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September  2021, 17(5): 2837-2853. doi: 10.3934/jimo.2020097

A dual-channel supply chain problem with resource-utilization penalty: Who can benefit from sales effort?

1. 

School of Management, Shanghai University, Shanghai 200444, China

2. 

School of Economics and Management, Tongji University, Shanghai, 200092, China

3. 

Edward P.Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695-7906, USA

* Corresponding author: Shu-Cherng Fang

Received  October 2019 Revised  February 2020 Published  September 2021 Early access  May 2020

Fund Project: This work is supported by National Natural Science Foundation of China (71502100, 71671125) and Humanities and Social Sciences Foundation of the Chinese Ministry of Education (20YJAZH135)

As manufacturers may engage in both direct sale and wholesale, the channel conflict between manufacturer and retailer becomes inevitable. This paper considers a dual-channel supply chain in which a retailer sells the product through store channel with sales effort while the manufacturer holds a direct channel and may provide an incentive measure to share the cost of sales effort. To meet social responsibility, a penalty on the total resource consumed is imposed on the manufacturer. We present a manufacturer-led decentralized model in which both members maximize individual profit, and then derive the corresponding optimal direct/store price and wholesale price. The dual-channel supply chain model without sales effort policy is also considered so as to explain the effects of sales effort policy and sharing cost measure on both parties. Special properties are presented to show (ⅰ) the influence of retailer's sales effort and manufacturer's sharing cost on the optimal strategies; (ⅱ) the resource-utilized penalty on the optimal decisions. Finally, numerical experiments are conducted to highlight the influence of various parameters on optimal solutions. We find that if the market response to retailer's sales effort is strong or the manufacturer's sharing portion of sales effort cost is increased, the retailer's profit and store selling price increase while the manufacturer's profit decreases and the direct sale and wholesale prices do not change. We also show that if the consumer's value on direct channel exceeds a threshold, the manufacturer's profit will be greater than that of the retailer. Moreover, if the market response to retailer's sales effort is strong, manufacturer's profit will be lesser than retailer's profit.

Citation: Lianxia Zhao, Jianxin You, Shu-Cherng Fang. A dual-channel supply chain problem with resource-utilization penalty: Who can benefit from sales effort?. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2837-2853. doi: 10.3934/jimo.2020097
References:
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A. AtasuV. D. R. Guide and L. N. Van Wassenhove, Product reuse economics in closed-loop supply chain research, Prod. Oper. Manag., 17 (2008), 483-496.  doi: 10.3401/poms.1080.0051.

[2]

A. Atasu and G. C. Souza, How does product recovery affect quality choice?, Prod. Oper. Manag., 22 (2013), 991-1010.  doi: 10.1111/j.1937-5956.2011.01290.x.

[3]

A. Atasu and R. Subramanian, Extended producer responsibility for e-waste: Individual or collective producer responsibility?, Prod. Oper. Manag., 21 (2012), 1042-1059.  doi: 10.1111/j.1937-5956.2012.01327.x.

[4]

S. Bernard, North–south trade in reusable goods: Green design meets illegal shipments of waste, J. Environmental Econ. Manag., 69 (2015), 22-35.  doi: 10.1016/j.jeem.2014.10.004.

[5]

E. Brouillat and V. Oltra, Extended producer responsibility instruments and innovation in eco-design: An exploration through a simulation model, Ecological Econ., 83 (2012), 236-245.  doi: 10.1016/j.ecolecon.2012.07.007.

[6]

B. Chen and J. Chen, When to introduce an online channel, and offer money back guarantees and personalized pricing?, European J. Oper. Res., 257 (2017), 614-624.  doi: 10.1016/j.ejor.2016.07.031.

[7]

X. ChenX. Wang and X. Jiang, The impact of power structure on the retail service supply chain with an O2O mixed channel, J. Oper. Res. Soc., 67 (2016), 294-301.  doi: 10.1057/jors.2015.6.

[8]

T. ChernonogT. Avinadav and T. Ben-Zvi, Pricing and sales-effort investment under bi-criteria in a supply chain of virtual products involving risk, European J. Oper. Res., 246 (2015), 471-475.  doi: 10.1016/j.ejor.2015.05.024.

[9]

W. K. ChiangD. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Manag. Sci., 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.

[10]

B. DanG. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, Internat. J. Prod. Econ., 139 (2012), 312-320.  doi: 10.1016/j.ijpe.2012.05.014.

[11]

B. Dan, S. Zhang and M. Zhou, Strategies for warranty service in a dual-channel supply chain with value-added service competition, Internat. J. Prod. Res., 56 (2018) 5677–5699. doi: 10.1080/00207543.2017.1377355.

[12]

Q. DingC. Dong and Z. Pan, A hierarchical pricing decision process on a dual-channel problem with one manufacturer and one retailer, Internat. J. Prod. Econ., 175 (2016), 197-212.  doi: 10.1016/j.ijpe.2016.02.014.

[13]

G. EsenduranE. Kemahlıoğlu-Ziya and J. M. Swaminathan, Impact of take-back regulation on the remanufacturing industry, Prod. Oper. Manag., 26 (2017), 924-944.  doi: 10.1111/poms.12673.

[14]

M. E. Ferguson and L. B. Toktay, The effect of competition on recovery strategies, Prod. Oper. Manag., 15 (2006), 351-368.  doi: 10.1111/j.1937-5956.2006.tb00250.x.

[15]

J. GaoH. HanL. Hou and H. Wang, Pricing and effort decisions in a closed-loop supply chain under different channel power structures, J. Cleaner Production, 112 (2016), 2043-2057.  doi: 10.1016/j.jclepro.2015.01.066.

[16]

D. Ghosh and J. Shah, A comparative analysis of greening policies across supply chain structures, Internat. J. Prod. Econ., 135 (2012), 568-583.  doi: 10.1016/j.ijpe.2011.05.027.

[17]

J. J. KacenJ. D. Hess and W. K. Chiang, Bricks or clicks? Consumer attitudes toward traditional stores and online stores, Global Econ. Manag. Rev., 18 (2013), 12-21.  doi: 10.1016/S2340-1540(13)70003-3.

[18]

H. Ke and J. Liu, Dual-channel supply chain competition with channel preference and sales effort under uncertain environment, J. Ambient Intell. Humanized Comput., 8 (2017), 781-795.  doi: 10.1007/s12652-017-0502-8.

[19]

B. LiP.-W. HouP. Chen and Q.-H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, Internat. J. Prod. Econ., 178 (2016), 154-168.  doi: 10.1016/j.ijpe.2016.05.010.

[20]

G. Martín-Herrán and S. P. Sigué, Prices, promotions, and channel profitability: Was the conventional wisdom mistaken?, European J. Oper. Res., 211 (2011), 415-425.  doi: 10.1016/j.ejor.2010.12.022.

[21]

K. Matsui, When should a manufacturer set its direct price and wholesale price in dual-channel supply chains?, European J. Oper. Res., 258 (2017), 501-511.  doi: 10.1016/j.ejor.2016.08.048.

[22]

B. NiuQ. Cui and J. Zhang, Impact of channel power and fairness concern on supplier's market entry decision, J. Oper. Res. Soc., 68 (2017), 1570-1581.  doi: 10.1057/s41274-016-0169-0.

[23]

A. Ovchinnikov, Revenue and cost management for remanufactured products, Prod. Oper. Manag., 20 (2011), 824-840.  doi: 10.1111/j.1937-5956.2010.01214.x.

[24]

X. PuL. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Res. Appl., 22 (2017), 1-12.  doi: 10.1016/j.elerap.2016.11.002.

[25]

B. Rodríguez and G. Aydın, Pricing and assortment decisions for a manufacturer selling through dual channels, European J. Oper. Res., 242 (2015), 901-909.  doi: 10.1016/j.ejor.2014.10.047.

[26]

T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects, Manag. Sci., 48 (2002), 992-1007.  doi: 10.1287/mnsc.48.8.992.168.

[27]

W. WangY. ZhangY. LiX. Zhao and M. Cheng, Closed-loop supply chains under reward-penalty mechanism: Retailer collection and asymmetric information, J. Cleaner Prod., 142 (2017), 3938-3955.  doi: 10.1016/j.jclepro.2016.10.063.

[28]

D. Xing and T. Liu, Sales effort free riding and coordination with price match and channel rebate, European J. Oper. Res., 219 (2012), 264-271.  doi: 10.1016/j.ejor.2011.11.029.

[29]

G. XuB. DanX. Zhang and C. Liu, Coordinating a dual-channel supply chain with risk-averse under a two-way revenue sharing contract, Internat. J. Prod. Econ., 147 (2014), 171-179.  doi: 10.1016/j.ijpe.2013.09.012.

[30]

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

[31]

L. Zhang and J. Wang, Coordination of the traditional and the online channels for a short-life-cycle product, European J. Oper. Res., 258 (2017), 639-651.  doi: 10.1016/j.ejor.2016.09.020.

[32]

P. ZhangY. He and C. V. Shi, Retailer's channel structure choice: Online channel, offline channel, or dual channels?, Internat. J. Prod. Econ., 191 (2017), 37-50.  doi: 10.1016/j.ijpe.2017.05.013.

show all references

References:
[1]

A. AtasuV. D. R. Guide and L. N. Van Wassenhove, Product reuse economics in closed-loop supply chain research, Prod. Oper. Manag., 17 (2008), 483-496.  doi: 10.3401/poms.1080.0051.

[2]

A. Atasu and G. C. Souza, How does product recovery affect quality choice?, Prod. Oper. Manag., 22 (2013), 991-1010.  doi: 10.1111/j.1937-5956.2011.01290.x.

[3]

A. Atasu and R. Subramanian, Extended producer responsibility for e-waste: Individual or collective producer responsibility?, Prod. Oper. Manag., 21 (2012), 1042-1059.  doi: 10.1111/j.1937-5956.2012.01327.x.

[4]

S. Bernard, North–south trade in reusable goods: Green design meets illegal shipments of waste, J. Environmental Econ. Manag., 69 (2015), 22-35.  doi: 10.1016/j.jeem.2014.10.004.

[5]

E. Brouillat and V. Oltra, Extended producer responsibility instruments and innovation in eco-design: An exploration through a simulation model, Ecological Econ., 83 (2012), 236-245.  doi: 10.1016/j.ecolecon.2012.07.007.

[6]

B. Chen and J. Chen, When to introduce an online channel, and offer money back guarantees and personalized pricing?, European J. Oper. Res., 257 (2017), 614-624.  doi: 10.1016/j.ejor.2016.07.031.

[7]

X. ChenX. Wang and X. Jiang, The impact of power structure on the retail service supply chain with an O2O mixed channel, J. Oper. Res. Soc., 67 (2016), 294-301.  doi: 10.1057/jors.2015.6.

[8]

T. ChernonogT. Avinadav and T. Ben-Zvi, Pricing and sales-effort investment under bi-criteria in a supply chain of virtual products involving risk, European J. Oper. Res., 246 (2015), 471-475.  doi: 10.1016/j.ejor.2015.05.024.

[9]

W. K. ChiangD. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Manag. Sci., 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.

[10]

B. DanG. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, Internat. J. Prod. Econ., 139 (2012), 312-320.  doi: 10.1016/j.ijpe.2012.05.014.

[11]

B. Dan, S. Zhang and M. Zhou, Strategies for warranty service in a dual-channel supply chain with value-added service competition, Internat. J. Prod. Res., 56 (2018) 5677–5699. doi: 10.1080/00207543.2017.1377355.

[12]

Q. DingC. Dong and Z. Pan, A hierarchical pricing decision process on a dual-channel problem with one manufacturer and one retailer, Internat. J. Prod. Econ., 175 (2016), 197-212.  doi: 10.1016/j.ijpe.2016.02.014.

[13]

G. EsenduranE. Kemahlıoğlu-Ziya and J. M. Swaminathan, Impact of take-back regulation on the remanufacturing industry, Prod. Oper. Manag., 26 (2017), 924-944.  doi: 10.1111/poms.12673.

[14]

M. E. Ferguson and L. B. Toktay, The effect of competition on recovery strategies, Prod. Oper. Manag., 15 (2006), 351-368.  doi: 10.1111/j.1937-5956.2006.tb00250.x.

[15]

J. GaoH. HanL. Hou and H. Wang, Pricing and effort decisions in a closed-loop supply chain under different channel power structures, J. Cleaner Production, 112 (2016), 2043-2057.  doi: 10.1016/j.jclepro.2015.01.066.

[16]

D. Ghosh and J. Shah, A comparative analysis of greening policies across supply chain structures, Internat. J. Prod. Econ., 135 (2012), 568-583.  doi: 10.1016/j.ijpe.2011.05.027.

[17]

J. J. KacenJ. D. Hess and W. K. Chiang, Bricks or clicks? Consumer attitudes toward traditional stores and online stores, Global Econ. Manag. Rev., 18 (2013), 12-21.  doi: 10.1016/S2340-1540(13)70003-3.

[18]

H. Ke and J. Liu, Dual-channel supply chain competition with channel preference and sales effort under uncertain environment, J. Ambient Intell. Humanized Comput., 8 (2017), 781-795.  doi: 10.1007/s12652-017-0502-8.

[19]

B. LiP.-W. HouP. Chen and Q.-H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, Internat. J. Prod. Econ., 178 (2016), 154-168.  doi: 10.1016/j.ijpe.2016.05.010.

[20]

G. Martín-Herrán and S. P. Sigué, Prices, promotions, and channel profitability: Was the conventional wisdom mistaken?, European J. Oper. Res., 211 (2011), 415-425.  doi: 10.1016/j.ejor.2010.12.022.

[21]

K. Matsui, When should a manufacturer set its direct price and wholesale price in dual-channel supply chains?, European J. Oper. Res., 258 (2017), 501-511.  doi: 10.1016/j.ejor.2016.08.048.

[22]

B. NiuQ. Cui and J. Zhang, Impact of channel power and fairness concern on supplier's market entry decision, J. Oper. Res. Soc., 68 (2017), 1570-1581.  doi: 10.1057/s41274-016-0169-0.

[23]

A. Ovchinnikov, Revenue and cost management for remanufactured products, Prod. Oper. Manag., 20 (2011), 824-840.  doi: 10.1111/j.1937-5956.2010.01214.x.

[24]

X. PuL. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Res. Appl., 22 (2017), 1-12.  doi: 10.1016/j.elerap.2016.11.002.

[25]

B. Rodríguez and G. Aydın, Pricing and assortment decisions for a manufacturer selling through dual channels, European J. Oper. Res., 242 (2015), 901-909.  doi: 10.1016/j.ejor.2014.10.047.

[26]

T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects, Manag. Sci., 48 (2002), 992-1007.  doi: 10.1287/mnsc.48.8.992.168.

[27]

W. WangY. ZhangY. LiX. Zhao and M. Cheng, Closed-loop supply chains under reward-penalty mechanism: Retailer collection and asymmetric information, J. Cleaner Prod., 142 (2017), 3938-3955.  doi: 10.1016/j.jclepro.2016.10.063.

[28]

D. Xing and T. Liu, Sales effort free riding and coordination with price match and channel rebate, European J. Oper. Res., 219 (2012), 264-271.  doi: 10.1016/j.ejor.2011.11.029.

[29]

G. XuB. DanX. Zhang and C. Liu, Coordinating a dual-channel supply chain with risk-averse under a two-way revenue sharing contract, Internat. J. Prod. Econ., 147 (2014), 171-179.  doi: 10.1016/j.ijpe.2013.09.012.

[30]

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

[31]

L. Zhang and J. Wang, Coordination of the traditional and the online channels for a short-life-cycle product, European J. Oper. Res., 258 (2017), 639-651.  doi: 10.1016/j.ejor.2016.09.020.

[32]

P. ZhangY. He and C. V. Shi, Retailer's channel structure choice: Online channel, offline channel, or dual channels?, Internat. J. Prod. Econ., 191 (2017), 37-50.  doi: 10.1016/j.ijpe.2017.05.013.

Figure 1.  Variation of profits with the change of $ \delta $ for $ c = 0.20, \lambda = 0.75, \beta = 0.15, a = 0.35 $
Figure 2.  Variation of profits with the change of $ a $ for $ c = 0.25, \lambda = 0.75, \beta = 0.08, \delta = 0.50 $
Figure 3.  Variation of profits with the change of $ \lambda $ for $ c = 0.15, \beta = 0.20, \delta = 0.55, a = 0.35 $
Figure 4.  Variation of profits with the change of $ \beta $ for $ c = 0.25, \lambda = 0.70, \delta = 0.55, a = 0.30 $
Table 1.  Notations
Notation Description
Parameters
$ \theta $ Willing-to-pay of the consumer for retailer channel, $ 0\leq\theta\leq1 $.
$ \delta $ Consumer's preference to select direct channel, $ 0\leq\delta<1 $.
$ c $ Marginal costs incurred by the retailer for the product sold through the store channel, $ 0\leq c <1 $.
$ \beta $ The coefficient of resource-utilization penalty, $ \beta\geq0 $.
$ a $ Market response to retailer's sales effort, $ 0\leq a\leq1 $.
$ \lambda $ Retailer's cost-sharing proportion for sales effort, $ 0\leq\lambda\leq1 $.
$ NS/S $ Without/with retailer's sales effort.
$ q_r, q_r^i $ Market demand for the retailer, $ i\in\{NS,S\} $.
$ q_d, q_d^i $ Market demand for the manufacturer, $ i\in\{NS,S\} $.
$ \Pi_r^i/\Pi_d^i $ Retailer's/Manufacturer's profit, $ i\in\{NS,S\} $.
$ CS^i $ Consumer surplus, $ i\in\{NS,S\} $.
Superscript $ * $ The optimal value of each decision variable.
Decision variables
$ p_r, p_r^i $ Retail price at the retail store channel, $ i\in\{NS,S\} $.
$ p_d, p_d^i $ Retail price at the manufacturer's online channel, $ i\in\{NS,S\} $.
$ s $ Sales effort level provided by the retailer.
$ w, w^i $ Manufacturer's wholesale price to retailer, $ i\in\{NS,S\} $.
Notation Description
Parameters
$ \theta $ Willing-to-pay of the consumer for retailer channel, $ 0\leq\theta\leq1 $.
$ \delta $ Consumer's preference to select direct channel, $ 0\leq\delta<1 $.
$ c $ Marginal costs incurred by the retailer for the product sold through the store channel, $ 0\leq c <1 $.
$ \beta $ The coefficient of resource-utilization penalty, $ \beta\geq0 $.
$ a $ Market response to retailer's sales effort, $ 0\leq a\leq1 $.
$ \lambda $ Retailer's cost-sharing proportion for sales effort, $ 0\leq\lambda\leq1 $.
