doi: 10.3934/jimo.2021087
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Optimal return and rebate mechanism based on demand sensitivity to reference price

1. 

School of Management, Management Science and Engineering, Xi'an Jiaotong University, ERC for Process Mining of Manufacturing Services in Shaanxi Province, Xi'an 710049, China

2. 

Department of Management, Driehaus College of Business, DePaul University, 1 E Jackson Blvd., Chicago, 60604, America

* Corresponding author: Nengmin Wang

Received  October 2020 Revised  March 2021 Early access April 2021

Fund Project: The first author is supported by the Key Project of National Natural Science Foundation of China under Grant 71732006; the National Natural Science Foundation of China under Grants 71572138, 71390331, 71401132, and 71371150

The acceleration of electronic products' upgrade affects consumers' purchase behaviour. How to encourage consumers to return old products in order to upgrade to new products and how to optimize such the closed-loop supply chain are important managerial topics. According to the theory of reference price, the closed-loop supply chain model with fixed rebate and variable rebate is established. The analysis results imply that, when consumers' willingness to return second-hand products depends on manufacturers' rebates and prices of new products, the profit of closed-loop supply chain decreases. In addition, when consumers are sensitive to price difference, enterprises can adopt low profit margin methods to increase new product demand. Furthermore, the profit of the manufacturer is closely related to whether consumers are loss-seeking or loss-averse. Finally, our analysis provides the insights of the relationship between the optimal return and rebate mechanism and the use time of the previous generation of products.

Citation: Junling Han, Nengmin Wang, Zhengwen He, Bin Jiang. Optimal return and rebate mechanism based on demand sensitivity to reference price. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021087
References:
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X. ChenK. LiF. Wang and X. Li, Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns, J. Ind. Manag. Optim., 16 (2020), 1389-1414.  doi: 10.2307/2152750.  Google Scholar

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Y. Chien, The effect of a pro-rata rebate warranty on the age replacement policy with salvage value consideration, IEEE Transactions on Reliability, 59 (2010), 383-392.  doi: 10.1109/TR.2010.2048677.  Google Scholar

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O. Kaya, Incentive and production decisions for remanufacturing operations, European Journal of Operational Research, 201 (2010), 442-453.  doi: 10.1016/j.ejor.2009.03.007.  Google Scholar

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show all references

References:
[1]

A. Atasu and G. C. Souza, How does product recovery affect quality choice?, Production and Operations Management, 22 (2013), 991-1010.  doi: 10.1111/j.1937-5956.2011.01290.x.  Google Scholar

[2]

A. AtasuL. N. Van Wassenhove and M. Sarvary, Efficient take-back legislation, Production and Operations Management, 18 (2009), 243-258.  doi: 10.1111/j.1937-5956.2009.01004.x.  Google Scholar

[3]

R. BatarfiM. Y. Jaber and S. M. Aljazzar, A profit maximization for a reverse logistics dual-channel supply chain with a return policy, Computer & Industrial Engineering, 106 (2017), 58-82.  doi: 10.1016/j.cie.2017.01.024.  Google Scholar

[4]

R. A. BrieschL. KrishnamurthiT. Mazumdar and S. P. Raj, A comparative analysis of reference price models, Journal of Consumer Research, 24 (1997), 202-214.  doi: 10.1086/209505.  Google Scholar

[5]

K. CaoQ. Bo and Y. He, Optimal trade-in and third-party collection authorization strategies under trade-in subsidy policy, Kybernetes, 47 (2018), 854-872.  doi: 10.1108/K-07-2017-0254.  Google Scholar

[6]

J. Chen and B. Chen, Competing with customer returns policies, International Journal of Production Research, 54 (2016), 2093-2107.  doi: 10.1080/00207543.2015.1106019.  Google Scholar

[7]

X. ChenK. LiF. Wang and X. Li, Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns, J. Ind. Manag. Optim., 16 (2020), 1389-1414.  doi: 10.2307/2152750.  Google Scholar

[8]

Y. Chien, The effect of a pro-rata rebate warranty on the age replacement policy with salvage value consideration, IEEE Transactions on Reliability, 59 (2010), 383-392.  doi: 10.1109/TR.2010.2048677.  Google Scholar

