A Weights Representation for Fuzzy Constraint-based AHP

Transcription

1 Weghts Represetato for Fuzzy Costrat-based HP Sh-ch OHNISHI Hoa-Gaue Uversty Sapporo JPN Taahro YMNOI Hoa-Gaue Uversty Sapporo JPN Hdeyu IMI Hoado Uversty Sapporo JPN bstract alytc Herarchy Process (HP) s very popular method the feld of decso-mag today Whle oe of the most atural extesos of HP usg fuzzy theory s to employ a recprocal matrx th fuzzy-valued etres some of the preset authors proposed the fuzzy costrat-based approach before I ths paper e cosder about eghts for fuzzy costrat-based approach ad propose a d of represetato for the eghts It employs ot oly fuzzy cocept but also results of sestvty aalyss Moreover t s also very useful for vestgatg data structure ad realzg results of HP Keyords: Decso Mag; HP; Fuzzy Sets Sestvty alyss; Weght represetato Itroducto The HP methodology as proposed by TL Saaty 977 [0] [] ad t has bee dely used the feld of decso mag It elcts eghts of crtera ad alteratves through rato udgmets of relatve mportace d fally the preferece for each alteratve ca be derved The classcal method requres the decso-maer (DM) to express hs or her prefereces the form of a precse rato matrx ecodg a valued preferece relato Hoever t ca ofte be dffcult for the DM to express exact estmates of the ratos of mportace Therefore may ds of methods employg tervals or fuzzy umbers as elemets of a parse recprocal data matrx have bee proposed to cope th ths problem Ths allos for a more flexble specfcato of parse preferece testes accoutg for the complete oledge of the DM fuzzy represetato expresses rch formato because the DM ca provde ) the core of the fuzzy terval as a rough estmate of hs perceved preferece ad ) the support set of the fuzzy terval as the rage that the DM beleves to surely cota the uo rato of relatve mportace Usually sce compoets of the parse matrx are locally obtaed from the DM by parse comparsos of crtera or alteratves ts global cosstecy s ot guarateed I classcal HP cosstecy s usually measured by a cosstecy dex (CI) based o the computato of a egevalue d very ofte the trastvty of prefereces betee the elemets to be compared s strogly related to the cosstecy of the matrx Usg fuzzy umbers as elemets of the recprocal matrces strct trastvty s too hard to preserve terms of equaltes betee fuzzy umbers Therefore the fuzzy costrat-based approach to HP[3][4] oly tres to mata cosstecy of precse matrces that ft the mprecse specfcatos provded by the DM e d of cosstecy dex for fuzzy-valued matrces s computed that correspods to the degree of satsfacto of the fuzzy specfcatos by the best fttg cosstet recprocal preferece matrces Importace or prorty eghts are the derved based o these precse preferece matrces O the other had to aalyze ho much the compoets of a parse comparso matrx flueces eghts ad cosstecy of a matrx e ca apply sestvty aalyss to HP[5] L Magdalea M Oeda-cego JL Verdegay (eds): Proceedgs of IPMU 08 pp Torremolos (Málaga) Jue

