Target-Following Online Resource Allocation Using Proxy Assignments

Abstract: We study a target-following variation of online resource allocation. As in classical resource allocation, the decision-maker must assign sequentially arriving jobs to one of multiple available resources. However, in addition to the assignment costs incurred from these decisions, the decision-maker is also penalized for deviating from exogenously given, nonstationary target allocations throughout the horizon. The goal is to minimize the total expected assignment and deviation penalty costs incurred throughout the horizon when the distribution of assignment costs is unknown. In contrast to traditional online resource allocation, in our setting the timing of allocation decisions is critical due to the nonstationarity of allocation targets. Examples of practical problems that fit this framework include many physical resource settings where capacity is time-varying, such as manual warehouse processes where staffing levels change over time, and assignment of packages to outbound trucks whose departure times are scheduled throughout the day. We first show that naive extensions of state-of-the-art algorithms for classical resource allocation problems can fail dramatically when applied to target-following resource allocation. We then propose a novel ``proxy assignment" primal-dual algorithm for the target-following online resource allocation problem that uses current arrivals to simulate the effect of future arrivals. We prove that our algorithm incurs the optimal $O(\sqrt{T})$ regret bound when the assignment costs of the arriving jobs are drawn i.i.d. from a fixed distribution. We demonstrate the practical performance of our approach by conducting numerical experiments on synthetic datasets, as well as real-world datasets from retail fulfillment operations.