A Fuzzy Weight Representation for Inner Dependence Method AHP

Transcription

1 A Fuzzy Wegh Represeao for Ier Depeece Meho AHP Sh-ch Ohsh 2 Taahro Yamao Heyu Ima 2 Faculy of Egeerg, Hoa-Gaue Uversy Sapporo, JAPAN 2 Grauae School of Iformao Scece a Techology, Hoao Uversy Sapporo, JAPAN Emal: {ohsh,yamao}@el.hoa-s-u.ac.p, ma@s.houa.ac.p Absrac AHP (Aalyc Herarchy Process) has bee ely use ecso mag. Ier epeece meho AHP s oe echque eve case of crera have epeecy. Hoever usg orgal AHP or Ier epeece meho, he resuls ofe lose relably because he comparso marx oes o alays have suffce cossecy. I hese cases, fuzzy represeao for eghg crera a aleraves usg resuls from a sesvy aalyss s useful. I hs paper, e prese alerave overall eghs by employg some assumpos. Sce a ea of less ambguy s employe, he resuls sho ho er epeece AHP has fuzzess he he comparso marx s o suffcely cosse.. Keyors Decso mag, AHP (Aalyc Herarchy Process), Fuzzy ses, Ier Depeece, Sesvy aalyss. Irouco AHP (Aalyc Herarchy Process) as propose by Saay T.L. 977 [], [2]. The meho has bee popular a ely use he oma of ecso mag, sce ca clue vagares such as humas feelgs. I ao, ca be evelope o ANP (Aalyc Neor Process) moels. Usually ormal AHP mus assume epeecy amog crera, alhough s ffcul o choose eough epee crera. Ier epeece meho AHP[0] s oe echque of solvg hs of problem eve case of crera have epeecy. Hoever, he comparso marx ofe oes o have eough cossecy he AHP or Ier epeece meho s use sce, for sace, a problem may coa oo may crera for ecso mag. I hese cases, e coser ha asers from ecso-maers (.e. compoes of he comparso marx) have ambguy or fuzzess. For resolvg hs ype of problem, fuzzy recprocal compoes have bee propose as compoes of he aa marx some research [2]. I hs paper, e coser ha eghs shoul also have ambguy or fuzzess. Therefore, s ecessary o represe hese eghs by use of fuzzy ses. Sesvy aalyss s apple o Ier epeece AHP o aalyze he amou he compoes of a parse comparso marx flueces he eghs a cossecy of a marx. Ths maes possble o sho he mague of he fuzzess he eghs. I prevous researches, e propose a e represeao for eghs of crera a aleraves ormal AHP. [7][8][]. I hs paper, a represeao of eghs of er epeece meho s propose. I s represee as L-R fuzzy umbers by usg he resuls from he sesvy aalyss. Ths paper ecompasses mehoology o represe eghs by fuzzy ses. I ao, a represeao of fuzzess as a resul of er epeece s presee he a comparso marx oes o have eough cossecy. 2 Ier epeece AHP 2. Process of Normal AHP (Process ) Represeao of srucure by a herarchy. The problem uer coserao ca be represee a herarchcal srucure. The hghes level of he herarchy cosss of a uque eleme ha s he overall obecve. A he loer levels, here are mulple acves (.e. elemes h a sgle level) h relaoshps amog elemes of he aace hgher level o be cosere. The acves are evaluae usg subecve ugmes of a ecso maer. Elemes ha le a he upper level are calle pare elemes hle hose ha le a loer level are calle chl elemes. Alerave elemes are pu a he loes level of he herarchy (Process 2) Pare comparso beee elemes a each level. A parse comparso marx A s creae from a ecso maer's asers. Le be he umber of elemes a a cera level. The upper 62

2 ragular compoes of he comparso marx a (< =,,) are 9, 8,.., 2,, /2,, or /9. These eoe eses of mporace from acvy o. The loer ragular compoes a are escrbe h recprocal umbers as follos a / a, ao, for agoal elemes, le a =. The loer ragular compoes a agoal elemes are occasoally ome from he re equao as hey are eve f upper ragular compoes are sho. The ecso maer shoul mae (-)/2 pare comparsos a a level h elemes. (Process 3) Calculaos of egh a each level. The eghs of he elemes, hch represe grae of mporace amog each eleme, are calculae from he parse comparso marx. The egevecor ha correspos o a posve egevalue of he marx s use calculaos hroughou hs paper. (Process 4) Prory of a alerave by a composo of eghs. The compose egh ca be calculae from he eghs of oe level loer. Wh repeo, he eghs of he alerave, hch are he prores of he aleraves h respec o he overall obecve, are fally fou. 2.2 Cossecy Sce compoes of he comparso marx are obae by comparsos beee o elemes, cohere cossecy s o guaraee. I AHP, he cossecy of he comparso marx A s measure by he follog cossecy ex (C.I.) C.I. A, (2) here s he orer of marx A, a A s s maxmum egevalue. I shoul be oe ha C.I. 0 hols. A f he value of C.I. becomes smaller, he he egree of cossecy becomes hgher, a vce versa. The comparso marx s cosse f he follog equaly hols. C.I. 0. Also cossecy rao (C.R.) s efe as C.I. C.R., M Where M s raom cossecy value. Hoever e oly employ C.I., sce e maly use 4 or 5-mesoal aa hose raom cossecy value s o far from. 2.3 Ier Depeece Meho Usually ormal AHP mus assume epeecy amog crera, alhough s ffcul o choose eough epee crera. Ier epeece meho AHP[0] s oe echque of solvg hs of problem eve case of crera have epeecy. I he meho, usg a epeecy marx F={ f }, e ca calculae real eghs as follos, =F (3) here s eghs from epee crero,.e. ormal eghs of ormal AHP a F s calculae as ege value of fluece marx. 3 Sesvy Aalyss Whe AHP s use, he comparso marx s ofe cosse or large ffereces amog he overall eghs of he aleraves o o appear. Thus, s very mpora o vesgae ho he compoes of a parse comparso marx fluece he cossecy or eghs. Sesvy aalyss s use o aalyze ho resuls are fluece he cera varables chage. Therefore, s ecessary o esablsh a sesvy aalyss of AHP. I our research, a prevously propose meho [7] s use o evaluae he flucuao of he cossecy ex a eghs he a comparso marx s perurbe. Ths meho s useful as oes o chage he srucure of he aa. Evaluag he cossecy ex a he eghs of a perurbe comparso marx are performe as follos. Perurbaos a are mpare o compoe a of a comparso marx, a he flucuao of he cossecy ex a he egh are expresse by he poer seres of. (2) Flucuaos of he cossecy ex a he eghs are represee by he lear combao of. (3) By he coeffce of, ca be sho ha ho he compoe of he comparso marx gves fluece o he cossecy ex a he egh. Sce he parse comparso marx A s a posve square marx, he follog Perro- 63

3 Frobeus heorem [4] hols. Theorem (Perro Frobeus) For a posve square marx A, he follog hols rue.. Marx A has a posve egevalue. If A s he larges egevalue he A s a smple roo. The posve egevecor, correspog o A, exss. A s calle he Frobeus roo of A. 2. Ay posve egevecors of A are he cosa mulples of. 3. The absolue value of he egevalues of A, excep for A, s smaller ha A. 4. The Frobeus roo of he raspose marx A' s equvale o he Frobeus roo of A. Ths heorem esures he exsece of a egh vecor a parse comparso marx. From Theorem, he follog heorem regarg a perurbe comparso marx hols rue [7]. Theorem2 Le A = (a ),, =,, be a comparso marx a le A() = A+D A, D A =(a ) be a marx ha has bee perurbe. Moreover, le A be he Frobeus roo of A h beg he correspog egevecor. Le 2 be he egevecor correspog o he Frobeus roo of A', he, he Frobeus roo () of A() a he correspog egevecor () ca be expresse as follos here ( ) o( ), (4) A ( ) o( ), (5) ' ( ) 2 DA, (6) ' 2 s a -meso vecor ha sasfes ( A A A I) ( D I), (7) here o() eoes a -meso vecor hch all compoes are o(). Proof of hs heorem ca be fou Ohsh s paper [7]. 3. Sesvy aalyss of cossecy ex Regarg a flucuao of he cossecy ex, he follog corollary ca be obae from Theorem 2. Corollary Usg a approprae p, e ca represe he cossecy ex C.I.() of he perurbe comparso marx as follos C.I.( ) C.I. o( ). (8) p (Proof) From he efo of he cossecy ex (3) a (4), C.I. ( ) C.I. o( ). Le =( ) a 2 =( 2 ) from (6). s ca o be represee as 2 herefore, he seco par of he rgh se s expresse by a lear combao of. (Q.E.D) p equao (8) Corollary shos he fluece of comparso marx compoes o he cossecy. O he oher ha, sce he comparso marx A() = (a ()) s recprocal, he a () = /a () a becomes a 2 a a o( ). (9) a a Here, sce a =/a, (0) s obae. The mpac o he cossecy ca be easly sho by use of hs propery. 3.2 Sesvy aalyss of eghs Wh regars o he flucuao eghs, he follog corollary ca also be obae from Theorem 2. Corollary 2 Usg a approprae q (), e ca represe he flucuao =( ) of he egh (.e. he egevecor correspog o he Frobeus roo) as follos h () ( ) (Proof) The -h ro compoe of he rgh se of (7)., 64

4 Theorem 2 s represee as 2 a 2 { (, ) a } a s expresse by a lear combao of. Here,(,) s Kroecer's symbol (, ) 0 ( ), ( ). I coras, sce A s a smple roo, Ra(A- A I) = -. Accorgly, he egh vecor s ormalze as ( ), he he coo s as follos. 0. (2) By usg a elemeary rasformao o formula (7) he coo above, e also ca represe by lear combaos of. (Q.E.D) As see equao (5) Theorem 2, he compoe ha has a grea fluece o egh () s he compoe hch has he greaes fluece o (). q equao () from Corollary 2 shos ho he fluece by he compoes of a comparso marx o he eghs ca be calculae. The fluece ca also be sho easly by use of equao (0). 3.3 Sesvy for er epeece meho We ca also calculae h regars o he flucuao eghs, he follo Corollary 3 Usg a approprae h (),e ca represe he flucuao =( ) of he eghs of er epeece AHP as follos ( ) h (Proof) From he equao (3), eghs of er epeece meho s calculae from lear rasformao of ormal eghs of ormal AHP, a from Corollary 2 s represee as sum of lear combao of.,.therefore eghs s also represee as sum of lear combao of.. (Q.E.D) 4 A Weghs Represeao The comparso marx ofe has poor cossecy (.e. 0.<C.I.<0.2) because ecompasses several acves. I hese cases, he compoes of a comparso marx are cosere o have fuzzess sce hey resul from he fuzzy ugme of humas. Therefore, eghs shoul be reae as fuzzy umbers. To represe fuzzess of egh, a L-R fuzzy umber s use. 4. L-R fuzzy umber L-R fuzzy umber M ( m,, ) LR s efe as fuzzy ses hose membershp fuco s as follos. x m R ( x m), M ( x) m x L ( x m). here L(x) a R(x) are shape fuco hch sasfes L(x) = L(-x), (2) L(0) =, (3) L(x) s a o creasg fuco 4.2 Fuzzy eghs of crera From he flucuao of he cossecy ex, he mulple coeffce g h () Corollary a 3 s cosere as he fluece o a. Sce g s alays posve, f he coeffce h () s posve, he real egh of crero s cosere o be larger ha. Coversely, f h () s egave, he real egh of acvy s cosere o be smaller. Therefore, he sg of h () represes he reco of he fuzzy umber sprea. The absolue value g h () represes he sze of he fluece. O he oher ha, f C.I. becomes bgger, he he ugme becomes more fuzzy. Cosequely, mulple C.I. g h () ca be regare as a sprea of a fuzzy egh cocere h 65

