Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 579 - 590

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2017.7120

Reliability Analysis of Water System Using

Intuitionistic Fuzzy Correlation Coefficient

M. K. Sharma

Department of Mathematics, R.S.S. (PG) College, Pilkhuwa, (Hapur) U.P., India

Copyright © 2017 M. K. Sharma. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

In the present research paper we have proposed formula for correlation coefficient

for intuitionistic fuzzy sets and applied it on the data of [Neera et al 2009], first

we convert these classical data into fuzzy data by multiply a numerical value such

that all data of any parameter lies between the interval [0, 1], and in this way we

get the membership function of an intuitionistic fuzzy sets, than we define the

non-membership function. After finding the intuitionistic fuzzy correlation

coefficient between the different parameters we also propose a concept of

proportion of reliability in common related to intuitionistic fuzzy correlation

coefficient (IFCC).

Mathematics Subject Classification: 60K10

Keywords: Fuzzy sets, Intuitionistic Fuzzy sets, Correlation, intuitionistic fuzzy

correlation coefficient (IFCC), Reliability

1 Introduction

It is very important in statistical analysis of data to find the correlation

coefficient between the variables or attributes, the correlation coefficient defined

on crisp sets have been studied in many works of conventional statistics. The

correlation analysis can also be employed to study the nature of relationship

between the variables or attributes. It is interesting to see how the notation of

correlation can be extended to fuzzy sets. In 1985, Murthy et.al. [1] (CFIS-4)

proposed a measure of correlation between two membership functions satisfying

some assumptions, which lie in the interval [-1, 1]. Ding-An Chiang, Nancy P.Lin

580 M. K. Sharma

[10] has introduced the correlation of fuzzy sets. Dug Hun Hong [5, 6], introduced

the correlation of fuzzy numbers.

Motivated by the above work, in this paper first we discuss the fuzzy

correlation of fuzzy data using the results of Ding-An Chiang, Nancy P.Lin

[10],[CFIS5], then we propose the parallel results of correlation for intuitionistic

fuzzy sets, from the results of correlation of fuzzy sets, because our focus is

evaluating the correlation between the vague values. The intuitionistic fuzzy

correlation coefficient is much better than the fuzzy correlation coefficient,

because in intuitionistic fuzzy sets we take the membership and non-membership

functions, instead of only membership function of fuzzy sets.

For applying the results of intuitionistic fuzzy correlation coefficient, we deal with

the intuitionistic fuzzy correlation between Physico-chemical characteristics of

lakes around Jaipur, Rajasthan. In present study source data were collected and

used from [Neera et al 2009], [11] in which authors have annually surveyed

Jalmahal, Amer, Nevta, and Ramgarh lakes to determine Physico-chemical

characteristic of water around Jaipur city in India. Non-degraded metals and

organic pollutants tend to accumulate in various vital organs of fishes and lead to

long term toxic effect, in which Zink is an essential trace element in living

organisms. This study is significance because fish from these water bodies are

consumed by people in the city, apparently healthy fishes may infact be carrier of

pollutants which will be biomagnified in health of the human bodies.

On the basis of author’s observation the study is performed and we applied

the intuitionistic fuzzy correlation in the data to verify the impact factor by change

in one parameter on the basis of temperature, as we know that all the

environmental factors are complementary to each other. As only temperature play

a vital role in environment because change in it characteristics, change the all

physical, chemical and biological life lives within all segments of environment.

2. Some Basic Definitions

2.1 Reliability: Reliability is defined as the probability that the system gives an

adequate performance for a specified period under prescribed operating

conditions.

Mathematically reliability of a system can be stated in the following

manner:

Let T denotes the time variable of the failure of a system. Then reliability of this

system is

R (T) =P (T>t)

=

t

duuf )(

= 1-F (t), t 0

Where f(.) and F(.) are pdf and cdf of T respectively.

Reliability analysis of water system 581

2.2 Correlation Coefficient of Fuzzy Sets:-

Definition 2.2.1:-Let ...........,,, 4321 nxxxxx be a random sample of size n from X,

alone with the membership grades of fuzzy set A~

, then the sample mean and

sample variance of the membership function of A~

, defined on X, can be define as:

,

)(1

~

~

n

xn

k

kA

A

1

)(

2

1

~~

2~

n

x

S

n

kAkA

A

Where

A~ denotes the average membership grades of fuzzy set A

~over the

random sample Xxxxxx n ...........,,, 4321 and 2

~A

S represent the degree of

variations of membership grades of fuzzy set A~

over Xxxxxx n ...........,,, 4321 ,

then standard deviation will be 2~~AA

SS . Now consider the two fuzzy sets

A~

and B~

, defined on crisp set X, then A~

and B~

can be expressed

as: ,:)(,~

~ XxxxAA

XxxxBB

:)(,~

~ ,

where ],1,0[:~ XA

].1,0[:~ XB

Definition 2.2.2:- Let there is a random sample Xxxxxx n ...........,,, 4321 along

with a sequence of paired data

,)(),(............................)(),(,)(),( ~~2~2~1~1~ nBnABABAxxxxxx which

correspond to the grads of membership function of fuzzy sets and B~

defined on

X, now let us Define the correlation

coefficient,

BA

n

kBkBAkA

BA SS

nxx

~~

1

~~~~

~~

)1/()())(

(1)

As defined in definition 1, A~ and

B~ denote the average membership grades of

fuzzy sets A~

and B~

over the random sample.

3. Correlation between Intuitionistic Fuzzy sets

Definition 3.1:- Let Xxxxxx n ...........,,, 4321 be a random sample of size n

from population X , along with the grades of an intuitionistic fuzzy set iA

~then the

sample mean and sample variance of intuitionistic fuzzy set iA

~defined on X , can

be given as:

n

xx

E

n

k

kAkA

A

ii

i

1

~~

~

)(1)(2

1

(2)

582 M. K. Sharma

And

1

)(1)(2

12

1

~~~

2~

n

Exx

S

n

kAkAkA

A

iii

i

(3)

Where iAE ~ represent the average membership grade of intuitionistic fuzzy set iA

~

over the random sample Xxxxxx n ...........,,, 4321 while 2~ iA

S represent the degree

of variation of intuitionistic fuzzy set iA~

over the random

sample ............,,, 4321 Xxxxxx n Then the sample standard deviation will be

written as, 2~~ ii AA

SS

Definition 3.2:- Let there is a random sample ,...........,,, 4321 Xxxxxx n along

with a sequence of paired intuitionistic fuzzy sets iA~

and iB

~ ie.,

)(),(,)(),(, 1~1~1~1~1 xxxxx iiii BBAA ,

)(),(,)(),(, 2~2~2~2~2 xxxxx iiii BBAA ,

)(),(,)(),(, 3~3~3~3~3 xxxxx iiii BBAA ,

.

)(),(,)(),(, ~~~~ nBnBnAnAn xxxxx iiii ,

Which correspond to the grade of membership and nonmember-ship of

intuitionistic fuzzy sets iA~

and iB

~ defined on the random sample

,...........,,, 4321 Xxxxxx n respectively?

Here iA~

and iB

~are two fuzzy sets in X, where

XxxxxA kkAkAk

iii :)(),(,

~~~

and

XxxxxB kkBkBk

iii :)(),(,

~~~

with the conditions

1)()(0 ~~ kAkAxx ii and 1)()(0 ~~ kBkB

xx ii must hold for all Xxk

Now the covariance of intuitionistic fuzzy sets iA

~and

iB~

defined on X can be

defined as

1

)(1)(2

1)(1)(

2

1

~,

~ cov

1

~~~

~~~

n

ExxExx

BA

n

k

BkBkBAkAkA

ii

iii

iii

(4)

And the correlation coefficient of intuitionistic fuzzy sets iA

~and

iB~

defined on X

can be defined as,

BA

ii

BA SS

BA

~~

~~

~,

~ cov

Reliability analysis of water system 583

ii

iii

iii

BA

n

k

BkBkBAkAkA

SS

nExxExx

~~

1

~~~

~~~ 1)(1)(

2

1)(1)(

2

1

(5)

But

1

)(1)(2

12

1

~~~

~

n

Exx

S

n

kAkAkA

A

iii

i

and

1

)(1)(2

12

1

~~~

~

n

Exx

S

n

kBkBkB

B

iii

i

Therefore

)1()(1)(2

1)1()(1)(

2

1

)1()(1)(2

1)(1)(

2

1

2

1

~~~

2

1

~~~

1

~~~

~~~

~~

nExxnExx

nExxExx

n

kBkBkB

n

kAkAkA

n

k

BkBkBAkAkA

BA

iiiiii

iii

iii

2

1

~~~

2

1

~~~

1

~~~

~~~

~~

)(1)(2

1)(1)(

2

1

)(1)(2

1)(1)(

2

1

n

kBkBkB

n

kAkAkA

n

k

BkBkBAkAkA

BA

iiiiii

iii

iii

ExxExx

ExxExx

(6)

Where iAE ~ and iB

E ~ denote the average membership grade of intuitionistic fuzzy

sets iA~

and iB

~ over the random sample ,...........,,, 4321 Xxxxxx n , respectively.

iAS ~ and iB

S ~ represent the sample standard deviations of intuitionistic fuzzy sets

iA~ and

iB~

over the random sample ,...........,,, 4321 Xxxxxx n respectively.

