💻 computer science

An Enhanced Memetic Algorithm for Emergency Outpatient Physician Scheduling Considering Patient Psychological Distress and Physician Fatigue in Fuzzy-CPT Framework

This study proposes an enhanced memetic algorithm within a fuzzy-CPT framework to optimize emergency outpatient physician scheduling by simultaneously minimizing patient psychological distress and physician fatigue perception, validated through real-world data from a Shanghai hospital during the COVID-19 pandemic.

Original authors: Junji Zhou, Jiawei Wu

Published 2026-09-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Junji Zhou, Jiawei Wu

Original paper licensed under CC BY 4.0 (https://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

In the high-stakes environment of a hospital emergency room, time is not just a metric; it is a psychological force. When patients wait, their anxiety does not simply rise in a straight line; it spikes sharply once the line grows too long, creating a state of distress that can feel overwhelming. Simultaneously, the doctors treating them are not immune to the pressure. Their exhaustion is not merely a matter of hours worked; it is a feeling that deepens disproportionately once they pass a personal limit, and it is heavily influenced by whether their colleagues are sharing the burden or if they feel they are the only ones left standing. For decades, hospital administrators have tried to solve the puzzle of who works when, often relying on rigid schedules that treat fatigue as a simple cost and waiting lines as a simple queue. However, these traditional methods often fail to capture the complex, human reality of how stress and tiredness actually feel during a crisis.

A new study by researchers Junji Zhou and Jiawei Wu tackles this problem by building a sophisticated scheduling tool that listens to the human mind. They focused on the weekly scheduling of emergency outpatient physicians during major public health emergencies, such as the peak of the COVID-19 pandemic. The researchers realized that to create a truly effective schedule, they had to account for two invisible but powerful factors: the psychological distress of waiting patients and the subjective fatigue of the doctors. They combined two behavioral concepts to model these feelings. First, they used a method to describe how fatigue feels fuzzy and gradual, rather than a sudden switch from "tired" to "exhausted." Second, they applied a theory of decision-making that recognizes people feel the pain of working too long much more intensely than the pleasure of working less, especially when they compare themselves to their peers. By weaving these human behaviors into a mathematical model, they created a system that seeks a balance between keeping patients calm and keeping doctors from burning out.

The researchers tested their approach using real data from a large tertiary hospital in Shanghai during the height of the pandemic. They compared their new method against older, standard scheduling algorithms. The results showed a clear advantage for their approach. In a series of tests involving twelve different weeks of patient data, their new algorithm consistently produced better outcomes. It was particularly successful at eliminating the psychological distress of waiting patients; in seven out of the twelve test cases, the new method managed to keep the waiting line short enough that patients felt no distress at all, whereas the older methods failed to achieve this in any of those cases. The new system also managed to reduce the overall "cost" of the schedule, which included the fatigue felt by the doctors, the money spent on bringing in extra help, and the anxiety of the waiting room. The algorithm achieved this not by simply working more hours, but by arranging the shifts more intelligently, matching the number of doctors on duty to the fluctuating arrival of patients throughout the day and night.

One of the most revealing aspects of the study was how cultural differences in human psychology might change the results. The researchers tested their model using two different sets of behavioral parameters: one based on studies conducted in the United States and another based on studies conducted in China. They found that the specific numbers used to describe how people value time and compare themselves to others significantly altered the final schedule. When using the parameters derived from Chinese studies, the model tended to create schedules that were more sensitive to the feeling of unfairness when only a few doctors worked overtime. This suggests that a one-size-fits-all approach to scheduling might not work globally; what feels like a fair and efficient schedule in one culture might feel stressful or inefficient in another. The study confirmed that understanding these subtle psychological nuances is essential for creating schedules that work in the real world.

To solve this complex puzzle, the researchers developed a specialized computer program, an enhanced version of a type of algorithm known as a memetic algorithm. Think of this program as a highly skilled planner that can test thousands of different schedule combinations in a short time. Unlike older planners that might get stuck in a local solution, this new program uses a multi-layered search strategy. It looks at the schedule from different angles: shifting the start time of a single shift by an hour, adding or removing a doctor for a specific day, or rearranging the entire team's rotation. It also uses a "two-stage" approach to start its work, first building a rough draft of the schedule using two different strategies—one focused on night shifts and one on day shifts—and then refining it. This allowed the program to find solutions that were both fair to the doctors and efficient for the patients, something that manual scheduling or simpler computer programs could not achieve.

The study also explored how changing specific rules would affect the outcome, a process known as sensitivity analysis. They found that the number of doctors available was a critical tipping point. When the hospital had fewer doctors, the system struggled to keep the waiting line short, leading to high patient distress and high doctor fatigue. However, once the number of doctors reached a certain threshold, the system could easily manage the flow, and the distress dropped to zero. Interestingly, they found that simply allowing doctors to work longer hours each day did not always help; there was an optimal range where flexibility was possible without causing extreme fatigue. They also discovered that the cost of bringing in extra doctors from other departments played a major role. If bringing in extra help was cheap, the system preferred to use them to keep the main team rested. If it was expensive, the system tried harder to optimize the existing team, though this sometimes led to higher fatigue.

Ultimately, this research provides a scientifically grounded tool for hospital managers facing the chaos of a public health crisis. It moves beyond simple math to include the human experience of waiting and working. By acknowledging that fatigue is a feeling that grows with comparison and that waiting is a psychological burden that spikes at certain points, the study offers a way to make scheduling decisions that are not just efficient, but humane. The findings suggest that during emergencies, the best way to protect both patients and doctors is to use scheduling systems that understand the mind as well as the clock. The researchers conclude that their method offers a robust way to handle the uncertainty of future health crises, ensuring that the front lines of healthcare remain sustainable even under the most intense pressure.

1. Problem Definition

The study addresses the weekly scheduling optimization problem for emergency outpatient physicians during major public health emergencies (e.g., pandemics). Unlike conventional outpatient settings, emergency clinics face highly stochastic and time-varying patient arrivals, strict infection control requirements, and the psychological vulnerability of patients.

The core challenge is a dual trade-off:

Patient Psychological Distress: Waiting patients experience nonlinear psychological distress (anxiety, suffering) that escalates sharply as queue lengths increase beyond a critical threshold, eventually plateauing at a tolerance limit.
Physician Fatigue Perception: Physicians working under high pressure do not experience fatigue linearly. Their perception is influenced by:
- Loss Aversion: They are significantly more sensitive to working hours exceeding a subjective psychological reference point than to hours within an acceptable range.
- Group Dynamics: Individual fatigue perception is amplified when only a few colleagues work overtime (unfairness) but dampened when overtime is widespread (group assimilation).
- Fuzzy Boundaries: Fatigue levels transition gradually rather than at rigid thresholds.

Existing models often treat fatigue as a linear penalty or ignore patient psychology, failing to capture these behavioral complexities. The goal is to minimize a unified objective function comprising physician fatigue perception costs, patient psychological distress costs, and physician secondment costs, subject to complex operational constraints (shift limits, rest periods, night shift regulations, and queue length thresholds).

2. Methodology

A. The "Fuzzy-CPT" Modeling Framework

The authors propose a unified framework integrating Fuzzy Set Theory and Cumulative Prospect Theory (CPT) to quantify costs:

Patient Psychological Distress Cost ( $C_{tor}$ ): Modeled using a modified ALO-S2 function (a sigmoid-like curve) to capture the nonlinear relationship between queue length and distress. Distress is zero below a threshold ( $L_c$ ), increases rapidly in a sensitive interval, and caps at an upper bound ( $L_{ul}$ ).
Physician Fatigue Perception Cost ( $C_{tir}$ ): Constructed via a three-layer logic:
1. Fuzzy Fatigue Grade Membership: Trapezoidal fuzzy membership functions map objective weekly working hours to three fuzzy fatigue levels (Mild, Moderate, Severe), capturing the continuous and ambiguous nature of fatigue.
2. CPT Value Function: A value function $\nu(T_n)$ models loss aversion. Working hours below a reference point ( $T_{ref}$ ) are in the "gain" domain, while hours above are in the "loss" domain, where the pain of overtime is amplified by a loss aversion coefficient ( $\lambda$ ).
3. Team Effect Weight: A probability weighting function $w(p)$ adjusts individual fatigue based on the proportion of the team working overtime ( $p$ ). This captures the social comparison effect where isolated overtime feels more burdensome than collective overtime.
Secondment Cost ( $C_{bor}$ ): A linear cost for physicians temporarily assigned from other departments to meet demand.

