Comparative State-of-the-Art Survey of Classical Fuzzy Set and Intuitionistic Fuzzy Sets in Multi-Criteria Decision Making

Abstract

Fuzzy sets extend deterministic multi-criteria decision-making (MCDM) methods to deal with uncertainty and imprecision in decision making. Over the years, many generalizations have been proposed to the classical Fuzzy sets to deal with different kinds of imprecise and subjective data. One such generalization is Atanassov’s Intuitionistic Fuzzy Set (IFS) which is becoming increasingly popular in MCDM research. Together, the two notions of uncertainty modeling: ‘classical fuzzy set’ (Zadeh) and intuitionistic fuzzy set (Atanassov) have been utilized in many real-world MCDM applications spanning diverse disciplines. As IFS grows in popularity by the day, this paper conducts a literature survey to (1) compare the trend of publications of ‘classical fuzzy’ set theory and its generalized form, the intuitionistic fuzzy set (IFS) as used in MCDM methods from 2000 to 2015; (2) classify their contributions into three novel tracks of applications, hybrid, and extended approaches; (3) determine which MCDM method is the most used together with the two forms of fuzzy modeling; and (4) report on other measures such as leading authors and their country affiliations, yearly scholarly contributions, and the subject areas where most of the two fuzzy notions in MCDM approaches are applied. Finally, the study presents trends and directions as far as the applications of classical fuzzy set and intuitionistic fuzzy sets in MCDM are concerned.

Full text

Comparative State-of-the-Art Survey of Classical Fuzzy Set
and Intuitionistic Fuzzy Sets in Multi-Criteria Decision Making
Eric Afful-Dadzie
1
•Zuzana Komı
´nkova
´Oplatkova
´
1
•Luis Antonio Beltran Prieto
1
Received: 6 July 2015 / Revised: 11 April 2016 / Accepted: 19 May 2016
Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2016
Abstract Fuzzy sets extend deterministic multi-criteria
decision-making (MCDM) methods to deal with uncer-
tainty and imprecision in decision making. Over the years,
many generalizations have been proposed to the classical
Fuzzy sets to deal with different kinds of imprecise and
subjective data. One such generalization is Atanassov’s
Intuitionistic Fuzzy Set (IFS) which is becoming increas-
ingly popular in MCDM research. Together, the two
notions of uncertainty modeling: ‘classical fuzzy set’
(Zadeh) and intuitionistic fuzzy set (Atanassov) have been
utilized in many real-world MCDM applications spanning
diverse disciplines. As IFS grows in popularity by the day,
this paper conducts a literature survey to (1) compare the
trend of publications of ‘classical fuzzy’ set theory and its
generalized form, the intuitionistic fuzzy set (IFS) as used
in MCDM methods from 2000 to 2015; (2) classify their
contributions into three novel tracks of applications,hy-
brid, and extended approaches; (3) determine which
MCDM method is the most used together with the two
forms of fuzzy modeling; and (4) report on other measures
such as leading authors and their country affiliations, yearly
scholarly contributions, and the subject areas where most
of the two fuzzy notions in MCDM approaches are applied.
Finally, the study presents trends and directions as far as
the applications of classical fuzzy set and intuitionistic
fuzzy sets in MCDM are concerned.
Keywords State of the art survey Classical fuzzy set 
Intuitionistic fuzzy sets Multi-criteria decision making 
Fuzzy generalization
1 Introduction
Decision making is considered a natural thought process
for all beings with cognitive abilities—we choose, sort, or
rank a range of alternatives measured by a set of criteria
[47]. In multi-criteria decision making (MCDM), the nat-
ure and level of complexity of the problem determine the
methodological approach adopted. The approach is often
either one of deterministic, stochastic, and fuzzy, or com-
binations of any of the above methods [56]. In most real-
life MCDM problems, however, the underlying informa-
tion in the decision problem tends to be uncertain, sub-
jective, or imprecise [22]. In such situations, linguistic
expressions are deemed appropriate in evaluating the
alternatives with respect to given criteria. When this hap-
pens, quantifying the linguistic statements using deter-
ministic or stochastic MCDM approaches can be difficult
and often inappropriate. In view of such challenges, the
notion of fuzzy set theory proposed by Zadeh [66] comes
very handy in dealing with such issues of uncertainty,
subjectivities, and imprecision in human judgments [32]. In
this paper, the original fuzzy set theory by Zadeh is con-
veniently referred to as ‘classical’ fuzzy sets to aid in its
comparison with the selected generalized form: intuition-
istic fuzzy set (IFS).
Fuzzy set as a mathematical tool has successfully been
applied in many disciplines especially in situations that
&Eric Afful-Dadzie
afful@fai.utb.cz
Zuzana Komı
´nkova
´Oplatkova
´
kominkovaoplatkova@fai.utb.cz
Luis Antonio Beltran Prieto
beltran_prieto@fai.utb.cz
1
Faculty of Applied Informatics, Tomas Bata University in
Zlin, Zlin, Czech Republic
123
Int. J. Fuzzy Syst.
DOI 10.1007/s40815-016-0204-y

demand efficient modeling of human decisions and judg-
ments [32,62]. Over the years, however, there have been
many extensions and generalizations of the original fuzzy
concept by Zadeh—aimed at improving the modeling of
‘uncertainty’ information. A few of the widely used fuzzy
generalizations are rough sets [44], intuitionistic fuzzy sets
[2], interval-valued fuzzy sets [20], hesitant fuzzy sets [55],
soft sets [39], and type-2 fuzzy sets [67,68] among others.
These generalized fuzzy forms, especially as used in
MCDM, normally deal with situations where the ‘classical’
fuzzy set appears inadequate in modeling a new kind of
vagueness and imprecision in data.
One such generalized form of the ‘classical’ fuzzy set
theory, gradually becoming popular with MCDM practi-
tioners, is Atanassov’s Intuitionistic Fuzzy Set (IFS) [2].
Since its introduction, IFS has gradually become one of the
most utilized fuzzy set generalizations after rough sets
especially as used in MCDM problems [65]. Together, the
two notions of uncertainty modeling—the classical (Zadeh)
and intuitionistic fuzzy sets (Atanassov)—have been
applied in many real-world MCDM applications spanning
diverse disciplines. With a continued global interest in
‘classical’ fuzzy MCDM methods and their generalized
variants in both industry and academia, this paper looks at
the trend as regards the applications of ‘classical’ fuzzy set
and its generalized form, the intuitionistic fuzzy set in
MCDM publications. To do this, the comparative state-of
the-art literature survey tracks the growing popularity of
the two Fuzzy constructs in MCDM problems by (1)
comparing the publication trend grouped into tracks of
applications,hybrid, and extended as used in classical
fuzzy and intuitionistic fuzzy MCDM (F-MCDM and IF-
MCDM) articles from 2000 to 2015; (2) determining which
MCDM method is the most used in the two forms of fuzzy
modeling; and (3) reporting on other measures such as
leading authors, country affiliations, and subject areas
where most of the two fuzzy notions in MCDM approaches
are concentrated. In addition, there is also a yearly com-
parison of research outputs between fuzzy set and intu-
itionistic fuzzy set as used in some popular MCDM
methods.
The rest of the paper is organized as follows. A brief
comparison of crisp, fuzzy, and intuitionistic fuzzy set is
provided. Further, the interrelationship existing among the
three theories is explained especially in terms of how one
theory evolved from the other. This is followed by a brief
review of the use of classical fuzzy and intuitionistic fuzzy
sets as used in MCDM approaches. In Sect. 3, the
methodology adopted for the literature review is explained.
Section 4presents the findings made in the study, followed
by a section on trends and directions in Sect. 5, and finally
by conclusions drawn from this study.
2 Crisp, Fuzzy, and Intuitionistic Fuzzy Sets
The three set constructs: crisp, classical fuzzy, and intuition-
istic fuzzy sets, which have been fundamental in multi-criteria
decision-making processes, evolved from one to the other. The
classical set theory gaverise to the fuzzy set, and subsequently,
to the intuitionistic fuzzy set. The evolution of these different
set notions is premised on proposing theories that deal with
different kinds of data. In the following sections, the three
theories are compared and their interrelationships explained.
2.1 Classical (Crisp) Sets
In mathematics and related disciplines, a set is considered
one of the main building blocks for representing informa-
tion. A classical set is informally defined as a collection of
objects (elements) which possess some properties (at-
tributes) that distinguish them from other objects which do
not have those characteristics [13]. Formally, a classical set
is one with crisp or sharp boundaries [53]. This implies that
such sets are bivalent, where elements either belong or do
not belong, as expressed in the following definitions:
Definition 1 Let Xand Abe a set and its subset,
respectively, with AX. Then
kAxðÞ¼ 1ifx2A
0ifx62 A
ð1Þ
where kAðxÞis referred to as the characteristic function of
set Ain X. Therefore, a classical or crisp set can be defined
as [31]
A¼\x;kAðxÞ
fg
[jx2A:ð2Þ
2.2 Fuzzy Sets
The fuzzy set theory is a generalization or a comprehensive
form of the crisp set. A fuzzy set A0of Xis a set of ordered
pairs x1;lA0ðx1ÞðÞ;x2;lA0ðx2ÞðÞ;...;xn;lA0ðxnÞðÞ
fg
, char-
acterized by a membership function lA0ðxÞthat maps each
element xin Xto a real number in the interval [0,1]. The
function value lA0ðxÞstands for the membership degree xin
A0[13,15,32,66].
Definition 2 A fuzzy set A0in X¼x
fg
is defined by
A0¼\x;lA0ðxÞ[jx2X
fg ð3Þ
where lA0:X!0;1½denotes the membership function of
the fuzzy set A0and further describes the degree of mem-
bership of xto fuzzy set A0. Since fuzzy set is a general-
ization of the classical crisp set defined in the interval [0,1],
full membership and full non-membership of xin A0occur
when lA0xðÞ¼1 and lA0ðxÞ=0, respectively. This implies
that every crisp set is in itself a fuzzy set.
International Journal of Fuzzy Systems
123

2.3 Intuitionistic Fuzzy Sets
As shown in Eq. (3), in fuzzy set, a set A0in Xis given by
A0¼\x;lA0ðxÞ[jx2X
fg
where lA0:X!0;1½
describes the membership function of the fuzzy set A0
within the interval of [0, 1]. In intuitionistic fuzzy sets,
however, a set A00 in Xis defined as [2,41,53]
A00 ¼\x;lA00 xðÞ;vA00 xðÞ[jx2X
fg
ð4Þ
where lA00 xðÞ;vA00 xðÞ:X!0;1½, respectively, represent
the membership and non-membership functions on condi-
tion that 0 lA00 ðxÞþvA00 ðxÞ1. In addition, IFS intro-
duces a third construct pA00 ðxÞ, i.e., the intuitionistic fuzzy
index which indicates whether or not xbelongs to A.
pA00 ¼1lA00 xðÞvA00 xðÞ ð5Þ
The intuitionistic index in Eq. 5measures the hesitancy
degree of element xin Awhere it becomes obvious that
0pA00 ðxÞ1 for each x2X. A small value of pA00 ðxÞ
implies that information about xis more certain [5]. On the
other hand, a higher value of the hesitancy degree pA00 ðxÞ
means the information that xholds is more uncertain. An
intuitionistic fuzzy set can therefore fully be defined as
A00 ¼\x;lA00 xðÞ;vA00 xðÞ;pA00 ðxÞ[jx2X
fg
ð6Þ
where
lA00 20;1½;vA00 20;1½;pA00 2½0;1
In summary, Figs. 1and 2give a pictorial relationship
among crisp sets, classical fuzzy sets, and intuitionistic
fuzzy sets (IFS). In Fig. 2in particular, the three sets are
compared on a coordinate system to demonstrate how
elements belonging to crisp sets, classical fuzzy sets, and
intuitionistic fuzzy sets interrelate [53]. In crisp set, since
an element either fully belongs or fully does not belong, the
coordinates l=1 and v=0 which express the idea of
‘fully belonging’, or the coordinates l=0 and v=1
denoting the idea of ‘fully not-belonging’, as shown in
Fig. 2a, adequately depict crisp set [53]. With fuzzy sets,
the coordinates l?v=1 signify that the whole area
connecting these points can embody elements in a fuzzy
set. Note that, this is besides the coordinates in the ‘fully
belonging’ and ‘fully not-belonging’ space as shown in
Fig. 2b. In intuitionistic fuzzy set, the condition
0l?v=1 ensures that the points in Fig. 2a and b,
together with the inner of the shaded triangle shown in
Fig. 2c, make up the elements in an intuitionistic fuzzy set
[53].
In general, intuitionistic fuzzy sets (IFSs) differ from
fuzzy sets in terms of the approach in that IFS introduces
three functions that express the degree of membership,
non-membership, and hesitancy [11,40,53]. The IFS
approach offers a different dimension to human decision
modeling by introducing three states of fuzzy constructs to
characterize the extent to which decision makers support,
oppose, and are hesitant or neutral about their decisions
[36]. This notion helps to quantify the extent of satisfac-
tion, dissatisfaction, and hesitancy in a decision maker’s
judgment. Therefore, intuitionistic fuzzy sets according to
Szmidt [53] offers an additional degrees of freedom for
decision makers through the introduction of non-member-
ship and the hesitancy degrees. The following section
reviews fuzzy and intuitionistic fuzzy sets as used in
MCDM methods.
2.4 Fuzzy and Intuitionistic Fuzzy MCDM Methods
Fuzzy set theory deals with imprecision in our natural
language used to communicate information [32]. Since its
introduction, fuzzy set approaches have been found a
suitable tool in modeling human knowledge especially in
decision-making problems that involve multiple subjective
criteria. Over the years, a range of decision support tech-
niques, methods, and approaches have been designed to
provide assistances in human decision-making processes
[56]. Some of these methods are analytic hierarchy process
(AHP) [49]; analytic network process (ANP) [50]; Tech-
nique for Order Preference by Similarity to Ideal Solution
(TOPSIS) [23]; VIseKriterijumska Optimizacija I Kom-
promisno Resenje, which translated in English means
multi-criteria optimization and compromise solution
(VIKOR) [42]; elimination Et choice translating reality
(ELECTRE) [46,48]; Preference Ranking Organization
Methods for Enrichment Evaluations (PROMETHEE)
[61]; Decision Making Trial and Evaluation Laboratory
(DEMATEL) [16,17,63]; and Axiomatic design (AD)
[33,51,52] among several others [12,45]. Many of these
methods and techniques were first proposed with a focus on
quantitative or deterministic multi-criteria decision making
(MCDM) but have since all been extended to deal with
situations of imprecise data using classical fuzzy sets.
Crisp set
Fuzzy set
Intuitionistic
fuzzy set
Fig. 1 Relationship among classical sets, fuzzy sets, and intuition-
istic fuzzy sets Source [53]
E. Afful-Dadzie et al.: Comparative State-of-the-Art Survey of Classical Fuzzy Set…
123

Consequently, fuzzy versions of AHP, TOPSIS, PRO-
METHEE, VIKOR, ELECTRE, and many others have
been developed and utilized in diverse decision problems
[25].
Similar to the use of classical fuzzy set theory in
MCDM, Attanasov’s introduction of intuitionistic fuzzy set
has also been extended to some of the major MCDM
techniques mentioned above. Besides the use of either
classical fuzzy set theory or intuitionistic fuzzy set in
MCDM, there are a host of research contributions that
combine a number of MCDM methods into what is simply
known as hybrid methods. These hybrid methods have
become popular in F-MCDM and IF-MCDM research in
diverse disciplines such as in Science, Engineering, Busi-
ness, Management, Accounting, Energy, Health, Environ-
mental Science, Arts, and Humanities among others. The
notion of hybrid fuzzy MCDM approach is investigated
together with other MCDM design perspectives such as
application articles and those that provide further exten-
sions as explained in the following sections.
3 Methodology
The research design sought to primarily compare the output
of fuzzy set theory with one of its generalized forms, the
intuitionistic fuzzy set, in terms of their trend of publica-
tions in MCDM methods. To do this, the first part of the
research created a basic classification scheme to group
publications from the two fuzzy constructs into tracks of
Applications,Hybrid, and Extended as used in classical
fuzzy and intuitionistic fuzzy MCDM (F-MCDM and IF-
MCDM) articles. Text mining techniques were further used
to affirm the accuracy of classified articles using term
frequency–inverse document frequency (tf–idf) and termi-
nology-extraction techniques. The concept behind the ter-
minology-extraction approach is shown in Fig. 3. In doing
so, the so-called ‘words of interest’ were created to search
through the articles (documents) in the corpus. For
instance, words of interest for hybrid also included words
like two-stage,combined, and integrated. The use of text
mining was necessary especially for articles where our
‘words of interest’ did not readily appear in the topic,
keywords, or in the abstract sections of the paper. Table 1
shows the inclusion and exclusion criteria proposed for the
comparative literature survey.
The research further presents results on the most popular
MCDM method(s) often used together with the two forms
of fuzzy modeling considered in this paper: classical fuzzy
set and intuitionistic fuzzy set. Finally, the literature survey
reports on leading authors, country affiliations as well as
areas or disciplines where most F-MCDM and IF-MCDM
articles are concentrated. The Scopus bibliographic data-
bases were used for the survey. The criteria for selecting
Scopus included among others (i) automated search avail-
ability, (ii) reputation and quality of journals indexed, and
(iii) the number and diversity of readership. Doctoral dis-
sertations, master’s theses, textbooks, conference
(a) crisp set (b) fuzzy set (c) intuitionistic fuzzy set
μμμ
ννν
ν
μ
≥
≥+0
1
ν
μ
=+
1
Fig. 2 Coordinate system comparison of elements in crisp, fuzzy, and intuitionistic fuzzy sets Source [53]
Step 1: Text Pre-processing
Articles manually
classified using
topic, keywords
and abstract s.
Corpus for each
classification of
application, hybrid
and extended
Pre-Processing eg.
Stoplists, tokenizaon etc.
Step 2: Text Analysis
Term frequency–inverse
document frequency
(tf–idf)
Document processing
Terminology
(Keywords) extraction
Confirmation of
classification
Fig. 3 Conceptual model for keywords search in the literature
International Journal of Fuzzy Systems
123

proceeding papers, erratum, and unpublished works were
not included in the review. Table 1summarizes the
methodological concept behind the research. Further in
Sect. 3.1, a detailed explanation and the motivation for the
choice of the three terms (applications,hybrid, and ex-
tended) used in the classification scheme are presented.
3.1 Classification Terms
3.1.1 Applications
The first track in the classification scheme focused on
grouping F-MCDM and IF-MCDM articles that apply
MCDM method(s) to solve real-world problems in specific
areas or disciplines. Most of such ‘application’ papers are
often fundamentally descriptive on the real-life problem it
solves using either F-MCDM or IF-MCDM method. In
view of this, they mostly do not address a theoretical
question or make a generalizable contribution to theory. A
cursory analysis found most of such articles in domain or
subject specific journals where the problem(s) the
author(s) address is/are most appreciated in the specific
area of application. For example, journals such as Safety
Science,International Journal of Advanced Manufacturing
Technology, Life Science Journal, International Journal of
Production Economics and Natural Hazards and Earth
System Sciences among others which would ordinarily not
have fuzzy logic as part of their core scope, published a lot
of these ‘application’ papers. It is also noted that some of
the hybrid category papers fall under the class of
application papers in instances where the particular hybrid
method was used to solve a real-world problem. In spite of
this, generally most of the articles that fall under the cat-
egory of application papers used only one F-MCDM or IF-
MCDM method in their solutions. For example, the use of
only fuzzy TOPSIS or Intuitionistic Fuzzy PROMETHEE.
3.1.2 Hybrid
In articles classified as hybrid, authors typically use words
such as hybrid,two-stage,multi-stage,combined,orinte-
grated to describe situations where more than one MCDM
method or a combination of an MCDM method and other
techniques were used to solve a problem. In typical
structured decision-making methods, there are a number of
processes and steps that are often followed. Some of these
are identifying the problem, constructing preferences,
weighting the criteria and decision makers, and ultimately
evaluating the alternatives to determine the best alternative
[22,29,31]. In view of this, most hybrid,two-stage,
combined,orintegrated methods tend to propose the use of
different MCDM methods for accomplishing different
tasks in the decision-making process. This would often
include but not limited to, the use of different MCDM
methods in such stages as, setting criteria weights, aggre-
gating decision makers’ preferences, analyzing consensus
in group decisions and ranking the alternatives. For
instance, a typical hybrid F-MCDM and IF-MCDM may
apply Fuzzy AHP or ANP to set the weights of criteria and
use fuzzy TOPSIS to accomplish the ranking.
Table 1 Inclusion and exclusion search criteria
Search criterion Inclusion Exclusion
Time period 2000–2015 Before 2000 and after 2015
Language English Non-English
Online database Scopus All other database
Document type Journal articles Editorial, Conference proceedings, Doctoral
dissertations, master’s theses, textbooks,
Letters, Erratum etc.
Survey Classical Fuzzy sets and intuitionistic Fuzzy set All other fuzzy generalized forms
Journal article classification Application
Hybrid
Extended
–
F-MCDM and IF-MCDM
methods (Words of
interest)
Classical Fuzzy (F-MCDM) and intuitionistic Fuzzy (IF-
MCDM) of {TOPSIS, AHP, ANP, VIKOR, ELECTRE,
PROMETHEE, DEMATEL, Axiomatic Design (AD)}
MCDM Methods outside this bracket
Leading F-MCDM and IF-
MCDM authors
Top (3-4) F-MCDM and IF-MCDM authors Authors with less than 2 articles
Country affiliations Top 10 Countries in each F-MCDM and IF-MCDM category Country affiliations outside the Top 10
Leading subject areas in
F-MCDM and IF-MCDM
publications
Top 5 subject areas with publications across all the selected
MCDM methods
Subject areas that are not representative of all
the selected F-MCDM and IF-MCDM
methods
E. Afful-Dadzie et al.: Comparative State-of-the-Art Survey of Classical Fuzzy Set…
123

This way, similar to articles classified as application,
hybrid F-MCDM and IF-MCDM methods also largely
contribute little in terms of theory as they rely on existing
methods but use them in some novel applications.
According to Mardani et al. [38], there is a rise in hybrid
MCDM methods especially in fuzzy environments. This
creates a research space to understanding for example, the
motive behind the choice of certain MCDM methods for
accomplishing certain tasks in the decision-making process
in hybrid method.
3.1.3 Extended
The third group in the classification of F-MCDM and IF-
MCDM publications from 2000 to 2015, focused on the so-
called extended approaches. We identified through a thor-
ough search, that articles in this category also tend to use
words such as improved,extended,novel,modified,new
and generalized to describe research contributions where
an original theoretical construct is modified, improved,
generalized or successfully extended into areas that have
not yet been explored with an F-MCDM and IF-MCDM
method. In the extended articles category, toy problems or
simply numerical examples are often used instead of real-
life applications to demonstrate the use of a new or mod-
ified methodology.
4 Findings
The motivation for the comparative literature review
adopted in this paper was to bring to attention the growing
diverse tracks of publications in F-MCDM and IF-MCDM
from 2000 to 2015 and to report generally on other inter-
estingness measures relative to their trend of publications.
The main classification scheme grouped articles into tracks
of applications,hybrid, and extended as used in fuzzy and
intuitionistic fuzzy MCDM (F-MCDM and IF-MCDM)
articles from 2000 to 2015. An attribute column called
‘specific mention’ was created, and subsequently, a
thorough screening was done to place the articles into
tracks of Applications,Hybrid, and Extended as seen in
Tables 2and 3, respectively, for F-MCDM and IF-MCDM.
In the following sections, the findings are discussed.
4.1 Application, Hybrid, and Extended
The results in Tables 2and 3reveal that in all instances of
comparing one fuzzy MCDM method to an intuitionistic
fuzzy MCDM method, the classical fuzzy set theory con-
tinues to be the preferred choice among practitioners and
researchers. This trend is corroborated by Tables 4and 5,
where comparative yearly contributions of F-MCDM and
IF-MCDM methods indicate that in all years from 2000 to
2015, F-MCDM publications far outweigh its generalized
form, the IF-MCDM. The results further reveal that articles
categorized as application were far more than those under
hybrid and extended. In addition, the most popular
F-MCDM method in terms of number of published articles,
happens to be the Fuzzy analytical hierarchy process
(FAHP) method. The intuitionistic fuzzy TOPSIS method,
on the other hand, is the most preferred choice among IF-
MCDM authors. It is furthermore realized, that there has
not yet been any article on intuitionistic fuzzy analytical
network process (IF-ANP) in any of the three categories.
This means there is yet to be an extension of intuitionistic
fuzzy set into ANP, to subsequently trigger the related
application and hybrid papers.
Incidentally in F-MCDM, the method that has been least
extended, improved, modified, or generalized, also happens
to be the fuzzy ANP method, with only 8 extended papers
out of 266 papers with specific mention of the method. This
trend is noteworthy compared to fuzzy PROMETHEE,
which with a total of 83 articles of specific ‘term’ mention,
already has 13 extended or generalized papers. The
research further reveals a growing ascendency in the
number of hybrid papers over the period. While most
hybrid articles use a 2-method MCDM solution [28], there
were cases where more than two methods were used in the
hybrids. Some of these instances of a 3-method fuzzy
Table 2 Fuzzy MCDM
publications grouped into
applications, hybrid, and
extended from 2000 to 2015
Methods All fields Specific mention Application Hybrid Extended
Fuzzy TOPSIS 5630 995 628 389 128
Fuzzy AHP 10,792 2159 1718 1103 117
Fuzzy VIKOR 1476 170 124 83 15
Fuzzy ANP 2200 266 228 124 8
Fuzzy PROMETHEE 1164 83 64 39 13
Fuzzy ELECTRE 1436 80 44 26 19
Fuzzy DEMATEL 995 167 131 79 19
Fuzzy axiomatic design 5352 341 128 16 78
International Journal of Fuzzy Systems
123

hybrid MCDM approach are [6] where a 3-method MCDM
framework involving fuzzy DEMATEL, fuzzy ANP and
fuzzy TOPSIS were applied in the evaluation of green
suppliers. Similarly in [24], a three-method MCDM
approach of Fuzzy ANP, Fuzzy TOPSIS, and Fuzzy
ELECTRE techniques was used in the selection of pro-
fessionals. In [69], the combination of fuzzy AHP, fuzzy
TOPSIS, and DEA were deemed necessary for supplier
selection and performance evaluation. Other
notable 3-method hybrid F-MCDM or IF-MCDM contri-
butions are [10,54,57,59,60,71] among others.
One unique trend is seen in the growing number of
methods traditionally outside the scope of MCDM meth-
ods, but used in hybrid articles. Some of these methods are
balanced scorecard (BSC) [54,58]; quality function
deployment (QFD) [4,18]; SERVQUAL [3,7]; SWOT
analysis [1,19], data envelopment analysis (DEA) [21,30]
among others. It was also realized that, among IF-MCDM
publications, the concentration has more been on devel-
opment of aggregation operators where many have been
developed over the period. The majority of such aggrega-
tion operators belong to the families of averaging opera-
tors, ordered weight aggregation (OWA), compensatory
operators, geometric operators, Shapley averaging opera-
tors, Sugeno integrals, Choquet operators, and many other
hybrid forms [37].
4.2 Yearly Contributions of F-MCDM and
IF-MCDM from 2000 to 2015
This section compares the yearly contributions of
F-MCDM and IF-MCDM publications in terms of their
frequency distribution from 2000 to 2015. To do this, the
tables were grouped conveniently into selected distance-
Table 3 Intuitionistic Fuzzy
MCDM publications grouped
into applications, hybrid, and
extended from 2000 to 2015
Methods All fields Specific mention Application Hybrid Extended
Intuitionistic fuzzy TOPSIS 1405 107 44 27 40
Intuitionistic fuzzy AHP 908 21 7 11 4
Intuitionistic fuzzy VIKOR 494 16 8 4 5
Intuitionistic fuzzy ANP 198 0 0 0 0
Intuitionistic fuzzy PROMETHEE 154 4 2 0 2
Intuitionistic fuzzy ELECTRE 227 7 3 0 3
Intuitionistic fuzzy DEMATEL 134 8 6 3 1
Intuitionistic fuzzy axiomatic design 602 37 2 0 35
Table 4 Yearly contributions of F-MCDM and IF-MCDM distance-
based methods from 2000 to 2015
Year TOPSIS VIKOR AHP ANP AD
F IF F IF F IF F IF F IF
2000 2 0 0 0 11 0 0 0 7 1
2001 1 0 0 0 24 0 0 0 7 0
2002 3 0 1 0 14 0 0 0 4 0
2003 3 0 0 0 18 0 1 0 11 0
2004 0 0 0 0 37 0 1 0 13 0
2005 9 0 0 0 61 0 2 0 13 0
2006 14 0 0 0 82 1 4 0 18 1
2007 29 1 1 0 111 0 4 0 20 3
2008 36 2 1 0 138 0 11 0 21 2
2009 74 5 3 0 151 2 16 0 36 3
2010 61 3 7 0 161 0 24 0 27 2
2011 113 15 22 4 220 1 31 0 24 3
2012 117 14 25 1 236 3 36 0 39 5
2013 152 15 34 4 278 4 43 0 34 6
2014 181 28 38 4 314 6 45 0 29 6
2015 200 24 38 3 303 4 48 0 38 5
Total 995 107 170 16 2159 21 266 0 341 37
Table 5 Yearly contributions of F-MCDM and IF-MCDM outrank-
ing methods from 2000 to 2015
Year PROMETHEE ELECTRE DEMATEL
F IF F IF F IF
2000 2 0 0 0 0 0
2001 2 0 1 0 0 0
2002 0 0 1 0 0 0
2003 1 0 1 0 0 0
2004 0 0 0 0 0 0
2005 1 0 1 0 0 0
2006 1 0 2 0 0 0
2007 4 0 1 0 3 0
2008 2 1 10 0 3 0
2009 2 0 3 0 5 0
2010 13 0 3 0 7 1
2011 7 0 14 1 18 1
2012 10 1 6 0 21 1
2013 15 0 9 2 27 0
2014 17 1 8 1 36 3
2015 6 1 17 3 47 2
Total 83 4 80 7 167 8
E. Afful-Dadzie et al.: Comparative State-of-the-Art Survey of Classical Fuzzy Set…
123

based (unique synthesis criterion) and outranking methods
in Tables 4and 5, respectively, where F-MCDM and IF-
MCDM were compared on the basis of one MCDM method
to the other. In all, there were a total of 4261 F-MCDM
publications to a paltry 200 for IF-MCDM across both
distance-based and outranking methods.
It is also realized that out of the total of 200 IF-MCDM
publications, 107 representing a majority of 53.5 % cen-
tered on intuitionistic fuzzy TOPSIS techniques, with the
next highest, IF-AHP registering 10.5 % of the total. On
the other hand, F-AHP and F-TOPSIS lead in F-MCDM
publications with 2159 and 995 representing 50.67 and
22.35 %, respectively, of the total share.
Though publications in IF-MCDM methods lag far
behind those of F-MCDM methods, increasingly,
IF-MCDM articles are also gradually gaining momentum.
In particular, intuitionistic fuzzy MCDM publications
(IF-MCDM), seem to have gained traction after 2009 even
though most of the publications center on only IF-TOPSIS
and IF-AHP. For example, there was a change of 340, and
158 % respectively, of IF-TOPSIS and IF-AHP publica-
tions from 2009 to 2015. Though the absolute figures
involved are insignificant compared to how classical
FuzzyMCDM(F-MCDM)methodsperformedinthe
same period, it points to a gradual interest in IF-MCDM
research.
4.3 Leading Authors
The section looked at leading authors in the area of
F-MCDM and IF-MCDM publications. Figures 4and 5
present leading authors in F-MCDM and IF-MCDM pub-
lications showing the frequency of their contributions from
2000 to 2015. The term leading authors as used in this
paper does not in any way imply the most influential which
has been addressed in past reviews [27]. In F-MCDM
publications, Kahraman, C. has contributed to almost all
the F-MCDM methods and leads in terms of number of
publications in F-TOPSIS, F-AHP, and PROMETHEE as
confirmed in [27]. Tseng, M.L and Tseng, G.H lead in
publications on fuzzy ANP and fuzzy DEMATEL,
respectively. In IF-MCDM, Fig. 5shows no particular
author dominance in terms of number of contributions
across the various MCDM methods. However, Li, D.F
leads in IF-TOPSIS publications, Abdullah, L in IF-AHP,
Xu, Z in IF-PROMETHEE, Deng, Y in IF-DEMATEL, and
Castineira, E.E leading in IF-Axiomatic publications.
4.4 Distribution of Country Affiliations of Authors
In Tables 6and 7are shown the frequency distributions of
country affiliations of authors across the spectrum of
popular MCDM methods. China leads in terms of the
number of published articles in both classical Fuzzy multi-
criteria decision making (F-MCDM) and its generalized
form, the Intuitionistic Fuzzy multi-criteria decision-mak-
ing (IF-MCDM) methods. China has a combined total of
1508 scholarly articles on F-MCDM representing 37.4 %
of the shared total among the top 10 country affiliations.
China also currently has 134 articles authored on IF-
MCDM representing 68.02 % of the total among the top 10
countries. Similarly, authors from Iran and Taiwan follow
China in terms of contributions to F-MCDM and its gen-
eralized form, IF-MCDM.
4.5 F-MCDM and IF-MCDM Publications Across
Subject Areas
This section compared subject areas or disciplines where
most F-MCDM and IF-MCDM publications are concen-
trated as shown in Figs. 6and 7. It is evident that in
F-MCDM approaches, Engineering and its related disci-
plines have a greater share of the number of published
articles with about 2103 articles across F-TOPSIS, F-AHP,
F-VIKOR, F-ANP, F-PROMETHEE, F-ELECTRE,
F-DEMATEL, and F-Axiomatic. However, a large number
of the Engineering-focused F-MCDM articles used F-AHP
followed by F-TOPSIS. On the other hand, Computer
0
5
10
15
20
25
30
35
40
45
Kahraman, C.
Zavadskas, E.K.
Vahdani, B.
Tavakkoli-…
Kahraman, C.
Tzeng, G.H.
Buyukozkan, G.
Mousavi, S.M.
Tavakkoli-…
Kahraman, C.
Tseng, M.L.
Tzeng, G.H.
Kahraman, C.
Kahraman, C.
Behzadian, M.
Hu, Y.C.
Fernandez, E.
Vahdani, B.
Kahraman, C.
Tzeng, G.H.
Tseng, M.L.
Liou, J.J.H.
Liu, X.
Kahraman, C.
Cebi, S.
F-TOPSIS F-AHP
F-VIKOR F-ANP
F-PROMETHEE
F-ELECTRE
F-DEMATEL
F-AXIOMATIC
Fig. 4 Leading F-MCDM authors from 2000 to 2015
International Journal of Fuzzy Systems
123

