A multicriteria decision making method using aggregation operators for simplified neutrosophic sets

IOS Press A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets Jun Ye∗ Department of Electrical and Information Engineering, Shaoxing Unive

IOS Press

A multicriteria decision-making method

using aggregation operators for simplified

neutrosophic sets

Jun Ye∗

Department of Electrical and Information Engineering, Shaoxing University, Shaoxing,

Zhejiang Province, P.R China

Abstract The paper introduces the concept of simplified neutrosophic sets (SNSs), which are a subclass of neutrosophic sets, and

defines the operational laws of SNSs Then, we propose some aggregation operators, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator Based on the two aggregation operators and cosine similarity measure for SNSs, a multicriteria decision-making method is established in which the evaluation values of alternatives with respective to criteria are represented by the form of SNSs The ranking order of alternatives is performed through the cosine similarity measure between an alternative and the ideal alternative and the best one(s) can be determined as well Finally, a numerical example shows the application of the proposed method

Keywords: Neutrosophic set, simplified neutrosophic set, operational laws, aggregation operator, cosine similarity measure, multicriteria decision-making

1 Introduction

Intuitionistic fuzzy sets [9] and interval-valued

intu-itionistic fuzzy sets [10] can only handle incomplete

information but not the indeterminate information and

inconsistent information which exist commonly in real

situations Then, the neutrosophic set proposed by

Smarandache is a powerful general formal framework

which generalizes the concept of the classic set, fuzzy

set [11], interval valued fuzzy set [6], intuitionistic

fuzzy set [9], interval-valued intuitionistic fuzzy set

[10], paraconsistent set [1], dialetheist set [1],

para-doxist set [1], and tautological set [1] So the notion

of neutrosophic sets is more general and overcomes the

∗Corresponding author Jun Ye, Department of Electrical and

Information Engineering, Shaoxing University, 508 Huancheng West

Road, Shaoxing, Zhejiang Province 312000, P.R China Tel.: +86 575

88327323; E-mail: yehjun@aliyun.com.

aforementioned issues In the neutrosophic set, indeter-minacy is quantified explicitly and truth-membership, indeterminacy-membership, and false-membership are independent This assumption is very important in many applications such as information fusion in which the data are combined from different sensors Recently, neutrosophic sets have been applied to image thresholding, image denoise applications, and image segmentation Cheng and Guo [3] proposed a threshold-ing algorithm based on neutrosophy, which could select the thresholds automatically and effectively Guo et al [18] defined some concepts and operators based on neu-trosophic sets and applied them for image denoising, which can process not only noisy images with different levels of noise, but also images with different kinds of noise well Guo and Cheng [19] applied neutrosophic sets to process the images with noise and proposed a novel neutrosophic approach for image segmentation

1064-1246/14/$27.50 © 2014 – IOS Press and the authors All rights reserved

In any multicriteria decision-making problem, the

final solution must be obtained from the synthesis of

performance degrees of criteria [2] To do this, the

aggregation of information is fundamental Therefore,

many researchers have developed a variety of

aggre-gation operators with assessment on [0, 1] [12–15,

20–22], proportional assessment on [1/9, 9] [2], and

linguistic assessment [16, 17] Then two of the most

common operators for aggregating arguments are the

weighted arithmetic average operator and the weighted

geometric average operator [20–22], which have been

wildly applied to decision-making problems

There-fore, these aggregation operators are important tools

for aggregating fuzzy information, intuitionistic fuzzy

information, interval-valued fuzzy information, and

interval-valued intuitionistic fuzzy information in the

decision-making problems Whereas the neutrosophic

set generalizes the above mentioned sets from

philo-sophical point of view From scientific or engineering

point of view, the neutrosophic set and set-theoretic

operators need to be specified Otherwise, it will be

difficult to apply it to the real applications

There-fore, Wang et al [4] proposed interval neutrosophic

sets (INSs) and some operators of INSs Then, Ye [7]

defined the Hamming and Euclidean distances between

INSs and developed the similarity measures between

INSs based on the relationship between similarity

mea-sures and distances and a multicriteria decision-making

method using the similarity measures between INSs in

interval neutrosophic setting, in which criterion

val-ues with respect to alternatives are evaluated by the

form of INSs Recently, Wang et al [5] proposed a

single valued neutrosophic set (SVNS), which is an

instance of the neutrosophic set, and provide the

set-theoretic operators and various properties of SVNSs

Furthermore, Ye [8] presented the information energy

of SVNSs, correlation of SVNSs, correlation

coeffi-cient of SVNSs, and weighted correlation coefficoeffi-cient

of SVNSs based on the extension of the correlation

of intuitionistic fuzzy sets and demonstrated that the

cosine similarity measure is a special case of the

corre-lation coefficient in single valued neutrosophic setting,

and then applied them to single valued neutrosophic

decision-making problems Meanwhile, motivated by

some intuitionistic fuzzy aggregation operators with

assessment on [0, 1] [20–22], we can also extend

them to neutrosophic sets Thus, it will be necessary

to develop some aggregation operators for

aggregat-ing neutrosophic information in the decision-makaggregat-ing

applications To do so, the main purposes of this paper

are to define the concept of simplified neutrosophic

sets (SNSs), which can be described by three real numbers in the real unit interval [0, 1], and some operational laws for SNSs and to propose two aggre-gation operators, including a simplified neutrosophic weighted arithmetic average operator and a simpli-fied neutrosophic weighted geometric average operator Then, a multicriteria decision-making method using the two aggregation operators of SNSs is established

in which the evaluation information of alternatives with respect to criteria is given by truth-membership degree, indeterminacy-membership degree, and falsity-membership degree under the simplified neutrosophic environment And then the ranking order of alterna-tives is performed through the cosine similarity measure between an alternative and the ideal alternative and the best choice can be obtained according to the measure values However, the main advantage of the proposed simplified neutrosophic multicriteria decision-making method can handle not only incomplete information but also the indeterminate information and inconsistent information which exist commonly in real situations The rest of paper is organized as follows Sec-tion 2 introduces the some concepts of neutrosophic sets SNSs and some operational laws are defined and two simplified neutrosophic weighted aggregation operators are proposed in Section 3 The two simpli-fied neutrosophic weighted aggregation operators and cosine similarity measure for SNSs are applied to a mul-ticriteria decision-making problem under the simplified neutrosophic environment and through the cosine sim-ilarity measure between each alternative and the ideal alternative, the ranking order of alternatives and the best one(s) can be obtained in Section 4 In Section 5, a numerical example demonstrates the application of the proposed decision-making method Finally, some final remarks and future research are offered in Section 6

2 Some concepts of neutrosophic sets

Neutrosophic set is a part of neutrosophy, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spec-tra [1], and is a powerful general formal framework Neutrosophic set permits one to incorporate indetermi-nacy, hesitation and/or uncertainty independent of the membership and non-membership information Thus the notion of neutrosophic set is a generalization of fuzzy, intuitionistic fuzzy, and interval-valued sets Smarandache [1] gave the following definition of a neutrosophic set

Definition 1 [1] Let X be a space of points (objects),

with a generic element in X denoted by x A

neutro-sophic set A in X is characterized by a truth-membership

function T A (x), an indeterminacy-membership function

I A (x) and a falsity-membership function F A (x) The

functions T A (x), I A (x) and F A (x) are real standard or

nonstandard subsets of ]0–, 1+[, that is T A (x): X−→

]0–, 1+[, I A (x): X−→ ]0–, 1+[, and F A (x): X−→ ]0–,

1+[.

There is no restriction on the sum of T A (x), I A (x) and

F A (x), so 0– ≤sup T A (x)+sup I A (x)+sup F A (x)≤3+.

Definition 2 [1] The complement of a neutrosophic

set A is denoted by A c and is defined as T c

A(x) =

{1+}  T A(x), I c

A(x) = {1+}  I A(x), and F A c(x)

= {1+}  F A(x) for every x in X.

