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doi: 10.3934/jimo.2021085
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The loss-averse newsvendor problem with quantity-oriented reference point under CVaR criterion

1. 

Department of Basic Science, Wuhan Donghu University, Wuhan 430212, China

2. 

Department of Automation, Tsinghua University, Beijing 100084, China

3. 

Financial Department, Wuhan Business University, Wuhan 430056, China

4. 

Department of Finance and Audit, Army Logistics University, Chongqing 401311, China

5. 

Department of Basic Science, Army Logistics University, Chongqing 401311, China

*Corresponding author: Ying Qiao

Received  December 2019 Revised  December 2020 Early access April 2021

Fund Project: The paper is supported by National Key Research and Development Project of China (No. 2018YFB1702903), Research Project of Hubei Provincial Department of Education (No. B2020240), and Youth Foundation of Wuhan Donghu University (No. 2020dhzk005)

This paper studies a single-period inventory problem with quantity-oriented reference point, where the newsvendor has loss-averse preferences and conditional value-at-risk (CVaR) measure is introduced to hedge against his risk. It is shown there exists a unique optimal order quantity maximizing the CVaR of utility. Moreover, it is decreasing in loss aversion level, confidence level and target unit profit, respectively. Then we establish the sufficient conditions under which the newsvendor's optimal order quantity may be larger than, equal to or less than the classical newsvendor solution. In particular, when the target unit profit is a convex combination of the maximum and minimum, the optimal order quantity is independent of price and cost parameters. Numerical experiments are conducted to illustrate our results and present some managerial insights.

Citation: Wei Liu, Shiji Song, Ying Qiao, Han Zhao, Huachang Wang. The loss-averse newsvendor problem with quantity-oriented reference point under CVaR criterion. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021085
References:
[1]

T. BaiM. Wu and S. X. Zhu, Pricing and ordering by a loss averse newsvendor with reference dependence, Transportation Research Part E: Logistics and Transportation Review, 131 (2019), 343-365.  doi: 10.1016/j.tre.2019.10.003.

[2]

A. O. Brown and C. S. Tang, The impact of alternative performance measures on single-period inventory policy, J. Ind. Manag. Optim., 2 (2006), 297-318.  doi: 10.3934/jimo.2006.2.297.

[3]

F. T. Chan and X. Xu, The loss-averse retailer's order decisions under risk management, Mathematics, 7 (2019), 595. doi: 10.3390/math7070595.

[4]

Y. ChenM. Xu and Z. G. Zhang, A risk-averse newsvendor model under the CVaR criterion, Operations Research, 57 (2009), 1040-1044. 

[5]

T. FengL. 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.

[6]

M. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99.  doi: 10.1287/opre.44.1.87.

[7]

C. Fulga, Portfolio optimization under loss aversion, European J. Oper. Res., 251 (2016), 310-322.  doi: 10.1016/j.ejor.2015.11.038.

[8]

C. Fulga, Portfolio optimization with disutility-based risk measure, European J. Oper. Res., 251 (2016), 541-553.  doi: 10.1016/j.ejor.2015.11.012.

[9]

A. Furnham and H. C. Boo, A literature review of the anchoring effect, The Journal of Socio-Economics, 40 (2011), 35-42.  doi: 10.1016/j.socec.2010.10.008.

[10]

J. Guo and X. D. He, Equilibrium asset pricing with Epstein-Zin and loss-averse investors, J. Econom. Dynam. Control, 76 (2017), 86-108.  doi: 10.1016/j.jedc.2016.12.008.

[11]

F. Herweg, The expectation-based loss-averse newsvendor, Econom. Lett., 120 (2013), 429-432.  doi: 10.1016/j.econlet.2013.05.035.

[12]

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

[13]

M. Khouja, The single-period (news-vendor) problem: Literature review and suggestions for future research, Omega, 27 (1999), 537-553.  doi: 10.1016/S0305-0483(99)00017-1.

[14]

B. LiP. HouP. Chen and Q. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, International Journal of Production Economics, 178 (2016), 154-168. 

[15]

W. LiuS. SongB. Li and C. Wu, A periodic review inventory model with loss-averse retailer, random supply capacity and demand, International Journal of Production Research, 53 (2015), 3623-3634.  doi: 10.1080/00207543.2014.985391.

