Coordination of a supply chain with a loss-averse retailer under supply uncertainty and marketing effort

Wenying Xie; Bin Chen; Fuyou Huang; Juan He

doi:10.3934/jimo.2020125

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November 2021, 17(6): 3393-3415. doi: 10.3934/jimo.2020125

Coordination of a supply chain with a loss-averse retailer under supply uncertainty and marketing effort

Wenying Xie ^1, , Bin Chen ^2, , Fuyou Huang ^2,, and Juan He ^3,

1.	School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610036, China
2.	Institute of Transportation Development Strategy & Planning of Sichuan Province, Chengdu 610041, China
3.	School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610036, China

^* Corresponding author: Fuyou Huang

Received January 2020 Revised May 2020 Published November 2021 Early access June 2020

Fund Project: The paper is supported by the National Natural Science Foundation of China (Grant No.71273214), the Humanities and Social Sciences of Education Ministry of China (Grant No. 18YJAZH024), the Science and Technology Project of Sichuan Province (Grant No. 20CXTD0081) and the Science and Technology Project of Transportation Department of Sichuan Province (Grant No. 2019-D-05)

This paper deals with a one-period two-stage supply chain, in which a loss-averse retailer facing stochastic demand orders products from a risk-neutral supplier subject to yield uncertainty. Marketing effort exerted by the retailer is employed to enhance the final market demand. We first establish a performance benchmark, and show that the wholesale price contract fails to coordinate the supply chain due to the effects of double marginalization and loss aversion. Then we propose a revenue-cost-sharing contract in order to achieve supply chain coordination. It is verified that a properly designed revenue-cost-sharing contract can achieve perfect coordination and a win-win outcome synchronously. Our results reveal that it is simple to implement and arbitrarily allocate the total channel profit between the retailer and the supplier. In addition, we examine the effect of the retailer's loss aversion degree on contract parameters and profit allocation, and we show that both the retailer and the supplier can benefit from marketing effort.

Keywords: Supply chain coordination, revenue-cost-sharing contract, loss aversion, supply uncertainty, marketing effort.

Mathematics Subject Classification: Primary: 90B50; Secondary: 91B30.

Citation: Wenying Xie, Bin Chen, Fuyou Huang, Juan He. Coordination of a supply chain with a loss-averse retailer under supply uncertainty and marketing effort. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3393-3415. doi: 10.3934/jimo.2020125

References:

[1]	M. Becker-Peth, E. Katok and U. W. Thonemann, Designing buyback contracts for irrational but predictable newsvendors, Management Science, 59 (2013), 1800-1816.
[2]	G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 1-150. doi: 10.1287/mnsc.1040.0215.
[3]	B. Chen, W. Xie, F. Huang and J. He, Quality competition and coordination in a vendor-managed inventory supply chain with two risk-averse manufacturers, Journal of Industrial & Management Optimization, http://dx.doi.org/10.3934/jimo.2020100.
[4]	K. Chen and T. Xiao, Reordering policy and coordination of a supply chain with a loss-averse retailer, Journal of Industrial & Management Optimization, 9 (2013), 827-853. doi: 10.3934/jimo.2013.9.827.
[5]	K. Chen and L. Yang, Random yield and coordination mechanisms of a supply chain with emergency backup sourcing, International Journal of Production Research, 52 (2014), 4747-4767. doi: 10.1080/00207543.2014.886790.
[6]	T. Feng, L. R. Keller and X. Zheng, Decision making in the newsvendor problem: A cross-national laboratory study, Omega, 39 (2011), 41-50. doi: 10.1016/j.omega.2010.02.003.
[7]	Y. Gerchak and Y. Wang, Revenue-sharing vs. wholesale-price contracts in assembly systems with random demand, Production & Operations Management, 13 (2004), 23-33.
[8]	I. Giannoccaro and P. Pontrandolfo, Supply chain coordination by revenue sharing contracts, International Journal of Production Economics, 89 (2004), 131-139.
[9]	B. C. Giri and S. Bardhan, Coordinating a supply chain under uncertain demand and random yield in presence of supply disruption, International Journal of Production Research, 53 (2015), 5070-5084. doi: 10.1080/00207543.2015.1030469.
[10]	A. Golmohammadi and E. Hassini, Capacity, pricing and production under supply and demand uncertainties with an application in agriculture, European Journal of Operational Research, 275 (2019), 1037-1049. doi: 10.1016/j.ejor.2018.12.027.
[11]	K. Govindan and M. N. Popiuc, Reverse supply chain coordination by revenue sharing contract: A case for the personal computers industry, European Journal of Operational Research, 233 (2014), 326-336. doi: 10.1016/j.ejor.2013.03.023.
[12]	S. E. Grasman, T. L. Olsen and J. R. Birge, Setting basestock levels in multi-product systems with setups and random yield, IIE Transactions, 40 (2008), 1158-1170. doi: 10.1080/07408170801967062.
[13]	M. G. G$\ddot{u}$ler and M. E. Keski, On coordination under random yield and random demand, Expert Systems with Applications, 40 (2013), 3688-3695.
[14]	H. Gurnani and Y. Gerchak, Coordination in decentralized assembly systems with uncertain component yields, European Journal of Operational Research, 176 (2007), 1559-1576. doi: 10.1016/j.ejor.2005.09.036.
[15]	Y. He and X. Zhao, Contracts and coordination: Supply chains with uncertain demand and supply: Contracts and coordination, Naval Research Logistics, 63 (2016), 305-319. doi: 10.1002/nav.21695.
[16]	Y. He, X. Zhao, L. Zhao and J. He, Coordinating a supply chain with effort and price dependent stochastic demand, Applied Mathematical Modelling, 33 (2009), 2777-2790. doi: 10.1016/j.apm.2008.08.016.
[17]	T. H. Ho, N. Lim and T. H. Cui, Reference dependence in multilocation newsvendor models: A structural analysis, Management Science, 56 (2010), 1891-1910. doi: 10.1287/mnsc.1100.1225.
[18]	B. Hu, M. Chao, X. Dong and Y. J. Son, Three-echelon supply chain coordination with a loss-averse retailer and revenue sharing contracts, International Journal of Production Economics, 179 (2016), 192-202.
[19]	F. Hu, C. C. Lim and Z. Lu, Coordination of supply chains with a flexible ordering policy under yield and demand uncertainty, International Journal of Production Economics, 146 (2013), 686-693. doi: 10.1016/j.ijpe.2013.08.024.
[20]	F. Huang, J. He and Q. Lei, Coordination in a retailer-dominated supply chain with a risk-averse manufacturer under marketing dependency, International Transactions in Operational Research, https://doi.org/10.1111/itor.12520. doi: 10.1111/itor.12520.
[21]	F. Huang, J. He and J. Wang, Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency, Journal of Industrial & Management Optimization, 15 (2019), 1753-1772. doi: 10.3934/jimo.2018121.
[22]	K. Inderfurth and J. Clemens, Supply chain coordination by risk sharing contracts under random production yield and deterministic demand, OR Spectrum, 36 (2014), 525-556. doi: 10.1007/s00291-012-0314-3.
[23]	K. Inderfurth and S. Vogelgesang, Concepts for safety stock determination under stochastic demand and different types of random production yield, European Journal of Operational Research, 224 (2013), 293-301. doi: 10.1016/j.ejor.2012.07.040.
[24]	P. C. Jones, T. J. Lowe and R. D. Traub, Matching supply and demand: The value of a second chance in producing seed corn, Manufacturing & Service Operations Management, 24 (2002), 222-238.
