doi: 10.3934/jimo.2019131

Loss-averse supply chain decisions with a capital constrained retailer

1. 

School of Marketing and Logistics Management, Nanjing University of Finance and Economics, Nanjing, Jiangsu Province 210023, China

2. 

School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China

3. 

Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México

4. 

School of Business, State University of New York at Oswego, Oswego, NY 13126, USA

* Corresponding author: ottoyang@126.com (Honglin Yang)

Received  February 2019 Revised  July 2019 Published  October 2019

Fund Project: This research is supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71521061 and 71790593 and the Ministry of Education in China of Humanities and Social Science Project under Grant No. 19YJC630242.

In real-world transactions, capital constraints restrict the rapid development of the enterprises in the supply chain. The loss aversion behaviors of enterprises directly affect the decision making. This paper investigates the optimal decisions of both the supplier and the capital constrained retailer being loss aversion decision makers under different financing strategies. The capital constrained retailer may borrow from a bank or use the supplier's trade credit to satisfy uncertain demand. With a wholesale price contract, we analytically solve the unique Stackelberg equilibrium under two financing schemes. We derive the critical wholesale price that determines the retailer's financing preference. We identify the impacts of the loss aversion coefficients and initial capital level on the operational and financing decisions. Numerical examples reveal that there exists a Pareto improvement zone regarding the retailer's loss aversion coefficient and initial capital level.

Citation: 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, doi: 10.3934/jimo.2019131
References:
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X. Chen, A model of trade credit in a capital-constrained distribution channel, Internat. J. Produc. Econ., 159 (2015), 347-357.  doi: 10.1016/j.ijpe.2014.05.001.  Google Scholar

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X. ChenG. Hao and L. Li, Channel coordination with a loss-averse retailer and option contracts, Internat. J. Produc. Econ., 150 (2014), 52-57.  doi: 10.1016/j.ijpe.2013.12.004.  Google Scholar

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X. Chen and A. Wang, Trade credit contract with limited liability in the supply chain with budget constraints, Ann. Oper. Res., 196 (2012), 153-165.  doi: 10.1007/s10479-012-1119-0.  Google Scholar

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M. Dada and Q. Hu, Financing newsvendor inventory, Oper. Res. Lett., 36 (2008), 569-573.  doi: 10.1016/j.orl.2008.06.004.  Google Scholar

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A. DiabataA. A. Taleizadehb and M. Lashgaric, A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items, J. Manufac. Systems, 45 (2017), 322-342.  doi: 10.1016/j.jmsy.2017.04.005.  Google Scholar

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A. Fitzpatrick and B. Lien, The use of trade credit by businesses, Rba Bulletin, 36 (2013), 39-46.   Google Scholar

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M. GhoreishiG. W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating products with partial backlogging, permissible delay in payments, inflation- and selling price-dependent demand and customer returns, Ann. Oper. Res., 226 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7.  Google Scholar

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A. HiassatA. Diabat and I. Rahwan, A genetic algorithm approach for location- inventory-outing problem with perishable products, J. Manufac. Systems, 42 (2017), 93-103.  doi: 10.1016/j.jmsy.2016.10.004.  Google Scholar

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B. HuC. MengD. Xu and Y. J. Son, Three-echelon supply chain coordination with a loss-averse retailer and revenue sharing contracts, Internat. J. Produc. Econ., 179 (2016), 192-202.  doi: 10.1016/j.ijpe.2016.06.001.  Google Scholar

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C. K. JaggiV. S. S. YadavalliM. Verma and A. Sharma, An EOQ model with allowable shortage under trade credit in different scenario, Appl. Math. Comput., 252 (2015), 541-551.  doi: 10.1016/j.amc.2014.12.040.  Google Scholar

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D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.  doi: 10.2307/1914185.  Google Scholar

[21]

P. Kouvelis and W. Zhao, Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts, Oper. Res., 60 (2012), 566-580.  doi: 10.1287/opre.1120.1040.  Google Scholar

[22]

P. Kouvelis and W. Zhao, The newsvendor problem and price only contract when bankruptcy costs exist, Produc. Oper. Manag., 20 (2011), 921-936.  doi: 10.1111/j.1937-5956.2010.01211.x.  Google Scholar

