Ordering Generalized Trapezoidal Fuzzy Numbers. Using Orthocentre of Centroids

Transcription

1 International Journal of lgera Vol. 6 no Ordering Generalized Trapezoidal Fuzzy Numers Using Orthoentre of Centroids Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar Dept. of pplied Mathematis GIS GITM University India drravi68@gmail.om (N. avi Shanar) Dept. of Mathematis GIT GITM University India strat Ordering fuzzy numers plays an important role in approximate reasoning optimization foreasting deision maing ontrolling sheduling and other usage. This paper illustrates a raning method for ordering fuzzy numers ased on rea Mode Spreads and Weights of generalized (non-normal)fuzzy numers. The area used in this method is otained from the non-normal trapezoidal fuzzy numer first y splitting the generalized trapezoidal fuzzy numers into three plane figures and then alulating the Centroids of eah plane figure followed y the Orthoentre of these Centroids and then finding the area of this Orthoentre from the original point. In this paper we also apply mode and spreads in those ases where the disrimination is not possile. This method is simple in evaluation and an ran various types of fuzzy numers and also risp numers whih are onsidered to e a speial ase of fuzzy numers. Keywords: aning funtion; Orthoentre; Centroid points; Generalized trapezoidal fuzzy numers. Introdution aning fuzzy numers plays an important role in deision maing. Most of the real world prolems that exist in nature are fuzzy than proailisti or deterministi. Prolems in whih fuzzy theory is used lie fuzzy ris analysis fuzzy optimization et. at one or other stage fuzzy numers must e raned efore an ation is taen y a deision maer. s fuzzy numers are represented y possiility distriutions they often overlap with eah other and disriminating them is a omplex tas than disriminating real numers where a natural order exist etween them. n effiient approah for ordering the fuzzy numers is defuzzifiation. For this we define a raning funtion from the set of all fuzzy

2 7 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar numers F() to the set of all real numers whih maps eah fuzzy numer into the real line where a natural order exists. Usually y reduing the whole of any analysis to a single numer muh of the information is lost and most of the raning methods onsider only one point of view on omparing fuzzy quantities. Hene an attempt is to e made to minimize this loss. Sine the ineption of fuzzy sets y Zadeh [] in 965 many authors have proposed different methods for raning fuzzy numers. However due to the omplexity of the prolem there is no method whih gives a satisfatory result to all situations. Most of the methods proposed so far are non- disriminating ounter-intuitive and some produe different ranings for the same situation and some methods annot ran risp numers. aning fuzzy numers was first proposed y Jain in the year 976 for deision maing in fuzzy situations y representing the ill-defined quantity as a fuzzy set. Jain [ ] proposed a method using the onept of maximizing set to order the fuzzy numers and the deision maer onsiders only the right side memership funtion. Sine then various proedures to ran fuzzy quantities are proposed y various researhers. Yager [ ] proposed four indies to order fuzzy quantities in [ ]. n damo [] fuzzy deision tree was an important reathrough in raning fuzzy numers. Duois and Prade [] proposed a omplete set of omparison indies in the frame wor of Zadeh s possiility theory. Bortolan and Degani [] reviewed some of these raning methods for raning fuzzy susets. Chen [5] presented raning fuzzy numers with maximizing set and minimizing set. Kim and Par [5] presented a method of raning fuzzy numers with index of optimism. Liou and Wang [7] presented raning fuzzy numers with integral value. Chooineh and Li [8] presented an index for ordering fuzzy numers. Sine then several methods have een proposed y various researhers whih inlude raning fuzzy numers using area ompensation y Fortemps and ouens [] distane method y Cheng [7]. Wang and Kerre [9 ] lassified the existing raning proedures into three lasses. The first lass onsists of raning proedures ased on fuzzy mean and spread and seond lass onsists raning proedures ased on fuzzy soring whereas the third lass onsists of methods ased on preferene relations and onluded that the ordering proedures assoiated with first lass are relatively reasonale for the ordering of fuzzy numers speially the raning proedure presented y damo [] whih satisfies all the reasonale properties for the ordering of fuzzy quantities. The methods presented in the seond lass are not doing well and the methods whih elong to lass three are reasonale. Later on raning fuzzy numers y area etween the entroid point and original point y Chu and Tsao [9] modifiation of the index of Liou and Wang y Garia and Lamata [] fuzzy ris analysis ased on raning of generalized trapezoidal fuzzy numers y Chen and Chen [] a new approah for raning trapezoidal fuzzy numers y asandy and Hajjari [] fuzzy ris analysis ased on raning generalized fuzzy numers with different heights and different spreads y Chen and Chen [6] ame into existene. mit Kumar et al. [6] presented a proedure on raning generalized trapezoidal fuzzy numers ased on ran mode divergene and spread. ao and Shanar

