Rank Inclusion in Criteria Hierarchies

Transcription

1 Mat-.08 deedet Research Project Aled Mathematcs Rak cluso Crtera Herarches Jue 00 Helsk Uversty of Techology ystems Aalyss Laboratory Att Pukka Deartmet of Egeerg Physcs ad Mathematcs 48484T

2 Cotets. NTRODUCTON...3. EARLER METHOD NCOMPLETE ORDNAL JUDGEMENT DEVELOPMENT OF THE MODEL REDUCTON OF THE FEABLE REGON DETERMNATON OF PREFERENCE TRUCTURE A COMPUTATONAL EXAMPLE POBLTE FOR DECON UPPORT MPLCATON FOR PREFERENCE ELCTATON ON GROUP DECON MAKNG ON DECON UPPORT YTEM CONCLUON REFERENCE...33

3 . troducto Ofte, the objectve of decso makg s to fd a good soluto for the roblem wth several attrbutes. These roblems volve, for stace, large vestmets of cororatos ad decsos for stace eergy olcy (see, e.g., Hämäläe 988). Most roblems volve more tha oe or two objectves ad attrbutes. Ths makes the roblem wde ad comlex whereby aalytc methods may hel the DM addressg the roblem. Aalytc models usually demad some kd of weghtg of the attrbutes ad, resece of may decso makers (DM), mortace weghtg of the decso makers s demaded. multle crtera decso makg (MCDM) the recse evaluato of the attrbute weghts may be dffcult. addto, ofte a dffcult task for the DM s to decde whch of the attrbutes s the most mortat oe or whether a attrbute s more mortat tha aother. These dffcultes may result from the urgecy of the decso, lmted resources, tagble attrbutes (Km ad Ah 999) or - case of grou decso makg - from artcats dfferet level of kowledge about the roblem or dffereces ther objectves. A commo way to hadle these dffcultes s to comare attrbutes wth each other or to set cocered lmts, absolute or relatve, to the attrbute weghts. deed, several studes have bee made MCDM wth artal formato (see, e.g., alo 995, alo ad Hämäläe 99, Km ad Ha 000, Carrzosa et. al 995). some cases, the assgmet of weght tervals to attrbutes ca be roblematc or too tme-cosumg. Thus, t may be easer to mark a subset whch cotas the most mortat attrbute or the most mortat attrbutes. Ths aer focuses o these kd of stuatos. ecfcally, a attrbute the set of most mortat attrbutes ca be, fact, the least mortat oe as log as there s a secfed umber () of most mortat attrbutes the marked set. ths sese, the DM does ot have to gve accurate statemets about hs/her attrbute refereces. Ths aer s orgazed as follows. ome of the decso makg methods ad studes uder artal formato are dscussed secto. ecto 3 troduces a method that ca be used a stuato, where the DM marks a set of the most mortat attrbutes ad a examle s gve secto 4. ome ossbltes decso suort are reseted secto 5 ad secto 6 summarzes the results. 3

4 . Earler methods ths aer, we cosder a stuato whch there are attrbutes. Each attrbute s relevat to the decso ad the degree of the fluece s determed by ts attrbute weght, w. The DM assesses score to every alteratve uder each attrbute, resectvely. These scores are set as tervals or recsely. By gvg statemets about the mortace of the attrbutes, the DM sets tervals for the attrbute weghts. Ths way the DM has defed a feasble rego for the attrbute weghts. The fal value of a alteratve s the weghted average of the scores. ce the scores ad the attrbute weghts have bee comletely secfed, there are fte umber of feasble alteratve s values. ce the DM may wat to seek for ossble domace relatoshs, ths leads to a otmzato roblem, whch s dscussed more detal secto 3.3. everal methods for MCDM wth artal formato have bee develoed. Usg tervals arwse comarsos ca be a effcet way to elct a feasble rego for the weght herarchy. Arbel (989) exlas how recse artculato of attrbute refereces ca be exteded to maage mrecse referece statemets through equaltes. These equaltes mark out the feasble rego for the weght herarchy. Arbel based hs aroach o the aalytc herarchy rocess (AHP) (aaty 980), whch s carred out by askg arwse comarso questos about the mortace of refereces. Arbel defes a lear rogrammg (LP) roblem subject to the DM s referece statemets ad structural costrats (o-egatvty ad ormg of referece weghts). The objectve fucto of the LP roblem s a artfcal varable, whose urose s to check whether the roblem has a soluto. Arbel s work has bee the groudwork for several aroaches later. A related LP aroach has bee used, for stace, PAR (alo ad Hämäläe 99). PAR, the DM evaluates uer ad lower bouds for the quotet of two attrbute weghts. t s also ossble to set absolute bouds drectly. After estmatg all the weght bouds ossble,.e., lower ad uer bouds, a feasble rego for the attrbute weghts has bee set. The DM also has to rovde boud for the alteratves scores uder each attrbute. The fal decso s clear f a alteratve s total value s greater tha ay other s over the feasble rego (arwse domace). f the decso s ot clear, the DM s requested to set addtoal bouds va arwse comarsos or 4

5 drectly utl the decso ca be made. PAR also allows the DM to chage her/hs earler refereces, sce a ew feasble rego s defed at every terato. mlar methodology s reseted later case of grou decso makg (alo 995). Ths aroach s exteds PAR to stuato wth several decso makers. Ths makes the stuato more comlex, sce artcats objectves dffer from each other. All the artcats are asked for mrecse rakg of attrbutes ad alteratves uder the attrbutes to get the lear costrats for the feasble rego, whch creases the umber of costrats. The rocedure after ths s very smlar to that of PAR. A method, whch resembles PAR ad uses matrx otato has also bee develoed by Mármol, Puerto ad Ferádez (998). The beeft of matrx otato s the coveece ad effcecy storg formato. The research dscusses dfferet stuatos that mght occur wth artal formato. The model has bee exteded (Puerto et al. 000) to cover three dfferet decso crteras: Lalace crtero, Otmstc/essmstc crtero ad Hurwcz crtero. PRME (Preferece Ratos Multattrbute Evaluato) (alo ad Hämäläe 00) the DM evaluates statemets, whch ca be holstc comarsos betwee alteratves, ordal stregth of referece judgmets or ratos of value dffereces. As PAR, PRME rovdes formato about cosstecy ad domace relatos betwee alteratves throughout the aalyss. PRME Decsos (Gustafsso, alo ad Gustafsso 00) s a suort system based o the PRME method (see htt:// The comutatoal algorthm PRME Decsos s based o a lear objectve fucto ad costrats due to the DM s refereces ad s thus smlar to several other aroaches (see, e.g., alo ad Hämäläe 99, Park ad Km 997, Arbel 989). Park ad Km (997) have roosed some techques makg deccos uder comlete formato. They dvded mrecse formato of attrbute weghts to fve tyes. weak rakg: { w w j },. strct rakg: { w α }, w j 3. rakg wth multles: { w α w }, j 5

6 4. terval form: { α α + }, w ε 5. rakg of dffereces: { w w w w } for j k l where α,ε 0. j k l,, a somewhat dfferet cotext, whch cocers also utlty ad robablty, Park ad Km allow both attrbute weghts ad scores to be kow mrecsely. Ths assumto leads to a o-lear, o-covex otmzato roblem. Two techques for solvg ths are descrbed. the frst oe, t s assumed that comlete formato of utlty values are fuctoally deedet for each attrbute. Ths assumto allows to costruct a set of LP:s to relace the orgal o-lear objectve fucto. The secod techque assumes that there s fuctoal deedece betwee the comlete formato of utlty values for some attrbutes. As the frst techque, a set of LP:s ca be costructed ths oe, too. t must be oted that these LP models are oly aroxmatos of the orgal objectve fucto. Km ad Park also reset a model whch ca be used uder comlete robablty formato ad dscuss dfferet tyes of domace. The study dscussed above (Park ad Km 997) has bee advaced to grou decso makg settgs (Km ad Ah 999). grou decso makg the stuato may become cosstet, meag that the feasble regos of decso makers volved dffer sgfcatly from each other. ths stuato, Park ad Km suggest that the DM:s wthdraw some of the earler statemets. The costrats are dvded to two subsets, the oes that the DM does ot wat to chace ( crtcal ) ad the oes that are ot so mortat ( ucrtcal ). After ths the feasble rego s comosed a way that mmzes the umber of ucrtcal costrats whch volate the rego. Usg ths secfed rego, each of the DM:s sets domace relatos betwee alteratves usg arwse comarsos. Usg the arwse comarso results, aggregated et referece testy ca be calculated. t s a measure whch s defed as dfferece betwee the degree of a alteratve beg referred to other alteratves ad the degree the other alteratves are beg referred to ths alteratve. The alteratve to be chose s the oe wth greatest aggregated et referece testy. Km ad Ha (000) reseted a mathematcal rogrammg model to establsh domace relatos a herarchcally structured attrbute tree. Ths model allows the 6

7 DM to set lear costrats o attrbute weghts ad scores. They also reset a algorthm for fdg a value terval for the tomost attrbute of the tree. The model assumes cosstecy of DM s lear costrats, so a mstake settg the costrats ca lead to a false soluto. The model dscussed ths aer exteds the study of decso makg uder comlete formato to a ew drecto whch there are geerally more tha oe ossble referece orders of the attrbutes. Ths meas that the DM does ot have to assume artal order, weak order, lear order or acyclc relato (Fshbur 970) to hold betwee the attrbutes. For stace, let us cosder a decso makg roblem wth three attrbutes ad the DM states that the most mortat attrbute s ether attrbute umber oe or umber two. Ths leads to a stuato whch the referece order ca be (,3,), (,,3), (,3,) or (,,3). Thus, there are two stuatos whch a attrbute that belogs to the set of the most mortat attrbute caddates, s actually the least mortat. 7

8 3. comlete ordal judgemets 3.. Develomet of the model ths aer, we focus o a stuato whch the DM ca mark a subset whch cotas some umber of the most mortat attrbutes. We cosder a roblem wth attrbutes of whch the DM ca ot out the most mortat oes. ce the feasble rego for the attrbute weghts, whch results from ths kd of statemets, s geerally o-covex, we reset a techque that ca be used to artto the feasble rego to covex subsets. Ths facltates solvg the otmzato roblem. Let the feasble rego for all attrbute weghts be w ( w, w,..., w ) R wm 0 m,...,, wm. () For stace, the feasble rego () a three-dmesoal case for all weght herarches,, s a lae restrcted by three vectors betwee ots (,0,0), (0,,0) ad (0,0,). the otato we use ths aer mortat) as b. Defto. a ~ b m meas that a s at least as good (or J ' such that ', a ~ a ', j J. j Thus, at least of the attrbutes dexed are at least as mortat as ay of the attrbutes J. Let ad J be a artto of the frst ostve tegers,.e., {, } J,..., ad J φ. () Moreover, let be a set of dexes to whch the most mortat oes belog. These assumtos mea that all dexes belog to or J ad statemets, the feasble rego ca be wrtte J. Caterg the revous { w ', ' w w ', j ( J ) \ '} ( ). (3) j 8

