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Group decision making with intuitionistic fuzzy preference relations
Hülya Behret
⇑
Erciyes University, Department of Industrial Engineering, 38039 Kayseri, Turkey
article info
Article history:
Available online 13 April 2014
Keywords:
Intuitionistic fuzzy preference relation
Group decision making
Fuzzy aggregation
Additive consistency
Multiplicative consistency
abstract
The capability of intuitionistic fuzzy preference relation in representing imprecise or not reliable
judgments which exhibit affirmation, negation and hesitation characteristics make it an attractive
research area in group decision making. As traditional fuzzy set theory cannot be used to express all
the information in a situation as such, its applications are limited. In Zadeh’s fuzzy set, the membership
degree of an element is defined by a real value, and nonmembership is expressed by a complement of
membership. This membership definition actually ignores the decision maker’s hesitation in the decision
making process. The advantage of Atanassov’s intuitionistic fuzzy sets is the capability of representing
inevitably imprecise or not totally reliable judgments and the capability of expressing affirmation,
negation and hesitation with the help of membership definitions. The consistency of intuitionistic fuzzy
preference relations and the priority weights of experts gathered from these preference relations play
an important role in group decision making problems in order to reach an accurate decision result. In this
paper, we propose a group decision making process with the usage of intuitionistic fuzzy preference
relations where we mainly focus our attention on the investigation of consistency of intuitionistic fuzzy
preference relations. Initially, we present two different optimization models to minimize the deviations
from additive and multiplicative consistency respectively. The optimal deviation values obtained from
the model results enable us to improve the consistency of considered preference relations. Then, based
on consistent collective preference relations, two mathematical programming models are established to
obtain the priority weights, of which the first is a linear programming model considering additive and
the second one is a nonlinear model considering multiplicative consistency. Furthermore, a number of
numerical illustrations are presented to observe the validity and practicality of the models. Finally, com-
parative analyses were performed in order to examine the differences between fuzzy and intuitionistic
fuzzy preference relations and the results of the analyses showed that the priority vectors and ranking
of the alternatives maintained from fuzzy or intuitionistic fuzzy preference relations change significantly.
2014 Elsevier B.V. All rights reserved.
1. Introduction
Decision making can be considered as the mental processes in
which we make a selection among several alternative choices.
Making a decision implies that there are alternative choices to be
considered, and in such a case we want to choose the one that
has the highest probability of success or effectiveness and best fits
with our goals, desires, lifestyle or values. In the decision making
process, a decision maker (DM) is usually asked to give his/her
preferences over alternatives. In this process, preference relations
(referred to as pairwise comparison matrices, judgment matrices)
help us to explain DM’s preference information in decision making
problems of several fields. During the last decades, the concept of
preference relations has received an increasing attention and
several studies have been developed on this subject. In 2007, Xu
presented a comprehensive survey of preference relations [54].
In decision making problems, the experts’ preferences on
decision alternatives are commonly described by multiplicative
preference relations [32,33,62,50,53], fuzzy preference relations
[29,37,23,7–12,21,22,15,5,24,28,46] or linguistic preference rela-
tions [18,19,50,51,13,40,38].
However, in most real life decision making problems, the DMs
may not be able to provide his/her preferences for alternatives to
a certain degree due to lack of precise or sufficient level of
knowledge related to the problem, or the difficulty in explaining
explicitly the degree to which one alternative is better than others.
In these situations, there is usually a degree of uncertainty in
providing their preferences over the alternatives considered, which
makes the result of the preference process exhibit the characteris-
tics of affirmation, negation and hesitation [60]. The voting exam-
ple is an appropriate example of such a case, where ‘‘yes’’, ‘‘no’’ and
http://dx.doi.org/10.1016/j.knosys.2014.04.001
0950-7051/2014 Elsevier B.V. All rights reserved.
⇑
Tel.: +90 3524374901; fax: +90 3524375784.
E-mail address: hbehret@erciyes.edu.tr
Knowledge-Based Systems 70 (2014) 33–43
Contents lists available at ScienceDirect
Knowledge-Based Systems
journal homepage: www.elsevier.com/locate/knosys
‘‘abstain’’ votes are possible. Abstention votes may be considered
as votes which are ‘‘unclassifiable’’ and indicate the hesitation
and indeterminacy of the voter over the alternatives. As traditional
fuzzy set theory, introduced by Zadeh [64] cannot be used to
express all the information in such a situation, its applications
are limited [60]. In Zadeh’s fuzzy set, the membership degree of
an element is defined by a real value
l
, where 0 6
l
61 and non-
membership degree is expressed by 1
l
. This expression of
membership provides a powerful framework to characterize
vagueness and uncertainty [41]. However, the representation of
nonmembership as a complement of membership actually ignores
the DM’s hesitation in the decision making process. In 1986, Ata-
nassov extended Zadeh’s fuzzy set, which only assigns a member-
ship degree to each element, and introduced the concept of
intuitionistic fuzzy sets which simultaneously consider the degrees
of membership and nonmembership with hesitation index [1,2].
The advantage of Atanassov’s intuitionistic fuzzy sets is the capa-
bility of representing inevitably imprecise or not totally reliable
judgments [30] and the capability of expressing affirmation, nega-
tion and hesitation with the help of membership definitions.
The consistency of intuitionistic fuzzy preference relations
(IFPRs) and the priority weights of experts gathered from these
preference relations play an important role in group decision mak-
ing problems in order to reach an accurate decision result. In the
present study, a group decision making model with intuitionistic
fuzzy preference relations considering both aggregation of individ-
ual preference relations and consistency aspects is proposed.
Throughout this study, the consequences of additive consistent
and multiplicative consistent IFPRs on priority weights is examined.
A linear programming model considering additive consistency and
a nonlinear model considering multiplicative consistency has been
developed to calculate the priority weights. These models also
enable us to improve the consistency of considered preference rela-
tions whereby consistent individual preference relations before
aggregation can be obtained. Furthermore, some illustrative exam-
ples are presented in order to examine the validity and practicality
of the developed models. Numerical analyses have shown that
although the priority weight vectors of the individual preference
relations of the experts differ, the ranking of the individual priority
weights do not differ significantly according to the additive consis-
tent or multiplicative consistent intuitionistic fuzzy preference
relations. If we derive consistent preference relations (additive or
multiplicative consistent), the ranking of the alternatives obtained
from collective preference relation or aggregated priority vectors
will generally be the same. Additionally, in the current literature,
the comparison of the usage of fuzzy and intuitionistic fuzzy pref-
erence relations in the group decision making problems has not
been investigated. The analysis of the ranking of the alternatives
in two cases (fuzzy and intuitionistic fuzzy ones) is an interesting
study area. In the present study, the usage of fuzzy and intuitionis-
tic fuzzy preference relations in the group decision making prob-
lems is investigated. The differences of fuzzy and intuitionistic
fuzzy preference relations, priority vectors and ranking of the alter-
natives obtained from these preference relations are analyzed.
The results of the numerical illustrations showed that intuition-
istic fuzzy preference relations provide more accurate priority vec-
tors and rankings of alternatives by taking into consideration the
DMs’ affirmation, negation and hesitation with the help of mem-
bership definitions.
The remainder of the paper has been organized as follows: In
Section 2, a literature review on the subject is presented. In Sec-
tion 3, some basic concepts about IFPR are explained. In Section 4,
the relation between FIPRs and IFPRs and the consistency issues of
IFPRs are analyzed. Sections 5 and 6 provide two optimization
models to calculate the priority weight of additive consistent and
multiplicative consistent collective IFPRs respectively. In Section 7,
numerical examples are given to illustrate the validity and practi-
cality of the proposed methods. Section 8provides comparative
analyses of fuzzy and intuitionistic fuzzy preference relations and
Section 9concludes this paper.
2. Literature review
In the literature, Atanassov’s intuitionistic fuzzy set theory has
been studied by many researchers dealing with decision making
concept [3,25,27,26,61,55,56,42–44,16,17,34,39,6]. Szmidt and
Kacprzyk [35,36] introduced the definition of the intuitionistic
fuzzy preference relation (IFPR). In addition, they also studied the
consensus reaching process, and analyzed the extent of agreement
in a group of experts. Atanassov et al. [3] proposed an algorithm for
solving the multi-person multi-attribute decision making prob-
lems, in which the attribute weights are given as exact numerical
values and the attribute values are expressed in intuitionistic fuzzy
numbers. Li [25] investigated multi-attribute decision making with
intuitionistic fuzzy information and established several linear pro-
gramming models to generate optimal weights for attribute. Lin
et al. [27] proposed a new method for handling multiple attribute
fuzzy decision making problems, where the characteristics of the
alternatives are represented by Atanassov’s intuitionistic fuzzy
sets. Li et al. [26] presented the fractional programming method
for multiple attribute group decision making using Atanassov’s
intuitionistic fuzzy sets. Xu [56] investigated group decision mak-
ing problems based on IFPR and incomplete IFPR. He used averag-
ing operators to aggregate intuitionistic preference information
and applied score and accuracy functions for the ranking and
selection of alternatives.
