Skip to main content

The optimal order decisions of a risk-averse newsvendor under backlogging


In this paper, we are concerned with the optimal order quantities for a risk-averse newsvendor with backlogging, where it is assumed that all or part of the unsatisfied demands can be fullfilled by backordering. The optimal order quantities are derived under the famous conditional value-at-risk (CVaR) criterion and mean-CVaR criterion on controlling the profit loss due to the uncertainty of market demands. With the optimal order quantities to the proposed models, several important monotone properties are obtained and their relationships with the existing results are discussed. Moreover, it is shown in this paper that low risk means low expected profit while high expected profit comes with high risk. Finally, we present some managerial insights for the risk-averse newsvendor model under backlogging.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.


  1. Agrawal, V., & Seshadri, S. (2000). Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manufacturing and Service Operations Management, 2, 410–423.

    Article  Google Scholar 

  2. Anastasios, X., Dimitrios, V., & Eleftherios, I. (2012). Optimal newsvendor policies for dual-sourcing supply chains: A disruption risk management framework. Computers & Operations Research, 39(2), 350–357.

    Article  Google Scholar 

  3. Bai, C., Govindan, K., Satir, A., & Yan, H. (2019). A novel fuzzy reference-neighborhood rough set approach for green supplier development practices. Annals of Operations Research.

  4. Biswajit, S., & Iikyeong, M. (2014). Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process. International Journal of Production Economics, 155, 204–213.

    Article  Google Scholar 

  5. Bolton, G. E., & Katok, E. (2008). Learning-by-doing in the newsvendor problem: A laboratory investigation of the role of experience and feedback. Manufacturing & Service Operations Management, 10(3), 519–538.

    Article  Google Scholar 

  6. Chen, S., Wang, H., Xie, Y., & Qi, C. (2014). Mean-risk analysis of radio frequency identification technology in supply chain with inventory misplacement: Risk-sharing and coordination. Omega, 46, 86–103.

    Article  Google Scholar 

  7. Chen, Y., Xu, M., & Zhang, Z. G. (2009). A risk-averse newsvendor model under the CVaR criterion. Operations Research, 57, 1040–1044.

    Article  Google Scholar 

  8. Choi, S., & Ruszczyński, A. (2011). A multiproduct risk-averse newsvendor with exponential utility function. European Journal of Operational Research, 214, 78–84.

    Article  Google Scholar 

  9. Choi, T. M., Li, D., & Yan, H. (2008). Mean-variance analysis for the newsvendor problem. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 38, 1169–1180.

    Article  Google Scholar 

  10. Eeckhoudt, L., Gollier, C., & Schlesinger, H. (1995). The risk averse (and prudent) newsboy. Management Science, 41, 786–794.

    Article  Google Scholar 

  11. Emilio, C., Alba, V. O. N., & Pepa, R. C. (2016). Robust newsvendor problem with autoregressive demand. Computers & Operations Research, 68, 123–133.

    Article  Google Scholar 

  12. Eric, T. A., Gavan, J. F., & Duancan, S. (2006). Measuring and mitigating the costs of stockouts. Management Science, 52, 1751–1763.

    Article  Google Scholar 

  13. Gan, X., Sethi, S. P., & Yan, H. (2004). Coordination of supply chains with risk-averse agents. Production and Operations Management, 13, 135–149.

    Article  Google Scholar 

  14. Gan, X., Sethi, S. P., & Yan, H. (2005). Channel coordination with a risk-neutral supplier and a downside-risk-averse retailer. Production and Operations Management, 14, 80–89.

    Article  Google Scholar 

  15. Ghiami, Y., & Beullens, P. (2016). Planning for shortages? Net present value analysis for a deteriorating item with partial backlogging. International Journal of Production Economics, 178, 1–11.

    Article  Google Scholar 

  16. Gruen, T., & Corsten, D. (2002). Rising to the challenge of out-of-stocks. ECR Journal, 2, 45–58.

    Google Scholar 

  17. Gurnani, H., & Tang, C. S. (1999). Note: Optimal ordering decisions with uncertain cost and demand forecast updating. Management Science, 45, 1456–1462.

    Article  Google Scholar 

  18. Jaksic, M., & Fransoo, J. C. (2015). Optimal inventory management with supply backordering. International Journal of Production Economics, 159, 254–264.