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P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Manag. Science, 62 (2016), 2341-2357.  doi: 10.1287/mnsc.2015.2248.  Google Scholar

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G. J. Kyparisis and C. Koulamas, The price-setting newsvendor problem with nonnegative linear additive demand, European J. Oper. Res., 269 (2018), 695-698.  doi: 10.1016/j.ejor.2018.02.019.  Google Scholar

[25]

C. Lee and B. Rhee, Coordination contracts in the presence of positive inventory financing costs, Internat. J. Produc. Econ., 124 (2010), 331-339.  doi: 10.1016/j.ijpe.2009.11.028.  Google Scholar

[26]

C. Lee and B. Rhee, Trade credit for supply chain coordination, European J. Oper. Res., 214 (2011), 136-146.  doi: 10.1016/j.ejor.2011.04.004.  Google Scholar

[27]

B. LiS. An and D. Song, Selection of financing strategies with a risk-averse supplier in a capital-constrained supply chain, Transpor. Res. Part E: Logistics and Transpor. Review, 118 (2018), 163-183.  doi: 10.1016/j.tre.2018.06.007.  Google Scholar

[28]

X. Li and Y. J. Li, On the loss-averse dual-sourcing problem under supply disruption, Comput. Oper. Res., 100 (2018), 301-313.  doi: 10.1016/j.cor.2016.12.011.  Google Scholar

[29]

W. H. LiuM. L. WangD. L. Zhu and L. Zhou, Service capacity procurement of logistics service supply chain with demand updating and loss-averse preference, Appl. Math. Model., 66 (2019), 486-507.  doi: 10.1016/j.apm.2018.09.020.  Google Scholar

[30]

Z. LiuL. ChenL. Li and X. Zhai, Risk hedging in a supply chain: Option vs. price discount, Internat. J. Produc. Econ., 151 (2014), 112-120.  doi: 10.1016/j.ijpe.2014.01.019.  Google Scholar

[31]

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[32]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Managemen Sci., 46 (2000), 404-420.  doi: 10.1287/mnsc.46.3.404.12070.  Google Scholar

[33]

S. TiwariM. Goh and A. A. Shaikh, Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, Internat. J. Produc. Econ., 200 (2018), 16-36.  doi: 10.1016/j.ijpe.2018.03.006.  Google Scholar

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C. X. Wang and W. Webster, The loss-averse newsvendor problem, Omega, 37 (2009), 93-105.  doi: 10.1016/j.omega.2006.08.003.  Google Scholar

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Y. Wei and T. M. Choi, Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes, European J. Oper. Res., 204 (2010), 255-262.  doi: 10.1016/j.ejor.2009.10.016.  Google Scholar

[36]

S. XiaoS. P. SethiM. Q. Liu and S. H. Ma, Coordinating contracts for a financially constrained supply chain, Omega, 72 (2017), 71-86.  doi: 10.1016/j.omega.2016.11.005.  Google Scholar

[37]

X. Xu and J. R. Birge, Joint production and financing decisions: Modeling and analysis, SSRN Electronic Journal, (2004). doi: 10.2139/ssrn.652562.  Google Scholar

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X. XuC. K. Chan and A. Langevin, Coping with risk management and fill rate in the loss-averse newsvendor model, Internat. J. Produc. Econ., 195 (2018), 296-310.  doi: 10.1016/j.ijpe.2017.10.024.  Google Scholar

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N. YanX. He and Y. Liu, Financing the capital-constrained supply chain with loss aversion: Supplier finance vs. supplier investment, Omega, 88 (2019), 162-178.  doi: 10.1016/j.omega.2018.08.003.  Google Scholar

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[41]

N. YanB. W. SunH. Zhang and C. Liu, A partial credit guarantee contract in a capital-constrained supply chain: Financing equilibrium and coordinating strategy, Internat. J. Produc. Econ., 173 (2016), 122-133.  doi: 10.1016/j.ijpe.2015.12.005.  Google Scholar

[42]

H. L. YangW. Y. Zhuo and L. S. Shao, Equilibrium evolution in a two-echelon supply chain with financially constrained retailers: The impact of equity financing, Internat. J. Produc. Econ., 185 (2017), 139-149.  doi: 10.1016/j.ijpe.2016.12.027.  Google Scholar