Priority weight generation from the preference relations is the
main issue of group decision making concept. Preference relation
presents a common format which provides the opportunity to
explain DM’s preference information in decision making problems
by pairwise comparisons [45]. However, in the process of decision
making it is very difficult for a DM to construct a consistent pref-
erence relation. Since an inconsistent preference relation may lead
to wrong conclusions, priority weight generation methods should
take into consideration the consistency of preference relations.
Most of the priority weight generation methods in the fuzzy set
theory papers in the literature are based on fuzzy interval prefer-
ence relations (FIPR), introduced by Xu [48]. Xu and Chen [61] pro-
posed a number of linear programming models for deriving the
priority weights from various fuzzy interval preference relations
considering additive and multiplicative consistency. Genç et al.
[15] showed that the consistency and the priority weights can be
derived by simple formulas based on interval multiplicative transi-
tivity rather than linear programming models proposed by Xu and
Chen [61]. Furthermore, these authors proposed two approaches in
order to estimate missing values of an incomplete FIPR. Xu [58]
investigated the consistency of fuzzy interval preference relations.
Initially he established a quadratic programming model to estab-
lish the importance weights of experts. He then proposed two
approaches to constructing additive and multiplicative consistent
fuzzy interval preference relations. Additionally, he showed the
relationship between the consistency of individual FIPRs and the
consistency of collective FIPR.
Gau and Buehrer [14] introduced the concept of vague sets
(interval valued fuzzy sets) and Bustince and Burillo [4] showed
that the notion of vague sets is actually that of intuitionistic fuzzy
sets. This argument assists researchers to construct priority weight
generation methods based on intuitionistic fuzzy preference rela-
tions. Xu [55] defined the concept of additive consistent intuition-
istic fuzzy preference relation (IFPR) and established a method for
estimating criteria weights from intuitionistic fuzzy preference
34 H. Behret / Knowledge-Based Systems 70 (2014) 33–43
relations. In another study, Xu [57] defined the concepts of additive
consistent intuitionistic judgment matrix, multiplicative consis-
tent intuitionistic judgment matrix and score matrix of intuitionis-
tic fuzzy decision matrix for the situations where the information
on attribute weights is incomplete. He established some simple
linear programming models using two transformation functions,
from which the attribute weights can be derived. In his study,
Wei [42] firstly proposed an optimization model based on the max-
imizing deviation method to determine the attribute weights
which are partially known. He also proposed another optimization
model for the special situations in which the information regarding
attribute weights is completely unknown. In another study, Wei
[43] established an optimization model based on gray relational
analyses (GRA) method to determine the attribute weights which
are partially known. In a more recent study, Wei [44] established
an optimization model based on the basic ideal of traditional gray
relational analysis (GRA) method, by which the attribute weights
can be determined. For the special situations in which the informa-
tion about attribute weights is completely unknown, he estab-
lished another optimization model. Additionally, he extended the
results to an interval-valued intuitionistic fuzzy environment,
and developed modified GRA method for interval-valued intuition-
istic fuzzy multiple attribute decision making with partially known
attribute weight information. Gong et al. [16] presented goal pro-
gramming models to derive the priority vector of the IFPR based
on the multiplicative consistent definition of the FIPR. More
recently, Gong et al. [17] proposed a least squares method and a
goal programming method to determine the priority vector of IFPR
based on additive consistency. Su et al. [34] developed an interac-
tive method to solve the dynamic intuitionistic fuzzy multi-attri-
bute group decision making problems. In addition, they
measured the consensus level of the group preferences by using
spearman correlation coefficient. Chen and Yang [6] established
some optimization models to determine the attribute weights for
the incomplete attribute weight information. In more recent stud-
ies, Wang and Li [41] extended the previous work of Wang et al.
[39]. They proposed a framework to manage multi-attribute group
decision making problems with incomplete pairwise comparison
preference over decision alternatives where qualitative and quan-
titative attribute values are furnished as linguistic variables and
crisp numbers, respectively. They then converted the attribute
assessments to interval-valued intuitionistic fuzzy numbers (IVI-
FNs). In their study, group consistency and inconsistency indices
are introduced for incomplete pairwise comparison preference
relations on alternatives and an auxiliary linear programming
model is developed to obtain unified attribute weights by minimiz-
ing the group inconsistency index under certain constraints. Zhang
et al. [65] proposed a new type of preference relation which is
intuitionistic fuzzy linguistic preference relation. They presented
an approach to group decision making based on this new type of
preference relation and applied the score and accuracy functions
to the ranking and selection of alternatives. Xu [59] investigated
the compatibility of intuitionistic preference relations. He pro-
posed some novel compatibility measures of intuitionistic fuzzy
information, and used them to derive a consensus reaching proce-
dure in group decision making with intuitionistic preference rela-
tions. Paternain et al. [31] presented a construction method for
Atanassov’s intuitionistic fuzzy preference relations based on fuzzy
preference relations. They considered the ignorance of the expert
in the construction of the intuitionistic fuzzy preference relations
and they proposed two generalizations of the weighted voting
strategy to work with Atanassov’s intuitionistic fuzzy preference
relations.
Since the usage of fuzzy preference relations in decision making
concept relies on very early studies, the usage of intuitionistic
fuzzy preference relations in this concept is relatively more recent.
When we investigate the current literature on the fuzzy group
decision making concept, we realize that the studies are mostly
conducted on fuzzy interval preference relations (FIPR), [61,15–
17,39,41] since a fuzzy interval preference relation is very suitable
for expressing the decision maker’s uncertain preference informa-
tion . However, as mentioned above, Atanassov’s intuitionistic
fuzzy set additionally gives the opportunity of representing inevi-
tably imprecise or not totally reliable judgments and also express-
ing affirmation, negation and hesitation with the help of
membership definitions. Therefore, in this paper we studied intui-
tionistic fuzzy preference relations (IFPRs) in the decision making
concept. In group decision making problems, the decisions are
made by a group of experts where the overall decision should be
reached from the aggregated values of individual decisions. How-
ever, consistency of the preference relations is an important prop-
erty in group decision making problems in order to draw the
correct conclusions. The priority weights of consistent preference
relations allow us to determine accurate ranking of the alterna-
tives. A group decision making problem should consider both
aggregation of expert decisions and consistency of preference
relations.
In this paper, we aim to propose a group decision making model
with intuitionistic fuzzy preference relations considering both
aggregation of individual preference relations and consistency
aspects. To achieve this, we aim to investigate the consequences
of additive consistent and multiplicative consistent IFPRs on prior-
ity weights. For this reason, we develop a linear programming
model considering additive consistency and a nonlinear model
considering multiplicative consistency to determine the priority
weights. These models also enable us to improve the consistency
of considered preference relations whereby we can determine con-
sistent individual preference relations before aggregation.
Furthermore, when we look at the current literature, we notice
that the comparison of the usage of fuzzy and intuitionistic fuzzy
preference relations in the group decision making problems has
not been investigated. The analyses of the ranking of the alterna-
tives in two cases (fuzzy and intuitionistic fuzzy ones) is an inter-
esting study area. In this paper we make this comparison by using
the proposed models for both preference relations. We analyze the
differences of fuzzy and intuitionistic fuzzy preference relations,
priority vectors and ranking of the alternatives obtained from these
preference relations.
3. Preliminaries
The notion of intuitionistic fuzzy sets is introduced as [1];
A¼fhx
i
;
l
A
ðx
i
Þ;
v
A
ðx
i
Þijx
i
2Xgð1Þ
which is characterized by a membership function
l
A
:X?[0,1] and
a nonmembership function
v
A
:X?[0,1] with the condition:
06
l
A
ðx
i
Þþ
v
A
ðx
i
Þ61;8x
i
2Xð2Þ
The value,
p
A
(x
i
)=1
l
A
(x
i
)
v
A
(x
i
) is called the indeterminacy
degree or hesitation degree of x
i
to A,06
p
A
(x
i
)61. Especially, if
p
A
(x
i
)=1
l
A
(x
i
)
v
A
(x
i
)=0,"x
i
2Xthen, the intuitionistic fuzzy
set A is reduced to a common fuzzy set.
Definition 2.1. [56]: An intuitionistic fuzzy preference relation B
on Xis defined as a matrix B=(b
ij
)
nn
XXwhere b
ij
=
h(x
i
x
j
),
l
(x
i
x
j
),
v
(x
i
x
j
)ifor all i,j=1,2,...,n. Let b
ij
=(
l
ij
,
v
ij
)isan
intuitionistic fuzzy value, composed by the certainty degree
l
ij
to
which x
i
is preferred to x
j
and the certainty degree
v
ij
to which x
i
is
non-preferred to x
j
, and
p
ij
=1
l
ij
v
ij
is interpreted as the
hesitation degree to which x
i
is preferred to x
j
. Moreover,
l
ij
and
v
ij
satisfy the following condition,
H. Behret / Knowledge-Based Systems 70 (2014) 33–43 35
06
l
ij
þ
v
ij
61;
l
ji
¼
v
ij
;
l
ii
¼
v
ii
¼0:5;
for all i;j¼1;2;...;nð3Þ
In the real life decision making problems, a decision is usually made
by a group of experts, E
k
(k=1,2,... ,m) with different weights
k
k
=(k
1
,k
2
,...,k
m
)
T
in the decision process. In such cases, the indi-
vidual preference relations of the experts are aggregated to derive a
collective preference relation B¼ðb
ij
Þ
nn
.
Definition 2.2 [63]. Let b
ðkÞ
ij
¼ð
l
ðkÞ
ij
;
v
ðkÞ
ij
Þ;k¼1;2;...;mbe a col-
lection of intuitionistic fuzzy values and let k
k
=(k
1
,k
2
,...,k
m
)
T
be the weight vector of b
ð1Þ
ij
;b
ð2Þ
ij
;...;b
ðmÞ
ij
;k
k
>0;P
m
k¼1
k
k
¼1, then
the aggregation B¼ðb
ij
Þ
nn
of B
ðkÞ
¼b
ðkÞ
ij
nn
is also an intuitionis-
tic fuzzy preference relation, where
b
ij
¼
l
ij
;
v
ij
;
l
ij
¼X
m
k¼1
k
k
l
ðkÞ
ij
;
v
ij
¼X
m
k¼1
k
k
v
ðkÞ
ij
;
06
l
ij
þ
v
ij
61;
l
ji
¼
v
ij
;
l
ii
¼
v
ii
¼0:5;
for all i;j¼1;2;...;n and k ¼1;2;...;mð4Þ
4. Consistency of intuitionistic preference relation
Consider that each intuitionistic fuzzy value b
ij
=(
l
ij
,
v
ij
) must
satisfy the condition 0 6
l
ij
+
v
ij
61, i.e.,
l
ij
61
v
ij
. This condition
is the same as the condition under which two real numbers
form an interval [20]. As a result, an intuitionistic fuzzy preference
relation can be transformed into an interval fuzzy preference
relation [50].
Definition 3.1 [48]. A fuzzy interval preference relation is defined
as R=(r
ij
)
nn
XXwhich satisfies,
r
ij
¼r
ij
;r
þ
ij
hi
;r
þ
ij
Pr
ij
P0;r
ij
þr
þ
ij
¼r
þ
ij
þr
ij
¼1;
r
ii
¼r
þ
ii
¼0:5for all i;j¼1;2;...;nð5Þ
where r
ij
represents the interval valued preference degree of the
alternative x
i
over x
j
,r
ij
and r
þ
ij
are lower and upper limits of r
ij
,
respectively.
Definition 3.2 [54]. A fuzzy interval preference relation R=(r
ij
)
nn
where r
ij
¼r
ij
;r
þ
ij
is additive consistent if a normalized priority
vector exists, w=(w
1
,w
2
,...,w
n
)
T
such that,
r
ij
60:5ðw
i
w
j
þ1Þ6r
þ
ij
;for all i ¼1;2;...;n1;
j¼iþ1;...;n;w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;ð6Þ
where Ris called an additive consistent fuzzy interval preference
relation.
Since an intuitionistic fuzzy preference relation can be
transformed into a fuzzy interval preference relation where
r
ij
¼
l
ij
;r
þ
ij
¼1
v
ij
, for all i,j=1,2,...,n, Eq. (6) can be trans-
formed as,
l
ij
60:5ðw
i
w
j
þ1Þ61
v
ij
for all i ¼1;2;...;n1;
j¼iþ1;...;nw
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1ð7Þ
Definition 3.3. An intuitionistic fuzzy preference relation
B=(b
ij
)
nn
is an additive consistent preference relation, where
b
ij
=[
l
ij
,
v
ij
] if there is a vector w=(w
1
,w
2
,...,w
n
)
T
such that
Eq. (7) holds.
Definition 3.4 [54]. A fuzzy interval preference relation R=(r
ij
)
nn
where r
ij
¼r
ij
;r
þ
ij
is multiplicative consistent if a normalized
priority vector w=(w
1
,w
2
,...,w
n
)
T
exists such that,
r
ij
6w
i
w
i
þw
j
6r
þ
ij
;for all i ¼1;2;...;n1;
j¼iþ1;...;n;w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;ð8Þ
where Ris called a multiplicative consistent fuzzy interval prefer-
ence relation.
We can use the transformation of intuitionistic preference
relation into fuzzy interval preference relation to check the
multiplicative consistency. Thus, we transform Eq. (8) as below,
l
ij
6w
i
w
i
þw
j
61
v
ij
;for all i ¼1;2;...;n1;
j¼iþ1;...;n;w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;ð9Þ
Definition 3.5. An intuitionistic fuzzy preference relation
B¼ðb
ij
Þ
nn
is a multiplicative consistent preference relation, where
b
ij
=[
l
ij
,
v
ij
] if there is a vector w=(w
1
,w
2
,...,w
n
)
T
such that Eq.
(9) holds.
5. Priority weight generation method for additive consistent
intuitionistic fuzzy preference relation
Let B=(b
ij
)
nn
be an intuitionistic fuzzy preference relation
where b
ij
=(
l
ij
,
v
ij
). If B=(b
ij
)
nn
is an additive consistent intuition-
istic fuzzy preference relation, then a priority vector w=(w
1
,w
2
,
...,w
n
)
T
of Bexists which satisfies Eq. (7). Since the preferences
of decision makers are very subjective and depend on personal
psychological aspects, this equation does not always hold [58].In
this situation, B=(b
ij
)
nn
will not be an additive consistent
intuitionistic fuzzy preference relation, then we relax Eq. (7) by
introducing the non-negative deviation variables d
ij
and d
þ
ij
,
i=1,2,...,n1; j=i+1,... ,n:
l
ij
d
ij
60:5ðw
i
w
j
þ1Þ61
v
ij
þd
þ
ij
for all i ¼1;2;...;n1;j¼iþ1;...;n
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1ð10Þ
As the deviation variables d
ij
and d
þ
ij
become smaller, B becomes
closer to an additive consistent intuitionistic fuzzy preference rela-
tion. As a result, in order to find the smallest deviation variables Xu
[55] developed the following linear optimization model:
d¼Min X
n1
i¼1
X
n
j¼iþ1
d
ij
þd
þ
ij
s:t:0:5ðw
i
w
j
þ1Þþd
ij
P
l
ij
0:5ðw
i
w
j
þ1Þd
þ
ij
61
v
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-1Þ
Solving this model (M-1), we determine the optimal deviation
values, _
d
ij
and _
d
þ
ij
;i;j¼1;2;...;n. Especially, if d
⁄
= 0, or in other
words, _
d
ij
¼_
d
þ
ij
¼0;for all;i;j¼1;2;...;n, then Bis an additive
consistent intuitionistic fuzzy preference relation. Otherwise, we
may use the nonzero deviation values to improve the additive con-
sistency as;
36 H. Behret / Knowledge-Based Systems 70 (2014) 33–43
_
B¼ð_
b
ij
Þ
nn
;_
b
ij
¼ð _
l
ij
;_
v
ij
Þ;_
l
ij
¼
l
ij
_
d
ij
;
_
v
ij
¼
v
ij
_
d
þ
ij
;i;j¼1;2;...;nð11Þ
Here, _
Bis the improved additive consistent intuitionistic fuzzy pref-
erence relation. Based on the improved preference relation, we
establish the following optimization models to calculate the priority
weight vector:
w
i
¼min w
i
s:t:0:5ðw
i
w
j
þ1ÞP_
l
ij
0:5ðw
i
w
j
þ1Þ61_
v
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-2Þ
w
þ
i
¼max w
i
s:t:0:5ðw
i
w
j
þ1ÞP_
l
ij
0:5ðw
i
w
j
þ1Þ61_
v
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-3Þ
Solving the models (M-2) and (M-3), we determine the weight
intervals w
i
;w
þ
i
,ifw
i
¼w
þ
i
for all i, and we determine a unique
priority vector w=(w
1
,w
2
,...,w
n
)
T
from the improved additive
consistent intuitionistic fuzzy preference relation.