    Article  Google Scholar 

  19. Johansen, S. G., & Thorstenson, A. (2014). Emergency orders in the periodic-review inventory system with fixed ordering costs and compound Poisson demand. International Journal of Production Economics, 157, 147–157.

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. Lodree, E. J, Jr. (2007). Advanced supply chain planning with mixtures of backorders, lost sales, and lost contract. European Journal of Operational Research, 181, 168–183.

    Article  Google Scholar 

  22. Mehmet, S., Jacqueline, M. B. R., Rene, H., & Jack, G. (2018). Modeling a green inventory routing problem for perishable products with horizontal collaboration. Computers & Operations Research, 89, 168–182.

    Article  Google Scholar 

  23. Montgomery, D. C., Bazaraa, M. S., & Keswani, A. K. (1973). Inventory models with a mixture of backorders and lostsales. Naval Research Logistics Quarterly, 20, 255–263.

    Article  Google Scholar 

  24. Pando, V., San José, L. A., García-Laguna, J., & Sicilia, J. (2013). A newsboy problem with anemergency order under a general backorder rate function. Omega, 41, 1020–1028.

    Article  Google Scholar 

  25. Pentico, D. W., Toews, C., & Drake, M. J. (2015). Approximating the EOQ with partial backordering at an exponential or rational rate by a constant or linearly changing rate. International Journal of Production Economics, 162, 151–159.

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26, 1443–1471.

    Article  Google Scholar 

  28. San josé, L. A., Sicilia, J., & Garciá-Laguna, J. (2006). Analysis of an inventory sys-tem with exponential partial backordering. International Journal of Production Economics, 100, 76–86.

    Article  Google Scholar 

  29. Schweitzer, M. E., & Cachon, G. P. (2000). Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence. Management Science, 46, 404–420.

    Article  Google Scholar 

  30. Sphicas, G. P. (2014). Generalized EOQ formula using a new parameter: Coefficient of backorder attractiveness. International Journal of Production Economics, 155, 143–147.

    Article  Google Scholar 

  31. Taleizadeh, A. A., Khanbaglo, M. P. S., & Cárdenas-Barrón, L. E. (2016). An EOQ inventory model with partial backordering and reparation of imperfect products. International Journal of Production Economics, 182, 418–434.

    Article  Google Scholar 

  32. Turgut, M., Serhan, D., & Ismail, S. B. (2013). Tactical inventory and backorder decisions for systems with predictable production yield. International Journal of Production Economics, 143, 294–303.

    Article  Google Scholar 

  33. Wang, Y. J., Sun, X. F., & Meng, F. X. (2016). On the conditional and partial trade credit policy with capital constraints: A Stackelberg model. Applied Mathematical Modelling, 40(1), 1–18.

    Article  Google Scholar 

  34. Weng, Z. K. (2004). Coordinating order quantities between the manufacturer and the buyer: A generalized newsvendor model. European Journal of Operational Research, 156, 148–161.

    Article  Google Scholar 

  35. Wouter, L., El-Houssaine, A., Khaled, Ha, & Bernard, P. (2018). Analysis of an improved branch-and-cut formulation for the inventory-routing problem with transshipment. Computers & Operations Research, 98, 137–148.

    Article  Google Scholar 

  36. Wu, J., Li, J., Wang, S., & Cheng, T. C. E. (2009). Mean-variance analysis of the newsvendor model with stockout cost. Omega, 37, 724–730.

    Article  Google Scholar 

  37. Wu, M., Zhu, S. X., & Teunter, R. H. (2014). A risk-averse competitive newsvendor problem under the CVaR criterion. International Journal of Production Economics, 156, 13–23.

    Article  Google Scholar 

  38. Xu, J., Zhou, X. H., Zhang, J. H., & Long, D. Z. Y. (2019). The optimal channel structure with retail costs in a dual-channel supply chain. International Journal of Production Research.

  39. Xu, M., & Li, J. (2010). Optimal decisions when balancing expected profit and conditional value-at-risk in newsvendor models. Journal of Systems Science and Complexity, 23, 1054–1070.

    Article  Google Scholar 

  40. Xu, X., Chan, C. K., & Langevin, A. (2018). Coping with risk management and fill rate in the loss-averse newsvendor model. International Journal of Production Economics, 195, 296–310.

