doi: 10.3934/jimo.2021126
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Inter-organizational contract control of advertising strategies in the supply chain

School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Yafei Zu

Received  September 2020 Revised  May 2021 Early access August 2021

Fund Project: The author is supported by the Youth Fund of the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Grant 20YJC630245), by the Fundamental Research Funds for the Central Universities (Grant 30919013229), by the Youth Fund of the School of Economics and Management, Nanjing University of Science and Technology (Grant JGQN2003), by the National Natural Science Foundation of China (Grant 61901105), and by the Natural Science Foundation of Jiangsu Province (Grant BK20190343)

Advertising has a crucial impact on a product's goodwill. To further improve a product's goodwill and make more profit, member firms in the supply chain use various contracts to coordinate the channel. Considering the dynamic effect of advertising, this paper studies a two-level supply chain consisting of one manufacturer and one retailer. The two members focus on maximizing their profits through advertising and pricing strategies under two types of contracts: the wholesale price contract and the consignment contract. The Stackelberg differential game is introduced, and the optimal advertising effort, wholesale and retail pricing strategies in the two situations are studied. Numerical examples and sensitivity analyses are conducted to explore the models further. The results show that the retailer's revenue proportion and the product's goodwill according to consumers significantly affect the strategies and the contract choice of the partner firms in the supply chain. A proportion of too high or too low revenue may lead to a contract selection conflict between the two partner firms. However, when consumers care more about the product's goodwill, this contract selection conflict can be weakened.

Citation: Yafei Zu. Inter-organizational contract control of advertising strategies in the supply chain. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021126
References:
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E. Adida and N. Ratisoontorn, Consignment contracts with retail competition, European J. Oper. Res., 215 (2011), 136-148.  doi: 10.1016/j.ejor.2011.05.059.  Google Scholar

[2]

N. AmroucheG. Martín-Herrán and G. Zaccour, Pricing and advertising of private and national brands in a dynamic marketing channel, J. Optim. Theory Appl., 137 (2008), 465-483.  doi: 10.1007/s10957-007-9340-8.  Google Scholar

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T. AvinadavT. Chernonog and Y. Perlman, Consignment contract for mobile apps between a single retailer and competitive developers with different risk attitudes, European J. Oper. Res., 246 (2015), 949-957.  doi: 10.1016/j.ejor.2015.05.016.  Google Scholar

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T. AvinadavT. Chernonog and Y. Perlman, Mergers and acquisitions between risk-averse parties, European J. Oper. Res., 259 (2017), 926-934.  doi: 10.1016/j.ejor.2016.11.030.  Google Scholar

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M. Ben-DayaE. HassiniM. Hariga and M. M. AlDurgam, Consignment and vendor managed inventory in single-vendor multiple buyers supply chains, Int. J. Prod. Res., 51 (2013), 1347-1365.  doi: 10.1080/00207543.2012.662725.  Google Scholar

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S. C. Hackett, Consignment contracting, J. Econ. Behav. Organ., 20 (1993), 247-253.  doi: 10.1016/0167-2681(93)90093-5.  Google Scholar

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X. He, A. Prasad and S. P. Sethi, Cooperative advertising and pricing in a dynamic stochastic supply chain: Feedback Stackelberg strategies, PICMET '08 - 2008 Portland International Conference on Management of Engineering & Technology, (2008), 1634–1649. doi: 10.1109/PICMET.2008.4599783.  Google Scholar

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W. HuY. Li and W. Wang, Benefit and risk analysis of consignment contracts, Ann. Oper. Res., 257 (2017), 641-659.  doi: 10.1007/s10479-015-1919-0.  Google Scholar

[26]

J. Huang, M. Leng and L. Liang, Recent developments in dynamic advertising research, European J. Oper. Res., 220 (2012), 591-609. doi: 10.1016/j.ejor.2012.02.031.  Google Scholar

[27]

