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The field of medical research has advanced remarkably in the twenty-first century; however, the healthcare industry faces significant challenges, including rising costs, increasing demand, and constrained resources. These factors make effective management of healthcare systems crucial. Efficient systems are those that better use resources, deliver services, and reduce waiting times.
Methods
The paper addresses the issue of healthcare staffing optimization by presenting a new model, the Fuzzy-Based Time-Dependent Multi-Server, Multi-Queueing (FB-TDMS-MQ) System. This model combines fuzzy logic with a genetic algorithm (GA) to optimize staffing levels based on real-time patient needs and service times, a capability that traditional fixed-staffing models cannot accommodate. To demonstrate the model's application in a real-world setting, a case study is conducted at Dhanwantri Hospital and Research Centre (DHRC) in Jaipur, a multi-specialty hospital. The model's performance is evaluated using simulation studies.
Results
The implementation of the FB-TDMS-MQ model led to significant improvements. Simulation studies demonstrated that the average “peak-hour waiting time” drastically reduced by 72.73% after incorporating fuzzy logic modifications. Additionally, the GA optimization approach resulted in a 50.2% reduction in average waiting time, showing superior performance compared to static staffing models. The model's sensitivity analysis proved highly useful in the case of unexpected events in the healthcare system.
Conclusions
The FB-TDMS-MQ model proved to be adaptable and effective in real-time healthcare staffing optimization. It demonstrated a reduction in waiting times and the ability to handle unexpected events, showing that the model could optimize staffing dynamically in fast-changing healthcare environments. The real-time control and modifications provided by the model have significant potential for improving hospital management and staff allocation.
The medical research has advanced a lot in the twenty-first century; however, the rising need for healthcare requires more effective management of healthcare systems. High cost of the economy, increasing demand and constrained resources are among many complex challenges for the healthcare industry. Traditional queueing models—even in the medical field—are based on the current techniques, but there are differences in the intensities of arrival of patients seeking care and the times of service of patients at the point of care [1–6]. Forecasting demand is susceptible to threats posed by predictive analytic models to investigate the variance of demand. To circumvent these problems, a hybrid FB-TDMS-MQ model is then developed in this paper, which will combine fuzzy logic and GA and time-varying multi-server queues. The model can efficiently reschedule the workforce in real-time depending upon the vagaries of the healthcare operational activities, and to generate answers to the present approach problem.
While the previous research gives us a good base case on the improvement of the healthcare system, there are a lot of missing, and quite a bit of it that should be settled to have a real understanding of real-time assessment system like fuzzy logic. Background information is also reviewed to help us to get a clear picture of the current idea.
Queueing theory in healthcare
Recent research by Kosiol et al. [1] and Amin et al. [2] present a systematic framework for fostering innovation in health care by defining its basic principles, critical success factors and hindrances. Elalouf & Wachtel [3] and Santos et al. [4] conducted a detailed look at the queueing problems in emergency departments and wait times, effective queueing strategies, and research methodology. Bai et al. [5] concluded that no amount of change made by human models will be able to increase the efficiency of staffing. While these review studies addressed significant issues, provided descriptive classifications, and outlined challenges, they were not adequately analyzed for fuzzy logic-type systems, which rely on current–time evaluations.
Queueing studies on the COVID-19 pandemic
The latest queueing models, created in adaptation to the situation caused by the isolation of the Coronavirus, emphasize the congestion issues mentioned by Zimmerman et al. [6], who used a queueing model to predict the ventilator capacity in the province of British Columbia during the initial stages of the outbreak of the coronavirus. Saini et al. [7] made an in-depth analysis of healthcare capacity during the "golden hours" of the pandemic. This study looked at the effectiveness of hospitals in providing treatment for the victims when they arrive. The hospital overcrowding was identified in the research through "Transient Act System (TAS)." This investigation reinforces the importance of the 'golden hour' concept in urgent medical attention, highlighting the refined prioritization of patients in an ever-changing setting. Still, the investigations do not show enough research into any platforms like fuzzy logic, which depends on real-time assessments.
Dynamic staffing levels and real-time adaptability
Although the studies on staffing level strategy have been published, they assumed the steady state, which is inevitably less relevant in a dynamic healthcare setting. Ala et al. [8] employ refined remedies within the mixed-integer linear programming framework to tackle the person in need picking challenge for prominent diagnosis-related group therapy. To find the shortest response time for routine and urgent patients, this study used the Poisson distribution and queueing theory. Damirchilo et al. [9] created a queueing theory-deep neural network system to improve manufacturing facility design and transporter selection. It addressed difficult challenges like stochastic facility service times, fluctuating demand, carrier capability, velocity, and batch capacity. Oskuei et al. [10] investigated single-machine scheduling that included varied processing times, sequence-dependent structures for consumable products, and an array of proactive care strategies. Konrad and Liu [11] conducted a real-time assessment of waiting-time allocation in dynamic lineups, a crucial aspect for the healthcare sector. It demonstrated how nonstationary demand and staffing methods influence wait times, driving healthcare simulation. In 2025, Aminah et al. [12] proposed the Accumulative Priority Queue with Finite Horizon (APQ-h), a dynamic scheduling system that used real-time queue management and simulation to optimise emergency department patient flow and wait times. Saini et al. [13] utilized advanced reinforcement learning to enhance online healthcare by elegantly managing doctor accessibility and prioritizing patient treatment according to need and arrival time. This study refined resource allocation through Q-learning, a sophisticated model-free reinforcement learning approach, enhancing patient experience by minimizing wait times and dynamically assigning doctors according to the current queue status. Qureshi et al. [14] created a dynamic queue algorithm to optimise healthcare appointment scheduling and patient flow management using real-time metrics, improving operational efficiency and reducing outpatient department wait times, so making healthcare systems smarter. Compared to a real-time dynamic staffing level strategy, some other studies have attempted to devise one; however, such strategies become irrelevant in changing situations because they are not adaptive and do not use fuzzy logic.
Fuzzy logic in staffing decisions
The integration of fuzzy logic in Queueing model may offer solutions to modern healthcare problem. Fuzzy logic is powerful in resolving agitations within the healthcare system that a conventional model finds difficulty coping with such as fluctuations in arrival time of patients, duration of the services, number of staffs. The studies [15, 16] have used fuzzy logic in staffing decisions to schedule healthcare; the studies have not combined them with the queueing models of varying time to tackle the health care flow and uncertainty. As well as the framework for the home healthcare service that incorporates heuristic search and fuzzy logic is discussed by Sinthamrongruk et al. [17], which is focused on the matching of qualified staff for patients with a high degree of efficiency. However, they do not address particularly a hybrid queueing model for manpower planning and system optimization.
Multi-queue integration
Using methods to set multi-queueing, Hillas et al. [18] gave a critical look into workforce planning to provide a more complete picture of hospital operations. Furthermore, Masmoudi et al. [19] examined routing and scheduling problems related home healthcare. Pereira et al. [20] found that concentrated training to improve internal procedures, infrastructure, instructive sessions and co-creation projects helped to improve healthcare management. These researches are healthcare management structure and assessment, which is missing the dynamic and time-varying components of fuzzy healthcare systems.
Fuzzy logic and Discrete Event Simulation (DES)
Fuzzy logic and optimization approaches have advanced in healthcare staffing optimization literature; however, real-time adaptation is lacking. Sarasu et al. [21] and Vázquez-Serrano et al. [22] highlight static and hybrid models for resource allocation, patient priority, and waiting time reduction without real-time dynamic staffing modifications. Fuzzy systems are employed to handle uncertainty (Azadeh et al., [23]; Lazarashouri & Najafi, [24]), but they are generally limited to emergency rooms or primary healthcare settings, lacking the flexibility needed in fast-changing contexts. A fuzzy decision support system for service quality management by Cui et al. [25] lacks real-time changes and genetic algorithms for optimization. The proposed model can address these limitations by dynamically responding to fluctuating healthcare demands and optimizing staffing levels based on patient needs. The hybrid approach presented in this research offers continuous optimization instead of predefined staffing schedules, making it adaptable to unpredictable healthcare environments, unlike previous works like Muñoz et al. [26]. Thus, this research addresses a significant gap by providing an optimal, real-time staffing solution that blends fuzzy logic's uncertainty management with genetic algorithms' continuous optimization.