[9]

X. ChuQ. Zhong and X. Li, Reverse channel selection decisions with a joint third-party recycler, International Journal of Production Research, 56 (2018), 5969-5981.  doi: 10.1080/00207543.2018.1442944.  Google Scholar

[10]

P. De Giovanni, State- and control-dependent incentives in a closed-loop supply chain with dynamic returns, Dyn. Games Appl., 6 (2016), 20-54.  doi: 10.1007/s13235-015-0142-6.  Google Scholar

[11]

P. De Giovanni, Closed-loop supply chain coordination through incentives with asymmetric information, Ann. Oper. Res., 253 (2017), 133-167.  doi: 10.1007/s10479-016-2334-x.  Google Scholar

[12]

P. De GiovanniP. V. Reddy and G. Zaccour, Incentive strategies for an optimal recovery program in a closed-loop supply chain, European J. Oper. Res., 249 (2016), 605-617.  doi: 10.1016/j.ejor.2015.09.021.  Google Scholar

[13]

P. De Giovanni and G. Zaccour, A two-period game of a closed-loop supply chain, European J. Oper. Res., 232 (2014), 22-40.  doi: 10.1016/j.ejor.2013.06.032.  Google Scholar

[14]

T. S. Genc and P. De Giovanni, Optimal return and rebate mechanism in a closed-loop supply chain game, European J. Oper. Res., 269 (2018), 661-681.  doi: 10.1016/j.ejor.2018.01.057.  Google Scholar

[15]

A. Goli, E. B. Tirkolaee and G.-W. Weber, A perishable product sustainable supply chain network design problem with lead time and customer satisfaction using a Hybrid Whale-Genetic algorithm, in Golinska-Dawson P (eds. Logistics Operations and Management for Recycling and Reuse). EcoProduction (Environmental Issues in Logistics and Manufacturing), Springer, Berlin, Heidelberg, (2020), 99–124. Google Scholar

[16]

J. Gönsch, Buying used products for remanufacturing: negotiating or posted pricing, Journal of Business Economics, 84 (2014), 715-747.   Google Scholar

[17]

K. Govindan and M. N. Popiuc, Reverse supply chain coordination by revenue sharing contract: A case for the personal computers industry, European J. Oper. Res., 233 (2014), 326-336.  doi: 10.1016/j.ejor.2013.03.023.  Google Scholar

[18]

K. GovindanH. Soleimani and D. Kannan, Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future, European J. Oper. Res., 240 (2015), 603-626.  doi: 10.1016/j.ejor.2014.07.012.  Google Scholar

[19]

V. D. R. GuideR. H. Teunter and L. N. V. Wassenhove, Matching demand and supply to maximize profits from remanufacturing, Manufacturing & Service Operations Management, 5 (2003), 303-316.  doi: 10.1287/msom.5.4.303.24883.  Google Scholar

[20]

Q. HeN. WangZ. YangZ. He and B. Jiang, Competitive collection under channel inco-nvenience in closed-loop supply chain, European J. Oper. Res., 275 (2019), 155-166.  doi: 10.1016/j.ejor.2018.11.034.  Google Scholar

[21]

C. Heath and M. G. Fennema, Mental depreciation and marginal decision making, Organizational Behavior and Human Decision Processes, 68 (1996), 95-108.  doi: 10.1006/obhd.1996.0092.  Google Scholar

[22]

T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European J. Oper. Res., 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038.  Google Scholar

[23]

S. HuZ.-J. Ma and J.-B. Sheu, Optimal prices and trade-in rebates for successive-generation products with strategic consumers and limited trade-in duration, Transportation Research Part E, 124 (2019), 92-107.  doi: 10.1016/j.tre.2019.02.004.  Google Scholar

[24]

M. Y. Jaber and A. M. A. El Saadany, The production, remanufacture and waste disposal model with lost sales, International Journal of Production Economics, 120 (2009), 115-124.  doi: 10.1016/j.ijpe.2008.07.016.  Google Scholar

[25]

M. Y. Jaber and M. A. Rosen, The economic order quantity repair and waste disposal model with entropy cost, European Journal of Operational Research, 188 (2008), 109-120.  doi: 10.1016/j.ejor.2007.03.016.  Google Scholar

[26]

M. U. KalwaniC. K. YimH. J. Rinne and Y. Sugita, A Price Expectations Model of Customer Brand Choice, Journal of Marketing Research, 27 (1990), 251-262.   Google Scholar

[27]

O. Kaya, Incentive and production decisions for remanufacturing operations, European Journal of Operational Research, 201 (2010), 442-453.  doi: 10.1016/j.ejor.2009.03.007.  Google Scholar

[28]

P. K. KopalleA. G. Rao and J. L. Assunção, Asymmetric reference price effects and dynamic pricing policies, Marketing Science, 15 (1996), 60-85.  doi: 10.1287/mksc.15.1.60.  Google Scholar

[29]

T. Maiti and B. C. Giri, Two-way product recovery in a closed-loop supply chain with variable markup under price and quality dependent demand, International Journal of Production Economics, 183 (2017), 259-272.  doi: 10.1016/j.ijpe.2016.09.025.  Google Scholar

[30]

A. Mardani, D. Kannan, R. E. Hooker, S. Ozkul, M. Alrasheedi and E. B. Tirkolaee, Evaluation of green and sustainable supply chain management using structural equation modelling: A systematic review of the state of the art literature and recommendations for future research, Journal of Cleaner Production, 249 (2020), 119383. doi: 10.1016/j.jclepro.2019.119383.  Google Scholar

[31]

Z. MiaoK. FuZ. Xia and Y. Wang, Models for closed-loop supply chain with trade-ins, Omega, 66 (2017), 308-326.  doi: 10.1016/j.omega.2015.11.001.  Google Scholar

[32]

S. MoonG. J. Russell and S. D. Duvvuri, Profiling the reference price consumer, Journal of Retailing, 82 (2006), 1-11.  doi: 10.1016/j.jretai.2005.11.006.  Google Scholar

[33]

J. ÖstlinE. Sundin and M. Björkman, Importance of closed-loop supply chain relationships for product remanufacturing, International Journal of Production Economics, 115 (2008), 336-348.  doi: 10.1016/j.ijpe.2008.02.020.  Google Scholar

[34]

D. S. Putler, Incorporating reference price effects into a theory of consumer choice, Marketing Science, 11 (1992), 287-309.  doi: 10.1287/mksc.11.3.287.  Google Scholar

[35]

K. N. Rajendran and G. J. Tellis, Contextual and temporal components of reference price, Journal of Marketing, 58 (1994), 22-34.   Google Scholar

[36]

V. Ramani and P. De Giovanni, A two-period model of product cannibalization in an atypical closed-loop supply chain with endogenous returns: The case of DellReconnect, European J. Oper. Res., 262 (2017), 1009-1027.  doi: 10.1016/j.ejor.2017.03.080.  Google Scholar

[37]

S. RayT. Boyaci and N. Aras, Optimal prices and trade-in rebates for durable, remanufacturable products, Manufacturing & Service Operations Management, 7 (2005), 208-228.  doi: 10.1287/msom.1050.0080.  Google Scholar

[38]

E. Rosch, Cognitive reference points, Cognitive Psychology, 7 (1975), 532-547.  doi: 10.1016/0010-0285(75)90021-3.  Google Scholar

[39]

R. C. SavaskanS. Bhattacharya and L. N. V. Wassenhove, Closed-loop supply chain models with product remanufacturing, Management Science, 50 (2004), 239-252.  doi: 10.1287/mnsc.1030.0186.  Google Scholar

[40]

A. A. Taleizadeh and R. Sadeghi, Pricing strategies in the competitive reverse supply chains with traditional and e-channels: a game theoretic approach, International Journal of Production Economics, 215 (2018), 48-60.  doi: 10.1016/j.ijpe.2018.06.011.  Google Scholar

[41]

R. H. Teunter, Economic order quantities for stochastic discounted cost inventory systems with remanufacturing, International Journal of Logistics Research & Applications, 5 (2002), 161-175.  doi: 10.1080/13675560210148669.  Google Scholar

[42]

