Coherent Risk Measures in Inventory Problems

Shabbir Ahmed; Ulas Cakmak; Alexander Shapiro

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SPEPS Preprint

Author(s):	Shabbir Ahmed, Georgia Insitute of Technology Ulas Cakmak, Georgia Insitute of Technology Alexander Shapiro, Georgia Insitute of Technology
Title:	Coherent Risk Measures in Inventory Problems
Date of Acceptance:	02.01.2006
Submission Date:	21.12.2005
Series Title:	Stochastic Programming E-Print Series (SPEPS)
Editors:	Julie L. Higle; Werner Römisch; Surrajeet Sen
Complete Preprint:	pdf (urn:nbn:de:kobv:11-10059972)
Keywords (eng):	dynamic programming, coherent risk measures, inventory models, newsvendor problem, mean-absolute deviation, conditional-value-at-risk
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Abstract (eng):

We analyze an extension of the classical multi-period, single-item, linear cost inventory problem where the objective function is a coherent risk measure. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min-max type formulations. For the single period newsvendor problem, we show that the structure of the optimal solution of the risk averse model is similar to that of the classical expected value problem. For a finite horizon dynamic inventory model, we show that, again, the optimal policy has a similar structure as that of the expected value problem. This result carries over even to the case when there is a fixed ordering cost. We also analyze monotonicity properties of the optimal order quantity with respect to the degree of risk aversion for certain risk measures.

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