Solving MAGDM Problems Using Singularly Perturbed Differential Equation of Reaction Diffusion Type

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Solving MAGDM Problems Using Singularly Perturbed

Differential Equation of Reaction Diffusion Type

Indhumathi M

1

, John Robinson P

2 Research Scholar1, Assistant Professor2 Bishop Heber College, Tiruchirappalli, India. [email protected], [email protected]

Abstract- In this paper, the Multiple Attribute Group Decision Making (MAGDM) problems with interval-valued intuitionistic fuzzy sets are considered. The exact solution of singularly perturbed differential equation is used to derive the decision maker weights and it is applied in MAGDM problems under interval valued

intuitionistic fuzzy environment. The Interval-Valued Intuitionistic Fuzzy Ordered Weighted Geometric (IIFOWG) operator is utilized for the decision making process and the decision maker weights to identify the best alternatives. The effectiveness of the proposed approach is done through numerical illustration.

Key Words: MAGDM, Intuitionistic Fuzzy Sets, Interval Valued Intuitionistic Fuzzy Sets, Interval Valued Intuitionistic Fuzzy Ordered Weighted Geometric (IIFOWG)operator, Interval Valued Intuitionistic Fuzzy Hybrid (IIFHG) operator, Singularly perturbed differential equations.

1. INTRODUCTION

Multiple attribute group decision making is an important task in human activities. It consists of finding the most preferred alternative from a given alternative set. It has been extensively applied to various areas such as society, economics, management, military and engineering technology. For example, investment decision making, project evaluation, economic evaluation, personnel evaluation, etc. Atanassov [1] introduced the concept of intuitionistic fuzzy set (IFS) consists of both a membership and non-membership function, which is a generalization of the concept of fuzzy set. Atanassov & Gargov [3], and Atanassov [2] developed the concept of interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of the IFS. The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. Chen & yang [4] investigated the MAGDM problem with intuitionistic fuzzy information is very useful for solving complicated decision problems under uncertain circumstance. Grzegorzewski [6] proposed the distance measures using Hausdroff space. Xu & Chen [23] developed some geometric aggregation operators, such as the interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator and the interval-valued intuitionistic fuzzy ordered weighted geometric (IIFOWG) operator, and the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG). Indumathi & Robinson [8], Robinson & Indumathi [18-19] and Robinson & Manjumari [16-17] developed the new concept for finding the

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method. In this work, we have investigated the MAGDM problem with interval valued intuitionistic fuzzy set for ranking the alternatives together with IIFOWG and IIFHG operators. A numerical illustration is given to show the effectiveness of the proposed approach.

2. PRELIMINARIES

In this section, some basic definitions and interval valued intuitionistic fuzzy weighted geometric operator are presented.

2.1. Basic Concepts of Interval Valued Intuitionistic Fuzzy Set

.

DEFINITION: (Interval-Valued Intuitionistic Fuzzy Set)

An interval-valued intuitionistic fuzzy set (IVIFS) A in X is an object having the form:

{ ,

_A

( ),

_A

( ) /

}

A



x



x



x



X

Where

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

A X A X







 with the condition

sup



_A

( ) sup

x





_A

( ) 1

x



, xX. The intervals

( )

A

x



and



A

( )

x

denote, the degree of

belongingness and the degree of non-belongingness of the element x to A, respectively. We denote by IVIFS(X) the set of all IVIFSs in X. Then for each

x



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





AU

( )

x





AU

( ) 1,

x



x



X

. DEFINITION:

For each

A



IVIFS X

( )

, the degree of an intuitionistic fuzzy interval of X in A is defined

and denoted by

( ) 1

( )

[1

( )

( ),1

( )

( )]

A A A

AU AU AL AL

x















 



 



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



_AL

( ) 1

x

 



_AU

( )

x





_AU

( )

x

and



_AU

 

1



_AL

( )

x





_AL

( )

x

, respectively.

3. DIFFERENT CLASSES OF

OPERATORS IN INTERVAL VALUED INTUITIONISTIC FUZZY SET

DEFINITION: Interval-Valued Intuitionistic Fuzzy Ordered Weighted Geometric

Operator(IIFOWG)

Let

a

_j

 





a b

_j

,

_j

 

 

,

c d

_j

,

_j







, for all j = 1, 2,….., n be a

collection of interval-valued intuitionistic fuzzy values.