Result3.2.1:- Let iA~

and iB~

be the two intuitionistic fuzzy sets on X, with

membership function iA~ and iB

~ and non-membership function

iA~ and iB

~ respectively, take a random sample ,...........,,, 4321 Xxxxxx n along

with a sequence of paired grades of intuitionistic fuzzy setsiA

~and

iB~

, then the

sample intuitionistic fuzzy correlation coefficient BA~~ defined by (6) will be lie in

[-1, 1], that is 1~~ BA

and

1~~

BA , if )(1)()(1)( ~~~~ kAkAkBkB

xxmxx iiii

4. A study based on correlation coefficient of intuitionistic fuzzy

sets

Present study deals with the intuitionistic fuzzy correlation between Physico-chemical characteristics of lakes around Jaipur city, in Rajasthan. In present

584 M. K. Sharma

study source data were collected and used from [Neera et al 2009], in which

authors have annually surveyed Jalmahal, Amer, Nevta, and Ramgarh lakes to

determine Physico-chemical charecterstic of water around Jaipur city in india.

Here all the water samples were collected (30 cm depth) for a period of 12

months (from January 2005 to December 2006). From these samples we take the

different parameter as atmospheric temperature (0C), water temperature (0C), free

CO2 (mg l-1), dissolved oxygen (mg l-1), Hardness (mg l-1), Zink (mg l-1) and

Endosulfan (mg l-1).

On the basis of above observation the study is performed and intuitionistic

fuzzy correlation is applied in above data to verify the impact factor by change in

one parameter on the basis of temperature. For applying intuitionistic fuzzy

correlation first we convert these data into a grade of membership and a grade of

non-membership such as 1)(0 ~~ AA

x .

1. Collected data:-

Collected data of Jalmahal lake, Amer lake, Mevtal lake and Ramgarh lake shown

in table 1, 2, 3 and 4 respectively, atmospheric temperature taking as the average

of maximum and minimum temperature during the photoperiod.

(1) Table 1 Collected data from Jalmahal lake during the year (2005-2006).

M

on

ths Atmos temp.(0C) Water

temp.

(0C)

Free

CO2

(mg l-1 )

Dessolved

Oxygen)

(mg l-1 )

Hard-

Ness

(mg l-1 )

Zink

(mg l-1 )

Endos-

ulfan

(mg l-1 ) Max Min Aver

age

Jan.

Feb.

Mar.

Apr.

May.

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

21.5

24.8

28.6

39.4

39.7

40.7

33.4

36

40

35.2

27.3

24.6

3.8

15.6

19

23.8

26.4

28.6

23.8

26

24.8

20.9

11.6

4.8

12.65

20.2

23.8

31.6

33.05

34.65

28.6

31

32.4

28.05

19.45

14.7

19.6

16.1

24

31

32

31

30

32

34.1

32

28.4

24.3

20.33

25

28

32

27

24

24.2

24.39

27.86

29.33

21.27

20.4

6.61

3.92

3.7

3.64

3.51

4.2

5.5

5.8

6.21

5.94

5.8

6.34

382

182

320

416

404.7

398.7

350

358

366

377.3

379.3

380.7

0.197

0.161

0.168

0.174

0.177

0.182

0.183

0.186

0.187

0.187

0.196

0.197

1.324

0.285

0.307

0.313

0.325

0.356

0.574

0.611

0.642

0.876

1.199

1.271

Reliability analysis of water system 585

(2) Table 2 Collected data from Amer lake during the year (2005-2006)

Mon

ths

Atmos temp.(0C) Water

temp.

(0C)

Free

CO2

(mg l-1 )

Dessolved

Oxygen)

(mg l-1 )

Hard-

Ness

(mg l-1 )

Zink

(mg l-1 )

Endos-

ulfan

(mg l-1 ) Max Min Aver

age

Jan.

Feb.

Mar.

Apr.

May.

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

21.5

24.8

28.6

39.4

39.7

40.7

33.4

36

40

35.2

27.3

24.6

3.8

15.6

19

23.8

26.4

28.6

23.8

26

24.8

20.9

11.6

4.8

12.65

20.2

23.8

31.6

33.05

34.65

28.6

31

32.4

28.05

19.45

14.7

17.2

16.5

23

30.5

31.1

31

30

31

33

32.4

24.6

22

16.13

22.2

20.2

20.53

20

20

20.4

20.53

23.46

23.46

29.33

19.07

7.15

4.86

5

4.99

5.4

5.81

5.7

5.53

5.67

5.54

5.4

6.34

157.3

170

204

182.7

192

182.7

155.3

163.3

176.7

180

182

156.7

0.167

0.141

0.147

0.143

0.143

0.147

0.152

0.155

0.158

0.158

0.164

0.167

1.176

0.149

0.151

0.165

0.212

0.276

0.278

0.381

0.483

0.514

0.742

1.161

(3) Table 3 Collected data from Mevtal lake during the year (2005-2006)

Mon

ths

Atmos temp.(0C) Water

temp.

(0C)

Free

CO2

(mg l-1 )

Dessolved

Oxygen)

(mg l-1 )

Hard-

Ness

(mg l-1 )

Zink

(mg l-1 )

Endos-

ulfan

(mg l-1 ) Max Min Aver

age

Jan.

Feb.

Mar.

Apr.

May.

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

21.5

24.8

28.6

39.4

39.7

40.7

33.4

36

40

35.2

27.3

24.6

3.8

15.6

19

23.8

26.4

28.6

23.8

26

24.8

20.9

11.6

4.8

12.65

20.2

23.8

31.6

33.05

34.65

28.6

31

32.4

28.05

19.45

14.7

15.2

14.1

22.4

28.4

28.4

30.2

29.3

30

31.1

30.2

22.6

18.7

14.67

19.5

20

20.5

20.5

18.5

18

19.6

23.46

24.93

27.86

20.53

7.29

5.67

5.5

5.4

5.4

5.54

6.21

5.94

6.61

6.21

6.08

6.61

158.7

162

250

158

148

144.7

139.3

146.7

155.3

156.7

156

156.7

0.156

0.132

0.136

0.136

0.139

0.14

0.142

0.146

0.146

0.148

0.148

0.152

0.198

0.111

0.115

0.116

0.117

0.117

0.118

0.118

0.12

0.13

0.155

0.186

586 M. K. Sharma

(4) Table 4 Collected data from Ramgarh lake during the year

Mon

ths

Atmos temp.(0C) Water

temp.

(0C)

Free

CO2

(mg l-1 )

Dessolved

Oxygen)

(mg l-1 )

Hard-

Ness

(mg l-1 )

Zink

(mg l-1 )

Endos-

ulfan

(mg l-1 ) Max Min Aver

age

Jan.

Feb.

Mar.

Apr.

May.

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

21.5

24.8

28.6

39.4

39.7

40.7

33.4

36

40

35.2

27.3

24.6

3.8

15.6

19

23.8

26.4

28.6

23.8

26

24.8

20.9

11.6

4.8

12.65

20.2

23.8

31.6

33.05

34.65

28.6

31

32.4

28.05

19.45

14.7

14.6

14

22

28.4

28.2

30

29

30

30.1

30

22

18

0

0

0

0

0

0

0

0

0

0

0

0

8.64

6.62

6.76

7.02

7.29

7.57

7.97

8.1

8.51

8.1

8.23

8.64

94.7

156.7

156.7

131.3

110.7

96.7

91.3

94

96.7

95.3

96.7

95.3

0.142

0.12

0.123

0.123

0.127

0.127

0.133

0.133

0.136

0.138

0.138

0.139

0.05

0.02

0.028

0.029

0.036

0.037

0.038

0.045

0.046

0.047

0.048

0.049

Now the graphical representation of atmospheric temperature (0C), water

temperature (0C), free CO2 (mg l-1), dissolved oxygen (mg l-1), Hardness (mg l-1),

Zink (mg l-1) and Endosulfan (mg l-1), of Jalmahal lake, Amer lake, Mevtal lake

and Ramgarh lake.

5. Intuitionistic Fuzzy Correlation Coefficient between different

parameters during the year (2005-2006).

(1) Table 1

Lakes

Intuitionistic

fuzzy

Correlation

Coefficient

Between

Jalmahal

lake

Amer lake

Mevtal lake

Ramgarh

lake

Atmos. temp. (oC)

& Water temp. (oC)

0.888053

0.876474

0.907773

0.938686061

Water temp. (oC) &

Dissolve

Oxygen(mg/lit)

-0.105767

-0.302258

-0.3895564

-0.085572

Reliability analysis of water system 587

Water temp. (oC) &

Free CO2 (mg/lit)

0.481121

-0.0554471

0.057366

0.473074

Water temp. (oC) &

Hardness (mg/lit)

0.426989

0.198680

-0.014044

0.025622

Water temp. (oC) &

Zink (mg/lit)

0.020166162

-0.2957026

-0.294693656

0.21872060

Water temp. (oC) &

Endosulfan (mg/lit)

-0.3977677

-0.571746

-0.46550952

0.15918532

6. Proportion of reliability in common related to intuitionistic

fuzzy correlation coefficient

Another important aspect of correlation is that a correlation coefficient is

important or not. We can find out the probability that it is a chance finding; just

we can see that in correlation coefficient the more interesting and ultimately more

important price of information is the proportion of variability in common between

the two variables. Basically, it can tell us whether the relationship is important or

no, we can find this proportion by squaring the correlation coefficient. For

example if we have a correlation coefficient of r =0.90, the square of that number

is 0.81 ie. 81% of the variability of two variables is in common between them. A

correlation coefficient of r =0.80 sounds not too different from a correlation

coefficient of .90, but the proportion of common variability is only 64% for the

first and 81% for the second. The following figure shows the proportion of

variability in common between two variables for different levels of correlation.

588 M. K. Sharma

7. Results and discussion

(1) IFCC between Atmos. temperature & Water temperature with

proportion of variability:

The temperature and Photoperiod was taken as main parameter. The intuitionistic

fuzzy correlation coefficient (IFCC) between atmospheric temperature and water

temperature is finding to be highly positive in all the seasons during all the

months for all lakes of Jaipur city, i.e. if atmospheric temperature is increase or

decrease in any season of any 4 lakes then the water temperature is also increase

or decrease. The maximum positive IFCC is 0.938686061 of Ramgarh Lake and

minimum positive IFCC is 0.876474 of Amer lake. Therefore the proportion of

common variability is only 88% in Ramgarh lake and 76% in Amer lake.

(2) IFCC between water temperature & Dissolve Oxygen with proportion of

Variability: The intuitionistic fuzzy correlation coefficient (IFCC) between water temperature

& Dissolve Oxygen is finding to be slightly negative in all the seasons during all

the months for all lakes of Jaipur city, i.e. if water temperature is increase or

decrease in any season of any 4 lakes then the Dissolve Oxygen is slightly

decrease or increase The most negative IFCC is -0.389556 of Mevtal lake and

minimum negative IFCC is -0.085572 in Ramgarh lake.But the proportion of

common variability is only 15% in Mevtal lake and 0.7% in Ramgarh lake.

(3) IFCC between water temperature & Free CO2with proportion of

variability: The intuitionistic fuzzy correlation coefficient (IFCC) between water temperature

& Free CO2 is finding to be slightly Positive in Jalmahal, Mevtal and Ramgarh

lakes and slightly negative in Amer lake. If water temperature is increase or

decrease in any season of these 3 lakes then the Free CO2 is also slightly increase

or decrease, but in Amer lake is not like so. The most positive IFCC is 0.481121

of Jalmahal lake and most negative IFCC is -0.05544 of Amer lake. But the

proportion of common variability is only 23% in Jalmahal lake and 0.3% in Amer

lake.

(4) IFCC between water temperature & Hardness with proportion of

variability: The intuitionistic fuzzy correlation coefficient (IFCC) between water temperature

& Hardness is finding to be slightly Positive in Jalmahal, Amer and Ramgarh

lakes and slightly negative in Mevtal lake. If water temperature is increase or

decrease in any season of these 3 lakes then the Hardness is also slightly increase

or decrease, but in Mevtal lake is not like so. The most positive IFCC is 0.4269 of

Jalmahal lake and most negative IFCC is -0.014044 of Mevtal lake.But the

proportion of common variability is only 17% in Jalmahal lake and 0.02% in

Mevtal lake.

