On the Borda Method for Multicriterial Decision-Making

Transcription

1 O the Brda Methd r Multcrteral Decs-Makg Radu A. Pău Iteratal Metary Fud Isttute th Street, N.W. Washgt, D.C rpau@m.rg r radupau@yah.cm Abstract The preset paper dscusses tw ssues wth multcrteral decs-makg methds Brda type (whe scres such as, -,, 2, are gve t the bects t be raked ad the herarchy s btaed based the ttals these scres). The rst ssue s related t the luece the result varus trasrmats the scres. We shw that a lear trasrmat the scres des t chage the al rakg ad that (almst) ay plymal secd degree r mre, wth pstve cecets, ca alter the slut (rakg). The same happes e chages the scres by emplyg the lgarthm, expetal, r square rt ucts. I the secd part the paper we csder a terated vers the Brda methd. We shw that ths methd s t rbust: there are cases whe deret sluts are retured at deret terats.. Itrduct It s well-kw that mst decss (ether persal, scal, r ecmc) are multcrteral: several pssbltes act (bects) are take t accut gve several crtera, r are csdered by several decs-makers. The gal s t select a partcular curse act (a uque bect), whch s te tmes the e at the tp a sythess rakg bects. There are umerus methds t address ths prblem the reader s guded twards Adraşu et al. (986) ad the reereces t prvdes. I the 8 th cetury, the Frech mathematca Jea-Charles de Brda has prpsed a methd whch s stll wdely used. I shrt, the methd requres that, usg a pre-deed scale, pts be allcated t bects that are t be raked, ad that these pts be summed; the sythess herarchy s btaed by srtg decreasg rder these scres. The pts that are allcated ca decrease cstat steps (r stace, the bect the th place a departg herarchy may receve m + pts, where m s the umber bects t be raked), r they ca belg t a specc set (r example, Ocescu (970) prpses that, ½, ¼,,½,,½ m-, ½ m be the pts allcated t bects the st, 2 d,, th,,m th place a departg herarchy). It has bee shw that such rakg methds ctradct e the ma ratalty cdts the Arrw therem : depedece. I shrt, a multcrteral decs-makg methd (a methd whch aggregates the departg herarches) satses the depedece prperty the al rderg ay tw bects des t deped the presece (r absece) a thrd bect. Fr stace, let us emply the example rm Pău (987, p. 30), wh csders ve decs-makers wh eed t rak three bects. The departg herarches are (a, a 2, a 3 ), (a, a 2, a 3 ), (a 3, a, a 2 ), (a 2, a 3, a ), (a 2, a 3, a ). I we allcate three pts t the rst bect a herarchy, tw pts t the secd bect ad e pt t the thrd, we bta the llwg scres: a 0, a 2, a 3 9, Fr a exhaustve vew, the reader s guded twards the semal wrk Arrw (963). 364

