TWO NEW WEIGHTED MEASURES OF FUZZY ENTROPY AND THEIR PROPERTIES

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1 merca. Jr. of Mathematcs ad Sceces Vol., No.,(Jauary 0) Copyrght Md Reader Publcatos TWO NEW WEIGTED MESURES OF FUZZY ENTROPY ND TEIR PROPERTIES R.K.Tul Departmet of Mathematcs S.S.M. College Daagar (Ida) Chader Sekhar Sharma Departmet of Mathematcs desh Isttute of Egg. & Tech. Fardkot (Ida) BSTRCT Measures of weghted formato are well kow the lterature of formato theory. I the preset commucato, we have developed some ew geeralzed measures of weghted fuzzy etropy, studed ther mportat propertes for ther authetcty ad have preseted them graphcally. Keywords: Fuzzy etropy, Fuzzy set, Symmetry, Cocavty, Trgoometrc etropy. INTRODUCTION The oto of fuzzy set stems from the observato made by Zadeh [6] that more ofte tha ot, the classes of objects ecoutered the real physcal world do ot have precsely defed crtera of membershp. Ths observato emphaszes the gap estg betwee metal represetatos of realty ad usual mathematcal represetatos thereof, whch are based o bary logc, precse umbers ad the lke. ccordg to Zadeh [6] classes of objects est oly through such metal represetatos through atural laguage terms such as hgh temperature, youg ma, bg sze, etc., ad also wth ous such as brd, char, etc. Whe proposg fuzzy sets, Zadeh s [6] cocers were eplctly cetered o ther potetal cotrbuto the domas of patter classfcato, processg ad commucato of formato, abstracto ad summarzato. Thus, keepg vew the dea of fuzzy sets, De Luca ad Term [] troduced the followg measure of fuzzy etropy correspodg to Shao s [5] measure. (.) ( ) log log fter ths developmet, a large umber of measures of fuzzy etropy were dscussed, characterzed ad geeralzed by varous authors. Kapur [7] troduced the followg measure of fuzzy etropy: ( ) ( ( )), ( ) log ;, ( ) ( ( )) Parkash [9] troduced a ew geeralzed fuzzy etropy volvg two real parameters, gve by ( ) = [( ) ] ( ) ( ( )) ; > 0,, 0 (.3) ad called t ( ) fuzzy etropy whch cludes some well kow measures of fuzzy etropy. Parkash ad Tul [] troduced the followg trgoometrc measure of etropy: (.) 07

2 R.K.Tul & Chader Sekhar Sharma k( ) k( ) s ( ) S ( ) S ( ( )) S (.4) where k ad are two parameters satsfyg 0, 0 k (.5) k ( ) S Parkash ad Tul [] have provded the applcatos of the fuzzy measures of etropy to the feld of trgoometry ad obtaed the followg trgoometrc equaltes: k ( ) S( k ) S (.6)... The equalty sg (.6) holds oly whe The equalty (.6) s a basc trgoometrc equalty volvg trgoometrc fuctos of the agles,,..., of a cove polygo of sdes. The equalty sg holds whe all agles are equal. Moreover, Parkash ad Tul [] obtaed the epressos for optmum fuzzess dfferet measures of fuzzy etropy. Parkash, Sharma ad Mahaja [0] developed some ew measures of weghted fuzzy etropy ad provded ther applcatos for the study of mamum weghted fuzzy etropy prcple. Some other measures of fuzzy etropy have bee studed by Ebaks [3], Rudas [4], Kapur [6], Guo ad X [4], Parkash ad Tul [3], u ad Yu [5], Pal ad Bezdek [8], Osma, bdel-fadel, El-Sersy, ad Ibraheem [7] etc. The cocept of weghted formato whch was troduced by Bels ad Guasu [], receved a good respose from formato theoretcas ad ths troducto opeed a ew feld for researchers who started workg over the cocept of weghted formato. Thus, o attachg weghts to the fuzzy values, measure of weghted fuzzy etropy ca be costructed parallel to the fuzzy etropy troduced by De Luca ad Term []. Ths measure s gve by (.7) ( ; W) w log log Usg the cocept of ths weghted, we have developed two ew measures of fuzzy etropy, the fdgs of whch have bee preseted the et secto.. TWO NEW WEGTED MESURES OF FUZZY ENTROPY I. Frstly, we propose a ew o-parametrc weghted measure of fuzzy etropy as gve by the followg mathematcal epresso: ( W : ) The result (.) ca be wrtte as ( W : ) w ( ) ( ) w ( ) 08 (.) (.)