Reliability analysis of water system 589

(5) IFCC between water temperature & Zink with proportion of variability: The intuitionistic fuzzy correlation coefficient (IFCC) between water temperature

& Zink is finding to be slightly positive in Jalmahal and Ramgarh lakes and

slightly negative in Amer and Mevtal lake. If water temperature is increase or

decrease in any season of Jalmahal and Ramgarh lakes then the zink is also

slightly increase or decrease, but if water temperature is increase or decrease in

any season of Amer and Mevtal lake then the zink is also slightly decrease or

increase. The most positive IFCC is 0.218720 of Ramgarh lake and minimum

negative IFCC is -0.294693 of Mevtal lake.But the proportion of common

variability is only 4.7% in Ramgarh lake and 8.6% in Mevtal lake.

(6) IFCC between water temperature & Endosulfan with proportion of

variability: The intuitionistic fuzzy correlation coefficient (IFCC) between water temperature

& Endosulfan is finding to be slightly negative in Jalmahal, Amer and Mevtal

lakes and slightly positive in Ramgarh lake. if water temperature is increase or

decrease in any season of these 3 lakes then the Endosulfan is also slightly

decrease or increase, but in Ramgarh lake if water temperature increase then

Endosulfan is also increase and vice-versa.The most positive IFCC is 0.159185 of

Ramgarh lake and most negative IFCC is -0.397767 of Jalmahal lake.But the

proportion of common variability is only 2.5% in JRamgarh lake and 15% in

jalmahal lake.

References

[1] C. A. Murthy, S.K. Pal, D. D. Majumder, Correlation between two fuzzy

membership functions, Fuzzy Sets and Systems, 17 (1985), 23–38.

https://doi.org/10.1016/0165-0114(85)90004-1

[2] C. H. Cheng, D. L. Mon, Fuzzy system reliability analysis by interval of

confidence, Fuzzy Sets and Systems, 56 (1993), no. 1, 29-35.

https://doi.org/10.1016/0165-0114(93)90182-h

[3] C. Y. Wen and K. Y. Cai, Fuzzy variables as a basis for a theory of fuzzy

reliability in the possibility context, Fuzzy Sets and Systems, 42 (1991), 145-

172. https://doi.org/10.1016/0165-0114(91)90143-e

[4] C. Y. Wen, K. Y. Cai and M. L. Zhang, Posbist reliability behaviour of

typical system with two type of failure, Fuzzy Sets and Systems, 43 (1991),

17-32. https://doi.org/10.1016/0165-0114(91)90018-l

[5] Dug Hun Hong, Seok Yoon Hwang, Correlation of intuitionistic fuzzy sets in

probability spaces, Fuzzy Sets and Systems, 75 (1995), 77-81.

https://doi.org/10.1016/0165-0114(94)00330-a

590 M. K. Sharma

[6] Dug Hun Hong, Seok Yoon Hwang, A note on the correlation of fuzzy

number, Fuzzy Sets and Systems, 79 (1996), 401-402.

https://doi.org/10.1016/0165-0114(95)00146-8

[7] G. Tadeusz and Jacek Manko, Correlation of intuitionistic fuzzy sets, Fuzzy

Sets and Systems, 44 (1991), 39-43.

https://doi.org/10.1016/0165-0114(91)90031-k

[8] M. L. Puri and D. Ralescu, Fuzzy random variables, Journal Math. Anal.

Appl., 114 (1986), 409-422. https://doi.org/10.1016/0022-247x(86)90093-4

[9] M. K. Sharma et al., Vague Reliability Analysis for Fault Diagnosis of

Cannula Fault in Power Transformer, Applied Mathematical Sciences, 8

(2014), no. 18, 851 – 863. https://doi.org/10.12988/ams.2014.310543

[10] Nancy P. Lin, Hung- Jen Chen, A fuzzy statistics based method for mining

fuzzy correlation rules, WSEAS Transactions on Mathematics, 11 (2007),

852-857.

[11] Neera Srivastava, Garima Harit and Ram Srivastava, A study of physico-

chemical characteristics of lakes around Jaipur, India, Journal of

Environmental Biology, 30 (2009), no. 5, 889-894.

[12] Wen Liang Hung, Jong Wuu Wu, Correlation of intuitionistic fuzzy sets by

centroid method, Information Sciences, 144 (2002), 219-225.

https://doi.org/10.1016/s0020-0255(02)00181-0

Received: January 30, 2017; Published: March 2, 2017