In this article I will present how to demonstrate, other than using the extension

principle, that the product of two trapezoidal intuitionistic fuzzy numbers (TrIFN) can be

approximated by a TrIFS. Here only one type of intuitionistic fuzzy sets, namely trapezoidal

intuitionistic fuzzy number is presented

Iuliana Carmen Bărbăcioru

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Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 4/2018  134    MULTIPLICATION OPERATION ON INTUITIONISTIC FUZZY NUMBERS   Iuliana Carmen Bărbăcioru, Lecturer PhD, “Constantin Brâncuși” University from Târgu Jiu, ROMANIA   ABSTRACT: In this article I will present how to demonstrate, other than using the extension principle, that the product of two trapezoidal intuitionistic fuzzy numbers (TrIFN) can be approximated by a TrIFS. Here only one type of intuitionistic fuzzy sets, namely trapezoidal intuitionistic fuzzy number is presented.    KEY WORDS: Intuitionistic fuzzy numbers, (α,β)-cut and core of a intuitionistic fuzzy sets, extension principle   1. INTRODUCTION  A generalized fuzzy number [8] , , , ;aa a b c d w   is  said to be a generalized trapezoidal fuzzy number if its membership function is given by:  ( )ax  {        x a   /ax a w b d   a x b    aw         b x c    /ad x w d c   c x d   d x   Where aw  represent the maximum membership degree of a  such that they satisfy the conditions: 0 1aw  . Atanassov [1] introduces for the first time the concept of intuitionistic fuzzy sets (IFS), an IFS A in X being       , ,A AA x x x x X   Where Aμ :X [0,1]  is degree of membership and A : X [0,1]   is degree of non-membership, with the condition    0 1A Ax x     ,  x X  . The concept of the trapezoidal intuitionistic fuzzy number   1 2, , , ; w , ua aa a a a a                (1) (TrIFS) is a generalization of the trapezoidal fuzzy number.  A TrIFS is a special intuitionistic fuzzy set on the real number set R, whose membership and non-membership functions are defined as follows:  ( )ax {                                 a   1/ax a w a a   1a x a    aw         1 2a x a    2/aa x w a a   2a x a                      a  (2)  ( )ax  {                                 a    1 1/aa x u x a a a         1a x a     au  1 2a x a     2 2/ax a u a x a a     2a x a                       a (3) Where aw  and au  represent the maximum membership degree and the minimum non-membership degree of a  such that they Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 4/2018  135 satisfy the conditions: 0 w 1a  , 0 1au  , and 0 w 1a au   . [a1; a2] ,     and  ̅   are called the mean interval and the lower and upper limits of the trapezoidal intuitionistic fuzzy number  a , respectively.    Figure 1. [5] A trapezoidal intuitionistic fuzzy number  1 2, , , ; w , ua aa a a a a        In [3] We introduced the notion of (α,β) -cut as the IFS A, α,β  [0.1], as [A](α,β) =              , ,0 1if   00 , 10,, 0, 0 1if   0   A AA AA AA Ax xx Xx xx X x xx x                               (4)   1.1. Cartesian Product of Intuitionistic Fuzzy Sets [5]  To consider   ̃ ,…,  ̃  be IFSs in X1,...,Xn with the corresponding membership functions 1( )Ax ,…, ( )nAx  and non-membership function 1( )Ax ,…, ( )nAx respectively.  The Cartesian product of the IFSs   ̃ ,…,  ̃  is defined as IFS in X1,...,Xn whose membership functions and non-membership functions are given by: 1... 1( ,..., )nnA Ax x   =  11min ( ),..., ( )nnA Ax x and  1... 1( ,..., )nnA Ax x   =  11max ( ),..., ( )nnA Ax x  .  1.2. Extension Principle in Cartesian Space [5]   We consider the function 1: ... nf X X Y    1( ,..., )nf x x y . The extension principle allows us to define the IFS  ̃ in Y induced by the IFS  ̃     ̃  in 1 ... nX X   through f as follows:      y, , ( )B BB y fy y x    whose membership and non-membership functions are defined as follows:  (y)B  {          B x                                                                      (5)  (y)B  {          B x                                                                      (6)  Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 4/2018  136 where        is the inverse image of y .   