3 Ordering generalized trapezoidal fuzzy numers 7 [8] presented a method on raning fuzzy numers using irumenter of entroids and index of modality. In this paper a new method is proposed whih is ased on Orthoentre of Centroids to ran fuzzy quantities. In a trapezoidal fuzzy numer first the trapezoid is split into three parts where the first seond and third parts are a triangle a retangle and a triangle respetively. Then the entroids of these three parts are alulated followed y the alulation of the Orthoentre of these entroids. Finally a raning proedure is defined whih is the area etween the Orthoentre of Centroids and the original point and also uses mode and spreads in those ases where the disrimination is not possile. In setion we riefly introdue the asi definitions arithmeti operations of fuzzy numers and some fuzzy raning methods. Setion presents the proposed raning method. In Setion some important results lie linearity of raning funtion and other properties are proved whih are useful for proposed approah. In Setion 5 the proposed method has een explained with examples whih desrie the advantages and the effiieny of the method. In Setion 6 the method demonstrates its power y omparing with other methods that exist in literature. Finally the onlusions of the wor are presented in Setion 7.. Basi definitions fuzzy arithmeti and raning of fuzzy numers. Basi Definitions Definition. Let U e a universe set. fuzzy set of U is defined y a memership funtion f : U [ ] where f ( x) is the degree of x in x U. Definition. fuzzy set of universe set U is normal if and only if sup f ( x) = x U Definition. fuzzy set of universe set U is onvex if and only if f ( λx ( λ ) y) min f ( x) f ( y) x y U and λ [ ]. Definition. fuzzy set of universe set U is a fuzzy numer iff is normal and onvex on U. Definition5. real fuzzy numer is desried as any fuzzy suset of the real line with memership funtion f ( x) possessing the following properties: () f ( x) is a ontinuous mapping from to the losed interval [ w ]. < w

4 7 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar () f ( x) () f ( x) () f ( x) (5) f ( x) (6) f ( x) Definition 6. = for all x ( a] is stritly inreasing on [ a ] = for all x [ ] is stritly dereasing on [ d] = for all x ( d ] where a d are real numers The memership funtion of the real fuzzy numer is given y L f a x w x f ( x) = f x d otherwise () where < w is a onstant a d are real numers and f L : [ a ] [ w] f : [ d] [ w] are two stritly monotoni and ontinuous funtions from to the losed interval[ w]. It is ustomary to write a fuzzy numer as = ( a If w = then = ( a ) is a normalized fuzzy numer otherwise is said to e a generalized or non-normal fuzzy numer If the memership funtion f ( x) is pieewise linear then is said to e a trapezoidal fuzzy numer. The memership funtion of a trapezoidal fuzzy numer is given y: w( x a) a x a w x f ( x) = () w( x d) x d d otherwise If w= then = ( a ) is a normalized trapezoidal fuzzy numer and is a generalized or non normal trapezoidal fuzzy numer if < w <. The image of = ( a is given y = ( d a;

5 Ordering generalized trapezoidal fuzzy numers 7 s a partiular ase if = the trapezoidal fuzzy numer redues to a triangular fuzzy numer given y = ( a The value of orresponds with the mode or ore and [a d] with the support. If w= then = ( a d). is a normalized triangular fuzzy numer is a generalized or non normal triangular fuzzy numer if < w <.. Fuzzy arithmeti operations If = ( a w ) and B = ( a w ) are two generalized trapezoidal fuzzy numers then (i) B = ( a a d min( w w )) Θ B = a d d a ;min w w (ii) ( ) (iii) = a d ; w ); > ( (iv) = ( d a ; w ); <.. Some methods of raning fuzzy numers In this setion some methods of raning fuzzy numers are reviewed. Yager fuzzy raning Yager [] proposed a entroid-index raning method for alulating the value * x for a fuzzy numer as follows : w( x) f ( x) dx * x = () f ( x) dx where w(x) denotes the value of a weighing funtion measuring the importane of the value x and f ( x) denotes the memership funtion of the fuzzy numer where f : X []. When w(x) =x the value the geometri enter of gravity (COG) [] shown as follows : xf ( x) dx * x = f ( x) dx () The larger the value of x the etter the raning of. * * x shown in Eq () eomes

