New centroid index for ordering fuzzy numbers

Transcription

1 Interntionl Sientifi Journl Journl of Mmtis New entroi inex for orering numers Tyee Hjjri Deprtment of Mmtis, Firoozkooh Brnh, Islmi z University, Firoozkooh, Irn Emil: tyeehjjri@yhooom strt Rnking numers is n importnt tool in eision proess In eision nlysis, quntities re use to esrie performne of lterntive in moeling rel-worl prolem Most of rnking proeures propose so fr in literture nnot isriminte quntities some re ounterintuitive s numers re represente y possiility istriutions, y my overlp with eh or, hene it is not possile to orer m It is true tht numers re frequently prtil orer nnot e ompre like rel numers whih n e linerly orere So fr, more thn rnking inies hve een propose sine 976 while ory of sets ws first introue y Zeh The most ommonly use pprohe for rnking numers is rnking inies se on entroi of numers Ever sine Yger presente entroi onept in rnking tehniques using entroi onept hve een propose investigte In pper y Cheng, entroi-se istne metho presente The metho utilize Eulien istnes from origin to entroi point of eh numers to ompre rnk numers Chu Tso foun tht istne metho oul not rnk numers orretly if y re negtive refore, suggeste using re etween entroi point origin to rnk numers Deng et l utilize entroi point of numer presente new re metho to rnk numers with rius of gyrtion (ROG) points to overome rwk of Cheng's istne metho Tso's re metho when some numers hve sme entroi point However, ROG metho nnot rnk negtive numers Reently, Wng et l pointe out tht entroi point formuls for numers provie y Cheng re inorret hve le to some mispplition suh s y Chu Tso, Pn Yeh Deng et l They presente orret entroi formule for numers justifie m from viewpoint of nlytil geometry Neverless, min prolem, out rnking numers methos, whih use entroi point, ws remine In 8, Wng Lee revise Chu Tso's metho suggeste new pproh for rnking numers se on Chu Tso's metho in wy to similr originl point However, re is shortoming in some situtions In, sy Hjjri improve heng's istne metho fterwr, Pni Bushn Ro et l presente new metho for rnking numers se on irumenter of entrois use n inex of optimism to reflet eision mker's optimisti ttitue lso n inex of molity tht represente neutrlity of eision mker However, re re some weknesses ssoite with se inies This pper proposes new entroi inex rnking metho tht is ple of effetively rnking vrious types of numers The ontents herein present severl omprtive exmples emonstrting usge vntges of propose entroi inex rnking metho for numers Inex Terms Centroi points, Comprison, Deisionmking, Defuzzifition, Fuzzy numers, Orering; I PRELIMINRIES In this setion, we riefly review some si onepts of generlize numers some existing methos for rnking numers We will ientify nme of numer with tht of its memership funtion for simpliity Throughout this pper, R sts for set of ll rel numers, E sts set of numers, " " expresses numer (x) for its memership funtion, x R Bsi nottions efinitions generlize numer " " is suset of rel line R, with memership funtion ( x) : R [, ] suh tht []: L( x) x x µ ( x) ( ) U ( x) x orwise, Where is onstnt, :,, :,, re L [ ] [ ] [ ] [ ] U two stritly monotonilly ontinuous mpping from R to lose intervl [,] If, n is norml numer; orwise, it is trpezoil numer is usully enote y (,,,, ) or (,,, ) if In prtiulr, when, trpezoil numer is reue to tringulr numer enote y (,,, ) or (,, ) if

2 Interntionl Sientifi Journl Journl of Mmtis Therefore, tringulr numers re speil ses of trpezoil numers We show set of generlize numers y F (R) or for simpliity y F (R) Sine L (x) U (x) re oth stritly monotonilly ontinuous funtions, ir inverse funtions exist shoul e ontinuous stritly monotonil Let L :,, U : [, ] [, ] L (x) U (x), [ ] [ ] e inverse funtions of L U shoul e integrle on lose intervl [, ] In or wors, oth respetively Then L ( y )y U ( y )y shoul exist In se of trpezoil numer, inverse funtions L U n e nlytilly expresse s L ( y) + ( ) y /, y U ( y) ( ) y /, y L (x) U (x) The funtions () re lso lle left right sie of numer [] In this pper, we ssume tht + () respetively ( x)x < + useful tool for eling with numers re ir α -uts The α -ut of numer is non set efine s α {x R : ( x) α }, for α (] ( l α (,] α ) oring to efinition of numer, it is seen t one tht every α -ut of numer is lose intervl Hene, for numer, we hve (α ) [L (α ), U (α )] where L (α ) inf {x R : ( x) α }, U (α ) sup{x R : ( x) α } If left right sies of numer re stritly monotone, s it is esrie, L U re inverse funtions of L (x) U (x), respetively The set of ll elements tht hve nonzero egree of memership in, it is lle support of, ie Supp ( ) x X ( x) (4) The set of elements hving lrgest egree of memership in, it is lle ore of, ie { Core ( ) x X } ( x) sup ( x) x X (5) In following, we will lwys ssume tht is ontinuous oune support Supp ( ) The strong support of shoul e Supp ( ) [, ] s : [,] [,] is s inresing s() Definition funtion reuing funtion if is s () We sy tht s is regulr funtion if s(r )r Definition If is numer with r-ut representtion, L (α ), U (α ) s is reuing [ ] funtion, n vlue of efine y (with respet to s); it is Vl ( ) s(α )[ U (α ) + L (α )]α (6) Definition If is numer with r-ut representtion L (α ), U (α ), s is reuing [ ] funtion n miguity of efine y (with respet to s) is m( ) s(α )[ U (α ) + L (α )]α (7) B rithmeti opertion In this susetion, rithmeti opertion etween two generlize trpezoil numers, efine on universl set of rel numers R, re reviewe [] Let (,,, ; ) (,,, ; ) e two generlize trpezoil numers n +, +, +, + ; min(, ),,, ; min(, ) (λ, λ, λ, λ ; ) λ > λ (λ, λ, λ, λ; ) λ < II REVIEW ON SOME CENTROID-INDEX FOR ORDERING FUZZY NUMBERS [] ws first reserher to propose entroiinex rnking metho to lulte vlue x for numer s x ( x) f ( x)x f ( x)x where (8) (x) is weighting funtion mesuring importne of vlue memership funtion of x f enotes numer

3 Interntionl Sientifi Journl Journl of Mmtis When ( x) x, x eomes geometri vlue Center of Grvity (COG) with x xf ( x)x f ( x)x (9) x The lrger vlue is of R ( ) x + y, x L L () U U L () U re respetive right left memership funtion of y, U L, re U L respetively The lrger vlue R ( ) etter rnking will e of inverse of is of Chu Tso [7] foun tht istne pproh y Cheng [6] h shortomings Hene to overome prolems, Ch Tso [7] propose new rnking inex funtion S ( ) x y, where x is efine in y yl ( y ) x + y U ( y ) y L ( y ) x + U ( y )y () U ( y ) x + L ( y )y (5) n efine s [4] (6) x ( + ) ( + ) (7) y + ( + ) ( + ) In speil se, when, trpezoil numer is reue to tringulr numer formuls (8) (9) will e simplifie s follows, respetively [ + + ] y x (8) (9) S ( ) etter rnking will e of In some speil ses, Ch Tso's [7] pproh lso hs sme shortoming of Cheng's [6] Ch Tso's entroi-inex re s follows For numers, B, C, B, C oring to R( ) x + y, wherey sme results re otine, tht is, if < B < C n < B < C This is lerly inonsistent with Cheng's entroi-inex mmtil logi For Chu Teso's entroi-inex S ( ) x y, if x, n vlue of S ( ) x y, is onstnt zero In or wors, numers with entroi (, y ) In this setion entroi point of numer orrespons to x vlue on horizontl xis y vlue on vertil xis The entroi point The lrger vlue is of y y ( x)x + x + U ( x) x, (4) III NEW CENTROID-INDEX METHODS yu ( y ) x + y L ( y )y Cheng [6] ritrry trpezoil numer (,,, ; ), entroi point ( x, y ) is L xl ( x) x + xx + xu ( x) x For xl ( x)x + xx + xu ( x)x, L ( x)x + x + U ( x)x y ( y)x + y ( y)y y ( y)x + ( y)y () where x etter rnking of Cheng [6] use entroi-se istne pproh to rnk numers For trpezoil numer (,,, ; ), istne inex n e efine s In stuy onute y Wng et l [4], entroi formule propose y Cheng [6] is shown to e inorret Therefore to voi mny mispplition, Wng et l [4] presente orret entroi formule s (, y ), re onsiere sme This is lso oviously unresonle for numer x is s efine [4]: xl ( x) x + xx + xu ( x) x L y ( x, y ) () ( x)x + x + U ( x)x yu ( y ) x + y L ( y )y U ( y ) x + L ( y )y () For trpezoil numer entroi point ( x, y ) (,,, ; ), is efine s in \ite{ywjw}: x ( + ) ( + ) y + ( + ) ( + ) () ()

