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12-2016
Fuzzy Pareto Solution in Multi-criteria Group
Decision Making with Intuitionistic Linguistic
Preference Relation
Bui Cong Cuong
Institute of Mathematics, Vietnam Academy of Science and Technology, [email protected]
Vladik Kreinovich
University of Texas at El Paso, [email protected]
Le Hoang Son
VNU University of Science, Vietnam National University, [email protected]
Nilanjan Dey
Techno India College of Technology, [email protected]
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Technical Report: UTEP-CS-16-88
To appear in International Journal of Fuzzy System Applications
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Cuong, Bui Cong; Kreinovich, Vladik; Son, Le Hoang; and Dey, Nilanjan, "Fuzzy Pareto Solution in Multi-criteria Group Decision
Making with Intuitionistic Linguistic Preference Relation" (2016). Departmental Technical Reports (CS). 1091.
http://digitalcommons.utep.edu/cs_techrep/1091
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Fuzzy Pareto Solution in multi-criteria group
decision making with intuitionistic linguistic
preference relation
Bui Cong Cuong
Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
[email protected]
Vladik Kreinovich
Department of Computer Science, University of Texas at El Paso, TX 79968, USA
[email protected]
Le Hoang Son *
VNU University of Science, Vietnam National University, Hanoi, Vietnam
[email protected]
Nilanjan Dey
Techno India College of Technology, Kolkata, India
[email protected]
*Corresponding author. Tel.: (+84) 904.171.284
Abstract: In this paper, we investigate the multi criteria group decision making with intuitionistic
linguistic preference relation. The concept of Fuzzy Collective Solution (FCS) is used to evaluate
and rank the candidate solution sets for modeling under linguistic assessments. Intuitionistic
linguistic preference relation and associated aggregation procedures are then defined in a new
concept of Fuzzy Pareto Solution. Numerical examples are presented to demonstrate computing
procedures. The results affirm efficiency of the proposed method.
Keywords: Fuzzy Collective Solution; Fuzzy Pareto Solution; Group Decision Making;
Intuitionistic Fuzzy Sets; Intuitionistic Linguistic Preference Relation.
1. Introduction
Group decision making (GDM) is useful to evaluate and judge the best solution
among alternatives through a social group. GDM, in a fuzzy environment, contains four
steps (Herrera, Herrera-Viedma & Verdegay, 1996): i) unifying evaluations from
experts; ii) aggregating their opinions to a final score for each alternative represented by a
linguistic label (Chen & Hwang, 1992; Cheng, 1999; Delgado, Verdegay & Vila, 1992;
Fodor & Roubens, 2013; Herrera & Herrera-Viedma, 1997; Hwang & Lin, 2012); iii)
ranking the labels; and iv) selecting the preferred alternative by group decision. It has
been the fact that using linguistic labels makes experts’ judgment more reliable and
informative than using numeric scores. Fuzzy Collective Solution (FCS) is widely used to
1
2
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
evaluate and rank the candidate solution for a model under linguistic assessments. The
existing algorithms for FCS provided the tool for aggregation in GDM with intuitionistic
preference relations.
Intuitionistic fuzzy set (IFS) is an extension of Zadeh’s fuzzy set (1965) characterized
by a membership and a non-membership function (Atanassov, 1986, 1999). In many
complex decision making problems, the decision can be represented by IFS for better
modelling of imprecise and uncertain data. Recently, some researchers applied IFS to
GDM. Gau and Buehrer (1993) introduced the vague set, which is an equivalence of IFS.
Hong and Choi (2000) developed approximate techniques for multi-attribute decision
making using minimum and maximum operators. Szmidt and Kacpzyk (2002) proposed
the intuitionistic fuzzy core and consensus winner in group decision making with
intuitionistic (individual and social) fuzzy preference relations. Xu and Yager (2006)
developed the intuitionistic fuzzy weighted geometric operator, fuzzy ordered and fuzzy
hybrid geometric operator extending the traditional and ordered weighted geometric
operator in the environment where given arguments are represented by IFSs.
Our contribution for GDM with intuitionistic preference relation is the new approach
using the Pareto solution of the optimization theory and some aggregation procedures
based on FCS to solve the model. This paper is organized in 8 sections with the first
section is the introduction. Section 2 briefly reviews group decision making model and
Low-operator. Section 3 introduces the concept of Fuzzy Collective Solution. Some
computing algorithms for FCS of the multi-criteria problem are given in the Section 4. In
section 5, intuitionistic linguistic preference relations are defined. Section 6 is devoted to
a new concept of Fuzzy Pareto Solution (FPS) and some aggregation procedures for the
FPS in the problems with intuitionistic linguistic preference relations. Section 7 shows an
example to illustrate the approach. The final section is conclusion of this paper.
2. A group decision making model
2.1. A GDM model under linguistic
In this paper, let us consider the following model under linguistic assessments. Assume
X  x1 , x2 ,..., xn  is a finite set of alternatives and E ={e1, …, em} is a finite set of
experts. We assume that there exists a distinguished person, say the manager, who
assigns an important degree w(ek )  w(k ) for each expert such that
0  w(k )  1,
and
k w(k )  1 .
Let
S  st , t  1,..., T  be a finite and totally ordered linguistic-labels set. Assume
that each expert ekE provides his/her opinions on X by mean of a linguistic preference
relation pk : X  X  S ,
where
pk (i, j )  pk ( xi , x j )  S ,
represents
the
linguistically assessed preference degree of alternative x i over xj. For example, we
consider the following linguistic labels S [12]:
Fuzzy Pareto Solution in multi-criteria group decision making
3
S = {I, EU ,VLC, SC, IM, MC, ML, EL, C}
where they are trapezoidal fuzzy numbers on [0,1].
I
Impossible
(0, 0, 0, 0);
EU
Extremely-unlikely
(0.00, 0.01, 0.02, 0.07);
VLC Very-low-chance
(0.04, 0.10, 0.18, 0.23);
SC
Small-chance
(0.17, 0.22, 0.36, 0.42);
IM
It- may
(0.32, 0.41, 0.58, 0.65);
MC Meaningful-chance
(0.58, 0.63, 0.80, 0.86);
ML Most-likely
(0.72, 0.78, 0.92, 0.97);
EL
Extremely-likely
(0.93, 0.98, 0.99, 1);
C
Certain
(1, 1, 1, 1);
2.2. A modification of LOWA operator
Yager (1998) defined the ordered weighted averaging (OWA) operator. The LOWA
operator is based on OWA and the convex combination of linguistic labels (Llamazares,
2007). However, in many real cases, the weights of experts were not considered. Thus,
we are concerning the aggregation problem in which the weights of experts are unknown.
Definition 1. Let a = {a1, ..., am} be a set of linguistic-labels to aggregate, and b is the
associated ordered labels vector, i.e. b = {aim, ai(m – 1), …, ai1}, such that aim  ai(m – 1)
 ... ai1.
The Low operator is defined as
Low(a, w)  C (wim , aim ), (1  wim , Low(a ' , w' ))
(1)
where w= [w1, ..., wm], is a weight vector such that, wi  [0,1] and I wi = 1,
a’={ai(m-1), ..., ai1}, w’={w’i(m-1), ...,w’i1}, w’j = wj/(1- wim) .
(2)
C is the convex combination operator of 2 labels sj , si, j  i with wj>0, wi>0, wj+wi =1,
C{(wj,sj).(wi,si)} = sk,
(3)
where k = i +round( wj.(j-i)), where round is the usual round operator.
2.3. Multi-criteria group decision making model
In our model, assume that there is a finite set of criteria: C= {C1, C2, ..., CL}. Respect to
each criterion Cl, each expert ek E provides his/her opinions on X by mean of a
linguistic preference relation
pkl : X  X  S ,
where
pkl (i, j )  pkl ( xi , x j )  S
(4)
is the linguistically assessed preference degree of alternative xi over xj. Moreover, assume
that there are important weights of criteria {l , l=1,2,..., L}, such that 0 l 1, and ll
= 1.
In our model we assume that pkl : X  X  S is soft-recipodal in the sense:
(i) pkl (i, i)  sT 1/ 2 ,for all i=1,…,n
(5)
4
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
(ii) pkl (i, j )  sT 1/ 2 , then
Let
P , k  1,..., m
k
pkl ( j, i)  sT 1/ 2
be
(i, j ), i  1,...n, j  1,..., n

linguistic
,
and
(6)
preference
for

Wi , j  st    k w(k ) : p k (i, j )  st .
each
relations.
For
every
st  S ,
we
define
(7)
This value is the sum of individual importance degrees of experts, who coincide to assign
the linguistic label st as preference value of the alternative xi over alternative
xj .
In the next section, we will define the concept of Fuzzy Collective Solution (FCS) and
apply to evaluate and to rank the alternatives set.
3. Fuzzy Collective Solution
3.1. Notions
The concept of fuzzy collective solution is similar to the concept of aggregated
dominance degrees of alternatives. For each pair (xi,xj), the linguistic dominance degree
is defined by
E ( xi , x j )  Low(S ,U )
where
(8)
U  uT ,..., u1  , ut  Wij  st  for t=1,...,T.
(9)
Definition 2. The fuzzy collective solution (FCS) is a fuzzy set on the alternatives set
X
FCS   fcs( x1 ) / x1 , fcs( x2 ) / x2 ,..., fcs( xn ) / xn 
(10)
where the membership degree of the alternative xi is calculated as follows:
fcs( xi )  Low( S ,V )
where
(11)
V  vT ,..., v1  , for t = 1,..., T, vt  # j : E ( xi , x j )  st  n  1 .
(12)
3.2. Aggregated Fuzzy Collective Solution
Now, we give a notion for aggregation of FCS in multi-criteria decision making
problems. Suppose that for each criterion Cl, we obtained the corresponding fuzzy
collective solution:
FCSl   fcsl ( x1 ) / x1 ,..., fcsl ( xn ) / xn 
l= 1, 2, ..., L.
Definition 3. The aggregated fuzzy collective solution (aFCS) is a fuzzy set on the
alternatives set X
aFCS  afcs( x1 ) / x1 , afcs( x2 ) / x2 ,..., afcs( xn ) / xn 
(13)
Fuzzy Pareto Solution in multi-criteria group decision making
5
where the membership degree of the alternative xi is calculated as
afcs( xi )  Low(S ,U  ) ,
where
i=1,…, n
(14)
U   uT ,..., u1  , U t  l l : fcs( xi )  st  , t=1, ...T
(15)
Note that aFCS is a type-2 fuzzy set on X.
4. Some computing procedures using FCS
In the following computing procedures, we use the linguistic preference relations
{ pkl , k  1,..., m, l  1,..., L} , the experts’ important weights w(k ) : ek  E , such
that
0  w(k )  1 , and
l , l  1,..., L , such that
k w(k )  1
and the criteria’s important weights
0  l  1 , and
l l  1. Now we consider some
computing procedures using FCS for the model.
Algorithm 1:
Step 1.
1a. For each criterion Cl, using { pkl , k  1,..., m} and
where
where
w(k ) : ek  E
El   El (i, j )   E ( xi , x j )  ,
calculate the linguistic dominance degrees
, j =1,…,n
the weights
i
(16)
El ( xi , x j )  Low( S ,Ul )
,
Ul  ulT ,..., ul1 
ult  Wij (st )  w(k ) : pkl (i, j )  st  ,
(17)
t=1,...,T
(18)
k
El   El ( xi , x j )  , calculate the fuzzy collective solution
FCSl   fcsl ( x1 ) / x1 , fcsl ( x2 ) / x2 ,..., fcsl ( xn ) / xn 
1b. Using the
(19)
as in the Definition 2.
Step 2. Using the
calculate
FCSl , l  1,..., L
and the important weights
the aggregated fuzzy collective solution (aFCS)
aFCS  afcs( x1 ) / x1 , afcs( x2 ) / x2 ,..., afcs (xn ) / xn  ,
as in the Definition 3.
Step 3.
l , l  1,..., L ,
Classify the set of alternatives X into subsets
(20)
6
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
X t   xi : afcs( xi )  st  , with t = 1, ... , T
Rank the non-empty sets
Xt
X t'
(21)
iff t  t ' , and therefore, the solution is the set
X t* , where t *  max t : X t  , st  S .
Algorithm 2:
Step 1. For each criterion Cl, calculate the linguistic dominance degrees as in (16).
El   El ( xi , x j )  , i,j =1,…,n,
l=1,…, L
Step 2.
2.a. Using {El , l  1,..., L} and the important weight
totally social opinion relation:
Q   q(i, j )  q( xi , x j )  , i , j = 1,…,n
where q( xi , x j )  Low(S ,U q ) ,
where
l , l  1,..., L , calculate the
U q  uqT ,..., uq1 
(22)
(23)
uqt  Wijq (st )  l l : El (i, j )  st  , t=1,...,T
(24)
2.b. Calculate the fuzzy collective solution according to the totally social opinion
relation Q
FCSQ   fcsQ ( x1 ) / x1 , fcsQ ( x2 ) / x2 ,..., fcsQ ( xn ) / xn 
(25)
as in Definition 2.
Step 3. Classify the alternative set X into subsets
X t   xi : fcsQ ( xi )  st  , t = 1, ..., T
(26)
and choose the solution as in the step 3 of the Algorithm 1.
Algorithm 3:
Step 1.
1a. For each
expert ek , using { pkl , l  1,..., L} and the important weights
l , l  1,..., L , calculate the relatively dominance degrees
F k according to expert
ek as
F k   F k (i, j )    F k ( xi , x j ) ,
where
where
i, j =1,…,n
F k ( xi , x j )  Low(S ,U k )
(27)
U k  [uTk ,..., u1k ] where utk  l l : pkl (i, j )  st  , t=1,...,T
k
k
1b. Using F , calculate the fuzzy evaluation FE according to the expert ek
(28)
Fuzzy Pareto Solution in multi-criteria group decision making
FE k   fek ( x1 ) / x1 ,..., fek ( xn ) / xn 
7
(29)
with the membership degree of the alternative xi is calculated as
fek ( xi ) Low(S ,Vk ) ,
where Vk  [vT ,..., v1 ] ,
i=1,…, n

