RICH method Simulation study

Transcription

1 HELSINKI UNIVERSITY OF TECHNOLOGY 3 Mrch 2002 RICH method Smulto study Pekk Ahljärv 48424P phljr@cc.hut.f

2 Cotets Itroducto Rk Icluso Crter Herrches (RICH) method Smulto Fesble extreme pots Decso rules Smulto results Dscusso d coclusos... 3

3 Itroducto It s typcl of mult-ttrbute decso mkg tht the decso mker (DM) s wllg or ble to provde oly complete vlue formto o prmeters such s ttrbute weghts d ltertve. The elctto of precsely defed ttrbute weghts s ofte dffcult d mprctcl. It tkes tme d resources d therefore the urgecy of the decso d lck of resources for the elctto of completely defed ttrbute weghts mke t hrd to crry out (Km d H 2000). Also possble mbgutes the defto of ttrbutes for tgble objectves my cuse problems (Km d Ah 999). RICH s ew method developed to the lyss of complete formto herrchcl weghtg models. I RICH the DM does ot hve to kow the complete rk order of the ttrbutes but sted he s llowed to specfy subsets of ttrbutes whch cot the most mportt ttrbute or to ssocte sets of rk umbers wth ttrbutes (Slo d Pukk 2002). I ths pper we study the computtol effectveess of the method wth extesve smulto. Results re compred wth the oes obted for complete rk order d equl weghts. Severl methods for delg wth complete formto hve bee developed over the pst decde. We wll shortly troduce three of them: PAIRS, PRIME d mthemtcl progrmmg model to estblsh domce reltos herrchclly structured ttrbute tree. The PAIRS (Preferece Assessmet by Imprecse Rto Sttemets) method (Slo d Hämäläe 992) llows the DM to express pproxmte preferece sttemets vlue trees s tervl judgmets. These dcte rge for the reltve mportce of the ttrbutes d correspod to ler costrts o the locl weghts. Domce results for the ltertves c be computed by solvg seres of ler progrmmg problems. The terctve decso support process provdes DM wth cosstecy tervl tht gudes the DM the specfcto of ew comprsos. PRIME (Prefece Rtos Multttrbute Evluto) (Slo d Hämäläe 200) seeks to reduce the elctto effort multttrbute evluto uder certty by cceptg mprecse preferece sttemets whch my be ether holstc comprsos betwee ltertves, ordl stregth of preferece judgmets or rto comprsos bout preferece dffereces. Preferece sttemets re modeled s ler costrts d these costrts become more restrctve s ew sttemets d refemets re troduced. More restrctve costrts permt more coclusve domce results to be ferred. 2

4 Km d H (2000) preset mthemtcl progrmmg model to estblsh domce reltos herrchclly structured ttrbute tree. The DM c provde complete formto o the ttrbute weght d the vlue of ltertves. These become set of costrts the model. Km d H preset lgorthm for obtg the vlue tervl o the topmost ttrbute. The vlue tervls o y ttrbute the ttrbute tree c be derved d the obted vlue tervl o y ttrbute s utlzed to obt the vlue tervl o hgher level ttrbute. The pper s rrged s follows. Frst we preset bscs of RICH method Secto 2. I Secto 3 we expl how the smulto ws crred out. Smulto results re showed Secto 4 d coclusos Secto 5. 2 Rk Icluso Crter Herrches (RICH) method I ths pper we oly use ddtve vlue model,..., x ) = wv ( x ) = V ( x, where V s the totl vlue of ltertve x, w s the weght of ttrbute, x s ltertve s x cosequece o ttrbute d v ( x ) s the vlue/score tht correspods to cosequece x. Attrbute s weght descrbes the chge totl vlue whe ttrbute chges from the worst possble cosequece to w the best oe. Usully weghts re ormlzed such tht the DM, bgger s the correspodg weght. w = =. More mportt the ttrbute s to RICH s bsed o the de tht we compose rego of the fesble ttrbute weghts bsed o ll the possble rk orders tht re comptble wth the DM s prefereces. The fesble rego for ll ttrbute weghts c be wrtte (Slo d Pukk 2002): S w = w = ( w, w2,..., w ) R w 0 =,..., w = =. () Here there re ttrbutes d the ttrbute weghts dd up to. Whe the DM defes set of ttrbutes, I, whose rks re set of rks, J, the umber of fesble rk orders s (Slo d Pukk 2002): 3

5 I!( J )!, f I J ( I J )! R( I, J ) =, (2) J!( I )!, f I < J ( J I )! where R ( I, J ) s the umber of complete rk orders the set of fesble rk orders R( I, J ) d I s the sze (.e. umber of ttrbutes) of the set of ttrbutes I d J s the sze of the set of rks. Complete rk order s defed s follows (Slo d Pukk 2002): Defto. Vector r = (r(),r(2),,r()) s complete rk order, f the elemets of r re permutto of {,2,,},.e., r() r(j) j,j {,,} d r() f r(+) =,,-. The fesble rego duced by complete rk order r s (Slo d Pukk 2002): { w S w w k =,2,..., } S( r ) = w r( k ) r( k+ ), (3) so the weght of the more mportt ttrbute must be t lest s bg s the weght of the ttrbute ext the complete rk order. Sce the set of fesble rk orders cludes R ( I, J ) complete rk orders the fesble rego for R( I, J ) s (Slo d Pukk 2002): S( I, J ) = U r R( I, J ) S( r ), (4) so the fesble rego for R( I, J ) c be expressed s the uo of the complete rk orders fesble regos. I wht follows S p (I) = S(I,{,,p}), where p s the umber of the most mportt ttrbutes I. 3 Smulto We smulted the RICH-method three dfferet cses: DM kows the most mportt ttrbute, DM kows two most mportt ttrbutes but ot ther order d DM kows tht two most mportt ttrbutes re group of three ttrbutes. We lso studed the cse where the weghts were lower bouded ( w, where s the umber of ttrbutes). The we compred ll these 3 results wth the results computed wth precse weghts. The results were lso compred wth the oes obted for complete rk order d equl weghts. Equl weghts correspod to the stuto where the DM hs o pror kowledge of the reltve mportce of the ttrbutes d therefore he uses the sme weght for every ttrbute. 4

6 The smulto ws crred out s follows:. Scores for every ltertve s ttrbutes were geerted rdomly from uform dstrbuto betwee (0,). The these scores were ormlzed (Slo d Hämäläe 200): v o v ( x) v ( x ) x) =, (5) v ( x ) v ( x ) N ( * o where v N (x) s ltertve s x ormlzed score for ttrbute, v ( o x ) s the lowest score ttrbute gets d v x ) s the hghest. Normlzto mps the scores oto the rge [0,]. ( * 2. Weght vector w ws geerted rdomly from uform dstrbuto such tht w = =. Ths s the rel weght vector wth whch we c compute wth the ddtve vlue model whch of the ltertves ws truly the best. 3. Extreme pots of the fesble rego (see secto 3.) were computed. Scores were multpled by the weghts mpled by extreme pots. 4. Prwse domces betwee ltertves were checked. Accordg to ths crtero ltertve x s preferred to ltertve y f d oly f the vlue of x exceeds tht of y for ll fesble scores (Slo d Hämäläe 995). Domted ltertves were removed. 5. If there ws oly oe ltertve we hd foud the soluto. Go to step 7. Otherwse go to step Decso rules mxmx, mxm, cetrl vlue d mmx (see secto 3.2) were ppled to the ltertves. Best ltertve wth ech ws chose. 7. Ws the rght ltertve foud? Expected loss of vlue d umber of ltertves were computed. The smulto descrbed bove ws crred out 5000 tmes d the verge of expected loss of vlue, percetge of correct choces d umber of ltertves were clculted. If the weghts were lower bouded ths ws tke to cosderto step Fesble extreme pots Let S be covex set R. X S s S s extreme pot ff X ', X '' S, X ' X '', λ ] 0,[ such tht X = λx ' + ( λ) X ''. It s geerl result tht f the fesble rego of stdrd LPproblem s bouded, the the objectve fucto (here the totl vlue) gets ts mxmum vlue 5

