ICAO Risk Tolerability Solution via Complex Indicators of Air Traffic Control Students’ Attitude to Risk

Abstract

The solution of the ICAO risk tolerability is proposed via complex indicators of air traffic control students’ attitude to risk. Physically tangible rates and characteristics are used to determine air traffic control students’ attitude to risk levels during flight separation minima violation. The following features of human factors expression are taken as corresponding indicators: main decision-making dominants, aspiration levels, and parameters of the fuzzy risk estimates. The final solution is received with the help of a multiplicative approach. Indicators developed in the paper are proposed to be received with special survey procedure and further results processing and normalization. The explained method is applicable for both acting air traffic controllers and students of the corresponding educational majors.

1. Introduction

Professional activity of “frontline” air operators (flight crew and air traffic controllers) can be considered a continuous decision-making chain in risk circumstances. This activity is part of the human factor, which is the main reason for air accidents for the last decades according to the statistics [1, 2]. Detrimental impact of the risk perception on flight safety is relevant for civil aviation. This is especially urgent for a complex system “flight crew—aircraft—environment—air traffic control authority” [3, 4, 5].

Results of researches dedicated to the development and operation of air transport management (ATM) system show that sufficient flight safety level support is impossible without efficient, proactive risk management activities. In turn, these activities are an integral component of the system and entirely correspond to the International Civil Aviation Organization (ICAO) safety paradigm.

According to the ICAO definition, flight safety is “the state, in which risks associated with aviation activities, related to, or in direct support of the operation of aircraft, are reduced and controlled to an acceptable level.” Thus, it is necessary to take into account risk estimates for the proper support of flight safety. Considering the definitions by Eurocontrol, International Civil Aviation Organization (ICAO), and other sources [6, 7, 8, 9, 10], let us regard risk as a probability of undesirable situation with harmful consequences. Its “severity” part can be determined using various methods, including the qualimetrical ones. They allow forecasting hazardous situations and performing necessary activities by the management and operator of the air transport system. It contributes to accident prevention and risk reduction.

Risk management-related tasks should be resolved. In order to do this, some necessary qualimetrical steps should be carried out. They should include the evaluation of quality-quantity indicators of the control process. This issue is relevant and complex for civil aviation. Indeed, hazards tend to accumulate during air transport system operation. Taking into account numerous objective and subjective factors, this might result in the so-called “factor resonance” phenomenon [11].

2. Risk tolerability

Generalizing worldwide experience of flight safety management, ICAO proposed to estimate civil aviation threats with special risk tolerability distribution [7]. It is composed of two aviation accidents parameters: likelihood and severity. All their possible combinations were considered. ICAO divided obtained results into three groups: Intolerable, tolerable, and acceptable (Figure 1).

Figure 1.

Risk cases distribution.

There are five qualitative levels of the air accident likelihood and severity proposed by ICAO. They are recommended for risk estimation and combined into the safety risk matrix. These levels can be described using the terms of fuzzy mathematics taken as corresponding fuzzy variables $\mathit{T}\left(\mathit{S}\right)$ and $\mathit{T}\left(\mathit{L}\right)$ [12, 13]:

$$\mathit{T}\left(\mathit{S}\right)={\mathit{R}}_{\mathit{C}}+{\mathit{R}}_{\mathit{H}}+{\mathit{R}}_{\mathit{Mj}}+{\mathit{R}}_{\mathit{Mn}}+{\mathit{R}}_{\mathit{N}};$$E1

$$\mathit{T}\left(\mathit{L}\right)={\mathit{R}}_{\mathit{F}}+{\mathit{R}}_{\mathit{O}}+{\mathit{R}}_{\mathit{R}}+{\mathit{R}}_{\mathit{I}}+{\mathit{R}}_{\mathit{EI}};$$E2

where fuzzy variables’ terms are ${\mathit{R}}_{\mathit{C}}$—catastrophic, ${\mathit{R}}_{\mathit{H}}$—hazardous, ${\mathit{R}}_{\mathit{Mj}}$—major, ${\mathit{R}}_{\mathit{Mn}}$—minor, ${\mathit{R}}_{\mathit{N}}$—negligible, ${\mathit{R}}_{\mathit{F}}$—frequent, ${\mathit{R}}_{\mathit{O}}$—occasional, ${\mathit{R}}_{\mathit{R}}$—remote, ${\mathit{R}}_{\mathit{I}}$—improbable, and ${\mathit{R}}_{\mathit{EI}}$—extremely improbable. Risk cases distribution across all possible likelihood and severity combinations is shown in Table 1.

Risk cases indicators	Risk level description
5A, 5B, 5C, 4A, 4B, 3A	Intolerable
5D, 5E, 4C, 4D, 4E, 3B, 3C, 3D, 2A, 2B, 2C, 1A	Tolerable
3E, 2D, 2E, 1B, 1C, 1D, 1E	Acceptable

Table 1.

ICAO risks cases [7].

Using the ICAO flight safety management recommendations, the US Federal Aviation Administration published circular with their own safety risk matrix. Combinations of severity and likelihood explained there have 62.5% of partially or totally acceptable levels [14]. However, they use four levels for both severity and likelihood. Moreover, the “acceptable risk level” is determined as a flexible value, which depends on the pilot’s particular opinion.

Risk estimation proposed by Eurocontrol is partial and concerns severity only [6]. Also, their recommendations delegate calculation of risk distribution combinations to the national authorities. Another risk matrix is proposed by the Korea Advanced Institute of Science (KAIS), Hongneung Campus, Seoul [15]. Some of these examples use four and five risk levels, while ICAO sticks to the three ones mentioned earlier. To keep up with ICAO, it’s definitions are used; and, therefore, 4-rate and 5-rate cases falls out of the analysis scope.

Providing general comments on risk tolerability, ICAO unfortunately gives no exact values. That is why various methods should be used to resolve risk tolerability distribution. Results of this kind can be implemented to enhance ATC learning process, to influence aircraft separation minima changes, to improve rules and instructions, etc.

The priority arrangement method (PRM) is the first one. It applies the normalized significance coefficient for each term of both fuzzy variables. Unfortunately, this led to a significant decrease in the number of generally acceptable cases that is unacceptable from the common-sense point of view [16]. Another method used for the same purpose is Harrington desirability function [17]. The results for all the mentioned approaches are shown in Table 2.

Approach	Risk level (%)
	Intolerable	Tolerable	Acceptable
ICAO proposal	24	44	32
FAA proposal	18.75	18.75	62.5
Harrington coefficients	40	36	24
PRM iteration 1	28	40	32
PRM iteration 2	68	20	12
PRM iteration 3	76	12	12
PRM iteration 4	76	20	4
PRM iteration 5	84	12	4
PRM iteration 6	88	8	4
PRM iteration 10	88	8	4

Table 2.

Risk tolerability distributions.

Another crucial point is that risk tolerability distribution solution should be performed with tangible and clear indicators and parameters. The “frontline” air operators should be primarily familiar with them. Such clarification problem is resolved with application of such ICAO safety concept components as the use of sound SOPs, hazard sources determination, risk factors control, personnel attitude to hazardous actions and conditions, etc. [18]. Considering the “attitude to risky actions or conditions” as the leading inbound marker to the problem, it is regarded as an explanatory link for flight safety within the human factor.

“Frontline” air operators’ professional activity is a continuous chain of decisions generated and implemented in apparent and latent forms. It is also influenced by multiple factors of stochastic and deterministic nature. Thus, it is possible to research the mentioned above attitude through the human factor indicators that influence decision making under risk circumstances:

• Main decision-making dominants;

• Aspiration levels;

• Fuzzy risk estimates.

Typical values of these indicators should be used to resolve risk tolerability distribution. It is worth mentioning that there are no similar studies of risk tolerability distribution resolution for presented rates.

Let us examine these indicators and their roles in more details. Researches performed so far deal with the risk of flight separation minima violation set by ICAO for the horizontal plane as at 2014.

3. Case study conditions

All methods proposed later on were implemented in the case study, which includes survey and data processing. In the performed survey, 132 air traffic controller students of fourth to fifth years of study from National Aviation University (Kyiv, Ukraine) and Kirovohrad Flight Academy (Kropyvnytskyi, Ukraine) were involved. By the time of the survey, all of them had completed at least 1 year of learning with more than 100 hours at ATC simulation facilities. In the survey, 11 flight separation minima were proposed including 8 km (1 minimum), 10 km (4 minima), 12 km (1 minimum), 20 km (4 minima), and 30 km (1 minimum ). All minima were proposed to the students one by one during the survey.

4. Main decision-making dominants

Main decision-making dominants [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] are parameters of human factor influence on decision making. They describe the attitude of “frontline” air operators to risk: whether the operator is inclined, not inclined, or indifferent to risky behavior. They also characterize motivation to achieve success or avoid failure. Dominants are found from utility estimation functions ${\mathit{f}}_{\mathit{UF}}\left(\mathit{L}\right)$ received from the distances between two aircraft within violated separation minimum.

In the simplest cases, the form of the utility function chart can be used to define the main decision-making dominant. However, for more detailed analysis, risk premium (RP) concept is introduced [31]. Risk premium is the difference between expected lottery reward, and it is determined equivalent.

The classical approach uses only one point ${\mathit{L}}_{0.5}$ for dominant determination:

$$\mathit{RP}=\overline{\mathit{L}}-{\mathit{L}}_{0.5}\left\{\begin{array}{l}<0-\mathit{\text{inclined\hspace{0.17em}to\hspace{0.17em}risk}}\\ >0-\mathit{\text{not\hspace{0.17em}inclined\hspace{0.17em}to\hspace{0.17em}risk}}\\ =0-\mathit{\text{indifferent\hspace{0.17em}to\hspace{0.17em}risk}}\end{array}\right.,$$E3

where $\overline{\mathit{L}}-$ is an expected lottery point:

$$\overline{\mathit{L}}=0.5\cdot \left({\mathit{L}}_0+{\mathit{L}}_1\right)=0.5\cdot \left(0+{\mathit{L}}_{\mathit{norm}}\right)=0.5\cdot {\mathit{L}}_{\mathit{norm}}.$$E4

Use of Eq. (4) for dominant determination makes results a bit rough. It can be show by the example, when $\overline{\mathit{L}}={\mathit{L}}_{0.5}$ (Figure 2, blue line). In this case, the respondent demonstrates an indifferent attitude to risk. But an example when this conclusion is wrong can be easily proposed (Figure 2, red line). It is achieved with introducing of two more points in the dominant analysis.

Figure 2.

Rough estimation example leading to wrong conclusion for $\mathit{L}=\mathit{20}\mathit{km}$. Blue line—rough estimate; red line—improved estimate.

Five points are used instead of three to increase accuracy. The analysis of the points can be performed using coordinates proportion method [20]. According to this method, the sum of coordinates $\sum \mathit{y}$, which is equal to $2.5\mathit{L}$, corresponds to the linear utility function of the respondent who is indifferent to risk. Thus, it is enough to compare coordinates of the sum of five points with $2.5\mathit{L}$. Risk-indifferent participants have $\sum \mathit{y}=2.5\mathit{L}$, the risk inclined ones have $\sum \mathit{y}>2.5\mathit{L}$, and the risk non-inclined respondents have $\sum \mathit{y}<2.5\mathit{L}$.

The key distances, taken as the points, are 0 km, distance for ¼ of utility, distance for half of the utility, distance for ¾ of utility, and full separation minimum (${\mathit{L}}_0$, ${\mathit{L}}_{0.25}$, ${\mathit{L}}_{0.5}$, ${\mathit{L}}_{0.75}$, ${\mathit{L}}_{\mathit{norm}}$). Such distances are chosen to support utility lotteries solution. Each distance possesses a particular utility $\mathit{u}\left(\mathit{L}\right)$. Border points obviously have utilities equal to 0 and 1. Intermediate points have utility values equal to 0.25, 0.5, and 0.75, correspondingly:

$$\begin{array}{l}\mathit{u}\left(\mathit{L}=0\right)={\mathit{f}}_{\mathit{UF}}\left(\mathit{L}=0\right)=0;\mathit{u}\left({\mathit{L}}_{0.25}\right)={\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{0.25}\right)=0.25;\\ \mathit{u}\left({\mathit{L}}_{0.5}\right)={\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{0.5}\right)=0.5;\mathit{u}\left({\mathit{L}}_{0.75}\right)={\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{0.75}\right)=0.75;\\ \mathit{u}\left({\mathit{L}}_1={\mathit{L}}_{\mathit{norm}}\right)={\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_1={\mathit{L}}_{\mathit{norm}}\right)=1.\end{array}$$E5

All intermediate distances are found with the help of lotteries. These lotteries are commonly implemented in economic proceedings [32]. However, they were applied for hardware performance as well [19], what makes them applicable for aviation risks assessment. The method of two-level lotteries application in aviation risk evaluation is already explained in details earlier [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].

