Entropy Differences of Arithmetic Operations with Shannon Function on Triangular Fuzzy Numbers

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1 Proceeding of the th WSES International onfenrence on PPLIED MTHEMTIS, Dalla, Texa, US, November -, 6 7 Entropy Difference of rithmetic Operation with Shannon Function on Triangular Fuzzy Number TIEN-HIN WNG, JI-LING LING, HSIO-LN HU () Department of Information Management I-Shou Univerity, Section, Hueh-heng Road, Ta-Hu Hiang, Kaohiung 84 TIWN tcwang@iuedutw (T Wang) () Department of Information Engineering; Department of omputer and Information Science I-Shou Univerity; ROM, Section, Hueh-heng Road, Ta-Hu Hiang, Kaohiung 84 ; drdeor@gmailcom btract: - The main purpoe of thi paper are to probe for the entropy difference between arithmetically manipulated Triangular Fuzzy Number (TFN) uing Shannon Function; and to tudy the relationhip between any two TFN Simultaneouly, we expand on the article of Wang et al [6 and Wang et al [5 The entropy difference between two TFN ubjected to different arithmetic operation are claified a even theorem Thi paper find that with the application of Shannon Function on triangular fuzzy number the grade of fuzzine are changed after arithmetic operation Key-Word: - Entropy, Triangular fuzzy number, Fuzzy et, Meaure of fuzzine, Shannon Function, rithmetic operation Introduction In the pat, reearcher have ued myriad of method to meaure the fuzzine of fuzzy et Kaufmann [ denoted the fuzzine of a fuzzy et by calculating the ditance between the fuzzy et and it nearet non-fuzzy et De Luca et al [ propoed uing the entropy to decribe the fuzzine of a fuzzy et Indeed, there are plenty more paper which dicu the entropy of fuzzy et [,, 6, 7 Pedrycz [4 ued the fuzzy et with a triangular memberhip function to demontrate the entropy change when the interval ize of the univeral et i changed Wang and hiu [6 extended the reult of Pedrycz finding to any type of fuzzy et It i neceary to bae the comparion of the fuzzine of everal fuzzy number on the ame definition of entropy computation In the pat, the tudy of the entropy difference of TFN with arithmetic operation ha alway relied on thi equation of entropy function: hx ( ) = 4 x( x) Shannon famou function, hx ( ) =x ln ( x) ln( x), i often ued to meaure fuzzine, yet it i difficult to find reearcher who ue the function to calculate entropy difference In thi paper, we ue Shannon Function on arithmetic operation of triangular fuzzy number and to examine the entropy difference after the calculation Moreover, we extend the tudie of Wang et al (), and Wang et al (5), which, uing hx ( ) = 4 x( ), found the entropy difference of triangular fuzzy number ubmitted to arithmetic operation Entropy and Fuzzy Set The Propertie of Entropy Suppoe i a fuzzy et and i defined in a univeral et U, where U i finite and real The memberhip function value of x in fuzzy et for x U i repreented a ( x ) and x ( ): x [, The meaurement of fuzzine of the fuzzy et i denoted a H( ) and it hold the following propertie (de Luca and Termini, 97 [; Zimmermann, 996 [8): H( )=, if i a crip et in U

