Pilot Fatigue Coefficient Based on Biomathematical Fatigue Model

Abstract

The routine assessment of pilot fatigue is paramount to ensuring aviation safety. However, current designs of pilot fatigue factors often lack the comprehensiveness needed to fully account for the dynamic and cumulative nature of fatigue. To bridge this gap, this study introduces a biomathematical fatigue model (BFM) that leverages system dynamics theory, integrating a dynamic feedback mechanism for fatigue information. The novelty of this approach lies in its capability to continuously capture and model fatigue fluctuations driven by varying operational demands. A comparative analysis with international methodologies for evaluating cumulative fatigue on weekly and monthly scales demonstrates that the proposed BFM effectively reproduces variations in pilot fatigue characteristics. Moreover, the pilot fatigue coefficient derived from the model provides a robust differentiation of fatigue profiles across diverse work types, making it particularly suitable for estimating cumulative fatigue over monthly intervals. This BFM-based approach offers valuable insights for the strategic planning of flight schedules and establishes an innovative framework for utilizing BFMs in fatigue management. By employing a scientifically grounded evaluation method rooted in system dynamics and the BFM, this study rigorously assesses cumulative pilot fatigue, confirming the model’s accuracy in replicating fatigue patterns and validating the efficiency and reliability of the derived fatigue coefficient.

1. Introduction

Fatigue has long been a significant issue in civil air transportation safety because it impairs pilot operational performance and frequently leads to aviation accidents [1]. According to the International Air Transport Association (IATA) safety report [2], fatigue inducement accounted for 11% of fatal aviation accidents that occurred from 2017 to 2021, making it the top human factor affecting aviation safety. Furthermore, the International Civil Aviation Organization (ICAO) [3] considers fatigue to be unavoidable and must be managed. The flight duty period usually consists of multiple task strings, and fatigue is not only a direct result of a single flight mission, but also an accumulation of multiple mission processes over time. Therefore, the regular assessment of fatigue levels is important to ensure flight safety and pilot health.

Cumulative fatigue refers to the accumulation of fatigue that occurs when an individual repeatedly experiences inadequate recovery from work over a long period. Unlike transient fatigue, cumulative fatigue is a long-term, progressive state. The essence of cumulative fatigue lies in the lack of sufficient recovery time during working periods, leading to a gradual deterioration in physical and psychological states. This cumulative fatigue affects an individual’s physiological functioning, cognitive ability, and emotional state, with specific manifestations including prolonged reaction time, reduced attention span, weakened judgment, memory loss, and increased mood swings, significantly increasing the risk of accidents. In the civil aviation industry, the problem of cumulative fatigue is particularly prominent among pilots. Due to the nature of flight missions, pilots often face cross-time zone flights, irregular working hours, and long standby periods. These factors disrupt the pilot’s biological clock and lead to sleep deprivation and poor sleep quality, aggravating cumulative fatigue levels. The European Aviation Safety Agency, in the Notice of Proposed Amendment (NPA) NO 200902C [4], first defined cumulative fatigue and required fatigue risk management systems (FRMSs) to control the associated risks. A regulation by the Federal Aviation Administration [5] outlines limits on cumulative flight time to prevent cumulative fatigue. Almost all major civil aviation regulatory agencies worldwide [6,7] require airlines to develop and implement fatigue management plans in the form of regulations, considering the work and rest patterns of pilots to reduce the risk of cumulative fatigue.

The BFM is a scientific method used to explain fatigue triggering and evolutionary rules using mathematical equations, which can quantify fatigue by scheduling information, and is a fatigue risk identification tool recommended by the ICAO. Borbély [8] proposed a two-process model of sleep regulation (TPMSR) that describes fatigue as the result of the interaction between the homeostatic and circadian processes of sleep. Over the past 40 years, TPMSR has been continuously developed and has led to a series of derived models. Åkerstedt, Folkard, and Portin [9] introduced the influence of sleep inertia in TPMSR and proposed a three-process model of alertness. Rajdev et al. [10] incorporated the previous process of sleep or wakefulness and the module of sleep debt in addition to the TPMSR to better describe the fatigue mechanism. Sun and Sun [11] proposed the modeling concepts of alertness energy, alertness kinetic energy, and alertness potential energy from the perspective of energy and developed model equations to quantify alertness. More recently, Wilson et al. [12] highlighted the need to establish kinetic models for diverse and dynamic fatigue variables in realistic production environments as a major direction in BFM research. In summary, current research on BFMs focuses on the expansion of fatigue-influencing factors, and there is a lack of research on the system dynamics modeling of fatigue evolution, which leads to a lack of interpretability in current model mechanisms. There is a dearth of exploratory research on the scientific application of BFMs to support fatigue management. Constructing a multidimensional fatigue coefficient for pilots based on the results of the BFM fatigue assessment will provide more accurate guidance for flight fatigue management in practical operations.

In this study, a dynamic model based on current BFMs and system dynamics techniques was constructed to quantify pilot fatigue levels throughout duty periods. The pilots’ cumulative fatigue was calculated by the model using the output fatigue levels recorded during the duty periods. A simulation analysis was conducted using pilots in different operating scenarios. This is intended to offer a new path for the routine assessment of pilots’ and airlines’ tiredness statuses, in conjunction with the assessment of pilots’ fatigue states based on real-world examples.

2. Materials and Methods

2.1. Description of the Problem

Fatigue is a complex physiological phenomenon that is influenced and regulated by multiple factors.