7 some of the extreme pots (Th 997). Sce the ddtve vlue model s ler we oly hve to clculte the extreme pots order to solve the mxmzto problem. I our cse extreme pots re the pots, whch dvde the rego of ll ttrbute weghts S w to fesble regos for dfferet rk orders. Let s exme the cse of three ttrbutes to mke the cocept of extreme pots eser to uderstd. w 3 (0,0,) S (( 3,, 2 )) S (( 3, 2, )) w 3 = w w 3 = w 2 S ((, 3, 2 )) S (( 2, 3, )) (,0,0) w (0,,0) w 2 w = S (( 2,, 3 )) 2 w S ((, 2, 3 )) Fgure. Extreme pots the cse of three ttrbutes The extreme pots re crculted fgure. The fesble regos for every possble complete rk order (7 possble complete rk orders) re lso show. The extreme pots re lsted below: Tble. Extreme pots whe there re three ttrbutes w w 2 w ,5 0,5 0 0,5 0 0,5 0 0,5 0,5 0,3333 0,3333 0,3333 Compoets of the extreme pots (weghts of the ttrbutes) re ll the sme or zero. The umber of ll possble extreme pots N of every possble complete rk order the cse of ttrbutes s: N = k=. (6) k 6

8 Equto 6 c be esly derved. Let s ssume there s oly oe ttrbute wth postve weght. Ths ttrbute c be chose = dfferet wys. Now let s ssume there re two ttrbutes wth postve weghts. These ttrbutes c be chose dfferet wys (wthout regrd to order). 2 The other terms of equto (6) re computed lkewse. For exmple the lst term = correspods to the stuto whe ll ttrbute weghts re equl. Fesble extreme pots re the pots tht fulfll the rk orders defed by R ( I, J ). For exmple f we kow tht the two most mportt ttrbutes re 2 d 3 ( R ({2,3},{,2}) ) the the fesble extreme pots the cse of three ttrbutes re the followg: Tble 2. Fesble extreme pots ( R ({2,3},{,2}) ) w w 2 w ,5 0,5 0,3333 0,3333 0,3333 If the weghts re lower bouded by w 3 = 9, the the fesble extreme pots re: Tble 3. Lower bouded fesble extreme pots w w 2 w 3 0, 0,7778 0, 0, 0, 0,7778 0, 0,4444 0,4444 0,3333 0,3333 0, Decso rules Decso rules my be ppled to obt suggesto whch ltertves c be recommeded. I ths smulto study we used four dfferet decso rules: mxmx, mxm, cetrl vlue d mmx (below v ( x) = mx v ( x ) d v x) = m = ( v ( x ) ) = Choose the ltertve x whose lrgest vlue s gretest over the set of fesble scores,.e., 7

9 v( x) v( x' ) x' X, where X s the cosequece spce (Cleme 996). s optmstc decso rule becuse t tkes the lrgest vlue of the best possble outcomes. Choose the ltertve x whose lest vlue s lrgest over the set of fesble scores,.e., v( x) v( x' ) x' X (Cleme 996). s pessmstc decso rule becuse t tkes the lrgest vlue of the worst possble outcomes. Vlues Choose the ltertve x for whch the md-pot of the fesble vlue tervl s gretest,.e., [ v x) + v( x) ] [ v( x' ) + v( x' )] x' X ( (Slo d Hämäläe 200). s combto of mxmx d mxm d therefore t mght be clled somewht eutrl decso rule. Choose the ltertve x for whch the mxmum, mesured s the lrgest dfferece betwee v(x) d the vlue of other ltertves, s smllest,.e., mx x'' x [ v( x' ' ) v( x) ] mx[ v( x'' ) v( x' )] x' X x'' x' 4 Smulto results (Slo d Hämäläe 200). Smulto results re represeted the ext three tbles. Tble 4 shows the expected loss of vlue, tble 5 the percetge of correct choces d tble 6 the umber of ltertves. Expected loss of vlue mesures the goodess of decso rule by computg the loss of vlue ssocted wth the possblty tht the rule leds to choce of ooptml ltertve (Slo d Hämäläe 200). Mthemtclly ths c be expressed s: where * ' ELV = w ( v ( x ) v ( x )), (7) = w s the rel weght of ttrbute, wth decso rule. * x s the best ltertve d ' x s the ltertve chose 8