S. Jørgensen and G. Zaccour, A survey of game-theoretic models of cooperative advertising, European J. Oper. Res., 237 (2014), 1-14.  doi: 10.1016/j.ejor.2013.12.017.  Google Scholar

[28]

S. Karray, Cooperative promotions in the distribution channel, Omega, 51 (2015), 49-58.  doi: 10.1016/j.omega.2014.07.009.  Google Scholar

[29]

S. Karray and G. Martín-Herrán, A dynamic model for advertising and pricing competition between national and store brands, European J. Oper. Res., 193 (2009), 451-467.  doi: 10.1016/j.ejor.2007.11.043.  Google Scholar

[30]

S. Li and Z. Hua, A note on channel performance under consignment contract with revenue sharing, European J. Oper. Res., 184 (2008), 793-796.  doi: 10.1016/j.ejor.2006.11.003.  Google Scholar

[31]

S. LiZ. Zhu and L. Huang, Supply chain coordination and decision making under consignment contract with revenue sharing, Int. J. Prod. Econ., 120 (2009), 88-99.  doi: 10.1016/j.ijpe.2008.07.015.  Google Scholar

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F. LuJ. Zhang and W. Tang, Wholesale price contract versus consignment contract in a supply chain considering dynamic advertising, Int. Trans. Oper. Res., 26 (2019), 1977-2003.  doi: 10.1111/itor.12388.  Google Scholar

[33]

L. LuQ. GouW. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, Int. J. Prod. Res., 54 (2016), 5250-5270.  doi: 10.1080/00207543.2016.1165878.  Google Scholar

[34]

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G. Martín-Herrán and S. Taboubi, Price coordination in distribution channels: A dynamic perspective, European J. Oper. Res., 240 (2015), 401-414.  doi: 10.1016/j.ejor.2014.07.022.  Google Scholar

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

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J. Wang and H. Shin, The impact of contracts and competition on upstream innovation in a supply chain, Prod. Oper. Manag., 24 (2015), 134-146.  doi: 10.1111/poms.12218.  Google Scholar

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W. XueY. Hu and Z. Chen, The value of buy-back contract under price competition, J. Clean. Prod., 27 (2019), 2679-2694.   Google Scholar

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J. YueJ. AustinZ. Huang and B. Chen, Pricing and advertisement in a manufacturer-retailer supply chain, European J. Oper. Res., 231 (2013), 492-502.  doi: 10.1016/j.ejor.2013.06.007.  Google Scholar

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Y. ZhangK. Donohue and T. H. Cui, Contract preferences and performance for the loss-averse supplier: Buy-back vs. revenue sharing, Manag. Sci., 62 (2016), 1734-1754.  doi: 10.1287/mnsc.2015.2182.  Google Scholar

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J. ZhangQ. GouL. Liang and Z. Huang, Supply chain coordination through cooperative advertising with reference price effect, Omega, 41 (2013), 345-353.  doi: 10.1016/j.omega.2012.03.009.  Google Scholar

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S. ZhangJ. ZhangJ. Shen and W. Tang, A joint dynamic pricing and production model with asymmetric reference price effect, J. Ind. Manag. Optim., 15 (2019), 667-688.  doi: 10.3934/jimo.2018064.  Google Scholar

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J. ZhaoY.-W. ZhouZ.-H. Cao and J. Min, The shelf space and pricing strategies for a retailer-dominated supply chain with consignment-based revenue sharing contracts, European J. Oper. Res., 280 (2020), 926-939.  doi: 10.1016/j.ejor.2019.07.074.  Google Scholar

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show all references

References:
[1]

E. Adida and N. Ratisoontorn, Consignment contracts with retail competition, European J. Oper. Res., 215 (2011), 136-148.  doi: 10.1016/j.ejor.2011.05.059.  Google Scholar

[2]

