EXPONENTIAL ENTROPY ON INTUITIONISTIC FUZZY SETS

K Y B E R N E T I K A V O L U M E 4 9 0 3, N U M B E R, P A G E S 4 7 EXPONENTIAL ENTROPY ON INTUITIONISTIC FUZZY SETS Rajkumar Vrma and Bhu Dv Sharma In th prsnt papr, basd on th concpt of fuzzy ntropy, an xponntial intuitionistic fuzzy ntropy masur is proposd in th stting of Atanassov s intuitionistic fuzzy st thory. This masur is a gnralizd vrsion of xponntial fuzzy ntropy proposd by Pal and Pal. A connction btwn xponntial fuzzy ntropy and xponntial intuitionistic fuzzy ntropy is also stablishd. Som intrsting proprtis of this masur ar analyzd. Finally, a numrical xampl is givn to show that th proposd ntropy masur for Atanassov s intuitionistic fuzzy st is consistnt by comparing it with othr xisting ntropis. Kywords: Classification: fuzzy st, fuzzy ntropy, Atanassov s intuitionistic fuzzy st, intuitionistic fuzzy ntropy, xponntial ntropy 94A7. INTRODUCTION Th thory of fuzzy sts proposd by Zadh [6 in 965 has gaind wid applications in many aras of scinc and tchnology. g. clustring, imag procssing, dcision making tc. bcaus of its capability to modl non-statistical imprcision or vagu concpts. Fuzzinss brings in a fatur of uncrtainty. Th first attmpt to quantify th fuzzinss was mad in 968 by Zadh [7, who introducd a probabilistic framwork and dfind th ntropy of a fuzzy vnt as wightd Shannon ntropy [ but this masur was not found adquat for masuring th fuzzinss of a fuzzy vnt. In 97, D Luca and Trmini [5 formulatd axioms which th fuzzy ntropy masur should comply, and thy dfind th ntropy of a fuzzy st basd on Shannon s function. It may b rgardd as th first corrct masur of fuzzinss of a fuzzy st. Atanassov [ introducd th notion of Atanassov s intuitionistic fuzzy st, which is a gnralization of th concpt of fuzzy st. Burillo and Bustinc [3 dfind th ntropy on Atanassov s intuitionistic fuzzy st and on intrval-valud fuzzy st. Vlachos and Srgiagis [3 proposd a masur of intuitionistic fuzzy ntropy and rvald an intuitiv and mathmatical connction btwn th notions of ntropy for fuzzy st and Atanassov s intuitionistic fuzzy st. Zhang and Jiang [8 dfind a masur of intuitionistic vagu fuzzy ntropy on Atanassov s intuitionistic fuzzy sts by gnralizing of th D Luca Trmini [5 logarithmic fuzzy ntropy. In this papr, w propos a nw information masur for Atanassov s intuitionistic

Exponntial ntropy on intuitionistic fuzzy sts 5 fuzzy sts. W call it xponntial intuitionistic fuzzy ntropy. It is basd on th concpt of xponntial fuzzy ntropy dfind by Pal and Pal [9. To dfin this ntropy function fuzzy st thortic approach has bn usd. Such an approach is found particularly usful in situations whr data is availabl in trms of intuitionistic fuzzy st valus but implmntation rquirmnts ar only fuzzy. So far th practic has bn to simply ignor th hsitation part. A bttr rsult has bn obtaind by not ignoring but by mrging th hsitation part suitably. W suggst a mathmatical mthod for it. This may hlp application of IFS data in industry, whr th tools usd ar of fuzzy st thory. Th papr is organizd as follows: In Sction som basic dfinitions rlatd to probability, fuzzy st thory and Atanassov s intuitionistic fuzzy st thory ar brifly discussd. In Sction 3 a nw information masur calld, xponntial intuitionistic fuzzy ntropy is proposd, which satisfis th axiomatic rquirmnts [. Som mathmatical proprtis of th proposd masur ar thn studid in this sction. In Sction 4 a numrical xampl is givn comparing our masur with othr ntropis proposd in [4 and [8.. PRELIMINARIES In this sction w prsnt som basic concpts rlatd to probability thory, fuzzy sts and Atanassov s intuitionistic fuzzy sts, which will b ndd in th following analysis. First, lt us covr probabilistic part of th prliminaris. Lt n = {P = p,..., p n : p i 0, n p i = }, n b a st of n-complt probability distributions. For any probability distribution P = p,..., p n n, Shannon s ntropy [, is dfind as HP = p i log p i. It is to b notd from th logarithmic ntropic masur that as p i 0, it s corrsponding slf information of this vnt, Ip i = logp i but Ip i = = log = 0. Thus w s that slf information of an vnt has concptual problm, as in practic, th slf information of an vnt, whthr highly probabl or highly unlikly, is xpctd to li btwn two finit limits. Som advantags for considring xponntial ntropy: In Shannon s thory, which is widly acclaimd, w find that th masur of slf information of an vnt with probability p i is takn as log/p i, a dcrasing function of p i. Th sam dcrasing charactr altrnativly may b maintaind by considring it as a function of p i rathr than of /p i. Th additiv proprty, which is considrd crucial in Shannon s approach, of th slf information function for indpndnt vnts may not hav a strong rlvanc impact in practic in som situations. Altrnativly, as in th cas of probability law, th joint slf information may b product rathr than sum of th slf informations in two indpndnt cass. Th abov considrations suggst th slf information as an xponntial function of p i.

6 R. K. VERMA AND B. D. SHARMA Basd on th ths considrations, Pal and Pal [9 proposd anothr masur calld xponntial ntropy givn by HP = p i pi. Ths authors point out that th xponntial ntropy has an advantag ovr Shannon s ntropy. For xampl, for th uniform probability distribution P = n, n,..., n, xponntial ntropy has a fixd uppr bound lim H n n, n,..., = 3 n which is not th cas for Shannon s ntropy. Dfinition.. Fuzzy St: A fuzzy st à dfind in a finit univrs of discours X = x,..., x n is givn by Zadh [6: à = { x, µãx x X}, 4 whr µã : X [0, is th mmbrship function of Ã. Th numbr µãx dscribs th dgr of mmbrship of x X to Ã. D Luca and Trmini [5 dfind fuzzy ntropy for a fuzzy st à corrsponding as Hà = n [ µãx i log µãx i µãx i log µãx i. 5 Fuzzy xponntial ntropy for fuzzy st à corrsponding to has also bn introducd by Pal and Pal [9 as Hà = n [ µãx i µãxi µãx i µãxi. 6 Furthr, Atanassov [ gnralizd th ida of fuzzy sts, by what is calld Atanassov s intuitionistic fuzzy sts, dfind as follows: Dfinition.. Atanassov s Intuitionistic Fuzzy St: An Atanassov s intuitionistic fuzzy st A in a finit univrs of discours X = x,..., x n is givn by: whr with th condition A = { x, µ A x, ν A x x X }, 7 µ A : X [0, and ν A : X [0, 8 0 µ A x ν A x, x X. 9 Th numbrs µ A x and ν A x dnot th dgr of mmbrship and dgr of nonmmbrship of x X to A, rspctivly.

Exponntial ntropy on intuitionistic fuzzy sts 7 Dfinition.3. Hsitation Margin: For ach Atanassov s intuitionistic fuzzy st A in X, if π A x = µ A x ν A x, 0 thn π A x is calld th Atanassov s intuitionistic indx or a hsitation dgr of th lmnt x X to A. For studying sts, thr is nd to considr st rlations and oprations, which in th study of Atanassov s intuitionistic fuzzy sts ar dfind as follows. Dfinition.4. St Oprations on Atanassov s Intuitionistic Fuzzy St[: Lt AIF SX dnot th family of all Atanassov s intuitionistic fuzzy sts in th univrs X, and lt A, B AIF SX givn by A = { x, µ A x, ν A x x X}, B = { x, µ B x, ν B x x X}, thn som st oprations can b dfind as follows: i A B iff µ A x µ B x and ν A x ν B x x X; ii A = B iff A B and B A; iii A C = { x, ν A x, µ A x x X}; iv A B = { x, µ A x µ B x, ν A x ν B x x X}; v A B = { x, µ A x µ B x, ν A x ν B x x X}; vi A = { x, µ A x, µ A x x X}; vii A = { x, ν A x, ν A x x X}; { viii A@B = x, µ A µ B, ν A ν B x X Mthod for Transforming AIFSs into FSs: Li, Lu and Cai [8, as brifly outlind blow, proposd a mthod for transforming Atanassov s intuitionistic fuzzy sts vagu sts into fuzzy sts by distributing hsitation dgr qually with mmbrship and non-mmbrship. Dfinition.5. Lt A = { x, µ A x, ν A x x X} b an Atanassov s intuitionistic fuzzy st dfind in a finit univrs of discours X. Thn th fuzzy mmbrship function µ A x to à à b th fuzzy st corrsponding to Atanassov s intuitionistic fuzzy st A is dfind as: µ A x = µ A x π Ax }. = µ Ax ν A x.

8 R. K. VERMA AND B. D. SHARMA This ara of study has attractd quit som attntion for applications in dcision-making. Finally w may as wll mntion som othr rlatd masurs with which w compar our study. Zhang and Jiang [8 prsntd a masur of intuitionistic vagu fuzzy ntropy basd on a gnralization of masur 5 as E ZJ A = [ µa x i ν A x i µa x i ν A x i log n νa x i µ A x i νa x i µ A x i log. Y [5 introducd two ffctiv masurs of intuitionistic fuzzy ntropy basd on a gnralization of th fuzzy ntropy dfind by Prakash t al. [0 givn by E JY A = n E JY A = n [{ µa x i ν A x i sin π 4 νa x i µ A x i sin π 4 [{ µa x i ν A x i cos π 4 νa x i µ A x i cos π 4 } }, 3. 4 Latr, Wi t al. [4 hav shown that th two ntropy functions 3 and 4 proposd by Y [5 ar mathmatically th sam and gav a simplifid vrsion as E W GG A = n [{ µa x i ν A x i cos π 4 }. 5 Throughout this papr, w dnot th st of all Atanassov s intuitionistic fuzzy sts in X by AIF SX. Similarly, F SX is th st of all fuzzy sts dfind in X. In th nxt sction w introduc an ntropy masur on Atanassov s intuitionistic fuzzy sts calld xponntial intuitionistic fuzzy ntropy corrsponding to 6 and vrify axiomatic basis of th sam. 3. EXPONENTIAL INTUITIONISTIC FUZZY ENTROPY Lt A b an Atanassov s intuitionistic fuzzy st dfind in th finit univrs of discours, X = x,..., x n. Thn, according to th Dfinition.5, an Atanassov s intuitionistic fuzzy st can b transformd into a fuzzy st to structur an ntropy masur of th intuitionistic fuzzy st by mans of µ A x i = µ A x i π Ax i = µ Ax i ν A x i.

Exponntial ntropy on intuitionistic fuzzy sts 9 Thn, in analogy with th dfinition of xponntial fuzzy ntropy givn in 6, w propos th xponntial intuitionistic fuzzy ntropy masur for Atanassov s intuitionistic fuzzy st A as follows: EA = n [ µa x i ν A x i µa x i ν A x i which can also b writtn as EA = n µ Ax i ν A x i [ µa x i ν A x i νa x i µ A x i µa x i ν A x i νa x i µ A x i µa x i ν A x i, 6. 7 In th nxt thorm, w stablish proprtis that according to Szmidt and Kacprzyk [, justify our proposd masur to b a bonafid/valid intuitionistic fuzzy ntropy : Thorm 3.. Th EA masur in 7 of th xponntial intuitionistic fuzzy ntropy satisfis th following propositions: P. EA = 0 iff A is a crisp st, i.., µ A x i = 0, ν A x i = or µ A x i =, ν A x i = 0 for all x i X. P. EA = iff µ A x i = ν A x i for all x i X. P3. EA = EA iff A B, i.., µ A x i µ B x i and ν A x i ν B x i, for µ B x i ν B x i or µ A x i µ B x i and ν A x i ν B x i, for µ B x i ν B x i for any x i X. P4. EA = EA C. P r o o f. P. Lt A b a crisp st with mmbrship valus bing ithr 0 or for all x i X. Thn from 7 w simply obtain that Now, lt EA = 0. 8 µ A x i ν A x i In viw of 9, xprssion in 7 can b writtn as EA = n = z A x i. 9 [ z A x i z Ax i z A x i z Ax i. 0

0 R. K. VERMA AND B. D. SHARMA From Pal and Pal [9, w know that 0 bcoms zro if and only if z A x i = 0 or, x i X i.., µ A x i ν A x i = 0 i.., ν A x i µ A x i =, x i X or µ A x i ν A x i = i.., µ A x i ν A x i =, x i X. And from Dfinition.,w hav Now solving quation with 3, w gt µ A x i ν A x i, x i X. 3 µ A x i = 0, ν A x i =, x i X. Nxt solving quation with 3, w gt µ A x i =, ν A x i = 0, x i X. Thrfor EA rducs to zro only if ithr µ A x i = 0, ν A x i = or µ A x i =, ν A x i = 0 for all x i X, proving th rsult. P. Lt µ A x i = ν A x i for all x i X. From 7 w obtain EA =. From quation 0, w hav EA = n fz A x i, whr [ za x i z Ax i z A x i z Ax i fz A x i = x i X. 4 Now, lt us suppos that EA =, i.. or n fz A x i = fz A x i = x i X. 5 Diffrntiating 5 with rspct to z A x i and quating to zro, w gt or f z A x i = z Ax i z A x i z Ax i z Ax i z A x i z Ax i = 0 z A x i z Ax i = z A x i z Ax i x i X. 6

Exponntial ntropy on intuitionistic fuzzy sts Using th fact that fx = x x is a bijction function, w can writ or and find z A x i = z A x i x i X 7 z A x i = 0.5 x i X 8 [ f z A x i z < 0 x i X. 9 A x i=0.5 Hnc fz A x i is a concav function and has a global maximum at z A x i = 0.5. Sinc EA = n n fz Ax i, So EA attains th maximum valu whn z A x i = 0.5 or µ A x i = ν A x i for all x i X. P3. In ordr to show that 7 fulfills P3, it suffics to prov that th function [ x y gx, y = y x y x x y whr x, y [0,, is incrasing with rspct to x and dcrasing for y. partial drivativs of g with rspct to x and y, rspctivly, yilds g x = [ y x y x x y x y g y = [ x y x y y x y x In ordr to find critical point of g, w st g g x = 0 and y = 0. This givs From 3 and 33, w hav and 30 Taking th 3 3 x = y. 33 g 0, whn x y 34 x g 0, whn x y 35 x for any x, y [0,, Thus gx, y is incrasing with rspct to x for x y and dcrasing whn x y. Similarly, w obtain that g 0, whn x y 36 y and g 0, whn x y. 37 y

R. K. VERMA AND B. D. SHARMA Lt us now considr two sts A, B IF SX with A B. Assum that th finit univrs of discours X = x,..., x n is partitiond into two disjoint sts X and X with X X. Lt us furthr suppos that all x i X ar dominatd by th condition µ A x i µ B x i ν B x i ν A x i, whil for all x i X µ A x i µ B x i ν B x i ν A x i. Thn from th monotonicity of gx, y and 7, w obtain that EA EB whn A B. P4. It is clar that A C = { x, ν A x i, µ A x i x X} for all x i X, i.., thn, from 7 w hav µ A C x i = ν A x i and ν A C x i = µ A x i EA = EA C. Hnc EA is a valid masur of Atanassov s intuitionistic fuzzy ntropy. This provs th thorm. Particular cas: It is intrsting to notic that if an Atanassov s intuitionistic fuzzy st is an ordinary fuzzy st, i.., for all x i X, ν A x i = µ A x i, thn th xponntial intuitionistic fuzzy ntropy rducs to xponntial fuzzy ntropy as proposd in [9. W now turn to study of proprtis of EA. Th proposd xponntial intuitionistic fuzzy ntropy EA, just lik fuzzy ntropy masur, satisfis th following intrsting proprtis. Thorm 3.. Lt A and B two Atanassov s intuitionistic fuzzy sts in a finit univrs of discours X = x,..., x n, whr Ax i = µ A x i, ν A x i, Bx i = µ B x i, ν B x i such that thy satisfy for any x i X ithr A B or A B, thn w hav EA B EA B = EA EB. P r o o f. Lt us sparat X into two parts X and X, whr X = {x i X : Ax i Bx i } and X = {x i X : Ax i Bx i }. That is, for all x i X µ A x i µ B x i and ν A x i ν B x i 38 and for all x i X µ A x i µ B x i and ν A x i ν B x i. 39

Exponntial ntropy on intuitionistic fuzzy sts 3 From dfinition in 7, w hav EA B = n [ µa B x i ν A B x i νa B x i µ A B x i νa B x i µ A B x i µa B x i ν A B x i = [ { n µb x i ν B x i x i X νb x i µ B x i { µa x i ν A x i x i X νa x i µ A x i Again from dfinition in 7, w hav EA B = n νb x i µ B x i µb x i ν B x i } νa x i µ A x i [ µa B x i ν A B x i νa B x i µ A B x i µa x i ν A x i }. 40 νa B x i µ A B x i µa B x i ν A B x i = [ { n Now adding 40 and 4, w gt µa x i ν A x i x i X νa x i µ A x i { µb x i ν B x i x i X νb x i µ B x i νa x i µ A x i µa x i ν A x i } νb x i µ B x i µb x i ν B x i EA B EA B = EA EB. }. 4 This provs th thorm.

4 R. K. VERMA AND B. D. SHARMA Thorm 3.3. For vry A AIF SX, i A@ A is a fuzzy st; ii A@ A = A@ A; iii A@ A = à ; whr à is th fuzzy st corrsponding to Atanassov s intuitionistic fuzzy st A. P r o o f. i From Dfinition.4, w hav Now taking @ with 4 and 43, w gt A@ A = It can b asily obsrvd that, A = { x, µ A x, µ A x x X}; 4 A = { x, ν A x, ν A x x X}. 43 { x, µ A ν A, ν A µ } A x X. 44 µ A ν A ν A µ A ii It obviously follows from quation 44. iii From quations 44 and, w hav { A@ A = x, µ A ν A { à = = x X., ν A µ A x, µ A ν A, ν A µ A } x X ; x X }. This provs th thorm. Thorm 3.4. For vry A AIF SX, EA = E A@ A. P r o o f. From quation 7, w hav EA = n [ µa x i ν A x i νa x i µ A x i νa x i µ A x i µa x i ν A x i

Exponntial ntropy on intuitionistic fuzzy sts 5 and E A@ A = n This provs th thorm. Thorm 3.5. For vry A AIF SX, [ µa x i ν A x i νa x i µ A x i E A@ A = E A@ A. νa x i µ A x i µa x i ν A x i. 45 P r o o f. It radily follows from Thorm 3.3iiand quation 45. Thorm 3.6. For vry A AIF SX, P r o o f. E A@ A = E A C @ A C C. From quation 45, w hav E A@ A = n [ µa x i ν A x i νa x i µ A x i νa x i µ A x i and E A C @ A C C [ = n µa x i ν A x i νa x i µ A x i This provs th thorm. µa x i ν A x i νa x i µ A x i µa x i ν A x i. In th nxt sction w considr an xampl to compar our proposd ntropy masur on Atanassov s intuitionistic fuzzy st, with othrs in and 5. 4. NUMERICAL EXAMPLE Exampl: Lt A = { x i, µ A x i, ν A x i x i X } b an AIFS in X = x,..., x n. For any positiv ral numbr n, D t al. [6 dfind th AIFS A n as follows: A n = { x i, [µ A x i n, [ ν A x i n x i X }. W considr th AIFS A on X = x,..., x n dfind as: A = { 6, 0., 0.8, 7, 0.3, 0.5, 8, 0.5, 0.4, 9, 0.9, 0.0, 0,.0, 0.0 }. By taking into considration th charactrization of linguistic variabls, D t al. [6 rgardd A as LARGE on X. Using th abov oprations, w hav

6 R. K. VERMA AND B. D. SHARMA A / for may b tratd as Mor or lss LARGE A for may b tratd as Vry LARGE A 3 for may b tratd as Quit vry LARGE A 4 for may b tratd as Vry vry LARGE Now w considr ths AIFSs to compar th abov ntropy masurs. It may b mntiond that from logical considration, th ntropis of ths AIFSs ar rquird to follow th following ordr pattrn: EA / > EA > EA > EA 3 > EA 4. 46 Calculatd numrical valus of th thr ntropy functions for ths cass ar givn in th tabl blow: A / A A A 3 A 4 E ZJ 0.589 0.570 0.4333 0.33 0.698 E W GG 0.4545 0.4377 0.309 0.59 0.709 E 0.553 0.5343 0.377 0.734 0.69 Tabl: Valus of th diffrnt ntropy masurs undr A /, A, A, A 3, A 4. Basd on th Tabl, w s that th ntropy masurs E ZJ and E W GG satisfy 46, and our proposd ntropy masur conforms to th sam, i.. EA / > EA > EA > EA 3 > EA 4. Thrfor, th bhavior of xponntial intuitionistic fuzzy ntropy EA is also consistnt for th viwpoint of structurd linguistic variabls. 5. CONCLUSIONS In this work, w hav proposd a nw ntropy masur calld xponntial intuitionistic fuzzy ntropy in th stting of Atanassov s intuitionistic fuzzy st thory. This masur can b considrd as a gnralizd vrsion of xponntial fuzzy ntropy proposd by Pal and Pal [0. This masur is imbud with svral proprtis. A numrical xampl is givn to illustrat th ffctivnss of proposd ntropy masur. Paramtric studis that introduc othr flxibility critria for th sam mmbrship functions, of this masur ar also undr study and will b rportd sparatly. ACKNOWLEDGEMENTS Th authors ar gratful to th associat ditor and anonymous rfrs for thir valuabl suggstions which hlpd in improving th prsntation of th papr. Rcivd Jun 9, 0

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