A New Measure of Divergence with its Application to Multi-Criteria Decision Making under Fuzzy Environment

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1 Mauscript Click here to dowload Mauscript: Shikha Maheshwari ad Rakumar Verma-Fuzzy Jese Expoetial Divergece Measure.doc Click here to view liked Refereces New Measure of Divergece with its pplicatio to Multi-Criteria Decisio Makig uder Fuzzy Eviromet Shikha Maheshwari, Rakumar Verma, mit Srivastava, Departmet of Mathematics, Jaypee Istitute of Iformatio Techology Noida-00, Uttar Pradesh, Idia Departmet of pplied Scieces, HMR Istitute of Techology ad Maagemet, Guru Gobid Sigh Idraprastha Uiversity, Hamidpur, Delhi-00, Idia maheshwari.shikha@gmail.com, rkver@gmail.com bstract: Divergece measure is a importat tool for determiig the amout of discrimiatio betwee two probability distributios. Sice the itroductio of fuzzy sets, divergece measures betwee two fuzzy sets have gaied attetio for their applicatios i various fields. Expoetial etropy measure has some advatages over Shao s etropy. I this paper, we used the idea of Jese Shao divergece to defie a ew divergece measure called fuzzy Jese-expoetial divergece (FJSD for measurig the discrimiatio/differece betwee two fuzzy sets. The measure is demostrated to satisfy some very elegat properties, which shows its stregth for applicatios i multi-criteria decisio makig problems. Further, we develop a method to solve multi-criteria decisio makig problems uder fuzzy pheomeo by utilizig the proposed measure ad demostrate by a umerical example. Keywords - etropy, fuzzy sets, fuzzy divergece, Jese-Shao divergece, multi-criteria decisio makig. Itroductio I the last six decades, divergece measures have bee extesively used to measure the differece betwee two probability distributios [,,, ad ] ad widely applied i various fields. The otio of divergece, itroduced by Kullback ad Leibler [] i, provides a measure of discrimiatio betwee two probability distributios. fter its itroductio, various geeralized measures of divergece have bee proposed by researchers [ ad ] ad studied their properties ad applicatio i details. I, Li [] defied a ew divergece measure amed as Jese-Shao divergece, which has bee gaied quite some attetio from researchers/practitioers ad successfully applied i variety of disciplies [,,, ad -]. Parallel to the cocept of probability theory, the otio of fuzzy sets (FSs itroduced by Zadeh [] i to deal with vagueess. Sice the, the theory of fuzzy set has become a vigorous area of research i differet disciplies such as egieerig, artificial itelligece, medical sciece, sigal processig, ad expert systems. I, hadari ad Pal [] forwarded the cocept of divergece measure from probabilistic to fuzzy pheomea ad defied a divergece measure betwee two fuzzy sets. Fuzzy divergece measure gives fuzzy iformatio measure for discrimiatio of a fuzzy set relative to some other fuzzy set. This fuzzy divergece measure has wide applicatios i may areas such as patter recogitio, fuzzy clusterig, sigal ad image processig etc. Some geeralized measures of

2 fuzzy divergece have bee studied by Hooda [], aa ad Hooda [], Shag ad Jiag []. s metioed above Jese-Shao divergece is a importat measure from applicatio poit of view. This divergece measure is based o Shao s etropy fuctio [0]. I, Pal ad Pal [, ] critically aalyze the Shao s fuctio ad discussed its some limitatios. To imbue these limitatios, Pal ad Pal [, ] defied a ew measure of etropy based o expoetial fuctio. Futher, Kvålseth [] ad Verma ad Sharma [] defied some geeralized versio of expoetial fuzzy etropy. I, Fa ad Xie [] itroduced the fuzzy divergece measure based o expoetial fuctio ad studied its relatio with divergece measure itroduced i []. Ghosh et al. [] developed some applicatios of fuzzy divergece measure i the area of automated leukocyte recogitio. I this paper, we propose a ew measure of divergece called fuzzy Jeso-expoetial divergece (FJED betwee two fuzzy sets. The ew divergece measure has elegat properties, which are stated ad proved i the paper to ehace the employability of this measure. The stregth of this extesio has bee demostrated by a example of multi-criteria decisio makig. The paper is orgaized as follows: I Sectio some basic defiitios related to probability ad fuzzy set theory are briefly give. I Sectio, the fuzzy Jeso-expoetial divergece measure is proposed. I Sectio some properties of the proposed fuzzy divergece measure are stated ad proved. I Sectio a fuzzy multi-criteria decisio makig method is proposed with the help of fuzzy Jeso-expoetial divergece measure. umerical example is also give to illustrate the solutio process ad our coclusios are preseted i Sectio. Let. Prelimiaries p, p,..., p : p 0, p ; deote the set of all complete fiite discrete probability distributios. For ay probability distributio P p p p,,...,, Shao [0] itroduced the etropy to measure the ucertaity associated with probability distributio H P p log p = as follows. ( Pal ad Pal [] aalyzed the classical Shao iformatio etropy ad itroduced a ew probabilistic etropy called expoetial etropy give by e p H P p e. ( These authors poit out that, the expoetial etropy has a advatage over Shao s etropy. For the uiform probability distributio fixed upper boud which is ot the case for Shao s etropy. P,,..., lim H,,..., e as e, expoetial etropy has a. (

3 Kullback ad Leibler [] measure of divergece of a probability distributioq ( q, q,, q give by to probability distributio P p p p,,...,, is p D( P Q p log. ( q I, Li [] itroduced the Jese Shao divergece betwee the two probability distributios P p, p,, p adq q, q,, q as P Q H ( P H ( Q JSD( P; Q H, ( where H( P is the Shao etropy. Sice H( P is a cocave fuctio, accordig to Jese s iequality [0], JSD( P; Q is oegative ad vaishes whe P Q. The JSD ca also be represets i terms of KL divergece as P Q P Q JSD( P; Q D P DQ. ( Defiitio. Fuzzy set []: fuzzy set defied i a fiite uiverse of discourse X x, x,..., x is mathematically represeted as x, x x X, ( where x: X 0, is measure of belogigess or degree of membership of a elemet x i. Defiitio Set Operatios o FSs []: Let FSX uiverse X, assume, FSX give as x x x X x x x X The some set operatios defied as follows: (i iff x x x X ; (ii iff ad ; C (iii x, x x X ; (iv x x x, x X; x, x x x X; (v where deote the family of all FSs i the,,,., stad for max ad mi operators, respectively. I, De Luca ad Termii [0] defied fuzzy etropy for a fuzzy set correspodig to Shao etropy [0], give by H( log ( log(. (

4 Fuzzy expoetial etropy for fuzzy set correspodig to ( has also bee itroduced by Pal ad Pal [] as x x H x e x e e e. ( Let ad be two fuzzy sets defied i X. Correspodig to the Kullback ad Leibler [] divergece measure, hadari ad Pal [] defied the fuzzy divergece measure of a fuzzy set with respect to fuzzy set as x x D x log x log. (0 x x Fa ad Xie [] proposed the fuzzy divergece measure based o expoetial fuctio as x x x x D( x e x e. ( e I the ext sectio, motivated by the idea of Jese-Shao divergece, we propose a ew measure called fuzzy Jese-expoetial divergece (FJED to measure the differece betwee two fuzzy sets. Some properties of the proposed FJED are also studied here.. Fuzzy Jese-Expoetial Divergece (FJED Defiitio.. Sigle Elemet Uiverse set: Let { x, x X} ad { x, x X} be the two fuzzy sets defied i a sigle elemet uiverse of discourse X {} x. We defie the fuzzy Jese-expoetial divergece measure betwee two FSs ad, as * ( ( eh eh FJED e H exp exp exp exp e. ( exp exp Theorem.: For all, FS( X, the fuzzy Jeso-expoetial divergece measure FJED * ( i ( satisfies the followig properties: (i FJED * ( 0, (ii FJED * ( 0 if ad oly if, (iii 0FJED * (. Proof: (i It follows from Jese iequality [0].

5 (ii Let, the it is obvious FJED * ( 0. Coversely, let FJED * ( 0 eh( eh( eh 0 ( Sice e H is a cocave fuctio [], the from Jese iequality [0] expressio ( holds if ad oly if (iii. FJED * ( attais the highest value for the followig cases: which gives the required results, i.e., This completes the proof.,0, 0, or 0,,,0, FJED (. * 0 I the above defiitio, we assumed sigle elemet uiverse set. Now, here we exteded this otio to fiite uiverse of set. Defiitio.: Let { x, x X} ad { x, x X} be two fuzzy sets i the uiverse of discourse X x x x,,..., measure betwee ad, is give by FJED, the the fuzzy Jeso-expoetial divergece exp exp e exp exp exp exp. ( I the ext sectio, we prove some elegat properties of the proposed measure give by (. For provig the properties, we will assume that the fiite uiverse of discourse X is partitioed ito two disoit sets X ad X, such that X X x x x x x x X, ( x x X,. (

6 Properties of Fuzzy Jese Expoetial Divergece The measure give by (0 has the followig importat properties: Theorem.: For all, FS( X, FJED FJED, (i (ii FJED FJED, (iii FJED FJED, (iv FJED FJED FJED (v FJED FJED FJED,. Proof: (i Cosiderig measure give by (, we have FJED e xx x exp exp e exp exp + exp exp exp exp exp exp + exp exp exp exp X exp exp + exp exp

7 ad exp exp e exp exp x X x x x x FJED e + ( exp ( ( exp ( exp exp. ( exp exp + exp exp exp exp + exp exp e exp exp x X

8 xx exp exp exp exp + exp exp exp exp e exp exp x X From ( ad (, we get + exp exp FJED FJED. (ii The proof follows o similar lies as part (i. (iii Usig measure give by (0, show as FJED exp e exp exp exp exp exp. (

9 exp exp + exp exp e exp exp x X FJED. xx (iv Cosiderig (0, we have FJED exp log exp exp + exp exp exp exp e exp exp x x x + ( exp ( ( exp

10 ad exp exp + exp exp e exp exp x X FJED exp exp ( x exp exp X + exp exp log log e exp exp + exp x ( exp 0

11 e exp exp x X xx exp exp + exp exp log log. exp exp + exp exp (0 O addig ( ad (0, we get FJED FJED FJED. (v The proof is as similar as part (iv. This completes the proof. Theorem.: For all, FS( X, (i C C FJED FJED, (ii FJSD FJSD C FJSD C C FJSD C. where C ad C are the complemet of fuzzy set ad, respectively. Proof: Cosider the expressio (i C C FJED FJED (

12 C C exp C C exp e exp exp + C exp C C exp C C C exp C C exp e x x x x C ( exp c ( C ( exp C ( + exp exp exp exp + exp exp ( x e exp exp

13 This proves (i. exp exp (ii The proof of (ii is o similar lies as part (i. Theorem.: For all,, C FS( X, exp exp + exp exp (i FJED C FJED C FJED C, (ii FJED C FJED C FJED C, (iii FJED C FJED C FJED C FJED C. Proof: (i Cosider the expressio FJSD C FJSD C FJSD C ( e exp exp C C exp C C exp + C exp C C exp C

14 C C exp C C exp + C exp C C exp C e exp exp C C exp C C exp e exp log + C exp C C exp C C C exp C C exp + C exp C C exp C e exp exp x X

15 This proves (i. C C log C C log exp exp xx + C exp C C exp C (ii The proof is o similar lies as part (i. (iii Let us assume FJED C C C exp C C exp C C exp C C exp exp exp x X + C exp C C exp C e exp log + Cexp C( x C exp C e

16 ad FJED C C C exp C C exp exp exp. xx + C exp C C exp C ( C C exp C C exp e exp exp + C exp C C exp C C C exp C C exp + C exp C C exp C e exp exp x X

17 C C exp C C exp xx exp exp. + C exp C C exp C ( ddig ( ad (, we obtai the required result. This completes the proof.. pplicatio of FJED to Fuzzy Multi-Criteria Decisio Makig. Fuzzy Multi-criteria Decisio Makig Problem: Decisio makig is a process of selectig the best optio (or optios from a fiite umber of feasible optios. It is a very commo activity that usually occurs i our daily life ad plays a importat role i busiess, fiace, maagemet, ecoomics, social ad political sciece, egieerig ad computer sciece, biology ad medicie etc. Multi criteria decisio makig (MCDM refers to select optimal optio from a fiite umber of feasible optios uder several criteria. To fid the most preferred optio, the decisio maker provides his/her preferece iformatio for the optios. I may real life decisio makig problems, the available iformatio is vague or imprecise. To adequately solve decisio problems with vague or imprecise iformatio, fuzzy set theory has become powerful tool. I the literature, a umber of multiple criteria decisio makig theories ad methods uder fuzzy eviromet have bee proposed for effectively solvig the multi criteria decisio makig problems ad various applicatios have bee cited i the literature [,, ]. Let M M, M,..., Mmbe a set of optios ad let C C, C,..., C be a set of criteria. For decisio makig, the decisio maker characterized each optio i terms of FSs by assigig appropriate values to fuctios. Suppose that the characteristics of the optio M i i terms of the set of criteria C are preseted by FS show as follows: FS M C, C C, i,,..., m ad,,...,, ( i i where idicates the degree with which the optio M satisfies the criterioc ad i 0,, i,,..., m ad,,...,. i We itroduce a four-step algorithm to solve above fuzzy multi-criteria decisio makig problem usig the measure defied i (. i

18 lgorithm: Step : Fidig the positive ideal ad egative ideal solutios deoted by M ad M, respectively give by: M,,...,, ( where M,,...,, ( max i mi Step : Calculatig the values of FJSD( M M i ad FJSD( M M i by the followig formulas respectively: i i exp i i exp FJED( M Mi exp exp i exp i i exp i ( ad i i exp i i exp FJED( M Mi exp exp. i exp i i exp i ( Step : Calculate the relative fuzzy Jese-expoetial divergece FJED M i of optios M i with respect to M ad M, where FJED M i i i i. ( FJED( M Mi. (0 FJED( M M FJED( M M i i

19 Step : Rak the optios M, i,,..., m i,,..., m, i ascedig order. The leadig i best optio. i, accordig to the values of i M, with smallest value of FJED M FJED M, i, is the I order to illustrate the applicability of the proposed algorithm, we cosider below a example of ivestmet compay decisio-makig problem. Example []: Cosider a ivestmet compay problem. ssume that a ivestmet compay desires to ivest a defiite amout of moey i the best optio amog five optios: (i a car compay M, (ii a food compay M, (iii a computer compay M, (iv a arms compay M ad (v a TV compay M. The ivestmet compay wats to take a decisio accordig to the followig four criteria: i. C, the risk aalysis, ii. C, the growth aalysis, iii. C, the social-political impact aalysis, iv. C, the evirometal impact aalysis. fter evaluatio of optios, the decisio maker forms the followig fuzzy sets, give by M C,0., C,0., C,0., C,0., M C,0., C,0., C,0., C,0., M C,0., C,0., C,0., C,0., M C,0., C,0., C,0., C,0., M C,0., C,0., C,0., C,0.. We eed to fid a rakig of the feasible optios, with a view to fid the best optio. Step : Obtaiig the M ad M give by M C,0., C,0., C,0., C,0., M C,0., C,0., C,0., C,0.. Step : Usig ( ad ( to calculate FJSD( M M i have the followig tables: ad FJSD( M M i, respectively, we

20 Table : ********************************************** Table : ********************************************** Step. Calculatig the relative fuzzy Jese-expoetial divergece measure FJED M i of M i s with respect to M ad M by usig (0. We obtai the followig table: Table ********************************************** ased o Table, the rakig order of optios is give by, M M M M M. Therefore, M is the most preferable optio, which is i agreemet as obtaied i [].. Coclusio I this paper, we proposed a ew iformatio measure called fuzzy Jese-expoetial divergece i the settig of fuzzy set theory. umber of properties of the proposed measure have bee stated ad proved. The, based o fuzzy Jese-expoetial divergece, we have developed a algorithm to deal the problems of multi-criteria decisio makig with fuzzy iformatio. Further study of this measure will be reported separately. Refereces. J.C. gulo, J. toli, S.L. Rosa ad R.O. Esquivel, Jese-Shao Divergece i cougated spaces: etropy excess of atomic systems ad sets with respect to their 0

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23 Table Click here to dowload Table: Tables.pdf Table : Values of FJSD( M M i FJSD( M M 0.00 FJSD( M M 0.00 FJSD( M M 0.0 FJSD( M M 0.0 FJSD( M M 0.0 Table : Values of FJSD( M M i FJSD( M M 0.00 FJSD( M M 0.0 FJSD( M M 0.0 FJSD( M M 0.0 FJSD( M M 0.0 Table : Values of Relative FJED M i FJED M 0. FJED M 0.

24 FJED M 0. FJED M 0. FJED M 0.0