In the models (M-1)–(M-3) we only consider an individual
intuitionistic fuzzy preference relation B=(b
ij
)
nn
, however a deci-
sion is usually made by a group of experts, E
k
(k=1,2,...,m) with
different weights k
k
=(k
1
,k
2
,...,k
m
)
T
in the decision process. In
such cases, the individual preference relations of the experts are
aggregated to provide a collective preference relation B¼ð_
b
ij
Þ
nn
.
In this situation, the consistency of the collective preference rela-
tion, as well as individual relations, need to be checked.
If we aggregate individual preference relations of the experts,
then Eq. (10) will be extended by using Eq. (4) as below for the
additive consistency of collective intuitionistic fuzzy preference
relation:
X
m
k¼1
kk
lðkÞ
ij
d
ij 60:5ðwiwjþ1Þ
61X
m
k¼1
kkvðkÞ
ij þdþ
ij for all
i¼1;2;...;n1;j¼iþ1;...;n;
k¼1;2;...;mw
iP0;i¼1;2;...;n;X
n
i¼1
wi¼1
ð12Þ
As the deviation variables d
ij
and d
þ
ij
become smaller, Bbecomes
closer to an additive consistent collective intuitionistic fuzzy prefer-
ence relation. We establish the following model to derive the small-
est deviation values:
d¼Min X
n1
i¼1
X
n
j¼iþ1
d
ij
þd
þ
ij
s:t:0:5ðw
i
w
j
þ1Þþd
ij
PX
m
k¼1
k
k
l
ðkÞ
ij
0:5ðw
i
w
j
þ1Þd
þ
ij
61X
m
k¼1
k
k
v
ðkÞ
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0
i¼1;2;...;n1;j¼iþ1;...;n;k¼1;2;...;m:
ðM-4Þ
Solving the model (M-4), we obtain the optimal deviation values. In
fact, if all the individual intuitionistic fuzzy preference relations are
additive consistent, then the collective preference relation
B¼ðb
ij
Þ
nn
will also be additive consistent which means
d
¼Min P
n1
i¼1
P
n
j¼iþ1
d
ij
þd
þ
ij
¼0. Based on the improved prefer-
ence relations, we develop the following optimization model to
obtain the priority vector for the collective preference relation:
w
i
¼min w
i
and w
þ
i
¼max w
i
s:t:0:5ðw
i
w
j
þ1ÞPX
m
k¼1
k
k
_
l
ðkÞ
ij
0:5ðw
i
w
j
þ1Þ61X
m
k¼1
k
k
_
v
ðkÞ
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
i¼1;2;...;n1;j¼iþ1;...;n;k¼1;2;...;m:
ðM-5Þ
The solution of (M-5) determines the priority vector
w¼ðw
1
;w
2
;...;w
n
Þ
T
for the additive consistent collective intui-
tionistic fuzzy preference relation.
6. Priority weight generation method for multiplicative
consistent intuitionistic fuzzy preference relation
We consider B=(b
ij
)
nn
is an intuitionistic fuzzy preference
relation where b
ij
=(
l
ij
,
v
ij
). If B=(b
ij
)
nn
is a multiplicative
consistent intuitionistic fuzzy preference relation, then there is a
priority vector w=(w
1
,w
2
,...,w
n
)
T
of Bwhich satisfies Eq. (9).
However B=(b
ij
)
nn
may not be a multiplicative fuzzy intuitionistic
preference relation and Eq. (9) does not always hold. In such cases,
we introduce non-negative deviation variables d
ij
and d
þ
ij
,
i=1,2,...,n1; j=i+1,... ,nto relax the condition Eq. (9):
l
ij
d
ij
6w
i
w
i
þw
j
61
v
ij
þd
þ
ij
;
for all i ¼1;2;...;n1;j¼iþ1;...;n;w
i
P0;
i¼1;2;...;n;X
n
i¼1
w
i
¼1;ð13Þ
Considering the smaller the deviation variables d
ij
and d
ij
+
in
Eq. (13), the closer B is to a multiplicative consistent fuzzy intui-
tionistic preference relation, we develop the nonlinear optimization
model:
c
¼Min X
n1
i¼1
X
n
j¼iþ1
d
ij
þd
þ
ij
s:t:w
i
w
i
þw
j
þd
ij
P
l
ij
w
i
w
i
þw
j
d
þ
ij
61
v
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-6Þ
Solving the model (M-6), we determine the optimal deviation
values, _
d
ij
and _
d
þ
ij
;i;j¼1;2;...;n. In particular, if
c
⁄
= 0, i.e.,
_
d
ij
¼_
d
þ
ij
¼0;for all;i;j¼1;2;...;n, then B is a multiplicative con-
sistent intuitionistic fuzzy preference relation, we can utilize the
nonzero deviation values to improve the multiplicative consistency
as;
H. Behret / Knowledge-Based Systems 70 (2014) 33–43 37
B¼ð
_
b
ij
Þ
nn
;_
b
ij
¼ð_
l
ij
;_
v
ij
Þ;_
l
ij
¼
l
ij
_
d
ij
;_
v
ij
¼
v
ij
_
d
þ
ij
;
i;j¼1;2;...;nð14Þ
Here, _
Bis the improved multiplicative consistent intuitionistic fuzzy
preference relation. In order to derive the priority weight vector
w=(w
1
,w
2
,...,w
n
)
T
, we establish the nonlinear programming
models:
w
i
¼min w
i
s:t:w
i
w
i
þw
j
P_
l
ij
w
i
w
i
þw
j
61_
v
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-7Þ
w
þ
i
¼max w
i
s:t:w
i
w
i
þw
j
P_
l
ij
w
i
w
i
þw
j
61_
v
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-8Þ
The solution to the models (M-7) and (M-8), give the weight inter-
vals w
i
;w
þ
i
,ifw
i
¼w
þ
i
for all i, then we determine a unique pri-
ority vector w=(w
1
,w
2
,...,w
n
)
T
from the improved multiplicative
consistent intuitionistic fuzzy preference relation.
The models (M-6)–(M-8) only consider an individual intuition-
istic fuzzy preference relation B=(b
ij
)
nn
, however for the decisions
made by a group of experts, E
k
(k=1,2,...,m) with different
weights k
k
=(k
1
,k
2
,...,k
m
)
T
, the individual preference relations of
the experts are aggregated to obtain a collective preference rela-
tion B¼ðb
ij
Þ
nn
. If we aggregate individual preference relations
of the experts, then Eq. (13) will be extended by using Eq. (4) for
the multiplicative consistency of collective intuitionistic fuzzy
preference relation as below:
X
m
k¼1
k
k
l
ðkÞ
ij
d
ij
6w
i
w
i
þw
j
61X
m
k¼1
k
k
v
ðkÞ
ij
þd
þ
ij
for all i ¼1;2;...;n1;j¼iþ1;...;n;
k¼1;2;...;mw
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1ð15Þ
Based on Eq. (14) we establish the following model to obtain the
smallest deviation values:
c
¼Min X
n1
i¼1
X
n
j¼iþ1
d
ij
þd
þ
ij
s:t:w
i
w
i
þw
j
þd
ij
PX
m
k¼1
k
k
l
ðkÞ
ij
w
i
w
i
þw
j
d
þ
ij
61X
m
k¼1
k
k
v
ðkÞ
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0
i¼1;2;...;n1;j¼iþ1;...;n;k¼1;2;...;m:
ðM-9Þ
Solving this model (M-4), we obtain the optimal deviation values.
As in the additive consistency, if all the individual intuitionistic
fuzzy preference relations are multiplicative consistent, then the
collective preference relation B¼ðb
ij
Þ
nn
will also be multiplicative
consistent which means c
¼Min P
n1
i¼1
P
n
j¼iþ1
d
ij
þd
þ
ij
¼0. Based
on the improved preference relations, we establish the following
nonlinear optimization model to obtain the priority vector for the
multiplicative consistent collective preference relation:
w
i
¼min w
i
and w
þ
i
¼max w
i
s:t:w
i
w
i
þw
j
PX
m
k¼1
k
k
_
l
ðkÞ
ij
w
i
w
i
þw
j
61X
m
k¼1
k
k
_
v
ðkÞ
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
i¼1;2;...;n1;j¼iþ1;...;n;k¼1;2;...;m:
ðM-10Þ
Solution of the model (M-10) determines the priority vector
w¼w
1
;w
2
;...;w
n
ðÞ
T
for the multiplicative consistent collective
intuitionistic fuzzy preference relation.
7. Numerical illustration
A group of decision makers compare alternative facade clothing
systems for the surface clothing of a building according to their
functional properties. The group involves three experts; E
1
= civil
engineer, E
2
= constructor and E
3
= architect, E
k
(k= 1, 2 and 3)
whose weight vector is k= (0.5, 0.2, 0.3)
T
. The experts compare five
alternative systems which are x
1
= plastic painting, x
2
= compact
laminate clothing, x
3
= wood clothing, x
4
= ceramic clothing and
x
5
= natural stone clothing, x
i
(i= 1, 2, 3, 4, 5). All the experts indi-
vidually compare each pair of criteria x
i
and x
j
, and give his/her
intuitionistic fuzzy preference value b
ðkÞ
ij
¼
l
ðkÞ
ij
;
v
ðkÞ
ij
, composed
by the certainty degree
l
ðkÞ
ij
to which x
i
is preferred to x
j
and the
certainty degree
v
ðkÞ
ij
to which x
i
is non-preferred to x
j
, and then
develop the following intuitionistic fuzzy preference relations
B
ðkÞ
¼b
ðkÞ
ij
55
:
B
ð1Þ
¼
ð0:5;0:5Þð0:3;0:7Þð0:3;0:6Þð0:5;0:5Þð0:1;0:9Þ
ð0:7;0:3Þð0:5;0:5Þð0:4;0:6Þð0:6;0:2Þð0:2;0:7Þ
ð0:6;0:3Þð0:6;0:4Þð0:5;0:5Þð0:7;0:1Þð0:4;0:6Þ
ð0:5;0:5Þð0:2;0:6Þð0:1;0:7Þð0:5;0:5Þð0:1;0:8Þ
ð0:9;0:1Þð0:7;0:2Þð0:6;0:4Þð0:8;0:1Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
B
ð2Þ
¼
ð0:5;0:5Þð0:6;0:3Þð0:1;0:8Þð0:5;0:5Þð0:3;0:7Þ
ð0:3;0:6Þð0:5;0:5Þð0:0;1:0Þð0:3;0:7Þð0:1;0:8Þ
ð0:8;0:1Þð1:0;0:0Þð0:5;0:5Þð0:6;0:3Þð0:5;0:3Þ
ð0:5;0:5Þð0:7;0:3Þð0:3;0:6Þð0:5;0:5Þð0:3;0:6Þ
ð0:7;0:3Þð0:8;0:1Þð0:3;0:5Þð0:6;0:3Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
B
ð3Þ
¼
ð0:5;0:5Þð0:0;0:9Þð0:1;0:7Þð0:3;0:6Þð0:5;0:5Þ
ð0:9;0:0Þð0:5;0:5Þð0:6;0:3Þð0:6;0:4Þð0:8;0:2Þ
ð0:7;0:1Þð0:3;0:6Þð0:5;0:5Þð0:5;0:2Þð0:7;0:2Þ
ð0:6;0:3Þð0:4;0:6Þð0:2;0:5Þð0:5;0:5Þð0:4;0:5Þ
ð0:5;0:5Þð0:2;0:8Þð0:2;0:7Þð0:5;0:4Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Based on B
ðkÞ
¼ðb
ðkÞ
ij
Þ
55
we first apply the model (M-1) to the indi-
vidual intuitionistic fuzzy preference relations in order to check the
additive consistency. Applying (M-1), we find the optimal objective
38 H. Behret / Knowledge-Based Systems 70 (2014) 33–43
values d
ðkÞ
as d
ð1Þ
¼0:733, d
ð2Þ
¼1:400, d
ð3Þ
¼0:927 and the opti-
mal nonzero deviation values ( _
d
ðkÞ
ij
and _
d
þ
ðkÞ
ij
):
_
d
þ
ð1Þ
12
¼_
d
ð1Þ
21
¼0:133;_
d
þ
ð1Þ
15
¼_
d
ð1Þ
51
¼0:133;_
d
þ
ð1Þ
42
¼_
d
ð1Þ
24
¼0:033;_
d
þ
ð1Þ
43
¼_
d
ð1Þ
34
¼0:033;_
d
þ
ð1Þ
45
¼_
d
ð1Þ
54
¼0:033;
_
d
þ
ð2Þ
21
¼_
d
ð2Þ
12
¼0:100;_
d
þ
ð2Þ
13
¼_
d
ð2Þ
31
¼0:050;_
d
þ
ð2Þ
41
¼_
d
ð2Þ
14
¼0:050;
_
d
þ
ð2Þ
23
¼_
d
ð2Þ
32
¼0:250;_
d
þ
ð2Þ
24
¼_
d
ð2Þ
42
¼0:150;_
d
þ
ð2Þ
25
¼_
d
ð2Þ
52
¼0:100;
_
d
þ
ð3Þ
12
¼_
d
ð3Þ
21
¼0:100;_
d
þ
ð3Þ
14
¼_
d
ð3Þ
41
¼0:100;_
d
þ
ð3Þ
23
¼_
d
ð3Þ
32
¼0:063;_
d
þ
ð3Þ
24
¼_
d
ð3Þ
42
¼0:200:
Since, d
ðkÞ
–0;for k ¼1;2;3 then none of the B
(k)
is additive consis-
tent. To improve the additive consistency of B
(k)
, we use optimal
deviation values _
d
ðkÞ
ij
and _
d
þ
ðkÞ
ij
and construct the improved additive
consistent intuitionistic fuzzy preference relations _
B
ðkÞ
by applying
Eq. (11):
_
Bð1Þ¼
ð0:5;0:5Þð0:3;0:567Þð0:3;0:6Þð0:5;0:5Þð0:1;0:767Þ
ð0:567;0:3Þð0:5;0:5Þð0:4;0:6Þð0:567;0:2Þð0:2;0:7Þ
ð0:6;0:3Þð0:6;0:4Þð0:5;0:5Þð0:667;0:1Þð0:4;0:6Þ
ð0:5;0:5Þð0:2;0:567Þð0:1;0:667Þð0:5;0:5Þð0:1;0:767Þ
ð0:767;0:1Þð0:7;0:2Þð0:6;0:4Þð0:767;0:1Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
_
B
ð2Þ
¼
ð0:5;0:5Þð0:5;0:3Þð0:1;0:75Þð0:45;0:5Þð0:3;0:7Þ
ð0:3;0:5Þð0:5;0:5Þð0:0;0:75Þð0:3;0:55Þð0:1;0:7Þ
ð0:75;0:1Þð0;75;0:0Þð0:5;0:5Þð0:6;0:3Þð0:5;0:3Þ
ð0:5;0:45Þð0:55;0:3Þð0:3;0:6Þð0:5;0:5Þð0:3;0:6Þ
ð0:7;0:3Þð0:7;0:1Þð0:3;0:5Þð0:6;0:3Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
_
B
ð3Þ
¼
ð0:5;0:5Þð0:0;0:8Þð0:1;0:7Þð0:3;0:5Þð0:5;0:5Þ
ð0:8;0:0Þð0:5;0:5Þð0:6;0:237Þð0:6;0:2Þð0:8;0:2Þ
ð0:7;0:1Þð0:237;0:6Þð0:5;0:5Þð0:5;0:2Þð0:7;0:2Þ
ð0:5;0:3Þð0:2;0:6Þð0:2;0:5Þð0:5;0:5Þð0:4;0:5Þ
ð0:5;0:5Þð0:2;0:8Þð0:2;0:7Þð0:5;0:4Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Since the optimal objective values of the improved preference rela-
tions will be equal to zero, ð_
d
ðkÞ
¼0 for k= 1, 2, 3, we establish mod-
els (M-2) and (M-3) to derive the individual priority weight vector
of each expert w
ðkÞ
¼w
ðkÞ
1
;w
ðkÞ
2
;w
ðkÞ
3
;w
ðkÞ
4
;w
ðkÞ
5
T
where k=1, 2, 3.
The results of the models are summarized in Table 1.
The results shown in Table 1 indicate that x
5
= natural stone
clothing is the best alternative for civil engineer x
3
= wood clothing
is the best for constructor and x
2
= compact laminate clothing is
the best for architect.