First, the duty period of pilots is closely related to flight and rest times, and their formulation must comply with the regulatory requirements. In addition, homeostatic and circadian processes are essential factors to consider, as they interact to regulate organism activities [13], and these two processes are often incorporated as important components in existing BFMs. Lastly, workload and rest quality are correlated with pilot performance and alertness [14,15]. Hence, in the process of modeling pilot fatigue, it is necessary to consider multiple factors, including duty duration, homeostatic and circadian processes, workload, and sleep quality.

The circadian rhythm is controlled by the “biological clock” genes in the suprachiasmatic nucleus (SCN) of the hypothalamus, and the expression level of these genes is also regulated by feedback from certain neurotransmitters in the neuroendocrine system [16]. Thus, fatigue is inevitably a manifestation of a complex dynamic system in organisms, and several studies have demonstrated that the process of neural activity is characterized by system dynamics [17]. Furthermore, it is more acceptable to use theoretical techniques of system dynamics to build a plausible information feedback mechanism of organism fatigue to establish BFMs because system dynamics offers advantages for investigating systems with nonlinear, dynamic feedback [18].

2.2. Establishment of BFM

2.2.1. Model Conception

The model proposed in this study draws on the concept of a “Sleep Reservoir” from the Sleep, Activity, Fatigue, and Task Effectiveness (SAFTE) model [19]. The SAFTE model suggests that individuals can accumulate sleep in the sleep reservoir when adequately rested or asleep; however, the sleep reservoir is depleted during wakefulness. Additionally, the model incorporates several concepts from the fatigue Audit InterDyne (FAID) model [20], which assumes that fatigue oscillates continuously between work and nonwork states, with fatigue accumulating during work and recovering during nonwork periods.

Based on the aforementioned model concepts and the theory of system dynamics, this study proposes a dynamic mechanism for fatigue information feedback. The causal relationships of fatigue information feedback are illustrated in Figure 1. Exogenous factors continuously influence endogenous mechanisms, ultimately affecting alertness and fatigue performance. A potential advantage of this model is its ability to incorporate more exogenous factors into the workload module and the rest quality module.

Exogenous factors continuously influence endogenous mechanisms, ultimately affecting alertness and fatigue performance. A potential advantage of this model is its ability to incorporate more exogenous factors into the workload module and the remaining quality module.

The assumptions include the following:

• Individuals store alertness resources.
• The alertness resources of an individual have an upper limit that depends on their capacity for alertness resources.
• During mental work, the consumption of alert resources dominates, whereas during the rest period, the recovery of alert resources plays a dominant role.
• At a certain moment, the consumption rate of alert resources is divided into endogenous and actual consumption rates. The endogenous consumption rate represents the inherent consumption or recovery rate of alert resources in an individual and is a function of the alert resource reserve, the alert resource capacity, homeostatic processes, and circadian processes. The actual consumption rate is influenced by external factors, such as workload, in addition to the endogenous consumption rate.
• The recovery rate of alert resources is likewise split into endogenous and real recovery rates at a certain point in time. In addition to the endogenous recovery rate, the actual recovery rate is also influenced by outside variables, such as the quality of rest.
• The alertness level of a pilot during a duty period at a certain moment is positively correlated with the endogenous consumption rate of alert resources.
• The pilot’s alertness level and weariness level are inversely connected during duty. The pilot’s level of weariness during the time off duty was denoted as 0.

2.2.2. Mathematical Representation of the Model

This study focused on pilots, assuming that alert resources were depleted during flight duty periods and recovered during off-duty periods, on the basis of the flight operation mode. Therefore, the expression for the human vigilance resource reserve is written as follows:

$$\left\{\begin{array}{c}{R}_1=R_0-S·∆T,duty\hfill \\ R_1=R_0+S·∆T,off\text{-}duty\hfill \end{array}\right.$$

(1)

Here, R₀ represents the initial reserve of alertness resources at the start of a time interval ∆T, whereas R₁ represents the reserve of alertness resources at the end of the ∆T interval. S represents the actual rate of alert resource consumption or recovery, which is influenced by external factors, such as workload and quality of rest, in addition to the endogenous consumption or recovery rate. Note that the reserve of alert resources R should not have a numerical value less than 0.

The actual rate of alert resource consumption or recovery S is modified by external factors, including workload and rest quality, in addition to the endogenous consumption or recovery rate. This is expressed as follows:

$$S=\alpha ·Q,$$

(2)

where α is the coefficient influence of exogenous influences on the actual consumption or recovery rate and Q is the endogenous consumption or recovery rate.

The endogenous consumption or recovery rate, Q, is the result of the combined effects of an alert resource reserve, an alert resource capacity, homeostatic processes within the human body, and circadian processes. Borbély and Achermann [21] provided detailed formula expressions for key processes in the TPMSR, where the main expression used for steady-state processes is e-t/τ, where t denotes the duration of the duty or non-duty period. τ is the time constant in the expression. In summary, the formula for endogenous consumption or recovery rate Q can be expressed as

$$Q=\left\{\begin{array}{cc}k·R·\exp \left(-{\displaystyle \frac{t}{{\tau }_d}}\right)·C\left(T\right)\hfill & ,duty\hfill \\ k·\left(R_m-R\right)·\exp \left(-{\displaystyle \frac{t}{{\tau }_r}}\right)·C\left(T\right)\hfill & ,non\text{-}duty\hfill \end{array}\right..$$

(3)

Here, k = 0.01 is the adjustment coefficient for the endogenous consumption or recovery rate. R represents the real-time alert resource reserve. R_m is the alertness resource capacity. C(T) represents the functional expression of the circadian process.

