# Linguistic Intuitionistic Fuzzy Sets and Application in MAGDM

Authors: Huimin Zhang
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

To better deal with imprecise and uncertain information in decision making, the definition of linguistic intuitionistic fuzzy sets (LIFSs) is introduced, which is characterized by a linguistic membership degree and a linguistic nonmembership degree, respectively. To compare any two linguistic intuitionistic fuzzy values (LIFVs), the score function and accuracy function are defined. Then, based ont-norm and t-conorm, several aggregation operators are proposed to aggregate linguistic intuitionistic fuzzy information, which avoid the limitations in exiting linguistic operation. In addition, the desired properties of these linguistic intuitionistic fuzzy aggregation operators are discussed. Finally, a numerical example is provided to illustrate the efficiency of the proposed method in multiple attribute group decision making (MAGDM).


---

## Body

## 1. Introduction

Intuitionistic fuzzy set (IFS) [1], which is characterized by a degree of membership and a degree of nonmembership, is a very powerful tool to process vague information. After the pioneering study of Atanassov [1], the IFS has captured much attention from researchers in various fields and many achievements have been made, such as entropy measure of IFS [2–7], distance, or similarity measure between IFSs [8–13] and aggregation operators of IFS [14–21]. In addition, related to IFS, some authors proposed several other tools to handle vague and imprecise information whereby two or more sources of vagueness appear simultaneously [22]. Atanassov and Gargov [23] introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by a membership function and a nonmembership function with interval values. Torra [24] and Torra and Narukawa [25] gave a definition of hesitant fuzzy set (HFS), which can better deal with the situations where several values are possible to determine the membership of an element. Zhu et al. [26] defined dual hesitant fuzzy set in terms of two functions that return two sets of membership values and nonmembership values, respectively, for each element in the domain.Although, the foregoing fuzzy tools are suitable for dealing with problems that are defined as quantitative situations [22], uncertainty is often because of the vagueness of meanings that is used by experts in problems whose nature is rather qualitative. For example, for reason of the increasing complexity of the decision making environment, time pressure, and the lack of data or knowledge about the problem domain, in the process of decision making under intuitionistic fuzzy environment, a decision maker may have difficulty in expressing the degree of membership and nonmembership as exact values, whereas he or she may think the use of linguistic values is more straightforward and suitable to express the degree of membership and nonmembership. Similar to IFS, linguistic intuitionistic fuzzy set (LIFS) is characterized by a linguistic membership degree and a linguistic nonmembership degree, respectively. By using the LIFS, decision makers are able to consider a linguistic hesitancy degree in the belongingness of an element to a set, where they cannot easily express their subjective judgment with a single linguistic term.The outline of the paper is organized as follows. The following section presents a brief introduction to the basic knowledge that will be used in the definition of LIFS. Section3 gives the concept of LIFS and constructs the score function and accuracy function for LIFS. Section 4 develops several aggregation operators for LIFS. Section 5 proposes a MAGDM method with linguistic intuitionistic fuzzy information. In Section 6, an application of the new approach is presented. Finally, conclusions are provided in Section 7.


## 2. Preliminaries

In the following, some basic concepts and knowledge related to IFS and linguistic approach are briefly described.Definition 1 (see [1]).
LetX be a universal set. An IFS A in X is given as
(1)
A
=
{
(
x
,
μ
A
(
x
)
,
v
A
(
x
)
)
∣
x
∈
X
}
,
where the functions μ
A
(
x
)
:
X
→
[
0
,
1
], v
A
(
x
)
:
X
→
[
0
,
1
] stand for the degree of membership and nonmembership of the element x to A, respectively. Any x
∈
X meets the condition 0
≤
μ
A
(
x
)
+
v
A
(
x
)
≤
1.π
A
(
x
) is called intuitionistic index or degree of indeterminacy of x to A
:
π
A
(
x
)
=
1
-
μ
A
(
x
)
-
v
A
(
x
). Obviously, if π
A
(
x
)
=
0, IFS A is reduced to a fuzzy set.Some basic definitions and operations on IFS are presented as follows.Definition 2 (see [14, 15]).
IfA and B are two IFSs of the set X, then(1)
A
=
B if and only if ∀
x
∈
X, μ
A
(
x
)
=
μ
B
(
x
), and v
A
(
x
)
=
v
B
(
x
);
(2)
A
c
=
{
(
x
,
v
A
(
x
)
,
μ
A
(
x
)
)
∣
x
∈
X
}, where A
c is the complement of A;
(3)
A
∧
B
=
(
min
⁡
(
μ
A
(
x
)
,
μ
B
(
x
)
)
,
max
⁡
(
v
A
(
x
)
,
v
B
(
x
)
)
);
(4)
A
∨
B
=
(
max
⁡
(
μ
A
(
x
)
,
μ
B
(
x
)
)
,
min
⁡
(
v
A
(
x
)
,
v
B
(
x
)
)
);
(5)
A
⊕
B
=
(
μ
A
(
x
)
+
μ
B
(
x
)
-
μ
A
(
x
)
μ
B
(
x
)
,
v
A
(
x
)
v
B
(
x
)
);
(6)
A
⊗
B
=
(
μ
A
(
x
)
μ
B
(
x
)
,
v
A
(
x
)
+
v
B
(
x
)
-
v
A
(
x
)
v
B
(
x
)
);
(7)
λ
A
=
(
1
-
(
1
-
μ
A
(
x
)
)
λ
,
v
A
(
x
)
λ
),  λ
>
0;
(8)
A
λ
=
(
μ
A
(
x
)
λ, 1
-
(
1
-
v
A
(
x
)
)
λ
), λ
>
0.In real world, many decision making problems present qualitative aspects that are complex to assess by means of numerical values. In such cases, it may be more suitable to consider them as linguistic variables.LetS
=
{
s
i
∣
i
=
0,1
,
…
,
t
} be a finite linguistic term set with odd cardinality, where s
i represents a possible linguistic term for a linguistic variable. For example, a set of seven terms S can be expressed as follows:
(2)
S
=
{
s
0
=
N
(
none
)
,
s
1
=
VL
(
very
low
)
,
s
2
=
L
(
low
)
,
s
3
=
M
(
medium
)
,
s
4
=
H
(
high
)
,
s
5
=
VH
(
very
high
)
,
s
6
=
P
(
perfect
)
}
.
It is required that the linguistic term set should satisfy the following characteristics [27–30].(1)
The set is ordered:s
i
>
s
j, if and only if i
>
j.
(2)
There is a negation operator:Neg
(
s
i
)
=
s
j such that j
=
t
-
i.
(3)
Max operator:max
⁡
(
s
i
,
s
j
)
=
s
i, if and only if i
≥
j.
(4)
Min operator:min
⁡
(
s
i
,
s
j
)
=
s
i, if and only if i
≤
j.To preserve all the given information, Xu [31] extended the discrete term set S to a continuous linguistic term set S
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
t
,
α
∈
[
0
,
t
]
}, where, if s
α
∈
S, then s
α is called the original linguistic term. Otherwise, s
α is called the virtual linguistic term.Definition 3 (see [31, 32]).
Consider any two linguistic termss
α, s
β
∈
S
¯, and μ
,
μ
1
,
μ
2
∈
[
0,1
], the add and multiply operations of linguistic variable are defined as follows:
(3)
s
α
⊕
s
β
=
s
β
⊕
s
α
=
s
α
+
β
,
s
α
⊗
s
β
=
s
β
⊗
s
α
=
s
α
β
.
t-norm and t-conorm have been widely used to construct operations for fuzzy sets and IFSs.Definition 4 (see [33, 34]).
At-norm is a mapping T
:
[
0
,
1
]
×
[
0
,
1
]
→
[
0
,
1
] satisfying, for all x
,
y
,
z
∈
[
0
,
1
],(1)
T
(
x
,
1
)
=
x;
(2)
T
(
x
,
y
)
=
T
(
y
,
x
);
(3)
T
(
x
,
T
(
y
,
z
)
)
=
T
(
T
(
x
,
y
)
,
z
);
(4)
T
(
x
,
y
)
≤
T
(
x
,
z
) whenever y
≤
z.The four basict-norms T
M,  T
P,  T
L, and T
D are given as follows:T
M
(
x
,
y
)
=
min
⁡
(
x
,
y
), (lattice operation);
T
P
(
x
,
y
)
=
x
·
y, (algebraic operation);
T
L
(
x
,
y
)
=
max
⁡
(
x
+
y
-
1,0
), (Lukasiewicz operation);
T
D
(
x
,
y
)
=
{
0
if
(
x
,
y
)
∈
[
0,1
)
2
min
⁡
(
x
,
y
)
otherwise
. (drastic operation).Definition 5 (see [33, 34]).
At-conorm is a mapping S
:
[
0
,
1
]
×
[
0
,
1
]
→
[
0
,
1
] satisfying, for all x
,
y
,
z
∈
[
0
,
1
],(
1
)  S
(
x
,
0
)
=
x;
(
2
)  S
(
x
,
y
)
=
S
(
y
,
x
);
(
3
)  S
(
x
,
S
(
y
,
z
)
)
=
S
(
S
(
x
,
y
)
,
z
);
(
4
)  S
(
x
,
y
)
≤
S
(
x
,
z
) whenever y
≤
z.The four basict-conorms S
M, S
P, S
L, and S
D are given as follows:S
M
(
x
,
y
)
=
max
⁡
(
x
,
y
), (lattice operation);
S
P
(
x
,
y
)
=
x
+
y
-
x
·
y, (algebraic operation);
S
L
(
x
,
y
)
=
min
⁡
(
x
+
y
,
1
), (Lukasiewicz operation);
S
D
(
x
,
y
)
=
{
max
⁡
(
x
,
y
)
if
(
x
,
y
)
∈
(
0,1
]
2
1
otherwise
. (drastic operation).


