Some power generalized aggregation operators based on the interval neutrosophic numbers

Transactions on Cybernetics

Some power generalized aggregation operators based on the interval neutrosophic numbers and their application to decision making

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Regular Paper 08-Aug-2013

liu, peide; shandong economic university, school of information management Interval neutrosophic set, power average, generalized weighted aggregation (GWA) operator, ordered weighted aggregation (OWA) operator, multiple attribute decision making

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Keywords:

CYB-E-2013-08-0791

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IEEE Transactions on Cybernetics

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Some power generalized aggregation operators based on the interval neutrosophic numbers and their application to decision making Peide Liu 

TA ( x), FA ( x)  [0,1] and 0  TA ( x )  FA ( x)  1 . Further, Atanassov and Gargov [4], Atanassov [5] proposed the interval-valued intuitionistic fuzzy set (IVIFS) in which the truth-membership function and falsity-membership function were extended to interval numbers. IFSs and IVIFSs can only handle incomplete information not the indeterminate information and inconsistent information. In IFSs, the indeterminacy is 1-TA ( x)-FA ( x) by default. Further, Smarandache [6] proposed the neutrosophic set (NS) by adding an independent indeterminacy-membership on the basis of IFS, which is a generalization of intuitionistic fuzzy set. In NS, the indeterminacy is quantified explicitly and truth-membership, indeterminacy membership, and false-membership are completely independent.

Recently, NSs have attracted widely attention, and made some applications. Wang et al. [7] proposed a single valued neutrosophic set (SVNS), which is an instance of the neutrosophic set. Ye [8] proposed the correlation coefficient and weighted correlation coefficient of SVNSs, and proved that the cosine similarity degree is a special case of the correlation coefficient in SVNS. Wang et al. [9] defined interval neutrosophic sets (INSs) in which the truth-membership, indeterminacy-membership, and false-membership were extended to interval numbers, and discussed various properties of INSs. Ye [10] defined the similarity measures between INSs on the basis of the Hamming and Euclidean distances, and a multicriteria decision-making method based on the similarity degree is proposed. However, so far, there has been no research on aggregation operators for INSs. The information aggregation operators are an interesting and important research topic, which are receiving more and more attention [11-29]. Yager [24] proposed a power-average (PA) operator and a power OWA (POWA) operator, which weighting vectors depend on the input data and allow values being fused to support and reinforce each other. Xu and Yager [25] developed an uncertain power ordered weighted geometric (UPOWG) operator. Xu and Wang [26] developed 2-tuple linguistic power average (2TLPA) operator, 2-tuple linguistic weighted power average (2TLWPA) operator, and 2-tuple linguistic power-ordered-weighted average (2TLPOWA) operator. Xu [27] developed a series of PA operators for aggregating the intuitionistic fuzzy numbers. Zhou et al. [28] proposed a generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy

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Abstract—The interval neutrosophic set (INS) can be better to express the incomplete, indeterminate and inconsistent information, and the power average (PA) operator can take all the decision arguments and their relationships into account, and the generalized weighted aggregation (GWA) operator can reflect the mentality of the decision-makers. In this paper, we combined PA and GWA operators to INS, and proposed some aggregation operators, include interval neutrosophic power generalized aggregation (INPGA) operator, interval neutrosophic power generalized weighted aggregation (INPGWA) operator and interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator. Firstly, we presented some new operational laws for interval neutrosophic numbers (INNs) and studied their properties. Then we proposed INPGA, INPGWA and INPGOWA operators, and studied some properties and special cases of them. Further, we gave a decision making method based on these operators. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness, and to illustrate the influence of the parameter on decision making results in INPGWA operator.

N REAL decision making, the decision information is often incomplete, indeterminate and inconsistent information. The fuzzy set theory proposed by Zadeh [1], is a good tool to process fuzzy information, however, it only has a membership, and cannot express non-membership. Atanassov [2,3] proposed the intuitionistic fuzzy set (IFS) by adding a non-membership function, i.e., the intuitionistic fuzzy sets consider both membership (or called truth-membership) TA ( x) and non-membership (or called falsity-membership) FA ( x) and satisfy the conditions This paper is supported by the National Natural Science Foundation of China (No. 71271124), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104 and .09YJA630088), the Natural Science Foundation of Shandong Province (No.ZR2011FM036), and Graduate education innovation projects in Shandong Province (SDYY12065).

Peide Liu are with the School of Management Science and Engineering, Shandong University of Finance and Economics, Ji’nan, Shandong 250014, China (E-mail: peide.liu@gmail.com). .

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I. INTRODUCTION

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Index Terms—Interval neutrosophic set; power average; generalized weighted aggregation (GWA) operator; ordered weighted aggregation (OWA) operator; multiple attribute decision making

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power ordered weighted averaging (GIFPOWA) operator. Xu et al. [29] proposed the linguistic weighted PA operator, the LPOWA operator, the uncertain linguistic weighted PA operator, and the ULPOWA operator. Obviously, these PA operators cannot aggregate the INNs. In this paper, we will study some PA aggregation operators based on INNs, and discuss some special cases and properties of them. Further, we give a decision making method for multiple attribute decision making (MADM) problems based on these operators.

In order to do so, the remainder of this paper is shown as follows. In section 2, we briefly review some basic concepts and operational rules of INS and propose the power generalized aggregation (PGA) operator and power generalized ordered weighted aggregation (PGOWA) operator. In Section 3, we propose an interval neutrosophic power generalized aggregation (INPGA) operator, an interval neutrosophic power generalized weighted aggregation (INPGWA) operator and an interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator, and discuss some desirable properties and special cases. In Section 4, we propose a decision making method based on the INPGWA and INPGOWA operators for the multiple attribute decision making problems in which attribute values take the form of INNs. In Section 5, we give an example to illustrate the application of proposed method, and compare the developed method with the existing methods. In Section 6, we conclude the paper.

A. The Interval Neutrosophic Set Definition 1 [6]. Let X be a universe of discourse, with a generic element in X denoted by x. A neutrosophic set A in X is (1) A  {x(TA ( x ), I A ( x ), FA ( x ))|x  X } function,

and

FA

is

the

falsity-membership function. TA ( x), I A ( x) and FA ( x) are real

where, TA is the truth-membership function, I A is the indeterminacy-membership function, and FA is the falsity-membership function. For each point x in X, we TA ( x), I A ( x), FA ( x)  [0,1] , have and 0  sup(TA ( x))  sup( I A ( x))  sup( FA ( x))  3 .

For convenience, we can use x  ([T L , T U ],[ I L , I U ],[ F L , F U ]) to represent an element in INS, and can call an interval neutrosophic number (INN).

In the following, we will discuss the distance and similarity degree between two INNs. Definition 4.

Let x  ([T1L , T1U ],[I1L , I1U ],[F1L , F1U ]) and y  ([T2L , T2U ],[ I 2L , I 2U ],[ F2L , F2U ]) be

two INNs, then the normalized Hamming distance between x and y is defined as follows: 1 d ( x, y )  T1L  T2L  T1U  T2U  I1L  I 2L 6 (4)

The neutrosophic set was presented from philosophical point of view. Obviously, it was difficult to apply in the real applications. Wang [7] further proposed the single valued neutrosophic set (SVNS) from scientific or engineering point of view, which is a generalization of classical set, fuzzy set, intuitionistic fuzzy set and paraconsistent sets etc., and it was defined as follows.

