Multi-criteria group decision making with picture linguistic numbers

VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

Multi-criteria Group Decision Making

with Picture Linguistic Numbers

Pham Hong Phong^1,∗, Bui Cong Cuong²

¹Faculty of Information Technology, National University of Civil Engineering,

55 Giai Phong Road, Hanoi, Vietnam

²Institute of Mathematics, Vietnam Academy of Science and Technology,

18 Hoang Quoc Viet Road, Building A5, Cau Giay, Hanoi, Vietnam

Abstract

In 2013, Cuong and Kreinovich deﬁned picture fuzzy set (PFS) which is a direct extension of fuzzy set (FS) and

intuitionistic fuzzy set (IFS). Wang et al. (2014) proposed intuitionistic linguistic number (ILN) as a combination of

IFS and linguistic approach. Motivated by PFS and linguistic approach, this paper introduces the concept of picture

linguistic number (PLN), which constitutes a generalization of ILN for picture circumstances. For multi-criteria

group decision making (MCGDM) problems with picture linguistic information, we deﬁne a score index and two

accuracy indexes of PLNs, and propose an approach to the comparison between two PLNs. Simultaneously, some

operation laws for PLNs are deﬁned and the related properties are studied. Further, some aggregation operations

are developed: picture linguistic arithmetic averaging (PLAA), picture linguistic weighted arithmetic averaging

(PLWAA), picture linguistic ordered weighted averaging (PLOWA) and picture linguistic hybrid averaging (PLHA)

operators. Finally, based on the PLWAA and PLHA operators, we propose an approach to handle MCGDM under

PLN environment.

Received 18 March 2016, Revised 07 October 2016, Accepted 18 October 2016

Keywords: Picture fuzzy set, linguistic aggregation operator, multi-criteria group decision making, linguistic group

decision making.

1. Introduction

types: “yes”, “abstain”, “no” and “refusal”.

Voting can be a good example of such

situation as the voters may be divided into

four groups: “vote for”, “abstain”, “vote

against” and “refusal of voting”. There

has been a number of studies that show

the applicability of PFSs (for example, see

[18, 19, 20]).

Cuong and Kreinovich [7] introduced the

concept of picture fuzzy set (PFS), which is

a generalization of the traditional fuzzy set

(FS) and the intuitionistic fuzzy set (IFS).

Basically, a PFS assigns to each element a

positive degree, a neural degree and a negative

degree. PFS can be applied to situations that

require human opinions involving answers of

Moreover, in many decision situations,

experts’ preferences or evaluations are given

by linguistic terms which are linguistic values

^∗Corresponding author. Email.: phphong84@yahoo.com

39

40 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

of a linguistic variable [32]. For example,

when evaluating a cars speed, linguistic terms

like “very fast”, “fast” and “slow” can be used.

To date, there are many methods proposed to

dealing with linguistic information. These

methods are mainly divided into three groups.

1) The methods based on membership

functions: each linguistic term is represented

as a fuzzy number characterized by a

accuracy of linguistic aggregation operators

by extending the linguistic term set,

n

o

S

=

s₀, s₁, . . . , s_g, to the continuous one,

¯

S

=

{ s_θ| θ ∈ [0, t]}, where

t

(

t > g) is a

¯

suﬃciently large positive integer. For s_θ∈ S

,

if s_θ∈ S

, s_θis called an original linguistic

term; otherwise, an extended (or virtual)

linguistic term. Based on this representation,

some aggregation operators were deﬁned:

linguistic averaging (LA) [26], linguistic

weighted averaging (LWA) [26], linguistic

ordered weighted averaging (LOWA) [26],

linguistic hybrid aggregation (LHA) [27],

induced LOWA (ILOWA) [26], generalized

ILOWA (GILOWA) [25] operators.

membership function.

These methods

compute directly on the membership

functions using the Extension Principle [13].

´

Herrera and Martınez [11] described an

aggregation operator based on membership

functions by

3) The methods based on 2-tuple

˜

app₁

F

S ⁿ−→ F (R) −→ S,

representation:

Herrera and Martınez

´

[11] proposed a new linguistic computational

where S ⁿdenotes the

the linguistic term set

n

-Cartesian product of

model using an added parameter to each

linguistic term. This new parameter is called

sybolic translation. So, linguistic information

˜

S , F symbolizes an

aggregation operator, F (R) denotes the set of

fuzzy numbers, and app₁is an approximation

is presented as a 2-tuple (s, α), where

a linguistic term, and is a numeric value

s is

function that returns a linguistic term in

S

α

whose meaning is the closest one to each

representing a sybolic translation. This model

makes processes of computing with linguistic

terms easily without loss of information.

Some aggregation operation for 2-tuple

representation were also deﬁned [11]: 2-tuple

arithmetic mean (TAM), 2-tuple weighted

averaging (TWA), 2-tuple ordered weighted

averaging (TOWA) operators.

obtained unlabeled fuzzy number in F (R)

In some early applications, linguistic terms

were described via triangular fuzzy numbers

.

[1, 4, 15], or trapezoidal fuzzy numbers

[5, 14].

2) The methods based on ordinal scales: the

main idea of this approach is to consider the

linguistic terms as ordinal information [28].

It is assumed that there is a linear ordering

Motivated by Atanassov’s IFSs [

2, 3],

Wang et al. [22 23] proposed intuitionistic

,

n

o

on the linguistic term set

S

=

s₀, s₁, . . . , s_g

linguistic number (ILN) as a relevant tool to

modelize decision situations in which each

assessment consists of not only a linguistic

term but also a membership degree and a

nonmembership degree. Wang also deﬁned

some operation laws and aggregation for

ILNs: intuitionistic linguistic arithmetic

such that s_i≥ s_jif and only if i ≥ j.

Based on elementary notions: maximum,

minimum and negation, many aggregation

operators have been proposed [9, 10, 12, 21,

24, 29, 30].

In 2008, Xu

[

24

]

introduced

a

computational model to improve the

averaging

[

22

]

(ILAA), intuitionistic

P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52 41

linguistic weighted arithmetic averaging

(ILWAA) [22], intuitionistic linguistic

For each x ∈ X

ν_A(x) is termed as the refusal degree of

. If ξ_A(x) = 0 for all x ∈ X is reduced to

an IFS [ ]; and if η_A(x) ξ_A(x) = 0 for all

x ∈ X, A is degenerated to a FS [31].

,

ξ_A(x) = 1−µ_A(x)−η_A(x)−

x

in

ordered weighted averaging (ILOWA) [23

and intuitionistic hybrid aggregation [23

]

A

, A

2,

3

=

(IHA) operators. Another concept, which

also generalizes both the linguistic term and

the intuitionistic fuzzy value at the same time,

is intuitionistic linguistic term [6, 8, 16, 17].

The rest of the paper is organized

Example 1. Let denotes the set of

A

all patients who suﬀer from “high blood

pressure”. We assume that, assessments of

20 physicians on blood pressure of the patient

as follows.

Section 2 recalls some

x

are divided into four groups: “high blood

relevant deﬁnitions: picture fuzzy sets

and intuitionistic fuzzy numbers. Section 3

introduces the concept of picture linguistic

number (PLN), which is a generalization

pressure” (7 physicians), “low blood pressure”

(4 physicians), “blood pressure disease” (3

physicians), “ not blood disease pressure” (6

physicians). The set

A can be considered as

of ILN for picture circumstances.

In

a PFS. The possitive degree, neural degree,

negative degree and refusal degree of the

patient x in A can be speciﬁed as follows.

Section 4, some aggregation operations

are developed: picture linguistic arithmetic

averaging (PLAA), picture linguistic

weighted arithmetic averaging (PLWAA),

picture linguistic ordered weighted averaging

(PLOWA) and picture linguistic hybrid

averaging (PLHA) operators. In Section 5,

based on the PLWAA and PLHA operators,

we propose an approach to handle MCGDM

under PLNs environment. Section 6 is an

illutrative example of the proposed approach.

Finally, Section 7 draws a conclusion.

7

20

3

20

µ_A(x) =

= 0.35, η_A(x) =

= 0.15,

4

ν_A(x) =

= 0.2, ξ_A(x) = 0.3.

20

Some more deﬁnitions, properties of PFSs

can be referred to [7].

2.2. Intuitionistic linguistic numbers

From now on, the continuous linguistic

¯

term set

S

=

{ s_θ| θ ∈ [0, t]} is used as

linguistic scale for linguistic assessments.

2. Related works

Let X , ∅, based on the linguistic term set

2.1. Picture fuzzy sets

and the intuitionistic fuzzy set [2, 3], Wang

Deﬁnition 1. [7] A picture fuzzy set (PFS)

A in a set X , ∅ is an object of the form

and Li [22] deﬁned the intuitionistic linguistic

number set as follows.

n

o

ꢄ

ꢀ ꢁ

ꢂꢃ

ꢄ

A = x, s_θ(x), µ_A(x) , ν_A(x) x ∈ X , (3)

which is characterized by a linguistic term

A = {(x, µ_A(x) , η_A(x) , ν_A(x)) |x ∈ X} , (1)

where µ_A

,

η_A

,

ν_A

:

X → [0, 1]. For each x ∈ X

,

s

_θ(x), a membership degree µ_A(x) and a non-

µ_A(x) η_A(x) and ν_A(x) are correspondingly

,

membership degree ν_A(x) of the element

s_θ(x), where

x to

called the positive degree, neutral degree and

negative degree of x in A, which satisfy

¯

µ_A: X → S → [0, 1] , x → s_θ(x)→ µ_A(x) ,

µ_A(x) + η_A(x) + ν_A(x) ≤ 1, ∀x ∈ X. (2)

(4)

42 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

¯

ν_A: X → S → [0, 1] , x → s_θ(x)→ ν_A(x) ,

arithmetic averaging [22], intuitionistic

linguistic ordered weighted averaging [23],

intuitionistic linguistic hybrid aggregation

operator [23] operators, and developed an

approach to deal with the MCGDM problems,

in which the criteria values are ILNs [23] .

(5)

with the condition

µ_A(x) + ν_A(x) ≤ 1, ∀x ∈ X.

(6)

ꢁ

ꢂ

Each

s

_θ(x), µ_A(x) , ν_A(x) deﬁned in (3) is

termed as an intuitionistic linguistic number

which exactly given in Deﬁnition 2.

3. Picture linguistic numbers

Deﬁnition 6. Let X , ∅, then a picture

Deﬁnition 2.

[

22

]

An

intuitionistic

linguistic number set

A in X is an object

linguistic number (ILN)

α

is deﬁned as

having the following form:

ꢁ

ꢂ

¯

α

=

s

_θ(α), µ (α) , ν (α) , where s_θ(α)∈ S

n

o

ꢄ

ꢀ ꢁ

ꢂꢃ

ꢄ

A = x, s_θ(x), µ_A(x) , η_A(x) , ν_A(x) x ∈ X ,

is a linguistic term, µ (α) ∈ [0, 1] (resp.

ν (α) ∈ [0, 1]) is the membership degree

(resp. non-membership degree) such that

(9)

which is characterized by a linguistic term

¯

s_θ(x)∈ S , a positive degree µ_A(x) ∈ [0, 1], a

µ (α)

denoted by Ω.

+

ν (α) ≤ 1. The set of all ILNs is

neural degree η_A(x) ∈ [0, 1] and a negative

degree ν_A(x) ∈ [0, 1] of the element

x to s_θ(x)

with the condition

Deﬁnition 3. [22] Let α, β ∈ Ω, then

D

(1) α ⊕ β = s_{θ(α)+θ(β)}

,

E

µ_A(x) + η_A(x) + ν_A(x) ≤ 1, ∀x ∈ X. (10)

θ(α)µ(α)+θ(β)µ(β) θ(α)ν(α)+θ(β)ν(β)

,

;

θ(α)+θ(β)

ꢁ

ꢂ

ξ_A(x) = 1 − µ_A(x) − η_A(x) − ν_A(x) is called

(2) λα

[0, 1].

=

s_λθ(α), µ (α) , ν (α) , for all λ ∈

the refusal degree of x to s_θ(x)for all x ∈ X.

In cases η_A(x) = 0 (for all x ∈ X), the

picture linguistic number set is returns to the

intuitionistic linguistic number set [22].

Deﬁnition 4.

[

23] For α ∈ Ω, the score

h (α) and the accuracy H (α) of

α

are

respectively given in Eqs. (7) and (8).

