Multiplication and Division of Triangular Fuzzy Numbers

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1 Dffodil Interntionl University Institutionl Repository DIU Journl of Science nd Technology Volume Issue July Multipliction nd Division of Tringulr Fuzzy Numbers Rhmn Md. Mosfiqur Dffodil Interntionl University Downloded from Copyright Dffodil Interntionl University Librry

2 DFFODIL INTERNTIONL UNIVERSITY JOURNL OF SCIENCE ND TECHNOLOGY VOLUME ISSUE JULY 6 9 MULTIPLICTION ND DIVISION OF TRINGULR FUZZY NUMERS Md. Mosfiqur Rhmn Lecturer in Mthemtics Deprtment of Nturl Sciences Dffodil Interntionl University E-mil: mosfiqur.ns@diu.edu.bd bstrct: In this rticle we use n exmple of two tringulr fuzzy numbers nd clculte their growth nd prtition pplying the rithmetic opertions of multipliction nd division respectively. This method hs been demonstrted with the help of numericl exmples. Lstly; computtions re done using MTL. Keywords: Fuzzy set fuzzy number distribution functions complementry distribution functions membership function vlution set nd rithmetic of fuzzy number.. Introduction Fuzziness mens different things depending upon the domin of ppliction nd the wy it is mesured. y mens of fuzzy sets usul notions cn be described mthemticlly in their very bstrctness. Fuzzy set theory hs been widely cclimed s offering greter richness in pplictions thn ordinry set theory. mong the vrious types of fuzzy sets those which re defined on the universl set R of rel numbers re of prticulr importnce. They my under certin conditions be viewed s fuzzy numbers which reflect the humn perception of uncertin numericl quntifiction. fter the successful pplictions of fuzzy sets theory on the controller systems this theory hve pplied in other res. In the most these pplictions fuzzy numbers re one wy to describe the dt vgueness nd imprecision. They cn be regrded s n extension of the rel numbers. Fuzzy numbers re of gret importnce in fuzzy systems. In the pplictions the tringulr nd the trpezoidl fuzzy numbers re usully used. It is now vigorous re of reserch with mnifold pplictions.. Preliminries.. Definition6: function m : X Æ is clled membership function... Definition6: function is clled fuzzy set on X where X is nonempty set of objects clled referenti set (the unit intervl) is clled vlution set nd represents we shll write the symbol for...() Exmple: Fuzzy sets with discrete non-ordered universe: Let X = {Sn Frncisco oston Los ngeles} be the set of cities one my choose to live in. The fuzzy set = "desirble city to live in" my be described s follows: = {(Sn Frncisco.9) (oston.8) (Los ngeles.6)}. pprently the universe of discourse X is discrete nd it contins non ordered objects - in this cse three big cities in the United Sttes. s one cn see the foregoing membership grdes listed bove re quite subjective; nyone cn come up with three different but legitimte vlues to reflect his or her preference...(b) Exmple: Fuzzy sets with discrete ordered universe: Let X = { 3 5 6} be the set of numbers of children fmily my choose to hve. Then the fuzzy set = "desirble number of children in fmily" my be described s follows: Dte of submission :.7.6 Dte of cceptnce : 6.8.6

3 5 = {(.) (.3) (.7) (3 ) (.7) (5.3) (6.)}. Here we hve discrete ordered universe X...(c) Exmple: Suppose is collection of vowels in English lphbets. Here the collection is well defined s which lphbet is vowel nd which is not vowel we cn specify. Then is clssicl set. 3. Fuzzy Number 3. Fuzzy Number 6: To qulify fuzzy number fuzzy set on R must possess t lest the following three properties:. must be norml fuzzy set.. must be closed intervl for every Œ( The support of must be bounded.. rithmetic of Fuzzy Number.. Multipliction of Fuzzy Numbers: tringulr fuzzy number (TFN) is defined by the following nottion: ~ x b c Then the membership function for this TFN is defined s Multipliction nd Division of Tringulr Fuzzy Numbers {( b - ) + }.{( b - ) + }{( b - c) + c}.{( b - c) + c} " Œ ( Let b p b - q b - c r nd b - c s - (. ) \ x {( p + ).( q + )}{( r + c ).( s + c )} { pq + q + p + }{ rs + sc + rc + c c } pq + q + p + pq + ( q + p ) + ( - nd x - ( q + p ) ± - ( q + p ) ± ( q + p ) ( q - p ) pq pq - ( sc + rc ) ± - ( sc + rc ) ± rs + sc + rc + c c - pq( - + pqx rs rs rs + sc + rc + ( c c - ( sc + rc ) ( sc - rc ) - rs( c c - + rsx ( The cuts of TFN define set of closed intervls. The intervls re ( b - ) + ( b - c) + c " Œ(. For two fuzzy numbers b nd b c we hve Æ( b Æ( b ( b ( b - ) + ( b - ) + - ) + ( b ( b c - c ) + c - c ) + c. - ) + ( b - c ) + c " Œ( Now (. ). - c ) + c " Œ(. (. )( Now fter putting the vlues of we hve (. )( p q r nd s Ï - ( b - + b ) ± ( b - b ) + x( b - )( b - ) x bb ( b - )( b - ) - ( bc - cc + bc ) ± ( bc - bc ) + x( b - c )( b - c) Ì bb x cc ( b - c)( b - c) Otherwise Ó