Definition 2 [7]. Let X be a universe of discourse, with a generic element in X denoted by x. A single valued neutrosophic set A in X is (2) A  {x(TA ( x ), I A ( x ), FA ( x ))|x  X } where, TA is the truth-membership function, I A is the indeterminacy-membership function, and FA is the falsity-membership function. For each point x in X, we



 I1U  I 2U  F1L  F2L  F1U  F2U

Definition



x  ([T , T ],[I , I ],[F1L , F1U ])

Let

5.

L 1

U 1

L 1

and

U 1

y  ([T , T ],[ I , I ],[ F , F ]) be two INNs, then the cosine of L 2

U 2

L 2

U 2

L 2

U 2

included angle between x and y is

T1LT2L T1UT2U  I1L I2L  I1U I2U  F1L F2L  F1U F2U cos(x, y)  (5) xy

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standard or nonstandard subsets of  0  ,1  .

There is no restriction on the sum of TA ( x), I A ( x) and FA ( x) , so 0   TA ( x)  I A ( x )  FA ( x )  3 .

,

and 0  TA ( x)  I A ( x)  FA ( x)  3 .

In the real applications, sometimes, it is not easy to express the truth-membership, indeterminacy-membership and falsity-membership by crisp numbers, and they can be expressed by interval numbers. Wang et al. [9] further defined interval neutrosophic sets (INSs) shown as follows.

Definition 3 [7]. Let X be a universe of discourse, with a generic element in X denoted by x. A interval neutrosophic set A in X is (3) A  {x(TA ( x ), I A ( x ), FA ( x ))|x  X }

On

where, TA is the truth-membership function, I A is the indeterminacy-membership

TA ( x), I A ( x), FA ( x)  [0,1]

have

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II. PRELIMINARIES

2

x  T1L   T1U    I1L    I1U    F1L    F1U  2

where,

2

2

2

2

y  T2L   T2U    I2L    I2U    F2L    F2U  2

2

2

2

2

2

,

2

Obviously, 0  cos( x, y )  1 . When y is the ideal solution I=([1,1],[0,0],[0,0]), the bigger the cos( x, I ) between x and I is, the more consistent the direction between x and I is. In this condition, T1L  T1U (6) cos* ( x, I )  2 2 2 2 2 2 2 T1L   T1U    I1L    I1U    F1L    F1U 





Definition

Let

6.

x  ([T1L , T1U ],[ I1L , I1U ],[ F1L , F1U ]) and

y  ([T , T ],[ I , I ],[ F2L , F2U ]) L 2

U 2

L 2

U 2

be

cos ( x , I )  cos ( y , I ) then x  y . *

*

two

INNs,

if

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Definition

[(I1L I 2L ) ,( I1U I 2U ) ],[( F1L F2L ) ,( F1U F2U ) ]

x  ([T1L , T1U ],[ I1L , I1U ],[ F1L , F1U ]) and

Let

7.

y  ([T2L , T2U ],[ I 2L , I 2U ],[ F2L , F2U ]) be two INNs, then the

operational laws are defined as follows.

(1) The complement of x is x  ([ F1L , F1U ],[1  I1U ,1  I1L ],[T1L , T1U ]) (2)

(3)



 I1L I 2L , I1U I 2U  ,  F1L F2L , F1U F2U 



U U 1 2

L L 1 2

U 1

U 2

L 2



L 1

L 2

U 1

U 2

nx  1  (1  T ) ,1  (1  T )  , L n 1

U 1

U 2

U U 1 2



U n 1





rR

x  ([T1L , T1U ],[ I1L , I1U ],[ F1L , F1U ])

Let

y  ([T , T ],[ I , I ],[ F , F ]) L 2

U 2

L 2

U 2

L 2

be

U 2

L 1 1

two

(3)  ( x  y )   x   y

U 2 1

L 2 1



U 2 1

L 2 1

U 1 1

(1 T )  , 1 (1 T ) (1 T ) 



 1 (1 T1L )1 2  , 1 (1 T1U )1 2  

(9)

(I1L )1 2 ,(I1U )1 2  , (F1L )1 2 ,(F1U )1 2   (1 2 )x



U 2 1

(I1L )1 (I1L )2 ,(I1U )1 (I1U )2  , (F1L )1 (F1L )2 ,(F1U )1 (F1U )2 





(5) For the left of equation (5)

 

(11)

1  (1  F1L ) ,1  (1  F1U )   (T2L ) , (T2U )  ,

and

1  (1  I 2L ) ,1  (1  I 2U )  , 1  (1  F2L ) ,1  (1  F2U ) 



 [(T1L ) (T2L ) ,(T1U ) (T2U ) ],

INNs,

[(1  (1  I1L ) )  (1  (1  I 2L ) )  (1  (1  I1L ) )(1  (1  I 2L ) ),

(12) (13)

[(1  (1  I1L ) )  (1  (1  I 2L ) )  (1  (1  I1L ) )(1  (1  I 2L ) )],

ev

and  ,1 ,2  0 , then we have (1) x  y  y  x (2) x  y  y  x

L 2 1

 1 (1 T ) ,1 (1 T )  , (I ) ,(I )  , (F ) ,(F ) 



1  (1  F1L ) n ,1  (1  F1U ) n  n  0

1.

U 2 1

x  y  (T1L ) , (T1U )  , 1  (1  I1L ) ,1  (1  I1U )  ,

x n   (T1L ) n , (T1U ) n  , 1  (1  I1L ) n ,1  (1  I1U ) n  ,

Theorem

   1 (1 T )

L 2 1

(10)



( I1L )n , ( I1U )n  , ( F1L ) n , ( F1U ) n  n  0

Fo

(5)

L 2

 F  F  F F , F  F  F F  L 1

(4)

L 1



(8)



x  y  [T T , T T ],  I  I  I I , I  I  I I  , L L 1 2

So, (3) is right.

(4) For equation (4) 1 x 2 x  ,1 (1 T1U )1  , (I1L )1 ,(I1U )1  , (F1L )1 ,(F1U )1 

(7)

x  y  T1L  T2L  T1LT2L , T1U  T2U  T1U T2U  ,

3

[(1  (1  F1L ) )  (1  (1  F2L ) )  (1  (1  F1L ) )(1  (1  F2L ) ),

(14)

(15)

[(1  (1  I1L ) )  (1  (1  F2L ) )  (1  (1  F1L ) )(1  (1  F2L ) )]

(16)

 [(T1LT2L ) ,(T1U T2U ) ], 1  ((1  I1L )(1  I 2L )) ,

(6) x  x  x (17) Proof.

(1) and (2) are obvious, the proofs are omitted here.