10 Tble 4. Problem sze d expected loss of vlue Most mportt ttrbute kow wthout border (weghts lower bouded) Altertves 5 5 0,0542 0,0299 0,026 0,0262 0,022 0,0290 0,0205 0, ,0728 0,038 0,0302 0,0329 0,0274 0,0343 0,0242 0, ,0846 0,0399 0,0322 0,0365 0,0304 0,0366 0,0266 0, ,0600 0,0327 0,038 0,0304 0,025 0,0297 0,0243 0, ,0765 0,043 0,0368 0,0386 0,036 0,0357 0,0274 0, ,088 0,0438 0,0386 0,04 0,0354 0,0386 0,0295 0, ,0637 0,0349 0,037 0,0349 0,0265 0,028 0,0245 0, ,078 0,044 0,0402 0,0430 0,0347 0,0364 0,0302 0, ,0908 0,0478 0,0435 0,0464 0,0364 0,039 0,032 0,0347 Two most mportt ttrbutes kow (ot ther order) wthout border (weghts lower bouded) Altertves 5 5 0,057 0,0348 0,0274 0,0389 0,0463 0,037 0,0246 0, ,246 0,030 0,026 0,0338 0,0505 0,0305 0,0234 0, ,346 0,0294 0,025 0,0337 0,053 0,0287 0,0224 0, ,0920 0,036 0,0282 0,036 0,0382 0,0285 0,02 0, ,36 0,0304 0,028 0,0339 0,048 0,029 0,0230 0, ,264 0,0287 0,0276 0,0320 0,0495 0,0290 0,027 0, ,0897 0,0306 0,030 0,038 0,0336 0,0265 0,0209 0, ,056 0,0304 0,0309 0,0344 0,0457 0,0274 0,0226 0, ,45 0,03 0,0303 0,0339 0,0492 0,0275 0,0228 0,0242 Two/Three most mportt ttrbutes kow (ot ther order) wthout border (weghts lower bouded) Altertves 5 5 0,636 0,0755 0,0647 0,0794 0,0604 0,0639 0,0505 0, ,758 0,0653 0,0555 0,0709 0,0645 0,0565 0,0480 0, ,758 0,0608 0,0506 0,0653 0,0659 0,055 0,0433 0, ,436 0,0657 0,057 0,072 0,055 0,0560 0,0452 0, ,559 0,0596 0,0533 0,0673 0,0607 0,0492 0,044 0, ,653 0,0545 0,0476 0,0626 0,0644 0,0488 0,047 0, ,257 0,0600 0,0530 0,0676 0,0468 0,0495 0,0383 0, ,376 0,053 0,0480 0,069 0,0530 0,0465 0,0387 0, ,503 0,0503 0,0465 0,0595 0,0596 0,0448 0,038 0,0465 Complete rk order & Equl weghts wthout border (weghts lower bouded) Altertves 5 5 0,0549 0,053 0,055 0,057 0,0247 0,089 0,030 0,023 0, ,0708 0,085 0,058 0,063 0,037 0,0204 0,042 0,038 0, ,085 0,09 0,062 0,059 0,0328 0,023 0,046 0,028 0,0590 Equl weghts 7 5 0,0583 0,035 0,085 0,068 0,0303 0,040 0,026 0,022 0, ,0774 0,056 0,0202 0,08 0,0388 0,075 0,050 0,035 0, ,097 0,076 0,020 0,078 0,0404 0,089 0,038 0,023 0, ,065 0,047 0,0240 0,0209 0,0377 0,042 0,047 0,040 0, ,0783 0,047 0,0229 0,098 0,045 0,049 0,048 0,028 0, ,0898 0,043 0,0232 0,0204 0,0487 0,058 0,06 0,030 0,0562 9

