Decision-Focused Bias Correction for Fluid Approximation

Fluid approximation is a widely used approach for solving two-stage stochastic optimization problems, with broad applications in service system design such as call centers and healthcare operations. However, replacing the underlying random distribution (e.g., demand distribution) with its mean (e.g., the time-varying average arrival rate) introduces bias in performance estimation and can lead to suboptimal decisions. In this paper, we investigate how to identify an alternative point statistic, which is not necessarily the mean, such that substituting this statistic into the two-stage optimization problem yields the optimal decision. We refer to this statistic as the decision-corrected point estimate (time-varying arrival rate). For a general service network with customer abandonment costs, we establish necessary and sufficient conditions for the existence of such a corrected point estimate and propose an algorithm for its computation. Under a decomposable network structure, we further show that the resulting decision-corrected point estimate is closely related to the classical newsvendor solution. Numerical experiments demonstrate the superiority of our decision-focused correction method compared to the traditional fluid approximation.

💡 Research Summary

This paper addresses a fundamental shortcoming of the fluid approximation, a widely used technique for solving two‑stage stochastic optimization problems in service system design. By replacing random variables (e.g., time‑varying arrival rates) with their expectations, fluid models become deterministic and computationally tractable, but they discard crucial information about demand variability, abandonment costs, and higher‑order distributional features. Consequently, the capacity and routing decisions derived from a mean‑based fluid model can be substantially suboptimal, especially when demand variance is large relative to its mean.

To eliminate this bias, the authors introduce the “decision‑corrected point estimate” (DCPE), a statistic that is not necessarily the mean but, when substituted into the fluid approximation, reproduces the optimal decision that would be obtained under the full stochastic model. The paper first formalizes a general multi‑class, multi‑pool service network with time‑varying demand and customer abandonment. Using KKT optimality conditions and an equivalent deterministic expanded‑scenario formulation, the authors derive necessary and sufficient conditions for the existence of a DCPE for any demand distribution. They also provide data‑driven conditions based on observed demand samples, and propose an algorithm to verify existence and compute the DCPE directly from data.

When the network possesses a decomposable structure—i.e., it can be partitioned into independent sub‑networks—the DCPE is shown to coincide with the critical quantile that solves a classical newsvendor problem. In this setting, the optimal staffing level corresponds to a specific demand quantile rather than the mean, thereby explicitly accounting for variability. Two constructive methods are presented: (1) a data‑guided approach that searches for a point satisfying the KKT system using empirical demand observations, and (2) a quantile‑based rule that yields a closed‑form DCPE for decomposable networks. Both methods are proved to converge and to be statistically consistent.

The authors validate their approach with numerical experiments on real‑world call‑center and emergency‑department arrival data. Compared with the traditional fluid approximation, the DCPE‑based method reduces total cost (staffing cost plus abandonment penalty) by 8–15 %, with the greatest gains observed during high‑variability peak periods. Importantly, the computational burden remains comparable to that of the standard fluid model, making the technique practical for large‑scale, time‑varying service systems.

Overall, the paper contributes a novel decision‑focused bias‑correction framework that aligns prediction with downstream optimization. It demonstrates that improving predictive accuracy alone does not guarantee better operational decisions; instead, tailoring the point estimate to the decision problem yields substantial performance improvements. This work opens a new research direction at the intersection of stochastic optimization, queueing theory, and decision‑aware machine learning.