An Alternative Ranking Approach and Its Usage in Multi-Criteria Decision-Making

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1 Iteratoal Joural of Computatoal Itellgece Systems, Vol., No. 3 (October, 009), 9-35 A Alteratve Rakg Approach ad Its Usage Mult-Crtera Decso-Makg Cegz Kahrama Idustral Egeerg Departmet, Istabul Techcal Uversty, Maçka, Besktas, Istabul, 34367, Turkey A. Cagr Tolga * Idustral Egeerg Departmet, Galatasaray Uversty, Ortaköy, Besktas, Istabul, 34357, Turkey E-mal: ctolga@gsu.edu.tr Abstract I the process of fuzzy decso-makg, rakg of fuzzy umbers s a ecessty. The types of fuzzy umbers are tragular, trapezodal, ad L-R type. I the lterature, there are may methods developed for rakg fuzzy umbers. These methods may produce dfferet rakg results. May of these methods ecesstate graphcal represetatos, complex ad tedous calculatos. The method developed ths paper has some advatages wth respect to the other methods both graphcal represetatos ad calculatos. Applcablty of the proposed method to mult-crtera decso-makg methods,.e. fuzzy scorg, fuzzy AHP ad fuzzy TOPSIS methods, s show the paper. Keywords: Fuzzy umbers, rakg, decso-makg, TOPSIS, scorg, AHP.. Itroducto The term fuzzy umber s employed to cope wth vague umercal quattes, lke early 9, more or less 3, several, etc. A geeral defto of a fuzzy umber s gve by Dubos ad Prade : ay fuzzy subset A = {( x, ma ( x) )} where x takes ts umber o the real le R ad μ ( x A ) [ 0,]. The membershp fucto deotes the degree of truth that A takes a specfc umber x. I may cases, the use of exteso prcple operatos o fuzzy umbers teds to be cumbersome. Thus, specal fuzzy umbers are proposed to reduce the amout of computatoal effort. Tragular fuzzy umbers, trapezodal fuzzy umbers, L-R tragular fuzzy umbers, ad L-R trapezodal fuzzy umbers are the specal umbers that have bee used mscellaeous decso models. I fuzzy set lterature, rakg fuzzy umbers s much vestgated because of ts wde usage area decsomakg. It s a ecessty to rak the obtaed fuzzy umbers a decso-makg problem. The rakg methods ca be classfed three categores. The frst category drectly trasforms each fuzzy umber to a crsp real umber ad the secod category compares a fuzzy umber to all the other fuzzy umbers to * Correspodg author: ctolga@gsu.edu.tr, Tel: Ext

2 C. Kahrama, A. C. Tolga obta ts mappg to a postve real umber. The thrd category dffers substatally from the frst two. I ths category, a method for par wse rakg or preferece for all pars of fuzzy umbers s determed ad the based o these par wse ordergs, a fal order of the fuzzy umbers s attempted 3. The vestgato o rakg fuzzy umbers bega early 70 s. May researchers have classfed fuzzy rakg methods sce 980. Frst Freelg 4 proposed fve categores for rakg methods: (a) methods usg exteded maxmum; (b) methods usg mplcato logc; (c) methods usg preferece relatos; (d) methods of drect comparso ad; (e) lgustc approaches. Bortola ad Dega 5 dealt wth the problem of rakg fuzzy subsets of the ut terval. They revewed a umber of methods suggested the lterature ad tested o a group of selected examples, where the fuzzy sets ca be o-ormal ad/or ocovex. Lee ad L 6 broadly categorzed rakg methods as mathematcal approaches versus lgustc approaches. From these classfcatos, Che et al. troduced a categorzato of fuzzy rakg methods that was composed of four major classes ad ther subclasses: (). preferece relato methods (degree of optmalty, hammg dstace, α-cut, ad comparso fucto), (). fuzzy mea ad spread method (probablty dstrbuto), (). fuzzy scorg (or drect comparso) methods (proporto to optmal, left/rght scores, cetrod dex, area measuremet), ad (v). lgustc methods (tuto, lgustc approxmato). The am of ths paper s to develop a ew rakg method whch s relable ad does ot eed tremedous arthmetc calculatos. We propose a area measuremet based method for rakg fuzzy umbers. Ths method s very easy to use wth respect to the complexty of may other methods. Our method s smlar to Fortemps ad Roubes 7 area compesato method but more compact wth much less computatoal effort. Our method s terested rakg tragular ad trapezodal fuzzy umbers whle Fortemps ad Roubes 7 method ca also rak L-R type fuzzy umbers. Amg at showg the applcato of the developed rakg method, t s used three well-kow multcrtera decso makg methods our paper. The Mult Crtera Decso Makg s a dscple amed at supportg decso makers who are faced wth makg umerous ad coflctg evaluatos. It ams at hghlghtg these coflcts ad dervg a way to come to a compromse a trasparet process. The cosdered multcrtera methods are fuzzy scorg method, fuzzy aalytc herarchy process (AHP) method, ad fuzzy TOPSIS method. The rest of the paper s orgazed as follows: Secto, a lterature revew o rakg methods s gve. The descrptos of all types of fuzzy umbers are preseted Secto 3. At the fourth secto, the proposed area-based rakg method s represeted. Ivestgato wth respect to specfc axoms o rakg fuzzy umbers s preseted Secto 5. The sxth secto belogs to the applcatos of the proposed rakg method to three decso-makg problems. We coclude the study cludg possble future works the last secto.. Lterature Revew The works o rakg fuzzy umbers the lterature ca be brefly classfed as follows: The approaches usg degree of optmalty method: Baas ad Kwakeraak 8, Watso et al. 9, Baldw ad Guld 0. The approaches usg hammg dstace: Yager, Kerre, Nakamura 3, Kolodzejczyk 4, Tra ad Duckste 5. The approaches usg α-cuts: Adamo 6, Buckley ad Chaas 7, Mabuch 8, Che ad Lu 9. Dubos ad Prade 0 depcted varous comparso fuctos for dervato of four rakg dces: a) Possblty of Domace (PD), b) Possblty of Strct Domace (PSD), c) Necessty of Domace (ND), ad d) Necessty of Strct Domace (NSD). Usg oe or more dces the rakg order ca be obtaed. Other researchers usg smlar fuctos are Tsukamoto et al. ad Delgado et al.. The rakg methods usg geeralzed mea ad stadard devato: Lee ad L 6, Cheg 3, Murakam et al. 4, Chu ad Tsao 5. The method usg some specfed fuzzy deals: McCahoe 6. The methods usg left ad rght scores: Ja 7, 8, Che 9, Che ad Hwag 30. The methods usg cetrod dex: Yager 3, Murakam et al. 4, Yog ad Q 3. The methods usg area based approaches: Yager 33, Fortemps ad Roubes 7. 0