## 3. Linguistic Intuitionistic Fuzzy Set

The concept of linguistic intuitionistic fuzzy set (LIFS) is given as follows.Definition 6.
LetX be a finite universal set and S
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
t
,
α
∈
[
0
,
t
]
} a continuous linguistic term set. A LIFS A in X is given as
(4)
A
=
{
(
x
,
s
θ
(
x
)
,
s
σ
(
x
)
)
∣
x
∈
X
}
,
where s
θ
(
x
)
,
s
σ
(
x
)
∈
S
¯ stand for the linguistic membership degree and linguistic nonmembership of the element x to A, respectively.For anyx
∈
X, the condition 0
≤
θ
+
σ
≤
t is always satisfied. π
(
x
) is called linguistic indeterminacy degree of x to A
:
π
(
x
)
=
s
t
-
θ
-
σ. Obviously, if θ
+
σ
=
t, then LIFS A has the minimum linguistic indeterminacy degree, that is, π
(
x
)
=
s
0, which means the membership degree of x to A can be precisely expressed with a single linguistic term and LIFS A is reduced to a linguistic variable. Oppositely, if θ
=
σ
=
0, then LIFS A
(
x
) has the maximum linguistic indeterminacy degree; that is, π
(
x
)
=
s
t. Similar to IFS, the LIFS A
(
x
) can be transformed into an interval linguistic variable [
s
θ
(
x
)
,
s
t
-
σ
(
x
)
], which indicates that the minimum and maximum linguistic membership degrees of the elements x to A are s
θ and s
t
-
σ, respectively.For notational simplicity, we suppose both LIFSA and B contain only one element, which stand for linguistic intuitionistic fuzzy values (LIFVs), that is, the pairs A
=
(
s
θ
,
s
σ
) and B
=
(
s
μ
,
s
ν
).To compare any two LIFVs, the score function and accuracy function are defined as follows.Definition 7.
LetA
=
(
s
θ
,
s
σ
) and B
=
(
s
μ
,
s
ν
) be two LIFVs, with s
θ
,
s
σ
,
s
μ
,
s
ν
∈
S
-
=
{
s
α
∣
s
0
<
s
α
≤
s
t
,
α
∈
[
0
,
t
]
}. The score function of A is defined as
(5)
S
(
A
)
=
s
(
t
+
θ
-
σ
)
/
2
,
and the accuracy function is defined as
(6)
H
(
A
)
=
s
θ
+
σ
.
Thus,A and B can be ranked by the following procedure:(1)
ifS
(
A
)
>
S
(
B
), then A
>
B;
(2)
ifS
(
A
)
=
S
(
B
) and
(a)
H
(
A
)
=
H
(
B
), then A
=
B;
(b)
H
(
A
)
>
H
(
B
), then A
>
B;
(c)
H
(
A
)
<
H
(
B
), then A
<
B.
It is easy to see that0
≤
(
t
+
θ
-
σ
)
/
2
≤
t and 0
≤
θ
+
σ
≤
t, which means s
(
t
+
θ
-
σ
)
/
2, s
θ
+
σ
∈
S
¯.Example 8.
LetA
=
(
s
1
,
s
3
),  B
=
(
s
0
,
s
4
), and C
=
(
s
1
,
s
5
) be LIFVs, which are derived from S
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
6
,
α
∈
[
0,6
]
}.
Applying formulas (5) and (6), we have
(7)
S
(
A
)
=
s
2
>
S
(
B
)
=
S
(
C
)
=
s
1
,
H
(
B
)
=
s
4
<
H
(
C
)
=
s
6
.
Thus, we obtainA
>
C
>
B.