6 7 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar Cheng s fuzzy raning Cheng [7] raned fuzzy numers with the distane method using the Eulidean distane etween the Centroid point and original point. For a generalized fuzzy numer = ( a the entroid is given y: w ( d a d a) ( ) x y = w d ( a) 6( ) w and the raning funtion i > j assoiated with as = x y Let i and j two fuzzy numers (i) If ( i ) > ( j ) then > i. j (ii) If ( ) < i (iii) If ( ) i = < i then j j = i then j j ( ) ( a d )( w) a d a d w He further improved Lee and Li s method y proposing the index of oeffiient σ of variation (CV) ascv = whereσ is standard error and μ is mean μ μ and σ > the fuzzy numer with smaller CV is raned higher. Liou and Wang s fuzzy raning Liou and Wang [7] raned fuzzy numers with total integral value. For a fuzzy α I = αi α I where numer the total integral value is defined as T L L I = g ( y)dy and I = g ( y)dy L of respetively and α [ ] are the right and left integral values is the index of optimism whih represents the degree of optimism of a deision maer. Ifα = the total integral value represents a pessimisti deision maer s view point whih is equal to left integral value. If α = the total integral value represents an optimisti deision maer s view point and is equal to the right integral value and whenα =. 5 the total integral value represents an moderate deision maer s view point and is equal to the mean of right and left integral values. For a deision maer the larger the value of α is the higher is the degree of optimism.. Proposed ordering of fuzzy numers The Centroid of a trapezoid is measured as the alaning point of the trapezoid (Fig.). Divide the trapezoid into three plane figures. These three plane figures are a triangle (PB) a retangle (BPQC) and a triangle (CQD) respetively. Let the Centroids of the three plane figures e G G& Grespetively. The

7 Ordering generalized trapezoidal fuzzy numers 75 orthoenter of these Centroids G G & G is taen as the point of position to define the raning of generalized trapezoidal fuzzy numers. The reason for seleting this point as a point of position is that eah Centroid point are alaning points of eah individual plane figure and the orthoentre of these Centroid points is a muh more alaning point for a generalized trapezoidal fuzzy numer. Therefore this point would e a etter position point than the Centroid point of the trapezoid. w P ( w) Q ( w) G G G O (a ) B CD (d ) Fig.. Generalized Trapezoidal fuzzy numer Consider a generalized trapezoidal fuzzy numer = ( a (Fig.). The a w w Centroids of the three plane figures are G = G = d w andg = respetively. Equation of the line G G is w y = and G does not lie on the line G G. Therefore G G and G are non-ollinear and they form a triangle. We define the Orthoentre O ( x y ) of the triangle with verties G G and G of the generalized trapezoidal fuzzy numer = ( a as a d w O x y = 6 w (5) s a speial ase for triangular fuzzy numer = ( a i.e. = the Orthoentre of Centroids is given y ( a)( d ) w O ( x y ) = (6) w The raning funtion of the generalized trapezoidal fuzzy numer = ( a whih maps the set of all fuzzy numers to a set of real numers is defined as:

8 76 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar a d w ( )( ) x ( ) = y = (7) 6w This is the rea etween the Orthoentre of the Centroids O ( x y ) as defined in Eq.(5) and the original point. The Mode (m) of the generalized trapezoidal fuzzy numer = ( a is defined as: w m = w ( ) dx = ( ) The Spread(s) of the generalized trapezoidal fuzzy numer = ( a is defined as: s w = ( d a) dx = w( d a) The Left spread (ls)of the generalized trapezoidal fuzzy numer = ( a is defined as: w ( a) dx = w( a) ls = () The ight spread (rs) of the generalized trapezoidal fuzzy numer = ( a is defined as: w ( d ) dx = w( d ) rs = () Using the aove definitions we now define the raning proedure of two generalized trapezoidal fuzzy numers. Let = a d ; w ) and B = a d ; w ) e two generalized trapezoidal ( ( fuzzy numers. The woring proedure to ompare and B is as follows: Step : Find and B Case (i) If > B then B > Case (ii) If < B then B < Case (iii) If = B omparison is not possile then go to step. Step : Find m and m B (8) (9)

