I study a general revenue management problem in which $ n $ customers arrive sequentially over $ n $ periods, and you must dynamically decide which to satisfy. Satisfying the period-$ t $ customer yields utility $ u_{t} \in \mathbb{R}_{+} $ and decreases your inventory holdings by $ A_{t} \in \mathbb{R}_{+}^{M} $. The customer vectors, $ (u_{t}, A_{t}')' $, are i.i.d., with $ u_{t} $ drawn from a finite-mean continuous distribution and $ A_{t} $ drawn from a bounded discrete or continuous distribution. I study this system's regret, which is the additional utility you could get if you didn't have to make decisions on the fly. I show that if your initial inventory endowment scales linearly with $ n $ then your expected regret is $ \Theta(\log(n)) $ as $ n \rightarrow \infty $. I provide a simple policy that achieves this $ \Theta(\log(n)) $ regret rate. Finally, I extend this result to Arlotto and Gurich's (2019) multisecretary problem with uniformly distributed secretary valuations.

翻译：我研究一个通用的收益管理问题：$n$个顾客在$n$个周期内依次到达，你必须动态决定满足哪些顾客。满足第$t$期顾客带来效用$u_{t} \in \mathbb{R}_{+}$，同时使库存持有量减少$A_{t} \in \mathbb{R}_{+}^{M}$。顾客向量$(u_{t}, A_{t}')'$是独立同分布的，其中$u_{t}$来自有限均值的连续分布，$A_{t}$来自有界离散或连续分布。我研究该系统的遗憾（即若无需实时决策所能获得的额外效用）。我证明：若初始库存禀赋随$n$线性增长，则当$n \rightarrow \infty$时，期望遗憾为$\Theta(\log(n))$。我给出一个简单策略实现该$\Theta(\log(n))$遗憾率。最后，我将此结果推广到Arlotto与Gurich（2019）提出的秘书估值服从均匀分布的多秘书问题。