[43]

H. L. YangW. Y. ZhuoY. Zha and H. Wan, Two-period supply chain with flexible trade credit contract, Expert Systems with Appl., 66 (2016), 95-105.  doi: 10.1016/j.eswa.2016.08.056.  Google Scholar

[44]

S. A. Yang and J. R. Birge, Trade credit, risk sharing, and inventory financing portfolios, Management Sci., 64 (2018), 667-3689.  doi: 10.1287/mnsc.2017.2799.  Google Scholar

[45]

B. F. ZhangD. Wu and L. Liang, Trade credit model with customer balking and asymmetric market information, Transpor. Res. Part E: Logistics and Transpor. Review, 110 (2018), 31-46.  doi: 10.1016/j.tre.2017.10.006.  Google Scholar

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Y. ZhangK. L. Donohue and T. H. Cui, Contract preferences and performance for the loss-averse supplier: Buyback versus revenue sharing, Management Sci., 62 (2016), 1734-1754.  doi: 10.1287/mnsc.2015.2182.  Google Scholar

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Y. ZhaoX. MengS. Y. Wang and T. C. E. Cheng, Buyback contracts with price-dependent demands: effects of demand uncertainty, European J. Oper. Res., 239 (2014), 663-673.  doi: 10.1016/j.ejor.2014.06.008.  Google Scholar

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show all references

References:
[1]

J. A. Buzacott and R. Q. Zhang, Inventory management with asset-based financing, Management Sci., 50 (2004), 1274-1292.  doi: 10.1287/mnsc.1040.0278.  Google Scholar

[2]

R. Caldentey and X. Chen, The role of financial services in procurement contracts, The Handbook of Integrated Risk Management in Global Supply Chains, John Wiley and Sons, Inc., New York, 2012. doi: 10.1002/9781118115800.ch11.  Google Scholar

[3]

C. T. ChangM. C. Cheng and L. Y. Ouyang, Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments, Appl. Math. Model., 39 (2015), 747-763.  doi: 10.1016/j.apm.2014.07.002.  Google Scholar

[4]

S. C. ChenL. E. Cárdenas-Barrón and J. T. Teng, Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity, Internat. J. Produc. Econ., 155 (2014), 284-291.  doi: 10.1016/j.ijpe.2013.05.032.  Google Scholar

[5]

X. Chen, A model of trade credit in a capital-constrained distribution channel, Internat. J. Produc. Econ., 159 (2015), 347-357.  doi: 10.1016/j.ijpe.2014.05.001.  Google Scholar

[6]

X. ChenG. Hao and L. Li, Channel coordination with a loss-averse retailer and option contracts, Internat. J. Produc. Econ., 150 (2014), 52-57.  doi: 10.1016/j.ijpe.2013.12.004.  Google Scholar

[7]

X. Chen and A. Wang, Trade credit contract with limited liability in the supply chain with budget constraints, Ann. Oper. Res., 196 (2012), 153-165.  doi: 10.1007/s10479-012-1119-0.  Google Scholar

[8]

K. C. ChiW. H. WongA. Langevin and Y. C. E. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, Internat. J. Produc. Econ., 189 (2017), 1-13.  doi: 10.1016/j.ijpe.2017.04.001.  Google Scholar

[9]

M. Dada and Q. Hu, Financing newsvendor inventory, Oper. Res. Lett., 36 (2008), 569-573.  doi: 10.1016/j.orl.2008.06.004.  Google Scholar

[10]

A. DiabataA. A. Taleizadehb and M. Lashgaric, A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items, J. Manufac. Systems, 45 (2017), 322-342.  doi: 10.1016/j.jmsy.2017.04.005.  Google Scholar

[11]

A. Fitzpatrick and B. Lien, The use of trade credit by businesses, Rba Bulletin, 36 (2013), 39-46.   Google Scholar

[12]

M. GhoreishiG. W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating products with partial backlogging, permissible delay in payments, inflation- and selling price-dependent demand and customer returns, Ann. Oper. Res., 226 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7.  Google Scholar

[13]

M. GiannettiM. Burkart and T. Ellingsen, What you sell is what you lend? Explaining trade credit contracts, Review of Financial Studies, 24 (2011), 1261-1298.   Google Scholar

