\((Q,r)\) model with \(CVaR_\alpha\) of costs minimization. (English) Zbl 1365.90011

Summary: In the classical stochastic continuous review, \((Q,r)\) model [Y.-S. Zheng, Manage. Sci. 38, No. 1, 87–103 (1992; Zbl 0764.90029); P. H. Zipkin, Foundations of inventory management. 2nd ed. New York: McGraw-Hill (2000; Zbl 1370.90005)], the inventory cost \(c(Q,r)\) has an averaging term which is given as an integral of the expected costs over the different inventory positions during the lead time on any given cycle. The main objective of the article is to study risk averse optimization in the classical \((Q,r)\) model using \(CVaR_{\alpha}\) as a coherent risk measure with respect to the probability distribution of the demand \(D\) on inventory position costs (the sum of the inventory holding and backorder penality cost), for any given (generic) confidence level \(\alpha\in[0,1)\).
We show that the objective function is jointly convex in \((Q,r)\). We also compare the risk averse solution and some other solutions in both analytical and computational ways. Additionally, some general and useful results are obtained.


90B05 Inventory, storage, reservoirs
Full Text: DOI


[1] S. Ahmed, Coherent risk measures in inventory problems,, European Journal of Operational Research, 1, 226, (2007) · Zbl 1128.90002
[2] P. Artzner, Coherent measure of risk,, Mathematical Finance, 9, 203, (1999) · Zbl 0980.91042
[3] X. Chen, Risk aversion in inventory management,, Operations Research, 55, 828, (2007) · Zbl 1167.90317
[4] L. Cheng, Bilevel newsvendor models considering retailer with CVaR objective,, Computers Industrial Engineering, 57, 310, (2009)
[5] A. Federgruen, A simple and efficient algorithm for computing optimal (r, Q) Policies in continuous-review stochastic inventory systems,, Operations Research, 40, 808, (1992) · Zbl 0764.90023
[6] J. Gotoh, Newsvendor solutions via conditional value-at-risk minimization,, EuropeanJournal of Operational Research, 179, 80, (2007) · Zbl 1275.90057
[7] G. Hadley, <em>Analysis of Inventory Systems</em>,, \(2^{nd}\) edition, (1963) · Zbl 0133.42901
[8] W. J. Hopp, <em>Factory Physics</em>,, \(2^{nd}\) edition, (2001)
[9] S. Moosa, A note on evaluating the risk in continuous review inventory systems,, International Journal of Production Research, 47, 5543, (2009) · Zbl 1198.90042
[10] J. G. Murillo, <em>Riesgo Operativo: Técnicas de modelación cuantitativa</em>,, \(1^{st}\) Sello Editorial Universidad de Medellín, (2014)
[11] G. Pflug, Some remarks on the value-at-risk and the conditional value-at-risk,, in Probabilistic Constrained Optimization, 272, (2000) · Zbl 0994.91031
[12] D. E. Platt, Tractable (Q, R) heuristic models for constrained service levels,, Management Science, 43, 951, (1997) · Zbl 0890.90051
[13] M. E. Puerta, <em>Matemáticas Aplicadas: Optimización de Inventarios Aleatorios</em>,, \(1^{st}\) Sello Editorial Universidad de Medellín, (2011)
[14] R. T. Rockafellar, Conditional Value-at-Risk for general loss distributions,, Journal of Banking and Finance, 23, 1443, (2002)
[15] H. N. Shi, The Schur-convexity of the mean of a convex function,, Applied Mathematics Letters, 22, 932, (2009) · Zbl 1180.26009
[16] R. Vinod, On incorporating business risk into continuous review inventory models,, European Journal of Operational Research, 75, 136, (1994) · Zbl 0809.90040
[17] X. M. Zhang, Convexity of the integral arithmetic mean of a convex function,, Rocky Mountain Journal of Mathematics, 40, 1061, (2010) · Zbl 1200.26021
[18] Y. Zheng, On properties of stochastic inventory systems,, Rocky Mountain Journal of Mathematics, 38, 87, (1992) · Zbl 0764.90029
[19] P. H. Zipkin, <em>Foundations of Inventory Management</em>,, \(2^{nd}\) edition, (2000) · Zbl 1370.90005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.