(30)

vt  j : F k ( xi , x j )  st , j  i
Step 2. Using the fuzzy evaluations
n  1 ,t=1,...,T
FE , k  1,..., m
k
(31)
and the weights
w(k ) : ek  E , calculate the aggregated fuzzy evaluation (aFE), which is a fuzzy set
on the alternatives set X:
aFE  afe( x1 ) / x1 ,..., afe( xn ) / xn  ,
(32)
as in Definition 3.
Step 3. Classify the set of alternatives X into subsets
X t   xi : afe( xi )  st  ,
t = 1, ..., T
(33)
and choose the solution as in the step 3 of the Algorithm 1.
5. Intuitionistic Linguistic Preference Relation
5.1. Intuitionistic Fuzzy Sets
Definition 4 (Atanassov, 1986). Let Y be a universe of discourse. Then an intuitionistic
fuzzy set (IFS)
A  (( A ( y), A ( y )) / y ) y  Y 
(34)
is characterized by a membership function  A : Y  [0,1], and a non-membership
function
 A : Y  [0,1],
where the numbers
with the condition 0   A ( y)  A ( y)  1 for all y  Y ,
 A ( y)
and
 A ( y) represent,
respectively, the degree of
membership and the degree of non-membership of the element y to the set A.
Definition 5 (Xu, 2007). Let
X   x1 , x2 ,..., xn  . An intuitionistic preference relation
B on the set X is represented by a matrix:
B  bij 
nn
where bij
 ( ( xi , x j ), ( xi , x j )) /( xi , x j ) , for all i, j  1, 2,..., n (35)
8
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
where
ij   ( xi , x j ) [0,1] is the preference degree of xi
 ij   ( xi , x j ) [0,1]
in comparing with x j and
is the non-preference degree of xi in comparing with
In [33], Xu supposed that
ij   ( xi , x j )
and
 ij   ( xi , x j )
xj .
satisfy the following
condition:
0  ij  ij  1,  ji   ij ,  ji  ij , ii   ii  0.5 ,
for all i, j  1, 2,..., n .
(36)
Next, we will define the intuitionistic linguistic preference relation and will present a new
approach to solve the group decision problems with intuitionistic linguistic preference
relations.
5.2. Intuitionistic linguistic preference relation
Definition 6. Let
X   x1 , x2 ,..., xn  be an alternatives set. Let S be a finite and
totally ordered linguistic labels set: S= {st , t=1,..., T} where T is an odd number. An
intuitionistic linguistic preference relation P on X is a matrix
P   pij 
nn


where pij  (  ( xi , x j ), ( xi , x j )) /( xi , x j ) ,for all i, j  1, 2,..., n
(37)
where
ij   ( xi , x j )  S
with x j and
is the linguistic preference degree of xi in comparing
 ij   ( xi , x j )  S
comparing with
is the linguistic non-preference degree of xi in
xj .
Definition 7. (Soft-reciprocal condition) An intuitionistic linguistic preference relation
P on the set X is said to be soft-reciprocal if
ij   ( xi , x j )  ST 1 , then  ji   ( x j , xi )  ST 1 and ij   ( xi , x j )  ST 1
2
,
2
2
for all
i, j  1, 2,..., n (38)
Note 1. This condition on preference relation is much weaker than other supposed
conditions in the papers on decision making problems using preference relations. For
example see (Xu, 2006).
6. Fuzzy Pareto Solution
In our approach, we need the Pareto Solution concept of Optimization Theory (Tuy,
1998).
Fuzzy Pareto Solution in multi-criteria group decision making
9
6.1. Pareto Solution in Optimization Theory
Let D   . Let f : D  R be a real function, i.e. for each
n
We have
xD ,
f ( x)  ( f1 ( x), f 2 ( x),..., f j ( x),..., f n ( x)) .
(39)
Definition 8 (Pareto Solution). Consider the following optimization problem
f(x)  max
subject to
xD .
x*  D is a Pareto solution, if there not exists x '  D such that
A
f ( x*)  f ( x ') and
for any
f ( x*)  f ( x ') , i.e.,
(40)
x '  D , if f j ( x*)  f j ( x ') for some j, then there is k  j such that
f k ( x ')  f k ( x*) .
A generalization of this concept is the following.
6.2. Generalized Pareto Solution
Let
D   , Si ,
for i  1, 2..., n be totally ordered sets, S  S1  S2  ...  Sn
(41)
g : D  S be a map from D to S .
Let
For each
x  D,
g ( x)  ( g1 ( x),..., g j ( x),..., gn ( x)) .
(42)
Definition 9 . (Generalized Pareto Solution)
A
x*  D is a generalized Pareto solution if there not exists x '  D such that
g ( x*)  g ( x ') and
for any
g ( x*)  g ( x '). i.e.,
(43)
x '  D , if g j ( x*)  g j ( x ') for some j, then there is k  j such that
gk ( x ')  gk ( x*) .
(44)
We will see that this new definition is a useful concept in the group decision making
problems with intuitionistic linguistic preference relations.
6.3. Fuzzy Pareto Solution
Definition 10. (Fuzzy Pareto Solution)
X   x1 , x2 ,..., xn  be an alternatives set. The evaluation information of the expert
Let
ek about X is a IFS on X
( p
k
M
( x1 ), pVk ( x1 )) / x1 ,...,( pMk ( xn ), pVk ( xn ) / xn 
, k=1, 2, ..., m
10
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
Let S1 , S2 be totally ordered linguistic-label sets.
Suppose that there is an aggregated evaluation map
g : X  S1  S2
g ( x)  ( g M ( x1 ), gV ( x1 )) / x1, ... , ( g M ( xn ), gV ( xn ) / xn  .
(45)
For each xi  X , i = 1,…,n
g ( xi )  ( gM ( xi ), gV ( xi )) is a linguistic IFS on X,
where g M ( xi )  S1 is the first membership degree and gV ( xi )  S2 is the second
membership degree. A
x*  X is an Fuzzy Pareto Solution (FPS ), if there not exists
x '  D such that:
gM ( x*)  gM ( x ') and
gM ( x*)  gM ( x ') and
gV ( x ')  gV ( x ) , or
(46)
gV ( x ')  gV ( x ).
(47)
In the case that S1  S2  S is a linguistic labels set defined as in Section 1, FPS is a
type-2 IFS.
6.4. Some aggregation procedures
We consider the following multi-criteria problem:
Given the alternatives set X   x1 , x2 ,..., xn  . Given the criteria set
The expert set
C1 , C2 ,..., CL  .
E  e1 ,..., em  with their importance w(k ) : ek  E such that
0  w(k )  1 ,
k w(k )  1 .Given criteria’s
0  l  1 , and  l l  1 .
and
important weights
1 , 2 ,...,  L 
such that
The linguistic labels set:
S  s1  I , s2  EU , s3  VLC, s4  SC, s5  IM , s6  MC, s7  ML, s8  EL, s9  C
We assume that each expert ekE provides his/ her opinions on X by mean of
intuitionistic linguistic preference relations
pkl  (M kl ,Vkl ), k  1,..., m, l  1,..., L
(48)
satisfying the soft-reciprocal condition.
The linguistic preference relations are:
M kl   kl ( xi , x j ) , for i, j  1,.., n,
kl ( xi , x j )  S
(49)
and the linguistic non-reference relations are
Vkl   kl ( xi , x j )  , for i, j  1,..., n,  kl ( xi , x j )  S ,
(50)
Fuzzy Pareto Solution in multi-criteria group decision making
11
We have to evaluate and to choose the solution sets based on the given intuitionistic
linguistic preference relations. Using the algorithms in Section 4, we present three
aggregation procedures for FPS.
Aggregation procedure 1:
Step 1.
1.1.a. For each criterion Cl, using pkl  (M kl ,Vkl ), k  1,..., m and the weights
w(k ) : ek  E ,
calculate the linguistic dominance degrees:
E1M ( xi , x j )  Low( S ,U M 1 ) ,
where U ML is calculated as (18) with
i, j= 1,…,n
pkl  M kl
1.1.b. For each criterion Cl , using the relation
(51)
.
EMl , calculate the FCS according to
each criterion Cl
FCSMl   fcsMl ( x1 ) / x1 ,..., fcsMl ( xn ) / xn  ,
l=1,…, L
(52)
as in Definition 2 .
1.2.a. For each criterion Cl , calculate the linguistic non-dominance degrees:
EVl ( xi , x j )  Low( S , UVl ) ,
where UVl is calculated as (18) with
i, j= 1,…,n
(53)
pkl  Vkl .
1.2.b. For each criterion Cl , using the relation
EVl , calculate the FCS according to each
criterion Cl
FCSVl   fcsVl ( x1 ) / x1 ,..., fcsVl ( xn ) / xn  ,
l=1,…, L
(54)
as in Definition 2 .
Now we obtain Intuitionistic Fuzzy Evaluation
IFE l  ( fcsMl ( x1 ), fcsVl ( x1 )) / x1 ,.., ( fcsMl ( xn ), fcsVl ( xn )) / xn  .
(55)
Step 2
Using the {( FCSMl , FCSVl ), l  1,..., L} and the important weights
1 , 2 ,...,  L  ,
calculate the aggregated fuzzy collective solutions (aFCS):
aFCSM  afcsM ( x1 ) / x1 , ..., afcsM ( xn ) / xn  and
aFCSV  afcsV ( x1 ) / x1 , ..., afcsV ( xn ) / xn 
(56)
(57)
as in Definition 3.
Finally, we obtain Aggregated Intuitionistic Fuzzy Evaluation (aIFE):
aIFE  (afcsM ( x1 ) / x1 , afcsV ( x1 )) ..., (afcsM ( xn ) / xn , afcsV ( xn ))
. (58)
12
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
Using aIFE, we choose Aggregated Fuzzy Pareto Solution (aFPS) of the problem.
Aggregation procedure 2:
Step 1.
pkl  (M kl ,Vkl ), k  1,..., m and the weights
For each criterion Cl, using
w(k ) : ek  E ,
calculate the linguistic dominance degrees:
E1M ( xi , x j )  Low( S ,U M 1 ) ,
i, j= 1,…,n
p  M kl
U
kl
Ml is calculated as (18) with
where
Calculate the linguistic non-dominance degrees:
EV1 ( xi , x j )  Low( S ,UV 1 ) ,
(59)
.
i, j= 1,…,n
(60)
pkl  Vkl .
where UVl is calculated as (18) with
Step 2.
2.a.
Using
E
1
M
the relations of linguistic dominance degrees
important weights
,..., EML  and the
  1 ,...,  L  , calculate the totally social opinion relation
QM  qM ( xi , x j )  , i, j  1,..., n
(61)
U qM  uqT ,..., uq1 
Where
qM ( xi , x j )  Low(S ,U qM ) ,
where
l
uqt  Wijq (st )  l l : EM
(i, j )  st ,


(62)
t=1,...,T .
Using the relations of linguistic non-dominance degrees
(63)
E ,..., E 
1
V
L
V
and the
important weights   1 ,...,  L  , calculate the totally social opinion relation
QV  qV ( xi , x j )  , i, j  1,..., n
where
qV ( xi , x j )  Low(S ,U qV )
uqt  Wijq (st )
where


(64)
U qM  uqT ,..., uq1 
,
l : EVl (i,
l

j )  st ,
t=1,...,T .
(65)
(66)
2.b. Using QM , QV , calculate the fuzzy collective solution according to the totally
social opinion relations

  fcs

(x ) / x 
FCSQM  fcsqM ( x1 ) / x1 ,..., fcsqM ( xn ) / xn and
(67)
FCSQV
(68)
qV
( x1 ) / x1 ,..., fcsqV
n
n
Fuzzy Pareto Solution in multi-criteria group decision making
as in Definition 2
From (67,68) we obtain Intuitionistic Fuzzy Evaluation

IFEQ  ( fcsqM ( x1 ), fcsqV ( x1 )) / x1 ,..,( fcsqM ( xn ), fcsqV ( xn )) / xn

13
(69)
Using IFEQ , we choose FPS of the problem.
Aggregation procedure 3:
Step 1. For each expert ek, for k=1,...,m, use pkl  (M kl ,Vkl ), l  1,..., L , and the
weights   1 ,...,  L  , calculate FMk ( xi , x j )  Low( S ,U Mk )
  : M (i, j)  s  t=1,...,T,
  fe ( x ) / x ,..., fe ( x ) / x  , k =1....,m
where U Mk  [uT ,..., u1 ] , where, ut 
FEMk
and the fuzzy evaluation
(70)
l
l
k
M
kl
1
t
k
M
1
n
n
fe ( xi ) Low(S ,V ) ,
k
M
where
where VMk  [vT ,..., v1 ] ,
(71)
k
M
(72)
for t=1,...,T , vt 
j:F
k
M

( xi , x j )  st , j  i
n 1
(73)
as (31) with F k  FMk .
Analogously, calculate
FVk ( xi , x j )  Low(S ,UVk ),
where U  [uT ,..., u1 ] , where ut 
k
V
and the fuzzy evaluation
 
l
(74)
: Vkl (i, j )  st  , t=1,...,T,
(75)
FEVk   feVk ( x1 ) / x1 ,..., feVk ( xn ) / xn  , k=1,..,m (76)
feVk ( xi ) Low( S ,V k ) , VVk  [vT ,..., v1 ] ,
where


vt  j : FVk ( xi , x j )  st , j  i
as (31) with
n  1 , t=1,…,T
F k  FVk .


Step 2. Using the fuzzy evaluation ( FEMk , FEVk ), k  1,..., m and the weights {w(k):
ekE},
calculate
the
aggregated
fuzzy
evaluation
(aFE):
aFEM  afeM ( x1 ) / x1 , ..., afceM ( xn ) / xn  and aFEV  afeV ( x1 ) / x1 , ..., afeV ( xn ) / xn 
as in Definition 3.
Then we obtain Aggregated Intuitionistic Fuzzy Evaluation (aIFE):
aIFE  (afeM ( x1 ), afeV ( x1 )) / x1 ,,...,(afeM ( xn ), afeV ( xn )) / xn  .
Using aIFE, we choose aFPS of the problem.
(78)
(77)
14
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
7. A numerical example
We consider the following multi criteria GDM problem with intuitionistic linguistic
preference relations:
The alternatives set: X   x1 , x2 , x3 , x4  .
The expert set: E  e1 , e2 , e3  with their importance w={0.2, 0.5, 0.3}.
The set of criteria: C  C1 , C2 , C3 with their importance weighs   0.35,0.4,0.25 .
The linguistic labels set:
S  s1  I , s2  EU , s3  VLC, s4  SC, s5  IM , s6  MC, s7  ML, s8  EL, s9  C
.
The intuitionistic linguistic preference relations are relations
pkl  (M kl ,Vkl ),
k  1, 2,3, l  1, 2,3
(79)
M kl   kl ( xi , x j )  , i, j  1, 2,3, 4
where
Vkl   kl ( xi , x j )  ,
(80)
i, j  1, 2,3, 4
(81)
are given in the Appendix.
7.1. Computing with the aggregation procedure 1
Step1.
Step 1.1. Computing for criterion C1
1.1.a. Use the linguistic preference relations
M 21
 IM ML MC
 SC IM MC

 SC SC IM

 ML IM EL
EU 
 IM MC
 SC IM
EU 
M 31  
 SC MC
EU 


IM 
 ML IM
,
MC
SC
IM
EL
EU 
 IM MC MC
 SC IM SC
EU 
M 31  
EU 
 SC MC IM


IM 
 ML IM EL
,
EU 
EU 
EU 

IM 
.
Calculate the linguistic dominance degrees:
E1M ( xi , x j )  Low(S ,U M 1 ) , i,j = 1,2,3, 4 as in (51).
E1M ( x1, x1 )  Low((s5 ),(1))  s5  IM .
E1M ( x1, x2 )  Low(( s6 , s7 , s6 ), (0.2, 0.5, 0.3))  C (0.5, s7 ), (0.5, Low(( s6 ), (1))
 C (0.5, s7 ), (0.5, s6 )  sk (1,2) ,
k (1, 2)  6  round ((0.5).(7  6))  6  1  7
 sk (1,2)  s7  ML .
E1M ( x1, x3 )  Low(( s7 , s6 , s6 ), (0.2, 0.5, 0.3))  C (0.2, s7 ), (0.8, Low(( s6 ), (1))
 C (0.2, s7 ), (0.8, s6 )  sk (1,3) ,
k (1,3)  6  round ((0.2).(8  6))  6  0  6
 sk (1,3)  s6  MC .
Fuzzy Pareto Solution in multi-criteria group decision making
15
E1M ( x1, x4 )  Low(( s2 , s2 , s2 ), (0.2, 0.5, 0.3))  C (0.2, s2 ), (0.8, Low(( s2 ), (1))
 C (0.2, s2 ), (0.8, s2 )  s2  EU .
E1M ( x2 , x1 )  Low((s4 , s4 , s4 ),(0.2,0.5,0.3))  C (0.2, s4 ),(0.8, s4 )  s4  SC.
E1M ( x2 , x2 )  Low((s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, s5 )  s5  IM .
E1M ( x2 , x3 )  Low(( s4 , s6 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s6 ), (0.5, Low(( s4 , s4 ),1))
 C (0.5, s6 ), (0.5, s4 )  sk (2,3) ,
k (2,3)  4  round ((0.5).(6  4))  4  1  5
E1M
 sk (2,3)  s5  IM
.
( x2 , x4 )  Low(( s3 , s2 , s2 ), (0.2, 0.5, 0.3))  C (0.2, s3 ), (0.8, Low(( s2 , s2 ), (1)
 C (0.2, s3 ), (0.8, s2 )  sk (2,4) ,
k (2, 4)  2  round ((0.2).(3 1))  2  0  2
 sk (2,4)  s2  EU .
E1M ( x3 , x1 )  Low(( s3 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s4 ), (0.5, Low(( s4 , s3 ), (3/ 5, 2 / 5))
 C (0.5, s4 ), (0.5, s4 )  s4  SC.
E1M ( x3 , x2 )  Low(( s6 , s4 , s6 ), (0.2, 0.5, 0.3))  C (0.2, s6 ), (0.8, Low(( s6 , s4 ), (3/ 8,5 / 8))
 C (0.2, s6 ), (0.8, s5 )  sk (3,2)  s5  IM .
since
Low((s6 , s4 ),(3/ 8,5/ 8))  sk , k  4  round ((3/ 8).(6  4))  4  1  5
E1M ( x3 , x3 )
 sk  s5
 Low((s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, s5 )  s5  IM .
E1M ( x3 , x4 )  Low(( s2 , s2 , s2 ), (0.2, 0.5, 0.3))  C (0.2, s2 ), (0.8, Low(( s2 ), (1))
 C (0.2, s2 ), (0.8, s2 )  s2  EU .
E1M ( x4 , x1 )  Low(( s8 , s7 , s7 ), (0.2, 0.5, 0.3))  C (0.2, s8 ), (0.8, Low(( s7 ), (1))
 C (0.2, s8 ), (0.8, s7 )  sk (4,1) ,
k (4,1)  7  round ((0.2).(8  7))  7
 sk (4,1)  s7  ML .
E1M ( x4 , x2 )  Low(( s5 , s5 , s5 ), (0.2, 0.5, 0.3))  C (0.2, s5 ), (0.8, Low(( s5 ), (1))
 C (0.2, s5 ), (0.8, s5 )  s5  IM .
E1M ( x4 , x3 )  Low(( s7 , s8 , s9 ), (0.2, 0.5, 0.3))  C (0.3, s9 ), (0.7, Low(( s8 , s7 ), (5 / 8, 2 / 8))
 C (0.3, s9 ), (0.7, s8 )  sk (4,1) ,
since
Low((s8 , s7 ),(5/ 7, 2 / 7))  sk , k  7  round ((5/ 7).(8  7))  7 1  8
k (4,3)  8  round ((0.3).(9  8))  8
 sk (4,3)  s8  EL .
 sk  s8
16
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
E1M ( x4 , x4 )  Low((s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, s5 )  s5  IM We
obtain the linguistic dominance degrees for criterion C1 :
 IM ML MC
 SC IM IM
EM1  
 SC IM IM

 ML IM EL
EU 
EU  .
EU 

IM 
(82)
Calculate the FCS by Definition 2.
fcs1M ( x1 )  Low(( s7 , s6 , s2 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3, s7 ), (2 / 3, Low(( s6 , s2 ), (0.5, 0.5))
 C (1/ 3, s7 ), (2 / 3, s4 )  sk ,
since
Low((s6 , s2 ),(0.5,0.5))  sk ' , k '  2  round ((0.5).(6  2))  2  2  4  sk '  s4 ,
k  4  round ((1/ 3).(7  4))  4  1  5
 sk  s5  IM .
fcs1M ( x2 )  Low(( s5 , s4 , s2 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3, s5 ), (2 / 3), Low(( s4 , s2 ), (1/ 2,1/ 2))
 C (1/ 3, s5 ), (2 / 3, s3 )  sk ,
since
Low((s4 , s2 ),(0.5,0.5))  sk ' , k '  2  round ((0.5).(4  2))  2  1  3
k  3  round ((1/ 3).(5  3))  3  1  4
 sk '  s3
 sk  s4  SC .
fcs1M ( x3 )  Low(( s5 , s4 , s2 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3, s5 ), (2 / 3, Low(( s4 , s2 ), (1/ 2,1/ 2))
 C (1/ 3, s5 ), (2 / 3, s3 )  s4  SC.
fcs1M ( x4 )  Low(( s7 , s5 , s8 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3), s8 , 2 / 3.Low(( s7 , s5 ), (0.5, 0.5)
 C (1/ 3, s8 ), (2 / 3, s6 )  sk ,
since
Low((s7 , s5 ),(0.5,0.5))  sk ' , k '  5  round ((0.5).(7  5))  5  1  6
k  6  round ((1/ 3).(8  6))  6  1  7
 sk '  s6
 sk  s7  ML .
We obtain FCS for criterion C1.
FCSM1  IM / x1 , SC / x2 , SC / x3 , ML / x4  .
(83)
1.1.b. Use the linguistic non-preference relations
 SC
 EU
V11  
 IM

 EU
SC VLC VLC 
 SC
 EU
SC VLC EU 
V21  
VLC
SC SC
SC 


SC VLC SC 
 EU
,
VLC SC 
 SC
 EU
SC VLC EU 
V31  
 EU
MC SC VLC 


EU VLC SC 
 EU
,
EU
SC
SC
SC
SC
SC
SC
SC
I
SC 
EU 
I 

SC 
Fuzzy Pareto Solution in multi-criteria group decision making
17
Calculate the linguistic dominance degrees using non-preference relations:
EV1 ( xi , x j )  Low(S ,UV 1 ) ,
i, j = 1,2,3,4
as in (53).
EV1 ( x1, x1 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1)
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV1 ( x1, x2 )  Low(( s4 , s2 , s4 ), (0.2, 0.3, 0.5))  C (0.2, s4 ), (0.8Low((s4 , s2 ), (3 / 8,5 / 8))
 C (0.2, s4 ), (0.8, s3 )  sk (1,2) ,
since
Low((s4 , s2 ),(3/ 8,5 / 8))  sk ' , k '  2  round ((3/ 8).(4  2))  2  1  3
k (1, 2)  3  round ((0.2).(4  3))  3  0  3
 sk '  s3
 sk (1,2)  s3  VLC
EV1 ( x1, x3 )  Low(( s3 , s3 , s4 ), (0.3, 0.5, 0.2))  C (0.3, s3 ), (0.7 Low(( s4 , s3 ), (2 / 7,5 / 7))
 C (0.3, s3 ), (0.7, s3 )  s3  VLC ,
since
Low((s4 , s3 ),(2 / 7,5 / 7))  sk ' , k '  3  round ((2 / 7)(4  3))  3  0  3
 sk '  s3
EV1 ( x1, x4 )  Low(( s3 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s4 ), (0.5 Low(( s4 , s3 ), (3/ 5, 2 / 5))
 C (0.5, s4 ), (0.5, s4 )  s4  SC ,
since
Low((s4 , s3 ),(3/ 5, 2 / 5))  sk ' , k '  3  round ((3/ 5).(4  3))  3  1  4
.
 sk '  s4
EV1 ( x2 , x1 )  Low((s2 , s2 , s2 ),(0.3,0.5,0.2))  s2  EU
EV1 ( x2 , x2 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV1 ( x2 , x3 )  Low(( s3 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.3, s4 ), (0.7 Low(( s3 ), (1))
 C (0.3, s4 ), (0.7, s3 )  sk (2,3) ,
k (2,3)  3  round ((0.3).(4  3))  3  0  3
EV1 ( x2 , x4 )
 sk (2,3)  s3  VLC
.
 Low((s2 , s2 , s2 ),(0.3,0.5,0.2))  s2  EU
EV1 ( x3 , x1 )  Low(( s5 , s3 , s2 ), (0.2, 0.5, 0.3))  C (0.2, s5 ), (0.8 Low(( s3 , s2 ), (5 / 8,3/ 8))
 C (0.2, s4 ), (0.8, s3 )  sk (3,1) ,
since
18
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
Low((s3 , s2 ),(5 / 8,3/ 8))  sk ' , k '  2  round ((5 / 8).(3  2))  2  1  3
 sk '  s3
..
k (3,1)  3  round ((0.2).(5  3))  3  0  3
 sk (3,1)  s3  VLC
.
 Low(( s4 , s6 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s6 ), (0.5 Low(( s4 ), (1))
EV1 ( x3 , x2 )
 C (0.5, s6 ), (0.5, s4 )  sk (3,2) ,
k (3, 2)  4  round ((0.5).(6  4))  4  1  5
EV1 ( x3 , x3 )
 sk (3,2)  s5  IM
.
 Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV1 ( x3 , x4 )  Low(( s4 , s3 , s1 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8 Low(( s3 , s1), (5 / 8,3/ 8))
 C (0.2, s4 ), (0.8, s2 )  sk (3,4) ,
since
Low((s3 , s1 ),(5 / 8,3/ 8))  sk ' , k '  1  round ((5 / 8).(3 1))  1  1  2
k (3, 4)  2  round ((0.2).(4  2))  2  0  2
EV1 ( x4 , x1 )
 sk (3,4)  s2  EU
 sk '  s2
.
 Low((s2 , s2 , s2 ),(0.3,0.5,0.2))  s2  EU .
EV1 ( x4 , x2 )  Low(( s4 , s2 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8 Low(( s4 , s2 ), (3/ 8,5 / 8))
 C (0.2, s4 ), (0.8, s3 )  sk (4,2) ,
since
Low((s4 , s2 ),(3/ 8,5 / 8))  sk , , k '  2  round ((3/ 8).(4  2))  2  1  3
k (4, 2)  3  round ((0.2).(4  3))  3  0  3
 sk '  s3
 sk (4,2)  s3  VLC .
EV1 ( x4 , x3 )  Low((s3 , s3 , s1 ),(0.2,0.5,0.3))  C (0.7, s3 ),(0.3, s1)  sk (4,3) ,
k (4,3)  1  round ((0.7).(3 1))  1  1  2
EV1 ( x4 , x4 )
 sk (4,3)  s2  EU
.
 Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
We obtain the linguistic dominance degrees using non-preference relations for criterion
C1.
Fuzzy Pareto Solution in multi-criteria group decision making
 SC
 EU
EV1  
VLC

 EU
VLC
VLC
SC
IM
VLC
SC
VLC
EU
SC 
EU 
EU 

SC 
19
(84)
Calculate the FCS as in (54).
fcsV1 ( x1 )  Low((s4 , s3 ),(1/ 3, 2 / 3))  C (1/ 3, s4 ),(2 / 3, s3 )  sk
k  3  round ((1/ 3).(4  3))  3  0  3
,
 sk  s3  VLC .
fcsV1 ( x2 )  Low((s3 , s2 ),(1/ 3, 2 / 3))  C (1/ 3, s3 ),(2 / 3), s2 )  sk
k  2  round ((1/ 3).(3  2))  2  0  2
,
 sk  s2  EU .
fcsV1 ( x3 )  Low(( s3 , s5 , s2 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3, s5 ), (2 / 3, Low(( s3 , s2 ), (1/ 2,1/ 2))
 C (1/ 3, s5 ), (2 / 3, s3 )  sk ,
since
Low((s3 , s2 ),(0.5,0.5))  sk ' , k '  2  round ((0.5).(3  2))  2  1  3  sk ,  s3
k  3  round ((1/ 3).(5  3))  3  1  4
 sk  s4  SC .
fcsV1 ( x4 )  Low((s3 , s2 ),(1/ 3, 2 / 3))  C (1/ 3, s3 ),(2 / 3, s2 )  sk ,
k  2  round ((1/ 3).(3  2))  2  0  2
 sk  s2  EU
.
We obtain FCS using non-preference relations for the criterion C1.
FCSV1  VLC / x1 , EU / x2 , SC / x3 , EU / x4  .
(85)
Finally, we obtain IFE for the criterion C1.
IFE1  ( IM ,VLC) / x1 , ( SC, EU ) / x2 , ( SC, SC) / x3 , ( ML, EU ) / x4  , (86)
and Fuzzy Pareto Solution (FPS) for criterion C1 is x4 .
Step 1.2. Computing for criterion C2.
1.2.a. Use the linguistic preference relations
 IM
VLC
M 12  
VLC

 SC
MC
ML
IM
SC
SC
IM
IM
MC
IM 
 IM
 SC

SC  ,
M 22  
VLC
EU 


IM 
 SC
ML
MC
IM
SC
IM
IM
MC
MC
MC 
 IM IM
 SC IM
SC  ,
M 32  
VLC SC
SC 


IM 
 SC SC
MC MC 
MC IM 
IM VLC 

IM IM 
20
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
Calculate the linguistic dominance degrees:
2
EM
( xi , x j )  Low(S ,U M 2 ) ,
U M 2  u2T ,..., u21  ,
with
i ,j =1,2,3,4
(87)
u2t  Wij (st )  w(k ) : M k 2 (i, j )  st  ,
k
t=1,...,T.
2
EM
( x1, x1 )
(88)
 Low((s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
2
EM
( x1, x2 )  Low(( s7 , s6 , s5 ), (0.5, 0.2, 0.3))  C (0.5, s7 ), (0.5, Low((s6 , s5 ),(2 / 5,3 / 5))
 C (0.5, s7 ), (0.5, s5 )  sk (1,2) ,
since
Low((s6 , s5 ),(2 / 5,3/ 5))  sk ' , k '  5  round ((2 / 5).(6  5))  5  0  5  sk ,  s5
.
k (1, 2)  5  round ((0.5).(7  5))  5  1  6
2
EM
 sk (1,2)  s6  MC
.
( x1, x3 )  Low(( s7 , s6 , s6 ), (0.2, 0.5, 0.3))  C (0.2, s7 ), (0.8, Low(( s6 ), (1))
 C (0.2, s7 ), (0.8, s6 )  sk (1,3) ,
k (1,3)  6  round ((0.2).(7  6))  6  0  6
2
EM
( x1, x4 )
 sk (1,3)  s6  MC
 Low((s5 , s6 , s6 ),(0.2,0.5,0.2))  C (0.8, s6 ),(0.2, s5 )  sk (1,4)
k (1, 4)  5  round ((0.8).(6  5))  5  1  6  sk (1,4)  s6  MC
2
EM
( x2 , x1 )
.
.
 Low((s3 , s4 , s4 ),(0.2,0.5,0.3))  C (0.8, s4 ),(0.2, s3 )  sk (2,1)
,
k (2,1)  3  round ((0.8).(4  3))  3  1  4  sk (2,1)  s4  SC
2
EM
( x2 , x2 )  Low(( s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
2
EM
( x2 , x3 )  Low(( s4 , s5 , s6 ), (0.2, 0.5, 0.3))  C (0.3, s6 ), (0.7, Low(( s5 , s4 ), (5 / 7, 2 / 7))
 C (0.3, s6 ), (0.7, s5 )  sk (2,3) ,
since
Low((s5 , s4 ),(5 / 7, 2 / 7))  sk ' , k '  4  round ((5 / 7).(5  4))  4  1  5
k (2,3)  5  round ((0.3).(6  5))  5  0  5
 sk (2,3)  s5  IM .
Fuzzy Pareto Solution in multi-criteria group decision making
21
2
EM
( x2 , x4 )  Low(( s4 , s4 , s5 ), (0.2, 0.5, 0.3))  C (0.3, s5 ), (0.7, Low(( s4 ), (1))
 C (0.3, s5 ), (0.7, s4 )  sk (2,4) .
k (2, 4)  4  round ((0.3).(5  4))  4  0  4
2
EM
( x3 , x1 )
 sk (2,3)  s4  SC
.
 Low((s3 , s3 , s3 ),(0.2,0.5,0.3))  C ( s3 ),(1)  s3  VLC.
2
EM
( x3 , x2 )  Low((s4 , s4 , s4 ),(0.2,0.5,0.3))  C (s4 ),(1)  s4  SC.
2
EM
( x3 , x3 )  Low(( s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low((s5 ,1)  s5  IM .
2
EM
( x3 , x4 )  Low(( s2 , s4 , s3 ), (0.2, 0.5, 0.3))  C (0.5, s4 ), (0.5) Low(( s3 , s2 )(3/ 5, 2 / 5))
 C (0.5, s4 ), (0.5, s3 )  sk (3,4) .
since
Low((s3 , s2 ),(3/ 5, 2 / 5))  sk ' , k '  2  round ((3/ 5).(3  2))  2  1  3
k (3, 4)  3  round ((0.5).(4  3))  3  1  4
 sk (3,4)  s4  SC
.
2
EM
( x4 , x1 )  Low((s4 , s4 , s4 ),(0.2,0.5,0.3))  C ( s4 ),(1)  s4  SC.
2
EM
( x4 , x2 )  Low(( s5 , s6 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s6 ), (0.5)Low((s5 , s4 )(2 / 5,3 / 5))
 C (0.5, s6 ), (0.5, s4 )  sk (4,2) .
since
Low((s5 , s4 ),(2 / 5,3/ 5))  sk ' , k '  4  round ((2 / 5).(5  4))  4  0  4
k (4, 2)  4  round ((0.5).(6  4))  1  1  5
 sk (4,2)  s5  IM .
2
EM
( x4 , x3 )  Low((s6 , s6 , s5 ),(0.2,0.5,0.3))  C (0.7, s6 ),(0.3, s5 )  s6  MC.
2
EM
( x4 , x4 )  Low(( s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
We obtain the linguistic dominance degrees for criterion C2 :
EM2
 IM
 SC

VLC

 SC
MC
MC
IM
SC
IM
IM
SC
MC
MC 
SC  .
SC 

IM 
(89)
22
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
Calculate the FCS
2
fcsM
( x1 )  Low(( s6 ),(1))  s6  MC .
2
fcsM
( x2 )  Low((s5 , s4 ),(1/ 3, 2 / 3))  C((1/ 3, s5 ),(2 / 3, s4 )}  sk ,
k  4  round ((1/ 3).(5  4))  4  0  4
 sk  s4  SC
2
fcsM
( x3 )  Low((s4 , s3 ),(2 / 3,1/ 3))  C{(2 / 3, s4 ),(1/ 3, s3 )}  sk ,
k  3  round ((2 / 3).(4  3))  3  1  4
 sk  s4  SC
2
fcsM
( x4 )  Low((s6 , s4 ),(1/ 3, 2 / 3))  C{(1/ 3, s6 ),(2 / 3, s4 )}  sk ,
k  4  round ((1/ 3).(6  4))  4  1  5
 sk  s5  IM
and the FCS FCSM2  MC / x1 , SC / x2 , SC / x3 , IM / x4  .
(90)
1.2.b. Use the linguistic non-preference relations
 SC
 SC VLC SC EU 
 IM
 SC SC

IM VLC 
V22  
V12  
 IM
 MC SC
SC MC 



 SC
 MC MC VLC SC  ,
VLC VLC 
 SC
 IM
VLC SC 
V32  
 SC
SC VLC 


VLC VLC SC 
 MC
,
EU
SC VLC
SC
SC
SC VLC
SC SC
SC
SC
EU 
SC 
SC 

SC 
Calculate the linguistic non-dominance degrees:
EV2 ( xi , x j )  Low( S ,UV 2 ) , for i,j =1,…,4
UV 2  u2T ,..., u21  ,with
(91)
u2t  Wij (st )  w(k ) : Vk 2 (i, j )  st  ,for each
k
t=1,...,T.
EV2 ( x1, x1 )
(92)
 Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV2 ( x1, x2 )  Low(( s3 , s2 , s4 ), (0.2, 0.5, 0.3))  C (0.3, s4 ), (0.7, Low((s3 , s2 ),(2 / 7,5 / 7)
 C (0.3, s4 ), (0.7, s2 )  sk (2,1) ,
since
Low((s3 , s2 ),(2 / 7,5 / 7))  sk ' , k '  2  round ((2 / 7).(3  2))  2  0  2,
k (1, 2)  2  round ((0.3).(4  2))  2  1  3
 sk (2,1)  s3  VLC.
Fuzzy Pareto Solution in multi-criteria group decision making
23
EV2 ( x1, x3 )  Low(( s4 , s3 , s3 ), (0.2, 0.5, 0.3))  C (0.3, s4 ), (0.7) Low(( s3 )(1))
 C (0.3, s4 ), (0.7, s3 )  sk (1,3) .
k (1,3)  3  round ((0.3).(4  3))  3  0  3
 sk (1,3)  s3  VLC.
EV2 ( x1, x4 )  Low(( s2 , s3 , s2 ), (0.2, 0.5, 0.3))  C (0.5, s3 ), (0.5) Low(( s2 )(1))
 C (0.5, s3 ), (0.5, s2 )  sk (1,4) .
k (1, 4)  2  round ((0.5).(3  2))  2  1  3
 sk (1,4)  s3  VLC.
EV2 ( x2 , x1 )  Low(( s4 , s5 , s5 ), (0.2, 0.5, 0.3))  C (0.5, s5 ), (0.5) Low(( s5 , s4 )(3/ 5, 2 / 5))
 C (0.5, s5 ), (0.5, s5 )  s5  IM ,
since
Low((s5 , s4 ),(3/ 5, 2 / 5))  sk ' , k '  4  round ((3/ 5).(5  4))  4  1  5.
EV2 ( x2 , x2 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV2 ( x2 , x3 )  Low(( s5 , s3 , s3 ), (0.2, 0.5, 0.3))  C (0.2, s5 ), (0.8) Low(( s3 )(1))
 C (0.2, s5 ), (0.8, s3 )  sk (2,3) ,
k (2,3)  3  round ((0.2).(5  3))  3  0  3
 sk (2,3)  s3  VLC.
EV2 ( x2 , x4 )  Low(( s3 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s4 ), (0.5) Low(( s4 , s3 ), (3/ 5, 2 / 5))
 C (0.5, s4 ), (0.5, s4 )  s4  SC ,
since
Low((s4 , s3 ),(3/ 5, 2 / 5))  sk ' , k '  3  round ((3/ 5).(4  3))  3  1  4.
EV2 ( x3 , x1 )
.
 Low(( s6 , s5 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s6 ), (0.8) Low(( s5 , s4 )(5 / 8,3/ 8))
 C (0.2, s6 ), (0.8, s5 )  sk (3,1) .
24
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
Since
Low((s5 , s4 ),(5 / 8,3/ 8))  sk ' , k '  4  round ((5 / 8).(5  4))  4  1  5,
k (3,1)  5  round ((0.2).(6  5))  5  0  5
 sk (3,1)  s5  IM .
EV2 ( x3 , x2 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV2 ( x3 , x3 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV2 ( x3 , x4 )  Low(( s6 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s6 ), (0.8) Low(( s4 , s3 )(3/ 8,5 / 8))
 C (0.2, s6 ), (0.8, s3 )  sk (3,4) .
since
Low((s4 , s3 ),(3/ 8,5 / 8))  sk ' , k '  3  round ((3/ 8).(4  3))  3  0  3,
EV2 ( x4 , x1 )  Low(( s4 , s4 , s6 ), (0.2, 0.5, 0.3))  C (0.3, s6 ), (0.7) Low(( s4 )(1))
 C (0.3, s6 ), (0.7, s4 )  sk (4,1) .
k (4,1)  4  round ((0.3).(6  4))  4  1  5
 sk (4,1)  s5  IM .
EV2 ( x4 , x2 )  Low(( s4 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8) Low(( s4 , s3 )(3/ 8,5 / 8))
 C (0.2, s4 ), (0.8, s3 )  sk (4,2) .
since
Low((s4 , s3 ),(3/ 8,5 / 8))  sk ' , k '  3  round ((3/ 8).(5  4))  3  0  3,
k (4, 2)  3  round ((0.2).(4  3))  3  0  3
 sk (4,2)  s3  VLC.
EV2 ( x4 , x3 )  Low(( s3 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.3, s4 ), (0.7) Low(( s3 )(1))
 C (0.3, s4 ), (0.7, s3 )  sk (4,3) .
k (4,3)  3  round ((0.3).(4  3))  3  0  3
 sk (4,3)  s3  VLC.
Fuzzy Pareto Solution in multi-criteria group decision making
25
EV2 ( x4 , x4 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
We obtain the linguistic non-dominance degrees using
V12 ,V 22 ,V32 
for criterion
C2 :
 SC
 IM
EV2  
 IM


 IM
VLC
VLC
SC
SC
VLC
SC
VLC
VLC
VLC 
SC 
.
SC 

SC 

(93)
Calculate FCS
fcsV2 ( x1 )  Low(( s3 ),(1))  s3  VLC
fcsV2 ( x2 )  Low(( s5 , s4 , s3 ), (1/ 3, 2 / 3))  C (1/ 3, s5 ), (2 / 3, Low(( s4 , s3 ), (0.5, 0.5))
 C (1/ 3, s5 ), (2 / 3, s4 )  sk ,
since
Low((s4 , s3 ),(0.5,0.5))  sk ' , k '  3  round ((0.5).(4  3))  3  1  4  sk ,  s4
..
k  4  round ((1/ 3).(5  4))  4  0  4
 sk  s4  SC.
fcsV2 ( x3 )  Low(( s5 , s4 ),(1/ 3, 2 / 3))  C (1/ 3, s5 ),(2 / 3), s4 )  sk ,
k  4  round ((1/ 3).(5  4))  4  0  4
 sk  s4  SC
.
fcsV2 ( x4 )  Low(( s5 , s4 , s3 ), (1/ 3, 2 / 3))  C (1/ 3, s5 ), (2 / 3, Low(( s4 , s3 ), (0.5, 0.5))
 C (1/ 3, s5 ), (2 / 3, s4 )  sk ,
since
Low((s4 , s3 ),(0.5,0.5))  sk ' , k '  3  round ((0.5).(4  3))  3  1  4  sk ,  s4
k  4  round ((1/ 3).(5  4))  4  0  4
 sk  s4  SC.
We obtain FCS using non-preference relations for criterion C2
FCSV2  VLC / x1 , SC / x2 , SC / x3 , SC / x4  .
Finally, we obtain IFE for criterion C2
(94)
26
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
IFE 2  ( MC,VLC) / x1 , ( SC, SC) / x2 , ( SC, SC) / x3 , ( IM , SC) / x4  . (95)
and FPS for criterion C2 is x1 .
Step 1.3. Computing for criterion C3.
1.3.a. Use the linguistic preference relations
 IM
 SC
M 13  
VLC

 SC
MC
ML
IM
SC
MC
IM
MC
SC
 IM
IM 
VLC

SC 
M 23  
 SC
IM 


IM 
VLC
,
ML
IM
IM
MC
IM
SC
SC
SC
ML 
 IM ML SC MC 
 SC IM MC IM 
IM 

M 33  
 IM SC IM SC 
SC 



IM 
VLC SC IM IM 
,
Calculate the linguistic dominance degrees
3
EM
( xi , x j )  Low(S ,U M 3 ) ,
i, j =1,2,3,4
U M 3  u3T ,..., u31  , with
(96)
u3t  Wij ( st )  w(k ) : M k 3 (i, j )  st  ,
k
t=1,...,T.
3
EM
( x1, x1 )
(97)
 Low((s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
3
EM
( x1, x2 )  Low(( s6 , s7 , s7 ), (0.2, 0.5, 0.3))  C (0.5, s7 ), (0.5, Low(( s7 , s6 ), (3/ 5, 2 / 5))
 C (0.5, s7 ), (0.5, s7 )  s7  ML,
since
Low((s7 , s6 ),(3/ 5, 2 / 5))  sk ' , k '  6  round ((3/ 5).(7  6))  6  1  7  sk ,  s7
3
EM
( x1, x3 )  Low(( s7 , s5 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s7 ), (0.8, Low(( s5 , s4 )(5 / 8,3/ 8))
 C (0.2, s7 ), (0.8, s5 )  sk (1,3) ,
since
Low((s5 , s4 ),(5 / 8,3/ 5))  sk ' , k '  4  round ((5 / 8).(5  4))  4  1  5  sk ,  s5 ,
k (1,3)  5  round ((0.2).(7  5))  5  0  5
 sk (1,3)  s5  IM
.
Fuzzy Pareto Solution in multi-criteria group decision making
27
3
EM
( x1, x4 )  Low(( s5 , s7 , s7 ), (0.2, 0.5, 0.3))  C (0.5, s7 ), (0.5, Low(( s7 , s5 ), (3/ 5, 2 / 5))
 C (0.5, s7 ), (0.5, s6 )  sk (1,4) ,
since
Low((s7 , s5 ),(3/ 5, 2 / 5))  sk ' , k '  5  round ((3/ 5).(7  5))  5  1  6  sk ,  s6 ,
k (1, 4)  6  round ((0.5).(7  6))  6  1  7
3
EM
 sk (1,4)  s7  ML
.
( x2 , x1 )  Low(( s4 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 , s3 )(3/ 8,5 / 8))
 C (0.2, s4 ), (0.8, s3 )  sk (2,1) ,
since
Low((s4 , s3 ),(3/ 8,5 / 8))  sk ' , k '  3  round ((3/ 8).(4  3))  3  0  3  sk ,  s3 ,
k (2,1)  3  round ((0.2).(4  3))  3  0  3
3
EM
( x2 , x2 )
 sk (2,1)  s3  VLC
.
 Low(( s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
3
EM
( x2 , x3 )  Low(( s6 , s5 , s6 ), (0.2, 0.5, 0.3))  C (0.2, s6 ), (0.8, Low((s6 , s5 ),(3 / 8,5 / 8))
 C (0.2, s6 ), (0.8, s5 )  sk (2,3) ,
since
Low((s6 , s5 ),(3/ 8,5 / 8))  sk ' , k '  5  round ((3/ 8).(6  5))  5  0  5  sk ,  s5
k (2,3)  5  round ((0.2).(6  5))  5  0  5
3
EM
 sk (2,3)  s5  IM .
.
( x2 , x4 )  Low(( s4 , s5 , s5 ), (0.2, 0.5, 0.3))  C (0.5, s5 ), (0.5, Low(( s5 , s4 )(3/ 5, 2 / 5))
 C (0.5, s5 ), (0.5, s5 )  s5  IM ,
28
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
since
Low((s5 , s4 ),(3/ 5, 2 / 5))  sk ' , k '  4  round ((3/ 5).(5  4))  4  1  5  sk ,  s5.
3
EM
( x3 , x1 )  Low(( s3 , s4 , s5 ), (0.2, 0.5, 0.3))  C (0.3, s5 ), (0.7, Low(( s4 , s3 )(5 / 7, 2 / 7))
 C (0.3, s5 ), (0.7, s4 )  sk (3,1) ,
since
Low((s4 , s3 ),(5 / 7, 2 / 7))  sk ' , k '  3  round ((5 / 7).(4  3))  3  1  4  sk ,  s4 ,
k (3,1)  4  round ((0.3).(5  4))  4  0  4
 sk (3,1)  s4  SC.
3
EM
( x3 , x2 )  Low(( s4 , s4 , s4 ),(0.2,0.5,0.3))  C (0.2, s4 ),(0.8, Low(( s4 ,1)  s4  SC.
3
EM
( x3 , x3 )  Low(( s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
2
EM
( x3 , x4 )  Low(( s5 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s5 ), (0.8, Low(( s4 )(1))
 C (0.2, s5 ), (0.8, s4 )  sk (3,4) ,
k (3, 4)  4  round ((0.2).(5  4))  4  0  4
 sk (3,4)  s4  SC.
3
EM
( x4 , x1 )  Low(( s4 , s3 , s3 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s3 )(1))
 C (0.2, s5 ), (0.8, s3 )  sk (4,1) ,
k (4,1)  3  round ((0.2).(4  3))  3  0  3
3
EM
 sk (4,1)  s3  VLC.
.
( x4 , x2 )  Low(( s6 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s6 ), (0.8, Low(( s4 )(1))
 C (0.2, s6 ), (0.8, s4 )  sk (4,2) ,
k (4, 2)  4  round ((0.2).(6  4))  4  0  4
 sk (4,2)  s4  SC.
3
EM
( x4 , x3 )  Low(( s4 , s4 , s5 ), (0.2, 0.5, 0.3))  C (0.3, s5 ), (0.7, Low(( s4 )(1))
 C (0.3, s5 ), (0.7, s4 )  sk (4,3) ,
Fuzzy Pareto Solution in multi-criteria group decision making
k (4,3)  4  round ((0.3).(5  4))  4  0  4
29
 sk (4,3)  s4  SC.
3
EM
( x4 , x4 )  Low(( s5 , s5 , s5 ),(0.2,0.5,0.3))  C (0.2, s5 ),(0.8, Low(( s5 ,1)  s5  IM .
We obtain the linguistic dominance degrees
 IM
VLC
EM3  
 SC

VLC
ML IM
IM
SC
IM
IM
IM
SC
ML 
IM  .
SC 

IM 
(98)
Calculate FCS by Definition 2.
3
fcsM
( x1 )  Low((s7 , s5 ),(2 / 3,1/ 3))  C (2 / 3, s7 ),(1/ 3, s5 )  sk
k  5  round ((2 / 3).(7  5))  5  1  6
 sk  s6  MC .
3
fcsM
( x2 )  Low((s5 , s3 ),(2 / 3,1/ 3))  C (2 / 3, s5 ),(1/ 3, s3 )  sk ,
k  3  round ((2 / 3).(5  2))  3  1  4
3
fcsM
( x3 )
 sk  s4  SC
.
 Low(( s4 ),(1))  s4  SC.
3
fcsM
( x4 )  Low(( s5 , s4 , s3 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3, s5 ), (2 / 3, Low(( s4 , s3 ), (0.5, 0.5)) 
 C (1/ 3, s5 ), (2 / 3, s4 )  sk ,
since
Low((s4 , s3 ),(0.5,0.5))  sk ' , k '  3  round ((0.5).(4  3))  3  1  4  sk ,  s4 ,
k  4  round ((1/ 3).(5  4))  4  0  4
We obtain
 sk  s4  SC .
3
FCSM
 MC / x1, SC / x2 , SC / x3 , SC / x4 .
(99)
1.3.b. Use the linguistic non-preference relations
 SC SC VLC
 SC SC
SC
V13  
 IM VLC SC

