Hospital Patient-to-Bed Assignment

• Hospital patient-to-bed assignment is the systematic allocation of inpatients to beds using mathematical optimization under clinical and operational constraints.
• Recent methods employ integer programming, constraint programming, and heuristic approaches to minimize patient transfers and balance occupancy.
• Integrating bed assignment with nurse scheduling, surge planning, and digital twin simulations enhances real-time efficiency and patient safety.

Hospital patient-to-bed assignment is a critical operational and tactical challenge in healthcare systems, involving the allocation of inpatients to specific beds or rooms subject to numerous clinical, logistical, and regulatory constraints. The problem is central to both daily hospital ward management and broader surge or emergency planning, and it interacts deeply with other processes such as nurse staffing, elective scheduling, infection control, and resource optimization. The formal study of this problem encompasses diverse algorithmic paradigms, ranging from integer programming and constraint optimization to stochastic dynamic programming and data-driven queueing models.

1. Formal Problem Definition and Core Constraints

Hospital patient-to-bed assignment typically involves matching a dynamically evolving set of admitted or soon-to-be-admitted patients to a fixed or time-varying configuration of beds, such that operational objectives are attained and policy constraints are satisfied. Standard problem formulations index patients $p \in P$, rooms or beds $r \in R$ (possibly with categories $C_r$ and capacities $B_r$), and, in dynamic settings, discrete time periods $t \in T$.

Typical variables and constraints include:

• Assignment variables: $x_{p,r,t}$ indicates whether patient $p$ occupies bed/room $r$ at time $t$; constraints enforce that each patient is assigned to exactly one compatible bed at all required times.
• Capacity constraints: Patient assignments never exceed the room/bed capacities per period.
• Resource and patient attributes: Assignments are feasible only if room or bed categories (e.g., monitoring level, isolation) meet or exceed the patient’s current clinical needs, e.g., $C_p \geq C_r$ for assignment.
• Cohorting, gender, and infection control: Constraints enforce separation or compatibility based on gender, infection status, or other cohorting variables. E.g., patients with a contagion flag must be isolated within a room.
• Transfer minimization: Assignment redesigns over time are penalized to reduce relocations, preserving patient comfort and workflow efficiency.
• Single-room or roommate preferences: Models often encode specific entitlement to single rooms, or, conversely, compatibility scores for roommate assignment (Brandt et al., 2023, Brandt et al., 2 Mar 2025).

Formally, a representative day-by-day optimization is: $$\min \sum_{p \in P_\text{prev} \cap P} c_p + \alpha \sum_{r \in R} \left|\sum_{p \in P} x_{p,r} - \frac{|P|}{|R|}\right|$$ subject to assignment and compatibility constraints, where $c_p$ denotes patient relocation indicators, and $\alpha \ll 1$ weights secondary occupancy balance (Sieve et al., 7 May 2025).

2. Optimization Methods and Solution Paradigms

The assignment problem is computationally intractable in general (NP-complete) but admits a range of exact and heuristic solution methods tailored to hospital scale and required response times.

• Integer Programming (IP/MIP): Multi-objective, multi-period MIPs can encode transfer minimization, gender separation, single-room entitlements, and pairing compatibility. Lexicographic objectives are common: minimize transfers first, then maximize private/compatible assignments (Brandt et al., 2023, Brandt et al., 2 Mar 2025).
• Constraint Programming/SMT: Satisfiability Modulo Theories (SMT) solvers can efficiently express and solve daily assignment instances given a quantifier-free encoding of assignment, cohorting, and movement constraints (Sieve et al., 7 May 2025).
• Rolling-horizon and heuristic approaches: Real-world implementations use dynamic, rolling-horizon MIPs, incremental feasibility checks, warm starts, and policy layering (e.g., static no-transfer models first, then relaxations only as needed). Typically, >95% of subdaily problems solve to optimality within one second on standard hardware (Brandt et al., 2023, Brandt et al., 2 Mar 2025).
• Approximate Dynamic Programming (ADP): Infinite-horizon patient assignment scheduling under stochastic arrivals and stays can be framed as a Markov Decision Process (MDP), with approximate solutions using simulation-based value function approximation. Such methods yield near-optimal policies with tractably short simulation runs (O'Reilly et al., 2024).
• Queueing-theoretic and chance-constrained models: Data-driven approaches modeled after FCFS or overflow routing maximize timely admission probabilities under bed uncertainty, admitting real-time, myopic IP solutions that match empirical demand patterns and support dynamic overflow control (Han et al., 2021).

3. Operational Objectives and Secondary Criteria

Hospital patient-to-bed assignment problems are typically multi-objective, reflecting both patient-centered and operational trade-offs. Common priorities include:

• Transfer minimization: Moving patients between rooms disrupts care and comfort; lexicographic or penalty-objective formulations strongly discourage moves (Brandt et al., 2023, Brandt et al., 2 Mar 2025).
• Private/single-room accommodation: Billing, clinical risk, and preference often require maximization or bounding of single-room allocations, subject to capacity (Brandt et al., 2023).
• Roommate compatibility: Empirical and survey-based scoring functions (absolute age-difference, co-morbidities, surgery status, etc.) can be used to minimize negative roommate interactions (Brandt et al., 2 Mar 2025).
• Occupancy balancing: Load balancing across wards (standard deviation or absolute deviation minimization) can reduce congestion and staff overload (Sieve et al., 7 May 2025).
• Cohorting, gender, infection: Hard constraints or prioritized objectives strictly enforce regulatory and infection-control separations (Sieve et al., 7 May 2025).
• Equipment and resource availability: Assignment must guarantee that essential equipment is available in target rooms (Brandt et al., 2023).
• Delay minimization: Especially for ED inflows, assignment models can explicitly minimize or bound patient boarding and waiting times, often through queueing-theoretic likelihood objectives (Han et al., 2021).

4. Dynamic, Stochastic, and Surge Environments

The assignment problem is fundamentally dynamic, with stochastically arriving patients, variable length-of-stay, room outages, and surge events:

• Stochastic MIP/SAA frameworks: Sample Average Approximation (SAA) over scenario trees enables robust assignment for bed, staff, and capacity planning with explicit uncertainty in arrivals, length-of-stay, and resources (Baas et al., 2023, Chouba et al., 2020).
• Robust optimization: Surge-planning models (e.g., for pandemics) solve multi-day, multi-hospital robust MILPs, integrating surge capacity setup, patient transfers, and hard bed-availability requirements under probabilistic (budget-of-uncertainty) future demands (Parker et al., 2024).
• Digital twin and simulation architectures: Executable formal models (e.g., in ABS) and digital-twin-based simulation with ontology-driven scenario generation enable exploration of "what-if" capacity, cohorting, and flow reconfiguration with rapid, automated feedback (Sieve et al., 7 May 2025).

In stochastic and surge settings, policy levers shift—delaying elective admissions, relaxing room cohorting, employing inter-hospital transfers—to prioritize shortage avoidance, surge absorption, or fairness across institutions (Parker et al., 2024, Baas et al., 2023).

5. Algorithmic and Computational Performance

State-of-the-art models achieve real-time or near-real-time solvability at hospital scale:

• IP/MIP-based methods: With careful variable selection, constraint aggregation, and rolling-horizon decomposition, 95–97% of subproblems for hospitals with up to 200 rooms and 150,000 patient-days/year are solved optimally in <1 s for daily subproblems; larger multi-room, multi-day models (surge planning, regional pandemics) remain tractable in minutes-horizon with parallel computing (Brandt et al., 2023, Sieve et al., 7 May 2025, Parker et al., 2024).
• SMT/OMT approaches: BedreFlyt's SMT-based pipeline solves daily assignments in ≈1 s/day for typical wards, with stress scenarios up to 2,000 patients solved in ≤80 min; unsatisfiable scenarios generate immediate feedback for policy fallback (Sieve et al., 7 May 2025).
• Heuristic and hybrid algorithms: Integrated greedy-assignment heuristics reach 0–1% gaps on transfer, gender-mix, and equipment-violation objectives, with order-of-magnitude faster runtimes compared to full MIP on realistic hospital datasets (Brandt et al., 2023).
• Adaptive rolling horizon and hot-starting: Modern systems precompute combinatorial feasibility, fix/merge stale assignments, and carry forward solution states to subsequent periods, further reducing computational load (Brandt et al., 2023, Brandt et al., 2 Mar 2025).

