Ranking Fuzzy Numbers based on Sokal and Sneath Index with Hurwicz Criterion

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1 Malaysan Journal of Mathematcal Scences 8(: 7-7 (04 MLYSIN JOURNL OF MTHEMTICL SCIENCES Journal homepage: Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron * Nazrah Raml Daud Mohamad Department of Mathematcs Statstcs Faculty of Computer Mathematcal Scences Unverst Teknolog MR Pahang 6400 Bar Jengka Pahang Malaysa Department of Mathematcs Faculty of Computer Mathematcal Scences Unverst Teknolog MR Malaysa Shah lam Selangor Malaysa E-mal: nazrahr@pahang.utm.edu.my *Correspondng author BSTRCT Rankng of fuzzy numbers s an mportant procedure for many applcatons n fuzzy theory n partcular decson-makng. In ths paper we propose a novel method for rankng fuzzy numbers usng Sokal Sneath set theoretc ndex. The fuzzy maxmum fuzzy mnmum fuzzy evdences fuzzy total evdences are obtaned n determnng the rankng. The Hurwcz crteron whch consders all types of decson makers perspectve s employed n aggregatng the fuzzy total evdences. The ratonalty propertes of the proposed method are presented. Moreover fve numercal examples are presented to llustrate the advantages of the proposed method. The rankng results show that the proposed method can overcome certan shortcomngs that exst n the prevous rankng methods. Keywords: Decson-makng Hurwcz crteron rankng fuzzy numbers Sokal Sneath ndex

2 Nazrah Raml & Daud Mohamad. INTRODUCTION Rankng of fuzzy numbers s an mportant procedure for many applcatons n fuzzy theory such as n approxmate reasonng decsonmakng optmzaton other usages. In fuzzy decson analyss fuzzy numbers are employed to descrbe the performance of alternatves the selecton of alternatves wll eventually lead to the rankng of correspondng fuzzy numbers. However rankng of fuzzy numbers s not an easy task snce fuzzy numbers are represented by possblty dstrbutons they can overlap wth each other. Snce Jan (976 frst presented the concept of rankng fuzzy numbers varous methods of rankng fuzzy numbers have been developed but no method can rank fuzzy numbers satsfactorly n all cases stuatons. Some methods produce non-dscrmnate non-ntutve results lmted to normal trangular types of fuzzy numbers only consder neutral decson makers perspectve. There are also methods that produce dfferent rankng results for the same stuatons some have the dffculty of nterpretaton. n early revew on rankng fuzzy numbers has been done by Bortolan Degan (985 followed by Chen Hwang (99 Wang Kerre (996. In 998 Cheng proposed a dstance ndex based on the centrod concept CV ndex. The dstance ndex has mproved Yagers ndex (980 whle the CV ndex has mproved Lee L s (988 approach. However n some stuatons the rankng result by the dstance ndex contradcts wth the result by the CV ndex. Thus to overcome the problems Chu Tsao (00 proposed an area between the centrod pont orgnal pont as the rankng ndex. Chen Chen (007 then found that Cheng s (998 dstance ndex Chu Tsao s (00 Yagers (980 methods cannot rank correctly two fuzzy numbers havng the same mode symmetrc spread. Thus Chen Chen (007 proposed a new rankng approach usng the score ndex concept. However Chen Chen s (007 method s only lmted to trapezodal type of fuzzy numbers does not cater for general fuzzy numbers. 8 Malaysan Journal of Mathematcal Scences

3 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron In other studes by Yao Wu (000 bbasby sady (006 they proposed sgned dstance method for rankng fuzzy numbers. Furthermore sady Zendehnam (007 proposed dstance mnmzaton method for rankng fuzzy numbers. However Yao Wu s (000 bbasby sady s (006 sady Zendehnam s (007 methods are only lmted to normal fuzzy numbers are found to produce non-dscrmnatve rankng result for fuzzy numbers havng the same mode symmetrc spread. In a dfferent study by Setnes Cross (997 they proposed Jaccard ndex wth mean aggregaton concept for rankng fuzzy numbers. However ther methods are only applcable for normal fuzzy numbers only consder neutral decson makers perspectve also cannot dstngush the rankng of fuzzy numbers havng the same mode symmetrc spread. In ths paper a new method for rankng fuzzy numbers based on Sokal Sneath ndex Hurwcz crteron s proposed. The Sokal Sneath s a set theoretc type of smlarty measure ndex whch s commonly used n pattern recognton classfcaton for populaton dversty Hurwcz s a crteron for decson-makng that compromses between the optmstc pessmstc crtera. Thus the proposed rankng method consders all types of decson makers perspectve such as optmstc neutral pessmstc whch s crucal n solvng decsonmakng problems. The proposed method can overcome certan shortcomngs that exst n the prevous rankng methods. The paper s organzed as follows. Secton brefly revews the prelmnary concepts defntons. In Secton we propose the Sokal Sneath ndex wth Hurwcz crteron for rankng fuzzy numbers. The ratonalty propertes of the proposed rankng method are presented n Secton 4. Secton 5 presents fve numercal examples to llustrate the advantages of the proposed method. Lastly the paper s concluded n Secton 6.. PRELIMINRIES In ths secton some basc concepts defntons on fuzzy numbers are revewed from the lterature. Defnton fuzzy number s a fuzzy set n the unverse of dscourse X wth the membershp functon defned as (Dubos Prade (980; Malaysan Journal of Mathematcal Scences 9

4 µ Nazrah Raml & Daud Mohamad ( x ( L µ x a x b w b x c = R µ ( x c x d 0 otherwse L a b w c d w 0 µ denote the left the rght membershp functons of the fuzzy number. L R where µ :[ ] [ 0 ] µ :[ ] [ 0 ] w ( ] The membershp functon propertes: R µ µ of a fuzzy number has the followng ( µ s a contnuous mappng from the unverse of dscourse X to 0 w. [ ] ( µ ( x = 0 for x < a x > d. ( µ ( x s monotonc ncreasng n [ b] (4 µ ( x = w for [ b c]. (5 µ ( x s monotonc decreasng n [ d] If the membershp functon ( x a. c. µ s a pecewse lnear then s called as a trapezodal fuzzy number wth membershp functon defned as µ ɶ ( x x a w a x b b a w b x c = d x w c x d d c 0 otherwse denoted as = ( a b c d; w. If b = c trangular fuzzy number denoted as ( a b d; w then the trapezodal becomes a =. 0 Malaysan Journal of Mathematcal Scences

5 Defnton Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron Let + [ b b ] + be two fuzzy numbers wth [ a a ] be ther α -cuts wth [ 0 ] = α α α = α α α α. The fuzzy maxmum of by the α -cuts method s defned as (Kaufmann Gupta (985; + + ( max ( α α max ( α α MX = a b a b. The fuzzy mnmum of s defned as Defnton α + + ( mn ( α α mn ( α α MIN = a b a b. Let = ( a b c d ; h ( a b c d ; h α = be two trapezodal fuzzy numbers. The fuzzy maxmum of by the second functon prncple s defned as (Chen Hseh (998; where ( = ( ; MX a b c d h a h = mn { h h} { max ( max ( max ( max ( } { max ( max ( max ( max ( } T = a a a d d a d d T = b b b c c b c c = mn T b = mn T c = mn T d = max T mn T mn T max T max T. The fuzzy mnmum of s defned as where ( = ( a b c d; h MIN h = mn { h h} T { mn ( a a mn ( a d mn ( d a mn ( d d } = Malaysan Journal of Mathematcal Scences

6 b Nazrah Raml & Daud Mohamad { ( ( ( ( } T = mn b b mn b c mn c b mn c c a = mn T = mn T c = max T d = max T mn T mn T Defnton 4 max T max T. The scalar cardnalty of a fuzzy number n the unverse of dscourse X s defned as (Zwck et al. (987 Defnton 5 = µ X ( x dx. For fuzzy numbers k a fuzzy preference P s called w- transtve f only f (Wang Ruan (995; ( > ( P( k P( k P P ( > ( P P. k k > mples. SOKL ND SNETH RNKING INDEX WITH HURWICZ CRITERION Based on the psychologcal rato model of smlarty from Tversky (977 whch s defned as ( X Y f ( X Y ( + α. ( + β. ( Sα β = f X Y f X Y f Y X varous ndex of smlarty measures have been proposed. For α = β = the rato model of smlarty becomes the Sokal Sneath ndex whch s defned as S ( X Y f ( X Y ( ( =. Typcally the. f X Y f X Y functon f s taken to be the cardnalty functon. The obects X Y descrbed by the features are replaced wth fuzzy numbers B whch are descrbed by the membershp functons. The fuzzy Sokal Sneath B ndex s defned as S ( B = where denotes the. B B Malaysan Journal of Mathematcal Scences