N. AmroucheG. Martín-Herrán and G. Zaccour, Pricing and advertising of private and national brands in a dynamic marketing channel, J. Optim. Theory Appl., 137 (2008), 465-483.  doi: 10.1007/s10957-007-9340-8.  Google Scholar

[3]

T. AvinadavT. Chernonog and Y. Perlman, Consignment contract for mobile apps between a single retailer and competitive developers with different risk attitudes, European J. Oper. Res., 246 (2015), 949-957.  doi: 10.1016/j.ejor.2015.05.016.  Google Scholar

[4]

T. AvinadavT. Chernonog and Y. Perlman, Mergers and acquisitions between risk-averse parties, European J. Oper. Res., 259 (2017), 926-934.  doi: 10.1016/j.ejor.2016.11.030.  Google Scholar

[5]

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, Int. J. Prod. Econ., 187 (2017), 85-101.  doi: 10.1016/j.ijpe.2017.02.012.  Google Scholar

[6]

Q. BaiM. Jin and X. Xu, Effects of carbon emission reduction on supply chain coordination with vendor-managed deteriorating product inventory, Int. J. Prod. Res., 208 (2019), 83-99.   Google Scholar

[7]

M. Ben-DayaE. HassiniM. Hariga and M. M. AlDurgam, Consignment and vendor managed inventory in single-vendor multiple buyers supply chains, Int. J. Prod. Res., 51 (2013), 1347-1365.  doi: 10.1080/00207543.2012.662725.  Google Scholar

[8]

W. H. Bolen, Contemporary Retailing, Prentice Hall, 1982. Google Scholar

[9]

A. BurattoR. Cesaretto and P. De Giovanni, Consignment contracts with cooperative programs and price discount mechanisms in a dynamic supply chain, Int. J. Prod. Res., 218 (2019), 72-82.  doi: 10.1016/j.ijpe.2019.04.027.  Google Scholar

[10]

G. P. Cachon, The allocation of inventory risk in a supply chain: Push, pull, and advance-purchase discount contracts, Manag. Sci., 50 (2004), 222-238.  doi: 10.1287/mnsc.1030.0190.  Google Scholar

[11]

G. P. Cachon, Supply chain coordination with contracts, Handbooks in Operations Research and Management Science, 11 (2003), 227-339.  doi: 10.1016/S0927-0507(03)11006-7.  Google Scholar

[12]

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

[13]

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

[14]

K.-S. ChoiJ. G. Dai and J.-S. Song, On measuring supplier performance under vendor-managed-inventory programs in capacitated supply chains, Manuf. Serv. Oper. Manag., 6 (2004), 53-72.  doi: 10.1287/msom.1030.0029.  Google Scholar

[15]

R. Dai and J. Zhang, Green process innovation and differentiated pricing strategies with environmental concerns of South-North markets, Transportation Research Part E: Logistics and Transportation Review, 98 (2017), 132-150.  doi: 10.1016/j.tre.2016.12.009.  Google Scholar

[16]

R. DaiJ. Zhang and W. Tang, Cartelization or Cost-sharing? Comparison of cooperation modes in a green supply chain, J. Clean. Prod., 156 (2017), 159-173.  doi: 10.1016/j.jclepro.2017.04.011.  Google Scholar

[17]

P. De GiovanniS. Karray and G. Martín-Herrán, Vendor management inventory with consignment contracts and the benefits of cooperative advertising, European J. Oper. Res., 272 (2019), 465-480.  doi: 10.1016/j.ejor.2018.06.031.  Google Scholar

[18]

P. De Giovanni and M. Roselli, Overcoming the drawbacks of a revenue-sharing contract through a support program, Ann. Oper. Res., 196 (2012), 201-222.  doi: 10.1007/s10479-012-1113-6.  Google Scholar

[19]

Y. Dong and K. Xu, A supply chain model of vendor managed inventory, Transportation Research Part E: Logistics and Transportation Review, 38 (2002), 75-95.  doi: 10.1016/S1366-5545(01)00014-X.  Google Scholar