[25]	D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291. doi: 10.2307/1914185.
[26]	B. Kazaz, Production planning under yield and demand uncertainty with yield-dependent cost and price, Manufacturing & Service Operations Management, 6 (2004), 209-224. doi: 10.1287/msom.1030.0024.
[27]	B. Kazaz and S. Webster, The impact of yield-dependent trading costs on pricing and production planning under supply and demand uncertainty, Manufacturing & Service Operations Management, 13 (2011), 404-417.
[28]	H. Krishnan, R. Kapuscinski and D. A. Butz, Coordinating contracts for decentralized supply chains with retailer promotional effort, Management Science, 50 (2004), 48-63.
[29]	M. Kunter, Coordination via cost and revenue sharing in manufacturercretailer channels, European Journal of Operational Research, 216 (2012), 477-486. doi: 10.1016/j.ejor.2011.07.001.
[30]	X. Li, Y. Li and X. Cai, Double marginalization and coordination in the supply chain with uncertain supply, European Journal of Operational Research, 226 (2013), 228-236. doi: 10.1016/j.ejor.2012.10.047.
[31]	P. Liao, F. Ye and X. Wu, A comparison of the merchant and agency models in the hotel industry, International Transactions in Operational Research, 26 (2019), 1052-1073. doi: 10.1111/itor.12365.
[32]	W. Liu, S. Song, Y. Qiao and H. Zhao, The loss-averse newsvendor problem with random supply capacity, Journal of Industrial & Management Optimization, 13 (2017), 1417-1429. doi: 10.3934/jimo.2016080.
[33]	S. K. Mukhopadhyay, X. Su and S. Ghose, Motivating retail marketing effort: Optimal contract design, Production and Operations Management, 18 (2009), 197-211. doi: 10.1111/j.1937-5956.2009.01014.x.
[34]	S. Ray, S. Li and Y. Song, Tailored supply chain decision making under price-sensitive stochastic demand and delivery uncertainty, Management Science, 51 (2005), 1733-1902. doi: 10.1287/mnsc.1050.0452.
[35]	A. N. Sadigh, S. K. Chaharsooghi and M. Sheikhmohammady, A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain, Journal of Industrial & Management Optimization, 12 (2016), 337-355. doi: 10.3934/jimo.2016.12.337.
[36]	M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 333-450. doi: 10.1287/mnsc.46.3.404.12070.
[37]	K. Taaffe, J. Geunes and H. E. Romeijn, Target market selection and marketing effort under uncertainty: The selective newsvendor, European Journal of Operational Research, 189 (2008), 987-1003. doi: 10.1016/j.ejor.2006.11.049.
[38]	T. A. Taylor, Sale timing in a supply chain: When to sell to the retailer, Manufacturing & Service Operations Management, 8 (2006), 23-42. doi: 10.1287/msom.1050.0089.
[39]	Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Computers & Operations Research, 39 (2012), 1872-1878. doi: 10.1016/j.cor.2011.07.009.
[40]	C. X. Wang, Random yield and uncertain demand in decentralised supply chains under the traditional and VMI arrangements, International Journal of Production Research, 47 (2009), 1955-1968. doi: 10.1080/00207540701666220.
[41]	C. X. Wang and S. Webster, Channel coordination for a supply chain with a risk-neutral manufacturer and a loss-averse retailer, Decision Sciences, 38 (2007), 361-389. doi: 10.1111/j.1540-5915.2007.00163.x.
[42]	X. Xu, C. K. Chan and A. Langevin, Coping with risk management and fill rate in the loss-averse newsvendor model, International Journal of Production Economics, 195 (2018), 296-310. doi: 10.1016/j.ijpe.2017.10.024.
[43]	X. Xu, F. T. S. Chan and C. K. Chan, Optimal option purchase decision of a loss-averse retailer under emergent replenishment, International Journal of Production Research, 57 (2019), 4594-4620. doi: 10.1080/00207543.2019.1579935.
[44]	X. Xu and Z. Meng, Coordination between a supplier and a retailer in terms of profit concession for a two-stage supply chain, International Journal of Production Research, 52 (2014), 2122-2133. doi: 10.1080/00207543.2013.854940.
[45]	X. Xu, Z. Meng, P. Ji, C. Dang and H. Wang, On the newsvendor model with conditional Value-at-Risk of opportunity loss, International Journal of Production Research, 54 (2016), 2449-2458. doi: 10.1080/00207543.2015.1100765.
[46]	X. Xu, H. Wang, C. Dang and P. Ji, The loss-averse newsvendor model with backordering, International Journal of Production Economics, 188 (2017), 1-10. doi: 10.1016/j.ijpe.2017.03.005.
[47]	Z. Yao, S. C. H. Leung and K. K. Lai, Manufacturer's revenue-sharing contract and retail competition, European Journal of Operational Research, 186 (2008), 637-651. doi: 10.1016/j.ejor.2007.01.049.
[48]	J. Zhang, Coordination of supply chain with buyer's promotion, Journal of Industrial & Management Optimization, 3 (2007), 715-726. doi: 10.3934/jimo.2007.3.715.
[49]	Y. Zhou, Z. Shen, R. Ying and X. Xu, A loss-averse two-product ordering model with information updating in two-echelon inventory system, Journal of Industrial & Management Optimization, 14 (2018), 687-705. doi: 10.3934/jimo.2017069.

