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Mathematics 2022, 10, 2923. https://doi.org/10.3390/math10162923 www.mdpi.com/journal/mathematics
Article
A New Similarity Measure of Fuzzy Signatures with a Case
Study Based on the Statistical Evaluation of Questionnaires
Comparing the Influential Factors of Hungarian and
Lithuanian Employee Engagement
László T. Kóczy 1,2,*, Dalia Susniene 3, Ojaras Purvinis 3 and Márta Konczosné Szombathelyi 4
1 Department of Informatics, Széchenyi István University, 9026 Győr, Hungary
2 Department of Telecommnnication and Media Informtics, Budapest University of Technology and
Economics, 1111 Budapest, Hungary
3 School of Economics and Business, Kaunas University of Technology, 51368 Kaunas, Lithuania;
dalia.susniene@ktu.lt (D.S.); opurvi@inbox.lt (O.P.)
4 Department of Marketing and Management, Széchenyi István University, 9026 Győr, Hungary;
kszm@sze.hu
* Correspondence: koczy@tmit.bme.hu
Abstract: Similarity between two fuzzy values, sets, etc., may be defined in various ways. The au
thors here attempt introducing a general similarity measure based on the direct extension of the
Boolean minimal form of equivalence operation. It is further extended to hierarchically structured
multicomponent fuzzy signatures. Two versions of this measure, one based on the classic min–max
operations and one based on the strictly monotonic algebraic norms, are proposed for practical ap
plication. A real example from management science is chosen, namely the comparison of employee
attitudes in two different populations. This example has application possibilities in the evaluation
and analysis of employee behaviour in companies as, due to the complex aspects in analysing mul
tifaceted behavioural paradigms in organizational management, it is difficult for companies to make
reliable decisions in creating processes for better social interactions between employees. In the pa
per, the authors go through the steps of building a model for exploring a set of different features,
where a statistical preprocessing step enables the identification of the interdependency and thus
the setup of the fuzzy signature structure suitable to describe the partially redundant answers given
to a standard questionnaire and the comparison of them with help of the (pair of the) new similarity
measures. As a side result in management science, by using an internationally applied standard
questionnaire for exploring the factors of employee engagement and using a sample of data ob
tained from Hungarian and Lithuanian firms, it was found that responses in Hungary and Lithua
nia were partially different, and the employee attitude was thus in general different although in
some questions an unambiguous similarity could be also discovered.
Keywords: fuzzy signature; similarity measure; correlation analysis; employee engagement; com
parison of two populations of questionnaire replies
MSC: 9010; 90B70
1. Introduction
There are plenty of phenomena in the real world where proper characterisation may
be done only by a large number of partly independent, partly redundant, often hierarchi
cally structured descriptors. Such are all problems where humans are part of the system,
Citation: Kóczy, L.T.; Susniene, D.,
Purvinis, O.;
Konczosné Szombathelyi, M. A New
Similarity Measure of Fuzzy
Signatures with a Case Study Based
on the Statistical Evaluation of
Questionnaires Comparing the
Influential Factors of Hungarian and
Lithuanian Employee Engagement.
Mathematics
2022, 10, 2923. https://
doi.org/10.3390/math10162923
Academic Editors: Michael
Voskoglou, Salvatore Sessa and
Georgios Tsekouras
Received: 5 July 2022
Accepted: 11 August 2022
Published: 14 August 2022
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This article is an open access article
distributed under the terms and con
ditions of the Creative Commons At
tribution (CC BY) license (https://cre
ativecommons.org/licenses/by/4.0/).
Mathematics 2022, 10, 2923 2 of 25
such as in management, social, and other humanrelated engineering and modelling prob
lems. The presence of human components means the presence of uncertainty. However,
uncertainty may be of various natures. The two main types are a statistical and probabil
istic uncertainty on one hand and a nondeterministic, illdefined type of uncertainty on
the other. The former may be modelled by classic probability theory and statistics, while
the latter needs the deployment of fuzzy modelling. In this paper, a combined approach
is presented and illustrated by a detailed case study. First, the statistical analysis helps
evaluate a collection of questionnaires on employee attitudes. This is the starting point of
the construction of the fuzzy signatures representing the complex replies to the questions
in the forms, where the hierarchical interdependence of the individual questions is deter
mined by the correlation coefficients among the answers. By calculating the arithmetic
means of the fuzzy signatures (see in the next section) for both populations’ replies, two
representative fuzzy signatures are obtained. The entirely new approach is the introduc
tion of a measure for the comparison of the similarity (extended logical equivalence) of
two FSigs, based on the fuzzy extension of a wellknown Boolean formula, whose exten
sion offers a multitude of novel formulae expressed by (negation, tnorm, tconorm) tri
plet. In this study, two such triplets are selected, and calculations on the average signa
tures of the two populations based on the two versions of this novel similarity measure
(degree) are presented.
The topic of the case study was chosen, as in our globalized world, businesses face
many challenges, where their goal is to enhance the performance in a sustainable and in
creasingly efficient way. It is becoming increasingly clear to organizations that their main
treasure is their staff. Therefore, an organizational policy should be pursued to align or
ganizational goals with the individual goals of employees in order to achieve common
goals. Thus, employee engagement is a crucial component of the businesses’ efficiency.
As a pair of original collections of employeeattituderelated questionnaires was available
to the authors, the comparison of the employee cultures of the origin countries of the two
collections, Hungary and Lithuania, was performed with help of the new similarity meas
ure.
Using the abovementioned combined statistical and fuzzysignaturebased ap
proach as the primary model, we examined the relationship between workplace features
as influential factors and the strategic role of employee engagement.
The case study is a complex enough problem to show how the two types of uncer
tainty may appear at the same time in the same problem. Here, the model buildup in such
a complex situation combines both statistical and fuzzy approaches in a single entity, as
both types of uncertainty are present. Computational intelligence and soft computing of
fer a very rich toolbox that includes various bioinspired modelling and learning tech
niques resulting in very efficient algorithmic solutions to extremely complex problems
with uncertainty, nondeterministic components, and vagueness. The case study is just an
example for such a complex problem.
The aim of this study is to present a complex computational intelligence approach
based on statistical analysis and fuzzy approaches. However, by the detailed analysis of
the case study, we also attempt to contribute to the knowledge on the relationship be
tween positive (organisational citizenship behaviour, OCB) and negative (counterproduc
tive work behaviour, CWB) employee attitudes. These two main types of factors are influ
encing the employee attitudes in business organizations. A secondary aim is also to point
out similarities and differences of two European nations’ management culture and tradi
tions.
In order to construct the model, first, it is necessary to determine the interdepend
ences and redundancies in the standard questionnaires and generally accepted OCB and
CWB factors. This will be done by analysing the correlations between these factors of cor
porate and employee communication, corporate culture, and management style, thus ob
taining a classification into closer connected subgroups and subsubgroups of the uncer
Mathematics 2022, 10, 2923 3 of 25
tain (vague) features. This way, building up the fuzzy signature (FSig) descriptors is pos
sible. By analysing the FSigs, it becomes possible to determine their respective impacts on
the employees’ engagement and satisfaction with their careers, etc. Thus, the contribution
of this paper is dedicated to presenting a novel comparison methodology based on the
preliminary analysis, but at the same time, the evaluation of the reallife data in the case
study may reveal some new facts for management science experts.
2. Fuzzy Signatures and Similarity
2.1. Fuzzy Signatures and Their Role in Modelling Employee Attitudes
The concept of fuzzy set was introduced by Zadeh [1]; in this definition, any member
x of a universal set X is assigned a membership degree within the unit interval:
]1,0[:,, XXA AA
Some time ago, we were motivated by an industrial project (classification of micro
scopic metallurgical images) to introduce and extension of the above, namely the concept
of vector valued fuzzy sets (VVF) [2]. This led to the simple generalization of Zadeh’s
definition into
k
AA XXA ]1,0[:,,
Here, the k traditional fuzzy sets are interpreted as orthogonal components of a single
kdimensional membership function; i.e., each member x is now assigned k different de
grees from the unit interval, with each of them expressing the degree of satisfying a certain
property or feature.
A further extension of the concept of VVF leads to a hierarchically structured gener
alization of the definition of VVF. This was later motivated by further reallife applications
(architectural and civil engineering problems, packaging, robot communication, etc.). This
new extended concept was called fuzzy signature [3]. Some mathematical properties of
FSigs were also investigated; for these, see, e.g., [4–7]. The definition of the FSig is as fol
lows:
,
...
:);,( 2
1
k
AAfsig
C
C
C
XXA
where
ij
iC
C]1,0[ or, where ij
C(j being the subsubsctript referring to the
corresponding subtree of Ci) is defined recursively, in the same manner.
Of course, fsig
A can be also represented by a rooted tree graph, where the hierarchi
cally nested subvectors are represented by subtrees. In order to calculate simpler mem
bership degrees, it is possible to reduce the subtrees to their respective roots (in reality,
intermediate nodes of the whole tree) and into a single membership degree assigned to
the given (sub)root by evaluating the aggregations assigned to each intermediate node
(including the root). In order to calculate the reduced fuzzy degrees, each intermediate
node of the tree graph (or each membership subvector) is assigned a fuzzy aggregation
operator (a monotonic operator preserving the extremal values 0 and 1), whose execution
combines the membership degrees at the leaves into a single value in [0,1]. This will be
associated with the former root, now becoming a leaf of the reduced tree after the reduc
tion. The complete FSig is defined by its tree graph (or nested vector) structure and the set
of fuzzy aggregations assigned to all nonleaf nodes. This way, executing the aggregations
on the values in the respective leaves, recursively, the whole FSig can be reduced to a
Mathematics 2022, 10, 2923 4 of 25
single node with a single membership grade (in the root of the whole tree) if it is necessary
for further processing, comparison, etc.
The case study discussed in this paper is based on some parts of a questionnaire on
employee attitudes. This questionnaire had been developed by an international university
research consortium for a worldwide crosscultural management research project con
ducted by a University Fellows International Research Consortium [8] during a survey on
communication styles used in the workplace. The questions and replies obtained by two
parallel surveys, one in a group of Hungarian and the other one in some Lithuanian com
panies, have a natural hierarchy and a suggested structure of interconnectedness accord
ing to the relevant literature [5]. This hierarchy may be represented by a rooted tree graph
where the root stands for the replying person, and the leaves of the tree contain the replies.
These are expressed originally by numbers according to the socalled Likert scale that can
be easily converted by a linear transformation into fuzzy membership degrees. The struc
ture is represented by the tree graph and the types of interdependence by the fuzzy oper
ators (aggregations). This complex combination of fuzzy graph and operations corre
sponds with the FSig definition, and thus, the case study can be very wellmodelled and
discussed in the language of FSigs.
It is a fortunate fact that two original and, so far in the major part, unpublished pop
ulations of responses to the questions, one obtained in Hungary and one in Lithuania,
were available. In addition, the necessary expert domain knowledge from the side of man
agement science—represented by the respective coauthors of this article—enabled the
modelling of the selected part represented by the questionnaires. Thus, an interesting and
consistent subset of the descriptors represented by the replies in both countries could be
identified. The idea of model setup based on FSigs incorporating the responses was al
ready published by the authors [9]. To avoid repetition of the argumentation in the cited
paper, here, the methodology of constructing the corresponding FSigs is straightfor
wardly applied. The procedure of modelling is only briefly presented in this paper for
both sets.
At this point, assuming that two similar sets of responses are available, the new chal
lenge emerges, namely whether these two populations of responses, reflecting the atti
tudes of two populations of employees coming from two different European countries,
reflect essentially similar behaviour or whether there is a noticeable difference between
the two. In order to carry out an educated comparison, a novel method is now proposed,
based on a new similarity measure suitable for the comparison of two fuzzy membership
degrees (or even membership functions), offering a straightforward generalisation of this
measure to FSigs. The approach will be introduced in the next subsection.
2.2. A New Similarity Measure of Two Fuzzy Memberships and of Two Fuzzy Signatures
The similarity or equivalence of two fuzzy sets has been discussed in the relevant
literature for several decades. (It often happened in the context of binary relations, cf., e.g.,
[10]). Usually, such definitions are based on the seemingly rather intuitive point of view
that two fuzzy membership degrees are equivalent if they are equal. However, looking at
the deeper semantics of fuzzy membership degrees, in a “philosophical” sense, it may be
also considered from another point of view. As fuzzy membership degrees and member
ship functions express uncertainty, this problem can also be viewed by starting from the
point that “the more uncertain is the belonging of a member to a set, i.e., the farther the
membership degree(s) is(are) from the extreme values 0 and 1, the less this membership
is certain itself”, and two values being equal but both being very uncertain may have a
different real meaning. This approach raises the question regarding how two uncertain,
illdefined values (or whole concepts) may be compared in a way that matches the spirit
of fuzzy sets and systems: the degree of similarity being expressed itself by a degree of
truth. In this paper, a very obvious new definition of similarity will be introduced and
applied, which can be, surprisingly, not found in the related literature.
Mathematics 2022, 10, 2923 5 of 25
Let us start with the consideration that fuzzy connectives (norms, aggregations, etc.)
are always defined as extensions of their Boolean counterparts, with the strict condition
that when substituting the arguments of any fuzzy operation OP by either one of the Bool
ean truth or membership values (the extremal values in the fuzzy truth degree range), the
result must be identical with the result of the Boolean operation OP. As the range of fuzzy
membership degrees is continuous, the number of possible extensions of any Boolean op
erator is infinite, and thus, in the fuzzy algebra, no canonical and minimal forms exist.
Nevertheless, the extension of any Boolean canonical or minimal form directly into its
fuzzy equivalent based on a set of corresponding simple fuzzy operations may be reason
able.
Similarity is a gradual concept, and thus, the measure or degree of similarity of two
objects can be directly expressed by a value from [0, 1], where 0 stands for “no similarity
at all”, and 1 expresses “absolute similarity = identity”.
In the previous literature, fuzzy similarity measures were defined in a way where the
inherent uncertainty expressed by the membership degrees was not taken into account,
and the following axiomatic properties were requested:
A fuzzy similarity measure Sim(A, B) is a mapping Sim:A×B→[0, 1], where Sim is
possessing the following properties [11]:
(S1) 0 ≤ Sim(A, B) ≤ 1 (boundedness by 0 and 1);
(S2) Sim(A, B) = Sim(B, A) (commutativity);
(S3) Sim(A, B) = 1 if A = B (selfidentity);
(S4) Sim(A, −A) = 0 if A is a crisp set (exclusion);
(S5) If A⊆B⊆C, then Sim(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C) (monotonicity in terms
of containment).
This required property set, however, can be debated. While (S1) is obvious, and (S2)
does not restrict the inherent fuzzy nature of the problem we are going to discuss, (S3) has
a “hidden crisp” semantics. It does not take into consideration that uncertain information
may not lead to a certain conclusion. (S4) and (S5), again, do not restrict the inherent un
certainty of the similarity we are going to discuss; thus, this axiom set may be accepted
without (S3).
Hence, by omitting (S3) but keeping the other properties, we propose that the new
fuzzy measure of similarity is defined as the truth degree of equivalence in the logical
sense. In the Boolean algebra, equivalence is expressed by the following minimal form
based on the triplet of disjunction, conjunction, and negation:
𝐴
≡
𝐵
=
(
𝐴
⋀
𝐵
)
⋁
(
⌝
𝐴
⋀
⌝
𝐵
)
(1a)
This expression can be extended in a general form to fuzzy operations:
𝐴
≋
𝐵
=
𝑠
(
𝑡
(
𝐴
,
𝐵
)
,
𝑡
(
−A,−B)) (1b)
where ≋ denotes the fuzzy extension of the logical equivalence operation, s stands for the
snorm (tconorm) operation (extended disjunction), t denotes tnorm (extended conjunc
tion), and—stands for fuzzy negation. The above formula has an infinite number of con
crete implementations depending on what triplet of fuzzy operations is chosen. In this
study, two alternative solutions are proposed. In the original crucial paper of Zadeh [1],
the {min, max, 1 − x} triplet was presented, and in the literature, this combination of oper
ations is rather popular and has been used in many applications. The formula thus be
comes
Sim
𝑍
(
𝐴
,
𝐵
)
=
𝐴
≋
𝐵
=
max{min{
A,B
}
,
min{1
−
A,
1
−
B
}}
(2)
It is less obvious how the other formula is derived. In the original paper of Zadeh, in
a footnote, an alternative definition of the binary fuzzy connectives was proposed under
the name of “interactive operations”. Later, the literature referred to these connectives as
“algebraic” conjunction and disjunction. Their usability has been shown in many applica
tions, and the algebraic structure of these two, with their respective generalisations, were
Mathematics 2022, 10, 2923 6 of 25
discussed in a number of theoretical papers. Let us just mention here that in [2], already,
the algebraic operations were applied, and their natural fitting with verbal statements and
their connections were shown already in [12]. These so called Hamacheroperations [13]
have a very interesting property: they are strictly monotonic wherever the arguments are
different from 0 and 1. This strict monotonicity allows the definition of a novel similarity
measure given below, which clearly expresses the uncertainty of the equivalence (or sim
ilarity) of two uncertain values in a stronger way:
Sim
𝐴𝑙𝑔
(
𝐴
,
𝐵
)
=
𝐴
≋
𝐵
=
A
*
B
+ (1
−
A
)*(1

B
)−
A
*
B
*(1
−
A
)*(1
−
B
)
(3)
as talg(A,B) =
A
*
B
and salg(A,B) = A + B − A*B.
As mentioned above, already very early, several studies were published supporting
the fact that the strictly monotonic norms have an important role in modelling reallife
behaviour, especially where human reasoning is involved (cf. Rödder [12], whose results
were based on real experiments). These observations were soon followed by the definition
of the parametric class of Hamacher operations and t and snorms [13], of which the al
gebraic norm pair is the simplest representative.
It should be mentioned that fuzzy similarity in the sense of (1b) assigns fuzzy mem
bership degrees the measure from (0, 1) unless the arguments A and B are crisp them
selves. Therefore, e.g., Sim𝑍(𝐴,𝐵)= 1,when 𝐵 = 𝐴 is only satisfied if either A = 1, or A =
0. Otherwise, the degree of similarity of A with itself is exactly A. This is an interesting
feature of the new measure that can be explained by the semantic interpretation of a fuzzy
membership degree: the expression of the degree of uncertainty. Namely, it is impossible
to tell with certainty anything about the similarity (or identity) of two uncertain features,
even having the same degree of membership, as they may not be identical or even very
similar in the case when the uncertainty of the two expresses different realities. It is even
more surprising when strictly monotonic norms are used (referring to the open interval
(0, 1) only), such as in (3). Here, if A = B, Sim𝐴𝑙𝑔(𝐴, 𝐵)= 𝐴2 + (1 − 2A + A2) − 𝐴2*(1 − 2A +
A2) SimAlg(A,B) = 0.4375 if A = B. This is an even more characteristic expression of the fact
“nothing certain can be said about uncertain features”.
Based on the above family of similarity measures, it is possible to compare more com
plex fuzzy objects, such as VVF elements, and also fuzzy signatures as defined above.
The aim of this study is to offer some theoretical–methodological contribution (the
new similarity measure) by presenting its application on a reallife case study. This appli
cation will be the analysis of the umbrella term “Employee engagement” as it is under
stood in the management science field. The analysis will be done by testing the proposed
model for comparing two populations of data obtained by questionnaires distributed at
companies in two different countries, Hungary and Lithuania. The model intends to en
compass the three employee engagement elements—trait engagement, state engagement,
and behavioural engagement—and explore their interplay to obtain a holistic view of the
phenomenon. For this purpose, the matching fuzzy signature structure of the available
data will be determined by using statistical evaluation and the basic fuzzy definitions, and
then, the average signatures of the two populations will be compared based on the above
introduced family of similarity measures.
3. Employee Engagement: Introduction to the Application Study
3.1. The Concept of Employee Engagement
The concept of employee commitment and engagement to work and the workplace
has been at the centre of business and research interest since the 1990s [14–19]. Kahn [20]
defined engagement as “the harnessing of organisation members’ selves to their work
roles; in engagement, people employ and express themselves physically, cognitively, and
emotionally during role performances”. The attitude of employees may be positively but
also negatively inclined.
Mathematics 2022, 10, 2923 7 of 25
The positive and negative aspects of engagement are referred to in the literature as
organizational citizenship behaviour (OCB) and counterproductive work behaviour
(CWB). OCB and CWB are extremes, and thus, they should have a strong negative corre
lation.
Konovsky and Organ [21] identified five dimensions of OCB: altruism, courtesy,
sportsmanship, civic virtue, and generalized compliance (conscientiousness). Altruism
means voluntary helping; courtesy includes helping others to prevent interpersonal prob
lems; sportsmanship denotes tolerating inconveniences without unnecessary complain
ing; civic virtue refers to a willingness to participate in organizational affairs. Finally, gen
eralized compliance is a discretionary behaviour going beyond the minimum requirement
level of the organization in areas of regulation and attendance.
The opposite is consciously destructive behaviour (CWB) that harms both the organ
ization and the individuals questioning the organizational goals and values, thereby de
grading employees’ and organizational performance. It has a negative effect on satisfac
tion and organizational culture and creates a bad mood [22].
In order to achieve, maintain, and increase employee engagement, we need to be
aware of the factors that influence engagement, especially OCB. Various researchers
[8,20,23–28] identified factors influencing employee behaviour, including organizational
factors (e.g., organizational culture, organizational communication, leadership style) and
individual characteristics (e.g., age, gender) that positively or negatively influence en
gagement towards the organization.
We argue that in order to understand employee engagement in an organisation,
many more influencing factors need to be identified and explored, and especially, the re
lationships among them need to be examined. In order to do this, the model set up in [8]
was tested (Figure 1).
Figure 1. Conceptual model of interlinked factors.
The application study in this paper aims at contributing to analysing how behav
ioural engagement (both OCB and CWB) correlates with the engagement and perception
of the organization.
3.2. The Methodology of Collecting Data
The method of the research in the application is based on a questionnaire developed
by an international university research consortium for a worldwide crosscultural man
agement research project conducted by a University Fellows International Research Con
sortium [8], as mentioned above, during a survey on communication styles used in the
workplace. The questionnaire was designed to include the issues on organizational cul
ture and trust in top management, job satisfaction, OCB/CWB, gender differences, and
styles of communication. The questionnaire was originally developed based on pilot test
ing. To accommodate the research questions, it consisted of the following eight sections:
1. Communication style;
2. Work experience, type, and size of a company and gender composition;
Mathematics 2022, 10, 2923 8 of 25
3. Personal characteristics;
4. Work environment and top management team;
5. Culture of organization;
6. OCB and CWB;
7. Career satisfaction;
8. Demographic data.
During the survey in Hungary, in the highly developed industrial region of Győr, a
total of 1038 valid responses were received, while the Lithuanian research, restricted to
the region of Panevezys, obtained 144 valid responses. In the survey, we tried to reflect all
age groups, seniority levels, genders, and positions at work as well as different types of
industries. The respondents were asked to respond on a ninepoint Likert scale {1…9}.
During the response analysis, several statistical methods were used such as descrip
tive statistics, correlation analysis, and ANOVA (analysis of variance), in order to estab
lish the structure of the fuzzy signatures.
4. Modelling Employee Engagement by Fuzzy Signatures
4.1. Transforming the Responses into Fuzzy Degrees
In the example presented in the previous section, which will serve as the case study
and a validation example of the proposed approach, answers to the questions in the ques
tionnaire were given using the widely deployed Likert scale, namely a scale from 1 to 9.
It was an obvious idea to linearly transform these values into the closedunit interval (i.e.,
to normalise the scale) in the absence of any argument supporting a nonlinear transfor
mation and thus to obtain fuzzy degrees expressing the degree of agreement with the
statement in the relevant question:
]1,0[]9,1[:
f
f(x) = (x − 1)/8
It is worth mentioning here that this application example illustrates how the theoret
ically continuous membership degrees are primarily chosen in practice only from a set of
rational values, in this case, with nine elements. Nevertheless, in the course of further cal
culations and evaluations, the range of the degrees in use will open widely to a finer gran
ulation scale although always remaining within the set of rational values in the unit inter
val.
After this transformation of the scale, further manipulations of the component de
grees in the questionnaires by fuzzy aggregation (and other) operations will be possible.
Here, the reader should be reminded the earlier remark that although first Zadeh and later
numerous authors extended the classic set (and logic) operations to fuzzy sets, these ex
tensions always had to conform with the rule that in the special case of binary membership
values, the operations must reduce to the original binary operations whose name the ex
tension bears. Thus, fuzzy complements (negations), fuzzy unions (disjunctions, t
conorms), and intersections (conjunctions, tnorms) have been defined in a plenitude;
however, a wider class of binary (and multiargument) operations was defined under the
name of fuzzy aggregations. This wider class of operations includes both tnorms and co
norms and, further, the broad class of mathematical means (including geometric,
weighted arithmetic, harmonic, etc., means), where the axiomatic conditions to satisfy are
only the two borderline conditions (the preservation of the extreme values 0 and 1 of the
binary membership degrees, as mentioned above) and monotonicity in terms of both (in
the case of multiple aggregations, all) arguments. Other Boolean operations such as im
plication, equivalence, antivalence, inhibition, etc., have also been extended to fuzzy sets
and concepts but usually not in accordance with the Boolean minimal canonical forms,
contrary to the similarity expression we proposed in Section 1.
Mathematics 2022, 10, 2923 9 of 25
4.2. The Structure of the Proposed Fuzzy Model
It may be interesting to point out that the earliest, rather straightforward, and very
general mathematical extension of the definition of fuzzy sets, the concept of Lfuzzy sets,
was proposed in [29] by Goguen. Recalling this definition, we have:
𝐴
=
{
𝑋
,
𝜇
}
,
𝜇
:
𝑋
→
𝐿
Here, L denotes a rather wide possible extension of the unit interval [0,1]: an arbitrary
algebraic lattice. Algebraic lattices [30] might be defined in two alternative ways, where
both lead to the same abstractions. One starts from a partial ordering relation, while the
other is based on a pair of binary operations with dual properties; namely, the operations
join and meet, which are more general versions of the Boolean logic operations “or” and
“and”. (This connection is not surprising, as Boolean algebra itself is a special lattice,
where, compared to the general definition of algebraic lattice, some additional properties
of the two operations are satisfied and, especially, the existence of the unary negation,
which has some joint properties with the former two binary ones, the most wellknown
being the pair of De Morgan equations.)
The definition of the algebraic lattice is given in the next section. Let Y = {yi} be a set
with a pair of binary operations over it, which are called join (∨) and meet (∧), for which
operations the following axiomatic properties hold:
𝑦
∨
𝑦
=
𝑦
∨
𝑦
,
𝑦
∧
𝑦
=
𝑦
∧
𝑦
(commutativity)
𝑦
∨
(
𝑦
∨
𝑦
)
=
(
𝑦
∨
𝑦
)
∨
𝑦
,
𝑦
∧
(
𝑦
∧
𝑦
)
=
(
𝑦
∧
𝑦
)
∧
𝑦
(associativity)
𝑦
∨
(
𝑦
∧
𝑦
)
=
𝑦
,
𝑦
∧
(
𝑦
∨
𝑦
)
=
𝑦
(absorption).
Then, Y is a laice for ∨ and ∧.
Lattices have further important properties (which, however, may be derived from
the above three pairs), and for bounded lattices, the axioms of idempotence, identity, and
boundary conditions also hold.
From the above definitions, join is the lowest upper bound, and meet is the greatest
lower bound of any pair of elements in Y in the sense of the partial ordering in Y. In most
applications, it is worthwhile considering a special class of bounded lattices called com
plete lattices. Here, all subsets of Y have both a supremum (the join of all respective ele
ments) and an infimum (meet of all elements of the subset).
As the concept of fuzzy sets was generalised to the idea of fuzzy signatures (FSig)
and fuzzy signature sets [3–7], as mentioned in Section 1, it could be interpreted that the
values assigned by an FSig to any element of the universe are nested vectors of member
ship degrees from the interval [0, 1]. This way, somewhat similar algebraic properties to
the originally defined fuzzy sets are obtained. It was an obvious question how the struc
ture of the FSigs compares to Goguen’s extension. While FSigs with no relation to each
other in any sense do not form any interesting algebra, it could be proven that FSigs de
duced from a single “mother FSig”, i.e., by considering the set of all possible subtrees
obtained by truncating the maximal FSig (the mother), form an Lfuzzy set, as the ele
ments of the “family set” form an algebraic lattice, where join and meet can be defined [7].
As mentioned earlier, in the FSig approach, certain features are arranged within sub
groups formally belonging to the same subvector (or subsubvector, etc.) when those
features (each of them assigned a membership degree) are closer related in semantics and
meaning. Semantics in the context of the application example refers mainly to the inter
pretation of these subfeatures by the responding employees and, at a higher level, by the
deeper meanings of the corresponding questions assigned to them by the creators of the
questionnaires. This may be expressed by the subtree within the FSig, where in the root
of the subtree, the fuzzy aggregation operation determines the way the degrees of uncer
tainty of such subfeatures are accumulated in a single degree (or, in the case of FSig sets,
in a single membership function). The key factor is here to determine the proper aggrega
tion in each “root of the subtrees”, i.e., each node within the graph that is not a leaf. In any
application, these aggregations may be derived from analysing the expert domain
Mathematics 2022, 10, 2923 10 of 25
knowledge related to those key behavioural patterns in the questions or may be deter
mined in an objective way by performing a statistical analysis and, based on the results,
applying some machine learning technique to optimally fit the parameters within the ag
gregations.
As this paper is focusing on the introduction and application of the new family of
similarity measures and does not argue for the usage of the fuzzy signature, as has been
done in [5] already, it is just mentioned that more traditional statistical methods have been
applied for building up the signature itself. The essential point in using FSig is the fact
that it allows multiple hierarchies with nested groups of features (subtrees in the repre
sentation), in which groups are first determined by the statistical interdependence of the
individual features (replies). If “brute force” statistical comparisons were applied, the
(partial) redundancies of the questions (having so many subjective elements and being
often not thoroughly thought over by the management experts constructing the question
naires) would be hidden, and often, replies with almost or partially identical semantics
would be treated as independent. That would largely distort the result of the statistical
analysis. This aspect also considers the fact that not only the respondents but the ques
tioning experts are humans with vagueness in the formulation of questions and answers.
In our proposed model, the respondents’ replies in the Likert scale are interpreted as
fuzzy degrees: they are necessarily fuzzy because these replies are unavoidably subjective
and imprecise or vague (even nondeterministic in the sense that they may not be repeated
exactly in another survey with the same respondents)—as all replies of this kind always
are. It is, however, not at all obvious which of the questions mentioned above should form
subvectors within the FSig model except the very general point in which management
science experts agree that OCB and CWBrelated answers should form separate groups.
This will be mapped into our model in the way that they form subtrees, one hierarchical
component of the FSig each. Thus, the overall structure of the FSig is
CWB
OCB
FSig Q
Q
Q
Thus, each respondent will be primarily characterized by a fuzzyvalued, 14 (10 + 4)
dimensional signature in the form of the above nested vector. The CWB part may not be
directly generated from the fuzzified answer because of the next considerations.
It should be stressed once again that fuzzy signature graphs (nested vectors) contain
fuzzy aggregations in the nonterminal (nonleaf) nodes. Aggregations, as mentioned
above, however, are always monotonic increasing functions of their arguments. Because
of this, components with a negative effect, such as the CWB attitudes, cannot be directly
combined with the ones having a positive effect (OCB) in the overall evaluation of the
attitude of the employee. As with aggregations in general, so the aggregations in the non
terminal nodes of the FSigs are also necessarily monotonic increasing functions of all ar
guments; it is necessary that instead of the fuzzified degrees obtained directly from the
Likert scale values of the responses, which are monotonic decreasing with increasing loy
alty and positive employee attitude, the complementary fuzzy membership degrees ob
tained from the original degrees assigned to the answers expressing the degree of “not
being negative in the attitude” should be aggregated with the positive answers of the OCB
components when the whole FSig is evaluated. The most commonly used negation satis
fying all properties of the Boolean negation and, in addition, having some further “nice”
symmetric and smoothness properties is the 1µ negation as it was originally defined in
[1]. Thus, in the case of CWB questions, a different function is used:
]1,0[]9,1[:'
f
f’ = (9 − x)/8
The crucial question now is how to construct the substructures and subsubstruc
tures of the FSig QFSig apart from the obvious division into the two positive and negative
semantics questions. We propose that, beyond the rather vague expert suggestions, the
Mathematics 2022, 10, 2923 11 of 25
further structures may be determined by applying a statistical analysis of the answers,
assuming here that especially the correlation analysis of the replies, including the cross
correlation of each pair, would reveal the deeper connections of the respondents’ inter
pretations of the questions. Thus, the closer relationships among the degrees assigned to
the answers would be revealed. This way, the proposed final FSig structure could be de
termined only after evaluating the correlation analysis of the replies. It is worthwhile men
tioning that such a correlation analysis could also deliver a feedback to the team that con
structed the original questionnaire by pointing out potential redundancies among the
questions.
Our proposed approach to construct a model that properly reflects the general em
ployee attitudes based on the obtained two national samples is to determine a fuzzy sig
nature that adequately describes the OCB features. This also includes the hierarchical in
terdependencies among the replies and higherlevel concepts within the OCB that can be
determined by subgroup aggregation. These latter are then reflected by the substruc
tures within the FSig. After stepbystep aggregation of the subgraphs or subvectors ac
cording to the hierarchical FSig structure, such as, e.g., the one that will be presented in
Figure 4, new, more concise descriptors based on higherlevel elements of the multicom
ponent OCB descriptor may be obtained. This happens as a result of executing the fuzzy
aggregation operations assigned to the nonterminal nodes. (The FSigs belonging to each
response form together a FSig Set (FSigS) over the universe of discourse consisting of all
(valid) responding persons.) This FSig models the complex structured problem of employ
ees’ engagement and attitude towards their respective employers via the hierarchically
structured FSig:
𝐴
=
{
𝑋
,
𝜇
}
,
A : XQFSig.
In the next section, two alternative FSig structures will be presented based on the
abovementioned considerations.
4.3. The Collected OCB vs. CWB Replies and Influential Factors
The following statements refer to the activities in which individuals may choose to
engage at work. For the bars to the right of each statement, the average score of responses
is given to indicate the degree to which each of the following statements is true about the
respondents (see Figure 2).
Figure 2. Average responses to the section Q6 questions about OCB and CWB (see Appendix A).
Mathematics 2022, 10, 2923 12 of 25
It can be seen that in Hungary, all average scores of CWB (Q611 through Q614) are
higher than in Lithuania. Moreover, OCB responses to the Q601 through Q607 in Hun
gary are higher, while Lithuanians responded with higher average scores to questions Q6
08 through Q610.
4.4. Clusters in the Responses
The basic purpose of cluster analysis is to group data, e.g., observation items, into
relatively homogeneous groups based on the variables involved in the analysis. Here, the
kmeans method was used for grouping the respondents [31]. The kmeans algorithm as
signs each record to the cluster with the least distance from the cluster centre. (Thus, the
number of clusters must be determined and specified before the algorithm starts, and this
may be done based on expert domain knowledge or a trialanderror search where the
best fit is chosen. There exist some estimation algorithms as well that may be used for
more complex data bases.) Each component of the cluster centre is equal to the mean of
the corresponding component of the records within the given cluster. The number of clus
ters was estimated here based on the expert domain knowledge of the participants of the
research and the information obtained from the literature cited earlier. For these investi
gation purposes, a threecluster solution seemed to be optimal. Hence, we obtained the
following clusters of both the Hungarian and Lithuanian data (Table 1).
Table 1. Comparison of clusters.
Questions
Hungary Lithuania
Cluster1
380Records
Cluster 2 329
Records:
Cluster 3 329
Records:
Cluster 1 60
Records
Cluster 2 57
Records
Cluster 3 30
Records
Q601 7.9 7.8 6.8 7.8 6.9 5.0
Q602 6.5 6.4 4.2 6.4 6.2 2.7
Q603 7.2 7.1 5.8 7.2 6.7 4.0
Q604 7.2 7.0 5.6 7.0 6.6 3.8
Q605 7.9 7.6 6.0 7.6 7.1 5.2
Q606: 7.3 6.7 5.0 6.6 6.1 3.1
Q607 6.9 6.5 4.7 6.2 6.1 2.9
Q608 6.9 7.1 4.0 7.3 6.3 4.8
Q609 6.5 6.3 3.6 7.0 6.1 3.8
Q610 6.3 6.6 3.5 7.7 6.9 5.0
Q611 4.0 6.0 3.2 2.1 5.6 1.8
Q612 5.3 7.1 6.1 5.0 6.4 2.6
Q613 2.2 5.3 3.2 1.8 4.7 1.9
Q614 2.4 5.9 4.2 2.6 5.3 2.8
4.4.1. Cluster 1
Respondents active and committed to the organization and to its staff: This group
has the highest average response rates to corporate OCB attitudes. The group members
also tend to have relatively low counterproductive features. This is the largest group in
both countries.
4.4.2. Cluster 2
Members of this cluster are true corporate citizens who are active and committed to
the organization and its staff and sensitive to the problems of others. In addition, however,
those belonging to this cluster also showed higher CWB as compared with cluster 1.
Mathematics 2022, 10, 2923 13 of 25
4.4.3. Cluster 3
This is a group of mediumactive Hungarian employees and passive Lithuanian em
ployees who are not too committed to the organization and its staff. Although the coun
terproductive behaviours are rather rejected, they are passive, just focusing on the man
datory/expected tasks. This group contains 32% of respondents in Hungary and only 20%
of respondents in Lithuania.
The cluster centre variables with scores equal to or higher than 5 in the clusters were:
Q601 (Willingly given of my time to help coworkers who have workrelated prob
lems);
Q605 (Encouraged others when they were down).
4.5. Correlation Analysis of the Responses
To investigate the relationship of the OCB and CWB attitudes vs. other abovemen
tioned questionnaire factors, we applied canonical correlation analysis (see Table 2). The
canonical correlation method seeks coefficients ai, i =1,2,.. n and bj, j =1, 2,.. m such that it
maximizes the pairwise determination coefficient R2 between the linear combinations of
x = a1 x1+ a2 x2+.. an xn
and
y = b1 y1 + b2 y2 +.. bm ym
of two sets of variables [32].
Table 2. Canonical correlations among the responses.
Groups of Factors
Hungary Lithuania
OCB
(Q601 through Q610)
CWB
(Q611 through Q614)
OCB
(Q601 through
Q610)
CWB
(Q6
11 through
Q614)
Communication styles
(Q101 through Q123) 0.64 0.40 0.69 0.59
Work experience
(Q201, Q202) 0.18 0.22 0.39 0.30
Personal characteristics
(Q301 through Q320) 0.64 0.34 0.67 0.59
Perception of the organization
(Q4a01 through Q4a7) 0.38 0.20 0.49 0.33
Perception of top management
(Q4b01 through Q4b23) 0.46 0.43 0.67 0.52
Culture of the organization
(Q501 through Q506) 0.36 0.35 0.69 0.50
Perception of career satisfaction
(Q701 through Q705) 0.35 0.19 0.59 0.32
Age (Q801) 0.17 0.20 0.37 0.28
𝑅 = √𝑅 is called the canonical correlation coefficient between two sets of random
variables xi, i=1,2,.. n and yj, j=1, 2,.. m.
It can be seen from Table 2 that the communication styles, personal characteristics,
and perception of top management have a medium impact (R ≥ 0.4) on OCB and CWB in
both countries. Additionally, the relationship between CWB and communication styles is
significantly greater in Lithuania than in Hungary.
The correlation between work experience and engagement (OCB and CWB) is weak
in both countries.
Mathematics 2022, 10, 2923 14 of 25
Perception of the organization has a significant correlation with OCB, namely R =
0.49 in Lithuania, while in Hungary, the same correlation is weaker: R = 0.38.
The culture of organization does not have much influence on OCB (R = 0.36) and
CWB (R = 0.35) in Hungary, while these correlations in Lithuania are quite stronger: R =
0.69 and R = 0.50, respectively.
Perception of career satisfaction has a weak influence on CWB in both countries (R ≤
0.32), and only in Lithuania does this perception have a medium correlation (R = 0.59)
with OCB.
It is worth noting that all abovementioned factors have less influence on CWB than
OCB in both countries except work experience in Hungary.
Finally, age has weak relationships with OCB and CWB in both countries.
4.6. ANOVA Analysis of the Data
Using analysis of variance (ANOVA), we tested whether the categorical variables
“category of the industry” (Q203), “size of the company” (Q204), “gender composition
of the organization” (Q2 05), “gender composition at job level” (Q206), “gender compo
sition of people one hierarchical level above” (Q207), “gender of the immediate supervi
sor” (Q208), “people more experienced than yourself who have positively influenced
your career” (Q209), and “gender of the mentor“ (Q210) had any influence on numerical
variables OCB (Q601 through Q610) and CWB (Q611 through Q614).
The oneway ANOVA is based on testing the hypothesis
H0 :µ1= µ2=...= µk
about the equality of the means µ1, µ2, ...µk of the subgroups into which the categorical
variable splits values of the numerical variables [33]. It should be noted that the alternative
hypothesis H0 just states that at least one of these equalities is not satisfied (see Figure 3).
This figure shows that the average response to question Q611 and its confidence interval
are split into 12 subgroups by Q23. It can be seen that the 10th subgroup is lower than
the average, while the confidence interval in the 11th subgroup is higher. Therefore, H0
does not hold.
Figure 3. The values belonging to Q611 are split into 12 subgroups by Q23. Means are denoted by
points and confidence intervals by bars.
For accepting or rejecting the null hypothesis H0, the socalled pvalue was used. The
lower the pvalue, the greater the statistical significance of the difference between sub
groups (see Table 3).
Mathematics 2022, 10, 2923 15 of 25
Table 3. pvalues obtained by ANOVA.
Q203 Q204 Q205 Q206 Q207 Q208 Q209 Q210
HU LT HU LT HU LT HU LT HU LT HU LT HU LT HU LT
Q601
0.01 0.73 0.34 0.58 0.23 0.75 0.08 0.05 0.08 0.67 0.58 0.24 0.01 0.75 0.01 0.65
Q602
0.01 0.47 0.06 0.77 0.04 0.01 0.28 0.05 0.28 0.12 0.27 0.26 0.01 0.04 0.01 0.71
Q603
0.33 0.47 0.18 0.12 0.19 0.11 0.38 0.16 0.38 0.91 1.00 0.39 0.00 0.02 0.00 0.50
Q604
0.75 0.86 0.22 0.92 0.02 0.01 0.06 0.01 0.06 0.35 0.04 0.84 0.43 0.31 0.43 0.33
Q605
0.00 0.70 0.27 0.64 0.00 0.71 0.00 0.47 0.00 0.01 0.01 0.11 0.17 0.30 0.17 0.59
Q606
0.22 0.9 0.31 0.86 0 0.71 0.01 0.17 0.01 0.79 0.38 0.38 0.45 0.68 0.45 0.37
Q607
0.28 0.85 0.19 0.91 0.14 0.83 0.13 0.16 0.13 0.52 0.63 0.16 0.24 0.71 0.24 0.27
Q608
0.36 0.96 0.21 0.12 0.87 0.43 0.74 0.15 0.74 0.54 0.6 0.79 0.02 0.16 0.02 0.21
Q609
0.77 0.97 0.10 0.60 0.00 0.78 0.03 0.39 0.03 0..89 0.70 0.07 0.00 0.05 0.00 0.86
Q610
0.29 0.66 0.00 0.40 0.06 0.13 0.06 0.50 0.06 0.86 0.87 0.10 0.00 0.25 0.00 0.40
Q611
0.04 0.61 0.06 0.32 0.14 0.15 0.52 0.93 0.52 0.61 0.63 0.42 0.74 0.03 0.74 0.28
Q612
0.02 0.71 0.01 0.32 0.4 0.14 0.01 0.05 0.01 0.85 0.04 0.9 0.01 0.34 0.01 0.44
Q613
0.00 0.01 0.02 0.36 0.65 0.15 0.71 0.37 0.71 0.84 0.86 0.45 0.93 0.19 0.93 0.85
Q614
0.03 0.67 0.65 0.57 0.46 0.27 0.19 0.27 0.19 0.85 0.14 0.43 0.46 0.36 0.46 0.71
It can be seen from the Table that all categorical factors Q203 through Q210 had an
impact on at least one of the OCB and CWB components (Q601 through Q614) at level p
< 0.05 in Hungary.
The Lithuanian case is different from the Hungarian one. Only Q205, Q206, Q207,
and Q209 had impact on the OCB components, while Q203 and Q209 had an impact on
at least one of the CWB components at level p < 0.05.
Similarly, it was found that the gender influence on OCB (Q601 through Q610) and
CWB (Q611 through Q614) had an impact at level p < 0.05 on Q605 (“Encouraged others
when they were down”), Q606 (“Acted as a “peacemaker” when others in the organiza
tion had disagreements”), Q612 (“Found fault with what the organization is doing”), and
Q614 (“Focused on what was wrong with my situation rather than the positive side of
it”) in Hungary. In contrary, in Lithuania, gender had an impact only on Q610 (“Attended
and actively participated in organizational meetings”) at level p < 0.05.
Based on the results, the structure of the FSigs can be determined, assigning to the
same subgraph the questions showing higher correlation with each other.
5. Comparison of the Fuzzy Signatures
5.1. Calcuations on the Data
As mentioned before, answers to questions with Likert scores are never precise, as
they depend on many subjective factors. In addition, the same question may be under
stood differently by different people. Therefore, answers are imprecise or, in other words,
fuzzy. As mentioned above, Likert scale values must be normalised and transformed into
fuzzy membership degrees in the unit interval [0, 1] [9].
As mentioned above, answers Q611 through Q614, indicating counterproductive
behaviour, were transformed into complementary membership degrees of “virtual posi
tive attitudes” using the formulae for calculating f and f’ in Sections 4.1 and 4.2.
The aim of this section was to develop the fuzzy signatures (see Figure 4) for the
Hungarian and Lithuanian responses and to compare the two.
Mathematics 2022, 10, 2923 16 of 25
Figure 4. Fuzzy signature structure according to [9].
Figure 4 shows the structure that is in accordance both with management theory in
general (cf. [34]) and our earlier results in [9]. From the statistics viewpoint, the nodes
above Q601 through Q614 (denoted by Q01 through Q14 in Figure 4) are latent or unob
served variables.
For instance, the node altruism can be considered as a latent factor, which is related
to or composed of the factors Q01 and Q02. Thus, Q601 and Q602 should contain some
common feature and should be thus correlated through this common feature.
To analyse correlations between all membership degrees, crosscorrelations were cal
culated for all variables (Tables 4 and 5).
Table 4. Crosscorrelation coefficients. Hungarian case.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Q601
1.0 0.4 0.4 0.3 0.4 0.3 0.3 0.2 0.2 0.2 −0.1
−0.1
0.1 0.1
Q602
0.4 1.0 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.3 −0.2
0.0 0.0 0.1
Q603
0.4 0.3 1.0 0.4 0.4 0.4 0.3 0.2 0.2 0.2 −0.2
−0.1
0.0 0.0
Q604
0.3 0.3 0.4 1.0 0.5 0.4 0.3 0.3 0.2 0.2 −0.1
0.0 0.0 0.0
Q605
0.4 0.3 0.4 0.5 1.0 0.6 0.5 0.3 0.3 0.3 −0.2
0.0 0.0 0.1
Q606
0.3 0.3 0.4 0.4 0.6 1.0 0.7 0.3 0.3 0.2 −0.2
0.1 0.0 0.1
Q607
0.3 0.3 0.3 0.3 0.5 0.7 1.0 0.4 0.4 0.3 −0.2
0.0 0.0 0.1
Q608
0.2 0.3 0.2 0.3 0.3 0.3 0.4 1.0 0.4 0.4 −0.2
−0.1
−0.1
0.0
Q609
0.2 0.4 0.2 0.2 0.3 0.3 0.4 0.4 1.0 0.5 −0.3
0.0 0.0 0.1
Q610
0.2 0.3 0.2 0.2 0.3 0.2 0.3 0.4 0.5 1.0 −0.3
−0.1
0.0 0.0
Q611
−0.1
−0.2
−0.2
−0.1
−0.2
−0.2
−0.2
−0.2
−0.3
−0.3
1.0 0.2 0.4 0.2
Q612
−0.1
0.0 −0.1
0.0 0.0 0.1 0.0 −0.1
0.0 −0.1
0.2 1.0 0.3 0.3
Q613
0.1 0.0 0.0 0.0 0.0 0.0 0.0 −0.1
0.0 0.0 0.4 0.3 1.0 0.5
Q614
0.1 0.1 0.0 0.0 0.1 0.1 0.1 0.0 0.1 0.0 0.2 0.3 0.5 1.0
Table 5. Crosscorrelation coefficients. Lithuanian case.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Q601
1.0 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.2 0.3 0.0 −0.2
0.0 0.0
Q602
0.5 1.0 0.5 0.5 0.3 0.4 0.3 0.2 0.2 0.2 −0.2
−0.3
−0.2
−0.2
Q603
0.5 0.5 1.0 0.6 0.5 0.4 0.3 0.3 0.3 0.2 −0.1
−0.2
−0.1
0.0
Q604
0.5 0.5 0.6 1.0 0.6 0.7 0.4 0.4 0.3 0.2 −0.1
−0.2
−0.1
−0.1
Q605
0.5 0.3 0.5 0.6 1.0 0.6 0.4 0.4 0.4 0.3 −0.1
−0.2
−0.1
0.0
Q606
0.4 0.4 0.4 0.7 0.6 1.0 0.7 0.3 0.3 0.2 −0.1
−0.3
−0.2
−0.1
Q607
0.4 0.3 0.3 0.4 0.4 0.7 1.0 0.4 0.4 0.3 −0.2
−0.4
−0.2
−0.2
Q608
0.4 0.2 0.3 0.4 0.4 0.3 0.4 1.0 0.3 0.4 0.0 −0.2
0.0 0.1
Q609
0.2 0.2 0.3 0.3 0.4 0.3 0.4 0.3 1.0 0.6 −0.1
−0.1
−0.1
0.1
Mathematics 2022, 10, 2923 17 of 25
Q610
0.3 0.2 0.2 0.2 0.3 0.2 0.3 0.4 0.6 1.0 −0.1
−0.3
0.0 0.1
Q611
0.0 −0.2
−0.1
−0.1
−0.1
−0.1
−0.2
0.0 −0.1
−0.1
1.0 0.3 0.6 0.5
Q612
−0.2
−0.3
−0.2
−0.2
−0.2
−0.3
−0.4
−0.2
−0.1
−0.3
0.3 1.0 0.3 0.3
Q613
0.0 −0.2
−0.1
−0.1
−0.1
−0.2
−0.2
0.0 −0.1
0.0 0.6 0.3 1.0 0.7
Q614
−0.0
−0.2
−0.0
−0.1
−0.0
−0.1
−0.2
0.1 0.1 0.1 0.5 0.3 0.7 1.0
The correlation coefficients do not contradict the structure given in Figure 4. For in
stance, the correlation coefficient between Q61 and Q62 equals 0.4 in the Hungarian and
0.5 in the Lithuanian case, while no other OCB correlation exceeds these values. This in
dicates that combining Q61 and Q62 into a single subtree representing a latent factor
(called altruism in the literature) is reasonable. Actually, Q61 has the same correlation
coefficient with Q63 and Q65 as well in the Hungarian case. This can be explained by the
fact that Q61 through Q610 have a stronger connection or closer relationship indicated
by the common features of citizenship behaviour. The same explanation applies to other
crosscorrelations. A similar potential subgrouping emerges for the Lithuanian answers.
In the next section, formal factor analysis will be used for identifying the subgroups
in the Q6 series questions. The term factor analysis is used both for exploratory and con
firmatory factor analysis, both being tools that enable the identification and evaluation of
latent factors based on the correlations between a group of the observed variables. Factor
analysis employs various algorithms that give similar but not always identical results.
We used confirmatory factor analysis, as it is recommended when the structure is
prespecified in [35] and confirmed by the results in [9]. At first, the confirmatory factor
analysis was applied to variables Q601 and Q6 02.
The given parameter estimates mean that
Q601 = 0.18 Altruism + δ1
Q602 = 0.10 Altruism + δ2
where the two estimation errors are denoted by δ1 and δ2 (cf. [32]).
To find the membership degrees of altruism, we applied the following approach.
Let the membership degrees of two ndimensional observed variables be denoted by
vectors x and y. Furthermore, let us denote the latent factor with factor loadings k1 and k2
by vector f. Then, the components of the abovementioned vectors are linked by the rela
tionship
xi = k1fi, + δ1, yi= k2fi, + δ2, i = 1,2,..n.
