Attribute Weight Determination using Sumudu Transform in Intuitionistic Triangular Fuzzy MAGDM Problems

(1)

Attribute Weight Determination using

Sumudu Transform in Intuitionistic Triangular Fuzzy

MAGDM Problems

Jeeva S

1

John Robinson P

2 Research Scholar1 , Assistant Professor2 Bishop Heber College, Tiruchirappalli, India.

[email protected], [email protected]

Abstract- In this paper, Multiple Attribute Group Decision Making (MAGDM) problems are investigated with intuitionistic triangular fuzzy sets for ranking the alternatives together with Intuitionistic Triangular Fuzzy Weighted Averaging (ITrFWA) and Intuitionistic Triangular Fuzzy Hybrid Aggregation (ITrFHA) operator. The attribute weights are obtain from Laplace and sumudu transform for determining and normalizing methods for fuzzy differential equations are studied. The weights obtained from the methods are applied in decision making problems. A numerical illustration is given to show the effectiveness of the proposed approach.

Keywords: MAGDM, Weighted averaging operator, Sumudu Transform, Initial Value Problem.

1. INTRODUCTION

The major challenge of decision making is uncertainty and the major goal of decision analysis is to reduce uncertainty. Robust decision efforts have formally integrated uncertainty and criterion subjectivity into the decision making process. To deal with this kind of qualitative, imprecise and incomplete information decision problems, Zadeh, [37] suggested employing the fuzzy set theory as a modelling tool for complex systems. Wang & Tong [31] investigated the consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations. Wan et al., [32] defined some new generalized aggregation operators for triangular intuitionistic fuzzy numbers and application to multi-attribute group decision making. Dong & Zhang, [4] discussed approaches to group decision making with incomplete information based on power geometric operators and triangular fuzzy AHP. Meng et al., [8] defined a new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations. Wang & Lin [33] discussed acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices. Ren & Liang [10] and Wu & Chiclana [35] discussed Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations. Chen & Li [2] discussed Dynamic multi- attribute decision making model based on triangular intuitionistic fuzzy numbers. Chen et al., [3] detailed Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making. Atanassov & Gargov [1] discussed Interval-valued intuitionistic fuzzy sets. Robinson & Amirtharaj [11-21], Jeeva & Robinson [6] and Robinson & Jeeva

[22-27] discussed the various decision making operators and conditions. Szmidt & Kacprzyk [28-30] found the distance between intuitionistic fuzzy sets. Wei et al. [34] and Xu & Yagar [36] discussed some different operators based on intuitionistic fuzzy sets. Greenberg & Michel [5] and Ray & Barrel [9] discussed a comprehensive, thorough, and up-to-date treatment of engineering mathematics. It is intended to introduce students of engineering, physics, mathematics, computer science, and related fields to those areas of applied mathematics that are most relevant for solving practical problems. Sumudu transform is very effective and reliable tool for solving the fuzzy differential equations. Khan et al., [7] discussed the solution of fuzzy differential equations by fuzzy sumudu transform.

In this paper, the attribute weights are derived from Initial value problems through Laplace and Sumudu transforms, where the weights are determined and normalized and utilized in decision making problems. In this work, we have investigated the MAGDM problem with intuitionistic triangular fuzzy set for ranking the alternatives together with ITrWA and ITrFHA operators. A numerical illustration is given to show the effectiveness of the proposed approach.

2. PRELIMINARIES

In this section, some basic definitions and arithmetic aggregation operators of Intuitionistic Fuzzy Numbers are presented.

DEFINITION 1: INTUITIONISTIC FUZZY SET

Let a set X be fixed. An IFS Ã in X is an object having the form

   

,

{

_Ã _Ã

}

,

Ã

 

x µ

x



x

 

x X

where the

 

:

 

0, 1

Ã

µ x X and

 

_:

 

_{0, 1}

Ã x E

(2)

of membership and degree of non-membership respectively, of the element x X to the set Ã, which is a subset of X, for every element x X,

 

0



µ

_Ã

x





_Ã

x

1.



DEFINITION 2: INTUITIONISTIC FUZZY NUMBER

An IFN Ã is defined as follows: 1. An intuitionistic fuzzy sub set of the real line. 2. Normal i.e there is any x0 R such that µÃ(x0)

= 1 (so



_Ã

 

x

₀



0

).

3. Convex for the membership function µÃ(x)











 



 

1 2 1 2

1, 2

. 1 , ,

, 0,1 .

Ã Ã Ã

i e µ x x min µ x µ x

x x R

 

 

  

4. Concave for the non- membership function γÃ(x)









 

 

1 2 1 2

1, 2

{ },

.