4 DFFODIL INTERNTIONL UNIVERSITY JOURNL OF SCIENCE ND TECHNOLOGY VOLUME ISSUE JULY Exmple: Let nd 356 be two tringulr fuzzy numbers with fmf " Œ( we choose x. ( = nd ( = Then (. ). ( + ).(3 + )( - ).(6 -)" Œ( " Œ( Now x ( ± 5 -.(3 - x ) - 5 ± + 8x x \ is cceptble " Œ(. " Œ( we choose x. nd for x ( - 6 ± 56 -.( - x ) 6 ± 6 + 8x 8 ± 6 + x x \ " Œ(. is cceptble (. )( Mtlb Function: point_n = ; min_x = -; mx_x = 3; x = linspce(min_x mx_x point_n)'; = trimf(x ); = trimf(x 3 5 6); C= fuzrith(x 'prod'); subplot(); plot(x 'b--' x 'm:' x C 'd'); title('fuzzy multipliction *'); Grph:.. Division of Fuzzy Numbers: tringulr fuzzy number (TFN) is defined by the following nottion: ~ x b c Then the membership function for this TFN is defined s ( The cuts of TFN define set of closed intervls. The intervls re ( b - ) + ( b - c) + c " Œ(. For two fuzzy numbers b c nd b we hve c Æ( b - Æ( b ) + ( b - ) + ( b - c ) + c - c ) + c.

5 5 ( b - ) + ( b - c ) + c " Œ ( ( b - ) + ( b - c ) + c " Œ (. ( b - ) + ( b - c ) +. c ( b ( b - ) + - c ) + c ( b - c) + c ( b - ) + Now ( / ) / ( b - ) + ( b - c) + c " Œ ( ( b - c ) + c ( b - ) + Therefore ( / )(.. Exmple: Let nd 356 be two tringulr fuzzy numbers with fmf ( = nd Multipliction nd Division of Tringulr Fuzzy Numbers Therefore ( / )( Mtlb Function: point_n = ; min_x = -; mx_x = 3; x = linspce(min_x mx_x point_n)'; = trimf(x ); = trimf(x 3 5 6); C= fuzrith(x 'div'); subplot(); plot(x 'b--' x 'm:' x C 'd'); title('fuzzy division /'); Grph: ( Then ( / ) / + - " Œ( Now x 6 - ( x + ) 6x - 6x - " Œ ( x + - nd x 3 + ( x + ) - 3x - 3x " Œ(. ( x + ) 5. Conclusion ppliction of TFN s is very potentil prticulrly in models; Mnipultion of TFN s is more menble thn tht of nonstndrd membership functions. The two rithmetic opertions multipliction nd division eliminte the computtionl complexity issue of discretiztion. References. Chou Chien-Chng; (9) The Squre Roots of Tringulr Fuzzy Number ICIC Express Letters. Vol. 3 Nos. pp De rr G.; (987) Mesure Theory nd Integrtion Wiley Estern Limited New Delhi.

6 DFFODIL INTERNTIONL UNIVERSITY JOURNL OF SCIENCE ND TECHNOLOGY VOLUME ISSUE JULY Kufmnn. nd M. M. Gupt; (98) Introduction to Fuzzy rithmetic Theory nd pplictions Vn Nostrnd Reinhold Co. Inc. Wokinghm erkshire.. Loeve M. (977) Probbility Theory Vol.I Springer Verlg New York. 5. Mhnt njn K. Fokrul. Mzrbhuiy nd Hemnt K. ruh; (8) Finding Clendr sed Periodic Ptterns Pttern Recognition Letters 9 (9) Klir George J. Yun o Fuzzy Sets nd Fuzzy Logic Theory nd pplictions Prencic-Hll Inc. N. J. U.S Mtlb (9). I m Md. Mosfiqur Rhmn. I pssed.sc (Hons.) nd Mster of Science in Mthemtics from Jhngirngr University. Then I joined in Dffodil Interntionl University s lecturer in Mthemtics under the dept. of Nturl Sciences nd fculty of science nd technology on th September.