The models (M-1)–(M-3) only consider individual intuitionistic
fuzzy preference relations, however in our group decision making
problem, the decision is made by three different experts with the
weights k= (0.5,0.2,0.3)
T
. In such cases, the individual preference
relations of the experts should be aggregated to determine a collec-
tive preference relation B¼ð
b
ij
Þ
nn
. In such a situation, the consis-
tency of the collective preference relation needs to be checked. We
aggregate the improved individual preference relations of the
experts by using Eq. (4) and apply (M-4) to obtain the smallest
deviation values for the collective preference relation. The result
of the model (M-4) implies that the optimal objective value is
equal to zero, d
¼Min P
n1
i¼1
P
n
j¼iþ1
d
ij
þd
þ
ij
¼0
which means
the collective preference relation is additive consistent. This is an
expected result since we aggregated the improved individual
preference relations to determine the collective preference rela-
tion. Following this, we apply (M-5) to calculate the priority vector
for the additive consistent collective intuitionistic fuzzy preference
relation and obtain
w¼ð0;0:232;0:417;0;0:351Þ
T
and the ranking
of the alternatives as x
3
x
5
x
2
x
4
x
1
. Thus, x
3
= wood cloth-
ing is the best alternative according to the aggregated preferences
of the three experts.
Secondly, based on B
ðkÞ
¼ðb
ðkÞ
ij
Þ
55
we apply the model (M-6) to
the individual intuitionistic fuzzy preference relations to check the
multiplicative consistency. Applying (M-6), we find the optimal
objective values ð
c
ðkÞ
Þ
as
c
ð1Þ
¼0:390,
c
ð2Þ
¼0:394,
c
ð3Þ
¼0:663
and the optimal nonzero deviation values ð_
d
ðkÞ
ij
and _
d
þ
ðkÞ
ij
Þ:
_
d
þ
ð1Þ
12
¼_
d
ð1Þ
21
¼0:060;_
d
þ
ð1Þ
31
¼_
d
ð1Þ
13
¼0:027;_
d
þ
ð1Þ
15
¼_
d
ð1Þ
51
¼0:100;_
d
þ
ð1Þ
25
¼_
d
ð1Þ
52
¼0:008;
_
d
þ
ð2Þ
23
¼_
d
ð2Þ
32
¼0:097;_
d
þ
ð2Þ
34
¼_
d
ð2Þ
43
¼0:100;
_
d
þ
ð3Þ
12
¼_
d
ð3Þ
21
¼0:100;_
d
þ
ð3Þ
24
¼_
d
ð3Þ
42
¼0:032;_
d
þ
ð3Þ
45
¼_
d
ð3Þ
54
¼0:200:
Since, d
ðkÞ
–0;for k ¼1;2;3 then none of the B
(k)
is multiplicative
consistent. To improve the additive consistency of B
(k)
, we use opti-
mal deviation values _
d
ðkÞ
ij
and _
d
þ
ðkÞ
ij
and develop the improved mul-
tiplicative consistent intuitionistic fuzzy preference relations _
B
ðkÞ
by
applying Eq. (14):
_
B
ð1Þ
¼
ð0:5;0:5Þð0:3;0:64Þð0:273;0:6Þð0:5;0:5Þð0:1;0:8Þ
ð0:64;0:3Þð0:5;0:5Þð0:4;0:6Þð0:6;0:2Þð0:2;0:692Þ
ð0:6;0:273Þð0:6;0:4Þð0:5;0:5Þð0:7;0:1Þð0:4;0:6Þ
ð0:5;0:5Þð0:2;0:6Þð0:1;0:7Þð0:5;0:5Þð0:1;0:8Þ
ð0:8;0:1Þð0:692;0:2Þð0:6;0:4Þð0:8;0:1Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
_
B
ð2Þ
¼
ð0:5;0:5Þð0:6;0:3Þð0:1;0:8Þð0:5;0:5Þð0:3;0:7Þ
ð0:3;0:6Þð0:5;0:5Þð0:0;0:903Þð0:3;0:7Þð0:1;0:8Þ
ð0:8;0:1Þð0:903;0:0Þð0:5;0:5Þð0:6;0:2Þð0:5;0:3Þ
ð0:5;0:5Þð0:7;0:3Þð0:2;0:6Þð0:5;0:5Þð0:3;0:6Þ
ð0:7;0:3Þð0:8;0:1Þð0:3;0:5Þð0:6;0:3Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
_
B
ð3Þ
¼
ð0:5;0:5Þð0:0;0:8Þð0:1;0:7Þð0:224;0:6Þð0:5;0:5Þ
ð0:8;0:0Þð0:5;0:5Þð0:6;0:206Þð0:6;0:368Þð0:8;0:2Þ
ð0:7;0:1Þð0:206;0:6Þð0:5;0:5Þð0:5;0:2Þð0:7;0:2Þ
ð0:6;0:224Þð0:368;0:6Þð0:2;0:5Þð0:5;0:5Þð0:4;0:5Þ
ð0:5;0:5Þð0:2;0:8Þð0:2;0:7Þð0:3;0:4Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Based on the improved preference relations, we utilize models
(M-7) and (M-8) to find the individual priority weight vector of each
Table 1
Optimal weights based on the improved additive consistent intuitionistic fuzzy
preference relations.
Experts Optimal weights Ranking of
alternatives
E
1
w
ð1Þ
1
¼0, w
ð1Þ
2
¼0:133, w
ð1Þ
3
¼0:333,
w
ð1Þ
4
¼0, w
ð1Þ
5
¼0:533
x
5
x
3
x
2
x
4
x
1
E
2
w
ð2Þ
1
¼0, w
ð2Þ
2
¼0, w
ð2Þ
3
¼0:5, w
ð2Þ
4
¼0:1,
w
ð2Þ
5
¼0:4
x
3
x
5
x
4
x
1
x
2
E
3
w
ð3Þ
1
¼0, w
ð3Þ
2
¼0:6, w
ð3Þ
3
¼0:4, w
ð3Þ
4
¼0,
w
ð3Þ
5
¼0
x
2
x
3
x
4
x
5
x
1
⁄
Where ‘‘’’ means ‘‘is preferred to’’ and ‘‘
00
means ‘‘is indifferent to’’.
H. Behret / Knowledge-Based Systems 70 (2014) 33–43 39
expert w
ðkÞ
¼w
ðkÞ
1
;w
ðkÞ
2
;w
ðkÞ
3
;w
ðkÞ
4
;w
ðkÞ
5
T
where k= 1, 2, 3. The
results of the models are summarized in Table 2.
The results shown in Table 2 show that x
5
= natural stone cloth-
ing is the best alternative for civil engineer, x
3
= wood clothing is
the best for constructor and x
2
= compact laminate clothing is the
best for architect.
We aggregate the improved individual preference relations of
the experts by using Eq. (4) and apply (M-9) to obtain the smallest
deviation values for the collective preference relation. The result of
the model (M-9) shows that the optimal objective value is equal
to zero, d
¼Min P
n1
i¼1
P
n
j¼iþ1
d
ij
þd
þ
ij
¼0
which means the
collective preference relation is multiplicative consistent. Subse-
quently we apply (M-10) to determine the priority vector for the
multiplicative consistent collective intuitionistic fuzzy preference
relation and obtain
w¼ð0:098;0:205;0:307;0:111;0:279Þ
T
and
the ranking of the alternatives as x
3
x
5
x
2
x
4
x
1
. This rank-
ing implies that x
3
= wood clothing is the best alternative according
to the aggregated preferences of the three experts.
From the above analyses, it is clear that although the priority
weight vectors of the individual preference relations of the experts
differ, the ranking of the individual priority weights do not differ
critically according to the additive consistent or multiplicative con-
sistent intuitionistic fuzzy preference relations. Additionally, we
would like to point out that, x
3
is the best alternative according
to the aggregated preferences of the three experts both for additive
consistent and multiplicative consistent cases.
As an alternative approach, if we aggregate the optimal weights
of the experts
w¼P
m
k¼1
k
k
w
ðkÞ
T
instead of aggregating preference
relations, we determine the aggregated priority vector of the addi-
tive consistent individual preference relations as
w¼ð0;0:247;0:387;0:020;0:347Þ
T
and the ranking of the alterna-
tives as x
3
x
5
x
2
x
4
x
1
. Subsequently, we determine the
aggregated priority vector of the multiplicative consistent individ-
ual preference relations as
w¼ð0:098;0:204;0:291;0:134;0:272Þ
T
and the ranking of the alternatives as x
3
x
5
x
2
x
4
x
1
. When
we observe the aggregated priority vectors, it can be seen that the
weights of the alternatives change according to the additive consis-
tent or multiplicative consistent intuitionistic preference relations,
although the rankings remain the same. Consequently, if we derive
consistent preference relations (additive or multiplicative consis-
tent), the ranking of the alternatives obtained from collective pref-
erence relation or aggregated priority vectors will generally be the
same.
8. Comparative analysis
In this section we compare the usage of fuzzy and intuitionistic
fuzzy preference relations in the group decision making problems.
A fuzzy preference relation Ron the set Xis represented by a com-
plementary matrix R=(
l
ij
)
nn
XXwith,
l
ij
P0;
l
ij
þ
l
ji
¼1;
l
ii
¼0:5for all i;j¼1;2;...;nð16Þ
where
l
ij
represents the preference degree of the alternative x
i
over
x
j
. A fuzzy preference relation R=(
l
ij
)
nn
is referred to as an addi-
tive consistent fuzzy preference relation, if it satisfies the following
property [47]:
l
ij
¼0:5ðw
i
w
j
þ1Þ;for all i ¼1;2;...;n1;j¼iþ1;...;n;
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;ð17Þ
If Ris an additive consistent fuzzy preference relation, then there is
a vector w=(w
1
,w
2
,...,w
n
)
T
of Rwhich satisfies Eq. (17). However
this property does not always hold. Thus we can introduce a
deviation variable d
ij
¼d
þ
ij
d
ij
to relax Eq. (17).
l
ij
¼0:5ðw
i
w
j
þ1Þþd
ij
d
þ
ij
;for all i ¼1;2;...;n1;j¼iþ1;...;n
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0ð18Þ
As the deviation variables d
ij
;d
þ
ij
become smaller, R=(
l
ij
)
nn
becomes closer to an additive consistent fuzzy preference relation.
In order to find the smallest deviation variables we can develop
the following optimization model:
d
f
¼Min X
n1
i¼1
X
n
j¼iþ1
d
þ
ij
þd
ij
s:t:0:5ðw
i
w
j
þ1Þþd
ij
d
þ
ij
¼
l
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-11Þ
The solution of this model determines the optimal deviation values
_
d
þ
ij
;_
d
ij
.Ifd
f
¼0 then Ris an additive consistent fuzzy preference
relation. Otherwise, the nonzero deviation values may be used to
improve the additive consistency as:
_
R¼ð_
l
ij
Þ
nn
;_
l
ij
¼
l
ij
þ_
d
þ
ij
_
d
ij
;
i¼1;2;...;n1;j¼iþ1;...;nð19Þ
And a fuzzy preference relation R=(r
ij
)
nn
is referred to as a
multiplicative consistent fuzzy preference relation if it satisfies
the condition below [49,52]:
l
ij
¼w
i
w
i
þw
j
;for all i ¼1;2;...;n1;j¼iþ1;...;n;
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;
ð20Þ
If Ris a multiplicative consistent fuzzy preference relation, then
there is a vector w=(w
1
,w
2
,...,w
n
)
T
of Rwhich satisfies Eq. (20).
However this property does not always hold. Thus we can introduce
a deviation variable d
ij
¼d
þ
ij
d
ij
d
ij
to relax Eq. (20):
l
ij
¼w
i
w
i
þw
j
þþd
ij
d
þ
ij
;for all i ¼1;2;...;n1;j¼iþ1;...;n
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0ð21Þ
Considering that the smaller the deviation variables d
ij
;d
þ
ij
in Eq.
(21), the closer Rto a multiplicative consistent fuzzy preference
relation, we develop the nonlinear optimization model:
Table 2
Optimal weights based on the improved multiplicative consistent intuitionistic fuzzy
preference relations.
Experts Optimal weights Ranking of
alternatives
E
1
w
ð1Þ
1
¼0:096, w
ð1Þ
2
¼0:170, w
ð1Þ
3
¼0:255,
w
ð1Þ
4
¼0:096, w
ð1Þ
5
¼0:383
x
5
x
3
x
2
x
4
x
1
E
2
w
ð2Þ
1
¼0:114, w
ð2Þ
2
¼0:049, w
ð2Þ
3
¼0:456,
w
ð2Þ
4
¼0:114, w
ð2Þ
5
¼0:267
x
3
x
5
x
4
x
1
x
2
E
3
w
ð3Þ
1
¼0:091, w
ð3Þ
2
¼0:364, w
ð3Þ
3
¼0:242,
w
ð3Þ
4
¼0:212, w
ð3Þ
5
¼0:091
x
2
x
3
x
4
x
5
x
1
⁄
Where ‘‘’’ means ‘‘is preferred to’’ and ‘‘
00
means ‘‘is indifferent to’’.
40 H. Behret / Knowledge-Based Systems 70 (2014) 33–43
c
f
¼Min X
n1
i¼1
X
n
j¼iþ1
d
þ
ij
þd
ij
s:t:w
i
w
i
þw
j
þd
ij
d
þ
ij
¼
l
ij
w
i
P0;i¼1;2;...;n;X
n
i¼1
w
i
¼1;d
ij
;d
þ
ij
P0
i¼1;2;...;n1;j¼iþ1;...;n:
ðM-12Þ
Solving the model (M-12), we determine the optimal deviation
values, _
d
þ
ij
;_
d
ij
. In particular, if c
f
¼0, then Ris a multiplicative con-
sistent fuzzy preference relation, otherwise the nonzero deviation
values can be utilized to improve the multiplicative consistency as;
_
R¼ð_
l
ij
Þ
nn
;_
l
ij
¼
l
ij
þ_
d
þ
ij
_
d
ij
;i¼1;2;...;n1;
j¼iþ1;...;nð22Þ
In fuzzy preference relation, all the information is expressed with
only membership functions and nonmembership is represented as
the complement of membership which actually ignores the DM’s
hesitation in the decision making process. However intuitionistic
fuzzy preference relation simultaneously considers the degrees of
membership and nonmembership with hesitation index. These
different preference relation definitions would result in different
decisions. For comparison of fuzzy and intuitionistic fuzzy
preference relations and their varying impacts on experts’ decisions,
following preference relation matrices are used.
In fuzzy preference relation, the experts individually compare
each pair of criteria x
i
and x
j
, and provide his/her fuzzy preference
value composed only by the certainty degree
l
ij
to which x
i
is
preferred to x
j
. The value which x
j
is preferred to x
i
,(=
l
ji
) will be
the complement of
l
ij
according to Eq. (16). Hence, fuzzy preference
relation of the expert will only include the membership values as
below;
R¼
ð0:5Þð0:3Þð0:3Þð0:2Þð0:5Þ
ð0:7Þð0:5Þð0:4Þð0:3Þð0:1Þ
ð0:7Þð0:6Þð0:5Þð0:7Þð0:4Þ
ð0:8Þð0:7Þð0:3Þð0:5Þð0:1Þ
ð0:5Þð0:9Þð0:6Þð0:9Þð0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Based on this preference relation we first apply the model (M-11) to
check the additive consistency. The solution of the model (M-11)
gives the optimal objective value as d
f
¼2:600 and the optimal
nonzero deviation values:
_
d
þ
12
¼_
d
þ
21
¼_
d
þ
24
¼_
d
þ
42
¼0:2;_
d
þ
14
¼_
d
þ
41
¼_
d
15
¼_
d
51
¼0:3;
_
d
23
¼_
d
32
¼_
d
þ
25
¼_
d
þ
52
¼_
d
þ
45
¼_
d
þ
54
¼0:1;
Since, d
f
–0 then Ris not additive consistent. To improve the addi-
tive consistency of R, we use optimal deviation values are used and
the improved additive consistent fuzzy preference relation _
Ris
determined by applying Eq. (19):
_
R¼
ð0:5Þð0:5Þð0:3Þð0:5Þð0:2Þ
ð0:5Þð0:5Þð0:3Þð0:5Þð0:2Þ
ð0:7Þð0:7Þð0:5Þð0:7Þð0:4Þ
ð0:5Þð0:5Þð0:3Þð0:5Þð0:2Þ
ð0:8Þð0:8Þð0:6Þð0:8Þð0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Since the optimal objective value of the improved preference
relation will be equal to zero we determine the priority weight
vector as w
1
=0, w
2
=0, w
3
= 0.4, w
4
=0,w
5
= 0.6 and the ranking
of alternatives as x
5
x
3
x
2
x
4
x
1
.