11 Tble 5. Problem sze d the percetge of correct choces Most mportt ttrbute kow wthout border (weghts lower bouded) Altertves 5 5 6,7 % 7,4 % 73,8 % 73,2 % 76,2 % 7,9 % 76,2 % 74,7 % 0 45,9 % 57,9 % 63,8 % 60,9 % 66,4 % 6,0 % 67,3 % 65,0 % 5 35,5 % 52,9 % 57,5 % 53,8 % 60,7 % 55,3 % 6,7 % 59,8 % ,8 % 68, % 68,5 % 69,4 % 72,3 % 68,9 % 72,2 % 7,0 % 0 4,2 % 53,8 % 58,5 % 55,8 % 6,2 % 57,9 % 63,2 % 6, % 5 32,3 % 48,6 % 52,9 % 50, % 56,6 % 52,4 % 58,8 % 55,9 % ,3 % 64, % 63,2 % 64,0 % 69,2 % 67,7 % 70, % 69,2 % 0 37,2 % 50,4 % 53,2 % 5,0 % 57, % 54,5 % 58,3 % 56,4 % 5 28,7 % 45,0 % 48,0 % 45,8 % 5,9 % 48,4 % 53,9 % 5,6 % Two most mportt ttrbutes kow (ot ther order) wthout border (weghts lower bouded) Altertves ,9 % 70,2 % 74,3 % 68,9 % 69,0 % 7,6 % 76,0 % 73,4 % 0 33,7 % 63,6 % 66,5 % 62,0 % 59,5 % 63,9 % 69,8 % 65,9 % 5 28,5 % 59,2 % 64,2 % 56,9 % 54,2 % 60,6 % 66,3 % 63,7 % ,7 % 69,3 % 7,0 % 67,7 % 68,2 % 70,2 % 74,7 % 7,3 % 0 33,7 % 60,8 % 63, % 59,8 % 56,4 % 62,0 % 67,5 % 64,9 % 5 25,6 % 57,7 % 59,8 % 56,5 % 50,6 % 58,2 % 64, % 6,2 % ,7 % 67,2 % 68,0 % 64, % 65,9 % 68,2 % 72,0 % 69,5 % 0 30,0 % 58, % 59,0 % 56,3 % 53,0 % 60,2 % 64,8 % 6,5 % 5 24,3 % 54,4 % 56,0 % 52,9 % 46,9 % 55,9 % 60,4 % 58,9 % Altertves Two/Three most mportt ttrbutes kow (ot ther order) wthout border (weghts lower bouded) ,0 % 56,6 % 60,5 % 55, % 62,8 % 58,8 % 63,6 % 59, % 0 26,4 % 48,5 % 53,6 % 46,7 % 54,7 % 52,5 % 57,4 % 52,7 % 5 22,9 % 45,6 % 50,9 % 43,2 % 49,8 % 50, % 55,2 % 50,0 % ,3 % 56,3 % 59,7 % 54,0 % 62,6 % 59,4 % 63,9 % 60, % 0 25,3 % 47,5 % 5,4 % 44,7 % 5,9 % 5,8 % 57,0 % 5,3 % 5 20,7 % 44,7 % 49,4 % 4,5 % 46,8 % 47,8 % 52,8 % 47,4 % ,6 % 55,3 % 58,2 % 52,7 % 6,4 % 58,8 % 63,5 % 58,2 % 0 24,4 % 47,4 % 50, % 43,2 % 50,5 % 50,3 % 55,2 % 49,6 % 5 8,8 % 44,7 % 48,2 % 4,0 % 45,0 % 46,8 % 5,4 % 46,0 % Complete rk order & Equl weghts wthout border (weghts lower bouded) Altertves 5 5 6,2 % 80,2 % 80,3 % 80,0 % 74,0 % 77,6 % 8,3 % 8,8 % 6,0 % 0 45,9 % 7,5 % 73,9 % 73, % 64,4 % 70,3 % 75,6 % 75,2 % 53,4 % 5 36,7 % 67,9 % 7,0 % 70,7 % 59, % 65,3 % 7,6 % 73,4 % 49,8 % Equl weghts ,8 % 79,4 % 76,9 % 78,0 % 69,7 % 79,3 % 80,8 % 80,8 % 59,9 % 0 4,0 % 7,2 % 68,8 % 70,2 % 57, % 70,7 % 72,8 % 74,3 % 49,7 % 5 3,8 % 66,2 % 64,8 % 66,5 % 55, % 66,2 % 72,3 % 72,6 % 46,6 % ,4 % 77,3 % 7,0 % 73,4 % 62,4 % 77,4 % 76,6 % 77,5 % 57,8 % 0 36,6 % 70,7 % 64,7 % 66,5 % 5,3 % 70,7 % 7, % 72,3 % 49, % 5 28,4 % 68,0 % 6,5 % 62,8 % 45, % 65,7 % 65,9 % 69,3 % 44,2 % 0