show all references

References:

[1]	M. Becker-Peth, E. Katok and U. W. Thonemann, Designing buyback contracts for irrational but predictable newsvendors, Management Science, 59 (2013), 1800-1816.
[2]	G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts: Strengths and limitations, Management Science, 51 (2005), 1-150. doi: 10.1287/mnsc.1040.0215.
[3]	B. Chen, W. Xie, F. Huang and J. He, Quality competition and coordination in a vendor-managed inventory supply chain with two risk-averse manufacturers, Journal of Industrial & Management Optimization, http://dx.doi.org/10.3934/jimo.2020100.
[4]	K. Chen and T. Xiao, Reordering policy and coordination of a supply chain with a loss-averse retailer, Journal of Industrial & Management Optimization, 9 (2013), 827-853. doi: 10.3934/jimo.2013.9.827.
[5]	K. Chen and L. Yang, Random yield and coordination mechanisms of a supply chain with emergency backup sourcing, International Journal of Production Research, 52 (2014), 4747-4767. doi: 10.1080/00207543.2014.886790.
[6]	T. Feng, L. R. Keller and X. Zheng, Decision making in the newsvendor problem: A cross-national laboratory study, Omega, 39 (2011), 41-50. doi: 10.1016/j.omega.2010.02.003.
[7]	Y. Gerchak and Y. Wang, Revenue-sharing vs. wholesale-price contracts in assembly systems with random demand, Production & Operations Management, 13 (2004), 23-33.
[8]	I. Giannoccaro and P. Pontrandolfo, Supply chain coordination by revenue sharing contracts, International Journal of Production Economics, 89 (2004), 131-139.
[9]	B. C. Giri and S. Bardhan, Coordinating a supply chain under uncertain demand and random yield in presence of supply disruption, International Journal of Production Research, 53 (2015), 5070-5084. doi: 10.1080/00207543.2015.1030469.
[10]	A. Golmohammadi and E. Hassini, Capacity, pricing and production under supply and demand uncertainties with an application in agriculture, European Journal of Operational Research, 275 (2019), 1037-1049. doi: 10.1016/j.ejor.2018.12.027.
[11]	K. Govindan and M. N. Popiuc, Reverse supply chain coordination by revenue sharing contract: A case for the personal computers industry, European Journal of Operational Research, 233 (2014), 326-336. doi: 10.1016/j.ejor.2013.03.023.
[12]	S. E. Grasman, T. L. Olsen and J. R. Birge, Setting basestock levels in multi-product systems with setups and random yield, IIE Transactions, 40 (2008), 1158-1170. doi: 10.1080/07408170801967062.
[13]	M. G. G$\ddot{u}$ler and M. E. Keski, On coordination under random yield and random demand, Expert Systems with Applications, 40 (2013), 3688-3695.
[14]	H. Gurnani and Y. Gerchak, Coordination in decentralized assembly systems with uncertain component yields, European Journal of Operational Research, 176 (2007), 1559-1576. doi: 10.1016/j.ejor.2005.09.036.
[15]	Y. He and X. Zhao, Contracts and coordination: Supply chains with uncertain demand and supply: Contracts and coordination, Naval Research Logistics, 63 (2016), 305-319. doi: 10.1002/nav.21695.
[16]	Y. He, X. Zhao, L. Zhao and J. He, Coordinating a supply chain with effort and price dependent stochastic demand, Applied Mathematical Modelling, 33 (2009), 2777-2790. doi: 10.1016/j.apm.2008.08.016.
[17]	T. H. Ho, N. Lim and T. H. Cui, Reference dependence in multilocation newsvendor models: A structural analysis, Management Science, 56 (2010), 1891-1910. doi: 10.1287/mnsc.1100.1225.
[18]	B. Hu, M. Chao, X. Dong and Y. J. Son, Three-echelon supply chain coordination with a loss-averse retailer and revenue sharing contracts, International Journal of Production Economics, 179 (2016), 192-202.
[19]	F. Hu, C. C. Lim and Z. Lu, Coordination of supply chains with a flexible ordering policy under yield and demand uncertainty, International Journal of Production Economics, 146 (2013), 686-693. doi: 10.1016/j.ijpe.2013.08.024.
