Dynamic Physician-to-Patient Assignment

• Dynamic physician-to-patient assignment is a decision-making framework that optimally matches physicians to patients in real time using stochastic and combinatorial methods.
• The framework leverages assortment optimization, queueing heuristics, and logic-based scheduling to balance match quality, fairness, and throughput in various healthcare settings.
• Models integrate offline menu planning with online adjustments by incorporating patient preference, provider availability, and capacity constraints to improve system performance.

The dynamic physician-to-patient assignment problem encompasses a class of operational and stochastic decision-making models aimed at optimally assigning physicians to patients over time, subject to dynamic constraints, uncertainty, and multifactorial objectives. Applications span outpatient rematching, emergency department dispatch, inpatient care with revisit/reentrance, and operating room/clinic scheduling. The problem is characterized by the interplay between system-level objectives (e.g., match quality, fairness, throughput), patient choice models, provider availability, and evolving data states. Approaches found in recent literature address both online and hybrid offline/online assignment settings, leveraging assortment optimization, queueing/prioritization heuristics, stochastic programming, and logic-based dynamic scheduling.

1. Mathematical Formulation and Core Model Structures

The dynamic physician-to-patient assignment problem is formalized using combinatorial, stochastic, and constraint-based methods:

• In assortment-based matching (Raman et al., 14 Feb 2025), patients $i=1,\dots,N$ and providers $j=1,\dots,M$ are matched, respecting an allocation matrix $X\in\{0,1\}^{N\times M}$ (the "menu" system), with each patient able to select at most one provider and each provider matched at most once per round. System dynamics are hybrid: menus $X$ are selected offline, but actual patient choices and provider availabilities evolve online as patients arrive in a random sequence.
• Queueing models for multi-stage, reentrant care (Liu et al., 2024) encode each patient as requiring a stochastic number of service stages, with reentrance between "needy" and "content" states. Decisions involve both service discipline (shortest-first vs longest-first) and stochastic assignment to available physicians (indexed by load, cost, and acuity).
• Scheduling models in emergency or operating room settings extend this to multi-resource allocation, with system state vectors $s_t$ recording physician availability, queue lengths, waiting times, and other operational metrics (Furian et al., 2022, Tsang et al., 2022). Objective functions combine waiting time, service-level targets, and operational cost metrics.

2. Assignment and Prioritization Policies

A diversity of assignment rules emerge, varying by problem structure, choice model, and system constraints:

Policy/Framework	Key Feature	Optimality/Trade-off
Greedy (Full-Menu) (Raman et al., 14 Feb 2025)	Maximize match-rate	Drags down match-quality; not constant-approx.
Pairwise Bipartite (Raman et al., 14 Feb 2025)	Max-weight matching	$p$-approximation for MQ; robust when $p$ high
Group-Based (Raman et al., 14 Feb 2025)	Pairwise clique merge	Preserves MR; tailored to $N/M$, $p$
Gradient-Descent (Raman et al., 14 Feb 2025)	Concave lower bound	Tight when providers scarce ($N\gg M$)
SF/LF Priority (Liu et al., 2024)	Static discipline	Threshold $a$: SF if low acuity penalty, LF if high
Myopic Assignment (Liu et al., 2024)	Immediate cost drop	Outperforms long-term, random heuristics
APQ (Furian et al., 2022)	Linear time/acuity	Simple; suboptimal to ML-based classifier
ML-Based (Furian et al., 2022)	Nonlinear features	Closes gap to hindsight-optimal schedule

Policies are selected based on specific objectives and system parameters, such as menu-size limits, provider scarcity (ratio $N/M$), and patient selectiveness ($p$ in uniform/multinomial logit choice).

3. Integrative Scheduling Under Uncertainty

In multi-resource environments with stochastic demand and supply (e.g., OR/anesthesiologist assignment (Tsang et al., 2022)), the assignment problem is embedded within two-stage stochastic programming (SP) or distributionally robust optimization (DRO) frameworks. Notable features:

• Scenario-based models solve for allocation and sequencing under sampled or distributionally ambiguous duration parameters $D$.
• Risk-neutral (expected cost) and risk-averse (CVaR) objectives are considered, with robust models employing ambiguity sets based on historical bounds and moments.
• Constraint-based formulations include symmetry-breaking, precedence/sequencing logic, and capacity constraints across providers and time slots.

Solution algorithms include sample average approximation (SAA, for SP) and column-and-constraint generation (C&CG, for DRO), enhanced by valid inequalities and constraint symmetry to improve tractability.

4. Dynamic Logic-Based and Real-Time Scheduling

Answer Set Programming (ASP) is leveraged for event-driven physician-to-patient assignment (Vozna et al., 7 Jan 2026), particularly in ambulatory and appointment environments with real-time availability, cancelations, and personalized needs:

• The logic model encodes hard constraints (no-double booking, load limits, prioritization on urgency) and simultaneously optimizes over multiple objectives (max assignments, urgency, load balancing).
• The dynamic component exhibits fast incremental solving: base logic is grounded once; incremental events (updated availabilities, new/cancelled requests) invoke only delta updates, yielding rapid response times on realistic data volumes.
• This framework supports micro-service and real-time integration with healthcare platforms.

5. Performance Metrics and Empirical Insights

Quantifiable system metrics and simulation studies reveal comparative effectiveness of assignment policies:

• In assortment optimization (Raman et al., 14 Feb 2025), the gradient-descent policy improves average match-quality (MQ) by 13% over greedy and 8% over group-based, particularly for high-comorbidity patients and under small menu constraints.
• Emergency department scheduling (Furian et al., 2022) observes that ML-based dispatch reduces the gap to hindsight-optimal by roughly one third (ML improvement of 17.5% vs APQ's 9.8%) on combined time and TTD objectives, with notable robustness to variable traffic and consultation times.
• Queueing/reentrance models (Liu et al., 2024) demonstrate cost reductions of 10–20% for myopic assignment over random, and threshold tuning between SF/LF priority rules based on acuity cost exponent $a$.
• Operating room scheduling with DRO (Tsang et al., 2022) yields lower waiting times and better performance under distribution shifts, justifying robust formulations when historical data are limited or distributions are uncertain.
• ASP-based frameworks (Vozna et al., 7 Jan 2026) achieve sub-0.02 s incremental assignment times in batch/event-driven deployments for up to hundreds of simultaneous requests.

6. Fundamental Trade-offs and Policy Recommendations

Multiple trade-offs are identified in policy selection and operational implementation:

• Menu-size vs match-quality: Large menus favor match-rate, but focused small menus (especially via gradient-descent or pairwise optimization) yield higher overall quality (Raman et al., 14 Feb 2025).
• Fairness vs average quality: Maximizing worst-case MQ (e.g., via pairwise) lowers the bottom quartile but may reduce overall MQ; gradient-descent maximizes average but exposes lowest quartile to poor matches.
• Immediate vs long-term cost balancing: Immediate cost-drop heuristics robustly outperform those attempting to balance aggregate future cost, especially under heavy congestion (Liu et al., 2024).
• Risk-neutral vs risk-averse optimization: DRO and CVaR objectives yield superior recovery under unexpected distributional shifts or high tail risk, trading increased resource activation for lower patient wait and operational disappointment (Tsang et al., 2022).
• Real-time responsiveness vs computational overhead: Incremental logic-based scheduling (ASP) achieves rapid updates with modest hardware investment, suitable for microservice integration and event-driven healthcare deployments (Vozna et al., 7 Jan 2026).

In practical terms, optimal policy should match realized system characteristics—provider scarcity ($N/M$), patient choice model or selectiveness ($p$), acuity grading, and operational risk tolerance.

7. Extensions and Research Directions

Recent works suggest multiple avenues for extension:

• Integration of multi-skill providers, time-varying rosters, and real-time retraining in ML-based approaches (Furian et al., 2022).
• Adaptive tuning of acuity penalty exponent $a$ in queueing/reentrance models based on monitored performance metrics—dynamically switching between SF/LF rules as required (Liu et al., 2024).
• Incorporation of persona modeling and patient preference stratification for vulnerable populations in centralized scheduling (Vozna et al., 7 Jan 2026).
• Hybrid offline-online models for episodic rematching, leveraging batching to balance optimization performance with patient response time (Raman et al., 14 Feb 2025).
• Scaling robust optimization models to larger hospital deployments through parallelization, symmetry reduction, and scenario aggregation (Tsang et al., 2022).

As operational data and electronic health records become more granular, adaptive and personalized dynamic physician-to-patient assignment algorithms can further improve care allocation efficiency, match quality, and system robustness.