E. B. Tirkolaee, P. Abbasian and G.-W. Weber, Sustainable fuzzy multi-trip location-routing problem for medical waste management during the COVID-19 outbreak, Science of the Total Environment, 756 (2021), 143607. doi: 10.1016/j.scitotenv.2020.143607.  Google Scholar

[43]

E. B. Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani and G.-W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, Journal of Cleaner Production, 250 (2020), 119517. doi: 10.1016/j.jclepro.2019.119517.  Google Scholar

[44]

J. N. Uhl and H. L. Brown, Consumer perception of experimental retail food price changes, Journal of Consumer Affairs, 5 (1971), 174-185.  doi: 10.1111/j.1745-6606.1971.tb00704.x.  Google Scholar

[45]

N. WangQ. He and B. Jiang, Hybrid closed-loop supply chains with competition in recycling and product markets, International Journal of Production Economics, 217 (2019), 246-258.  doi: 10.1016/j.ijpe.2018.01.002.  Google Scholar

[46]

W. WangP. ZhangJ. DingJ. LiH. Sun and L. He, Closed-loop supply chain network equilibrium model with retailer-collection under legislation, J. Ind. Manag. Optim., 15 (2019), 199-219.   Google Scholar

[47]

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Figure 1.  $ \ \pi _{1}^{*} $ when$ \ \beta $ varies
Figure 2.  $ \ \pi _{R_1}^{*} $ when$ \ k $ varies
Figure 3.  $ \ \pi _{M_1}^{*} $ when$ \ k $ varies
Figure 4.  $ \ \pi _{R_1}^{*} $ when$ \ k $ varies
Figure 5.  $ \ \pi _{M_1}^{*} $ when$ \ k $ varies
Figure 6.  $ \ \pi _{1}^{*} $ when$ \ k $ varies
Figure 7.  $ \ \pi _{2}^{*} $ when$ \ \beta $ varies
Figure 8.  $ \ \pi _{R_2}^{*} $ when$ \ t $ varies
Figure 9.  $ \ \pi _{M_2}^{*} $ when$ \ t $ varies
Figure 10.  $ \ \pi _{R_2}^{*} $ when$ \ k $ varies
Figure 11.  $ \ \pi _{M_2}^{*} $ when$ \ k $ varies
Figure 12.  $ \ \pi _{2}^{*} $ when$ k $ varies
Figure 13.  $ \ \pi _{R_2}^{*} $ when$ \ k $ varies
Figure 14.  $ \ \pi _{M_2}^{*} $ when$ \ k $ varies
Figure 15.  $ \ \pi _{2}^{*} $ when$ k $ varies
Figure 16.  $ \ (\pi _{R_1}^{*}-\pi _{R_2}^{*}) $ when$ \ t $ varies
Figure 17.  $ \ (\pi _{M_1}^{*}-\pi _{M_2}^{*}) $ when$ \ t $ varies
Figure 18.  $ \ \pi _{1}^{*}-\pi _{2}^{*} $ when$ \ t $ varies
Table 1.  The description of the symbols
Notations Description
$ \ p_0 $ The price at which the previous generation of products are sold
$ \ p $ Sales price of new generation products
$ \ \omega $ The unit wholesale price of the new product
$ \ c $ Unit cost of new product
$ \ T $ Product life cycle
$ \ t $ Product usage time
$ \ a $ Market demand scale
$ \ b $ Price sensitivity coefficient
$ \ \alpha $ The quantity of used products returned by consumers independent
of price and rebate
$ \ \beta $ Consumers' willingness to return used products
$ \ \delta $ Discount factor according to product quality
$ \ k $ Price difference (price difference between the new product and