9 Ordering generalized trapezoidal fuzzy numers 77 Case (i) If > m m B then B > Case (ii) If < m m B then B < Case (iii) If m = m B omparison is not possile then go to step. Step : Find s and s B Case (i) If > s s B then B < Case (ii) If s < s B then B > Case (iii) If s = s B omparison is not possile then go to step. Step : Find ls and ls B Case (i) If ls > ls B then B > Case (ii) If < ls ls B then B < Case (iii) If ls = ls B omparison is not possile then go to step 5. Step 5: Examine w and w Case (i) If w > w then B > Case (ii) If w < w then B < Case (iii) If w = w then B. Some important results In this setion some important results whih are the asis for defining the raning proedure in setion are disussed and proved. The proposed raning verifies all the reasonale properties[7] stated elow: Let S e an aritrary finite suset of E. : For any aritrary finite suset S of E and S

10 78 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar : ( B) S B and B B. : ( B C) S B and B C. : ( B) S inf { Supp ( ) } Sup { Supp ( B)} B. ' : ( B) S inf { Supp ( ) } > Sup { Supp ( B)} > B. 5 : Let S and S' e two aritrary finite susets of E and B S S' > B on S iff > B on S'. 6 : Let B C B C e elements of E. If B then C B C. ' 6 : Let B C B C e elements of E. If B then C B C where C O. Proposition. The raning funtion defined in setion y means of Eq. (7) is a linear funtion for normalized trapezoidal fuzzy numer ). ; ( d a = i.e. If ) ; ( d a = and ) ; ( d a B = are two normalized trapezoidal fuzzy numers then (i) ; B B = are non negative real numers (ii) = (iii) = Proof (i): ase (i) Let e two non-negative real numers d d a a B = = 6 d d a a B

11 Ordering generalized trapezoidal fuzzy numers 79 = ( ) (( a )( d) ) 6 ( ) (( a )( d ) ) 6 = B. Similarly the result an e proved for ase (ii) and ase (iii). = a d = d a Proof (ii): Let ( a )( d ) = 6 ( a )( d ) = 6 ( ) = ( ) (y (i)) = Θ (y (ii)) =. ( ) = Proof (iii): 5. Numerial Examples In this setion the proposed method is first explained y raning some fuzzy numers. Example 5. 5 Let = ( 57; ) and B = 5 ; e triangular fuzzy numers 8 Then O ( x y ) = ( ) O B ( x y ) = ( 5.79) Therefore ( ) = 8. and ( B ) =. 95 Sine > ( B) > B. Example 5. 7 Let = ( ; ) and B = ; e triangular fuzzy numers 5 Then O ( x y ) = (.6666) and O B ( x y ) = (.5) Therefore ( ) = and ( B ) =. 5 Sine > ( B) > B Example Let = ( ; ) = ( ; ) and B = ; B = ; 5 5

12 8 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar Step : O ( x y ) = (.6666) and ( x y ) = (.5) O B Therefore ( ) = and ( B ) =. 5 Sine ( ) < ( B) < B From examples 5. and 5. we see that the proposed method an ran fuzzy numers and their images as it is proved that > B < B. Example 5. Let = (...5; ) and B = (...; ) Then O ( x y ) = (..66) and O B ( x y ) = (..66) Therefore ( ) =. 9 and ( B ) =. 99 Sine > ( B) > B Example 5.5 Let = (...5;.8 ) and B = (...5; ) e non-normal triangular fuzzy numers. Then O ( x y ) = (..8) and O B ( x y ) = (..6) Therefore ( ) =. 67 and ( B ) =. 8 Sine < ( B) < B From example 5.5 it is lear that the proposed method an ran fuzzy numers with different height and same spreads. Example 5.6 = = Let (...5; ) and B (...5; ) Then O ( x y ) = (..) and ( x y ) = (..66) Therefore ( ) =. and ( B ) =. 98 Sine > ( B) > B ll the aove examples are proved using step only. Example 5.7 Let = (...5; ) and B = (...7; ) Step : ( x y ) = (..66) O B x y =..66 Therefore ( ) =. 9 and ( B ) =. 9 Sine ( ) = ( B) So goto step. =. and m ( B ) =. O B O and Step : m Sine m ( ) = m Step : =. s ( ) and s ( B ) Sine s ( ) = s Step : B So goto step. =. B So go to step.