(3) For the left of equation (3)  ( x  y)   T1L  T2L  T1LT2L , T1U  T2U  T1U T2U  ,

1  ((1  I1U )(1  I 2U ))   1  ((1  F1L )(1  F2L )) ,

1  ((1  F1U )(1  F2U ))  and for the right of equation (5) (x  y)

 I1L I 2L , I1U I 2U  ,  F1L F2L , F1U F2U 

 [T1LT2L ,T1UT2U ], I1L  I2L  I1L I2L , I1U  I2U  I1U I2U  ,

(5)

x  y  ( x  y ) 1

2

1 2







 F  F  F F , F  F   (T T ) ,(T T )  ,



L 2

L L 1 2

U 1

U 2

U U 1 2

( I1L I 2L ) ,( I1U I 2U )  , ( F1L F2L ) ,( F1U F2U ) 



L 1



L 2

L L  1 2

L L 1 2

U 1

U 2

ly

 1  (1  (T  T  T T )) ,1  (1  (T  T  T T ))  , L 1



On







iew

(4) 1 x  2 x  (1  2 ) x

 F1U F2U 





U U  1 2

 1  ((1  T1L )(1  T2L )) ,1  ((1  T1U )(1  T2U ))  ,

1 (1 (I1L  I2L  I1L I2L )) ,1 (1 (I1U  I2U  I1U I2U ))  ,

( I1L I 2L )n ,( I1U I 2U )  , ( F1L F2L )n ,( F1U F2U ) 

1 (1 (F1L  F2L  F1L F2L )) ,1 (1 (F1U  F2U  F1U F2U )) 





and for the right of equation (3)



 [(T1LT2L ) ,(T1UT2U ) ], 1 ((1 I1L )(1 I2L )) ,1 ((1 I1U )(1 I2U ))  ,

 x  y  1  (1  T1L ) ,1  (1  T1U )  , ( I1L ) ,( I1U )  ,

 

1 ((1 F1L )(1 F2L )) ,1 ((1 F1U )(1 F2U ))  So, (5) is right.

(6) For equation (6)

( F1L ) ,( F1U )   1  (1  T2L ) ,1  (1  T2U )  , ( I 2L ) ,( I 2U )  , ( F2L ) ,( F2U ) 









 [(1  (1  T ) )  (1  (1  T ) )  (1  (1  T1L ) )(1  (1  T2L ) ),

x1  x2   (T1L )1 , (T1U )1  , 1  (1  I1L )1 ,1  (1  I1U )1  ,

[(1  (1  T1L ) )  (1  (1  T2L ) )  (1  (1  T1L ) )(1  (1  T2L ) )],

1  (1  F1L )1 ,1  (1  F1U )1    (T1L )2 , (T1U )2  ,

L  1

L  2

[( I1L ) ( I 2L ) ,( I1U ) ( I 2U ) ],[( F1L ) ( F2L ) ,( F1U ) ( F2U ) ]



 1  ((1  T1L )(1  T2L )) ,1  ((1  T1U )(1  T2U ))  ,

 

L 2 1



1  (1  I ) ,1  (1  I )  , 1  (1  F1L )2 ,1  (1  F1U )2  



L 1 1

U 2 1

L 2

U 1 1

U 2 1

  (T ) (T1 ) , (T ) (T )  ,

Transactions on Cybernetics

Page 4 of 12

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1  (1  I1L )1 (1  I1L )2 ,1  (1  I1U )1 (1  I1U )2  , 1  (1  F1L )1 (1  F1L )2 ,1  (1  F1U )1 (1  F1U )2 



T ( aindex ( i ) ) denotes the support of the i th largest argument

by all the other arguments, i.e.,



n

T ( aindex (i ) )   Sup ( aindex ( i ) , aindex ( j ) )

  (T1L )1 2 , (T1U )1 2  , 1  (1  I1L )1 2 ,1  (1  I1U )1 2  , 1  (1  F1L )1 2 ,1  (1  F1U )1 2 



where Sup ( aindex ( i ) , aindex ( j ) ) indicates the support of j th

B. GWA operator Definition 8. A GWA operator of dimension n is a mapping GWA: R n  R . Such that, 1

Fo

 n  GWA(a1 , a2 ,, an )    w j a j  (18)  j 1  where W  ( w1 , w2 ,  , wn )T is a weight vector of (a1 , a2 , , an ) , and satisfying w j  [0,1]( j  1, 2,  , n) and

n

w j 1

j

rR

n

i

i n

ev

C. The power generalized aggregation operators Definition 9 [24]. A power average (PA) operator proposed by Yager [24] can be defined as follows:

 (1  T (a ))a

where n

T (ai )   Sup (ai , a j ) j 1 j i

i

(19)

(20)

2) Sup (a, b)  Sup (b, a ) .

Further, Yager [24] also defined a power ordered weighted average (POWA) operator.

Definition 10 [24]. A POWA operator of dimension n is a mapping POWA: R n  R . Such that, n

(21)

i 1

where i  R  R  ui  g  i   g  i 1  , Ri   Vindex ( j )  TV   TV  j 1 g :[0, 1]→[0, 1] is a basic unit-interval monotonic (BUM) function which satisfies the following properties.

1) g (0)  0 ; 2) g (1)  1 ; 3) g ( x)  g ( y ) , if x  y .

n

V i 1

i

,

Vi  1  T (ai )

and

n

T (ai )   Sup (ai , a j ) , then PGA is called the power j 1 j i

generalized aggregation (PGA) operator. In addition,  is a parameter such that   (, 0)  (0, ) .

Some properties of the PGA operator can be shown as follows.

(1) When    , then PGA(a1 , a2 , , an )  min(a1 , a2 ,  , an ) . n

(2) When   0 , PGA(a1 , a2 , , an )   a j j . The PGA w

j 1

operator can reduce to the power geometric (PG) operator. n

(3) When   1 , PGA(a1 , a2 , , an )   w j a j . The PGA j 1

operator can reduce to the PA operator.

(4) When    , then PGA(a1 , a2 , , an )  max(a1 , a2 , , an ) .

It’s easy to prove that the PGA operator has the following properties: (1) Theorem 2(Commutativity).

Let (a1, a2 , , an ) be any permutation of (a1 , a2 ,, an ) , then PGA( a1, a2 , , an )  PGA(a1 , a2 ,  , an ) .

(2) Theorem 3 (Idempotency) Let a j  a, j  1, 2,  , n , then PGA( a1 , a2 , , an )  a (3) Theorem 4 (Boundedness) The PGA operator lies between the max and min operators: min(a1 , a2 , , an )  PGA(a1 , a2 ,, an )  max(a1 , a2 ,, an ) .

Definition 12. A PGOWA operator of dimension n is a mapping PGA: R n  R . Such that, 1

n

i 1

wi  Vi

(24)

ly

3) Sup (a, b)  Sup ( x, y ) , if| a  b  x  y .

TV  Vindex (i ) , Vindex ( i )  1  T ( aindex ( i ) )

where

On

and Sup (a, b) is regarded as the support for a from b , which satisfies the following properties.

1) Sup (a, b)  [0,1] .

POWA(a1 , a2 , , an )   ui aindex (i )

1

 n  PGA(a1 , a2 ,, an )    w j a j   j 1 

iew

 (1  T (a )) i 1

largest argument for the i th largest argument.

In the following, we will combine the PA and POWA operators with GWA operator respectively, and propose power generalized aggregation (PGA) operator and power generalized ordered weighted aggregation (PGOWA) operator which can generalize PA and POWA operators.

Definition 11. A PGA operator of dimension n is a mapping PGA: R n  R . Such that,

 1 , then

GWA is called the generalized weighted aggregation (GWA) operator.  is a parameter such that   (, 0)  (0, ) .

i 1

(23)

j 1 j i

 x1 2

PA(a1 , a2 , , an ) 

4

(22)

 n   PGOWA(a1 , a2 , , an )     j aindex  j   j 1 

(25)

Page 5 of 12

Transactions on Cybernetics

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < where index is an indexing function such that aindex ( j ) is the jth largest of the a j ( j  1, 2  , n) .  j ( j  1, 2  , n) is a collection of weights such that R  R   j  g  j   g  j 1   TV   TV 

(26)

j

n

i 1

i 1

where, R j   Vindex ( i ) , TV  Vindex (i ) , Vindex ( i )  1  T ( aindex ( i ) )

generalized aggregation (INPGA) operator, an interval neutrosophic power generalized weighted aggregation (INPGWA) operator and an interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator.

Definition 13.

Let x j  ([T jL , T jU ],[ I Lj , I Uj ],[ FjL , FjU ]) ( j  1, 2, n) be a collection of the INNs, and INPGA : n   , if

g :[0, 1]  [0, 1] is a basic unit-interval monotonic (BUM) function which satisfies the following properties.

1) g (0)  0 ; 2) g (1)  1 ; 3) g ( x)  g ( y ) , if x  y .

 n    (1  T ( x j )) x j j 1 INPGA( x1 , x2 ,  , xn )   n    (1  T ( x j ))  j 1  is the set of Where

T ( aindex ( i ) ) indicates the support of the i th largest

argument by all the other arguments, i.e., n

T (aindex (i ) )   Sup (aindex ( i ) , aindex ( k ) )

(27)

Fo

k 1 k i

5

n

and T ( x j )   Sup ( x j , xi ) ,

 is

a

1

     

(28)

all

INNs,

parameter

such

i 1 i j

that   (0, ) (because we cannot define x  when   0 ),

PGOWA is called the power generalized ordered weighted aggregation (PGOWA) operator.

Some properties of the PGOWA operator can be shown as follows.

(1) When    , then PGOWA(a1 , a2 ,, an )  min(a1 , a2 ,  , an ) .

then INPGA is called the interval neutrosophic power generalized aggregation (INPGA) operator.

Theorem 8.

Let L U L U L U x j  ([T j , T j ],[ I j , I j ],[ Fj , Fj ]) ( j  1, 2 , n) be a

ev

rR

 is a parameter such that   (, 0)  (0, ) , then

n

j (2) When   0 , PGOWA( a1 , a2 ,, an )   aindex  j .



j 1

n

(3) When   1 , PGOWA(a1 , a2 ,, an )    j aindex  j  . j 1

,

(2) Theorem 6 (Idempotency) Let a j  a, j  1, 2,  , n , then PGOWA(a1 , a2 , , an )  a (3) Theorem 7 (Boundedness) The PGOWA operator lies between the max and min operators: min(a1 , a2 ,, an )  PGOWA(a1 , a2 ,, an )  max(a1 , a2 ,, an )

III. THE INTERVAL NEUTROSOPHIC POWER GENERALIZED AGGREGATION OPERATORS

The prominent characteristic of PGA and PGOWA operators is that they can take all the decision arguments and their relationships into account. In the following, we will extend PGA and PGOWA operators to the interval neutrosophic numbers (INNs), and propose an interval neutrosophic power

(1T ( x j )) 1  (1T ( x j )) 1         n n   n   n (1T ( x j ))  (1T ( x j ))    1 1 (1 (1 F L ) ) j1  U  j1    ,1 1 (1 (1 Fj ) ) j        j 1   j 1            

ly

Similiarly, the PGOWA operator also has the following properties: (1) Theorem 5(Commutativity).

Let (a1, a2 , , an ) be any permutation of (a1 , a2 ,, an ) , then PGOWA(a1, a2 , , an )  PGOWA(a1 , a2 , , an ) .

(1T ( x j )) 1  (1T ( x j )) 1        n n   n   n (1T ( x j ))  (1T ( x j ))    1 1 (1 (1 I L ) ) j1 , U  j1    ,1 1 (1 (1 I j ) ) j      j 1   j 1            

On

The PGOWA operator can reduce to the POWA operator.

(4) When    then PGOWA(a1 , a2 , , an )  max(a1 , a2 , , an ) .

1 1 (1T ( x j )) (1T ( x j ))       n n       n n (1T ( x j )) (1T ( x j ))   , 1   (1  (T U ) ) j1  ,   1   (1  (TjL ) ) j1 j    j 1     j 1           

iew

The PGOWA operator can reduce to the power ordered weighted geometric (POWG) operator.

collection of the INNs, then, the result aggregated from Definition 13 is still an INN, and even INPGA( x1 , x2 ,, xn )

Proof.

According to the operational rules of INNs, we have



x j  (T jL )  , (T jU )  , 1  (1  I Lj ) ,1  (1  I Uj )  , 1  (1  F jL ) ,1  (1  FjU )   (1  T ( x j )) x j n



(1  T ( x j )) n

 (1  T ( x ))  (1  T ( x )) j 1

j

j 1



,

x j

j

(1 T ( x j )) (1 T ( x j ))   n n  T x (1  ( )) (1 T ( x j ))  j   ,   1  (1  (T jL )  ) j 1 ,1  (1  (T jU )  ) j 1      

Transactions on Cybernetics

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < INPGA( x1, x2 ,  , xn )  INPGA( x1 , x2 ,  , xn ) Proof. According to definition 13, we have

(1 T ( x j )) (1 T ( x j ))   n n  (1 T ( x j )) (1 T ( x j ))      L U , , (1  (1  I j ) ) j 1  (1  (1  I j ) ) j1      

 n    (1  T ( x j )) x j j 1 INPGA( x1 , x2 , , xn )   n    (1  T ( x j ))  j 1

(1 T ( x j )) (1 T ( x j ))   n n  (1 T ( x j )) (1 T ( x j ))      , (1  (1  F jU )  ) j 1  (1  (1  F jL )  ) j 1       and n

 j 1

j

(1  T ( x j )) x j 

n

 j 1

j

(1  T ( x j ))

n



j 1

 j (1  T ( x j )) n

 j 1

j

(1  T ( x j ))

rR

Fo

 j (1T ( x j ))

ev

  n n  n  j (1T ( x j ))  j (1T ( x j ))  n   ,  (1  (1  I Lj ) ) j1 ,  (1  (1  I Uj ) ) j 1  j 1  j 1    

 n     (1  T ( xj )) xj  j 1  INPGA( x1, x2 , , xn )   n      (1  T ( x j ))   j 1     Since ( x1 , x2 ,  , xn ) is any permutation of ( x1 , x2 ,  , xn ) , we have n

n

 (1  T ( x ))   (1  T ( x )) , j

j 1

j

j 1

n

n

 (1  T ( x )) x   (1  T ( x )) x j

j 1

j

j 1

j

j

thus INPGA( x1, x2 , , xn )  INPGA( x1 , x2 ,  , xn ) .

(2) Theorem 10 (Idempotency) Let x j  x, j  1, 2,  , n , then INPGA( x1 , x2 ,  , xn )  x

(31)

Proof. Since x j  x , for all j , we have

iew

 j (1 T ( x j ))  j (1T ( x j ))   n n  n  (1  T ( x ))  j (1 T ( x j ))   n j j    ,  (1  (1  FjU ) ) j 1  (1  (1  FjL ) ) j1  j 1  j 1     then 1

1

     

1

x j

(1T ( xj )) 1  (1T ( x j )) 1         n n   n    (1T ( xj )) (1T ( x j ))  n   1 1 (1 (1 FL ) ) j1  U  j1        ,1 1 (1 (1 F ) ) j j         j 1   j 1            

which completes the proof.

The INPGA operator satisfies the following properties: (1) Theorem 9 (Commutativity).

Let ( x1, x2 ,  , xn ) be any permutation of ( x1 , x2 ,  , xn ) , then

1

1

     

1

 n   n      (1  T ( x j )) x   x  (1  T ( x j ))  j 1    j 1    x  1   x  n n       (1  T ( x j ))    (1  T ( x j ))   j 1   j 1  (3) Theorem 11 (Boundedness) The INPGA operator lies between the max and min operators: min( x1 , x2 ,, xn )  INPGA( x1 , x2 ,, xn )  max( x1 , x2 ,, xn ) (32)

ly

(1T ( x j )) 1  (1T ( xj )) 1        n n   n    (1T ( x j )) (1T ( xj ))  n   1 1 (1 (1 I L ) ) j1 , U  j1        ,1 1 (1 (1 I ) ) j j         j 1 j 1            

 n    (1  T ( x j )) x j j 1 INPGA( x1 , x2 , , xn )   n    (1  T ( x j ))  j 1

On

 n   (1 T(xj ))xj  j 1    n   (1T(xj ))   j 1  (1T ( x j )) 1  (1T ( xj )) 1        n n      (1T ( x j )) (1T ( xj ))  n n     , 1 (1 (TjU ) ) j1  ,   1 (1 (TjL ) ) j1    j 1     j 1           

(30)

and

 j (1  T ( x j ))  j (1  T ( x j ))   n n  T x   j (1 T ( x j ))  (1  ( )) n n j j   ,   1   (1  (T jL )  ) j 1 ,1   (1  (T jU )  ) j 1   j 1 j 1    

 j (1T ( x j ))

6

Proof. Let a  min( x1 , x2 , , xn ) , b  max( x1 , x2 , , xn ) .

Since a  a  b , the j

1

1

1

 n  n  n        (1 T ( x j ))a   (1 T ( x j )) x j   (1 T ( x j ))b  j 1 j 1 j 1        n   n   n   (1 T ( x j ))   (1 T ( x j ))   (1 T ( x j ))   j 1   j 1   j 1  That is

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 n     (1  T ( x j )) x j  j 1  b a n    (1 ( )) T x j     j 1  i.e. min( x1 , x2 , , xn )  INPGA( x1 , x2 , , xn )  max( x1 , x2 , , xn ) .

In the following, we will discuss some cases of the INPGA operator.

(1) When   0 ,

INPGA( x1 , x2 ,  , xn )

Definition 14.

Let x j  ([T jL , T jU ],[ I Lj , I Uj ],[ FjL , FjU ]) ( j  1, 2, n) be a

collection of the INNs, and INPGWA : n   , if 1

 n      j (1  T ( x j )) x j  j 1  INPGWA( x1 , x2 ,, xn )   n      j (1  T ( x j ))   j 1  Where  is the set of all INNs,

  1 , 2 ,  , n 

T

is

the

weight

vector

n

n

j 1

i 1 i j

(35)

and of

x j ( j  1, 2,  , n) ,  j  [0,1],   j  1 . T ( x j )   Sup ( x j , xi ) ,

(1 T ( x j )) (1 T ( x j ))   n n  n (1 T ( x j )) (1 T ( x j ))  n   , ,  (T jU ) j 1     (T jL ) j 1   j 1  j 1    

 is a parameter such that   (0, ) , then INPGWA is called

Fo

ev

rR

(1 T ( x j )) (1 T ( x j ))   n n   (1 T ( x j )) (1 T ( x j ))  n n    , (33) ,1   (1  I Uj ) j 1 1   (1  I Lj ) j 1   j 1 j 1     (1 T ( x j )) (1 T ( x j ))   n n  (1 T ( x j )) (1 T ( x j ))   n n   1   (1  F jL ) j 1  ,1   (1  F jU ) j 1   j 1 j 1    

neutrosophic power generalized weighted (INPGWA) operator.

Specially, T

1 1 1 if    , ,  ,  , INPGWA operator should be n n n INPGA operator.

Theorem 12.

Let L U L U L U x j  ([T j , T j ],[ I j , I j ],[ Fj , Fj ]) ( j  1, 2, n) be a

collection of the INNs, then, the result aggregated from Definition 14 is still an INN, and even INPGWA(x1, x2 ,, xn ) 1 1   n n     L  j U  j    1 (1 (Tj ) )  , 1 (1 (Tj ) )   ,   j 1   j 1    1 1 n n       (36) 1 1 (1 (1 I Lj ) )j  ,1 1 (1 (1 I Uj ) )j   ,   j 1   j 1  

1 1 n n       1 1 (1 (1 FjL ) )j  ,1 1 (1 (1 FjU ) )j      j 1   j 1     (1  T ( x )) j j where   j n   (1  T ( x )) j j j 1 The proof of this theorem is similar with theorem 8, it’s omitted here.

In the following, we will discuss some properties of the INPGWA operator. Similar to Theorems 10 and 11, it can be easily proved that the INPGWA operator has the following properties (1) Theorem 13(Idempotency) Let x j  x, j  1, 2,  , n , then

(34)

(1T ( x j )) (1T ( x j ))   n n  n T x (1  ( )) (1T ( x j ))   n j    , ( FjU ) j1 ( FjL ) j1  j 1  j 1     The INPGA operator can reduce to the interval neutrosophic power aggregation (INPA) operator.

In definition 13, we assumed that all of the objects ( x1 , x2 , , xn ) being aggregated were of equal importance. However, in many real cases, the importance degrees are not equal; thus, we can assign different weights for different objects, and further define a new aggregation operator to process this case.

ly

On

(1T ( x j )) (1T ( x j ))      n n   n      (1 T ( x )) (1 T ( x )) n j j    ,1   (1  TjU ) j1  ,   1   (1  TjL ) j1    j 1   j 1            

the interval aggregation

iew

The INPGA operator can reduce to the interval neutrosophic power geometric (INPG) operator.

(2) When   1 , INPGA( x1 , x2 ,, xn )

(1T ( x j )) (1T ( x j ))   n n  n  (1 T ( x )) (1T ( x j ))  n j   , , (I Uj ) j1 ( I Lj ) j1  j 1  j 1    

7

INPGWA( x1 , x2 , , xn )  x (37) (3) Theorem 14 (Boundedness) The INPGWA operator lies between the max and min operators: min( x1 , x2 ,, xn )  INPGWA( x1 , x2 ,, xn )  max( x1 , x2 ,, xn ) (38) Note that The INPGWA operator has not the commutativity.

Transactions on Cybernetics

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < In the following, we will discuss some cases of the INPGWA operator.

(1) When   0 , n  n    INPGWA( x1 , x2 ,  , xn )    (T jL ) j ,  (T jU ) j  ,  j 1 j 1   n n     1   (1  I Lj ) j ,1   (1  I Uj ) j  , j 1 j 1  

(39)

Fo

The INPGWA operator can reduce to the interval neutrosophic power weighted geometric (INPWG) operator.

(2) When   1 , INPGWA( x1 , x2 , , xn )

  n U j  ,1    (1  T j )  j 1  

  ,  

(40)

ev

rR

n   n L j n U j   n L j U j   ( I j ) ,  ( I j )  ,  ( Fj ) ,  ( Fj )   j 1 j 1  j 1   j 1  The INPGWA operator can reduce to the interval neutrosophic power weighted aggregation (INPWA) operator.

Theorem 15. Letting Sup ( xi , x j )  k , for all i  j , then

1

     

1

(41)

(43)

n

where, TV  Vindex (i ) , Vindex ( i )  1  T ( xindex ( i ) ) i 1

g :[0, 1]  [0, 1] is a basic unit-interval monotonic (BUM) function which satisfies the following properties.

1) g (0)  0 ; 2) g (1)  1 ; 3) g ( x)  g ( y ) , if x  y .

T ( xindex ( i ) ) indicates the support of the i th largest argument

by all the other arguments, i.e., n

T ( xindex ( i ) )   Sup ( xindex ( i ) , xindex ( k ) )

(44)

k 1 k i

 is a parameter such that   (0, ) , then INPGOWA is

called the interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator.

Theorem 16.

Let x j  ([T jL , T jU ],[ I Lj , I Uj ],[ FjL , FjU ]) ( j  1, 2, n) be a

collection of the INNs, then, the result aggregated from Definition 15 is still an INN, and even INPGOWA(x1, x2 ,, xn )

ly 

i 1 i j

Thus

 n     j (1  T ( x j )) x j j 1  INPGWA( x1 , x2 , , xn )   n     j (1  T ( x j ))  j 1 1

1

     

      1    j (1  (n  1)k ) x j    j x j   n j 1 j 1            j x j   n   n   j 1    (1 ( 1) )  n k    j j     j j   1 1     The INPGWA operator can weight all the given INNs themselves, and the weighting vectors depend on the input arguments. However, in many decision making problems, we need to consider the weight of the argument positions, for n

j  Rj   R j 1  g  , R j   Vindex ( i ) i 1  TV   TV 

 j  g 

1 1      j    j    n    n  L U      1  1 Tindex j     , 1  1 Tindex j      ,        j 1        j 1    

n

T ( x j )   Sup ( x j , xi )  (n  1)k

1

x j ( j  1, 2  , n) .  j ( j  1, 2  , n) is a collection of weights

such that

On

 n      j x j   j 1  which indicates that when all the supports are the same, the INPGWA operator becomes an interval neutrosophic generalized weighted average (INGWA) operator.

Proof. if Sup ( xi , x j )  k , for all i  j , then

n

1

 n   INPGOWA( x1 , x2 , , xn )     j xindex (42)  j   j 1  where  is the set of all INNs, and index is an indexing function such that xindex ( j ) is the jth largest of the ILNs

iew

 n     j (1  T ( x j )) x j j 1  INPGWA( x1 , x2 , , xn )   n     j (1  T ( x j ))  j 1

example, diving games in Olympic Games. Base on the PGOWA operator, we define a new aggregation operator called an interval neutrosophic power generalized ordered weighted averaging (INPGOWA) operator.

Definition 15.

Let x j  ([T jL , T jU ],[ I Lj , I Uj ],[ FjL , FjU ]) ( j  1, 2, n) be a collection of the INNs, and INPGOWA : n   , if

n n     1   (1  F jL ) j ,1   (1  F jU ) j    j 1 j 1  

  n    1    (1  T jL ) j    j 1 

8







1 1 n n    j   j    L U     1 1 1 1 Iindex j    ,1 1 1 1 Iindex j    ,        j 1     j 1









1 1 n n    j    j    L  1 1 FU  ,1 1 1 1 1 1 Findex        index j   j          j 1   j 1   (45) The proof of this theorem is similar with theorem 8, it’s omitted here.

Theorem 17. Especially, if g ( x )  x , then the INPGOWA





operator can reduce to the INPGA operator.

Proof:





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 n   INPGOWA(x1 , x2 ,, xn )     j xindex  j   j 1  1

 n   Rj    R j 1    x  g    g      index j       TV    j 1   TV  

n

INPGOWA( x1 , x2 , , xn )    j xindex  j 



j 1

n n  j j  L U  1   (1  Findex ( j ) ) ,1   (1  Findex ( j ) )  ,  j 1    j 1

1

 n Vindex ( j )   xindex j     TV  j 1 

n  n L  j  U   ( I index ) j ,  ( I index j ( ) ( j) )  ,  j 1 j 1  

1

1

 n  n       (1  T ( x j )) x j    (1  T ( xindex ( j ) )) xindex j   j 1 j 1  =INPGA    n  n      (1  T ( x j ))    (1  T ( xindex ( j ) ))  j 1  j 1    So, INPGA operator is a special case of INPGOWA operator.

Similar to Theorem 4, we have the following result.

Theorem 18. Letting Sup ( xi , x j )  k , for all i  j , then 1

ev

rR

Fo

1 n  ILPGOWA( x1 , x2 , , xn )    x j  (46) n  j 1  which indicates that when all the supports are the same, the INPGOWA operator becomes an interval neutrosophic generalized average (INGA) operator.

Similar to Theorems 9, 10 and 11, it can be easily proved that the INPGOWA operator has the following properties: (1) Theorem 19 (Commutativity).

Let ( x1, x2 , , xn ) be any permutation of ( x1 , x2 , , xn ) , then INPGOWA( x1, x2 , , xn )  INPGOWA( x1 , x2 , , xn ) (2) Theorem 20 (Idempotency) Let x j  x, j  1, 2,  , n , then

IV. A MULTIPLE ATTRIBUTE DECISION MAKING METHOD BASED ON INPGWA OPERATOR In this section, we will use INPGWA or INPGOWA operators to the multiple attribute decision making problems in which attribute values take the form of INNs, and propose a decision making method.

For a multiple attribute decision making problem, let A   A1 , A2 ,, Am  be the collection of alternatives,

 j j  L U    (Tindex ( j ) ) ,  (Tindex ( j ) )   j 1 j 1   n n  j j  L U 1   (1  I index ( j ) ) ,1   (1  I index ( j ) )  , j 1  j 1 

that

xij  ([TijL , TijU ],[ I ijL , I ijU ],[ FijL , FijU ]) is

the

evaluation

information of the alternative Ai on the criteria C j which is

represented by the form of INNs. L U L U L U U U U where Tij , Tij , I ij , I ij , Fij , Fij  [0,1] and Tij  I ij  Fij  3 .

Then we can rank the order of the alternatives.

If the attribute weight is known, we can use the INPGWA operator to aggregate all attribute values, otherwise, we can use INPGA operator or INPGOWA operator aggregate them. When the attribute weight is known, we can suppose weight vector of attribute set C  C1 , C2 , , Cn  is w  ( w1 , w2 ,, wn ) ,

ly

j 1

n

and C  C1 , C2 , , Cn  be the collection of attributes. Suppose

On

INPGOWA( x1 , x2 ,, xn )   xindex  j  j

(48)

n   n    L U ) j ,  ( Findex ) j    ( Findex j j ( ) ( )  j 1 j 1    The INPGOWA operator reduces to the interval neutrosophic power ordered weighted arithmetic (INPOWA) operator.

Clearly, both the INPGWA and INPGOWA operators can consider the given arguments and their relationships; the difference between two operators is that the ILPGWA operator emphasizes their own importance of these arguments, while the ILPGOWA operator stresses their ordered position importance.

iew

INPGOWA( x1 , x2 ,, xn )  x (3) Theorem 21 (Boundedness) The INPGOWA operator lies between the max and min operators: min( x1 , x2 ,, xn )  INPGOWA( x1 , x2 ,, xn )  max( x1 , x2 ,, xn ) Similarly, some special cases of the INPGOWA operators are shown as follows.

(1) When   0 , n

9

n

(47)

n n  j j  L U 1   (1  Findex ( j ) ) ,1   (1  Findex ( j ) )    j 1  j 1  The INPGOWA operator reduces to the interval neutrosophic power ordered weighted geometric (INPOWG) operator.

(2) When   1 ,

n

and w j  [0,1],  w j  1 . So, we can only introduce the j 1

decision making method based on the INPGWA operator.

Then the decision steps are shown as follows.

Step 1. Consider that types of attribute have benefit type and cost type, and then we firstly convert cost type to benefit type by the complement function in equation (7).

If C j is a cost type, then we can replace xij with xij for all i .

Step 2. Calculate the supports.

Sup( xij , xil )  1  d ( xij , xil ), i  1, 2, , m; j , l  1, 2, , n

(49)

which satisfies the support conditions defined in definition 9,where d ( xij , xil ) is Hamming distance between two interval

Transactions on Cybernetics

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < neutrosophic numbers xij and xil , which is defined by formula (4).

Step 3. Calculate T ( xij ) n

l 1 l j

Step 4. Calculate the weight  ij associated with the interval neutrosophic number xij w j (1  T ( xij )) n

 w (1  T ( x j 1

j

ij

))

i  1, 2, , m; j  1, 2,  , n (51)

Fo

Step 5. Calculate the comprehensive evaluation value of each alternative xi  INPGWA( xi1 , xi 2 ,, xin )

rR

1 1   n n         1   (1  (TijL ) ) ij  , 1   (1  (TijU ) ) ij   ,   j 1 j 1     

1 1 n n       L   ij U   ij   1  1   (1  (1  Iij ) )  ,1  1   (1  (1  Iij ) )   ,   j 1 j 1    

Step 6: Calculate the cos* ( xi , I ) , where I=([1,1],[0,0],[0,0]) by equation (6).

Step 7: Rank all the alternatives  A1 , A2 , , Am  and select the

TABLE 1 THE EVALUATION VALUES OF FOUR POSSIBLE ALTERNATIVES WITH RESPECT TO THE THREE CRITERIA

([0.4,0.6],[0.1,0.3], [0.2,0.4]) ([0.6,0.7],[0.1,0.2], [0.2,0.3])

We adopt the proposed method to rank the alternatives.

To get the best alternative(s), the following steps are involved: (1) Convert the cost criterion to benefit criterion. Since C3 is a cost criterion, we can replace xi 3 (i  1, 2, 3, 4) with xi 3 (i  1, 2, 3, 4) , and get the decision matrix X:

([0.4,0.5],[0.2,0.3],[0.3,0.4]) ([0.6,0.7],[0.1,0.2],[0.2,0.3]) X  ([0.3,0.6],[0.2,0.3],[0.3,0.4])  ([0.7,0.8],[0.0,0.1],[0.1,0.2]) ([0.4,0.5],[0.7,0.8],[0.7,0.9]) ([0.8,0.9],[0.5,0.7],[0.3,0.6])  ([0.7,0.9],[0.6,0.8],[0.4,0.5])  ([0.8,0.9],[0.6,0.7],[0.6,0.7])

([0.4,0.6],[0.1,0.3],[0.2,0.4]) ([0.6,0.7],[0.1,0.2],[0.2,0.3]) ([0.5,0.6],[0.2,0.3],[0.3,0.4]) ([0.6,0.7],[0.1,0.2],[0.1,0.3])

(2)Calculate the supports Sup( xij , xil ) j, l  1, 2,3.i  1, 2,3, 4 by formula (49) (for simplicity, we denote Sup( xij , xil ) with Sij ,il  ). We can get

S11,12 =S12,11 =0.950 , S11,13 =S13,11 =0.683 , S12,13 =S13,12 =0.633

S 21,22 =S22,21 =1.000 , S21,23 =S23,21 =0.717 , S 22,23 =S23,22 =0.717 S31,32 =S32,31 =0.967 , S31,33 =S33,31 =0.700 , S32,33 =S33,32 =0.733

S 41,42 =S42,41 =0.917 , S41,43 =S 43,41 =0.600 , S 42,43 =S43,42 =0.617

(3) Calculate T ( xij )(i,  1, 2,3, 4. j  1, 2,3) by formula (50)

C3 ([0.7,0.9],[0.2,0.3], [0.4,0.5]) ([0.3,0.6],[0.3,0.5], [0.8,0.9])

T ( x21 )  1.717,T ( x22 )  1.717,T ( x23 )  1.433

T ( x31 )  1.667,T ( x32 )  1.700,T ( x33 )  1.433

ly

In order to demonstrate the application of the proposed method, we will cite an example about the investment selection of a company (adapted from [10]). There is a company, which wants to invest a sum of money to an industry. There are 4 alternatives which can be considered by a panel, including: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is an arms company. The evaluation on the alternatives is based on three criteria: (1) C1 is the risk; (2) C2 is the growth; (3) C3 is the environmental impact. where C1 and C2 are benefit criteria, and C3 is a cost criterion. The weight vector of the criteria is given by W =(0.35, 0.25, 0.4).

The final decision information can be obtained by the INNs, and shown in table 1.

([0.4,0.5],[0.2,0.3], [0.3,0.4]) ([0.6,0.7],[0.1,0.2] A2 ,[0.2,0.3])

([0.4,0.5],[0.2,0.4], [0.7,0.9]) ([0.6,0.7],[0.3,0.4], [0.8,0.9])

T ( x11 )  1.633,T ( x12 )  1.583,T ( x13 )  1.317

V. AN APPLICATION EXAMPLE

C2

([0.5,0.6],[0.2,0.3], [0.3,0.4]) ([0.6,0.7],[0.1,0.2], [0.1,0.3])

On

best one(s) by the cosine similarity degree in definition 6.

Step 8: End.

A1

([0.3,0.6],[0.2,0.3], [0.3,0.4]) ([0.7,0.8],[0.0,0.1], A4 [0.1,0.2])

C3

iew

ev

1 1 n n       1  1   (1  (1  FijL ) ) ij  ,1  1   (1  (1  FijU ) ) ij      j 1 j 1      (52) where i  1, 2,  , m .

C1

C2

A3

T ( xij )   Sup ( xij , xil )i  1, 2,  , m; j  1, 2,  , n (50)

 ij 

C1

10

T ( x41 )  1.517,T ( x42 )  1.533,T ( x43 )  1.217

(4) Calculate the weights  ij (i, j  1, 2, 3, 4) by formula (51)

 11  0.370, 12  0.259, 13  0.372  21  0.365, 22  0.261, 23  0.374  31  0.362, 32  0.261, 33  0.377  41  0.367, 42  0.264, 43  0.369 (5) Calculate the comprehensive evaluation value of each alternative by formula (52), suppose   1 x1  ([0.400,0.528],[0.266,0.432],[0.370,0.541]) x2  ([0.691,0.801],[0.278,0.446],[0.239,0.432]) x3  ([0.534,0.763],[0.384,0.564],[0.340,0.440]) x4  ([0.721,0.828],[0.307,0.419],[0.333,0.462]) (6) Calculate the cos* ( xi , I ) by formula (6), we can get

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> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < cos* ( x1 , I )  0.619, cos* ( x2 , I )  0.824, cos ( x3 , I )=0.716, cos ( x4 , I )  0.817 (7) Rank the alternatives According to the cos* ( xi , I ) *

*

,

the

ranking

is A2  A4  A3  A1 .

In order to illustrate the influence of the parameter  on decision making results of this example, we use the different value  in INPGWA operator in step 5 to rank the alternatives.

The ranking results are shown in Table 2.

TABLE 2 ORDERING OF THE ALTERNATIVES BY UTILIZING THE DIFFERENT  IN INPGWA OPERATOR



cos value cos* ( xi , I )(i  1, 2,3, 4)

  0.1

  0.8

  2.0

  5.2

  6.5

  10.2   10.3

  16.9

  17.0

  30.0

cos* ( x3 , I )  0.718, cos* ( x4 , I )  0.832 cos* ( x1 , I )  0.613, cos* ( x2 , I )  0.828 cos* ( x3 , I )  0.718, cos* ( x4 , I )  0.826 cos* ( x1 , I )  0.619, cos* ( x2 , I )  0.824

A4  A2  A3  A1

A2  A4  A3  A1

A2  A4  A3  A1

cos* ( x3 , I )  0.716, cos* ( x4 , I )  0.817 cos* ( x1 , I )  0.645, cos* ( x2 , I )  0.804 cos* ( x3 , I )  0.708, cos* ( x4 , I )  0.776

A2  A4  A3  A1

cos* ( x1 , I )  0.691, cos* ( x2 , I )  0.764 cos* ( x3 , I )  0.693, cos* ( x4 , I )  0.716 cos* ( x1 , I )  0.692, cos* ( x2 , I )  0.763 cos* ( x3 , I )  0.693, cos* ( x4 , I )  0.714 cos* ( x1 , I )  0.6934, cos* ( x2 , I )  0.762 cos* ( x3 , I )  0.6927, cos* ( x4 , I )  0.713 cos* ( x1 , I )  0.7031, cos* ( x2 , I )  0.755 cos* ( x3 , I )  0.691, cos* ( x4 , I )  0.7034 cos* ( x1 , I )  0.704, cos* ( x2 , I )  0.755 cos* ( x3 , I )  0.691, cos* ( x4 , I )  0.703 cos* ( x1 , I )  0.722, cos* ( x2 , I )  0.745 cos* ( x3 , I )  0.6885, cos* ( x4 , I )  0.6886 cos* ( x1 , I )  0.723, cos* ( x2 , I )  0.745 cos* ( x3 , I )  0.6884, cos* ( x4 , I )  0.6883 cos* ( x1 , I )  0.7407, cos* ( x2 , I )  0.7408 cos* ( x3 , I )  0.6877, cos* ( x4 , I )  0.6804 cos* ( x1 , I )  0.7409, cos* ( x2 , I )  0.7408 cos* ( x3 , I )  0.6877, cos* ( x4 , I )  0.6804 cos* ( x1 , I )  0.7570, cos* ( x2 , I )  0.7396 cos* ( x3 , I )  0.6873, cos* ( x4 , I )  0.6777

A2  A4  A3  A1 A2  A4  A3  A1 A2  A4  A1  A3

A2  A4  A1  A3 A2  A1  A4  A3 A2  A1  A4  A3 A2  A1  A3  A4 A2  A1  A3  A4

A1  A2  A3  A4

A1  A2  A3  A4

ordering of the alternatives is A2  A4  A1  A3 .

When 6.6    10.2 , the ordering of the alternatives is A2  A1  A4  A3 . When10.3    16.9 , the ordering of the alternatives is A2  A1  A3  A4 .

(3) When   17.0 , the ordering of the alternatives is A1  A2  A3  A4 and the best alternative is A1 .

So,  can be used to express the mentality of the decision-makers, the more the  is, the more optimistic decision-makers are. On the contrary, the smaller the  is, the more pessimistic decision-makers are. So, the organization or individual can properly select the desirable alternative according to his/her interest and the actual needs. Generally speaking, if they don’t have special preference, we can consider the rankings of some special values when  gets 0, 1 and 2 because these values have some special meanings. In this example, we can think the best alternative is A2 .

In order to verify the effective of the proposed method in this paper, we can compare with the method proposed by Ye [10].

Firstly, the same ranking results were produced by these methods. Secondly, the method proposed by Ye [10] was based on similarity measure, it cannot realize the information aggregation. The method proposed in this paper was based on the aggregation operators, and it can provide the more general and more flexible features as  is assigned different values.

VI. CONCLUSION

In real decision making, the decison information is often incomplete, indeterminate and inconsistent information, and the interval neutrosophic set (INS) can be better to express this kind of information. The traditional power average operators and the generalized aggregation operators are generally suitable for aggregating the crisp numbers, and yet they will fail in dealing with interval neutrosophic numbers (INNs). In this paper, we have given some aggregation operators based on INNS, include interval neutrosophic power generalized aggregation (INPGA) operator, interval neutrosophic power generalized weighted aggregation (INPGWA) operator and interval neutrosophic power generalized ordered weighted aggregation (INPGOWA) operator, and studied some properties and some special cases of them. Further, we have given a decision making method based on these operators.

Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness, and to illustrate the influence of the parameter  on decision

ly

  6.6

cos* ( x1 , I )  0.610, cos* ( x2 , I )  0.830

(2) When 0.8    16.9 , the best alternative is A2 , but the ordering of the alternatives is different with respect to different  . When 0.8    5.2 , the ordering of the alternatives is A2  A4  A3  A1 . When 5.3    6.5 , the

On

  5.3

A4  A2  A3  A1

cos* ( x3 , I )  0.723, cos* ( x4 , I )  0.866

As we can see from Table 2, the ordering of the alternatives may be different for the different value  in INPGWA operator.

(1) When   0.7 , the ordering of the alternatives is A4  A2  A3  A1 and the best alternative is A4 .

iew

  5.0

cos* ( x1 , I )  0.592, cos* ( x2 , I )  0.841

ev

  1.0

A4  A2  A3  A1

cos* ( x3 , I )  0.723, cos* ( x4 , I )  0.871

rR

  0.7

*

Fo

0

Ranking

cos ( x1 , I )  0.589, cos ( x2 , I )  0.843 *

11

Transactions on Cybernetics

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < making results in INPGWA operator. The prominent characteristic of the developed approaches is that they can take all the decision arguments and their relationships into account, and can reflect the mentality of the decision-makers, and the proposed method are more scientific to do decision making.

Finally, an illustrative example has been given to show the steps of the developed methods. In further research, it is necessary and meaningful to study some applications of these operators such as pattern recognition, medical diagnosis and inconsistent analysis, etc.

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iew

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On