# American Institute of Mathematical Sciences

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

 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.
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 and 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), 813-818. doi: 10.1287/mnsc.1120.1626. [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), 599-626. doi: 10.1111/j.1540-5414.2005.00104.x. [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), 209-227. doi: 10.1111/j.1475-3995.2006.00545.x. [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), 467-482. doi: 10.1016/j.ijpe.2006.10.007. [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), 598-618. doi: 10.1016/j.ijpe.2007.10.014. [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), 252-263. doi: 10.1016/j.ejor.2011.06.044. [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, Springer, New York, (2008), 237-270. [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), 1328-1349. doi: 10.1287/opre.1100.0798. [9] G. P. Cachon and S. Netessine, Game theory in supply chain analysis, Handbook of Quantitative Supply chain Analysis, (2004), 13-65. [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), 1131-1140. doi: 10.1287/mnsc.1050.0391. [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), 667-680. doi: 10.1002/nav.20239. [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), 81-91. doi: 10.1111/j.1937-5956.2010.01159.x. [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), 426-446. doi: 10.1111/j.1937-5956.2009.01035.x. [14] O. C. Demirag, Y. Chen and J. Li, Customer and retailer rebates under risk aversion, International Journal and Production Economics, 133 (2011), 736-750. [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), 88-108. doi: 10.1002/nav.20390. [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), 778-794. doi: 10.1111/j.1937-5956.2011.01298.x. [17] Q. Geng and S. Mallik, Joint mail-in rebate decisions in supply chains under demand uncertainty, Production and Operations Management, 20 (2011), 587-602. doi: 10.1111/j.1937-5956.2010.01171.x. [18] D. Granot and S. Yin, On the effectiveness of returns policies in the price-dependent newsvendor models, Naval Research Logistics, 52 (2005), 765-779. doi: 10.1002/nav.20114. [19] B. Grow, The great rebate runaround, Business Week, 52 (2003), November 23. [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), 262-274. doi: 10.1007/s11424-011-9063-7. [21] S. Karlin and C. R. Carr, Prices and optimal inventory policy, Studies in applied probability and management science, Arrow, S. Karlin and H. Scarf (Editors), Stanford Press, Stanford, CA. 11 (1962), 159-172. [22] M. Khouja, A joint optimal pricing, rebate value, and lot sizing model, European Journal of Operational Research, 174 (2006), 706-723. doi: 10.1016/j.ejor.2005.02.041. [23] M. Khouja and J. Zhou, The effect of delayed incentives on supply chain profits and consumer surplus, Production and Operations Management, 19 (2009), 172-197. doi: 10.1111/j.1937-5956.2009.01076.x. [24] M. Lariviere and E. L. Porteus, Selling to the newsvendor: an analysis of price-only contract, Manufacturing & Service Oprations Management, 3 (2001), 293-305. doi: 10.1287/msom.3.4.293.9971. [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), 208-219. doi: 10.1016/j.ijpe.2013.04.037. [26] Q. Lu and S. Moorthy, Coupons versus rebates, Marketing Science, 26 (2007), 67-82. doi: 10.1287/mksc.1050.0129. [27] H. Mills, Customers tire of excuses for rebates that never arrive, The NewYork Times, 73 (2003), 17 April. [28] E. S. Mills, Uncertainty and price theory, Quartly Journal of Economics, 73 (1959), 116-130. doi: 10.2307/1883828. [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), 719-745. doi: 10.1016/j.ejor.2006.05.045. [30] A. J. Nevins, Some effects of uncertainty: Simulation of a model of price, Quartly Journal of Production Economics, 80 (1966), 73-87. doi: 10.2307/1880580. [31] N. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194. doi: 10.1287/opre.47.2.183. [32] L. H. Polatoglu, Optimal order quantity and pricing decisions in single period inventory systems, International Journal of Production Economics, 23 (1991), 175-185. doi: 10.1016/0925-5273(91)90060-7. [33] R. Shi, J. Zhang and J. Ru, Impacts of power structure on supply chains with uncertain demand, Production and Operations Management, 22 (2013), 1232-1249. doi: 10.1111/poms.12002. [34] Y. Song, S. Ray and S. Li, Impacts of power structure on supply chains with uncertain demand, Manufacturing & Service Oprations Management, 10 (2008), 1-18. [35] T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects, Management Science, 48 (2002), 992-1007. doi: 10.1287/mnsc.48.8.992.168. [36] T. A. Taylor and W. Xiao, Incetives for retailer forecasting: rebates vs returns, Management Science, 22 (2001), 461-476. [37] G. T. Thowsen, A dynamic, nonstationary inventory problem for a price/quantity setting firm, Naval Research Logistics, 22 (1975), 461-476. doi: 10.1002/nav.3800220306. [38] R. W. Tresch, Principles of Economics, West Publishing Company, 2001. [39] X. Zhao and D. R. Atkins, Newsvendors under simultaneous price and inventory competition, Manufacturing & Service Oprations Management, 10 (2008), 539-546. doi: 10.1287/msom.1070.0186.