9 whch ad J s a artto of the frst ostve tegers ad 0. Ths feasble rego s ot geerally covex, whch comlcates the use of otmzato techques. w 3 ({,}) w w Fgure. ({,}) Fgure llustrates a stuato where the DM states that attrbute or attrbute s the most mortat oe. 9

10 w 3 ({,}) w w Fgure. ({,}) the stuato llustrated Fgure the DM sets the attrbutes ad both to be more mortat tha attrbute 3. Let ad J be a artto of the frst ostve tegers, ad a set such that ', ' ad { w w w ', j ( J ) \ '} ', ). (4) j As (4) defes, the attrbute weghts for the attrbutes are greater tha or equal to the weghts of those ot. Thus, ', ) s the feasble rego uder a assumto that the most mortat attrbutes are. Ths meas that ', ) ca be used to exame a stuato where the DM states the attrbutes are the most mortat oes. the otato we use ths aer, " A B" meas that A s a roer subset of B. Lemma Let ad J be a artto of the frst ostve tegers ad let A ad B be sets such that A, B, A B ad A B, where <. The, t( A, )) t( B, )) φ. 0

11 f A B φ, we say that A ad B are searated. Thereby ', ) :s have searated terors. Proof We ck a arbtrary w t( A, )) ad show that ths w s ot t( B, )). Let w t( A, )) be arbtrary. ce w t( A, )), accordg to (4), w 0 ( J ) ad w > w A, j ( J ), j A. addto, sce j A B ad A B, t A such that t B ad u B such that u A. Ths mles that w > w. Thus, w ( B, )) ad addto ths mles that t u w t( B, )). ce w s arbtrary, t( A, )) t( B, )) φ. Lemma Let be a ostve teger such that ( ) ', ). '. The, Proof Accordg to (4), f w belogs to ', ), the w w ', j ( J) \ '. f we take a uo of ', ) :s over all :s ossble, for a arbtrary w () there must be a set * such that w ' ', ) w '* ad w w '*, j ( J ) \ '*. Thus, ( ). j j O the other had, f w (), there s a subset such that w w ', j ( J) \ '. Thus, w j ' ', ). addto, sce accordg to Lemma, t( A, )) t( A', )) φ, A A', A A', the orgal feasble rego () has bee searated to subsets ', ) wth searated terors. Ths esures that dvdg the feasble rego ths way does ot

12 cause dsaearace of ay of the feasble ots. addto, oe of the teror ots ', ) :s are more tha oe subset. ca be chose ways, so there are ', ) sets. dfferet w 3 {}, {,}) {}, {,}) w w Fgure 3. Dvso of the feasble rego to covex regos Fgure 3 llustrates the dvso of ({, }) to two covex subsets defed (4). Theorem Proof ', ) s covex. Let [ 0,] λ. ', ) s covex, f x, x ', ) λ x + ( λ) x ', ). (5) Let x w, w,..., w ), x w, w,..., w ) ad x ( ( 3 λx + ( λ) x ( λw + ( λ) w,..., λw + ( λ) w ), λ [ 0,]. Frst we must check that w x 3. 0 m,...,, sce w, w, λ,( λ) 3m m m 0 (6)

13 m w 3 m ( λw m + ( λ) wm ) λ w m + ( λ) wm λ + λ m m m (7) Accordg to (6) ad (7) ', ) too. t ca be see that x 3. The ext ste s to check whether x 3 belogs to w3 λ w + ( λ) w λw j + ( λ) w j w3 j ', j ( J) \ ', sce w w w w ad λ,( λ) 0. (8) j, j Thus, x ', ). Ad accordg to (5), ', ) s covex. 3 ce, accordg to Theorem, ', ) s covex, t s ossble to use dfferet methods that have bee develoed for solvg otmzato roblems a covex rego. The ext two lemmas show how two ', ) :s dffer from each other f the sze of s dssmlar or cotas dfferet umber of the most mortat attrbutes. Lemma 3 Assume that ad k are tegers, such that 0, k ad, J s a artto of the frst ostve tegers. The, ( ) ( ) k. k > Proof We rove Lemma 3 two ways, ad. Both drectos are dscussed searately. Frst we have to show that f > k, every ot () must also be (). We do ths by focusg o two cosecutve ad k,.e. + ad. k Accordg to (3) ', ' + such that w w j ', j ( J ) \ '. Ths mles that '' ', ' ' such that w w j ' ', j ( J ) \ sgle would be eough to show that x + ( ) x ( ). ''. fact a Next we have to show that there s a ot (), whch does ot belog to + ( ). For stace, f we choose w to be a ot such that w ( '' ' + w ( ), w ( ) + { k} ), k J, '', ' ' ad w 0 ' { k}. Thus, () s also a roer subset of + ( )., 3

