⚡ electrical engineering

Integrated Framework for Long-term Elective Surgery Management under Uncertainty: From Strategic to Tactical Planning

This paper proposes a novel integrated framework that combines strategic queue-control policies with tactical operating theatre scheduling under uncertainty to stabilize waiting lists, manage cancellation risks, and optimize long-term elective surgery management in public healthcare systems.

Original authors: Joao P. F. da Silva, Lucas Bortoletto, Rafael C. S. Schouery, Edilson F. Arruda, Fabricio F. Santos, Ana Paula C. I. Alves

Published 2026-07-29

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CC BY 4.0

Original authors: Joao P. F. da Silva, Lucas Bortoletto, Rafael C. S. Schouery, Edilson F. Arruda, Fabricio F. Santos, Ana Paula C. I. Alves

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a massive, chaotic ship called "The Hospital." This ship has a limited number of engines (operating rooms) and a never-ending stream of passengers (patients) waiting to get medical treatment. The tricky part is that you don't know exactly how many new passengers will jump on board tomorrow, nor do you know exactly how long each person's journey will take. Some trips are quick sprints; others are marathon runs that might spill over into the night. If you try to pack too many people into an engine room, the ship might run out of fuel (time), forcing you to kick people off the boat at the last minute (cancellations). If you pack too few, the engines sit idle, wasting precious fuel. This is the daily struggle of public healthcare: how to keep the ship moving smoothly without leaving anyone stranded or crashing the engine.

This paper dives into the world of elective surgery planning, which is basically the art of scheduling non-emergency operations like hip replacements or eye surgeries. The authors are tackling a classic problem: how do you balance the need to treat as many people as possible with the reality that things are unpredictable? They are building on two big ideas. First, queue management, which is just a fancy way of saying "managing the line." Think of it like a water tank: if the tank fills up too fast, it overflows; if it drains too fast, it runs dry. Second, optimization under uncertainty, which means making the best possible plan when you can't see the future. The paper asks: Can we create a smart system that doesn't just look at today's schedule but also predicts how the line will grow or shrink over months, ensuring everyone gets their turn without the hospital running out of steam?

The authors propose a clever, three-part "traffic control" system to solve this, tested at a large university hospital in Brazil. Instead of trying to solve the whole problem in one giant, confusing math equation, they broke it down into three connected modules that talk to each other.

First, there's the Queue Manager. Imagine a smart thermostat for the patient line. It doesn't just count heads; it watches the "temperature" of the waiting list. If the line gets too long (reaching a specific "reorder point"), it signals the system to start scheduling more surgeries for that specific type of medicine. If the line is short, it says, "Hold off, we're good for now." This prevents the hospital from overworking itself when the line is already under control, or ignoring a crisis when the line is exploding. It uses a modified version of a classic inventory rule (called an (R, Q) policy) to decide exactly how many patients to pull from the line each month, ensuring that no one waits longer than six months.

Second, there's the Risk Calculator. This is where the paper gets really interesting about the "unknowns." In the past, hospitals often assumed a surgery would take exactly 60 minutes. But in reality, it might take 45 or 90. The authors used a model inspired by a Newsvendor (a person selling newspapers who has to guess how many to buy before knowing the demand). They asked: "If we schedule three surgeries in a six-hour block, what are the odds we'll finish them all on time?" They found that if you try to cram too many surgeries in, the risk of running overtime skyrockets, leading to cancellations. Their model calculates the "sweet spot"—the perfect number of surgeries to plan so that you get the most done without triggering a chaotic overtime mess. For example, in one specialty, they found that planning three surgeries was the limit; trying to squeeze in a fourth made cancellations too likely.

Third, there's the Scheduler. This is the final piece that takes the instructions from the first two modules and actually builds the monthly timetable. It's like a puzzle solver that fits the right medical teams into the right rooms, making sure that if a specialty needs two rooms at once (like for organ transplants), it gets them, and that no one is double-booked. Crucially, this scheduler doesn't just look at what's available today; it looks at the "target" set by the Queue Manager and the "safe capacity" set by the Risk Calculator.

When the authors tested this system using real data from the hospital, the results were striking. They ran a simulation over a year and compared their smart system against a "baseline" approach (which is basically the old way of just trying to fill every slot every month). The old way led to chaos: some waiting lists grew uncontrollably because the system didn't prioritize the ones that were getting too long, while others were treated even though they didn't need it yet. The new system, however, kept the waiting lists stable. It successfully kept the patient lines hovering around a safe, manageable level, ensuring that patients were treated within the six-month target without the hospital burning out its staff or cancelling surgeries due to overtime.

The paper suggests that by linking these three steps—watching the line, calculating the risk of overtime, and then scheduling the rooms—hospitals can stop reacting to crises and start managing them. It's not about having more operating rooms; it's about using the ones you have much smarter. The simulation showed that this approach could stabilize the system, reduce the risk of last-minute cancellations, and make the whole process fairer for patients, all without needing to build new hospitals or hire endless new staff. It's a reminder that sometimes, the best way to solve a traffic jam isn't to build a wider road, but to install a better traffic light.

Technical Summary: Integrated Framework for Long-term Elective Surgery Management under Uncertainty

Problem Statement
Public healthcare systems face persistent challenges in managing elective surgery waiting lists, where hospitals must balance limited operating theatre (OT) capacity against uncertain and evolving patient demand. Current planning approaches often suffer from a disconnect between strategic and tactical levels. Strategic decisions (Case Mix Planning) typically define long-term targets but are frequently treated as deterministic inputs for tactical scheduling (Master Surgery Scheduling). Conversely, tactical models often optimize for a single period without explicitly controlling the long-term evolution of specialty-specific queues or accounting for stochastic patient arrivals and surgery durations. This isolation can lead to schedules that misalign with actual demand, resulting in excessive waiting times, resource underutilization, or high risks of overtime-induced cancellations. The core problem addressed is the lack of a unified approach that links strategic queue-control mechanisms with tactical OT allocation while explicitly modeling uncertainty in both demand and surgery duration.

Methodology
The authors propose a novel, modular integrated framework that decomposes the long-term elective surgery planning problem into three interconnected modules. This design avoids the computational intractability of a monolithic stochastic model while ensuring alignment between decision layers.

Module 1: Long-Term Waiting List Management (Strategic)
- Approach: Utilizes a modified $(R, Q)$ inventory control policy adapted for surgery queues.
- Mechanism: The policy determines the target number of patients to be scheduled in each planning cycle (e.g., monthly) for each medical subspecialty. It establishes lower ( $\underline{Q}$ ) and upper ( $\overline{Q}$ ) bounds on the batch size of surgeries to be processed.
- Theoretical Basis: The model derives analytical bounds based on the steady-state distribution of a stochastic process representing the queue. It ensures that the probability of a patient waiting longer than a target time $\tau$ remains below a specified threshold $\alpha_{rq}$ , guaranteeing long-term queue stability and adherence to waiting-time requirements despite stochastic demand.
Module 2: Overtime and Cancellation Risk Management (Strategic/Tactical Interface)
- Approach: Addresses the uncertainty in surgery duration using a Newsvendor-based model and an inventory-based formulation.
- Mechanism:
  - Newsvendor Model: Determines an optimal time budget ( $t^*_s$ ) for the last surgery in a block. If the remaining time in the session is less than $t^*_s$ , the surgery is cancelled to avoid overtime. This balances the cost of idle time against the penalty of overtime.
  - Inventory Model: Uses the resulting probability distribution of completed surgeries to determine the optimal number of surgeries ( $n^*$ ) to plan in a block. It maximizes the net value (gain from additional surgeries minus expected cancellation costs).
- Output: Generates a probability distribution of the number of surgeries that can be realistically completed in a block, accounting for cancellation risks.
Module 3: Operating Theatre Scheduling Problem (Tactical)
- Approach: Formulates a Mixed-Integer Linear Programming (MILP) model to allocate OT blocks to surgical subspecialties.
- Inputs: The model takes the strategic bounds ( $\underline{Q}, \overline{Q}$ ) from Module 1 and the effective block capacity distributions from Module 2. These are preprocessed (via successive convolutions) to determine the minimum and maximum number of blocks required for each subspecialty to meet targets with a specific confidence level.
- Optimization: The MILP solves for a monthly schedule that respects theatre compatibility, team availability, anaesthetist constraints, and sequential block requirements. It employs a hierarchical objective function to:
  1. Minimize violations of team and anaesthetist availability.
  2. Minimize allocations to non-preferred theatres.
  3. Maximize allocation to priority subspecialties.
  4. Maximize total block utilization.
  5. Maximize sequential block allocations for logistical efficiency.

Key Contributions

Integrated Framework: The paper presents the first framework, to the authors' knowledge, that explicitly aligns long-term waiting-list management with medium-term resource allocation while accounting for both demand uncertainty and surgery duration uncertainty.
Modular Decomposition: Instead of a single complex stochastic optimization, the authors propose a tractable, modular system. This preserves managerial interpretability and computational efficiency, making it suitable for real-world implementation.
Stochastic Capacity Modeling: By treating surgery duration as stochastic and modeling cancellation risks, the framework moves beyond deterministic capacity assumptions. It provides a probabilistic estimate of "effective capacity" rather than nominal capacity.
Strategic-Tactical Linkage: The framework translates strategic queue targets into tactical constraints (block limits) for the MILP, ensuring that daily scheduling decisions are driven by long-term stability goals rather than just immediate resource availability.

Results
The framework was evaluated using a case study at University Hospital Unicamp (HC-Unicamp), a large public referral hospital in Brazil. The study simulated a one-year horizon with 100 independent runs, comparing the proposed framework against a baseline policy (a standard MILP without strategic queue controls).

Queue Stability: The proposed framework successfully stabilized waiting lists. Queues for various subspecialties converged to the reorder threshold ( $R$ ) and fluctuated within a controlled region. In contrast, the baseline policy led to uncontrolled queue growth for several subspecialties, indicating a mismatch between allocated capacity and actual demand.
Cancellation Risk Management: The Overtime and Cancellation module demonstrated that increasing the number of planned surgeries per block increases the probability of cancellations. For the "Foot Orthopaedics" subspecialty, the model identified an optimal planning level of 3 surgeries per block, balancing throughput against the rising cost of cancellations.
Resource Utilization: While the baseline policy showed higher average theatre occupancy, this was achieved by scheduling all subspecialties regardless of queue status. The proposed framework exhibited lower occupancy in certain months because it strategically withheld capacity from subspecialties whose queues were already within the target range. This "controlled flexibility" allowed resources to be redirected to subspecialties with growing queues, improving overall system efficiency without requiring additional resources.
Computational Efficiency: The modular approach allowed the strategic components to be computed once, with the tactical MILP solving efficiently at each monthly cycle, demonstrating practical applicability.

Significance and Claims
The paper claims that its primary significance lies in bridging the gap between theoretical optimization and practical hospital management. By integrating strategic queue control with tactical scheduling, the framework supports:

Stabilization of Waiting Lists: Ensuring that waiting times remain within legal and ethical bounds (e.g., Brazil's SUS requirements) over the long term.
Realistic Planning: Acknowledging that surgery durations are uncertain and that over-scheduling leads to cancellations, thereby providing more achievable targets.
Transparent Decision-Making: Offering a structured, data-driven approach to theatre allocation that aligns with institutional priorities and resource constraints.
Efficiency without Expansion: Demonstrating that better coordination and demand management can improve access and reduce waiting times without the need for costly infrastructure expansion.

The authors note that the framework is adaptable to different hospital settings and that the modular design allows for the incorporation of alternative demand models or additional constraints (such as bed management) in future work. They emphasize that the approach is particularly relevant for public healthcare systems facing high demand and resource constraints.

Technical Summary: Integrated Framework for Long-term Elective Surgery Management under Uncertainty

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