For convenience, each 4-tuple

_θ(α), µ (α) , η (α) , ν (α) is called a picture

linguistic number (PLN), where s_θ(α)is a

α

=

ꢁ

ꢂ

s

h (α) = θ (α) (µ (α) − ν (α)) ,

H (α) = θ (α) (µ (α) + ν (α)) .

(7)

(8)

linguistic term, µ (α) ∈ [0, 1]

,

+

η (α) ∈ [0, 1]

ν (α) ≤ [0, 1]

η (α) and ν (α) are membership, neutral

and nonmembership degrees of an evaluated

object to _θ(α), respectively. Two PLNs and

are said to be equal, θ (α)

µ (α) µ (β) η (α) ν (β)

Let ∆ denotes the set of all PLNs.

,

.

ν (α) ∈ [0, 1] and µ (α)

µ (α)

+

η (α)

Deﬁnition 5.

said to be greater than

one of the following conditions is satisﬁed.

(1) If h (α) > h (β);

(2) If h (α) = h (β), and H (α) > H (β).

[23] Consider

α

,

β ∈ Ω,

α is

,

β

, denoted by α > β, if

s

α

=

β

α

=

β

, if θ (α)

η (β) and ν (α)

,

.

=

,

=

Based on basic operators (Deﬁnition 3)

and order relation (Deﬁnition 5), Wang et al.

deﬁned the intuitionistic linguistic weighted

Example 2. α

hs₄, 0.3, 0.3, 0.2i is a

=

PLN, and from it, we know that the positive

P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52 43

degree, neural degree, negative degree and

the refusal degree of evaluated object to s₄

are 0.3, 0.3, 0.2 and 0.2, respectively.

and

ν ((α ⊕ β) ⊕ γ)

θ α ν α + θ β ν β + θ γ ν γ

( ) ( )

=

.

θ (α) + θ (β) + θ (γ)

Hence,

In the following, some operational laws of

PLNs are introduced.

(α ⊕ β) ⊕ γ = hθ (α) ⊕ θ (β) ⊕ θ (γ) ,

θ (α) µ (α) + θ (β) µ (β) + θ (γ) µ (γ)

θ (α) + θ (β) + θ (γ)

θ (α) η (α) + θ (β) η (β) + θ (γ) η (γ)

θ (α) + θ (β) + θ (γ)

θ (α) ν (α) + θ (β) ν (β) + θ (γ) ν (γ)

θ (α) + θ (β) + θ (γ)

Deﬁnition 7. Let α, β ∈ ∆, then

D

θ(α)µ(α)+θ(β)µ(β)

(1) α ⊕ β = s_{θ(α)+θ(β)}

,

θ(α)+θ(β)

,

E

θ(α)η(α)+θ(β)η(β) θ(α)ν(α)+θ(β)ν(β)

+

,

;

θ(α)+θ(β)

ꢁ

ꢂ

.

(2) λα

=

s

_λθ(α), µ (α) , η (α) , ν (α) , for all

λ ∈ [0, 1].

(11)

By the same way, α ⊕ (β ⊕ γ) equals to the right of

Eq. (11). Therefore, (α ⊕ β) ⊕ γ = α ⊕ (β ⊕ γ).

It is easy to prove that both α ⊕ β and λα

λ ∈ [0, 1]) are PLNs. Proposition 1 further

(

(3) We have

D

examines properties of aforesaid notions.

λ (α ⊕ β) = s_{λ(θ(α)+θ(β))}

,

θ (α) µ (α) + θ (β) µ (β) θ (α) η (α) + θ (β) η (β)

,

Proposition 1. Let

[0, 1], we have:

α,

β

,

γ ∈ ∆, and

λ,

ρ ∈

θ (α) + θ (β)

θ (α) ν (α) + θ (β) ν (β)

θ (α) + θ (β)

+

(1) α ⊕ β = β ⊕ α;

*

(2) (α ⊕ β) ⊕ γ = α ⊕ (β ⊕ γ);

(3) λ (α ⊕ β) = λα ⊕ λβ;

λθ (α) µ (α) + λθ (β) µ (β)

λθ (α) + λθ (β)

= s_{λθ(α)+λθ(β)}

,

(4) If λ + ρ ≤ 1, (λ + ρ) α = λα ⊕ ρα.

λθ (α) η (α) + λθ (β) η (β)

λθ (α) + λθ (β)

,

+

Proof. (1) It is straightforward.

(2) We have

λθ (α) ν (α) + λθ (β) ν (β)

λθ (α) + λθ (β)

ꢁ

ꢂ

= s_λθ(α), µ (α) , η (α) , ν (α)

D

E

θ ((α ⊕ β) ⊕ γ) = θ (α) ⊕ θ (β) ⊕ θ (γ) .

⊕ s_λθ(β), µ (β) , η (β) , ν (β)

=λα ⊕ λβ.

µ ((α ⊕ β) ⊕ γ)

(4) We have

θ (α) η (α) + θ (β) η (β)

θ (α) + θ (β)

D

E

= (θ (α) + θ (β))

(λ + ρ) α = s_{(λ+ρ)θ(α)}, µ (α) , η (α) , ν (α)

*

λθ (α) µ (α) + ρθ (α) µ (α)

+ θ (γ) µ (γ)) / (θ (α) + θ (β) + θ (γ))

= s_{λθ(α)+ρθ(α)}

,

λθ (α) + ρθ (α)

θ (α) µ (α) + θ (β) µ (β) + θ (γ) µ (γ)

λθ (α) η (α) + ρθ (α) η (α)

λθ (α) + ρθ (α)

=

.

,

θ (α) + θ (β) + θ (γ)

+

λθ (α) ν (α) + ρθ (α) ν (α)

λθ (α) + ρθ (α)

Similarly,

ꢁ

ꢂ

= s_λθ(α), µ (α) , η (α) , ν (α)

η ((α ⊕ β) ⊕ γ)

D

E

⊕ s_ρθ(α), µ (α) , η (α) , ν (α)

=λα ⊕ ρα.

θ (α) η (α) + θ (β) η (β) + θ (γ) η (γ)

θ (α) + θ (β) + θ (γ)

=

,

44 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

and

In order to compare two PLNs, we deﬁne the score,

ﬁrst accuracy and second accuracy for PLNs.

