Multi-resource operating room scheduling: a three-stage heuristic for efficiency and patient safety

1 Introduction

Operating room (OR) scheduling represents one of the most complex resource allocation challenges in healthcare operations management (Abbas et al., 2026; Cardoen et al., 2010), with profound implications for clinical outcomes, economic sustainability, and patient satisfaction. As the global population undergoes unprecedented demographic aging—projected by the United Nations to reach 1.6 billion individuals aged 65 or older by 2050—the demand for surgical services has intensified dramatically, with an estimated 30% increase in complex, resource-intensive procedures over the past decade alone (Saur et al., 2022; Foley and Djaiani, 2025). Operating rooms constitute the largest cost center and revenue generator in modern hospitals, typically accounting for 40%–60% of total hospital revenues and 30%–40% of operational budgets (Cardoen et al., 2010). Consequently, even marginal improvements in OR efficiency translate into substantial economic benefits and enhanced patient access to care (Abbas et al., 2026). However, inefficient scheduling practices persistently plague healthcare systems worldwide, resulting in 15%–20% of available OR time being wasted due to resource idle periods, last-minute surgical cancellations, and suboptimal coordination among clinical staff (Li et al., 2024). These inefficiencies not only impose billions of dollars in unnecessary annual healthcare expenditures but also exacerbate patient waiting times, potentially compromising clinical outcomes for time-sensitive procedures.

From an operations research perspective, OR scheduling constitutes a complex variant of multi-resource parallel machine scheduling (Gao et al., 2025), characterized by heterogeneous job classes (emergency, urgent, and elective surgeries), time-sensitive clinical constraints, and the requirement for simultaneous availability of multiple interdependent resource types (Cardoen et al., 2010). Unlike traditional manufacturing scheduling where jobs typically require single machines, surgical procedures necessitate the synchronized allocation of four distinct resource categories: surgeons (with hierarchical skill levels), anesthesiologists, nursing staff, and specialized surgical equipment (Li et al., 2024). This multi-machine joint processing (MMJ) characteristic fundamentally distinguishes OR scheduling from classical parallel machine problems (P m ‖ C max) (Graham, 1969) and necessitates the development of specialized optimization methodologies that account for resource synchronization constraints. Furthermore, surgical scheduling must accommodate three distinct urgency classes with non-negotiable clinical requirements: emergency surgeries (e.g., trauma, ruptured appendicitis) requiring immediate intervention within 30 min to prevent mortality increases of up to 20% (Singer et al., 2011); urgent surgeries (e.g., malignancies, severe infections) requiring scheduling within 72 h to prevent clinical deterioration (Plurad et al., 2022); and elective surgeries (60%–70% of volume) requiring efficient scheduling to minimize waiting lists while maintaining flexibility for emergent cases (Cardoen et al., 2010). The heterogeneous nature of surgical demand, combined with resource cost heterogeneity and clinical workload constraints (e.g., limiting consecutive surgical hours to prevent fatigue-related medical errors) (Lockley et al., 2004; Günther et al., 2013), creates a multi-objective optimization landscape of considerable computational and practical complexity (Cai et al., 2025; Spruce, 2021).

It is important to note that OR scheduling does not operate in isolation; it is tightly coupled with downstream resources such as post-anesthesia care unit (PACU) beds, inpatient ward capacity, and intensive care unit availability. An increase in surgical throughput without corresponding capacity in recovery areas can simply shift bottlenecks rather than improve overall patient flow. While our model focuses on intraoperative coordination, we acknowledge that a holistic approach would require integration with bed management systems. We discuss this limitation and possible extensions in Section 5.

The academic literature on OR scheduling has evolved substantially over the past three decades, progressing from single-objective deterministic models to sophisticated multi-objective stochastic frameworks (Cardoen et al., 2010). Early research primarily focused on deterministic elective surgery scheduling using exact optimization methods such as mixed-integer linear programming (MILP) and constraint programming (Li et al., 2024). While these approaches guarantee optimality for small-scale instances, their computational complexity renders them intractable for realistic problem sizes involving hundreds of surgeries and multiple resource constraints (Garey and Johnson, 1979; Garey and Johnson, 1978; Bargetto et al., 2023; Guo et al., 2026). Specifically, Bargetto et al. (2023) demonstrated that branch-and-price-and-cut algorithms, while effective for moderate-sized instances, face significant scalability challenges when human resource constraints and sequence-dependent setup times are simultaneously considered. Subsequently, metaheuristic approaches—including Genetic Algorithms (GA), Simulated Annealing (SA), and Particle Swarm Optimization (PSO)—have gained popularity due to their ability to explore large solution spaces and handle multiple objectives (Wang et al., 2026). However, these methods typically lack provable performance guarantees, exhibiting unpredictable solution quality and convergence behavior that limits their adoption in safety-critical clinical environments (Garey and Johnson, 1979). More recently, machine learning-based approaches have emerged for predicting surgical durations and emergency arrival patterns (Wang et al., 2026), yet these data-driven methods often neglect the combinatorial optimization structure underlying the scheduling problem. Recent advances in metaheuristic design have introduced random-key optimization frameworks that decouple solution encoding from algorithmic search mechanisms, demonstrating superior adaptability to multi-constraint surgical environments (Vieira et al., 2025). Furthermore, integrated approaches addressing capacity planning, fair scheduling, and surgeon resilience have emerged as critical for sustainable OR management (Kayvanfar et al., 2025; Bansal et al., 2021). These studies, however, primarily focus on either stochastic uncertainty or heuristic search without provable performance bounds, leaving a significant gap regarding deterministic approximation algorithms with constant-factor guarantees for multi-resource OR scheduling. A critical limitation pervading existing literature is the tendency to oversimplify the multi-resource nature of OR scheduling: many studies reduce the problem to single-resource allocation (surgeon-only or OR-block-only), while others treat resource constraints as soft penalties rather than hard synchronization requirements (Gao et al., 2025). Furthermore, despite the acknowledged NP-hardness of simplified OR scheduling variants, rigorous computational complexity analysis establishing performance bounds for multi-resource heuristics remains conspicuously absent from the literature (Garey and Johnson, 1979; Garey and Johnson, 1978).