the previous one) sensitivity coefficient ($ \ k\geq0 $)
$ \ g $ Cost of the unit of remanufactured product
(including remanufacturing, transportation and detection costs)
$ \ l $ Fixed rebate
$ \ d $ The demand for new products
$ \ r_i $ The quantity of previous generation products returned
$ \ \pi_{M_i} $ Manufacturer's profit
$ \ \pi_{R_i} $ Retailer's profit
$ \ \pi_i $ Profits of closed loop supply chains
$ \ i=1,2 $ Represent the corresponding parameters under fixed rebate and
variable rebate respectively
Notations Description
$ \ p_0 $ The price at which the previous generation of products are sold
$ \ p $ Sales price of new generation products
$ \ \omega $ The unit wholesale price of the new product
$ \ c $ Unit cost of new product
$ \ T $ Product life cycle
$ \ t $ Product usage time
$ \ a $ Market demand scale
$ \ b $ Price sensitivity coefficient
$ \ \alpha $ The quantity of used products returned by consumers independent
of price and rebate
$ \ \beta $ Consumers' willingness to return used products
$ \ \delta $ Discount factor according to product quality
$ \ k $ Price difference (price difference between the new product and
the previous one) sensitivity coefficient ($ \ k\geq0 $)
$ \ g $ Cost of the unit of remanufactured product
(including remanufacturing, transportation and detection costs)
$ \ l $ Fixed rebate
$ \ d $ The demand for new products
$ \ r_i $ The quantity of previous generation products returned
$ \ \pi_{M_i} $ Manufacturer's profit
$ \ \pi_{R_i} $ Retailer's profit
$ \ \pi_i $ Profits of closed loop supply chains
$ \ i=1,2 $ Represent the corresponding parameters under fixed rebate and
variable rebate respectively
Table 2.  If consumers are loss-seeking
$ \ \beta $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 0.989 0.693 0.225 0.092 0.317 0.989 0.693 0.193 0.092 0.285
0.2 0.987 0.688 0.215 0.094 0.309 0.989 0.692 0.191 0.093 0.284
0.4 0.986 0.683 0.205 0.095 0.301 0.988 0.691 0.189 0.093 0.282
0.6 0.982 0.679 0.196 0.097 0.293 0.988 0.690 0.187 0.093 0.280
0.8 0.980 0.674 0.186 0.098 0.285 0.987 0.689 0.185 0.094 0.279
1 0.977 0.669 0.177 0.100 0.277 0.987 0.688 0.183 0.094 0.277
$ \ \beta $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 0.989 0.693 0.225 0.092 0.317 0.989 0.693 0.193 0.092 0.285
0.2 0.987 0.688 0.215 0.094 0.309 0.989 0.692 0.191 0.093 0.284
0.4 0.986 0.683 0.205 0.095 0.301 0.988 0.691 0.189 0.093 0.282
0.6 0.982 0.679 0.196 0.097 0.293 0.988 0.690 0.187 0.093 0.280
0.8 0.980 0.674 0.186 0.098 0.285 0.987 0.689 0.185 0.094 0.279
1 0.977 0.669 0.177 0.100 0.277 0.987 0.688 0.183 0.094 0.277
Table 3.  If consumers are loss-averse
$ \ \beta $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 0.989 0.860 0.445 0.018 0.463 0.989 0.860 0.471 0.018 0.489
0.2 0.963 0.807 0.341 0.026 0.367 0.963 0.808 0.369 0.026 0.395
0.4 0.937 0.755 0.240 0.035 0.275 0.937 0.756 0.269 0.035 0.304