1 ,

, 0,1

.

Ã Ã Ã

i e x x max x x

x x R

           .

DEFINITION 3: Triangular Fuzzy Number (TrFN)

( , , )

A



a b c

is called a triangular fuzzy number, if the membership function



A:R

 

0,1 is expressed as:

( )

0 otherwise

A

x a

a x b

b a x c

x b x c

b c



  _{ }      _{ }     

where

x



R

, 0

   

a

b

c

1.

DEFINITION 4: Let a_j  



a b c, ,



;



 

_j, _j



_ for all

1, 2,...,

j



n

be a collection of intuitionistic fuzzy values. The Intuitionistic Triangular Fuzzy Ordered Weighted Averaging Operator (ITrFOWA),

:

n

ITrFOWA Q



Q

is defined as



1 2









 



 

1 1

, ,...., , , ; 1 1 j, j

j j

n w n w

n a a

j j

ITrFOWA a a a a b c  _  _

 

  

_    _

 







Where





   

1 , 2 ,...,

 

n



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

a

__{ }_j_₁



a

__{ }_j for all j=2,....n. where

W





w w

_{1 ,} _{2 , ... ,}

w

_n



Tbe the weighting vector

of

a

_j for all



w

j



0

and

1

1.

n j j

w





3. AN APPROACH TO GROUP DECISION

MAKING WITH INTUITIONISTIC

TRANGULAR FUZZY INFORMATION

Step 1: Utilize the decision information given in the intuitionistic triangular fuzzy decision matrix

R

_k, and the ITrFWA operator,

 



   





     



1 2

,

,...,

,

1, 2,...

;

1, 2,...

n

k k k k k k

i i i i i i

r

u

v

ITrFWA

r

i

m k



t



To derive the individual overall preference intuitionistic fuzzy values

r

_i kof the alternative Ai.

Step 2: Utilize the IFHA operator,







 1  2  



,

, , ,..., t , 1, 2,...

i i i v w i i i

r 

 

IFHA  r r r i mTo

derive the collective overall

r

i



i



1, 2,



..

m



preference intuitionistic fuzzy values of the alternative Ai where

v





v v

1

,

2



v

n



be the weighting vector of

decision makers, with:

V

k



 

0,1 ,

1 1; t k k V  





1, 2 n



w w w w is the associated weighting vector of the IFHA operator with

 

1

0,1 , 1.

n j j j w w  





Step 3: To calculate the distance between collective overall values

r

_i





 

_i

,

_i



and intuitionistic fuzzy positive ideal solution.





































1 2 1 , 8 i i i i i i i i i d r r

                          _ _ _ _    _ _ _ _     _ _ _ _           

Step 4: Rank all the alternatives A ii



1, 2, ,m



and

select one(s) in accordance with



i,





1, 2, ,



d r r i  m . The smallerd r



_i , r



is the

better alternatives Ai.

4. DETERMINING EXPERTS WEIGHTS

FOR MAGDM PROBLEMS USING

SUMUDU TRANSFORM

DEFINITION: 5 SUMUDU TRANSFORM

We consider functions in the set

A

, defined by

 





| |/

1 2

| , , / 0, suchthat | | ,

if 1 0,

j

t j

f t M and or f t Me

A t                     

(3)

argument of the function

f

. Specifically for

f t

 

in

A

, the sumudu transform is defined by

 

2 0

1 0

, 0 ,

, 0.

t

f ut e dt u

G u S f t

f ut e dt u



 _

 _{ }

  _  _

   





DEFINITION: 6 FUZZY SUMUDU TRANSFORM

Let f t

 

be a continuous fuzzy-valued function. Suppose that

f ut e

 

t is improper fuzzy-Riemann-integrable on



0,



, then

 

0

t

f ut e dt

 _



is

called the fuzzy Sumudu transform and it is denoted by

 

₀

 

,



1

,

2



t

G u



S f t















f ut e dt u





 

DEFINITION 7: FUZZY LAPLACE TRANSFORM

Let f t

 

be a continuous fuzzy-valued function. Suppose that

f t e

 

ut is improper fuzzy-Riemann-integrable on



0,





, then

 

0

ut

f t e dt

 _



is

called the fuzzy Laplace transform and it is denoted by

 

₀

 

ut

,

0

F u



L f t















f t e dt u





DEFINITION 8: Let

t y

,



R

, if there exists

z



R

such that

t

 

y z

, then

z

is called H-difference of

t

and

y

and it is denoted by

t



y

.

EXAMPLE

Consider the following initial value problem

 

   

, 0 1.

y t  y t y  To solve by using Fuzzy Laplace and Fuzzy Sumudu Transform.

By using fuzzy sumudu transform, we have

 

S y t_  _S_y t _and

 

0 .

t

S y t_  _



y ut e dt 

We know that, S y t

 

S y t

 

y

 

0 .

u



 

 

 

 

 

Therefore,_S _{y t}

 

S y t

 

y

 

0 _;

u



 

 

 

 

 

   

1

(1) ( by y(0)=1). 1

S y t

u

 

 

  _

Taking inverse fuzzy sumudu transform on both sides, we get,

y t

 



e

t

.

By using Fuzzy Laplace Transform and simplify then we get,

 

_{( )} ₍₀₎ 1

1

L y t y

s



 .

Taking inverse Laplace Transform then we get,

( ) t ( by (0) 1).

y t e  y 

By using fuzzy sumudu transform we get the exact solution and the weight vector by normalizing the exact solution which is given in the following table:

Table 1: Exact solution ofy t

 

 y t

 

X

( ) t

y t e ( )

( )

i y t w

y t

 

0.2 0.980198673306755 0.257549241778004 0.4 0.960789439152323 0.252449425101961 0.6 0.941764533584249 0.247450591561995 0.8 0.923116346386636 0.242550741558039

5. NUMERICAL ILLUSTRATION

Suppose that a tele-communication company intends to choose a manager for R & D department from five volunteers. The decision making committee assess the five concerned volunteers based on four attributes shown as follows:

C1-Proficiency in identifying research areas; C2-Proficiency in administration; C3-Personality; C4-Self Confidence.

The five possible alternatives A i_i



1, 2, 3, 4, 5



are to be evaluated using intuitionistic trapezoidal fuzzy numbers by the three decision makers whose weighting vectors are 



0.51, 0.31, 0.18



T,w



0.19, 0.35, 0.22



T

under the above four attributes sumudu transform weighting vector

0.257549241778004, 0.252449425101961,

0.247450591561995, 0.242550741558039

T

_{ } _

 

and

construct, respectively, the decision matrices as listed in the following matrices R

 

r₂ ijk 5* 4



k1, 2, 3



As follows:

















































































1

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

0.6, 0.7, 0.8 ; 0.7, 0.3 0.5, 0.6, 0.7 ; 0.7, 0.2

0.1, 0.2, 0.4 ; 0.6, 0.4 0.2, 0.3, 0.5 ; 0.5, 0.4

0.3, 0.4, 0.5 ; 0.8, 0.1 0.1, 0.3, 0.4 ; 0.6, 0.3

0.2, 0.3, 0.4 ; 0.6, 0.2 0.3, 0.4, 0.5 ; 0.4, 0.3

R 



       





































































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

0.4, 0.5, 0.7 0.7, 0.2 0.5, 0.6, 0.7 0.4, 0.5

0.5, 0.6, 0.7 0.5, 0.3 0.3, 0.5, 0.7 0.2, 0.3

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

0.2, 0.3

; ;

;

, 0.4

;

; ;

; 0.7,



0.1







0.5, 0.6, 0.7 0.1,



; 0.3



(4)





































































2

0.4, 0.5, 0.6 0.4, 0.3 0.1, 0.2, 0.3 0.5, 0.2

0.5, 0.6, 0.7 0.6, 0.2 0.4, 0.5, 0.6 0.6, 0.1

0.1, 0.2, 0.3 0.5, 0.3 0.1, 0.2, 0.4 0.4, 0.3

0.2, 0.3, 0.4 0.7, 0.1 0.1, 0.2, 0.3 0.5, 0.2

0.1

; ;

;

, 0.2, 0. 0.

;

3 ;

R 



5, 0.1







0.2, 0.3, 0.4 0.3,



; 0.2



        









































































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

0.3, 0.4, 0.6 0.6, 0.1 0.4, 0.5, 0.6 0.3, 0.4

0.4, 0.5, 0.6 0.4, 0.2 0.2, 0.4, 0.6 0.5, 0.2

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

0.1, 0.2, 0.3 0.

; ;

6, 0

; ;

; .2





0.4, 0.5, 0.6 0.4,



; 0.2



        





































































3

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

0.7, 0.8, 0.9 0.6, 0.4 0.6, 0.7, 0.8 0.6, 0.3

0.2, 0.3, 0.5 0.5, 0.5 0.3, 0.4, 0.6 0.4, 0.5

0.4, 0.5, 0.6

; ;

;0.7, 0.2 0.2, 0.4, 0.5 0.5, 0.4;

0.3, 0.4, 0.5 0.5, 0.;

R 



3







0.4, 0.5, 0.6 0.



; 3, 0.4



        









































































0.6, 0.7, 0.9 0.2, 0.7 0.5, 0.6, 0.7 0.1, 0.8 0.5, 0.6, 0.8 0.6, 0.3 0.6, 0.7, 0.8 0.3, 0.6 0.6, 0.7, 0.8 0.4, 0.4 0.4, 0.6, 0.8 0.5, 0.4 0.2, 0.4, 0.6 0.2, 0.5 0.7, 0.8, 0.9 0.1, 0.7 0.3, 0.4, 0.5 0.

; ;

6, 0

; ;

; .2





0.6, 0.7, 0.8 0.4,



; 0.4



        

By using the algorithm we obtain:

1 0.738052

( , ) 1589444;

d r r 

2 0.676905

( , ) 5080126;

d r r 

3 0.708179

( , ) 7671513;

d r r 

4 0.702990

( , ) 9641018;

d r r 

5 0.6927662

( , ) 2089317.

d r r 

Rank all the alternatives A ii



1, 2, 3, 4, 5



1 3 4 5 2

A A A A A

Hence, the best alternative is

2 A .

6. CONCLUSION

In this paper, Laplace and Sumudu Transforms are used to obtain the solution of differential equations and it is utilized to derive the decision maker weights in MAGDM problems under intuitionistic triangular fuzzy sets. In the process of determining weights, multi criteria are explicitly considered, the numerical solutions are decomposed, and the decision maker’s weights for attributes and corresponding decision making methods have also been proposed.

REFERENCE

[1] Atanassov, K., & Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31(3), 343–349.

[2] Chen, Y., & Li, B. (2011). Dynamic multi- attribute decision making model based on triangular intuitionistic fuzzy numbers, Scientia Iranica, 18(2), 268-274.

[3] Chen, Z., Chin, K., & Li, Y., Yang, Y. (2016). Proportional hesistant fuzzy linguistic term set for multiple criteria group decision making,

Information Sciences, 357, 61-87.

[4] Dong, M., Li, S., & Zhang, H. (2015). Approaches to group decision making with incomplete information based on power geometric operators and triangular fuzzy AHP, Expert Systems with Applications, 42(21), 7846-7857.

[5] Greenberg, Michel, D. (1988). “Advanced Engineering Mathematics”, New Jersey, Prentice-Hall Publishers, ISBN: PREPUNE J-52-5.

[6] Jeeva, S., Robinson, J.P. (2017). Application of Sumudu Transform in Intuitionistic Fuzzy MAGDM Problems, International Journal of Pure and Applied Mathematics, 119(11), 109-117. 7.

[7] Khan, N.A.A., Razzaq, O.A., & Ayyaz, M. (2015). On the solution of fuzzy differential equations by fuzzy Sumudu transform. Nonlinear Eng. 2015, 4, 49–60.

[8] Meng, F., Lin, J., Tan, C., & Zhang, Q. (2017). A new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations, Fuzzy Sets and Systems, 315, 1-25. [9] Ray, W.C., & Barrel, L.C. (1982). Advanced

Engineering Mathematics, 5th edition, Auckland, MC GRAW HILL, ISBN: MC GRAW-88-2.

[10] Ren, J., & Liang, H.(2017). Measuring the sustainability of marine fuels: A Fuzzy group multi-criteria decision making approach, Transporation Research Part D: Transport and Environment, 54, 12-29.

[11] Robinson, J.P., & Amirtharaj, E.C.H., (2011a). A short primer on the correlation coefficient of Vague sets. International Journal of Fuzzy System Applications, 1(12), 55-69. Doi:10.4018/ijfsa.2011040105.

[12] Robinson, J.P., & Amirtharaj, E.C.H., (2011b). Extended TOPSIS with correlation coefficient of Triangular Intuitionistic fuzzy sets for Multiple Attribute Group Decision Making. International Journal of Decision Support System Technology, 3(3), 15-40. Doi:10.4018/jdsst.2011070102.

(5)

[14] Robinson, J.P., & Amirtharaj, E.C.H., (2012b). A search for the Correlation coefficient of Triangular and Trapezoidal intuitionistic Fuzzy sets for Multiple Attribute Group Decision Making, Communications in computer and Information Sciences-283, Springer –Verlag, 333-342.

[15] Robinson, J.P., & Amirtharaj, E.C.H., (2013). A strategic TOPSIS algorithm with correlation Coefficient of Trapezoidal Fuzzy Intuitionistic Fuzzy Sets, Advances in Decision Sciences, Volumes 2014, 1-10.

[16] Robinson, J.P., & Amirtharaj, E.C.H., (2014a). MADM Problems with Correlation Coefficient of Trapezoidal Fuzzy Intuitionistic Fuzzy Sets,

Advances in Decision Sciences, Volume 2014, 1-10.

[17] Robinson, J.P., & Amirtharaj, E.C.H., (2014b). MAGDM-Miner: A New Algorithm for Mining Trapezoidal Intuitionistic Fuzzy Correlation Rules, International Journal of Decision Support System Technology. 6(1), 34-58.

[18] Robinson, J.P., & Amirtharaj, E.C.H., (2014c). Efficient Multiple Attribute Group Decision Making Models with Correlation Coefficient of Vague sets, International Journal of Operations Research and Information Systems. 5(3), 27-51.

[19] Robinson, J.P., & Amirtharaj, E.C.H., (2015). MAGDM Problems with Correlation coefficient of Triangular Fuzzy IFS,

International Journal of Fuzzy Applications,

4(1), 1-32.

[20] Robinson, J.P., & Amirtharaj, E.C.H., (2016a). Multiple Attribute Group Decision Analysis for Intuitionistic Triangular and Trapezoidal Fuzzy Numbers. International Journal of Fuzzy System Applications, 5(3), 42-76. [21] Robinson, J.P., & Amirtharaj, E.C.H.,

(2016b). Contrasting Correlation Coefficient with Distance Measure in Interval Valued Intuitionistic Trapezoidal Fuzzy Numbers.

International Journal of Fuzzy System Applications, 5(3), 42-76.

[22] Robinson, J.P., & Jeeva, S., (2016). Mining Trapezoidal Intuitionistic Fuzzy Correlation Rules for Eigen Valued Magdm Problems.

International Journal of Control Theory and Applications, 9(7), 585-616.

[23] Robinson, J.P., & Jeeva, S. (2017). Efficient Intuitionistic Fuzzy Decision Making Problems with Sumudu Transform, Emerging Trends in Mathematics and Mathematics Education, ISBN: 978-93-80693-88-0. [24] Robinson, J.P., & Jeeva, S. (2017).

Application of Jacobian & Sor Iteration

process in Intuitionistic Fuzzy MAGDM Problems, Mathematical Sciences International Research Journal, 6(2), pp. 130-134.

[25] Robinson, J.P, & Jeeva, S. (2017). MAGDM problems with sumudu transform for interval valued intuitionistic triangular fuzzy sets,

IEEE Digital Library, 958-963.

[26] Robinson, J.P., & Jeeva, S. (2017). Application Of Integro-Differential Equations Using Sumudu Transform In Intuitionistic Trapezoidal Fuzzy MAGDM Problems”, Advances In Algebra And Analysis, 2.

[27] Robinson, J.P., & Jeeva, S. (2018). Application of Double Sumudu Transform in MAGDM Problems with Intuitionistic Triangular Fuzzy Sets”, International Journal of Research In Advent Technology, Vol-6(7), 1620-1628.

[28] Szmidt, E., & Kacprzyk, J. (2000). Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114(3), 505–518. Szmidt, E., & Kacprzyk, J. (2002). Using intuitionistic fuzzy sets in group decision making. Control and Cybernetics, 31, 1037–1053.

[29] Szmidt, E., & Kacprzyk, J. (2003). A consensus-reaching process under intuitionistic fuzzy preference relations.

International Journal of Intelligent Systems,

18(7), 837–852. doi:10.1002/int.10119. [30] Wang, Z.J., & Tong. , X. (2016). Consistency

analysis and group decision making based on triangular fuzzy additive reciprocal preference relations, Information Sciences, 361-362, 29-47.

[31] Wan, S.P., Wang, F., Lin, L.L., & Dong, J.Y. (2016). Some new generalized aggregation operators for triangular intuitionistic fuzzy numbers and application to multi-attribute group decision making, Computer & Industrial Engineering, 93, 286-301.

[32] Wang, Z.J., & Lin, J. (2017). Acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices, Information Sciences, 402, 105-123.

[33] Wei, G., Zhao, X., Lin, R., & Wang, H. J. (2012). Generalized triangular fuzzy correlated averaging operator and their application to multiple attribute decision making. Applied Mathematical Modelling,

36(7), 2975–2982.

(6)

complementary preference relations, Information Sciences, 279, 716-734.

[35] Xu, Z. S., & Yager, R. R. (2006). Some geometric aggregation operators based on

Intuitionistic Fuzzy sets. International Journal of General Systems, 35(4), 417–433.

Proportional hesitant fuzzy linguistic term set for multiple criteria group decision