Science and its related disciplines featured largely in IF-
MCDM scholarly articles with a total of 133 articles.
The most utilized method in the IF-MCDM Computer
Science articles was intuitionistic fuzzy TOPSIS (IF-
MCDM).
5 Trends and Directions
Many of the results presented in the study show that clas-
sical fuzzy set theory is still the most preferred choice
among MCDM practitioners in spite of the growing number
0
1
2
3
4
5
6
IF-TOPSIS
IF-AHP IF-VIKOR IF-PROMETHEE
IF-ELECTRE
IF-DEMATEL
IF-AXIOMATIC
Fig. 5 Leading IF-MCDM authors from 2000-2015
Table 6 Distribution of Top 10 F-MCDM Publications by country Affiliation
Country F-TOPSIS F-AHP F-VIKOR F-ANP F-PROMETHEE F-ELECTRE F-DEMATEL F-AD Total
China 295 922 33 35 17 12 29 165 1508
Iran 213 242 44 60 13 15 40 2 629
Taiwan 98 251 25 80 10 4 61 1 530
Turkey 130 214 21 49 12 12 13 40 491
India 94 157 26 16 9 5 14 13 334
USA 46 95 5 15 6 5 8 20 200
Malaysia 30 46 7 7 5 2 2 1 100
Canada 28 40 3 3 4 4 3 10 95
South Korea 17 35 4 2 1 1 0 0 60
United Kingdom 18 48 2 7 2 1 0 4 82
Table 7 Distribution of Top 10 IF-MCDM Publications by country affiliation
Country IF-TOPSIS IF-AHP IF-VIKOR IF-ANP IF-PROMETHEE IF-ELECTRE IF-DEMATEL IF-AD Total
China 84 9 10 0 3 2 4 22 134
Iran 3 2 2 0 0 1 3 1 12
Taiwan 2 0 1 0 1 1 1 0 6
Turkey 10 2 0 0 0 0 1 1 14
India 2 2 1 0 0 2 0 2 9
USA 3 0 0 0 0 0 1 1 5
Malaysia 1 4 1 0 0 0 0 0 6
Canada 3 1 1 0 0 0 0 0 5
South Korea 1 0 2 0 0 0 0 0 3
United Kingdom 2 0 0 0 1 0 0 0 3
E. Afful-Dadzie et al.: Comparative State-of-the-Art Survey of Classical Fuzzy Set…
123

of generalized forms. This observation is seen in Tables 4
and 5. One striking observation, however, is that, while
intuitionistic fuzzy set has been utilized in almost all the
MCDM methods, one piece noticeably missing is the
Analytical Hierarchy Network method (ANP). There are
virtually no record of such either in application,hybrid or
extended types of articles in literature. This opens a research
gap into why there are no such use of intuitionistic fuzzy set
in ANP. Further, the classification scheme also pointed to a
rise in hybrid fuzzy MCDM solutions as seen in Tables 2
and 3. However, what is generally missing is the motivation
for opting for a hybrid MCDM solution rather than a one-
method solution. In particular, are cases where as many as 3
or 4 MCDM methods were used to solve a decision prob-
lem. In some cases, a reduction in the number of methods
combined in the hybrid and tested on the same decision data
also realizes the same result. This brings to the fore the
issues about when to apply such hybrid MCDM methods
and in what kinds of decision problems.
Another trend worth mentioning, is the unique combi-
nation of two different generalized forms in one solution.
For instance in [26], intuitionistic and hesitant fuzzy sets
are conveniently combined to perform an economic anal-
ysis. Finding ideal situations to utilize two or more fuzzy
generalized forms can be rare given their different theo-
retical foundations. Another combination of fuzzy gener-
alized forms is seen in [34] where fuzzy axiomatic design
was combined with intuitionistic fuzzy sets in MCDM
setting. Others are [8,9,11,33,40,41,64]. We also
realized a general trend in hybrid MCDM articles where
one of the methods in the hybrid was chosen outside the
traditional scope of MCDM. For instance, concepts and
methods such as SERVQUAL, QFD, BSC, and DEA were
found to have been combined with either TOPSIS, AHP or
ANP in real-world applications. The successful imple-
mentation of such hybrid methods, give an indication that
more other methods outside the scope of MCDM can be
explored in useful applications.
Finally, there is a growing popularity of classical fuzzy
sets and their generalized forms in many domain-specific
journals where hitherto articles using fuzzy sets would not
be considered. This is an indication of how popular and
0
200
400
600
800
1000
1200
Engineering Computer Science Mathemacs BUS., MGMT. & ACCT. Decision Sciences
Fig. 6 Distribution of subject area publications on F-MCDM
0
10
20
30
40
50
60
70
Engineering Computer Science Mathemacs BUS, MGMT & ACCT Decision Sciences
Fig. 7 Distribution of subject area publications on IF-MCDM
International Journal of Fuzzy Systems
123

widespread the extensions of MCDM using fuzzy mathe-
matics have become across various disciplines.
6 Conclusion
In uncertainty modeling, especially in multi-criteria deci-
sion-making settings, Zadeh’s original fuzzy construct
remains the most utilized theory. However, with several
forms of generalizations to the original fuzzy set theory,
this paper set out to track and compare publications of
classical Fuzzy multi-criteria decision-making techniques
with one of its promising generalized forms, the intu-
itionistic fuzzy set. To do this, the paper first demonstrated
in theory, the interrelationships among classical set, fuzzy
set, and the intuitionistic fuzzy set, and how one theory gets
‘generalized’ into the other. A classification scheme de-
vised in this study, grouped scholarly articles of F-MCDM
and IF-MCDM into tracks of applications,hybrid, and
extended approaches. Such classification scheme helped to
track the various forms of solutions and styles in F-MCDM
and IF-MCDM articles. The result showed enormous
amount of articles in the category of applications. The
result further revealed that while F-AHP is the most pre-
ferred MCDM method among F-MCDM researchers, IF-
TOPSIS leads in terms of publications in IF-MCDM arti-
cles. Most notably, while there are scholarly articles on
classical fuzzy analytical network process (F-ANP), there
has not yet been any article published on intuitionistic
fuzzy Analytical Network Process, whether in the form of
applied, hybrid, or extended. This creates a research space
for researchers to explore the potential extension of intu-
itionistic fuzzy sets into ANP.
The research also explored other interestingness mea-
sures such as yearly contributions of F-MCDM and IF-
MCDM across the spectrum of popular distance-based and
outranking MCDM methods. Here, we get a full picture of
how intuitionistic fuzzy MCDM techniques compare with
fuzzy MCDM. It was realized that, although publications in
IF-MCDM methods lag far behind those of F-MCDM
methods, IF-MCDM is showing huge promise through the
gradual momentum it is gaining especially since 2009. The
comparative literature survey further revealed that author
Kahraman,Chas the most published F-MCDM articles
while in IF-MCDM, there is no dominant author in terms of
leads in the number of publications. Engineering is the
most featured subject area among F-MCDM articles while
Computer Science leads in IF-MCDM subject area publi-
cations. Finally, China followed by Iran and Taiwan are
where most F-MCDM and IF-MCDM authors’ institutions
are based.
The general approach adopted in the paper was to help
create attention among MCDM researchers and
practitioners about the trend of publications relating to
fuzzy sets and their generalized form, the intuitionistic
fuzzy set. Further, the research has shown areas where
F-MCDM and IF-MCDM articles have not been forth-
coming to trigger interests among researchers. Future
research hopes to compare fuzzy sets with other
notable generalizations such as hesitant sets, rough sets,
and interval-valued sets among others.
Acknowledgments This work was supported by the ministry of
education, youth and sports of the Czech Republic within the national
sustainability programme project No. LO1303 (MSMT-7778/2014)
and also by the European regional development fund under the project
CEBIA-Tech No. CZ.1.05/2.1.00/03.0089, further it was supported by
grant agency of the Czech Republic—GACR 588 P103/15/06700s
and by Internal Grant Agency of Tomas Bata University in Zlin under
the project No. GA/CEBIATECH/2016/007.
References
1. Amin, S.H., Razmi, J., Zhang, G.: Supplier selection and order
allocation based on fuzzy SWOT analysis and fuzzy linear pro-
gramming. Expert Syst. Appl. 38(1), 334–342 (2011)
2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1),
87–96 (1986)
3. Awasthi, A., Chauhan, S.S., Omrani, H., Panahi, A.: A hybrid
approach based on SERVQUAL and fuzzy TOPSIS for evaluat-
ing transportation service quality. Comput. Ind. Eng. 61(3),
637–646 (2011)
4. Bevilacqua, M., Ciarapica, F.E., Giacchetta, G.: A fuzzy-QFD
approach to supplier selection. J. Purch. Supply Manag. 12(1),
14–27 (2006)
5. Boran, F.E., Genc¸, S., Kurt, M., Akay, D.: A multi-criteria
intuitionistic fuzzy group decision making for supplier selection
with TOPSIS method. Expert Syst. Appl. 36(8), 11363–11368
(2009)
6. Bu
¨yu
¨ko
¨zkan, G., C¸ifc¸ i, G.: A novel hybrid MCDM approach
based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to
evaluate green suppliers. Expert Syst. Appl. 39(3), 3000–3011
(2012)
7. Carrasco, R.A., Mun
˜oz-Leiva, F., Sa
´nchez-Ferna
´ndez, J., Lie
´-
bana-Cabanillas, F.J.: A model for the integration of e-financial
services questionnaires with SERVQUAL scales under fuzzy
linguistic modeling. Expert Syst. Appl. 39(14), 11535–11547
(2012)
8. Cebi, S., Kahraman, C.: Extension of axiomatic design principles
under fuzzy environment. Expert Syst. Appl. 37(3), 2682–2689
(2010)
9. Chen, C.T.: Extensions of the TOPSIS for group decision-making
under fuzzy environment. Fuzzy Sets Syst. 114(1), 1–9 (2000)
10. Chen, J.K., Chen, I.S.: Using a novel conjunctive MCDM
approach based on DEMATEL, fuzzy ANP, and TOPSIS as an
innovation support system for Taiwanese higher education.
Expert Syst. Appl. 37(3), 1981–1990 (2010)
11. Chen, S.M., Cheng, S.H., Chiou, C.H.: Fuzzy multiattribute group
decision making based on intuitionistic fuzzy sets and evidential
reasoning methodology. Inf. Fusion 27, 215–227 (2016)
12. Chu, T.C., Lin, Y.: An extension to fuzzy MCDM. Comput.
Math. Appl. 57(3), 445–454 (2009)
13. Czogała, E., Łe˛ski, J.: Classical sets and fuzzy sets Basic defi-
nitions and terminology. In: Kacprzyk J (ed) Fuzzy and Neuro-
Fuzzy Intelligent Systems, pp. 1–26. Physica-Verlag HD (2000)
E. Afful-Dadzie et al.: Comparative State-of-the-Art Survey of Classical Fuzzy Set…
123