Based on queueing model analysis, fuzzy optimization, and healthcare system evaluation, traditional models should be improved to handle uncertainty, inefficiency, and complexity. This research is motivated by the need to address the current healthcare staffing models, especially by addressing particular areas of its priorities. Unlike other methods, such as OptQueue, that rely on static or periodic staffing adjustments, this approach integrates real-time optimization based on fuzzy logic and a GA to make staffing-level decisions that adapt to changing patient demand and uncertainty. This flexibility allows the model to appropriately respond to the real world setting of healthcare. The model's effectiveness in reducing patient wait times as well as improving care efficiency while keeping operations cost effective and financially sustainable is assessed in the study. While traditional Markovian queueing models assume labor parameters, in the proposed approach, fuzzy logic is used to deal with the uncertainties and GA is used for the dynamic optimization of staffing which provides a more robust solution. The model periodically makes updates based on real-time data that helps ensure that it is responsive to patient flow fluctuations. The paper efforts delivering improvement staffing optimization methods in healthcare facilities by proposing hybrid fuzzy-genetic algorithm approach addressing the limitations of conventional models to control the real-time variability.
Gaps in existing research and importance of the FB-TDMS-MQ model
Table 1 compares the gaps in the existing OptQueue model and the importance of the proposed FB-TDMS-MQ system across various healthcare settings. By evaluating the current literature, this table highlights the limitations of traditional static or periodic scheduling models and demonstrates the advantages of FB-TDMS-MQ in real-world healthcare applications, including emergency care, ICU management, and telemedicine resource allocation.
Table 1.
Comparison Result Between the Proposed FB-TDMS-MQ System and the Existing OptQueue Models
As shown in Table 1, the FB-TDMS-MQ model offers significant advantages over the OptQueue model, including real-time dynamic staffing adjustments, handling of uncertainties via fuzzy logic, and a multi-queue system designed for different patient types. Unlike OptQueue, which relies on static staffing schedules and parameters, the GA used by FB-TDMS-MQ enables continuous optimization and has been validated using real-world data from Dhanwantri Hospital. This makes the proposed dynamic healthcare model more adaptable, flexible, and efficient, with features that outperform existing models in terms of performance and computational speed in real-world healthcare settings.
Comparative justification
In this study, we have chosen Fuzzy Logic and Genetic Algorithm (GA) over machine learning approaches for several key reasons. While machine learning models excel in handling large, labeled datasets, they require significant computational resources and may not effectively address the inherent uncertainties and variability present in healthcare staffing. Fuzzy Logic is particularly suited for managing the imprecise nature of healthcare data, such as fluctuating patient arrivals and service times, by using fuzzy sets and inference rules. This enables real-time staffing adjustments without the need for precise data. Moreover, GA provides an optimization framework that dynamically adjusts staffing levels based on real-time data, making it well-suited for situations with multiple variables and constraints. Unlike machine learning models, which often require extensive training and a large volume of data, Fuzzy Logic and GA offer a more interpretable, flexible, and computationally efficient solution for optimizing staffing in fast-changing healthcare environments.
Notations and assumptions
Notations
The following notations are used in the proposed method:
: Time-dependent arrival rate for queue i at time t.
: Time-dependent service rate for queue i at time t.
s: Number of servers (healthcare workers) available for queue i
: Average number of patients waiting in the queue for queue i at time t.
L(t): Total number of patients in the system at time t
: Utilization factor for queue i at time t, indicating the proportion of time servers are busy.
: Fuzzy membership function for the arrival rate λi(t) at time t, reflecting fuzzy categories (Low, Medium, High).
: Fuzzy membership function for the service time μi(t) at time t, reflecting fuzzy categories (Short, Medium, Long).
: Fuzzy membership function for staffing level S, indicating categories (Insufficient, Sufficient, Overstaffed).
: Fuzzy membership function for the arrival rate at time t (Low, Medium, High).
: Fuzzy membership function for the service time at time t (Short, Medium, Long).
: Fuzzy membership function for staffing level S at any given point x.
Wavg: Average waiting time across all patients in the system
: Defuzzified staffing level based on fuzzy logic, used in the system optimization model.
f(t): Function defining the time-dependent arrival rate λi(t), including factors such as time of day and season
g(t): Function defining the time-dependent service rate μi(t), influenced by staffing and workload
: Fuzzy membership function for staffing level S, representing categories like insufficient, sufficient, and overstaffed.
T: Total time period considered in the system optimization.
Assumptions
The following assumptions are grounded in the proposed method:
In this model, the upper limit of patient service time was used as 15 min which is based on empirical data as well as expert opinion.
We assume that there are at least n distinct patient categories or lines, including those for emergencies, routine, and specialty treatment.
Each queue i (where i = 1, 2, …, n) has its own arrival rate λi(t) and service rate μi(t).
There are servers (healthcare workers), and the number of servers supporting each queue might be dynamically altered based on the arrival rate and service time.
At time t, the arrival rate λi(t) for queue i and the service rate μi(t) can vary due to factors like time of day, disease outbreaks, or workload.
Methodology
This section presents the process of creating the proposed FB-TDMS-MQ model that is aimed at optimizing workforce planning in fuzzy healthcare systems. This methodology was about the structure of the proposed FB-TDMS-MQ model. This model uses a dynamic strategy, where healthcare workers are not fixed resources but the decision variables. This enables the system to adjust staffing levels as and when demand for patients fluctuates. This is especially important in order to cope with hours of peak demand, such as in emergency situations, where more resources are needed. Unlike conventional static staffing models that are based on set workforce numbers, the current staffing model is guided by the concept of tailoring staffing to optimize the queueing system to minimize waiting times and optimize patient queueing.
In this paper, the term "real-time" optimization means that the system has the ability to dynamically adjust the number of staffing needs according to fluctuations in patient demand and service times. However, instead of continuous updating, the model is based on periodic updates at discrete times of the day usually at the beginning of each shift (for example every 6 to 8 h) This way it is possible to make adjustments according to some actual measurements such as the number of patients coming in, the service time of every consultation, which are recorded and analyzed at short intervals of time. Thus, although there can be periodic updating, it is based on real-time data obtained during each shift, giving the system an ability to adapt to changing conditions. The full description and mathematical formulation are separately described in Sect. "Model Description and Mathematical Formulation".
Structure of the methodology
Consider the following structure for the proposed model to increase its readability:
Phase 1: data collection and analysis
The data required for the model include real-world healthcare data, such as.
Time-varying Patient Arrival Rates: These can be extracted from past data of patients visiting to the doctor and factoring in the time of day, the day of the week, and seasonal trends.
Service Time Data: Metrics based on the patient service time by patient complexity and treatment type.
Workforce Availability: Staffing, patterns of shifts, vacations, emergency personnel needed etc.
Phase 2: multi-queueing model implementation
The next step will be to use the multi-server, multi-queueing model. The medical system is modeled by M/G/s queueing model, where M represents the Markovian (Poisson) type arrival process, G represents the general distribution of the service duration, and s represents the server counts.
Arrival Rate (λi(t)): Each queue i has a time-dependent arrival rate that may vary according to the time of day, season, and other factors.
Service Rate (μi(t)): Each queue i has a time-dependent service rate based on patient complexity and healthcare worker efficiency.
Average Waiting Time: To express the overall efficiency and fairness of a system, we compute the average waiting time over all patients, thus providing a more meaningful estimate for the characterization of a healthcare system. In addition, to take into account the distribution and variability of waiting times to determine the equity of service provision in systems with time-varying patient volumes, we may also analyze the average squared waiting time.
Queue Length Calculation (Lq(t)): For the M/G/s system the mean waiting time in the queue (Wq) is best approximated by using the Pollaczek-Khinchin (P-K) formula for multi-server systems which accounts for the general service time.
System Length Calculation (L(t)): The system length can be determined using the total patient count in the system (waiting + being served).
Phase 3: fuzzy logic integration
We have easily incorporated fuzzy logic in the queueing model to handle the unexpected nature of arrival rates, service durations, and staff levels in a classy manner. Fuzzy logic integration involves some indispensable steps, such as the ones listed below:
A
Define fuzzy sets for input variables:
Arrival Rate (λi(t)): Low, Medium, and High.
Service Time (μi(t)): Short, Medium, and Long.
Staffing Level (S): Insufficient, Sufficient, and Overstaffed.
B
Define Fuzzy Rules: Fuzzy rules are developed to adjust staffing levels based on arrival rates and durations of service such as:
If the arrival rate is High and the service time is Long then the staffing should be overstaffed.
If the rate of arrival is Low and the service time is Short, then the staffing should be Insufficient.
C
Fuzzy Inference System (FIS): The fuzzy output (staffing level) is obtained by using the fuzzy rules in fuzzy inference system. Defuzzying method is used to convert fuzzy output to crisps staffing level recommendation.
D
Defuzzification: Defuzzification is usually performed by centroid method in which the output staffing level is the centroid of the fuzzy set.
Phase 4: system optimization objective
After implementing the fuzzy logic system into the time-dependent multi-server multi-queueing system, we are able to optimize the system. This is done to effectively plan manpower by reducing patient waiting times, depleting maximum efficiency from services and assessing the correct level of staffing as dependent upon demand at a given time. The optimization problem aims at reducing patient waiting times while at the same time dynamically adapting staffing levels in order to accommodate varying patient demand. In addition, it attempts to optimize staffing levels using the fuzzy outputs of the FIS.
Phase 5: genetic algorithm
The algorithm 1 (presented in Sect. "Genetic Algorithm for Staffing Optimization") is used to improve staffing levels using GA. This strategy effectively eliminates patient waiting time and dynamically optimizes the number of multiple staffing levels to match the varying demands.
Phase 6: validation and simulation
We describe how the model was implemented using real world data and how the effectiveness was assessed. To this end, the proposed FB-TDMS-MQ model is validated and simulated with a realistic case study on a multi-specialty hospital, i.e., DHRC, Jaipur. We analyze the time-dependent multi-server, multi-queueing model using simulation techniques in which we assume that patients arrive at the facility according to a given distribution, the arrival rate of service is normally heterogeneous, and we can decide on the number of staffs to use at the same time. The results are analyzed for the model performance in terms of:
Queue length and waiting times: How effectively does the system reduce patient waiting times and optimize server utilization?
Staffing optimization: Whether fuzzy logic-based staffing decisions lead to efficient manpower usage.
Phase 7: results and discussion
After validation and simulation, the study results are analyzed, and a discussion around the results has been conducted in relation to the practical application of the model, which has been validated by using real data from the healthcare facility. At the same time, we monitor system duration, waiting time, and queue length indicators, as well as other performance indicators.
Flow chart of the methodology
The flowchart in Fig. 1 illustrates the FB–TDMS–MQ model, which integrates real-time data acquisition, fuzzy decision-making, and genetic algorithm (GA) optimization for staffing adjustments in a simulation environment. The process begins with the collection of patient and service data, followed by rule evaluation through a fuzzy inference system that determines recommended staffing changes. These changes are further optimized using GA, which generates feasible staffing plans. The staffing plan is then applied in a simulation model, where real-time effects are observed and updated data is collected for the next cycle. The process repeats until the termination criteria are met. Key feedback loops ensure continuous improvement and adherence to staffing and performance targets. The chart also incorporates the execution of fuzzy rules and defines the role of medical staff, queues, and decision intervals in the model.
Flow chart of the FB-TDMS-MQ model from real-time data acquisition to fuzzy decision making, GA optimization, and simulation
Arrival rate justification
In this study, we assume that the patient arrivals are Poisson distribution (M) based on the available arrival data of DHRC Jaipur. However, we have added a detailed statistical analysis to check this assumption. The arrival data of patients were gathered during few months and they were recorded with hourly intervals which enabled us to analyze arrival rate distribution. In order to check the validity of the Poisson assumption, the Fig. 2 shows a comparison of observed and expected frequencies for the Poisson distribution.
Comparison of observed and expected patient arrival rates (Poisson distribution)
Figure 2 compares the Poisson distribution models of actual and predicted patient arrival rates for day and night shifts at a hospital facility. The blue and green bars show the observed daytime and nighttime arrival frequency, while the red and orange lines show the Poisson distribution's anticipated values. The night shift has 4 patients every hour, which is distinct from the day shift's 10 patients per hour. This comparison shows how closely the observed data matches the theoretical Poisson distribution, which demonstrates that patient arrivals follow a Poisson process with some natural variability for both time periods.
Service time modeling and upper bound justification
Service times for patient consultations are modeled by fuzzy sets i.e. triangular distribution. This choice is common in the case of exact data sparsity, but good approximations can be made using observations and expert opinion. In this model, the maximum number of minutes for patient service time was set to 15 min, which was derived from empirical and expert opinion.
The upper limit of 15 min was based on the history from DHRC, Jaipur which showed that from an analysis of the service time by the patients for routine consultations, 95% of the cases have been completed within this time. This data was gathered over a period of months and was helpful in determining the limits of patient interaction times. In consultation with healthcare professionals at DHRC, Jaipur including doctors and administration staff it was confirmed that 15 min is taken as the typical upper limit of the duration of consultations for non-complex cases. This expert consensus was used to ensure that the service time values were realistic and aligned with the practices in the hospital. By integrating both empirical information and the expert information, we reached a reasonable and effective upper bound of service time which was then modeled by using triangular fuzzy membership functions.
Model description and mathematical formulation
In this section we present in details all the main components and the mathematical formulation of the proposed FB-TDMS-MQ model. The key components of the proposed model are explained in detail and their mathematical formulations are given below:
Queueing system framework
The health care system is structured as a dynamic multi-equipment multi-queue system, in which the rate of the patients’ arrival, the service delivery duration, and the server count are continuously time-dependent and adaptable. So far, the classical combinatorial view of queueing theory has been enhanced to accommodate the dynamic behavior of healthcare departments with time-varying parameters to depict the dynamic nature of the system in a beautifully elegant manner. In this paper, we utilize an advanced M/G/s queueing network with fuzzy-based time-sensitive elements and servers and time-sensitive queueing system to model a health care system. The system has immense functionalities that can be displayed in a detailed mathematical formulation.
Key components of the queueing model (M/G/s)
A
Arrival Rate (λi(t)): The patients who come to the healthcare facility and join the queue at arrival time t are referred to as the arrival rate.
1
where f is a function that will seem to describe the fluctuation of patient arrival with regard to numerous variables such as time of the day, holidays, and emergency conditions.
B
Service Rate (μi(t)): The service rate is the rate at which the healthcare workers can process the patients at time t in queue i. This rate is also time-dependent and may vary due to healthcare worker fatigue, complexity of medical cases, and other operational conditions:
2
where g is a function describing how staffing and workload affect service speed.
C
Traffic Intensity : The traffic intensity signifies the proportion of time during which the servers are engaged. It is elegantly calculated as:
3
D
Average waiting time (Wavg): In order to evaluate the efficiency of the healthcare system and the fairness of it further, we will use the average waiting time among all patients. The average squared waiting time can also be investigated to cater waiting time fluctuation and distribution to establish service delivery fairness particularly in systems with fluctuating patient loads. Average waiting time across all patients in the system, calculated as:
4
where Wi is the waiting time of patient i, and N is the total patients’ count in the system at time t.
E
Queue Length (Lq(t)): For M/G/s system, the average waiting time in queue Wq is best approximated using the Pollaczek-Khinchin (P-K) formula for multi-server systems, which accounts for the general service time:
5
Where the queue length Lq is approximated as:
6
Where:
λ = Arrival rate.
Var(S) = Variance of the service time.
E[S2] = Second moment of the service time distribution.
ρ = Traffic intensity, ρ = .
F
Fitness Function:
The fitness function assesses how well employees decrease average waiting time (Wavg) across all lines and time periods. This is how we rate each person's performance. For M/G/s queues, the GA fitness function is estimated using the Pollaczek-Khinchin method. The average waiting time for each queue is Wq = Lq/λ, where Lq is calculated using the first and second seconds of general service time. This approach works well for μ around λ, gives a consistent way to estimate GA staffing levels, and is theoretically sound for general service time distributions. The fitness function is given by
7
where Wavg is the average waiting time and given by
8
This factor shows how much work the system is doing and how well the servers are being used at the moment.
Integration of fuzzy logic for decision making
We use fuzzy logic in an elegant way to deal with healthcare system complexity and unpredictability (in this queueing model). Fuzzy logic is used to predict patient attendance and service hours and staff numbers.
A
Fuzzy Inputs
Fuzzy Arrival Rate (λi(t)): The arrival rate varies by time of day, flu season, and other factors. Use membership functions to show patient arrival rates as fuzzy sets like Low, Medium, and High. Let denote the membership function for the arrival rate at time t:
9
Fuzzy Service Time (μi(t)): Service time is calculated based on patient complexity and healthcare practitioner competency. Service time is estimated using Short, Medium, and Long membership functions. Let denote the membership function for the service time at time t:
10
Fuzzy Staffing Level (S): The staffing level is mixed up into Insufficient, Sufficient, and Overstaffed. Based on demand and service time, membership functions determine the right quantity of healthcare personnel. Let represent the fuzzy membership function for staffing levels:
11
B
Fuzzy Rules
A set of fuzzy inference rules is designed to calculate the best staffing level for each queue i at time t. Examples of fuzzy rules include:
Rule 1: If the arrival rate is High and the service time is Long, then the staffing level should be overstaffed.
Rule 2: If the arrival rate is Medium and the service time is Medium, then the staffing level should be Sufficient.
Rule 3: If the arrival rate is Low and the service time is Short, then the staffing level can be Insufficient.
C
Fuzzy Inference System (FIS) Setup
A fuzzy inference system (FIS) defines the time-dependent multi-server, multi-queueing system's elusive parameters. The FIS elegantly converts fuzzy input variables into proper output values using richer fuzzy sets, membership functions, and language limitations. These rules allow the fuzzy inference system calculate an output staffing level to improve healthcare. Fuzzy inference alters worker numbers based on 6- to 8-h shifts, not every hour. This shift-based method helps the algorithm make staffing proposals that meet real-world conditions, when labor limits need shift-start modifications. The technology uses each shift's arrival rate and service time to determine the optimal amount of staff to enhance performance without midshift modifications.
i.
Justifying the Membership Functions: The fuzzy inference system (FIS) implemented some easy triangular input and output variables. Arrival rates, service times, and wait times, as well as staff, are all considerations. Triangular shapes are very easy to use and make a visual representation for the meaning of low, medium and high. Shapes have a role in the control of input data uncertainty and imprecision. Triangles represent the time of waiting because they merge simplicity and details to allow smooth transitions between fuzzy sets and to be simple to compute.
ii.
Fine-Tuning the Membership Functions: In order to test the fuzzy inference approach, we manipulated the membership functions of the triangle using the data from some early case studies. Altering these settings gained better forecasts of wait time from the system. In order to match the observations, low, medium and high fuzzy set breakpoints were carefully adjusted. Experience improved the input variable response of the system.
The fuzzy inference system used in the experiments is based on the following configuration:
Input membership functions: Triangular membership functions for arrival rate and service rate with breakpoints fine-tuned based on initial data.
Output membership functions: Triangular membership functions for waiting time, with breakpoints adjusted to ensure smooth transitions between fuzzy sets.
Fuzzy rules: A set of IF–THEN rules based on domain knowledge, where the condition part uses arrival rate and service rate to determine the waiting time.
D
Defuzzification
Defuzzification involves transforming fuzzy outputs, expressed through linguistic labels such as Overstaffed or Sufficient, into precise numeric values suitable for decision-making purposes. Following the application of the fuzzy rules, we transform the fuzzy output (staffing level) into an accurate value through centroid defuzzification:
12
where is the crisp staffing level recommendation is based on the fuzzy system, μi(x) is the membership value of fuzzy set i, and x is the corresponding staffing level for that fuzzy set.
System optimization objective
After applying fuzzy logic on time dependent model of multi server, multi queueing system, here we can optimize the manpower planning by minimizing the waiting time of the patient and the optimized staffing levels based on the demand. The aim is to reduce waiting times for all patient lines and times of day:
13
To optimize staffing level under real-time environment, we define the objective function (Eq. 13), which will reduce the mean waiting time for all queues. Equation 13 ensures that staffing decisions are optimized to minimize patient waiting time across different types of queuing, such as emergency, outpatient, and specialty care.
Genetic algorithm for staffing optimization
GA is used to optimize healthcare staffing. The GA regulates the staffing to decrease wait times. After examining past 6 to 8 h of arrival rate information, the system will adjust staffing levels and make recommendations for the start of the shifts. This allows for the plans to be practical and falls within existing healthcare practice. The GA is used to choose the best crossover and mutation crew for innovation. Preventive maintenance is an improvement technique that reduces the wait time of patients by meeting the demand over a period. The process is iterated until optimal performance of healthcare workers will be reached.
GA-based staff assignment and constraints
The GA-based staff assignment in the current study is to reduce average waiting times in various queues of patients. However, in order to guarantee the feasibility of the generated staffing solutions, several hard constraints are added in the optimization process. These constraints are created to represent limitations of real-world situations in terms of staff and hospital operations.
Staffing Limits: Each queue in the system has a maximum value of the number of staff that can be assigned based on hospital policy and operational limits. These limits are to guarantee that staffing configurations will not surpass the capability of the hospital or cause inefficiencies by having too many staff. The GA takes these staffing limits into account by limiting the number of healthcare workers assigned to each queue up to predefined limits.
Skills Requirements: Since every queue like emergency, outpatient require specialized staff, a constraint is set so that healthcare workers with suitable skills are allocated to the right queue. For instance, emergency care certifications are reserved for only healthcare workers and not for patients who are in it. This constraint ensures staffing configurations serve operational requirements of each queue.
Shift Rules: The model follows the rules of the hospital, such as the maximum number of hours a working individual can work, the time a working individual is allowed to take a break and the time a working person will be on shift. These rules are programmed into the GA by ensuring that the staffing configurations adhere to normal shift patterns used in the hospital. The GA algorithm takes these constraints into account and produces solutions which comply with these rules of operation.
GA Process for Staffing Optimization
A
Population Initialization: Let be the population of N individuals (chromosomes), where each individual pi represents a staffing configuration across all queues. Each chromosome pi is a vector where sj represents the staffing level for queue j, and m is the total number of queues.
B
Fitness Evaluation: Let the fitness function be , representing the average waiting time for the staffing configuration pi. The waiting time Wi for each queue i is calculated using Eq. (14):
14
Where is the arrival rate for queue i, is the service rate for queue i, and Lq is the queue length for queue i.
Thus, the total fitness for an individual pi is given by Eq. (15):
15
C
Selection of Parents (Tournament Selection): Select two individuals such that (lower fitness means better solution). Tournament selection involves randomly choosing two individuals and selecting the one with the lower fitness.
D
Crossover (Recombination): For two parent chromosomes and , a single-point crossover occurs at position c, where . The offspring chromosomes and are and .
E
Mutation: For a given chromosome , select a random position j and change the staffing level sj to a new value within the allowable range , as given in Eq. (16).
16
F
New Population Formation: Let E ⊂ P be the elite set (best individuals based on fitness). The new population is formed as given in Eq. (17):
17
G
Convergence Check: If the difference between the best and worst fitness values in the population is smaller than a defined threshold , the algorithm is considered to have converged as given in Eq. (18):
18
H
Solution Extraction: The best solution for the individual that minimizes the fitness function is given by Eq. (19):
19
The method involves the generation of a population of staffing configurations, the evaluation of those configurations based on waiting time, and the utilization of genetic operations (selection, crossover, and mutation) to develop the population in the direction of an optimal solution. All of these processes are carried out through the utilization of these processes.
