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1 Joural of Ucerta Systems Vol.8, No.2, pp , 2014 Ole at: Hestato Degree of Itutostc Fuzzy Sets a New Cose Smlarty Measure Lazm bdullah *, Wa Khadjah Wa Ismal Departmet of Mathematcs, Faculty off Scece ad Techology, Uversty Malaysa Tereggau Kuala Tereggau, Tereggau, Malaysa Receved 19 October 2012 Revsed 4 ugust 2013 bstract Cose smlarty measure betwee fuzzy sets was made a breakthrough based o the deaa of hattacharya o a measure of dvergece betwee twoo multomal populatos. The measure wass exteded by Ye 2011 specfcally for measurg smlarty betwee tutostc fuzzy sets (IFSs). The IFS s characterzed by the otos of membershp degree, o membershp degree ad the degree of hestato as vector represetatos vector space. Ye proposed a cose smlarty measuree ad weghted d cose smlarty measure for IFSs. However, the hestato degree of IFS s excluded Ye smlarty measure. Ths paper proposes a ew cose smlarty measure ad weghted cose smlarty measure for IFSs by cosderg membershp degree, o membershp degree ad hestato degree cocurretly. The hestato degree s added to thee ew cose smlarty measures wthout compromsg thee otos of membershp degree ad o membershp degree. Four umercal examples patter recogto are provded to llustrate the feasblty of the proposed methods World cademc Press, UK. ll rghts reserved. Keywords: tutostc fuzzy sets, cose c smlartyy measure, patter recogto, degree of hestato 1 Itroducto The theory of smlarty measure. smlarty measure s a mportat tool for determg the degree of smlarty betwee two objects. Sce taassov [1] exteded fuzzy sets to IFS, mayy dfferet smlarty measures betwee IFSs have bee proposed the lterature. L ad Cheg [6] dscussed some smlarty measures o IFSs ad a proposed a smlarty measure betwee IFSs whch s the frst oe to be appled to patter recogto problems. bout a year later, Lag ad Sh [7] proposed several s smlarty measures to dfferetate dfferett IFSs ad dscussed the relatoshps betwee these measures. Mtchell [8] terpreted IFSs as esembles of ordered fuzzy sets from a statstcal vewpot to modfy L ad Cheg s measures [6]. ased o the exteso of the Hammg dstace to fuzzy sets, Szmdt ad Kacprzyk [11] troduced the Hammg dstace betwee IFSs ad proposed a smlarty measure betwee IFSs based o the dstace. I other research, Hug ad Yag [4] proposed aother method to calculate the dstace betwee IFSs based o the Hausdorff dstace ad the used ths dstace to geerate several smlarty measures betwee IFSs that are suted to be used lgustc varables. Hug add Yag [5] also proposed a method to calculate the degree of smlarty betwee IFSs, whch the proposed smlartyy measures are duced by Lp metrc. The latest developmet these t measures s cose smlarty measure. Formulato of the problem. Ye [15] troduced cose smlarty measure m ad weghted cose smlarty measure for IFSs after cosderg the t advatages of membershp degree ad a o membershp degree as vector represetatos. Ideed, t s a exteso of cose smlarty measure of fuzzy sets. Despte the success of hs smlarty measures, the role of hestato degree IFSs s eglected. The mportace of hestato degree. The ew cose smlarty measure proposed ths paper does cosder the hestato degree as a vector represetato the formula. The cluso of o hestato degree mght produce more comprehesve judgmet due to the represetator s of complete kowledgee defg the membershp fucto. The mportace of hestato degree IFS ad the recet developmet cose smlartyy measures motvate a ew dea the possblty of tegratg of these two otos. Szmdt ad Kacpryzk [10] alsoo stressed the ecessty of * Correspodg author. Emal: lazm_m@umt.edu.my (L. bdullah).