[14]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Society, 36 (1985), 335-338.  doi: 10.1057/jors.1985.56.  Google Scholar

[15]

A. HiassatA. Diabat and I. Rahwan, A genetic algorithm approach for location- inventory-outing problem with perishable products, J. Manufac. Systems, 42 (2017), 93-103.  doi: 10.1016/j.jmsy.2016.10.004.  Google Scholar

[16]

B. HuC. MengD. Xu and Y. J. Son, Three-echelon supply chain coordination with a loss-averse retailer and revenue sharing contracts, Internat. J. Produc. Econ., 179 (2016), 192-202.  doi: 10.1016/j.ijpe.2016.06.001.  Google Scholar

[17]

C. K. JaggiV. S. S. YadavalliM. Verma and A. Sharma, An EOQ model with allowable shortage under trade credit in different scenario, Appl. Math. Comput., 252 (2015), 541-551.  doi: 10.1016/j.amc.2014.12.040.  Google Scholar

[18]

B. JingX. Chen and G. Cai, Equilibrium financing in a distribution channel with capital constraint, Produc. Oper. Manag., 21 (2012), 1090-1101.  doi: 10.1111/j.1937-5956.2012.01328.x.  Google Scholar

[19]

B. Jing and A. Seidmann, Finance sourcing in a supply chain, Decision Support Systems, 58 (2014), 15-20.  doi: 10.1016/j.dss.2013.01.013.  Google Scholar

[20]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291.  doi: 10.2307/1914185.  Google Scholar

[21]

P. Kouvelis and W. Zhao, Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts, Oper. Res., 60 (2012), 566-580.  doi: 10.1287/opre.1120.1040.  Google Scholar

[22]

P. Kouvelis and W. Zhao, The newsvendor problem and price only contract when bankruptcy costs exist, Produc. Oper. Manag., 20 (2011), 921-936.  doi: 10.1111/j.1937-5956.2010.01211.x.  Google Scholar

[23]

P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Manag. Science, 62 (2016), 2341-2357.  doi: 10.1287/mnsc.2015.2248.  Google Scholar

[24]

G. J. Kyparisis and C. Koulamas, The price-setting newsvendor problem with nonnegative linear additive demand, European J. Oper. Res., 269 (2018), 695-698.  doi: 10.1016/j.ejor.2018.02.019.  Google Scholar

[25]

C. Lee and B. Rhee, Coordination contracts in the presence of positive inventory financing costs, Internat. J. Produc. Econ., 124 (2010), 331-339.  doi: 10.1016/j.ijpe.2009.11.028.  Google Scholar

[26]

C. Lee and B. Rhee, Trade credit for supply chain coordination, European J. Oper. Res., 214 (2011), 136-146.  doi: 10.1016/j.ejor.2011.04.004.  Google Scholar

[27]

B. LiS. An and D. Song, Selection of financing strategies with a risk-averse supplier in a capital-constrained supply chain, Transpor. Res. Part E: Logistics and Transpor. Review, 118 (2018), 163-183.  doi: 10.1016/j.tre.2018.06.007.  Google Scholar

[28]

X. Li and Y. J. Li, On the loss-averse dual-sourcing problem under supply disruption, Comput. Oper. Res., 100 (2018), 301-313.  doi: 10.1016/j.cor.2016.12.011.  Google Scholar

[29]

W. H. LiuM. L. WangD. L. Zhu and L. Zhou, Service capacity procurement of logistics service supply chain with demand updating and loss-averse preference, Appl. Math. Model., 66 (2019), 486-507.  doi: 10.1016/j.apm.2018.09.020.  Google Scholar

[30]

Z. LiuL. ChenL. Li and X. Zhai, Risk hedging in a supply chain: Option vs. price discount, Internat. J. Produc. Econ., 151 (2014), 112-120.  doi: 10.1016/j.ijpe.2014.01.019.  Google Scholar

[31]

F. Modigliani and M. H. Miller, The cost of capital, corporation finance and the theory of investment, Amer. Econ. Rev., 48 (1958), 261-297.   Google Scholar

[32]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Managemen Sci., 46 (2000), 404-420.  doi: 10.1287/mnsc.46.3.404.12070.  Google Scholar