12 Tble 6. Problem sze d the umber of ltertves Most mportt ttrbute kow wthout border (weghts lower bouded) Altertves Averge percetge of ltertves oe ltertve less th hlf of ltertves Averge percetge of ltertves oe ltertve less th hlf of ltertves ,7 % 5,3 % 24,3 % 53,9 % 4,2 % 44,3 % 0 48,7 % 2, % 4,4 % 37,2 % 8,4 % 68,2 % 5 40,9 % 0,9 % 74,5 % 29,6 % 6,8 % 92, % ,8 %,0 % 5,9 % 68,6 % 4,8 % 20,4 % 0 67,9 % 0,2 % 0,6 % 52, % 2,5 % 35,7 % 5 60,3 % 0, % 24,7 % 44,5 %,3 % 6,7 % ,4 % 0, % 0,7 % 83,6 % 0,8 % 5,2 % 0 86,7 % 0,0 % 0,7 % 7,4 % 0,5 % 8,9 % 5 8,6 % 0,0 % 2,0 % 64,2 % 0,2 % 8,9 % Two most mportt ttrbutes kow wthout border (weghts lower bouded) Altertves Averge percetge of ltertves oe ltertve less th hlf of ltertves Averge percetge of ltertves oe ltertve less th hlf of ltertves ,8 % 7,5 % 3,0 % 52,5 % 6, % 47,3 % 0 45,9 % 2,7 % 48,2 % 35,5 % 9,6 % 73, % 5 37,6 %,2 % 80,7 % 27,2 % 7,9 % 94,9 % 7 5 7,8 % 3,0 % 6,0 % 62,0 % 8,4 % 30,7 % 0 58,5 % 0,6 % 23,3 % 46, % 4,3 % 48,3 % 5 50,8 % 0,2 % 48,0 % 37,0 % 3,0 % 79,5 % ,8 % 0,5 % 3, % 74,8 % 2,9 % 4,2 % 0 76,5 % 0, % 4,5 % 6, %,0 % 20,5 % 5 69,4 % 0,0 %,4 % 52,8 % 0,6 % 42,7 % Two/Three most mportt ttrbutes kow wthout border (weghts lower bouded) Altertves Averge percetge of ltertves oe ltertve less th hlf of ltertves Averge percetge of ltertves oe ltertve less th hlf of ltertves ,2 %,7 %,3 % 65,2 % 6,7 % 25,0 % 0 63,6 % 0,2 % 4,7 % 48,5 % 3,6 % 42,0 % 5 55,7 % 0, % 34,7 % 39,9 % 2,6 % 73, % , % 0,6 % 4,3 % 74,0 % 3,3 % 4,0 % 0 75,6 % 0, % 4,5 % 60,2 %,2 % 20,6 % 5 69,4 % 0,0 % 0,9 % 5,4 % 0,7 % 45,7 % ,6 % 0,0 % 0,9 % 84,7 % 0,8 % 5, % 0 88,8 % 0,0 % 0,7 % 74,7 % 0,2 % 6, % 5 84,4 % 0,0 %,3 % 67,5 % 0, % 4,9 % Complete rk order wthout border (weghts lower bouded) Altertves Averge percetge of ltertves oe ltertve less th hlf of ltertves Averge percetge of ltertves oe ltertve less th hlf of ltertves ,3 % 20,2 % 58, % 4,4 % 3,4 % 69,9 % 0 32,4 % 0,9 % 80,9 % 26,3 % 22,4 % 90,8 % 5 25,2 % 6,9 % 97,8 % 9,3 % 9,2 % 99,6 % ,5 % 5,5 % 47, % 46,3 % 24, % 60, % 0 37,4 % 6,8 % 69,9 % 30,4 % 4,7 % 84,0 % 5 30,3 % 3,9 % 92,7 % 23, % 3,6 % 97,7 % ,4 % 0, % 36,6 % 5,2 % 7,2 % 49,3 % 0 42,9 % 4,0 % 56,4 % 35,3 % 0,5 % 72,5 % 5 35,4 % 2,6 % 84,3 % 27,7 % 7,3 % 94,9 %