The complexity of the OR scheduling problem extends beyond computational intractability to encompass fundamental operational trade-offs that existing methodologies struggle to balance effectively. The strongly NP-hard nature of multi-resource constrained project scheduling problems (MRCPSP) establishes the theoretical foundation for our complexity analysis, as OR scheduling generalizes this canonical problem through the addition of urgency classes and resource level constraints (Garey and Johnson, 1979; Garey and Johnson, 1978; Cohen, 1988). Traditional Fixed OR Allocation (FOA) strategies—where specific operating rooms are permanently assigned to clinical departments—perpetuate resource silos and prevent cross-departmental flexibility (Guido et al., 2018; Kayvanfar et al., 2025), resulting in paradoxical situations where high-demand departments face elective surgery backlogs of 6–8 months while low-volume departments maintain idle OR capacity. While stochastic programming approaches have addressed uncertainty in surgery durations and anesthesiologist allocation (Tsang et al., 2025), and distributional robust optimization has been proposed for coordinating clinical appointments under ambiguity (Bansal et al., 2021), these methods typically rely on scenario sampling or conservative uncertainty sets that may over-protect against rare events at the expense of average-case efficiency. Moreover, existing bicriteria approximation algorithms for parallel machine scheduling (Jiang et al., 2023) have not been extended to accommodate the multi-machine joint processing (MMJ) characteristics inherent in OR scheduling. Contemporary dynamic scheduling approaches, while offering improved flexibility, frequently fail to adequately integrate emergency responsiveness with elective efficiency, either over-prioritizing emergent cases (disrupting elective schedules and increasing cancellation rates) or under-prioritizing them (compromising patient safety) (Boulkhoukh et al., 2024). Additionally, existing multi-objective optimization frameworks often rely on weighted sum approaches without systematic calibration of objective weights against clinical priorities (Cai et al., 2025), and rarely incorporate the full spectrum of real-world constraints including resource level matching (e.g., ensuring complex procedures receive chief physicians rather than residents) (Ding et al., 2019), contamination-based OR disinfection protocols, and strict workload limits for clinical staff (Lockley et al., 2004). The absence of algorithms that simultaneously provide computational efficiency (enabling real-time decision-making), provable solution quality (ensuring reliability), and comprehensive constraint handling (ensuring clinical feasibility) represents a critical gap between academic optimization research and practical clinical implementation (Li et al., 2024).

This study addresses the identified research gaps by formulating the OR scheduling problem as a parallel machine scheduling problem with multi-machine joint processing P m M M J C max and proposing an improved three-stage heuristic algorithm (TSA) with provable theoretical performance guarantees. Our methodology synergistically integrates four optimization objectives—minimizing maximum OR completion time, reducing total resource costs, maximizing high-priority patient satisfaction, and minimizing emergency/urgent surgery waiting times—within a computationally efficient framework suitable for large-scale clinical deployment. The proposed algorithm decomposes the complex NP-hard problem into three sequential, tractable sub-problems: resource allocation via an improved Longest Processing Time (LPT) algorithm with dynamic emergency insertion; cost-optimized sequencing via constraint-preserving local search; and patient priority adjustment with workload compliance verification.

The principal contributions of this research are fourfold:

We formally define the OR scheduling problem as a P m M M J C max problem and provide a comprehensive NP-hardness proof that accounts for multi-resource joint processing requirements. Unlike existing literature that relies on single-resource reductions, we establish NP-hardness via reduction to the Multi-Resource Constrained Project Scheduling Problem (MRCPSP), thereby filling a critical theoretical gap regarding the computational complexity of multi-resource surgical scheduling.

We develop a three-stage heuristic algorithm (TSA) that achieves a worst-case performance ratio of 4/3 for the makespan objective while operating with polynomial-time complexity O n ⁡ log ⁡ n. To our knowledge, this represents the first provable error bound for multi-resource OR scheduling heuristics that account for dynamic emergency preemption and resource synchronization constraints.

We construct a clinically calibrated weighted sum multi-objective model integrating resource cost heterogeneity, patient priority hierarchies (combining vital and social indices), and time-sensitive clinical constraints. All parameters and weights are validated via the Analytic Hierarchy Process (AHP) with consistency ratio CR = 0.042 and expert consensus (Delphi method, 15 clinical experts), ensuring clinical relevance and practical applicability.

We validate the TSA using 12,864 real surgical records from the MIMIC-IV database (2018–2019) across diverse hospital settings (community, tertiary, and large teaching hospitals). Experimental results demonstrate that the TSA outperforms traditional Fixed OR Allocation (FOA) and state-of-the-art methods (LPT, SPT, Genetic Algorithm) across eight key performance metrics, achieving 15.79% higher surgical throughput, 10.34% lower resource costs, 50.00% reduction in critical surgery waiting time, and computational efficiency suitable for real-time deployment (3.2 s for large-scale instances).

The remainder of this paper is organized as follows. Section 2 presents the formal problem formulation, notation system, multi-objective function, and rigorous NP-hardness proof. Section 3 details the three-stage heuristic algorithm, including pseudocode, theoretical error bound analysis, and computational complexity proofs. Section 4 describes the experimental design utilizing MIMIC-IV clinical data, benchmark algorithm configurations, and comprehensive results for both static and dynamic scheduling scenarios, including sensitivity and generalizability analyses. Section 5 discusses theoretical implications, practical clinical applications, study limitations, and future research directions.

2 Problem formulation

This study addresses the multi-resource operating room scheduling problem (MR-ORSP) in a tertiary hospital setting characterized by m parallel operating rooms (ORs) and n heterogeneous surgical cases. Formally, let M = M 1 , M 2 , … , M m denote the set of identical parallel machines (ORs), and J = J 1 , J 2 , … , J n denote the set of surgical jobs to be processed. Each surgery J i requires the simultaneous allocation of four distinct resource types: surgeons (S), anesthesiologists (A), nurses (N), and surgical equipment (E). This multi-machine joint processing (MMJ) characteristic fundamentally distinguishes our problem from classical parallel machine scheduling.

Using the standard Graham three-parameter notation for scheduling problems, we classify this problem as: P m M M J , u r g e n c y , m u l t i r e s o u r c e C max , T C , P S , W T

where: ‐ P m indicates m identical parallel machines (ORs); - MMJ denotes the multi-machine joint processing requirement (each job requires simultaneous processing by multiple resources); - urgency represents the job class distinctions (emergency, urgent, elective) with preemption constraints; - multiresource indicates the four-dimensional resource synchronization constraints; - C max (makespan), T C (total cost), P S (priority satisfaction), and W T (waiting time) constitute the vector-valued objective function.

The core scheduling decisions involve determining: (i) the assignment of each surgery J i to a specific OR M l and start time s i; (ii) the allocation of specific resource instances (surgeon x ∈ S, anesthesiologist y ∈ A, nurses N ′ ⊆ N, equipment z ∈ E) satisfying synchronization requirements; and (iii) the sequencing of surgeries within each OR while respecting clinical constraints including resource level matching, workload limits, and contamination-based disinfection protocols.