2 [5] Ths allos us possble to o ho fuzzy ther eghts are Later part of ths paper e propose a d represetato of eghts for fuzzy costratbased approach They are represeted as L-R fuzzy umbers by use of the result from the sestvty aalyss It ca sho ot oly ho to represet eghts by fuzzy sets but also a represetato of fuzzess of results from HP 2 fuzzy costrat-based approach to the alytc Herarchy I ths secto e sho a fuzzy costratbased approach [3][4] to the HP that some of preset authors proposed before Sce usg fuzzy umbers as elemets of a parse matrx s more expressve tha usg crsp values or tervals e hope that the fuzzy approach allos a more accurate descrpto of the decso mag process Rather tha forcg the DM to provde precse represetatos of mprecse perceptos e suggest usg a mprecse represetato stead I the tradtoal method the obtaed matrx does ot exactly ft the HP theory ad thus must be modfed so as to respect mathematcal requremets Here e let the DM be mprecse ad chec f ths mprecse data ecompasses precse preferece matrces obeyg the HP requremets 2 Fuzzy recprocal data matrx t frst e employ a fuzzy parse comparso ~ recprocal matrx R = { ~ r } pertag to elemets (crtera alteratves) I the HP model etry r of a preferece matrx reflects the rato of mportace eghts of elemet over elemet I the fuzzy-valued matrx dagoal elemets are sgletos (= ) ad the other etres r~ ( ) have membershp fucto μ hose support s postve: ~ r = supp(μ ) (0 + ) = Moreover f elemet s preferred to elemet the supp(μ ) les [ + ) hle supp(μ ) les (0 ] f the cotrary holds The DM s supposed to supply the core (modal value) r of r~ ad ts support set [l u ] for < The support set s the rage that the DM beleves surely cotas the uo rato of relatve mportace The DM may oly supply etres above the dagoal le the classcal HP We assume recprocty ~ r = / ~ as follos [8] r μ ( r) = μ (/ r) Therefore core / ~ r = / r (2) ( ) (3) supp( / ) [/ u / l ] ~ r = We may assume all etres hose core s larger tha or equal to form tragular fuzzy sets e f r e assume r~ s a tragular fuzzy umber deoted as (4) = l r u r~ ( ) Δ but the the symmetrc etry r~ s ot tragular lteratvely oe may suppose that f r < r~ s ( / u / r / l ) Δ Therefore the follog trastvty codto herted from the HP theory ll ot hold (5) ~ r ~ r ~ = r partcular because multplcato of fuzzy tervals does ot preserve tragular membershp fuctos Hoever eve th tervals ths equalty s too demadg (sce t correspods to requestg to usual equaltes stead of oe) ad mpossble to satsfy For stace tae = the above equalty O the left had sde s a terval o the rght-had sde s a scalar value (=) So t maes o sese to cosder a fuzzy HP theory here fuzzy tervals ould smply replace scalar etres the preferece rato matrx 22 Cosstecy I ths approach a fuzzy-valued rato matrx s cosdered to be a fuzzy set of cosstet ofuzzy rato matrces Each fuzzy etry s veed as a flexble costrat rato matrx s cosstet the sese of HP (or HPcosstet) f ad oly f there exsts a set of eghts 2 summg to such that for all r = / Usg a fuzzy recprocal matrx some d of cosstecy dex of the data matrx s ecessary[7] Ths dex ll ot measure the HP- cosstecy of a o-fully cosstet 262 Proceedgs of IPMU 08

3 matrx but stead ll measure the degree to hch a HP-cosstet matrx R exsts that satsfes the fuzzy costrats expressed the fuzzy recprocal matrx R ~ More precsely ths degree of satsfacto ca be attached to a - tuple of eghts = ( 2 ) sce ths - tuple defes a HP-cosstet rato matrx Ths degree s defed as (6) α ( ) = m μ ( / ) maxmzeα st μ α = = ad e ca express the frst costrat as follos It s the degree of cosstecy of the eght patter th the fuzzy rato matrx R ~ the sese of fuzzy costrat satsfacto problems (FCSPs) [5] The coeffcet α() s some sese a emprcal valdty coeffcet measurg to hat extet a eght patter s close to or compatble th the DM revealed preferece The best fttg eght patters ca thus be foud by solvg the follog FCSP: maxmze α m μ 0 = = here s the eght of alteratve ad s the total umber of alteratves Maxmzg α correspods