Mathematcal and omputatonal pplcatons, Vol 6, No, pp 359-369, ssocaton for Scentfc Research METHOD FOR RNKING OF FUZZY NUMERS USING NEW WEIGHTED DISTNE T llahvranloo, S bbasbandy, R Sanefard Department of Mathematcs Islamc zad Unversty, Scence and Research ranch, Tehran, Iran tofgh@allahvranloocom bstract- In ths paper, the researchers proposed a modfed new weghted dstance method to rank fuzzy numbers The modfed method can effectvely rank varous fuzzy numbers, ther mages and overcome the shortcomngs of the prevous technques The proposed model s studed for a broad class for fuzzy numbers and class of functons the membershp of whch s formed on the bass of the template ( max ( x ) Ths artcle also used some comparatve examples to llustrate the advantage of the proposed method Key Words : Rankng, Fuzzy numbers, Fuzzy dstance, Defuzzfcaton INTRODUTION In many applcatons, rankng of fuzzy numbers s an mportant component of the decson process In addton to a fuzzy envronment, rankng s a very mportant decson makng procedure Snce Jan [4, 5] employed the concept of maxmzng set to order the fuzzy numbers n 976, many authors have nvestgated varous rankng methods Some of these rankng methods have been compared and revewed by ortolan and Degan [6], and more recently by hen and Hwang [7] Other contrbutons n ths feld nclude: an ndex for orderng fuzzy numbers defned by hoobneh and L [8], rankng alternatves usng fuzzy numbers studed by Das [9], automatc rankng of fuzzy numbers usng artfcal neural networks proposed by Requena et al [], rankng fuzzy values wth satsfacton functon nvestgated by Lee et al [], rankng and defuzzfcaton methods based on area compensaton presented by Fortemps and Roubens [], and rankng alternatves wth fuzzy weghts usng maxmzng set and mnmzng set gven by Ra and Kumar [3] However, some of these methods are computatonally complex and dffcult to mplement, and others are counterntutve and not dscrmnatng Furthermore, many of them produce dfferent rankng outcomes for the same problem In 988, Lee and L [4] proposed a comparson of fuzzy numbers by consderng the mean and dsperson (standard devaton) based on the unform and the proportonal probablty dstrbutons heng [5] proposed the coeffcent of varance (V nde, e V (standard error) / ( mean ),, In ths approach, the fuzzy number wth smaller V ndex s ranked hgher, therefore heng's V ndex also contans shortcomngs To mprove Murakam et al's method, heng [5] proposed the dstance method for rankng fuzzy numbers; e, R( ) y x For any two fuzzy numbers ; f R( ) R( ), then ( ) R( and, f R, R( ) R( ), then and f ) then Moreover, the dstance method contradcts the V ndex n rankng some s
36 T llahvranloo, S bbasbandy and R Sanefard fuzzy numbers onsder the three fuzzy numbers, (,3,5 ), (7,3,58), (5,4,7) from [5] In heng's dstance method, R()=59, R()=64, and R()=66, produce the rankng order From ths result, the researchers can logcally nfer the rankng order of the mages of these fuzzy numbers as However, n the dstance method, the rankng order remans Obvously, the dstance method also has shortcomngs Moreover, n [] a method based on "Sgn Dstance" was ntroduced and a new method based on "Dstance Mnmzaton" was ntroduced by sady et al's [3] Ths method has some drawbacks, e, for all trangular fuzzy numbers u ( x,, ) where x ( ) / 4 and also trapezodal fuzzy numbers u x, y,, ), such that x y ( ) /, ( gves the same results However t s clear that these fuzzy numbers do not place n an equvalence class Recently, a new method based on "the left and the rght spreads at some -levels of trapezodal fuzzy numbers" was ntroduced [] Ths method has some shortcomng too, because, for any two symmetrc trapezodal fuzzy numbers, gves