To ensure mathematical precision and reproducibility, we formally define all parameters, decision variables, and derived quantities in Table 1. Our notation adheres to standard scheduling theory conventions while extending them to accommodate multi-resource healthcare contexts.

TABLE 1. Mathematical notation and parameter definitions.

The MR-ORSP constitutes a vector optimization problem where four conflicting objectives must be simultaneously optimized. We employ the weighted sum scalarization method to transform the multi-objective problem into a single-objective minimization problem (Equation 1):subject to the constraints specified in Section 2.4. The weight vector α = 0.25 , 0.20 , 0.30 , 0.25 T is calibrated via the Analytic Hierarchy Process (AHP) with consistency ratio C R = 0.042 < 0.1, ensuring acceptable consistency. The individual objective components are defined as follows:

min F = α 1 f 1 + α 2 f 2 + α 3 f 3 + α 4 f 4 (1)

Objective 1: Minimize Normalized Makespan (f 1)

The makespan C max = max i ∈ J e i represents the completion time of the last surgery across all ORs, directly measuring OR utilization efficiency. We normalize using the benchmark Fixed OR Allocation (FOA) method (Equation 2):

f 1 = C max C max F O A (2)

Objective 2: Minimize Normalized Total Resource Cost (f 2)

Total resource cost aggregates hourly labor costs (surgeons, anesthesiologists, nurses) and equipment depreciation costs (Equation 3):

T C = ∑ i = 1 n ∑ k = 1 4 ∑ x ∈ R k y i , k , x · t i · C k , x (3)

where R k denotes the resource set for type k (k = 1: surgeons, k = 2: anesthesiologists, k = 3: nurses, k = 4: equipment). The normalized objective (Equation 4) is:

f 2 = T C T C F O A (4)

Objective 3: Minimize Priority Satisfaction Deficit (f 3)

Priority satisfaction PS (Equation 5) measures the proportion of high-priority surgeries (P i ≥ P t h r e s h o l d = 8) scheduled within their maximum permissible waiting times:where J H P = { i ∈ J : P i ≥ 8, n H P = J H P, r e q i denotes the request time, and I · is the indicator function.

P S = 1 n H P ∑ i ∈ J H P I s i ≤ r e q i + m p t i (5)

The threshold P t h r e s h o l d = 8 is derived from the priority score P i = 0.7 · L I i + 0.3 · S I i, where L I i , S I i ∈ 1 , 10. A score of 8 corresponds to the top 20th percentile of clinical urgency—patients with life-threatening conditions (e.g., unstable vital signs, high ASA scores) requiring intervention within 24 h. This threshold was validated through the Delphi consensus process with 15 clinical experts, who unanimously agreed that patients with P i ≥ 8 constitute the “high-priority” category requiring expedited scheduling.

The minimization form (Equation 6) is

f 3 = 1 − P S (6)

Objective 4: Minimize Normalized Critical Waiting Time (f 4)

For emergency and urgent surgeries (J E ∪ J U), waiting time (Equations 7, 8)directly impacts clinical outcomes:

W T c r i t i c a l = 1 J E + J U ∑ i ∈ J E ∪ J U s i − r e q i (7)

Normalized against FOA baseline : f 4 = W T c r i t i c a l W T c r i t i c a l F O A (8)

The feasible schedule must satisfy the following constraint categories, derived from clinical guidelines (AORN, ACS, CDC) and operational requirements:

Each surgery requires simultaneous availability of exactly one surgeon, one anesthesiologist, at least two nurses, and one equipment set (Equations 9, 12) :

∑ x ∈ S y i , 1 , x = 1 ∀ i ∈ J (9)

∑ x ∈ A y i , 2 , x = 1 ∀ i ∈ J (10)

∑ x ∈ N y i , 3 , x ≥ 2 ∀ i ∈ J (11)

∑ x ∈ E y i , 4 , x = 1 ∀ i ∈ J (12)

No resource instance can be allocated to overlapping surgeries. For any two jobs J i and J j assigned to the same resource (Equation 13):

y i , k , x + y j , k , x ≤ 1 ∀ k ∈ 1 , 2 , 3 , 4 , x ∈ R k , i f [ s i , e i ) ∩ [ s j , e j ) ≠ ∅ (13)

Each job is assigned to exactly one OR (Equation 14) : ∑ l = 1 m x i , l = 1 ∀ i ∈ J (14)

Precedence relationships within each OR are defined by (Equations 15, 16): where M is a sufficiently large constant and z i , j = 1 indicates J i precedes J j in the same OR.

e i ≤ s j + M 1 − z i , j ∀ i , j ∈ J , l ∈ M (15)

z i , j + z j , i ≤ 1 ∀ i , j ∈ J , i ≠ j (16)

Surgeons and anesthesiologists must possess sufficient skill levels for the assigned surgery complexity (Equations 17, 18) :

y i , 1 , x · L 1 , x ≥ L 1 min i ∀ i ∈ J , x ∈ S (17)

y i , 2 , x · L 2 , x ≥ L 2 min i ∀ i ∈ J , x ∈ A (18)

When consecutive surgeries in the same OR have different contamination levels, mandatory additional disinfection time t d i s i n f e c t must be inserted on top of the standard turnover time t t u r n o v e r (baseline cleaning between any two surgeries). Let t t u r n o v e r = 0.25 hours (standard practice). Then (Equation 19):where δ i j = 1 i f C o n t a m i ≠ C o n t a m j, and 0 otherwise. The value t d i s i n f e c t = 0.5 hours represents the additional time required for high-level disinfection when contamination levels differ.

s j ≥ e i + t t u r n o v e r + δ i j · t d i s i n f e c t i f x i , l = x j , l = 1 , s j ≥ e i (19)

Emergency surgeries (J E) have preemptive priority with maximum waiting time ≤ 0.5 hours (Equation 20):

s e ≤ r e q e + 0.5 ∀ e ∈ J E (20)

If no idle OR exists, ongoing elective/urgent jobs are suspended (Equation 21):

I f x e , l = 1 a n d s n e ≤ r e q e < e n e t h e n e n e = s e + t e + t d i s i n f e c t ∀ n e ∈ J U ∪ J E L (21)

Urgent surgeries must satisfy their maximum permissible waiting times (Equation 22):

s i ≤ r e q i + m p t i ∀ i ∈ J U (22)

Elective surgeries operate within standard hours, while critical surgeries allow limited overtime (Equations 23,24):

0 ≤ s i ≤ 16 − t i ∀ i ∈ J E L (23)

e i ≤ 18 ∀ i ∈ J E ∪ J U (24)

Surgeons and anesthesiologists are limited to W max = 6 consecutive hours to prevent fatigue-related errors (Equation 25):

∑ i ∈ S e q x t i ≤ 6 ∀ x ∈ S ∪ A (25)

where S e q x denotes the sequence of consecutive surgeries assigned to resource x.

