2.3.1 Fuzzy set of membership function

This section explains the class of objects where elements have certain degrees of membership. The extension of a classical or numerical set is termed a ‘fuzzy set’. The universe of real numbers discourse is the most common basis for the membership functions used for fuzzy sets, as shown in Equation (1):

(1)\begin{equation}A\textrm{ = }\{{x,\,{\mu_A}(x )\textrm{|}x \in X} \}\end{equation}

where A is a fuzzy set in the X universe of discourse, x is the fuzzy element and ${\mu _A}(x )$ is the membership function (MF) of x in A. The university of discourse is defined as the complete range of objects, the so-called domain. In fuzzy logic, a fuzzy set consists of a university of discourse (domain) and a membership function which identifies the degree to which a particular element of the domain belongs to the set or not. The fuzzy set is a pair (A, ${\mu _A}$), where A is a set and ${\mu _A}\textrm{:}A \to [{0,1} ]$ is a membership function. For each $x \in A$, the value ${\mu _A}(x )$ is called the ‘degree of membership’ of x in (A, ${\mu _A}$). For example, let (A, ${\mu _A}$) be the fuzzy set of the close objects, and then an object x with ${\mu _A}(x )$ = 1 is in (A, ${\mu _A}$) (absolutely close); x with ${\mu _A}(x )$ = 0 is not in (A, ${\mu _A}$) (absolutely not close), x with 0 < ${\mu _A}(x )$ < 1 is partially in (A, ${\mu _A}$) (somewhat close).

The three most common membership functions are triangular, trapezoidal and Gaussian, as shown in fuzzy set B, C and D, respectively. The former two membership functions are simpler as they are shaped in straight lines forming triangular and trapezoidal forms for fuzzy sets B and C, as shown in Equations (2) and (3), respectively:

(2)\begin{equation}B({x;\alpha ,\beta ,\gamma } )= \left\{ {\begin{array}{@{}ll} 0& {\textrm{for}\,x\mathrm{\ < }\alpha }\\ {({x - \alpha } )\textrm{/}({\beta - \alpha } ) }& {\textrm{for}\,\alpha \le x \le \beta }\\ {({\gamma - x} )\textrm{/}({\gamma - \beta } ) }& {\textrm{for}\,\beta \le x \le \gamma }\\ 0& { \textrm{for}\,x\mathrm{\ > }\gamma } \end{array}} \right.\end{equation}

where parameters $\alpha ,\beta ,\gamma$ denote the lower limit, mode and upper limit of support, respectively, that determine the size and position of the triangular membership function.

(3)\begin{equation}C({x;\alpha ,\beta ,\gamma ,\delta } )\textrm{ = }\left\{ {\begin{array}{@{}ll} {0 }& {\textrm{for}\,x\mathrm{\ < }\alpha }\\ {({x - \alpha } )\textrm{/}({\beta - \alpha } )}& {\textrm{for}\,\alpha \le x \le \beta }\\ 1& {\textrm{for}\,\beta \le x \le \gamma }\\ {({\delta - x} )\textrm{/}({\delta - \gamma } ) }& {\textrm{for}\,\gamma \le x \le \delta }\\ {0 }& {\textrm{for}\,x\mathrm{\ > }\delta } \end{array}} \right.\end{equation}

The Gaussian membership function as shown in fuzzy set D in Equation (4) is built on the Gaussian distribution curve. This membership function is popular due to its smoothness and conciseness in the notation:

(4)\begin{equation}D({x;\beta ,\gamma } )\textrm{ = }\left\{ {\begin{array}{@{}ll} {G\left( {x;\gamma - \beta ,\gamma - \dfrac{\beta }{2},\gamma } \right)}& {\textrm{for}\,x \le \gamma }\\ {1 - G\left( {x;\gamma ,\gamma + \dfrac{\beta }{2},\gamma + \beta } \right)}& {\textrm{for}\,x\mathrm{\ > }\gamma } \end{array}} \right.\end{equation}

2.3.2 Rule generation and evaluation

Rule generation and evaluation of fuzzy inference covers the process of implementing certain rules to the fuzzy input and subsequently evaluating each rule across the inputs obtained from the process of fuzzification. To control the output variable, an IF-THEN rule is constructed based with one or more conditions and a conclusion. The relationships are formed by fuzzy expressions associated with its logical operators (Wang and Kerre, Reference Wang and Kerre2001).

Two logical operators typically used for a fuzzy relation are intersection (AND) and union (U). The intersection of n fuzzy sets is represented in Equation (5) with the operator of Min indicating the lowest membership value within the expression and is given as the membership value of the expression. The union of n fuzzy sets is computed as shown in Equation (6):

(5)\begin{gather}membership[{({{x_i}\,\textrm{is}\,A_1^i} )AND \ldots AND({{x_n}\,\textrm{is}\,A_n^i} )} ]\textrm{ = }Min({{\mu_{A_1^i}}({{x_1}} ), \ldots ,{\mu_{A_n^i}}({{x_n}} )} )\end{gather}

(6)\begin{gather}membership[{({{x_i}\,is\,A_1^i} )\,OR \ldots OR({{x_n}\,is\,A_n^i} )} ]\textrm{ = }Max({{\mu_{A_1^i}}({{x_1}} ), \ldots ,{\mu_{A_n^i}}({{x_n}} )} )\end{gather}

2.3.3 Defuzzification

Defuzzification is the reverse process of fuzzification that provides the result in a scalar value. In the present study, a centre of gravity or centroid method is applied due to its extensive use (Lin et al. Reference Lin, Lai and Yeh2007) and is calculated as shown in Equation (7):

(7)\begin{equation}y\textrm{ = }\frac{{\mathop \sum \nolimits_i [{\mu ({{y_i}} ){y_i}} ]}}{{\mathop \sum \nolimits_i \mu ({{y_i}} )}}\end{equation}

where i shows each linguistic term of the output variable CDR performance, ${y_i}$ is the value of each linguistic term i, and $\mu ({{y_i}} )$ is the aggregated output membership function.

3.2.1 The input, output linguistic variables and membership functions

In the fuzzy logic model, parameters are fuzzified and their value ranges were identified and determined, respectively. The parameters were transformed into fuzzy sets and their memberships were defined, as detailed in Table 2. There are some justifications for the selection of the membership function for each parameter. For the continuous parameters, such as situation awareness, experience and workload, Gaussian membership was used to better reflect the distribution of the set. For the discrete parameters, such as tools (VSD, CRA, notification mode), task, shift and sector, triangular or trapezoidal membership was used due to its nature in explaining categorical levels. Regarding trust, in particular, the triangular membership was used because Mirchi et al. (Reference Mirchi, Vu, Miles, Sturre, Curtis and Strybel2015) found that the median of trust level in ATC context was 4⋅14. Knowing a specified median number, it is easier to categorise trust to three levels of low, medium and high. This is because trust is a state at cognitive level and it is challenging to see its distribution. The fuzzy sets were generated to represent the factors in modelling CDR performance that were adopted from and categorised into the SHELL model, as follows.

Table 2.

The linguistic variables and the respective membership functions

3.2.1.1 Liveware-Liveware (LL)

Liveware is the central component in the SHELL model reflecting the personal attributes of the human element. In the context of CDR, three factors were identified for LL interface: situation awareness (SA), experience and trust. Situation awareness is an important factor in maintaining and updating a mental picture of the air traffic situation during CDR tasks (Pekela and Hilburn, Reference Pekela and Hilburn1997; Chiappe et al., Reference Chiappe, Vu and Strybel2012; Edwards et al., Reference Edwards, Homola, Mercer and Claudatos2017; Trapsilawati et al., Reference Trapsilawati, Herliansyah, Nugraheni, Fatikasari and Tissamodie2020). Experience was perceived as the factor measuring the level of robustness of human actions and had been researched in air traffic control tasks (Kang and Landry, Reference Kang and Landry2014; Klomp et al., Reference Klomp, Borst, van Paassen and Mulder2015; Kearney et al., Reference Kearney, Li and Lin2016; Sanda, Reference Sanda2018; Wang et al., Reference Wang, Wang, Lin, Cong, Xue and Ochieng2021).

Situation awareness was fuzzified using a Gaussian membership function for the corresponding fuzzy sets consisting of three levels (i.e. low, medium, high) within a range of 0–100, as shown in Figure 2. This value indicates the SA measure of response accuracy. The range of experience was adopted from Wang et al. (Reference Wang, Wang, Lin, Cong, Xue and Ochieng2021) where the levels of ATC working experience consist of level 2 to 5. The levels were further categorised into three categories: novice, skilled (level 2 and 3) and expert (level 4 and 5). For the novice, skilled and expert levels, the mean and the standard deviation were 0 and 1⋅5; 4⋅6 and 1⋅5; 11⋅5 and 1⋅5 years, respectively.

Figure 2.

The membership function of situation awareness

The trust level towards automation in ATC significantly influences the use of ATC automation provided in the workstation (Mirchi et al., Reference Mirchi, Vu, Miles, Sturre, Curtis and Strybel2015). The factor was fuzzified using a triangular membership function with the range of 0–5. For the trust level, the median of 4⋅14 was used to represent the medium trust as adopted from Mirchi et al. (Reference Mirchi, Vu, Miles, Sturre, Curtis and Strybel2015).

3.2.2 The fuzzy rules construction and defuzzification

The fuzzy rules were constructed to represent the IF-THEN relationships of input and output variables according to the literature. A notable point in constructing the rules was the high number of input variables (i.e. 15 variables) for predicting the CDR performance. Therefore, the modelling was conducted by creating five fuzzy subsets according to each category in the SHELL model, including: L-L, L-S, L-H, L-E, and L-O. The rules for each fuzzy subset were generated using factorial analysis of input variables in each category in the SHELL model, yielding 89 rules in total. The rules were developed from the combination of the respective membership functions of each variable in the fuzzy subsets. In this research, the CDR performance for each rule was defined according to the effects of each input parameter in affecting the CDR performance in the prior literature. For instance, SA is correlated with CDR performance; although an operator is skilled, when he has a medium SA, he will show medium performance if he has low trust in ATC supporting tools. The low trust will lead to disuse of the tools and thus prevent the operator from increasing his SA, which can be increased by the support of ATC tools.

To reflect the CDR process affected by several parameters, the fuzzy rules with multiple antecedents were used, particularly using the following intersection function:

Rule: IF $({{x_i} is A_1^i} ) AND \ldots AND ({{x_n} is A_n^i} )$ THEN Y is B,

where $A_1^i$, $A_2^i$, … $A_n^i$ and B are the fuzzy linguistics terms reflecting the input linguistic variables ${x_{1, }}{x_{2, }}$ … ${x_{n, }}$ and the output linguistic variable of Y, respectively. In this study, the terms were represented by a triangular, trapezoidal or Gaussian fuzzy number.

The number of fuzzy rules for each fuzzy subsystem is detailed in Table 3. The rules for all subsets are given as follows:

Table 3.

The number of fuzzy rules for each SHELL interface

Rules for L-L

1. 1. If (SA is low) AND (experience is novice) AND (trust is low) THEN (CDR performance is poor);

2. 5. If (SA is medium) AND (experience is skilled) AND (trust is low) THEN (CDR performance is average);

3. 20. If (SA is high) AND (experience is skilled AND (trust is medium) THEN (CDR performance is good);

   …

4. 27. If (SA is high) AND (experience is expert) AND (trust is high) THEN (CDR performance is good)

Rules for L-S

1. 1. If (SD is auditory) AND (notification is shadow mask) THEN (CDR performance is poor);

2. 5. If (SD is auditory) AND (notification is box-animation) THEN (CDR performance is good);

3. 6. If (SD is semantic) AND (notification is shadow mask) THEN (CDR performance is medium);

   …

4. 10. If (SD is semantic) AND (notification is box-animation) THEN (CDR performance is good);

Rules for L-H

1. 1. If (VSD is no) AND (CRA is no) THEN (CDR performance is poor);

2. 4. If (VSD is yes) AND (CRA is no) THEN (CDR performance is medium);

3. 7. If (VSD is no) AND (CRA is no) AND (MCD is yes) THEN (CDR performance is medium);

   …

4. 12. If (VSD is yes) AND (CRA is yes) AND (MCD is yes) THEN (CDR performance is good);

Rules for L-E

1. 1. If (CG is converging) AND (TD is high) AND (Region is irrelevant) AND (MD is Low) THEN (CDR performance is poor);

2. 8. If (CG is converging) AND (TD is low) AND (Region is relevant) AND (MD is High) THEN (CDR performance is good);

3. 9. If (CG is crossing) AND (TD is high) AND (Region is irrelevant) AND (MD is Low) THEN (CDR performance is poor);

4. 16. If (CG is crossing) AND (TD is low) AND (Region is relevant) AND (MD is High) THEN (CDR performance is good);

5. 17. If (CG is overtaking) AND (TD is high) AND (Region is irrelevant) AND (MD is Low) THEN (CDR performance is medium);

   …

6. 24. If (CG is overtaking) AND (TD is low) AND (Region is relevant) AND (MD is High) THEN (CDR performance is good)

Rules for L-O

1. 1. If (WL is high) AND (task is CD) AND (shift is N) THEN (CDR performance is poor);

2. 8. If (WL is high) AND (task is CD + routine) AND (shift is M) THEN (CDR performance is medium);

3. 9. If (WL is low) AND (task is CD) AND (shift is N) THEN (CDR performance is med);

   …

4. 16. If (WL is low) AND (task is CD + routine) AND (shift is M) THEN (CDR performance is good)

The execution of all fuzzy rules is done in parallel during the inference process where the input indicates the degree of all corresponding parameters in each rule. The instance of graphical representation of the rules for the L-L interface is shown in Figure 3. Consequently, the degree of values as the result of the rule mapping is the same or lower than the degree of its variables. To obtain the result of each rule, the fuzzy number is truncated according to the minimum degree of its variables. This study adopts the intersection or so-called minimum operation of the trimmed fuzzy numbers for all fuzzy rules to establish the output of inference in the fuzzy number that is further aggregated to represent the contribution of each input variable given the set of fuzzy rules (Lin et al., Reference Lin, Lai and Yeh2007). Next, a defuzzification method is implemented to translate the inference result into the original crisp values.

Figure 3.

Graphical representation of the rules for the L-L interface