[20]

F. El Ouardighi and B. Kim, Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition, European J. Oper. Res., 206 (2010), 329-340.  doi: 10.1016/j.ejor.2010.02.035.  Google Scholar

[21]

G. FeichtingerR. F. Hartl and S. P. Sethi, Dynamic optimal control models in advertising: Recent developments, Manag. Sci., 40 (1994), 195-226.  doi: 10.1287/mnsc.40.2.195.  Google Scholar

[22]

S. C. Hackett, Consignment contracting, J. Econ. Behav. Organ., 20 (1993), 247-253.  doi: 10.1016/0167-2681(93)90093-5.  Google Scholar

[23]

X. He, A. Prasad and S. P. Sethi, Cooperative advertising and pricing in a dynamic stochastic supply chain: Feedback Stackelberg strategies, PICMET '08 - 2008 Portland International Conference on Management of Engineering & Technology, (2008), 1634–1649. doi: 10.1109/PICMET.2008.4599783.  Google Scholar

[24]

J. HeydariK. Govindan and Z. Basiri, Balancing price and green quality in presence of consumer environmental awareness: A green supply chain coordination approach, Int. J. Prod. Res., 59 (2021), 1957-1975.  doi: 10.1080/00207543.2020.1771457.  Google Scholar

[25]

W. HuY. Li and W. Wang, Benefit and risk analysis of consignment contracts, Ann. Oper. Res., 257 (2017), 641-659.  doi: 10.1007/s10479-015-1919-0.  Google Scholar

[26]

J. Huang, M. Leng and L. Liang, Recent developments in dynamic advertising research, European J. Oper. Res., 220 (2012), 591-609. doi: 10.1016/j.ejor.2012.02.031.  Google Scholar

[27]

S. Jørgensen and G. Zaccour, A survey of game-theoretic models of cooperative advertising, European J. Oper. Res., 237 (2014), 1-14.  doi: 10.1016/j.ejor.2013.12.017.  Google Scholar

[28]

S. Karray, Cooperative promotions in the distribution channel, Omega, 51 (2015), 49-58.  doi: 10.1016/j.omega.2014.07.009.  Google Scholar

[29]

S. Karray and G. Martín-Herrán, A dynamic model for advertising and pricing competition between national and store brands, European J. Oper. Res., 193 (2009), 451-467.  doi: 10.1016/j.ejor.2007.11.043.  Google Scholar

[30]

S. Li and Z. Hua, A note on channel performance under consignment contract with revenue sharing, European J. Oper. Res., 184 (2008), 793-796.  doi: 10.1016/j.ejor.2006.11.003.  Google Scholar

[31]

S. LiZ. Zhu and L. Huang, Supply chain coordination and decision making under consignment contract with revenue sharing, Int. J. Prod. Econ., 120 (2009), 88-99.  doi: 10.1016/j.ijpe.2008.07.015.  Google Scholar

[32]

F. LuJ. Zhang and W. Tang, Wholesale price contract versus consignment contract in a supply chain considering dynamic advertising, Int. Trans. Oper. Res., 26 (2019), 1977-2003.  doi: 10.1111/itor.12388.  Google Scholar

[33]

L. LuQ. GouW. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, Int. J. Prod. Res., 54 (2016), 5250-5270.  doi: 10.1080/00207543.2016.1165878.  Google Scholar

[34]

B. Lus and A. Muriel, Measuring the impact of increased product substitution on pricing and capacity decisions under linear demand models, Prod. Oper. Manag., 18 (2009), 95-113.  doi: 10.1111/j.1937-5956.2009.01001.x.  Google Scholar

[35]

G. Martín-Herrán and S. P. Sigué, Manufacturer defensive and offensive advertising in competing distribution channels, Int. Trans. Oper. Res., 27 (2020), 958-983.  doi: 10.1111/itor.12693.  Google Scholar

[36]