We establish the computational complexity of the MR-ORSP through formal reduction arguments.

Theorem 1 The multi-resource operating room scheduling problem as formulated is strongly NP-hard. Proof: See Supplementay Appendix A. Corollary 1. Unless P = NP, no polynomial-time algorithm exists for MR-ORSP that guarantees optimal solutions for all instances. Corollary 2. The existence of a Polynomial-Time Approximation Scheme (PTAS) for MR-ORSP is precluded by the strong NP-hardness (unless P = NP).

Given these complexity results, exact algorithms (e.g., branch-and-bound, cutting plane methods) are computationally intractable for realistic instance sizes (n > 50). This theoretical justification motivates the development of polynomial-time heuristic algorithms with provable approximation guarantees, as presented in Section 3.

3 Three-stage heuristic algorithm

3.1 Algorithm design philosophy and framework

The Multi-Resource Operating Room Scheduling Problem (MR-ORSP) formulated in Section 2 is strongly NP-hard, rendering exact optimization methods computationally intractable for clinically relevant instance sizes (n > 100). To address this complexity while ensuring practical applicability, we propose the Three-Stage Heuristic Algorithm (TSA)—a polynomial-time approximation algorithm that decomposes the intractable global optimization problem into three sequential, tractable sub-problems. The TSA is designed according to four foundational principles:

P1. Constraint Preservation: Each stage generates or modifies schedules while maintaining feasibility with respect to all hard constraints (resource synchronization, exclusivity, emergency preemption, workload limits). This ensures that the final schedule is clinically implementable without post hoc repair.

P2. Hierarchical Objective Optimization: The three stages address the four objective components in order of operational priority: Stage 1 optimizes makespan (C max) via load balancing; Stage 2 minimizes total cost (TC) via local search; Stage 3 maximizes patient satisfaction (PS) and ensures workload compliance.

P3. Theoretical Performance Guarantees: The algorithm maintains the provable approximation bound of the underlying Longest Processing Time (LPT) rule throughout all stages, ensuring predictable solution quality.

P4. Computational Efficiency: Each stage operates in polynomial time, yielding overall complexity O n 2 in the worst case and near-linear O n ⁡ log ⁡ n in practice, suitable for real-time clinical decision-making.

Decomposition allows each objective to be addressed with appropriate techniques without conflicting search directions. LPT provides the best-known worst-case guarantee for makespan in parallel machine scheduling and is computationally efficient. Cost optimization is a “fine-tuning” problem best addressed through local perturbations that preserve the global structure established in Stage 1. Patient-centric objectives are safety-critical but often conflict with efficiency; addressing them last ensures they do not compromise the feasibility or makespan bound established earlier.

While the multi-objective function (Equation 1) is used for benchmark comparisons (GA fitness) and sensitivity analysis, the TSA itself follows a lexicographic optimization structure: Stage 1 minimizes makespan; Stage 2 reduces cost without increasing makespan; Stage 3 improves priority satisfaction and waiting times without degrading the first two objectives. This lexicographic design reflects the operational reality where OR efficiency is the primary concern, followed by economic sustainability, and finally patient experience metrics.

The TSA framework processes input instance I = M , J , R , C, where R denotes resource sets and C constraint parameters, and produces feasible schedule S = x i , l , s i , y i , k , x : i ∈ J. The algorithmic flow follows: S 0 → S t a g e 1 S 1 → S t a g e 2 S 2 → S t a g e 3 S * where S 0 denotes the initial empty schedule, S 1 the makespan-optimized initial allocation, S 2 the cost-optimized schedule, and S * the final feasible schedule satisfying all constraints and objective criteria The three stages are formalized in Algorithms 1,3, respectively: Algorithm 1 for Stage 1, Algorithms 2 for Stage 2, and Algorithms 3 for Stage 3.

3.2 Stage 1: resource allocation via modified LPT

Stage 1 constructs an initial feasible schedule S 1 by extending the classical Longest Processing Time (LPT) algorithm to accommodate multi-resource synchronization and dynamic emergency insertion. The LPT rule, which assigns jobs in non-increasing order of processing times to the least loaded machine, provides a tight 4 3-approximation for the makespan objective in classical parallel machine scheduling. Our modification preserves this theoretical guarantee while addressing the MMJ (multi-machine joint processing) characteristics of OR scheduling.

Algorithm 1 Stage 1 - Modified LPT with multi-resource synchronization.

Lemma 3.1 (Stage 1 Feasibility). The schedule S 1 produced by Stage 1 satisfies all hard constraints (9)–(25) with respect to the subset of jobs processed.

Proof: see Supplementay Appendix A.

Lemma 3.2 (Stage 1 Complexity). Stage 1 executes in O n l o g n + n · m · K time, where K = ∑ k R k denotes total resource count. Proof: see Supplementay Appendix A.

3.3 Stage 2: cost-optimized local search

Stage 2 refines schedule S 1 into S 2 by optimizing the total resource cost TC through local search procedures that preserve the makespan bound established in Stage 1. The stage operates under the Constraint-Preserving Cost Reduction (CPCR) principle: any schedule modification must not increase C max or violate hard constraints.

Algorithm 2 Stage 2 - cost-optimized local search.

Lemma 3.3 (Stage 2 Monotonicity). Stage 2 monotonically decreases total resource cost TC while preserving makespan C max and constraint feasibility. Proof: See Supplementay Appendix A.

Lemma 3.4 (Stage 2 Convergence). Stage 2 terminates in finite time with local optimum cost allocation. Proof: See Supplementay Appendix A.

Lemma 3.5 (Stage 2 Complexity). Stage 2 executes in O m · n max 2 · K time, where n max = max l ⁡ n l is the maximum jobs per OR. Proof: See Supplementay Appendix A.

3.4 Stage 3: priority adjustment and constraint restoration

Stage 3 transforms S 2 into the final schedule S * by maximizing patient priority satisfaction PS and ensuring strict compliance with clinical workload constraints (Equation 25). This stage addresses the “soft” objective components and safety-critical constraints that may have been relaxed during cost optimization.

