# American Institute of Mathematical Sciences

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 & Management Optimization, doi: 10.3934/jimo.2021085
##### References:

show all references

##### References:
Optimal order quantity vs loss aversion level
Optimal order quantity vs target unit profit for different confidence levels
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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