American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2020097

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

Lianxia Zhao 1, , Jianxin You 2, and  Shu-Cherng Fang 3,,

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  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.

Keywords: Supply chain management, dual-channel, pricing strategy, sales effort, resource-utilization penalty.
Mathematics Subject Classification: Primary: 90B06, 90B60; Secondary: 91B06.
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 & Management Optimization, doi: 10.3934/jimo.2020097
References:
[1]

A. Atasu, V. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

X. Chen, X. 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.  Google Scholar

[8]

T. Chernonog, T. 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.  Google Scholar

[9]

W. K. Chiang, D. 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.  Google Scholar

[10]

B. Dan, G. 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.  Google Scholar

[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.  Google Scholar

[12]

Q. Ding, C. 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.  Google Scholar

[13]

G. Esenduran, E. 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.  Google Scholar

[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.  Google Scholar

[15]

J. Gao, H. Han, L. 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.  Google Scholar

[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.  Google Scholar

[17]

J. J. Kacen, J. 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.  Google Scholar

[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.  Google Scholar

[19]

B. Li, P.-W. Hou, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

B. Niu, Q. 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.  Google Scholar

[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.  Google Scholar

[24]

X. Pu, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Wang, Y. Zhang, Y. Li, X. 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.  Google Scholar

[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.  Google Scholar

[29]

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, Internat. J. Prod. Econ., 147 (2014), 171-179.  doi: 10.1016/j.ijpe.2013.09.012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

P. Zhang, Y. 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.  Google Scholar

show all references

References:
[1]

A. Atasu, V. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

X. Chen, X. 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.  Google Scholar

[8]

T. Chernonog, T. 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.  Google Scholar

[9]

W. K. Chiang, D. 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.  Google Scholar

[10]

B. Dan, G. 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.  Google Scholar

[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.  Google Scholar

[12]

Q. Ding, C. 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.  Google Scholar

[13]

G. Esenduran, E. 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.  Google Scholar

[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.  Google Scholar

[15]

J. Gao, H. Han, L. 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.  Google Scholar

[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.  Google Scholar

[17]

J. J. Kacen, J. 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.  Google Scholar

[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.  Google Scholar

[19]

B. Li, P.-W. Hou, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

B. Niu, Q. 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.  Google Scholar

[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.  Google Scholar

[24]

X. Pu, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Wang, Y. Zhang, Y. Li, X. 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.  Google Scholar

[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.  Google Scholar

[29]

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, Internat. J. Prod. Econ., 147 (2014), 171-179.  doi: 10.1016/j.ijpe.2013.09.012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

P. Zhang, Y. 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.  Google Scholar

Figure 1.  Variation of profits with the change of $ \delta $ for $ c = 0.20, \lambda = 0.75, \beta = 0.15, a = 0.35 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Variation of profits with the change of $ a $ for $ c = 0.25, \lambda = 0.75, \beta = 0.08, \delta = 0.50 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Variation of profits with the change of $ \lambda $ for $ c = 0.15, \beta = 0.20, \delta = 0.55, a = 0.35 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Variation of profits with the change of $ \beta $ for $ c = 0.25, \lambda = 0.70, \delta = 0.55, a = 0.30 $
Figure Options

Download full-size image

Download as PowerPoint slide

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\} $.
Table Options

Download as excel

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
Table Options

Download as excel

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
[1]

Xi Zhao, Teng Niu. Impacts of horizontal mergers on dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020173
[2]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167
[3]

Hongxia Sun, Yao Wan, Yu Li, Linlin Zhang, Zhen Zhou. Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours. Journal of Industrial & Management Optimization, 2021, 17 (2) : 601-631. doi: 10.3934/jimo.2019125
[4]

Honglin Yang, Jiawu Peng. Coordinating a supply chain with demand information updating. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020181
[5]

Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165
[6]

Wenyan Zhuo, Honglin Yang, Leopoldo Eduardo Cárdenas-Barrón, Hong Wan. Loss-averse supply chain decisions with a capital constrained retailer. Journal of Industrial & Management Optimization, 2021, 17 (2) : 711-732. doi: 10.3934/jimo.2019131
[7]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158
[8]

Wei Chen, Yongkai Ma, Weihao Hu. Electricity supply chain coordination with carbon abatement technology investment under the benchmarking mechanism. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020175
[9]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[10]

Xiao-Xu Chen, Peng Xu, Jiao-Jiao Li, Thomas Walker, Guo-Qiang Yang. Decision-making in a retailer-led closed-loop supply chain involving a third-party logistics provider. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021014
[11]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018
[12]

Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140
[13]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180
[14]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011
[15]

Yuanshi Wang. Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 963-985. doi: 10.3934/dcdsb.2020149
[16]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103
[17]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[18]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285
[19]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038
[20]

David W. K. Yeung, Yingxuan Zhang, Hongtao Bai, Sardar M. N. Islam. Collaborative environmental management for transboundary air pollution problems: A differential levies game. Journal of Industrial & Management Optimization, 2021, 17 (2) : 517-531. doi: 10.3934/jimo.2019121

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (44)
  • HTML views (244)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences