Recursive Aggregation of OWA Operators for Fuzzy Number Based on Non-additive Measure with $\sigma-\delta$ Rules

(1)

Vol. 12, 2018, 79-87

ISSN: 2349-0632 (P), 2349-0640 (online) Published 20 March 2018

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/jmi.v12a8

79

Journal of

Recursive Aggregation of OWA Operators for Fuzzy

Number Based on Non-additive Measure with

σσσσ

-

λλλλ

Rules

Zengtai Gong and Huiting Li

College of Mathematics and Statistics, Northwest Normal University

Lanzhou, 730070, P.R. China, Email: [email protected]

Received 2 March 2018; accepted 19 March 2018

Abstract. In this paper, considering of the fuzziness and uncertainty of the objective things

and the connection of the weight vector, the OWA operators for fuzzy number based on a

non-additive measure with

σ

−

λ

rules and its recursive aggregation theory are proposed

and investigated by means of the

σ

−

λ

rules of a non-additive measure. In addition, the

calculation methods are present and an illustrative example is designed.

Keywords: Fuzzy measure, OWA operator, fuzzy number, recursive aggregation AMS Mathematics Subject Classification (2010): 28E10, 04A72

1. Introduction

Since Yager [13] proposed Ordered Weighted Averaging operators in 1988, the OWA operators has been widely used in many fields [2,14]. In 2005, Yager proposed the recursive forms of OWA operators [11], and after that, Jin investigated the discrete and continuous recursive forms of OWA operator [7], However, the recursive forms of OWA operators based on the classical probability measure and Lebesgue integral integration operator. We must note that when using the existing aggregation results, an indisputable fact is that the attribute index is mostly related with each other, and not mutually independent. In other words, the weight vector is not able to be measured independently, and it may also not satisfy countable additivity of the classical probability measure. In 1974, Sugeon [9] proposed the concept of fuzzy measure, which can be used to study the correlation problem of attribute index [3, 5, 6, 12, 15]. On the other hand, due to the fuzziness and uncertainty of the objective things, the evaluation values involved in the decision problems are not always expressed as crisp numbers, and some of them are more suitable to be denoted by fuzzy number. So, the fuzzy sets theory introduced by Zadeh was a very good tool to deal with vagueness and uncertainty in real decision problems[15]. The fuzzy number, as a special fuzzy sets, has been applied to many aspects [1, 10]. Based on the above consideration, in this article, the OWA operators for fuzzy number based on a

non-additive measure with

σ

−

λ

rules and its recursive aggregation theory are proposed

and investigated. In addition, the calculation methods are put forward and an illustrative example is given.

2. Preliminaries

(2)

80

A set function

µ

is called a regular fuzzy measure if

(1)

µ

(∅)=0;

(2)

µ

( X)=1;

(3) for every A and _B∈A_{such that} _A⊆_B_,

µ

₍_A₎≤

µ

₍_B_).

A regular fuzzy measure

µ

is called Sugeno measure if

µ

satisfies

σ

−

λ

rules, briefly denoted as g_λ. The fuzzy measure shown in this paper is Sugeno measure.

Definition 2. [5, 6, 12, 15] g_λ is called a fuzzy measure based on

σ

−

λ

rules if

     

≠

   

 

− +

     

∑

∏

∞

0, = ), (

0, ,

1 )] ( [1 1

=

1 =

1

=

λ

λ λ

i i

A g

A g A

g

∪

where , )

{ }

0

sup 1

(− ∞

∪

∈

µ

λ

,

{ }

_A_i ⊂A _, A_i∩A_j =∅ for all i, j=1,2,⋯ and

j i≠ .

Particularly, if

λ

=0, then g_λ is a classic probability measure. If X is a finite set, for any subset A of X , then

( )

    

≠

   

 

− +

∑

∏

∈ ∈

0. = }), ({

0, ,

1 })] ({ [1

1

=

λ

λ λ

x g

x g A

g

A x

If X is a finite set, then the parameter

λ

of a regular Sugeno measure based on

σ

−

λ

rules is determined by the equation

)). ( (1 = 1

1 =

i n

i

x g_λ

λ

+

∏

Definition 3. [12] Fuzzy set A~∈E~ is called a fuzzy number if A~ is a normal, convex fuzzy set, upper semi-continuous and supp A0 ={x∈R:A(x)>0} is compact. We use

E~ to denote the fuzzy number space .