B. The Enhanced Memetic Algorithm (EMA)

Since the problem is NP-hard with nonlinear constraints, the authors design an Enhanced Memetic Algorithm (EMA) to solve the model. The EMA improves upon a Basic Memetic Algorithm (BMA) through three key enhancements:

Hierarchical Encoding & Multi-Objective Decoding:
- Uses a hierarchical chromosome structure where each physician is a gene unit containing shift sequences and metadata.
- Includes "cost-cache genes" to store pre-calculated fatigue, distress, and secondment costs, enabling real-time fitness evaluation without redundant computation.
Adaptive Two-Stage Population Initialization:
- Stage 1: Generates a scheduling skeleton using two distinct greedy strategies (one focusing on night-shift distribution, the other on day-shift balance).
- Stage 2: Refines the skeleton via iterative adjustment with a reset mechanism to avoid local optima and ensure diversity.
Multi-Granularity Collaborative Neighborhood Search:
- Integrates Variable Neighborhood Descent (VND), Variable Neighborhood Search (VNS), and Generalized VNS (GVNS).
- Defines four levels of neighborhood operators: Hour-level (Move, Shorten, Extend), Shift-level (Increase, Decrease), Physician-level (Add, Reduce), and Group-level (Rearrange).
- Uses an exponential decay acceptance criterion to balance convergence speed and solution quality.

3. Key Contributions

The paper claims four primary contributions:

Theoretical Framework: Development of a "Fuzzy-CPT" perspective that constructs a three-dimensional fatigue assessment framework (physiological load, subjective perception, and group influence), moving beyond linear penalty models.
Optimization Model: Formulation of a weekly scheduling model that simultaneously minimizes physician fatigue perception, patient psychological distress, and secondment costs, incorporating complex behavioral constraints.
Algorithmic Innovation: Design of the EMA, which features hierarchical encoding, a two-stage initialization strategy, and multi-granularity collaborative search to effectively solve the NP-hard model.
Empirical Validation & Cultural Adaptability: Validation using real data from a Shanghai tertiary hospital during the COVID-19 peak. The study includes a cross-cultural comparison of CPT parameters (Chinese vs. U.S. settings), demonstrating the model's adaptability to different behavioral contexts.

4. Experimental Results

Numerical experiments were conducted using 12 weeks of real data from a large Shanghai hospital.

Algorithm Performance: The EMA significantly outperformed the Basic Memetic Algorithm (BMA) and an Improved Variable Neighborhood Search (IVNS) algorithm.
- EMA achieved the lowest total objective values ( $Z$ ) across all test cases.
- EMA successfully reduced patient psychological distress cost ( $C_{tor}$ ) to zero in 7 out of 12 cases, whereas BMA achieved this in only 1 case and IVNS in none.
- EMA demonstrated superior dynamic responsiveness to patient arrival fluctuations compared to rigid actual scheduling and lagging IVNS results.
Parameter Sensitivity:
- Cultural Differences: Scheduling results using Chinese CPT parameters (higher risk sensitivity $\alpha, \beta$ ) generally yielded higher fatigue costs than U.S. parameters, highlighting the impact of cultural behavioral traits on optimization outcomes.
- Thresholds ( $L_c, L_{ul}$ ): Increasing queue length thresholds expanded the feasible region, allowing for better fatigue management without increasing distress.
- Physician Count ( $N_1$ ): A critical scale effect was observed; increasing physicians up to a critical point drastically reduced costs, while further increases yielded diminishing returns.
- Reference Hours ( $T_{ref}$ ): Raising the psychological reference point shifted the optimization space, allowing for better fatigue distribution.
- Secondment Cost ( $P_{bor}$ ): Higher secondment costs forced the algorithm to optimize internal scheduling, sometimes increasing fatigue but maintaining queue control.

5. Significance and Claims

The authors assert that this research provides a scientifically grounded decision-support tool for emergency outpatient scheduling during public health crises.

Behavioral Realism: By integrating fuzzy logic and prospect theory, the model captures the "irrational" but predictable psychological responses of both patients and physicians, which linear models miss.
Practical Applicability: The use of real-world data and the demonstration of the EMA's ability to handle complex, time-varying demand validate its utility for hospital administrators.
Cross-Cultural Insight: The study highlights that scheduling parameters (specifically risk attitudes and group effects) are culturally dependent, suggesting that global scheduling tools must be calibrated to local behavioral contexts.
Managerial Insights: The sensitivity analyses offer actionable guidance, such as identifying critical physician staffing levels and the trade-offs between internal scheduling flexibility and external resource secondment.

The paper concludes that the proposed framework and algorithm effectively balance service efficiency, patient experience, and physician well-being, offering a robust solution for the complex, high-pressure environment of emergency medicine.