β ≯ α ⇔





h (β) ≤ h (α)





(16)

h (β) , h (α) or H₁(β) ≤ H₁(α)

Deﬁnition 8. We deﬁne the score h (α), ﬁrst



h β , h α or H₁β , H₁

or H₂(β) ≤ H₂(α).

( )

α

( )

accuracy H₁(α) and second accuracy H₂(α) for α ∈

∆





as in Eqs. (12), (13) and (14).

Combining (15) and (16), we get h (α)

= h (β),

h (α) = θ(α) (µ (α) − ν (α)) ,

(12)

H₁(α) = H₁(β) and H₂(α) = H₂(β). Thus α ∼ β.

(2) Taking account of Deﬁnition 9, we get

H₁(α) = θ(α) (µ (α) + ν (α)) ,

(13)

(14)



h (α) > h (β)



H₂(α) = θ (α) (µ (α) + η (α) + ν (α)) .

h (α) = h (β) and H₁(α) > H₁(β)

h (α) = h (β) and H₁(α) = H₁(β)

and H₂(α) > H₂(β),



(17)





Deﬁnition 9. For

α

,

β ∈ ∆,

α is said to be greater

than , denoted by α > β, if one of following three

β

and

cases is satisﬁed:

(1) h (α) > h (β);

(2) h (α) = h (β) and H₁(α) > H₁(β);

(3) h (α) = h (β), H₁(α) = H₁(β) and H₂(α) > H₂(β).



h (β) > h (γ)



h (β) = h (γ) and H₁(β) > H₁(γ)

h (β) = h (γ) and H₁(β) = H₁(γ)

and H₂(β) > H₂(γ).



(18)





It is easy seen that there exist pairs of PLNs

which are not comparable by Deﬁnition 9. For

Pairwise combining conditions of (17) and (19), we

obtain

example, let us consider

hs₄, 0.2, 0.1, 0.1i. We have h (α)

H₁(β) and H₂(α)

α

=

hs₂, 0.4, 0.2, 0.2i and



h (α) > h (γ)



β

=

h (β) H₁(α)

,

=



h (α) = h (γ) and H₁(α) > H₁(γ)

h (α) = h (γ) and H₁(α) = H₁(γ)

and H₂(α) > H₂(γ).



=

H₂(β). Then, neither α ≥ β nor

(19)





β ≥ α occurs. In these cases,

α

and

β

are said to be

equivalent.

Then, α > γ.

ꢀ

Let (α₁, . . . , α_n) be a collection of PLNs, we denote:

Deﬁnition 10. Two PLNs

α

and

β

are termed as

n

ꢅ

ꢆ

o

ꢄ

equivalent, denoted by α ∼ β, if they have the same

score, ﬁrst accuracy and second accuracy, that is

h (α) = h (β), H₁(α) = H₁(β) and H₂(α) = H₂(β).

ꢄ

arcmin_h(α₁, . . . , α_n) = α_jh α_j= min {h (α_i)} ,

n

ꢅ

ꢆ

o

ꢄ

arcmin_H(α₁, . . . , α_n) = α_jH₁α_j= min {H₁(α_i)} ,

1

n

ꢅ

ꢆ

o

ꢄ

arcmin (α₁, . . . , α_n) = α_jH₂α_j= min {H₂(α_i)} ,

Proposition 2. Let us consider α, β, γ ∈ ∆, then

(1) There are only three cases of the relation between

α and β: α > β, β > α or α ∼ β.

H₂

n

ꢅ

ꢆ

o

ꢄ

arcmax_h(α₁, . . . , α_n) = α_jh α_j= max {h (α_i)} ,

n

ꢅ

ꢆ

o

ꢄ

(2) If α > β and β > γ, then α > γ;

arcmax_H(α₁, . . . , α_n) = α_jH₁α_j= max {H₁(α_i)} ,

1

n

ꢅ

ꢆ

o

ꢄ

arcmax_H(α₁, . . . , α_n) = α_jH₂α_j= max {H₂(α_i)} .

2

Proof. (1) We assume that α ≯ β and β ≯ α. By

Deﬁnition 9,

Deﬁnition 11. Lower bound and upper bound of

the collection of PLNs (α₁, . . . , α_n) are respectively

deﬁned as

α ≯ β ⇔





h (α) ≤ h (β)



ꢀ

ꢃ





α⁻= arcmin_Harcmin_H(arcmin_h(α₁, . . . , α_n)) ,

(15)

h (α) , h (β) or H₁(α) ≤ H₁(β)

2

1



h α , h β or H₁α , H₁

or H₂(α) ≤ H₂(β),

( )

β

( )

ꢀ

ꢃ



α⁺= arcmax_Harcmax_H(arcmax_h(α₁, . . . , α_n)) .





2

1

P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52 45

Based on Deﬁnitions 9, 10 and 11, the following

proposition can be easily proved.

Proof. By Deﬁnition 7, aggregated value by using

PLWAA is also a PLN. In the next step, we prove (23)

by using mathematical induction on n.

Proposition 3. For each collection of PLNs

(α₁, . . . , α_n),

1) For n = 2: By Deﬁnition 7,

α⁻. α_i. α⁺, ∀i = 1, . . . , n.

(20)

ꢁ

ꢂ

w₁α₁= s_{w θ(α )}, µ (α₁) , η (α₁) , ν (α₁) ,

(24)

(25)

1

The

.

in the left of Eq. (20) means that for all α_j∈ α⁻

,

and

we have α_j< α_ior α_j∼ α_i. Similar for the

. in the

right.

ꢁ

ꢂ

w₂α₂= s_{w θ(α )}, µ (α₂) , η (α₂) , ν (α₂) .

2

4. Aggregation operators of PLNs

We thus obtain

ꢁ

In this section some operators, which aggregate

PLNs, are proposed: picture linguistic arithmetic

averaging (PLAA), picture linguistic weighted

arithmetic averaging (PLWAA), picture linguistic

ordered weighted averaging (PLOWA) and picture

linguistic hybrid aggregation (PLHA) operators.

Throughout this paper, each weight vector is with

respect to a collection of non-negative number with the

total of 1.

w₁α₁⊕ w₂α₂= s_{w θ(α )+w θ(α )}

,

1

2

w₁θ (α₁) µ (α₁) + w₂θ (α₂) µ (α₂)

w₁θ (α₁) + w₂θ (α₂)

,

w₁θ (α₁) η (α₁) + w₂θ (α₂) η (α₂)

w₁θ (α₁) + w₂θ (α₂)

w₁θ (α₁) ν (α₁) + w₂θ (α₂) ν (α₂)

(26)

+

w₁θ (α₁) + w₂θ (α₂)

i. e., (23) holds for n = 2.

2) Let us assume that (23) holds for

is

Deﬁnition 12. Picture linguistic arithmetic

n

=

k

(

k ≥ 2), that

averaging (PLAA) operator is

a

mapping

PLAA : ∆ⁿ→ ∆ deﬁned as

1

n

w₁α₁⊕ . . . ⊕ w_kα_k=

PLAA (α₁, . . . , α_n) = (α₁⊕ · · · ⊕ α_n) ,

(21)

k

P

*

w_iθ (α_i) µ (α_i)

where (α₁, . . . , α_n) is a collection of PLNs.

i=1

s _k

i=1

,

P

Deﬁnition 13. Picture

linguistic

weighted

k

w_iθ(α_i)

P

w_iθ (α_i)

arithmetic averaging (PLWAA) operator is a mapping

(27)

i=1

PLWAA : ∆ⁿ→ ∆ deﬁned as

k

P

+

w_iθ (α_i) η (α_i)

w_iθ (α_i) ν (α_i)

PLWAA_w(α₁, . . . , α_n) = w₁α₁⊕ · · · ⊕ w_nα_n, (22)

i=1

,

.

k

P

where

w =

(w₁, . . . , w_n) is the weight vector of the

w_iθ (α_i)

collection of PLNs (α₁, . . . , α_n).

i=1

Proposition 4. Let (α₁, . . . , α_n) be a collection of

Then,

PLNs, and

w =

(w₁, . . . , w_n) be the weight vector of

this collection, then PLWAA_w(α₁, . . . , α_n) is a PLN

and

w₁α₁⊕ . . . ⊕ w_kα_k⊕ w_k+1α_k+1

k

P

*

w_iθ (α_i) µ (α_i)

PLWAA_w(α₁, . . . , α_n) =

i=1

n

= s _k

,

P

k

*

w_iθ(α_i)

w_iθ (α_i) µ (α_i)

P

i=1

w_iθ (α_i)

i=1

n

s

,

P

i=1

n

w_iθ(α_i)

P

i=1

k

w_iθ (α_i)

P

(23)

i=1

+

w_iθ (α_i) η (α_i)

w_iθ (α_i) ν (α_i)

i=1

n

P

,

⊕

+

w_iθ (α_i) η (α_i)

w_iθ (α_i) ν (α_i)

k

P

i=1

w_iθ (α_i)

,

.

n

i=1

P

ꢁ

ꢂ

w_iθ (α_i)

s_w

₎, µ (α_k+1) , η (α_k+1) , ν (α_k+1

)

_k+1θ(α_k+1

i=1

46 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

*

!

= s

,

(5) Associativity: Consider an added collection of

k

P

w_iθ(α_i) +w_k+1α_k+1

PLNs (γ₁, . . . , γ_m) with the associated weight vector

i=1

ꢅ

ꢆ

!

P

w⁰= w⁰₁, . . . , w⁰_m

,

k

w_iθ (α_i) µ (α_i) + w_k+1θ (α_k+1) µ (α_k+1

)

i=1

!

,

PLWAA_u(α₁, . . . , α_n, γ₁, . . . , γ_m)

=PLWAA_v(PLWAA_w(α₁, . . . , α_n) ,

k

P

w_iθ (α_i) + w_k+1θ (α_k+1

)

i=1

!

k

0

PLWAA_w(γ₁, . . . , γ_m)) ,

P

w_iθ (α_i) η (α_i) + w_k+1θ (α_k+1) η (α_k+1

)

i=1

ꢇ

ꢈ

ꢅ

ꢆ

w⁰

2

w_m⁰

2

!

,

where u = ¹, . . . , ^wⁿ

,

¹, . . . ,

and v =

,

.

w

1

2

1

2

k

P

2

w_iθ (α_i) + w_k+1θ (α_k+1

)

i=1

!

k

P

Deﬁnition 14. Picture linguistic ordered weighted

w_iθ (α_i) ν (α_i) + w_k+1θ (α_k+1) ν (α_k+1)+

averaging (PLOWA) operator is a mapping PLOWA

:

i=1

!

∆ⁿ→ ∆ deﬁned as

k

P

w_iθ (α_i) + w_k+1θ (α_k+1

)

i=1

PLOWA_ω(α₁, . . . , α_n) = ω₁β₁⊕ · · · ⊕ ω_nβ_n, (28)

k+1

P

*

w_iθ (α_i) µ (α_i)

where

ω

=

(ω₁, . . . , ω_n) is the weight vector of the

= 1, . . . , n) is the -th

largest of the totally comparable collection of PLNs

i=1

= s_k+1

,

P

PLOWA operator and β_j∈ ∆ (

j

k+1

w_iθ(α_i)

P

i=1

w_iθ (α_i)

i=1

(α₁, . . . , α_n).

k+1

P

+

w_iθ (α_i) η (α_i)

w_iθ (α_i) ν (α_i)

i=1

,

.

Deﬁnition 14 requires that all pairs of PLNs of the

collection (α₁, . . . , α_n) are comparable. We further

consider the cases when the collection (α₁, . . . , α_n) is

k+1

P

w_iθ (α_i)

i=1

ꢅ

ꢆ

This implies that, (23) holds for

completes the proof.

According to Deﬁnitions 9, 10, 13, Propositions

3 and 4, it can be easily proved that the PLWAA

operator has the following properties. Let (α₁, . . . , α_n)

be a collection of PLNs with the weight vector

(w₁, . . . , w_n), we have:

n

=

k

+ 1, which

not totally comparable. If α_i∼ α_jand θ (α_i) < θ α_j

,

ꢀ

we assign α_jto α_i. It is reasonable since α_iand α_jhave

the same score, ﬁrst accuracy and second accuracy.

Example 3. Let us consider α₁

hs₄, 0.2, 0.3, 0.3i hs₂, 0.1, 0.2, 0.6i

α₃

(0.2, 0.4, 0.15, 0.25). Taking

=

hs₂, 0.2, 0.4, 0.4i

,

=

w =

α₂

=

,

=

,

α₄

hs₄, 0.1, 0.2, 0.2i and

ω =

(1) Idempotency: If α_i= α for all i = 1, . . . , n,

Deﬁnitions 9 and 10 into account, we get

PLWAA_w(α₁, . . . , α_n) = α.