to gettg as close as possble to the deal preferece patters of the DM ( the sese of the Chebychev orm) Let (7) α max m μ α s a degree of cosstecy dfferet from Saaty's dex but hch ca be used as a atural substtute to the HP-cosstecy dex for evaluatg the DM s degree of ratoalty he expressg hs or her prefereces Solvg ths flexble costrat satsfacto problem terms of α eables the fuzzy rato matrx to be tured to a terval-valued matrx defg crsp costrats for the ma problem of calculatg local eghts as sho the ext subsecto s usual the FCSP problem ca be re-stated as follos (8) / [ μ ( α) μ ( α) ] here μ ( α) ad μ ( α) are the loer ad upper boud of the α-cut of μ ( / ) respectvely Ths becomes (9) μ ( α) μ ( α) Here f all μ are tragular fuzzy umbers ( l r u ) Δ the problem becomes a olear programmg problem as follos [NLP] maxmze α { l + α r l )} { u + α( r u )} ( = = The problem s oe of fdg a soluto to a set of lear equaltes f e fx the value of α Hece t ca be solve by the dchotomy method 23 Ucty of the optmal eght patter Results obtaed by Dubos ad Fortemps [6] o best solutos to maxm optmzato problems th covex domas ca be appled here Ideed t s obvous that the set of eght patters obeyg (9) for all s covex Call ths doma D α If ad 2 are D α so s 2 ther covex combato = λ + ( λ) Note that the ratos / le betee / ad 2 / 2 Hece f for all / dffers from 2 / 2 t s clear that (0) 2 μ m 2 μ μ > So partcular suppose there are to optmal eght patters ad 2 the optmal Proceedgs of IPMU

4 α-cuts hose rato matrces dffer for all odagoal compoets It mples that for 2 = ( + )/2 () μ > α = hch s cotradctory I ths case there s oly oe eght patter coheret th the tervalvalued matrx hose etres are tervals ( r ) α I the case he there are at least to optmal eght patters ad 2 the optmalα-cuts ther rato matrces cocde for at least oe odagoal compoet ( / = 2 / 2 ) I [6] t s sho that ths case (2) 2 μ 2 μ = = α So the procedure s the terated: For such etres of the matrx the fuzzy umbers place ( ) must be replaced by the α -cut of r~ It ca be doe usg the best eght patter obtaed from the dchotomy method checg for ( ) such that (3) μ α = The problem [NLP] s solved aga th the e fuzzy matrx It yelds a optmal cosstecy degree β>α (by costructo) If the same lac of ucty pheomeo reappears some fuzzy matrx etres are aga tured to tervals ad so o utl all etres are terval The obtaed soluto s called dscrm - optmal soluto [6] ad s provably uque from theorem 5 the paper The global (aggregated) evaluato of a decso f s gve by meas of the uque (dscrm) optmal eght patter D α : (4) V = u ( f) f here u ( f ) s the utlty of decso f uder crtero 3 Sestvty alyss of HP Whe e actually use HP t ofte occurs that a comparso matrx s ot cosstet or that there s ot great dfferece amog the overall eghts of the alteratves Thus t s very mportat to vestgate ho compoets of a parse comparso matrx fluece o ts cosstecy or o eghts So as to aalyze ho results are flueced he a certa varable has chaged e use the sestvty aalyss O the bass of the reasos metoed above e also proposed sestvty aalyss of HP [5][6] It evaluates a fluctuato of the cosstecy dex ad eghts he a comparso matrx s perturbed ad t s useful because t does ot chage a structure of the data The evaluatos of cosstecy dex ad the eghts of a perturbed comparso matrx are performed as follos Perturbatos ε a d are mparted to compoet a of a comparso matrx ad the fluctuato of the cosstecy dex ad the eght are expressed by the poer seres of ε 2 Fluctuatos of