Newsvendor problem with random shortage cost under a risk criterion

We study profit maximization vs risk approaches for the standard newsvendor problem with uncertainty in demand as well as a generalized version with uncertainty in the shortage cost (as often applies in practice). We consider two well-known risk approaches: Value-at-Risk (VaR) included as a constraint and Conditional Value-at-Risk (CVaR). We first derive the explicit expressions of the optimal solution with uncertainty of shortage cost under different risk measures and then perform a numerical analysis to quantify the effect of uncertainty in shortage cost and risk measures. For the standard newsvendor problem, we find that the optimal quantity under CVaR is always lower than that under the VaR constraint, which in turn is lower than the order quantity that maximizes the expected profit. Insightful explanations for this result are that: (a) a higher degree of risk aversion drives the newsvendor to order fewer products, increasing the likelihood that all will be sold; (b) this effect is stronger for the CVaR approach as this does not consider the expected profit at all. Another interesting and counter-intuitive observation for the CVaR approach is that a higher retail price may lead to a smaller order quantity, as fewer items need to be sold in order to attain a sufficient profit. These results show that one should be careful in employing the CVaR risk measure for newsvendor type problems. The results remain valid if the shortage cost becomes uncertain. However, increased uncertainty of this type does improve the relative profitability under the CVaR approach by increasing the order quantity under that criterion whilst there is no effect under the other criteria.