2 110 L. bdullah ad W.K.W. Ismal: Hestato Degree of Itutostc Fuzzy Sets takg to cosderato a thrd parameter (hestato degree), whch arses due to the lack of kowledge or persoal error [3]. What we do ths paper. I ths paper, we propose a ew cose smlarty measure ad a ew weghted cose smlarty measure for IFSs by cosderg hestato degree. Ths ovel cose smlarty measure for IFSs s proposed based o the precedece research o cose smlarty measure (agular coeffcet) betwee fuzzy sets ad IFSs. The rest of the paper s orgazed as follows. I the ext secto, basc otos ad deftos o fuzzy sets, IFSs ad cose measure fuzzy sets ad cose smlarty measures IFSs are elucdated. ew smlarty measure s proposed Secto 3. The applcatos of the proposed methods patter recogtos are furshed Secto 4. Fally, cocluso s appeared the last secto. 2 Prelmares Theory of fuzzy sets ad tutostc fuzzy sets (IFS). Fuzzy set theory, a well-kow theory was proposed by Zadeh [16] ad defes set membershp as a possblty dstrbuto. The geeral rule for ths ca expressed as: f :[0,1] [0,1] (1) where s some umber of possbltes. Ths bascally states that we ca take possble evets ad use f to geerate as sgle possble outcome. I fuzzy set theory, the degree of belogg of elemet to the set s represeted by a membershp value the real terval [0, 1] ad there exsts degree of o-membershp whch s complemetary ature. Oe of the extesos of fuzzy sets s IFSs. IFS have bee foud to be hghly useful to deal wth vagueess ad IFSs X s defed by taassov [1] as: { x, ( x), ( x) x X}, (2) where ( x ): X [0,1] ad ( x ): X [0,1] wth the codto 0 ( x) ( x) 1. The umbers ( x ) ad ( x ) represet respectvely the membershp degree ad o-membershp degree of the elemet x to the set. For each IFSs X, f ( x) 1 ( x) ( x), x X, the ( x ) s called the tutostc dex of the elemet x the set. It s a hestacy degree of x to. It s obvous that 0 ( x ) 1, x X. For two IFS { x, ( x), ( x) xx} ad { x, ( x), ( x) x X}, two relatos are defed as follows [1]: 1. f ad oly f ( x) ( x) ad ( x) ( x) for ay x X 2. f ad oly f ( x) ( x) ad ( x) ( x) for ay x X. The exstg cose smlarty measure for fuzzy sets ad IFSs. Cose smlarty measures are defed as the er product of two vectors dvded by the product of ther legths [2, 9]. ssume that { ( x1), ( x2),..., ( x)} ad { ( x1), ( x2),..., ( x)} are two fuzzy sets the uverse of dscourse X { x1, x2,..., x}, x X. cose smlarty measure (agular coeffcet) based o attacharya s dstace [2, 9] betwee the fuzzy sets ad ca be defed as C ( x ) ( x ) F (, ). 2 2 ) ) The cose smlarty measure takes value the terval [0, 1]. It s udefed f ( x ) 0 ad/or ( x ) 0 ( 1,2,..., ). ased o the exteso of the cose measure betwee fuzzy sets, a cose smlarty measure betwee two IFSs ad s defed. ssume that there are two IFSs ad a uverse dscourse X { x1, x2,..., x } [15], (3)