In intuitionistic preference relation, the experts individually
compare each pair of criteria x
i
and x
j
, and provide his/her intui-
tionistic fuzzy preference value b
ij
=(
l
ij
,
v
ij
), composed by the cer-
tainty degree
l
ij
to which x
i
is preferred to x
j
and the certainty
degree
v
ij
to which x
i
is non-preferred to x
j
, and then develop the
following intuitionistic fuzzy preference relation B=(b
ij
)
55
;
B¼
ð0:5;0:5Þð0:3;0:3Þð0:3;0:3Þð0:2;0:5Þð0:5;0:5Þ
ð0:3;0:3Þð0:5;0:5Þð0:4;0:3Þð0:3;0:2Þð0:1;0:6Þ
ð0:3;0:3Þð0:30:4Þð0:5;0:5Þð0:7;0:0Þð0:4;0:4Þ
ð0:5;0:2Þð0:2;0:3Þð0:0;0:7Þð0:5;0:5Þð0:1;0:5Þ
ð0:5;0:5Þð0:6;0:1Þð0:4;0:4Þð0:5;0:1Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Based on Bwe first apply the model (M-1) to the individual intui-
tionistic fuzzy preference relation to check the additive consistency.
Applying (M-1), we find the optimal objective value as d
⁄
= 0.400
and the optimal nonzero deviation values:
_
d
þ
14
¼_
d
41
¼0:045;_
d
þ
51
¼_
d
15
¼0:083;_
d
þ
32
¼_
d
23
¼0:023;
_
d
þ
43
¼_
d
34
¼0:049:
Since, d
⁄
–0, Bis not additive consistent. In order to improve the
additive consistency, optimal deviation values can be used and
the improved additive consistent intuitionistic fuzzy preference
relations can be calculated by applying Eq. (11):
_
B¼
ð0:5;0:5Þð0:3;0:3Þð0:3;0:3Þð0:2;0:455Þð0:417;0:5Þ
ð0:3;0:3Þð0:5;0:5Þð0:377;0:3Þð0:3;0:2Þð0:1;0:6Þ
ð0:3;0:3Þð0:3;0:377Þð0:5;0:5Þð0:651;0:0Þð0:4;0:4Þ
ð0:455;0:2Þð0:2;0:3Þð0:0;0:651Þð0:5;0:5Þð0:1;0:5Þ
ð0:5;0:417Þð0:6;0:1Þð0:4;0:4Þð0:5;0:1Þð0:5;0:5Þ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Since the optimal objective value of the improved preference rela-
tion will be equal to zero, we establish models (M-2) and (M-3)
to determine the individual priority weight vector of the expert as
w
1
= 0.149, w
2
= 0.115, w
3
= 0.361, w
4
= 0.059, w
5
= 0.315 and the
ranking of alternatives as x
3
x
5
x
1
x
2
x
4
.
When the results of additive consistent fuzzy and intuitionistic
fuzzy priority relations are examined, we realize that the priority
vector and ranking of the alternatives change significantly in the
two cases. In the intuitionistic fuzzy case, the third alternative is
determined as the best choice while in the fuzzy case the fifth
alternative is calculated as the best. Furthermore when we exam-
ine the rankings of alternatives in two cases, significant differences
between the two results are observed.
We can also check the multiplicative consistency of both prefer-
ence relations and compare the priority vectors. Based on fuzzy
preference relation, model (M-12) is applied to check the multipli-
cative consistency. The solution of the model (M-12) gives the
optimal objective value as
c
f
¼1:070. Since
c
f
–0 then Ris not
multiplicative consistent. In order to improve the multiplicative
consistency of R, optimal deviation values are used and the
improved multiplicative consistent fuzzy preference relation _
Ris
determined by applying Eq. (22). The optimal objective value of
the improved preference relation will be equal to zero and we cal-
culate the priority weight vector as w
1
= 0.033, w
2
= 0.076,
w
3
= 0.304, w
4
= 0.130, w
5
= 0.457 and the ranking of alternatives
as x
5
x
3
x
4
x
2
x
1
. In the intuitionistic case, we first apply
the model (M-6) to the individual intuitionistic fuzzy preference
relation to check the multiplicative consistency. Applying (M-6),
we find the optimal objective value as
c
⁄
= 0.385 and the optimal
nonzero deviation values. Since,
c
⁄
–0, Bis not multiplicative con-
sistent. In order to improve the multiplicative consistency, optimal
deviation values are used and the improved multiplicative consis-
tent intuitionistic fuzzy preference relation is determined by
applying Eq. (14). The optimal objective value of the improved
H. Behret / Knowledge-Based Systems 70 (2014) 33–43 41
preference relation will be equal to zero and we establish models
(M-7) and (M-8) to calculate the individual priority weight vector
of the expert as w
1
= 0.194, w
2
= 0.129, w
3
= 0.290, w
4
= 0.194,
w
5
= 0.194 and the ranking of alternatives as x
3
x
5
x
1
x
4
x
1
.
When we examine the results of multiplicative consistent fuzzy
and intuitionistic fuzzy priority relations, we observe that the pri-
ority vector and ranking of the alternatives change significantly in
two cases as in the additive consistent situation. Although the best
alternative does not change in two cases, the ranking of the
remaining alternatives changes significantly.
9. Conclusion and further study
In the process of group decision making problems, the decision
makers sometimes may not provide their preferences for alterna-
tives to a certain degree and there is usually a degree of uncer-
tainty in providing their preferences over the alternatives
considered. Intuitionistic fuzzy preference relations have the capa-
bility of representing imprecise or not totally reliable judgments
which exhibit affirmation, negation and hesitation characteristics.
The consistency of intuitionistic fuzzy preference relations and
the priority weights of experts gathered from these preference
relations play an important role in group decision making prob-
lems in order to reach an accurate decision result. In this study,
we have proposed a group decision making process with the usage
of intuitionistic fuzzy preference relations. The suggested process
is based on the evaluation of the consistency of intuitionistic fuzzy
preference relations. We have constructed two different optimiza-
tion models to minimize the deviations from additive or multipli-
cative consistency respectively. The optimal deviation values
obtained from the model results enable us to improve the consis-
tency of considered preference relations. Following this, we aggre-
gated individual improved (which means consistent) fuzzy
intuitionistic preference relations in order to determine a collective
consistent fuzzy intuitionistic preference relation. Based on the
consistent collective preference relation, we have developed a lin-
ear programming model considering additive consistency and a
nonlinear model considering multiplicative consistency to obtain
the priority weights. The priority weights of the experts also enable
us to determine the ranking of the alternatives. Furthermore, we
have given some illustrative examples in order to examine the
validity and practicality of the developed models. Numerical anal-
yses have shown that although the priority weight vectors of the
individual preference relations of the experts differ, the ranking
of the individual priority weights do not differ significantly accord-
ing to the additive consistent or multiplicative consistent intui-
tionistic fuzzy preference relations. And additionally, if we derive
consistent preference relations (additive or multiplicative consis-
tent), the ranking of the alternatives obtained from collective pref-
erence relation or aggregated priority vectors will generally be the
same.
In the final section, we presented comparative analyses
between fuzzy and intuitionistic fuzzy preference relations. When
we examine the results of both additive and multiplicative consis-
tent fuzzy and intuitionistic fuzzy priority relations, we realize that
the priority vector and ranking of the alternatives change signifi-
cantly in two cases. Furthermore when we look at the rankings
of alternatives in two cases (fuzzy and intuitionistic fuzzy), we
observe significant differences between two results. The main rea-
son for this difference is the capability of intuitionistic fuzzy pref-
erence relations simultaneously considering the degrees of
membership and nonmembership with hesitation index while
fuzzy preference relations can only consider membership values.
The results of the numerical illustrations showed that intuitionistic
fuzzy preference relations provide more accurate priority vectors
and rankings of alternatives by taking into consideration the
DMs’ affirmation, negation and hesitation with the help of mem-
bership definitions.
The proposed group decision making process and models may
be used in many real-world applications in which the DMs may
not be able to provide his/her preferences for alternatives to a cer-
tain degree due to lack of precise or sufficient level of knowledge
related to the problem, or the difficulty in explaining explicitly
the degree to which one alternative is better than others. Possible
application areas may be supply chain management, project eval-
uation, risk management, pattern recognition, medical diagnosis,
investment, personnel examination and military system efficiency
evaluation. Although the focus of this study is mainly on the
consistency of intuitionistic preference relations; the process
developed can be extended to include group consensus and prior-
ity weight generation from incomplete intuitionistic preference
relations.
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