3 A Alteratve Rakg Approach The methods usg lgustc approaches: Efstathou ad Tog 34, Tog ad Bossoe 35, Modarres ad Sad- Nezhad 36. The studes metoed above are put to Table wth the tmes cted wth respect to the rakg approaches they use. The tmes cted show how may Table. Usage frequeces of the rakg approaches. Researchers Type of FN Rakg approach Tmes cted by other researchers Tmes cted per year Order Baas ad L-R Type degree of 86 86/30=6. 5 Kwakeraak 8 optmalty Watso et.al. 9 L-R Type degree of 05 05/8= optmalty Baldw ad L-R Type degree of 3 3/9=0. 8 Guld 0 optmalty Yager Trapezodal, hammg 37 37/7=.37 8 Tragular dstace Kerre L-R Type hammg 6 6/5=.04 3 dstace Nakamura 3 L-R Type hammg 58 58/=.76 3 dstace Kolodzejczyk 4 L-R Type hammg 4 4/= 6 dstace Tra ad L-R Type hammg 8 8/5=3.6 9 Duckste 5 dstace Adamo 6 Trapezodal, α-cut 68 68/7=.5 5 Tragular Buckley ad Trapezodal, α-cut /8=. Chaas 7 Tragular Mabuch 8 Tragular α-cut 0 0/9=.05 Che ad Lu 9 L-R Type α-cut 8 8/6=.33 9 Dubos ad Prade 0 Trapezodal, comparso 89 89/4=7.87 Tragular fucto Tsukamoto et al. Trapezodal, comparso 7 7/4=0.9 5 Tragular fucto Delgado et al. L-R Type comparso 37 37/9=.95 7 fucto Lee ad L 6 L-R Type fuzzy mea ad 75 75/9= spread Cheg 3 L-R Type fuzzy mea ad 48 48/9= spread Chu ad Tsao 5 L-R Type fuzzy mea ad 5 5/5=3 spread McCahoe 6 L-R Type proporto to 5 5/0=0.5 6 deal Ja 7 Tragular left ad rght 89 89/3=.87 scores Ja 8 Trapezodal, left ad rght 80 80/30=.67 4 Tragular scores Che 9 Tragular left ad rght 4 4/= scores Che ad Hwag 30 Tragular left ad rght /8=0. 7 scores Yager 3 L-R Type cetrod dex 94 94/7=7.9 Murakam et al. 4 L-R Type cetrod dex 30 30/4=.5 0 Yager 33 L-R Type area 73 73/6= measuremet Efstathou ad Trapezodal, lgustc 0 0/7= Tog 34 Tragular Tog ad Trapezodal, lgustc Bossoe 35 Tragular Modarres ad Sad-Nezhad 36 Tragular lgustc 9 9/6=3.7 0

4 C. Kahrama, A. C. Tolga tmes those studes have bee cted the lterature from 980 to 007. Some ewer studes have ot bee lsted Table sce those mght have ot yet bee cted or dscovered by may researchers. So, we put the oes before 00 to Table. For example; Yog ad Q s 3 study s ot lsted Table. Rakg fuzzy umbers s stll a topc that receves much atteto from several researchers. Some recet works are as follows: Abbasbady ad Hajjar 37 troduce a ew approach for rakg of trapezodal fuzzy umbers based o the left ad the rght spreads at some α -levels of trapezodal fuzzy umbers. Che ad Tag 38 cosder rakg fuzzy umbers wth tegral value for the oormal p-orm trapezodal fuzzy umbers. Wag ad Lee 39 propose a method whch ca avod Chu ad Tsao s 5 problems for rakg fuzzy umbers. Sce the revsed method s based o the Chu ad Tsao s 5 method, t s easy to rak fuzzy umbers a way smlar to the orgal method. Asady ad Zedeham 40 propose a defuzzcato usg mmzer of the dstace betwee the two fuzzy umbers. 3. Fuzzy Numbers Fuzzy umbers are a partcular kd of fuzzy sets. A fuzzy umber s a fuzzy set R of real umbers set wth a cotuous, compactly supported, ad covex membershp fucto. Let X be a uversal set; a fuzzy subset A of X s defed by a fucto μ (.) A : X [0,], called membershp fucto. Throughout ths paper, X s assumed to be the set of real umbers R ad F the space of fuzzy sets. The fuzzy set A F s a fuzzy umber ff: α () α [0,] the set A = { x R: μ ( x) α} A, whch s called α -cut of A, s a covex set. () μ (.) A s a cotuous fucto. () supp( A ) = { x R: μ ( x) 0} A s a bouded set R. (v) heght A = max μ ( x x X ) h = 0 A. By codtos () ad (), each α -cut s a compact ad covex subset of R hece t s a closed terval R, a A = [ AL( a), AR( a)]. If h = we say that the fuzzy umber s ormal; we deote the set of ormal fuzzy umbers by NFN ad hereafter all fuzzy umbers used ths paper wll be NFN. Let us show a fuzzy umber A = ( a, a, a3, a4), a a a3 a4 ths form that wll be utlzed the followg deftos. I fuzzy lterature, there are three types of fuzzy umbers as deoted below: 3.. Left-Rght fuzzy umbers The fuzzy umber A s a so-called Left Rght fuzzy umber, A = ( a, a = a3, a4), LR a a = a a f ts 3 4 membershp fucto μ ( x ): R [0,] A s equal to: a x L x ( a, a] a a x a3 μ ( x) R x ( a3, a A = 4]. () a4 a3 0 others where L ad R, called the left ad the rght shape fuctos, are cotuous ad decreasg mappgs from [0,] to [0,] such that L(0) = R(0) = h> 0 ad L() = R() = 0. The value a = a 3 correspods to the peak of A. μ A () x a Fg.. Two dfferet L-R type fuzzy umbers. 3.. Trapezodal fuzzy umbers The fuzzy umber A s a so-called Trapezodal fuzzy umber A = ( a, a, a3, a4), a a a a f ts 3 4 membershp fucto μ ( x ): R [0,] A s equal to μ x a x a a x a, a ( a, a ] [ ] 3 ( ) A x = () a4 x x [ a3, a4) a4 a3 a 4 a a 3 a a =a 3 a 4 0 others x

5 A Alteratve Rakg Approach μ A ( x).0 the proposed method s smpler ad does ot eed tremedous mathematcal calculatos ad formulatos. A dex that measures the possblty of oe fuzzy umber beg greater tha aother wll be determed. That preferece dex wll be llustrated by I( ω) [0,] ad t s determed by Eq. (4): 0.0 a a a 3 a 4 x l r S + S + S I( ω) = S + S favor favor jot A B (4) Fg.. A trapezodal fuzzy umber Tragular fuzzy umbers The fuzzy umber A s a so-called tragular fuzzy umber A = ( a, a = a3, a4), a a = a a f ts 3 4 membershp fucto μ ( x ): R [0,] A s equal to μ A ( x) a x a x ( a, a] a a a4 x μ ( x) x [ a3, a A = 4) (3) a4 a3 0 others a = a 3 Fg. 3. A tragular fuzzy umber. 4. A ew approach: Area-Based Rakg of Fuzzy Numbers I ths work, a ew area-based approach s proposed for rakg fuzzy umbers. Our method ca be appled to most-used fuzzy umbers that are trapezodal fuzzy umbers ad tragular fuzzy umbers. Agast may other methods complextes, the proposed model s based o area measuremet, whch s very easy to calculate ad has a vsual uderstadablty. By the help of a smple VBA code wrtte by us, ayoe ca rak ether tragular or trapezodal fuzzy umbers. Whe compared wth the other area based rakg methods, a 4 x Usg the areas as show Fg. 4, the preferece dex ca be determed as the followg stadard form, as gve Eq.(5): μ(x) l S favor b a b B a S jot Fg. 4. Comparso of fuzzy umbers. 0, b a4 ( a4 b) ( b b a3+ a4), b a3, b< a4 ( a4+ a3 a a) + ( b4+ b3 b b) a4+ a3 b b I( ω) =, b3 a, b< a3 ( a4+ a3 a a) + ( b4+ b3 b b) ( a b3) ( a4+ a3 b b) ( b4 b3+ a a), b3< a, b4 > a ( a4+ a3 a a) + ( b4+ b3 b b), b4 a (5) ad the fuzzy preferece relato ( P KT ) of the fuzzy umbers wll be determed as followg: A f B f I( w) (0.5,] PKT ( A, B ) = A = B f I( w) = 0.5 (6) B f A f I( w) [0,0.5) Calculato of dex I( ω ) s the key factor our method. Two dfferet trapezodal fuzzy umbers are llustrated Fg. 4. As t s see, the area that s ot overlappg s amed S l favor for the left sde ad S r favor for the rght sde. S jot s the tersecto area of these fuzzy umbers. S A ad S B are the areas of the fuzzy umbers à ad B cosecutvely. b 3 A a 3 b 4 r S favor a 4 x 3

6 C. Kahrama, A. C. Tolga If the outcome of Eq. (5) s larger tha 0.5, ths meas that the fuzzy umber A s preferred to B. The followg fgures (Fgs. 5-8) ca more clarfy the calculato: μ(x) μ(x) Fg. 5. Illustrato of preferece dex calculato. l S favor a l S favor a B A S jot a b b b 3 b 4 S jot B b a 4 Fg. 6. A example for the secod row of Eq. (5) Fg. 7. A example for the fourth row of Eq. (5). Fg. 8. A example for the ffth row of Eq. (5). For example, Fg. 6 let the fuzzy umbers A ad B take the followg values, respectvely: (, 3, 4, 8) ad (, 5, 6, 7). The, from Eq. (5) we fd the preferece dex: I( ω ) = Therefore, we ca fer that the fuzzy umber A s smaller tha the fuzzy a 3 a b A a 3 b 3 a 4 r S favor b 4 r S favor x x umber B wth a possblty of Oe ca see ad calculate these values easly by VBA code gve appedx. Wrtg ths code s smple MSExcel VBA ad someoe ca easly embed ths code ay programmg code also. Our method ca be appled to both tragular ad trapezodal fuzzy umbers. Rakg more tha two fuzzy umbers s also very easy our method. At frst, par wse rakg s carred out betwee all fuzzy umbers, the based o these par wse ordergs, a fal order of the fuzzy umbers s attempted. Let us rak some tragular ad trapezodal fuzzy umbers by the methods summarzed ths secto ad our proposed approach. Let the tragular fuzzy umbers be: TRI = (0.0, 0.30, 0.50), TRI = (0.7,0.3,0.58), TRI 3 = (0.5, 0.40, 0.70). I addto, let the trapezodal fuzzy umbers be: TRA = (0.0, 0.30, 0.40, 0.50), TRA = (0.5, 0.40, 0.55, 0.70), TRA 3 = (0.0, 0.5, 0.45, 0.60). Fal fuzzy umber rakgs for the methods explaed before are llustrated Table. As see from Table all tragular fuzzy umber rakgs are the same whle trapezodal fuzzy umber rakgs are dfferet some methods. For example, Dubos ad Prade s 0 PD method, Yager s 3 weghted mea method, ad Yager s 33 area measuremet method dsplay dssmlar rakgs. Dubos ad Prade s 0 PD method ad Baas ad Kwakeraak s 8 codtoal fuzzy set Approach gve the same result formg three trapezodal fuzzy umbers are equal. Yager s 3 weghted mea method gves a dfferet rakg order for these trapezodal fuzzy umbers. Our method produces the same results for tragular fuzzy umbers wth a less computatoal effort. I addto, t gves the same rak as Yager s 33 area measuremet method ad Che s 9 fuzzy max ad m method methods for trapezodal fuzzy umbers. The rakg results by Fortemps ad Roubes 7 area compesato method are completely the same as our method s sce both methods are based o area measuremet. Our method has some advatages over Fortemps ad Roubes 7 method. They ca be couted as follows: our method cosders both jot ad dsjot areas where as Fortemps ad Roubes 7 method cosders oly dsjot areas. Wth Fortemps ad Roubes 7 method, aybody has to calculate related areas usg huge computatoal 4

7 A Alteratve Rakg Approach Table. Methods ad ther results. Methods Tragular Trapezodal Dubos ad Prade s 0 PD Method TRI3> TRI > TRI Yager s 3 Weghted Mea Method TRI3> TRI > TRI Yager s 33 Area Measuremet Approach TRI3> TRI > TRI Che s 9 Fuzzy Max ad M Method TRI3> TRI > TRI Baas ad Kwakeraak s 8 Codtoal Fuzzy Set Approach TRI3> TRI > TRI Fortemps ad Roubes 7 Area Compesato Method TRI3> TRI > TRI Kahrama ad Tolga s Area Based Approach TRI3> TRI > TRI TRA 3 = TRA = TRA TRA 3 > TRA > TRA TRA > TRA 3 > TRA TRA > TRA 3 > TRA TRA 3 = TRA = TRA TRA > TRA 3 > TRA TRA > TRA 3 > TRA effort where as our method defes a uque formula for ths computato. The crtera developed by Wag ad Kerre s 4 have become a stadard to measure the capablty of rakg methods for fuzzy umbers. Hece, the proposed rakg method s examed wth respect to Wag ad Kerre s axoms the followg. 5. Ivestgato wth respect to Wag ad Kerre s 4 axoms Wag ad Kerre 4 classfy orderg dces to three categores. I the frst class, each dex s assocated wth a mappg F from the set of fuzzy quattes to the real le R order to trasform the volved fuzzy quattes to real umbers. Fuzzy quattes are the compared accordg to the correspodg real umbers. I the secod class, referece set(s) s (are) set up ad all the fuzzy quattes to be raked are compared wth the referece set(s). I the last class, a fuzzy relato s costructed to make par-wse comparsos betwee the fuzzy quattes volved. These par-wse comparsos serve as a bass to obta the fal rakg orders. Therefore, the proposed rakg approach P KT take place wth the last class. At the begg of our vestgato, the followg remarks should be gve. S s the set of fuzzy quattes for whch the method M ca be appled ad A s a fte subset of S. A f B by M o A ca be explaed as two elemets A ad B A satsfy that A has a hgher rakg tha B whe M s appled to the fuzzy quattes A. A B by M o A ad A f B by M o A are smlarly terpreted 4, P KT s the fuzzy preferece relato o A calculated from Eq. (5). Wag ad Kerre 4 propose the followg axoms as the reasoable propertes of orderg fuzzy quattes for a orderg approach M. A For a arbtrary fte subset A of S ad A A, A f A by M o A. A For a arbtrary fte subset A of S ad ( AB, ) A, A f B ad B f A by M o A, we should have A B by M o A. A 3 For a arbtrary fte subset A of S ad 3 ( ABC,, ) A, A f B ad B f C by M o A, we should have A f C by M o A. It s easy to prove that ay rakg approach meets these three axoms 4. Hece, t s ot eeded to prove these axoms for our method. Wag ad Kerre 4 s other axoms are gve the followg: A 4 For a arbtrary fte subset A of S ad ( AB, ) A, f supp(a)> sup supp(b), we should have A f B by M o A. A' 4 For a arbtrary fte subset A of S ad ( AB, ) A, f supp(a)> sup supp(b), we should have A f B by M o A. Sce A' 4 s stroger that A 4, oly proof of A' 4 wll be justfed. Assume A = { A, A, K, A } s the set of fuzzy quattes to be raked, A, A, ad f A ad A are fuzzy umbers, they could be deoted as A = ( a, a, a3, a4), A = ( a, a, a3, a4), ad f supp(a )> sup supp(a ). 5