The Interval-Valued Intuitionistic Fuzzy Ordered Weighted Geometric (IIFOWG) operator,

: n

IIFOWG Q Q is defined as:









( ) ( )

1 2 ( )

1

( ) ( )

1 1 1 1

( , ,..., )

, , 1 1 ,1 1 ,

j

j j j j

j j

n w

w n j

j

n n n n

w w w w

j j

j j j j

IIFOWG a a a a

a b c d



   



   



   

_       _

   

 





where w



w w₁, ₂,...,wn



Tis the associated

weight vector such that

w

_i



0

and

1

n

j j

w





.

Furthermore,



(1), (2),..., ( )  n



is a permutation of (1, 2, ……., n), such that

a

_₍_j_₁₎



a

__{( )}_j for all j = 2, …….., n.

DEFINITION: Interval Valued Intuitionistic Fuzzy Hybrid Geometric Operator(IIFHG)

Let



_, _, _,



j j j j j

a  a b  _{ }c d _ , for all j = 1,

2,….., n be a collection of interval-valued intuitionistic fuzzy values. The Interval-Valued Intuitionistic Fuzzy Hybrid Geometric (IIFHG) operator,

IIFHG Q

:

n



Q

is defined as:

1 2 1

(1) (2) (1)

,

(1) (2) (1)

(

,

,....,

)

(

)

(

)

...

(

)

ij ij ij ij

ij ij ij

r

IIFHG

r

 

  

  





=





 









( ) ( )

1 1

( ) ( )

1 1

, ,

,

1 1 ,1 1

k k

n n

k k

ij ij

k k

n n

k k

ij ij

k k

a b

c d

 

 

 

 

 

 

 

 

 

   

 

 



where







 

₁, ₂,....,



₁



Tis a weight vector of IIFHG operator with_k0 (k= 1, 2,…, n) and

1

n

k k

w





and

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







is the kth largest of the weighted IIFHG ( )k

 

( )k lk

ij ij

r  r  ,

i=1,2,…,m, j=1,2,…,n. In this case that the information about attribute weights is completely known, that is, the weight vector

1 2

(

,

,...,

m

)

T

w



w w

w

of the attributes can be completely determined in advance, then bases on the collective interval valued intuitionistic fuzzy decision matrix



1, 2,...,



k

(3)

we can use the IIFWG operator to obtain the overall value of the alternative.

3.2. Linguistic (RIM) Quantifiers For Determining Unknown Expert-Weights

The problem of determining weights can be addressed in different ways, for example with the use of the so-called ‘Linguistic Quantifiers’, introduced by Zadeh, [25]. A relative linguistic quantifier Q, such as ‘most’, ‘few’, ‘many’, and ‘all’, can be represented as a fuzzy subset of the unit interval, where for a given proportion

r



[0,1]

of the total of the values to aggregate, Q(r) indicates the extent to which this proportion satisfies the semantics defined in Q. For example, given Q = ‘most’, if Q(0.7) = 1 then it would mean that a proportion of 70% totally satisfies the idea conveyed by the quantifier ‘most’, whereas Q(0.55) = 0.25 indicates that the proportion 55% is barely compatible with this concept ( i.e., only 25%). Regular Increasing Monotone (RIM) quantifiers are especially interesting for their use in OWA operators. These quantifiers present the following properties:

i. Q(0) = 0 ii. Q(1) = 1

iii. If

r

₁



r

₂ then

Q r

( )

₁



Q r

( )

₂ .

Yager, [27] suggested the following method to compute weights

w

_i, with the use of a RIM quantifier Q:

1

i

w

Q

n



 







 







 





, i = 1, 2, …., n.

Where the membership function of a linear RIM quantifier Q(r) is defined by two parameters

,

[0,1]

a b



, as:

0

( ) .

1

if r a

r a

Q r if a x b

b a

if r b

 

  

_{ }  



 

4. AN APPROACH TO GROUP DECISION MAKING WITH INTERVAL-VALUED INTUITIONISTIC FUZZY

INFORMATION

Step: 1 Utilize the IIFOWG 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 Calculate the score matrix S=(sij)mxn of the

collective interval-valued intuitionistic fuzzy decision matrix R.





1 ( )

2

ij ij ij ij ij ij

s s r  a   c b d , i = 1, 2,….., m

and j = 1, 2,….., n.

Step: 3 To derive the weights by Quantifier (RIM) by using

1

i

i i

w Q Q

n n



   

 _{ } _ _

   

i =1,2,…, 6.

0 0.1

0.1

( ) 0.1 0.9 .

0.9 0.1

1 0.9

if r

r

Q r if r

if r

 

  

 _  



 

Step: 4 Utilize the IIFHG operator, to derive the collective overall preference intuitionistic fuzzy values

r

_i



i



1, 2,



..

m



of the alternative Ai,where



1

,

2 n



v



v v



v

be the weighting vector of decision

makers, with:

 





1

1 2

0,1 , 1; ,

t

k n

k

w w w w

V V







  is the

associated weighting vector of the IIFHG operator with

 

1

0,1 , 1

n

j j

j

w w







 .

Step: 5 To calculate the distance between collective overall values

r

_i







a b

_i

,

_i

 

,

c d

_i

,

_i









1

1 ,

4

n _A _B _A _B _A _B

i

A B A B A B

d A B n

     

     

     

     





 _ _ _ _ _ _

 

 _ _

    

 

 



.

Step: 6 Rank all the alternatives

A i

_i





1, 2,



,

m



and select one(s) in accordance with d(A,B).

5. DETERMINING EXPERTS WEIGHTS FOR MAGDM PROBLEMS USING SINGULARLY PERTURBED DIFFERENTIAL EQUATION OF REACTION-DIFFUSION TYPE

Singular perturbation problems are of common occurrence in all branches of applied mathematics and engineering. These problems are encountered in various fields such as solid mechanics, fluid dynamics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics etc.

DEFINITION: Singular Perturbation Problem Singularly perturbed differential equations are characterized by a small positive parameter ɛ multiplying the highest order derivative and/or the lower order derivatives of the differential equation.

 

ij _{m n}

R



r

_

 

( 1, 2,3, 4)

k k

ij m n

R r k



(4)

We consider a class of linear singular perturbation problems of the form

( )

u

_

x

b x u x

_

f x











,

x



(0,1)

, with

0 1

(0)

,

(1)

u

_



u u

_



u

.

Where ε is a small parameter 0 <ε≤ 1; b(x) and f(x) are continuous on [0,1].

DEFINITION: Reaction-Diffusion Problem A Singular perturbation problem is said to be reaction-diffusion type if the order of the differential equation is reduced by two when the perturbation parameter ɛ is set to equal to zero.

Problem proposed by the decision maker

The decision maker proposed the decision making problem in the form of a SPP.

( )

( ) 1

u x

x









 

,

x



(0,1)

,with

(0)

2, (1) 1

u



u



. The exact solution is

1 1

1 1 1 1

1

( )

1

x x

e

y x

e

x

e

 

   



 















































 





























.

By normalizing the exact solution for ɛ = 0.01 which is given in the following table:

Table 1: Exact solution for





u



   

x



u x

 

1

x

.

X

y

(x)

( )

y x

w

y x





0.2 8.4861 0.40867 0.4 6.1682 0.29703 0.6 4.0537 0.19521 0.8 2.0578 0.09909

6. NUMERICAL ILLUSTRATION

A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning systems should be installed in the library. The contractor offers eight feasible alternatives, which might be adapted to the physical structure of the library. The alternatives

x

_j(j = 1,2,3,4,…,8) are to be evaluated by using four decision makers

d

_k(k = 1,2,3,4) under the six attributes. Two monetary alternatives and six non-monetary alternatives (that is,

G

₁: owning cost,

G

₂: operating cost ,

G

₃: performance,

G

₄: noise level,

G

₅: maintainability,

G

₆: reliability,

G

₇: flexibility,

G

₈:

safety, where three alternatives G1, G2 and G4 are cost

attributes, and the other five alternatives are benefit alternatives).The eight possible alternatives