Validation analysis
This section verifies the proposed FB-TDMS-MQ model using the case study information. To see how well the model works, we calculate queue lengths, average wait times, and staffing levels.
Case study overview
The case study, which we have done using data from a multi-speciality hospital named DHRC, Jaipur, illustrates the problem of integrating authentic data from a healthcare system in the form of an emergency department (Queue 1) and an outpatient (Queue 2) respectively within 24 h. The hospital has different types of employees. The system monitors the amount of personnel, length of service and the commended number of weekly patients attended. The data consists of the following information:
Key assumptions:
Time granularity: 1-h intervals.
Staffing flexibility: Adjustable per hour.
Realistic data: Based on historical hospital records.
Validation process
The validation process is conducted in the following phases:
Step 1: define time-dependent parameters
Queue 1 (Emergency Department)
Arrival rate λ1(t):
Day (8:00–20:00): 10 patients/hour
Night (20:00–8:00): 4 patients/hour
Fuzzy sets: Low (0–4), Medium (3–10), High (8–15)
Service rate μ1(t):
Per server: 6 patients/hour (day), 4 patients/hour (night)
Fuzzy sets: Short (10–15 min), Medium (10 min), Long (5–10 min)
Queue 2 (Outpatient Clinic)
Arrival rate (λ2(t)):
Peak (9:00–12:00): 8 patients/hour
Moderate (12:00–15:00): 6 patients/hour
Low (other times): 2 patients/hour
Service rate (μ2(t)): Constant: 5 patients/hour per server
Initial Staffing:
Queue 1: s1 = 2 servers
Queue 2: s2 = 2 servers
Step 2: calculate queue metrics at t = 10 (10:00 AM)
Queue 1:
λ1(10) = 10, μ1(10) = 6, s1 = 2
Traffic Intensity:
Assumed Service Time Parameters:
Variance Var(S1) = 0.01 hours2 (example value).
Second Moment E[] = 0.02 hours2 (example value).
Queue length (Lq): The queue length for the M/G/s system was determined using the Pollaczek-Khinchin formula that allows for the general service time distribution in the system.
Queue 2:
λ2(10) = 8, μ2(10) = 5, s2 = 2
Traffic Intensity:
Assumed Service Time Parameters:
Variance Var(S2) = 0.015 hours2 (example value).
Second Moment E[] = 0.03 hours2 (example value).
Queue length (Lq):
Step 3: calculate average waiting time using Little’s Law
Using Little's Law, we calculate the average waiting time for both queues:
For Queue 1 (Emergency): W =
For Queue 2 (Outpatient): W =
Step 4: calculate average waiting time across the system
If we assume 10 patients per queue:
Step 5: fuzzy logic adjustment for staffing at t = 10
Fuzzify Inputs:
Queue 1 Arrival Rate (10 patients/hr): High (membership = 0.5), Medium (= 0.5)
Service Time (10 min): Medium (= 1.0)
Fuzzy Rules:
If Arrival = High and Service = Medium → Overstaffed (strength = min (0.5, 1.0) = 0.5)
If Arrival = Medium and Service = Medium → Sufficient (strength = 0.5)
Defuzzified Staffing:
Staffing membership functions:
Insufficient: △(0, 1, 2)
Sufficient: △(1, 3, 5)
Overstaffed: △(3, 5, 6)
Aggregated output: μagg(x) = 0.5 for all Staff Level x = 2, 3, 4, 5
Centroid defuzzification:
Recommendation: Increase staffing to s1 = 4 servers.
This recalculation optimized Queue 1 (Emergency Department) staffing by means of increasing the number of servers from 2 to 4. The primary KPI's for this personnel setup are as shown below:
Arrival Rate (λ1) at 10 AM: 10 patients/hour
Service Rate (μ1) at 10 AM for s1 = 4 servers: 24 patients/hour
Traffic Intensity (ρ): 0.4167 (indicating moderate server utilization)
Queue Length (Lq): 0.900 patients
Average Waiting Time (W): 5.4 min
Step 7: reduction in average waiting time after optimized staffing (s1 = 4)
Now, to calculate the reduction after optimization, we compare the average waiting time before and after optimizing the staffing.
Before optimization (with s1 = 2), the average waiting time was 19.8 min.
After optimization (with s1 = 4), the average waiting time was recalculated as 5.4 min.
The reduction in average waiting time is calculated as:
Substitute the values:
The optimal staffing setup with s1 = 4 servers reduced Queue 1 (Emergency Department) wait time by 72.73% at 10 AM Dynamic staffing optimization reduces patient wait times by adjusting workforce levels to demand.
Step 8: genetic algorithm process for staffing optimization
A
Initialization: In our experiments we used the uniform random initialization strategy for creating initial population. Each individual in the population is given a randomly generated mix of arrival rates, service rates or staff levels within the allowable range. All experiments were set to a population size of 100 individuals in order to assure diversity in the initial solution space.
B
Population Creation: The following configuration was used in the experiments:
Population size: 100 individuals.
Number of generations: 50 generations (chromosomes).
Crossover operator: Single-point crossover (crossover probability = 0.9).
Mutation operator: Gaussian mutation (mutation probability = 0.1).
Random seed: The random number generator was seeded with a fixed seed (42) for all the trials. This implies that algorithm execution can produce consistent results.
Representation: Each chromosome encodes staffing levels for all queues (e.g., [s₁, s₂, …, sₙ] for n queues).
Example: [2, 3] for 2 servers in Queue 1 (Emergency) and 3 in Queue 2 (Outpatient).
Range: Staffing levels per queue constrained to 1–10 servers (realistic for healthcare).
C
Chromosome Encoding: We have used a real coding scheme of each individual, where the genes are actual values of some important parameters, e.g. arrival rate for each queue, service rate for each queue and staffing levels of each queue. Each gene present in the chromosome represents a specific parameter, and each individual is a possible staffing configuration. The values are represented as real numbers so that the evolutionary process can be fine-tuned.
Algorithm 1 Genetic Algorithm for Optimizing Staffing Configuration in Healthcare Systems
The GA (Algorithm 1) to optimize staffing configuration in healthcare systems is important for better operational efficiency and service provision. This algorithm helps minimize patient wait times by dynamically adjusting staffing levels at different queues based on real-time information, such as arrival rates, service rates, and queue lengths.
E
GA Process Results Summary for Staffing Optimization
Table 2 displays a summary of the results obtained from the GA process for optimizing staffing roles. A comprehensive mathematical description of how the genetic algorithm works to optimize staffing was provided in Algorithm 2, which may be found in the Appendix.
Table 2.
Showing the results summary of GA process for staffing optimization
Step
Individual/chromosome
Fitness value
Explanation
Initial population
[5, 4, 6, 3]
Invalid (Negative or Undefined)
Infeasible due to invalid waiting times (service rate < arrival rate)
[7, 5, 4, 8]
Valid
Fitness calculated, awaiting crossover and mutation
[5, 4, 7, 8]
Valid
Fitness calculated, awaiting crossover and mutation
[3, 8, 5, 6]
Valid
Fitness calculated, awaiting crossover and mutation
[4, 7, 8, 4]
Valid
Fitness calculated, awaiting crossover and mutation
Crossover
Parent 1: [7, 5, 4, 8], Parent 2: [3, 8, 5, 6]
N/A
Crossover at position 2
Offspring 1: [7, 5, 5, 6]
N/A
After crossover
Offspring 2: [3, 8, 4, 8]
N/A
After crossover
Mutation
Offspring 1 (mutated): [6, 5, 5, 6]
N/A
Mutation applied to Offspring 1 at gene 1 (staffing for Queue 1)
New population
[5, 4, 7, 8]
10
Best individual after elitism and mutation
[6, 5, 5, 6]
12
New individual after crossover and mutation
[3, 8, 4, 8]
15
New individual after crossover and mutation
[7, 5, 4, 8]
14
Parent selected in new population
[5, 4, 4, 8]
16
Another parent selected in new population
Convergence check
N/A
6
Difference between highest (16) and lowest (10) fitness values
In this section, we conduct simulations that are based on the Pollaczek-Khinchin formula for the queue length and the average waiting time calculation. This analysis is used to validate the proposed FB-TDMS-MQ model, based on a realistic example of a hospital system over a period of 24 h.