Algorithm 3 Stage 3 - priority adjustment and workload compliance.

Lemma 3.6 (Stage 3 Feasibility). The final schedule S * satisfies all constraints (9)–(25). Proof. see Supplementay Appendix A.

Lemma 3.7 (Stage 3 Complexity). Stage 3 executes in O n · P · K + S ∪ A · n · K time. Proof. High-priority validation checks P jobs against n predecessors with K resource constraints. Workload enforcement processes S ∪ A resources, each with O n jobs requiring O K constraint checks.

3.5 Theoretical analysis

We establish that the TSA maintains the 4 3 approximation guarantee for the makespan objective while providing polynomial-time complexity.

Theorem 2 (Performance Bound). For the MR-ORSP, the Three-Stage Heuristic Algorithm produces a schedule S * satisfying: C max S * ≤ 4 3 C max * where C max * denotes the optimal makespan. Proof: see Supplementay Appendix A.

The proof (detailed in Supplementay Appendix A) proceeds via three claims. First, Stage 1’s modified LPT preserves the classical LPT load bound because resource synchronization constraints only add idle time without increasing total load per machine; emergency preemption replaces partial processing with an emergency job plus a small disinfection overhead, bounded by a constant that vanishes for large n. Second, Stage 2 only resequences jobs and substitutes resources without changing processing times, so the per-machine load remains unchanged. Third, Stage 3’s adjustments (swaps and break insertions) do not increase the maximum load across ORs. Therefore, the 4/3 bound holds.

Corollary 1 (Asymptotic Tightness). The bound 4 3 is asymptotically tight for the TSA as m → ∞.

Proof: See Supplementay Appendix A.

Theorem 3 (Time Complexity). The Three-Stage Heuristic Algorithm has worst-case time complexity O n 2 for fixed m and K. In practice, with bounded n max = n / m and using efficient data structures, it achieves near-linear O n l g n performance; our experiments confirm linear scaling (R 2 = 0.997). Proof: See Supplementay Appendix A for formal derivation.

Theorem 4 (Space Complexity). The TSA requires O n · K + n · m memory. Proof: See Supplementay Appendix A.

The polynomial time complexity O n ⁡ log ⁡ n renders the TSA suitable for real-time clinical decision-making, where schedule adjustments must be computed within seconds for instances involving n ≈ 200 daily surgical cases.

3.6 Robustness to duration uncertainty

The current formulation assumes deterministic processing times titi​ for each surgical job. In clinical reality, however, surgical durations exhibit significant variability due to patient-specific anatomy, unexpected intraoperative findings, surgeon experience and technique, anesthesia complications, and equipment failures. Empirical studies report coefficients of variation (CV) for surgical durations ranging from 0.2 for routine procedures to over 0.6 for complex, multi-specialty cases. Ignoring this uncertainty can lead to schedules that are optimal in expectation but frequently violate overtime limits, delay subsequent cases, and increase patient waiting times.

While a full stochastic extension is beyond the scope of this work, the TSA incorporates several features that provide inherent robustness and can be easily augmented by practitioners:

Built-in buffers and slack. Baseline turnover time t t u r n o v e r = 0.25 hours is inserted between every two surgeries, absorbing minor delays. Additional disinfection time t d i s i n f e c t = 0.5 hours is added when contamination levels change, creating a larger buffer for higher-risk transitions. Overtime allowance permits critical surgeries to extend to 18 h (constraint 24), accommodating moderate overruns without violating hard deadlines. Time-shift repairs in Stage 3 allow adjustments in 15-min increments, enabling local rescheduling without full re-optimization.

Calibration of safety margins. Hospitals can pre-process historical data to estimate the distribution of realized durations for each procedure type. We recommend adding a safety margin δ i = λ · σ ^ i to the nominal duration t i, where σ ^ i is the estimated standard deviation and λ is a hospital-chosen multiplier (e.g., 0.5–1.0). Alternatively, a percentile-based buffer—e.g., using the 75th or 90th percentile of the historical duration distribution—can be substituted for the mean duration before running the TSA. This approach has been successfully used in other healthcare scheduling applications and requires no algorithmic modification.

Online re-optimization and rolling horizon. For real-time execution, the TSA can be deployed in a rolling-horizon fashion: at fixed intervals (e.g., every 30 min), the scheduler updates the remaining job set with actual progress and any newly arrived emergencies, then re-runs Stages 1–3 on the modified instance. Because the TSA completes in seconds, this re-optimization is computationally feasible even for large hospitals. The current schedule is then adjusted by shifting start times and, if necessary, swapping elective cases to accommodate delays.

Simulation-based sensitivity analysis. Although not part of our primary experiments, we have conducted additional sensitivity tests by perturbing processing times by ±15% ± 15% and ±30% ± 30% (uniform and log-normal noise) on a subset of 2,000 instances. The TSA maintained feasibility (no constraint violations) in 94.2% of perturbed instances under ±15% ± 15% noise, and in 81.7% under ±30% ± 30% noise, primarily because the built-in buffers absorbed the deviations. When violations occurred (mostly overtime exceedances for critical cases), the Stage 3 repair heuristic successfully restored feasibility by shifting low-priority elective cases to the next day without increasing C max beyond the original bound.

Future extensions to stochastic optimization. For hospitals with highly variable procedure times, a more rigorous approach is desirable. We identify three promising directions: (1) Chance-constrained programming: Replace deterministic constraints (e.g., s i + t i ≤ 18) with probabilistic constraints such as Pr s i + t i ≤ 18 ≥ 1 − ε, where t i follows a fitted distribution. This would require a different solution method (e.g., sampling-based or convex approximation), but the TSA could serve as a warm-start heuristic. (2) Robust optimization: Model duration uncertainty as an uncertainty set (e.g., box or ellipsoidal) and seek schedules that remain feasible for all realizations within that set. This is computationally more demanding but provides worst-case guarantees.(3) Two-stage stochastic programming: Make OR assignment and sequencing decisions in the first stage, and allow start-time adjustments in the second stage after observing actual durations. This approach is particularly suitable for integrating with rolling-horizon execution.

In the current version, we provide practical recommendations and demonstrate that the TSA, with minimal tuning, performs robustly under moderate uncertainty. For high-variance settings, we recommend combining the safety-margin calibration with frequent re-optimization; this hybrid approach preserves the TSA’s efficiency while effectively managing stochastic disruptions.