14. Dalman, H., Gu
¨zel, N., Sivri, M.: A fuzzy set-based approach to
multi-objective multi-item solid transportation problem under
uncertainty. Int. J. Fuzzy Syst. (2015). doi:10.1007/s40815-015-
0081-9
15. Dubois, D., Prade, H.: The three semantics of fuzzy sets. Fuzzy
Sets Syst. 90(2), 141–150 (1997)
16. Fontela, E., Gabus, A.: The DEMATEL observer (1976)
17. Fontela, E., Gabus, A.: DEMATEL, innovative methods, 2,
67–69 (1974) Report
18. Fung, R.Y., Chen, Y., Tang, J.: Estimating the functional rela-
tionships for quality function deployment under uncertainties.
Fuzzy Sets Syst. 157(1), 98–120 (2006)
19. Ghazinoory, S., Esmail Zadeh, A., Memariani, A.: Fuzzy SWOT
analysis. J. Intell. Fuzzy Syst. Appl. Eng. Technol. 18(1), 99–108
(2007)
20. Grattan-Guinness, I.: Fuzzy membership mapped onto intervals
and many-valued quantities. Math. Log. Q. 22(1), 149–160
(1976)
21. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation
method. Fuzzy Sets Syst. 119(1), 149–160 (2001)
22. Gwo-Hshiung, T.: Multiple Attribute Decision Making: Methods
and Applications. Springer, Berlin (2010)
23. Hwang, C.L., Yoon, K.: Multiple Criteria Decision Making.
Lecture Notes in Economics and Mathematical Systems, vol. 186.
Springer, Heidelberg (1981)
24. Kabak, M., Burmaog
˘lu, S., Kazanc¸og
˘lu, Y.: A fuzzy hybrid
MCDM approach for professional selection. Expert Syst. Appl.
39(3), 3516–3525 (2012)
25. Kahraman, C.: Fuzzy Multi-criteria Decision Making: Theory
and Applications with Recent Developments, vol. 16. Springer,
Berlin (2008)
26. Kahraman, C., C¸ evik Onar, S., O
¨ztays¸i, B.: Engineering eco-
nomic analyses using intuitionistic and hesitant fuzzy sets. J. In-
tell. Fuzzy Syst. (Preprint), 1–18 (2015)
27. Kahraman, C., Onar, S.C., Oztaysi, B.: Fuzzy multicriteria
decision-making: a literature review. Int. J. Comput. Intell. Syst.
8(4), 637–666 (2015)
28. Kaya, I., O
¨ztays¸i, B., Kahraman, C.: A two-phased fuzzy multi-
criteria selection among public transportation investments for
policy-making and risk governance. Int. J. Uncertain. Fuzziness
Knowl. Based Syst. 20(supp01), 31–48 (2012)
29. Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives:
Preferences and Value Trade-offs. Cambridge University Press,
Cambridge (1993)
30. Khalili-Damghani, K., Taghavifard, M., Olfat, L., Feizi, K.: A
hybrid approach based on fuzzy DEA and simulation to measure
the efficiency of agility in supply chain: real case of dairy
industry. Int. J. Manag. Sci. Eng. Manag. 6(3), 163–172 (2011)
31. Kleindorfer, P.R., Kunreuther, H.C., Schoemaker, P.J.H.: Deci-
sion Sciences: An Integrative Perspective. Cambridge University
Press, Cambridge (1993)
32. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic, vol. 4. Prentice
Hall, New Jersey (1995)
33. Kulak, O., Kahraman, C., O
¨ztaysi, B., Tanyas, M.: Multi-attribute
information technology project selection using fuzzy axiomatic
design. J. Enterp. Inf. Manag. 18(3), 275–288 (2005)
34. Lee, A.H., Chen, W.C., Chang, C.J.: A fuzzy AHP and BSC
approach for evaluating performance of IT department in the
manufacturing industry in Taiwan. Expert Syst. Appl. 34(1),
96–107 (2008)
35. Li, M.: Extension of axiomatic design principles for multicri-
teria decision making problems in intuitionistic fuzzy envi-
ronment. Math. Probl. Eng. 2013 (2013). doi:10.1155/2013/
813471
36. Li, D.F.: Decision and Game Theory in Management with Intu-
itionistic Fuzzy Sets, pp. 421–441. springer, Berlin (2014)
37. Li, M., Jin, L., Wang, J.: A new MCDM method combining QFD
with TOPSIS for knowledge management system selection from
the user’s perspective in intuitionistic fuzzy environment. Appl.
Soft Comput. 21, 28–37 (2014)
38. Mardani, A., Jusoh, A., Zavadskas, E.K.: Fuzzy multiple criteria
decision-making techniques and applications—two decades review
from 1994 to 2014. Expert Syst. Appl. 42(8), 4126–4148 (2015)
39. Molodtsov, D.: Soft set theory—first results. Comput. Math.
Appl. 37(4), 19–31 (1999)
40. Montes, I., Pal, N.R., Janis, V., Montes, S.: Divergence measures
for intuitionistic fuzzy sets. IEEE Trans. Fuzzy Syst. 23(2),
444–456 (2015)
41. Onar, S.C., Oztaysi, B., Otay, I
˙., Kahraman, C.: Multi-expert
wind energy technology selection using interval-valued intu-
itionistic fuzzy sets. Energy 90, 274–285 (2015)
42. Opricovic, S., Tzeng, G.H.: Compromise solution by MCDM
methods: a comparative analysis of VIKOR and TOPSIS. Eur.
J. Oper. Res. 156(2), 445–455 (2004)
43. Ouyang, Y., Pedrycz, W.: A new model for intuitionistic fuzzy
multi-attributes decision making. Eur. J. Oper. Res. 249(2),
677–682 (2016)
44. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356
(1982)
45. Ribeiro, R.A.: Fuzzy multiple attribute decision making: a review
and new preference elicitation techniques. Fuzzy Sets Syst. 78(2),
155–181 (1996)
46. Roy, B.: Classement et choix en pre
´sence de points de vue
multiples (la me
´thode ELECTRE). Rev. Franc¸aise Automat.,
Informat. Recherche Ope
´rationnelle 8, 57–75 (1968)
47. Roy, B.: Paradigms and challenges. In: Figueira J, Greco S,
Ehrogott M (eds) Multiple Criteria Decision Analysis: State of
the Art Surveys, pp. 3–24. Springer, New York (2005)
48. Roy, B., Vincke, P.: Multicriteria analysis: survey and new
directions. Eur. J. Oper. Res. 8(3), 207–218 (1981)
49. Saaty, T.L.: Decision Making for Leaders: The Analytic Hierar-
chy Process for Decisions in a Complex World. RWS Publica-
tions, Pittsburgh (1990)
50. Saaty, T.L.: Fundamentals of the analytic network process. In:
Proceedings of the 5th International Symposium on the Analytic
Hierarchy Process, pp. 12–14 (1999)
51. Suh, N.P.: The Principles of Design, vol. 990. Oxford University
Press, New York (1990)
52. Suh, N.P.: Axiomatic Design: Advances and Applications (The
Oxford Series on Advanced Manufacturing). Oxford University
Press, Oxford (2001)
53. Szmidt, E.: Distances and Similarities in Intuitionistic Fuzzy Sets.
Studies in Fuzziness and Soft Computing. Springer International
Publishing, New York (2014)
54. Tadic
´, S., Zec
ˇevic
´, S., Krstic
´, M.: A novel hybrid MCDM model
based on fuzzy DEMATEL, fuzzy ANP and fuzzy VIKOR for
city logistics concept selection. Expert Syst. Appl. 41(18),
8112–8128 (2014)
55. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010)
56. Triantaphyllou, E.: Multi-criteria Decision Making Methods: A
Comparative Study, vol. 44. Springer, New York (2000)
57. Tsai, W.H., Chou, W.C.: Selecting management systems for
sustainable development in SMEs: a novel hybrid model based on
DEMATEL, ANP, and ZOGP. Expert Syst. Appl. 36(2),
1444–1458 (2009)
58. Tseng, M.L.: Implementation and performance evaluation using
the fuzzy network balanced scorecard. Comput. Educ. 55(1),
188–201 (2010)
59. Tzeng, G.H., Huang, C.Y.: Combined DEMATEL technique with
hybrid MCDM methods for creating the aspired intelligent global
manufacturing & logistics systems. Ann. Oper. Res. 197(1),
159–190 (2012)
International Journal of Fuzzy Systems
123

60. Vahdani, B., Hadipour, H., Tavakkoli-Moghaddam, R.: Soft
computing based on interval valued fuzzy ANP-A novel
methodology. J. Intell. Manuf. 23(5), 1529–1544 (2012)
61. Vincke, J.P., Brans, P.: A preference ranking organization
method. The PROMETHEE method for MCDM. Manag. Sci.
31(6), 647–656 (1985)
62. Wang, L.X.: A Course in Fuzzy Systems, pp. 258–265. Prentice-
Hall Press, USA (1999)
63. Warfield, J.N.: Societal Systems: Planning, Policy, and Com-
plexity. Wiley, New York (1976)
64. Yang, X., Song, X., Qi, Y., Yang, J.: Constructive and axiomatic
approaches to hesitant fuzzy rough set. Soft. Comput. 18(6),
1067–1077 (2014)
65. Yu, D., Shi, S.: Researching the development of Atanassov
intuitionistic fuzzy set: using a citation network analysis. Appl.
Soft Comput. 32, 189–198 (2015)
66. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)
67. Zadeh, L.A.: The Concept of a Linguistic Variable and Its
Application to Approximate Reasoning, pp. 1–10. Springer, New
York (1974)
68. Zadeh, L.A.: Fuzzy logic and approximate reasoning. Synthese
30(3–4), 407–428 (1975)
69. Zeydan, M., C¸ olpan, C., C¸ obanog
˘lu, C.: A combined methodol-
ogy for supplier selection and performance evaluation. Expert
Syst. Appl. 38(3), 2741–2751 (2011)
70. Zhang, Q.: Fuzzy multiple attribute TOPSIS decision making
method based on intuitionistic fuzzy set theory. In Proceedings on
IFSA, pp. 1602–1605, (2005)
71. Zolfani, S.H., Sedaghat, M., Zavadskas, E.K.: Performance
evaluating of rural ICT centers (telecenters), applying fuzzy
AHP, SAW-G and TOPSIS Grey, a case study in Iran. Technol.
Econ. Dev. Econ. 18(2), 364–387 (2012)
Eric Afful-Dadzie holds a Ph.D. in Informatics. His current research
focuses on Decision Sciences with a special interest in Fuzzy multi-
criteria decision making (MCDM). His other research interests
include Knowledge Management, Management information systems,
and Data Analytics for Business Decisions.
Zuzana Komı
´nkova
´Oplatkova
´is an Associate Professor at Tomas
Bata University in Zlin. Her research interests include artificial
intelligence, soft computing, evolutionary techniques, symbolic
regression, and neural networks. She is the author of around 100
papers in journals, book chapters, and conference proceedings.
Luis Antonio Beltran Prieto received his Engineering degree in
Computational Systems from the Technological Institute of Celaya
(Mexico). Subsequently, he worked as a Lecturer in the Department
of Systems and Computing at Technological Institute of Celaya for 7
years. He is currently a Ph.D. student at Tomas Bata University in
Zlı
´n, Czech Republic. His scientific interest and fields of research
include artificial intelligence, evolutionary computing, cloud com-
puting, machine learning, and mobile applications development.
E. Afful-Dadzie et al.: Comparative State-of-the-Art Survey of Classical Fuzzy Set…
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