G. Martín-Herrán and S. Taboubi, Price coordination in distribution channels: A dynamic perspective, European J. Oper. Res., 240 (2015), 401-414.  doi: 10.1016/j.ejor.2014.07.022.  Google Scholar

[37]

M. Nerlove and K. Arrow, Optimal advertising policy under dynamic conditions, In: Mathematical Models in Marketing. Lecture Notes in Economics and Mathematical Systems (Operations Research), vol 132. Springer, Berlin, Heidelberg. (1976), 167–168. doi: 10.1007/978-3-642-51565-1_54.  Google Scholar

[38]

Annual Shampoo Market Overview and Trend Insight Report, 2020. Available from: http://yu-fa.net. Google Scholar

[39]

R. ShiJ. Zhang and J. Ru, Impacts of power structure on supply chains with uncertain demand, Prod. Oper. Manag., 22 (2013), 1232-1249.  doi: 10.1111/poms.12002.  Google Scholar

[40]

S. P. Sigué and P. Chintagunta, Advertising strategies in a franchise system, European J. Oper. Res., 198 (2009), 655-665.  doi: 10.1016/j.ejor.2008.09.027.  Google Scholar

[41]

N. Singh and X. Vives, Price and quantity competition in a differentiated duopoly, Rand J. Econ., 15 (1984), 546-554.   Google Scholar

[42]

A. R. Thomas and T. J. Wilkinson, The Distribution Trap: Keeping Your Innovations from Becoming Commodities: Keeping Your Innovations from Becoming Commodities, ABC-CLIO, 2009. Google Scholar

[43]

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

[44]

J. Wang and H. Shin, The impact of contracts and competition on upstream innovation in a supply chain, Prod. Oper. Manag., 24 (2015), 134-146.  doi: 10.1111/poms.12218.  Google Scholar

[45]

W. XueY. Hu and Z. Chen, The value of buy-back contract under price competition, J. Clean. Prod., 27 (2019), 2679-2694.   Google Scholar

[46]

J. YueJ. AustinZ. Huang and B. Chen, Pricing and advertisement in a manufacturer-retailer supply chain, European J. Oper. Res., 231 (2013), 492-502.  doi: 10.1016/j.ejor.2013.06.007.  Google Scholar

[47]

G. Zaccour, On the coordination of dynamic marketing channels and two-part tariffs, Automatica J. IFAC, 44 (2008), 1233-1239.  doi: 10.1016/j.automatica.2007.10.009.  Google Scholar

[48]

Y. ZhangK. Donohue and T. H. Cui, Contract preferences and performance for the loss-averse supplier: Buy-back vs. revenue sharing, Manag. Sci., 62 (2016), 1734-1754.  doi: 10.1287/mnsc.2015.2182.  Google Scholar

[49]

J. ZhangQ. GouL. Liang and Z. Huang, Supply chain coordination through cooperative advertising with reference price effect, Omega, 41 (2013), 345-353.  doi: 10.1016/j.omega.2012.03.009.  Google Scholar

[50]

J. ZhangL. LeiS. Zhang and L. Song, Dynamic vs. static pricing in a supply chain with advertising, Comput. Ind. Eng., 109 (2017), 266-279.  doi: 10.1016/j.cie.2017.05.006.  Google Scholar

[51]

S. ZhangJ. ZhangJ. Shen and W. Tang, A joint dynamic pricing and production model with asymmetric reference price effect, J. Ind. Manag. Optim., 15 (2019), 667-688.  doi: 10.3934/jimo.2018064.  Google Scholar

[52]

J. ZhaoY.-W. ZhouZ.-H. Cao and J. Min, The shelf space and pricing strategies for a retailer-dominated supply chain with consignment-based revenue sharing contracts, European J. Oper. Res., 280 (2020), 926-939.  doi: 10.1016/j.ejor.2019.07.074.  Google Scholar