2. Multiplication of Two TrIFN by (α, β)-cut Method  2.1. Arithmetic Operations of Intuitionistic Fuzzy Numbers based on Extension Principle [5]  [5] It looked as if  ̃ and  ̃  are IFN then arithmetic operations (*) of  IFN are determined by the formula:    ̃   ̃               1 2 1 21 21 21 2 1 2 1 2**y, min , , max ,p , ,su infA A A Ay x xy x xx x x x x x y                 The membership function and non-membership function as follows:      1 211 2* 1 2*m( ip ,y) su nn AAy xA Axx x        1 211 2* 1 2*m( af ,y) in xn Ay xA AA xx x      2.2. Arithmetic Operations of Intuitionistic Fuzzy Numbers based on   (α, β)-cut Method [5]   As we have seen in the relationship (4), if  1 2, , , ; ,A AA a a a a w u is a TrIFN then (α,β) -cut is given for instance by  [A](α,β) =  1 2, ; ,A A A Aa w a w a u a u            ,  0 1, , 0,1        . Property 2.1 If   1 2, , , ; ,A AA a a a a w u  and  1 2, , , ; ,B BB b b b b w u  are two TrIFN’s, then   1 1 2 2, , , ; ,A B A BA B ab a b a b ab w w u u   is approximated TrIFN. Proof: We are transforming z=x×y.  [5] proposes to demonstrate this property using (α,β) -cut method.     Figure 2. [5] TrIFN’s  1 2, , , ; ,A AA a a a a w u  and  1 2, , , ; ,B BB b b b b w u    First determine the membership function of acceptance (membership) IFS A B  by α-cut method. α-cut for membership function of A :  Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 4/2018  137 ( )Ax     , 0,1A Aa w a w         ,A Aa w a wx      .  α-cut for membership function of B : ( )Bx     , 0,1B Bb w b w         ,B Bb w b wy      . So we will have:  z=x×y      ,A B A Ba w b w a w b w        . Therefore:  ( )A Bz={                                                                  ab   21 1 1142B B A ab zA    1 1ab z a b    A Bw w         1 1 2 2a b z a b    22 2 2242B B A ab zA   2 2a b z ab                                              ab  Where        12 2 212 2 2A BB AA w wA a a b bB aw bwB b a a a b b       .  Next we will determine the non-membership function of acceptance (membership) IFS A B  by β-cut method.  β-cut for membership function of A : ( )Ax     1 , 0,1A Aa u a u          1 ,A Aa u a ux       .  β-cut for membership function of B : ( )Bx     1 , 0,1B Bb u b u        1 ,B Bb u b uy       . So we will have:  z=x×y      1 1 ,A B A Ba u b u a u b u         .  Therefore:  ( )A Bz={                                                                  ab   23 3 33412B B A ab zA     1 1ab z a b    A Bu u                                             1 1 2 2a b z a b    24 4 44412B B A ab zA     2 2a b z ab                                              ab  Where        34 2 234 2 2A BA BA u uA a a b bB au buB b a a a b b       .  Consequently A B  has a trapezoidal shape of IFN which can be approximated by TrIFS.  3. CONCLUSION  Starting from [7] and [5] we showed that membership functions and non-membership function when the TrIFN’s have no particular form of [5]. Of course,  1 1 2 2, , , ; ,A B A BA B ab a b a b ab w w u u  is very much influenced by the positioning of numbers  1 2, , , ; ,A AA a a a a w u  and  1 2, , , ; ,B BB b b b b w u  the system axis.  REFERENCE  [1] Atanasov, K.T., Intuitionistic fuzzy sets, Fuzzy Sets Syst., vol. 20, pp. 87–96, August 1986.  Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 4/2018  138 [2] Atanassov, K.T., My personal view on intuitionistic fuzzy sets theory. Stud. Fuzziness Soft Comput. 220, 23–43 (2008) [3] Bărbăcioru, I. C., A Note on (α,β)-cut in Intuitionistic Fuzzy Sets Theory , Fiability & Durability 1/2014, Editura “ Academica Brâncuşi”, Târgu Jiu, ISSN 1844-640X, 200-206.  [4] D.-F. Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Studies in Fuzziness and Soft Computing 308, DOI: 10.1007/978-3-642-40712-3_1, Springer-Verlag Berlin Heidelberg 2014 [5] Mahapatra1,G.S., Roy, T.K., Intuitionistic Fuzzy Number and Its Arithmetic Operation with Application on System Failure, Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013. [6] Rezvani, S., Multiplication Operation on Trapezoidal Fuzzy Numbers, Journal of Physical Sciences, Vol. 15, 17-26, 2011. [7] Taleshian, A., Rezvani, S., Multiplication Operation on Trapezoidal Fuzzy Numbers, Journal of Physical Sciences, Vol. 15, 2011, 17-26,ISSN: 0972-8791. [8] Zadeh, L.A., Fuzzy sets, Inf. Control 8(3), 338–353 (1965). 

MULTIPLICATION OPERATION ON INTUITIONISTIC FUZZY NUMBERS

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MULTIPLICATION OPERATION ON INTUITIONISTIC FUZZY NUMBERS

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