[4] Let (a1,a2,⋯,an)∈Rn, if ai E ~

~ _∈ _{, and its membership function is}

  

   

 

+ ≤ −

+

≤ ≤ − +

−

, else 0

, <

,

= )

( ,2

,2 ,2

,1 ,1

,1

~ _i _i _i

i i i

i i

i

a a x a

x a

a x a

a x

x

u

δ

(3)

Measure with σ-λ Rules

81

then ~a_i =(a_i−

δ

_i_,1,a_i,a_i+

δ

_i_,2) is said to be triangle fuzzy number.

For any two triangle fuzzy numbers ai

~_and j

a~ , i≠ j, then

) ,

, (

= ~ ~

,1 ,1

,1

,1 j j i j i i j j

i i j

i a a a a a a a

a + −

δ

+ −

δ

+ +

δ

+ +

δ

). ,

, (

= ~

,2 ,1 i i i i

i

i ka k ka ka k

a

k⋅ −

δ

+

δ

[8] Let u~,v~∈E~ , the partial order u ~~v means that it satisfy the conditions

r

r v

u~] [~]

[ ≤ , r∈[0,1], i.e. u~r−≤v~r−,

+ + _≤

r

r u

u~ ~ , r∈[0,1].

Denoting Ai ={x1,x2,⋯,xi} , i=1,2,⋯,n , let gλ({xi})=gi , then

. = }) ({ = )

(A1 g x1 g1

g_λ _λ

Definition 4. Let g_λ be fuzzy measure satisfying

σ

−

λ

rules. Denote }

, , , {

= ₁ ₂ _i

i x x x

A ⋯ , i=1,2,⋯,n. An OWA operator of dimension n for fuzzy number based on a non-additive measure with

σ

−

λ

rules is a mapping

E E E

E

Fn n

~ ~ ~

~ : ~

2

1× ×⋅ ⋅⋅× → defined as

, ~ )) ( ) ( ( = ) ~ , , ~ , ~ ( ~

1 1

= 2

1 i i i

n

i n

n a a a g A g A b

F ⋯

∑

_λ − _λ ₋ ⋅

where b~_i is the i-th largest value out of a~=(~a₁,a~₂,⋯,a~_n) (i.e.,b~1≥b~2 ≥⋯≥b~n).

Corollary 1. When

λ

=0 and bi ~

is a crisp number, then the definition 4 degenerates to the classic OWA operator.

Definition 5. The measure of orness (andness) associated with an OWA operator Fn ~

of dimension n for fuzzy number based on a non-additive measure with

σ

−

λ

rules is defined as

), ( 1

1 = )

( ( )

1

1 = )

(

i n n

i n

A g n

g

orness _λ

∑

_λ

−

). ( 1

1 1 = ) ( 1

= )

( ( )

1

1 = )

( )

(

i n n

i n

n

A g n

g orness g

andness _λ _λ

∑

_λ

−

− − −

3. Recursive forms of the OWA operator for fuzzy number based on a non-additive measure with

σ

−

λ

rules

Theorem 1. Let g_λ be fuzzy measure satisfying

δ

−

λ

rules. Denote _A_i ₌_{_x₁_,_x₂_,⋯, }

i

x , i=1,2,⋯,n_.₍ ( )₍ ₎ ( )₍ ₁₎₎ −

− i

n i n

A g A

g_λ _λ is the i-th element for the weighting vector of

(4)

82

be written as

n n n n n i i n i n n i n L

n P g A g A b g A g A b

F~ = ( ( ( ) ( )) ~) ( ( )( ) ( )( ₁)) ~

1 1) ( 1) ( 1 1 = ) ( ⋅ − ⋅ + − ⋅ − − − − −

∑

λ λ λ λ , ~ )) ( (1 ~ = 1 ) ( 1 ) ( n n n n n

L F g A b

P ⋅ ₋ + − _λ ₋ ⋅

where ), ( = )) ( (1 1

= ( ) ₁ ( ) ₁

) ( − − − − n n n n n

L g A g A

P _λ _λ

)), ( ) ( ( = ) ( ) (

( ) ( 1) ( 1) ₁

1 ) ( ) ( − − − − ⋅ − − i n i n n L i n i n A g A g P A g A

g_λ _λ _λ _λ

1. = )) ( ) ( ( 1 1) ( 1) ( 1 1 = − − − − −

∑

i n i n n i A g A

g_λ _λ

For a fixed level of orness

α

, we get

, ) ( 1 1) ( = 1) ( 2 1 = ) ( i n n i n L A g n P − −

∑

+ − λ

α

, ) ( = )

( 1 ( )

1) ( 1 ) ( n L n n P A g A

g_λ _λ− ⋅

⋯ , ) ( = )

( 2 ( )

1) ( 2 ) ( n L n n n n P A g A

g_λ ₋ _λ− ₋ ⋅

. = )

( 1 ( )

) ( n L n n P A

g_λ ₋

Proof: The simplest aggregation is for two elements, as

. = ) ( = ) ( , ~ )) ( (1 ~ ) ( = ~ 1 (2) (2) 2 1 (2) 1 1 (2)

2 gλ A b gλ A b orness gλ gλ A

α

F ⋅ + − ⋅

Let us now consider the aggregation 3

~

F . In this case

. = ) ( 2 1 = ) ( (3) 2 1 = (3)

α

λ λ i i A g g orness

∑

This leads to the system of independent equations

       ⋅ − − ⋅ − + − + − + . )) ( (1 = )) ( ) ( ( , ) ( = ) ( 1, = )) ( ) ( ( )) ( ) ( ( 0) ) ( ( , 2 = ) ( ) ( (3) 1 (2) 1 (3) 2 (3) (3) 1 (2) 1 (3) 2 (3) 3 (3) 1 (3) 2 (3) 1 (3) 2 (3) 1 (3) L L P A g A g A g P A g A g A g A g A g A g A g A g A g λ λ λ λ λ λ λ λ λ λ λ λ

α

The solution is

, ) ( 1 2 = 1 (2) (3) A g PL λ

α

+ , ) ( = )

( 1 (3)

(2) 1 (3) L P A g A

g_λ _λ ⋅

. = ) ( 2 (3) (3)

L P A g_λ

More generally, in the case of n arguments, we can get theorem 1.

(5)

83

. 1 2) (

1) ( = ) , ( = ) (

+ −

−

α

n n n

P

P L

n L

δ

−

λ

rules. Denote ={ 2, , 1)

( _⋯

x Ain

−

} i

x , i=2,⋯,n , A₁(n−1)=∅ in F~_n₋₁ . ={ 1, 2, , )

( _⋯

x x

Ain , xi} , i=1,2,⋯,n ,

∅

= ) ( 0 n

A . ( ( ) ( 1))

) ( )

(

−

− i

n i n

A g A

dimension n. P_L(n) denotes correlation coefficient. Right Recursive form (RRF) of the OWA operator for fuzzy number based on a non-additive measure with

σ

−

λ

rules can be written as

) ~ )) ( )

( ( ( ~

)) ( ) ( ( =

~ ( 1)

1 1) ( 1) ( 1) (

2 = ) ( 1 ) ( 0 ) ( ) ( 1 ) (

i n i n n

i n n

i n R n

n n n

n g A g A b P g A g A b

F _λ − _λ ⋅ + ⋅

∑

_λ− − − _λ− ₋− ⋅

, ~ ~

) (

= 1

) ( 1 ) ( 1 ) (

−

⋅ +

⋅ n

n R n

n

F P b A g_λ

where

), ( 1

= 1( )

) ( )

(n n n

R g A

P − _λ

)), (

) ( ( = ) ( ) (

g_λ(n) A_i(n) −g_λ(n) A_i(₋n₁) P_R(n)⋅ g_λ(n−1) A_i(n−1) −g_λ(n−1) A_i(₋n₁−1)

1. = )) ( )

(

( ( 1) ( 1) ( 1) (₁1) 2

=

− − − −

− ₋

∑

n

i n n

i

A g A

g_λ _λ

For a fixed level of andness

α

, we get

, 1) ( ) (

1) ( =

1) ( 1) ( 1

2 = ) (

− + −

−

− − −

∑

g A n

n P

n i n n

i n R

λ

α

, 1 = )

( 1( ) ( ) )

( n

R n

n

P A

g_λ −

1, 1)

) ( ( = )

( 2( 1) ( )

1) ( ) ( 2 )

( − − − ⋅ n +

R n

n n

n

P A

g A

g_λ _λ

⋯

1. 1)

) ( ( = )

( ( )₁ ( 1) ( ₁1) ( ) )

( − − ⋅ +

− − −

n R n

n n n

n n

P A

g A

g_λ _λ

Proof: The simplest aggregation is between two elements

, = ) ( 1 = ) (

, ~ )) ( (1 ~ ) ( =

~ (2)

1 (2) (2)

2 (2) 1 (2) 1

(2) 1 (2)

2 gλ A b gλ A b andness gλ gλ A

α

F ⋅ + − ⋅ −

Let us now consider the aggregation F~₃. In this case

. = ) ( 2

1 1 = )

( (3) (3)

2

1 =

(3)

α

λ

λ i

i

A g g

andness −

∑

(6)

84        ⋅ − − − − + − + − + − − , )) ( ) ( ( = )) ( ) ( ( , 1 = ) ( 1, = )) ( ) ( ( )) ( ) ( ( )) ( ) ( ( , 2 = 2 ) ( ) ( (3) (2) 1 (2) (2) 2 (2) (3) 1 (3) (3) 2 (3) (3) (3) 1 (3) (3) 2 (3) (3) 3 (3) (3) 1 (3) (3) 2 (3) (3) 0 (3) (3) 1 (3) (3) 2 (3) (3) 1 (3) R R P A g A g A g A g P A g A g A g A g A g A g A g A g A g λ λ λ λ λ λ λ λ λ λ λ λ λ

α

the solution is

, 2 ) ( 2 = ₍₂₎ 2 (2) (3) +

−g A

PR λ

α

, 1 = )

( 1(3) (3) (3)

R P A

g_λ −

1. 1) ) ( ( = )

( ₂(3) (2) ₂(2) (3)

(3) − ⋅ +

R P A

g A

g_λ _λ

More generally, in the case of n arguments, we can get theorem 2.

Corollary 3. It is interesting to notice that P_L(n) depends on n and

α

, as . ) 1)(1 ( ) 1)(1 ( = ) , ( = ) (

α

− − + − − n n n P

Pn _L

R

δ

−

λ

rules. Denote A_i ={x₁,x₂,⋯, }

i

x , i=1,2,⋯,n. ( ( )( ) ( )( ₁)) − − i n i n A g A

dimension n. PL(n) denotes correlation coefficient. A more general form of the OWA operator for fuzzy-number based on a non-additive measure with

σ

−

λ

rules can be written as k k n k n i i n i n n k i i

n g A g A b g A g A b

F~ =( ( ( ) ( )) ~) ( ( ) ( 1)) ~

) ( ) ( 1 ) ( ) ( 1 1, = ⋅ − + ⋅ − − − − ≠

∑

λ λ λ λ k k n k n i i n i n n i n

k g A g A b g A g A b

P ( ( ( ) ( )) ~) ( ( ) ( )) ~

= 1 ) ( ) ( 1 1) ( 1) ( 1 1 = ) ( ⋅ − ⋅ + − ⋅ − − − − −

∑

λ λ λ λ , ~ )) ( ) ( ( ~ = 1 ) ( ) ( 1 ) ( k k n k n n n

k F g A g A b

P ⋅ ₋ + _λ − _λ ₋ ⋅

where

( ) ( 1) ( 1)

( ) ( ) 1

1 ( ) ( 1) ( 1)

1 2

(

)

(

)),

i<k,

(

)

(

))

=

(

)

(

)),

i>k.

n n n

n n k i i

i i i k n n n

k i i

P

g

A

g

A

g

A

g

A

P

g

A

g

A

λ λ λ λ λ λ − − − − ≠ − − − −



_⋅

₋

−



⋅

−



=1 ( ( ) ( ))= ( ( ) ( 1)).

) ( ) ( 1 1 = 1 ) ( ) ( ) ( − − − − −

−

∑

i

n i n n i k n k n n

k g A g A g A g A

P _λ _λ _λ _λ

For a fixed level of orness

α

, we get

, ) ( ) ( 2) ( ) ( 1) ( = 1 1) ( ) ( − − + − + − − + − k n n k A g n k n n k n P λ

α

), ( = )

( ( ) ( 1) ₁

1 ) ( A g P A

g n n

k

n ⋅ −

λ λ

(7)

Measure with σ-λ Rules 85 ), ( = )

( ( ) ( 1) ₁

1 ) (

− − − ⋅ k

n n k k n A g P A

g_λ _λ

1, 1) ) ( ( = ) ( 1 1) ( ) ( ) ( ⋅ − + − − k n n k k n A g P A

g_λ _λ

⋯ 1. 1) ) ( ( = )

( ₁ ( ) ( 1) ₂

) ( ⋅ − + − − − n n n k n n A g P A

g_λ _λ

Proof: According to the given conditions, we have the linear equation system

                 − ⋅ − − ⋅ − − − − ⋅ − − ⋅ − − − − − − − − − − − + − − − − − − − − − − −

∑

). ( ) ( ( = ) ( ) ( )), ( ) ( ( = ) ( ) ( 1 = ) ( ) ( )), ( ) ( ( = ) ( ) ( )), ( ) ( ( = ) ( ) ( 1, = )) ( ) ( ( , 1) ( = ) ( 1 2) ( 1 1) ( ) ( 1 ) ( ) ( 1 1) ( 1) ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 2 1) ( 1 1) ( ) ( 2 ) ( 1 ) ( 0 1) ( 1 1) ( ) ( 0 ) ( 1 ) ( 1 ) ( ) ( 1 = ) ( 1 1 = n n n n n k n n n n k n k n n k k n k n n k k n k n k n k n n k k n k n n n n k n n i n i n n i i n n i A g A g P A g A g A g A g P A g A g P A g A g A g A g P A g A g A g A g P A g A g A g A g n A g λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ

α

⋯ ⋯

The solution is theorem 3.

Corollary 4. For a more general recursive form of the OWA operator for fuzzy number

based on a non-additive measure with

σ

−

λ

rules. When

(1) k =n, we get the LRF of the OWA operator for fuzzy number based on a non-additive measure with

σ

−

λ

rules.

(2) k=1, we get the RRF of the OWA operator for fuzzy number based on a non-additive measure with

σ

−

λ

rules.

4. Illustrative example

A management selects 4 experts to evaluate for a scheme. Information evaluation values as shown in table 1, where expert is xi, evaluation value is denoted as

) , , ( = ~ ,2 ,1 i i i i

i

i a a a

a −

δ

+

δ

which is a triangle fuzzy number.

Table 1:

expert evaluation

λ g

1

x (0.20,0.30,0.40) 0.1524

2

x (0.80,0.90,1.00) 0.1586

3

x (0.50,0.60,0.70) 0.2428

4

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86

Step 1: According to 1 = (1 ({ })) 1

=

i n

i

x g_λ

λ

+

∏

, we know

λ

=0.05. Meanwhile

0.1524, =

}) ({ = )

( 1

(4) 1 (4)

x g A

g_λ _λ

0.3122, =

1] })) ({ }))(1

({ [(1

1 = )

( (4) ₂

1 (4) 2

(4) + + −

x g x

g A

g_λ

λ

_λ

λ

_λ

λ

0.5588. =

1] })) ({ }))(1

({ }))(1

({ [(1

1 = )

( 3

(4) 2

(4) 1

(4) 3

(4) ₊ ₊ ₊ ₋

x g x

g x

g A

g_λ

λ

_λ

λ

_λ

λ

_λ

λ

Step 2: According to definition 4, we can know the comprehensive evaluation 4 ~ F of 4 experts.

,0.70) (0.50,0.60 ))

( ) ( ( ,1.00) (0.80,0.90 ))

( ( = ~

1 (4) 2 (4) 1

(4)

4 g A ⋅ + g A −g A ⋅

F _λ _λ _λ

,0.30) (0.10,0.20 ))

( (1 ,0.40) (0.20,0.30 ))

( ) (

( 3

(4) 2

(4) 3

(4) − ⋅ + − ⋅

+ g_λ A g_λ A g_λ A

3). 3953,0.495 (0.2953,0.

=

Step 3: When a new expert x5 gives a value for this scheme which is (0.30,0.40,0.50).

According to Definition 4 and Theorem 1, we know that

0.3411, =

) ( 3

1 = ) (

= (4)

3

1 = (4)

j j

A g g

orness _λ _λ

α

∑

0.6743, =

1 3

4 = 1 2) (

1) ( = (5)

+ +

− −

α

n

n PL

0.6743, =

= ) ( ₄ (5) (5)

L P A g_λ

8). 3968,0.496 (0.2968,0.

= ,0.50)) (0.30,0.40 ))

( (1 ~ = ~

4 (5) 4

(5)

5 P ⋅F + −g A ×

F L λ

Step 4: When a new expert x6 gives a value for this scheme which is (0.40,0.50,0.60),

we can know the comprehensive evaluation F~₆ of 4 experts.

0.7213 =

1 4

5 = (6)

+

α

L

P

0.7213 =

= ) ( ₅ (6) (6)

L P A g_λ

6). 4256,0.525 (0.3256,0.

= ,0.60) (0.40,0.50 ))

( (1 ~ =

~

5 (6) 5

(6)

6 P ⋅F + −g A ×

F L λ

REFERENCES

1. J.J.Buckley, Fuzzy eign value and imput-output analysis, Fuzzy Sets and Systems, 34(1990) 187-195.

2. J.J.Dujmovic, Continuous preference logic for system evaluation, IEEE Transactions

on Fuzzy Systems, 15(2008) 1082-1099.

3. Z.T.Gong and W.J.Lei, The weighted moving averages based on a non-additive

measures with

σ

−

λ

rules, Lanzhou University Press, 52(2016) 686-689 (in

(9)

87

4. Z.T.Gong and L.Chen, On sugeno measure based on

σ

−

λ

rules and its defect of additivity, Fuzzy Sets and Systems, 22 (2008) 126-129.

5. M.Grabisch, T.Murofushi and M.Sugeno, Fuzzy measures and integrals: Theory and

Application, Heidelberg: Physica-verlag, 2000.

6. M.H.Ha and C.X.Wu, Fuzzy Measure and Fuzzy Integral Theory, Science Press,

Beijing, 1998 (in Chinese).

7. L.S.Jin, M.Kalina and G.Qian, Discrete and continuous recursive forms of OWA

operators, Fuzzy Sets and Systems, 308 (2016).

8. S.Nanda and K.Kar, Convex fuzzy mappings, Fuzzy Sets and Systems, 48 (1992)

129-132.

9. M.Sugeno, Theory of fuzzy integral and its application, Doctorial Dissertation, Tokyo

Institute of Technology, 1974.

10. H.Tanaka, Fuzzy data analysis by possibilistic linear models, Fuzzy Sets and Systems,

24 (1987) 363-375.

11. L.Troiano and R.R.Yager, Recursive and iterative OWA operators, International

Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 13 (2005) 579-599.

12. C.X.Wu and M.Ma, The Fundation of Fuzzy Analysis, National Defend Industy Press,

Beijing, 1991 (in Chinese).

13. R.R.Yager, On ordered weighted averaging aggregation operators in muiticriteria

decision making, IEEE Transactions on Systems, Man. and Cybernetics, 18 (1988) 183-190.

14. R.R.Yager, Families of OWA operators, Fuzzy Sets and Systems, 59 (1993) 125-148.