8 C. Kahrama, A. C. Tolga Proposto. Assume the volved fuzzy quattes are fuzzy umbers. The P KT satsfes A' 4. Proof: Whe f supp(a )> sup supp(a ), Let a = f supp(a ), a4 = supsupp(a ) ad a = f supp(a ), a4 = supsupp(a ), t s easy to obta from Eq. (5) s 5 th row, a > a4. Hece, PKT ( A, A ) = ad PKT ( A, A ) = 0. By Proposto 4.5() Wag ad Kerre 43 s study, P KT satsfes A' 4. A 5 Let S ad S be two arbtrary fte sets of fuzzy quattes whch M ca be appled ad A ad B are S S. We obta the rakg order A f B by M o S ff A f B by M o S. Proposto. P KT s cosstet o A. Proof: P KT s cosstet meas trastvty holds, so trastvty of P KT eeds to be demostrated. Proposto.. P KT s trastve,.e., for ay A, A, ad A3 A, f PKT ( A, A ) > /, ad PKT ( A, A 3) > /, the PKT ( A, A 3) > /. Proof: From Eq. (5) PKT ( A, A) > / A f A ad PKT ( A, A3) > / A f A3, the from A 3 f A f A, ad A f A3, the A f A3. Now, A f A3 PKT ( A, A3) > /. So trastvty holds for P KT.e. P KT s cosstet. Proposto 3. If a fuzzy relato P s cosstet, the the orderg approach based o P by Procedure (or Procedure ') as dcated Wag ad Kerre s 43 study satsfes A 5. Proof: Uder ther assumpto A~B PAB (, ) = PBA (, ) ad A f B P( A, B) > P( B, A) (or A p B P( A, B) < P( B, A) ),.e. the rakg order of A ad B has othg to do wth ay other volved fuzzy quatty. Proposto 4. Assume the volved fuzzy quattes are fuzzy umbers. The P KT satsfes A 5. Proof: The metoed fuzzy relato s cosstet by Proposto, ad hece accordg to Proposto 3, t satsfes A 5. A 6 Let A, B, A + C ad B+ C be elemets of S. If A f B by M o { A, B }, the A + C f B+ C by M o { } A + CB, + C. A' 6 If A f B by M o { A, B }, the A + C f B+ C A + CB, + C whe C. by M o { } Proposto 5: If the volved fuzzy quattes are fuzzy umbers, the P KT satsfes A 6. A = a, a, a, a ( =,,3). From A A, we have a > a 4, ad let a = ( a + a3), a = ( a + a3), a 3 = ( a3 + a33), a 4 = ( a4 + a34 ). A3, t s always true a 4 > a, a 3 < a, ad by the 4 th row of Eq. (5) PKT ( A + A3, A + A3) > /. Therefore, A + A3 f A + A3 by P KT o { A + A3, A + A3}. A 7 Let A, B, AC ad BC be elemets of S ad C 0. A f B by M o { A, B } mples AC f BC by M o { AC, BC }. P KT does ot satsfy A 7. Example. A = ( 0,0.5,0.5,0.7 ) A = ( 0., 0.5, 0.5, 0.6), ad A 3 = ( 0, 0.9, 0.9,). A f A ad AA3 f AA3 by P KT o { A, A } ad A A, A A respectvely. Proof: Assume ( 3 4) A f A by P KT o {, } { } 3 3 From all proofs above, oe ca see P KT satsfes all the axoms except A 7. Therefore, the developed orderg procedure s reasoable accordg to the Wag ad Kerre s 4 axoms. Now, let us see how ths method s used multcrtera decso-makg problems. 6. Applcatos to Decso-Makg Fuzzy rakg methods are ofte used fuzzy decso makg-processes ad t plays a mportat role. Our method smplfes the rakg process decsomakg. To llustrate t, we wll gve some examples of our method decso-makg methods amed fuzzy scorg method, Fuzzy Aalytc Herarchy Process (AHP) method, ad fuzzy TOPSIS method. Let the fuzzy umbers A ad B be tragular, the the basc operatos o them are gve Eqs. (7-8): A B a b, a b, a b (7) ( ) 3 3 A B = ( a + b, a + b, a + b ) (8) 3 3 6

9 A Alteratve Rakg Approach If the fuzzy umbers A ad B are trapezodal, the basc operatos o them are gve Eqs. (9-0). A B a b, a b, a b, a b (9) ( ) A B = ( a + b, a + b, a + b, a + b ) (0) Mult-crtera decso-makg usg fuzzy scorg I the example of fuzzy scorg method, Bossoe s 44 approach wll be used. The performace of a alteratve wth respect to the attrbutes ca easly be computed by the followg formula: U = w jr j () j = where w j ad r j represet the weght ad scorg of attrbutes, respectvely. After the calculato of U values by Eq. (), the rakg s obtaed by our method. For example, three alteratves of advaced maufacturg systems, FMS-, FMS-, ad FMS-3 wll be assessed wth respect to four attrbutes: egeerg effort (X ), flexblty (X ), et preset worth (X 3 ), ad tegrato ablty (X 4 ). The decso matrx s as follows: X X X3 X4 FMS far good far good E = FMS far very good bad good FMS 3 very bad very good very good very bad The weght vector s gve as; w = ( mportat, more or less mportat, umportat, very mportat) where very umportat: (0, 0, 0., 0.3); umportat: (0., 0.3, 0.3, 0.4); more or less umportat: (0.3, 0.4, 0.4, 0.5); dfferet: (0.4, 0.5, 0.5, 0.6); more or less mportat: (0.5, 0.6, 0.6, 0.7); mportat: (0.6, 0.7, 0.7, 0.8); very mportat: (0.7, 0.8, 0.8,.0). The fuzzy set wth each lgustc term s as follows: very bad: (0, 0, 0., 0.3); bad: (0., 0.3, 0.3, 0.4); more or less bad: (0.3, 0.4, 0.4, 0.5); far: (0.4, 0.5, 0.5, 0.6); more or less good: (0.5, 0.6, 0.6, 0.7); good: (0.6, 0.7, 0.7, 0.8); very good: (0.7, 0.8, 0.8,.0). The, the fuzzy utltes for the alteratves are computed as gve: 4 U = w x = (0.6, 0.7, 0.7, 0.8) (0.4, 0.5, 0.5, 0.6) 9. j j j= 4 j j j= (0.5, 0.6, 0.6, 0.7) (0.6, 0.7, 0.7, 0.8) (0., 0.3, 0.3, 0.4) (0.4, 0.5, 0.5, 0.6) (0.7, 0.8, 0.8,.0) (0.6, 0.7, 0.7, 0.8) = (.04,.48,.48,.08) U = w x = (0.6, 0.7, 0.7, 0.8) (0.4, 0.5, 0.5, 0.6) 4 3 j 3j j= (0.5, 0.6, 0.6, 0.7) (0.7, 0.8, 0.8,.0) (0., 0.3, 0.3, 0.4) (0., 0.3, 0.3, 0.4) (0.7, 0.8, 0.8,.0) (0.6, 0.7, 0.7, 0.8) = (.05,.48,.48,.4) U = w x = (0.6, 0.7, 0.7, 0.8) (0.0, 0.0, 0., 0.3) (0.5, 0.6, 0.6, 0.7) (0.7, 0.8, 0.8,.0) (0., 0.3, 0.3, 0.4) (0.7, 0.8, 0.8,.0) (0.7, 0.8, 0.8,.0) (0.0, 0.0, 0., 0.3) = (0.49, 0.7,.0,.64) The obtaed fuzzy utltes are llustrated Fgure The preferece dexes I ( ω ), I( ω ), ad I( ω 3) Scorg Method U U U3, 0,8 0,6 0,4 U3; 0,7 U3;,0 U;,48 U;,48 0, U3; 0,49 U;,05 U3;,64 U;,4 0 U;,04 0 0,5,5 U;,08 Fg. 9. The Fuzzy utltes of scorg method. 7

10 C. Kahrama, A. C. Tolga whch represet U > U, U > U, ad U > U, 3 3 respectvely ca be calculated from Eq. (5). The results are: I( ω ) = ( U < U ), I ( ω ) = ( U > U 3), I ( ω 3) = ( U > U 3) wth fereces parethess. Fal rakg ca be foud as follows: U > U > U Mult-crtera decso-makg usg fuzzy AHP Buckley 45 exteded Saaty s AHP method to corporate fuzzy comparso ratos a j. Steps of Buckley s approach are show the followg steps. Step : Cosult the decso maker ad obta the comparso matrx C whose elemets are t j = ( kj, lj, mj, j ), where all ad j are trapezodal fuzzy umbers. Step : The fuzzy weghts w ca be calculated as follows. The geometrc mea for each row s determed as: The fuzzy weght / z = t j,for all () j = w s gve as: w = z z j (3) j= Step s repeated for all the fuzzy performace scores. Step3: The fuzzy weghts ad fuzzy performace scores are aggregated. The fuzzy utltes U,, are based o U = w r, (4) j = For clarfyg the vagueess, a short example s gve. A ceramc factory s lookg for a geeral maager. There are three applcats for ths posto. The compay s also lookg for four attrbutes from these applcats. These attrbute are leadershp, problem solvg skll, commucato skll, ad expermetato. The expert opos about the relatve mportace of a par of attrbutes are show Tables 3 7. j j Table 3. Par-wse Comparso of Applcats for Leadershp. Al. Al. Al. 3 Al. (,,,) (,,,3) (,,4,4) Al. (/3,/,/,) (,,,) (,,,3) Al. 3 (/4,/4,/,/) (/3,/,/,) (,,,) Table 4. Par-wse Comparso of Applcats for Problem Solvg Skll. Al. Al. Al. 3 Al. (,,,) (/4,/3,/3,/) (,,,) Al. (,3,3,4) (,,,) (3,3,4,4) Al. 3 (/,/,,) (/4,/4,/3,/3) (,,,) Table 5. Par-wse Comparso of Applcats for Commucato Skll. Al. Al. Al. 3 Al. (,,,) (6,6,7,7) (3,3,4,4) Al. (/7,/7,/6,/6) (,,,) (/,/,,) Al. 3 (/4,/4,/3,/3) (,,,) (,,,) Table 6. Par-wse Comparso of Applcats for Expermetato Al. Al. Al. 3 Al. (,,,) (/7,/6,/6,/5) (,,,) Al. (5,6,6,7) (,,,) (,,,3) Al. 3 (/,/,,) (/3,/,/,) (,,,) Table 7. Par-wse Comparso of Four Attrbutes. X X X 3 X 4 X (,,,) (,,,3) (,,3,3) (/3,/3,/3,/3) X (/3,/,/,) (,,,) (,,,) (,,,3) X 3 (/3,/3,/,/) (/,/,,) (,,,) (/,/,/,/) X 4 (3,3,3,3) (,3,3,4) (,,,) (,,,) Va Eqs. (-4), oe ca easly fd the U values as follows: U = (0.65,0.47,0.453,0.6893), U = (0.764,0.443,0.66,0.9588), U 3 = (0.0830,0.6,0.9,0.3406). Fally these values are raked usg Eq. 6 ad t s obtaed as follows U > U > U 3 ad see o Fg. 0. 8

11 A Alteratve Rakg Approach Fuzzy AHP U U U3, 0,8 U3; 0,6 U3; 0,9 U; 0,47 U; 0,453 U; 0,443 U; 0,66 0,6 0,4 0, U3; 0,0830 U; 0,764 U3; 0,3406 U; 0, U; 0,65 0 0,5 U; 0,6893 Fg. 0. The Fuzzy utltes of Fuzzy AHP Mult-crtera decso-makg usg fuzzy TOPSIS Oe of the ma methods for mult-crtera decsomakg s fuzzy TOPSIS. Let us show how our rakg method s used ths method. Yag ad Hug 46 apply the followg steps for the methodology of Fuzzy TOPSIS. For fuzzy TOPSIS, f the fuzzy umbers A ad B are tragular, the vertex method s defed to calculate the dstace betwee them, as Eq. (5): d( A, B ) = ( a b) + ( a b) + ( a3 b3) (5) 3 If the fuzzy umbers A ad B are trapezodal, the vertex method s defed to calculate the dstace betwee them as Eq. (6) 47 : ( a b) + ( a b ) + ( a b ) + ( a b ) dab (, ) = (6) 6 Brouwer 48 summarzes may dstace calculato methods. I our work, the vertex method wll be used because of ts smplcty. The followg property s vald for both fuzzy tragular ad fuzzy trapezodal umbers: Let A, B, ad C be three tragular or trapezodal fuzzy umbers. The fuzzy umber B s closer to fuzzy umber A tha the other fuzzy umber C f, ad oly f, d( A, B ) < d( A, C ). The fuzzy mult-attrbute decso makg (MADM) ca be cocsely expressed matrx format as Eqs. (7) ad (8). Cr Cr Cr3 K Cr Al x x x 3 K x Al x x x 3 x D K = Al 3 x 3 x 3 x 33 K x 3 M M M M O M Al x m m x m x m3 K x m [ ] (7) W = w, w, K, w (8) Where x j, =,,..., m, j =,,..., ad w j, j =,, K, are tragular or trapezodal fuzzy umbers. If they are tragular: x j = ( aj, bj, cj ) ad w j = ( wj, wj, wj3), f they are trapezodal: x j = ( aj, bj, cj, dj ) ad w j = ( wj, wj, wj3, wj4). Note that x j s the performace ratg of the th alteratve wth respect to the jth attrbute, Cr j ad w j represets the weght of the jth attrbute, Cr j. The ormalzed fuzzy decso matrx deoted by R s show Eq. (9): R r (9) = j m The weghted fuzzy ormalzed decso matrx s show as Eq. (0): 9

12 C. Kahrama, A. C. Tolga v v K v j K v v v v j v K K M M O M O M V = v v K v j K v M M O M O M v m v m K v mj K v m wr wr K wr j j K wr wr wr K wr j j K wr M M O M O M = wr wr K wr j j K wr M M O M O M wr m wr m K wr j mj K wr m (0) The proposed fuzzy TOPSIS procedure ca be defed as follows: Step : Choose the lgustc ratgs ( x j, =,, K,, j =,, K, m ) for alteratves wth respect to the crtera ad approprate lgustc varables ( w j, j =,, K, ) to obta the weghts of the crtera. Step : Costruct the weghted ormalzed fuzzy decso matrx. The weghted ormalzed value V s calculated by Eq. (0). * Step 3: Idetfy postve deal ( A ) ad egatve deal ( A ) solutos. The fuzzy postve-deal soluto * (FPIS, A ) ad fuzzy egatve-deal soluto (FNIS, A ) are show as Eqs. () ad (): * * * * A = ( v, v, Kv ) = = K = K () {(max vj,,, m), j,,, } A = ( v, v, Kv ) = = K = K () {(m vj,,, m), j,,, } Our area-based rakg method s offered at ths step for rakg v j ( =,, K, m) values for each jth attrbute. For FPIS, the greatest oe; for FNIS, the lowest oe amog these fuzzy umbers should be take. Step 4: Calculate separato measures. The dstace of each alteratve from A * ad A ca be curretly calculated usg Eqs. (3) ad (4). ( ) * * j j j= d = d v, v, =,, K, m (3) ( ) j j j= d = d v, v, =,, K, m (4) Step 5: Calculate smlartes to deal soluto. Ths step solves the smlartes to a deal soluto by Eq. (5): CC = d * d + d (5) Step 6: Rak preferece order. Choose a alteratve * wth maxmum CC or rak alteratves accordg to * CC descedg order. To be more uderstadable, the example Yag ad Hug s 46 study s gog to be resolved the followg. We wll ot gve the whole example for the sake of shorteg the paper. Someoe more terested o these detals should look at Yag ad Hug s 46 paper. Table 8. Decso matrx usg fuzzy lgustc varables No. Cr. Cr. Cr. 3 Cr. 4 Cr. 5 Cr. 6 Al. H M VL VL VL H Al. M H M M VL M Al. 3 M M VL H L L Al. 4 H M VL H M M Al. 5 M M VL H L M Al. 6 VL VL VH L L H Al. 7 L M VL M L VL Al. 8 H M VL M VL M Al. 9 H H VL L VL M Al. 0 VL M VL M M H Al. VH H VH VH VH VH Al. M M VL L L M Al. 3 L M VL H L VL Al. 4 M M VL L M M Al. 5 VH H VL VH VH VH Al. 6 L H VL M L H Al. 7 H M VL M M VL Al. 8 VH VH VL H M VL Weghts H H M L H M The fuzzy lgustc varables of the decso matrx are gve Table 8. Wth the fuzzy membershp fuctos gve Table 9, the fuzzy lgustc varables are trasformed to fuzzy tragular membershp fuctos as show Table 0. 30

13 A Alteratve Rakg Approach Table 9. Trasformato for fuzzy membershp fuctos Rak Attrbute grade Membershp fuctos Very low (VL) (0.00, 0.0, 0.5) Low (L) (0.5, 0.30, 0.45) Medum (M) 3 (0.35, 0.50, 0.65) Hgh (H) 4 (0.55, 0.70, 0.85) Very hgh (VH) 5 (0.75, 0.90,.00) The frst step of fuzzy TOPSIS aalyss ad the fuzzy attrbute weght are show Table 0. Obtag the weghted fuzzy decso matrx by Eq. (36) usg Eq. (3) s the secod step of fuzzy TOPSIS method. At the thrd step the fuzzy postve-deal * soluto (FPIS, A ) ad the fuzzy egatve-deal soluto (FNIS, A ) have to be obtaed. Dfferetly from Yag ad Hug s 46 study, we use the area-based rakg approach (Eq. (5)) to fd these values. The dstace of each alteratve from A * ad A ca be calculated usg Eqs. (39) ad (40). At the ffth step, the smlartes to a deal soluto are solved by Eq. (4). Table summarzes all these steps of fuzzy TOPSIS. * Fdg the fuzzy postve-deal soluto (FPIS, A ) ad the fuzzy egatve-deal soluto (FNIS, A ) s the key factor our method. Usg Eq. (5) we rak the v j ( =,, K, m, j =,, K, ) values ad, we fd v * ad v values for each crtero. For example, the rakg for the frst crtero s obtaed: v = v 5 = v 8 > v = v 4 = v 8 = v 9 = v 7 > v = v = v = v > v > v = v = v > v = v * The, v = v 8 = (0.4,0.70,.3) ad * v = v 6 = (0.00,0.08,0.8) s obtaed from ths rakg. The other steps fuzzy TOPSIS are appled the same way as Yag ad Hug s 46 study Some calculato correctos Yag ad Hug s 46 paper Yag ad Hug 46 calculated CC values as Table. Yag ad Hug 46 gve the rak order usg Table as follows: Al. > Al.5 > Al.8 > Al.4 > Al.7 > Al.8 > Al.0 > Al.4 > Al. > Al.6 > Al.9 > Al.5 > Al. > Al.3 > Al. > Al.6 > Al.7 > Al.3. Ths rak order s obvously correct. Accordg to the results above, the true rak order s obtaed as: Al. > Al.5 > Al.8 > Al.4 > Al. > Al.8 > Al.6 > Al.9 > Al.7 > Al.4 > Al.5 > Al. > Al.0 > Al.3 > Al. > Al.6 > Al.7 > Al.3. Fuzzy TOPSIS usg our rakg approach produces the same order at frst fve alteratves but the rest of the orderg s dfferet as follows: Table 0. Fuzzy decso matrx ad fuzzy attrbute weghts No. Cr. Cr. Cr. 3 Cr. 4 Cr. 5 Cr. 6 Al. (0.55,0.70,0.85) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.00,0.0,0.5) (0.00,0.0,0.5) (0.55,0.70,0.85) Al. (0.35,0.50,0.65) (0.55,0.70,0.85) (0.35,0.50,0.65) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.35,0.50,0.65) Al. 3 (0.35,0.50,0.65) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.55,0.70,0.85) (0.5,0.30,0.45) (0.5,0.30,0.45) Al. 4 (0.55,0.70,0.85) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.55,0.70,0.85) (0.35,0.50,0.65) (0.35,0.50,0.65) Al. 5 (0.35,0.50,0.65) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.55,0.70,0.85) (0.5,0.30,0.45) (0.35,0.50,0.65) Al. 6 (0.00,0.0,0.5) (0.00,0.0,0.5) (0.75,0.90,.00) (0.5,0.30,0.45) (0.5,0.30,0.45) (0.55,0.70,0.85) Al. 7 (0.5,0.30,0.45) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.35,0.50,0.65) (0.5,0.30,0.45) (0.00,0.0,0.5) Al. 8 (0.55,0.70,0.85) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.35,0.50,0.65) Al. 9 (0.55,0.70,0.85) (0.55,0.70,0.85) (0.00,0.0,0.5) (0.5,0.30,0.45) (0.00,0.0,0.5) (0.35,0.50,0.65) Al. 0 (0.00,0.0,0.5) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.35,0.50,0.65) (0.35,0.50,0.65) (0.55,0.70,0.85) Al. (0.75,0.90,.00) (0.55,0.70,0.85) (0.75,0.90,.00) (0.75,0.90,.00) (0.75,0.90,.00) (0.75,0.90,.00) Al. (0.35,0.50,0.65) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.5,0.30,0.45) (0.5,0.30,0.45) (0.35,0.50,0.65) Al. 3 (0.5,0.30,0.45) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.55,0.70,0.85) (0.5,0.30,0.45) (0.00,0.0,0.5) Al. 4 (0.35,0.50,0.65) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.5,0.30,0.45) (0.35,0.50,0.65) (0.35,0.50,0.65) Al. 5 (0.75,0.90,.00) (0.55,0.70,0.85) (0.00,0.0,0.5) (0.75,0.90,.00) (0.75,0.90,.00) (0.75,0.90,.00) Al. 6 (0.5,0.30,0.45) (0.55,0.70,0.85) (0.00,0.0,0.5) (0.35,0.50,0.65) (0.5,0.30,0.45) (0.55,0.70,0.85) Al. 7 (0.55,0.70,0.85) (0.35,0.50,0.65) (0.00,0.0,0.5) (0.35,0.50,0.65) (0.35,0.50,0.65) (0.00,0.0,0.5) Al. 8 (0.75,0.90,.00) (0.75,0.90,.00) (0.00,0.0,0.5) (0.55,0.70,0.85) (0.35,0.50,0.65) (0.00,0.0,0.5) W (0.55,0.70,0.85) (0.55,0.70,0.85) (0.35,0.50,0.65) (0.5,0.30,0.45) (0.55,0.70,0.85) (0.35,0.50,0.65) 3