0.6 0.911 0.702 0.141 0.046 0.187 0.911 0.704 0.172 0.045 0.217
0.8 0.885 0.650 0.046 0.058 0.104 0.886 0.652 0.078 0.057 0.135
1 0.858 0.598 -0.047 0.071 0.024 0.860 0.600 -0.013 0.071 0.058
$ \ \beta $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 0.989 0.860 0.445 0.018 0.463 0.989 0.860 0.471 0.018 0.489
0.2 0.963 0.807 0.341 0.026 0.367 0.963 0.808 0.369 0.026 0.395
0.4 0.937 0.755 0.240 0.035 0.275 0.937 0.756 0.269 0.035 0.304
0.6 0.911 0.702 0.141 0.046 0.187 0.911 0.704 0.172 0.045 0.217
0.8 0.885 0.650 0.046 0.058 0.104 0.886 0.652 0.078 0.057 0.135
1 0.858 0.598 -0.047 0.071 0.024 0.860 0.600 -0.013 0.071 0.058
Table 4.  If consumers are loss-seeking
$ \ k $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 1.088 0.746 0.168 0.081 0.249 1.095 0.761 0.157 0.078 0.235
0.2 1.035 0.714 0.181 0.087 0.268 1.041 0.726 0.169 0.084 0.253
0.4 0.998 0.690 0.194 0.093 0.287 1.003 0.701 0.182 0.090 0.272
0.6 0.970 0.673 0.207 0.099 0.307 0.975 0.682 0.194 0.096 0.290
0.8 0.949 0.659 0.221 0.106 0.327 0.953 0.667 0.208 0.103 0.311
1 0.931 0.648 0.235 0.112 0.347 0.935 0.656 0.221 0.109 0.330
$ \ k $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 1.088 0.746 0.168 0.081 0.249 1.095 0.761 0.157 0.078 0.235
0.2 1.035 0.714 0.181 0.087 0.268 1.041 0.726 0.169 0.084 0.253
0.4 0.998 0.690 0.194 0.093 0.287 1.003 0.701 0.182 0.090 0.272
0.6 0.970 0.673 0.207 0.099 0.307 0.975 0.682 0.194 0.096 0.290
0.8 0.949 0.659 0.221 0.106 0.327 0.953 0.667 0.208 0.103 0.311
1 0.931 0.648 0.235 0.112 0.347 0.935 0.656 0.221 0.109 0.330
Table 5.  If consumers are loss-averse
$ \ k $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 1.123 0.818 0.155 0.065 0.220 1.124 0.820 0.186 0.065 0.251
0.2 1.024 0.773 0.173 0.053 0.226 1.024 0.775 0.203 0.052 0.255
0.4 0.952 0.741 0.185 0.044 0.229 0.953 0.743 0.215 0.043 0.259
0.6 0.899 0.717 0.194 0.037 0.231 0.899 0.719 0.225 0.037 0.262
0.8 0.857 0.699 0.202 0.032 0.234 0.858 0.700 0.232 0.032 0.263
1 0.824 0.684 0.208 0.028 0.236 0.825 0.685 0.237 0.027 0.265
$ \ k $ with fixed rebate mechanism with variable rebate mechanism
$ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 1.123 0.818 0.155 0.065 0.220 1.124 0.820 0.186 0.065 0.251
0.2 1.024 0.773 0.173 0.053 0.226 1.024 0.775 0.203 0.052 0.255
0.4 0.952 0.741 0.185 0.044 0.229 0.953 0.743 0.215 0.043 0.259
0.6 0.899 0.717 0.194 0.037 0.231 0.899 0.719 0.225 0.037 0.262
0.8 0.857 0.699 0.202 0.032 0.234 0.858 0.700 0.232 0.032 0.263
1 0.824 0.684 0.208 0.028 0.236 0.825 0.685 0.237 0.027 0.265
Table 6.  If consumers are loss-seeking
$ \ t $ $ \ p_{2}^{*} $ $ \ \omega_{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 1.099 0.912 -0.531 0.037 -0.494
0.2 1.075 0.864 -0.308 0.047 -0.261
0.4 1.051 0.817 -0.122 0.058 -0.064
0.6 1.027 0.769 0.026 0.070 0.096
0.8 1.004 0.721 0.136 0.084 0.220
1 0.980 0.674 0.209 0.098 0.307