the cosstecy dex ad the eghts are represeted by the lear combato of d 3 From the coeffcet of d t foud that ho the compoet of the comparso matrx gves fluece o the cosstecy dex ad the eght Sce a parse comparso matrx s a postve square matrx Perro- Frobeus theorem holds From ths theorem the follog theorem about a perturbed comparso matrx holds Theorem Let = (a ) = be a comparso matrx ad let (ε) = +εd D =(a d ) be a matrx that has bee perturbed Moreover let λ be the Frobeus root of be the egevector correspodg to t ad 2 be the egevector correspodg to the Frobeus root of ' the the Frobeus root λ ( ε ) of ( ε ) ad the correspodg egevector (ε) ca be expressed as follos (5) λ( ε ) = λ + ελ o( ε ) + (6) ( ε) = + ε + o( ) here ε 264 Proceedgs of IPMU 08

5 ' 2 (7) λ = D ' 2 s a -dmeso vector that satsfes (8) λ I) = ( D λ I) ( here o(ε) deotes a -dmeso vector hch all compoets are o(ε) (Proof) From Perro-Frobeus theorem the Frobeus root λ s the smple root Thus expasos (5) ad (6) are vald Therefore characterstc equatos become ( + εd )( + ε + o( ε)) = ( λ + ελ + o( ε))( + ε = λ + o( ε)) From these to equatos (8) ca be obtaed Further by Perro-Frobeus theorem 2 ' =λ 2 ' holds ad t becomes λ ' ' 2 = 2 D Thus equato (7) holds (QED) 3 Sestvty aalyss of the cosstecy dex bout a fluctuato of the cosstecy dex the follog corollary ca be obtaed from Theorem Corollary Usg a approprate g e ca represet the cosstecy dex CI(ε) of the perturbed comparso matrx as follos (9) CI ( ε ) = CI + ε g d o( ε ) (Proof) + From the defto of the cosstecy dex ad (5) λ CI( ε ) = CI + ε + o( ε ) holds Here let =( ) ad 2 =( 2 ) from (7) λ s represeted as λ = 2 so the secod part of the rght sde s expressed by a lear combato of d (QED) To see g the equato (9) Corollary ho the compoets of a comparso matrx mpart fluece o ts cosstecy ca be foud O the other had sce the comparso matrx (ε) = (a (ε)) s recprocal; a (ε) = /a (ε) holds ad t becomes 2 a d (20) a + εa d = ε + o( ε ) a a Here sce a =/a (2) d = d s obtaed I fact e ca see fluece more easly f e use ths property 32 Sestvty aalyss of eghts bout the fluctuato of the eght the follog corollary also ca be obtaed from Theorem Corollary 2 Usg a approprate h () e ca represet the fluctuato =( ) of the eght (e the egevector correspodg to the Frobeus root) as follos ( ) (22) = h d (Proof) The -th ro compoet of the rght sde of (8) Theorem s represeted as 2 a { δ ( ) a } d 2 ad s expressed by a lear combato of d Hereδ() s Kroecer's symbol d Proceedgs of IPMU

6 δ ( ) = 0 ( = ) ( ) O the other had sce λ s a smple root Ra(-λ I) = - ccordgly the eght vector s ormalzed as ( + ε ) = = the the follog codto follos (23) = 0 Usg elemetary trasformato to formula (9) the codto above e also ca represet by lear combatos of d (QED) From the equato (6) Theorem the compoet that has a great fluece o eght (ε) s the compoet hch has the greatest fluece o ccordgly from Corollary 2 ho compoets of a comparso matrx mpart fluece o the eghts ca be foud to see h () the equato (22) Of course e ca also see fluece more easly by use of equato (2) 4 d of eght represetato Wth the fuzzy compoet matrx data e cosder that compoets of a comparso matrx are results from fuzzy udgmet of huma Therefore eghts could be treated as fuzzy umbers () The multple of coeffcets g h Corollary ad 2 s cosdered as the fluece cocered th a from the fluctuato of the cosstecy dex Sce g s alays postve f the coeffcet () h s postve the real eght of crtero s cosdered as bgger tha crsp eghts