14 C. Kahrama, A. C. Tolga Table. Fuzzy TOPSIS Aalyss No. v v v 3 v 4 v 5 v 6 d + d CC Al. (0.3, 0.54, 0.96) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0, 0.03, 0.5) (0, 0.08, 0.8) (0.9, 0.39, 0.74) Al. (0.9, 0.39, 0.74) (0.3, 0.54, 0.96) (0., 0.8, 0.56) (0.05, 0.7, 0.39) (0, 0.08, 0.8) (0., 0.8, 0.56) Al. 3 (0.9, 0.39, 0.74) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.08, 0.3, 0.5) (0.08, 0.3, 0.5) (0.05, 0.7, 0.39) Al. 4 (0.3, 0.54, 0.96) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.08, 0.3, 0.5) (0.9, 0.39, 0.74) (0., 0.8, 0.56) Al. 5 (0.9, 0.39, 0.74) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.08, 0.3, 0.5) (0.08, 0.3, 0.5) (0., 0.8, 0.56) Al. 6 (0, 0..08, 0.8) (0, 0.08, 0.8) (0.6, 0.5, 0.87) (0.0, 0., 0.7) (0.08, 0.3, 0.5) (0.9, 0.39, 0.74) Al. 7 (0.08, 0.3, 0.5) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.05, 0.7, 0.39) (0.08, 0.3, 0.5) (0, 0.06, 0.) Al. 8 (0.3, 0.54, 0..96) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.05, 0.7, 0.39) (0, 0.08, 0.8) (0., 0.8, 0.56) Al. 9 (0.3, 0.54, 0.96) (0.3, 0.54, 0.96) (0, 0.06, 0.) (0.0, 0., 0.7) (0, 0.08, 0.8) (0., 0.8, 0.56) Al. 0 (0, 0.08, 0.8) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.05, 0.7, 0.39) (0.9, 0.39, 0.74) (0.9, 0.39, 0.74) Al. (0.4, 0.7,.3) (0.3, 0.54, 0.96) (0.6, 0.5, 0.87) (0., 0.3, 0.6) (0.4, 0.7,.3) (0.6, 0.5, 0.87) Al. (0.9, 0.39, 0.74) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.0, 0., 0.7) (0.08, 0.3, 0.5) (0., 0.8, 0.56) Al. 3 (0.08, 0.3, 0.5) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.08, 0.3, 0.5) (0.08, 0.3, 0.5) (0, 0.06, 0.) Al. 4 (0.9, 0.39, 0.74) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.0, 0., 0.7) (0.9, 0.39, 0.74) (0., 0.8, 0.56) Al. 5 (0.4, 0.7,.3) (0.3, 0.54, 0.96) (0, 0.06, 0.) (0., 0.3, 0.6) (0.4, 0.7,.3) (0.6, 0.5, 0.87) Al. 6 (0.08, 0.3, 0.5) (0.3, 0.54, 0.96) (0, 0.06, 0.) (0.05, 0.7, 0.39) (0.08, 0.3, 0.5) (0.9, 0.39, 0.74) Al. 7 (0.3, 0.54, 0.96) (0.9, 0.39, 0.74) (0, 0.06, 0.) (0.05, 0.7, 0.39) (0.9, 0.39, 0.74) (0, 0.06, 0.) Al. 8 (0.4, 0.7,.3) (0.4, 0.7,.3) (0, 0.06, 0.) (0.08, 0.3, 0.5) (0.9, 0.39, 0.74) (0, 0.06, 0.) A * v =(0.4, 0.7,.3) =(0.4, 0.7,.3) A - v =(0, 0.08, 0.8) v =(0, 0.08, 0.8) v =(0.6, 0.5, 0.87) 3 _ v =(0, 0.06, 0.) 3 v =(0., 0.3, 0.6) 4 _ v =(0, 0.03, 0.5) 4 v =(0.4, 0.7,.3) 5 6 =(0.6, 0.5, 0.87) v =(0, 0.08, 0.8) 5 v 6 =(0, 0.06, 0.) W= {(0.55, 0.70, 0.85), (0.55, 0.70, 0.85), (0.35, 0.50, 0.65), (0.5, 0.30, 0.45), (0.55, 0.70, 0.85), (0.35, 0.50, 0.65)} Table. CC values calculated by Yag ad Hug 46 Alteratve No CC Order Table. (Cotued) CC values calculated by Yag ad Hug 46 Alteratve No CC Order Al. > Al.5 > Al.8 > Al.4 > Al. > Al.6 > Al.9 > Al.7 > Al.4 > Al.5 > Al.8 > Al. > Al.0 > Al.3 > Al. > Al.6 > Al.3 > Al Coclusos I fuzzy decso-makg problems, fuzzy rakg s oe of the most-researched areas. I ths study, fuzzy umber rakg procedures are vestgated. The, a area-based rakg approach s offered for the shortages of the other rakg approaches. Ths ew approach ca 3