A

_i for i = 1, 2, 3, …., 8 are to be evaluated using the interval valued intuitionistic fuzzy numbers, whose weighting vector is derived by using singular perturbation problem by the four decision makers is

(0.40867, 0.29703, 0.19521, 0.09909)

T

 

are

construct respectively, the decision matrices in terms of the IVIFS as listed in the following matrices

( ) ( ) ( ) ( )

8 6 8 6

(

)

(

,

)

k k k k

ij x ij ij x

R



r







(k=1, 2, 3, 4), As follows:



 











 









 







 











 









 







 











 









 







 











 









 







 







(1)

0.5,0.6 , 0.3,0.4 0.6,0.7 , 0.2,0.3 0.2,0.3 , 0.4,0.5 0.3,0.4 , 0.1,0.2 0.5,0.6 , 0.3,0.4 0.3,0.4 , 0.2,0.3 0.6,0.7 , 0.1,0.2 0.1,0.2 , 0.6,0.7 0.2,0.4 , 0.5,0.6 0.1,0.2 , 0.3,0.4 0.5,0.6 , 0.2,0.3 0.7,0.8 , 0.1,0.2 0.5,0.6 , 0.1,0.3

R 



 











 







 











 









 







 











 









 







 











 









 





0.4,0.5 , 0.2,0.3 0.1,0.3 , 0.6,0.7 0.2,0.3 , 0.4,0.5 0.6,0.7 , 0.1,0.2 0.6,0.7 , 0.2,0.3 0.7,0.8 , 0.1,0.2 0.5,0.6 , 0.2,0.3 0.3,0.4 , 0.5,0.6 0.3,0.4 , 0.1,0.2 0.5,0.7 , 0.2,0.3 0.6,0.7 , 0.2,0.3 

            



 





 





 





 



( 0.7, 0.8 , 0.1, 0.2 ) ( 0.3, 0.4 , 0.2, 0.3 ) ( 0.8, 0.9 , 0.0, 0.1 ) ( 0.6, 0.7 , 0.1, 0.2 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.4, 0.3 , 0.1, 0.2 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.5, 0.6 , 0.3, 0.4 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.2, 0.3 , 0.4, 0.5 ) ( 0.3, 0.4 , 0.5, 0.6 ) ( 0.6, 0.8 , 0.1,







 





 





 





 



0.2 ) ( 0.5, 0.6 , 0.1, 0.2 ) ( 0.5, 0.6 , 0.2, 0.3 ) ( 0.8, 0.9 , 0.0, 0.1 ) ( 0.4, 0.5 , 0.4, 0.5 ) ( 0.4, 0.5 , 0.1, 0.2 ) ( 0.3, 0.4 , 0.3, 0.5 ) ( 0.6, 0.7 , 0.1, 0.3 ) ( 0.2, 0.3 , 0.4, 0.6 ) ( 0.5, 0.6 , 0.2, 0.3 ) ( 0.3, 0.4 , 0.5, 0.6 ) ( 0.5, 0.7 , 0.2, 0.3 ) ( 0.6, 0.7 ,



0.1, 0.3 )



            

   







   





   



   







   





   



   







   





   



   







   





   



   





(2)

0.5, 0.6 , 0.2, 0.3 0.7, 0.8 , 0.1, 0.2 0.3, 0.4 , 0.4, 0.5 0.5, 0.6 , 0.3, 0.4 0.5, 0.7 , 0.2, 0.3 0.3, 0.5 , 0.1, 0.2 0.7, 0.8 , 0.1, 0.2 0.4, 0.5 , 0.4, 0.5 0.3, 0.4 , 0.5, 0.6 0.3, 0.4 , 0.5, 0.6 0.7, 0.8 , 0.1, 0.2 0.5, 0.6 , 0.1, 0.3 0.5, 0.6 , 0.1, 0.3

R 

   







   



   







   





   



   







   





   



   







   





   



0.6, 0.7 , 0.2, 0.3 0.2, 0.3 , 0.4, 0.5 0.4, 0.6 , 0.2, 0.3 0.6, 0.7 , 0.2, 0.3 0.5, 0.7 , 0.2, 0.3 0.6, 0.7 , 0.2, 0.3 0.5, 0.6 , 0.3, 0.4 0.5, 0.6 , 0.3, 0.4 0.3, 0.5 , 0.3, 0.4 0.6, 0.7 , 0.2, 0.3 0.5, 0.6 , 0.2, 0.3 