equal orderng Havng revewed the prevous methods, ths artcle proposes here a method to use the concept of fuzzy dstance, so as to fnd the order of fuzzy numbers Ths method can dstngush the alternatves clearly The man purpose of ths artcle s to present a new method for rankng of fuzzy numbers In addton to ts rankng features, ths method removes the ambgutes resulted from the comparson of prevous rankng The paper s organzed as follows: In Secton, we recall some fundamental results on fuzzy numbers Proposed method for rankng of fuzzy numbers s n the Secton 3 In ths Secton some theorems and remarks are proposed and llustrated Dscusson and comparson of ths work and other methods are carred out n Secton 4 The paper ends wth conclusons n secton 5 SI DEFINITIONS ND NOTTIONS The basc defntons of a fuzzy number are gven n [8, 9, ] as follows: Defnton fuzzy number s a mappng ( : [,] wth the followng propertes: s an upper sem-contnuous functon on, ( outsde of some nterval a, ], [ b 3 There are real numbers a,b such as a a b b and ( s a monotonc ncreasng functon on a, ], [ b ( s a monotonc decreasng functon on b, ], ( for all x n a, ] [ b [ b Let be the set of all real numbers The researchers assume a fuzzy number that can be expressed for all x n the form
Method for Rankng of Fuzzy Numbers 36 g( when x[ a, b), when x [ b, c], ( () h( when x ( c, d], otherwse where a, b, c, d are real numbers such as a b c d and g s a real valued functon that s ncreasng and rght contnuous and h s a real valued functon that s decreasng and left contnuous The trapezodal fuzzy number x, y,, ), wth two ( defuzzfer x, y and left fuzzness and rght fuzzness s a fuzzy set where the membershp functon s as ( xx ) x xx, x x y, ( ( y y x y, otherwse Support functon s defned as follows: supp ( ) x ( x ( s closure of set x ( where In ths paper, the researchers wll always refer to fuzzy number descrbed by () 3 NEW PPROH FOR RNKING OF FUZZY NUMERS In ths secton, the researchers wll propose the rankng of fuzzy numbers assocated wth the metrc D n F, that F denotes the space of fuzzy numbers We wll assume that the fuzzy number F s represented by means of the followng LR - representaton: (, ) [,] where [,] : [ L ( ), R ( )] (, ) Here, L : [,] (, ) s a monotoncally non-decreasng and R : [,] (, ) s a monotoncally nonncreasng left-contnuous functons The functons L () and R () express the left and rght sdes of a fuzzy number, respectvely Defnton 3 The followng values consttute the weghted averaged representatve and weghted wdth, respectvely, of the fuzzy number : I ( ) ( c L ( ) ( c) R ( )) d, ()
36 T llahvranloo, S bbasbandy and R Sanefard and D( ) ( R ( ) L ( )) f ( ) d Here c denotes an "optmsm/pessmsm" coeffcent n conductng operatons on fuzzy numbers The functon f ( ) s nonnegatve and ncreasng functon on [,] wth f ( ), f ( ) and f ( ) d The functon f ( ) s also called weghtng functon In actual applcatons, functon f ( ) can be chosen accordng to the actual stuaton In ths artcle, n practcal case, we assume that f ( ) Defnton 3 Let : F, be a functon that s defned as follows: when I( ), F : ( ) (4) when I ( ) Remark 3 If nf supp( ) or nf L ( ), then ( ) Remark 3 If sup supp( ) or sup R ( ), then ( ) Defnton 33 For arbtrary fuzzy numbers and the quantty TRD(, ) s called the TRD dstance between the fuzzy numbers and It s easly proved that the TRD dstance satsfes the followng propertes: TRD (, ), TRD (, ), TRD (, ) TRD (, ) TRD (, ), TRD (, ) TRD (, ) The membershp functon of a s a ( x ), f x a, and a ( x ), f x a Hence [ I ( ) I( )] [ D( ) D( )], f a, there s ( when x, when x Ths artcle consders as a fuzzy orgn and snce F, so for each F, TRD(, ) [ I( )] [ D( )] (5) Defnton 34 For each F and wth "optmsm/pessmsm" coeffcent equal to 5, TR (, ) ( ) TRD (, ), s called weghted dstance The steps of TR algortm are: Step : omputng the left and rght sdes of each fuzzy number ( L () and R () ) Step : Usng Eqs () and (3) to fnd weghted averaged and weghted wdth ( I () and D () ) Step 3 omputng TRD between fuzzy numbers and () by Eqs (4) and (5) (3)