(2) Boundary:

α₂> α₁∼ α₄> α₃.

(29)

α₄is assigned to α₁

th position of weight vector

. By adding the 2-th and 3-

ω

, we obtain ω⁰

=

α⁻. PLWAA_w(α₁, . . . , α_n) . α⁺.

(0.2, 0.55, 0.25). Hence,

ꢅ

ꢆ

(3) Monotonicity: Let α^∗₁, . . . , α_n^∗be a collection of

PLNs such that α^∗_i≤ α_ifor all i = 1, . . . , n, then

0

PLOWA_ω(α₁, α₂, α₃, α₄) = PLOWA_ω(α₁, α₂, α₃) .

In this case, β₁= α₂, β₂= α₁and β₃= α₃.

ꢀ

ꢃ

PLWAA_wα^∗₁, . . . , α_n^∗≤ PLWAA_w(α₁, . . . , α_n) .

(4) Commutativity:

In the same way as in Proposition 4, we have the

following proposition.

ꢀ

ꢃ

0

PLWAA_w(α₁, . . . , α_n) = PLWAA_w

α

_σ(1), . . . , α_σ(n)

,

where

σ

is any permutation on the set {1, . . . , n} and

Proposition 5. Let (α₁, . . . , α_n) be a collection of

PLNs, and ω =

(ω₁, . . . , ω_n) be the weight vector of

ꢀ

ꢃ

w⁰= w_σ(1), . . . , w_σ(n)

.

P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52 47

ꢅ

ꢆ

the PLOWA, then PLOWA_ω(α₁, . . . , α_n) is a PLN and

PLOWA_ω(α₁, . . . , α_n) =

(3) Monotonicity: Let α^∗₁, . . . , α_n^∗be a totally

comparable collection of PLNs such that α^∗_i≤ α_ifor

all i = 1, . . . , n, then

ꢅ ꢆ ꢅ ꢆ

ω_jθ β_j

n

P

ꢀ

ꢃ

ω_jθ β_jµ β_j

PLOWA_ωα^∗₁, . . . , α_n^∗≤ PLOWA_ω(α₁, . . . , α_n) ;

*

j=1

n

s

,

j

,

P

ꢅ ꢆ

ω_jθ β_jν β_j

n

ω θ β

j

P

( )

j=1

(4) Commutativity:

(30)

j=1

ꢀ

ꢃ

ꢅ ꢆ ꢅ ꢆ

ω_jθ β_j

ꢅ ꢆ ꢅ ꢆ

ω_jθ β_j

n

P

PLOWA_ω(α₁, . . . , α_n) = PLOWA_ω

α

_σ(1), . . . , α_σ(n)

,

ω_jθ β_jη β_j

+

j=1

,

where σ is any permutation on the set {1, . . . , n}.

ꢅ ꢆ

n

P

(5) Associativity:

Consider an added totally

j=1

comparable collection of PLNs (γ₁, . . . , γ_m) with the

ꢅ

ꢆ

associated weight vector ω⁰

. . . ≥ α_n≥ γ₁≥ . . . ≥ γ_m,

=

ω⁰₁, . . . , ω_m⁰. If α₁≥

with β_j

(

j

= 1, . . . , n) is the

j

-th largest of the collection

(α₁, . . . , α_n).

PLOWA_ꢀ(α₁, . . . , α_n, γ₁, . . . , γ_m)

=PLOWA_δ(PLOWA_ω(α₁, . . . , α_n) ,

Example 4. (Continuation of Example 3) We have

0

PLOWA_ω(α₁, α₂, α₃) = α¯,

(31)

0

PLOWA_ω(γ₁, . . . , γ_m)) ,

ꢇ

ꢈ

ꢅ

ꢆ

where α¯ is determined as follows.

ω⁰

ω_m⁰

2

, . . . , ^ωⁿ

,

¹, . . . ,

and δ =

,

.

ω₁

2

1

2

1

2

where ꢀ =

2

θ (α¯) = ω⁰₁× θ (β₁) + w₂⁰× θ (β₂) + w₃⁰× θ (β₃)

= 0.2 × 4 + 0.55 × 2 + 0.25 × 2 = 2.4,

Proposition 6 shows some special cases of the

PLOWA operator.

Proposition 6. Let (α₁, . . . , α_n) be

comparable collection of PLNs, and

be the weight vector, then

a

totally

µ (α¯)

ꢀ

= w⁰₁× θ (β₁) × µ (β₁) + w₂⁰× θ (β₂) × µ (β₂)

ω =

(ω₁, . . . , ω_n)

ꢆ

+w⁰₃× θ (β₃) × µ (β₃) /θ (α¯)

(1) If

ω

=

(1, 0, . . . , 0), then PLOWA_ω(α₁, . . . , α_n)

=

max {α_i};

0.2 × 4 × 0.2 + 0.55 × 2 × 0.2 + 0.25 × 2 × 0.2

i=1,...,n

=

2.4

(2) If

ω =

(0, . . . , 0, 1), then PLOWA_ω(α₁, . . . , α_n)

=0.2.

min {α_i};

i=1,...,n

As a similarity, η (α¯) = 0.325 and ν (α¯) = 0.408. We

(3) If ω_j= 1, and ω_i= 0 for all i , j, then

ﬁnally get

PLOWA_ω(α₁, . . . , α_n)

of the collection of PLNs (α₁, . . . , α_n).

= β_jwhere β_jis the j-th largest

PLOWA_ω(α₁, α₂, α₃, α₄) = hs_2.4, 0.2, 0.325, 0.408i .

Deﬁnition 15. Picture Linguistic hybrid averaging

(PLHA) operator for PLNs is a mapping PLHA : ∆ⁿ

∆ deﬁned as

→

The PLOWA can be shown to satisfy the

properties of idempotency, boundary, monotonicity,

commutativity and associativity. Let (α₁, . . . , α_n) be

a totally comparable collection of PLNs, and

(ω₁, . . . , ω_n) be the weight vector of the PLOWA

operator, then

PLHA_w,ω(α₁, . . . , α_n) = ω₁β⁰₁⊕ · · · ⊕ ω_nβ_n⁰;

ω =

where

PLHA operator, and β⁰_jis the

comparable collection of ILNs (nw₁α₁, . . . , nw_nα_n)

with

(w₁, . . . , w_n) is the weight vector of the

ω

is the associated weight vector of the

(1) Idempotency: If α_i= α for all i = 1, . . . , n, then

j-largest of the totally

PLOWA_ω(α₁, . . . , α_n) = α;

(2) Boundary:

w

=

collection of PLNs (α₁, . . . , α_n).

The Proposition 7 gives the explicit formula for

PLHA operator.

min {α_i} ≤ PLOWA_ω(α₁, . . . , α_n) ≤ max {α_i} ;

i=1,...,n

48 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

Proposition 7. Let (α₁, . . . , α_n) be a collection of

PLNs,

PLHA operator, and

ω

=

(ω₁, . . . , ω_n) be the associated vector of the

(w₁, . . . , w_n) be the weight

vector of (α₁, . . . , α_n), then PLHA_w,ω(α₁, . . . , α_n) is a

w

=

PLHA_u,ꢀ(α₁, . . . , α_n, γ₁, . . . , γ_m)

=PLHA_v,δPLHA_w,ω(α₁, . . . , α_n) ,

ꢀ

PLNs and

ꢃ

0

PLHA_{w ,ω}(γ₁, . . . , γ_m) ,

PLHA_w,ω(α₁, . . . , α_n) =

ꢅ ꢆ ꢅ ꢆ

n

ꢅ

ꢆ

P

w⁰

2

w_m⁰

ω_jθ β⁰_jµ β⁰_j

where

u

=

, . . . , ^wⁿ

,

¹, . . . ,

,

ꢀ

=

w₁

2

¹, . . . ,

*

2

j=1

ꢅ

ꢆ

ꢅ

ꢆ

P

ω⁰

ω_m⁰

ꢅ

ꢆ

n

s

,

, . . . , ^ωⁿ

,

and v = δ =

,

.

ω₁

ꢅ ꢆ

1

2

ω_jθ β⁰_j

P

n

ω_jθ β⁰_j

j=1

²We can prove that the PLWAA and PLOWA

operators are two special cases of the PLHA

operator as in Proposition 8.

2

(32)

j=1

ꢅ ꢆ ꢅ ꢆ

n

P

ω_jθ β⁰_jη β⁰_j

ω_jθ β⁰_jν β_j

+

j=1

,

ꢅ ꢆ

n

P

ω_jθ β⁰_j

j=1

ω_jθ β⁰_j

j=1

ꢅ

ꢆ

Proposition 8. If

ω

=

¹_n, . . . , ¹_n, the

where β⁰_jis the

j

-largest of the totally comparable

PLHA operator is reduced to the PLWAA

ꢅ

ꢆ

operator; and if

w

=

¹_n, . . . , ¹_n, the PLHA

collection of ILNs (nw₁α₁, . . . , nw_nα_n).

operator is reduced to the PLOWA operator.

Similar to PLWAA and PLOWA operators, the

PLHA operator is idempotent, bounded, monotonous,

commutative and associative. Let (α₁, . . . , α_n) be a

5. GDM

assessments

under

picture

linguistic

collection of PLNs,

vector of the PLHA operator, and

the weight vector of (α₁, . . . , α_n), then

ω

=

(ω₁, . . . , ω_n) be the associated

(w₁, . . . , w_n) be

w

=

Let us consider a hypothetical situation,

in which

{A₁, . . . , A_m} is the set of

alternatives, and

(1) Idempotency: If α_i= α for all i = 1, . . . , n, then

A

=

PLHA_w,ω(α₁, . . . , α_n) = α;

C

=

{C₁, . . . , C_n} is the

set of criteria with the weight vector

c

=

(2) Boundary:

n

o

(c₁, . . . , c_n). We assume that

D

=

d₁, . . . , d_p

α⁻. PLHA_w,ω(α₁, . . . , α_n) . α⁺;

is a set of decision makers (DMs), and

w

=

ꢅ

ꢆ

ꢅ

ꢆ

(3) Monotonicity: Let α^∗₁, . . . , α_n^∗be a collection of

PLNs such that α^∗_i. α_ifor all i = 1, . . . , n, then

w₁, . . . , w_pis the weight vector of DMs.

Each DM d_kpresents the characteristic of

the alternative A_iwith respect to the criteria

ꢀ

ꢃ

PLHA_w,ωα^∗₁, . . . , α_n^∗. PLHA_w,ω(α₁, . . . , α_n) ;

ꢉ

ꢊ

(k)

ij

ꢅ

ꢆ

(k)

C_jby the PLN

= 1, . . . , m

α

=

s_{θ α}, µ_α, η_α, ν_α

ij

(4) Commutativity:

(i

,

j

= 1, . . . , n

,

k

= 1, . . . , p). The

ꢅ

ꢆ

(k)

ꢀ

ꢃ

decision matrix R_kis given by R_k

=

α

PLHA_w,ω(α₁, . . . , α_n) = PLHA_w,ω

α

_σ(1), . . . , α_σ(n)

,

ij

m×n

(k

= 1, . . . , p). The alternatives will be ranked

where

σ

is any permutation on the set

by the following algorithm.

ꢀ

ꢃ

{1, . . . , n} and w⁰= w_σ(1), . . . , w_σ(n)

.

Step 1. Derive the overall values α⁽_i^k)of the

alternatives A_i, given by the DM d_k:

(5) Associativity: Consider an added

collection of PLNs (γ₁, . . . , γ_m) with the

ꢅ

ꢆ

ꢅ

ꢆ

α⁽_i^k)= PLWAA_cα^(k), . . . , α^(k)

,

(33)

associated weight vector w⁰

=

w⁰, . . . , w⁰_m

such that nw₁α₁≥ · · · ≥ nw_nα_n≥¹mw₁⁰γ₁≥

· · · ≥ mw⁰_mγ_m. We have

i1

in

for i = 1, . . . , m, and k = 1, . . . , p.