    Article  Google Scholar 

  41. Xu, X., & Chan, F. T. S. (2019). Optimal option purchasing decisions for the risk-averse retailer with shortage cost. Asia-Pacific Journal of Operational Research, 36(2), 1–25.

    Article  Google Scholar 

  42. Xu, X., Chan, F. T. S., & Chan, C. K. (2019). Optimal option purchase decision of a loss-averse retailer under emergent replenishment. International Journal of Production Research, 57(4), 4594–4620.

    Article  Google Scholar 

  43. Xu, X., Meng, Z., Ji, P., Dang, C., & Wang, H. (2016). On the newsvendor model with conditional value-at-Risk of opportunity loss. International Journal of Production Research, 54(8), 2449–2458.

    Article  Google Scholar 

  44. Xu, X., Meng, Z., Shen, R., Jiang, M., & Ji, P. (2015). Optimal decisions for the loss-averse newsvendor problem under CVaR. International Journal of Production Economics, 164, 146–159.

    Article  Google Scholar 

  45. Xu, X., Wang, H., Dang, C., & Ji, P. (2017). The loss-averse newsvendor model with backordering. International Journal of Production Economics, 188, 1–10.

    Article  Google Scholar 

  46. Xue, W., Ma, L., & Shen, H. (2015). Optimal inventory and hedging decisions with CVaR consideration. International Journal of Production Economics, 162, 70–82.

    Article  Google Scholar 

  47. Zhang, J., Liu, H., Yu, G., Ruan, J., & Chan, F. T. S. (2019). A three-stage and multi-objective stochastic programming model to improve the sustainable rescue ability by considering secondary disasters in emergency logistics. Computers & Industrial Engineering, 135, 1145–1154.

    Article  Google Scholar 

  48. Zhang, J., Liu, Y., Zhao, Y., & Deng, T. (2018). Emergency evacuation problem for a multi-source and multi-destination transportation network: Mathematical model and case study. Annals of Operations Research.

  49. Zhang, Y., & Yang, X. (2016). Online ordering policies for a two-product, multi-period stationary newsvendor problem. Computers & Operations Research, 74, 143–151.

    Article  Google Scholar 

  50. Zheng, M., Wu, K., & Shu, Y. (2016). Newsvendor problems with demand forecast updating and supply constraints. Computers & Operations Research, 67, 193–206.

    Article  Google Scholar 

  51. Zhou, Y. J., Chen, X. H., & Wang, Z. R. (2008). Optimal ordering quantities for multi-products with stochastic demand: Return-CVaR model. International Journal of Production Economics, 112, 782–795.

    Article  Google Scholar 

  52. Zhou, Y. W., & Wang, S. D. (2009). Manufacturer-buyer coordination for newsvendor-type-products with two ordering opportunities and partial backorders. European Journal of Operational Research, 198, 958–974.

    Article  Google Scholar 

Download references


This work was supported by the National Natural Science Foundation of China with Grant 71871026, the Shandong Social Science Planning Foundation with Grant 20CJJJ06 and the Innovation Method Fund of China with Grant 2018IM020200. The authors would also like to thank the Qilu Young Scholars and Tang Scholars of Shandong University for financial and technical support.

Author information



Corresponding author

Correspondence to Xinsheng Xu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



Proof of Theorem 4.1


$$\begin{aligned} h(q,v) = v-\frac{1}{1-\alpha }\texttt {E\;}[v-P(q, \xi )]^+. \end{aligned}$$

By the definition of the profit function \(P(q, \xi )\) in (1), it can be easily checked that

$$\begin{aligned} h(q,v)= & {} v -\frac{1}{1-\alpha }\int _0^{q}[v-(p-r)t +(c-r)q]^+dF(t)\nonumber \\&-\frac{1}{1-\alpha }\int _q^{+\infty }[v-(p-c+b -w(p_1)(p_1-c_s+b))q\nonumber \\&-(w(p_1)(p_1-c_s+b)-b )t]^+dF(t). ~~ \end{aligned}$$

It follows from Rockafellar and Uryasev (2002) that h(qv) is jointly concave in (qv).