13 Expected loss of vlue Lower bouded weghts gve better results lmost every cse. Especlly otceble the dfferece s whe usg mxmx decso rule (see e.g. mxmx wth most mportt ttrbute kow, 7 ttrbutes d 5 ltertves). It s turl to demd tht every ttrbute s weght must be t lest /3 sce rtol DM does ot tke sgfct ttrbutes to the decso mkg process. s clerly the worst decso rule whe weghts re ot lower bouded (mrked wth border the bove tbles). The best decso rules re cetrl d mmx. Comprg the results wth the oes obted for complete rk order t s otceble how close the results re whe most mportt ttrbute s kow d whe two most mportt ttrbutes re kow. Results re clerly worse whe two most mportt ttrbutes re the group of three ttrbutes but ths s uderstdble sce the thrd ttrbute ws chose rdomly. Greter ucertty leds to worse results. The results obted whe two most mportt ttrbutes re the group of three ttrbutes re worse or oly slghtly better th the oes obted wth equl weghts so there does ot seem to be much sese usg t. Whe the most mportt ttrbute s kow the expected loss of vlue creses whe the umber of ltertves creses (see e.g. mmx wth weghts lower bouded, 5ttrbutes d wth 5 or 5 ltertves). The expected loss of vlue lso creses whe the umber of ttrbutes creses. Both re uderstdble sce t s more lkely to ed up wth ooptml ltertve whe the umber of ltertves or ttrbutes creses. Wth two most mportt ttrbutes kow the expected loss of vlue geerlly decreses whe the umber of ttrbutes creses (see e.g. mxm wth weghts lower bouded, 5 ltertves d wth 5 or 0 ttrbutes). Ths s uderstdble sce kowg the two most mportt ttrbutes s reltvely more regultg whe the umber of ttrbutes creses. The sme result pples lso to the cse whe two most mportt ttrbutes re group of three ttrbutes, but ths cse s ot terestg sce the results re so poor s metoed before. correct choces correct choces shows the percetge of problem stces whch the decso rule detfed the ltertve wth the hghest vlue (Slo d Hämäläe 200). Lower bouded weghts gve better results every cse whe the complete rk order s ot kow d lso lmost every cse whe t s kow. Oce g the dfferece s gretest whe usg mxmx 2

14 decso rule (see e.g. mxmx wth two most mportt ttrbutes kow, 7 ttrbutes d 0 ltertves). Results re very good whe most mportt ttrbute or two most mportt ttrbutes re kow. Whe two most mportt ttrbutes re the group of three ttrbutes the result re oce g worse or oly slghtly better th the oes obted wth equl weghts. d mmx re the best decso rules but mxm s lmost s good. s ot fr behd ether whe weghts re lower bouded (see e.g. mxmx d cetrl wth the most mportt ttrbute kow, weghts lower bouded, 0 ttrbutes d 5 ltertves). Nturlly the results re better whe the complete rk order s kow but the dfferece s ot bg oe whe usg the best decso rules. The percetge of correct choces decreses whe the umber of ltertves creses d lso whe the umber of ttrbutes creses. Both re uderstdble sce t s more lkely to ed up wth ooptml ltertve whe the umber of ltertves or ttrbutes creses. Number of ltertves The verge percetges of ltertves re clerly hgher every cse whe complete rk order s ot kow. Ths s uderstdble sce the umber of fesble extreme pots s extremely bgger the cse o complete rk order d therefore t s less lkely to fd prwse domces. Percetges of oe ltertve re smll so decso rules re relly eeded. Oce g results re better whe weghts re lower bouded. The verge percetge of ltertves decreses whe the umber of ltertves creses. Ths s uderstdble sce whe the umber of ltertves creses t s more lkely tht domces occur d therefore the porto of ltertves decreses. 5 Dscusso d coclusos I ths pper we studed the computtol effectveess d propertes of RICH method wth extesve smulto. The results were ecourgg: the expected loss of vlue ws oly modertely hgher th the cse of complete rk order. Also the percetges of correct choces were hgh. It s obvous from the results tht the ttrbute weghts should be lower bouded. Ths s lso tutvely uderstdble s expled before. Four decso rules were used: mxmx, mxm, cetrl d mmx. The ltter two proved to be the best oes lthough the dffereces were ot tht bg. Decso rules were relly 3

15 eeded sce oly smll percetge of problems there ws oly oe ltertve. Number of ltertves s geerlly hgher whe complete rk order s ot kow but ths does ot seem to ffect the fdg of the best ltertve too much. Accordg to the smulto results RICH method ppers to be computtolly lble. Oe c get good results wthout kowg the complete rk order. Therefore t wll probbly be terest of further studes. 4

16 Refereces. Cleme, R. T. (996). Mkg Hrd Decsos A Itroducto to Decso Alyss, 2 d Edto. Duxbury Press, Cmbrdge. 2. Km, S. H. d Ah, B. S. (999). Iterctve Group Decso Mkg Procedure uder Icomplete Iformto. Europe Jourl of Opertol Reserch, vol. 6, pp Km, S. H. d H, C. H. (2000). Estblshg Domce Betwee Altertves wth Icomplete Iformto Herrchclly Structured Attrbute Tree. Europe Jourl of Opertol Reserch, vol. 22, pp Slo, A. d Hämäläe, R. P. (992). Preferece Assessmet by Imprecse Rto Sttemets. Opertos Reserch, vol. 40, pp Slo, A. d Hämäläe, R. P. (995). Preferece Progrmmg Through Approxmte Rto Comprsos. Europe Jourl of Opertol Reserch, vol. 82, pp Slo, A. d Hämäläe, R. P. (200). Preferece Rtos Multttrbute Evluto (PRIME) Elctto d Decso Procedures uder Icomplete Iformto. IEEE Trsctos o Systems, M, d Cyberetcs. (to pper) 7. Slo, A. d Pukk, A. (2002). Rk Icluso Crter Herrches. (muscrpt, Februry 2002) 8. Th, H. A. (997). Opertos Reserch: A Itroducto, 6 th Edto. Pretce-Hll, Ic., New Jersey 5