13 Ordering generalized trapezoidal fuzzy numers 8 ls ( ) =. and ls ( B ) =. Sine ls ( ) > ls ( B ) > B 6. esults and disussion In this setion the advantages of the proposed method is shown y omparing with other existing methods in literature where the methods annot disriminate fuzzy numers. The results are shown in Tale I and Tale II. Example 6. Consider two fuzzy numers = ( 5 ) and B = ( 6) By Liou and Wang Method [7]it is lear that the two fuzzy numers are equal for all the deision maers as α I T =.5α ( α ). 5 and α I T ( B) =.5α ( α ). 5 Whih is not even true y intuition. By using our method we have O ( x y ) = (.) and O B ( x y ) = (.) Therefore ( ) = 5. and ( B ) = > B Sine > ( B) > B Example 6. Let = (...5; ) B = ( ; ) Cheng [7] raned fuzzy numers with the distane method using the Eulidean distane etween the Centroid point and original point. Where as Chu and Tsao [9] proposed a raning funtion whih is the area etween the entroid point and original point.their entroid formulae are given y w ( d a d a) ( ) x y = w d ( a) 6( ) w ( ) ( a d )( w) a d a d w w ( d a d a) ( ) y = ( ) w x w d a 6 a d Both these entroid formulae annot ran risp numers whih are a speial ase of fuzzy numers as it an e seen from the aove formulae that the denominator in the first oordinate of their entroid formulae is zero and hene entroid of risp numers are undefined for their formulae. By using our method we have

14 8 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar O ( x y ) = (..66) and ( x y ) = (.) O B Therefore ( ) =. 98 and ( B ) =. Sine < ( B) < B From this example it is proved that the proposed method an ran risp numers whereas other methods failed to do so. Example 6. Consider four fuzzy numers = (...; ) = (..5.8; ) = (...9; ) (.6.7.8; ) = Whih were raned earlier y Yager[] Fortemps and ouens[] Liou and Wang[7] and Chen [5] as shown in Tale I. Tale I. Comparison of various raning methods Fuzzy raning Yager [] Fortemps & ouens[] α = aning order > > > > > > Liou & Wang[7] α = > > α = Chen [5] β = β = β = Proposed > > > > > > > > It an e seen from Tale that none of the methods disriminates fuzzy numers. Yager[] and Fortemps and ouens [] methods failed to disriminate the fuzzy numers and Whereas the methods of Liou and Wang[7] and Chen [5]annot disriminate the fuzzy numers and. By using our method we have

15 Ordering generalized trapezoidal fuzzy numers 8 O ( ) (..66) x y = O (.5.6) x y = O (..5) x y = (.7.66) x y = Therefore =. O 67 ( ) =. 865 ( ) =. ( ) =. 56 > > > Example 6. In this we onsider seven sets of fuzzy numers availale in literature and the omparative study is presented in Tale II. = = Set : (...6.8;.5) and B (...;.7 ) Set : = (...5; ) and B = (...5; ) Set : = (...5; ) and B = ( ; ) Set : = (.5...; ) and B = (...5; ) Set 5: = (..5.5; ) and B = ( ; ) Set 6: = (..6.8; ) = ( ; ) C = Set 7: = (...5; ) and B = ( ; ) B and ( ; ) Tale II. Comparison of proposed fuzzy raning with existing raning methods Methods Set Set Set Set Set 5 Set 6 Set 7 Cheng [7] < B B *** B > B B C < < *** Chu and Tsao [9] < B B *** < B > B B C < < *** Chen and Chen [] < B < B < B < B > B C B < < > B asandy and *** B Hajjari [] < B B < B B C < < > B Chen and Chen [6] < B < B < B < B > B B C < < > B Kumar et al.[6] < B < B < B < B > B B C < < > B Proposed method > B > B < B < B < B B C < < > B *** not omparale 7. Conlusions and future wor This paper reommends a method that rans fuzzy numers whih is simple and onrete. This method rans trapezoidal as well as triangular fuzzy numers and their images. This method also rans risp numers whih are speial ase of fuzzy numers whereas some methods proposed in literature annot ran risp numers. This method whih is simple and easier in alulation not only gives