[20]	F. Huang, J. He and Q. Lei, Coordination in a retailer-dominated supply chain with a risk-averse manufacturer under marketing dependency, International Transactions in Operational Research, https://doi.org/10.1111/itor.12520. doi: 10.1111/itor.12520.
[21]	F. Huang, J. He and J. Wang, Coordination of VMI supply chain with a loss-averse manufacturer under quality-dependency and marketing-dependency, Journal of Industrial & Management Optimization, 15 (2019), 1753-1772. doi: 10.3934/jimo.2018121.
[22]	K. Inderfurth and J. Clemens, Supply chain coordination by risk sharing contracts under random production yield and deterministic demand, OR Spectrum, 36 (2014), 525-556. doi: 10.1007/s00291-012-0314-3.
[23]	K. Inderfurth and S. Vogelgesang, Concepts for safety stock determination under stochastic demand and different types of random production yield, European Journal of Operational Research, 224 (2013), 293-301. doi: 10.1016/j.ejor.2012.07.040.
[24]	P. C. Jones, T. J. Lowe and R. D. Traub, Matching supply and demand: The value of a second chance in producing seed corn, Manufacturing & Service Operations Management, 24 (2002), 222-238.
[25]	D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291. doi: 10.2307/1914185.
[26]	B. Kazaz, Production planning under yield and demand uncertainty with yield-dependent cost and price, Manufacturing & Service Operations Management, 6 (2004), 209-224. doi: 10.1287/msom.1030.0024.
[27]	B. Kazaz and S. Webster, The impact of yield-dependent trading costs on pricing and production planning under supply and demand uncertainty, Manufacturing & Service Operations Management, 13 (2011), 404-417.
[28]	H. Krishnan, R. Kapuscinski and D. A. Butz, Coordinating contracts for decentralized supply chains with retailer promotional effort, Management Science, 50 (2004), 48-63.
[29]	M. Kunter, Coordination via cost and revenue sharing in manufacturercretailer channels, European Journal of Operational Research, 216 (2012), 477-486. doi: 10.1016/j.ejor.2011.07.001.
[30]	X. Li, Y. Li and X. Cai, Double marginalization and coordination in the supply chain with uncertain supply, European Journal of Operational Research, 226 (2013), 228-236. doi: 10.1016/j.ejor.2012.10.047.
[31]	P. Liao, F. Ye and X. Wu, A comparison of the merchant and agency models in the hotel industry, International Transactions in Operational Research, 26 (2019), 1052-1073. doi: 10.1111/itor.12365.
[32]	W. Liu, S. Song, Y. Qiao and H. Zhao, The loss-averse newsvendor problem with random supply capacity, Journal of Industrial & Management Optimization, 13 (2017), 1417-1429. doi: 10.3934/jimo.2016080.
[33]	S. K. Mukhopadhyay, X. Su and S. Ghose, Motivating retail marketing effort: Optimal contract design, Production and Operations Management, 18 (2009), 197-211. doi: 10.1111/j.1937-5956.2009.01014.x.
[34]	S. Ray, S. Li and Y. Song, Tailored supply chain decision making under price-sensitive stochastic demand and delivery uncertainty, Management Science, 51 (2005), 1733-1902. doi: 10.1287/mnsc.1050.0452.
[35]	A. N. Sadigh, S. K. Chaharsooghi and M. Sheikhmohammady, A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain, Journal of Industrial & Management Optimization, 12 (2016), 337-355. doi: 10.3934/jimo.2016.12.337.
[36]	M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 333-450. doi: 10.1287/mnsc.46.3.404.12070.
[37]	K. Taaffe, J. Geunes and H. E. Romeijn, Target market selection and marketing effort under uncertainty: The selective newsvendor, European Journal of Operational Research, 189 (2008), 987-1003. doi: 10.1016/j.ejor.2006.11.049.
[38]	T. A. Taylor, Sale timing in a supply chain: When to sell to the retailer, Manufacturing & Service Operations Management, 8 (2006), 23-42. doi: 10.1287/msom.1050.0089.
[39]	Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Computers & Operations Research, 39 (2012), 1872-1878. doi: 10.1016/j.cor.2011.07.009.