Borbély and Achermann [21] described the expression of the circadian rhythm function as a complex superposition of multiple sine functions. Moreover, some scholars [22,23] have used simple cosine function superposition to describe the circadian rhythm process and have achieved good performance. Therefore, to balance scientificity and convenience, the model of this study uses the two-harmonic quantity cosine function as the endogenous human circadian rhythm function. The expression C(T) is as follows:

$$C\left(T\right)=C_1·\mathrm{c}\mathrm{o}\mathrm{s}\left[{\displaystyle \frac{2\pi ·\left(T-{\phi }_1\right)}{24}}\right]+C_2·\mathrm{c}\mathrm{o}\mathrm{s}\left[{\displaystyle \frac{2\pi ·\left(T-{\phi }_2\right)}{12}}\right]$$

(4)

where C₁ and C₂ represent the cosine function amplitudes of the 24 and 12 h rhythm cycles, respectively. The first represents the typical 24 h circadian cycle, which is usually associated with physiological phenomena such as the human sleep—wake cycle, hormone secretion, and temperature fluctuations. The second cycle represents a 12 h semicircadian cycle, which may be associated with other physiological phenomena (e.g., secondary fluctuations in midday body temperature or hormone secretion). φ₁ and φ₂ represent the cosine function phases for the two rhythm cycles, and their parameters can be personalized according to the pilot’s personal sleep type [24].

Alertness at a certain moment in the duty period was positively correlated with the endogenous depletion rate of alert resources, and the degree of fatigue was negatively correlated with the degree of alertness. The degree of fatigue F is expressed as follows:

$$F=100-\beta ·Q$$

(5)

Here, β is the fatigue value adjustment coefficient, which is used to adjust the alertness value output to an appropriate numerical range.

The kinetic model of the dynamic feedback of an organism on fatigue information is based on this model. It is adjusted by setting parameters such as the influence coefficient (α) of the exogenous influencing factors on the model’s actual consumption or recovery rate, the pilot’s alert resource capacity (R_m), the time constants (τ_d and τ_r) in the expression of the homeostatic process function, the amplitudes of the cosine function (C₁ and C₂), the phases (φ₁ and φ₂) in the diurnal function, and the adjustment coefficient of the fatigue value β. By setting the pilot duty times, the model can be used to mimic how pilot fatigue values change during the duty period. This model allows for a good simulation of the fatigue value trend of the pilot duty period; for example, the fatigue value curves in Figure 2 illustrate changes in the fatigue coefficient under different working durations, revealing the impact of the dynamic feedback mechanism on fatigue. These curves help validate the model’s application value in fatigue management.

2.3. Cumulative Fatigue Quantification Model

Based on the BFM mentioned above, a cumulative fatigue quantification model is further proposed. The cumulative fatigue score for each duty period is numerically equal to the area between the fatigue level curve and the x-axis, as indicated by the colored shaded area in Figure 2.

Consequently, the formula for calculating the cumulative fatigue score of the pilots is as follows:

$$FC={\sum }_{i=1}^n{\int }_{ti1}^{ti2}F\left(t\right)dt.$$

(6)

3. Results of Fatigue Simulation

3.1. Assigning Parameter Values of BFM

Referring to the parameter settings for the biological mathematical model by Peng et al. [23], the time constants of the homeostatic process function expression in this model are set as τ_d = 18.2 and τ_r = 4.2 by default. The phase moments of the cosine function for a 24 and 12 h circadian rhythm period in the expression of the circadian process function are set by default to φ₁ = 18 points and φ₂ = 21 points, respectively, with the amplitudes set as C₁ = 0.97 and C₂ = 0.23, respectively.

The α value of external factors on the actual consumption or recovery rate of alertness resources is determined on the basis of task type and sleep (rest) quality, and other factors. This study categorizes flights into three categories: short-haul flights, typically performed by a single or augmented flight crew, consisting of two or three pilots, which involve no long layovers at the destination for rest; long-haul non-overnight flights, performed by three crews (six pilots) with only brief layovers at the destination; and long-haul overnight flights, conducted by two crews (four pilots) where the rests are taken overnight at the destination. Under the multiple-set flight crew operating mode, pilots have the opportunity for onboard rotation rest, resulting in a relatively lower average workload but often involving a long duty duration. Considering short-haul flights as the standard, the influence coefficients for both depletion and recovery were assumed to be one. For long-haul overnight flights, the crew consists of four pilots, and the overall workload for the duty period is decreased; however, an overnight stay at the destination alters the quality of the remaining composition. For long-haul, non-overnight flights, the crew consists of six pilots, further reducing the overall workload for the duty period. There is no need for overnight stays abroad, and the rest quality is considered equivalent to that of short-haul flights.

The assigned values for the α coefficient for short-haul flight tasks, long-haul overnight flight tasks, and long-haul non-overnight flight tasks that are consistent with the aforementioned facts are listed in

Table 1. Assignment of α for three task types.