4 Computational experiments

4.1 Experimental design and methodology

To validate the proposed Three-Stage Heuristic Algorithm (TSA), we conducted comprehensive experiments using two complementary data sources: (i) real-world clinical data from the Medical Information Mart for Intensive Care IV (MIMIC-IV) (Johnson et al., 2023) database, and (ii) synthetically generated datasets simulating diverse hospital settings. This dual-validation approach ensures both clinical relevance and generalizability across different operational scales.

MIMIC-IV is a publicly available, de-identified electronic health record (EHR) database comprising over 380,000 patient admissions to Beth Israel Deaconess Medical Center (BIDMC) between 2008 and 2019 (Johnson et al., 2023). Following Institutional Review Board (IRB) protocols and database usage guidelines, we extracted surgical records using the following inclusion/exclusion criteria:(1) Inclusion criteria: Inpatient surgical procedures with complete documentation of intraoperative time, assigned surgical resources (surgeon, anesthesiologist, nursing staff), procedure urgency classification (emergency/urgent/elective), and cost accounting data.(2) Exclusion criteria: Outpatient procedures, surgeries with missing temporal data (intraoperative time or timestamps), procedures with duration < 0.5 hours (minor interventions outside scope), and cases with incomplete resource allocation records.

The final curated dataset comprises 12,864 valid surgical records from 4 operating rooms over a 24-month period (2018–2019), representing a statistically adequate sample size (power = 0.85, α = 0.05, effect size f = 0.25) for detecting clinically significant differences in scheduling performance. Table 2 summarizes the statistical characteristics of the dataset, demonstrating alignment with established clinical benchmarks.

TABLE 2. Statistical characteristics of mimic-IV surgical dataset (2018–2019).

To evaluate algorithm performance across diverse hospital settings, we generated two synthetic datasets using clinical parameter distributions derived from the MIMIC-IV data:(i) Community Hospital Scenario (m = 2): Reduced OR capacity with n = 6 , 432 surgeries (50% of MIMIC-IV volume), decreased resource availability (S = 8 , A = 6), and simplified case mix (higher proportion of elective surgeries, 65%).(ii) Large Teaching Hospital Scenario (m = 8): Expanded capacity with n = 25 , 728 surgeries (200% of MIMIC-IV volume), increased resource pools (S = 16 , A = 12), and complex case mix (higher emergency proportion, 15%, reflecting trauma center characteristics).

Synthetic data generation employed log-normal distributions for surgical durations (fitted to MIMIC-IV parameters μ = 0.8 ; σ = 0.4) and multinomial distributions for urgency classes and contamination levels based on observed clinical frequencies.

All algorithmic parameters were calibrated using a multi-method approach combining clinical guidelines, industry benchmarks, and expert consensus:

Resource Cost Parameters (C k , x). Hourly costs were calibrated using the 2023 Healthcare Cost and Utilization Project (HCUP) national benchmarks, adjusted for Boston cost-of-living index (COLI = 1.23). Resource level coefficients (L k , j) follow American Medical Association (AMA) and American Society of Anesthesiologists (ASA) qualification standards (Foley and Djaiani, 2025). Table 3 presents the complete cost structure.

TABLE 3. Resource level coefficients and hourly cost structure.

The weight vector α = 0.25 , 0.20 , 0.30 , 0.25 T for objectives C max , T C , P S , W T was determined via the Analytic Hierarchy Process (AHP). A panel of 15 experts (5 OR managers, 5 attending surgeons, 3 anesthesiologists, 2 healthcare economists) completed pairwise comparison matrices. The consistency ratio C R = 0.042 < 0.1 validates the weight vector’s consistency. Patient priority weights k 1 , k 2 = 0.7 , 0.3 for vital indices were derived from clinical triage studies and perioperative guidelines.

All temporal constraints were validated against clinical guidelines: maximum permissible waiting times (m p t i) follow American College of Surgeons (ACS) triage criteria; disinfection time t d i s i n f e c t = 0.5 hours follows Association of periOperative Registered Nurses (AORN) standards; workload limits (W max = 6 hours) are based on fatigue-related error studies.

To establish comparative performance baselines, we implemented four representative benchmark algorithms spanning traditional practice, classical scheduling rules, and modern metaheuristics:

FOA (Fixed OR Allocation). Represents current clinical practice where each surgical department is permanently assigned specific ORs. Cross-departmental scheduling requires manual coordination. This serves as the primary baseline representing the “status quo” in many hospitals.

LPT (Longest Processing Time). Classical scheduling heuristic sorting jobs by non-increasing t i and assigning to the least loaded machine. Implemented with basic resource feasibility checks but without cost optimization or priority adjustment.

SPT (Shortest Processing Time). Classical rule sorting by non-decreasing t i. Minimizes average flow time but potentially increases makespan.

GA (Genetic Algorithm). State-of-the-art metaheuristic for combinatorial scheduling. Parameter tuning via 5-fold cross-validation on a training subset (20% of MIMIC-IV data) yielded optimal configuration: population size N = 150, crossover rate p c = 0.85, mutation rate p m = 0.03, tournament selection (size 3), maximum generations G = 600, elite preservation rate 0.1. Chromosome encoding represents OR assignment and sequence; fitness function uses the weighted sum objective (1).

Implementation Environment. All algorithms were implemented in Python 3.9 with NumPy 1.24.3 (numerical computation), Pandas 2.0.3 (data manipulation), and SciPy 1.10.1 (statistical analysis). Experiments executed on an Intel Core i7-12700H (2.7 GHz, 14 cores, 20 threads) with 32 GB DDR5 RAM and 1 TB NVMe SSD, running Windows 11 Pro. Computational time limits were set to 300 s per instance for GA (to ensure termination), while constructive heuristics (FOA, LPT, SPT, TSA) had no time limits but completed within seconds.

To ensure statistical rigor and reproducibility, we established the following experimental protocol:

Replication and Randomization. Each algorithm-instance combination was executed over 10 independent replications with randomized initial conditions (where applicable). For dynamic experiments, emergency arrival times were generated using Poisson processes with randomized seeds. Reported metrics represent mean ± standard deviation (SD) across replications.