[40]	C. X. Wang, Random yield and uncertain demand in decentralised supply chains under the traditional and VMI arrangements, International Journal of Production Research, 47 (2009), 1955-1968. doi: 10.1080/00207540701666220.
[41]	C. X. Wang and S. Webster, Channel coordination for a supply chain with a risk-neutral manufacturer and a loss-averse retailer, Decision Sciences, 38 (2007), 361-389. doi: 10.1111/j.1540-5915.2007.00163.x.
[42]	X. Xu, C. K. Chan and A. Langevin, Coping with risk management and fill rate in the loss-averse newsvendor model, International Journal of Production Economics, 195 (2018), 296-310. doi: 10.1016/j.ijpe.2017.10.024.
[43]	X. Xu, F. T. S. Chan and C. K. Chan, Optimal option purchase decision of a loss-averse retailer under emergent replenishment, International Journal of Production Research, 57 (2019), 4594-4620. doi: 10.1080/00207543.2019.1579935.
[44]	X. Xu and Z. Meng, Coordination between a supplier and a retailer in terms of profit concession for a two-stage supply chain, International Journal of Production Research, 52 (2014), 2122-2133. doi: 10.1080/00207543.2013.854940.
[45]	X. Xu, Z. Meng, P. Ji, C. Dang and H. Wang, On the newsvendor model with conditional Value-at-Risk of opportunity loss, International Journal of Production Research, 54 (2016), 2449-2458. doi: 10.1080/00207543.2015.1100765.
[46]	X. Xu, H. Wang, C. Dang and P. Ji, The loss-averse newsvendor model with backordering, International Journal of Production Economics, 188 (2017), 1-10. doi: 10.1016/j.ijpe.2017.03.005.
[47]	Z. Yao, S. C. H. Leung and K. K. Lai, Manufacturer's revenue-sharing contract and retail competition, European Journal of Operational Research, 186 (2008), 637-651. doi: 10.1016/j.ejor.2007.01.049.
[48]	J. Zhang, Coordination of supply chain with buyer's promotion, Journal of Industrial & Management Optimization, 3 (2007), 715-726. doi: 10.3934/jimo.2007.3.715.
[49]	Y. Zhou, Z. Shen, R. Ying and X. Xu, A loss-averse two-product ordering model with information updating in two-echelon inventory system, Journal of Industrial & Management Optimization, 14 (2018), 687-705. doi: 10.3934/jimo.2017069.

Figure 1. Wholesale price with respect to revenue-sharing coefficient under different loss aversion degrees

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Figure 2. Cost-sharing coefficient with respect to wholesale price under different loss aversion degrees

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Figure 3. Allocation of additional expected profit under coordination

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Figure 4. Effect of loss aversion on order quantity

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Figure 5. Effect of loss aversion on marketing effort level

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Figure 6. Effect of loss aversion on expected profit

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Figure 7. Effect of loss aversion on profit allocation

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Figure 8. Production quantity with respect to $ k $

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Figure 9. Marketing effort level with respect to $ k $

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Figure 10. Expected profit with respect to $ k $

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2021 Impact Factor: 1.411

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2021 Impact Factor: 1.411

American Institute of Mathematical Sciences

Coordination of a supply chain with a loss-averse retailer under supply uncertainty and marketing effort

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American Institute of Mathematical Sciences

Coordination of a supply chain with a loss-averse retailer under supply uncertainty and marketing effort

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