Application of Markov Chains in Interval Valued Intuitionistic Fuzzy MAGDM Problems

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International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019

E-ISSN: 2321-9637

Available online at www.ijrat.org

International Conference on Applied Mathematics and Bio-Inspired Computations

10

th

& 11

th

January 2019

Application of Markov Chains in Interval Valued

Intuitionistic Fuzzy MAGDM Problems

John Robinson P

1

, Usha S

2

Assistant Professor1, Research Scholar2

Department of Mathematics, Bishop Heber College, Tiruchirappalli, India.

[email protected]1, [email protected]

Abstract: The MAGDM problem is based on finding the weights using Markov chains especially when the attribute

weights are completely unknown. The IIFOWA operator and the IIFHA operator are utilized to aggregate the decision matrices provided by the decision-makers. Correlation coefficient of IVIFS is utilized to rank the alternatives and select the most desirable one. A numerical illustration is presented to demonstrate the proposed approach.

Keywords— MAGDM, Ordered Weighted Averaging, Correlation of Interval valued intuitionistic fuzzy sets,

Markov chain.

1. INTRODUCTION

Atanassov and Atanassov & Gargov [1,2] introduced the concept of intuitionistic fuzzy set (IFS) and Interval valued intuitionistic fuzzy sets (IVIFS) which are a generalization of the concept of fuzzy set. Yager [14] developed the Ordered Weighted Averaging (OWA) operator and applied in decision making problems. Xu & Yager [12] developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. Xu [11,13] also developed some arithmetic aggregation operators, such as the intuitionistic fuzzy

weighted averaging (IFWA) operator, the

intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid averaging (IFHA) operator. The interval-valued

intuitionistic fuzzy sets (IVIFSs), which is

characterized by a membership function and a non-membership function whose values are intervals rather than exact numbers, are a very useful means to describe the decision information in the process of decision making. Wei & Wang [10], developed some geometric aggregation operators for MAGDM problems.

Using the approach as in Park et al. [3] we investigate MAGDM problems in which all the information provided by the decision-makers is presented as interval valued intuitionistic fuzzy decision matrices where each of its elements is characterised by Interval Valued Intuitionistic Fuzzy Number (IVIFN). Park et al. [3] proposed an Ordered

Weighted Geometric (OWG) model to aggregate all individual Interval Valued Intuitionistic Fuzzy decision matrices provided by the decision makers into the collective Interval Valued Intuitionistic Fuzzy decision matrix. In the proposed model, from the Markov chain method, the weight information of the decision makers is established to determine the unknown weights. Correlation coefficient is used as a tool to rank alternatives since it preserves the linear relationship between the variables. Robinson & Amirtharaj [4-9] defined correlation coefficient for vague sets Interval vague sets and triangular and trapezoidal intuitionistic fuzzy sets and proposed different MAGDM algorithms. Park et al. [3] have also adopted correlation coefficient as a ranking tool for deciding the best alternatives. In this paper, the correlation coefficient for IVIFSs is utilized for ranking the alternatives. Correlation coefficient of the overall Interval Valued Intuitionistic Fuzzy values and the ideal Interval Valued Intuitionistic Fuzzy Numbers (IVIFN) value is calculated and the ranking of the most desirable alternatives is done based on the obtained correlation coefficients. A MAGDM model based on the Markov chain model for computing weights is presented, and a numerical illustration is given.

2. PREMINARILES

In this section, some basic concepts related to IFSs and IVIFSs are presented.

2.1 Intuitionistic Fuzzy Set

Let X is the universe of discourse. An intuitionistic fuzzy set A in X is an object having the

form



_, _{( ),} _{( )}



A A

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( ), ( ) : [0,1]

A x A x x

   denote membership function and

non-membership function, respectively, of A and

satisfy 0_A( ),x _A( ) 1x  for everyxX.For each

IFS A in X we call _{( )} ₁ _{( )} _{( )}

A x A x A x

    as the

intuitionistic index of x in A. It is hesitation degree (or degree of indeterminacy) of x to A. It is obvious that 0A( ) 1x  for eachxX.