$ NS/S $ Without/with retailer's sales effort.
$ q_r, q_r^i $ Market demand for the retailer, $ i\in\{NS,S\} $.
$ q_d, q_d^i $ Market demand for the manufacturer, $ i\in\{NS,S\} $.
$ \Pi_r^i/\Pi_d^i $ Retailer's/Manufacturer's profit, $ i\in\{NS,S\} $.
$ CS^i $ Consumer surplus, $ i\in\{NS,S\} $.
Superscript $ * $ The optimal value of each decision variable.
Decision variables
$ p_r, p_r^i $ Retail price at the retail store channel, $ i\in\{NS,S\} $.
$ p_d, p_d^i $ Retail price at the manufacturer's online channel, $ i\in\{NS,S\} $.
$ s $ Sales effort level provided by the retailer.
$ w, w^i $ Manufacturer's wholesale price to retailer, $ i\in\{NS,S\} $.
Table 2.  Equilibrium decisions under different strategies
Variables $ i=NS $ $ i=S $
$ p_r^i $ $ \frac{(\beta+2\delta)(1+c)+\beta\delta}{2(\beta+2\delta)} $ $ \frac{\lambda(1-\delta)[(\beta+2\delta)(c+1)+\delta\beta ]-a^2[\delta(\beta+\delta)+c(\beta+2\delta)]}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]} $
$ p_d^i $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $
$ w^i $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $
$ s_i $ N/A $ \frac{a(1-c-\delta)}{2\lambda(1-\delta)-a^2} $
$ q_r^i $ $ \frac{1-c-\delta}{2(1-\delta)} $ $ \frac{\lambda(1-c-\delta)}{2\lambda(1-\delta)-a^2} $
$ q_d^i $ $ \frac{2c\delta-\beta(1-c-\delta)}{2(1-\delta)(\beta+2\delta)} $ $ \frac{\lambda[2c\delta-\beta(1-c-\delta)]-a^2\delta}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]} $
$ \Pi_r^i $ $ \frac{(1-c-\delta)^2}{4(1-\delta)} $ $ \frac{\lambda(1-c-\delta)^2}{2[2\lambda(1-\delta)-a^2]} $
$ \Pi_d^i $ $ \frac{\delta^2}{2(\beta+2\delta)} $ $ \frac{\delta^2[2\lambda(1-\delta)-a^2]^2 +a^2(\beta+2\delta)(\lambda-1)(1-c-\delta)^2}{2(\beta+2\delta)[2\lambda(1-\delta)-a^2]^2} $
NS=No sales effort, S=Sales effort
Variables $ i=NS $ $ i=S $
$ p_r^i $ $ \frac{(\beta+2\delta)(1+c)+\beta\delta}{2(\beta+2\delta)} $ $ \frac{\lambda(1-\delta)[(\beta+2\delta)(c+1)+\delta\beta ]-a^2[\delta(\beta+\delta)+c(\beta+2\delta)]}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]} $
$ p_d^i $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $
$ w^i $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $ $ \frac{\delta(\beta+\delta)}{\beta+2\delta} $
$ s_i $ N/A $ \frac{a(1-c-\delta)}{2\lambda(1-\delta)-a^2} $
$ q_r^i $ $ \frac{1-c-\delta}{2(1-\delta)} $ $ \frac{\lambda(1-c-\delta)}{2\lambda(1-\delta)-a^2} $
$ q_d^i $ $ \frac{2c\delta-\beta(1-c-\delta)}{2(1-\delta)(\beta+2\delta)} $ $ \frac{\lambda[2c\delta-\beta(1-c-\delta)]-a^2\delta}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]} $
$ \Pi_r^i $ $ \frac{(1-c-\delta)^2}{4(1-\delta)} $ $ \frac{\lambda(1-c-\delta)^2}{2[2\lambda(1-\delta)-a^2]} $
$ \Pi_d^i $ $ \frac{\delta^2}{2(\beta+2\delta)} $ $ \frac{\delta^2[2\lambda(1-\delta)-a^2]^2 +a^2(\beta+2\delta)(\lambda-1)(1-c-\delta)^2}{2(\beta+2\delta)[2\lambda(1-\delta)-a^2]^2} $
NS=No sales effort, S=Sales effort
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