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1 Goupe de Taal Euopée Ade Multctèe à la Décso Euopea Wog Goup Multple Ctea Decso Adg Sée 3, º8, autome 008. Sees 3, º 8, Fall 008. Opo Maes Secto Hamozg poty weghts ad dffeece judgmets alue fucto mplemetato Mauel A. Matos INESC Poto, Fac Egehaa, U Poto, Potugal Itoducto Value fuctos ae oe of the most popula ways of dealg wth multattbute decso mag poblems eal wold poblems, amely stuatos le publc tedes whee the taspaecy of the ealuato methodology s a mpotat aspect. The fact s that, may tmes, alue fuctos used pactce ae bult a aïe way, begg wth the defto of weghts by the Decso Mae (DM), complemeted by a moe o less abtay specfcato of the ddual alue fuctos fo each attbute. A addte model s geeally assumed, sometmes wthout eally checg the coespodet equemets. The epeece of the autho suppotg ths d of pocesses, amely the publc taspot ad powe systems sectos, showed that DM (ad commttees actg the DM, the most fequet case) adhee wthout dffculty to a fomal pocess of buldg coectly a multattbute alue fucto ad ae able to epess the pefeeces tems of tade-offs o dffeece judgmets, afte some dscusso ad hag gaed sght to the poblem. Howee, thee ae fequetly coflcts betwee the paametes of the multattbute alue fucto ( weghts ) detemed by the fomal pocess ad the ute oto of elate mpotace of the ctea the DM eep the mds, as a cosequece of the pesoal qualtate aalyss of the stuato, o wat to mpose, due to poltcal o socal costas the decso famewo. These coflcts must be tae seously, sce they may compomse the success of the ete decso-ad pocess. Page Ths pape s deoted to the dscusso of the meas that ca be used to dssole these coflcts, ode to help buldg alue fuctos that ae techcally coect ad at the same tme ae compatble to the pecepto the DM has about the elate mpotace of the ctea. I a fst appoach to the poblem, t may seem that the cosstecy esdes the DM, sce all the fomato comes fom hm. Of couse ths may happe: a DM that hs md pleges eomet potecto esus cost may show the opposte spt specfc stuatos, whe faced wth eal costs ad eal eometal cosequeces. But, most of the tmes, the cosstecy s elated wth the defto of the age of the ddual alue fucto of each attbute, a aspect that geeally s ot the ma coce the decso-ad pocess. I the et secto we wll eew the ma cocepts assocated to the use of alue fuctos, cludg the usual pocedue to deteme the paametes, ad dscuss the og of the coflct. Secto 3 oduces the methodology we adocate to deal wth the poblem, fo the most fequet case of lea addte fuctos, followg ou pesetato the 67 th meetg of the Euopea Wog Goup Multple Ctea Decso Adg. A complete methodology fo the lea case s peseted secto 4. Illustate eamples ae dstbuted alog these sectos. Coclusos ad efeeces coclude the pape. The poblem Sce the topc of alue fuctos s well ow - see, fo stace, Keeey ad affa (976), Keeey (99), Cleme ad elly (00) o Belto ad Stewat (00) we wll just eew the ma deftos, ode to establsh the omeclatue. Howee, elato wth the poblem had, we must po out that methodologes le AHP (Saaty, 980) ad smla pocedues do espect the pecepto of the DM of the elate mpotace of the attbutes. I echage, they do ot use eal ddual alue fuctos ( ), sce the scoe of a alteate each attbute depeds o all the emag alteates (leadg sometmes to a eesal whe a ew alteate s oduced). Dffeetly, the Macbeth appoach (Baa e Costa ad Vasc, 994) taes steps towads the ageemet of the pocess of buldg the multattbute alue fucto wth the DM s pecepto of the elate mpotace of the ctea, by usg absolute efeece leels all the attbutes. ecetly, Baa e Costa et al (008) show how to use Macbeth stuatos close to the oes dscussed ths pape, wth smla objectes. Fally, the methods of the Fech School, le ELECTE (oy, 99) ad elated methodologes, ae

2 Goupe de Taal Euopée Ade Multctèe à la Décso Euopea Wog Goup Multple Ctea Decso Adg Sée 3, º8, autome 008. Sees 3, º8, Fall 008. based a etely dffeet appoach to decso-ad, whee weghts do ot hae the compesatoy meag they hae alue fucto theoy, so they ae out of ths dscusso. We wll assume that the equemets fo a addte fucto ae fulflled. I these ccumstaces, a decso mag stuato wth attbutes, the omalzed alue fucto wll be: (,, K, ). ( ) whee the ddual (o patal) alue fuctos ( ) may be lea, polyomal, epoetal, o ay othe specfc shape that eflects the aato of the pefeece of the DM alog the coespodet attbute. We wll cosde though the tet that all the patal alue fuctos ae omalzed, betwee (,wost )0 ad (,best ), so also. Note that wost ad best ae used hee to defe the age lmts, wthout ay coecto wth possbly estg alteates. The typcal pocedue to buld a addte alue fucto fo a specfc poblem has the followg steps: ) Defe sepaately each of the ddual alue fuctos ( ). Ths meas establshg fst the age of the fucto attbute, ad the detemg (o choosg) the appopate shape fo the fucto, afte eacto wth the DM; ) Obta addtoal fomato fom the DM ode to deteme the paametes of the multattbute alue fucto. Ths may be doe a umbe of ways, the moe fequet of whch s based o judgmets of dffeece betwee (atfcal) alteates. Each dffeece geeates a equato o the paametes, so - depedet judgmets ae ecessay (ad suffcet) the geeal case to calculate the totalty of the paametes (sce ). We llustate ths pocess wth a smple eample that s also used to show the d of coflct we ae speag about. Note that the method used to calculate the paametes s depedet of the shape of the ddual alue fuctos. Eample We wat to buld a methodology to ealuate caddates egadg cost, mllos of ( ) ad duato, wees ( ) of a specfc tas. Iteacto wth the DM led to the defto of the two ddual alue fuctos, ths case lea: ( ) ( ), best, best, wost, wost, wost, wost Now, ode to deteme the paametes, two atfcal alteates A(.6, 50) ad B(3., 30) ae peseted to the DM. Sce he pefes B, we chage B utl we each a dffeece, say A~(3., 35). Ths allows us to buld the equato (A)(3., 35) ad, sce,.0.( 0.75) ( ) , 0.57 At ths po, the DM ejected the alue fucto, because hs md he was coced that duato was less mpotat tha cost, ad the weghts just detemed dsplay the opposte, sce <. The po hee s: what to do? Of couse we may ty to coce the DM he s wog ad should accept ou model ayway (geeally a ot ey successful opto), but we also may eew the pocess ad chec f we stll hae some degees of feedom to accommodate the DM s stog feelg that cost s moe mpotat. Now, befoe gog to ou poposal to deal wth ths poblem, t s woth emembeg, fo the sae of geealty, that, wth moe tha two attbutes, the pocess s smla ad subject to same commets. Fdg dffeece judgmets depats geeally fom atfcal alteates whee all the attbutes ecept oe ae the wost alue. Fo stace: (, best, wost 3, wost wost ) ( ) A L, B, wost, best 3, wost L, wost Theefoe, whe compag A ad B, the DM has just to cosde the fst two ctea, sce A ad B ae equal all the othes. If A fb, we mmedately ow that >, but would eed aga to mpoe the fst attbute of B utl eachg a dffeece. It should also be oted that, lea alue fuctos, the judgmet of dffeece s equalet to settg tadeoffs betwee the attbutes. Fo stace, the case of Eample, the tade-off s 0.6/50.04 M /wee. Ths judgmet must of couse be peseed by ay methodology we choose. Of couse, whe the ddual fuctos ae ot lea, tade-offs ae oly local ad caot be geealzed to the whole age of the attbutes, so the tem tade-offs betwee the attbutes maes o sese that case. A cosstet appoach fo the lea case I ode to ease the eplaato, we aalyze, wthout loss of geealty, the case of a two-attbute lea fucto. We stat fom the ddual alue fuctos, whee s the age of attbute (ote that t ca be egate, mmzato attbutes): Page