$ \ t $ $ \ p_{2}^{*} $ $ \ \omega_{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 1.099 0.912 -0.531 0.037 -0.494
0.2 1.075 0.864 -0.308 0.047 -0.261
0.4 1.051 0.817 -0.122 0.058 -0.064
0.6 1.027 0.769 0.026 0.070 0.096
0.8 1.004 0.721 0.136 0.084 0.220
1 0.980 0.674 0.209 0.098 0.307
Table 7.  If consumers are loss-averse
$ \ t $ $ \ p_{2}^{*} $ $ \ \omega_{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 0.980 0.841 0.080 0.020 0.100
0.2 0.968 0.817 0.127 0.024 0.151
0.4 0.956 0.793 0.165 0.028 0.193
0.6 0.944 0.769 0.194 0.032 0.226
0.8 0.932 0.745 0.213 0.037 0.250
1 0.920 0.721 0.223 0.042 0.265
$ \ t $ $ \ p_{2}^{*} $ $ \ \omega_{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
0 0.980 0.841 0.080 0.020 0.100
0.2 0.968 0.817 0.127 0.024 0.151
0.4 0.956 0.793 0.165 0.028 0.193
0.6 0.944 0.769 0.194 0.032 0.226
0.8 0.932 0.745 0.213 0.037 0.250
1 0.920 0.721 0.223 0.042 0.265
Table 8.  If consumers are loss-seeking
$ \ t $ $ \ p_{1}^{*}-p_{2}^{*} $ $ \ \omega_{1}^{*}-\omega_{2}^{*} $ $ \ \pi _{M_1}^{*}-\pi _{M_2}^{*} $ $ \ \pi _{R_1}^{*}-\pi _{R_2}^{*} $ $ \ \pi _{1}^{*}-\pi _{2}^{*} $
$ t\in [0, 1-\frac{l}{p}) $ 0.1 -0.104 -0.207 0.615 0.055 0.670
0.3 -0.080 -0.160 0.410 0.044 0.454
0.5 -0.060 -0.112 0.243 0.032 0.275
$ t\in [1-\frac{l}{p}, 1] $ 0.98 0.001 0.002 -0.004 -0.001 -0.005
0.99 0.002 0.005 -0.006 -0.002 -0.008
1 0.004 0.007 -0.009 -0.002 -0.011
$ \ t $ $ \ p_{1}^{*}-p_{2}^{*} $ $ \ \omega_{1}^{*}-\omega_{2}^{*} $ $ \ \pi _{M_1}^{*}-\pi _{M_2}^{*} $ $ \ \pi _{R_1}^{*}-\pi _{R_2}^{*} $ $ \ \pi _{1}^{*}-\pi _{2}^{*} $
$ t\in [0, 1-\frac{l}{p}) $ 0.1 -0.104 -0.207 0.615 0.055 0.670
0.3 -0.080 -0.160 0.410 0.044 0.454
0.5 -0.060 -0.112 0.243 0.032 0.275
$ t\in [1-\frac{l}{p}, 1] $ 0.98 0.001 0.002 -0.004 -0.001 -0.005
0.99 0.002 0.005 -0.006 -0.002 -0.008
1 0.004 0.007 -0.009 -0.002 -0.011
Table 9.  If consumers are loss-averse
$ \ t $ $ \ p_{1}^{*}-p_{2}^{*} $ $ \ \omega_{1}^{*}-\omega_{2}^{*} $ {\mbox{$ \ \pi _{M_1}^{*}-\pi _{M_2}^{*} $ $ \ \pi _{R_1}^{*}-\pi _{R_2}^{*} $ $ \ \pi _{1}^{*}-\pi _{2}^{*} $
$ t\in [0, 1-\frac{l}{p}) $ 0.1 -0.050 -0.100 0.085 0.018 0.103
0.3 -0.038 -0.076 0.043 0.014 0.057
0.5 -0.026 -0.052 0.009 0.010 0.019
$ t\in [1-\frac{l}{p}, 1] $ 0.98 0.002 0.005 -0.032 -0.001 -0.033
0.99 0.003 0.006 -0.032 -0.001 -0.033
1 0.004 0.007 -0.033 -0.002 -0.035
$ \ t $ $ \ p_{1}^{*}-p_{2}^{*} $ $ \ \omega_{1}^{*}-\omega_{2}^{*} $ {\mbox{$ \ \pi _{M_1}^{*}-\pi _{M_2}^{*} $ $ \ \pi _{R_1}^{*}-\pi _{R_2}^{*} $ $ \ \pi _{1}^{*}-\pi _{2}^{*} $
$ t\in [0, 1-\frac{l}{p}) $ 0.1 -0.050 -0.100 0.085 0.018 0.103
0.3 -0.038 -0.076 0.043 0.014 0.057
0.5 -0.026 -0.052 0.009 0.010 0.019
$ t\in [1-\frac{l}{p}, 1] $ 0.98 0.002 0.005 -0.032 -0.001 -0.033
0.99 0.003 0.006 -0.032 -0.001 -0.033
1 0.004 0.007 -0.033 -0.002 -0.035
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