Lottery method is applied three times to get three lottery equivalents. Here, a lottery equivalent is a result that represents the distance between two aircraft. This distance is such that operator does not care whether to get it with 100% probability or to participate in the lottery. In other words, it is used to find the distance of lottery equivalent ${\mathit{L}}_{0.5}$ with the utility of 0.5. The lottery has 50% of receiving any marginal results. For the lottery of the first level, these results are 0 km and full flight separation minimum.

The first received lottery equivalent is used to find two more lottery equivalents for ${\mathit{L}}_{0.75}$ and ${\mathit{L}}_{0.25}$ (Figure 3). Considering two initial points and three point received from lotteries, it is possible to build the desired utility estimate function.

Figure 3.

Lotteries used to determine utility function points for flight separation minima.

The example of generalized utility estimate function for all participants plotted for $\mathit{L}=20\mathit{km}$ is given in Figure 4.

Figure 4.

Generalized utility estimate function for all participants with the flight separation minimum $\mathit{L}=\mathit{20}\mathit{km}$.

Normalized utility estimate function for all participants concerning all proposed minima is given in Figure 5.

Figure 5.

Normalized utility estimate function for all participants and all flight separation minima.

Figures 4 and 5 show that utility rise in a non-linear way. Utility function data are taken from case study survey. In both graphs, a fundamental understanding of risk for all involved ATC students concerning single $\mathit{L}=20\mathit{km}$ separation minimum (Figure 4) and all mentioned minima taken together (Figure 5) is presented. According to the graph points, it can be stated that, in general, ATC students possesses non-inclined to risk behavior.

5. Aspiration level

Aspiration level is one of the main psychological features and participants’ typical peculiarities, fundamental for personality. It is recommended to be determined during the medical investigation of air accident [33]. Basically, aspiration level is the stable characteristic of an identity, which is used: (a) for defining the complexity level of tasks wanted to be resolved, (b) for the target selection of further actions depending on the previous success/failure, and (c) for determining the desired self-image. Aspiration level demonstrates the correspondence between personal goals and capabilities. Thus, aviation operators with high aspiration level are characterized by high confidence level, persistence, high productivity, and healthy criticism in achievements estimation [34, 35].

Given researches are related to the of human factor expression qualimetry during flight separation minima violation. Considering recommendations of the proceedings [5], hereafter, the aspiration level is defined as a point of distance L^* on the flight separation minimum. The L^* point corresponds to the highest utility increase from the air traffic controller’s point of view. In other words, it corresponds to ATC operator’s highest performance during support of proper flight safety level at given distance between two aircraft. The proceedings [16, 36, 37] allow plotting and analyzing utility chart by a formally unlimited number of points for open decision-making task.

Since the aspiration level L_AL is the relatively stable indicator of personal air traffic controller commitments [16, 38, 39, 40, 41], then L_AL = L if and only if.

$$\left\{\begin{array}{l}\mathit{\Delta }{\mathit{f}}_{\mathit{UF}}\left(\mathit{L}\right)={\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{r}}\right)-{\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{r}-1}\right)>{\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{i}}\right)-{\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{i}-1}\right);\\ \mathit{i}=\overline{2,\left(\mathit{r}-1\right)},\end{array}\right.$$E6

or if

$$\left\{\begin{array}{l}\mathit{\Delta }{\mathit{f}}_{\mathit{UF}}\left(\mathit{L}\right)={\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{r}}\right)-{\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{r}-1}\right)\Rightarrow max;\\ {\mathit{f}}_{\mathit{UF}}\left({\mathit{L}}_{\mathit{r}}\right)>0.\end{array}\right.$$E7

The overall contribution from this utility function includes three more reference points. They are ${\mathit{L}}^{-}$_, which corresponds to maximum utility increase in lower semi plane (−100; 0), ${\mathit{L}}^0$, which corresponds to distance with 0 utility for (−100; 100) scale, and ${\mathit{L}}^{+}$, which corresponds to the maximum utility increase in top semi plane (0; 100).

After data analysis, a series of charts for all 11 separation minima were plotted. The examples of these charts are presented in Figure 6. Each chart here represents a single aspiration indicator distribution for one of four $\mathit{L}=10\mathit{km}$ minima. Each of the presented four plots shows how many participants consider each particular distance between 0 km and separation minimum as delivering maximum utility. In other words, every plot shows aspiration level distribution for all respondents. For all the taken minima, the distance chosen most often is 10 km, which is the separation minimum itself. However, many ATC students choose other distances to provide maximum utility growth.

Figure 6.

The aspiration levels distribution of the respondents for four flight separation minima of the cross-aircraft aircraft $\mathit{L}=\mathit{10}\mathit{km}.$ Distances as at 2014. Red line—under IFR (instrument flight rules) procedure with continuous radar monitoring in the approach area APP (local ATC) (TMA (terminal control area)) using ATC automated system except approach segment; Blue line—at take-off phase (within control zone (CTR (control zone) at altitudes 1700 m and below) when medium aircraft follows heavy; Green line—for lateral separation for the IFR flights under continuous radar monitoring when crossing the level occupied by the same direction traffic in ACC (general ATC) (CTA (control area)) and APP (TMA) at the moment of crossing on conditions that no tracks converging; Purple line—under IFR procedure with continuous radar monitoring when crossing the same direction level occupied by another aircraft in approach area APP (TMA) using ATC automated system at the moment of crossing on conditions that no tracks converging.

Interestingly, all the taken minima have peak point close to the middle of the separation minimum range. In Figure 6, such middle peaks coincide for all $\mathit{L}=10\mathit{km}$ separation minima. The same effect is observed for the group of $\mathit{L}=20\mathit{km}$ separation minima as well.

6. Fuzzy estimates

Main decision-making dominants and aspiration levels do not cover the whole totality of human factors expression during flight separation minima violation. The experience of earlier researches witnesses that the human factor qualimetry can be significantly improved by fuzzy models of risk level estimation [42, 43, 44, 45, 46, 47, 48, 49, 50]. These models implementation conforms to the human mental process property of providing qualitative estimates rather than quantitative.

Considering all mentioned above and applying Miller’s “magic number” [51], the following risk severity scale can be presented as the fuzzy variable $\mathit{T}$:

$$\mathit{T}={\tilde{\mathit{R}}}_{\mathit{C}}+{\tilde{\mathit{R}}}_{\mathit{VB}}+{\tilde{\mathit{R}}}_{\mathit{B}}+{\tilde{\mathit{R}}}_{\mathit{AV}}+{\tilde{\mathit{R}}}_{\mathit{S}}+{\tilde{\mathit{R}}}_{\mathit{VS}}+{\tilde{\mathit{R}}}_{\mathit{D}}.$$E8

where ${\tilde{\mathit{R}}}_{\mathit{C}}$—critical, ${\tilde{\mathit{R}}}_{\mathit{VB}}$—very big, ${\tilde{\mathit{R}}}_{\mathit{B}}$—big, ${\tilde{\mathit{R}}}_{\mathit{AV}}$—average, ${\tilde{\mathit{R}}}_{\mathit{S}}$—small, ${\tilde{\mathit{R}}}_{\mathit{VS}}$—very small, and ${\tilde{\mathit{R}}}_{\mathit{D}}$—disappearing.

Using the proposed scale (Eq. (8)), air traffic control students as respondents expressed their opinions about hazard severity for all distances between two aircraft during flight separation minima violation [45, 52]. Their answers gave data for the fuzzy variable membership function of “risk severity” [53, 54]. After the initial data are collected, they are normalized using the “supportive matrix” method [55]. The final values are used to plot the family of membership functions charts for all terms of “risk severity” fuzzy variable (Figure 7).

Figure 7.

The values of the membership function for “risk severity” fuzzy variable terms: Blue—“Critical,” red—“Very big,” green—“Big,” purple—“Average,” light blue—“Small,” orange—“Very small,” and teal—“Disappearing.”

Starting from the left side, each line represents a separate fuzzy variable term of the membership function value (catastrophic, very big, big, average, small, very small, and negligible) concerning every possible distance between two aircraft.

Every line in Figure 7 shows the integral opinion of cross-aircraft distance categorized as one of the seven severity levels. For example, the distance of 6 km is considered to have a “very big” severity level with the membership value of 1. At the same time, the nature of fuzzy values also possesses the severity of “catastrophic,” “big,” and “average” levels with the correspondent membership values. Such plot allows finding aggregated ATC students’ opinion about the distances belonging to the particular severity levels.