2.2 Interval-Valued Intuitionistic Fuzzy Set

An interval-valued intuitionistic fuzzy set (IVIFS) A in _X_, _X_{ } and card(X) = n, is an object

having the form:



_, _{( ),} _{( ) :}



_,

A A

A x  x  x xX where

: [0,1], : [0,1]

A X D A X D

    with the condition

sup



_A( )x sup



_A( )x 1 for anyxX. Here for eachxX,



A( )x and A( )x are closed intervals and their lower and upper end points are denoted by

( ), ( ), ( ) and ( )

AL x AU x AL x AU x





, respectively, and

thus we can replace with



,[ AL( ), AU( )],[ AL( ), AU( )] :



,

A x  x  x  x  x xX

where ₀ _{( )} _{( ) 1for any}

AU x AU x x X

 

    . For each

( )

AIVIFS X , we call

( ) 1 ( ) ( ) [1 ( ) ( ),1 ( ) ( )]

A x A x A x AU x AU x AL x AL x

        

an intuitionistic fuzzy interval of X in A. Its lower and upper points are



AL( ) 1x  



AU( )x 



AU( )x and

1 ( ) ( )

AU AL x AL x

    , respectively.

2.3 Correlation of Interval-Valued Intuitionistic Fuzzy Sets

Let X={x1, x2,…, xn} be the finite universal

set and be given by

.

For each , Park et al, (2009) defined the

informational intuitionistic energy of A as follows:

.

Correlation of A and B is defined by:

.

Correlation coefficient of A and B is defined by:

Theorem-2.1:

For all , the correlation

coefficient satisfies:

1) .

2) .

3) .

2.4 DEFINITION:

Interval Valued Intuitionistic Fuzzy Ordered

Weighted Average Operator :

where is a weight

vector of IIFOWA operator with (k=1,2,..,l)

and , and s

is the kth

largest of the weighted IVIFN’s ,

i=1,2,..,m, j=1,2,…,n.

2.5 DEFINITION:

Interval Valued Intuitionistic Fuzzy Hybrid Average Operator :

 

1 1 1 1

1 1 i,1 1 i , _i, _i

m m _w m m

w _w _w

ij ij ij ij ij

i i i i

r a b c d

   

   

 _    _{ } _ 



 

 



where of the attributes

can be completely determined in

advance, then bases of the collective interval-valued intuitionistic fuzzy decision matrix

For the ranking order of the alternatives in accordance with the decision making problem, we

give the largest IVIFN as the value

of the ideal alternative for excellent. Then the informational intuitionistic energies of the IVIFN rj

of the alternative Oj and the IVIFN r* of the ideal

alternative O* can be presented, respectively, by

Where

, ( )

A BIVIFS X



 





_i, _AL( ),_i _AU( ) ,_i _AL( ),_i _AU( ) :_i _i



A x  x  x  x  x xX

( )

A IVIFS X

2 2 2 2 2 2

1

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 N

AL i AU i AL i AU i AL i AU i IVIFS

I

x x x x x x

E A      

      





1 1 ( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 n

IVIFS AL i BL i AU i BU i AL i BL i AU i BU i i

C A B  x  x  x  x  x  x  x  x





  



( ) ( ) ( ) ( )

AL xi BL xi AU xi BU xi

      ( , ) ( , ) ( ) ( ) IVIFS IVIFS IVIFS IVIFS

C A B K A B

E A E B





, ( )

A BIVIFS X

( , ) ( , ) IVIFS IVIFS

K A B K B A

0K_IVIFS( ,A B)1 ( , ) 1 IVIFS

A B K A B 



(1) (2) ( )

    

(1) 1 (2) 2

 

( )

, , ,..., ...