14 Now, sce ) ( ) ( )... ( ), ( ) ( ) too, whe k. f 0 ( ( ) ( ), the x ( ) x ( ). ce w (), k k k > ', ' w w j ', j ( J ) \ '. Ths mles that ' ', '' k w w '', j ( J ) '' oly f k for a arbtrary w. t stll has to be show that j \ k. ( ) ( ) x ( ), x ( ), so there has to be a ot (), whch s k k ot (). f k ( ) ( ), so k, whch mles that > k. k k Lemma 4 Let us assume that there are two arttos of the frst ostve tegers,, J ad, J. The, (. ) ( ) Lemma 3 ad Lemma 4 show how chagg the sze of ad the umber of the most mortat attrbutes affect o the feasble rego. The urose of reducg the feasble rego s dscussed later ths aer. Proof Accordg to Lemma, ( ) ', ) ad ( ) ',. ' ) ' ce, t s obvous that ' ', '. addto, sce s a roer subset of, there are subsets of every sze whch are ot. Thus, ( ) ', ) ', ) ', ) ( ) ', ) ' ' ', ' ', ' whch mlcates that ( ) ( ). Frst we have to show that every dex s as well. After that a addtoal dex from must exclude to comlete the roof., 4

15 ce ( ) ( ), every arbtrary w ( ) must also belog to ( ). For ths w there s a set of dexes ', ', so that w w j ', j ( J ) \ '. Ths same set must also be a subset of, sce otherwse w ( ). addto, because w s arbtrary, ca cota ay. Thus,. Let w ), w ( ). ce w ), ( ( ', ' such that w w j ', j ( J) \ '. addto, sce w ( ), a set, ' such that w w ', j ( J ) \ ' caot ' j exst. Ths set does exst f, whch mles that must hold. Thus,. 3.. Reducto of the feasble rego As Lemma 4 suggests, reducg the set of the most mortat attrbutes reduces the feasble rego. The reducto of the feasble rego becomes mortat, f the gve referece statemets are suffcet for domace structures. However, oe should ot restrct the feasble rego f the orgal referece statemets seem correct after rethkg. To get a llustratve cture of the effcecy of reducg the feasble rego, we eed a quatty that measures the sze of the rego. f we hold costat, we have ', ) sets, where '. ce all the attrbutes are equal osto before weghtg, t s reasoable to defe a outer measure such that every ', ) has the same sze. We defe the measure, ϕ, such a way that t roduces the feasble rego s share of the whole feasble sace. ce, accordg to Lemma ad Lemma, ', ) :s have searated terors ad ( ) s a uo of these ', ) :s, the followg measure ca also be used to evaluate the sze of ( ). For stace, f the weght herarchy s three-dmesoal, ϕ 3 s the quotet of the 5

16 feasble rego s area ad the area of. The ossble values for ϕ 3 ths case are 0, /3, /3 ad (see Fgure whch ϕ 3 3 ad Fgure whch ϕ 3 3 ). Defto Let U Y. ϕ : Y A s a outer measure, f t fulflls the followg codtos (Garey ad Zemer 995):. ϕ ( U ) 0, whe U φ,. 0 ϕ ( U ) U Y, 3. ϕ (, U ) ϕ( U ) U U 4. ϕ( U ) ϕ( U ) for ay coutable collecto of sets { U } Lemma 5 Y Let U be a collecto of dsjot ', ) :s ad U be the umber of elemets,.e., ', ) :s, U. Furthermore, let Y be the sace whch cotas all U :s ad ( U ϕ ) a fucto such that ϕ : Y {0,,..., }, for every U Y.. Proof ϕ ( U ) U s a outer measure. We have to check that ϕ Lemma 5 fulflls the four codtos reseted above. The frst codto holds, sce f U s emty, U 0. The secod oe holds too, sce 0 < ad the maxmum of U s. the thrd oe, sce U U, 6

17 $#""! $#""! U U ad ϕ ( U ) U U ϕ ( U ). Hece, the thrd codto holds. Cosder a uo of m arbtrary dssmlar elemets, ϕ ( U ) U :s. case these U :s cota oly ϕ ( U m m m m U U other had, f there are elemets, whch belog to more tha oe dssmlar elemets m U m U ). O the U, the umber of s smaller tha the total umber of elemets m U m m m m. Ths dcates, that ϕ ( U ) U < U ϕ ( U ). Although m s arbtrary, the evet of fte umber of subsets must be dscussed searately. ce there ca ot be more tha searate ', ) :s, eve f there was a fte umber of subsets the uo, the maxmum of the left had sde s ad, resectvely, the rght had sde has o maxmum. Ths meas that the fourth codto s fulflled., ce ( ) calculate s a uo of ', ) :s. Lemma 6 ϕ for the feasble rego by defg searate ', ) :s, such that ', we ca U to be these dsjot Assume that the DM has dcated that k attrbutes defe the set ad that cotas the most mortat attrbutes; thus, the feasble rego s ). Furthermore, let U to be the set of all dsjot ( ', ) :s. ' ( 7

18 f the DM removes a attrbute from the set, there are k- attrbutes the ew set,. The sze of the feasble rego relatve to the earler feasble rego s k q( k, ). k Proof Accordg to Lemma 5, ϕ ( U ). We ca study the sze of the ew feasble rego by studyg the quotet of the rego measures. Let U ' be the set of all dsjot ( ', ) :s. ' ϕ ( U q( k, ) ϕ ( U ') ) ( k )! k ( k )!( k ) k k k. k k k ( k )!!( k )! k!!( k )! Lemma 6 ca be geeralzed to maage also bgger reductos. Let us assume a stuato whch the DM wats to reduce by s attrbutes. Usg Lemma 6 s tmes cosecutvely, we get s k ( k )( k )... ( k s + ) Q( s, k, ) k k( k )( k )... ( k s + ( k )!( k s)! ( k s)! k! 0 ) k s k s. (9) 8

19 t must be oted that dervg (9), the sze of gets smaller by oe ut at every ste. As a examle, f the orgal sze of s 7 ad, reducg by 4 uts reduces the feasble rego to of the orgal sze Determato of referece structures ce we focus o a stuato, whch the DM s objectve s to fd the otmal soluto for the roblem uder dscusso, a objectve fucto for the aalytc decso makg model s requred. The objectve fucto deeds o the decso crtero that has bee chose ad t ca be both lear or o-lear. Here, we assume that the objectve fucto s lear. The mrecsely estmated set ad the arameter defe a feasble rego, (), for the attrbute weghts wth lear costrats (3). The roblem ca be wrtte subject to max/m Z( x) w v ( x) (0) w (), whch w s the attrbute weght for attrbute ad v (x) s the score of alteratve x uder attrbute. ce () defed ths model s ot geerally covex, t s reasoable to dvde t to ', ) :s (4). f (0) s maxmzed, the DM uses a crtero called maxmax crtero (alo ad Hämäläe 00). Ths ca be terreted to mea that the DM s otmstc ad assumes the best result ossble to hae. Vce versa, f (0) s frst mmzed every ', ) ad the maxmum of these mma s selected, the DM uses maxm crtero (alo ad Hämäläe 00). f Z( x) Z( y) through the feasble rego, the x domates y (arwse domace) (alo ad Hämäläe 99). A aroach that dffers somewhat from the oe reseted above s arwse comarsos betwee alteratves (alo ad Hämäläe 99). The DM erforms arwse comarsos to solve arwse bouds, µ ( x, y), uder each attrbute, 9

20 resectvely. Ths s doe by mmzg the score dfferece betwee the alteratves, [ v ( x) v ( )] µ ( x, y) m y. () t s oteworthy that alteratve scores ca also be set as tervals, sce µ :s are mma of score dffereces. Parwse comarsos are reroduced betwee each ar of alteratves (x,y). The score dffereces should be ormalzed so that they are comarable wth other attrbutes score dffereces before weghtg. Alteratve x domates y ff µ ( x, y) w µ ( x, y) 0 w ( ). () t s otable, that m w µ ( x, y) ca be solved each ', ) wth smle LP methods ad f oe or more of these mma s egatve, () does ot hold. Clearly, a stuato whch () does ot hold betwee ay ar of alteratves s also ossble. ths stuato o further decsos ca be made ad the DM has to reduce the set of the most mortat attrbutes, whch case, (see Lemma 4) the feasble rego becomes smaller. Aother way to restrct the feasble rego, accordg to Lemma 3, s to elarge. f () holds betwee some x ad every other alteratve, x s the best alteratve sce t domates all other alteratves. More geerally, a LP roblem ca also be wrtte geeral form. Thus, t ca be wrtte subject to max / m x, Z T k x (3) whch k s a coeffcet vector, x varable ad the feasble rego. There are several techques ad comuter software for solvg ths LP a covex rego. ce s ot ecessarly covex, a method to acheve covexty s eeded. Brach ad Boud algorthm (B&B) (see, e.g., Taha 997) s geerated for solvg teger rogrammg roblems but t ca be modfed to solve o-covex otmzato roblems as a set of covex roblems. The dea of B&B s to dvde the 0

21 feasble rego to smaller regos such that the o-teger otmum, that has already bee foud, s always left betwee the ew regos. addto, the ew costrats are the revous ad the ext teger from the otmum for oe of the varables. otmum ew rego ew rego Fgure 4. The dea of Brach ad Boud algorthm Fgure 4 gves a examle o the dvso of the feasble rego to ew regos a two-dmesoal roblem. The orgal (o-lear) objectve fucto s aga otmzed these ew regos ad a ew dvso s made. Ths rocedure s cotued as far as the otmum s a teger every rego or every o-teger otmum s smaller ( case of maxmzato) tha oe of the teger otma. The soluto to the roblem s the most otmal teger otmum. ths case B&B s used a way that dvdes the feasble rego to covex regos. each of these covex regos, the otmum of the objectve fucto s solved. The overall soluto s the maxmum of the maxma (or mmum of the mma) these covex regos,.e., max Z T max{max k x x '}. (4) ' ce :s are covex, the maxmum of the objectve fucto ca be easly solved by usg ths B&B techque.

22 4. A comutatoal examle Let us assume a decso makg roblem wth seve attrbutes, whch are umbered from to 7. The DM decdes that the two most mortat attrbutes are the set {,,3,4}. Ths leaves J{5,6,7}. The feasble rego ca ow be wrtte { w ' {,,3,4}, ' w w ', j ( J ) \ '} ({,,3,4}) j. Ths rego s dvded to covex regos wth the hel of B&B. Ths leaves us wth sx covex regos otated as (4). Usg ths otato, we have {,}, ), {,3}, ), {,4}, ), {,3}, ), {,4}, ) ad {3,4}, ). ths examle, the roblem s aroached usg arwse comarsos as dscussed secto 3.3. ({,,3,4}) {,}, ) {,3}, ) {,4}, ) {,3}, ) {,4}, ) {3,4}, ) Fgure 5. Use of B&B dvso of the feasble rego ad solvg the roblem Fgure 5 llustrates the use of B&B solvg the roblem wth arwse comarsos. Next the DM erforms arwse comarsos betwee alteratves uder each attrbute. These comarsos do ot have to be accurate, stead, mrecse formato s allowed. Ths meas that the DM ca set arwse comarsos as score tervals. Table troduces the DM s decsos.