2 leadg t sythess herarchy (a 2, a, a 3 ). Hwever, we elmate bect a 2 rm all departg herarches, ad thus have the ve decs-makes dcate (a, a 3 ), (a, a 3 ), (a 3, a ), (a 3, a ), (a 3, a ), the, a smlar maer as abve, a btas 7 pts whle a 3 btas 8 pts, ad thus the sythess herarchy s (a 3, a ). Clearly, the relatshp betwee a ad a 3 s ppste w t the e btaed whe a 2 was preset. Ather weakess may methds r herarchy aggregat s the pssblty t mapulate the al result by adustg sme dvdual herarches (vtes). Ths pt has bee emphaszed the early 70s see Gbbard (973) ad Satterthwate (975). Oe way t cuter ths pssblty s t make the methd mre cmplcated, s that recastg ad luecg the result are less lkely. A classc example s the Cpelad methd, thrughly preseted by Nurm (989). The rst degree Cpelad methd s act a Brda methd: the bect the th place s allcated m pts (ad t m + as abve). The methd ca be mapulated plymal tme. The secd degree Cpelad methd s mre evlved. Ater the rst step, whch bects receve pts as descrbed abve, the secd step the ttal scre r each bect s btaed by summg the rst step pts thse bects dmated by ths partcular bect. Mapulat ths methd s sgcatly mre dcult; ths has bee shw t be a NPcmplete prblem. The dea t terate a Brda-lke methd urther appears aturally. Hwever, ths rases tw mprtat ssues: Frst, ttal scres at each step crease rapdly, whch may lead t dcultes whe mplemetg large, real-le applcats cmputer. A usual slut such cases s data rmalzat, btaed by applyg a specc trasrmat the tal gures. We wll prve that such trasrmats may alter the al result, ad s ths shuld be avded. Precsely, we shw that a lear trasrmat des t alter the result, but sme plymal trasrmats d chage the herarchy. A smlar stuat s ud whe e uses the expetal, lgarthmc, r square rt trasrmats. Secd, a desrable eature terated methds s that herarches at deret terats rema uchaged. A umercal example wll shw that ths gal s smetmes mssed. 2. Ntat I what llws, wll always represet the umber decs-makers (r the umber crtera), whch we dete d, d 2,,d, ad m wll represet the umber bects t be raked, deted, 2,, m. Each decs-maker cveys a herarchy bects (a ttal, lear rder) ad bects these departg herarches are allcated pts accrdg t ther pst the herarchy (mre pts beg allcated t bects raked hgher). Let d, ), where, m, dete the umber pts receved by bect decs-maker s d herarchy. We ca w dee the qualty a bect: ( ) = d, ). Let us csder a uct : R R. I the bects are allcated pts such as (d, )), hece trasrmed by applyg uct, the the crrespdg qualty a bect wll be deted ( ). 3. The sestvty the Brda methd t pt trasrmats Lemma. (The addtvty lemma) Let, 2 : R R be tw ucts. Let us assume we have a multcrteral decs-makg prblem such that, r tw bects, k, we have 365

3 ( ) > ( ), ( ) k 2 > 2 ( k Let uct (x) = a (x) + a 2 2 (x) + a 3, where a ad a 2 are tw pstve umbers. The, ). Pr. ( ) > ( k ). ( ) = ( a ( )) + a2 2( )) + a3) = a ( )) + a2 2( )) + a3 ( ) + a2 ( ) + a3 ( k ) + a2 ( k ) + a3 = a, 2 2 ( k ) = a. Clearly, ( ) > ( k ). The llwg example wll be used several tmes belw. We csder ur decs-makers wh have t decde ve bects, ad let the departg herarches be: d d 2 d 3 d 4 st place d place 4 3 rd place th place th place I we allcate 5 pts t the bect the rst place, 4 t the bect the secd, ad s, pt t the th placed bect, the the ttal scres the ve bects bta are 5, 4, 3,, 7 ad thus the Brda herarchy s (, 2, 3, 4, 5 ). ( ). Lemma 2. Gve the prevus example, r ay uct (x) = x k, wth k 2, we bta ( 2 ) > Pr. It s bvus that ( ) = 3 4 k + 3 k, ( 2 ) = 2 5 k + 3 k +. It suces t shw that 2 5 k > 3 4 k. We ca wrte 5 k = (4 + ) k ad use the Newt bmal seres. The, 2 (4 + ) k = 0 k k k k 0 2 ( C k 4 + Ck Ck 4 + Ck 4 ) = 2 4 k + 2 k 4 k- + α, where α s a strctly pstve umber. Sce k 2, we bta 2 4 k + 2 k 4 k- 3 4 k, whch meas that 2 5 k > 3 4 k. Observat. I the prevus setup, t ca be shw that r k 3 a eve strger result emerges: ( 3 ) > ( ). The equalty s clearly true r k = 3 whle, r k greater tha 3, t ca be 366

4 prved by duct ad by relyg ce aga the Newt bmal seres. We ca w demstrate the ma result ths sect. Therem. () A lear trasrmat, wth pstve cecets, the allcated pts des t alter the rakg prduced by the Brda methd. () Let us csder a plymal secd degree r mre, wth pstve cecets, such that the cecet the rst degree term s ether zer r equal t the cecet the secd degree term. Fr ay such plymal (x), there are multcrteral decsmakg prblems r whch the slut btaed by the Brda methd s altered e apples the (x) trasrmat. Pr. The Brda methd btas the sythess herarchy by rderg ( ) = d, ), m. A lear trasrmat (x) = ax + b, wth a 0, leads t ( ) whch, accrdg t Lemma, wll keep the herarchy uchaged 2. Let us csder a plymal (x) = a x k + a 2 x k- + + a k x + a k+, wth a 0, k +, ad ether a k = 0 r a k = a k-. I we dete (x) = a x k-+, k +, we have (x) = (x) + 2 (x) + + k-2 (x) + k+ (x) + ( k- (x) + k (x)). Accrdg t Lemma 2, r ay =, 2,, k 2, we have that ( 2) > ( ). Usg Lemma repeatedly, we get that g ( 2 ) > g ( ), where g(x) = (x) + 2 (x) + + k-2 (x) + k+ (x). Let us aalyze separately the case trasrmat h(x) = x 2 + x. Usg ths uct, 30 pts are allcated t the bect the rst place, 20 t the bect the secd place, 2 t the thrd placed bect, 6 t the bect the urth place ad 2 pts t the last bect a herarchy. Usg the prevus example departg herarches, h ( ) = 72 < h ( 2 ) = 74. Accrdg t Lemma, the same s the case we apply trasrmat g (x) = a k- g(x). We emply Lemma ce aga r ucts h(x) ad g (x), whch, by addt, lead precsely t plymal (x). Observat 2. The cdt we mpsed the cecets the rst ad secd degree terms the plymal s eeded the abve pr, gve the partcular example we were wrkg wth. Ideed, let us csder the plymal (x) = x 2 + 4x. The pts t be allcated t bects are w 45, 32, 2, 2, ad 5. Accrdgly, ( ) = 7 > ( 2 ) = 6, s the rder the tw bects reversed. As expected, ther lear trasrmats als chage the herarchy. Therem 2. Let us csder trasrmats (x) = 2 x, 2 (x) = l x, ad 3 (x) = x ½. There are multcrteral decs-makg prblems r whch the slut btaed by the Brda methd s altered e apples these trasrmats. Pr. I we csder the prevus example ad trasrmat (x) = 2 x, the the pts t be allcated are 32, 6, 8, 4, ad 2, whch leads t ( ) = 56, ( 2) = 74, ( 3) = 70, ( 4) = 28, ( 5) = 6, ad thus t herarchy ( 2, 3,, 4, 5 ). Ths s cmpletely deret rm the herarchy btaed whe 5, 4, 3, 2, pts were allcated. Fr trasrmat 2 (x) = l x we csder the example belw: d d 2 d 3 d 4 pts st place d place rd place th place th place Ths s a partcular case Lemma, whch (x) = x, 2 (x) = x, a = a 2 = a/2, ad a 3 = b. 367

5 Fr ths example, the Brda methd, usg the stadard pts 5, 4, 3, 2,, leads t herarchy (, 2, 3, 5, 4 ). We w allcate l 5, l 4, l 3, l 2, ad l pts (usg the apprxmate values we have lsted the abve table), whch leads t ttal scres 4.37, 4.65, 3.68, 2.99, ad 3.39 ad thus t herarchy ( 2,, 3, 5, 4 ), deret rm the prevus e. Lastly, r trasrmat 3 (x) = x ½ we csder the llwg example: d d 2 d 3 d 4 d5 pts st place d place rd place th place th place The Brda methd usg the stadard pt system leads t herarchy ( 3, 4,, 2, 5 ). By trasrmg the pts t be allcated usg uct 3 (see the apprxmate values the table abve), the ve bects get the scres 8.236, 8.292, 8.936, 8.74, ad 7.732, whch leads t herarchy ( 2,, 3, 5, 4 ), deret rm the prevus e. 4. Iteratg the Brda methd As meted the trduct, the Cpelad methd secd degree s a tw-step prcedure: the rst step pts are allcated t bects a smlar ash t the Brda methd, whle the secd step the ttal r each bect s btaed by summg the rst step pts the bects that partcular bect dmates. We w prceed t terate ths prcedure urther. Frmally, r a multcrteral decs-makg prblem the type we have aalyzed s ar, where bects qualty s evaluated by ( ), wth m, we dee ( ), wth 0, ths maer: ( ) = ( ), + ( ) = ( l ), k= l D ( ) where D k () s the set bects decs-maker s d k herarchy whch are dmated by bect. The, at each step, the decreasg rder ( ), wth m, dcates the herarchy at that partcular terat. Naturally, the quest s whether, gve a partcular prblem, herarches rema uchaged at varus steps. I s, ths wuld mea the prcedure s rbust, ad thus ull cdece shuld be placed that (cstat) herarchy. Urtuately, ths s t always the case, ad we wll demstrate ths by emplyg a umercal example. Fur decs-makers eed t rak ve bects, ad ther departg herarches are prvded the table belw: k, d d 2 d 3 d 4 st place d place rd place th place th place

6 Gve the abve, the results the rst te terats (retured by a smple cmputer prgram) are the llwg: Scres at terat : 2,, 5, 9, 3 Herarchy at terat : (, 2, 4, 3, 5 ) Scres at terat 2: 72, 75, 29, 59, 25 Herarchy at terat 2: ( 2,, 4, 3, 5 ) Scres at terat 3: 484, 483, 209, 38, 63 Herarchy at terat 3: (, 2, 4, 3, 5 ) Scres at terat 4: 326, 3227, 353, 2567, 073 Herarchy at terat 4: ( 2,, 4, 3, 5 ) Scres at terat 5: 2292, 24, 903, 6937, 747 Herarchy at terat 5: ( 2,, 4, 3, 5 ) Scres at terat 6: 4336, 4875, 59789, 2475, 4736 Herarchy at terat 6: ( 2,, 4, 3, 5 ) Scres at terat 7: , 94547, , , 3439 Herarchy at terat 7: ( 2,, 4, 3, 5 ) Scres at terat 8: , , , , Herarchy at terat 8: ( 2,, 4, 3, 5 ) Scres at terat 9: , , 74440, , Herarchy at terat 9: ( 2,, 4, 3, 5 ) Scres at terat 0: , , , , Herarchy at terat 0: ( 2,, 4, 3, 5 ) As dcated abve, herarches at the rst ur steps alterate ( ad 2 swtch ther places). Frm the urth step the herarchy remas cstat (hwever, we have t explred past the 0 th terat). 5. Ccludg remarks The tw ma results the preset paper are: () Trasrmg the tal pts allcated t bects a multcrteral decs-makg prblem slved by the Brda methd may alter the al result. Ths s the case r may the usual lear ucts e may emply: plymal secd degree r mre (wth sme restrcts placed cecets), lgarthm, expetal, square rt. The herarchy s t mded e emplys a lear trasrmat the tal pts. (2) Iteratg the Brda methd may lead t deret herarches at deret terats. We eed t emphasze ce aga that Brda-lke methds are the mst cmmly used r multcrteral decs prblems. The, the sgcace the abve results s bvus. Frst, caut shuld be exerted whe mdyg the tal data (by rmalzat r ther perats mpsed, r stace, by lmtats cmputg capacty). Als, e shuld always bear md that the al result may deped the actual methd emplyed, r eve the vers that partcular methd. 369

7 Reereces Adraşu M., Bacu A., Pascu A., Puşcaş E., Tasad Al. (986) Metde de decz multcrterale, Edtura Tehcă, Bucureşt Arrw K. (963) Scal Chce ad Idvdual Values, Jh Wley, New Yrk Gbbard A. (973) Mapulat Vtg Schemes: A Geeral Result, Ecmetrca, 4, Nurm H. (989) Cmputatal Appraches t Bargag ad Chce, Jural Theretcal Pltcs,, Ocescu O. (970) Prcedee de estmare cmparatvă a ur becte purtăatare de ma multe caracterstc, Revsta de Statstcă, 4 Pău Gh. (987) Paradxurle clasametelr, Edtura Ştţcă ş Ecclpedcă, Bucureşt Satterthwate M. (975) Strategy-Press ad Arrw's Cdts, Jural Ecmc Thery, 0,