## $$(Q,r)$$ 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.

### MSC:

 90B05 Inventory, storage, reservoirs

### Citations:

Zbl 0764.90029; Zbl 1370.90005
Full Text:

### References:

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