7 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron scalar cardnalty of are the t-norm s-norm respectvely. We propose fuzzy Sokal Sneath rankng ndex wth Hurwcz crteron as follows: Step : For each par of the fuzzy numbers fuzzy mnmum of fnd the fuzzy maxmum. The fuzzy maxmum fuzzy mnmum can be obtaned by the α -cuts method for normal fuzzy numbers the second functon prncple for non-normal fuzzy numbers. Step : Calculate the evdences of E ( E ( E E ( ( ( = ( ( E ( = S MIN ( ( = ( ( whch are defned based on fuzzy Sokal Sneath ndex as E S MX ( E S MX S ( MIN ( where S ( fuzzy Sokal Sneath ndex fuzzy number. To smplfy E ( E ( to denote E ( E ( Step : ( = E = s the denotes the scalar cardnalty of C c are used to represent respectvely. Lkewse respectvely. Calculate the total evdences E ( E ( total total C c are used whch are defned based on the Hurwcz crteron concept as E = βc + β c E = βc + β c. total ( ( ( ( [ 00.5 β = 0. 5 β ( 0.5 ] total β represent pessmstc neutral optmstc crtera respectvely. To smplfy E ( E ( are used to represent E ( E ( respectvely. total total Malaysan Journal of Mathematcal Scences

8 Step 4: Nazrah Raml & Daud Mohamad For each par of the fuzzy numbers compare the total evdences n Step whch wll result the rankng of the two fuzzy numbers as follows: ( f only f E ( > E (. ( f only f E ( < E (. ( f only f E ( = E (. Step 5: Check the transtvty of ( Wang Ruan (995. Step 6: E by usng the w-transtvty from For n fuzzy numbers wth transtve par wse rankng do the total orderng. Whle for non-transtve par wse rankng use the sze of domnated class method from Cross Setnes ( RTIONLITY PROPERTIES We consder the ratonalty propertes for the orderng approaches by Wang Kerre (00. The propertes are presented n Table wth M be the orderng ndex S s the set of fuzzy quanttes for whch ndex M can be appled X s a fnte subset of S X. TBLE : Ratonalty Propertes for Orderng Indces (Wang Kerre (00 xoms Propertes W For an arbtrary fnte subset X of S X W For an arbtrary fnte subset X of S ( by M on X we should have by M on X. by M on X. X W For an arbtrary fnte subset X of S ( W 4 X by M on X we should have by M on X. For an arbtrary fnte subset X of S ( X nf supp( > sup supp( we should have by M on X. 4 Malaysan Journal of Mathematcal Scences

9 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron TBLE (contnued: Ratonalty Propertes for Orderng Indces (Wang Kerre (00 xoms 4 Propertes W For an arbtrary fnte subset X of S ( nf supp( sup ( W 5 X > supp we should have by M on X. Let S S be two arbtrary fnte sets of fuzzy quanttes n whch M can be appled are n S S. We obtan the rankng order by M on S f only f by M on S. W 6 Let W be elements of S. If by M on { } then + + by M on { + + }. Let + + be elements of S. If by M on { } then + + by M on { + + } for 0. be elements of S 0. If by M on { } then by M on { }. W 7 Let Note: W 4 s stronger than W 4 whch means that the rankng ndex meets W 4 f t meets W 4 (Wang Kerre (00. Theorem The functon ( Proof. Let E has the propertes of W W W 4 be two fuzzy numbers wth ( evdences E ( of Sokal Sneath ndex. Then E ( total MX ( ( MX ( ( = β + β MX MIN = β + β ( β = β ( + ( ( = Obvously E ( E ( Hence thus E satsfes axom W. W4 W 5. E be the total by MIN( ( MIN( E. Malaysan Journal of Mathematcal Scences 5

10 Now Nazrah Raml & Daud Mohamad mples E ( E ( mples E ( E (. By ant-symmetrc rules clearly E ( E ( by E axom W s satsfed. = whch mples Two cases are consdered for showng that axoms W 4 W 4 are satsfed. Frstly assume X s the unverse of dscourse wth hgt( hgt( where hgt( denotes the heght of fuzzy number. If supp( sup supp( nf > clearly we obtan the followng: MX ( snce ( x ( x (Dubos Prade (980. µ for x X MX µ ( MX ( = φ MIN ( = φ. MIN = Thus E ( ( β MX ( ( ( MIN ( ( ( = β MX MX + MIN MIN ( ( MX = β MX + ( β MX ( ( MX ( ( ( ( β + β = β + β =. MX 6 Malaysan Journal of Mathematcal Scences

11 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron Obvously E ( > 0 thus E ( ( E ( β 0 <. MX ( ( ( MIN ( ( ( = β MX MX + Thus E ( E ( W 4 s satsfed. MIN MIN ( ( β ( = β = 0. > whch mples by Hence axom W 4 s also satsfed. Next assume ( hgt( hgt <. If supp( sup supp( nf > clearly we obtan the followng: MIN MX ( = ( snce MIN ( x µ ( x MX ( = φ MIN ( = φ. µ for x X. E axom Thus MIN ( E ( β ( ( β MIN ( MIN ( result follows as n the case of hgt( hgt(. + = the ssume that are two fuzzy numbers n S S where S S are two arbtrary fnte sets of fuzzy numbers. The rankng of ( E where s solely nfluenced by ( E Malaysan Journal of Mathematcal Scences 7

12 E ( Nazrah Raml & Daud Mohamad MX ( ( MX ( ( = β + β MX MIN MIN( ( MIN ( E ( MX ( ( MX ( ( = β + β MX MIN The operatons only nvolved fuzzy numbers MIN ( ( MIN ( (not nvolve any other fuzzy numbers n S or S ths ensures the same rankng order f t s based on S S thus axom W 5 s satsfed. 5. NUMERICL EXMPLES In ths secton fve sets of numercal examples are presented to llustrate the valdty advantages of the proposed method. Example Consder the data used n Sun Wu (006.e. two fuzzy numbers = = 9 as shown n Fgure. 4 Fgure : Fuzzy Numbers n Example Intutvely the rankng order s. However by the fuzzy smulaton analyss from Sun Wu (006 the rankng order s whch s 8 Malaysan Journal of Mathematcal Scences

13 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron unreasonable. By the proposed method ( = ( E E = therefore the rankng order s regardless of the decson makers perspectve. Ths result s consstent wth human ntuton. Example Consder the data used n Wang et al. (009.e. two trangular fuzzy as shown n Fgure. numbers ( ( = = Fgure : Fuzzy Numbers n Example Snce fuzzy numbers have the same mode symmetrc spread a number of the exstng rankng methods cannot dscrmnate them such as Chen (985 Setnes Cross (997 Yao Wu (000 Chu Tsao (00 bbasby sady (006 wth p = sady Zendehnam (007 Wang Lee (008. The nconsstent results are also produced usng dstance ndex CV ndex of Cheng s (998 method. Moreover Wang et al. s (005 method produces whle Wang et al. s (009 produces. By the proposed method we obtan E ( = 0.67β 0. E ( = 0.67β Thus the rankng order s [ β β = 0.5 ( ] β 0.5 where for pessmstc decson makers for neutral decson makers for optmstc decson makers. The rankng result s affected by decson makers perspectve ths shows that the proposed Malaysan Journal of Mathematcal Scences 9

14 Nazrah Raml & Daud Mohamad method has strong dscrmnaton power. The result s also consstent wth Wang Luo s ndex (009. Example Consder the data used n Wang Lee (008.e. two trapezodal fuzzy numbers = ( 6790;0.6 = ( 5790; as shown n Fgure. Some of the exstng rankng methods such Setnes Cross (997 Yao Wu (000 Wang et al. (005 bbasby sady (006 sady Zendehnam (007 Wang et al. (009 Wang Luo (009 can only rank normal fuzzy numbers thus fal to rank the fuzzy numbers. Moreover Chu Tsao (00 rank them as whle Cheng s dstance ndex Wang Lee s (008 ndex rank them as. By the proposed method we have ( ( = β [ 00.9 for = 0. 9 for β ( 0.9] E = β E thus obtan the rankng result as for β β. Thus the rankng result s affected by decson makers perspectve ths shows that the equal rankng result does not necessarly occur for neutral decson makers. Fgure : Fuzzy Numbers n Example Fgure 4: Fuzzy Numbers n Example 4 Example 4 Consder the data used n bbasby Haar (009.e. a trapezodal fuzzy number two trangular fuzzy numbers = ( as shown n Fgure 4. ( ( = = TBLE : Rankng Results of Example 4 0 Malaysan Journal of Mathematcal Scences