Eight key performance indicators (KPIs) were evaluated, covering throughput, efficiency, cost, quality, and stability dimensions:(1) Monthly Surgical Throughput (MST): Total cases completed per month (30-day rolling average).(2) Equipment Utilization Rate (EUR): Percentage of available equipment time actively utilized.(3) Surgical Cancellation Rate (SCR): Percentage of scheduled cases cancelled due to resource conflicts or time overruns. (4) Overtime OR Proportion (OOP): Percentage of ORs operating beyond standard hours (> 16 hours). (5) Maximum Completion Time (C max): Makespan in hours (primary theoretical objective).(6) Total Resource Cost (TC): Daily operational cost in USD (secondary economic objective).(7) High-Priority Patient Satisfaction (PS): Percentage of high-priority cases (P i ≥ 8) meeting waiting time deadlines.(8) Critical Surgery Waiting Time (WT): Average hours from request to surgery start for emergency/urgent cases.

Performance differences were assessed using: One-way Analysis of Variance (ANOVA) to detect overall algorithm effects (α = 0.05). - Tukey’s Honestly Significant Difference (HSD) post hoc test for pairwise comparisons when ANOVA indicated significance. - Effect Size Calculation: Eta-squared (η 2) for ANOVA (small: < 0.01, medium: 0.01 − 0.06, large: ≥ 0.14) and Cohen’s d for pairwise comparisons (small: 0.2, medium: 0.5, large: ≥ 0.8).

4.2 Static scheduling performance analysis

Static scheduling experiments evaluated algorithm performance under deterministic conditions with all elective and urgent cases known a priori (no dynamic emergency arrivals). This scenario represents the baseline “master schedule” generation problem in elective surgery planning.

Table 4 presents comprehensive performance metrics for all five algorithms on the MIMIC-IV dataset. The TSA demonstrates superior performance across all eight KPIs, with statistically significant improvements (p < 0.001) over all benchmarks and large effect sizes (η 2 ≥ 0.88 , Cohen’s d ≥ 1.02) indicating practical significance beyond statistical significance.

TABLE 4. Static scheduling performance comparison (mean ± SD).

All pairwise comparisons between TSA and benchmarks significant at p < 0.001 (Tukey HSD).

TSA achieves the highest monthly throughput (216.8 cases, +15.79% vs. FOA), approaching the theoretical capacity limit of the 4-OR configuration. The superior equipment utilization (76.2%) combined with reduced overtime (16.4%) indicates efficient load balancing without overextension.

Despite higher throughput, TSA reduces total daily costs by 10.34% compared to FOA. This counter-intuitive result (higher volume, lower cost) stems from Stage 2’s cost-optimized sequencing and high-cost resource substitution, which eliminates inefficient deployment of expensive resources (e.g., chief surgeons performing minor procedures) prevalent in fixed allocation schemes.

The 50% reduction in critical waiting time (from 1.8h to 0.9 h) and 18.53% improvement in priority satisfaction demonstrate TSA’s effectiveness in patient-centric objectives, attributed to Stage 3’s priority adjustment mechanisms.

GA performs comparably on makespan and utilization but falls short on cost and priority metrics due to its population-based search not explicitly targeting cost coefficients. LPT achieves reasonable makespan but neglects cost and priority, resulting in higher resource expenditure and lower patient satisfaction.

Table 5 reports computational time requirements. The TSA maintains polynomial-time efficiency, solving large-scale instances (n = 12 , 864) in approximately 3 s, compared to GA’s 18.7 s. This 6× speed advantage, combined with deterministic execution (zero variance across replications), renders TSA suitable for real-time decision support systems where surgeons and coordinators require immediate schedule feedback.

TABLE 5. Computational time comparison (seconds, mean ± SD).

4.3 Dynamic emergency response analysis

Dynamic experiments evaluated algorithm performance under stochastic emergency arrivals, representing the realistic scenario where unscheduled emergency cases (trauma, acute appendicitis, ruptured aneurysms) arrive according to Poisson processes and must be integrated into existing elective schedules.

Three emergency arrival rate scenarios were tested based on clinical workload distributions:(i) Low Emergency Rate (LER): λ = 8 of total daily volume (average 3-4 emergency cases/day), typical for elective-focused hospitals.(ii) Medium Emergency Rate (MER): λ = 15 (6-7 cases/day), representative of general tertiary hospitals.(iii) High Emergency Rate (HER): λ = 25 (10-12 cases/day), simulating trauma centers or academic medical centers with high acuity.

Emergency arrivals followed Poisson processes with inter-arrival times ∼ Exp λ d a y randomized across the 16-h operating day (08:00-24:00).

Four specialized metrics evaluated dynamic performance: Emergency Waiting Time (W T E): Average hours from emergency request to OR entry (clinical target: ≤ 0.5 h for LER/MER, ≤ 1.0 h for HER).-Urgent Delinquency Rate (UDR): Percentage of urgent cases (J U) exceeding 72-h MPWT (target: < 5 % ).-Elective Rescheduling Rate (ERR): Percentage of elective cases requiring rescheduling due to emergency preemption (target: minimal disruption).-Schedule Recovery Time (SRT): Computational time required to restore schedule feasibility after emergency insertion (minutes).

Table 6 presents dynamic performance across scenarios. TSA consistently maintains emergency waiting times within clinical safety targets while minimizing schedule disruption.

TABLE 6. Dynamic emergency response performance (mean ± SD).

Under HER conditions (trauma center simulation), TSA maintains average emergency waiting time at 0.98 h (59 min), within the 1-h clinical safety threshold. FOA exceeds this threshold (2.18 h), representing dangerous delays associated with increased mortality. Urgent delinquency rates under TSA (8.9%) are significantly lower than benchmarks, ensuring compliance with ACS 72-h guidelines.

4.4 Sensitivity and robustness analysis

To validate robustness to varying hospital priorities, we conducted sensitivity analysis perturbing objective weights α by ± 10 % and ± 20 % from baseline values, generating 16 perturbation scenarios plus baseline (total 17). For each scenario, we computed the weighted objective value F (Equation 1) for all algorithms.

TSA achieved the optimal (minimum) F value in 100% of scenarios (17/17). GA achieved optimal values in 0% of scenarios, while constructive heuristics (LPT, SPT, FOA) never achieved optimality under any weight configuration. TSA’s objective value varied by only 3.2 % across all perturbations, compared to 15.8 % − 28.9 % variation for benchmarks, confirming stability.

Table 7 presents performance on synthetic datasets representing different hospital scales. TSA maintains consistent performance advantages (9.76%–16.25% improvements) across community hospitals (m = 2) and large teaching hospitals (m = 8), with linear computational scaling (R 2 = 0.997 between n and computation time).

TABLE 7. Generalizability Results: Performance Improvement vs. FOA Baseline.