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Figure 1.  The overall mechanism
Figure 2.  Comparison of situations with a different $ \varphi $
Table 1.  Notations and definitions
Notation Definition
Decision variables
$ A(t) $ Manufacturer's advertising effort for the product, $ A(t)>0 $.
$ s(t) $ Retailer's markup (i.e., retail margin) under the wholesale
price contract, $ s(t)>0 $.
$ w(t) $ Wholesale price of the product under the wholesale price contract,
$ w(t)>0 $.
$ p(t) $ Retail price of the product under the consignment contract, $ p(t)> 0 $.
Parameters and other variables
$ \alpha $ The coefficient associated with the product's retail price in the
demand function, $ \alpha> 0 $.
$ \beta $ The coefficient associated with the product's goodwill in the
demand function, $ \beta>0 $.
$ \theta $ The coefficient associated with the manufacturer's advertising effort
in the product goodwill function, $ \theta>0 $.
$ \varphi $ Retailer's proportion of sales revenue under the consignment contract,
$ 0< \varphi< 1 $.
$ \delta $ The decay rate of the product's goodwill, $ \delta> 0 $.
$ \mu $ Cost parameter associated with the manufacturer's advertising effort,
$ \mu>0 $.
$ D(t) $ Demand for the product at time $ t $, with initial demand $ D_0> 0 $.
$ G(t) $ Goodwill of the product at time $ t $, with $ G_{0}\geq 0 $.
$ J_R, J_M $ Objective functions (expressed in net profit) of the retailer and
the manufacturer, respectively, for $ t \in [0, +\infty) $.
Notation Definition
Decision variables
$ A(t) $ Manufacturer's advertising effort for the product, $ A(t)>0 $.
$ s(t) $ Retailer's markup (i.e., retail margin) under the wholesale
price contract, $ s(t)>0 $.
$ w(t) $ Wholesale price of the product under the wholesale price contract,
$ w(t)>0 $.
$ p(t) $ Retail price of the product under the consignment contract, $ p(t)> 0 $.
Parameters and other variables
$ \alpha $ The coefficient associated with the product's retail price in the
demand function, $ \alpha> 0 $.
$ \beta $ The coefficient associated with the product's goodwill in the
demand function, $ \beta>0 $.
$ \theta $ The coefficient associated with the manufacturer's advertising effort
in the product goodwill function, $ \theta>0 $.
$ \varphi $ Retailer's proportion of sales revenue under the consignment contract,
$ 0< \varphi< 1 $.
$ \delta $ The decay rate of the product's goodwill, $ \delta> 0 $.
$ \mu $ Cost parameter associated with the manufacturer's advertising effort,
$ \mu>0 $.
$ D(t) $ Demand for the product at time $ t $, with initial demand $ D_0> 0 $.
$ G(t) $ Goodwill of the product at time $ t $, with $ G_{0}\geq 0 $.
$ J_R, J_M $ Objective functions (expressed in net profit) of the retailer and
the manufacturer, respectively, for $ t \in [0, +\infty) $.
Table 2.  Discussion of the function
Parameter condition Condition of $ \varphi $1 Sign of $ y $
$ bc>ad $ $ \varphi< -b/a $ or $ \varphi> - d/c $ $ y>0 $
$ -b/a< \varphi< -d/c $ $ y<0 $
$ \varphi = -b/a $ $ y=0 $
$ bc<ad $ $ \varphi< -d/c $ or $ \varphi> - b/a $ $ y>0 $
