Robust Dynamic Staffing with Predictions

TL;DR

This work studies robust dynamic staffing where a planner hires across a planning horizon to meet an end-of-horizon demand that is revealed only at the operating day. Predictions arrive as intervals that become tighter over time while worker availability declines, creating a fundamental trade-off between hiring early (more supply but less accurate forecasts) and hiring late (better forecasts but scarcer supply). The authors propose a simple online algorithm, the LP-based emulator, and prove it is minimax optimal by first solving a polynomial-size offline LP against a restricted adversary and then emulating its solution online; they extend the framework to multiple stations, costly hiring/releasing, and joint hiring-staffing objectives, with a practical resolving variant. Numerical experiments demonstrate the approach outperforms Bayesian heuristics in both cost and speed and is competitive with Bayesian-optimal policies when computable. Overall, the paper provides a distribution-agnostic, robust online decision framework for sequential staffing under evolving information and diminishing supply, with broad applicability beyond last-mile logistics.

Abstract

We consider a natural dynamic staffing problem in which a decision-maker sequentially hires workers over a finite horizon to meet an unknown demand revealed at the end. Predictions about demand arrive over time and become increasingly accurate, while worker availability decreases. This creates a fundamental trade-off between hiring early to avoid understaffing (when workers are more available but forecasts are less reliable) and hiring late to avoid overstaffing (when forecasts are more accurate but availability is lower). This problem is motivated by last-mile delivery operations, where companies such as Amazon rely on gig-economy workers whose availability declines closer to the operating day. To address practical limitations of Bayesian models (in particular, to remain agnostic to the underlying forecasting method), we study this problem under adversarial predictions. In this model, sequential predictions are adversarially chosen uncertainty intervals that (approximately) contain the true demand. The objective is to minimize worst-case staffing imbalance cost. Our main result is a simple and computationally efficient online algorithm that is minimax optimal. We first characterize the minimax cost against a restricted adversary via a polynomial-size linear program, then show how to emulate this solution in the general case. While our base model focuses on a single demand, we extend the framework to multiple demands (with egalitarian/utilitarian objectives), to settings with costly reversals of hiring decisions, and to inconsistent prediction intervals. We also introduce a practical "re-solving" variant of our algorithm, which we prove is also minimax optimal. Finally we conduct numerical experiments showing that our algorithms outperform Bayesian heuristics in both cost and speed, and are competitive with (approximate or exact) Bayesian-optimal policies when those can be computed.

Figures (4)

• Figure 1: The fundamental tradeoff between using supply availability and prediction information.
• Figure 2: Geometric interpretation of a run of \ref{['alg:opt simple instance']} when $\Gamma\in[0,C\cdot(\rho_1\cdot s-L_1)]$. The heights of red rectangles represent hiring decisions before the supply constraint binds. The height of the blue rectangle represents hiring on day $t^\dagger$, the last day of hiring, when all remaining workers are hired. The area of each rectangle corresponds to the (effective) amount of initial supply used on that day; hence, the total area of all rectangles equals the initial supply $s$.
• Figure 3: Graphical illustration of the minimax optimal algorithm facing the prediction sequence $\boldsymbol{\mathcal{P}}$ in example instances of the single-pool problem (with perfectly consistent predictions). We set $T= 10$, $c = C = 1$, $\rho_t = 1 - 0.8^{T - t + 1}$ for all $t$, $L_0 = 0,R_{0} = 1$. In (a), (b), $\Delta_t = 1 - 0.5^{T - t}$ for all $t\in[T]$; while in (c) $\Delta_t = 1$ for all $t\in[T - 1]$ and $\Delta_T = 0.3$. Supply $s = 4, 0.6, 2$ in (a), (b), (c), respectively. The red dotted bar is the prediction revealed in $\boldsymbol{\mathcal{P}}$. The black (gray) bar is the number of workers hired on (before) that day. The optimal minimax cost $\Gamma^*$ is reported in each subfigure.
• Figure EC.1: Graphical illustration of the experiment setup in \ref{['apx:numerical:predictions from samples']}.

Theorems & Definitions (35)

• Definition 1: Cost guarantee
• Remark 1
• Lemma 1
• Proposition 1
• Remark 2
• Lemma 2
• Lemma 3
• Theorem 1
• Lemma 4
• Theorem 2