6. Integration with Broader Hospital Operations

Patient-to-bed assignment interacts with multiple adjacent processes:

• Integrated resource optimization: Bed assignment is often embedded within broader staff (e.g., nurse) scheduling, operating room scheduling, and resource-reconfiguration models, benefitting from shared constraints and integrative objectives that capture walking distances, patient-nurse continuity, and simultaneous staff-patient allocation (Brandt et al., 2023, Dodaro et al., 2021).
• Digital twin and knowledge-driven frameworks: Ontology-based knowledge bases (e.g., in RDF/OWL) integrate static and dynamic asset models, allowing mappings of treatment protocols, room availability, and infection-control policies, supporting both short-term assignment and long-term infrastructure scenario analysis (Sieve et al., 7 May 2025).
• Queueing-based performance management: Real-time or near-real-time data-driven models (e.g., the P-model) support robust control of patient flow, overflow, and waiting-time minimization in ED and acute admissions, under empirically calibrated arrival, service, and discharge processes (Han et al., 2021).
• Policy sensitivity: Assignment policies expose tuning knobs—priority weighting of transfers vs. single rooms, overflow vs. waiting, and day-to-day volatility reduction—which can be matched to managerial or regulatory objectives (Brandt et al., 2023, Baas et al., 2023).

7. Empirical Evidence and Practical Impact

Large-scale simulation and real-world computational experiments have validated the efficacy of the latest models:

• Short-term operations: Digital-twin-based systems and rolling-horizon IPs have demonstrated a 70% reduction in bed moves compared to naïve daily-resolving baselines, and occupancy maximization over 90% under moderate to severe bed shortages (Sieve et al., 7 May 2025, Dodaro et al., 2021).
• Compatibility and entitlement: New roommate-compatibility models, when applied with preference scoring and fast rolling-horizon IPs, achieve full assignment feasibility, zero transfers, and high single-room allocation rates in >95% of days within sub-second solving time (Brandt et al., 2 Mar 2025, Brandt et al., 2023).
• Surge and pandemic management: In COVID-19 case studies, joint robust allocation of surge units and limited patient transfers reduced required surge capacity by nearly 90% relative to no-transfer policies (Parker et al., 2024, Baas et al., 2023).
• Queueing performance: Data-driven chance-constrained approaches have reduced mean ED boarding by 4–6 h in peak hours, ensuring service level compliance while controlling overflow rates within 1–2% of manual policies (Han et al., 2021).
• Heuristic versus exact methods: Greedy heuristics, when appropriately constructed and embedded, deliver near-optimal assignment quality for transfers and mix penalties, at 1–2 orders of magnitude shorter computation times compared to fully integrated MIPs (Brandt et al., 2023).

• "BedreFlyt: Improving Patient Flows through Hospital Wards with Digital Twins" (Sieve et al., 7 May 2025)
• "Structural Insights and an IP-based Solution Method for Patient-to-room Assignment under Consideration of Single Room Entitlements" (Brandt et al., 2023)
• "Optimal Hospital Capacity Management During Demand Surges" (Parker et al., 2024)
• "A stochastic programming approach for dynamic allocation of bed capacity and assignment of patients to collaborating hospitals during pandemic outbreaks" (Baas et al., 2023)
• "Markov Decision Process and Approximate Dynamic Programming for a Patient Assignment Scheduling problem" (O'Reilly et al., 2024)
• "Combinatorial and Computational Insights about Patient-to-room Assignment under Consideration of Roommate Compatibility" (Brandt et al., 2 Mar 2025)
• "Data-Driven Inpatient Bed Assignment Using the P Model" (Han et al., 2021)
• "Integrated patient-to-room and nurse-to-patient assignment in hospital wards" (Brandt et al., 2023)
• "Operating Room (Re)Scheduling with Bed Management via ASP" (Dodaro et al., 2021)
• "A Mixed Integer Linear Program For Human And Material Resources Optimization In Emergency Department" (Chouba et al., 2020)