l

l l

ij ij ij ij ij ij ij

r IIFOWA  r r r  r   r    r   







 



 

1 1 1 1

1 1 k,1 1 k , k, k

n n n n

k k k k

ij ij ij ij

k k k k

a  b  c  d 

   

   

 _    _{ } _





 





1 2

( , ,..., _l)T





 



0 k   1 1 l k k   



r_ij a b_ij, _ij _{ }, c d_ij, _ij_   ( ) ( ) ( ) ( )

,

k k k k k

ij ij ij ij ij

r



 



a



b



 

 

c



d







 

( )k ( )k lk

ij ij

r  r 

1 2

1 2 1 2

(

,

,...,

)

w w

...

wm

ij j j mj j j mj

r



IIFHA r r

r



r



r

 

r

1 2

(

,

,...,

_m

)

T

w



w w

w

( 1, 2,.. )

k

U k m

 

ij _{m n}

R r





   

* _{1,1 , 0, 0}

r 

*

o

2 2 2 2 2 2

1

( ) ( (1 ) (1 ) ),

2

IVIFS j j j j j j j j j

E r  a      b c d a c   b d

*

( ) 1.

IVIFS

E r 

1 1 1

, , , , i, i, 1 (1 )i

m m m

w w w

j j j j j j ij j ij j ij

i i i

r a b c d a a b b c c

  

   

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International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019

E-ISSN: 2321-9637

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and , is the aggregating IVIFN

for Oj (j=1, 2,…, n), which is obtained by applying

Eq (7). To calculate the correlation

between the IVIFNs r* and rj(j=1, 2, …,n):

And then we

calculate the correlation coefficient of r*

and rj(j=1, 2,…,n):

The

greater the value of , the better the

alternative Oj is closer to the ideal alternative O*.

Therefore, the alternative Oj (j=1, 2,…,n) can be

ranked according to the correlation coefficient so that the best alternative can be selected.

3. WEIGHTS FOR MAGDM PROBLEMS FROM MARKOV CHAIN

The Markov process is a stochastic system

capable of assuming one of n states

s s

₁

, ,..., ,

₂

s

_n and the states change only at discrete points in time. The state at the kth instant depends only on the state of the



k



1



th instant and not on any of the previous states. In other words, in a successive sequence of trials the outcome of the kth trail depends only on the outcome of the



k1



th trial, and not on any of preceding ones.

Transition Probabilities: To describe a Markov

process, we must specify for each state,

s

_i, the probability of making the next transition to each other

of the n states. The transition probability

p

_ijis the probability that if the present state of the process is

s

_i

, the next state will be

s

_j. These probabilities

,

p

_ij

,

must satisfy ₀ ₁

ij

p

  and

1

1 n

ij j

p





Transition Matrix: The

n

2transition probabilities

describing a Markov process can most conveniently be given in the form of an n by n transition matrix

 

ij

P p .any square matrix with real, non-negative

elements in which the sum of each row is 1 is called a stochastic matrix. Thus every stochastic matrix is the transition matrix of some Markov process, and vice versa. Let us look at some properties of a stochastic matrix P.

1. If P is the stochastic matrix, its

k

thpower

P

kis also a stochastic matrix, for k0,1, 2,3...

(matrix 0

1,

P  the identity matrix).

2. If all rows of P are identical, then

1 2 3 4

....,

P  P P P 

3. Since each row of stochastic matrix adds up to

1, only n-1 columns need be given: the

remaining column can be derived from them. In addition to the transition matrix, we also need to know the initial probabilities

 

0

[ (0),

1 2

(0),...,

n

(0)],







For

i



1, 2,..., ,

n

0



_i(0) 1 and 0

(0) 1. n

i i







Consider the following graph: The transition matrix for the graph is given as

follows: 1

1 (1 ) i

m

w

j ij

i

d d



 





*

( , )

IVIFS j

C

r r

* 1

( , ) ( ), 1, 2,..., . 2

IVIFS j j j

C r r  a b j n

*

( , )

IVIFS j

K r r

* *

*

( , )

( , ) , 1, 2,..., .

( ) ( )

IVIFS j IVIFS j

IVIFS IVIFS j

C r r

K r r j n

E r E r

 



*

( , )

IVIFS j

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Similarly calculating the product of this transition matrix after some stage we get:

The entries of the above state are considered as the weights of the attributes described earlier in the stochastic graph.

W= (0.0475, 0.1584, 0.7443, 0.0380, 0.0119)

4. ALGORITHM FOR MAGDM PROBLEMS

Let be the set of n

alternatives, be the set of

decision-maker, and be the

weight vector of decision-makers, where

. Let

be the set of m attributes.

Step: 1 Utilize the IIFOWA operator to

aggregate all individual interval-valued

intuitionistic fuzzy decision matrices

 

(

1, 2,3, 4)

k k

ij m n

R

r

k





into a collective

interval-valued intuitionistic fuzzy decision matrix

R=(rij)mxn.

Step: 2 Derive the weights by the Markov chain

process.

Step: 3 Use the IIFHA operator to get the overall

values rj of the alternatives Oj .

Step: 4 Calculate the informational intuitionistic

energies of the obtained values rj and the

correlation between the value r* of the ideal alternative O* and the value rj .

Step: 5 Calculate the correlation coefficients

KIVIFS(r*,rj) , between the values r* and rj.

Step: 6 Utilize the obtained correlation

coefficients KIVIFS(r*,rj), (j=1.,2,..., n) to rank the

alternatives Oj (j=1, 2,…, n), and then select the

most desirable one(s).

5. NUMERICAL ILLUSTRATION

A problem concerning with a

manufacturing company is discussed, searching the best global supplier for one ofits most critical parts used in assembling process. The attributes which are considered here in selection of four potential global suppliers O_j(j=1, 2, 3, 4) are: U1:

Overall cost of the product; U2: Quality of the

product; U3: Service performance of supplier; U4:

Supplier’s profile; U5: Risk factor.

1 2 3 4

1 2 (1)

3 4 5

[0.5, 0.6],[0.2, 0.3] [0.3, 0.4],[0.4, 0.6] [0.4, 0.5],[0.3, 0.5] [0.3, 0.5],[0.4, 0.5] [0.3, 0.5],[0.4, 0.5] [0.1, 0.3],[0.2, 0.4] [0.7, 0.8],[0.1, 0.2] [0.1, 0.2],[0.7, 0.8] [0.6, 0.7],[0.2, 0.3] [0.3, 0

O O O O

u u R u u u

 .4],[0.4, 0.5] [0.5, 0.8],[0.1, 0.2] [0.1, 0.2],[0.5, 0.8] [0.5, 0.7],[0.1, 0.2] [0.2, 0.4],[0.5, 0.6] [0.4, 0.6],[0.2, 0.3] [0.2, 0.3],[0.4, 0.6] [0.1, 0.4],[0.3, 0.5] [0.7, 0.8],[0.1, 0.2] [0.5, 0.6],[0.2, 0.3] [0.2, 0.3],[0.5, 0.6]

 

 

 

1 2 3 4

1 2 (2)

3 4 5

[0.4, 0.5],[0.2, 0.4] [0.3, 0.5],[0.4, 0.5] [0.4, 0.6],[0.3, 0.4] [0.3, 0.4],[0.4, 0.6] [0.3, 0.4],[0.4, 0.6] [0.1, 0.3],[0.3, 0.7] [0.6, 0.8],[0.1, 0.2] [0.1, 0.2],[0.6, 0.8] [0.6, 0.7],[0.1, 0.2] [0.3, 0

O O O O

u u

R u

u u

 .4],[0.4, 0.5] [0.7, 0.8],[0.1, 0.2] [0.1, 0.2],[0.7, 0.8] [0.5, 0.6],[0.1, 0.3] [0.2, 0.3],[0.6, 0.7] [0.4, 0.6],[0.3, 0.4] [0.3, 0.4],[0.4, 0.6] [0.1, 0.3],[0.3, 0.5] [0.6, 0.8],[0.1, 0.2] [0.5, 0.6],[0.2, 0.4] [0.2, 0.4],[0.5, 0.6]

 

 

 

1 2 3 4

1 2 (3)

3 4 5

[0.4, 0.7],[0.1, 0.2] [0.4, 0.5],[0.2, 0.4] [0.2, 0.4],[0.3, 0.4] [0.3, 0.4],[0.2, 0.4] [0.3, 0.5],[0.3, 0.4] [0.2, 0.4],[0.4, 0.5] [0.6, 0.8],[0.1, 0.2] [0.1, 0.2],[0.6, 0.8] [0.6, 0.7],[0.1, 0.2] [0.4, 0

O O O O

u u

R u

u u

 .5],[0.3, 0.4] [0.5, 0.7],[0.1, 0.3] [0.1, 0.3],[0.5, 0.7] [0.5, 0.6],[0.1, 0.3] [0.1, 0.2],[0.7, 0.8] [0.5, 0.7],[0.2, 0.3] [0.2, 0.3],[0.5, 0.7] [0.3, 0.5],[0.4, 0.5] [0.6, 0.7],[0.2, 0.3] [0.6, 0.8],[0.1, 0.2] [0.1, 0.2],[0.6, 0.8]

 

 

 



1

,

2

,...,

n



O



o o

o



1, 2,..., l



D d d d

l

1 2

( ,

,...,

_l

)

T





 



1

0, 1, 2,..., , 1

l

k k

k

k l and





 







1

,

2

,...,

m



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1 2 3 4

1 2 (4)

3 4 5

[0.6, 0.7],[0.2, 0.3] [0.4, 0.5],[0.4, 0.5] [0.4, 0.5],[0.3, 0.4] [0.3, 0.4],[0.4, 0.5] [0.3, 0.4],[0.3, 0.4] [0.1, 0.2],[0.2, 0.3] [0.6, 0.7],[0.1, 0.3] [0.1, 0.3],[0.6, 0.7] [0.7, 0.8],[0.1, 0.2] [0.3, 0

O O O O

u u

R u

u u

 .4],[0.5, 0.6] [0.5, 0.8],[0.1, 0.2] [0.1, 0.2],[0.5, 0.8] [0.5, 0.6],[0.1, 0.3] [0.2, 0.3],[0.4, 0.6] [0.4, 0.5],[0.2, 0.3] [0.2, 0.3],[0.4, 0.5] [0.1, 0.2],[0.5, 0.7] [0.6, 0.7],[0.1, 0.2] [0.5, 0.6],[0.3, 0.4] [0.3, 0.4],[0.5, 0.6]

 

 

 

STEP:1 Utilize the IIFOWA operator to aggregate all individual

interval-valued intuitionistic fuzzy decision matrices

into a collective interval-valued

intuitionistic fuzzy decision matrix R=(rij)mxn.

STEP: 2 To derive a weight vector wby using the

Markov chain process:

W= (0.0475, 0.1584, 0.7443, 0.0380, 0.0119)

STEP: 3

Using IIFHA operator to obtain the overall value r_j



j1, 2,3, 4,5



of the alternative

O

_j(j=1,2,3,4):

Now the collective interval-valued intuitionistic fuzzy decision matrix R is as follows:



 



1 0.3681, 0.5298 , 0.0404, 0.2840

r   _



 



2 0.4519, 0.5758 , 0.3639, 0.4979

r   _



 



3 0.4453, 0.6368 , 0.1121, 0.2234

r   _



 



4 0.2561, 0.3844 , 0.5625, 0.7111

r   _

Step: 4 To calculate the informational

intuitionistic energy EIVIFS(rj) of the obtained

values rj (j=1, 2, 3, 4) and to calculate the

correlation CIVIFS(r*,rj) between the value r* of the

ideal alternative O* and the value rj (j=1, 2, 3, 4).