[33]

S. TiwariM. Goh and A. A. Shaikh, Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, Internat. J. Produc. Econ., 200 (2018), 16-36.  doi: 10.1016/j.ijpe.2018.03.006.  Google Scholar

[34]

C. X. Wang and W. Webster, The loss-averse newsvendor problem, Omega, 37 (2009), 93-105.  doi: 10.1016/j.omega.2006.08.003.  Google Scholar

[35]

Y. Wei and T. M. Choi, Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes, European J. Oper. Res., 204 (2010), 255-262.  doi: 10.1016/j.ejor.2009.10.016.  Google Scholar

[36]

S. XiaoS. P. SethiM. Q. Liu and S. H. Ma, Coordinating contracts for a financially constrained supply chain, Omega, 72 (2017), 71-86.  doi: 10.1016/j.omega.2016.11.005.  Google Scholar

[37]

X. Xu and J. R. Birge, Joint production and financing decisions: Modeling and analysis, SSRN Electronic Journal, (2004). doi: 10.2139/ssrn.652562.  Google Scholar

[38]

X. XuC. K. Chan and A. Langevin, Coping with risk management and fill rate in the loss-averse newsvendor model, Internat. J. Produc. Econ., 195 (2018), 296-310.  doi: 10.1016/j.ijpe.2017.10.024.  Google Scholar

[39]

N. YanX. He and Y. Liu, Financing the capital-constrained supply chain with loss aversion: Supplier finance vs. supplier investment, Omega, 88 (2019), 162-178.  doi: 10.1016/j.omega.2018.08.003.  Google Scholar

[40]

N. Yan and B. Sun, Coordinating loan strategies for supply chain financing with limited credit, OR Spectrum, 35 (2013), 1039-1058.  doi: 10.1007/s00291-013-0329-4.  Google Scholar

[41]

N. YanB. W. SunH. Zhang and C. Liu, A partial credit guarantee contract in a capital-constrained supply chain: Financing equilibrium and coordinating strategy, Internat. J. Produc. Econ., 173 (2016), 122-133.  doi: 10.1016/j.ijpe.2015.12.005.  Google Scholar

[42]

H. L. YangW. Y. Zhuo and L. S. Shao, Equilibrium evolution in a two-echelon supply chain with financially constrained retailers: The impact of equity financing, Internat. J. Produc. Econ., 185 (2017), 139-149.  doi: 10.1016/j.ijpe.2016.12.027.  Google Scholar

[43]

H. L. YangW. Y. ZhuoY. Zha and H. Wan, Two-period supply chain with flexible trade credit contract, Expert Systems with Appl., 66 (2016), 95-105.  doi: 10.1016/j.eswa.2016.08.056.  Google Scholar

[44]

S. A. Yang and J. R. Birge, Trade credit, risk sharing, and inventory financing portfolios, Management Sci., 64 (2018), 667-3689.  doi: 10.1287/mnsc.2017.2799.  Google Scholar

[45]

B. F. ZhangD. Wu and L. Liang, Trade credit model with customer balking and asymmetric market information, Transpor. Res. Part E: Logistics and Transpor. Review, 110 (2018), 31-46.  doi: 10.1016/j.tre.2017.10.006.  Google Scholar

[46]

B. F. ZhangD. WuL. Liang and D. L. Olson, Supply chain loss-averse newsboy model with capital constraint, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 46 (2016), 646-658.  doi: 10.1109/TSMC.2015.2475720.  Google Scholar

[47]

Y. ZhangK. L. Donohue and T. H. Cui, Contract preferences and performance for the loss-averse supplier: Buyback versus revenue sharing, Management Sci., 62 (2016), 1734-1754.  doi: 10.1287/mnsc.2015.2182.  Google Scholar

[48]

Y. ZhaoT. M. ChoiT. C. E. Cheng and S. Y. Wang, Supply option contracts with spot market and demand information updating, European J. Oper. Res., 266 (2018), 1062-1071.  doi: 10.1016/j.ejor.2017.11.001.  Google Scholar

[49]

Y. ZhaoX. MengS. Y. Wang and T. C. E. Cheng, Buyback contracts with price-dependent demands: effects of demand uncertainty, European J. Oper. Res., 239 (2014), 663-673.  doi: 10.1016/j.ejor.2014.06.008.  Google Scholar

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Figure 1.  The retailer's order quantity changes with $ \lambda_R $ under $ U(0,250) $
Figure 2.  The retailer's order quantity changes with $ \lambda_R $ under $ N(100,60) $
Figure 3.  The retailer's order quantity changes with $ \lambda_S $ under $ U(0,250) $ with $ \lambda_R = 2 $
Figure 4.  The retailer's order quantity changes with $ \lambda_S $ under $ N(100,60) $ with $ \lambda_R = 10 $
Figure 5.  The difference of player's expected utility changes with $ \lambda_R $ under $ U(0,250) $
Figure 6.  The difference of player's expected utility changes with $ \lambda_S $ under $ U(0,250) $
Figure 7.  The supplier's expected utility changes with $ w_j $ under $ U(0,250) $
Figure 8.  The supplier's expected utility changes with $ w_j $ under $ N(100,60) $
Figure 9.  The retailer's order quantity changes with $ \Omega $ under $ U(0,250) $ with $ \lambda_R = 2 $, $ \lambda_S = 1.5 $
Figure 10.  The retailer's order quantity changes with $ \Omega $ under $ N(100,60) $with $ \lambda_R = 10 $, $ \lambda_S = 2 $
Figure 11.  The difference of player's expected utility changes with $ \Omega $ under $ U(0,250) $
Table 1.  Comparisons of three models with loss aversion
Literature Financing scheme Loss-averse player Decision objective
Upstream Downstream
Zhang et al.(2016) BCF Retailer EPM EUM
Yan et al.(2018) TCF or SI Retailer EPM EUM
This paper BCF or TCF Retailer and Supplier EUM EUM
Literature Financing scheme Loss-averse player Decision objective
Upstream Downstream
Zhang et al.(2016) BCF Retailer EPM EUM
Yan et al.(2018) TCF or SI Retailer EPM EUM
This paper BCF or TCF Retailer and Supplier EUM EUM
Table 2.  Notation
Notation Definition
$ p $ Retail price
$ c $ Production cost
$ w_j $ Wholesale price, where $ j=B,T $ denotes bank credit financing or trade credit
financing, respectively (the supplier's decision variable)
$ X $ Random demand, defined over continuous interval$ [0,+∞) $
$ f(X) $ Probability density function of $ X $
$ F(X) $ Cumulative distribution function of $ X $
$ z(X) $ Failure rate of the demand distribution, $ z(x)=\frac{f(x)}{\bar{F}(x)} $
$ q_j $ Order quantity, where $ j=B,T $ (the retailer's decision variable)
$ \lambda_i $ Loss-aversion coefficient, where $ i=S,R $ denotes the supplier or retailer,
respectively
$ \pi_{ij} $ Profit of player $ i $ under financing scheme $ j $, where $ i=S,R $ and $ j=B,T $
$ U(\pi) $ Utility function
$ EU(\pi_{ij}) $ Expected utility function
$ \Omega $ Retailer's initial capital level
$ r_f $ Risk-free interest rate
$ r_B $ Interest rate of bank loans
$ r_T $ Interest rate of trade credit (the supplier's decision variable)
For notation purposes, we use the symbols "BCF" and "TCF" to represent "bank credit financing" and "trade credit financing", respectively. In addition, for convenience, we refer to the supplier as "she" and the retailer as "he".
Notation Definition
$ p $ Retail price
$ c $ Production cost
$ w_j $ Wholesale price, where $ j=B,T $ denotes bank credit financing or trade credit
financing, respectively (the supplier's decision variable)
$ X $ Random demand, defined over continuous interval$ [0,+∞) $
$ f(X) $ Probability density function of $ X $
$ F(X) $ Cumulative distribution function of $ X $
$ z(X) $ Failure rate of the demand distribution, $ z(x)=\frac{f(x)}{\bar{F}(x)} $
$ q_j $ Order quantity, where $ j=B,T $ (the retailer's decision variable)
$ \lambda_i $ Loss-aversion coefficient, where $ i=S,R $ denotes the supplier or retailer,
respectively
$ \pi_{ij} $ Profit of player $ i $ under financing scheme $ j $, where $ i=S,R $ and $ j=B,T $
$ U(\pi) $ Utility function
$ EU(\pi_{ij}) $ Expected utility function
$ \Omega $ Retailer's initial capital level
$ r_f $ Risk-free interest rate
$ r_B $ Interest rate of bank loans
$ r_T $ Interest rate of trade credit (the supplier's decision variable)
For notation purposes, we use the symbols "BCF" and "TCF" to represent "bank credit financing" and "trade credit financing", respectively. In addition, for convenience, we refer to the supplier as "she" and the retailer as "he".
Table 3.  Sensitivity analysis with respect to $ {\lambda _R}$
$ \lambda_R $ $ w_B $ $ w_T $ $ q_B $ $ q_T $ $ \Delta EU(\pi_{R}) $ $ \Delta EU(\pi_{S}) $ $ \Delta EU(\pi_{SC}) $
1.0 56.0 64.7 105.8 112.9 -681.9 921.6 239.7
1.5 53.1 59.3 104.3 110.4 -512.6 690.1 177.5
2.0 50.5 55.1 102.7 108.5 -379.6 536.9 157.3
2.5 48.3 51.7 101.2 106.7 -280.4 426.7 146.3
3.0 46.3 48.8 99.9 105.2 -207.1 343.6 136.5
3.5 44.5 46.4 98.8 103.6 -149.3 277.3 128.0
4.0 43.0 44.2 97.3 102.4 -75.8 224.0 148.2
4.5 41.6 42.4 95.9 100.7 -37.0 180.0 143.0
5.0 40.3 40.7 94.6 99.0 -8.5 142.9 134.3
5.5 39.1 39.2 93.4 98.1 38.4 111.3 149.7
6.0 38.0 37.8 92.1 96.9 73.4 84.1 157.5
6.5 37.0 36.6 90.8 95.5 95.0 60.3 155.3
7.0 36.0 35.5 89.9 94.0 118.3 39.9 158.2
7.5 35.1 34.4 88.7 93.1 135.3 21.2 156.5
8.0 34.3 33.4 87.3 91.9 153.1 4.6 157.7
8.5 33.5 32.5 86.4 90.8 155.4 -10.4 145.0
9.0 32.8 31.7 85.2 89.5 157.0 -23.8 133.2
9.5 32.2 30.9 83.6 88.4 175.9 -35.9 140.0
10.0 31.6 30.2 82.3 87.3 187.2 -47.1 140.1
$ \lambda_R $ $ w_B $ $ w_T $ $ q_B $ $ q_T $ $ \Delta EU(\pi_{R}) $ $ \Delta EU(\pi_{S}) $ $ \Delta EU(\pi_{SC}) $
1.0 56.0 64.7 105.8 112.9 -681.9 921.6 239.7
1.5 53.1 59.3 104.3 110.4 -512.6 690.1 177.5
2.0 50.5 55.1 102.7 108.5 -379.6 536.9 157.3
2.5 48.3 51.7 101.2 106.7 -280.4 426.7 146.3
3.0 46.3 48.8 99.9 105.2 -207.1 343.6 136.5
3.5 44.5 46.4 98.8 103.6 -149.3 277.3 128.0
4.0 43.0 44.2 97.3 102.4 -75.8 224.0 148.2