Period	Type of Operation	Assign a Value to α
Alertness resource consumption (duty)	Short-haul flight	1
	Long-haul overnight flight	0.8
	Long-haul non-overnight flight	0.6
	Reaching the critical state of monthly cumulative fatigue	1.5
Alertness resource recovery (off duty)	Short-haul flight	1
	Long-haul overnight flight	0.8
	Long-haul non-overnight flight	1
	Reaching the critical state of monthly cumulative fatigue	0.6

.

Assuming that the pilot’s alertness resource capacity is R_m = 100, research has shown that under the given or default parameters, when simulating the schedule of long-term day-shift work (9:00–17:00), the initial alertness resource reserve at the start of the shift tend to be approximately 53.05. Accordingly, the pilot’s monthly initial alertness resource reserve is set to R₀ = 53.05. In the same scenario, when the fatigue value adjustment coefficient is set to β = 10, the fatigue value generally stabilizes within the range of approximately 20 to 80.

The biological mathematical model parameters used in this study are listed in

Table 2. Model parameter settings.

Parameters	Definition	Assignment
τd	Time constant of the homeostatic function for the duty period	18.2
τr	Time constant of the homeostatic function for the off-duty period	4.2
φ1	Phase of the circadian process 24 h rhythm	18
φ2	Phase of the circadian process 12 h rhythm	21
C1	Amplitude of 24 h rhythm of circadian process	0.97
C2	Amplitude of 12 h rhythm of circadian process	0.23
Rm	Alertness resource capacity	100
R0	Reserve of alert resources for initial moments of duty	53.05
β	Adjustment coefficient for fatigue value	10

.

3.2. Verification of the Alertness Simulation Effect of BFM

3.2.1. Participants

A total of 146 pilots from a Chinese airline participated in this study, the average age of which was 34.66 ± 6.29 years, and the age distribution was as follows: under 30 years, 25.3%; 30–39 years, 52.1%; and more than 40 years, 22.6%. For captains, 39%; for copilots, 61%.

Pilots were required to use a subjective fatigue scale and a psychomotor vigilance task (PVT) during flight to select time points to test and record actual fatigue conditions. The real schedule information of 146 pilots was subsequently imported into the aforementioned BFMs to simulate and quantify the alertness level of the pilots during the flight duty period, and the model simulation alertness corresponding to the subjective and objective fatigue observation time points was obtained.

The pilot information and test data collected during this study were strictly confidential. Regulators, airlines, and pilots supported this research process. The study followed the principles of informed consent. Prior to data collection, pilots received sufficient training to make them aware of the scope and purpose of the data to be tested, and verbal informed consent was obtained from the pilots. Furthermore, the test does not have a negative impact on flight safety or the physical and mental safety of pilots, and it was approved by the ethics committee. If the pilot still had any doubts, they could freely choose whether to conduct the test.

3.2.2. Actual Fatigue Data Collection

The pilots’ subjective fatigue was characterized using the Karolinska sleepiness scale (KSS) and the Samn–Perelli fatigue scale (SP). The KSS is a nine-point Likert-type scale of subjective sleepiness ranging from 1 = “very alert” to 9 = “very sleepy, fighting sleep”. Its effectiveness was verified via various physiological measurement methods. The SP is a seven-point Likert-type scale of subjective fatigue ranging from 1 = “fully alert, wide awake” to 7 = “completely exhausted, unable to function effectively”. It is widely used in aircrew fatigue research and has high reliability and validity [25]. Both the KSS and the SP are scales recommended by the ICAO, and they are two common types of data that should be collected during international flight fatigue data collection [26].

Objective fatigue is characterized by the mean response time (MRT) in a continuous performance test (CPT), a classical objective performance test method that presents three symbols each time as one stimulus through a display panel [27]. By asking subjects to compare the presented stimuli and make judgments and continuous keystroke responses, quantitative values such as the MRT and error rate can be calculated. The use of symbols {╤╟ ╤ ╢╩╠ ╦ ╣} as stimuli in this study, which do not have evident semantic features, allows for better control of the learning and practicing effects and a more reliable assessment of subjects’ alertness levels [28].

In this study, the KSS, SP, and CPT tests were combined in an Android system application to assess the subjective and objective fatigue of the pilots during actual flights. Before the experiment, the pilots were provided with a tablet device (Model HONOR ViewPad 6, designed and manufactured by HONOR in Shenzhen, China.) with the app installed and received sufficient training to familiarize them with the testing tasks. All tests were conducted voluntarily by pilots during their spare time, ensuring flight safety was not compromised. The decision to conduct testing relied on the pilot’s sense of responsibility, ensuring that the pilots involved in data collection were proactive and serious. Additionally, quality checks were conducted on the collected data. Any data with abnormal reactions or high error rates were removed to further ensure their quality.

3.2.3. Overview of Actual Fatigue Data

In this study, unreasonable observation data with CPT accuracies lower than 80% or average responses higher than 5000 ms were eliminated. A total of 382 sets of effective data were collected from 146 pilots. Each set of data included the KSS and SP values and the MRT of the CPT observed by the pilot at a given time. Additionally, the alertness value of the corresponding time point was simulated by BFMs. The results were obtained for 142 overnight flights and 43 non-overnight flights in North America, 77 non-overnight flights in Oceania, and 41 overnight flights and 79 non-overnight flights in Europe. The specific data sources and result information are summarized in

Table 3. Summary of source information for observations.