Since one of the main requirements is to be as close as possible to the ICAO terms, the number of given terms should be reduced. It is performed by the removal of the modifier “very” [9, 51, 55]. After all, the seven use terms were reduced to five in the following way:

$$\begin{array}{ccccc}\underset{\Downarrow }{{\tilde{\mathit{R}}}_{\mathit{C}}}& \underset{\Downarrow }{{\tilde{\mathit{R}}}_{\mathit{VB}}\cup {\tilde{\mathit{R}}}_{\mathit{В}}}& \underset{\Downarrow }{{\tilde{\mathit{R}}}_{\mathit{AV}}}& \underset{\Downarrow }{{\tilde{\mathit{R}}}_{\mathit{S}}\cup {\tilde{\mathit{R}}}_{\mathit{VS}}}& \underset{\Downarrow }{{\tilde{\mathit{R}}}_{\mathit{D}}}\\ {\tilde{\mathit{R}}}_{\mathit{C}}& {\tilde{\mathit{R}}}_{\mathit{B}}& {\tilde{\mathit{R}}}_{\mathit{AV}}& {\tilde{\mathit{R}}}_{\mathit{S}}& {\tilde{\mathit{R}}}_{\mathit{D}}\end{array}.$$E9

7. Aggregation

Since three different parameters are used to define the opinions of ATC students about risk, it would be convenient to combine them into one single indicator. Such an indicator should include all three parameters with reasonable proportions. In current research, the widely applicable aggregation function is taken [9]:

$$\mathit{f}={\left(\frac{1}{\mathit{k}}\sum _{\mathit{i}=1}^{\mathit{k}}{\mathit{\alpha }}_{\mathit{i}}\times {\mathit{R}}_{\mathit{i}}^{\mathit{p}}\right)}^{\frac{1}{\mathit{p}}},$$E10

where р is conditional compromise coefficient which is used to define the acceptable compensation rate of small values with big ones, k is number of risk indicators (in current case k = 3), R_i is an indicator, determined by risk level, and α is a weight coefficient. For main decision-making dominant, R_D is used, R_AL is used for aspiration level, and R_F for fuzzy estimates. Since there is no preliminary information about their significance, they are considered to be equally important. Taking into account the same assumption, р → 0 for the “careful” aggregation policy and thus:

$$\mathit{\varphi }=\prod _{\mathit{i}=1}^{\mathit{k}}{\mathit{R}}_{\mathit{i}}.$$E11

The multiplicative approach is clear, applied with ease, and has an extensive application history among technical and humanistic systems research [51, 55, 56, 57, 58, 59]. However, since data should be normalized to the [0, 1] range, it should be changed in the following way:

$$\mathit{\varphi }=\sqrt[\mathit{k}]{{\prod }_{\mathit{i}=1}^{\mathit{k}}{\mathit{R}}_{\mathit{i}}}.$$E12

Thus, for a single flight separation minimum, aggregated estimate takes the following form:

$$\mathit{R}=\sqrt[3]{{\mathit{R}}_{\mathit{D}}\cdot {\mathit{R}}_{\mathit{AL}}\cdot {\mathit{R}}_{\mathit{F}}}=\sqrt[3]{\frac{{\mathit{L}}_{\mathit{D}}}{{\mathit{L}}_{\mathit{norm}}}\cdot \frac{{\mathit{L}}_{\mathit{AL}}}{{\mathit{L}}_{\mathit{norm}}}\cdot \frac{{\mathit{L}}_{\mathit{F}}}{{\mathit{L}}_{\mathit{norm}}}}.$$E13

Here, $\left({\mathit{L}}_{\mathit{D}},{\mathit{L}}_{\mathit{AL}},{\mathit{L}}_{\mathit{F}}\right)$ are generalized and normalized distances found for main decision-making dominant, aspiration level, and fuzzy estimates, correspondingly. The ${\mathit{L}}_{\mathit{norm}}$ distance stands for the separation minimum distance taken for reference.

The last thing to do is to select the proper key points of all three methods. During the detailed analysis, the following rules were reached:

• All 11 flight separation minima should be taken into account;

• Dominants should be used for all risk inclination categories;

• Lottery equivalent in use is 0.75 as it strongly correlates with the aspiration level;

• The aspiration level itself is taken for all minima;

• A fuzzy estimate is considered as the severity level changing from minor to major in the ICAO concept (from average to small in authors’ terms).

These rules allowed to receive separate formulas for each risk level indicator and the general formula for integral calculations. The correspondent results of generalized and aggregated indicators overall calculations are presented in Table 3. Given results show that air traffic controllers, in general, consider distances more than 0.73 of flight separation minima as acceptable.

No	Separation minimum	Particular methods indicators			Integral indicator $R$
		$R_D$	$R_{\mathit{AL}}$	$R_F$
1	L = 8 km	0.74	0.77	0.74	0.75
2	L = 10 km	0.78	0.75	0.66	0.73
3		0.78	0.72	0.66	0.72
4		0.79	0.75	0.70	0.75
5		0.80	0.77	0.69	0.75
Generalize within distance		0.79	0.75	0.68	0.74
6	L = 12 km	0.75	0.72	0.76	0.74
7	L = 20 km	0.70	0.68	0.60	0.66
8		0.65	0.75	0.73	0.71
9		0.72	0.73	0.62	0.69
10		0.70	0.71	0.69	0.70
Generalize within distance		0.69	0.71	0.66	0.69
11	L = 30 km	0.68	0.73	0.71	0.71
Final estimate		0.73	0.74	0.71	0.73

Table 3.

Aggregated indicators for risk level estimation (yellow cells designate final value for a single separation minimum or generalized minima with the same distances).

Table 3 shows the final point, which may be called severity separator. It can be found in the right bottom cell. In the opinion of ATC students, all distances to the left from this point are more likely to be risky, and vice versa, all distances to the right from this point are more likely to be riskless. Such a result can be also considered as an integral reserved value for flight separation minima.

8. Risk tolerability distribution solution

To resolve the ICAO risk tolerability distribution, the following approach was applied. Since there are five levels of severity, four key points are required.

• Concerning main decision-making dominants, three lottery key points were considered as an intermediary between the severity levels. The last fourth point was taken as flight separation minimum distance.

• Concerning aspiration levels, three key utility points were used with the flight separation minimum distance as well.

• Concerning fuzzy estimates, the reduced intersection points were used, as shown in Eq. (9).

The final results with all three presented methods are presented in Table 4. Here, ${\mathit{R}}_{\mathit{C}}$—catastrophic risk level, ${\mathit{R}}_{\mathit{H}}$—hazardous risk level, ${\mathit{R}}_{\mathit{Mj}}$—major risk level, ${\mathit{R}}_{\mathit{Mn}}$—minor risk level, ${\mathit{R}}_{\mathit{N}}$—negligible risk level, ${\mathit{L}}_{\mathit{C}}$—distance equivalent to catastrophic risk level, ${\mathit{L}}_{\mathit{H}}$—distance equivalent to hazardous risk level, ${\mathit{L}}_{\mathit{Mj}}$—distance equivalent to major risk level, ${\mathit{L}}_{\mathit{Mn}}$—distance equivalent to minor risk level, ${\mathit{L}}_{\mathit{N}}$—distance equivalent to negligible risk level, ${\mathit{L}}_{0.25}$—distance equivalent to 0.25 lottery determinant, ${\mathit{L}}_{0.5}$—distance equivalent to 0.5 lottery determinant, ${\mathit{L}}_{0.75}$—distance equivalent to 0.75 lottery determinant, ${\mathit{L}}^{-}$, ${\mathit{L}}^0$_, and ${\mathit{L}}^{+}$ were explained earlier, ${\tilde{\mathit{L}}}_{\mathit{C}}$—distance where “critical” term ends, ${\tilde{\mathit{L}}}_{\mathit{B}}$—distance where “big” term ends, ${\tilde{\mathit{L}}}_{\mathit{AV}}$—distance where “average” term ends, and ${\tilde{\mathit{L}}}_{\mathit{S}}$—distance where “small” term ends.