            

           

         

( 0.7, 0.8 , 0.1, 0.2 ) ( 0.2, 0.3 , 0.2, 0.3 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.5, 0.6 , 0.2, 0.3 ) ( 0.5, 0.6 , 0.1, 0.3 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.3, 0.4 , 0.5, 0.6 ) ( 0.6, 0.8 , 0.1, 0.2 ) ( 0.3, 0.4 , 0.5, 0.6 ) ( 0.5, 0.6 , 0.2, 0.4 ) ( 0.6, 0.7 , 0.1, 

           

         

(5)

   







   





   



   







   





   



   







   





   



   







   





   



   





(3)

0.5, 0.7 , 0.1, 0.2 0.6, 0.7 , 0.1, 0.2 0.2, 0.4 , 0.3, 0.4 0.6, 0.7 , 0.2, 0.3 0.4, 0.5 , 0.3, 0.4 0.4, 0.6 , 0.2, 0.3 0.5, 0.6 , 0.1, 0.2 0.8, 0.9 , 0.0, 0.1 0.2, 0.3 , 0.6, 0.7 0.4, 0.5 , 0.4, 0.5 0.7, 0.8 , 0.1, 0.2 0.5, 0.6 , 0.2, 0.3 0.4, 0.5 , 0.3, 0.5

R 

   







   



   







   





   



   







   





   



   







   





   



0.6, 0.7 , 0.2, 0.3 0.1, 0.3 , 0.5, 0.6 0.7, 0.8 , 0.1, 0.2 0.4, 0.5 , 0.3, 0.5 0.7, 0.8 , 0.1, 0.2 0.8, 0.9 , 0.0, 0.1 0.3, 0.4 , 0.5, 0.6 0.4, 0.6 , 0.3, 0.4 0.4, 0.6 , 0.3, 0.4 0.7, 0.8 , 0.1, 0.2 0.5, 0.6 , 0.1, 0.2 

            

           

         

( 0.7, 0.8 , 0.1, 0.2 ) ( 0.4, 0.5 , 0.1, 0.2 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.6, 0.7 , 0.1, 0.3 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.4, 0.5 , 0.1, 0.3 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.5, 0.6 , 0.3, 0.4 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.3, 0.4 , 0.5, 0.6 ) ( 0.4, 0.6 , 0.2, 0.3 ) ( 0.7, 0.8 , 0.1, 

           

         

0.2 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.4, 0.5 , 0.3, 0.4 ) ( 0.6, 0.7 , 0.1, 0.3 ) ( 0.5, 0.6 , 0.3, 0.4 ) ( 0.3, 0.5 , 0.1, 0.2 ) ( 0.4, 0.6 , 0.2, 0.3 ) ( 0.5, 0.6 , 0.2, 0.3 ) ( 0.5, 0.6 , 0.2, 0.3 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.4, 0.5 , 0.4, 0.5 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.6, 0.7 ,0.2, 0.3 )             

   







   





   



   







   





   



   







   





   



   







   





   



   





(4)

0.6, 0.7 , 0.2, 0.3 0.5, 0.6 , 0.3, 0.4 0.2, 0.3 , 0.5, 0.6 0.6, 0.7 , 0.1, 0.3 0.4, 0.5 , 0.2, 0.4 0.4, 0.5 , 0.2, 0.3 0.5, 0.6 , 0.1, 0.2 0.7, 0.8 , 0.1, 0.2 0.3, 0.4 , 0.4, 0.6 0.4, 0.5 , 0.3, 0.4 0.6, 0.8 , 0.1, 0.2 0.6, 0.7 , 0.2, 0.3 0.4, 0.6 , 0.2, 0.4

R 

   







   



   







   





   



   







   





   



   







   





   