2 Proceeding of the th WSES International onfenrence on PPLIED MTHEMTIS, Dalla, Texa, US, November -, 6 74 If ( x) =, x U, then H( ) ha a unique maximum For two fuzzy et and B, if B ( x) ( x) for ( x) and H ( ) H ( B) H ( ) = H ( ) B ( x) ( x) 4, where for ( x) then ( x ) i the tandard complement of ( x ), that i, ( x) = ( x ) The reearcher ue the integration of the following equation to calculate the global entropy meaure of the fuzzy et independent of x over the univeral et U [4 H ( ) h( ( x)) p( x) = () x X where px ( ) i the probability denity function of the available data in U, h( ( x)) i the entropy function, hx ( ):[, [, i monotonically increaing in [, and monotonically decreaing in [,, hx ( ) =, a x = and x = ; hx ( ) =, a x = The following equation are the well-known entropy function ued to meaure the fuzzine of H( ) which can be regarded a an entropy of a fuzzy et ( x ) : x if x [, hx () = () ( ) if x, [ hx ( ) = 4 x( ) () and hx ( ) =x ln ( x) ln( x) (4) where Eq(4) i called Shannon function [8, which i applied to calculate the entropy of the TFN in thi paper Definition of Fuzzy Set Definition The TFN i denoted by a triplet ( a ; it i uual to repreent thi TFN a, a, a) = ( a,, ) a a The memberhip function can be defined a follow [, 8:, x < a, a a, a x a μ ( x) = a a (5) a, a x a a a where μ ( x ) i the degree of memberhip or the memberhip function value of x in fuzzy et, U and μ ( x ) repreent a univeral et and memberhip function, repectively In the expreion below ( x μ ) and fuzzy et have the form: μ ( x ): U [, Definition Two TFN and = ( a,, ) a a and B = ( b,, ) b b, o [ () ddition: B are defined a B = ( a, a, a ) ( b, b, b ) = ( a b, a b, a b ) () Subtraction: B = ( a, a, a ) ( b, b, b ) = ( a b, a b, a b ) () Multiplication: (a) n = ( na, na, na) n N N : denote a natural number (b) m = ( ma, ma, ma) m I, m < I : denote an integer (c) r = ( ra, ra, ra) r R, r R : denote a real number = ( a, a, a ) R, < (d) n, m, r, and are triangular fuzzy number (4) i the image of triangular fuzzy number, defined a: = = ( ) ( a, a, a ) The reult are till triangular fuzzy number through the arithmetic operation of addition to, ubtraction from and multiplication by two triangular fuzzy number The Entropy Difference on TFN through rithmetic Operation We will dicu and prove the entropy difference of TFN through arithmetic operation in thi ection ccording to Definition, the reult of putting two TFN through arithmetic operation i alway a TFN Suppoe that we have two triangular fuzzy number and B, briefly indicated a = ( a,, ) a a and B = ( b,, ) b b, repectively Their memberhip function are hown a below:, x < a, a a, a x a (6) μ ( x) = a a a, a x a a a

3 Proceeding of the th WSES International onfenrence on PPLIED MTHEMTIS, Dalla, Texa, US, November -, 6 75, x < b, b b, b x b μ ( x) = b B b b, b x b b b Theorem For two TFN and (7), expreed a B a a a = (,, ) and B= ( b, b, b ), uppoing px ( ) where i a contant, the relationhip of the entropy of and B ha : H( ) = kh( B), where a a a k = b b b Proof pplying Eq(4) to Eq(6), we obtain, x < a, a x a x a x a x a ln ln, a x a h( μ ( x)) = a a a a a a a a a a a a ln ln, a x a a a aa aa aa pplying Eq() to the above equation, and computing the entropy of TFN, the reult i hown a a a a a a H( ) = ln ln p( x) a a a a a a a a a a a a a a ln ln p( x) a a a a a a a a a Let x a y =, then = ( a a) dy Let a z =, then = ( a a) a a a a H a a y y y y dy a a z z z z d ( ) = ( )( ) [ ln ( )ln( ) ( )( ) [ ln ( )ln( ) z y ( y) ( y) z ( z) ( z) = ( )( a a) y ln y ln( y) ( )( aa) z ln z ln( z) = ( )( a a) ln ( )( a a) ln = ( a a a) ln = ( a aa) =, Hence, H ( ) = ( a aa ) (8) Uing the ame method to integrate the entropy of B, the reult i H ( B) = ( b bb ) (9) H( ) a aa = H( ) = kh( B), where H( B) b bb a aa k = b b b Lemma For the two TFN and, if the following condition are atified, then B H( ) = H( B) () When a = a b = b, that i ( x) and B ( x) are left-facing right triangle and they have the ame bae width () When a = a, that i ( x) and B ( x) are b = b right-facing right triangle and they have the ame bae width () When a a = a, that i ( x) and B ( x) are the b b = b iocele triangle (4) When a a a = b b b Theorem Suppoing = B, being a TFN, then the entropy of i the ummation of the

