American Institute of Mathematical Sciences

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October  2016, 12(4): 1587-1611. doi: 10.3934/jimo.2016.12.1587

Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands

Jianbin Li 1, , Niu Yu 1, , Zhixue Liu 1, and  Lianjie Shu 2,

1. 

School of Management, Huazhong University of Science and Technology, Wuhan, China
2. 

Faculty of Business, University of Macau, Macau, China

Received  February 2014 Revised  July 2015 Published  January 2016

We examine the pure rebate strategies in a two-echelon supply chain under stochastic demand with multiplicative error. Given exogenous wholesale price and retail price, we characterize the unique Nash equilibrium when both manufacturer and retailer provide rebate policy to consumers under nonlinear and linear price-dependent demand functions, including iso-elastic multiplicative demand function (EMDF) and linear multiplicative demand function (LMDF). Based on a game theoretical framework, we prove that there still exists a unique equilibrium when the price elasticity is rather small with constraint conditions in the former case. We also find that in this case the retailer(manufacturer) may increase its rebate value in reaction to the manufacturer's(retailer's) rebate value in order to stimulate sales, which is contrary to the conventional wisdom that the retailer(manufacturer) will shrink its rebate value to gain an ``extra advantage" unfairly. As a result, both parties share the same profit at equilibrium. Further, we compare the expected profit outcomes at equilibrium among joint-rebate game, single-party rebate game and no-rebate game by using numerical examples. It is shown that the joint-rebate policy is not always dominates the others unless the price elasticity is sufficiently flexible.
Keywords: Supply chain, equilibrium., newsvendor model, rebate, Nash game.
Mathematics Subject Classification: Primary: 90B05, 91B06; Secondary: 91A1.
Citation: Jianbin Li, Niu Yu, Zhixue Liu, Lianjie Shu. Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1587-1611. doi: 10.3934/jimo.2016.12.1587
References:
[1]

A. Arya and B. Mittendorf, Managing strategic inventories via manufacturer-to-consumer rebates,, Management Science, 59 (2013), 813.  doi: 10.1287/mnsc.1120.1626.  Google Scholar

[2]

F. J. Arcelus, S. Kumar and G. Sirinivasan, Retailer's response to alternate manufacturer's incentives under a single-period, price-dependent, stochastic-demand framework,, Decision Science, 36 (2005), 599.  doi: 10.1111/j.1540-5414.2005.00104.x.  Google Scholar

[3]

F. J. Arcelus, S. Kumar and G. Sirinivasan, Pricing, rebate, advertising, and ordering policies of a retailer facing price-dependent demand in newsvendor framework under different risk preferences,, International Transportation and Operations Research, 13 (2006), 209.  doi: 10.1111/j.1475-3995.2006.00545.x.  Google Scholar

[4]

F. J. Arcelus, S. Kumar and G. Sirinivasan, Pricing and rebate policies for the newsvendor problem in the presence of a stochastic redemption rate,, International Journal and Production Economics, 107 (2007), 467.  doi: 10.1016/j.ijpe.2006.10.007.  Google Scholar

[5]

F. J. Arcelus, S. Kumar, G. Sirinivasan, Pricing and rebate policies in two-echelon supply chain with asymmetric information under price-dependent, stochastic demand,, International Journal and Production Economics, 113 (2008), 598.  doi: 10.1016/j.ijpe.2007.10.014.  Google Scholar

[6]

F. J. Arcelus, S. Kumar, G. Sirinivasan, The effectiveness of manufacturer vs. retailer rebates within a newsvendor framework,, International Journal and Production Economics, 219 (2012), 252.  doi: 10.1016/j.ejor.2011.06.044.  Google Scholar

[7]

G. Aydin, E. L. Porteus, N. Agrawal and S. Smith, Manufacturer-to-retailer versus manufacturer-to-consumer rebates in a supply chain,, Retail supply Chain Management, (2008), 237.   Google Scholar

[8]

T. Boyaci and O. Ozer, Information acquisition for capacity planning via pricing and advance selling: when to stop and act?,, Operations Research, 58 (2010), 1328.  doi: 10.1287/opre.1100.0798.  Google Scholar

[9]

G. P. Cachon and S. Netessine, Game theory in supply chain analysis,, Handbook of Quantitative Supply chain Analysis, (2004), 13.   Google Scholar

[10]

Y. Chen, S. Moorthy and Z. J. Zhang, Research note-price discrimination after the purchase: Rebates as state-dependent discounts,, Managment Science, 51 (2005), 1131.  doi: 10.1287/mnsc.1050.0391.  Google Scholar

[11]

X. Chen, C. L. Li, B. D. Rhee and D. Simchi-Levi, The impact of manufacturer rebates on supply chain profits,, Naval Research Logistics, 54 (2007), 667.  doi: 10.1002/nav.20239.  Google Scholar

[12]

C. H. Chiu, T. M. Choi and C. S. Tang, Price, rebate, and returns supply contracts for coordinating supply chains with price-dependent demands,, Production and Operations Management, 20 (2011), 81.  doi: 10.1111/j.1937-5956.2010.01159.x.  Google Scholar

[13]

S. Cho, K. F. McCardle and C. S. Tang, Optimal pricing and rebate strategies in a two-level supply chain,, Production and Operations Management, 18 (2009), 426.  doi: 10.1111/j.1937-5956.2009.01035.x.  Google Scholar

[14]

O. C. Demirag, Y. Chen and J. Li, Customer and retailer rebates under risk aversion,, International Journal and Production Economics, 133 (2011), 736.   Google Scholar

[15]

O. C. Demirag, O. Baysar, P. Keskinocak and J. L. Swann, The effects of customer rebates and retailer incentives on a manufacturer's profits and sales,, Naval Research Logistics, 57 (2010), 88.  doi: 10.1002/nav.20390.  Google Scholar

[16]

F. Gao, O. C. Demirag and F. Y. Chen, Early sales of seasonal products with weather-conditional rebates,, Production and Operations Management, 21 (2012), 778.  doi: 10.1111/j.1937-5956.2011.01298.x.  Google Scholar

[17]

Q. Geng and S. Mallik, Joint mail-in rebate decisions in supply chains under demand uncertainty,, Production and Operations Management, 20 (2011), 587.  doi: 10.1111/j.1937-5956.2010.01171.x.  Google Scholar

[18]

D. Granot and S. Yin, On the effectiveness of returns policies in the price-dependent newsvendor models,, Naval Research Logistics, 52 (2005), 765.  doi: 10.1002/nav.20114.  Google Scholar

[19]

B. Grow, The great rebate runaround,, Business Week, 52 (2003).   Google Scholar

[20]

W. Hu and J. Li, Disposing the leftovers under the consignment contract with revenue sharing: Retailer vs supplier,, Journal of System Science Complex, 25 (2012), 262.  doi: 10.1007/s11424-011-9063-7.  Google Scholar

[21]

S. Karlin and C. R. Carr, Prices and optimal inventory policy,, Studies in applied probability and management science, 11 (1962), 159.   Google Scholar

[22]

M. Khouja, A joint optimal pricing, rebate value, and lot sizing model,, European Journal of Operational Research, 174 (2006), 706.  doi: 10.1016/j.ejor.2005.02.041.  Google Scholar

[23]

M. Khouja and J. Zhou, The effect of delayed incentives on supply chain profits and consumer surplus,, Production and Operations Management, 19 (2009), 172.  doi: 10.1111/j.1937-5956.2009.01076.x.  Google Scholar

[24]

M. Lariviere and E. L. Porteus, Selling to the newsvendor: an analysis of price-only contract,, Manufacturing & Service Oprations Management, 3 (2001), 293.  doi: 10.1287/msom.3.4.293.9971.  Google Scholar

[25]

D. Liang, G. Li, L. Sun and Y. Chen, The role of rebates in the hybrid competition between a national brand and a private label with present-biased consumers,, International Journal and Production Economics, 145 (2013), 208.  doi: 10.1016/j.ijpe.2013.04.037.  Google Scholar

[26]

Q. Lu and S. Moorthy, Coupons versus rebates,, Marketing Science, 26 (2007), 67.  doi: 10.1287/mksc.1050.0129.  Google Scholar

[27]

H. Mills, Customers tire of excuses for rebates that never arrive,, The NewYork Times, 73 (2003).   Google Scholar

[28]

E. S. Mills, Uncertainty and price theory,, Quartly Journal of Economics, 73 (1959), 116.  doi: 10.2307/1883828.  Google Scholar

[29]

M. Nagarajan and G. Sosic, Game-theoretic analysis of cooperation among supply chain agents: Review and extensions,, European Journal of Operational Research, 187 (2008), 719.  doi: 10.1016/j.ejor.2006.05.045.  Google Scholar

[30]

A. J. Nevins, Some effects of uncertainty: Simulation of a model of price,, Quartly Journal of Production Economics, 80 (1966), 73.  doi: 10.2307/1880580.  Google Scholar

[31]

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

[32]

L. H. Polatoglu, Optimal order quantity and pricing decisions in single period inventory systems,, International Journal of Production Economics, 23 (1991), 175.  doi: 10.1016/0925-5273(91)90060-7.  Google Scholar

[33]

R. Shi, J. Zhang and J. Ru, Impacts of power structure on supply chains with uncertain demand,, Production and Operations Management, 22 (2013), 1232.  doi: 10.1111/poms.12002.  Google Scholar

[34]

Y. Song, S. Ray and S. Li, Impacts of power structure on supply chains with uncertain demand,, Manufacturing & Service Oprations Management, 10 (2008), 1.   Google Scholar

[35]

T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects,, Management Science, 48 (2002), 992.  doi: 10.1287/mnsc.48.8.992.168.  Google Scholar

[36]

T. A. Taylor and W. Xiao, Incetives for retailer forecasting: rebates vs returns,, Management Science, 22 (2001), 461.   Google Scholar

[37]

G. T. Thowsen, A dynamic, nonstationary inventory problem for a price/quantity setting firm,, Naval Research Logistics, 22 (1975), 461.  doi: 10.1002/nav.3800220306.  Google Scholar

[38]

R. W. Tresch, Principles of Economics,, West Publishing Company, (2001).   Google Scholar

[39]

X. Zhao and D. R. Atkins, Newsvendors under simultaneous price and inventory competition,, Manufacturing & Service Oprations Management, 10 (2008), 539.  doi: 10.1287/msom.1070.0186.  Google Scholar

show all references

References:
[1]

A. Arya and B. Mittendorf, Managing strategic inventories via manufacturer-to-consumer rebates,, Management Science, 59 (2013), 813.  doi: 10.1287/mnsc.1120.1626.  Google Scholar

[2]

F. J. Arcelus, S. Kumar and G. Sirinivasan, Retailer's response to alternate manufacturer's incentives under a single-period, price-dependent, stochastic-demand framework,, Decision Science, 36 (2005), 599.  doi: 10.1111/j.1540-5414.2005.00104.x.  Google Scholar

[3]

F. J. Arcelus, S. Kumar and G. Sirinivasan, Pricing, rebate, advertising, and ordering policies of a retailer facing price-dependent demand in newsvendor framework under different risk preferences,, International Transportation and Operations Research, 13 (2006), 209.  doi: 10.1111/j.1475-3995.2006.00545.x.  Google Scholar

[4]

F. J. Arcelus, S. Kumar and G. Sirinivasan, Pricing and rebate policies for the newsvendor problem in the presence of a stochastic redemption rate,, International Journal and Production Economics, 107 (2007), 467.  doi: 10.1016/j.ijpe.2006.10.007.  Google Scholar

[5]

F. J. Arcelus, S. Kumar, G. Sirinivasan, Pricing and rebate policies in two-echelon supply chain with asymmetric information under price-dependent, stochastic demand,, International Journal and Production Economics, 113 (2008), 598.  doi: 10.1016/j.ijpe.2007.10.014.  Google Scholar

[6]

F. J. Arcelus, S. Kumar, G. Sirinivasan, The effectiveness of manufacturer vs. retailer rebates within a newsvendor framework,, International Journal and Production Economics, 219 (2012), 252.  doi: 10.1016/j.ejor.2011.06.044.  Google Scholar

[7]

G. Aydin, E. L. Porteus, N. Agrawal and S. Smith, Manufacturer-to-retailer versus manufacturer-to-consumer rebates in a supply chain,, Retail supply Chain Management, (2008), 237.   Google Scholar

[8]

T. Boyaci and O. Ozer, Information acquisition for capacity planning via pricing and advance selling: when to stop and act?,, Operations Research, 58 (2010), 1328.  doi: 10.1287/opre.1100.0798.  Google Scholar

[9]

G. P. Cachon and S. Netessine, Game theory in supply chain analysis,, Handbook of Quantitative Supply chain Analysis, (2004), 13.   Google Scholar

[10]

Y. Chen, S. Moorthy and Z. J. Zhang, Research note-price discrimination after the purchase: Rebates as state-dependent discounts,, Managment Science, 51 (2005), 1131.  doi: 10.1287/mnsc.1050.0391.  Google Scholar

[11]

X. Chen, C. L. Li, B. D. Rhee and D. Simchi-Levi, The impact of manufacturer rebates on supply chain profits,, Naval Research Logistics, 54 (2007), 667.  doi: 10.1002/nav.20239.  Google Scholar

[12]

C. H. Chiu, T. M. Choi and C. S. Tang, Price, rebate, and returns supply contracts for coordinating supply chains with price-dependent demands,, Production and Operations Management, 20 (2011), 81.  doi: 10.1111/j.1937-5956.2010.01159.x.  Google Scholar

[13]

S. Cho, K. F. McCardle and C. S. Tang, Optimal pricing and rebate strategies in a two-level supply chain,, Production and Operations Management, 18 (2009), 426.  doi: 10.1111/j.1937-5956.2009.01035.x.  Google Scholar

[14]

O. C. Demirag, Y. Chen and J. Li, Customer and retailer rebates under risk aversion,, International Journal and Production Economics, 133 (2011), 736.   Google Scholar

[15]

O. C. Demirag, O. Baysar, P. Keskinocak and J. L. Swann, The effects of customer rebates and retailer incentives on a manufacturer's profits and sales,, Naval Research Logistics, 57 (2010), 88.  doi: 10.1002/nav.20390.  Google Scholar