SC
 IM SC
SC 
 SC VLC SC VLC 
 SC VLC



 IM SC
SC 
IM
SC VLC VLC 
V23  
V33  
 IM IM
 SC SC
SC 
SC
SC 




SC 
MC
VLC
VLC
SC


 SC SC

,
,
SC VLC 
SC VLC 
SC IM 

SC SC 
Calculate the linguistic non-dominance degrees:
EV3 ( xi , x j )  Low(S ,UV 3 ) , i,j =1,..,4 .
(100)
30
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
UV 3  u3T ,..., u31 
, with
u3t  Wij (st )  w(k ) : Vk 3 (i, j )  st  ,
k
t=1,..., T. (101)
EV3 ( x1, x1 )
 Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV3 ( x1, x2 )  Low(( s4 , s3 , s3 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8) Low(( s3 )(1))
 C (0.2, s4 ), (0.8, s3 )  sk (1,2) .
k (1, 2)  3  round ((0.2).(4  3))  3  0  3
 sk (2,1)  s3  VLC.
EV3 ( x1, x3 )  Low(( s3 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s4 ), (0.5) Low(( s4 , s3 )(3/ 5, 2 / 5))
 C (0.5, s4 ), (0.5, s4 )  s4  SC ,
since
Low((s4 , s3 ),(3/ 5, 2 / 5))  sk ' , k '  3  round ((3/ 5).(4  3))  3  1  4  sk ,  s4.
EV3 ( x1, x4 )  Low(( s4 , s3 , s3 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8) Low(( s3 )(1))
 C (0.2, s4 ), (0.8, s3 )  sk (1,4) .
k (1, 4)  3  round ((0.2).(4  3))  3  0  3
 sk (1,4)  s3  VLC.
EV3 ( x2 , x1 )  Low(( s4 , s5 , s5 ), (0.2, 0.5, 0.3))  C (0.5, s5 ), (0.5) Low(( s5 , s4 )(3/ 5, 2 / 5))
 C (0.5, s5 ), (0.5, s5 )  s5  IM ,
since
Low((s5 , s4 ),(3/ 5, 2 / 5))  sk ' , k '  4  round ((3/ 5).(5  4))  4  1  5.
EV3 ( x2 , x2 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
Fuzzy Pareto Solution in multi-criteria group decision making
31
EV3 ( x2 , x3 )  Low(( s4 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s4 ), (0.5, s3 )  sk (2,3) ,
k (2,3)  3  round ((0.5).(4  3))  3  1  4,  sk (2,3)  s4  SC.
EV3 ( x2 , x4 )  Low((s4 , s3 , s3 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8)Low ((s3 )(1))
 C (0.2, s4 ), (0.8, s3 )  sk (2,4) .
k (2, 4)  3  round ((0.2).(4  3))  3  0  3
 sk (2,4)  s3  VLC.
EV3 ( x3 , x1 )  Low(( s5 , s5 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s5 ), (0.8) Low(( s5 , s4 )(5 / 8,3/ 8))
 C (0.2, s5 ), (0.8, s5 )  sk (3,1) .
since
Low((s5 , s4 ),(5 / 8,3/ 8))  sk ' , k '  4  round ((5 / 8).(5  4))  4  1  5,
k (3,1)  5  round ((0.2).(5  5))  5  0  5
 sk (3,1)  s5  IM .
EV3 ( x3 , x2 )  Low(( s3 , s5 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s3 ), (0.8, Low(( s5 , s4 ), (5 / 8,3/ 8))
 C (0.2, s4 ), (0.8, s5 )  sk (3,2) ,
since
Low((s5 , s4 ),(5 / 8,3/ 8))  sk ' , k '  4  round ((5 / 8).(5  4))  4  1  5,
k (3, 2)  4  round ((0.8).(5  4))  4  1  5
 sk (3,2)  s5  IM .
EV3 ( x3 , x3 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
EV3 ( x3 , x4 )  Low(( s4 , s4 , s5 ), (0.2, 0.5, 0.3))  C (0.3, s5 ), (0.8) Low(( s4 )(1))
 C (0.2, s5 ), (0.8, s4 )  sk (3,4) .
k (3, 4)  4  round ((0.2).(5  4))  4  0  4
 sk (3,4)  s4  SC.
EV3 ( x4 , x1 )  Low(( s5 , s6 , s4 ), (0.2, 0.5, 0.3))  C (0.5, s6 ), (0.5) Low(( s5 , s4 )(2 / 5,3/ 5))
 C (0.5, s6 ), (0.5, s4 )  sk (4,1) .
32
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
since
Low((s5 , s4 ),(2 / 5,3/ 5))  sk ' , k '  4  round ((2 / 5).(5  4))  4  0  4,
k (4,1)  4  round ((0.5).(6  4))  4  1  5
 sk (4,1)  s5  IM .
EV3 ( x4 , x2 )  Low(( s4 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8) Low(( s4 , s3 )(3/ 8,5 / 8))
 C (0.2, s4 ), (0.8, s3 )  sk (4,2) .
since
Low((s4 , s3 ),(3/ 8,5 / 8))  sk ' , k '  3  round ((3/ 8).(5  4))  3  0  3,
k (4, 2)  3  round ((0.2).(4  3))  3  0  3
 sk (4,2)  s3  VLC.
EV3 ( x4 , x3 )  Low(( s4 , s3 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8) Low(( s4 , s3 )(3/ 8,5 / 8))
 C (0.2, s4 ), (0.8, s3 )  sk (4,3) .
since
Low((s4 , s3 ),(3/ 8,5 / 8))  sk ' , k '  3  round ((3/ 8).(5  4))  3  0  3,
k (4, 2)  3  round ((0.2).(4  3))  3  0  3
 sk (4,2)  s3  VLC.
EV2 ( x4 , x4 )  Low(( s4 , s4 , s4 ), (0.2, 0.5, 0.3))  C (0.2, s4 ), (0.8, Low(( s4 ), (1))
 C (0.2, s4 ), (0.8, s4 )  s4  SC.
We obtain the linguistic non-dominance degree
 SC VLC SC VLC 
 IM SC VLC VLC 
.
EV3  
 IM SC
SC
SC 


 IM VLC VLC SC 
(102)
Calculate FCS by Definition 2.
fcsV3 ( x1 )  Low((s4 , s3 ),(1/ 3, 2 / 3))  C (1/ 3, s4 ),(2 / 3, s3 )  sk ,
k  3  round ((1/ 3).(4  3))  3  0  3
 sk  s3  SC  fcsV3 ( x1)  SC.
Fuzzy Pareto Solution in multi-criteria group decision making
33
fcsV3 ( x2 )  Low(( s5 , s4 , s3 ), (1/ 3, 2 / 3))  C (1/ 3, s5 ), (2 / 3, Low(( s4 , s3 ), (0.5, 0.5))
 C (1/ 3, s5 ), (2 / 3, s4 )  sk ,
since
Low((s4 , s3 ),(0.5,0.5))  sk ' , k '  3  round ((0.5).(4  3))  3  1  4  sk ,  s4
k  4  round ((1/ 3).(5  4))  4  0  4
 sk  s4  SC.
fcsV3 ( x3 )  Low((s5 , s4 ),(2 / 3,1/ 3))  C (2 / 3, s5 ),(1/ 3), s4 )  sk ,
k  4  round ((2 / 3).(5  4))  4  1  5
 sk  s5  IM  fcsV3 ( x)  IM .
fcsV3 ( x4 )  Low((s5 , s3 ),(1/ 3, 2 / 3))  C (1/ 3, s5 ),(2 / 3, s3 )  sk ,
k  3  round ((1/ 3).(5  4))  3  0  3
We obtain
 sk  s3  VLC  fcsV3 ( x4 )  VLC.
FCSV3  VLC / x1 , SC / x2 , IM / x3 ,VLC / x4 
(103)
Finally, we obtain IFE for criterion C3
IFE 3  ( MC, VLC) / x1 , ( SC, SC) / x2 , ( SC, IM ) / x3 , ( SC, VLC) / x4 
(104)
and the FPS for criterion C3 is x1 .
Step2. Use
{( FCSMl , FCSVl ), l  1, 2,3} and the criteria’s important weights
  0.35,0.4,0.25 , calculate the aggregated fuzzy collective solutions (aFCS) as in
Definition 3.
where
where
aFCS  afcs( x1 ) / x1 ,..., afcs( xn ) / xn  ,
afcsM ( xi )  Low(S ,U M  ) ,
(105)
i=1,…,n

(106)

l
U M   uM T ,..., uM 1  , t=1, ...T, U M t  l l : fcsM
( xi )  st (107)
Calculate afcsM ( x1 ). Because
2
3
fcs1M ( x1 )  s5 , fcsM
( x1 )  s6 , fcsM
( x1 )  s6 ,
34
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
afcsM ( x1 )  Low((s6 , s5 ),(0.65,0.35)  C (0.65, s6 ),(0.35, s5 )  sk ,
k  5  round ((0.65).(6  5))  5  1  6
 sk  s6  MC  afcsM ( x1 )  MC.
Calculate afcsM ( x2 ). Because
2
3
fcs1M ( x2 )  s4 , fcsM
( x2 )  s4 , fcsM
( x2 )  s4 ,
afcsM ( x2 )  Low(( s4 ),(1)  s4  SC.
Calculate afcsM ( x3 ). Because
2
3
fcs1M ( x3 )  s4 , fcsM
( x3 )  s4 , fcsM
( x3 )  s4 ,
afcsM ( x3 )  Low((s4 ),(1)  s4  SC.
Calculate afcsM ( x4 ). Because
2
3
fcs1M ( x4 )  s7 , fcsM
( x4 )  s5 , fcsM
( x4 )  s4 ,
afcsM ( x4 )  Low(( s7 , s5 , s4 ), (0.35, 0.4, 0.25) 
 C (0.35, s7 ), (0.65, Low(( s5 , s4 ), (0.4 / 0.65, 0.25 / 0.65))  C (0.35, s7 ), (0.65, s5 )  sk ,
since
Low((s5 , s4 ),(0.4 / 0.65,0.25 / 0.65))  sk ' , k '  4  round ((0.4 / 0.65).(5  4))  4  1  5  sk ,  s5 ,
k  5  round ((0.35).(7  5))  5  1  6
 sk  s6  MC  afcsM ( x4 )  MC. Fina
lly, we obtain
Calculate
where
where
aFCSM  {MC / x1 , SC / x2 , SC / x3 , MC / x4 }.
(108)
aFCSV  afcsV ( x1 ) / x1 ,..., afcsV ( x4 ) / x4  ,
(109)
afcsV ( xi )  Low(S ,UV  ) ,
UV   uV T ,..., uV 1  ,
i=1,2,3,4
t=1, ...T,


UV t  l l : fcsVl ( xi )  st .
Calculate afcsV ( x1 ). Because
(110)
(111)
2
3
fcs1M ( x1 )  s3 , fcsM
( x1 )  s3 , fcsM
( x1 )  s3 ,
afcsV ( x1 )  Low((s3 ),(1))  s3  VLC.
Calculate afcsV ( x2 ). Because
fcsV1 ( x2 )  s4 , fcsV2 ( x2 )  s4 , fcsV3 ( x2 )  s3 ,
afcsM ( x2 )  Low((s4 , s2 ),(0.65,0.35)  sk ,
Fuzzy Pareto Solution in multi-criteria group decision making
k  2  round ((0.65).(4  2))  2  1  3
35
 sk  s3  VLC  afcsV ( x2 )  VLC.
fcsV1 ( x3 )  s4 , fcsV2 ( x3 )  s4 , fcsV3 ( x3 )  s5 ,
afcsV ( x3 ). Because
afcsV ( x3 )  Low(( s5 , s4 ),(0.25,0.75)  C (0.25, s5 ),(0.75, s4 ) sk ,
Calculate
k  4  round ((0.25).(5  4))  4  0  4
Calculate afcsV ( x4 ). Because
 sk  s4  SC  afcsV ( x3 )  SC.
fcsV1 ( x4 )  s2 , fcsV2 ( x4 )  s4 , fcsV3 ( x4 )  s3 ,
afcsV ( x4 )  Low(( s4 , s3 , s2 ), (0.35, 0.4, 0.25) 
 C (0.35, s4 ), (0.65, Low(( s3 , s2 ), (0.4 / 0.65, 0.25 / 0.65))  C (0.35, s4 ), (0.65, s3 )  sk ,
since
Low((s3 , s2 ),(0.4 / 0.65,0.25 / 0.65))  sk ' , k '  2  round ((0.4 / 0.65).(3  2))  2  1  3  sk ,  s3 ,
k  3  round ((0.35).(4  3))  3  0  3
We obtain
 sk  s3  VLC  afcsV ( x4 )  VLC.
aFCSV  {VLC / x1 , VLC / x2 , SC / x3 , VLC / x4}.
(112)
Finally, we obtain Aggregated Intuitionistic Fuzzy Evaluation (aIFE)
aIFE  {( MC,VLC) / x1 ,( SC, VLC) / x2 ,( SC, SC) / x3 ,( MC, VLC) / x4 }.
The Aggregated Fuzzy Pareto Solution (aFPS) are
(113)
x1 , x4  .
7.2. Computing with the aggregation procedure 2
Step 1.
l
For each criterion C , l=1,2,3 calculate the linguistic dominance degrees as in (51, 53)
we obtain:
E1M
 IM ML MC
 SC IM IM