4 Proceeding of the th WSES International onfenrence on PPLIED MTHEMTIS, Dalla, Texa, US, November -, 6 76 entropie of i, H ( ) = H ( B) = H ( ) HB ( ) and B That Proof For x ( ) = x ( ) Bx ( ), the memberhip function can be expreed a, x < a b, a b ( a b), a b x a b μ ( x) = ( a b) ( a b) ( a b) x, a b x a b ( a b) ( a b) Let ( a b) y =, ( a b) x z = ( a b) ( a b) ( a b) ( a b) pplying Eq(4) to the above equation, we obtain h μ = ( ( x)), x < a b, a b ( y)ln( y) ( y)ln( y), a b x a b ( z)ln( z) ( z)ln( z), a b x a b pplying Eq() to the above equation, and computing the entropy of TFN, the integration reult i hown a H ( ) ( y)ln( y) ( y)ln( y) p( x) a b a b a b = [ a b [ ( z)ln( z) ( z)ln( z) p( x) ( a b) (( ) ( b)) dy y= = a b a ( a b) ( a b) ( a b ) x z = = (( a b) ( a b)) ( a b) ( a b) H ( ) = ( )[( ) ( ) [ ln ( )ln( ) ( )[( ) ( ) [ ln ( )ln( ) = ( ) [( a b) ( a b) ln ()( [ a b) ( a b) ln = ( ( a b) ( a b) ( a b) ) = ( a a a) ( b b b) = H( ) H( B) Hence, H ( ) = ( a a a ) ( b b b ) = H( ) H( B ) a b a b y y y y dy a b a b z z z z Theorem Suppoing = B, being a TFN, then the entropy of i the difference of the entropie of and B That i, H ( ) = H ( B) = H ( ) HB ( ) Proof For x ( ) = x ( ) Bx ( ), the memberhip function can be expreed a, x < ab, a b ( ab), ab x a b μ ( x) = ( a b) ( a b) ( a b) x, a b x ab ( a b) ( a b) Let ( a b) y =, ( ab) x z = ( a b) ( ab) ( ab) ( a b) pplying Eq(4) to the above equation, we obtain h μ = ( ( x)), x < ab, a b ( y)ln( y) ( y)ln( y), a b x a b ( z)ln( z) ( z)ln( z), a b x a b pplying Eq() to the above equation, and computing the entropy of TFN, the reult i hown a H ( ) ( y)ln( y) ( y)ln( y) p( x) ab ab ab = [ ab [ ( z)ln( z) ( z)ln( z) p( x) ( a b ) y= = (( a b) ( ab)) dy ( a b) ( ab) ( a b) z = = ab a b ( ab) ( a b) H ( ) = [ [ [ [ ( ( a b ) ( a b ) ( a b) ) (( ) ( )) ( ) ( a b ) ( a b ) yln y ( y)ln( y) dy ()( a b) ( a b ) zln z ( z)ln( z) = = ( a aa) ( b b b) = H( ) H( B) Hence, H ( ) = ( a aa) ( b b b) = H( ) H( B)

5 Proceeding of the th WSES International onfenrence on PPLIED MTHEMTIS, Dalla, Texa, US, November -, 6 77 Theorem 4 Suppoing i the image of TFN, then the entropy of i equal to the negative entropy of That i, H ( ) = H( ) = H( ) Proof For ( x) =( x), the memberhip function can be expreed a, x <a, a x a, a x a μ ( x) a = a a, a x a a a Let x a y =, a x z = a a a a pplying Eq(4) to the above equation, we obtain h μ = ( ( x)), x <a, a ( y)ln( y) ( y)ln( y), a x a ( z)ln( z) ( z)ln( z), a x a pplying Eq() to the above equation, and computing the entropy of TFN, the reult i hown a a H ( ) = [ ( y)ln( y) ( y)ln( y) p( x) a a [ ( z)ln( z) ( z)ln( z) p( x) a x a y= = a a a a x ( a ) dy, z = =( a a) a a [ [ H ( ) = ( )( a a ) yln y ( y)ln( y) dy ()( a a ) zln z ( z)ln( z) = ( ) a a a = ( ) a a a = ( ) H( ) Hence, H ( ) = ( ) ( a a a) = H ( ) [ ( ) Remark Suppoing i the image of TFN, then the entropy of i equal to the negative entropy of, uing Shannon Function to compute their entropie It can be expreed a H( ) H( ) =, Theorem 5 Suppoing = n n N, being a TFN, then the entropy of i n time the entropy of That i, H ( ) = Hn ( ) = nh ( ) Proof H (*) = n ( a aa), n N n =, H(*) = ( a a a) = H( ), i etablihed Suppoe n= k i etablihed, H (*) = k ( a a a) = kh( ) Then when n= k, H(*) = ( k ) ( a a a) = k ( a a a) ( a a a) = ( k) H( ) H( ) i etablihed By uing mathematical induction, we can prove that when = n, n N, then H ( ) = nh ( ) i etablihed Remark If the bae width of the TFN i n ( n N) time the bae width of the TFN, then the entropy of i n time the entropy of, uing Shannon Function to compute their entropie, Theorem 6 Suppoing =n n N, being a TFN, then the entropy of i n time the entropy of That i, H ( ) = H( n) =( nh ) ( ) Proof For x ( ) =nx ( ), the memberhip function can be expreed a, x <na, na x na, na x na μ ( x) = na na na, na x na na na Let x na y =, na z = na na na na pplying Eq(4) to the above equation, we obtain, x <na, na h( μ ( x)) = ( )ln( ) ( )ln( ), y y y y na x na ( z)ln( z) ( z)ln( z), na x na

6 Proceeding of the th WSES International onfenrence on PPLIED MTHEMTIS, Dalla, Texa, US, November -, 6 78 pplying Eq() to the above equation, and computing the entropy of TFN, the reult i hown a H ( ) = ( y)ln( y) ( y)ln( y) p( x) na na na [ na [ ( z)ln( z) ( z)ln( z) p( x) x na y= = na na na na x ( na ) dy, z = =( na na) na na [ [ H ( ) = ( )( na na) yln y ( y)ln( y) dy ()( na na ) zln z ( z)ln( z) = [( n) a na na = ( n) ( a a a = ( nh ) ( ) Hence, H ( ) = H( n) = ( nh ) ( ) H ( ) = Hm ( ) = mh ( ) Remark ombining Theorem 5 and 6, the reult i for any integer m, m I, = m, then the following relationhip i alway true: Theorem 7 Suppoe,, and B are three TFN, and they exit the linear relation a follow: = n mb, n, m I The entropie of,, and B exit the relation a: H ( ) = Hn ( mb) = nh ( ) mhb ( ) Proof We et and From = n = mb Theorem 7 and 8, we know that H ( ) = nh ( ), H ( () ) = mhb ( ) We et = From Theorem we know that H ( ) = H ( ) H ( ) (B) From formula () and (B), we obtain Hence, the theorem i true H ( ) = nh ( ) mhb ( ) 4 oncluion In thi paper, we ued Shannon Function to find the entropy difference of triangular fuzzy number with the entropy of arithmetic operation and we ummarized the above theorem The main concluion are a follow ) Firtly, when two TFN are added or ubtracted, the entropy i equal to the addition or ubtraction of the entropie of each individual TFN Secondly, no matter whether two TFN are left-facing or right-facing right triangle, if they have the ame bae width, then their entropie are the ame Thirdly, if two TFN are iocele triangle, their entropie are the ame Fourthly, if TFN are multiplied by poitive integer number or poitive real number, the entropy i poitive Fifthly, if TFN are multiplied by negative integer number or negative real number, the entropy i negative Sixthly, the entropy of a TFN i the ame a the negative image of it TFN entropy Seventhly, if a poitive or negative linear relationhip exit between TFN, a poitive or negative linear relationhip alway exit between their entropie Thi paper prove that whether the entropie of triangular fuzzy number increae or decreae depend on the operation of arithmetic Reference: [ Kaufmann, Introduction to the Theory and Fuzzy Subet, cademic Pre, NY, 975 [ Kaufmann and Madan M GUPT, Fuzzy Mathematical Model in Engineering and Management Science, ELSEVIER Science Publiher BV, 988 [ De Luca and S Termini, Definition of Non-probabilitic Entropy in the Setting Fuzzy Set Theory, Information and ontrol, Vol, 97, pp- [4 W Pedrycz, Why Triangular Memberhip Function, Fuzzy Set and Sytem, Vol64, 994, pp- [5 Tien-hin Wang and Hiao-Lan hu, The Entropy Difference of Triangular Fuzzy Number through rithmetic, The 9th World Multi-onference on Sytemic, ybernetic and Informatic(WMSI 5), July -, 5 - Orlando, Florida, US [6 Wen-June Wang and hih-hui hiu, The Entropy hange of Fuzzy Number with rithmetic Operation, Fuzzy Set and Sytem, Vol,, pp [7 R R Yager, On the Meaure of Fuzzine and Negation, Part I: Memberhip in Unit Interval, International Journal of General Sytem, Vol5, 979, pp-9 [8 H-J Zimmermann, Fuzzy Set Theory and It pplication, Kluwer-Nijhoff, Boton, 996

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