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arxiv: 1907.03219 · v1 · pith:BWX4PLYFnew · submitted 2019-07-07 · 🧮 math.OC

Data-Driven Distributionally Robust Appointment Scheduling over Wasserstein Balls

Ruiwei Jiang , Minseok Ryu , Guanglin Xu This is my paper

Pith reviewed 2026-05-25 01:51 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributionally robust optimizationappointment schedulingWasserstein distancedata-driven optimizationcopositive programmingsemidefinite programmingstochastic programming
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The pith

A data-driven distributionally robust model for appointment scheduling converges to the true optimum as the number of historical samples grows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers single-server appointment scheduling where service durations are random and no-shows may occur, but the probability distribution of these uncertainties is unknown and only observed through historical data treated as independent samples. It constructs distributionally robust optimization models that minimize the worst-case expected total cost over all distributions inside a Wasserstein ball centered at the empirical distribution from the data. A central result is that both the optimal cost value and the optimal appointment schedules converge to those of the underlying true stochastic model as the sample size increases. The models are reformulated into copositive programs that admit high-quality semidefinite programming approximations and, under mild conditions, into polynomial-sized linear programs. Numerical tests indicate that the resulting schedules achieve better out-of-sample performance than existing benchmark methods.

Core claim

For two stochastic appointment models—one with random service times and one that also includes random no-shows—the authors formulate data-driven DRO problems over Wasserstein balls. They prove that the optimal value and the set of optimal schedules converge to those of the true probability distribution as the number of samples tends to infinity. The resulting infinite-dimensional problems are recast as copositive programs, which are then approximated by tractable semidefinite programs, and under additional conditions reduced exactly to finite linear programs of polynomial size.

What carries the argument

The Wasserstein-ball ambiguity set in the distributionally robust formulation, which simultaneously encodes robustness to distributional uncertainty and guarantees asymptotic consistency with the true distribution.

If this is right

  • Optimal appointment schedules obtained from the model become arbitrarily close to the true optimum for sufficiently large data sets.
  • The models can be solved to high accuracy via standard semidefinite programming solvers.
  • Under mild conditions the same models reduce to ordinary linear programs whose size grows only polynomially with the number of appointments.
  • The schedules produced by the approach exhibit improved performance on new, unseen realizations compared with two existing methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Wasserstein DRO construction could be applied to multi-server or networked appointment systems while preserving the convergence property.
  • If data arrive from a process that is not i.i.d., the convergence guarantee would no longer apply and a different ambiguity set might be needed.
  • The copositive reformulation technique may transfer to other appointment or queueing problems that admit similar cost structures.

Load-bearing premise

The historical data consist of independent samples drawn from the unknown true distribution.

What would settle it

An experiment in which the gap between the DRO optimal cost and the true-model optimal cost fails to shrink toward zero as the number of historical samples is increased while keeping the true distribution fixed.

Figures

Figures reproduced from arXiv: 1907.03219 by Guanglin Xu, Minseok Ryu, Ruiwei Jiang.

Figure 1
Figure 1. Figure 1: An Example of the Network (G, E) with n = 2 and K = 1. The nodes in layers 1 and 2 are arranged along the coordinates L(1) and L(2), respectively. This indicates that, by associating appropriate length to each arc, solving the longest-path prob￾lem in the network produces an optimal solution to the DP. We summarize this observation in the following proposition. Proposition 5. Suppose that Assumptions 2–3 h… view at source ↗
Figure 2
Figure 2. Figure 2: Out-of-sample performance as a function of the Wasserstein radius [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Out-of-sample performance of optimal W-DRAS, CM, and SAA appointment schedules [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Out-of-sample performance of the W-DRAS and SAA optimal appointment schedules [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reliability of W-DRAS, CM, and SAA as a function of data size [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Out-of-sample performance of the optimal W-DRAS, CM, and SAA appointment sched [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Out-of-sample performance and reliability of the optimal W-NS, MM, and SAA appoint [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Out-of-sample performance of the optimal W-NS, MM, and SAA appointment schedules [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
read the original abstract

We study a single-server appointment scheduling problem with a fixed sequence of appointments, for which we must determine the arrival time for each appointment. We specifically examine two stochastic models. In the first model, we assume that all appointees show up at the scheduled arrival times yet their service durations are random. In the second model, we assume that appointees have random no-show behaviors and their service durations are random given that they show up at the appointments. In both models, we assume that the probability distribution of the uncertain parameters is unknown but can be partially observed via a set of historical data, which we view as independent samples drawn from the unknown distribution. In view of the distributional ambiguity, we propose a data-driven distributionally robust optimization (DRO) approach to determine an appointment schedule such that the worst-case (i.e., maximum) expectation of the system total cost is minimized. A key feature of this approach is that the optimal value and the set of optimal schedules thus obtained provably converge to those of the true model, i.e., the stochastic appointment scheduling model with regard to the true probability distribution of the uncertain parameters. While our DRO models are computationally intractable in general, we reformulate them to copositive programs, which are amenable to tractable semidefinite programming problems with high-quality approximations. Furthermore, under some mild conditions, we recast these models as polynomial-sized linear programs. Through an extensive numerical study, we demonstrate that our approach yields better out-of-sample performance than two state-of-the-art methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a data-driven distributionally robust optimization (DRO) approach for single-server appointment scheduling under two stochastic models: (i) random service durations with all appointees showing up, and (ii) random no-shows combined with conditional random service durations. Historical data are treated as i.i.d. samples from an unknown distribution; Wasserstein balls centered at the empirical distribution are used to minimize the worst-case expected total cost. The paper proves that both the optimal value and the set of optimal schedules converge to those of the true distribution as the sample size grows. The resulting DRO models are reformulated as copositive programs (approximable by SDPs) and, under mild conditions, as polynomial-sized linear programs. Numerical experiments claim superior out-of-sample performance relative to two benchmark methods.

Significance. If the convergence results and reformulations are correct, the work supplies a theoretically grounded, asymptotically consistent method for robust scheduling that directly addresses distributional ambiguity via Wasserstein distance. The explicit consistency guarantee and the reduction to tractable LPs under stated conditions are strengths that distinguish the contribution within data-driven DRO literature for stochastic programming. The numerical validation, if reproducible, supports practical relevance in healthcare operations research.

major comments (2)
  1. [§4] §4 (convergence analysis): The proof that optimal schedules converge to the true-model optimum is conditioned on the i.i.d. sampling assumption stated in the abstract and §2; while this is standard, the argument does not address whether the Wasserstein radius selection (which must shrink with n) remains valid or how the rate degrades under mild dependence, making the load-bearing asymptotic claim sensitive to this modeling choice.
  2. [§5.2, Theorem 5.3] §5.2, Theorem 5.3 (LP reformulation): The 'mild conditions' under which the copositive program reduces to a polynomial-sized LP are not explicitly verified against the piecewise-linear or indicator-based cost functions arising from the no-show model; without this check the claim that the LP is available for both stochastic models is not fully supported.
minor comments (2)
  1. [§2] Notation for the total cost function (waiting, idle, and overtime components) is introduced in §2 but the precise functional form used in the numerical instances should be restated in the caption of Table 1 or Figure 2 for clarity.
  2. [§6] The description of how the Wasserstein radius is chosen in the numerical study (§6) is brief; adding the explicit functional form (e.g., scaling with n^{-1/2}) would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments and the recommendation for minor revision. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [§4] §4 (convergence analysis): The proof that optimal schedules converge to the true-model optimum is conditioned on the i.i.d. sampling assumption stated in the abstract and §2; while this is standard, the argument does not address whether the Wasserstein radius selection (which must shrink with n) remains valid or how the rate degrades under mild dependence, making the load-bearing asymptotic claim sensitive to this modeling choice.

    Authors: The convergence analysis in §4 is explicitly conditioned on the i.i.d. sampling assumption stated in the abstract and §2, which is the standard setting for such consistency results in data-driven DRO. The Wasserstein radius is chosen to shrink with n (e.g., on the order of n^{-1/2} via concentration inequalities) to ensure the optimal value and solutions converge to the true stochastic optimum. The proof relies on this i.i.d. structure and does not claim validity under dependence. Extending the analysis to dependent samples would require different concentration tools and is beyond the paper's scope. We will add a remark in §4 noting the i.i.d. assumption and the corresponding radius selection. revision: partial

  2. Referee: [§5.2, Theorem 5.3] §5.2, Theorem 5.3 (LP reformulation): The 'mild conditions' under which the copositive program reduces to a polynomial-sized LP are not explicitly verified against the piecewise-linear or indicator-based cost functions arising from the no-show model; without this check the claim that the LP is available for both stochastic models is not fully supported.

    Authors: We thank the referee for this observation. The cost functions in the no-show model are piecewise linear (waiting, idle, and overtime costs) combined with indicators for no-shows, which admit a finite piecewise-linear representation. These satisfy the mild conditions of Theorem 5.3. We will add an explicit verification in the revised §5.2 confirming that the conditions hold for both models, supporting the availability of the polynomial-sized LP reformulation. revision: yes

standing simulated objections not resolved
  • Analysis of convergence results and radius selection under mild dependence in the sampling process

Circularity Check

0 steps flagged

No circularity; convergence claim rests on standard i.i.d. assumption and external DRO duality

full rationale

The paper states the i.i.d. sampling assumption explicitly as the basis for both the Wasserstein ball construction and the asymptotic convergence of optimal values/schedules to the true model. Reformulations to copositive programs and polynomial LPs are presented as consequences of standard DRO duality arguments (no equations reduce a prediction to a fitted input by construction). No self-citation is invoked as the sole justification for a uniqueness theorem or ansatz that would force the result. The derivation chain remains independent of the paper's own fitted quantities and is self-contained against external DRO theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the i.i.d. sampling assumption for historical data and on the definition of the Wasserstein ball (radius and ground metric not specified in abstract); these are standard DRO inputs rather than new entities invented by the paper.

axioms (1)
  • domain assumption Historical data consist of independent samples drawn from the unknown distribution
    Explicitly invoked in the abstract as the foundation for the data-driven approach and convergence result.

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