a complete pattern recognition approach under atanassov’s intuitionistic fuzzy sets

Accepted Manuscript

A complete pattern recognition approach under Atanassov’s intuitionistic fuzzy

sets

Chun-Hsiao Chu, Kuo-Chen Hung, Peterson Julian

PII: S0950-7051(14)00136-1

DOI: http://dx.doi.org/10.1016/j.knosys.2014.04.014

Reference: KNOSYS 2807

To appear in: Knowledge-Based Systems

Received Date: 11 November 2012

Revised Date: 7 April 2014

Accepted Date: 10 April 2014

Please cite this article as: C-H. Chu, K-C. Hung, P. Julian, A complete pattern recognition approach under

Atanassov’s intuitionistic fuzzy sets, Knowledge-Based Systems (2014), doi: http://dx.doi.org/10.1016/j.knosys.

2014.04.014

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http://dx.doi.org/10.1016/j.knosys.2014.04.014

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http://dx.doi.org/http://dx.doi.org/10.1016/j.knosys.2014.04.014

A complete pattern recognition approach under Atanassov’s intuitionistic fuzzy sets

Chun-Hsiao Chua, Kuo-Chen Hungb,*, Peterson Julianc

aDepartment of Tourism, Aletheia University

bDepartment of Computer Science and Information Management, Hungkuang University cDepartment of Traffic Science, Central Police University

Abstract

This research is aimed at developing a method for solving pattern recognition problems

under the Atanassov’s intuitionistic fuzzy sets based on similarity measures. First we

proposed two similarity measures and then developed a method based on our similarity

measures. We also proved that our method is able to solve the pattern recognition problems.

Finally, a fault diagnosis example of the turbine vibration has been examined by our method.

The example demonstrates that the proposed method can not only diagnose the main faults of

the turbine generator but also it can detect useful information for future trends and multi-fault

analysis. In addition, for the convenience of computing and ranking processes, a computer

interface decision support system is also developed to help decision maker make diagnoses

more efficiently.

Keywords: Fault diagnosis, Atanassov’s intuitionistic fuzzy sets (AIFSs), similarity measure,

pattern recognition

* Corresponding author. Tel.: +886-4-26318652 ext. 5411; fax: +886-4-26521921. E-mail address: [email protected]

2

1. Introduction

Fault diagnosis has recently become a very popular research issue. A well performed

diagnosis technique has substantial economic benefits by scheduling preventive maintenance

and avoiding extensive downtime periods caused by failure. However, when problems arise

in a very complex mechanism it can be very tricky to diagnose because a similar set of

failure signs may be caused by multiple sets of sources. The characteristics of state failure

extracted from failure detection often have features of fuzziness, coupled with the vagueness

of the signs, such as the ambiguity of the sign, the uncertainty of the diagnosis data and so on,

making the process of giving an accurate diagnosis even more difficult. This complex

problem can be well characterized using fuzzy relative theory. Similarity measure can then be

used to compute and transform the vague sets for fault diagnosis and reasoning solving fault

diagnosis problems.

Failures in a steam turbine generator can usually be specified according to the different

amplitude ratio of vibration signal in several different frequency ranges [17]. There are many

different approaches in dealing with fault diagnosis problem, for examples, parallel system

[20], control system [8], expert system [11] and neural network [10]. In the problem at hand,

the estimation of precise values becomes very difficult in many cases. The methods

mentioned above would obtain inaccurate results under these uncertain circumstances since a

precise value of the amplitude ratio of vibration signals would be unobtainable. Therefore,

the fuzzy approach would be appropriate for illuminating these problems.

Fuzzy sets (FSs) [25] and AIFSs [1] are two specific implementations used to handle

uncertainty, and each of the mentioned sets has its advantages. The FSs demonstrate the

uncertainty in the membership degree. Moreover, AIFSs describe the uncertainty by both

membership degree and non-membership degree. The fundamental characteristics of AIFSs

3

are that the values of its membership function and non-membership are vague values rather

than crisp numbers. Therefore, it could be used to simulate many activities and processes

requiring human expertise and knowledge, which are inevitably imprecise.

In the following paragraph, we will point out four major features while selecting the best

pattern for the given sample to provide a motivation for our proposed method.

The first feature is that in the beginning of our algorithm, we have to consider all elements

in the disclose of universe to use all available information. If we cannot decide the best

pattern, then we must only concentrate on a small domain to execute our pattern selection.

It may be the rule of thumb. If we cannot solve a complex problem then the most natural way

is to concentrate on a small domain that is the most important, simplifying complex problem

to a simpler one. Hence, in our algorithm, after the first sixth steps, if the best pattern for the

given sample can not be decided, then we will concentrate on single element, one after one,

in the disclose of universe sequentially according to their relative important weight.

The second feature is to eliminate the influence of weights, because the proposed distances

for intuitionistic fuzzy sets are weighted sums.

For example, after the domain is shrunk to an element and then we are only concerned with

the membership function for the moment. For two patterns A and B with one sampleC ,

the distance between A and C can be abstractly expressed as ( )( ) ∑=

=n

kkkww wCAd

n1

,..., ,1

α .

Similarly, the distance between B and C is ( )( ) ∑=

=n

kkkww wCBd

n1

,..., ,1

β where

( )nww ,...,1 are related to weights.

If patterns A and B are identical then any distance measure will imply that

( ) ( )CBdCAd ,, = . Our second feature is to find a process to reveal that BA = .

4

For example, if 3=n and we have ( ) ( )CBdCAd ,, = implying that

( ) ( ) 0,,,, 332211321 =−−−⋅ βαβαβαwww . (1)

Our goal is to derive that BA = so that we need to construct at least three sets of weights

that are linearly independent: aW , bW and cW where aW looks like ( )321 ,, www .

If a recognition dilemma occurs so that ( ) =CAdaW , ( )CBd

aW , , ( ) =CAdbW , ( )CBd

bW ,

and ( ) =CAdcW , ( )CBd

cW , then we know that ( )332211 ,, βαβαβα −−− is orthogonal to

three independent vectors aW , bW and cW implying that BA = where ( )321 ,, ααα=A

and ( )321 ,, βββ=B .

Hence, when dealing with weights, if we sophisticatedly select n independent nWW ,...,1

with ( ) ( )CBdCAdjj WW ,, = for nj ,...,1= then we can derive that BA = .

If our algorithm contains a family of independent weights, then the iterative algorithm will

clearly point out that the recognition dilemma was caused by the same data for two patterns

at a given point in the disclose of universe.

The benefit of this feature is that we can individually consider membership, non-membership

and hesitancy in future steps of our algorithm, for a given element in the disclose of universe,

and then and subsequently do the same for the rest of the elements in the disclose of universe ,

according to their relative importance.

The third feature is to detect that the original data had the same value for a component. For

example, the universal set is a singleton as { }1xX = and three AIFSs consisting of two

patterns, ( )3.0,2.0,5.0=A and ( )1.0,4.0,5.0=B with one sample ( )aaC −= 7.0,3.0, ,

where ( ) 5.01 =xAμ , ( ) 2.01 =xvA , and ( ) 3.01 =xAπ . Similar definition for B and C .

5

Owing to ( ) 5.01 =xAμ and ( ) 5.01 =xBμ , if we only consider the distance of membership

functions, then for any distance we will always have ( ) ( )( ) =11 , xxd CA μμ ( ) ( )( )11 , xxd CB μμ ,

regardless of the value of a . It reveals that sometimes one component of two patterns have

the same value that will cause recognition problem.

The fourth feature is illustrated by the non-membership function of the above example as

( ) ( ) ( )2

111

xvxvxv BA

C

+= so that for a common distance, we will derive that

( ) ( )( ) =11 , xvxvd CA ( ) ( )( )11 , xvxvd CB pointing out the arithmetic mean of two values will

cause recognition problem when the sample is the arithmetic mean of two patterns.

We will develop a method that contains many iterative steps which has the ability to apply

the first feature to examine the entirety of the disclose of universe to each element in the

disclose of universe, and then separately consider membership, non-membership and

hesitancy, to finally distinguish the type of recognition problem that was caused by the third

feature, the duplicated data or the fourth feature, the arithmetic mean of two data. Hence, we

must construct a new distance that is independent of arithmetic mean.

This study developed a fault diagnosis method for turbine generator under the intuitionistic

fuzzy circumstance. In order to handle the uncertain situation and vague information, we

developed a method consisting of similarity measures under the AIFSs environment. The

similarity between two AIFSs was defined by taking into account the intuitionistic fuzzy

values where the similarity measures were constructed based on the Minkowski distance.

Then, a new method of the fault diagnosis of turbine generator was presented. The similarity

degree between our detecting sample and fault patterns was evaluated by the proposed

similarity measure method. A certain type of fault pattern with bigger value of similarity

6

measure implies that the detecting sample is more likely to belong to this type of fault. Hence,

the type of fault for the sample is determined according to the value of similarity measure.

We also presented numerical examples to explain the validity and reasonability of applying

our method in solving the fault diagnosis problems.

2. Preliminaries

2.1. AIFSs

AIFS [1] is an extension of FSs [25] and it has been proved that AIFS coincides with

vague sets [3]. An AIFS A in a fixed set E is an objective with the expression

( ) ( ){ }, ,A AA x x x x Eμ ν= ∈ where the functions [ ]: 0,1A Eμ → and [ ]: 0,1A Eν →

denote the degree of membership and the degree of non-membership of the element x E∈ ,

respectively. For every x E∈ , ( ) ( )0 1A Ax xμ ν≤ + ≤ . For each AIFS A, “hesitation degree”

is considered. The hesitation degree of an element x E∈ satisfies the following expression:

( ) ( ) ( )xvxx AAA −−= μπ 1 .

2.2. Fault diagnosis based on AIFSs operations

Essentially, the fault diagnostic technique for equipments is a pattern recognition problem.

In other word, the operating status of the machine can be divided into normal and abnormal

status. The signal sample of the abnormality belongs to which fault is a pattern recognition

problem. A common way for solving the pattern recognition problem is to construct

similarity measures to identify the samples.

We assume that there exists m fault patterns, which are represented by AIFSs Pi (i = 1,

2, …, m), and a testing sample, B, which is also an AIFS, to be recognized. Then the

7

diagnosed pattern, Pk , should be the one with highest value of similarity measure value i.e.,

( ) ( )BPSBPS ikimi

k ,max,,,...,1 ≠=

> . Furthermore, several investigations of similarity measures

applied in pattern recognition problem based on AIFSs have been addressed extensively. The

readers can refer to [2], [4] , [7], [9], [13], [19], [24] and [26].

3. Two similarity measures

For two objectives A and B , a well-defined similarity measure ( )BAS , to appraise the

resemblance between them, the following four properties developed by [16] must be held:

P1: ( ) 1,0 ≤≤ BAS .

P2: ( ) 1, =BAS if and only if BA = .

P3: ( ) ( )ABSBAS ,, = .

P4: If three AIFSs A, B and C that satisfies CBA ⊆⊆ , then ( ) ( )BASCAS ,, ≤ and

( ) ( )CBSCAS ,, ≤ .

In this section, we will construct two similarity measures for two AIFSs

( ) ( ){ }XxxvxxA iiAiAi ∈= ,,μ and ( ) ( ){ }XxxvxxB iiBiBi ∈= ,,μ . After that, we will

verify that our similarity measures are well-defined by examining the above four properties.

3.1. First similarity measure and its properties

The first similarity measure we proposed is based on Minkowski distance. The Minkowski

distance between two vectors ( )mxxX ,...,1= and ( )myyY ,...,1= is defined as

( )YXd ,pm

i

p

ii yx1

1

⎟⎠⎞⎜

⎝⎛ −= ∑

=. Hence, for two AIFSs A and B, the distance is defined as:

8

( ) ( ) ( ) ( ) ( ) ( )( )ααααππμμ

1

1

),( ∑=

−+−+−=n

jjBjAjBjAjBjA xxxvxvxxBAd (2)

with ∞<≤α1 that is the sum of n Minkowski distances.

However, our generalization of Minkowski distance in equation (1) does not take into

account the decision makers' preference among Aμ , Av , and Aπ and the difference for

elements in the disclose of universe. When the preferences are considered, we introduce the

preference rates 1δ , 2δ , and 3δ for Aμ , Av , and Aπ respectively, with 10 << iδ for

3,2,1=i , ∑=

=3

1

1i

iδ and the weight wj for xj in the disclose of universe with jw<0 for

nj ...,1= and 1=∑=

n

jijw . Accordingly, our first similarity measure is defined as follows:

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) αααα

ππδδμμδ1

13211, ⎥

⎦

⎤⎢⎣

⎡−+−+−−= ∑

=

n

jjBjAjBjAjBjAj xxxvxvxxwBAS . (3)

An anonymous reviewer informed us that there is a more general expression of restricted

equivalence functions and aggregation functions that ( )yxREF , ( ) ( )( )yx 221

1 1 ϕϕϕ −−= −

αyx −−= 1 , with ( ) ( ) αϕ 1

1 11 xx −−= , ( ) xx =2ϕ and the aggregation function is the

weighted arithmetic mean then we can directly refer to the previous results to show our first

similarity measure satisfying the four properties developed by [20]. There are three papers

[4], [5] and [6] that are useful to develop restricted equivalence functions for measuring

similarities.

3.2. Second similarity measure and its properties

The second similarity measure is also defined based on Minkowski distance between

AIFSs A and B. Motivated by the maximizing deviation method ([21], [22] and [23]), the

9

second similarity measure is defined as:

( )( ) ( )

( ) ( ){ }( ) ( )

( ) ( ){ }( ) ( )

( ) ( ){ }

αααα

ππππ

δδμμ

μμδ

1

1321 ,,,

1,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

−= ∑=

n

j jBjA

jBjA

jBjA

jBjA

jBjA

jBjA

j xxMax

xx

xvxvMax

xvxv

xxMax

xxwBAS

(4)

Remark. We accepted the convention of 00

0 = to simplify the expression.

The following proves the four properties proposed by Mitchell [16] of the second

similarity measure of (4).

Property 1 ( ) 1,0 ≤≤ BAS

Proof:

Since ( ) ( ) ( ) ( ) ( ) ( ) 1,,,,,0 ≤≤ xxxvxvxx BABABA ππμμ , ∑=

=n

jjw

1

1 and 1321 =++ δδδ , we

have

( )

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }∑

∑

=

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

≥

++=

n

j jBjA

jBjA

jBjA

jBjA

jBjA

jBjA

j

n

jj

xxMax

xx

xvxvMax

xvxv

xxMax

xxw

w

1321

3211

,,,

1

ααα

ππππ

δδμμ

μμδ

δδδ

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }

αααα

ππππ

δδμμ

μμδ

1

1321 ,,, ⎟

⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

≥ ∑=

n

j jBjA

jBjA

jBjA

jBjA

jBjA

jBjA

j xxMax

xx

xvxvMax

xvxv

xxMax

xxw

(5)

implying ( ) 1,0 ≤≤ BAS which proves Property 1.

Property 2. ( ) 1, =BAS if and only if BA =

Proof:

10

If A=B, ( ) 1, =BAS is obviously held.

If ( ) 1, =BAS , we have

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }∑

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −n

j jBjA

jBjA

jBjA

jBjA

jBjA

jBjA

j xxMax

xx

xvxvMax

xvxv

xxMax

xxw

1321 ,,,

ααα

ππππ

δδμμ

μμδ (6)

equals to zero implying ( ) ( )jBjA xx μμ = , ( ) ( )jBjA xvxv = , and ( ) ( )jBjA xx ππ = for all

j=1,2,...,n, since 0>iδ , for 3,2,1=i and 0>jw for nj ,....,1= . Hence, ( ) 1, =BAS if

and only if BA = gets proved.

Property 3. ( ) ( )ABSBAS ,, =

Proof:

Owing to ( )BAS , is symmetric with respect to A and B, we have ( ) ( )ABSBAS ,, = such

that Property 3 is proved.

Property 4. For three AIFSs A, B and C satisfying CBA ⊆⊆ , then ( ) ( )BASCAS ,, ≤ and

( ) ( )CBSCAS ,, ≤ .

Proof:

For three AIFSs A, B and C satisfying CBA ⊆⊆ with ( ) ( ) ( ),xxx CBA μμμ ≤≤

( ) ( ) ( ),xvxvxv CBA ≥≥ and ( ) ( ) ( ),xxx CBA πππ ≥≥ we have

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }∑

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −n

j jBjA

jBjA

jBjA

jBjA

jBjA

jBjA

j xxMax

xx

xvxvMax

xvxv

xxMax

xxw

1321 ,,,

ααα

ππππ

δδμμ

μμδ

( )( )

( )( )

( )( )∑

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−=n

j jA

jB

jA

jB

jB

jAj x

x

xv

xv

x

xw

1321 111

ααα

ππ

δδμμ

δ (7)

11

and

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }

( ) ( )( ) ( ){ }∑

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −n

j jCjA

jCjA

jCjA

jCjA

jCjA

jCjA

j xxMax

xx

xvxvMax

xvxv

xxMax

xxw

1321 ,,,

ααα

ππππ

δδμμ

μμδ

( )( )

( )( )

( )( )∑

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−=n

j jA

jC

jA

jC

jC

jAj x

x

xv

xv

x

xw

1321 111

ααα

ππ

δδμμ

δ . (8)

Due to

( )( )

( )( )

( )( )∑

=⎟⎟

⎠

⎞⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−n

j jA

jB

jA

jB

jB

jAj x

x

xv

xv

x

xw

1321 111

ααα

ππ

δδμμ

δ

( )( )

( )( )

( )( )∑

=⎟⎟

⎠

⎞⎜⎜

⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−≥n

j jA

jC

jA

jC

jC

jAj x

x

xv

xv

x

xw

1321 111

ααα

ππ

δδμμ

δ , (9)

we have ( ) ( )BASCAS ,, ≤ . By the analogy of this process, we also have ( ) ( )CBSCAS ,, ≤ .

Hence, Property 4 is proved.

Due to the fact that the second similarity measure has the four properties proposed by [16],

we know that it is also well-defined.

4. A proposed integration method for the fault diagnosis of turbine generators

For a diagnosis problem, with patterns { }mPP ,...,1 and a sample B, we will develop a

method that consists of 66 +n steps of iterations, where n is the cardinal number of the

disclose of universe, which apply the two similarity measures of (6) and (12) repeatedly. The

flowchart of proposed integration method has been presented as Fig. 1.

In addition, our procedure is gradually shrinking the set of possible patterns to reserve

those patterns attaining the maximum similarity measure. To simplify the expression, we will

assume that ( ) ( )jBjPji xxi

μμ −=Δ , ( ) ( )jBjPji xvxvi

−=Ω and ( ) ( )jBjPji xxi

ππ −=Ψ

12

when comparing the sample B with the pattern iP .

The detailed steps for our proposed method are explained in the following.

Step 1. For mi ,...1= , we compute

( ) ( )α

ααα δδδ1

13211 1, ⎥

⎦

⎤⎢⎣

⎡Ψ+Ω+Δ−= ∑

=

n

jjijijiji wBPS . (10)

If ( ) ( ){ }timiBPSBPS it ≠=> and ,,...,1:,max, 11 , then sample B will be assigned to pattern

tP . Otherwise, we will assume that

( ) ( ){ }{ }0111 :,max,: Θ∈==Θ iikk PBPSBPSP (11)

where 0Θ is the original set of patterns { }mPP ,...,1 .

Step 2. Depending on the preference of the decision maker, we select a small positive number,

θ such that θδ 21 + , θδ +2 and θδ 33 − are still in the open interval ( )1,0 . For iP in

1Θ , we evaluate

( ) ( ) ( ) ( )( )α

ααα θδθδθδ1

13212 321, ⎥

⎦

⎤⎢⎣

⎡Ψ−+Ω++Δ+−= ∑

=

n


If there is a pattern, sP in 1Θ satisfying ( ) ( ){ }siPBPSBPS iis ≠Θ∈> ,:,max, 122 , then

13

( )f kx

Fig. 1. The flowchart for our proposed method.

14

sample B will be assigned to pattern sP . Otherwise, we will assume that

( ) ( ){ }{ }1222 :,max,: Θ∈==Θ iikk PBPSBPSP . (13)

Step 3. For iP in 2Θ , we compute

( ) ( ) ( ) ( )( )α


13213 321, ⎥

⎦

⎤⎢⎣

⎡Ψ−+Ω++Δ+−= ∑

=

n


If there is a pattern, αP in 2Θ satisfying ( ) ( ){ }αα ≠Θ∈> iPBPSBPS ii ,:,max, 233 , then

sample B will be assigned to pattern αP . Otherwise, we will assume that


Step 4. For iP in 3Θ , we evaluate

( ) ( )α

ααα δδδ1

13214 1, ⎥

⎦

⎤⎢⎣

⎡++−= ∑

=

n

jjijijiji PNMwBPS (16)

with ( ) ( )

( ) ( ){ }max ,

i

i

P j B j

ij

P j B j

x xM

x x

μ μ

μ μ

−= ,

( ) ( )( ) ( ){ }jBjP

jBjP

ji xvxv

xvxvN

i

i

,max

−= and

( ) ( )( ) ( ){ }max ,

i

i

P j B j

ij

P j B j

x xP

x x

π π

π π

−= .

If there is a pattern, βP in 3Θ satisfying ( ) ( ){ }ββ ≠Θ∈> iPBPSBPS ii ,:,max, 344 , then

sample B will be assigned to pattern βP . Otherwise, we will assume that


Step 5. Following the above procedure, for iP in 4Θ , we compute

( ) ( ) ( ) ( )( )α


13215 321, ⎥

⎦

⎤⎢⎣

⎡−++++−= ∑

=

n

jjijijiji PNMwBPS . (18)

15

If there is a pattern, rP in 4Θ satisfying ( ) ( ){ }riPBPSBPS iir ≠Θ∈> ,:,max, 455 , then

sample B will be assigned to pattern rP . Otherwise, we will assume that


Step 6. For iP in 5Θ , we evaluate

( ) ( ) ( ) ( )( )α


13216 321, ⎥

⎦

⎤⎢⎣

⎡−++++−= ∑

=

n

jjijijiji PNMwBXS . (20)

If there is a pattern, σP in 5Θ satisfying ( ) ( ){ }σσ ≠Θ∈> iPBPSBPS ii ,:,max, 566 , then

sample B will be assigned to pattern σP . Otherwise, we will assume that


Step 7. In the above process, we use all elements in the disclose of universe to derive a

synthesized result, but we cannot decide the pattern for the given sample. In the next section,

we will explain that there are two phenomena causing the undifferentiated findings such that

we will separate the disclose of universe, { }nxx ,...,1 to a series of singleton as follows. From

the preference of decision maker, we assume that

( ) ( ) ( )nfff xxx �� 21 (22)

to indicate among elements in the disclose of universe, ( )1fx is the most important and

( )nfx is the least important.

Now we concentrate on ( ){ }1fx . For any pattern, say iP in 6Θ , we compute ( )BPS i ,7 as

( ) ( ) ( ) ( )( )[ ] αααα δδδ 1

1312117 1, fififii BPS Ψ+Ω+Δ−= . (23)

16

If there is a pattern, aP in 6Θ satisfying ( ) ( ){ }aiPBPSBPS iia ≠Θ∈> ,:,max, 677 , then

sample B will be assigned to pattern aP . Otherwise, we will assume that


Step 8. For iP in 7Θ , we evaluate ( )BPS i ,8 as

( ) ( ) ( ) ( ) ( ) ( ) ( )( )[ ] αααα θδθδθδ 1

1312118 321, fififii BPS Ψ−+Ω++Δ+−= . (25)

If there is a pattern, bP in 7Θ satisfying ( ) ( ){ }biPBPSBPS iib ≠Θ∈> ,:,max, 788 , then

sample B will be assigned to pattern bP . Otherwise, we will assume that



( ) ( ) ( ) ( ) ( ) ( ) ( )( )[ ] αααα θδθδθδ 1

1312119 321, fififii BPS Ψ−+Ω++Δ+−= . (27)

If there is a pattern, cP in 8Θ satisfying ( ) ( ){ }ciPBPSBPS iic ≠Θ∈> ,:,max, 899 , then

sample B will be assigned to pattern cP . Otherwise, we will assume that



( ) ( ) ( ) ( )( )[ ] αααα δδδ 1

33221110 1, fififii PNMBPS ++−= . (29)

If there is a pattern, dP in 9Θ satisfying ( ) ( ){ }diPBPSBPS iid ≠Θ∈> ,:,max, 91010 , then

sample B will be assigned to pattern dP . Otherwise, we will assume that

17



( ) ( ) ( ) ( ) ( ) ( ) ( )( )[ ] αααα θδθδθδ 1

13121111 321, fififii PNMBPS −++++−= . (31)

If there is a pattern, eP in 10Θ satisfying ( ) ( ){ }eiPBPSBPS iie ≠Θ∈> ,:,max, 101111 , then

sample B will be assigned to pattern eP . Otherwise, we will assume that



( ) ( ) ( ) ( ) ( ) ( ) ( )( )[ ] αααα θδθδθδ 1

13121112 321, fififii PNMBPS −++++−= . (33)

If there is a pattern, gP in 11Θ satisfying ( ) ( ){ }giPBPSBPS iig ≠Θ∈> ,:,max, 111212 , then

sample B will be assigned to pattern gP . Otherwise, we will assume that


Following this trend, if the procedure does not stop, then we will execute the following

steps such that Step jk +6 is similar to Step j+6 except that (a) j+Θ6 is replaced by

jk+Θ6 , (b) ( )BPS ij ,6+ is replaced by ( )BPS ijk ,6 + and (c) ( )1fx is replaced by ( )kfx . In

the next section, we will prove that if our method does not stop during the process then at the

final Step 66 +n , there must be a pattern, say zP satisfying

( ) ( ){ }ziPBPSBPS niinzn ≠Θ∈> +++ ,:,max, 566666 , (35)

then sample B will be assigned to pattern zP .

18

5. The validation of our method

In this section, we will prove that our method can solve pattern recognition problems. By

way of contradiction, there are two patterns, say ηP and λP satisfying

( ) ( ) ( ){ }56666666 :,max,, ++++ Θ∈== niinnn PBPSBPSBPS λη . (36)

For a given { }ni ,...,1∈ , we know that ηP and λP both in ji+Θ6 for 3,2,1=j then

( ) ( )BPSBPS jiji ,, 66 λη ++ =

(37)

to imply that

( ) ( ) ( )αη

αη

αη δδδ ififif Ψ+Ω+Δ 321 ( ) ( ) ( )

αλ

αλ

αλ δδδ ififif Ψ+Ω+Δ= 321 , (38)

( ) ( ) ( ) ( ) ( ) ( )αη

αη

αη θδθδθδ ififif Ψ−+Ω++Δ+ 32 321 ( ) ( ) ( ) ( ) ( ) ( )

αλ

αλ

αλ θδθδθδ ififif Ψ−+Ω++Δ+= 32 321 , (39)

and

( ) ( ) ( ) ( ) ( ) ( )αη

αη

αη θδθδθδ ififif Ψ−+Ω++Δ+ 32 321 ( ) ( ) ( ) ( ) ( ) ( )

αλ

αλ

αλ θδθδθδ ififif Ψ−+Ω++Δ+= 32 321 . (40)

We rewrite equations (38-40) as follows

( )321 ,, δδδ ( ) 0,, 321 =• YYY , (41)

( )θδθδθδ 3,,2 321 −++ ( ) 0,, 321 =• YYY , (42)

and

( )θδθδθδ 3,2, 321 −++ ( ) 0,, 321 =• YYY , (43)

where • is the inner product and ( )αη ifY Δ=1 ( )

αλ ifΔ− , ( )

αη ifY Ω=2 ( )

αλ ifΩ− and ( )

αη ifY Ψ=3 ( )

αλ ifΨ− .

Next, we will explain the existence of θ to guarantee θδ +< 10 , θδ 21 + , θδ +2 ,

θδ 22 + , and 133 <− θδ .

19

If 0>iδ for 3,2,1=i , and 13

1

=∑=i

iδ , then there is a θ , for example,

{ }3,2,1:1,min4

1 =−= kkk δδθ . (44)

that satisfies

1330 <+<−< θδθδ ii . (45)

for 3,2,1=i . We need the next theorem for our further verification.

Theorem 1. The three vectors ( )θδθδθδ 3,2, 3213 −++=V , ( )θδθδθδ 3,,2 3212 −++=V

and ( )3211 ,, δδδ=V are linearly independent.

Proof:

We compute the determinant of ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−++−++

θδθδθδθδθδθδ

δδδ

32

32

321

321

321

. If we consider the

operation of “ ( )1− first row” adding it to the second row that is the third kind elementary

row operation so this operation will not influence the determinant value. Similarly, we use

“ ( )1− first row” adding it to the third row and then we apply the first row expansion to

imply that

det

32

32det

321

321

321

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−++−++

θδθδθδθδθδθδ

δδδ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

θθθθθθ

δδδ

32

32321

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=θθθθ

δθθθθ

δθθθθ

δ2

2det

3

32det

32

3det 321

( ) 033 2321

2 ≠=++= θδδδθ . (46)

20

From the fact that the determinant is non zero, we know that the three vectors, 1V , 2V and

3V are linearly independent.

Owing to ( )321 ,, YYY is orthogonal to three independent vectors in 3-dimentional space, then

we derive that for equations (41-43), ( ) ( )0,0,0,, 321 =YYY .

For the same i , we know that ηP and λP both in ki+Θ6 for 6,5,4=k then

( ) ( )BPSBPS kiki ,, 66 λη ++ = (47)

to imply that

( ) ( ) ( )αη

αη

αη δδδ ififif PNM 321 ++ ( ) ( ) ( )

αλ

αλ

αλ δδδ ififif PNM 321 ++= , (48)

( ) ( ) ( ) ( ) ( ) ( )αη

αη

αη θδθδθδ ififif PNM 32 321 −++++ ( ) ( ) ( ) ( ) ( ) ( )

αλ

αλ

αλ θδθδθδ ififif PNM 32 321 −++++= , (49)

and

( ) ( ) ( ) ( ) ( ) ( )αη

αη

αη θδθδθδ ififif PNM 32 321 −++++ ( ) ( ) ( ) ( ) ( ) ( )

αλ

αλ

αλ θδθδθδ ififif PNM 32 321 −++++= . (50)

We rewrite equations (48-50) as follows

( )321 ,, δδδ ( ) 0,, 321 =• ZZZ , (51)

( )θδθδθδ 3,,2 321 −++ ( ) 0,, 321 =• ZZZ , (52)

and

( )θδθδθδ 3,2, 321 −++ ( ) 0,, 321 =• ZZZ , (53)

where ( )αη ifMZ =1 ( )

αλ ifM− , ( )

αη ifNZ =2 ( )

αλ ifN− and ( )

αη ifPZ =3 ( )

αλ ifP− .

Owing to ( )321 ,, ZZZ is orthogonal to three independent vectors in 3-dimentional space,

then we derive that ( ) ( )0,0,0,, 321 =ZZZ .

21

Owing to 01 =Y and 01 =Z , we obtain that ( )αη ifΔ ( )

αλ ifΔ= and ( )

αη ifM ( )

αλ ifM= such that

( )ifηΔ ( )ifλΔ= and ( )ifMη ( )ifMλ= to imply

( )( ) ( )( )ifBifP xx μμη

− ( )( ) ( )( )ifBifP xx μμλ

−= (54)

and

( )( ) ( )( )( )( ) ( )( ){ }ifBifP

ifBifP

xx

xx

μμ

μμ

η

η

,max

− ( )( ) ( )( )( )( ) ( )( ){ }ifBifP

ifBifP

xx

xx

μμμμ

λ

λ

,max

−= . (55)

We need the next theorem for further verification.

Theorem 2. For real numbers ba, and c if cbca −=− then (i) ba = or (ii)

ba ≠ with ( ) 2bac += .

Proof.

We square on both sides to imply that 2222 22 cbcbcaca +−=+− and then simply the

expression to derive that ( )cbaba −=− 222 . Depending on ba = or ba ≠ , we can

finish the verification.

From equation (54), based on Theorem 2, we derive that

( )( )=ifP xη

μ ( )( )ifP xλ

μ (56)

or

( )( )=ifB xμ2

1( )( )[ +ifP x

ημ ( )( )]ifP x

λμ . (57)

22

We combine equations (54, 55) to imply that

( )( ) ( )( ){ }ifBifP xx μμη

,max ( )( ) ( )( ){ }ifBifP xx μμλ

,max= . (58)

We assume that ( )( )>ifP xη

μ ( )( )ifP xλ

μ . Using equation (57), we rewrite equation (58) as

( )( )ifP xη

μ ( )( )[ ( )( )]ifPifP xxλη

μμ +=2

1, (59)

and then ( )( )=ifP xη

μ ( )( )ifP xλ

μ that is contradicted with ( )( )>ifP xη

μ ( )( )ifP xλ

μ to imply

that ( )( )>ifP xη

μ ( )( )ifP xλ

μ will not happen.

Similarly, we know that ( )( )<ifP xη

μ ( )( )ifP xλ

μ will not happen, either. Hence, we conclude

that equation (56) must happen.

Following this trend, from 02 =Y and 02 =Z , we prove that

( )( )=ifP xvη ( )( )ifP xv

λ. (60)

Moreover, from 03 =Y and 03 =Z , we prove that

( )( )=ifP xη

π ( )( )ifP xλ

π . (61)

Equations (56, 60, 61) are valid for ni ,...,1= . Therefore, for ni ,...,1= , we derive that

( ) =iP xη

μ ( )iP xλ

μ (62)

( ) =iP xvη

( )iP xvλ

(63)

and

( ) =iP xη

π ( )iP xλ

π . (64)

From equations (62-64), it yields that λη PP = . However, for a well constructed pattern

recognition problem, two possible patterns cannot be identical. Hence, equation (36) cannot

23

happen to prove that our method can solve the pattern recognition problems.

6. Illustrative example for fault diagnosis of turbine generators

In this section, the proposed method is applied to fault diagnosis of steam turbine

generator under intuitionistic fuzzy circumstance. The fault pattern and detecting sample

adopted from [12] are listed. We select the amplitude ratio of vibration signal in six different

frequency ranges, such as less than 0.4f, 0.5f, f, 2f, 3f and more than 3f, as the characteristic

value. Therein, f is the fundamental frequency of the turbine generator. Three typical failures

are selected as failure patterns. Here, P1 is oil whip, P2 is unbalance, P3 is misalignment. And

B1, B2 are the two samples to be evaluated.

Table 1 presents the six different frequency ranges for the failure patterns. Another, the six

different frequency ranges for two samples are be evaluated in Table 2. Each element of the

tables, is given in the form of a pair of numbers corresponding to the membership, non-

membership values, respectively, e.g., the pattern (P1) vs. frequency (< 0.4f) is described by

(μ, ν) = (0.06, 0.84) on the second column of the Table 1.

In order to find a proper diagnosis, we calculate the similarity measure for each pair of

sample Bi, and the system fault pattern Pj, where { }1,2 ,i∈ and { }1,2,3 .j∈ According to

the calculation, we then assign the ith sample to the most similar pattern according to the

calculations of similarity measure. To be compatible with [12], we use 61=iw for

6,...,1=i and 31=iδ for 3,2,1=i . Moreover, we recall [9] with 2=p , [19] with 2=p ,

[24] and [26] with 61=iw for 6,...,1=i to compare their findings to indicate all have the

same results. The results of the diagnostic technique for the considered samples are given in

Table 3.

24

Table 1 The fault patterns

Patterns Frequency range (f: denote operation frequency)

< 0.4f 0.5f f 2f 3f >3f Oil whip

(P1) (0.06, 0.84) (0.84, 0.02) (0.20, 0.75) (0.02, 0.89) (0.20, 0.75) (0.01, 0.92)

Unbalance (P2)

(0.01, 0.93) (0.02,0.90) (0.90, 0.01) (0.08, 0.85) (0.01, 0.89) (0.02, 0.93)

Misalignment (P3)

(0.01, 0.94) (0.01, 0.94) (0.40, 0.42) (0.40, 0.44) (0.28, 0.56) (0.10, 0.61)

Table 2 The considered samples

Samples Frequency range (f: denote operation frequency)

< 0.4f 0.5f f 2f 3f >3f B1 (0.01, 0.96) (0.00, 0.97) (0.37, 0.60) (0.46, 0.51) (0.31, 0.66) (0.21, 0.75) B2 (0.00, 0.98) (0.05, 0.92) (0.69, 0.27) (0.04, 0.93) (0.03, 0.94) (0.00, 0.97)

Table 3 A comparison with existing methods and our proposed method

Sample 1B Sample 2B

Patterns Patterns 1P 2P 3P 1P 2P 3P

Our method 0.779 0.827 0.918

(1) 0.797

0.939 (1)

0.822

[9] 0.163 0.393 0.839

(1) 0.185

0.795 (1)

0.481

[19] 0.582 0.696 0.920

(1) 0.593

0.897 (1)

0.741

[12] 0.554 0.704 0.926

(1) 0.582

0.893 (1)

0.721

[24] 0.670 0.745 0.953

(1) 0.713

0.933 (1)

0.787

[26] 0.582 0.697 0.923

(1) 0.593

0.898 (1)

0.747

6.1. Diagnosis result for sample B1

From the results are shown in the Table 3, we know that the degree of similarity between

failure pattern Pj , j = 1, 2, 3, and sample B1 has been calculated and we have S(P1,B1)<

S(P2,B1)< S(P3,B1). It is clear that the degree of similarity between P3 and B1 is the highest

25

one, the sample B1 belongs to fault P3 and the most possible type of vibration fault is

misalignment. Thus the check process for fault diagnosis should be P3�P2� P1. The result

is coincided with [12].

6.2. Diagnosis result for sample B2

From the results of Table 3, we obtain that S(P1,B2)=0.7967, S(P2,B2)=0.9389 and

S(P3,B2)=0.8222, respectively. Thus it shows that the degree of similarity between B2 and P2

is the highest one, which means that sample B2 belongs to fault P2 and the most possible type

of vibration fault is unbalance. Thus the sequence of fault checking should be P2�P3� P1.

The result is also coincided with [12].

From the above fault diagnostic results, it can be observed that our proposed diagnosis

method is useful. Therefore, it can offer effective and reasonable diagnosis information for

fault diagnosis.

6.3 Comparison with different approaches

In the following, we will provide two examples to illustrate that our proposed approach

can solve previous unsolved dilemmas. [15] suggested the following similarity measure,

( ) ( ) ( ) ( ) ( ) ( ) ( )( )pn

i

p

iBiA

p

iBiA

p

iBiApL xxxvxvxxn

BAT1

1, 2

11, ⎟

⎠⎞⎜

⎝⎛ −+−+−−= ∑

=

ππμμ . (65)

We proposed an example with four AIFSs, with a sample { }XxxA ii ∈= 0,0, , and three

patterns { }XxxB ii ∈= 3.0,4.0,1 , { }XxxB ii ∈= 4.0,3.0,2 , and { }XxaaxB ii ∈= ,,3 ,

with 30037=a . The special value of a is to create a counter example for any value of

p , the similarity measure proposed by [15] will imply that

26

( ) =ABT pL ,1, ( ) =ABT pL ,2, ( )ABT pL ,3, . (66)

We list the computation results between [15] and our proposed approach in the next Table 4.