Method for Rankng of Fuzzy Numbers 363 Defnton 35 For each arbtrary fuzzy numbers, F, defne the rankng of and by TR on F, e TR, ) TR (, ) f and only f, ( (, ) TR (, (, ) TR (, TR ) f and only f, TR ) f and only f ~ Ths artcle formulates the order and as f and only f or ~, f and only f or ~ Ths artcle consders the followng reasonable axoms that Wang and Kerre [] proposed for fuzzy quanttes rankng Let TR be an orderng method, S the set of fuzzy quanttes for whch the method TR can be appled, and a fnte subset of S The statement "two elements and n satsfy that has a hgher rankng than when TR s appled to the fuzzy quanttes n " wll be wrtten as " by TR on ", " ~ by TR on ", and " by TR on " are smlarly nterpreted [], the axoms as the reasonable propertes of orderng fuzzy quanttes for an orderng approach TR are as follows: - For an arbtrary fnte subset of S and ; - For an arbtrary fnte subset of S and (, ) ; and by TR on, ths method should have ~ -3 For an arbtrary fnte subset of S and (,, ) 3 ; and by TR on, ths method should have -4 For an arbtrary fnte subset of S and (, ) ; nf supp( ) > sup supp( ), ths method should have '- 4 For an arbtrary fnte subset of S and (, ) ; nf supp( ) > sup supp( ), ths method should have - 5 Let S, S be two arbtrary fnte sets of fuzzy quanttes n whch TR can be appled and and are n S S Ths method obtan the rankng order on S ff on S -6 Let,, and be elements of S If by TR on,, then by TR on and '- 6 Let,, and be elements of S If by TR on,, then by TR on and Remark 33 The functon TR has the propertes (-), (-),, (-5) Remark 34 If, then Hence, ths approach can nfer rankng order of the mages of the fuzzy numbers 4 NUMERIL EXMPLES Now, the authors compare proposed method wth the others n [8,,4,5] Throughout ths secton, we assum that f ( x ) x, and "optmsm/pessmsm" coeffcent s 5 Example onsder the followng sets n [7] (see Fgs )
364 T llahvranloo, S bbasbandy and R Sanefard Set : =(5,,5), =(7,3,3), =(9,5,) Set : =(4,7,,), =(7,4,), =(7,,) Set 3: =(5,,), =(5,8,,) (trapezodal fuzzy number), =(5,,4) Set 4: =(4,7,4,) (trapezodal fuzzy number), =(5,3,4), =(6,5,) To compare wth other methods authors refer the reader to Table uthors Fgs Table omparatve results of Example Set Set Set3 Set4 Fuzzy number Proposed method 6946 687 554 463 786 6576 656 5377 Sng Dstance method wth p= Sng Dstance method wth p= 873 73 559 477 5 95 4 3 5 5 6 4 5 ~ 8869 8756 757 7853 94 95 946 7958 65 33 865 8386 Dstance Mnmzaton 6 575 5 475 7 65 65 55 9 7 55 55
Method for Rankng of Fuzzy Numbers 365 Result ~ bbasbandy and Haar (Magntude method) 5334 5584 5 55 7 6334 646 584 8666 7 566 575 Result hoobneh and L 3333 548 333 5 5 583 464 5833 667 667 547 6 Yager 6 575 5 45 7 65 55 55 8 7 65 55 hen 3375 435 375 5 5 565 45 57 667 65 55 65 aldwn and Guld 3 7 7 4 33 7 37 4 44 37 45 4 ~ ~ hu and Tsao 99 847 5 44 35 347 35 64 3993 35 747 69 Yao and Wu 6 575 5 475 7 65 65 55 8 7 55 55 ~ heng Dstance 79 7577 77 76 86 849 837 756 968 86 7458 74 heng V unform dstrbuton 7 38 33 693 4 46 34 385 5 95 75 433 heng V proportonal dstrbuton 83 6 8 47 8 46 34 36 37 57 73 55