We note that the CVaR model (4) is equivalent to

$$\begin{aligned} \max _{q\ge 0}[\max _{v \in R} h(q,v)].~~~~~~~~~~~~~~ \end{aligned}$$

To solve this problem, we distinguish between two cases: \( b\le \frac{ w(p_1)(p_1-c_s) }{1-w(p_1)}\) and \(b\ge \frac{ w(p_1)(p_1-c_s) }{1-w(p_1)} \) respectively.

(i) \( b\le \frac{ w(p_1)(p_1-c_s) }{1-w(p_1)}\). The proof below aims to find the optimal solution of the inner problem of (9) when a particular q is given. We consider the following three cases.

Case 1\(v<(r-c)q\). For this case, it follows from (8) that

$$\begin{aligned} h(q, v) = v, \end{aligned}$$

and hence

$$\begin{aligned} \frac{\partial h(q,v)}{\partial v} = 1>0. \end{aligned}$$

It follows from the concavity of \(h(q, \cdot )\) that the maximum of \(h(q, \cdot )\) cannot be attained for this case.

Case 2\((r-c)q \le v < (p-c)q\). For this case, it follows from (8) that

$$\begin{aligned} h(q,v)=v-\frac{1}{1-\alpha }\int _{0}^{\frac{v+(c-r)q}{p-r}}[v-(p-r)t+(c-r)q] dF(t) \end{aligned}$$


$$\begin{aligned} \frac{\partial h(q,v)}{\partial v}=1-\frac{1}{1-\alpha }F \left( \frac{v+(c-r)q}{p-r} \right) . \end{aligned}$$

Obviously, it satisfies \(\frac{\partial h(q,v)}{\partial v}|_{v=(r-c)q}=1>0\). If \(q\ge F^{-1}(1-\alpha )\), we must have

$$\begin{aligned} \frac{\partial h(q,v)}{\partial v}|_{v=(p-c)q}=1-\frac{1}{1-\alpha }F(q)\le 0. \end{aligned}$$

By the continuous differentiability of \(h(q, \cdot )\), there exists \(v^* \in [(r-c)q, (p-c)q)\) such that

$$\begin{aligned} \frac{\partial h(q,v)}{\partial v}|_{v= v^*} = 1-\frac{1}{1-\alpha }F\left( \frac{v^*+(c-r)q}{p-r}\right) =0, \end{aligned}$$

which implies

$$\begin{aligned} v^* = (p-r)F^{-1}(1-\alpha )-(c-r)q. \end{aligned}$$

It follows from the concavity of \(h(q, \cdot )\) that the maximum of \(h(q, \cdot )\) is attained at \(v^*\) when \(q\ge F^{-1}(1-\alpha )\).

Case 3.\(v \ge (p-c)q \). For this case, it follows from (8) that

$$\begin{aligned} h(q,v)= & {} v-\frac{1}{1-\alpha }\int _0^{q}[v-(p-r)t+(c-r)q] dF(t) -\frac{1}{1-\alpha }\int _q^{\frac{v-(p-c+b-w(p_1)(p_1-c_s+b))q }{w(p_1)(p_1-c_s+b)-b}}[v- \\&(p-c+b-w(p_1)(p_1-c_s+b))q-(w(p_1)(p_1-c_s+b)-b)t]dF(t),~~~~~~~~~~~~~~~~ \end{aligned}$$


$$\begin{aligned} \frac{\partial h(q,v)}{\partial v}=1-\frac{1}{1-\alpha } F \left( \frac{v-(p-c+b-w(p_1)(p_1-c_s+b))q }{w(p_1)(p_1-c_s+b)-b} \right) . \end{aligned}$$

Note that \(\alpha >0\), we have

$$\begin{aligned} \lim _{v \rightarrow +\infty } \frac{\partial h(q,v)}{\partial v} < 0. \end{aligned}$$

If \(q\le F^{-1}(1-\alpha )\), we have

$$\begin{aligned} \frac{\partial h(q,v)}{\partial v}|_{v=(p-c)q} = 1-\frac{1}{1-\alpha }F(q) \ge 0. \end{aligned}$$

Due to the continuous differentiability of \(h(q, \cdot )\), there must exist \(v^*\) such that

$$\begin{aligned} \frac{\partial h(q,v)}{\partial v}|_{v= v^*} =1-\frac{1}{1-\alpha } F \left( \frac{v^*-(p-c+b-w(p_1)(p_1-c_s+b))q }{w(p_1)(p_1-c_s+b)-b} \right) =0, \end{aligned}$$

which implies

$$\begin{aligned} v^*=(w(p_1)(p_1-c_s+b)-b)F^{-1}(1-\alpha )+(p-c+b-w(p_1)(p_1-c_s+b))q. \end{aligned}$$

It follows from the concavity of \(h(q, \cdot )\) that the maximum of \(h(q, \cdot )\) is attained at \(v^*\) when \(q\le F^{-1}(1-\alpha )\).