3 Joural of Ucerta Systems, Vol.8, No.2, pp , C 1 ( x ) ( x ) ( x ) ( x ) IFS (, ) ) ) ) ) (4) The cose smlarty measures of two IFSs ad satsfes the followg propertes: 1. 0 (, ) 1 2. (, ) (, ) 3. (, ) 1 f =,.e., ) ) ad ) ) for 1,2,...,. If we cosder the weghts of x, a weghted cose smlarty measure betwee IFSs ad s defed as follows [15]: 1 ) ) ) ) (, ) w (5) ( x ) ( x ) ( x ) ( x ) where w [0,1], 1, 2,..., ad w 1. If w 1/, 1,2,...,, the there s C (, ) (, ). The weghted cose smlarty measure of two IFSs ad also satsfes the followg propertes: 1. 0 C (, ) 1 2. C (, ) C (, ) 3. C (, ) 1 f =,.e., ) ) ad ) ) for 1,2,...,. Clearly the equato (4) ad equato (5) do cosder membershp degree ad o membershp degree IFSs. The two equatos seem complete whe the mportace of hestato degree s eglected. 3 New Cose Smlarty Measure for IFSs The ew cose smlarty measure. ssume that there are two IFSs ad the uverse of dscourse X { x1, x2,..., x }. The IFS s characterzed by the degree of membershp, ), degree of o-membershp, ) ad degree of hestato, ) for 1, 2,3,...,, whch ca be cosdered as vector represetatos wth elemets: ( ( x1 ), ( x2),.., ( x)), ( ( x1), ( x2)..., ( x)) ad ( ( x1), ( x2),..., ( x)). For the IFS, t s characterzed by the degree of membershp, ), degree of o membershp, ) ad degree of hestato, ( x ) for 1, 2,3,...,, whch ca be cosdered as vector represetatos wth elemets: ( ( x1), ( x2),..., ( x)), ( ( x1), ( x2),..., ( x)) ad ( ( x1), ( x2),..., ( x)). ovel cose smlarty measure betwee ad s proposed as follows: C 1 ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) IFS (, ) ) ) ) ) ) ) The cose smlarty measure of two IFS ad satsfes the followg propertes: 1. 0 (, ) 1 2. (, ) (, ) 3. (, ) 1 f =,.e., ) ), ) ) ad ) ) for 1, 2,...,. Proof: 1. It s obvous that the property s true accordg to cose value for equato (6). 2. It s obvous that the property s true. Therefore, a (6)

4 112 L. bdullah ad W.K.W. Ismal: Hestato Degree of Itutostc Fuzzy Sets 3. Whe =, there are ) ), ) ), ad ) ) for 1,2,...,. So there s (, ) 1. Whe (, ) 1, there are ) ) ) ), ad ) ) for 1,2,...,. So there s =. If we cosder the weght of x, a weghted cose smlarty measure betwee IFSs ad s proposed as follows: ) ) ) ) ) ) (, ) w (7) ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) where w [0,1], 1,2,..., ad w 1. If w 1/, 1,2,...,, the there s C (, ) (, ). Hestato degree π for upper ad lower boudares are cosdered the equato (6) ad equato (7). The proposed methods are dfferet from equato (4) ad equato (5) based o the usage of vector represetato the equato. From the equato (6) ad equato (7), t ca be see that the hestato degree s also cosdered whle equato (4) ad equato (5), there s o hestato degree. The weghted cose smlarty measure of two IFSs ad also satsfes the followg propertes: () 0 C (, ) 1 () C (, ) C (, ) () (, ) 1 f =,.e., ), ( x ), ) ) ad ) ( x ) for 1,2,...,. Proof: () It s obvous that the property s true accordg to cose value for equato (7). () It s obvous that the property s true. () Whe =, there are ) ), ) ) ad ( x ) ) for 1,2,...,. So there s C (, ) 1. Whe C (, ) 1, there are ) ), ) ) ad ) ) for 1,2,...,. So there s =. s the ew cose smlarty measure satsfes the smlarty measure propertes, the proposed weghted cose smlarty measure also satsfes the codtos. 