5 a. here Defo (fuzzy egh) Le be a crsp egh of crero of er epeece moel, a g h () eoe he coeffces fou Corollary a 3. If 0.<C.I.<0.2, he a fuzzy egh s efe by l v f ( x ) v f supp( ~ v ) r sup supp( v ~ ) v here (,, ) LR (3) I he above equaos, f supp, sup supp are loer a upper lms of suppor ses a are calculae as follos. C.I.s(, h ) g h, (4) C.I.s(, h ) g h, (5), ( h 0),( h 0) s(, h) s(, h) 0.( h 0) 0.( h 0) 4.2 Fuzzy eghs of aleraves Usg he fuzzy eghs of crera efe above a local crsp eghs of aleraves h respec o cera crero, e ca calculae overall eghs from he vepo of he overall obecve by exeso. Hoever, he resuls from he operao of fuzzy umbers are frequely oo ambguous o erpre. Fuzzy eghs of acves are ormalze hus her sum s, herefore e ca avo much ambguy sce hs coo has bee cosere [9] I geeral, operag h cosras s ffcul bu ca be accomplshe f every fuzzy membershp fuco s lear. Especally for every ormal ragular fuco h a core u he cosra u hols, a he orer of sgleo coeffces s assume. Thus, he upper a loer lm of -cu ses of lear sum ca be easly calculae. Le f ( x ) be a crsp local egh of alerave h respec o acvy, a hs paper, assume 0 f. The, he overall ( x) f ( x2 ) f ( x ) egh of a alerave s also he L-R fuzzy umber a s represee as follos. v ( v, l, r) LR f supp( v ~ ) max( ) f ( x ) ( ) supsupp( v ~ ) m( ) f ( x ) ( ) ( ( ( ( 5. Coclusos ) f ( x ) ) f ( x ) f ( x ) ) f ( x ) ) We propose a represeao for he er epeece overall eghs of aleraves by use of fuzzy ses a he resul of a sesvy aalyss for cases hch cossecy of he comparso marx s poor. Our approach shos ho o represe eghs, as ell as ho he resul of AHP has fuzzess, he cossecy exss. Ths as ue o reuce ambguy he represeao presee hs or compare o prevous ormal fuzzy operaos. Acolegemes Ths research s parly suppore by a gra from he msry of eucao, spors, scece, a echology o he aoal proec of Avace mprovemes of vso, mage, speech a laguage formao processg a he applcao o he echologes for he ellge srume a corol he Hgh-ech Research Ceer of Hoa-Gaue Uversy. 66

6 Refereces [] T. L. Saay. A scalg meho for prores herarchcal srucures. J. Mah. Psy., 5(3): , 977. [2] T. L. Saay. The Aalyc Herarchy Process. McGra-Hll, Ne Yor, 980. [3] T. L. Saay. Scalg he membershp fuco. Europea J. of O.R., 25: , 986. [4] M. Sao. A Irouco o Lear Algebra. Toyo Uversy Press, 966. [5] Y. Taaa. Rece avace sesvy aalyss mulvarae sascal mehos. J. Japaese Soc. Comp. Sa., 7:--25, 994. [6] K. Toe. The Game Feelg Decso Mag. Na-gre Press, Toyo, 986. [7] S. Ohsh, H. Ima, a M. Kaaguch. Evaluao of a Sably o Weghs of Fuzzy Aalyc Herarchy Process usg a sesvy aalyss. J. Japa Soc. for Fuzzy Theory a Sys., 9:40--47, [8] S. Ohsh, H. Ima, T. Yamao. Weghs Represeao of Aalyc Herarchy Process by use of Sesvy Aalyss, IPMU 2000 Proceegs, [9] D. Dubos a H. Prae. Possbly Theory A Approach o Compuerze Processg of Uceray,. Pleum Press, Ne Yor, 988. [0].T. L. Saay. Ier a Ouer Depeece AHP. Uversy of Psburgh, 99. [] S. Ohsh, T. Yamao, H. Ima, A Fuzzy Represeao for Weghs of Aleraves AHP, Ne Dmesos Fuzzy Logc a Relae Techologes, Vol., pp [2] S. Ohsh, D. Dubos, H.Prae, T.Yamao, A Fuzzy Cosra-base Approach o he Aalyc Herarchy Process, Uceray a Iellge Iformao Sysems, pp ,