Table 4 The comparison between [15] and our proposed approach

[15] ( ) 3917.0,12, =ABTL ( ) 3917.0,22, =ABTL ( ) 3917.0,32, =ABTL

Our proposed approach ( ) 5033.0,11 =ABS ( ) 5033.0,21 =ABS ( ) 5033.0,31 =ABS

( ) 6021.0,12 =ABS ( ) 6095.0,22 =ABS ( ) 6074.0,32 =ABS

From Table 4, it reveals that [15] can not solve the pattern recognition problem. By our

approach, at the second step, we find that the sample A should be belonged to pattern 2B .

For the second example, we recall that [14] defined a similarity measure as follows:

( ) ( ) ( )( )p

n

i

pBAvBAp

pe ii

nBAS ∑

=

+−=1

11, ϕϕμ (67)

where ( ) ( ) ( ) 2iBiABA xxi μμϕμ −= , ( ) ( ) ( ) 2iBiABAv xvxvi −=ϕ , for Xxi ∈ , and

∞<≤ p1 .

Park et al. [18] generalized equation (67) to include the hesitation to provide their new

similarity measures as follows:

( ) ( ) ( ) ( )( )p

n

i

pBABAvBAp

pg iii

nBAS ∑

=

++−=1

11, πμ ϕϕϕ (68)

where ( ) ( )( ){ }nixvxxA iAiA ,...,1:,, == μ and ( ) ( )( ){ }nixvxxB iBiB ,...,1:,, == μ ,

( ) ( ) ( )2

iiBA

xxi ππ

π

μμϕ

−= , with ( )iBAμϕ and ( )iBAvϕ in equation (67).

In Example 3 of [18], there are three patterns 1A , 2A and 3A with

( ) ( ) ( ){ }6.0,0.0,,7.0,1.0,,6.0,2.0, 3211 xxxA = (69)

27

( ) ( ) ( ){ }8.0,2.0,,6.0,0.0,,6.0,2.0, 3212 xxxA = (70)

and

( ) ( ) ( ){ }8.0,2.0,,7.0,2.0,,5.0,1.0, 3213 xxxA = . (71)

We will construct a sample

( ) ( ) ( ){ }7.0,15.0,,65.0,1.0,,55.0,15.0, 321 xxxC = (72)

to illustrate that the similarity measure proposed by [18] sometimes can not solve the pattern

recognition for the sample C . The computation results for [18] and our approach are listed

in the next table.

Table 5 The comparison between [18] and our proposed approach

[18]

( )1

1, 0.8667gS A C = ( )1

2 , 0.8667gS A C = ( )1

3 , 0.8667gS A C =

Our proposed approach

( )1 1, 0.9111S A C = ( )1 2 , 0.9111S A C = ( )1 3 , 0.9111S A C =

( )2 1, 0.9278S A C = ( )2 2 , 0.9278S A C = ( )2 3 , 0.9278S A C =

( )3 1, 0.9278S A C = ( )3 2 , 0.9278S A C = ( )3 3 , 0.9278S A C =

( )4 1, 0.6993S A C = ( )4 2 , 0.6118S A C = ( )4 3 , 0.6421S A C =

From Table 5, it points out that [18] can not solve the pattern recognition problem. By our

approach, at the fourth step, we find that the sample C should belong to pattern A1.

We demonstrate that our mechanism can solve the pattern problem of the counter example

with the given patterns that [15] and [18] failed. It is consistent with our proof that our

proposed algorithm can solve any pattern recognition problem.

7. Computer interface

Applying AIFSs to fault diagnosis analysis to deal with imprecision, uncertainty and

28

fuzziness in decision-making may become a popular research topic in the current uncertain

environment. The application of AIFSs in supporting engineers can provide a useful way to

help them make decisions more efficiently.

In this paper, in order to make the process of computing and ranking much easier, we have

developed an information system called Atanassov’s intuitionistic fuzzy fault diagnosis

decision support system (AIFFDDSS) as shown in Fig. 2. This prototype system is developed

by Visual Basic 6 and ACCESS on a N-tier client server architecture. In Fig. 3, the engineer

needs to key in the parameters setting data, such as the numbers of pattern, the characters for

each pattern and the numbers of sample, respectively. The intuitive values for each pattern

have input, as illustrated in Fig. 4. And the input interface for each sample vs. character is

shown in Fig. 5. The system can calculate the final values of similarity measures between the

sample and each pattern. The fault diagnosis results are shown in Fig. 6. If a pattern has the

highest similarity value, then the sample is more likely belonging to the pattern.

Fig. 2. The functional interface of AIFFDDSS.

29

Fig. 3. The parameter setting for AIFFDDSS.

Fig. 4. Input AIFS for each patterns and input weights.

30

Fig. 5. Input AIFS for samples.

Fig. 6. The outcomes for the fault diagnosis.

8. Conclusion and further study

In this paper, by taking into account the AIFSs, a method based on similarity measures is

proposed. The proposed method has been proved to be effective and been applied to the fault

diagnosis of turbine generator. This method can overcome the situation that the value of the

amplitude ratio of vibration signal is uncertain and thus has a promising future in engineering

applications. Furthermore, ranking of the similarity degree can not only reveal the most

possible pattern of fault occurred but also point out the rank of fault diagnosis. An illustrative

31

example has been provided to illustrate the validity and reasonability of the developed

method in this study. Furthermore, to make the process of computing and ranking much

easier and more efficient, we have developed a computer-based AIFFDDSS system to

effectively aid the decision makers in handling multi-criterion decision making problems

under AIFSs.

Acknowledgements

The authors thank the anonymous referees for several valuable and helpful suggestions

that improved the presentation of the paper. In addition, we also thank Sophia Liu for her

assistance in improving English. This research was supported by National Science Council of

the Republic of China under Grant No. NSC 102-2410-H-241-010-MY2 and NSC 102-2410-

H-015-012.

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a complete pattern recognition approach under atanassov’s intuitionistic fuzzy sets

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