On the optimal solution of a distributionally robust multi-product newsvendor problem
研究了在产能约束下仅知需求均值和方差的多产品报童问题，给出了最优产能分配方案，发现存在阈值决定产品是否被分配产能，且优先顺序由单位成本、缺货成本和需求统计量决定。
This paper derives the optimal solution for a distributionally robust multi-product newsvendor problem, in which different products are produced under a capacity constraint, while only the mean and variance of the product demand are known. The problem aims to find a capacity allocation scheme to minimize the system cost of the worst-case among all possible demand distributions. When the total capacity is among certain ranges, the optimal solution has a closed-form. For other ranges, the optimal solution can be derived by solving one equation. The solution shows that there exist a number of threshold values for the capacity, below which some products are neglected (allocated with zero capacity). Besides, in the optimal solution, which products are prioritized for production is determined by the order of an index measured by the unit production cost, unit shortage cost, demand mean, and demand variance. For a special case where the cost structures for different products are identical, a closed-form solution is derived for any total capacity value. For this special case, the index order is simply determined by the coefficient of variation of each product demand. Sensitivity analysis shows that when the capacity is abundant, a larger demand variability of one product may cause a higher production quantity for this product; while if the capacity is tight, a larger demand variability cause a lower production quantity. For a set of test problems, the performance of the robust optimization solution is quite close to that of the stochastic optimization solution.