3 TWO NEW WEIGTED MESURES OF FUZZY ENTROPY ND TEIR PROPERTIES where ( );,,..., Dfferetatg equato (.) twce, we get ( W : ) So ( W : ) s a cocave fucto of. Thus, we ote that ( W : ) satsfes the followg propertes:. () ( W : ) s a cocave fucto of ( ) () ( W : ) s a creasg fucto of ( ) whe < 0 0 ( ) () ( W : ) s a decreasg fucto of ( ) whe ( ) (v) ( W : ) does ot chage whe ( ) s chaged to - ( ). (v) ( W : ) = 0 whe ( ) Thus, ( : ) = 0 or W s a vald measure of weghted fuzzy etropy. Note: To obta mamum value of ( W : ), we have ( W : ) ( ) Thus ( : ) = 0 ( ) = W ma w = 4 w Net, wth the help of umercal data, we have preseted the weghted measure (.) graphcally. For ths purpose, W correspodg to dfferet fuzzy values ( ) uder the W 8,8.,8.,8.3,8.4,8.5,8.6,8.7,8.8,8.9,9 as show the followg Table-.: we have computed dfferet values of ( : ) weghted dstrbuto Table-. ( ) w ( W : )

4 R.K.Tul & Chader Sekhar Sharma Net, we have preseted ( W : ) graphcally ad obtaed Fg.-. whch shows that the weghted measure (.) s a cocave fucto ad ths proves the authetcty of the proposed measure. (:W) µ() Fg.-. II. Net, we propose aother ew weghted measure of fuzzy etropy as follow: ( W : ) = w ( ( ) ) { ( ( )) } ( ) ( ) (.3) Proceedg as above, we have checked that ( W : ) () ( W : ) () ( W : ) () ( W : ) (v) ( W : ) s a cocave fucto of ( ). does t chage whe ( ) s a creasg fucto of ( ) satsfes the followg propertes: s replaced by ( ) for s a decreasg fucto of ( ) (v) ( W : ) = 0 whe ( ) = 0 or. ece, ( W : ) s a vald measure of fuzzy etropy. Net, we have preseted the values of ( W : ) the measure troduced equato (.3) s a cocave fucto.. 0 ( ). for ( ). graphcally ad obtaed the followg Fg.-. whch shows that (:W) µ() Fg.-. REFERENCES [] Bels, M. ad Guasu, S. (968). quattatve-qualtatve measure of formato cyberetc systems. IEEE Trasactos o Iformato Theory 4:

5 TWO NEW WEIGTED MESURES OF FUZZY ENTROPY ND TEIR PROPERTIES [] De Luca,. ad Term, S. (97). defto of o-probablstc etropy settg of fuzzy set theory. Iformato ad Cotrol 0: [3] Ebaks, B.R. (983). O measures of fuzzess ad ther represetatos. Joural of Mathematcal alyss ad pplcatos 94: [4] Guo, X. Z. ad X, X. L. (006). Some ew geeralzed etropy formulas of fuzzy sets. Joural of the Northwest Uvewrsty 36: [5] u, Q.ad Yu, D. (004). Etropes of fuzzy dscerblty relato ad ts operatos. Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systems : [6] Kapur, J.N. (997). Measures of Fuzzy Iformato. Mathematcal Sceces Trust Socety, New Delh. [7] Osma, M.S.., bdel-fadel,.m.., El-Sersy, K.F. ad Ibraheem, N.M.. (008). O the soluto of fuzzy parametrc bcrtero programmg problems. OPSERC, 45. [8] Pal, N.R. ad Bezdek, J.C. (994). Measurg fuzzy ucertaty. IEEE Trasacto o Fuzzy Systems : [9] Parkash, O. (998). ew parametrc measure of fuzzy etropy. Iformato Processg ad Maagemet of Ucertaty : [0] Parkash, O., Sharma, P. K. ad Mahaja, R (008). New measures of weghted fuzzy etropy ad ther applcatos for the study of mamum weghted fuzzy etropy prcple. Iformato Sceces 78: [] Parkash, O. ad Tul, R.K. (005): Some Trgoometrc equaltes va fuzzy formato. Proc. 7 th. Cof. Id. Soc, If. Theory & pp., [] Parkash, O. ad Tul, R.K. (006). Mamum fuzzess dfferet measures of etropy. Iteratoal Joural of Maagemet ad Systems : [3] Parkash, O. ad Tul, R.K. (007). O geeralzed avrada-charvat fuzzy etropy. Qualty, Relablty ad Ifocom Tech. (Ed.P.K.Kapur &.K.Verma), MacMlla Ida Ltd., [4] Rudas, I. J. (00). Measures of fuzzess: theory ad applcatos. dvaces fuzzy systems ad evolutoary computato. pp87-9. World Scetfc ad Egeerg Socety Press, thes. [5] Shao, C. E.(948). mathematcal theory of commucato. Bell System Techcal Joural 7: , [6] Zadeh, L.. (965). Fuzzy sets. Iformato ad Cotrol 8: