Abstract: The problem of inventory optimization for a risk-averse retailer with uncertain discrete demand distribution is studied in this paper. A mean-risk inventory model which can balance the retailer's expected profit and the conditional value-at-risk (CVaR) of the profit by a pessimistic coefficient is developed. To overcome the difficulty of obtaining an inventory policy caused by the demand distribution uncertainty, two robust counterparts based on max-min robust criterion are proposed. The former which maximizes the trade-off between the worst-case expected profit and the worst-case CVaR is pareto efficient but more conservative; while the latter optimizes the worst-case trade-off between the expected profit and the CVaR, and then is non-pareto efficient but less conservative. For uncertain demand distribution, only some historical demand data are assumed to be known. Using statistical inference theory, an uncertain set to which the unknown demand probability belongs is constructed with a certain confidence level based upon the likelihood estimation. Such an uncertain set is then integrated into the above two robust counterparts and regarded as a constraint. By Lagrange dual theory, the two robust counterparts with an uncertain set constraint are transformed into two tractable concave optimization problems which can be solved efficiently. Moreover, a proof is presented to show the equivalence of the transformed tractable models with original ones. At last, some case-oriented numerical examples are executed to analyze the impact of the different system parameters and the demand sample size on the optimal inventory strategy and the operational performance of the retailer. A Pareto frontier between retailer's expectation profit and its conational value-at-risk is also proposed. The results show that the uncertainty in demand distribution will inevitably lead to the inventory performance loss, however, the loss value is relatively small, which indicates the retailer's inventory strategy based on the likelihood estimation is robust, and can effectively restrain the impact of the uncertain demand distribution on the retailer inventory performance. Besides, the more the historical demand samples, the closer the retailer's operational performance under robust inventory strategy to its optimal level. Furthermore, it can be found that the optimal inventory strategies for the above two robust counterpart models are qualitatively equal, although they are different in conservation.

Key words: inventory, uncertainty, mean-risk model, robust optimization, likelihood estimation