36, 253, 199 120, 829, 598
$ \Pi_{S_{3}}^{*} $ 795, 757 40, 129, 372 157, 021, 459 924, 292 46, 611, 256 182, 384, 299
$ \Pi_{S_{4}}^{*} $ 795, 757 51, 594, 907 117, 766, 094 924, 292 59, 928, 757 136, 788, 224
$ \Pi_{S C}^{*} $ 4, 206, 146 213, 548, 114 2, 237, 555, 793 4, 885, 542 248, 041, 403 2, 598, 976, 261
Range of $ \phi^{*} $ [0.029646773, 0.868565973]; set at 0.7 [0.029646773, 0.868565973]; set at 0.7
Table 9.  Numerical Analysis Results for the Pure Channel Mode
POC Mode PFC Mode
Decentralized strategy Coordination strategy Decentralized strategy Coordination strategy
Simultaneous actions Sequential actions Simultaneous actions Sequential actions
$ {z_ * } $ 98 98 98 98 98 98
$ {p_ * } $ 2, 655 1, 296 531 4, 662 2, 276 932
$ {q_ * } $ 744 26, 812 2, 325, 562 45 1, 606 139, 293
$ w_1^ * $ 567 254 104 883 333 104
$ w_2^ * $ 471 184 37 787 282 37
$ w_3^ * $ 498 243 56 814 367 56
$ w_4^ * $ 478 264 42 794 418 42
$ \Pi_{A}^{*} $ 388, 834 6, 840, 438 170, 114, 826 40, 896 719, 457 17, 892, 172
$ \Pi_{S_{1}}^{*} $ 311, 067 2, 801, 843 31, 763, 295 32, 717 294, 690 3, 340, 769
$ \Pi_{S_{2}}^{*} $ 311, 067 3, 502, 304 11, 298, 353 32, 717 368, 362 1, 188, 327
$ \Pi_{S_{3}}^{*} $ 311, 067 4, 377, 880 17, 054, 118 32, 717 460, 452 1, 793, 701
$ \Pi_{S_{4}}^{*} $ 311, 067 5, 472, 350 12, 790, 588 32, 717 575, 566 1, 345, 276
$ \Pi_{S C}^{*} $ 1, 633, 102 22, 994, 817 243, 021, 181 171, 765 2, 418, 526 25, 560, 245
Range of $ \phi^{*} $ [0.028147498, 0.871647411]; set at 0.7 [0.028147498, 0.871647411]; set at 0.7
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POC Mode PFC Mode
Decentralized strategy Coordination strategy Decentralized strategy Coordination strategy
Simultaneous actions Sequential actions Simultaneous actions Sequential actions
$ {z_ * } $ 98 98 98 98 98 98
$ {p_ * } $ 2, 655 1, 296 531 4, 662 2, 276 932
$ {q_ * } $ 744 26, 812 2, 325, 562 45 1, 606 139, 293
$ w_1^ * $ 567 254 104 883 333 104
$ w_2^ * $ 471 184 37 787 282 37
$ w_3^ * $ 498 243 56 814 367 56
$ w_4^ * $ 478 264 42 794 418 42
$ \Pi_{A}^{*} $ 388, 834 6, 840, 438 170, 114, 826 40, 896 719, 457 17, 892, 172
$ \Pi_{S_{1}}^{*} $ 311, 067 2, 801, 843 31, 763, 295 32, 717 294, 690 3, 340, 769
$ \Pi_{S_{2}}^{*} $ 311, 067 3, 502, 304 11, 298, 353 32, 717 368, 362 1, 188, 327
$ \Pi_{S_{3}}^{*} $ 311, 067 4, 377, 880 17, 054, 118 32, 717 460, 452 1, 793, 701
$ \Pi_{S_{4}}^{*} $ 311, 067 5, 472, 350 12, 790, 588 32, 717 575, 566 1, 345, 276
$ \Pi_{S C}^{*} $ 1, 633, 102 22, 994, 817 243, 021, 181 171, 765 2, 418, 526 25, 560, 245
Range of $ \phi^{*} $ [0.028147498, 0.871647411]; set at 0.7 [0.028147498, 0.871647411]; set at 0.7
Table 10.  Numerical Analysis Results for the Pure Channel Mode
Parameters Original Value $ \pm $ Increment Range
$ \delta $ Offline channel subsidy factor 0.1 0.01 [0, 0.5]
$ b $ Price-elasticity index of the expected demand 5.0 0.01 [4.5, 5.5]
$ \theta $ Mutual fusion coefficient between channels 0.5 0.01 [0.1, 0.9]
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Parameters Original Value $ \pm $ Increment Range
$ \delta $ Offline channel subsidy factor 0.1 0.01 [0, 0.5]
$ b $ Price-elasticity index of the expected demand 5.0 0.01 [4.5, 5.5]
$ \theta $ Mutual fusion coefficient between channels 0.5 0.01 [0.1, 0.9]
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