Omitting errors δ1 and δ2, we obtain the approximate estimates
𝒙
𝒊
=
𝒌
𝟏
𝒇
𝒊
,
𝒚
𝒊
=
𝒌
𝟐
𝒇
𝒊
To find the values (scores) fi of factor f, we have to find the minimum of the difference
between the observed values and their estimates, i.e.,
𝒅
=
∑
(
𝒙
𝒊
−
𝒙
𝒊
)
𝟐
𝒏
𝒊
𝟏
+
∑
(
𝒚
𝒊
−
𝒚
𝒊
)
𝟐
→
𝒎𝒊𝒏
𝒏
𝒊
𝟏
Substituting the expressions of 𝑥 and 𝑦 into 𝑑, we obtain
𝒅
=
(
𝒌
𝟏
𝒇
𝒊
−
𝒙
𝒊
)
𝟐
𝒏
𝒊
𝟏
+
(
𝒌
𝟐
𝒇
𝒊
−
𝒚
𝒊
)
𝟐
→
𝒎𝒊𝒏
𝒏
𝒊
𝟏
Applying the conditions for the minimum of d, we obtain a system of n equations
∂d/(∂fi )=0, I = 1,2,..n,
which has the solution
fi=p1 xi+p2 yi, I = 1,2,..n,
where
𝒑
𝟏
=
𝒌
𝟏
𝒌
𝟏
𝟐
𝒌
𝟐
𝟐
,
𝒑
𝟐
=
𝒌
𝟐
𝒌
𝟏
𝟐
𝒌
𝟐
𝟐
.
However, the factor scores fi found this way may not belong to the interval [0, 1] since
the sum p1+ p2 is not equal to one.
Mathematics 2022, 10, 2923 18 of 25
Therefore, for normalisation, we used the weighted average to find the membership
degrees mfi of the latent factor scores fi.
𝒎
𝒇𝒊
=
𝒑
𝟏
𝒙
𝒊
𝒑
𝟐
𝒚
𝒊
𝒑
𝟏
𝒑
𝟐
=
𝒑
𝟏
𝒑
𝟏
𝒑
𝟐
𝒙
𝒊
+
𝒑
𝟐
𝒑
𝟏
𝒑
𝟐
𝒚
𝒊
,
i=1,2,..n
Applying this algorithm, we obtain the membership degrees (weights) for the aggre
gation operation generating the membership degree in the root of the subgroup altruism:
Altruism = 0.64Q601 + 0.36Q602.
This relationship is presented graphically in Figure 5.
Figure 5. The subtree altruism.
Similarly, weights of the weighted mean aggregations in the root nodes of the sub
trees courtesy, sportsmanship, civic virtue, and compliance were found. The abovemen
tioned method was also applied to the complementary semantics questions and corre
sponding leaves Q11, Q12, Q13, and Q14. The subtree obtained this way is denoted as
1Counterproductive behaviour in Figures 6 and 7.
Figure 6. Fuzzy signature of the Hungarian responses.
Figure 7. Fuzzy signature of the Lithuanian responses.
Mathematics 2022, 10, 2923 19 of 25
The research revealed that the fuzzy signatures obtained by analysing the Hungarian
and the Lithuanian responses were more or less similar (cf. Figures 6 and 7). However,
there exist also some significant differences. The weight of question Q607 (“Acted as a
stabilizing influence in the organization when dissension occurs”) equals 0.59 in the Hun
garian case and equals only 0.30 in the Lithuanian case. On the other hand, the weight of
question Q608 (“Attended functions that were not required but which helped the organ
ization’s image”) equals 0.41 for Hungarian responses, while it is 0.70 for the Lithuanian
answers. Hence, the structure of the consistence of the civic virtue factor is different in the
two countries. Similarly, differences in the structure of compliance also occur between the
two national groups of respondents.
Despite these differences, the distribution of the membership degrees for engage
ment is similar in both countries (see Figure 8). The largest difference here equals 6%.
Figure 8. Histogram of the membership degrees assigned to engagement.
The average of the engagement membership degrees in the Hungarian responses
equals 0.63, and it equals 0.67 in the Lithuanian responses.
Therefore, there is no essential difference in the engagement in work between Hun
gary and Lithuania. The maximal difference between the frequencies is 6% (Hungary 9%
and Lithuania 3%) in the membership degree interval from 0.45 to 0.50.
Additionally, to evaluate the similarities and differences between the typical (aver
age) responses of the two countries, the calculation of the similarity measures leaf by leaf
in the two fuzzy signature trees is possible. These similarities were obtained by averaging
all replies and applying Formulas (1a) and (1b) of the fuzzy similarity given above in both
its implementations, namely (2) and (3), while the variables are replaced by the leaf mem
bership degrees:
Sim
𝐴𝑙𝑔
(
𝑚
,
𝑚
)
=
𝑚
≋
𝑚
=
m
HU
*
m
LT
+(1
−
m
HU
)(1
−
m
LT
)
−
m
HU
*
mLT
(1
−
m
HU
)(1
−
mLT
)
𝝐
[0, 1]
and
𝐒𝐢𝐦
𝒁
(
𝒎
𝑯𝑼
,
𝒎
𝑳𝑻
)
= max( min(mHU, mLT), min(1 − mHU, 1 − mLT) )
𝝐
[0, 1],
where mHU is the average membership degree of the leaf containing the Hungarian an
swers, and mLT is the average of the Lithuanian answers.
For instance, the average membership degree of Hungarian answers to the question
Q01 equals to 0.82, while the membership degree of the Lithuanian answers to Q01 is 0.73.
The application of the above formulas yields
𝐒𝐢𝐦
𝑨𝒍𝒈
(
𝒎
𝑯𝑼
,
𝒎
𝑳𝑻
)
= 0.82×0.73 + (1 − 0.82)(1 − 0.73) − 0.82×0.73(1 − 0.82)(1
− 0.73) = 0.62,
0%
5%
10%
15%
20%
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
frequency
Hungary Lithuania
Mathematics 2022, 10, 2923 20 of 25
𝐒𝐢𝐦𝐙
(
𝒎
𝑯𝑼
,
𝒎
𝑳𝑻
)
= max(min(0.82, 0.73), min(1 − 0.82, 1 − 0.73)) = 0.73.
The similarity SimZ = 0.73 is rather close to 1 and is quite significant. The other for
mula is based on strictly monotonic norms, and because of that, it cannot be compared
with the result of the corresponding max–min formula. Indeed, the results obtained by
the algebraic formulas are always closer to the degree of indifference (or full uncertainty),
i.e., 0.5, and thus, algebraic similarity measures of partially similar degrees should be al
ways less than in the other case; 0.62 > 0.5, and thus, it reflects the presence of similarity.
This difference corresponds to the philosophical fact that uncertain values combined lead
to even more uncertain results (a wellknown principle in mechanical engineering toler
ance calculations). Because of this, algebraic similarity degrees should be evaluated in
comparison with each other.
On the other hand, if. for instance, we have very different membership degrees such
as 0.1 and 0.8, both formulas would show clear dissimilarity:
𝐒𝐢𝐦
𝑨𝒍𝒈
(
𝟎
.
𝟏
,
𝟎
.
𝟖
)
= 0.1 × 0.8 + (1 − 0.1)(1 − 0.8) − 0.1 × 0.8(1 − 0.1)(1 −
0.8) =
0.25,
SimZ(0.1, 0.8) = max(min(0.1, 0.8), min(1 − 0.1, 1 − 0.8)) = 0.20.
Here, the rule mentioned above is illustrated again: two dissimilar degrees are eval
uated by low membership; however, dissimilarity is just as uncertain as similarity, and
thus, the algebraic version yields a higher value (closer to 0.5).
The similarity measures for each leaf of the signature tree were calculated, and the
results are given in Figure 9.
Figure 9. Complex fuzzy signature containing the calculated similarity measures of Hungarian and
Lithuanian answers. Denotations: H is average of the Hungarian membership degrees, L is the av
erage of the Lithuanian ones, z is the similarity measure calculated by SimZ, and a is the similarity
measure obtained by SimAlg.
5.2. Some Evaluation Remarks
Values where SimZ or SimAlg > 0.5 in Figure 9 indicate existing (maybe slight) simi
larity of the overall pool of answers belonging to the respective subtree (in extreme case,
a single leaf) of the given node. In the results, application of the SimZ measure shows
more values above 0.5, and for all questions Q01 through Q14, this value exceeds the neu
tral degree at the leaves. Q01 and Q05 show very good similarities, even being > 0.7. All
aggregated (verbally labelled) subtrees also have greater than 0.5 but only the components
of citizenship behaviour except compliance are above 0.6, while none exceeds 0.7, which
is similar to the case of aggregated citizenship behaviour. The root overall value that de
scribes the general attitude of the employees’ average is similarly less than 0.7 even in the
Mathematics 2022, 10, 2923 21 of 25
weaker SimZ form. None of these are < 0.5, which means that the application of the SimZ
measure did not reveal any direct dissimilarity between the Hungarian and the Lithua
nian employees’ attitudes.
The SimAlg measure, on the other hand, shows a more interesting set of results. It
approves this similarity of the two answer pools only for Q01, Q03, Q04, Q05, and Q13
and further for the subsets altruism, courtesy, sportsmanship. and citizenship behaviour.
As it was mentioned above, values where SimZ < 0.5 indicate dissimilarity, and such is
the case for Q11, Q12, Q14, Q02, Q06…Q10, counterproductive behaviour, civic virtue,
and compliance and the overall root degree as well. For citizenship behaviour, the result
is just 0.5, meaning in this case “No decision on similarity can be made”. However, all
values around the neutral degree 0.5 have generally a similar interpretation. Using this
measure, only Q01 and Q05 may be determined as really having a certain similarity when
applying this measure. Q01 and Q05 show really interesting similarity in this measure; all
the others are close to “No real decision is possible”. Comparison with expert evaluation
and a more detailed mathematical comparison (see below) could reveal further facts.
As it could be seen, the less “drastic” min–max (Zadeh style) measure revealed much
more similarity between the two populations. From the point of view of the similarity
measures, it can be concluded that the two implementations may result in different se
mantic interpretations.
Of course, further implementations of (1b) may bring further recognitions in the fu
ture. It also seems a good direction to continue these investigations on the whole popula
tions rather than on the averages only. If both histograms are compared value for value
according to the transformed Likert scale values, the similarities of the two populations
may be described in a more informative way, showing where within the answers the high
est matches and the greatest differences occur. At present, the size of the available statis
tical populations is quite different, so this investigation did not seem reasonable. We hope
to obtain more Lithuanian answers to more closely approach the number of the Hungar
ian ones, and then, the similarity measures between the two histograms will also be feasi
ble and meaningful.
Comparing the influential factors of employee engagement from the two countries’
perspective, the study provides insights into understanding employee engagement fac
tors relating them to the country’s specifics, in our case, national and organizational cul
tures. Therefore, the managerial implications are related to a better understanding of the
importance of culture on employee engagement, which is critical in supporting the deci
sionmaking process, especially at international companies.
It should be stressed that these results represent a small contribution only to the man
agement science aspects of the case study. However, they may be further investigated in
a wider management context, e.g., when personnel mobility is analysed, which is defi
nitely connected with the employee attitudes. (For this area, see, e.g., [36].)
6. Discussion and Conclusions
In this paper, twofold results have been presented, the main novelty being the intro
duction of a new fuzzy similarity measure that could be easily extended to similarity of
fuzzy signatures as well and the analysis of employee attitude data, which were collected
in two regions (Győr in Hungary and Panvezys in Lithuania).
The first and main result is the presentation of an inherently fuzzy similarity measure
that differs from the earlier generally accepted fuzzy similarity measures with hidden
crisp semantics (expressed by property (S3) in reference [10]). The new family of measures
is based on the rather intuitive semantic assumption that no certain statements can be
deducted from uncertain premises; i.e., two fuzzy degrees with equal value may express
different realities, and thus, their similarity may be limited. We showed by two different
implementations of this measure that, indeed, crisp results may be only obtained for crisp
arguments and also that, depending on the chosen pair of tnorm and snorm, the degree
Mathematics 2022, 10, 2923 22 of 25
of similarity may be even less than the degree of uncertainty in the objects compared. (This
also excludes transitivity in any sense.)
Further, we showed that the similarity measure family proposed could be easily ex
tended to fuzzy signatures independently from the structure and the depth of the struc
ture of the FSigs.
The new fuzzy similarity measure class thus proposed was based on the extension of
the minimal form of the Boolean equivalence relationship in a rather straightforward and
intuitive way. This class was then extended to fuzzy signatures (multicomponent hierar
chical fuzzy descriptors). The two proposed implementations were based on the classical
max–min norms originally proposed by Zadeh on one hand, while the other one was
based on the algebraic norms, the most popular and simplest representations of the
Hamacher norm. After having been defined, these two norms were implemented and
tested for a reallife problem example. An infinite number of further members of this fam
ily can be similarly defined, and future research may show which of them is more suitable
in what context.
To summarise the reallife example, first, a model recently proposed by the authors
for fuzzifying questionnaires was presented, where the Likert scale responses to the ques
tions in an internationally widely applied standard questionnaire were transformed line
arly to fuzzy membership degrees. In this questionnaire, employee replies to various
groups of questions concerning their attitudes towards their respective companies were
collected in two regions of the two countries Hungary and Lithuania.
Some of these attitudes represent positive and some negative tendencies of behav
iour. The Likert scale values were transformed directly in the case of positive factors
(OCB), and in complemented form in the negative cases (CWB). By applying correlation
calculation and factor analysis, subgroups in the set of questions were identified, and
based on them, the structure of the fuzzy signature representing the subset of the ques
tions under investigation (Q6) was determined. In the next section, an analysis of the Q6
question subgroup in connection with replies to other subsets of questions was carried
out.
By exploring the model using samples of data from both Hungarian and Lithuanian
firms, we found that these correlations between the Q6 section and other section re
sponses, represented by the average values, in Hungary and Lithuania were significantly
different. Perception of the organization had a significant correlation with the OCB group,
namely with correlation coefficient R = 0.49 in Lithuania, while in Hungary, the same cor
relation was only R = 0.38, i.e., slightly weaker. On the other hand, the culture of the or
ganization greatly influenced neither the OCB (R = 0.36) nor the CWB (R = 0.35) value
groups in the data coming from Hungary, while these correlations in Lithuania were
somewhat stronger: R = 0.69 and R = 0.50. Perception of career satisfaction had a weak
influence (R ≤ 0.35) on CWB in both countries, and only in Lithuania did this perception
have a medium correlation (R = 0.59) with the OCB subgroup. The gender composition
of the employees had an impact on several factors of engagement in Hungary, while it
only had an impact on one single component of engagement in Lithuania.
The composition of the signatures revealed differences in the structure of the inter
mediate nodes between the two countries. Nevertheless, the final node engagement in
work had no essential differences in the distribution of membership degrees in Hungary
and Lithuania. It is worthwhile continuing this research towards revealing the effects of
these differences on the employee culture, mobility, etc., and how they may be used for
improving the management quality in companies.
In the next section, the average attitudes were calculated for each of the questions
separately for the two populations, and then, the two versions of the novel similarity
measures were both applied separately for comparison. It should be stressed that in this
case, values further away from 0.5 indicate similarity or dissimilarity, namely S > 0.5
means similarity, with the closer S is to 1, the higher the former, while S < 0.5 means the
higher dissimilarity occurs the closer S is to 0.
Mathematics 2022, 10, 2923 23 of 25
From the mathematical point of view, it was presented that the strictly monotonic
(except in 0 and 1) algebraic measure showed a fast tendency of getting closer to 0.5. The
conclusion can be drawn that the min–maxbased similarity measure is easier to handle,
while the algebraic one more intensively points out real, deep similarity or the lack of any
articulated similarity/dissimilarity.
From the management point of view, based on the results of the analysis, it can be
considered proven that the differences in the national company cultures are reflected in
the employees’ engagement. All these differences have to be taken into consideration in
managing diversity in the cultures in company organisations.
Author Contributions: Conceptualization, D.S. and M.K.S.; methodology, O.P. and L.T.K.; valida
tion, O.P. and L.T.K.; formal analysis, O.P. and L.T.K.; resources, D.S. and M.K.S.; data curation,
O.P.; writing—original draft preparation, D.S. and O.P.; writing—review and editing, M.K.S.; visu
alization, O.P.; final review and editing, supervision, L.T.K. The authors declare that they all have
read and agreed to the published version of the manuscript. Authorship is limited to those who
have contributed substantially to the work reported. All authors have read and agreed to the pub
lished version of the manuscript.
Funding: The first author acknowledges the support given by National Research, Development, and
Innovation Office. (NKFIH), grant no. K108405.
Informed Consent Statement: Informed consent was obtained from all subjects involved in the
study.
Data Availability Statement: Data supporting reported results can be found in the authors’ PCs;
there are not publicly archived datasets.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
(Questions 6.1–6.14) OCB—CWB
INSTRUCTIONS: The following statements refer to activities in which individuals
may choose to engage at work. Please indicate the extent to which you have personally
engaged in the following activities.
In the space before each statement, write the number (1, 2, 3, 4, 5, 6, 7, 8, or 9) to
indicate the degree to which each of the following statements is true about you. When
responding, please try to use the full range of numbers on this scale (1 to 9). There are no
right or wrong answers to these questions.
1. Willingly given of my time to help coworkers who have workrelated problems.
2. Taken time out of my own busy schedule to help with recruiting or training new
employees.
3. “Touched base” with others before initiating actions that might affect them.
4. Taken steps to try to prevent problems with coworkers and any other personnel in
the organization.
5. Encouraged others when they were down.
6. Acted as a “peacemaker” when others in the organization have disagreements.
7. Acted as a stabilizing influence in the organization when dissention occurs.
8. Attended functions that were not required but which helped the organization’s im
age.
9. Attended training/information sessions that employees were encouraged but not re
quired to attend.
10. Attended and actively participated in organizational meetings.
11. Consumed time complaining about trivial matters.
12. Found fault with what the organization is doing.
13. Tended to make “mountains out of molehills” (make problems bigger than they are).
14. Focused on what was wrong with my situation rather than the positive side of it.
Mathematics 2022, 10, 2923 24 of 25
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