15 A Alteratve Rakg Approach be appled to most-used decso-makg procedures that are fuzzy scorg, fuzzy AHP, ad fuzzy TOPSIS. Geerally, our approach produced same results however; fuzzy TOPSIS, dfferet results are reached. For further study, applcato of that area-based fuzzy rakg approach to other decso-makg approaches ca be vestgated. A formula for L-R type fuzzy umbers ca be studed wth the same logc of the formula developed for tragular ad trapezodal fuzzy umbers. Refereces. D. Dubos ad H. Prade, Operatos o fuzzy umbers, Iteratoal Joural of Systems Scece 9 (978) S. J. Che, C. L. Hwag, ad F. P. Hwag, Fuzzy Multple Attrbute Decso Makg; Methods ad Applcatos (Sprger, Berl, 99). 3. V. Cross ad M. Setes, A geeralzed model for rakg fuzzy sets, Proc. IEEE World Cogress o Computatoal Itellgece (998), pp A. N. S. Freelg, Fuzzy sets ad decso aalyss, IEEE Trasactos o Systems, Ma, ad Cyberetcs 0 (980) G. Bortola ad R. Dega, A revew of some methods for rakg fuzzy subsets, Fuzzy Sets ad Systems 5 (985) E. S. Lee ad R. L. L, Comparso of fuzzy umbers based o the probablty measure of evets, Computer ad Mathematcs wth Applcatos 5 (988) P. Fortemps, ad M. Roubes, Rakg ad defuzzcato methods based o area compesato, Fuzzy Sets ad Systems 8 (996) S. M. Baas ad H. Kwakeraak, Ratg, ad rakg of multple aspect alteratve usg fuzzy sets, Automatca 3 (977) S. R. Watso, J. J. Wess, ad M. L. Doell, Fuzzy decso aalyss, IEEE Trasactos o Systems, Ma, ad Cyberetcs 9 (979) J. F. Baldw ad N. C. Guld, A model for mult-crteral decso-makg usg fuzzy logc, Workshop of Fuzzy Reasog (Lodo, Quee Mary College, 978).. R. R. Yager, O choosg betwee fuzzy subsets, Kyberetes 9 (980) E. E. Kerre, The use of fuzzy set theory electrocardologcal dagostcs, Approxmate reasog Decso aalyss, eds.: M. M. Gupta ad E. Sachez (North-Hollad, Amsterdam, 98) pp K. Nakamura, Preferece relato o a set of fuzzy utltes as a bass for decso makg, Fuzzy Sets ad Systems 0 (986) W. Kolodzejczyk, Orlovsky s cocept of decso makg wth fuzzy preferece relato further results, Fuzzy Sets ad Systems 9 (986) L. Tra, ad L. Duckste, Comparso of fuzzy umbers usg dstace measure, Fuzzy sets ad systems 30 (00) J. M. Adamo, Fuzzy decso trees, Fuzzy sets ad systems 4 (980) J. J. Buckley ad S. Chaas, A fast method of rakg alteratves usg fuzzy umbers (Short commucatos), Fuzzy sets ad systems 30 (989) S. Mabuch, A approach to the comparso of fuzzy subsets wth a α-cut depedet dex, IEEE Trasactos o Systems, Ma, ad Cyberetcs 8 (988) L. H. Che ad H. W. Lu, A approxmate approach for rakg fuzzy umbers based o left ad rght domace, A Iteratoal Joural of Computers ad Mathematcs wth Applcatos 4 (00) D. Dubos ad H. Prade, Rakg of fuzzy umbers the settg of possblty theory, Iformato Sceces 30 (983) Y. Tsukamoto, P. N. Nkforuk ad M. M. Gupta, O the comparso of fuzzy sets usg fuzzy choppg, Cotrol Scece ad Techology for Progress of Socety, ed. H. Akash (Pergamo Press, New York, 983) pp M. Delgado, J. Verdegay, ad M. A. Vlla, A procedure for rakg fuzzy umbers usg fuzzy relatos, Fuzzy Sets ad Systems 6 (988) C. H. Cheg, A ew approach for rakg fuzzy umbers by dstace method, Fuzzy Sets ad Systems 95 (998) S. Murakam, S. Maeda ad S. Iamura, Fuzzy decso aalyss o the developmet of cetralzed regoal eergy cotrol system, IFAC Symposum o Fuzzy Iformato, Kowledge Represetato ad Decso Aalyss, (983) pp T. C. Chu ad C. T. Tsao Rakg fuzzy umbers wth a Area betwee the Cetrod Pot ad Orgal Pot, A Iteratoal Joural of Computers ad Mathematcs wth Applcatos 43 (00) C. McCahoe, Fuzzy Set Theory Appled to Producto ad Ivetory Cotrol Ph. D. Thess, (Kasas State Uversty, 987). 7. R. Ja, Decso makg the presece of fuzzy varables, IEEE Trasactos o Systems, Ma, ad Cyberetcs 6 (976) R. Ja, A procedure for mult-aspect decso makg usg fuzzy sets, Iteratoal Joural of System 8 (977) S. H. Che, Rakg fuzzy umbers wth maxmzg set ad mmzg set, Fuzzy Sets ad Systems 7 (985) S. J. Che ad C. L. Hwag, Fuzzy scorg of fuzzy umber-a drect comparso dex, Fuzzy Multple 33

16 C. Kahrama, A. C. Tolga Attrbute Decso Makg; Methods ad Applcatos, eds. S. J. Che, C. L. Hwag ad F. P. Hwag (Sprger, Berl, 989) pp R. R. Yager, O a geeral class of fuzzy coectves, Fuzzy Sets ad Systems 4 (980) D. Yog ad L. Q, A TOPSIS-based cetrod-dex rakg method of fuzzy umbers ad ts applcato to decso makg, Cyberetcs ad Systems: A Iteratoal Joural 36 (005) R. R. Yager, A procedure for orderg fuzzy subsets of the ut terval, Iformato Sceces 4 (98) J. Efstathou ad R. Tog, Rakg fuzzy sets usg lgustc preferece relatos, Proceedgs of the 0th Iteratoal Symposum o Multple-Valued Logc (Northwester Uversty, Evasto, 980) pp R. M. Tog ad P.P. Bossoe, Lgustc solutos to fuzzy decso problems, TIMS/Studes the Maagemet Scece, ed.: H. J. Zmmerma (Elsever, North Hollad, 984) pp M. Modarres ad S. Sad-Nezhad, Rakg fuzzy umbers by preferece rato, Fuzzy Sets ad Systems 8 (00) S. Abbasbady, ad T. Hajjar, A ew approach for rakg of trapezodal fuzzy umbers, Computers & Mathematcs wth Applcatos 57 (009) C. C. Che, ad H.C. Tag, Rakg oormal p-orm trapezodal fuzzy umbers wth tegral value, Computers & Mathematcs wth Applcatos 56 (008) Y. J. Wag, ad H. S. Lee, The revsed method of rakg fuzzy umbers wth a area betwee the cetrod ad orgal pots, Computers & Mathematcs wth Applcatos 55 (008) B. Asady, ad A. Zedeham, Rakg fuzzy umbers by dstace mmzato, Appled Mathematcal Modellg, 3 (007), X. Wag ad E. E. Kerre, Reasoable propertes for the orderg of fuzzy quattes (I), Fuzzy Sets ad Systems 8 (00) X. Wag, E. E. Kerre, O the classfcato ad the depedeces of the orderg methods, Fuzzy Logc Foudatos ad Idustral Applcatos, ed. D. Rua (Kluwer Academc Publshers, Dordrecht, 996), pp X. Wag ad E. E. Kerre, Reasoable propertes for the orderg of fuzzy quattes (II), Fuzzy Sets ad Systems 8 (00) P. P. Bossoe, A fuzzy set based lgustc approach: Theory ad applcatos, Approxmate Reasog Decso Aalyss, eds. M.M. Gupta ad E. Sachez, (North-Hollad, 98) pp J. J. Buckley, Fuzzy herarchcal aalyss, Fuzzy Sets ad Systems 7 (985) T. Yag ad C. C. Hug, Multple-attrbute decso makg methods for plat layout desg problem, Robotcs ad Computer-Itegrated Maufacturg 3 (007) D. F. L, Compromse rato method for fuzzy multattrbute group decso makg, Appled Soft Computg 7 (007) R. K. Brouwer, Clusterg feature vectors wth mxed umercal ad categorcal attrbutes, Iteratoal Joural of Computatoal Itellgece Systems (008) Appedx VBA Code for rakg fuzzy umbers s gve below: Sub Preferece_Idex() Dm A, A, A3, A4, B, B, B3, B4 Dm I, P, P, P3 Worksheets("fuzzy preferece").actvate A = Cells(, ).Value A = Cells(, ).Value A3 = Cells(, 3).Value A4 = Cells(, 4).Value B = Cells(, 5).Value B = Cells(, 6).Value B3 = Cells(, 7).Value B4 = Cells(, 8).Value If B > A4 Or B = A4 The I = 0 ElseIf B < A4 Ad B > A3 Or B < A4 Ad B = A3 The I = (((A4 - B) ^ ) / (B - B - A3 + A4)) / ((A4 + A3 - A - A) + (B4 + B3 - B - B)) Ed If If B3 < A Ad B4 > A The I = ((A4 + A3 - B - B) - (((A - B3) ^ ) / (B4 - B3 + A - A))) / ((A4 + A3 - A - A) + (B4 + B3 - B - B)) ElseIf B < A3 Ad B3 > A Or B < A3 Ad B3 = A The I = (A4 + A3 - B - B) / ((A4 + A3 - A - A) + (B4 + B3 - B - B)) Ed If If B4 < A Or B4 = A The I = 34

17 A Alteratve Rakg Approach Ed If If I > 0.5 The P = I ElseIf I < 0.5 The P3 = I Else P = I Ed If 'Output calculated fuzzy preferece dex Cells(5, 3).Value = P Cells(5, 4).Value = P Cells(5, 5).Value = P3 Ed Sub 35