Simulation design
Table 3 lists the necessary parameters used in the simulation that define the way the healthcare queueing model works and performs.
Table 3.
Simulation parameters
Parameter
Description
Arrival Rate (λi(t))
Time-dependent rate at which patients arrive at each queue (varies by time of day, week, etc.)
Service Rate (μi(t))
Time-dependent rate at which patients are processed in each queue
Number of Servers (s)
Dynamic adjustment of healthcare workers per queue based on demand and service times
Queue Length (Lq(t))
The number of patients waiting in the queue
System Length (L(t))
Total number of patients in the system (including both those waiting and being served)
Utilization Factor (ρ)
The proportion of time the servers are busy, calculated for each queue
Average Waiting Time (Wavg)
The average time spent by patients across all queues
Defuzzified Staffing Level (S)
Optimized staffing levels after fuzzification and defuzzification of fuzzy inputs
Run Length
24-h simulation to capture day and night dynamics
Warm-Up Period
4-h warm-up to stabilize the system before data collection
Replications
30 independent replications to ensure reliability and account for variance
The simulation was performed under the following conditions:
Arrival Rate for Emergency Queue: 10 patients per hour during peak hours, adjusted dynamically for night shifts.
Service Rate for Outpatient Queue: 5 patients per hour, fixed across all time periods.
Staffing Levels: After a baseline of 2 servers per queue, fuzzy logic suggestions were employed to make dynamic staffing modifications.
The input data given to the simulation such as patient arrival rates, service rates and initial staffing levels for the various queues are shown in Table 4. These parameters were calculated using historical data collected from DHRC, Jaipur, and these parameters are used as a foundation to perform testing of the efficacy of the current real-time staffing optimization model.
Due to dynamic arrival rates, service rates, and server availability, we adjust personnel levels, line length, and waiting times.
Key results from GA validation
Table 5 compares the baseline and optimal staffing arrangements after applying the GA to a multi-queue healthcare system. The following metrics are evaluated:
Queue metrics and adjustments for system optimization
Table 6 presents the essential metrics for patient flow, wait times, and service levels during simulation, along with modifications made following staffing optimization.
The simulation analysis proved that the hybrid model is effective in lowering waiting times by 50.2% for all queues in optimizing staffing decision in response to changing patient demand. In particular, the role of GA was a key factor in the dynamic optimization of staffing levels, whereas fuzzy logic provided a further guarantee that decisions about staffing were sensitive to the complexity of real-time healthcare environments. The simulation showed that queue 1 (Emergency) and queue 2 (Outpatient) benefited greatly from the increase in staffing that led to reduced waiting times and effective use of resources. Meanwhile, queue 3 (Specialty), because of low demand, had less effect on staffing without any negative effect on system performance.
Statistical validation of the proposed FB-TDMS-MQ model
The proposed FB-TDMS-MQ model will be used to improve statistical validation through effective minimization of patient waiting time, optimization of staffing, and overall system efficiency compared to existing models. There will be statistical tests and calculation of KPIs (key performance indicators) such as average waiting time and staffing levels to determine if there is improvement with this. This can be done using a t-test to compare average performance before and after the model has been implemented. We have calculated 95% confidence intervals for the KPIs in order to give a range of plausible values. The average waiting time from several simulation runs will have a CI of [10.5, 12.3] minutes which implies that we are 95% confident that the true average waiting time lies within this interval.
Step 1: define key performance indicators (KPIs)
Based on the simulation analysis, the important KPIs used to confirm the effectiveness of FB-TDMS-MQ model are:
A.
Average Waiting Time (Wavg): Wavg is defined as the average time spent by patients in the system (both in the queue and being served).
B.
Staffing Levels (S): The number of servers (healthcare workers) allocated to each queue.
C.
System Utilization (ρ): ρ is the proportion of time servers are busy, indicating the level of efficiency in resource utilization.
Step 2: calculation of average waiting time (Wavg)
From the simulation analysis, the FB-TDMS-MQ model was found to reduce the average waiting time by 50.2% across all queues. The initial and optimized waiting times were as follows:
Initial Average Waiting Time (before optimization):
Queue 1 (Emergency): 20.25 min
Queue 2 (Outpatient): 18.5 min
Queue 3 (Specialty): 15.4 min
Overall Average:
Optimized Average Waiting Time (after optimization with FB-TDMS-MQ model):
Queue 1 (Emergency): 10.11 min
Queue 2 (Outpatient): 9.8 min
Queue 3 (Specialty): 8.2 min
Overall Average:
Percentage Reduction in Average Waiting Time:
Step 3: calculation of staffing levels (S)
Staffing levels before and after optimization were calculated based on the GA optimization results:
Initial Staffing Levels:
Queue 1 (Emergency): 5 servers
Queue 2 (Outpatient): 4 servers
Queue 3 (Specialty): 3 servers
Total Staffing: 5 + 4 + 3 = 12 servers
Optimized Staffing Levels (after GA optimization):
Queue 1 (Emergency): 7 servers
Queue 2 (Outpatient): 8 servers
Queue 3 (Specialty): 2 servers
Total Staffing: 7 + 8 + 2 = 17 servers
Percentage Change in Staffing Levels
Step 4: statistical analysis—t-test
To statistically validate the effectiveness of the FB-TDMS-MQ model, we perform a paired t-test to compare the initial and optimized waiting times. The t-test will determine if the difference in waiting times before and after optimization is statistically significant.
Step 5: t-test calculation
The formula for the paired t-test is:
20
Where:
is the mean of the differences between a pair of data.
Sd is the "standard deviation" of the differences between paired data.
n is the number of pairings (in this case, the number of patient records).
Table 7 shows the waiting times for each queue before and after optimization.
Table 7.
Waiting times for each queue before and after optimization
For df = 2, at α = 0.05 significance level, the t-critical value is approximately 4.303.
Step 6: conclusion of statistical validation
The Paired t-test results showed that the FB-TDMS-MQ model was significant in decreasing waiting times for all the queues. The t-statistic was 8.61 which is much larger than the critical value for t = 4.303, so the difference in waiting times is statistically significant. Reduction of wait times was significant with the FB-TDMS-MQ model where wait times were reduced by 50.2%. The number of workers having the optimum value increased by 41.67% which resulted in an improvement in resource utilization. The t-test showed that the reduction of waiting times is statistically significant, indicating that it is not a random effect, but actually an improvement as a result of model-adjustment. The FB-TDMS-MQ model has been both statistically tested and simulated to show the potential of improving the efficiency of the health care system by making adjustments to staffing levels and reducing wait times.
Sensitivity analysis
To test the robustness of the proposed staffing optimization model in a range of healthcare settings, we performed a sensitivity analysis in which we tested the performance of the model to variations in the scale of the hospital, patient volume and socio-economic context. The analysis took variations in the size of the different hospitals into account, varying from small clinics to large hospitals, and looked at the effect of different budgetary constraints and the availability of staffing resources. The model recommends adjustments to staffing every 6 to 8 h to ensure that labor laws and realistic staffing issues are addressed. For purposes of sensitivity analysis, the following factors will be of interest:
Sensitivity analysis of arrival rates (λi(t))
Table 8 shows the changes in the arrival rate for the Emergency line, and hence the effect on average waiting time and number of staff required.
Sensitivity analysis for integration of fuzzy logic for decision making
Table 11 shows the effect of several fuzzy membership functions when making staffing decisions. The system is able to adjust to variations in the healthcare environment, because the arrival rates, service hours, and staff levels are all variable.
Table 11.
