针对具有风险厌恶的零售商，建立了权衡期望利润和条件风险值（CVaR）的均值-风险库存优化模型，给出了离散需求分布不确定条件下能实现帕累托最优但具有较高保守性和非帕累托最优但具有较低保守性的两种鲁棒对应。针对不确定需求分布，在仅知历史需求样本数据情况下，应用统计推断理论构建了满足一定置信水平的基于似然估计的需求概率分布不确定集。在此基础上，运用拉格朗日对偶理论，将上述两种鲁棒对应模型转化为易于求解的凹优化问题，并证明了其与原问题的等价性。最后，针对实际案例进行了数值计算，分析了不同系统参数和样本规模对零售商最优库存决策及其运作绩效的影响，并给出了零售商期望利润和条件风险值两个目标权衡的帕累托有效前沿。结果表明，采用基于似然估计的鲁棒优化方法得到的零售商库存策略具有良好鲁棒性，能够有效抑制需求分布不确定性对零售商库存绩效的影响。而且，历史需求样本规模越大，鲁棒库存策略下的零售商运作绩效越接近最优情况。进一步，通过对比发现，两种鲁棒对应模型虽然保守性不同，但在最终库存策略上保持一致。
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.
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