MAX-MIN AND MIN-MAX VALUES OF VARIOUS MEASURES OF FUZZY DIVERGENCE

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1 merca Jr of Mathematcs ad Sceces Vol, No,(Jauary 0) Copyrght Md Reader Publcatos wwwjouralshubcom MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES OF FUZZY DIVERGENCE RKTul Departmet of Mathematcs SSM College Daagar (Ida) marjt Sgh Departmet of Mathematcs desh Isttute of Egg & Tech Fardkot (Ida) STRCT I the preset mauscrpt, we have cosdered some ew o-probablstc (fuzzy) measures of drected dvergece, ad keepg vew the mportace ad areas of applcatos of these measures, we have vestgated ther optmum values Keywords: Dstace measure, Fuzzy etropy, Fuzzy drected dvergece, Covexty INTRODUCTION The measure of dstace s a mportat term that descrbes the dfferece betwee fuzzy sets ad ca be cosdered as a dual cocept of smlarty measure May researchers have used dstace measure to defe fuzzy etropy Usg the axom defto of dstace measure, Fa, Ma ad Xe [] developed some ew formulas of fuzzy etropy duced by dstace measure ad studed some ew propertes of dstace measure Rosefeld [0] defed the shortest dstace betwee two fuzzy sets as a desty fucto o the o - egatve reals Correspodg to the probablstc measure of dvergece due to Kullback ad Lebler [5], hadar ad Pal [] troduced the followg measure of fuzzy drected dvergece: µ ( x ) µ ( x ) I( : ) µ ( x )log ( µ ( x ))log µ ( x ) µ ( x ) () Correspodg to Rey s [9] ad Havrada ad Charvat s [3] dvergece measures, Kapur [4] took the followg expressos of measures of fuzzy drected dvergece: D ( : ) = log ( x ) ( x ) ( ( x )) ( ( x )) () D x x x x ( : ) ( ) ( ) ( ( )) ( ( )) (3) Tra ad Duckste [] developed a ew approach for rakg fuzzy umbers based o a dstace measure Parkash [6] troduced a geeralzed fuzzy dvergece, gve by (, ) [( ) ] ( ) ( ) ( ( )) ( ( )) D x x x x (4) May measures of fuzzy dvergece alog wth ther detaled propertes ad mportat applcatos have bee dscussed by varous authors cludg those of Kapur [4], Parkash [6], Parkash ad Sharma [7], Parkash ad Tul [8] etc I fact, Kapur [4] has developed may expressos for the measures of fuzzy drected dvergece correspodg to probablstc measures of dvergece due to Harvada ad Charvat [3], Rey [9] etc OPTIMIZTION OF VRIOUS MESURES OF DIVERGENCE 3

2 RKTul & marjt Sgh I ths secto, we cosder Rey s [9] measure of fuzzy drected dvergece gve () ad exame t for ts maxmum ad mmum values I Mmum values of Rey s [9] measure of fuzzy drected dvergece We ow fd the mmum value of D ( : ) Sce D ( : ) s a covex fucto, ts mmum value exsts For mmum value, we put D ( : ) 0 ( x ) whch gves Now ( x) k gves ( x ) ( x ) ( x ) k k ( x) M D x x Case-I Whe k 0, the k k ( : ) log ( ) ( ) M D ( : ) log ( x ) log ( x ) Case-II Whe k, the ( x ) ( x ) M D ( : ) log Case-III Whe k, the M D ( : ) log ( x ) log ( x ) Illustrato To llustrate the above process, we cosder x For k 0, M D ( : ) log ( x ) For 3 4 log log log log log log( ) 3 k,,,,, log log log log ( ) 3 M D ( : ) log log log log For k, ( )! log log 3 4

3 MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES M D ( : ) log ( x ) log ( x ) log 34( ) log( )! II Maxmum values of Rey s [9] measure of fuzzy drected dvergece We ow fd maxmum value of D ( : ) Case-I: Whe k s ay +ve teger, the we ca choose k values of x others k as 0, that s, x,,,,,0,0,,0 Now, we ca wrte D ( : ) m log ( x ) ( x ) ( x ) ( x ) Thus, the maxmum value of D ( : ) s gve as Max D x x as uty ad log ( x ) ( x ) ( x ) ( x ) m m ( : ) log ( ) log ( ) m Illustrato To llustrate the above process, we cosder x For 0,,, 3 4 m, M D ( : ) log ( x ) log ( x ) For m, 3 4 log log( ) 3 M D ( : ) log ( x ) log ( x ) 3 4 log log log log log log( ) 3 For m, 4 5 Max D ( : ) log log 3 log log log log! log( ) 3 4 For m 3, Max D ( : ) log log3 log 4 log log log For m, 5 6 log log 3 log 4 log log log log 3! log( ) 4 5 5

4 RKTul & marjt Sgh Max D ( : ) log ( x ) log ( x ) log log log log 3 4 log( )! The above values show that Max D ( : ) s pecewse covex fucto Case-II: If k s ay fracto, the, we ca wrte k 6 log34( ) m, where m s a +ve teger ad s a postve fracto We ca choose m fuzzy values of x as uty, th of x as ad remag m values of x x,,,,,0,0,,0 Thus, m value as 0, that s, D x x x x ( : ) log ( ) log ( m ) ( m ) log ( ) m m log ( x ) log ( x ) log ( x ) ( x ) m m m To llustrate the above process, we cosder x,,, 3 4 m 3 m 4 Max D ( : ) log log3 log( m) log log log m m3 m log m m log( m)! log log m m log( ) m! ( ) We, ow check the covexty of ( ) where ( ) log Take ( ) h m, d h d m Thus ( ) m d ( ) d lso h m If, the m d h( ) 0 h( ) s a covex fucto of d log h( ) s a covex fucto of log h ( ) s a covex fucto of for,

5 MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES If, the d h( ) 0 h( ) s a cocave fucto of d log h( ) s a cocave fucto of log h ( ) s a covex fucto of for 0, Hece, ( ) s a covex fucto of for each Its mmum value exsts at m lso, the mmum value of Max D ( : ) s gve as M Max D ( : ) log( ) m! m m m log( ) m! log m m For 0 log( ) m! log m m m m! m! log( ) log log( ) log m m m!, M Max D ( : ) log( ) log! m, M Max D ( : ) log( ) log 3 m 3!, M Max D ( : ) log( ) log 4 m 4!, M Max D ( : ) log( ) log 5 For For For 3 For m, For ( )! m, M Max D ( : ) log( ) log Coclusos: Whe x fucto It ca also be proved that f x s a decreasg fucto lso both cases, Max D : s a mootocally decreasg, the 7 Max cross etropy s a creasg s a mootocally creasg, the Max cross etropy s a pecewse covex fucto of k I the lterature of dstace measures, there exst may parametrc ad o-parametrc measures of fuzzy dvergece troduced by varous researchers Proceedg o smlar way as doe secto, the optmum values of other dvergece measures ca be studed REFERENCES [] hadar, D ad Pal, NR (993) Some ew formato measures for fuzzy sets Iformato Sceces 67: 09-8 [] Fa, JL, Ma, YL ad Xe, WX (00) O some propertes of dstace measure Fuzzy Sets ad Systems 7:

6 RKTul & marjt Sgh [3] Havrada, JH ad Charvat, F (967) Quatfcato methods of classfcato process:cocept of structural -etropy Kyberetka 3: [4] Kapur, JN (997) Measures of Fuzzy Iformato Mathematcal Sceces Trust Socety, New Delh [5] Kullback, S ad Lebler, R (95) O formato ad suffcecy als of Mathematcal Statstcs : [6] Parkash, O (000) O fuzzy symmetrc dvergece The Fourth sa Fuzzy System Symposum : [7] Parkash, O ad Sharma, P K (005) Some ew measures of fuzzy drected dvergece ad ther geeralzato Joural of the Korea Socety of Mathematcal Educato Seres : [8] Parkash, O ad Tul, RK (005) O geeralzed measures of fuzzy drected dvergece Ultra Scetst of Physcal Sceces 7: 7-34 [9] Rey, (96) O measures of etropy ad formato Proceedgs 4th erkeley Symposum o Mathematcal Statstcs ad Probablty : [0] Rosefeld, (985) Dstace betwee fuzzy sets Patter Recogto Letters 3: 9-3 [] Tra, L ad Duckste, L (00) Comparso of fuzzy umbers usg a fuzzy dstace measure Fuzzy Sets ad Systems 30: