---
title: Dynamic Physician-to-Patient Assignment
url: https://www.emergentmind.com/topics/dynamic-physician-to-patient-assignment-problem
type: topic
---

# Dynamic Physician-to-Patient Assignment

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 [2502.10353], 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 [2406.12135] 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 [2206.03752, 2204.11374]. 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) [2502.10353]   | Maximize match-rate   | Drags down match-quality; not constant-approx. |
| Pairwise Bipartite [2502.10353]   | Max-weight matching   | $p$-approximation for MQ; robust when $p$ high |
| Group-Based [2502.10353]          | Pairwise clique merge | Preserves MR; tailored to $N/M$, $p$     |
| Gradient-Descent [2502.10353]     | Concave lower bound   | Tight when providers scarce ($N\gg M$)   |
| SF/LF Priority [2406.12135]       | Static discipline     | Threshold $a$: SF if low acuity penalty, LF if high |
| Myopic Assignment [2406.12135]    | Immediate cost drop   | Outperforms long-term, random heuristics |
| APQ [2206.03752]                  | Linear time/acuity    | Simple; suboptimal to ML-based classifier|
| ML-Based [2206.03752]             | 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 [2204.11374]), 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 [2601.04274], 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 [2502.10353], 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 [2206.03752] 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 [2406.12135] 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 [2204.11374] 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 [2601.04274] 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 [2502.10353].
- 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 [2406.12135].
- 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 [2204.11374].
- 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 [2601.04274].

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 [2206.03752].
- Adaptive tuning of acuity penalty exponent $a$ in queueing/reentrance models based on monitored performance metrics—dynamically switching between SF/LF rules as required [2406.12135].
- Incorporation of persona modeling and patient preference stratification for vulnerable populations in centralized scheduling [2601.04274].
- Hybrid offline-online models for episodic rematching, leveraging batching to balance optimization performance with patient response time [2502.10353].
- Scaling robust optimization models to larger hospital deployments through parallelization, symmetry reduction, and scenario aggregation [2204.11374].

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.

Source: https://www.emergentmind.com/topics/dynamic-physician-to-patient-assignment-problem