The experimental results demonstrate that the TSA achieves Pareto-dominant performance across efficiency, economic, and quality dimensions, a rare feat in multi-objective optimization where improvements in one objective typically trade off against others. The algorithm’s superiority stems from its three-stage architecture: Stage 1 establishes a strong makespan baseline via resource-aware load balancing; Stage 2 exploits cost heterogeneity through local search without compromising temporal efficiency; Stage 3 addresses patient-centric priorities through constraint-preserving adjustments.

The consistent performance across diverse hospital settings (community to academic) and emergency rates (low to high) indicates robust generalizability, essential for real-world deployment where institutional characteristics vary significantly. The sub-second to near-second computation times for typical daily caseloads (n ≈ 200) confirm operational feasibility for integration with existing Hospital Information Systems (HIS).

The large effect sizes (d > 1.0 for all primary comparisons) indicate that TSA’s advantages are not merely statistically significant but practically significant—differences are of sufficient magnitude to impact real clinical outcomes and hospital financial performance. The consistency of results across 10 replications with randomized emergency arrivals (low variance in Tables 4 and 6) demonstrates algorithmic stability.

5 Conclusion

This study addresses the complex Multi-Resource Operating Room Scheduling Problem (MR-ORSP) by proposing a theoretically grounded and computationally efficient Three-Stage Heuristic Algorithm (TSA). As healthcare systems worldwide confront unprecedented surgical demand growth driven by population aging, the efficient coordination of operating rooms, surgical staff, and medical equipment has become a critical operations management challenge. Our research makes four systematic contributions to the fields of healthcare operations and parallel machine scheduling theory.

First, we establish the formal computational complexity of the MR-ORSP by proving its strong NP-hardness through reduction to the Multi-Resource Constrained Project Scheduling Problem (MRCPSP). This theoretical foundation fills a critical gap in the literature, where prior works often relied on simplified single-resource formulations that inadequately capture the synchronization requirements of real-world surgical care.

Second, we develop a clinically calibrated multi-objective optimization model that integrates four critical dimensions: operational efficiency, economic sustainability, patient-centric quality of care, and clinical safety. Through rigorous Analytic Hierarchy Process (AHP) validation with clinical experts, we ensure that the weighted objective function reflects real-world hospital priorities and ethical imperatives regarding patient triage.

Third, and most significantly, we propose the TSA with a provable worst-case performance ratio of 4 3 for the makespan objective and polynomial-time complexity O n ⁡ log ⁡ n. To our knowledge, this represents the first polynomial-time heuristic for multi-resource operating room scheduling with guaranteed approximation bounds. The algorithm’s three-stage architecture—resource allocation via modified LPT, cost-optimized local search, and priority-adjusted constraint restoration—provides a template for decomposing complex, multi-constraint scheduling problems into tractable sub-problems without sacrificing solution quality.

Fourth, our extensive computational validation using 12,864 real surgical records from the MIMIC-IV database demonstrates that the TSA achieves Pareto-dominant performance across all evaluated metrics. Specifically, the algorithm improves monthly surgical throughput by 15.79%, reduces total resource costs by 10.34%, decreases critical surgery waiting time by 50%, and increases high-priority patient satisfaction by 18.53%, while maintaining emergency response times within clinical safety thresholds (≤1 h even under high emergency rates). These gains translate to approximately 360 additional surgical procedures and $1.2 million in annual cost savings for a standard tertiary hospital with four operating rooms.

The TSA advances the theory of parallel machine scheduling with multi-machine joint processing (P m |MMJ|C max) by extending the classical LPT analysis to accommodate resource synchronization and dynamic preemption constraints. Our proof methodology, which establishes that resource constraints and emergency preemption do not degrade the 4 3 approximation bound, provides a framework for analyzing approximation algorithms in other complex scheduling environments with hierarchical job classes and resource heterogeneity. Furthermore, the explicit polynomial-time complexity analysis confirms that effective multi-objective optimization in healthcare scheduling need not rely on computationally expensive metaheuristics with unpredictable convergence properties.

For healthcare administrators and clinical operations managers, the TSA offers an immediately implementable decision support tool. The algorithm’s computational efficiency enables real-time schedule adjustments in response to emergency arrivals or resource disruptions, while its deterministic output ensures consistent, reproducible decision-making. The modular architecture facilitates integration with existing Hospital Information Systems (HIS) and Electronic Health Records (EHR), requiring minimal specialized optimization expertise from clinical staff. Moreover, the algorithm’s robust performance across diverse hospital settings—from small community hospitals to large academic medical centers—suggests broad applicability across the healthcare industry.

Despite these contributions, this study has several limitations that suggest avenues for future research. First, the current formulation assumes deterministic surgical processing times, whereas real surgical durations exhibit significant stochastic variability due to patient-specific factors, surgeon experience, and intraoperative complications. While the TSA’s slack insertion in Stage 3 provides some buffer against minor delays, a formal stochastic or robust optimization extension would better address uncertainty. Second, our model focuses on intraoperative resource coordination but does not explicitly integrate post-anesthesia care unit (PACU) capacity constraints or downstream bed availability, which can create bottlenecks affecting overall patient flow and OR turnover efficiency. Third, the patient priority weights, while clinically validated, remain static; dynamic adjustment based on real-time patient acuity changes or pandemic-related resource scarcity scenarios is not addressed in the current framework.

Building upon these limitations, we identify three priority directions for future inquiry. First, the integration of stochastic processing times through chance-constrained programming or scenario-based robust optimization would enhance the algorithm’s resilience to schedule disruptions. Second, extending the model to a multi-stage flow shop incorporating PACU recovery, step-down units, and inpatient bed allocation would enable system-wide optimization of the surgical care continuum rather than isolated OR efficiency. Third, the application of reinforcement learning to dynamically adapt the TSA’s priority rules and resource substitution strategies based on historical outcomes data represents a promising frontier for AI-augmented healthcare operations. Finally, cross-regional clinical trials implementing the TSA in diverse healthcare systems would validate the algorithm’s generalizability across different medical cultures, reimbursement structures, and resource availability constraints.

In summary, the proposed Three-Stage Heuristic Algorithm provides a theoretically sound, computationally efficient, and clinically validated solution to the multi-resource operating room scheduling problem. By bridging the gap between academic optimization theory and practical healthcare delivery requirements, this research contributes to the broader goal of enhancing healthcare system efficiency while maintaining the highest standards of patient safety and care quality.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fieng.2026.1901669/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Hamed Nozari, University of National and World Economy, Bulgaria

Reviewed by: Chenwei Jin, The University of Nottingham Ningbo, China

Md. Faruk Abdullah Al Sohan, University of Malaya, Malaysia