4 Interntionl Sientifi Journl Journl of Mmtis Sine tringulr numers re speil ses of trpezoil numer with for ny tringulr numers with piewise liner memership funtion, its entroi n e etermine y [ + + ] y x (4) (5) Definition For generlize trpezoil numer (,,, ; ) with entroi point ( x, y ), entroi-inex ssoite with rnking B if n B, where δ, δ B re spres of B respetively R( ) + δ > R(B) + δ B R( ) + δ < R(B) + δ B Cse () In suh ses we use Definitions 4 4 to rnk numers s Definition 4 lone is not suffiient to isriminte in ll ses, tht is, if B, n if Iαβ ( ) > Iαβ ( B ), Iαβ ( ) < Iαβ ( B ), n B, Remrk For two ritrry trpezoil numers B, we hve is efine s Iαβ β ( x + y ) where α, β [,] R( + B) R( ) + R( B) + ( β ) Iα I αβ is (6) molity whih represents importne of e ntrl vlue ginst extreme vlues x, y I αβ Here, β represent β is weight ssoite with extreme vlues x y Moreover, Iα αy + ( α ) x is inex of optimism whih weight of entrl vlue represents egree of optimism of eision-mkers If α, we hve pessimisti eision mker's view point whih is equl to istne of entroi point from Y -xis If α, we hve optimisti eision mker's view point whih is equl to istne of entroi point from X -xis, when α 5, we hve moerte eision mker's view point is equl to men of entroi point from Y X xis The lrger vlue of α is, higher egree of eision mker The inex of optimism is not lone suffiient to isriminte numers s this uses only extreme of iumenterr of entrois Hene, we upgre this y using n inex Definition For generlize trpezoil numer (,,, ; ) with entroi point ( x, y ), rnking funtion of trpezoil numer whih mps set of ll numers to set of rel numers is efine s R ( ) x + y,, whih is Eulien istne from entroi point originl point Using ove efinitions we efine rnking etween numers s follows: let B re two numers, n () R( ) > R( B) if only if B, () () n R( ) < R( B) if only if B, if R( ) R( B) n in this se isrimintion of numers is not possile Cse () let B re two symmetri tringulr numers with sme ore, if Remrk 4 For two symmetri tringulr numers B, with sme ore B, iff δ < δb Hene R(risp``numer ) R(tringulr ``simmetri`` ``numer ) For exmple, onsier risp numer (,,) two symmetri numers B (,,), C (,,) s we sw y pplying Rezvni's pproh results will e R( ) R( B) R(C ), rnking orer is B C However, s δ δ B, δ C, rnking orer is R( ) > R( B) > R(C ) In ition, to ompre risp numer (,,) two symmetri numers B (,,), C (,,),we hve sme result Remrk 5 For ll numers, B, C D we hve () () () (4) Exmple B n C B C B n B B C ~ B n C ~ B C B, C D n C B D 6 The two numers (,,4,5) B (,,5) use in this exmple re pte from Chen Sngunst's [5] Fig shows grphs of two numers The results otine y propose pproh or pprohes re shown in Tle It is worth mentioning tht Yger's [6] pproh, Cheng's [6] pproh, Chu Tso's [7] pproh, Chen Sngunst's [5] nnot ifferentite B, tht is, ir rnking re lwys sme, ie ~ B Note tht rnking B otine y Murkmi et l's [7] pproh, Chen Chen's [] pproh Chen Chen's [] pproh, re thought of s unresonle not onsistent with humn intuition ue to ft tht