## 4. Aggregation Operators for Linguistic Intuitionistic Fuzzy Sets

Since the definition of LIFS is given, it is necessary to introduce the operations and computations between them.Definition 9.
LetA
=
(
s
θ
,
s
σ
) and B
=
(
s
μ
,
s
ν
) be two LIFVs; then(1)
A
=
B if and only if s
θ
=
s
μ and s
σ
=
s
ν;
(2)
A
c
=
(
s
σ
,
s
θ
), where A
c is the complement of A;
(3)
the intersection ofA and B: A
∧
B
=
(
min
⁡
(
s
θ
,
s
μ
)
,
max
⁡
(
s
σ
,
s
ν
)
);
(4)
the union ofA and B: A
∨
B
=
(
max
⁡
(
s
θ
,
s
μ
)
,
min
⁡
(
s
σ
,
s
ν
)
).
Motivated byt-norm and t-conorm, we propose the following operation laws for linguistic variables.Definition 10.
Considering any two linguistic termss
α
,
s
β
∈
S
¯
=
{
s
γ
∣
s
0
<
s
γ
≤
s
t
,
γ
∈
[
0
,
t
]
}, the add and multiply operations of linguistic variable are defined as follows:
(8)
s
α
⊕
s
β
=
s
β
⊕
s
α
=
s
t
S
(
α
/
t
,
β
/
t
)
,
(9)
s
α
⊗
s
β
=
s
β
⊗
s
α
=
s
t
T
(
α
/
t
,
β
/
t
)
,
where S
(
α
/
t
,
β
/
t
) and T
(
α
/
t
,
β
/
t
) are t-conorm and t-norm, respectively.SinceS
(
α
/
t
,
β
/
t
), T
(
α
/
t
,
β
/
t
)
∈
[
0,1
], we have t
S
(
α
/
t
,
β
/
t
), t
T
(
α
/
t
,
β
/
t
)
∈
[
0
,
t
], which indicate the operation results match the original linguistic term set S
¯; that is, s
t
S
(
α
/
t
,
β
/
t
),  s
t
T
(
α
/
t
,
β
/
t
)
∈
S
¯. In addition, it is worth noting that, because of the monotonicity of t-conorm and t-norm, the values of function t
S
(
α
/
t
,
β
/
t
) and t
T
(
α
/
t
,
β
/
t
)are monotonically increasing with the increasing of α and β, which means the operation results obtained by (8) and (9) are in accord with our intuition.If we take the well-knownS
P
(
α
/
t
,
β
/
t
) and T
P
(
α
/
t
,
β
/
t
) into (8) and (9), respectively, then they can be rewritten as follows:
(10)
s
α
⊕
s
β
=
s
β
⊕
s
α
=
s
t
(
α
/
t
+
β
/
t
-
α
β
/
t
2
)
=
s
α
+
β
-
α
β
/
t
,
s
α
⊗
s
β
=
s
β
⊗
s
α
=
s
t
(
α
β
/
t
2
)
=
s
α
β
/
t
.Example 11.
LetS
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
6
,
α
∈
[
0,6
]
}. Applying (10), we have s
3
⊕
s
4
=
s
3
+
4
-
3
×
4
/
6
=
s
5, s
4
⊕
s
5
=
s
4
+
5
-
4
×
5
/
6
=
s
5.667, s
3
⊗
s
4
=
s
3
×
4
/
6
=
s
2, and s
4
⊗
s
5
=
s
4
×
5
/
6
=
s
3.333.
Thus, we obtains
3
⊕
s
4
<
s
4
⊕
s
5 and s
3
⊗
s
4
<
s
4
⊗
s
5. Such results seem to be intuitive and can be easily accepted.Alternatively, if we take the operation laws of Definition3, we have s
3
⊕
s
4
=
s
7
∉
S
¯, s
4
⊗
s
5
=
s
20
∉
S
¯, where the subscripts of s
7 and s
20 are bigger than the cardinality of linguistic term set S. In addition, if we extend the discrete term set S to a continuous term set S
¯
′
=
{
s
α
∣
α
∈
[
0
,
q
]
}, [35] where q  (
q
>
t
) is a sufficiently large positive integer, there is an unavoidable question on how to define the semantics for s
7 and s
20. Obviously, s
7 or s
20 has different semantics in different linguistic term set S with different cardinalities. As a result, it is unrealistic to assign semantics to a given linguistic value s
α derived from linguistic term set with variable cardinality. If we follow the method of s
α
⊕
s
β
=
min
⁡
{
s
α
+
β
,
s
t
} and s
α
⊗
s
β
=
min
⁡
{
s
α
β
,
s
t
} [36] for s
α,s
β
∈
S
¯
=
{
s
γ
∣
s
0
<
s
γ
≤
s
t
,
γ
∈
[
0
,
t
]
}, then we have s
3
⊕
s
4
=
s
4
⊕
s
5
=
s
6 and s
3
⊗
s
4
=
s
4
⊗
s
5
=
s
6. Such results seem to be counter-intuitive and may not be easily accepted. Applying (10), we can overcome the limitations that the subscripts of the linguistic variable are bigger than the cardinality of the corresponding linguistic term set S and obtain results agreed with our intuition.Based on (10), we can get the following operation laws for LIFVs.Definition 12.
LetA
=
(
s
θ
,
s
σ
) and B
=
(
s
μ
,
s
ν
) be two LIFVs, where s
θ, s
σ, s
μ, s
ν
∈
S
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
t
,
α
∈
[
0
,
t
]
} with λ
>
0; then
(11)
A
⊕
B
=
(
s
t
(
θ
/
t
+
μ
/
t
-
θ
μ
/
t
2
)
,
s
t
(
σ
ν
/
t
2
)
)
=
(
s
θ
+
μ
-
θ
μ
/
t
,
s
σ
ν
/
t
)
,
(12)
A
⊗
B
=
(
s
t
(
θ
μ
/
t
2
)
,
s
t
(
σ
/
t
+
ν
/
t
-
σ
ν
/
t
2
)
)
=
(
s
θ
μ
/
t
,
s
σ
+
ν
-
σ
ν
/
t
)
,
(13)
λ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
)
,
(14)
A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
)
.
Some special cases of λ
A and A
λ are obtained as follows.
IfA
=
(
s
θ
,
s
σ
)
=
(
s
t
,
s
0
), then
(15)
λ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
)
=
(
s
t
(
1
-
(
1
-
t
/
t
)
λ
)
,
s
t
(
0
/
t
)
λ
)
=
(
s
t
,
s
0
)
,
A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
)
=
(
s
t
(
t
/
t
)
λ
,
s
t
(
1
-
(
1
-
0
/
t
)
λ
)
)
=
(
s
t
,
s
0
)
.
IfA
=
(
s
θ
,
s
σ
)
=
(
s
0
,
s
t
), then
(16)
λ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
)
=
(
s
t
(
1
-
(
1
-
0
/
t
)
λ
)
,
s
t
(
t
/
t
)
λ
)
=
(
s
0
,
s
t
)
,
A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
)
=
(
s
t
(
0
/
t
)
λ
,
s
t
(
1
-
(
1
-
t
/
t
)
λ
)
)
=
(
s
0
,
s
t
)
.
IfA
=
(
s
θ
,
s
σ
)
=
(
s
0
,
s
0
), then
(17)
λ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
)

=

(
s
t
(
1
-
(
1
-
0
/
t
)
λ
)
,
s
t
(
0
/
t
)
λ
)

=

(
s
0
,
s
0
)
,
A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
)

=

(
s
t
(
0
/
t
)
λ
,
s
t
(
1
-
(
1
-
0
/
t
)
λ
)
)

