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Decision-Focused Bias Correction for Fluid Approximation

Can Er, Mo Liu

TL;DR

This work identifies and remedies biases introduced by fluid approximations in two-stage service-system optimization. It defines a decision-corrected arrival rate that preserves optimal staffing decisions under the full demand distribution, and establishes necessary and sufficient conditions for its universal or data-dependent existence. The authors develop two constructive corrections—a data-driven method and a quantile-based approach for decomposable networks—proving consistency and linking to classical newsvendor insights in structured cases. Numerical experiments on real arrival data show substantial cost reductions versus traditional mean-based fluid inputs. The framework enables decision-focused forecasting that aligns prediction with downstream staffing and routing decisions while retaining the tractability of fluid models.

Abstract

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.

Decision-Focused Bias Correction for Fluid Approximation

TL;DR

This work identifies and remedies biases introduced by fluid approximations in two-stage service-system optimization. It defines a decision-corrected arrival rate that preserves optimal staffing decisions under the full demand distribution, and establishes necessary and sufficient conditions for its universal or data-dependent existence. The authors develop two constructive corrections—a data-driven method and a quantile-based approach for decomposable networks—proving consistency and linking to classical newsvendor insights in structured cases. Numerical experiments on real arrival data show substantial cost reductions versus traditional mean-based fluid inputs. The framework enables decision-focused forecasting that aligns prediction with downstream staffing and routing decisions while retaining the tractability of fluid models.

Abstract

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.

Paper Structure

This paper contains 31 sections, 10 theorems, 79 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Fluid approximation methods in service system design
  4. Predict-then-optimize methods
  5. Two Stage Decision-making Model
  6. Problem setup
  7. Application 1: emergency department operations.
  8. Application 2: multi-period multi-item newsvendor problem.
  9. Application 3: call centers.
  10. Fluid approximation
  11. Existence of Unbiased Fluid Approximation
  12. Universal existence of decision-corrected fluid approximation
  13. Distribution-dependent existence of decision-corrected fluid approximation
  14. Correction Method for Fluid Approximation
  15. Data-guided correction for general network
  16. ...and 16 more sections

Key Result

Lemma 1

Fix an arrival profile $\boldsymbol{\lambda} = (\lambda_1,\dots,\lambda_T) \in (\mathbb{R}^n)^T$. For the fluid model eq:fa, there exist dual variables $y_t\in\mathbb{R}^m_+$ (for $A x_t\le b$) and $z_t\in\mathbb{R}^n_+$ (for $R x_t\le \lambda_t$) such that a tuple $(b,\{x_t\}_{t=1}^T;\{y_t,z_t\}_{t (D) Dual feasibility: (CS) Complementary slackness:

Figures (11)

  • Figure 1: Example of demand arrivals for one customer type.
  • Figure 2: Bipartite representation of $n$ customer types and $m$ server types. Edges indicate feasible job assignments. The binary routing matrix $\mathbf{R}$ captures possible assignments; the capacity-consumption matrix $\mathbf{A}$ captures resource usage.
  • Figure 3: Comparison between the traditional fluid approximation and the decision-corrected fluid approximation for one type of customer
  • Figure 4: Counterexample: Non-existence of constant unbiased corrected arrival rate.
  • Figure 5: Representation of a decomposable network
  • ...and 6 more figures

Theorems & Definitions (25)

  • Example 1: One customer type, one server system
  • Definition 1: Decision-corrected arrival rate
  • Example 2: Non-existence of constant decision-corrected arrival rate
  • Example 3: Non-existence of the decision-corrected arrival rate
  • Lemma 1: KKT conditions for the two-stage fluid model
  • proof
  • Lemma 2: Scenario–expansion equivalence
  • proof
  • Theorem 1
  • proof
  • ...and 15 more