 SC IM IM

 ML IM EL
EU 
 IM
 IM ML MC MC 
VLC
 SC IM IM SC 
EU 
3
2
 EM

EM

 SC
EU 
VLC IM IM SC 




IM 
SC
SC
MC
IM
VLC

 ,
,
ML IM
VLC VLC VLC 
 SC

SC VLC SC  , 3 VLC
EV 
 SC
SC
SC
SC 


SC VLC SC 
VLC
ML IM
IM
SC
IM
IM
IM
SC
ML 
IM 
SC 

IM 
and
 SC VLC VLC
 EU
SC VLC
1
EV  
VLC IM
SC

 EU VLC EU
 SC
SC 


EU  , 2  IM
EV 
 IM
EU 


SC 
 IM
Step 2.
2.a1. Using these relations of linguistic relative dominance degrees
the important weights
relation:
  0.35,0.4,0.25 ,
SC
SC
IM
SC
IM
SC
E
1
M
ML 
IM 
SC 

SC 
, EM2 , EM3  and
calculate the totally social opinion
36
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
QM  qM ( xi , x j )  , i  1,..., 4, j 1,..., 4.
(114)
where
qM ( xi , x j )  Low( S ,U qM ) , U qM  uqT ,..., uq1  ,
(115)
where
l
uqt  Wijq (st )  l l : EM
(i, j )  st ,t=1,...,T .
(116)


qM ( x1, x1)  Low((s5 , s5 , s5 ),(0.35,0.4,0.25))  Low(( s5 ),(1))  s5  IM .
qM ( x1, x2 )  Low((s7 , s6 , s7 ),(0.35,0.4,0.25))  Low((s7 ,0.6),( s6 ,0.4))  sk (1,2) ,
k (1, 2)  6  round ((0.6).(7  6))  6  1  7
 sk (1,2)  s7  ML  qM ( x1, x2 )  ML.
.
qM ( x1, x3 )  Low((s6 , s6 , s5 ),(0.2,0.5,0.3))  C (0.7, s6 ),(0.3, s5 )  sk (1,3) ,
k (1,3)  5  round ((0.7).(6  5))  5  1  6
 sk (1,3)  s6  MC.
qM ( x1 , x4 )  Low(( s2 , s6 , s7 ), (0.35, 0.4, 0.25))  C (0.25, s7 ), (0.75, Low (( s6 , s2 ), (40 / 75,35 / 75))
 C (0.25, s7 ), (0.75, s4 )  sk (1,4) ,
since
Low((s6 , s2 ),(40 / 75, 25 / 75))  sk ' , k '  2  round ((40 / 75).(6  2))  2  2  4  sk ,  s4 ,
k (1, 4)  4  round ((0.25).(7  4))  4  1  5
 sk (1,4)  s5  IM .
.
qM ( x2 , x1 )  Low((s4 , s4 , s3 ),(0.35,0.4,0.25))  C (0.75, s4 ),(0.25, s3 )  sk (2,1) ,
k (2,1)  3  round ((0.75).(4  3))  3  1  4
 sk (2,1)  s4  SC.
.
qM ( x2 , x2 )  Low((s5 , s5 , s5 ),(0.35,0.4,0.25))  Low((s5 ),(1))  s5  IM .
qM ( x2 , x3 )  Low((s5 , s5 , s5 ),(0.35,0.4,0.25))  Low(( s5 ),(1))  s5  IM .
qM ( x2 , x4 )  Low(( s2 , s4 , s5 ), (0.35, 0.4, 0.25))  C (0.25, s5 ), (0.75, Low((s4 , s2 ), (40 / 75,35 / 75))
 C (0.25, s5 ), (0.75, s3 )  sk (2,4) ,
since
Low((s4 , s2 ),(40 / 75, 25 / 75))  sk ' , k '  2  round ((40 / 75).(4  2))  2  1  3  sk ,  s3 ,
k (2, 4)  3  round ((0.25).(5  3))  3  1  4
 sk (2,4)  s4  SC.
qM ( x3 , x1 )  Low((s4 , s3 , s4 ),(0.35,0.4,0.25))  C (0.6, s4 ),(0.4, s3 )  sk (3,1) ,
Fuzzy Pareto Solution in multi-criteria group decision making
k (3,1)  3  round ((0.6).(4  3))  3  1  4
37
 sk (3,1)  s4  SC.
qM ( x3 , x2 )  Low((s5 , s4 , s4 ),(0.35,0.4,0.25))  C (0.35, s5 ),(0.65, s4 )  sk (3,2) ,
k (3, 2)  4  round ((0.35).(5  4))  4  0  4
 sk (3,2)  s4  SC.
qM ( x3 , x3 )  Low((s5 , s5 , s5 ),(0.35,0.4,0.25))  Low(( s5 ),(1))  s5  IM .
qM ( x3 , x4 )  Low((s2 , s4 , s4 ),(0.35,0.4,0.25))  C (0.65, s4 ),(0.35, s2 )  sk (3,4) ,
k (3, 4)  2  round ((0.65).(4  2))  2  1  3
 sk (3,4)  s3  VLC.
qM ( x4 , x1 )  Low(( s7 , s4 , s3 ), (0.35, 0.4, 0.25))  C (0.25, s7 ), (0.75, Low((s4 , s3 ), (40 / 75,35 / 75))
 C (0.25, s7 ), (0.75, s4 )  sk (4,1) ,
since
Low((s4 , s3 ),(40 / 75, 25 / 75))  sk ' , k '  3  round ((40 / 75).(4  3))  3  1  4  sk ,  s4 ,
k (4,1)  4  round ((0.25).(7  4))  4  1  5
 sk (1,4)  s5  IM .
qM ( x4 , x2 )  Low((s5 , s4 , s5 ),(0.35,0.4,0.25))  C (0.6, s5 ),(0.4, s4 )  sk (4,2) ,
k (4, 2)  4  round ((0.6).(5  4))  4  1  5
 sk (4,2)  s5  IM .
qM ( x4 , x3 )  Low(( s8 , s6 , s4 ), (0.35, 0.4, 0.25))  C (0.35, s8 ), (0.65, Low((s6 , s4 ), (40 / 65, 25 / 65))
 C (0.35, s8 ), (0.65, s5 )  sk (4,3) ,
Low(( s6 , s4 ), (40 / 65, 25 / 65))  sk ' , k '  4  round ((40 / 75).(6  4))  4  1  5  sk ,  s5 ,
k (4,3)  5  round ((0.35).(8  5))  5  1  6  sk (4,3)  s6  MC.
qM ( x4 , x4 )  Low((s5 , s5 , s5 ),(0.35,0.4,0.25))  Low(( s5 ),(1))  s5  IM .
We obtain the totally social opinion relation
QM
 IM
 SC

 SC

 IM
IM 
SC  .
VLC 

MC IM 
ML MC
IM
IM
IM
IM
IM
2.a2. Using these relations of linguistic non-dominance degrees
(117)
E , E , E  and the
1
V
2
V
3
V
important weights   0.35,0.4,0.25 , calculate the totally social opinion relation
38
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
QV  qV ( xi , x j )  , i  1,..., 4, j  1,..., 4.
U qM  uqT ,..., uq1 
where
qV ( xi , x j )  Low(S ,U qV ) ,
where
uqt  Wijq (st )  l l : EVl (i, j )  st ,

(118)

t=1,...,T .
(119)
(120)
qV ( x1, x1 )  Low(( s4 , s4 , s4 ),(0.35,0.4,0.25))  Low(( s4 ),(1))  s4  SC.
qV ( x1, x2 )  Low((s3 , s3 , s3 ),(0.35,0.4,0.25))  Low(( s3 ),(1))  s3  VLC.
qV ( x1, x3 )  Low(( s3 , s3 , s4 ), (0.35, 0.4, 0.25))  C (0.25, s4 ), (0.75, s3 )  sk (1,3) ,
k (1,3)  3  round ((0.25).(4  3))  3  0  3  sk (1,3)  s3  VLC.
qV ( x1 , x4 )  Low((s4 , s3 , s3 ), (0.35, 0.4, 0.25))  C (0.35, s4 ), (0.65, s3 )  sk (1,4) ,
sk (1,4)  3  round ((0.35).(4  3))  3  0  3  sk (1,4)  s3  VLC .
qV ( x2 , x1 )  Low(( s2 , s5 , s5 ), (0.35, 0.4, 0.25))  C (0.75, s5 ), (0.25, s2 )  sk (2,1) ,
k (2,1)  2  round ((0.75).(5  2))  2  2  4,  sk (2,1)  s4  SC.
qV ( x2 , x2 )  Low((s4 , s4 , s4 ),(0.35,0.4,0.25))  Low(( s4 ),(1))  s4  SC.
qV ( x2 , x3 )  Low((s3 , s3 , s3 ),(0.35,0.4,0.25))  Low(( s3 ),(1))  s3  VLC.
qV ( x2 , x4 )  Low(( s2 , s4 , s3 ), (0.35, 0.4, 0.25))  C (0.4, s4 ), (0.6, Low(( s3 , s2 ), (25 / 60,35 / 60))
 C (0.25, s5 ), (0.6, s2 )  sk (2,4) ,
since
Low((s3 , s2 ),(25 / 60,35 / 60))  sk ' , k '  2  round ((25 / 60).(3  2))  2  0  2  sk ,  s2 ,
k (2, 4)  2  round ((0.25).(5  2))  2  1  3
 sk (2,4)  s3  VLC.
qV ( x3 , x1 )  Low(( s3 , s5 , s5 ), (0.35, 0.4, 0.25))  C (0.65, s5 ), (0.35, s3 )  sk (3,1) ,
k (3,1)  3  round ((0.65)(5  3))  3  1  4,  sk (3.1)  s4  SC.
qV ( x3 , x2 )  Low(( s5 , s4 , s4 ),(0.35,0.4,0.25))  C (0.35, s5 ),(0.65, s4 )  sk (3,2) ,
k (3, 2)  4  round ((0.35).(5  4))  4  0  4
 sk (3,2)  s4  SC.
qV ( x3 , x3 )  Low((s4 , s4 , s4 ),(0.35,0.4,0.25))  Low(( s4 ),(1))  s4  SC.
Fuzzy Pareto Solution in multi-criteria group decision making
39
qV ( x3 , x4 )  Low(( s2 , s4 , s4 ), (0.35, 0.4, 0.25))  C (0.65, s4 ), (0.35, s2 )  sk (3,4) ,
k (3, 4)  2  round ((0.65).(4  2))  2  1  3,  sk (3,4)  s3  VLC.
qV ( x4 , x1 )  Low(( s2 , s5 , s5 ), (0.35, 0.4, 0.25))  C (0.65, s5 ), (0.35, s2 )  sk (4,1) ,
k (4,1)  2  round ((0.65).(5  2))  2  2  4,  sk (4,1)  s4  SC.
qV ( x4 , x2 )  Low((s3 , s3 , s3 ),(0.35,0.4,0.25))  Low(( s3 ),(1))  s3  VLC.
qV ( x4 , x3 )  Low(( s2 , s3 , s3 ), (0.35, 0.4, 0.25))  C (0.65, s3 ), (0.35, s2 )  sk (4,3) ,
k (4,3)  2  round ((0.65).(3  2))  2  1  3,  sk (4,3)  s3  VLC.
qV ( x4 , x4 )  Low((s4 , s4 , s4 ),(0.35,0.4,0.25))  Low(( s4 ),(1))  s4  SC.
We obtain the totally social opinion relation
preference relations:
 SC SC SC
VLC SC SC
QV  
 SC SC SC

VLC SC VLC
using intuitionistic linguistic nonSC 
SC 
.
VLC 

SC 
2.b. Using QM , QV ,calculate the fuzzy collective solution , calculate

(121)

FCSQM  fcsQM ( x1 ) / x1, fcsQM ( x2 ) / x2 , fcsQM ( x3 ) / x3 , fcsQM ( x4 ) / x4 ,
(122)
fcsQM ( x1 )  Low(( s7 , s6 , s5 ), (1/ 3,1/ 3,1/ 3))  C (1/ 3, s7 ), (2 / 3, Low(( s6 , s5 ), (0.5, 0.5))
 C (1/ 3, s7 ), (2 / 3, s6 )  k (1),
since
Low((s6 , s5 ),(0.5,0.5))  C ((0.5, s6 ),(0.5, s5 )}  sk ' , k '  5  round ((0.5).(6  5))  5  1  6,
k (1)  6  round ((1/ 3).(7  6))  6  0  6
 sk (1)  s6  MC.
fcsQM ( x2 )  Low(( s5 , s4 ), (1/ 3, 2 / 3))  C ((1/ 3, s5 ), (2 / 3, s4 )}  sk (2) ,
k (2)  4  round ((1/ 3).(5  4))  4  0  4,  sk (2)  s4  SC.
fcsQM ( x3 )  Low(( s5 , s4 , s3 ), (1/ 3,1/ 3,1/ 3))  C{(1/ 3, s5 ), (2 / 3, Low(( s4 , s3 ), (0.5, 0.5))}
 C (1/ 3, s5 ), (2 / 3, s4 )  sk , k  4  round ((1/ 3).(5  4))  4  0  4, sk  s4  SC.
fcsQM ( x4 )  Low(( s6 , s5 ), (1/ 3, 2 / 3))  C{(1/ 3, s6 ), (2 / 3, s5 )}  sk ,
k  5  round ((1/ 3).(6  5))  5  0  5,  sk  s5  IM .
40
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
and
FCSQM  MC / x1 , SC / x2 , SC / x3 , IM / x4  .
Calculate the


FCSQV  fcsQV ( x1 ) / x1, fcsQV ( x2 ) / x2 , fcsQV ( x3 ) / x3 , fcsQV ( x4 ) / x4 ,
.
fcsQV ( x1 )  Low((s3 ), (1)))  s3  VLC.
fcsQV ( x2 )  Low(( s4 , s3 ), (2 / 3,1/ 3))  C ((2 / 3, s4 ), (1/ 3, s3 )}  sk (2) ,
k (2)  3  round ((2 / 3).(4  3))  3  1  4,  sk (2)  s4  SC.
fcsQV ( x3 )  Low((s4 , s3 ),(2 / 3,1/ 3))  C{(2 / 3, s4 ),(1/ 3, s3 )}  sk (3) ,
k (3)  3  round ((2 / 3).(4  3))  3  1  4
 sk (3)  s4  SC.
fcsQV ( x4 )  Low((s4 , s3 ),(1/ 3, 2 / 3))  C{(1/ 3, s4 ),(2 / 3, s3 )}  sk (4) ,
k (4)  3  round ((1/ 3).(4  3))  3  0  3
We obtain
 sk (4)  s3  VLC.
FCSQV  VLC / x1 , SC / x2 , SC / x3 , VLC / x4 .
Finally, we obtain Intuitionistic Fuzzy Evaluation:
IFEQ  ( MC,VLC) / x1 ,( SC, SC) / x2 ,( SC, SC) / x3 ,( IM , VLC) / x4  .
From
IFEQ we choose the FPS of the problem. The Fuzzy Pareto Solution of the
problem is
 x1 .
7.3. Computing with the Aggregation procedure 3
Step 1.1. Computing for expert e1
Use
 IM
 SC
M 11  
VLC

 EL
MC
IM
MC
IM
with the weights
 IM
EU 
VLC
SC VLC  ,
M 12  
VLC
IM EU 


ML IM 
 SC
ML
MC
ML
IM
SC
SC
IM
IM
MC
  (0.35,0.4,0.25), calculate
IM 
 IM
 SC
SC  ,
M 13  
VLC
EU 


IM 
 SC
MC
ML
IM
SC
MC
IM
MC
SC
IM 
SC 
IM 

IM 
FMk ( xi , x j )  Low(S ,U Mk ), i
,j=1,2,3,4
using (70, 71), we obtain the relative dominance degrees according to e1
Fuzzy Pareto Solution in multi-criteria group decision making
 IM
 SC
FM1  
VLC

 IM
MC
IM
IM
IM
41
SC 
SC 
,
VLC 

MC IM 
ML
SC
IM
1
and using (72, 73) calculate the fuzzy evaluation FEM
FEM1  MC / x1 , SC / x2 , SC / x3 , IM / x4  .
Use
 SC
 EU
V11  
 IM

 EU
SC VLC VLC 
 SC VLC SC EU 
 SC SC VLC

 SC SC


SC VLC EU  ,
IM
VLC
SC
,
 V   SC SC
V12  
13
 MC SC
 IM VLC SC
SC SC
SC 
SC MC 




SC VLC SC 
SC
 MC MC VLC SC 
 IM SC
with the weights
SC 
SC 
SC 

SC 
  (0.35,0.4,0.25) . Calculate FVk ( xi , x j )  Low(S ,UVk ),
,j=1,2,3,4 .
Using (74, 75), we obtain the relative non-dominance degree according to e1
 SC
VLC
FV1  
 IM

 SC
SC VLC VLC 
SC SC VLC  ,
SC SC
IM 

SC VLC SC 
and using (76) calculate the fuzzy evaluation
FEV1
FEV1  VLC / x1 , VLC / x2 , IM / x3 , SC / x4  .
Finally, we obtain IFE according to e1
(MC,VLC) / x1,(SC,VLC) / x2 ,(SC, IM ) / x3 ,( IM , SC) / x4  ,
and the FPS according to e1 is x1 .
Step 1.2. Computing for expert e2
Use
M 21
 IM ML MC
 SC IM MC

 SC SC IM

 ML IM EL
EU 
 IM
 SC
EU  ,
M 22  
VLC
EU 


IM 
 SC
ML
MC
IM
SC
IM
IM
MC
MC
MC 
 IM
VLC
SC  ,
M 23  
 SC
SC 


IM 
VLC
ML
IM
IM
MC
IM
SC
SC
SC
using (70, 71), we obtain the relative dominance degrees according to e 2
ML 
IM  .
SC 

IM 
i
42
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
 IM
 SC
FM2  
 SC

 IM
and the fuzzy evaluation
SC 
SC  ,
VLC 

MC IM 
ML MC
IM
IM
IM
IM
IM
FEM2 is
FEM2  MC / x1 , SC / x2 , SC / x3 , IM / x4  .
Use
 SC
 EU
V21  
VLC

 EU
SC 
 SC VLC SC VLC 
 SC EU VLC VLC 
 IM
 IM SC VLC SC 
SC VLC EU  ,
SC VLC VLC  .
,
 V23  
V22  
 IM IM
MC SC VLC 
 IM SC
SC
SC 
SC VLC 





EU VLC SC 
 MC VLC VLC SC 
 SC VLC VLC SC 
EU
VLC
Using (74, 75), we obtain the relative non-dominance degree according to e2
 SC EU VLC VLC 
 SC SC VLC VLC 
 ,
FV2  
 SC IM
SC VLC 


 SC VLC VLC SC 
and the fuzzy evaluation
FEV2
FEV2  VLC / x1 , VLC / x2 , SC / x3 , VLC / x4  .
Finally, we obtain IFE according to e2
(MC,VLC) / x1, (SC,VLC) / x2 , (SC, SC) / x3 , ( IM ,VLC) / x4 
and using (76), the FPS according to e2 is x1 .
Step 1.3. Computing for expert e3
Use
 IM MC
 SC IM
M 31  
 SC MC

 ML IM
MC
SC
IM
EL
EU 
 IM
 SC
EU  ,
M 32  
VLC
EU 


IM 
 SC
IM
IM
SC
SC
MC MC 
 IM
 SC
MC IM  ,
M 33  
 IM
IM VLC 


IM IM 
VLC
ML
SC
IM
SC
MC
IM
SC
IM
using (70, 71), we obtain the relative dominance degrees according to e 3
MC 
IM  .
SC 

IM 
Fuzzy Pareto Solution in multi-criteria group decision making
 IM
 SC
FM3  
 SC

 IM
and the fuzzy evaluation
IM 
SC  ,
VLC 

MC IM 
MC
MC
IM
IM
IM
IM
SC
FEM3
FEM3  MC / x1 , SC / x2 , SC / x3 , IM / x4  .
Use
 SC
 EU
V31  
 EU

 EU
SC
SC
SC
SC
SC
SC
SC
I
SC 
 SC
 IM

EU  ,
V32  
 SC
I 


SC 
 MC
SC VLC
SC VLC
SC SC
SC
SC
EU 
 SC VLC
 IM SC

SC  ,
V33  
 SC SC
SC 


SC 
 SC SC
SC VLC 
SC VLC  .
SC IM 

SC SC 
Using (74, 75), we obtain the relative non-dominance degree according to e3
 SC
 SC
FV3  
VLC

 SC
and the fuzzy evaluation
VLC 
SC SC VLC  ,
SC SC VLC 

SC VLC SC 
SC
SC
FEV3 is FEV3  SC / x1 , SC / x2 , VLC / x3 , SC / x4 .
Finally, we obtain IFE according to e3
(MC, SC) / x1, (SC, SC) / x2 , (SC,VLC) / x3 , ( IM , SC) / x4  ,
and the FPS according to e3 is
x1, x3 .
Step 2. Use the fuzzy evaluation
(FE
k
M
, FEVk ), k  1, 2,3 and the weights {w(k):
ekE}, calculate the aggregated fuzzy evaluation (aFE).
aFEM  MC / x1 , SC / x2 , SC / x3 , IM / x4  ,
aFEV  VLC / x1 ,VLC / x2 , SC / x3 , SC / x4  .
We obtain the aggregated intuitionistic fuzzy evaluation (aIFE).
aIFE  (MC,VLC ) / x1 ,( SC,VLC) / x2 ,( SC, SC) / x3 ,( IM , SC) / x4  .
Finally, the aFPS of the problem is x1 .
43
44
Bui Cong Cuong, Vladik Kreinovich, Le Hoang Son, Nilanjan Dey
8. Conclusions
This paper focuses on multi criteria group decision making problem under linguistic
assessments. The linguistic preference relations model forms a useful tool in representing
decision makers’ choices. Some aggregation operators and computing processes using
Fuzzy Collective Solution were given in Sections 3 and 4. Then, we proposed a new
approach based on the new concept of Fuzzy Pareto Solution for the group decision
making based on intuitionistic linguistic preference relations. Some aggregation
procedures for the FPS and a computing example were also proposed. In the future, we
will further investigate various aggregation procedures in the situations with type-2
intuitionistic fuzzy preference information.
APPENDIX
 IM
 SC
M 11  
VLC

 EL
 IM
VLC
M 12  
VLC

 SC
 IM
 SC
M 13  
VLC

 SC
 SC
 EU
V11  
 IM

 EU
MC
IM
MC
IM
MC
IM
SC
IM
MC
IM
SC
MC
SC
SC
SC
SC
EU 
 IM
 SC
SC VLC  ,
M 21  
 SC
IM EU 


ML IM 
 ML
ML IM 
 IM
 SC
SC SC  ,
M 22  
VLC
IM EU 


MC IM 
 SC
ML IM 
 IM
VLC
MC SC  ,
M 23  
 SC
IM IM 


SC IM 
VLC
VLC VLC 
 SC
 EU
VLC EU  ,
V21  
VLC
SC
SC 


VLC SC 
 EU
ML
 IM
EU 
 SC
EU  ,
M 31  
 SC
EU 


EL IM 
 ML
MC MC 
 IM
 SC
IM SC  ,
M 32  
VLC
IM SC 


MC IM 
 SC
IM ML 
 IM
 SC
IM IM  ,
M 33  
 IM
SC SC 


SC IM 
VLC
VLC SC 
 SC
 EU
VLC EU  ,
V31  
 EU
SC VLC 


VLC SC 
 EU
ML MC
MC
MC
IM
SC
IM
MC
SC
IM
IM
EL
IM
ML
IM
SC
MC
ML
IM
MC
SC
EU
SC
MC
EU
MC
IM
 SC VLC SC EU 
 SC EU VLC
 SC SC


IM
VLC
 , V   IM SC VLC
V12  
22
 MC SC
 IM SC
SC MC 
SC



 MC MC VLC SC 
 SC VLC VLC
 SC SC VLC SC 
 SC VLC SC
 SC SC


SC
SC
,
 V   IM SC VLC
V13  
23
 IM VLC SC SC 
 IM IM
SC



SC SC 
 IM SC
 MC VLC VLC
VLC 
 SC
 IM
SC  ,
V32  
 SC
VLC 


SC 
 MC
 SC
VLC 
 IM

VLC  ,
V33  
 SC
SC 


SC 
 SC
IM
MC
IM
SC
MC
IM
SC
IM
ML
SC
IM
SC
MC
IM
SC
IM
EU 
EU  ,
EU 

IM 
MC 
IM  ,
VLC 

IM 
MC 
IM  ,
SC 

IM 
SC 
EU  ,
I 

SC I
SC 
SC VLC EU 
SC VLC SC  ,
SC SC SC 

SC SC SC 
SC
SC
SC
SC
SC
SC
VLC
SC
SC
SC
SC VLC 
SC VLC  .
SC IM 

SC SC 
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