=

(
s
0
,
s
0
)
.
Ifλ
→
0, then
(18)
λ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
)
=
(
s
0
,
s
t
)
,
A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
)
=
(
s
t
,
s
0
)
.
Ifλ
→
+
∞, then
(19)
λ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
)
=
(
s
t
,
s
0
)
,
A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
)
=
(
s
0
,
s
t
)
.Theorem 13.
LetA
=
(
s
θ
,
s
σ
) and B
=
(
s
μ
,
s
ν
) be two LIFVs, where s
θ
,
s
σ
,
s
μ
,
s
ν
∈
S
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
t
,
α
∈
[
0
,
t
]
} with λ
,
λ
1
,
λ
2
>
0. Then, one has(1)
λ
(
A
⊕
B
)
=
λ
A
⊕
λ
B,
(2)
λ
1
A
⊕
λ
2
A
=
(
λ
1
+
λ
2
)
A,
(3)
A
λ
⊗
B
λ
=
(
A
⊗
B
)
λ,
(4)
A
λ
1
⊗
A
λ
2
=
A
λ
1
+
λ
2.Proof.
(
1
) By (11), we have A
⊕
B
=
(
s
θ
+
μ
-
θ
μ
/
t
,
s
σ
ν
/
t
). Thus, based on (13), we have
(20)
λ
(
A
⊕
B
)
=
(
s
t
(
1
-
(
1
-
(
θ
+
μ
-
θ
μ
/
t
)
/
t
)
λ
)
,
s
t
(
(
σ
ν
/
t
)
/
t
)
λ
)
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
(
1
-
μ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
(
ν
/
t
)
λ
)
.
Similarly, sinceλ
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
) and λ
B
=
(
s
t
(
1
-
(
1
-
μ
/
t
)
λ
)
,
s
t
(
ν
/
t
)
λ
), then
(21)
λ
A
⊕
λ
B
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
)
+
t
(
1
-
(
1
-
μ
/
t
)
λ
)
-
t
(
1
-
(
1
-
θ
/
t
)
λ
)
(
1
-
(
1
-
μ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
t
(
ν
/
t
)
λ
/
t
)
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
(
1
-
μ
/
t
)
λ
)
,
s
t
(
σ
/
t
)
λ
(
ν
/
t
)
λ
)
.
Hence, we obtainλ
(
A
⊕
B
)
=
λ
A
⊕
λ
B.
(
2
) By (13), we have λ
1
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
1
)
,
s
t
(
σ
/
t
)
λ
1
) and λ
2
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
2
)
,
s
t
(
σ
/
t
)
λ
2
); thus, we obtain
(22)
λ
1
A
⊕
λ
2
A
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
λ
1
)
+
t
(
1
-
(
1
-
θ
/
t
)
λ
2
)
-
t
(
1
-
(
1
-
θ
/
t
)
λ
1
)
(
1
-
(
1
-
θ
/
t
)
λ
2
)
,
s
t
(
σ
/
t
)
λ
1
(
σ
/
t
)
λ
2
)
=
(
s
t
(
1
-
θ
/
t
)
λ
1
(
1
-
θ
/
t
)
λ
2
,
s
t
(
σ
/
t
)
λ
1
+
λ
2
)
=
(
s
t
(
1
-
θ
/
t
)
λ
1
+
λ
2
,
s
t
(
σ
/
t
)
λ
1
+
λ
2
)
=
(
λ
1
+
λ
2
)
A
.
(
3
) By (14), we get A
λ
=
(
s
t
(
θ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
) and B
λ
=
(
s
t
(
μ
/
t
)
λ
,
s
t
(
1
-
(
1
-
ν
/
t
)
λ
)
); thus, based on (12), we have
(23)
A
λ
⊗
B
λ
=
(
s
t
(
θ
/
t
)
λ
(
μ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
)
+
t
(
1
-
(
1
-
ν
/
t
)
λ
)
-
t
(
1
-
(
1
-
σ
/
t
)
λ
)
(
1
-
(
1
-
ν
/
t
)
λ
)
)
=
(
s
t
(
θ
/
t
)
λ
(
μ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
(
1
-
ν
/
t
)
λ
)
)
.
SinceA
⊗
B
=
(
s
θ
μ
/
t
,
s
σ
+
ν
-
σ
ν
/
t
), then, by (14), we have
(24)
(
A
⊗
B
)
λ
=
(
s
t
(
θ
μ
/
t
/
t
)
λ
,
s
t
(
1
-
(
1
-
(
σ
+
ν
-
σ
ν
/
t
)
/
t
)
λ
)
)
=
(
s
t
(
θ
/
t
)
λ
(
μ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
-
ν
/
t
-
σ
ν
/
t
2
)
λ
)
)
=
(
s
t
(
θ
/
t
)
λ
(
μ
/
t
)
λ
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
(
1
-
ν
/
t
)
λ
)
)
.
Hence, we obtainA
λ
⊗
B
λ
=
(
A
⊗
B
)
λ.
(
4
) By (14), we have A
λ
1
=
(
s
t
(
θ
/
t
)
λ
1
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
1
)
) and A
λ
2
=
(
s
t
(
θ
/
t
)
λ
2
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
2
)
); thus, we have
(25)
A
λ
1
⊗
A
λ
2
=
(
s
t
(
θ
/
t
)
λ
1
(
θ
/
t
)
λ
2
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
1
)
+
t
(
1
-
(
1
-
σ
/
t
)
λ
2
)
-
t
(
1
-
(
1
-
σ
/
t
)
λ
1
)
(
1
-
(
1
-
σ
/
t
)
λ
2
)
)
=
(
s
t
(
θ
/
t
)
λ
1
+
λ
2
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
1
)
(
1
-
σ
/
t
)
λ
2
)
)
=
(
s
t
(
θ
/
t
)
λ
1
+
λ
2
,
s
t
(
1
-
(
1
-
σ
/
t
)
λ
1
+
λ
2
)
)
=
A
λ
1
+
λ
2
,
which completes the proof of Theorem 13.Motivated by the intuitionistic fuzzy aggregation operators [14, 15], in what follows, we define some aggregation operators for LIFVs.Definition 14.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (i
=
1,2
,
…
,
n) be a set of LIFVs. Then, the linguistic intuitionistic fuzzy weighted averaging (LIFWA) operator is defined as
(26)
LIFWA
(
A
1
,
A
2
,
…
,
A
n
)
=
w
1
A
1
⊕
w
2
A
2
⊕
⋯
⊕
w
n
A
n
=
(
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
i
/
t
)
w
i
)
,
s
t
∏
i
=
1
n
(
σ
i
/
t
)
w
i
)
,
where w
=
(
w
1
,
w
2
,
…
,
w
n
)
T is the weight vector of A
i  (i
=
1,2
,
…
,
n), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1.In Particular, ifw
=
(
1
/
n
,
1
/
n
,
…
,
1
/
n
)
T, then the LIFWA operator is reduced to a linguistic intuitionistic fuzzy averaging (LIFA) operator; that is,
(27)
LIFWA
(
A
1
,
A
2
,
…
,
A
n
)
=
1
n
A
1
⊕
1
n
A
2
⊕
⋯
⊕
1
n
A
n
=
(
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
i
/
t
)
1
/
n
)
,
s
t
∏
i
=
1
n
(
σ
i
/
t
)
1
/
n
)
.Based on Definition14, we get some properties of the LIFWA operator.Theorem 15.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (i
=
1,2
,
…
,
n) be a set of LIFVs and w
=
(
w
1
,
w
2
,
…
,
w
n
)
T the weight vector of A
i  (i
=
0,1
,
2
,
…
,
n), with w
i
∈
[
0,1
] and  ∑
i
=
1
n
w
i
=
1; then one has the following.(1)
Idempotency. If allA
i  (i
=
1,2
,
…
,
n) are equal, that is, A
i
=
(
s
θ
i
,
s
σ
i
)
=
(
s
θ
,
s
σ
), for any i, then
(28)
L
I
F
W
A
(
A
1
,
A
2
,
…
,
A
n
)
=
(
s
θ
,
s
σ
)
.
(2)
Monotonicity. LetA
i
′
=
(
s
θ
i
′
,
s
σ
i
′
)  (i
=
1,2
,
…
,
n) be a set of LIFVs. If s
θ
i
′
≥
s
θ
i and s
σ
i
′
≤
s
σ
i, for any i, then
(29)
L
I
F
W
A
(
A
1
′
,
A
2
′
,
…
,
A
n
′
)
≥
L
I
F
W
A
(
A
1
,
A
2
,
…
,
A
n
)
,
for any w.
(3)
Boundary. Consider(30)
(
min
⁡
i
(
s
θ
i
)
,
max
⁡
i
(
s
σ
i
)
)
≤
L
I
F
W
A
(
A
1
,
A
2
,
…
,
A
n
)
≤
(
max
⁡
i
(
s
θ
i
)
,
min
⁡
i
(
s
σ
i
)
)
.Proof.
(
1
) Since A
i
=
(
s
θ
i
,
s
σ
i
)
=
(
s
θ
,
s
σ
), for any i, then
(31)
LIFWA
(
A
1
,
A
2
,
…
,
A
n
)
=
(
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
/
t
)
w
i
)
,
s
t
∏
i
=
1
n
(
σ
/
t
)
w
i
)
=
(
s
t
(
1
-
(
1
-
θ
/
t
)
)
,
s
t
(
σ
/
t
)
)
=
(
s
θ
,
s
σ
)
.
(
2
) If s
θ
i
′
≥
s
θ
i, that is, θ
i
′
≥
θ
i, for any i, then we have
(32)
θ
i
′
≥
θ
i
⟹
0
≤
1
-
θ
i
′
t
≤
1
-
θ
i
t
≤
1
⟹
(
1
-
θ
i
′
t
)
w
i
≤
(
1
-
θ
i
t
)
w
i
⟹
∏
i
=
1
n
(
1
-
θ
i
′
t
)
w
i
≤
∏
i
=
1
n
(
1
-
θ
i
t
)
w
i
⟹
1
-
∏
i
=
1
n
(
1
-
θ
i
′
t
)
w
i
≥
1
-
∏
i
=
1
n
(
1
-
θ
i
t
)
w
i
⟹
t
(
1

-

∏
i
=
1
n
(
1
-
θ
i
′
t
)
w
i

)
≥
t
(
1

-

∏
i
=
1
n
(
1
-
θ
i
t
)
w
i

)
.
Similarly, whens
σ
i
′
≤
s
σ
i, for any i, we can get t
∏
i
=
1
n
(
σ
i
′
/
t
)
1
/
n
≤
t
∏
i
=
1
n
(
σ
i
/
t
)
1
/
n.
According to Definition7, we obtain
(33)
(
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
i
′
/
t
)
1
/
n
)
,
s
t
∏
i
=
1
n
(
σ
i
′
/
t
)
1
/
n
)
≥
(
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
i
/
t
)
1
/
n
)
,
s
t
∏
i
=
1
n
(
σ
i
/
t
)
1
/
n
)
;
that is,
(34)
LIFWA
(
A
1
′
,
A
2
′
,
…
,
A
n
′
)
≥
LIFWA
(
A
1
,
A
2
,
…
,
A
n
)
.
(
3
) Since min
⁡
i
(
s
θ
i
)
≤
s
θ
i
≤
max
⁡
i
(
s
θ
i
)and max
⁡
i
(
s
σ
i
)
≤
s
σ
i
≤
min
⁡
i
(
s
σ
i
), for any i, then, based on the monotonicity of Theorem 15, we derive
(35)
(
min
⁡
i
(
s
θ
i
)
,
max
⁡
i
(
s
σ
i
)
)
≤
LIFWA
(
A
1
,
A
2
,
…
,
A
n
)
≤
(
max
⁡
i
(
s
θ
i
)
,
min
⁡
i
(
s
σ
i
)
)
.Definition 16.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (i
=
1,2
,
…
,
n) be a set of LIFVs. Then, the linguistic intuitionistic fuzzy ordered weighted averaging (LIFOWA) operator is defined as
(36)
LIFOWA
(
A
1
,
A
2
,
…
,
A
n
)
=
w
1
A
(
1
)
⊕
w
2
A
(
2
)
⊕
⋯
⊕
w
n
A
(
n
)
=
(
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
(
i
)
/
t
)
w
i
)
,
s
t
∏
i
=
1
n
(
σ
(
i
)
/
t
)
w
i
)
,
where A
(
i
)
=
(
s
θ
(
i
)
,
s
σ
(
i
)
) is the ith largest of A
1
,
A
2
,
…
,
A
n and w
=
(
w
1
,
w
2
,
…
,
w
n
)
T is the associated weight vector of A
(
i
)
(
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1.Similar to Theorem15, we have some properties of the LIFOWA operator.Theorem 17.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (
i
=
1,2
,
…
,
n
) be a set of linguistic intuitionistic fuzzy values and w
=
(
w
1
,
w
2
,
…
,
w
n
)
T the associated weight vector of A
(
i
)  (
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1; then one has the following.(1)
Idempotency. If allA
i  (
i
=
1,2
,
…
,
n
) are equal, that is, A
i
=
(
s
θ
i
,
s
σ
i
)
=
(
s
θ
,
s
σ
), for any i, then
(37)
L
I
F
O
W
A
(
A
1
,
A
2
,
…
,
A
n
)
=
(
s
θ
,
s
σ
)
.
(2)
Monotonicity. LetA
i
′
=
(
s
θ
i
′
,
s
σ
i
′
)  (
i
=
1,2
,
…
,
n
) be a set of linguistic intuitionistic fuzzy values. If s
θ
i
′
≥
s
θ
i and s
σ
i
′
≤
s
σ
i, for any i, then
(38)
L
I
F
O
W
A
(
A
1
′
,
A
2
′
,
…
,
A
n
′
)
≥
L
I
F
O
W
A
(
A
1
,
A
2
,
…
,
A
n
)
,
for any w.
(3)
Boundary. Consider(39)
(
min
⁡
i
(
s
θ
i
)
,
max
⁡
i
(
s
σ
i
)
)
≤
L
I
F
O
W
A
(
A
1
,
A
2
,
…
,
A
n
)
≤
(
max
⁡
i
(
s
θ
i
)
,
min
⁡
i
(
s
σ
i
)
)
.
(4)
Commutativity. LetA
i
′
=
(
s
θ
i
′
,
s
σ
i
′
)  (
i
=
0,1
,
2
,
…
,
n
) be a set of linguistic intuitionistic fuzzy values, then for any w
(40)
LIFOWA
⁡
(
A
1
′
,
A
2
′
,
…
,
A
n
′
)
=
LIFOWA
⁡
(
A
1
,
A
2
,
…
,
A
n
)
,
where (
A
1
′
,
A
2
′
,
…
,
A
n
′
) is any permutation of (
A
1
,
A
2
,
…
,
A
n
).Definition 18.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (
i
=
1,2
,
…
,
n
) be a set of LIFVs. Then, the linguistic intuitionistic fuzzy weighted geometric (LIFWG) operator is defined as
(41)
LIFWG
(
A
1
,
A
2
,
…
,
A
n
)
=
A
1
w
1
⊗
A
2
w
2
⊗
⋯
⊗
A
n
w
n
=
(
s
t
∏
i
=
1
n
(
θ
i
/
t
)
w
i
,
s
t
(
1
-
∏
i
=
1
n
(
1
-
σ
i
/
t
)
w
i
)
)
,
where w
=
(
w
1
,
w
2
,
…
,
w
n
)
Tis the weight vector of A
i  (
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1.The LIFWG operator has the following properties.Theorem 19.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (
i
=
1,2
,
…
,
n
) be a set of LIFVs and w
=
(
w
1
,
w
2
,
…
,
w
n
)
T the weight vector of A
i  (
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1; then one has the following.(1)
Idempotency. If allA
i  (
i
=
1,2
,
…
,
n
) are equal, that is, A
i
=
(
s
θ
i
,
s
σ
i
)
=
(
s
θ
,
s
σ
), for any i, then
(42)
L
I
F
W
G
(
A
1
,
A
2
,
…
,
A
n
)
=
(
s
θ
,
s
σ
)
.
(2)
Monotonicity. LetA
i
′
=
(
s
θ
i
′
,
s
σ
i
′
)  (
i
=
1,2
,
…
,
n
) be a set of linguistic intuitionistic fuzzy values. If s
θ
i
′
≥
s
θ
i and s
σ
i
′
≤
s
σ
i, for any i, then
(43)
L
I
F
W
G
(
A
1
′
,
A
2
′
,
…
,
A
n
′
)
≥
L
I
F
W
G
(
A
1
,
A
2
,
…
,
A
n
)
,
for any w.
(3)
Boundary. Consider(44)
(
min
⁡
i
(
s
θ
i
)
,
max
⁡
i
(
s
σ
i
)
)
≤
L
I
F
W
G
(
A
1
,
A
2
,
…
,
A
n
)
≤
(
max
⁡
i
(
s
θ
i
)
,
min
⁡
i
(
s
σ
i
)
)
.Definition 20.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (
i
=
1,2
,
…
,
n
) be a set of LIFVs. Then, the linguistic intuitionistic fuzzy ordered weighted geometric (LIFOWG) operator is defined as
(45)
LIFOWG


(
A
1
,
A
2
,
…
,
A
n
)
=
A
(
1
)
w
1
⊗
A
(
2
)
w
2
⊗
⋯
⊗
A
(
n
)
w
n
=
(
s
t
∏
i
=
1
n
(
θ
(
i
)
/
t
)
w
i
,
s
t
(
1
-
∏
i
=
1
n
(
1
-
σ
(
i
)
/
t
)
w
i
)
)
,
where A
(
i
)
=
(
s
θ
(
i
)
,
s
σ
(
i
)
) is the ith largest of A
1
,
A
2
,
…
,
A
n and w
=
(
w
1
,
w
2
,
…
,
w
n
)
T is the associated weight vector of A
(
i
)  (
i
=
1,2
,
…
,
n), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1.Similar to Theorem15, we have some properties of the LIFOWG operator.Theorem 21.
LetA
i
=
(
s
θ
i
,
s
σ
i
)  (
i
=
1,2
,
…
,
n
) be a set of LIFVs and w
=
(
w
1
,
w
2
,
…
,
w
n
)
T the associated weight vector of A
(
i
)  (
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1; then one has the following.(1)
Idempotency. If allA
i  (
i
=
1,2
,
…
,
n
) are equal, that is, A
i
=
(
s
θ
i
,
s
σ
i
)
=
(
s
θ
,
s
σ
), for any i, then
(46)
L
I
F
O
W
G
(
A
1
,
A
2
,
.
.
.
,
A
n
)
=
(
s
θ
,
s
σ
)
.
(2)
Monotonicity. LetA
i
′
=
(
s
θ
i
′
,
s
σ
i
′
)   (
i
=
1,2
,
…
,
n
) be a set of linguistic intuitionistic fuzzy values. If s
θ
i
′
≥
s
θ
i and s
σ
i
′
≤
s
σ
i, for any i, then
(47)
L
I
F
O
W
G
(
A
1
′
,
A
2
′
,
.
.
.
,
A
n
′
)
≥
L
I
F
O
W
G
(
A
1
,
A
2
,
.
.
.
,
A
n
)
,
for any w.
(3)
Boundary. Consider(48)
(
min
⁡
i
(
s
θ
i
)
,
max
⁡
i
(
s
σ
i
)
)
≤
L
I
F
O
W
G
(
A
1
,
A
2
,
.
.
.
,
A
n
)
≤
(
max
⁡
i
(
s
θ
i
)
,
min
⁡
i
(
s
σ
i
)
)
.
(4)
Commutativity. LetA
i
′
=
(
s
θ
i
′
,
s
σ
i
′
)   (
i
=
1,2
,
…
,
n
) be a set of linguistic intuitionistic fuzzy values; then for any w
(49)
L
I
F
O
W
G
(
A
1
′
,
A
2
′
,
.
.
.
,
A
n
′
)
=
L
I
F
O
W
G
(
A
1
,
A
2
,
.
.
.
,
A
n
)
,
where (
A
1
′
,
A
2
′
,
.
.
.
,
A
n
′
) is any permutation of (
A
1
,
A
2
,
.
.
.
,
A
n
).Lemma 22 (see [37, 38]).
Letx
i
>
0,  λ
i
>
0,  i
=
1,2
,
…
,
n, and ∑
i
=
1
n
λ
i
=
1; then
(50)
∏
i
=
1
n
x
i
λ
i
≤
∑
i
=
1
n
λ
i
x
i
,
with equality if and only if x
1
=
x
2
=
⋯
=
x
n.Based on Lemma22, we have the following theorem.Theorem 23.
LetA
i
=
(
s
θ
i
,
s
σ
i
)
(
i
=
1,2
,
…
,
n
) be a set of LIFVs; then one has
(51)
L
I
F
W
A
(
A
1
,
A
2
,
.
.
.
,
A
n
)
≥
L
I
F
W
G
(
A
1
,
A
2
,
.
.
.
,
A
n
)
,
L
I
F
O
W
A
(
A
1
,
A
2
,
.
.
.
,
A
n
)
≥
L
I
F
O
W
G
(
A
1
,
A
2
,
.
.
.
,
A
n
)
,
with equality if and only if A
1
=
A
2
=
⋯
=
A
n.Proof.
Letw
=
(
w
1
,
w
2
,
…
,
w
n
)
T be the weight vector of A
i  (
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1; then, by Lemma 22, we have 1
-
∏
i
=
1
n
(
1
-
θ
i
/
t
)
w
i
≥
1
-
∑
i
=
1
n
w
i
(
1
-
θ
i
/
t
)
=
1
-
∑
i
=
1
n
w
i
+
∑
i
=
1
n
w
i
θ
i
/
t
=
∑
i
=
1
n
w
i
θ
i
/
t
≥
∏
i
=
1
n
(
θ
i
/
t
)
w
i, with equality if and only if θ
1
=
θ
2
=
⋯
=
θ
n; that is, t
(
1
-
∏
i
=
1
n
(
1
-
θ
i
/
t
)
w
i
)
≥
t
∏
i
=
1
n
(
θ
i
/
t
)
w
i, with equality if and only if θ
1
=
θ
2
=
⋯
=
θ
n; and ∏
i
=
1
n
(
σ
i
/
t
)
w
i
≤
∑
i
=
1
n
w
i
σ
i
/
t
=
1
-
∑
i
=
1
n
w
i
(
1
-
σ
i
/
t
)
≤
1
-
∏
i
=
1
n
(
1
-
σ
i
/
t
)
w
i, with equality if and only if σ
1
=
σ
2
=
⋯
=
σ
n; that is, t
(
∏
i
=
1
n
(
σ
i
/
t
)
w
i
)
≤
t
(
1
-
∏
i
=
1
n
(
1
-
σ
i
/
t
)
w
i
), with equality if and only if σ
1
=
σ
2
=
⋯
=
σ
n.
Consequently, by Definition7, we obtain (
s
t
(
1
-
∏
i
=
1
n
(
1
-
θ
i
/
t
)
w
i
)
,
s
t
∏
i
=
1
n
(
σ
i
/
t
)
w
i
)
≥
(
s
t
∏
i
=
1
n
(
θ
i
/
t
)
w
i
,
s
t
(
1
-
∏
i
=
1
n
(
1
-
σ
i
/
t
)
w
i
)
), with equality if and only if θ
1
=
θ
2
=
⋯
=
θ
n and σ
1
=
σ
2
=
⋯
=
σ
n; that is,
(52)
LIFWA
(
A
1
,
A
2
,
…
,
A
n
)
≥
LIFWG
(
A
1
,
A
2
,
…
,
A
n
)
,
with equality if and only ifA
1
=
A
2
=
⋯
=
A
n.
Similarly, we can also proveLIFOWA
(
A
1
,
A
2
,
…
,
A
n
)
≥
LIFOWG
(
A
1
,
A
2
,
…
,
A
n
), with equality if and only if A
1
=
A
2
=
⋯
=
A
n.
Besides the above properties, we can derive the following desirable results of the LIFOWA and LIFOWG operators.Theorem 24.
LetA
i
=
(
s
θ
i
,
s
σ
i
)   (
i
=
1,2
,
…
,
n
) be a set of LIFVs and w
=
(
w
1
,
w
2
,
.
.
.
,
w
n
)
T the associated weight vector of A
(
i
)   (
i
=
1,2
,
…
,
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1. Then one has the following.(1)
Ifw
=
(
1,0
,
…
,
0
)
T, then
(53)
L
I
F
O
W
A
(
A
1
,
A
2
,
…
,
A
n
)
=
L
I
F
O
W
G
(
A
1
,
A
2
,
…
,
A
n
)
=
max
⁡
i
(
A
i
)
.
(2)
Ifw
=
(
0,0
,
…
,
1
)
T, then
(54)
L
I
F
O
W
A
(
A
1
,
A
2
,
…
,
A
n
)
=
L
I
F
O
W
G
(
A
1
,
A
2
,
…
,
A
n
)
=
min
⁡
i
(
A
i
)
.
(3)
Ifw
i
=
1 and w
j
=
0, for j
≠
i, then
(55)
L
I
F
O
W
A
(
A
1
,
A
2
,
…
,
A
n
)
=
L
I
F
O
W
G
(
A
1
,
A
2
,
…
,
A
n
)
=
A
(
i
)
,
where A
(
i
)
=
(
s
θ
(
i
)
,
s
σ
(
i
)
) is the ith largest of A
1
,
A
2
,
…
,
A
n.Example 25.
LetA
1
=
(
s
1
,
s
2
), A
2
=
(
s
1
,
s
3
), A
3
=
(
s
2
,
s
4
), and A
4
=
(
s
4
,
s
1
) be LIFVs, which are derived from S
¯
=
{
s
α
∣
s
0
<
s
α
≤
s
6
,
α
∈
[
0,6
]
}, and let w
=
(
0.2,0.3,0.4,0.1
)
T be the weight vector of A
i   (
i
=
1,2
,
3,4
).Applying (26) and (41), we obtain the aggregated LIFVs as follows:
(56)
LIFWA
(
A
1
,
A
2
,
…
,
A
4
)
=
(
s
6
(
1
-
∏
i
=
1
4
(
1
-
θ
i
/
6
)
w
i
)
,
s
6
∏
i
=
1
4
(
σ
i
/
6
)
w
i
)
=
(
s
1.827
,
s
2.781
)
,
LIFWG
(
A
1
,
A
2
,
…
,
A
4
)
=
(
s
6
∏
i
=
1
4
(
θ
i
/
6
)
w
i
,
s
6
(
1
-
∏
i
=
1
4
(
1
-
σ
i
/
6
)
w
i
)
)
=
(
s
1.516
,
s
3.157
)
.It is easy to see that(57)
LIFWA
(
A
1
,
A
2
,
…
,
A
4
)
=
(
s
1.827
,
s
2.781
)
>
LIFWG
(
A
1
,
A
2
,
…
,
A
4
)
=
(
s
1.516
,
s
3.157
)
.By Definition7, we calculate the following values of score function and accuracy function. 						(58)
S
(
A
1
)
=
s
(
6
+
1
-
2
)
/
2
=
s
2.5
,
S
(
A
2
)
=
s
(
6
+
1
-
3
)
/
2
=
s
2
,
S
(
A
3
)
=
s
(
6
+
2
-
4
)
/
2
=
s
2
,
S
(
A
4
)
=
s
(
6
+
4
-
1
)
/
2
=
s
4.5
,
H
(
A
2
)
=
s
1
+
3
=
s
4
,
H
(
A
3
)
=
s
2
+
4
=
s
6
.SinceS
(
A
4
)
>
S
(
A
1
)
>
S
(
A
2
)
=
S
(
A
3
) and H
(
A
3
)
>
H
(
A
2
), then
(59)
A
(
1
)
=
A
4
=
(
s
4
,
s
1
)
,
A
(
2
)
=
A
1
=
(
s
1
,
s
2
)
,
A
(
3
)
=
A
3
=
(
s
2
,
s
4
)
,
A
(
4
)
=
A
2
=
(
s
1
,
s
3
)
.Assume thatw
=
(
0.155,0.345,0.345,0.155
)
T, which are determined by the normal distribution based method [39], is the associated weight vector of A
(
i
). Then, by (36) and (45), we have
(60)
LIFOWA
(
A
1
,
A
2
,
…
,
A
n
)
=
(
s
6
(
1
-
∏
i
=
1
4
(
1
-
θ
(
i
)
/
6
)
w
i
)
,
s
6
∏
i
=
1
n
(
σ
(
i
)
/
6
)
w
i
)
=
(
s
1.983
,
s
2.430
)
,
LIFOWG
(
A
1
,
A
2
,
…
,
A
n
)
=
(
s
6
∏
i
=
1
n
(
θ
(
i
)
/
6
)
w
i
,
s
6
(
1
-
∏
i
=
1
n
(
1
-
σ
(
i
)
/
6
)
w
i
)
)
=
(
s
1.575
,
s
2.882
)
.It is easy to see that(61)
LIFOWA
(
A
1
,
A
2
,
…
,
A
4
)
=
(
s
1.983
,
s
2.430
)
>
LIFOWG
(
A
1
,
A
2
,
…
,
A
4
)
=
(
s
1.575
,
s
2.882
)
.


## 5. MAGDM Method with Linguistic Intuitionistic Fuzzy Information

In the following, we present a handling method for MAGDM problems, where the weight vector of attributes is known and the attribute performance values take the form of LIFVs.LetA
=
(
A
1
,
A
2
,
…
,
A
m
) be the set of alternatives and C
=
(
C
1
,
C
2
,
…
,
C
n
) the set of attributes, whose weight vector is w
=
(
w
1
,
w
2
,
…
,
w
n
), with w
i
∈
[
0,1
] and ∑
i
=
1
n
w
i
=
1. Let D
=
(
D
1
,
D
2
,
…
,
D
l
) be the set of decision makers. Suppose R
k
=
(
r
i
j
k
)
m
×
n is the decision matrix, where r
i
j
k
=
(
s
θ
i
j
k
,
s
σ
i
j
k
) denotes the preference value and takes the form of LIFV, which is given by decision makers D
k for alternative A
i with respect to attribute C
j and s
θ
i
j
k
,
s
σ
i
j
k
∈
S
=
{
s
i
∣
i
=
0,1
,
…
,
t
}.The proposed method is described as follows.Step 1.
Construct the linguistic intuitionistic fuzzy decision matrixR
k
=
(
r
i
j
k
)
m
×
n.Step 2.
Utilize the LIFOWA or LIFOWG operator to derive the aggregated decision matrixR
=
(
r
i
j
)
m
×
n:
(62)
r
i
j
=
(
s
θ
i
j
,
s
σ
i
j
)
=
LIFOWA
(
r
i
j
1
,
r
i
j
2
,
…
,
r
i
j
l
)
,
or
r
i
j
=
(
s
θ
i
j
,
s
σ
i
j
)
=
LIFOWG
(
r
i
j
1
,
r
i
j
2
,
…
,
r
i
j
l
)
,
where the LIFOWA and LIFOWG weights are determined by the normal distribution based method [39].Step 3.
Aggregater
i
j  (j
=
1,2
,
…
,
n) to yield the collective overall preference values r
i for each alternative A
i   (i
=
1,2
,
…
,
m) based on the LIFWA or LIFWG operator.Step 4.
Rank the alternatives in accordance withr
i, according to Definition 7.


## 6. Numerical Example

In this section, we consider an example adapted from Herrera and Herrera-Viedma [40]. Suppose an investment company, which wants to invest a sum of money in the best option. There is a panel with four possible alternatives of where to invest the money: A
1 is a car industry; A
2 is a food company; A
3 is a computer company; A
4 is an arms industry. The investment company must make a decision according to four criteria: C
1 is the risk analysis; C
2 is the growth analysis; C
3 is the social-political impact analysis; C
4 is the environmental impact analysis. The weight vector of attributes is w
=
(
0.3,0.1,0.2,0.4
). Three experts are invited to provide their preferences for each alternative on each attribute with the linguistic term set S
=
{
s
0
=
extremely
poor
,
s
1
=
very
poor
,
s
2
=
poor
,
s
3
=
slightly
poor
,
s
4
=
fair
,
s
5
=
slightly
good
,
s
6
=
good
,
s
7
=
very
good
,
s
8
=
extremely
good
}.Step 1.
The decision makers provide their evaluation values and construct the linguistic intuitionistic fuzzy decision matrixR
k
=
(
r
i
j
k
)
m
×
n  (k
=
1,2
,
3) as shown in Tables 1, 2, and 3, respectively.Table 1
The decision matrixR
1 given by D
1.
C
1
C
2
C
3
C
4
A
1
(
s
6
,
s
1
)
(
s
3
,
s
1
)
(
s
3
,
s
3
)
(
s
1
,
s
6
)
A
2
(
s
3
,
s
4
)
(
s
3
,
s
4
)
(
s
2
,
s
5
)
(
s
2
,
s
4
)
A
3
(
s
1
,
s
3
)
(
s
2
,
s
3
)
(
s
3
,
s
2
)
(
s
6
,
s
1
)
A
4
(
s
6
,
s
2
)
(
s
4
,
s
3
)
(
s
5
,
s
1
)
(
s
7
,
s
1
)Table 2
The decision matrixR
2 given by D
2.
C
1
C
2
C
3
C
4
A
1
(
s
3
,
s
2
)
(
s
4
,
s
1
)
(
s
3
,
s
4
)
(
s
2
,
s
3
)
A
2
(
s
5
,
s
2
)
(
s
2
,
s
1
)
(
s
3
,
s
4
)
(
s
2
,
s
5
)
A
3
(
s
2
,
s
3
)
(
s
3
,
s
3
)
(
s
1
,
s
2
)
(
s
3
,
s
3
)
A
4
(
s
5
,
s
2
)
(
s
3
,
s
3
)
(
s
5
,
s
2
)
(
s
4
,
s
1
)Table 3
The decision matrixR
3 given by D
3.
C
1
C
2
C
3
C
4
A
1
(
s
3
,
s
3
)
(
s
3
,
s
5
)
(
s
6
,
s
1
)
(
s
2
,
s
6
)
A
2
(
s
3
,
s
2
)
(
s
2
,
s
4
)
(
s
2
,
s
1
)
(
s
3
,
s
4
)
A
3
(
s
6
,
s
1
)
(
s
2
,
s
5
)
(
s
3
,
s
4
)
(
s
1
,
s
3
)
A
4
(
s
5
,
s
1
)
(
s
4
,
s
4
)
(
s
6
,
s
2
)
(
s
5
,
s
2
)Step 2.
By the normal distribution based method [39], the associated weight vector is determined as ω
=
(
0.243,0.514,0.243
)
T. Then, utilize the LIFOWA operator to derive the aggregated decision matrix R
=
(
r
i
j
)
4
×
4, which is shown in Table 4. Alternatively, if the LIFOWG operator instead of LIFOWA operator is applied in Step 2, the aggregated decision matrix R
=
(
r
i
j
)
m
×
n can be derived as shown in Table 5.Table 4
The aggregated decision matrixR by LIFOWA operator.
C
1
C
2
C
3
C
4
A
1
(
s
3.998
,
s
1.865
)
(
s
3.264
,
s
1.479
)
(
s
3.998
,
s
2.463
)
(
s
1.771
,
s
5.070
)
A
2
(
s
3.584
,
s
2.367
)
(
s
2.537
,
s
2.856
)
(
s
2.537
,
s
3.051
)
(
s
2.260
,
s
4.223
)
A
3
(
s
3.230
,
s
2.297
)
(
s
2.260
,
s
3.397
)
(
s
2.574
,
s
2.856
)
(
s
3.657
,
s
2.297
)
A
4
(
s
5.282
,
s
1.401
)
(
s
3.777
,
s
3.478
)
(
s
5.282
,
s
1.401
)
(
s
4.720
,
s
1.184
)Table 5
The aggregated decision matrixR by LIFOWG operator.
C
1
C
2
C
3
C
4
A
1
(
s
3.550
,
s
2.041
)
(
s
3.217
,
s
2.303
)
(
s
3.550
,
s
2.860
)
(
s
1.690
,
s
5.501
)
A
2
(
s
3.397
,
s
2.563
)
(
s
2.463
,
s
3.417
)
(
s
2.462
,
s
3.727
)
(
s
2.207
,
s
4.270
)
A
3
(
s
2.207
,
s
2.574
)
(
s
2.207
,
s
3.584
)
(
s
2.719
,
s
3.129
)
(
s
2.719
,
s
2.774
)
A
4
(
s
5.227
,
s
1.505
)
(
s
3.730
,
s
3.542
)
(
s
5.227
,
s
1.505
)
(
s
4.838
,
s
1.257
)Step 3.
Aggregater
i
j   (j
=
1,2
,
…
,
4) to yield the collective overall preference values r
i for each alternative A
i  (
i
=
1,2
,
…
,
4
) based on the LIFWA or LIFWG operator with the weight vector w
=
(
0.3,0.1,0.2,0.4
), as shown in Table 6.Table 6
The collective overall preference values and the rankings of alternatives.
A
1
A
2
A
3
A
4
Ranking
LIFOWALIFWA
(
s
3.142
,
s
2.874
)
(
s
2.772
,
s
3.199
)
(
s
3.198
,
s
2.495
)
(
s
4.938
,
s
1.435
)
A
4>A
3>A
1>A
2
LIFOWGLIFWG
(
s
2.612
,
s
3.932
)
(
s
2.596
,
s
3.619
)
(
s
2.501
,
s
2.876
)
(
s
4.990
,
s
1.650
)
A
4>A
3>A
2>A
1Step 4.
By rankingr
i  (i
=
1,2
,
3,4) based on Definition 7, the priorities of the alternatives can be obtained, which are listed in Table 6. Obviously, both the methods come to the same conclusion that the best alternative is A
4.In the above linguistic intuitionistic fuzzy aggregation operators, the weights of the arguments are supposed to be crisp numbers. However, for reason of complexity or uncertainty of decision making, decision makers may have difficulty in assigning weights of the attributes with crisp numbers and may be inclined to appoint the weights of the attributes as linguistic values, interval linguistic values, or LIFVs. The existing linguistic aggregation operators cannot deal with such cases. According to the operation laws described by (11) and (12), we can also solve such decision making problems. For example, suppose the decision makers take the weights of attributes as linguistic values in this example, that is, w
1
=
(
s
3
,
s
2
),  w
2
=
s
1,  w
3
=
[
s
2
,
s
4
], and w
4
=
[
s
4
,
s
7
], which can be transformed to LIFVs; that is, w
1
=
(
s
3
,
s
2
), w
2
=
(
s
1
,
s
7
), w
3
=
(
s
2
,
s
4
), and w
4
=
(
s
4
,
s
1
). Thus, by (11) and (12), the collective overall preference values r
i for each alternative A
i   (
i
=
1,2
,
…
,
4
) with linguistic weights can be obtained by the following:
(63)
r
i
=
(
w
1
⊗
r
i
1
)
⊕
(
w
2
⊗
r
i
2
)
⊕
(
w
3
⊗
r
i
3
)
⊕
(
w
4
⊗
r
i
4
)
.Without loss of generalization, on the basis of Table4, we have the following results:
(64)
r
1
=
(
(
s
3
,
s
2
)
⊗
(
s
3.998
,
s
1.865
)
)
⊕
(
(
s
1
,
s
7
)
⊗
(
s
3.264
,
s
1.479
)
)
⊕
(
(
s
2
,
s
4
)
⊗
(
s
3.998
,
s
2.463
)
)
⊕
(
(
s
4
,
s
1
)
⊗
(
s
1.771
,
s
5.070
)
)
=
(
s
3.200
,
s
1.245
)
,
r
2
=
(
(
s
3
,
s
2
)
⊗
(
s
3.584
,
s
2.367
)
)
⊕
(
(
s
1
,
s
7
)
⊗
(
s
2.537
,
s
2.856
)
)
⊕
(
(
s
2
,
s
4
)
⊗
(
s
2.537
,
s
3.051
)
)
⊕
(
(
s
4
,
s
1
)
⊗
(
s
2.260
,
s
4.223
)
)
=
(
s
2.931
,
s
1.221
)
,
r
3
=
(
(
s
3
,
s
2
)
⊗
(
s
3.230
,
s
2.297
)
)
⊕
(
(
s
1
,
s
7
)
⊗
(
s
2.260
,
s
3.397
)
)
⊕
(
(
s
2
,
s
4
)
⊗
(
s
2.574
,
s
2.856
)
)
⊕
(
(
s
4
,
s
1
)
⊗
(
s
3.657
,
s
2.297
)
)
=
(
s
3.355
,
s
0.882
)
,
r
4
=
(
(
s
3
,
s
2
)
⊗
(
s
5.282
,
s
1.401
)
)
⊕
(
(
s
1
,
s
7
)
⊗
(
s
3.777
,
s
3.478
)
)
⊕
(
(
s
2
,
s
4
)
⊗
(
s
5.282
,
s
1.401
)
)
⊕
(
(
s
4
,
s
1
)
⊗
(
s
4.720
,
s
1.184
)
)
=
(
s
5.109
,
s
0.424
)
,
by which we can obtain the rankings of alternatives: that is, A
4
>
A
3
>
A
1
>
A
2.Such results do not exceed the cardinality of the corresponding linguistic term set and can be easily accepted. By contrast, if we transform the above LIFVs into interval linguistic values and take the operation laws defined by Definition3, we may have similar problems discussed in Example 11.


## 7. Conclusions

In this paper, we introduce the concept of LIFS, which can be seen as a generalization of IFS and is suitable to deal with imprecise and uncertain information in decision making. We further define the score function and accuracy function to compare the LIFVs. Moreover, we propose several aggregation operators for LIFSs, such as the LIFWA operator, LIFOWA operator, LIFWG operator, and LIFOWG operator, together with their desired properties. Comparing with the existing linguistic operation laws, we can obtain more intuitive and acceptable results by these aggregation operators proposed.



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*Source: 432092-2014-04-29.xml*