Sensitivity to fuzzy logic parameters
Fuzzy Set Variation
Queue 1 (Emergency)
Queue 2 (Outpatient)
Queue 3 (Specialty)
Arrival Rate
High: Increased staffing to Overstaffed
Medium: Adjusted staffing to Sufficient
Low: No change in staffing
Service Time
Long: Increased staffing to Overstaffed
Medium: Optimized staffing to Sufficient
Short: No change in staffing
Staffing Level
Insufficient: Adjust staffing upward to Sufficient
Sensitivity analysis for GA process for staffing optimization
This section discusses the sensitivity of the GA parameters used to optimize real-time healthcare staffing. The GA changes the population depending on population size, mutation rate, crossover rate, and generation.
Sensitivity to population size (GA)
Table 12 displays the performance of the GA in terms of fitness (overall performance) and convergence rate as a function of population size for optimizing the staffing levels.
Table 15 considers the effect of a growing number of generations upon the optimization process of the GA, particularly the convergence and staffing level optimization.
The sensitivity analysis used gave a deeper insight into the behavior of the model under different conditions. Fluctuations in the levels of arrival rate, service time, and staff were found to directly affect the waiting time, queue length, and system utilization. The analysis showed that a decrease in service time and an increase in staffing levels led to an improvement in system performance, validating the adaptability of the model accounting for other patient flow situations. Furthermore, the sensitivity analysis of GA process helped the key parameters like population size, mutation rate, crossover rate and number of generations. It was discovered that these factors strongly affect convergence rate and quality of staffing optimization, which highlights the importance of GA parameters tuning for maximizing their performance.
However, we can also interpret these sensitivity analysis results by further visualization.
Visual representations
The sensitivity analysis of the model of FB-TDMS-MQ with the data shown in Tables 8, 9, 10, 11, 12, 13, 14 and 15 is then illustrated in the graphs presented (Figs. 3, 4, 5, 6, 7 and 8).
Performance comparison of queues for different staffing levels
Impact of arrival rate on waiting time
Figure 3 shows that as the number of arrivals increases, waiting time is increased for Emergency, Outpatient and Specialty Queues 1–3. This further re-emphasizes the need for flexible staffing based on patient arrivals to avoid delay.
Impact of service time on waiting time
Figure 4 shows that for all-line waiting times, shorter service times decrease all-line waiting times. Smaller service times are better for queues 1 and 2. From this it is shown that quicker service may enhance system performance.
Impact of mutation rate on average waiting time
As can be seen in Fig. 5, the higher the mutation rates, the smaller the waiting times. Increasing the mutation rate gives more alternatives to the GA to find the best workforce and system performance.
Impact of crossover rate on average waiting time
Figure 6 shows that higher crossover rates result in faster searches for staffing, resulting in reduced staffing wait times. This indicates that higher crossover rates are favorable for wait times of patients as larger the variability in solution.
Impact of population size on average waiting time
Figure 7 shows that as population sizes increase, GA convergence increases, and the waiting times decrease. With more people, the algorithm can search more possibilities and come up with more optimized staffing patterns more quickly.
Performance comparison of queues for different staffing levels
Figure 8 shows the effects that changes in staffing in Queue 1 (Emergency), Queue 2 (Outpatient), and Queue 3 (Specialty) have on queue length, waiting time, and utilization. Queues 1 and 2 increase wait times and utilization as staff increase, while showing better allocation of resources during heavy demand times. Queue 3 shows a more consistent demand, which means less variances. This represents the flexibility of this system and demonstrates the way staffing optimization improves the efficiency of operation throughout patient flows.
This graphic representation (Figs. 3, 4, 5, 6, 7 and 8) provides evidence of the importance of diligently managing personnel levels, the length of service, and arrival rates in order to minimize wait times and the efficiency of the healthcare system. By adjusting these factors, the workload of the system could be balanced, and congestion could be reduced, as well as patient treatment could be expedited.
Results and discussion
In this section, the research findings and conclusions of the study, validation, simulation, sensitivity, and case study results will be discussed. The discussion shows the relevance of the results in the context of real life.
After the introduction of the FB-TDMS-MQ model, the decision was made on how to test the Fuzzy Backbone structure of TDMS Model using a real-life case study from "DHRC, Jaipur," a diversified medical team hospital. The operations using the GA method can have a dynamic staffing level, which is a more flexible and efficient scheme than the traditional fixed staffing models as shown in the case study of DHRC, Jaipur. The GA-based staffing optimization resulted in the reduction of average waiting times at all patient queues. During the optimization process, the model considered clear restrictions of staffing limits, skills, and shifts to ensure that all staffing configurations generated by the GA were possible. The penalty mechanism effectively ensured that infeasible solutions such as those that violate staffing limitations or shift rules are discarded and that as a result, optimal staffing assignments were generated that followed hospital operational policies.
The validation of the assumption of arrival rate through statistical analysis gave an approval to use the Poisson distribution in modeling the arrival of patients. This helps to add credibility to the proposed FB-TDMS-MQ model, that will ensure its practical relevance in dynamic settings of the healthcare environment. The assumption on upper limit of 15 min for service time was carefully found out from empirical data and expert judgement from DHRC, Jaipur. In this way, the service time distribution model could build on the actual healthcare operation to ensure that realistic healthcare operation could be represented in the model to optimize staffing and patient flow.
Simulation and validation tests show that the FB-TDMS-MQ model is an effective aid in optimizing hospital staff work and reducing patient waiting time. Fuzzy logic for real-time staff adjustment reduced the peak-hour waiting time by 72.73%, which showed its demand management efficacy. The best hospital staffing strategy, which reduced wait time by 50.2%, was selected using GA. The effectiveness of the model to evaluate the operational performance of the hospital is argued. The system flexibly allocated the number of workers to improve the performance throughout unexpected changes in the demand. The simulation results outperformed FB-TDMS-MQ model. The findings showed that FB-TDMS-MQ has a positive impact on maximizing resources and providing better healthcare to patients. The results of the simulation was validated by comparing it with the real life data from DHRC, Jaipur. The simulated waiting times, patient flow and staffing levels were in close agreement with the real data and can be confident that the model accurately depicts the real healthcare system. In addition, through sensitivity analysis, model robustness was demonstrated under different input parameters, which further proved the model's good performance under different scenarios.
Sensitivity studies have shown that the model is able to accommodate different numbers of arrivals, different service times and different number of staff members. Greater arrival rates and longer service durations result in longer queueing times and waits, therefore, there needs to be time-consistent adjustments to staffing appropriate to demand. Figures 3, 4, 5, 6, 7 and 8 show the effect of arrival times and service times on line, system and wait time. The graphs clearly show different trends that are well in line with reality regarding the functioning of the healthcare system.
Thus, there will be several advantages of FB-TDMS-MQ in terms of healthcare. The comparison with benchmark problems and state-of-the-art algorithms proves that this hybrid fuzzy-genetic algorithm is able to bring a significant improvement in the quality of solutions and also in the computational efficiency. The combination of fuzzy logic with GA offers better flexibility in coping with the uncertainty/variability which exists in healthcare systems, not well represented in other methods such as the Simulated Annealing (SA) or the Particle Swarm Optimization (PSO). With the current system upgrades, this technique of management maximizes queue management and scheduling of staff groups through the patient flow, hence boosting patient flow and service efficiency. While it may prevent data processing from carrying on, the method may be helpful for both healthcare professionals and individuals. The proposed hybrid fuzzy-genetic algorithm approach outperforms the traditional stochastic models and other meta-heuristics approaches in terms of take uncertainty to the next level as well as provide real-time staffing optimization. The model's adapting capability to dynamic situations where patient flow is changing makes it a major advantage over simple models which are static and cannot adjust to variable demands in real life healthcare settings.
Limitations and future research directions
The proposed model FB-TDMS-MQ is a novel approach to optimize the manpower of healthcare personnel and decrease the patient wait time using fuzzy logic and GA. However, it has certain limitations:
Data Dependency: The model is highly dependent on the quality and accuracy of the input data (e.g. arrival rates, service times), which may differ from one healthcare system to another and impact performance.
External Factors: A number of external factors, such as unforeseen circumstances, extreme emergencies, or change in patient behaviour, may affect the effectiveness of the model since these impacts were not fully integrated in the initial model setup.
Fuzzy Logic and GA Tuning: The success of the model is dependent upon the accurate setting of the fuzzy inference system and the GA. Incorrect tuning may result in poor levels of staffing.
Cost-effectiveness: This study primarily focused on time-based metrics and operational efficiency, and did not include an analysis of cost-effectiveness, labor costs, or economic trade-offs, which are important considerations for evaluating the fiscal sustainability of the proposed intervention.
Resource-Intensive: The use of the model and its adaptation in real time might entail a demand to a significant computational resource, especially when the model is used at a large scale.
The following are some potential directions for future research:
Advanced machine learning algorithms: There is still room for more research into the use of advanced machine learning algorithms such as deep learning to discover further applications for improving the accuracy and utility of fuzzy systems.
Patient involvement: Impact of having patients involved in triage in fuzzy environments, especially in cases of refusing to participate in the process, need to be explored further.
Cost-effectiveness: Future research will address the economic aspects of the intervention by incorporating cost-effectiveness analysis, marginal utility considerations, and labor costs to provide a more comprehensive understanding of its financial viability and long-term sustainability.
Global Application: The proposed approach has great potential for application on a global scale, especially in countries with limited resources. The power and utility of the fuzzy-based hybrid queueing model for healthcare systems is something to further explore.
Conclusion
The FB-TDMS-MQ is a combination of fuzzy logic and GA that is suggested as a way to optimize healthcare staffing in real time. The real-time optimization strategy lets hospitals modify their personnel as needed based on real-time data collected throughout each shift. This method keeps the staffing model flexible enough to handle fluctuations in patient demand without having to modify it every day, which would be problematic for operations.
Fuzzy logic, GAs, and advanced queueing systems for staffing are considered more effective than deterministic ones. Fuzzy sets, fuzzy rules, and a Fuzzy Inference System that operates in real time using live data have revolutionized the workforce in the healthcare field. The GA-based staff assignment approach was able to achieve the objective of minimization of patient waiting times while meeting important operational constraints. By taking into account staffing limitations, skills and shift rules the modeling made sure that all staffing configurations were not only optimized for efficiency, but feasible given real-world hospital conditions. This makes it possible for the system to adapt to new situations fast enough. This new approach involves the use of several types of medical information to help humans make better decisions.
The FB-TDMS-MQ model was tested at a multispecialty hospital, "DHRC, Jaipur," India. Validation and simulation activities show that the model successfully reduces average wait times, optimizes healthcare resource use, and improves operational flow. Notably, applying fuzzy logic to optimize the workforce in real time resulted in an impressive 72.73% reduction in average waiting times, demonstrating effective demand management. The average patient wait time decreased by 50.2% after the optimal staffing strategy, determined by GA, was implemented. This indicates that the model provides a reliable prediction of hospital performance. The simulation results exceeded initial expectations of the FB-TDMS-MQ model. Sensitivity analysis confirmed that the model can adapt to hospitals of various sizes and socio-economic conditions, making it a versatile tool to enhance healthcare system efficiency. It also shows that the model can handle uncertainties in arrival rates, service times, and staff fluctuations. Adjusting staff levels when necessary may be essential to maintaining smooth operations, especially when service request start and end times vary. The proposed solution offers flexible support in dynamic healthcare environments by effectively managing patient flow and staffing based on specific situations.
Thus, the FB-TDMS-MQ model has a number of advantages over current techniques such as OptQueue. A combination of Fuzzy logic and GA can adjust staffing in real-time, considering the uncertainties and dynamic patient supply and demand, by integrating the two planning methods, so it is more applicable to the practical healthcare environments, and more flexible than static optimization models. This study shows the potential for the FB-TDMS-MQ model to be used for optimal staffing for various healthcare environments.
Acknowledgements
Manipal University Jaipur
Abbreviations
FB TDMS-MQ
Fuzzy–Based Time -Dependent Multi-Server, Multi-Queueing
Algorithm 2 Genetic Algorithm Process for Staffing Optimization
Authors’ contributions
All authors have made substantial contributions to the conception or design of the work. The content preparation, data collecting, and writing responsibilities were all carried out by Balveer Saini [First author]. The supervision and technical evaluation of the project were carried out by Dr. Dinesh Kumar Saini. All contributors have thoroughly reviewed and provided their consent for the version designated for publication.
Funding
Open access funding provided by Manipal University Jaipur. Open access funding is provided by Manipal University, Jaipur.
Data availability
The data utilized in this study were acquired from a multi-specialty hospital, “DHRC, Jaipur, via an official request and a subsequent approval process. The analyzed region is based on information from 5000 patients at DHRC, Jaipur, spanning multiple medical fields. In ever-changing and uncertain environments, employing simulations, sensitivity analysis, and empirical data improves the efficiency of healthcare operations. The data was adjusted to meet the research objectives.
Declarations
Ethics approval and consent to participate
Not applicable.
The participants were informed about the study's purpose, procedures, and their voluntary participation, and they provided written consent to participate in the research.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s Note
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The data utilized in this study were acquired from a multi-specialty hospital, “DHRC, Jaipur, via an official request and a subsequent approval process. The analyzed region is based on information from 5000 patients at DHRC, Jaipur, spanning multiple medical fields. In ever-changing and uncertain environments, employing simulations, sensitivity analysis, and empirical data improves the efficiency of healthcare operations. The data was adjusted to meet the research objectives.
: Time-dependent arrival rate for queue i at time t.
: Time-dependent service rate for queue i at time t.
: Average number of patients waiting in the queue for queue i at time t.
: Utilization factor for queue i at time t, indicating the proportion of time servers are busy.
: Fuzzy membership function for the arrival rate λi(t) at time t, reflecting fuzzy categories (Low, Medium, High).
: Fuzzy membership function for the service time μi(t) at time t, reflecting fuzzy categories (Short, Medium, Long).
: Fuzzy membership function for staffing level S, indicating categories (Insufficient, Sufficient, Overstaffed).
: Fuzzy membership function for the arrival rate at time t (Low, Medium, High).
: Fuzzy membership function for the service time at time t (Short, Medium, Long).
: Fuzzy membership function for staffing level S at any given point x.
: Defuzzified staffing level based on fuzzy logic, used in the system optimization model.
: Fuzzy membership function for staffing level S, representing categories like insufficient, sufficient, and overstaffed.
: The traffic intensity signifies the proportion of time during which the servers are engaged. It is elegantly calculated as:
.
denote the membership function for the arrival rate at time t:
denote the membership function for the service time at time t:
represent the fuzzy membership function for staffing levels:
is the crisp staffing level recommendation is based on the fuzzy system, μi(x) is the membership value of fuzzy set i, and x is the corresponding staffing level for that fuzzy set.
be the population of N individuals (chromosomes), where each individual pi represents a staffing configuration across all queues. Each chromosome pi is a vector 
where sj represents the staffing level for queue j, and m is the total number of queues.
, representing the average waiting time for the staffing configuration pi. The waiting time Wi for each queue i is calculated using Eq. (14):
is the arrival rate for queue i, 
is the service rate for queue i, and Lq is the queue length for queue i.
such that 
(lower fitness means better solution). Tournament selection involves randomly choosing two individuals and selecting the one with the lower fitness.
and 
, a single-point crossover occurs at position c, where 
. The offspring chromosomes 
and 
are 
and 
.
, select a random position j and change the staffing level sj to a new value 
within the allowable range 
, as given in Eq. (16).
, the algorithm is considered to have converged as given in Eq. (18):
that minimizes the fitness function is given by Eq. (19):
] = 0.02 hours2 (example value).
] = 0.03 hours2 (example value).
Algorithm 1 Genetic Algorithm for Optimizing Staffing Configuration in Healthcare Systems
is the mean of the differences between a pair of data.