3 Goupe de Taal Euopée Ade Multctèe à la Décso Euopea Wog Goup Multple Ctea Decso Adg Sée 3, º8, autome 008. Sees 3, º8, Fall 008. ( ), best, wost, wost, wst We also ow that A~B, A(a, a ) ad B(b, b ). Theefoe, to calculate the paametes we must sole (A)(B): a, wst a, wst b o, wst a a b b ad a b a b b, wst Ths tus mafest a well-ow fact, sometmes fogotte: the elato betwee the paametes depeds, ot oly o the tade-off assumed (dectly o dectly) by the DM, but also o the ages of the ddual alue fuctos. Fo stace, f all pefeece fomato emas uchaged, but we modfy the age of the fst attbute to α., the esult s: α The po hee s: do you hae eally ths degee of feedom egadg the age of alues to cosde? The aswe s: most of the tmes, yes. All pacttoes ow that, may stuatos, age lmts ae fuzzy ad some dscetoay judgmet s used to set the alues fo the ddual alue fucto of the specfc attbute. So, wth ceta lmts, we ca eted o estct the age cosdeed fo the attbute wthout compomsg the epesetato of the decso stuato o clude uealstc alues the age. A dffeet dscusso egads the absolute meag of the alue a specfc attbute. Oe could say that (cost)0.5 should always mea that the coespodg alteate s halfway the eal alue scale of the DM. The fact, howee, s that alue fuctos (o utlty fuctos) ae ald up to a lea tasfomato of ts scale, so we ae just tyg to buld a mathematcal model that espects the DM s elatos betwee alue dffeeces betwee pas of alteates. Comg aga to the model (ad tag α to ease the eplaato), the ew multattbute alue fucto would be (ote that, wst may be dffeet fom the fome wost alue, wst, sce the age has chaged): (, ) (, ) ( ),,, wst wst, wst, wst ad, wst (, ) M ( ) N, so, fo eey alteates X, Y: ( X ) ' '( Y) costat ( X ) ( Y), wst (, wst, wst ) ( ) I summay, the modfcato of the age of a attbute leads to a chage the paametes, but the ew alue fucto s just a lea tasfomato of the pecedg oe, so t s equalet to t, because t duces the same ode ay set of alteates ad espects the ato of dffeece of alue betwee pas of alteates (ths cocluso f ald fo ay alue of α, as the eade may chec easly). Ths ca be used to adapt the weghts of the alue fucto to the deses of the DM, wthout chagg the ode duced by the pefeece fomato he peously deleed. We ow eted Eample to llustate the dea. Eample Followg hs dsageemet wth the weghts, the DM dcated that 0.6 ad 0.4 would be close to hs pecepto of the elate mpotace of the two ctea. Ths allowed us to calculate the equed chage though a b a b 0.06 leadg to Splttg the suplus two equal pats (othe hypotheses ae possble), the modfed ( ) would be: ( ) So, the old uacceptable (fo the DM) multattbute alue fucto: ( ) , Page 3

4 Goupe de Taal Euopée Ade Multctèe à la Décso Euopea Wog Goup Multple Ctea Decso Adg Sée 3, º8, autome 008. Sees 3, º8, Fall 008. s substtuted by: (, ) ' that fully agees wth the DM s pecepto of the weghts ad also espects the pefeece fomato (dffeece judgmet) ge by hm. I ths case, t s easy to see that: (, ) 0.7 (, ) 0. 