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:我研究了一个一般收入管理问题,在这个问题中, 客户按顺序在n美元周期内到达, 您必须动态地决定要满足哪个问题。 满足这一时期- 美元客户生产通用美元 u{ t} 美元, 并在\ mathbb{ R} 美元 中减少你的存货持有量。 客户矢量, $ (u ⁇ t}, A}} $) 美元, 是i. d. d. 美元, 由一定数量的连续分配和从约束的离散或连续分配中抽取 $ 美元 。 我研究这个系统的遗憾, 这是如果你不必在飞上做决定, 你可以得到的额外效用。 我表明,如果你最初的库存储备量以 n美元线度计算, 那么你的遗憾是 $ ( log (n) 美元 美元, 以 n\\ rightrow 美元 美元 。 我提供一个简单的政策, 实现这个 美元\\ The log\ rental ralal exprecreal ral 结果, 20美元。