4 pplcato of Patter Recogto I ths secto, the ovel cose smlarty measure for IFSs s appled to patter recogto to demostrate the feasblty. Example 1: The followg example dscusses the medcal dagoss problem retreved from [11]. Let us cosder a set of dagoss Q = {Q 1 (Vral fever), Q 2 (Malara), Q 3 (Typhod), Q 4 (Stomach problem), Q 5 (Chest problem)} ad a set of symptoms S = {s 1 (Temperature), s 2 (Headache), s 3 (Stomach pa), s 4 (Cough), s 5 (Chest pa)}. Suppose a patet wth respect to all the symptoms ca be represeted by the followg IFSs: P(Patet) = {<s 1, 0.8, 0.1, 0.1>, < s 2, 0.6, 0.1, 0.3>, < s 3, 0.2, 0.8, 0>, < s 4, 0.6, 0.1, 0.3>, < s 5, 0.1, 0.6, 0.3>} The each dagoss Q ( 1,2,3,4,5) ca also be vewed as IFSs wth respect to all the symptoms as follows: Q 1 (Vral fever) = {<s 1, 0.4, 0, 0.6>, < s 2, 0.3, 0.5, 0.2>, < s 3, 0.1, 0.7, 0.2>, < s 4, 0.4, 0.3, 0.3>, < s 5, 0.1, 0.7, 0.2>} Q 2 (Malara) = {<s 1, 0.7, 0, 0.3>, < s 2, 0.2, 0.6, 0.2>, < s 3, 0, 0.9, 0.1>, < s 4, 0.7, 0, 0.3>, < s 5, 0.1, 0.8, 0.1>} Q 3 (Typhod) = {<s 1, 0.3, 0.3, 0.4>, < s 2, 0.6, 0.1, 0.3>, < s 3, 0.2, 0.7, 0.1>, < s 4, 0.2, 0.6, 0.2>, < s 5, 0.1, 0.9, 0>} Q 4 (Stomach problem) = {<s 1, 0.1, 0.7, 0.2>, < s 2, 0.2, 0.4, 0.4>, < s 3, 0.8, 0, 0.2>, < s 4, 0.2, 0.7, 0.1>, < s 5, 0.2, 0.7, 0.1>} Q 5 (Chest problem) = {<s 1, 0.1, 0.8, 0.1>, < s 2, 0, 0.8, 0.2>, < s 3, 0.2, 0.8, 0>, < s 4, 0.2, 0.8, 0>, < s 5, 0.8, 0.1, 0.1>}. The am s to classfy patter P to oe of the classes Q 1, Q 2, Q 3, Q 4 ad Q 5. Smlarly, applyg the equato (6) the followg result s obtaed: ( P, Q1 ) , IFS 2 C ( P, Q ) , ( P, Q3 ) , ( P, Q4 ) , ( P, Q5 ) The above result shows that the degree of smlarty betwee Q 2 ad P s greater tha others. The, t ca assg the patet to the dagoss Q 2 (Malara) accordg to the recogto prcple. I order to valdate the results, two

5 Joural of Ucerta Systems, Vol.8, No.2, pp , other methods from [12] ad [15] are compared wth the proposed method. comparso result betwee the proposed method ad other methods s dscussed ad lsted Table 1: Table 1: Dagoss of patet P symptom usg dfferet methods Methods Patet P symptom Normalzed Hammg dstace [11] Malara Symmetrc dscrmato measure for IFSs [12] Vral fever Cose smlarty measure [15] Vral fever The proposed cose smlarty measure Malara From Table 1, t ca be see that the result from proposed method s cosstet wth the result from [11]. However, t dffers from the result of [12] ad cose smlarty measure wthout cosderg hestato degree proposed by Ye [15]. Example 2: The followg example dscusses the patter recogto problem about the classfcato of buldg materals retreved from [13]. Gve four classes of buldg materal each s represeted by the tutostc fuzzy sets the feature space ad there s a ukow buldg materal : 1 = {<x 1, 0.173, 0.524, 0.303>, <x 2, 0.102, 0.818, 0.080>, <x 3, 0.530, 0.326, 0.144>, <x 4, 0.965, 0.008, 0.027>, <x 5, 0.420, 0.351, 0.229>, <x 6, 0.008, 0.956, 0.036>, <x 7, 0.331, 0.512, 0.157>, <x 8, 1.000, 0.000, 0.000>, <x 9, 0.215, 0.625, 0.160>, <x 10, 0.432, 0.534, 0.034>, <x 11, 0.750, 0.126, 0.124>, <x 12, 0.432, 0.432, 0.136>} 2 = {<x 1, 0.510, 0.365, 0.125>, <x 2, 0.627, 0.125, 0.248>, <x 3, 1.000, 0.000, 0.000>,<x 4, 0.125, 0.648, 0.227>, <x 5, 0.026, 0.823, 0.151>, <x 6, 0.732, 0.153, 0.115>, <x 7, 0.556, 0.303, 0.141>, <x 8, 0.650, 0.267, 0.083>, <x 9, 1.000, 0.000, 0.000>, <x 10, 0.145, 0.762, 0.093>, <x 11, 0.047, 0.923, 0.030>, <x 12, 0.760, 0.231, 0.009>} 3 = {<x 1, 0.495, 0.387, 0.118>, <x 2, 0.603, 0298, 0.099>, <x 3, 0.987, 0.006, 0.007>,<x 4, 0.073, 0.849, 0.078>, <x 5, 0.037, 0.923, 0.040>, <x 6, 0.690, 0.268, 0.042>, <x 7, 0.147, 0.812, 0.041>, <x 8, 0.213, 0.653, 0.134>, <x 9, 0.501, 0.284, 0.215>, <x 10, 0.000, 1.000, 0.000>, <x 11,0.324, 0.483, 0.193>, <x 12, 0.045, 0.912, 0.043>} 4 = {<x 1, 1.000, 0.000, 0.000>, <x 2, 1.000, 0.000, 0.000>, <x 3, 0.857, 0.123, 0.020>,<x 4, 0.734, 0.158, 0.108>, <x 5, 0.021, 0.896, 0.083>, <x 6, 0.076, 0.912, 0.012>, <x 7, 0.152, 0.712, 0.136>, <x 8, 0.113, 0.756, 0.131>, <x 9, 0.489, 0.389, 0.122>, <x 10, 1.000, 0.000, 0.000>, <x 11, 0.386, 0.485, 0.129>, <x 12, 0.028, 0.912, 0.06>} = {<x 1, 0.978, 0.003, 0.019>, <x 2, 0.980, 0.012, 0.008>, <x 3, 0.798, 0.132, 0.070>,<x 4, 0.693, 0.213, 0.094>, <x 5, 0.051, 0.876, 0.073>, <x 6, 0.123, 0.756, 0.121>, <x 7, 0.152, 0.721, 0.127>, <x 8, 0.113, 0.732, 0.155>, <x 9, 0.494, 0.368, 0.138>, <x 10, 0.987, 0.000, 0.013>, <x 11, 0.376, 0.423, 0.201>, <x 12, 0.012, 0.897, 0.091>}. Our am s to justfy whch class the ukow patter belogs to. Smlarly, applyg the equato (6) the followg result obtaed: (, 1 ) , (, 2 ) , (, 3 ) , (, 4 ) The above result shows that the degree of smlarty betwee 4 ad s greater tha others. The, t ca assg the ukow buldg materal to the buldg materal 4 accordg to the recogto prcple. I order to valdate the results, two other methods from [5] ad [14] are compared wth the proposed method. comparso result betwee the proposed method ad other methods s dscussed ad lsted Table 2: Table 2: Classfcato of ukow buldg materal usg dfferet methods Method Ukow Patter elogg Prcple of mmum degree of dfferece betwee IFSs [13] 4 Smlarty measure of IFSs based o L p metrc [5] 4 Etropy measure for IVIFSs [14] 4 The proposed cose smlarty measure 4 From Table 2, t ca be see that the result obtaed from the proposed method s cosstet wth the other methods. The ukow patter belogs to the buldg materal 4. Example 3: The followg example also dscusses the patter recogto problem about the classfcato of hybrd meral retreved from [13].

6 114 L. bdullah ad W.K.W. Ismal: Hestato Degree of Itutostc Fuzzy Sets Gve fve kds of meral felds, each s featured by the cotet of sx merals ad has oe kd of typcal hybrd meral. The fve kds of typcal hybrd meral ca be express by fve IFSs C 1, C 2, C 3, C 4 ad C 5 the feature space X={x 1,x 2,., x 6 } ad there s ukow hybrd meral : C 1 = {<x 1, 0.739, 0.125, 0.136>, <x 2, 0.033, 0.818, 0.149>, <x 3, 0.188, 0.626, 0.186>, <x 4, 0.492, 0.358, 0.150>, <x 5, 0.020, 0.628, 0.352>, <x 6, 0.739, 0.125, 0.136>} C 2 = {<x 1, 0.124, 0.665, 0.211>, <x 2, 0.030, 0.825, 0.145>, <x 3, 0.048, 0.800, 0.152>, <x 4, 0.136, 0.648, 0.216>, <x 5, 0.019, 0.823, 0.158>, <x 6, 0.393, 0.553, 0.054>} C 3 = {<x 1, 0.449, 0.387, 0.164>, <x 2, 0.662, 0.298, 0.040>, <x 3, 1.000, 0.000, 0.000>, <x 4, 1.000, 0.000, 0.000>, <x 5, 1.000, 0.000, 0.000>, <x 6, 1.000, 0.000, 0.000>} C 4 = {<x 1, 0.280, 0.715, 0.005>, <x 2, 0.521, 0.368, 0.111>, <x 3, 0.470, 0.423, 0.107>, <x 4, 0.295, 0.658, 0.047>, <x 5, 0.188, 0.806, 0.006>, <x 6, 0.735, 0.118, 0.147>} C 5 = {<x 1, 0.326, 0.452, 0.222>, <x 2, 1.000, 0.000, 0.000>, <x 3, 0.182, 0.725, 0.093>, <x 4, 0.156, 0.765, 0.079>, <x 5, 0.049, 0.986, 0.055>, <x 6, 0.675, 0.263, 0.062>} = {<x 1, 0.629, 0.003, 0.068>, <x 2, 0.524, 0.356, 0.120>, <x 3, 0.210, 0.689, 0.101>, <x 4, 0.218, 0.753, 0.029>, <x 5, 0.069, 0.876, 0.055>, <x 6, 0.658, 0.256, 0.086>}. Our am s to justfy whch kd of meral the ukow hybrd meral belogs to. ssume the weghts of x 1, x 2, x 3, x 4, x 5 ad x 6 are 1/6. y applyg equato (7), the followg result obtaed: C ( C, ) , 1 C ( C, ) , 2 C ( C, ) , 3 C ( C, ) , 4 C ( C5, ) The above result shows that the degree of smlarty betwee C 5 ad s the largest. Therefore, t s clear that hybrd meral should be classfed to C 5. Ths result s agreemet wth the oes obtaed from [13]. Example 4: I order to demostrate the applcatos of the proposed weghted cose smlarty measures for IFSs to patter recogto, the problem retreved from [6] s dscussed. There are three kow patters 1, 2 ad 3, respectvely. The patters are represeted by the followg IFSs the gve fte uverse X { x1, x2, x3} : 1 = {<x 1, 1, 0, 0>, < x 2, 0.8, 0, 0.2>, <x 3, 0.7, 0.1, 0.2>} 2 = {<x 1, 0.8, 0.1, 0.1>, < x 2, 1, 0, 0 >, <x 3, 0.9, 0, 0.1>} 3 = {<x 1, 0.6, 0.2, 0.2>, < x 2, 0.8, 0, 0.2>, <x 3, 1, 1, 0>}. Gve a ukow patter Q whch s represeted by the IFS: Q = {<x 1, 0.5, 0.3, 0.2>, < x 2, 0.6, 0.2, 0.2>, <x 3, 0.8, 0.1, 0.1}>}. Our am s to classfy the patter Q to oe of the classes 1, 2 ad 3. ssume the weghts of x 1, x 2 ad x 3 are 0.5, 0.3 ad 0.2. y applyg equato (7), the followg result obtaed: C ( 1, Q) , C ( 2, Q) , C ( 3, Q) The above result shows that the degree of smlarty betwee 3 ad Q s the largest. Therefore, t s clear that patter Q should be classfed to 3. Ths result s agreemet wth the oes obtaed from [6, 8, 12, 15, 17]. comparso result betwee the proposed method ad the other methods s lsted Table 3. From Table 3, t ca be see that the result from the proposed method s cosstet wth other methods. Table 3: Classfcato of ukow patter usg dfferet methods Methods Ukow Patter elogg Degree of smlarty betwee IFSs [6] 3 Modfed Degfeg-Chuta smlarty measure [8] 3 Symmetrc dscrmato measure for IFSs [12] 3 Cross etropy of IVIFSs [17] 3 Cose smlarty measure [15] 3 The proposed weghted cose smlarty measure 3

7 Joural of Ucerta Systems, Vol.8, No.2, pp , Cocluso I ths paper, the kowledge of degree of membershp, o membershp ad degree of hestato of IFSs are cosdered cocurretly as the vector represetatos vector multplcato. The ew cose smlarty measure ad weghted cose smlarty measure for IFSs was proposed. It s mportat to ote that the presece of degree of hestato the smlarty measure has become a ew cotrbuto ths paper. Fally, the umercal examples have successfully demostrated the feasblty of the proposed cose smlarty measure ad weghted cose smlarty measure patter recogto. Refereces [1] taassov, K., Itutostc fuzzy sets, Fuzzy Sets ad Systems, vol.20, o.1, pp.87 96, [2] hattacharya,., O a measure of dvergece of two multomal populatos, The Ida Joural of Statstcs, vol.7, o.4, pp , [3] Chara, T., ad.k. Ray, ew measure usg tutostc fuzzy set theory ad ts applcato to edge detecto, ppled Soft Computg, vol.8, o.2, pp , [4] Hug, W.L., ad M.S. Yag, Smlarty measures of tutostc fuzzy sets based o Hausdorff dstace, Patter Recogto Letters, vol.25, o.14, pp , [5] Hug, W.L., ad M.S. Yag, Smlarty measures of tutostc fuzzy sets based o Lp metrc, Iteratoal Joural of pproxmate Reasog, vol.46, o.1, pp , [6] L, D., ad C. Cheg, New smlarty measures of tutostc fuzzy sets ad applcato to patter recogto, Patter Recogto Letters, vol.23, os.1-3, pp , [7] Lag, Z., ad P. Sh, Smlarty measures o tutostc fuzzy sets, Patter Recogto Letters, vol.24, o.15, pp , [8] Mtchell, H.., O the Degfeg-Chuta smlarty measure ad ts applcato to patter recogto, Patter Recogto Letters, vol.24, o.16, pp , [9] Salto, G., ad M.J. McGll, Itroducto to Moder Iformato Retreval, McGraw-Hll, ucklad, [10] Szmdt, E., ad J. Kacprzyk, Etropy for tutostc fuzzy sets, Fuzzy Sets ad Systems, vol.118, o.3, pp , [11] Szmdt, E., ad J. Kacprzyk, Itutostc fuzzy sets some medcal applcatos, Ffth Iteratoal Coferece o Itutostc Fuzzy Sets, vol.4, pp.58 64, [12] Vlachos, I.K., ad G.D. Sergads, Itutostc fuzzy formato-applcato to patter recogto, Patter Recogtos Letters, vol.28, o.2, pp , [13] Wag, W., ad X. X, Dstace measure betwee tutostc fuzzy sets, Patter Recogto Letters, vol.26, o.13, pp , [14] We, C.P., Wag, P., ad Y.Z. Zhag, Etropy, smlarty measure of terval-valued tutostc fuzzy sets ad ther applcatos, Iformato Sceces, vol.181, o.19, pp , [15] Ye, J., Cose smlarty measures for tutostc fuzzy sets ad ther applcatos, Mathematcal ad Computer Modellg, vol.53, os.1-2, pp.91 97, [16] Zadeh, L.., Fuzzy sets, Iformato Cotrol, vol.8, o.3, pp , [17] Zhag, Q.S., Jag, S., Ja,., ad S. Luo, Some formato measures for terval-valued tutostc fuzzy sets, Iformato Sceces, vol.180, o.24, pp , 2010.