1 2 3

4

( ) 0.4413; ( ) 0.4777; ( ) 0.4409;

( ) 0.5388.

IVIFS IVIFS IVIFS

IVIFS

E r E r E r

E r

  



.

 

* * *

1 2 3

* 4

, 0.2081; , 0.5139; , 0.5411; , 0.3203.

IVIFS IVIFS IVIFS

IVIFS

C r r C r r C r r

C r r

  



Step: 5 To calculate the correlation coefficient

KIVIFS(r*,rj) between the values r* and rj (j=1, 2, 3,

4).

 

* * *

1 2 3

* 4

, 0.3133; , 0.7435; , 0.8149;

, 0.4364.

IVIFS IVIFS IVIFS

IVIFS

K r r K r r K r r

K r r

  



Ranking the alternatives based on correlation

coefficients, we get:

3 2 4 1

O O O O

Hence the most desirable global supplier is O3.

6. CONCLUSION

We have investigated the MAGDM problems under interval-valued intuitionistic fuzzy environment, and proposed an approach to handling the situations where the attribute values are characterized by IVIFNs, and the information about attribute weights completely unknown. The proposed approach first fuses all individual

interval-valued intuitionistic fuzzy decision

matrices into the collective interval-valued intuitionistic fuzzy decision matrix by using the IIFOWA operator. Then we have used the weights calculated from markov chain method and the IIFHA operator to get the overall interval-valued intuitionistic fuzzy values of alternatives and have used the proposed method for calculating correlation coefficients between IVIFNs to rank the alternatives and then to select the most desirable one. The proposed approach in this work not only can comfort the influence of unjust arguments on the decision results, but also avoid losing or distorting the original decision information in the process of aggregation. Thus, the proposed approach provides us an effective and practical way to deal with person multi-attribute decision making problems, where the attribute values are characterized by IVIFNs and the information about attribute weights is partially known.

 

(

1, 2,3, 4)

k k

ij m n

R

r

k





1 2

1 2 1 2

( ,

,...,

)

w w

,...,

wm

j j j mj j j mj

r



IIFHA r r

r



r



r



r

1 2

1 2 3 4 5

[0.4279, 0.6079],[0.1088, 0.2103] [0.4521, 0.5613],[0.3967, 0.5523] [0.2359, 0.3499],[0.2499, 0.4121] [0.2025, 0.3869],[0.3513, 0.5058] [0.5829, 0.7503],[0.0718, 0.1562] [0.4182, 0.5153],[0.4719, 0.567

O O

u u R u u u

 8]

[0.4354, 0.5607],[0.0631, 0.2187] [0.2386, 0.3656],[0.6061, 0.7333] [0.1233, 0.2961],[0.3281, 0.5003] [0.6800, 0.7820],[0.1919, 0.3086]

       

3 4

[0.2716, 0.4204],[0.2357, 0.3469] [0.3818, 0.4965],[0.3968, 0.5524] [0.5607, 0.7308],[0.0631, 0.1714] [0.1586, 0.3151],[0.6774, 0.8062] [0.4753, 0.7308],[0.0631, 0.1714] [0.1586, 0.3151],[0.6079, 0.8062] [0.3699, 0.

O O

5514],[0.1562, 0.2488] [0.2939, 0.3984],[0.5109, 0.6594] [0.4743, 0.6352],[0.1287, 0.2369] [0.2808, 0.4033],[0.6039, 0.7195]

(6)

International Journal of Research in Advent Technology (IJRAT) Special Issue, January

2019

E-ISSN: 2321-9637

10

th

& 11

th

January 2019

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