15 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron Index Fuzzy numbers Index value Rankng results E ( E ( β β Proposed β [ 0] ndex E ( E ( E ( E ( β β bbasby Haar (009 sady Zendehnam (007 Barkhordary et al. (007 bbasby sady (006 p = Chu Tsao (00 Yao Wu (000 Cheng (998 Chen ( The rankng values of the proposed method are shown n Table. Thus the rankng order of the fuzzy numbers s regardless of the decson makers perspectve. However Yao Wu s (000 bbasby sady s (006 sady Zendehnam s (007 ndces produce Malaysan Journal of Mathematcal Scences

16 Nazrah Raml & Daud Mohamad the rankng as whch cannot dscrmnate the rankng of. Obvously the results obtaned by Yao Wu s (000 bbasby sady s (006 sady Zendehnam s (007 are unreasonable. Moreover Chen s (985 Barkhordary et al. s (007 methods produce the rankng as whle bbasby Haar (009 rank the fuzzy numbers as. Other rankng methods such as Cheng s (998 dstance Chu Tsao s (00 produce smlar rankng results wth the proposed method. Example 5 Consder the data used n Wang et al. (009.e. a trangular fuzzy number 5 4 as shown n Fgure 5. ( a fuzzy number ( = = The membershp functon of s defned as µ ( x ( x [ ] = ( x [ 4 ]. 4 0 else Fgure 5: Fuzzy Numbers n Example 5 Some of the exstng rankng methods such as Chen Chen (00 Chen Chen (007 Chen Chen (009 can only rank trapezodal fuzzy numbers thus fal to rank the fuzzy numbers. By usng the proposed method we have E ( = β Malaysan Journal of Mathematcal Scences

17 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron ( = β E +. Therefore the rankng order s regardless of the decson makers perspectve as shown n Table. Based on Table Deng et al. s (006 ndex produces the rankng order as whch s unreasonable. The rankng result of the proposed method s consstent wth human ntuton other methods n Table. Table : Rankng Results of Example 5 Index Fuzzy numbers Index value Rankng results E β β 0 Proposed ndex ( ( Chen Chen (00 Chen Chen (007 Chen Chen (009 Nead Mashnch (0 Wang et al. (009 sady Zendehnam (007 Deng et al. (006 E β 0 [ ].6 * * * * * 0.7 * * Chu Tsao.45 (00.8 Cheng ( Setnes Cross ( * : the rankng method cannot calculate the rankng value 6. CONCLUSION Ths paper presents a new method for rankng fuzzy numbers usng Sokal Sneath ndex Hurwcz crteron. The new method takes nto consderaton all types of decson makers perspectve whch s crucal n Malaysan Journal of Mathematcal Scences

18 Nazrah Raml & Daud Mohamad solvng decson-makng problems. The proposed method can overcome certan shortcomngs that exst n the prevous rankng methods such as can rank both non-normal general types of fuzzy numbers can dscrmnate the rankng of fuzzy numbers havng the same mode symmetrc spreads whch fals to be ranked by the prevous ones. Besdes the results of the proposed method are also consstent wth human ntuton most of other prevous methods. REFERENCES bbasby S. sady B. (006. Rankng of fuzzy numbers by sgn dstance. Informaton Scences. 76: bbasby S. Haar T. (009. new approach for rankng of trapezodal fuzzy numbers. Computers Mathematcs wth pplcatons. 57: sady B. Zendehnam. (007. Rankng fuzzy numbers by dstance mnmzaton. ppled Mathematcal Modellng. : Barkhordary M. llahvranloo T. Haar T. (007. Novel rankng method of fuzzy numbers. Proceedngs of the Frst Jon Congress on Fuzzy Intellgent Systems. Ferdows Unversty of Mashhad Iran. Bortolan G. Degan R. (985. revew of some methods for rankng fuzzy subsets. Fuzzy Sets Systems. 5: -9. Chen S. H. (985. Rankng fuzzy numbers wth maxmzng set mnmzng set. Fuzzy Sets Systems. 7: -9. Chen S. J. Chen S. M. (00. new method for hlng multcrtera fuzzy decson makng problems usng FN-IOW operators. Cybernatcs Systems. 4: Chen S. J. Chen S. M. (007. Fuzzy rsk analyss based on the rankng of generalzed trapezodal fuzzy numbers. ppled Intellgence. 6(: -. 4 Malaysan Journal of Mathematcal Scences