Risk levels		Models in use
		Dominants	Aspiration levels	Fuzzy estimates
Unacceptable	$R_C$	$\begin{array}{l}{L}_C<L_{0.25}\iff \\ \iff L_C<0.31\end{array}$	$\begin{array}{l}{L}_C<L^{-}\iff \\ \iff L_C<0.46\end{array}$	$\begin{array}{l}0<L_C<{\tilde{L}}_C\iff \\ \iff 0<L_C<0.42\end{array}$
	$R_H$	$\begin{array}{l}{L}_{0.25}<L_H<L_{0.5}\iff \\ \iff 0.31<L_H<0.53\end{array}$	$\begin{array}{l}{L}^{-}<L_H<L^0\iff \\ \iff 0.46<L_H<0.65\end{array}$	$\begin{array}{l}{\tilde{L}}_C<L_H<{\tilde{L}}_B\iff \\ \iff 0.42<L_H<0.56\end{array}$
	$R_{\mathit{Mj}}$	$\begin{array}{l}{L}_{0.5}<L_{\mathit{Mj}}<L_{0.75}\iff \\ \iff 0.53<L_{\mathit{Mj}}<0.73\end{array}$	$\begin{array}{l}{L}^0<L_{\mathit{Mj}}<L^{\ast }\iff \\ \iff 0.65<L_{\mathit{Mj}}<0.74\end{array}$	$\begin{array}{l}{\tilde{L}}_B<L_{\mathit{Mj}}<{\tilde{L}}_{\mathit{AV}}\iff \\ \iff 0.56<L_{\mathit{Mj}}<0.71\end{array}$
Acceptable	$R_{\mathit{Mn}}$	$\begin{array}{l}{L}_{0.75}\le L_{\mathit{Mn}}<L_{\mathit{norm}}\iff \\ \iff 0.73<L_{\mathit{Mn}}<L_{\mathit{norm}}\end{array}$	$\begin{array}{l}{L}^{\ast }<L_{\mathit{Mn}}<L_{\mathit{norm}}\iff \\ \iff 0.74<L_{\mathit{Mn}}<L_{\mathit{norm}}\end{array}$	$\begin{array}{l}{\tilde{L}}_{\mathit{AV}}<L_{\mathit{Mn}}<{\tilde{L}}_S\iff \\ \iff 0.71<L_{\mathit{Mn}}<0.83\end{array}$
	$R_N$	$L_N\ge L_{\mathit{norm}}$	$L_N\ge L_{\mathit{norm}}$	$L_N\ge 0.83$

Table 4.

Partial solutions of ІСАО risk tolerability distribution for flight separation minima.

Finally, the application of a multiplicative approach allows to resolve the ІСАО risk tolerability distribution (Table 5) with integral estimates.

Risk levels		Integral estimates
Unacceptable	Catastrophic	$L_C<0.39$
	Hazardous	$0.39\le L_H<0.58$
	Major	$0.58\le L_{\mathit{Mj}}<0.73$
Acceptable	Minor	$0.73\le L_{\mathit{Mn}}<0.94$
	Negligible	$L_N\ge 0.94$

Table 5.

The integral solution of ICAO risk tolerability distribution with risk estimates.

9. Conclusions

It is possible to make general conclusions based on the presented scientific results. These conclusions concern the development of a new methodology. It is dedicated to the qualimetry of human factor regularities expression during the decision making in aeronautical systems. The ICAO recommendations were taken into account during the correspondent indicators development. They were implemented by the composition of fuzzy models applied to air traffic control students’ attitude to flight separation minima violation in a horizontal plane. Other components of such attitude include well-grounded key points of utility estimate functions for the mentioned minima continuum plotted within formally closed and open decision-making tasks. The first group of points is used to find respondents’ main decision-making dominants (inclination, indifference, and non-inclination to risk). The second group of points is used to find aspiration levels that correctly characterize respondents’ self-image.

Important scientific results include:

1. For the first time, the multiplicative approach is grounded and implemented to determine the integral estimate of air traffic control students’ attitude both to sole flight separation minimum and minima totality. The correspondent cent is equal to 0.73 of flight separation minima.

2. The new method of main decision-making dominant determination is proposed. It differs from the widely known one by more key points being used and a novel algorithm submitted for their analysis.

3. The results of the main decision-making dominants analysis show that non-inclination is a major attitude among air traffic control students. It allows changing the professional education programs, taking into account the received results.

4. Especially important feature of the received results is their proactivity. It will enable preventing potentially harmful consequences of air traffic controllers’ work by implementing personalized training on various simulators.

All the results form strong premises for further researches, which should be performed in the following areas:

1. The study of decision-making indicators, taking into account age, academic performance, and other factors;

2. The analysis of the mentioned indicators dynamics during the whole professional activity period of air traffic control personnel;

3. The complex research of the proposed indicators for three dimensions with space utility functions plot and integral indicators estimation for such conditions.

It should be mentioned that further research areas are not limited to the proposed ones but merely demonstrate opinion on primaries.