5 ' A geeal pocedue to ecocle pefeece fomato the lea case I ode to tacle moe geeal stuatos, we defe the followg statg po: a) A multattbute lea alue fucto s defed, but the ages of each attbute ae ot ow adace. Howee, a eal ad a eteal age ae defed fo each attbute. The eal ages should clude all the possble alues that may appea the alteates, whle the eteal age lmts the alue fuctos to cedble alues of the attbute. b) The DM defes a set of weghts fo the alue fucto; c) Thee ae - depedet judgmets of dffeece (o tade-offs) suppled by the DM. Moe fomally, ou data ae the paametes,,,, judgmets A j ~B j, j.., ad eal ad eteal et et ages [ ] ad [ ], espectely. The am of the eecse s to deteme the fal ages [ ] fo each attbute that espect smultaeously the pefeeces of the DM ad the eal ad eteal ages. The geeal fom of the multattbute alue fucto s, assumg mmzg attbutes (cludg mamzg attbutes would be staghtfowad):. (,, K, ) Wthout loss of geealty, we wll assume that the judgmets of dffeece ole oly two ctea at a tme (as metoed the peous secto), but also that cteo s pat of all the dffeeces, so A ~ B j j a b j j j j j a b 0 a b ( ) K ( ) 0 j j j j j j a j b j Besdes these costas (j..), we must clude (..): et et Ay soluto ( L ) that espects the peous costas would lead to the desed multattbute alue fucto. If the soluto set s empty, epadg the eteal ages would be eeded ode to ega feasblty, uless the specfcatos of the DM ae so coflctg that o soluto ests eally plausble ages of the attbutes. I ode to sole the poblem of fdg a specfc soluto, the smplest way s to buld a lea pogam wth a coeet objecte fucto, le fo stace: m f m f y, wth et o y, y We ow llustate the pocedue usg aga the stuato of Eamples ad. Eample 3 Besdes the dffeece stated Eample ad the weghts dcated Eample, we obtaed fom the DM the followg eal ad eteal ages: We may ow establsh the set of costas addg the equato: o ( ) 0 leadg to the equato that elates the age lmts (ote that K j s a costat): Page 4 ( ) Usg ow the objecte fucto f metoed befoe, we fd multple solutos, as epected due to degees of feedom of the poblem. I fact, all the feasble solutos wth. ad 0, le ( )

5 Goupe de Taal Euopée Ade Multctèe à la Décso Euopea Wog Goup Multple Ctea Decso Adg Sée 3, º8, autome 008. Sees 3, º8, Fall 008. o ( ), ca be used, because they ae equalet all seses, as dscussed befoe. Wth f, we would obta the soluto of Eample, ceteed the eal age. The et fgue shows the ages ad depcts ths soluto Coclusos The pape shows how to deal wth the possble cosstecy betwee alue fuctos bult by the boo ad the pecepto of the DM egadg the elate mpotace of the ctea, eal decso-ad stuatos. The dscusso tus clea that the aswe esdes chagg the ages of the ddual alue fuctos, sce epadg the age of a attbute leads to a cease of the coespodet weght, whle espectg the emag pefeece fomato (judgmets of dffeece) poded by the DM. A complete pocedue was peseted to lea multattbute fuctos, ad ca easly be eteded to clude epoetal fuctos. The autho belees that the mateal peseted the pape s useful fo pacttoes, sce ths d of stuato appeas moe ofte tha ot. efeeces Baa e Costa, C, Vasc, JC. MACBETH. A eacte path towads the costucto of cadal alue fuctos. Iteatoal Tasactos Opeatoal eseach 994; 3: Baa e Costa, C et al. Deelopmet of eusable Bd Ealuato Models fo the Potuguese Electc Tasmsso Compay. Decso Aalyss 008; 5; : 4. Belto, V, Stewat, T. Multple Ctea Decso Aalyss. A Itegated Appoach. Kluwe Academc: Dodecht; 00. Cleme, T, elly, T. Mag had decsos wth decso tools. Dubuy Pess: Pacfc Goe, CA; 00. Keeey, L. Value-Focused Thg. A Path to Ceate Decso Mag. Haad Uesty Pess: Cambdge; 99; Keeey, L, affa, H. Decsos wth multple objectes: Pefeeces ad alue tadeoffs. Wley: New Yo; 976; oy, B. The outag appoach ad the foudatos of the ELECTE methods. Theoy ad Decso 99; 3: Saaty, TL. The Aalytc Heachy Pocess. Plag, Poty Settg, esouce Allocato. McGaw-Hll: New Yo; 980. Page 5