16 8 Y. L. P. Thorani P. Phani Bushan ao and N. avi Shanar satisfatory results to well defined prolems ut also gives a orret raning order to prolems. Comparative examples are used to illustrate the advantages of the proposed method. ppliation of this raning proedure in various fuzzy deision maing prolems fuzzy ris analysis fuzzy optimization prolems and fuzzy networ analysis fuzzy transportation prolems et. is left as future researh wor. eferenes [] asandy S. and Hajjari T. 9. new approah for raning of trapezoidal fuzzy numers. Computers and Mathematis with ppliations 57(): -9. [] damo J. M. 98. Fuzzy deision trees Fuzzy Sets and Systems : 7-9. ] Bortolan G. and Degani review of some methods for raning fuzzy susets. Fuzzy Sets and Systems 5: -9. [] Chen S.J. and Chen S.M. 7. Fuzzy ris analysis ased on the raning of generalized trapezoidal fuzzy numers pplied Intelligene 6 (): -. [5] Chen S. H aning fuzzy numers with maximizing set and minimizing set Fuzzy Sets and Systems 7(): -9. [6] Chen S.M. and Chen J.H. 9. Fuzzy ris analysis ased on raning Generalized fuzzy numers with different heights and different spreads. Expert Systems with ppliations 6 (): [7] Cheng C. H new approah for raning fuzzy numers y distane method Fuzzy Sets and Systems 95(): 7-7. [8] Chooineh F. and Li H. 99.n index for ordering fuzzy numers Fuzzy Sets and Systems 5: [9] Chu T. C. and Tsao C. T.. aning fuzzy numers with an area etween the Centroid point and original point Computers and Mathematis with ppliations : -7. [] Duois D. and Prade H. 98. aning fuzzy numers in the setting of possiility theory Information Sienes : 8-. [] Fortemps P. and ouens M aning and defuzzifiation methods ased on area ompensation Fuzzy Sets and Systems 8: 9-.

17 Ordering generalized trapezoidal fuzzy numers 85 [] Garia M. S. and Lamata M. T. 7. modifiation of the index of liou and wang for raning fuzzy numer International Journal of Unertainty Fuzziness and Knowledge- Based Systems 5 (): -. [] Jain Deision maing in the presene of fuzzy variales IEEE Transations on Systems Man and Cyernetis 6: [] Jain proedure for multi aspet deision maing using fuzzy sets International Journal of systems siene 8: -7. [5] Kim K.and Par K. S. 99. aning fuzzy numers with index of optimism Fuzzy Sets and Systems 5: -5. [6] Kumar. Singh P. Kaur. and Kaur P.. aning of generalized trapezoidal fuzzy numers ased on ran mode divergene and spread. Turish Journal of Fuzzy Systems (): -5. [7] Liou T. S. and Wang M. J. 99. aning fuzzy numers with integral value Fuzzy Sets and Systems 5: [8] ao P. P. B. and Shanar N.. aning fuzzy numers with a distane method using irumenter of entroids and index of modality dvanes in Fuzzy Systems rtile ID 788 doi:.55//788: -7. [9] Wang X. and Kerre E. E.. easonale properties for the ordering of fuzzy quantities (I) Fuzzy Sets and Systems 8: [] Wang X. and Kerre E. E.. easonale properties for the ordering of fuzzy quantities (II) Fuzzy Sets and Systems 8: [] Yager.. On hoosing etween fuzzy susets Kyernetes 9 (98) 5-5. [] Yager.. On a general lass of fuzzy onnetives. Fuzzy sets and systems () [] Zadeh L.. Fuzzy sets Information and ontrol 8 () eeived: May