show all references

##### References:
 [1] A. Arya and B. Mittendorf, Managing strategic inventories via manufacturer-to-consumer rebates, Management Science, 59 (2013), 813-818. doi: 10.1287/mnsc.1120.1626. [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), 599-626. doi: 10.1111/j.1540-5414.2005.00104.x. [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), 209-227. doi: 10.1111/j.1475-3995.2006.00545.x. [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), 467-482. doi: 10.1016/j.ijpe.2006.10.007. [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), 598-618. doi: 10.1016/j.ijpe.2007.10.014. [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), 252-263. doi: 10.1016/j.ejor.2011.06.044. [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, Springer, New York, (2008), 237-270. [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), 1328-1349. doi: 10.1287/opre.1100.0798. [9] G. P. Cachon and S. Netessine, Game theory in supply chain analysis, Handbook of Quantitative Supply chain Analysis, (2004), 13-65. [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), 1131-1140. doi: 10.1287/mnsc.1050.0391. [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), 667-680. doi: 10.1002/nav.20239. [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), 81-91. doi: 10.1111/j.1937-5956.2010.01159.x. [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), 426-446. doi: 10.1111/j.1937-5956.2009.01035.x. [14] O. C. Demirag, Y. Chen and J. Li, Customer and retailer rebates under risk aversion, International Journal and Production Economics, 133 (2011), 736-750. [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), 88-108. doi: 10.1002/nav.20390. [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), 778-794. doi: 10.1111/j.1937-5956.2011.01298.x. [17] Q. Geng and S. Mallik, Joint mail-in rebate decisions in supply chains under demand uncertainty, Production and Operations Management, 20 (2011), 587-602. doi: 10.1111/j.1937-5956.2010.01171.x. [18] D. Granot and S. Yin, On the effectiveness of returns policies in the price-dependent newsvendor models, Naval Research Logistics, 52 (2005), 765-779. doi: 10.1002/nav.20114. [19] B. Grow, The great rebate runaround, Business Week, 52 (2003), November 23. [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), 262-274. doi: 10.1007/s11424-011-9063-7. [21] S. Karlin and C. R. Carr, Prices and optimal inventory policy, Studies in applied probability and management science, Arrow, S. Karlin and H. Scarf (Editors), Stanford Press, Stanford, CA. 11 (1962), 159-172. [22] M. Khouja, A joint optimal pricing, rebate value, and lot sizing model, European Journal of Operational Research, 174 (2006), 706-723. doi: 10.1016/j.ejor.2005.02.041. [23] M. Khouja and J. Zhou, The effect of delayed incentives on supply chain profits and consumer surplus, Production and Operations Management, 19 (2009), 172-197. doi: 10.1111/j.1937-5956.2009.01076.x. [24] M. Lariviere and E. L. Porteus, Selling to the newsvendor: an analysis of price-only contract, Manufacturing & Service Oprations Management, 3 (2001), 293-305. doi: 10.1287/msom.3.4.293.9971. [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), 208-219. doi: 10.1016/j.ijpe.2013.04.037. [26] Q. Lu and S. Moorthy, Coupons versus rebates, Marketing Science, 26 (2007), 67-82. doi: 10.1287/mksc.1050.0129. [27] H. Mills, Customers tire of excuses for rebates that never arrive, The NewYork Times, 73 (2003), 17 April. [28] E. S. Mills, Uncertainty and price theory, Quartly Journal of Economics, 73 (1959), 116-130. doi: 10.2307/1883828. [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), 719-745. doi: 10.1016/j.ejor.2006.05.045. [30] A. J. Nevins, Some effects of uncertainty: Simulation of a model of price, Quartly Journal of Production Economics, 80 (1966), 73-87. doi: 10.2307/1880580. [31] N. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194. doi: 10.1287/opre.47.2.183. [32] L. H. Polatoglu, Optimal order quantity and pricing decisions in single period inventory systems, International Journal of Production Economics, 23 (1991), 175-185. doi: 10.1016/0925-5273(91)90060-7. [33] R. Shi, J. Zhang and J. Ru, Impacts of power structure on supply chains with uncertain demand, Production and Operations Management, 22 (2013), 1232-1249. doi: 10.1111/poms.12002. [34] Y. Song, S. Ray and S. Li, Impacts of power structure on supply chains with uncertain demand, Manufacturing & Service Oprations Management, 10 (2008), 1-18. [35] T. A. Taylor, Supply chain coordination under channel rebates with sales effort effects, Management Science, 48 (2002), 992-1007. doi: 10.1287/mnsc.48.8.992.168. [36] T. A. Taylor and W. Xiao, Incetives for retailer forecasting: rebates vs returns, Management Science, 22 (2001), 461-476. [37] G. T. Thowsen, A dynamic, nonstationary inventory problem for a price/quantity setting firm, Naval Research Logistics, 22 (1975), 461-476. doi: 10.1002/nav.3800220306. [38] R. W. Tresch, Principles of Economics, West Publishing Company, 2001. [39] X. Zhao and D. R. Atkins, Newsvendors under simultaneous price and inventory competition, Manufacturing & Service Oprations Management, 10 (2008), 539-546. doi: 10.1287/msom.1070.0186.
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