4.5 41.6 42.4 95.9 100.7 -37.0 180.0 143.0
5.0 40.3 40.7 94.6 99.0 -8.5 142.9 134.3
5.5 39.1 39.2 93.4 98.1 38.4 111.3 149.7
6.0 38.0 37.8 92.1 96.9 73.4 84.1 157.5
6.5 37.0 36.6 90.8 95.5 95.0 60.3 155.3
7.0 36.0 35.5 89.9 94.0 118.3 39.9 158.2
7.5 35.1 34.4 88.7 93.1 135.3 21.2 156.5
8.0 34.3 33.4 87.3 91.9 153.1 4.6 157.7
8.5 33.5 32.5 86.4 90.8 155.4 -10.4 145.0
9.0 32.8 31.7 85.2 89.5 157.0 -23.8 133.2
9.5 32.2 30.9 83.6 88.4 175.9 -35.9 140.0
10.0 31.6 30.2 82.3 87.3 187.2 -47.1 140.1
Table 4.  Sensitivity analysis with respect to $ {\lambda _S}$
$ \lambda_S $ $ w_B $ $ w_T $ $ q_B $ $ q_T $ $ \Delta EU(\pi_{R}) $ $ \Delta EU(\pi_{S}) $ $ \Delta EU(\pi_{SC}) $
1 65.2 77.8 104.7 116.2 -1642.9 1649.4 6.5
5 65.2 78.6 104.7 114.3 -1711.1 1521.3 -189.8
10 65.2 79.6 104.7 112.0 -1788.8 1369.2 -419.6
15 65.2 80.5 104.7 109.7 -1859.1 1225.7 -633.4
20 65.2 81.3 104.7 107.5 -1922.8 1090.5 -832.3
25 65.2 82.1 104.7 105.3 -1980.4 963.1 -1017.3
30 65.2 82.8 104.7 103.1 -2032.5 843.3 -1189.2
35 65.2 83.4 104.7 100.9 -2079.6 730.6 -1349.0
40 65.2 84.1 104.7 98.9 -2122.0 624.8 -1497.2
45 65.2 84.6 104.7 96.8 -2160.3 525.5 -1634.8
50 65.2 85.1 104.7 94.8 -2194.6 432.3 -1762.3
55 65.2 85.6 104.7 92.9 -2225.5 344.8 -1880.7
60 65.2 86.0 104.7 91.0 -2253.2 262.8 -1990.4
65 65.2 86.3 104.7 89.2 -2278.1 185.9 -2092.2
70 65.2 86.7 104.7 87.5 -2300.4 113.6 -2186.8
75 65.2 87.0 104.7 85.8 -2320.5 45.8 -2274.7
80 65.2 87.3 104.7 84.2 -2338.6 -17.9 -2356.5
85 65.2 87.5 104.7 82.7 -2354.9 -77.9 -2432.8
90 65.2 87.7 104.7 81.2 -2369.6 -134.3 -2503.9
95 65.2 88.0 104.7 79.8 -2382.9 -187.5 -2570.4
100 65.2 88.2 104.7 78.5 -2394.9 -237.7 -2632.6
$ \lambda_S $ $ w_B $ $ w_T $ $ q_B $ $ q_T $ $ \Delta EU(\pi_{R}) $ $ \Delta EU(\pi_{S}) $ $ \Delta EU(\pi_{SC}) $
1 65.2 77.8 104.7 116.2 -1642.9 1649.4 6.5
5 65.2 78.6 104.7 114.3 -1711.1 1521.3 -189.8
10 65.2 79.6 104.7 112.0 -1788.8 1369.2 -419.6
15 65.2 80.5 104.7 109.7 -1859.1 1225.7 -633.4
20 65.2 81.3 104.7 107.5 -1922.8 1090.5 -832.3
25 65.2 82.1 104.7 105.3 -1980.4 963.1 -1017.3
30 65.2 82.8 104.7 103.1 -2032.5 843.3 -1189.2
35 65.2 83.4 104.7 100.9 -2079.6 730.6 -1349.0
40 65.2 84.1 104.7 98.9 -2122.0 624.8 -1497.2
45 65.2 84.6 104.7 96.8 -2160.3 525.5 -1634.8
50 65.2 85.1 104.7 94.8 -2194.6 432.3 -1762.3
55 65.2 85.6 104.7 92.9 -2225.5 344.8 -1880.7
60 65.2 86.0 104.7 91.0 -2253.2 262.8 -1990.4
65 65.2 86.3 104.7 89.2 -2278.1 185.9 -2092.2
70 65.2 86.7 104.7 87.5 -2300.4 113.6 -2186.8
75 65.2 87.0 104.7 85.8 -2320.5 45.8 -2274.7
80 65.2 87.3 104.7 84.2 -2338.6 -17.9 -2356.5
85 65.2 87.5 104.7 82.7 -2354.9 -77.9 -2432.8
90 65.2 87.7 104.7 81.2 -2369.6 -134.3 -2503.9
95 65.2 88.0 104.7 79.8 -2382.9 -187.5 -2570.4
100 65.2 88.2 104.7 78.5 -2394.9 -237.7 -2632.6
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