from secto 2 ad f h () s egatve the real eght of crtero s cosdered as smaller () Therefore a sg of h represets a drecto the spread of the fuzzy umber Of course the absolute value g h () represets the amout of the fluece O the other had α becomes bgger the the udgmet becomes more fuzzess Cosequetly multple α g h () ca be regarded as a spread of v~ a fuzzy eght of crtero cocered th a Defto (fuzzy eght) Let be a crsp eght of crtero fuzzy costrat-based HP ad g h () deote the coeffcets foud Corollary ad 2 a fuzzy eght of crtero v~ s defed by (24) here (25) (26) ( p q ) LR v = ( ) ( ) s( ) g h p = α h ( ) ( ) q = α s( + h ) g h s ( + h) = 0 0 s ( h) = ( h > 0) ( h 0) ( h > 0) ( h 0) We ca calculate the fuzzy eghts of actvtes usg defto above The the fuzzy eghts of alteratves are calculated by operatos of fuzzy umbers They sho ho the result from HP has fuzzess 5 Coclusos I classcal HP t s ofte dffcult for the DM to provde a exact parse data matrx because t s hard to estmate ratos of mportace a precse ay Therefore e use fuzzy recprocal matrces ad propose a e d of cosstecy dex ad eghts Ths dex s cosdered as a emprcal valdty coeffcet evaluatg to hat extet a eght patter s close to the DM revealed preferece The from these ad results from sestvty aalyss e propose a d of eghts represetato for the fuzzy costrat-based approach to HP 266 Proceedgs of IPMU 08

7 I the ext step e ll be able to sho examples of HP etrely Moreover e pla to mplemet these results o actual data We ll also try to refe the search for approprate eghts that employs the DM s subectve dstace Refereces [] C G E Boeder J G de Graa F Lootsma (989) Mult crtera decso aalyss th fuzzy parse comparsos Fuzzy Sets ad Systems [2] D Bouyssou T Marchat M Prlot P Pery Tsouas P Vce (2000) P Evaluato Models: a Crtcal Perspectve Kluer cad Pub Bosto [3] J J Bucley (985) Rag alteratves usg fuzzy umbers Fuzzy Sets ad Systems [4] J J Bucley (985) Fuzzy herarchcal aalyss Fuzzy Sets ad Systems [5] D Dubos H Farger H Prade (996) Possblty theory costrat satsfacto problems: Hadlg prorty preferece ad ucertaty ppled Itellgece [6] D Dubos P Fortemps (999) Computg mproved optmal solutos to max m flexble costrat satsfacto problems Europea Joural of Operatoal Research [7] M Fedrzz R Marques Perera(995) Postve fuzzy matrces domat egevalues ad a exteso of Saaty's alytcal Herarchy Process Proceedgs of the VI Iteratoal Fuzzy Systems ssocato World Cogress IFS' [8] P J M va Laarhove W Pedrycz (983) fuzzy exteso of Saaty s prorty theory Fuzzy Sets ad Systems [9] L C Leug D Cao (2000) O cosstecy ad rag of alteratves fuzzy HP Europea Joural of Operatoal Research [0] T L Saaty(977) scalg method for prortes herarchcal structures J Math Psy 5(3) [] T L Saaty (980) The alytc Herarchy Process McGra-Hll Ne Yor [2] Salo (996) O fuzzy rato comparso herarchcal decso models Fuzzy Sets ad Systems [3] S Ohsh D Dubos H Prade T Yamao (2006) Study o the alytc Herarchy Process Usg a Fuzzy Recprocal Matrx Proceedg of the coferece IPMU 2006 pp [4] S Ohsh D Dubos H Prade T Yamao (2008) fuzzy costrat-based approach to the alytc Herarchy Process Ucertaty ad Itellget Iformato Systems World Scetfc chapter 6 [5] S Ohsh H Ima T Yamao (2000) Weghts Represetato of alytc Herarchy Process by use of Sestvty alyss Proceedg of the coferece IPMU2000 Vol3 pp [6] S Ohsh H Ima T Yamao (2007) Fuzzy Represetato for Weghts of lteratves HP Ne Dmesos Fuzzy Logc ad Related Techologes Proceedg 5th Eusflat Coferece VolⅡ pp3-36 Proceedgs of IPMU