[16]

W. LiuS. SongY. Qiao and H. Zhao, The loss-averse newsvendor problem with random supply capacity, J. Ind. Manag. Optim., 13 (2017), 1417-1429.  doi: 10.3934/jimo.2016080.

[17]

W. LiuS. Song and C. Wu, Impact of loss aversion on the newsvendor game with product substitution, International Journal of Production Economics, 141 (2013), 352-359.  doi: 10.1016/j.ijpe.2012.08.017.

[18]

W. LiuS. Song and C. Wu, The loss-averse newsvendor problem with random yield, Transactions of the Institute of Measurement and Control, 36 (2014), 312-320. 

[19]

X. Long and J. Nasiry, Prospect theory explains newsvendor behavior: The role of reference points, Management Science, 61 (2015), 3009-3012.  doi: 10.1287/mnsc.2014.2050.

[20]

L. MaW. XueY. Zhao and Q. Zeng, Loss-averse newsvendor problem with supply risk, Journal of the Operational Research Society, 67 (2016), 214-228. 

[21]

L. MaY. ZhaoW. XueT. C. E. Cheng and H. Yan, Loss-averse newsvendor model with two ordering opportunities and market information updating, International Journal of Production Economics, 140 (2012), 912-921.  doi: 10.1016/j.ijpe.2012.07.012.

[22]

P. MandalR. Kaul and T. Jain, Stocking and pricing decisions under endogenous demand and reference point effects, European J. Oper. Res., 264 (2018), 181-199.  doi: 10.1016/j.ejor.2017.05.053.

[23]

Y. QinR. WangA. J. VakhariaY. Chen and M. M. H. Seref, The newsvendor problem: Review and directions for future research, European J. Oper. Res., 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.

[24]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. 

[25]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of banking & finance, 26 (2002), 1443-1471. 

[26]

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

[27]

J. Sun and X. Xu, Coping with loss aversion in the newsvendor model, Discrete Dyn. Nat. Soc., (2015), Art. ID 851586, 11 pp. doi: 10.1155/2015/851586.

[28]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertain, Journal of Risk and Uncertainty, 5 (1992), 297-323. 

[29]

B. Vipin and R. K. Amit, Loss aversion and rationality in the newsvendor problem under recourse option, European J. Oper. Res., 261 (2017), 563-571.  doi: 10.1016/j.ejor.2017.02.012.

[30]

C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452.  doi: 10.1016/j.ijpe.2009.12.007.

[31]

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

[32]

R. Wang and J. Wang, Procurement strategies with quantity-oriented reference point and loss aversion, Omega, 80 (2018), 1-11.  doi: 10.1016/j.omega.2017.08.007.

[33]

M. WuT. Bai and S. X. Zhu, A loss averse competitive newsvendor problem with anchoring, Omega, 81 (2018), 99-111.  doi: 10.1016/j.omega.2017.10.003.

[34]

M. WuS. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, International Journal of Production Economics, 156 (2014), 13-23.  doi: 10.1016/j.ijpe.2014.05.009.

[35]

X. XuF. 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.

[36]

X. XuC. 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.

[37]

X. XuZ. MengR. ShenM. Jiang and P. Ji, Optimal decisions for the loss-averse newsvendor problem under CVaR, International Journal of Production Economics, 164 (2015), 146-159. 

[38]

X. XuH. WangC. 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.

[39]

H. YuJ. Zhai and G.-Y. Chen, Robust optimization for the loss-averse newsvendor problem, J. Optim. Theory Appl., 171 (2016), 1008-1032.  doi: 10.1007/s10957-016-0870-9.

[40]

X.-B. Zhao and W. Geng, A note on "Prospect theory and the newsvendor problem", J. Oper. Res. Soc. China, 3 (2015), 89-94.  doi: 10.1007/s40305-015-0072-4.

show all references

References:
[1]

T. BaiM. Wu and S. X. Zhu, Pricing and ordering by a loss averse newsvendor with reference dependence, Transportation Research Part E: Logistics and Transportation Review, 131 (2019), 343-365.  doi: 10.1016/j.tre.2019.10.003.