0.6, 0.7 , 0.2, 0.3 0.2, 0.3 , 0.6, 0.7 0.7, 0.8 , 0.1, 0.2 0.4, 0.5 , 0.4, 0.5 0.6, 0.7 , 0.2, 0.3 0.7, 0.8 , 0.1, 0.2 0.4, 0.6 , 0.3, 0.4 0.5, 0.6 , 0.2, 0.4 0.4, 0.5 , 0.2, 0.3 0.7, 0.8 , 0.1, 0.2 0.4, 0.5 , 0.2, 0.4

             

           

         

( 0.5, 0.7 , 0.1, 0.2 ) ( 0.2, 0.3 , 0.1, 0.3 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.6, 0.7 , 0.1, 0.3 ) ( 0.4, 0.5 , 0.2, 0.3 ) ( 0.5, 0.6 , 0.1, 0.2 ) ( 0.4, 0.5 , 0.4, 0.5 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.3, 0.4 , 0.5, 0.6 ) ( 0.4, 0.6 , 0.2, 0.3 ) ( 0.8, 0.9 , 0.0, 

           

         

0.1 ) ( 0.6, 0.7 , 0.2, 0.3 ) ( 0.4, 0.5 , 0.3, 0.4 ) ( 0.7, 0.8 , 0.1, 0.2 ) ( 0.5, 0.6 , 0.3, 0.4 ) ( 0.3, 0.5 , 0.1, 0.2 ) ( 0.4, 0.6 , 0.2, 0.3 ) ( 0.6, 0.7 , 0.1, 0.3 ) ( 0.5, 0.7 , 0.2, 0.3 ) ( 0.5, 0.7 , 0.1, 0.2 ) ( 0.4, 0.5 , 0.4, 0.5 ) ( 0.5, 0.7 , 0.1, 0.2 ) ( 0.5, 0.6 ,0.2, 0.4 )             

By using the algorithm we obtain:

0.

(

,

)

3

7

714;

d A B



d A B

(

,

)



0.

3

9

984;

0.

(

,

)

3

9

218;

d A B



d A B

(

,

)



0.

3

9

754;

0.

(

,

)

4

0

819;

d A B



d A B

( , )



0.41750;

( , )

0.38487;

d A B



d A B

( , )



0.38201

. Rank all the alternatives

A i

i





1, 2, 3, 4, 5, 6, 7,8



6 5 2 4 3 7 8 1

A



A



A



A



A



A



A



A

.

Hence, the best alternative is

A

₆.

7. CONCLUSION

In this paper, the exact solution of singularly perturbed differential equation of reaction diffusion type problem is used to derive the decision maker weights and it is applied in MAGDM problems under interval valued intuitionistic fuzzy environment. 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 IIFOWG operator. Next, in the situations where the information about attribute weights is incomplete, we have constructed the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix, and the quantifier (most) are used to determine the attribute weights. Then we have used the obtained attribute weights and the IIFHG operator to get the overall interval-valued intuitionistic fuzzy values of alternatives and have used the proposed method for calculating distance between IVIFNs to rank the alternatives and then to select the most desirable one.

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interval-valued intuitionistic fuzzy sets. FuzzySets and Systems, 64, 159–174. [3] Atanassov, K., & Gargov, G. (1989):

Interval valued intuitionistic fuzzy sets. FuzzySets and Systems, 31, 343–349. [4] Chen, Y., & Yang, W. (2011): A new

multiple attribute group decision making method in intuitionistic fuzzy setting. Applied Mathematical Modelling, 35, 4424-4437. [5] Gorzalczany, M. B. (1987): A method of

inference in approximate reasoning basedon interval-valued fuzzy sets. Fuzzy Sets and Systems, 21(1), 1–17.

[6] Grzegorzewski, P. (2004): Distances between intuitionistic fuzzy sets and interval valued fuzzy sets based on Hausdorff metric. Fuzzy Sets and Systems, 148, 319-328.

[7] Hong, D. H., & Lee, S. (2002): Some algebraic properties and a distance measure for interval-valued fuzzy numbers. Information Sciences, 148(1), 1–10.

[8] Indhumathi, M., & Robinson, J.P. (2018): A Hybrid Scheme for Solving Singularly Perturbed Delay Differential Equations and its Applications to MADM Problems”, International Journal of Pure and Applied Mathematics, 20(6), 7653-7664, (2018). [9] Jeeva, S., & Robinson, J.P. (2017).

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