19 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron Chen S. M. Chen J. H. (009. Fuzzy rsk analyss based on rankng generalzed fuzzy numbers wth dfferent heghts dfferent spreads. Expert Systems wth pplcatons. 6: Chen S. J. L. Hwang C. (99. Fuzzy Multple ttrbute Decson Makng Methods pplcatons. New York: Sprnger. Chen S. H. Hseh C. H. (998. Graded mean representaton of generalzed fuzzy numbers. Proceedngs of the Sxth Conference on Fuzzy Theory Its pplcatons Tawan Republc of Chna -5. Cheng C. H new approach for rankng fuzzy numbers by dstance method. Fuzzy Sets System. 95: Chu T. C. Tsao C. T. (00. Rankng fuzzy numbers wth an area between the centrod pont orgnal pont. Computers Mathematcs wth pplcatons. 4: -7. Cross V. Setnes M. (998. generalzed model for rankng fuzzy sets. IEEE World Congress on Computatonal Intellgence. : Deng Y. Zhenfu Z. Q L. (006. Rankng fuzzy numbers wth an area method usng radus of gyraton. Computers Mathematcs wth pplcatons. 5: 7-6. Dubos D. Prade H. (980. Fuzzy Sets Systems: Theory pplcatons. New York: cademc Press. Jan R. (976. Decson makng n the presence of fuzzy varables. IEEE Trans. Systems Man Cybernetcs. 6: Kaufmann. Gupta M. M. (985. Introducton to Fuzzy rthmetc: Theory pplcatons. New York: Van Nostr Renhold Company. Lee E. S. L R. L. (988. Comparson of fuzzy numbers based the probablty measure of fuzzy events. Computers Mathematcs wth pplcatons. 5: Malaysan Journal of Mathematcal Scences 5

20 Nazrah Raml & Daud Mohamad Nead.M. Mashnch M. (0. Rankng fuzzy numbers based on the areas on the left the rght sdes of fuzzy numbers. Computers Mathematcs wth pplcatons. 6: Setnes M. Cross V. (997. Compatblty based rankng of fuzzy numbers. Proceedng of Fuzzy Informaton Processng Socety (NFIPS 997 Syracuse New York: Sun H. Wu J. (006. new approach for rankng fuzzy numbers based on fuzzy smulaton analyss method. ppled Mathematcs Computaton. 74: Tversky. (977. Features of smlarty. Psychologcal Revew. 84: 7-5. Wang X. Kerre E. (996. On the classfcaton dependences of the orderng methods. In D. Ruan (Ed. Fuzzy Logc Foundatons Industral pplcatons (pp Dordrecht: Kluwer cademc Publsher. Wang X. Kerre E. E. (00. Reasonable propertes for the orderng of fuzzy quanttes (I. Fuzzy Sets Systems. 8: Wang Y. J. Lee H. S. (008. The revsed method of rankng fuzzy numbers wth an area between the centrod orgnal ponts. Computers Mathematcs wth pplcatons. 55: Wang Z. X. Lu Y. J. Fan Z. P. Feng B. (009. Rankng L-R fuzzy number based on devaton degree. Informaton Scences. 79: Wang Y. M. Luo Y. (009. rea rankng of fuzzy numbers based on postve negatve deal ponts. Computers Mathematcs wth pplcatons. 58: Wang X. Z. Ruan D. (995. On the transtvty of fuzzy preference relatons n rankng fuzzy numbers. In D. Ruan (Ed. Fuzzy Set Theory dvanced Mathematcal pplcatons (pp Boston: Kluwer. 6 Malaysan Journal of Mathematcal Scences

21 Rankng Fuzzy Numbers based on Sokal Sneath Index wth Hurwcz Crteron Wang M. L. Wang H. F. Lung L. C. (005. Rankng fuzzy number based on lexcographc screenng procedure. Internatonal Journal of Informaton Technology Decson Makng. 4(4: Yager R. R. (980. On a general class of fuzzy connectves. Fuzzy Sets Systems. 4(6: 5-4. Yao J. S. Wu K. (000. Rankng fuzzy numbers based on decomposton prncple sgned dstance. Fuzzy Sets Systems. 6: Zwck R. Carlsten E. Budescu D. V. (987. Measures of smlarty between fuzzy concepts: comparatve analyss. Internatonal Journal of pproxmate Reasonng. : -4. Malaysan Journal of Mathematcal Scences 7