P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52 49

Table 1. Decision matrix R₁

C₁

C₂

C₃

A₁hs₄, 0.6, 0.1, 0.2i hs₄, 0.4, 0.2, 0.2i hs₅, 0.2, 0.3, 0.5i

A₂hs₅, 0.7, 0.2, 0.1i hs₄, 0.4, 0.1, 0.4i hs₄, 0.5, 0.2, 0.3i

A₃hs₅, 0.3, 0.1, 0.4i hs₅, 0.4, 0.3, 0.3i hs₆, 0.7, 0.1, 0.2i

A₄hs₄, 0.6, 0.1, 0.2i hs₄, 0.6, 0.1, 0.2i hs₅, 0.3, 0.1, 0.5i

Step 2. Derive the collective overall values

α_iby aggregating the individual overall values

α⁽_i¹⁾, . . . , α⁽_i^p):

A_i(i = 1, 2, 3, 4) by the PLHA operator with

associated weight vector ω = (0.2, 0.5, 0.3).

α₁= hs_4.40, 0.3965, 0.2045, 0.3438i ,

α₂= hs_4.57, 0.3481, 0.1428, 0.4040i ,

α₃= hs_5.32, 0.3628, 0.1666, 0.4050i ,

α₄= hs_5.16, 0.4098, 0.1510, 0.3948i .

ꢅ

ꢆ

α_i= PLHA_w,ωα⁽_i¹⁾, . . . , α⁽_i^p)

,

(34)

ꢅ

ꢆ

where

ω

=

ω₁, . . . , ω_pis the weight vector

of the PLHA operator (i = 1, . . . , m).

Step 3. Calculate the scores h (α_i), ﬁrst

accuracies H₁(α_i) and second accuracies

Step 3. By eq. (12),

h (α₁) = 0.2318, h (α₂) = −0.2556

h (α₃) = −0.2246, h (α₄) = 0.078.

H₂(α_i)

(i = 1, . . . , m), rank the alternatives

by using Deﬁnition 9 (the alternative A_iis

1

called to be better than the alternative A_i

,

2

By Deﬁnition 9,

denoted by A_i> A_i, iﬀ α_i> α_i, for all

1

2

1

2

i₁, i₂= 1, . . . , m).

h (α₁) > h (α₄) > h (α₃) > h (α₂)

then A₁> A₄> A₃> A₂.

6. An illutrative example

This situation concerns four alternative

enterprises, which will be chosen by

7. Conclusion

three DMs whose weight vector is

(0.3, 0.4, 0.3)

The enterprises will be

considered under three criteria C₁C₂and C₃

Assume that the weight vector of the criteria

is

(0.37, 0.35, 0.28). Three decision

w

=

In this paper, motivated by picture fuzzy

sets and linguistic approaches, the notion

of picture linguistic numbers are ﬁrst

deﬁned. We propose the score, ﬁrst accuracy

and second accuracy of picture linguistic

numbers, and propose a simple approach

for the comparison between two picture

linguistic numbers. Simultaneously, the

operation laws for picture linguistic numbers

are given and the accompanied properties are

studied. Further, some aggregation operators

are developed: picture linguistic arithmetic

averaging, picture linguistic weighted

.

,

.

c

=

matrices are listed in Tabs. 1, 2 and 3.

Step 1. Using explicit form of the PLWAA

operation given in Eq. 23, we obtain overall

values α⁽_i^k)of the alternatives A_igiven by the

DMs d_k(i = 1, 2, 3, 4 and k = 1, 2, 3) as

in Tab. 4.

Step 2. Aggregate all the individual overall

values α⁽_i¹⁾α⁽_i²⁾and α⁽_i³⁾of the alternatives

,

50 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52

Table 2. Decision matrix R₂

C₁

C₂

C₃

A₁hs₄, 0.7, 0.1, 0.2i hs₆, 0.2, 0.2, 0.5i hs₄, 0.7, 0.2, 0.1i

A₂hs₃, 0.2, 0.2, 0.6i hs₅, 0.5, 0.1, 0.2i hs₅, 0.3, 0.1, 0.4i

A₃hs₄, 0.2, 0.1, 0.5i hs₇, 0.2, 0.2, 0.6i hs₅, 0.1, 0.2, 0.6i

A₄hs₅, 0.7, 0.2, 0.1i hs₅, 0.2, 0.1, 0.7i hs₄, 0.6, 0.1, 0.2i

Table 3. Decision matrix R₃

C₁

C₂

C₃

A₁hs₄, 0.6, 0.3, 0.1i hs₆, 0.2, 0.3, 0.5i hs₅, 0.2, 0.1, 0.7i

A₂hs₃, 0.2, 0.2, 0.5i hs₅, 0.2, 0.1, 0.6i hs₆, 0.2, 0.2, 0.6i

A₃hs₅, 0.3, 0.2, 0.5i hs₇, 0.8, 0.1, 0.1i hs₅, 0.2, 0.2, 0.5i

A₄hs₃, 0.7, 0.1, 0.2i hs₅, 0.2, 0.2, 0.5i hs₆, 0.3, 0.1, 0.6i

Table 4. Overall values α⁽_i^k)of the alternatives A_igiven by the DMs d_k(i = 1, 2, 3, 4; k = 1, 2, 3)

d₁

d₂

d₃

A₁hs_4.28, 0.4037, 0.1981, 0.2981i hs_4.70, 0.4766, 0.1685, 0.3102i hs_4.98, 0.3189, 0.2438, 0.4373i

A₂hs_4.37, 0.5526, 0.1680, 0.2474i hs_4.26, 0.3561, 0.1261, 0.3700i hs_4.54, 0.2000, 0.1615, 0.5756i

A₃hs_5.28, 0.4604, 0.1663, 0.3032i hs_5.33, 0.1737, 0.1722, 0.5722i hs_5.70, 0.4904, 0.1570, 0.3281i

A₄hs_4.28, 0.5019, 0.1000, 0.2981i hs_5.28, 0.4070, 0.1682, 0.3917i hs_4.54, 0.3593, 0.1385, 0.4637i

arithmetic averaging, picture linguistic

ordered weighted averaging and picture

linguistic hybrid aggregation operators.

Finally, based on the picture linguistic

weighted arithmetic averaging and the picture

linguistic hybrid aggregation operators, we

propose an approach to handle multi-criteria

group decision making problems under

picture linguistic environment.

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This research is funded by the Vietnam

National Foundation for Science and

Technology Development (NAFOSTED)

under grant number 102.01- 2017.02.

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