$ -d/c< \varphi< -b/a $ $ y<0 $
$ \varphi = -b/a $ $ y=0 $
$ bc=ad $ Arbitrary $ y=a/c>0 $
1 Parameter $\varphi$ needs to meet the condition $\varphi \in (0,1)$ first to ensure the practical implications.
Parameter condition Condition of $ \varphi $1 Sign of $ y $
$ bc>ad $ $ \varphi< -b/a $ or $ \varphi> - d/c $ $ y>0 $
$ -b/a< \varphi< -d/c $ $ y<0 $
$ \varphi = -b/a $ $ y=0 $
$ bc<ad $ $ \varphi< -d/c $ or $ \varphi> - b/a $ $ y>0 $
$ -d/c< \varphi< -b/a $ $ y<0 $
$ \varphi = -b/a $ $ y=0 $
$ bc=ad $ Arbitrary $ y=a/c>0 $
1 Parameter $\varphi$ needs to meet the condition $\varphi \in (0,1)$ first to ensure the practical implications.
Table 3.  Sensitivity analysis: wholesale price contract
$ p(0) $ $ p(T) $ $ w(0) $ $ w(T) $ $ A(0) $ $ A(T) $ $ J_R $ $ J_M $
$ \alpha $ 0.8 9.3750 9.5367 3.1250 3.1789 1.0427 0.0000 16.0238 7.8110
0.7 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
0.6 12.5003 12.7892 4.1667 4.2630 1.3959 0.0000 21.5476 10.4131
$ \beta $ 0.6 10.7143 11.0215 3.5714 3.6738 1.4413 0.0000 18.6181 8.9239
0.5 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
0.4 10.7143 10.8489 3.5714 3.6163 0.9502 0.0000 18.1885 8.9277
$ \theta $ 0.9 10.7143 10.9834 3.5714 3.6611 1.3479 0.0000 18.5230 8.9250
0.8 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
0.7 10.7143 10.8756 3.5714 3.6252 1.0411 0.0000 18.2548 8.9273
$ \delta $ 0.6 10.7143 10.8901 3.5714 3.6300 1.0869 0.0000 18.3113 8.9269
0.5 10.7143 10.9070 3.5714 3.6357 1.1385 0.0000 18.3439 8.9267
0.4 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
$ G_0 $ 0 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
5 13.3929 12.7621 4.4643 4.2540 1.4428 0.0000 26.7325 12.9890
10 16.0714 14.5983 5.3571 4.8661 1.6918 0.0000 36.6622 17.8197
$ T $ 1 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
2 10.7143 11.3267 3.5714 3.7756 2.0493 0.0000 39.0824 17.8136
3 10.7143 11.7422 3.5714 3.9141 2.6798 0.0000 62.9809 26.5708
$ p(0) $ $ p(T) $ $ w(0) $ $ w(T) $ $ A(0) $ $ A(T) $ $ J_R $ $ J_M $
$ \alpha $ 0.8 9.3750 9.5367 3.1250 3.1789 1.0427 0.0000 16.0238 7.8110
0.7 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
0.6 12.5003 12.7892 4.1667 4.2630 1.3959 0.0000 21.5476 10.4131
$ \beta $ 0.6 10.7143 11.0215 3.5714 3.6738 1.4413 0.0000 18.6181 8.9239
0.5 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
0.4 10.7143 10.8489 3.5714 3.6163 0.9502 0.0000 18.1885 8.9277
$ \theta $ 0.9 10.7143 10.9834 3.5714 3.6611 1.3479 0.0000 18.5230 8.9250
0.8 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
0.7 10.7143 10.8756 3.5714 3.6252 1.0411 0.0000 18.2548 8.9273
$ \delta $ 0.6 10.7143 10.8901 3.5714 3.6300 1.0869 0.0000 18.3113 8.9269
0.5 10.7143 10.9070 3.5714 3.6357 1.1385 0.0000 18.3439 8.9267
0.4 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
$ G_0 $ 0 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
5 13.3929 12.7621 4.4643 4.2540 1.4428 0.0000 26.7325 12.9890
10 16.0714 14.5983 5.3571 4.8661 1.6918 0.0000 36.6622 17.8197
$ T $ 1 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264
2 10.7143 11.3267 3.5714 3.7756 2.0493 0.0000 39.0824 17.8136
3 10.7143 11.7422 3.5714 3.9141 2.6798 0.0000 62.9809 26.5708
Table 4.  Sensitivity analysis: consignment contract
$ p(0) $ $ p(T) $ $ w(0) $ $ w(T) $ $ A(0) $ $ A(T) $ $ J_R $ $ J_M $
$ \alpha $ 0.8 6.2500 6.4241 5.0000 5.1393 1.6806 0.0000 6.5087 25.5112
0.7 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.6 8.3333 8.6454 6.6667 6.9164 2.2555 0.0000 8.7988 34.2495
$ \beta $ 0.6 7.1429 7.4757 5.7143 5.9805 2.3344 0.0000 7.6410 29.5481
0.5 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.4 7.1429 7.2874 5.7143 5.8299 1.5285 0.0000 7.3570 28.9960
$ \theta $ 0.9 7.1429 7.4339 5.7143 5.9471 2.1798 0.0000 7.5775 29.4257
0.8 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.7 7.1429 7.3164 5.7143 5.8531 1.6765 0.0000 7.4004 29.0811
$ \delta $ 0.6 7.1429 7.3321 5.7143 5.8657 1.7515 0.0000 7.4374 29.1536
0.5 7.1429 7.3505 5.7143 5.8804 1.8358 0.0000 7.4589 29.1955
0.4 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
$ G_0 $ 0 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
5 8.9286 8.6119 7.1429 6.8896 2.3277 0.0000 10.8800 42.5342
10 10.7143 9.8528 8.5714 7.8823 2.7293 0.0000 14.9184 58.3372
$ T $ 1 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
2 7.1429 7.8186 5.7143 6.2549 3.3651 0.0000 16.5499 61.4808
3 7.1429 8.3091 5.7143 6.6473 4.4927 0.0000 28.0220 97.9091
$ \varphi $ 0.2 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.4 7.1429 7.3128 4.2857 4.3877 1.4365 0.0000 14.7899 21.8029
0.6 7.1429 7.2553 2.8571 2.9021 0.9523 0.0000 21.9276 14.4509
$ p(0) $ $ p(T) $ $ w(0) $ $ w(T) $ $ A(0) $ $ A(T) $ $ J_R $ $ J_M $
$ \alpha $ 0.8 6.2500 6.4241 5.0000 5.1393 1.6806 0.0000 6.5087 25.5112
0.7 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.6 8.3333 8.6454 6.6667 6.9164 2.2555 0.0000 8.7988 34.2495
$ \beta $ 0.6 7.1429 7.4757 5.7143 5.9805 2.3344 0.0000 7.6410 29.5481
0.5 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.4 7.1429 7.2874 5.7143 5.8299 1.5285 0.0000 7.3570 28.9960
$ \theta $ 0.9 7.1429 7.4339 5.7143 5.9471 2.1798 0.0000 7.5775 29.4257
0.8 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.7 7.1429 7.3164 5.7143 5.8531 1.6765 0.0000 7.4004 29.0811
$ \delta $ 0.6 7.1429 7.3321 5.7143 5.8657 1.7515 0.0000 7.4374 29.1536
0.5 7.1429 7.3505 5.7143 5.8804 1.8358 0.0000 7.4589 29.1955
0.4 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
$ G_0 $ 0 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
5 8.9286 8.6119 7.1429 6.8896 2.3277 0.0000 10.8800 42.5342
10 10.7143 9.8528 8.5714 7.8823 2.7293 0.0000 14.9184 58.3372
$ T $ 1 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
2 7.1429 7.8186 5.7143 6.2549 3.3651 0.0000 16.5499 61.4808
3 7.1429 8.3091 5.7143 6.6473 4.4927 0.0000 28.0220 97.9091
$ \varphi $ 0.2 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415
0.4 7.1429 7.3128 4.2857 4.3877 1.4365 0.0000 14.7899 21.8029
0.6 7.1429 7.2553 2.8571 2.9021 0.9523 0.0000 21.9276 14.4509
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