Robust Dynamic Staffing with Predictions

Abstract

We consider a natural dynamic staffing problem in which a decision-maker sequentially hires staff over a finite time horizon to meet an unknown target demand at the end. The decision-maker also receives a sequence of predictions about the demand that become increasingly more accurate over time. Consequently, the decision-maker prefers to delay hiring decisions to avoid overstaffing. However, workers’ availability decreases over time, resulting in a fundamental trade-off between securing staff early (thus risking overstaffing) versus hiring later based on more accurate predictions (but risking understaffing). This problem is primarily motivated by the staffing challenges that arise in last-mile delivery operations. A company such as Amazon has access to flexible gig economy workers whose availability decreases closer to the target operating day, but they can be hired at any time before that day if they are available.

 We study the above problem when predictions take the form of uncertainty intervals that encompass the true demand. The goal of the decision-maker is to minimize the staffing imbalance cost at the end of the horizon against any sequence of prediction intervals being chosen by an adversary. Our main result is the characterization of a simple and computationally efficient online algorithm that obtains the optimal worst-case imbalance cost; said differently, it is minimax optimal. At a high level, our algorithm relies on identifying a restricted adversary against which we can characterize the minimax optimal cost by solving a certain offline LP. We then show how to emulate the LP solution in a general instance (i.e., when facing an unrestricted adversary) to obtain a cost bounded above by the LP’s objective. As our base model, we consider staffing for one target demand. We also consider generalizations to multiple target demands with either an egalitarian cost objective (i.e., the worst cost across demands) or a utilitarian cost objective (i.e., sum of costs), to the case where the hiring decisions can be reversed in given discharging costs, and to the case where the prediction intervals are not consistent. We show how to extend our LP-based emulator minimax optimal policy to these settings.

Keywords: Last-mile delivery, online algorithms, robust decision making, sequential staffing, predictions