[2]

A. O. Brown and C. S. Tang, The impact of alternative performance measures on single-period inventory policy, J. Ind. Manag. Optim., 2 (2006), 297-318.  doi: 10.3934/jimo.2006.2.297.

[3]

F. T. Chan and X. Xu, The loss-averse retailer's order decisions under risk management, Mathematics, 7 (2019), 595. doi: 10.3390/math7070595.

[4]

Y. ChenM. Xu and Z. G. Zhang, A risk-averse newsvendor model under the CVaR criterion, Operations Research, 57 (2009), 1040-1044. 

[5]

T. FengL. 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.

[6]

M. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99.  doi: 10.1287/opre.44.1.87.

[7]

C. Fulga, Portfolio optimization under loss aversion, European J. Oper. Res., 251 (2016), 310-322.  doi: 10.1016/j.ejor.2015.11.038.

[8]

C. Fulga, Portfolio optimization with disutility-based risk measure, European J. Oper. Res., 251 (2016), 541-553.  doi: 10.1016/j.ejor.2015.11.012.

[9]

A. Furnham and H. C. Boo, A literature review of the anchoring effect, The Journal of Socio-Economics, 40 (2011), 35-42.  doi: 10.1016/j.socec.2010.10.008.

[10]

J. Guo and X. D. He, Equilibrium asset pricing with Epstein-Zin and loss-averse investors, J. Econom. Dynam. Control, 76 (2017), 86-108.  doi: 10.1016/j.jedc.2016.12.008.

[11]

F. Herweg, The expectation-based loss-averse newsvendor, Econom. Lett., 120 (2013), 429-432.  doi: 10.1016/j.econlet.2013.05.035.

[12]

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

[13]

M. Khouja, The single-period (news-vendor) problem: Literature review and suggestions for future research, Omega, 27 (1999), 537-553.  doi: 10.1016/S0305-0483(99)00017-1.

[14]

B. LiP. HouP. Chen and Q. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, International Journal of Production Economics, 178 (2016), 154-168. 

[15]

W. LiuS. SongB. Li and C. Wu, A periodic review inventory model with loss-averse retailer, random supply capacity and demand, International Journal of Production Research, 53 (2015), 3623-3634.  doi: 10.1080/00207543.2014.985391.

[16]

W. LiuS. SongY. Qiao and H. Zhao, The loss-averse newsvendor problem with random supply capacity, J. Ind. Manag. Optim., 13 (2017), 1417-1429.  doi: 10.3934/jimo.2016080.

[17]

W. LiuS. Song and C. Wu, Impact of loss aversion on the newsvendor game with product substitution, International Journal of Production Economics, 141 (2013), 352-359.  doi: 10.1016/j.ijpe.2012.08.017.

[18]

W. LiuS. Song and C. Wu, The loss-averse newsvendor problem with random yield, Transactions of the Institute of Measurement and Control, 36 (2014), 312-320. 

[19]

X. Long and J. Nasiry, Prospect theory explains newsvendor behavior: The role of reference points, Management Science, 61 (2015), 3009-3012.  doi: 10.1287/mnsc.2014.2050.

[20]

L. MaW. XueY. Zhao and Q. Zeng, Loss-averse newsvendor problem with supply risk, Journal of the Operational Research Society, 67 (2016), 214-228. 

[21]

L. MaY. ZhaoW. XueT. C. E. Cheng and H. Yan, Loss-averse newsvendor model with two ordering opportunities and market information updating, International Journal of Production Economics, 140 (2012), 912-921.  doi: 10.1016/j.ijpe.2012.07.012.

[22]

P. MandalR. Kaul and T. Jain, Stocking and pricing decisions under endogenous demand and reference point effects, European J. Oper. Res., 264 (2018), 181-199.  doi: 10.1016/j.ejor.2017.05.053.

[23]

Y. QinR. WangA. J. VakhariaY. Chen and M. M. H. Seref, The newsvendor problem: Review and directions for future research, European J. Oper. Res., 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.

[24]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. 

[25]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of banking & finance, 26 (2002), 1443-1471. 

[26]

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

[27]

J. Sun and X. Xu, Coping with loss aversion in the newsvendor model, Discrete Dyn. Nat. Soc., (2015), Art. ID 851586, 11 pp. doi: 10.1155/2015/851586.

[28]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertain, Journal of Risk and Uncertainty, 5 (1992), 297-323. 

[29]

B. Vipin and R. K. Amit, Loss aversion and rationality in the newsvendor problem under recourse option, European J. Oper. Res., 261 (2017), 563-571.  doi: 10.1016/j.ejor.2017.02.012.

[30]

C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452.  doi: 10.1016/j.ijpe.2009.12.007.

[31]

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

[32]

R. Wang and J. Wang, Procurement strategies with quantity-oriented reference point and loss aversion, Omega, 80 (2018), 1-11.  doi: 10.1016/j.omega.2017.08.007.

[33]

M. WuT. Bai and S. X. Zhu, A loss averse competitive newsvendor problem with anchoring, Omega, 81 (2018), 99-111.  doi: 10.1016/j.omega.2017.10.003.

[34]

M. WuS. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, International Journal of Production Economics, 156 (2014), 13-23.  doi: 10.1016/j.ijpe.2014.05.009.

[35]

X. XuF. 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.

[36]

X. XuC. 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.

[37]

X. XuZ. MengR. ShenM. Jiang and P. Ji, Optimal decisions for the loss-averse newsvendor problem under CVaR, International Journal of Production Economics, 164 (2015), 146-159. 

[38]

X. XuH. WangC. 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.

[39]

H. YuJ. Zhai and G.-Y. Chen, Robust optimization for the loss-averse newsvendor problem, J. Optim. Theory Appl., 171 (2016), 1008-1032.  doi: 10.1007/s10957-016-0870-9.

[40]

X.-B. Zhao and W. Geng, A note on "Prospect theory and the newsvendor problem", J. Oper. Res. Soc. China, 3 (2015), 89-94.  doi: 10.1007/s40305-015-0072-4.

Figure 1.  Optimal order quantity vs loss aversion level
Figure 2.  Optimal order quantity vs target unit profit for different confidence levels
Table 1.  Summary of the notations
Notation Description
$ p $ Selling price per unit,
$ c $ Purchasing cost per unit,
$ s $ Salvage value per unit, $ p>c>s $,
$ Q $ Order quantity,
$ D $ Random demand,
$ f(x) $ Probability density function of $ D $,
$ F(x) $ Cumulative distribution function of $ D $,
$ \bar F(x) $ Tail distribution of $ F(x) $, i.e., $ \bar F(x)=1-F(x) $,
$ w_0 $ Target profit per unit,
$ \pi_0 $ Reference point,
$ \pi $ Newsvendor's profit,
$ \lambda $ Loss aversion level, $ \lambda\geq1 $,
$ \alpha $ Confidence level, $ 0\leq\alpha<1 $,
$ \beta $ Coefficient of convex combination, $ 0\leq\beta\leq1 $,
$ Q^* $ Optimal order quantity maximizing CVaR of utility,
$ Q^*_u $ Optimal order quantity maximizing expected utility,
$ Q^*_p $ Optimal order quantity maximizing expected profit
(with reference point),
$ Q^*_0 $ Classical newsvendor solution.
Notation Description
$ p $ Selling price per unit,
$ c $ Purchasing cost per unit,
$ s $ Salvage value per unit, $ p>c>s $,
$ Q $ Order quantity,
$ D $ Random demand,
$ f(x) $ Probability density function of $ D $,
$ F(x) $ Cumulative distribution function of $ D $,
$ \bar F(x) $ Tail distribution of $ F(x) $, i.e., $ \bar F(x)=1-F(x) $,
$ w_0 $ Target profit per unit,
$ \pi_0 $ Reference point,
$ \pi $ Newsvendor's profit,
$ \lambda $ Loss aversion level, $ \lambda\geq1 $,
$ \alpha $ Confidence level, $ 0\leq\alpha<1 $,
$ \beta $ Coefficient of convex combination, $ 0\leq\beta\leq1 $,
$ Q^* $ Optimal order quantity maximizing CVaR of utility,
$ Q^*_u $ Optimal order quantity maximizing expected utility,
$ Q^*_p $ Optimal order quantity maximizing expected profit
(with reference point),
$ Q^*_0 $ Classical newsvendor solution.
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