Operation Area	Overnight or Not	Air Route (IATA Three-Letter Designator)	No. of Crew Members	No. of Outbound Results	No. of Returning Results	Total	KSS	SP	MRT
North America	Yes	PVG-LAX-PVG	4	64	78	142	3.51 ± 1.65	2.44 ± 0.96	1433.20 ± 493.20
		PKX-YVR-PKX
		PVG-YYZ-PVG
	No	PVG-LAX-PVG	6	41	2	43	3.12 ± 1.16	2.21 ± 0.71	1267.95 ± 275.36
Oceania	No	PKX-AKL-PKX	6	65	12	77	3.29 ± 1.37	2.44 ± 0.83	1292.92 ± 302.74
		PVG-MEL-PVG
		PKX-SYD-PKX
Europe	Yes	PKX-AMS-PKX	4	26	15	41	3.88 ± 1.29	2.68 ± 0.76	1530.42 ± 471.64
		PVG-FRA-PVG
		PVG-STN-PVG
	No	PKX-AMS-PKX	6	74	5	79	3.38 ± 1.39	2.38 ± 0.76	1368.23 ± 396.12
		PVG-FRA-PVG
		PVG-LHR-PVG
		PVG-STN-PVG

, and the data results are written in the form of “Mean ± SD”.

Two-factor analysis of variance was performed on the fatigue characterization data for different operation types.

Table 4. Results from analysis of variance.

Type of Operation	KSS		SP		MRT
	F	p	F	p	F	p
Operation Area	1.708	0.183	2.450	0.088	4.947	0.008
Overnight or Not	1.976	0.161	1.332	0.249	1.254	0.264
Outbound-Return Trip	15.907	0.000	11.989	0.001	6.129	0.014
Overnight or Not and Outbound-Return Trip	13.308	0.000	7.482	0.007	12.323	0.001
Operation Area and Overnight or Not	0.015	0.904	0.019	0.890	0.190	0.663
Operation Area and Outbound-Return Trip	0.459	0.633	1.431	0.240	2.822	0.061

summarizes the ANOVA results of the analysis of variance.

As is evident from the analysis of variance in Table 4, the pilots’ fatigue between the “outbound” trips and “returning” trips shows a significant difference. (KSS: F = 15.907, p = 0.000; SP: F = 11.989, p = 0.001; overall MRT: F = 6.129, p = 0.014.) Simultaneously, the interaction between “overnight or not” and “outbound-return trip” significantly affects pilot fatigue. (KSS: F = 13.308, p = 0.000; SP: F = 7.482, p = 0.007; overall MRT: F = 12.323, p = 0.001). To better present the probability density and overall distribution of the data under the two operation types of “overnight or not” and “outbound-return”, the data are presented by the split-violin diagram, as illustrated in Figure 3.

As shown in Figure 3, the overall distributions of the three types of fatigue data during the outbound phase were extremely similar for overnight and non-overnight flights. However, the overall data distributions of the overnight and non-overnight operation types clearly differ in the return phase, with a greater proportion of high-value data in non-overnight operations.

3.2.4. Comparison of Actual Observations with Model Simulations

A preliminary validation of the previous model was conducted based on a previous study that compared simulated and observed BFM values [29]. In this study, the sum of the “KSS + SP” values was used as the quantitative representation of subjective fatigue, and the average reaction time was used as the quantitative representation of objective fatigue. The following formula was used to assign all the observation data used in the comparison in this study to a range of 0–100 because of the significant disparity in the subjective and objective tiredness quantization characterization values. The fatigue characteristics of the three types of data remained the same after data redistribution: the higher the value is, the greater the tiredness.

$$y^{\prime }=a+(b-a)(y-\mathrm{m}\mathrm{i}\mathrm{n}Y)/\left(\mathrm{m}\mathrm{a}\mathrm{x}Y-\mathrm{m}\mathrm{i}\mathrm{n}Y\right).$$

(7)

Here, y and y’ are the data before and after data allocation, respectively; maxY and minY are the maximum and minimum values of the set Y where the data are located; and [a,b] is the target range interval of data allocation.

The average data of the three types of fatigue characteristics of the 146 pilots were compared using Pearson’s correlation coefficient, and the data analysis results are shown in

Table 5. Correlation of three kinds of fatigue data.

	No. of Cases	Correlation (r)	Significance (p)	Type of Correlation
Objective observations—model simulation values	146	0.491	<0.001	medium positive correlation
Subjective observations—model simulation values	146	0.736	<0.001	strong positive correlation
Subjective observations—objective observations	146	0.661	<0.001	strong positive correlation

.

There was a significant correlation between the objective observation value and the simulated value of fatigue (r = 0.491, p < 0.001). Although the model was able to reflect objective fatigue, its correlation was slightly lower than that of subjective fatigue, which may be because objective metrics (e.g., reaction time) are more influenced by external factors. The correlation between the subjective fatigue observations and the simulated values was stronger (r = 0.736, p < 0.001). This shows that the model can better predict the subjective fatigue state of pilots, which is an important reference value for pilot fatigue management. The observed subjective and objective fatigue correlations were significant between the two pairs (r = 0.661, p < 0.001). The strong positive correlation between the subjective and objective observations suggests a degree of consistency. The mean absolute error (MAE) and root mean squared error (RMSE) were 22.85 and 27.08%, respectively. The MAE between the subjective observations and the simulated values was 14.01, and the RMSE was 17.02.

The relationships between the subjective and objective fatigue observations and model simulations are shown in Figure 4.

3.3. Simulation of Cumulative Fatigue

Three categories of pilots were selected for the N airline: class A, class B, and class C. Class A pilots, belonging to the A330 fleet, fly short-haul flights for extended periods. Class B pilots, part of the B777 fleet, fly intercontinental overnight flights. Class C pilots, who belong to the B787 fleet, fly intercontinental non-overnight flights. The duty schedules for representatives A1, B1, and C1 from each category of pilots for a particular month are shown in Figure 5.

3.3.1. Data Acquisition

The fatigue test uses established test protocols, tools, and processes recommended by the ICAO and the IATA, combining the KSS, SP, and CPT into an Android application to assess subjective and objective levels of pilot fatigue during actual flights. To ensure data monitoring and quality, the pilots were provided with tablet devices with the specified apps installed before data collection. The pilots were also trained on the significance and purpose of data collection, fatigue test methods, test procedures, and precautions to familiarize themselves with the test tasks. The test was conducted voluntarily by the pilots in their spare time, and flight safety was prioritized.

Thus, the test was implemented in a real operating environment, the test method was mature, the test process was in line with international standards, there was sufficient mobilization before the test, and safeguards were in place during the test. The data collected were quality checked after the test, and any data that showed abnormal responses or error rates were removed to improve data quality.

3.3.2. Simulation Results of the Weekly Cumulative Fatigue

Numerous approaches have been used globally to measure pilots’ cumulative fatigue levels over a given amount of time. In its Advisory Circular on FRMS requirements, Canada’s civil aviation standards described a method for simulating potential fatigue dangers in the work schedule called the “Fatigue Likelihood Scoring Matrix” (for further information, see (c) of AC 700-046) [30]. As indicated in

Table 6. Fatigue likelihood scoring matrix.

Fatigue Factor	Risk Score
	0	1	2	4	8
Total duty hours in 7 days	≤36	36.1–43.9	44–47.9	48–54.9	≥55
Maximum duration of a single duty period (h)	≤8	8.1–9.9	10–11.9	12–13.9	≥14
Minimum duration of a short break (h)	≤16	15.9–3	12.9–10	9.9–8	<8
Total hours of night work in 7 days (21:00–09:00)	0	0.1–8	8.1–16	16.1–24	>24
Frequency of long breaks (two-night sleeps with a non-working day)	>1 in 7 days	≤1 in 7 days	≤1 in 14 days	≤1 in 21 days	≤1 in 28 days

, each row of the matrix denotes a different state for the five schedule-based fatigue factors, each of which was assigned a risk score. The total score corresponding to each fatigue element yields the overall risk score. Using this procedure, a pilot’s “fatigue likelihood score” within a week’s time unit can be quantitatively calculated.

Using the schedule of nine pilots shown in Figure 5 as the study subjects, the fatigue risk for the initial week (days 1–7) and the final week (days 22–28 for Pilot A1, days 22–28 for Pilot B1, and days 20–26 for Pilot C1) was evaluated using the scoring matrix in Table 6. Consequently, Pilot A1 scored 3 for the initial week and 5 for the final week, Pilot B1 obtained 11 and 16, and Pilot C1 obtained 20 and 14 for fatigue likelihood, respectively.

Next, the cumulative fatigue scores for these 18 time intervals were quantified using the model from this study and compared with the outcomes of the two approaches. Interestingly, there was perfect consistency between the ranks of the 18 scores determined by the two approaches. Figure 6 illustrates the specific outcomes.

3.3.3. Simulation Results of the Monthly Cumulative Fatigue

The fatigue curve of nine pilots during a one-month flight mission was simulated. Next, using Equation (6), the average monthly cumulative fatigue scores were calculated for class A, B, and C pilots as 7133.48, 13,107.0, and 12,285.44, respectively. Based on the current fatigue coefficient algorithm [7], the airline had fatigue coefficients of 0.2, 0.4, and 0.24 for the respective fleets of the three pilots during the same period. The specific monthly cumulative fatigue and fatigue coefficients are shown in Figure 7.

The current calculation logic for the pilot fatigue factor used by the CAAC is the ratio of the average flight time of the pilots’ to the critical fatigue flight time, as follows:

$$f=ft/(num\times 64).$$

(8)

Here, f represents the fatigue coefficient, ft is the total monthly flying time of the airline, and num represents the number of available airline crews. The model sets the standard for the critical flight time at 64 h. This model is typically used to calculate the overall fatigue coefficient for pilots in an airline or fleet. If the average monthly flying time for a pilot exceeds 64 h, then the fatigue coefficient is greater than one. In practice, when num is 1, it can also be used to calculate the fatigue coefficient of an individual.