Putting together the three cases, we have proved that the optimal solution \(v^*\) to the inner problem of (9) is given by

$$\begin{aligned} v^{*} = \left\{ \begin{array}{llll} (p-r)F^{-1}(1-\alpha )-(c-r)q&{}~~q\ge F^{-1}(1-\alpha ) ,\\ (w(p_1)(p_1-c_s+b)-b)F^{-1}(1-\alpha )\\ +(p-c+b-w(p_1)(p_1-c_s+b))q &{}~~ q \le F^{-1}(1-\alpha ). \\ \end{array}\right. \end{aligned}$$

Therefore, the problem (9) is reduced to

$$\begin{aligned} \max _{q \ge 0} h(q, v^*).~~ \end{aligned}$$

Since \(v^*\) is a linear function of q, \(h(q, v^*)\) is concave in q due to the fact that h(qv) is jointly concave in (qv). The rest of the proof is to show that the maximum of \(h(q, v^*)\) is attained.

We consider two situations. One is \(q\ge F^{-1}(1-\alpha )\). It follows from (8) and (10) that

$$\begin{aligned}&h(q,v^*) = (p-r)F^{-1}(1-\alpha )- (c-r)q\\&-\frac{1}{1 - \alpha }\int _{0 }^{ F^{-1}(1-\alpha )} [(p-r)(F^{-1}(1-\alpha )-t)]dF(t) \end{aligned}$$


$$\begin{aligned} \frac{\partial h(q,v^*)}{\partial q}=-(c-r)<0. \end{aligned}$$

By the concavity of \(h(q, v^*)\) in q, the maximum of \(h(q, v^*)\) cannot be achieved for this situation.

The remaining situation is \( q\le F^{-1}(1-\alpha ) \). It follows from (8) and (10) that

$$\begin{aligned} h(q,v^*)= & {} (w(p_1)(p_1-c_s+b)-b)F^{-1}(1-\alpha )+(p-c+b-w(p_1)(p_1-c_s+b))q~~~~~~~~~~~~~~~~ \\&- \frac{1}{1-\alpha }\int _{0 }^{q} [(w(p_1)(p_1-c_s+b)-b)F^{-1}(1-\alpha )\\&-(p-r)t+(p-r+b-w(p_1)(p_1-c_s+b))q]dF(t) \\&- \frac{1}{1-\alpha }\int _{q }^{F^{-1}(1-\alpha )}[ (w(p_1)(p_1-c_s+b)-b)(F^{-1}(1-\alpha )-t)]dF(t) \end{aligned}$$


$$\begin{aligned} \frac{\partial h(q,v^*)}{\partial q}=p-c+b-w(p_1)(p_1-c_s+b)-\frac{p-r+b-w(p_1)(p_1-c_s+b)}{1-\alpha }F(q). \end{aligned}$$

Since it satisfies

$$\begin{aligned} \frac{\partial h(q,v^*)}{\partial q}|_{q=0}= & {} p-c+b-w(p_1)(p_1-c_s+b)\ge 0,\\&\frac{\partial h(q,v^*)}{\partial q}|_{q=F^{-1}(1-\alpha )}= -(c-r)\le 0 . \end{aligned}$$

There must exist \(q_1^*\in (0, F^{-1}(1-\alpha ))\) such that

$$\begin{aligned}&\frac{\partial h(q_1^*,v^*)}{\partial q}=p-c+b-w(p_1)(p_1-c_s+b)\\&\qquad -\frac{p-r+b-w(p_1)(p_1-c_s+b)}{1-\alpha }F(q_1^*)=0, \end{aligned}$$

which gives

$$\begin{aligned} q_1^*=F^{-1} \left( \frac{(1-\alpha )(p-c+b-w(p_1)(p_1-c_s+b)}{p-r+b-w(p_1)(p_1-c_s+b)} \right) . \end{aligned}$$

It follows from the concavity of \(h(q, v^*)\) that \(q^*_1\) is the optimal solution of (11), and hence is also the optimal solution of (9) for the case of i).