[16]

F. Gao, O. C. Demirag and F. Y. Chen, Early sales of seasonal products with weather-conditional rebates,, Production and Operations Management, 21 (2012), 778.  doi: 10.1111/j.1937-5956.2011.01298.x.  Google Scholar

[17]

Q. Geng and S. Mallik, Joint mail-in rebate decisions in supply chains under demand uncertainty,, Production and Operations Management, 20 (2011), 587.  doi: 10.1111/j.1937-5956.2010.01171.x.  Google Scholar

[18]

D. Granot and S. Yin, On the effectiveness of returns policies in the price-dependent newsvendor models,, Naval Research Logistics, 52 (2005), 765.  doi: 10.1002/nav.20114.  Google Scholar

[19]

B. Grow, The great rebate runaround,, Business Week, 52 (2003).   Google Scholar

[20]

W. Hu and J. Li, Disposing the leftovers under the consignment contract with revenue sharing: Retailer vs supplier,, Journal of System Science Complex, 25 (2012), 262.  doi: 10.1007/s11424-011-9063-7.  Google Scholar

[21]

S. Karlin and C. R. Carr, Prices and optimal inventory policy,, Studies in applied probability and management science, 11 (1962), 159.   Google Scholar

[22]

M. Khouja, A joint optimal pricing, rebate value, and lot sizing model,, European Journal of Operational Research, 174 (2006), 706.  doi: 10.1016/j.ejor.2005.02.041.  Google Scholar

[23]

M. Khouja and J. Zhou, The effect of delayed incentives on supply chain profits and consumer surplus,, Production and Operations Management, 19 (2009), 172.  doi: 10.1111/j.1937-5956.2009.01076.x.  Google Scholar

[24]

M. Lariviere and E. L. Porteus, Selling to the newsvendor: an analysis of price-only contract,, Manufacturing & Service Oprations Management, 3 (2001), 293.  doi: 10.1287/msom.3.4.293.9971.  Google Scholar

[25]

D. Liang, G. Li, L. Sun and Y. Chen, The role of rebates in the hybrid competition between a national brand and a private label with present-biased consumers,, International Journal and Production Economics, 145 (2013), 208.  doi: 10.1016/j.ijpe.2013.04.037.  Google Scholar

[26]

Q. Lu and S. Moorthy, Coupons versus rebates,, Marketing Science, 26 (2007), 67.  doi: 10.1287/mksc.1050.0129.  Google Scholar

[27]

H. Mills, Customers tire of excuses for rebates that never arrive,, The NewYork Times, 73 (2003).   Google Scholar

[28]

E. S. Mills, Uncertainty and price theory,, Quartly Journal of Economics, 73 (1959), 116.  doi: 10.2307/1883828.  Google Scholar

[29]

M. Nagarajan and G. Sosic, Game-theoretic analysis of cooperation among supply chain agents: Review and extensions,, European Journal of Operational Research, 187 (2008), 719.  doi: 10.1016/j.ejor.2006.05.045.  Google Scholar

[30]

A. J. Nevins, Some effects of uncertainty: Simulation of a model of price,, Quartly Journal of Production Economics, 80 (1966), 73.  doi: 10.2307/1880580.  Google Scholar

[31]

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

[32]

L. H. Polatoglu, Optimal order quantity and pricing decisions in single period inventory systems,, International Journal of Production Economics, 23 (1991), 175.  doi: 10.1016/0925-5273(91)90060-7.  Google Scholar

[33]

R. Shi, J. Zhang and J. Ru, Impacts of power structure on supply chains with uncertain demand,, Production and Operations Management, 22 (2013), 1232.  doi: 10.1111/poms.12002.  Google Scholar

[34]

Y. Song, S. Ray and S. Li, Impacts of power structure on supply chains with uncertain demand,, Manufacturing & Service Oprations Management, 10 (2008), 1.   Google Scholar

[35]

T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects,, Management Science, 48 (2002), 992.  doi: 10.1287/mnsc.48.8.992.168.  Google Scholar

[36]

T. A. Taylor and W. Xiao, Incetives for retailer forecasting: rebates vs returns,, Management Science, 22 (2001), 461.   Google Scholar

[37]

G. T. Thowsen, A dynamic, nonstationary inventory problem for a price/quantity setting firm,, Naval Research Logistics, 22 (1975), 461.  doi: 10.1002/nav.3800220306.  Google Scholar

[38]

R. W. Tresch, Principles of Economics,, West Publishing Company, (2001).   Google Scholar

[39]

X. Zhao and D. R. Atkins, Newsvendors under simultaneous price and inventory competition,, Manufacturing & Service Oprations Management, 10 (2008), 539.  doi: 10.1287/msom.1070.0186.  Google Scholar

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