Owing to the significant difference in magnitude between the calculated results of the monthly cumulative fatigue model and those of the CCAC’s method, an improved method for calculating the monthly fatigue coefficient of pilots was proposed on the basis of the existing BFM for ease of presentation. This method first simulates the cumulative fatigue score of each flight crew member during the flight duty period on the basis of BFMs. The cumulative fatigue scores for each flight duty period within a month are subsequently added to obtain the monthly cumulative fatigue scores. Finally, by comparing the calculated fatigue score with the critical fatigue state in the original fatigue coefficient formula, a hypothetical cumulative fatigue score for reaching the critical state is assumed. The monthly cumulative fatigue score is calculated using Equation (6). This score is then compared with the cumulative fatigue score at the critical state of monthly cumulative fatigue (FC_Criticality) to obtain the pilot’s monthly fatigue coefficient, f, which is calculated via Equation (7). The monthly pilot fatigue coefficients of class A, class B, and class C, which are calculated using this improved method, are also shown in Figure 7.

$$f=FC/{FC}_{Criticality}.$$

(9)

The timetable for assuming monthly fatigue reaching the critical state is referred to in CCAR121 [7]. As stated in provision 487(c) of CCAR121, air operators are prohibited from assigning flight duty periods to a flight crew member that exceed 210 h in any calendar month. Provision 495(b) dictates that if a flight crew member’s duty period reaches four consecutive working days, they should not be scheduled to execute any flight duties on the fifth calendar day. This study provides a duty time schedule based on the maximum monthly flight duty period (210 h) and maximum number of consecutive workdays (four days) per the given provisions, which determines when the monthly cumulative weariness reaches the critical stage. The specific duty time schedule is depicted in Figure 8, and the coefficient α, which represents the impact of external factors on actual alertness resource consumption or the recovery rate, can be observed in Table 6.

The calculation reveals that the “average flight time/64” of the three pilots is greater than the “fatigue coefficient of their respective fleets”. One explanation could be that although the airline provided support for this study, the collection of biological data was also a part of the airline’s proactive activity to identify tiredness risk. Consequently, when selecting test routes and pilots, those who are more fatigued are often chosen. As a result, the chosen pilots tend to have fatigue levels higher than the overall average. Considering the presence of this potential impact, it would be preferable to compare the “fatigue coefficient in this study” with the “average flight time/64”.

4. Discussion

4.1. Principal Findings

This study compared the constructed model with the fatigue quantification methods of the Civil Aviation Administration of Canada and the CAAC. In the evaluation process of cumulative fatigue using the Canadian method, highly consistent results have demonstrated the generalizability of the BFM-based cumulative fatigue quantification model in flight fatigue assessment. In the evaluation process of cumulative fatigue using China’s method, which uses the BFM-based fatigue coefficient, the following results were obtained: for pilots flying short-haul flights, the fatigue coefficient from the proposed model is close to the fatigue coefficient obtained from “average flight duration/64” (0.41 < 0.45); however, the fatigue coefficient from the proposed model is significantly greater (0.76 > 0.5, 0.71 > 0.34) for pilots flying long-haul flights. These findings suggest that the BFM-based pilot fatigue index can better differentiate the fatigue characteristics of different types of tasks. Long-haul flights often have longer duty periods than short-haul flights and are more likely to be performed at night or during the body’s circadian period. Caldwell et al. [31] suggested that the circadian trough usually occurs between 2:00 and 6:00, a period when human alertness is reduced and reaction times are slower. Rodrigues et al. [32] suggested that the number of night shifts and work during circadian low points are potential root causes of fatigue. These perspectives and theories can help explain why pilots flying long-haul flights have higher fatigue coefficients. In addition, the fatigue measurement research conducted by Sammito et al. [33] revealed that pilots who regularly perform long-haul flights tend to become increasingly fatigued. Therefore, although the current fatigue coefficient algorithm is not sensitive to the additional fatigue burden created by long-haul flights, the results of the proposed model may accurately reflect the higher level of cumulative fatigue experienced by pilots on long-haul flights.

The results demonstrated that the monthly cumulative fatigue level of B-class pilots, who flew long-term intercontinental overnight flights, was similar to that of C-class pilots, who flew intercontinental non-overnight flights, with B-class pilots scoring slightly higher (0.76 > 0.71). This suggests that, based on the fatigue coefficient derived from the BFMs used in this study, it can be concluded that pilots under both types of operations have similar overall cumulative fatigue conditions. This finding coincides with a fatigue observation study based on sleep monitoring and CPTs according to Li et al. [34], which reported no significant differences in sleep efficiency or sustained attention among crew members under either operating condition. However, the “average flight duration/64” of C-class pilots is far less than that of B-class pilots (0.34 < 0.54), indicating that Pilot C accumulates higher levels of fatigue in shorter flight times. When the simulation effects of the BFMs proposed in this research were validated, the actual fatigue data also revealed a significant difference in the fatigue characterization data between overnight and non-overnight operations during the return stage, especially when the proportion of high-fatigue data under non-overnight operation was relatively high. These findings suggest that although the pilots’ monthly cumulative fatigue rates under both types of operations were comparable, non-overnight pilots may have experienced larger fatigue peaks during a single round-trip operation. This could be because stopovers and rotational rests during non-night operations limit the amount of time and the quality of rest that can be taken onboard. According to Valdez et al. [35], weariness tends to worsen as wakefulness increases. It is possible that pilots who did not perform nocturnal operations remained awake for an extended period on the way back. Pilots have more opportunities to sleep on board because of the increasing crew size and non-overnight operation duties. Nevertheless, there is no guarantee that pilots will take advantage of these opportunities for slumber; consequently, there is no guarantee that they will promptly regain their attentiveness. Additionally, non-overnight flights are more likely to have higher levels of weariness toward the end of flight time, according to Li et al. [34]. Airlines can prioritize pilots with higher onboard sleep rates for longer flights or factors affecting onboard sleep rates when evaluating pilot performance. By encouraging pilots to actively seize rotation rest opportunities, these actions may lower the risk of fatigue.

In the statistical analysis of the actual observed fatigue characterization data used to verify the simulation effect of the BFMs, it was also found that there were no significant differences in the fatigue characterization data of the pilots in different operating areas. Regardless of whether an overnight or non-overnight operation is used, the performance of the pilot fatigue characterization data is very similar for outbound flights, which may reflect the consistency in training and preparation among pilots conducting intercontinental flights under various operating types.

This study introduced a dynamic feedback mechanism for the ongoing interaction between endogenous and exogenous fatigue factors in BFMs. The simulation results are strongly correlated with the actual observed subjective and objective fatigue, confirming the feasibility and applicability of this modeling method in the field of aviation. The correlation of the model fatigue simulation value with subjective fatigue is relatively strong, and the difference from the correlation with objective fatigue is large, which indirectly indicates that subjective fatigue and objective fatigue are strongly related but independent of each other, which is consistent with some research conclusions [36,37]. Wilson et al. [12] reported that expanding the factor parameters of BFMs was important. A potential advantage of the proposed BFMs is that they can conveniently expand the module of the exogenous factors, replacing the impact coefficient α of the external influencing factors on the actual alertness resource consumption or recovery rate in this study. The expanded module can influence alertness and fatigue performance by affecting alertness resource consumption and recovery rates. This can help improve the accuracy of the predictions.

4.2. Limitations

In civil aviation, it is important to construct reliable fatigue prediction models for pilot fatigue management. Under the parameter settings of the BFMs in this study, a certain degree of deficiency exists between the model predictions and the actual situation. “The causes of fatigue are heterogeneous” is indeed an important reason, but there are also shortcomings in the model settings: (1) In the parameter settings of exogenous factors, this study does not describe workload and rest quality as a function equation of multiple factor inputs; however, it simplistically categorizes them into three types and sets up corresponding coefficients on the basis of the research objects. In reality, the exogenous factor module can be broadened to include more exogenous factor sets as a function of multiple variables; this will be an important direction for future work. (2) The model neglects the influence of individual differences such as age, gender, and work experience. The parameters that could be customized for the kinetic mechanism of the model include the alertness resource capacity and the phase of circadian rhythm processes [38]. Theoretically, customized parameters should yield a consumption or recovery rate of endogenous alertness resources that are more suited to specific groups or individuals. The cumulative fatigue model of this study only personalized the circadian rhythm phase parameters on the basis of individual differences in sleep type among pilots. (3) Current BFMs are highly valued in categorizing the relief and intensification of fatigue on the basis of sleep-wake behavior [39], whereas this study assumes that alertness resources are consumed during duty periods and recovered during non-duty periods. Evidently, the method based on sleep-wake behavior is apparently closer to actual physiological processes. Thus, this model could be supplemented with a reliable sleep estimation module, which might help enhance the accuracy of fatigue simulations.

Despite the aforementioned shortcomings, this does not hinder its effectiveness as a fatigue model tool for identifying potential fatigue risks. In fatigue risk management, no single fatigue measure can serve as the gold standard, as Dawson et al. [40] noted that BFMs cannot be the sole decision-making basis. Wilson et al. [12] also reported that clarifying the limitations of BFMs does not prevent their continued use; rather, it increases users’ and researchers’ certainty about their real-world effectiveness. Pilkington-Cheney et al. [41] proposed a BFM application guide in which users must understand the background, assumptions, and limitations of the model and understand the output, as well as the relationship between the output and fatigue. Undeniably, in the construction of FRMSs in the civil aviation industry, a series of BFMs have played a role in preliminarily identifying scheduling-related fatigue risks. This is a widely recognized analytical method, and the continuous improvement of the pilot cumulative fatigue model based on the BFM will contribute to achieving more precise and personalized fatigue management.

5. Conclusions

5.1. Summary of Existing Research

The pilot cumulative fatigue model based on BFMs can more comprehensively and accurately reflect the accumulated fatigue status of pilots, helping to distinguish the fatigue differences between different types of tasks. This study also represents a new attempt to improve the fatigue coefficient calculation method for CCAC’s approach. The research expands the application scenarios of BFMs and demonstrates good feasibility and practicality, with certain application prospects in pilot fatigue monitoring and management.

Based on systems dynamics theory, this study introduces a BFM that incorporates a dynamic mechanism for fatigue information feedback. Validation results demonstrate the model’s ability to objectively simulate trends in pilots’ fatigue levels during duty periods, affirming its feasibility as a fatigue modeling approach. This model provides airlines with a scientific tool for comprehensive crew fatigue assessment and offers substantial support for fatigue risk management.

5.2. Research Prospects

Promising directions for future research include factor expansion and sleep estimation. For exogenous factors, additional parameters will be integrated into the BFM, enabling it to function as a multi-factor input equation or module adaptable to diverse scenarios. The model’s dynamic properties will also be a focus, with personalized parameter settings introduced within the endogenous mechanism to account for individual differences, such as age, gender, and work experience. Expanding these factors could facilitate more personalized and precise predictions, marking a key direction for future research.

A reliable sleep estimation module represents another valuable area of investigation. With the advancement of sleep-monitoring technologies based on mobile phones and wearable devices, collecting real-world sleep data has become feasible. In addition to improving the accuracy of sleep estimation, physiological data gathered from these devices can help determine individual circadian rhythms. The interaction between BFMs and these devices holds significant potential for enhancing fatigue prediction accuracy, an area poised for further exploration.