(ii) \(b\ge \frac{w(p_1)(p_1-c_s) }{1-w(p_1)}\). The proof is similar to the case of i), which is omitted herein. This completes the proof. \(\square \)

Proof of Corollory. 4.2

Since \(F(\cdot )\) is monotonically increasing, it follows from the formula of \(q_1^*\) that \(q_1^*\), as a function of c, \(p_1\), b, r and \(c_s\), has the corresponding increasing or decreasing properties as stated in the corollary. This completes the proof. \(\square \)

Proof of Corollory. 4.3

The proof is similar to that of Corollary. 4.2, which is omitted herein. \(\square \)

Proof of Corollory 4.4

It follows from (1) that

$$\begin{aligned} \texttt {E\;}[P(q, \xi )]= & {} (p-c+b-w(p_1)(p_1-c_s+b))q+(w(p_1)(p_1-c_s+b)-b)E(\xi )\\&-(p-r+b-w(p_1)(p_1-c_s+b))\int _0^q(q-t)dF(t).~~ \end{aligned}$$

We calculate

$$\begin{aligned} \frac{\partial \texttt {E\;}[P(q^*_1,\xi )]}{\partial \alpha }= & {} [p-c+b-w(p_1)(p_1-c_s+b)\nonumber \\&-(p-r+b-w(p_1)(p_1-c_s+b))F(q^*_1)]\frac{\partial q^*_1}{\partial \alpha }. ~ \end{aligned}$$

If \(b \le \frac{w(p_1)(p_1-c_s) }{1-w(p_1)} \), it follows with Theorem 4.1 that \(q^*_1=F^{-1} \Big ( \frac{(1-\alpha )(p-c+b-w(p_1)(p_1-c_s+b))}{p-r+b-w(p_1)(p_1-c_s+b)}\Big )\) and \(\frac{\partial q^*_1 }{\partial \alpha }\le 0\). We recall that

$$\begin{aligned} q^*_0=F^{-1} \left( \frac{p-c+b-w(p_1)(p_1-c_s+b)}{p-r+b-w(p_1)(p_1-c_s+b)} \right) , \end{aligned}$$

which gives

$$\begin{aligned} p-c+b -w(p_1)(p_1-c_s+b)-(p-r+b-w(p_1)(p_1-c_s+b))F(q^*_0)=0. \end{aligned}$$

We also recall \(q_1^* \le q_0^*\). The monotonicity of \(F(\cdot )\) implies

$$\begin{aligned}&p-c+b -w(p_1)(p_1-c_s+b)-(p-r+b-w(p_1)(p_1-c_s+b))F(q^*_1)\ge ~~\\&p-c +b-w(p_1)(p_1-c_s+b)-(p-r+b-w(p_1)(p_1-c_s+b))F(q^*_0)=0. \end{aligned}$$

It follows with (12) and \(\frac{\partial q^*_1 }{\partial \alpha }\le 0\) that

$$\begin{aligned} \frac{\partial \texttt {E\;}[P(q^*_1, \xi )]}{\partial \alpha } \le 0, \end{aligned}$$

which proves that \(\texttt {E\;}[P(q^*_1, \xi )]\) is decreasing in the confidence level \(\alpha \) when \(b \le \frac{w(p_1)(p_1-c_s) }{1-w(p_1)} \). The proof for the case of \(b\ge \frac{w(p_1)(p_1-c_s) }{1-w(p_1)}\) is similar to that for the case of \(b \le \frac{w(p_1)(p_1-c_s) }{1-w(p_1)} \), which is omitted herein. This completes the proof. \(\square \)

Proof of Theorem 4.5

The proof is similar to that for Theorem 4.1, which is omitted herein. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Chan, F.T.S. & Xu, X. The optimal order decisions of a risk-averse newsvendor under backlogging. Ann Oper Res (2020).

Download citation


  • Logistics
  • Risk management
  • Newsvendor model
  • Conditional value-at-risk
  • Backordering
  • Optimal order quantity