23 Table. The DM s mrecse evaluato of alteratves uder each attrbute attr. attr. attr. 3 attr. 4 attr. 5 attr. 6 attr. 7 A-A [3, 6] [,.] [-, -0.5] [0, 0.5] [-3, -] [, ] [0, 3] A-A3 [, ] [-, 0,8] [0.5, ] [-., -.8] [-4, -] [,.5] [-, 0] A-A3 [-3, 0] [-, 0] [4, 5] [-3, -] [-0.5, 0.5] [-0.5, 0.3] [-4, -] The attrbutes are stored colums ad the altervatves, A, A ad A3, are comared o rows. The score tervals of the arwse comarsos are ormalzed a way that t s ossble to comare the tervals betwee dfferet attrbutes. The scale of the scores s chose such a way that the scores are easy to deal wth. Usg (), arwse bouds ca be solved from the results of Table. Table resets these arwse bouds. Table. Parwse bouds attr. attr. attr. 3 attr. 4 attr. 5 attr. 6 attr. 7 µ (,) µ (,) µ (,3) µ (3,) µ (,3) µ (3,) The arwse bouds of Table are calculated from the mma or maxma of the arwse comarsos. For stace, µ (, ) uder attrbute s the mmum of the comarso betwee alteratves ad uder attrbute. Vce versa, µ (,) uder attrbute s mus maxmum of the same comarso. Now that the arwse bouds have bee solved, the ext task s to seek for arwse domace. As (), ths s doe by calculatg a weghted average of the arwse bouds. ce arwse domace s desred, the objectve of the LP roblem s to fd the mmum of all feasble µ ( x, y) :s. The ext ste s to check out, whether there s arwse domace betwee some of the alteratves. As oted secto 3.3., f m µ ( x, y) s egatve some ', ), x does ot domate y. 3

24 {,}, ) for examle, we have the followg LP for determg domace betwee alteratves ad : m 3w + w w3 3w5 + w6 subject to w w {3,4,5,6,7}, w w {3,4,5,6,7}, w 0 {,...,7} ad w + w + w + w + w + w + w The coeffcets of the varables are the arwse bouds uder each attrbute (see Table ). The costrats result from the defto (4) of {,}, ). The soluto of ths LP s 0 wth varable values w w w w w 0.. Ths meas that alteratve deed domates alteratve {,}, ). fact, the mmum attrbutes ad have the exact same weght as attrbutes 3, 5 ad 7 do. A smlar rocedure to ths has to be reeated other ', ) :s as far, as a egatve soluto aears, to fd out whether there s absolute domace betwee alteratves ad. Table 3. olutos of the LP:s {,}, ) {,3}, ) {,4}, ) {,3}, ) {,4}, ) {3,4}, ) µ (,) µ (,) µ (,3) µ (3,) µ (,3) µ (3,) Table 3 resets the mma of the arwse bouds betwee each ar every rego ', ). As Table 3 dcates, there are o domace ars. Moreover, the mma are egatve excet for oe case. As Table 3 llustrates, alteratve s qute close to domatg alteratve ad, vce versa, alteratve s far away from domatg alteratve. As wll be dscussed later secto 5.., the reaso for ot achevg 4

25 domace structures ca result from some of the attrbutes havg weghts equal to zero the LP mma. 5

26 5. Possbltes for decso suort 5.. mlcatos for referece elctato Though the model demads oly a smle set to whch the most mortat attrbutes belog, the set ad arameter should be carefully ad thoroughly defed. By dog ths, the feasble rego should become smaller. addto, the DM ca restrct the feasble rego by gvg addtoal statemets cocerg attrbutes. For stace, f the DM sets frst that the 5 most mortat attrbutes belog to ad the that the most mortat belog to ', the feasble rego becomes smaller. Reducg the feasble rego s mortat, sce whe the rego s smaller, there s smaller rego whch a alteratve has to domate all other alteratves. several earler aroaches the DM s able to set relatve tervals for the attrbute weghts. ths model, these lmts have to be set a dfferet, less effectve way. Fgure a stuato whch there are three attrbutes ad the DM has determed that attrbutes ad 3 are more mortat tha attrbute, but ot more tha twce as mortat. Ths ca be wrtte w w, w3 w. The DM s uable to secfy ay relatos betwee attrbutes ad 3. Ths kd of stuato ca be aroxmated as follows: the two most mortat attrbutes are umbers oe ad three. However, ths does ot take to accout the statemet that attrbutes ad 3 are ot more tha twce as mortat as attrbute. Thus, the feasble rego s too wde, whch may cause a gratutous rejecto of a arwse domace hyothess. t s obvous that ths model s ot arorate for stuatos whch the DM secfes relatve tervals uless these tervals are set searately after the dvso of the feasble rego. f the DM s urose s to exame arwse domace betwee alteratves, the above model s a lttle roblematc. ce the mma of arwse comarsos may be egatve, the mmzato soluto s a ot whch all other attrbute weghts are zero excet for the attrbute that has the greatest egatve arwse boud. Ths stuato ca aturally oly hae f these attrbutes weght s a feasble soluto (.e., the dscussed attrbute belogs to the set ). To avod ths kd of stuato, the DM ca set addtoal costrats for the attrbute weghts. For stace, the DM ca set a restrcto, that every attrbute must have at least a re-secfed mmum weght. Ths makes sese ad s mortat, sce every crtero volved the 6

27 roblem must also fluece the decso. Otherwse, the attrbute s suerfluous ad should have bee left out from the model already the begg. t s lausble that the attrbute weghts should be ket geuely ostve every ossble case regardless of the decso crtero. To get a thorough cocet of the decso roblem soluto, the objectve fucto has to be otmzed over the whole feasble rego. As oted above secto 3.., the umber of covex subsets, ', ), equals to. Thus, the DM has to solve otmzato roblems. t s also oteworthy that f () s egatve some ', ), arwse domace assumto ca be rejected. Thus, t s ot always ecessary to solve every otmzato roblem, f the DM oly wats to get formato about ossble domace. 5.. O grou decso makg the case of several decso makers, the roblems ted to become more dffcult to solve wth the hel of a aalytc model sce the decso makers oos ad level of kowledge dffer from each other. Because the DM:s attrbute refereces may dffer from each other, the feasble rego for the weght herarchy may be uequal betwee the DM:s. ths kd of stuato there are two ways to defe the feasble rego:. the whole grou has the same feasble rego,. every DM has ther ow feasble rego. the latter oe, there s o roblem defg the feasble rego. O the cotrary, t ca be defed as (3). the revous oe there are a few ways to determe the feasble rego for the weght herarchy. ome ossble ways are dscussed ext. The feasble rego ca be defed such that every DM s ersoal feasble rego must clude the commo feasble rego. Thus, the commo feasble rego s the tersecto of the DM:s feasble regos. Here, we assume that there are m DM:s. The feasble rego ca ow be wrtte ( {,..., m} ), 7

28 whch s the arameter of DM ad ad J are the corresodg dex sets. ce {,..., m} whch {,..., m} ( ) ' ( '), ' ad ' m {m { }, '}, there s a ossblty that a commo feasble rego caot be defed. Moreover, f DM states that {s}, the oly ossble commo feasble rego s a rego defed uder a assumto that attrbute umber s s the most mortat oe. addto, ths requres that the attrbute s belogs to each of the :s. All other referece statemets are useless. f the grou s able to agree o the arameter, ths kd of stuato s avoded but stll ths aroach requres commo oos at least regardg to some attrbutes. Ths aroach should oly be used stuatos whch every DM must totally accet the decso. Aother way s to defe the commo feasble rego as the uo of the DM:s ersoal feasble regos. Thus, the feasble rego ca be wrtte {,..., m} whch ' ' ad {,..., m} ( ) ' ( '), ' max{ }. The roblem costructg the commo feasble rego ths way, s that the feasble rego easly becomes very loose. The weghtg the fal decso may also ed u a ot, whch s feasble for oe of the ersoal feasble regos. ce both of the methods above have roblems costructg the feasble rego or keeg t arrow eough, a dfferet kd of aroach s useful. Here, we cosder that each of the m DM:s defes a set wth the same sze. Each of these sets cotas 8

29 the most mortat of attrbutes. Ths ca be uderstood as a votg rocess: each DM has to vote for a re-secfed umber of attrbutes. The attrbutes wth most votes defe the feasble rego. f some attrbutes have the same umber of votes tha the attrbute wth least votes the set of the attrbutes wth the most votes, the most mortat attrbutes caot be decded uambguously. ths stuato, these attrbutes are cluded to the set f the growth of the set s smaller tha the dmuto of the set f all these attrbutes wth the same amout of votes were left out. Vce versa, f the growth of the set s bgger, they are all left out. By defg the feasble rego ths way, the grou has decded the set of the most mortat attrbutes. Because of ths, the feasble rego defed by these attrbutes s covex. addto to settg the set of the most mortat attrbutes, each of the DM:s also has to set scores or score tervals for the alteratves uder each attrbute. Parwse comarsos are allowed too. f the scores are set as tervals, the lmts for the commo tervals are determed by the average values of the ersoal lmts. other case, the scores are the mea values of the ersoal scores. Whe the commo feasble rego for the weght herarchy ad the commo scores have bee solved, the roblem ca be solved lkewse secto O decso suort systems Nowadays decso suort systems (D) are wdely used to hel the DM to fd a soluto for the roblem uder dscusso. Maly these systems are software (e.g., Web-HPRE (Mustajok ad Hämäläe 000), PRME Decsos (Gustafsso, alo ad Gustafsso 00), Logcal Decsos (see, e.g., Cleme 996)), whch are based o aalytc models. ma, these software clude sestvty aalyss, whch ca be laborous to erform ad, because of the cotous growth comutatoal ower, they maage to hadle large roblems. For effcet use of ths model, a decso suort software would be useful. Frst the DM would eter the attrbutes ad the alteratves. The elctato rocedure would cosst of two arts. the frst art the software would ask the set of attrbutes whch cotas the most mortat oes. Parameter ad the sze of the asked set would be defed by the DM. the secod art, the DM s would be asked to eter score tervals for the alteratves uder each attrbute. After the elctato, the software would solve the otmzato roblems usg B&B modfcato as reseted 9

30 secto 3.3. These results are usually reseted both umercally ad grahcally. grahcal resetato the results are ofte reseted as bar grahs (see, e.g., PRME Decsos, Logcal Decsos, Web-HPRE). Value Alteratve Alteratve Alteratve 3 Alteratves Fgure 6. Grahcal resetato of results Fgure 6 llustrates a examle of decso roblem s results as a bar grah. Bar grahs are llustratve, sce the DM s able to see the results for dfferet crteros easly at the same tme. the examle of Fgure 6, clearly alteratve 3 s the best oe sce ts mmum value (0.55) s greater tha the maxmum values of the other alteratves. Thus, t domates alteratves ad. Whle comarg alteratves ad, alteratve s better f otmstc crtero s used (0.53 > 0.4) ad alteratve s better f essmstc crtero s used (0.5 > 0.). These results mea that there s o arwse domace betwee alteratves ad. As Fgure 6 llustrates, ths formato s easy to see wthout eve otcg the umbers. ths model, the software would reduce the DM s work f the DM wats to secfy the statemets cocerg the attrbute refereces or the scores attached to each alteratve uder the attrbutes. t s also llustratve to see the chages the grahs, f the set of the most mortat attrbutes s modfed. estvty aalyss eables the DM to study how much would the scores attached to alteratves have to chage uder each of the attrbutes to geerate domace betwee the alteratves or 30

31 whether there would be a sgfcat dfferece the soluto f the DM reduced the set. 3

32 6. Cocluso The model dscussed ths aer focuses o a decso makg stuato, where the DM s allowed to gve mrecse statemets about the weght herarchy by defg a set of attrbutes whch cotas the most mortat oes. Ths formato defes a feasble rego for the weght herarchy whch s ot geerally covex. No-covexty of the feasble rego results from the statemet that oly some of the attrbutes a secfed set are guarateed to be more mortat tha the attrbutes left out from ths secfc set. The feasble rego ca be dvded to covex subsets wth the hel of a modfcato of Brach ad Boud algorthm. The Brach ad Boud modfcato s effcet dvso of the covex rego, sce t creates the smallest ossble umber of covex subsets wth searated terors. The uformty of these covex regos hels the formulato of the otmzato roblems. order to make the feasble rego smaller ad more realstc, addtoal lear costrats ca be stated each of the covex regos searately. ce the feasble rego s dvded to subsets uder a assumto that some secfed set of dexes defes the set of most mortat attrbutes, ths model allows accurate examato of dfferet combatos. addto, f the most mortat attrbutes set s reduced after calculato, otmzato solutos are stll avalable wthout further calculatos. The model reseted ths aer does ot allow the DM to gve statemets about the least mortat attrbutes,.e., the least mortat attrbutes belog to A. The roblem s smlar to ths ad ca be resumably solved usg aalogous methodology. Eve statemets from the mddle of the referece order should be ossble to hadle a smlar way. 3

33 7. Refereces Arbel, A. (989). Aroxmate Artculato of Preferece ad Prorty Dervato. Euroea Joural of Oeratoal Research. 43, Carrzosa, E., Code, E., Feradez, F.R. ad Puerto, J. (995). Mult-crtera Aalyss wth Partal formato About the Weghtg Coeffcets. Euroea Joural of Oeratoal Research. 8, Cleme, R. T. (996). Makg Hard Decsos A troducto to Decso Aalyss, d Edto. Duxbury Press. Fshbur, P. C. (970). Utlty Theory for Decso Makg. Joh Wley & os, c. Garey, R. F. ad Zemer, W. P. (995). Moder Real Aalyss. PW Publshg Comay. Gustafsso J., alo A. ad Gustafsso T. (00). PRME Decsos: A teractve Tool for Value Tree Aalyss. Proceedgs of the XV Coferece o Multle Crtera Decso Makg, Akara, Turkey, July 000. Hämäläe, R. P. (988). Comuter Asssted Eergy Polcy Aalyss the Parlamet of Flad, terfaces. 8, -3. Km, J. K. ad Cho,. H. (00). A Utlty Rage-based teractve Grou uort ystem for Multattrbute Decso Makg. Comuters & Oeratos Research. 8, Km,. H. ad Ah, B.. (999). teractve Grou Decso Makg Procedure uder comlete formato. Euroea Joural of Oeratoal Research. 6, Km,. H. ad Ha, C. H. (000). Establshg Domace Betwee Alteratves wth comlete formato a Herarchcally tructured Attrbute Tree. Euroea Joural of Oeratoal Research., Mármol, A. M., Puerto, J. ad Ferádez, F. R. (998). The Use of Partal formato o Weghts Multcrtera Decso Problems. Joural of Mult-crtera Decso Aalyss. 7,

34 Mustajok, J. ad Hämäläe, R. P. (000). Web-HPRE: Global Decso uort by Value Tree ad AHP Aalyss. formato ystems ad Oeratoal Research. 38, Park, K.. ad Km,. H. (997). Tools for teractve Decso Makg wth comletely detfed formato. Euroea Joural of Oeratoal Research. 98, -3. Puerto, J., Mármol, A. M., Moroy, L., Feradez, F.R. (000). Decso Crtera wth Partal formato. teratoal Trasactos Oeratoal Research. 7, aaty, T. L. (980). The Aalytc Herarchy Process. New York: McGraw-Hll. alo, A. (995). teractve Decso Adg for Grou Decso uort. Euroea Joural of Oeratoal Research. 84, alo, A. ad Hämäläe, R. P. (99). Preferece Assessmet by mrecse Rato tatemets. Oeratos Reseach. 40, alo, A. ad Hämäläe, R. P. (995). Preferece Programmg Through Aroxmate Rato Comarsos. Euroea Joural of Oeratoal Research. 8, alo, A. ad Hämäläe, R. P. (00). Preferece Ratos Multattrbute Evaluato (PRME) Elctato ad Decso Procedures uder comlete formato. EEE Trasactos o ystems, Ma, ad Cyberetcs. (to aear). Taha, H. A. (997). Oeratos Research: A troducto, 6 th Edto. Pretce-Hall. 34