366 T llahvranloo, S bbasbandy and R Sanefard Example Let fuzzy numbers and descrbed by the membershp functons (Fg ) g( when x[,), when x[,], ( x when x(,], otherwse Where x when x[,9), g( 7x 6 x57 when x[9,] nd x ( x when when when otherwse x [, ), x [,], x (, ], Fg TR (, ) 8 y usng ths methodtr (, ) 39 and 57 Thus, the rankng order s s you may see n Table, the results of hu and Tsao method and heng dstance method are unreasonable The result of Sgn Dstance method wth p and wth p [], are the same as ths new approach Table omparatve results of Example fuzzy number Wegthed Dstance Sgn Dstance Sgn Dstance heng Dstance hu and Tsao (new approach) wth p= wth p= 39 68 63 75 4 857 65 87 76 45 Example 3 onsder the trangular fuzzy number (,,3 ), and the general number, (,,,), shown n Fgure 3 The membershp functon of s defned by
Method for Rankng of Fuzzy Numbers 367 ( x ) ( ( x ) 4 when x[,], when x[,4], otherwse Fg 3 In Lou and Wang's rankng method, dfferent rankngs are produced for the same problem when applyng dfferent ndces of optmsm In Sgn Dstance method wth p, d p (, ) 5, d p (, ) 4 78, and wth p, d p (, ) 3 957, d p (, ) 3845, the rankng order s obtaned In hu and Tsao rankng method, there s S ( ) 445 and S ( ) 8, therefore, y usng ths new approach, there s TR (, ) 593 and TR (, ) 5873 Thus, the rankng order s, too lso, the result of Dstance Mnmzaton method, was smlar to ths method Obvously, ths method can also rank fuzzy numbers other than trangular and trapezodal, and compared to Lou and Wang's method long wth method of hu and Tsao, ths method produces a smpler rankng result Example 4 The two trangular fuzzy numbers (3,,) and (3,, ) shown n Fg 4, taken from paper [5] To compare wth other methods ths artcle refers the reader to Table 3 Fg 4
368 T llahvranloo, S bbasbandy and R Sanefard fuzzy number new approach Table 3 omparatve results of Example 4 Magntude Sgn Dstance Sgn Dstance method Wth p= wth p= Dstance Mnmzaton hen Mn 373 3 6 4546 3 5 384 3 6 43 3 5 ~ ~ ~ ~ Max- In Table 3, ~ s the results of Sgn Dstance method wth p=, Magntude method, Dstance Mnmzaton and hen method, whch s unreasonable The results of ths method s the same as Sgn Dstance method wth p=, e, ll the above examples show that the results of ths method are reasonable results The proposed method can be overcome the shortcomng of "Magntude" method and "Dstance Mnmzaton" method 5 ONLUSION In ths artcle, the researchers proposed a new weghted dstance between the two fuzzy numbers and a rankng method for the fuzzy numbers The new method can effectvely rank varous fuzzy numbers, ther mages and overcome the shortcomngs of the prevous technques The calculatons of the proposed method are smpler than the other approaches Ths artcle also used comparatve examples to llustrate the advantages of the proposed method 6 REFERENES S bbasbandy and sady, Rankng of fuzzy numbers by sgn dstance, Informaton Scence 76, 45-46, 6 S bbasbandy and T Haar, new approach for rankng of trapezodal fuzzy numbers, omputer and Mathematcs wth pplcatons, 57, 43 49, 9 3 sady and Zendehnam, Rankng fuzzy numbers by dstance mnmzaton, ppl Math Model, 3, 589 598, 7 4 R Jan, Decson-makng n the presence of fuzzy varable, IEEE Trans Systems Man and ybernetsm, 6, 698 73, 976 5 R Jan, procedure for mult-aspect decson makng usng fuzzy sets, Internat J Systems Sc, 8, 7, 978 6 G ortolan and R Degan, revw of some methods for rankng fuzzy numbers, Fuzzy Sets and Systems, 5, 9, 985 7 S J hen and L Hwang, Fuzzy Multple ttrbute Decson Makng, Sprnger- Verlag, erln, 97 8 F hoobneh and H L, n ndex for orderng fuzzy numbers, Fuzzy Sets and Systems, 54, 87 94,993 9 O Das, Rankng alternatves usng fuzzy numbers: computatonal approach, Fuzzy Sets and Systems, 56, 47-5, 993 I Requena, M Delgado and J I Verdegay, utomatc rankng of fuzzy numbers wth the crteron of decson maker leant by an artfcal neural network, Fuzzy Sets and Systems, 64, 9, 994
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