5 Interntionl Sientifi Journl Journl of Mmtis enter of grvity of is lrger thn enter of grvity of B on Y -xis Fig Fuzzy numer B in exmple 6 IV CONCLUSIONS In spite of mny rnking methos, no one n rnk numers with humn intuition onsistently in ll ses Here, we pointe out shortoming of some reent entroi-inex methos presente new entroi-inex metho for rnking numers The propose formule re simple hve osistent expression on horizontl xis vertil xis lso e use for some espeil ses in mny entroiinex methos The pper herein presents severl omprtive exmples to illustrte vliity vntges of propose entroi-inex rnking metho It shows tht rnking orer otine y propose entroi-inex rnking metho is more onsistent with humn intuitions thn existing methos Furrmore, propose rnking metho n effetively rnk mix of vroius types of numers, whih is nor vntge of propose metho over or existing rnking pprohes CKNOWLEDGMENT This work ws supporte in prt y grnt from Firoozkooh Brnh of Islmi z University REFERENCES [] L Zeh, Fuzzy sets, Informtion Control 8 (965) 8-5 [] R Jin, Deision-mking in presene of vrile, IEEE Trns Systems Mn Cyernet, 6 (976) [] R Jin, proeure for multi-spet eision mking using sets Internt J Systems Si, 8 (977) -7 [4] X Wng E E Kerre, Resonle properties for orering of quntities (I), Fuzzy Sets Syst 8 () [5] X Wng E E Kerre, Resonle properties for orering of quntities (II), Fuzzy Sets Syst 8 () [6] C H Cheng, new pproh for rnking numers y istne metho, Fuzzy Sets Syst 95 (998) 7-7 [7] T Chu C Tso, Rnking numers with n re etween entroi point orginl point, Comput Mth ppl 4 () -7 [8] X W Liu S L Hn, Rnking numers with preferene weighting funtion expettion, Comput Mth ppl 49 (5) [9] Y Deng Q Liu, TOPSIS-se entroi inex rnking metho f numers its pplition in eision-mking, yerneti Systems, 6 (5) [] Y Deng, ZF Zhu Q Liu, Rnking numers with n re etho using of gyrtion, Comput Mth ppl 5 (6) 7-6 [] S J Chen S M Chen, Fuzzy risk nlysis se on rnking of generlize trpezoil numers, pplie Intelligene 6 (7) - [] MS Gri MT Lmt, moifition of inex of Liou Wng for rnking numers,intj Uner Fuzz Know Bse Syst 4 (4)(7) [] YJ Wng HSh Lee, The revise metho of rnking numers with n ere etween entroi originl points, Comput Mth ppl 55 (8) - 4 [4] S sy T Hjjri, new pproh for rnking of trpezoil numers, Comput Mth ppl 57 (9) 4-49 [5] T Hjjri S sy, note on " The revise metho of rnking LR numer se on evition egree", Expert Syst with pplitions, 8 () [6] S sy T Hjjri, n improvement on entroi point metho for rnking of numers, J Si IU 78 () 9-9 [7] T Hjjri, Rnking of numers se on miguity egree, ustrlin Journl of Bsi pplie Sienes 5 () 6-69 [8] T Hjjri, On evition egree methos for rnking numers ustrlin Journl of Bsi pplie Sienes, 5 () [9] LQ Dt, FY Vinent SY hou, n improve rnking metho for numers se on entroiinex, Interntionl Fuzzy Systems, 4 () () 4-49 [] FY Vinent LQ Dt, n improve rnking metho for numers with integrl vlues, ppl Soft Comput, 4 (4) 6-68 [] RR Yger, On generl lss of onnetive, Fuzzy Sets Syst 4 (98) 5-4 [] D Duios H Pre, Opertions on numers, Internt J System Si 9 (978) 6-66 [] S M Chen J H Chen, Fuzzy risk nlysis se on rnking generlize numers with ifferent heights ifferent spres, Expert Systs with pplitions 6 (9)

6 Interntionl Sientifi Journl Journl of Mmtis [4] Y-M Wng, J-B Yng, D-L Xu, K-S Chin, On entrois numers, Fuzzy Sets Syst 57 (6) [5] S M Chen K Sngunst, nlysing risk se on new rnking generlize numers with ifferent heights ifferent spres, Expert Systs with pplitions 8 () 6-7 [6] R R Yger, On hoosing etween susets, Kyernetes, 9 (98) 5-54 [7] S Murkmi, H Me, S Immur, Fuzzy eision nlysis on evelopment of entrlize regionl energyontrol system, Proeeing of IFC Symposium Mrseille,(98) 6-68 [8] P Phni Bushn Ro R Shnkr, Rnking numers with istne metho using irumenter of entrois n inex of molity, vne in Fuzzy Systems, () Tyeeh Hjjri is ssistnt professor of pplie mmtis t Firoozkooh rnh of IU (Irn) She reeive her PhD in pplie mmtis- numeril nlysis Her min reserh interests re in res of pplie mmtis suh s numeril nlysis, systems its pplitions, neurl networks, risk nlysis She is referee of severl interntionl journls in frme of pplie mmtis systems