Definition 3 [1] A neutrosophic set A is contained in

the other neutrosophic set B, A ⊆ B if and only if inf

T A (x)≤inf T B (x), sup T A (x)≤sup T B (x), inf I A (x)≥inf

I B (x), sup I A (x)≥sup I B (x), inf F A (x)≥inf F B (x), and

sup F A (x)≥sup F B (x) for every x in X.

Definition 4 [1] The union of two neutrosophic sets

A and B is a neutrosophic set C, written as C = A ∪B,

whose truth-membership, indeterminacy membership

and false-membership functions are related to those of

A and B by T C (x)=T A (x) ⊕ T B (x)  T A (x) T B (x),

I C (x) = I A (x)⊕ I B (x)  I A (x) I B (x), and F C (x) = F A (x)

⊕ F B (x)  F A (x) F B (x) for any x in X.

Definition 5 [1] The intersection of two neutrosophic

sets A and B is a neutrosophic set C, written as C = A ∩ B,

whose truth-membership, indeterminacy-membership

and false-membership functions are related to those of

A and B by T C (x) = T A (x) T B (x), I C (x) = I A (x) I B (x),

and F C (x) = F A (x) F B (x) for any x in X.

3 Operational relations of SNSs

Smarandache [1] has provided a variety of real-life

examples for possible applications of his neutrosophic

sets, which are comprised of three subsets of the

non-standard interval ]0–, 1+[, i.e., T

A (x)⊆]0–, 1+[, I

A (x)

⊆]0–, 1+[, and F

A (x)⊆]0–, 1+[ However, it is

diffi-cult to apply neutrosophic sets to practical problems

Therefore, we will reduce neutrosophic sets of

non-standard intervals into a kind of SNSs of non-standard

intervals that will preserve the operations of the

neu-trosophic sets In this section, we introduce the concept

of SNSs, which are a subclass of neutrosophic sets, and the operational laws of SNSs, then propose two weighted aggregation operators of SNSs for the infor-mation aggregation in neutrosophic decision making problems

Definition 6 Let X be a space of points (objects),

with a generic element in X denoted by x A neutro-sophic set A in X is characterized by a truth-membership function T A (x), a indeterminacy-membership function

I A (x) and a falsity-membership function F A (x) If the functions T A (x), I A (x) and F A (x) are singleton subinter-vals/subsets in the real standard [0, 1], that is T A (x): X

−→ [0, 1], I A (x): X −→ [0, 1], and F A (x): X−→ [0,

1] Then, a simplification of the neutrosophic set A is

denoted by

A = {x, T A(x), I A(x), F A(x) |x ∈ X}

which is called a SNS It is a subclass of neutrosophic sets

In this paper, we shall use the SNS whose T A (x),

I A (x) and F A (x) values are single points in the real

standard [0, 1] instead of subintervals/subsets in the real standard [0, 1] Thus, each SNS can be described

by three real numbers in the real unit interval [0, 1] Therefore, the sum ofT A(x) ∈ [0, 1], I A(x) ∈ [0, 1]

andF A(x) ∈ [0, 1] satisfies the condition 0 ≤ T A(x) +

I A(x) + F A(x) ≤ 3 For the sake of simplicity, the SNS

A = {x, T A(x), I A(x), F A(x) |x ∈ X} is denoted by

the simplified symbolA = T A(x), I A(x), F A(x) In

this case, we can give the following definitions

Definition 7 The SNS A is contained in the other SNS

B, A ⊆B if and only if T A (x) ≤T B (x), I A (x) ≥I B (x), and

F A (x) ≥F B (x) for every x in X.

Definition 8 Let A, B are two SNSs Operational

rela-tions are defined by

A + B = T A(x) + T B(x) − T A(x)T B(x),

I A(x) + I B(x) − I A(x)I B(x),

F A(x) + F B(x) − F A(x)F B(x) (1)

A · B = T A(x)T B(x), I A(x) I B(x),

λA =
1− (1 − T A(x)) λ , 1 − (1 − I A(x)) λ ,

1− (1 − F A(x)) λ

A λ=
T A λ(x), I A λ(x), F A λ(x), λ > 0 (4)

Based on the above operational laws, we can propose

the following weighted arithmetic aggregation

opera-tor and weighted geometric aggregation operaopera-tor for

SNSs

Definition 9 Let A j (j = 1, 2, , n) be a SNS The

simplified neutrosophic weighted arithmetic average

operator is defined by

F w(A1, A2, , A n)=n

j=1

w j A j (5)

where W = (w1, w2, , w n) is the weight vector

ofA j(j = 1, 2, , n), w j ∈ [0, 1] and n

j=1 w j= 1

Especially, Assume W = (1/n, 1/n, , 1/n), then G w

is called as an arithmetic average operator for SNSs

Definition 10 Let A j (j = 1, 2, , n) be a SNS The

simplified neutrosophic weighted geometric average

operator is defined by

G w(A1, A2, , A n)=n

j=1

A w j

where W = (w1, w2, , w n) is the weight vector ofA j

(j = 1, 2, , n), w j∈ [0, 1] andn

j=1 w j = 1 Especially,

Assume W = (1/n, 1/n, , 1/n), then G w is called as

an geometric average operator for SNSs

Theorem 1 For a SNS A j (j = 1, 2, , n), we have the

following result by use of Equation (5):

F w(A1, A2, , A n)=



1−

n



j=1

(1− T A j(x)) w j ,

1−

n



j=1

(1− I A j(x)) w j ,

1−n

j=1

(1− F A j(x)) w j

 (7)

where W=(w1, w2, , w n ) is the weight vector of A j

(j = 1, 2, , n), w j ∈ [0, 1] andn

j=1 w j= 1

Proof The proof of Equation (7) can be done by means

of mathematical induction

(1) When n = 2, then,

w1A1=
1− (1 − T A1(x)) w1, 1 − (1 − I A1(x)) w1,

1− (1 − F A1(x)) w1

,

w2A2=
1− (1 − T A2(x)) w2, 1 − (1 − I A2(x)) w2,

1− (1 − F A2(x)) w2

.

Thus,

F w(A1, A2)= w1A1+ w2A2

=
2− (1 − T A1(x)) w1− (1 − T A2(x)) w2

−(1 − (1 − T A1(x)) w1)(1− (1 − T A2(x)) w2),

2− (1 − I A1(x)) w1− (1 − I A2(x)) w2

−(1 − (1 − I A1(x)) w1)(1− (1 − I A2(x)) w2),

2− (1 − F A1(x)) w1− (1 − F A2(x)) w2

−(1 − (1 − F A1(x)) w1)(1− (1 − F A2(x)) w2)

=
1− (1 − T A1(x)) w1(1− T A2(x)) w2,

1− (1 − I A1(x)) w1(1− I A2(x)) w2,

1− (1 − F A1(x)) w1(1− F A2(x)) w2

(8)

(2) When n = k, by applying Equation (7), we get

F w(A1, A2, , A k)=



1−k

j=1

(1− T A j)w j ,

1−k

j=1

(1− I A j)w j ,

1−

k



j=1

(1− F A j)w j

 (9)

(3) When n = k+1, by applying Equations (8) and (9),

we can get

F w(A1, A2, , A k+1)

=



1−k

j=1

(1− T A j(x)) w j

+(1 − (1 − T A k+1(x)) w k+1)

−(1 −k

j=1

(1− T A j(x)) w j) (1− (1 − T A k+1(x)) w k+1),

k



j=1

(1− I A j(x)) w j

+(1 − (1 − I A k+1(x)) w k+1)

−(1 −k

j=1

(1− I A j(x)) w j) (1− (1 − I A k+1(x)) w k+1),

1−k

j=1

(1− I A j(x)) w j

+(1 − (1 − F A k+1(x)) w k+1)

−(1 −

k



j=1

(1− F A j(x)) w j) (1− (1 − F A k+1(x)) w k+1)

=



1−

k+1



j=1

(1− T A j(x)) w j ,

1−k+1

j=1

(1− I A j(x)) w j ,

1−k+1

j=1

(1− F A j(x)) w j



(10)

Therefore, considering the above results, we have

Equation (7) for any n This completes the proof.

It is obvious that the F w operator has the following

properties:

(1) Idempotency: Let A j (j = 1, 2, , n) be a collection

of SNSs If A i (j = 1, 2, , n) is equal, i.e A j = A for

j = 1, 2, , n, then F w(A1, A2, , A n)= A.

(2) Boundedness: Let A j(j = 1, 2, , n)

be a collection of SNSs and let A−=



min

j T A j(x), max

j I A j(x), max

j F A j(x)

and

A+=



max

j T A j(x), min

j I A j(x), min

j F A j(x)

for j = 1, 2, , n, then A−⊆ F w(A1, A2, ,

A n)⊆ A+.

(3) Monotonity: Let A j(j = 1, 2, , n) be

a collection of SNSs If A j ⊆A j∗ for

j = 1, 2, , n, then F w(A1, A2, , A n)⊆

F w A∗, A∗, , A∗

Theorem 2 For a SNS A j (j = 1, 2, , n), we have the

following result by applying Equation (6):

G w(A1, A2, , A n)

=

 n



j=1

T w j

A j(x),n

j=1

I w j

A j(x),n

j=1

F w j

A j(x)

 (11)

where W = (w1, w2, , w n ) is the weight vector of A j

(j = 1, 2, , n), w j ∈ [0, 1] andn

j=1 w j= 1

By a similar proof manner, we can give the proof of Theorem 2 (omitted)

It is obvious that the G w operator has the following properties:

(1) Idempotency: Let A j (j = 1, 2, , n) be a collection of SNSs If A i (j = 1, 2, , n)

is equal, i.e A j = A for j = 1, 2, , n, then

G w(A1, A2, , A n)= A.

(2) Boundedness: Let A j (j = 1, 2, , n) be

a collection of SNSs and let A−=

 min

j T A j(x), max

j I A j(x), max

j F A j(x)

and

A+=

 max

j T A j(x), min

j I A j(x), min

j F A j(x)

for j = 1, 2, , n, then A−⊆ G w(A1,

A2, , A n)⊆ A+.

(3) Monotonity: Let A j (j = 1, 2, , n) be

a collection of SNSs If A j ⊆A j∗ for

j = 1, 2, , n, then G w(A1, A2, , A n)⊆

G w A∗

1, A∗

2, , A∗

n

The aggregation results F w and G w are still SNSs Obviously, there are different focal points between Equations (7) and (11) The weighted arithmetic aver-age operator emphasizes group’s major points, and then the weighted geometric average operator emphasizes personal major points

4 Decision-making method based on the simplified neutrosophic weighted aggregation operators

In this section, we present a handling method for multicriteria decision-making problems by means of the two aggregation operators and cosine similarity measure for SNSs under the simplified neutrosophic environment

Let A = {A1, A2, , A m } be a set of alternatives and

let C = {C1, C2, , C n } be a set of criteria Assume

that the weight of the criterion C j (j = 1, 2, , n),

entered by the decision-maker, is w j , w j ∈ [0, 1] and

n

j=1 w j= 1 In the decision process, the evaluation

information of the alternative A ion the criteria is

rep-resented by the form of a SNS:

A i=
C j , T A i(C j), I A i(C j), F A i(C j)

|C j ∈ C ,

where 0≤ T A i(C j)+ I A i(C j)+ F A i(C j)≤ 3, T A i(C j)

≥ 0, I A i(C j)≥ 0, F A i(C j)≥ 0, j = 1, 2, , n, and

i = 1, 2, , m For convenience, the value of SNSs

is denoted byα ij =t ij , i ij , f ij (i = 1, 2, , m; j = 1, 2,

, n) Therefore, we can get a simplified neutrosophic

decision matrix D = ( α ij)m ×n:

D = (α ij)m×n=

⎡

⎢

⎢

⎢

t11, i11, f11 t12, i12, f12 · · · t1n , i1n , f1n

t21, i21, f21 t22, i22, f22 · · · t2n , i2n , f2n

. . .

t m1 , i m1 , f m1 t m2 , i m2 , f m2 · · · t mn , i mn , f mn

⎤

⎥

⎥

⎥.

Then, the aggregating simplified neutrosophic value

α i for A i (i = 1, 2, , m) is α i=t i , i i , f i = F iw(α i 1,α i 2,

,α i n) orα i=t i , i i , f i = G iw(α i 1,α i 2, ,α i n) is

obtained by applying Equations (7) or (11) according to

each row in the simplified neutrosophic decision matrix

D = (a ij)m ×n

To rank alternatives in the decision-making process,

we define an ideal SNS value as the ideal alternative

␣∗=1, 0, 0, and then based on the cosine similarity

measure between SVNSs proposed by Ye [8], the cosine

similarity measure between SNSs␣i (i = 1, 2, , m) and

␣∗can be defined as follows:

S i(α i , α∗)=  t i t∗i + i i i∗i + f i f i∗

t2

i + i2

i + f2

i

2

+ i∗

i

2

+ f∗

i

2

=  t i

t2

i + i2

i + f2

i

(12)

Then, the bigger the measure value S i(␣i,␣∗) (i = 1,

2, , m) is, the better the alternative A i is, because

the alternative A i is close to the ideal alternative␣∗.

Through the cosine similarity measure between each

alternative and the ideal alternative, the ranking order

of all alternatives can be determined and the best one

can be easily identified as well

In summary, the decision procedure for the proposed

method can be summarized as follows:

Step 1: Calculate the weighted arithmetic average values by using Equation (7) or the weighted geometric average values by using Equation (11)

Step 2: Calculate the cosine similarity measure between each alternative and the ideal alter-native by using Equation (12)

Step 3: Give the ranking order of the alternatives from the obtained measure values, and then get the best choice

Step 4: End

5 Numerical example

In this section, an example for a multicriteria decision-making problem of engineering alternatives

is used as a demonstration of the application of the proposed decision-making method in a realistic sce-nario, as well as the application and effectiveness of the proposed decision-making method

Let us consider the decision-making problem adapted from [8] There is an investment company, which wants to invest a sum of money in the best option There is a panel with four possible alternatives to invest

the money: (1) A1 is a car company; (2) A2 is a food

company; (3) A3is a computer company; (4) A4is an arms company The investment company must take a decision according to the following three criteria: (1)

C1is the risk; (2) C2is the growth; (3) C3is the environ-mental impact Then, the weight vector of the criteria is

given by W = (0.35, 0.25, 0.4), which is adopted from

the literature [8]

For the evaluation of an alternative A i (i = 1, 2, 3, 4) with respect to a criterion C j (j = 1, 2, 3), it is obtained

from the questionnaire of a domain expert For example, when we ask the opinion of an expert about an

alterna-tive A1with respect to a criterion C1, he or she may say that the possibility in which the statement is good is 0.4

and the statement is poor is 0.3 and the degree in which

he or she is not sure is 0.2 For the neutrosophic

nota-tion, it can be expressed as␣11=0.4, 0.2, 0.3 Thus,

when the four possible alternatives with respect to the

above three criteria are evaluated by the expert, we can

obtain the following simplified neutrosophic decision

matrix D:

D=

⎡

⎢

⎢

⎣

0.4, 0.2, 0.3 0.4, 0.2, 0.3 0.2, 0.2, 0.5

0.6, 0.1, 0.2 0.6, 0.1, 0.2 0.5, 0.2, 0.2

0.3, 0.2, 0.3 0.5, 0.2, 0.3 0.5, 0.3, 0.2

0.7, 0.0, 0.1 0.6, 0.1, 0.2 0.4, 0.3, 0.2

⎤

⎥

⎥

⎦.

The proposed method is applied to solve this

problem according to the following computational

procedure:

Step 1: We can obtain the weighted arithmetic

average value (aggregating simplified

neu-trosophic value)␣i for A i (i = 1, 2, 3, 4) by

using Equation (7):

␣1=0.3268, 0.2000, 0.3881, ␣2=

0.5627, 0.1414, 0.2000, ␣3=0.4375,

0.2416, 0.2616, and ␣4= 0.5746, 0.1555,

0.1663

Step 2: By applying Equation (12), we can

com-pute each cosine similarity measure S i(␣i,

␣∗) (i = 1, 2, 3, 4) as follows:

S1(␣1, ␣∗) = 0.5992, S2(␣2, ␣∗) = 0.9169,

S3(␣3, ␣∗) = 0.7756, and S4(␣4, ␣∗) =

0.9297

Step 3: From the measure value S i(␣i,␣∗) (i = 1, 2, 3,

4) between an alternative and the ideal

alter-native, the ranking order of four alternatives

is A4 A2 A3 A1

Therefore, we can see that the alternative A4is the

best choice among all the alternatives

On the other hand, we can also utilize the weighted

geometric average operator as the following

computa-tional procedure:

Step 1’: We compute the weighted geometric

aver-age values by applying Equation (11) for A i

(i = 1, 2, 3, 4), each aggregating simplified

neutrosophic value␣i (i = 1, 2, 3, 4) is as

follows:

␣1=0.3031, 0.2000, 0.3680, ␣2=

0.5578, 0.1320, 0.2000, ␣3=0.4181,

0.2352, 0.2551, and ␣4=0.5385, 0,

0.1569

Step 2’: By using Equation (12), we can compute

each cosine similarity measure S i(␣i,␣∗)

(i = 1, 2, 3, 4) as follows:

S1(␣1, ␣∗) = 0.5863, S2(␣2, ␣∗) = 0.9188,

S3(␣3, ␣∗) = 0.7696, and S4(␣4, ␣∗) =

0.9601

Step 3’: The ranking order of four alternatives is A4

 A2 A3 A1 Thus, we can see that the

alternative A4is still the best choice among all the alternatives

Obviously, we can see that the above two kinds of ranking orders and the best alternative are the same The method proposed in this paper differs from existing approaches for fuzzy multi-criteria decision making not only due to the fact that the proposed method uses the SNS concept and two simplified neu-trosophic aggregation operators, but also due to the consideration of the indeterminacy information besides truth and falsity information in the evaluation of the alternative with respect to criteria, which makes it have more feasible and practical than other traditional decision making methods in real decision-making prob-lems Therefore, its advantage is easily reflecting the ambiguous nature of subjective judgments because SNSs are suitable for capturing imprecise, uncertain, and inconsistent information in the multicriteria deci-sion analysis

6 Conclusion

This paper introduced the concept of SNSs, which are a subclass of neutrosophic sets, and defined some operational laws of SNSs Then, we proposed two aggregation operators for SNSs, including a simplified neutrosophic weighted arithmetic average operator and

a simplified neutrosophic weighted geometric average operator The two aggregation operators were applied

to multicriteria decision-making problems under the simplified neutrosophic environment, in which crite-rion values with respect to alternatives are evaluated

by the form of simplified neutrosophic values and the criterion weights are known information We utilized the cosine similarity measure between an alternative and the ideal alternative to rank the alternatives and

to determine the best one(s) according to the measure values Finally, a numerical example is provided to illustrate the application of the developed approach The proposed simplified neutrosophic multicriteria decision-making method is more suitable for real

scien-tific and engineering applications because it can handle

not only incomplete information but also the

indeter-minate information and inconsistent information which

exist commonly in real situations The techniques

pro-posed in this paper can provide more useful way for

decision-makers In the future, we shall deal with group

decision making problems with incomplete decision

contexts and preference relations performed in the

selection process under the simplified neutrosophic

environment and apply the simplified neutrosophic

aggregation operators to solve practical applications in

other areas such as expert system, information fusion

system, and medical diagnoses

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