Introduction to Decision Making Methods

Transcription

1 Introducton to Decson Makng Methods János Fülöp Laboratory of Operatons Research and Decson Systes, Coputer and Autoaton Insttute, Hungaran Acadey of Scences 1. Decson Makng Process Decson akng s the study of dentfyng and choosng alternatves based on the values and preferences of the decson aker. Makng a decson ples that there are alternatve choces to be consdered, and n such a case we want not only to dentfy as any of these alternatves as possble but to choose the one that best fts wth our goals, objectves, desres, values, and so on. (Harrs (1980)) Accordng to Baker et al. (2001), decson akng should start wth the dentfcaton of the decson aker(s) and stakeholder(s) n the decson, reducng the possble dsagreeent about proble defnton, requreents, goals and crtera. Then, a general decson akng process can be dvded nto the followng steps: Step 1. Defne the proble Ths process ust, as a nu, dentfy root causes, ltng assuptons, syste and organzatonal boundares and nterfaces, and any stakeholder ssues. The goal s to express the ssue n a clear, one-sentence proble stateent that descrbes both the ntal condtons and the desred condtons. Of course, the one-sentence lt s often exceeded n the practce n case of coplex decson probles. The proble stateent ust however be a concse and unabguous wrtten ateral agreed by all decson akers and stakeholders. Even f t can be soetes a long teratve process to coe to such an agreeent, t s a crucal and necessary pont before proceedng to the next step. Step 2. Deterne requreents Requreents are condtons that any acceptable soluton to the proble ust eet. Requreents spell out what the soluton to the proble ust do. In atheatcal for, these requreents are the constrants descrbng the set of the feasble (adssble) solutons of the decson proble. It s very portant that even f subjectve or judgental evaluatons ay occur n the followng steps, the requreents ust be stated n exact quanttatve for,.e. for any possble soluton t has to be decded unabguously whether t eets the requreents or not. We can prevent the ensung debates by puttng down the requreents and how to check the n a wrtten ateral. Step 3. Establsh goals Goals are broad stateents of ntent and desrable prograatc values... Goals go beyond the nu essental ust have s (.e. requreents) to wants and desres. In atheatcal for, the goals are objectves contrary to the requreents that are constrants. The goals ay be conflctng but ths s a natural concotant of practcal decson stuatons. Step 4. Identfy alternatves Alternatves offer dfferent approaches for changng the ntal condton nto the desred condton. Be t an exstng one or only constructed n nd, any alternatve ust eet the requreents. If the nuber of the possble alternatves s fnte, we can check one by one f t 1

2 eets the requreents. The nfeasble ones ust be deleted (screened out) fro the further consderaton, and we obtan the explct lst of the alternatves. If the nuber of the possble alternatves s nfnte, the set of alternatves s consdered as the set of the solutons fulfllng the constrants n the atheatcal for of the requreents. Step 5. Defne crtera Decson crtera, whch wll dscrnate aong alternatves, ust be based on the goals. It s necessary to defne dscrnatng crtera as objectve easures of the goals to easure how well each alternatve acheves the goals. Snce the goals wll be represented n the for of crtera, every goal ust generate at least one crteron but coplex goals ay be represented only by several crtera. It can be helpful to group together crtera nto a seres of sets that relate to separate and dstngushable coponents of the overall objectve for the decson. Ths s partcularly helpful f the eergng decson structure contans a relatvely large nuber of crtera. Groupng crtera can help the process of checkng whether the set of crtera selected s approprate to the proble, can ease the process of calculatng crtera weghts n soe ethods, and can facltate the eergence of hgher level vews of the ssues. It s a usual way to arrange the groups of crtera, subcrtera, and sub-subcrtera n a tree-structure (UK DTLR (2001)). Accordng to Baker et al. (2001), crtera should be able to dscrnate aong the alternatves and to support the coparson of the perforance of the alternatves, coplete to nclude all goals, operatonal and eanngful, non-redundant, few n nuber. In soe ethods, see Keeney and Raffa (1976), non-redundancy s requred n the for of ndependency. We enton that soe authors use the word attrbute nstead of crteron. Attrbute s also soetes used to refer to a easurable crteron. Step 6. Select a decson akng tool There are several tools for solvng a decson proble. Soe of the wll be brefly descrbed here, and references of further readngs wll also be proposed. The selecton of an approprate tool s not an easy task and depends on the concrete decson proble, as well as on the objectves of the decson akers. Soetes the spler the ethod, the better but coplex decson probles ay requre coplex ethods, as well. Step 7. Evaluate alternatves aganst crtera Every correct ethod for decson akng needs, as nput data, the evaluaton of the alternatves aganst the crtera. Dependng on the crteron, the assessent ay be objectve (factual), wth respect to soe coonly shared and understood scale of easureent (e.g. oney) or can be subjectve (judgental), reflectng the subjectve assessent of the evaluator. After the evaluatons the selected decson akng tool can be appled to rank the alternatves or to choose a subset of the ost prosng alternatves. 2

3 Step 8. Valdate solutons aganst proble stateent The alternatves selected by the appled decson akng tools have always to be valdated aganst the requreents and goals of the decson proble. It ay happen that the decson akng tool was sappled. In coplex probles the selected alternatves ay also call the attenton of the decson akers and stakeholders that further goals or requreents should be added to the decson odel. 2. Sngle crteron vs. ultple crtera, fnte nuber of alternatves vs. nfnte nuber of alternatves It s very portant to ake dstncton between the cases whether we have a sngle or ultple crtera. A decson proble ay have a sngle crteron or a sngle aggregate easure lke cost. Then the decson can be ade plctly by deternng the alternatve wth the best value of the sngle crteron or aggregate easure. We have then the classc for of an optzaton proble: the objectve functon s the sngle crteron; the constrants are the requreents on the alternatves. Dependng on the for and functonal descrpton of the optzaton proble, dfferent optzaton technques can be used for the soluton, lnear prograng, nonlnear prograng, dscrete optzaton, etc. (Nehauser et al. (1989)). The case when we have a fnte nuber of crtera but the nuber of the feasble alternatves (the ones eetng the requreents) s nfnte belongs to the feld of ultple crtera optzaton. Also, technques of ultple crtera optzaton can be used when the nuber of feasble alternatves s fnte but they are gven only n plct for (Steuer, R. E. (1986)). Ths bref survey focuses on decson akng probles when the nuber of the crtera and alternatves s fnte, and the alternatves are gven explctly. Probles of ths type are called ultattrbute decson akng probles. 3. Mult-attrbute decson akng ethods Consder a ult-attrbute decson akng proble wth crtera and n alternatves. Let C 1,,C and A 1,..,A n denote the crtera and alternatves, respectvely. A standard feature of ult-attrbute decson akng ethodology s the decson table as shown below. In the table each row belongs to a crteron and each colun descrbes the perforance of an alternatve. The score a j descrbes the perforance of alternatve A j aganst crteron C. For the sake of splcty we assue that a hgher score value eans a better perforance snce any goal of nzaton can be easly transfored nto a goal of axzaton. As shown n decson table, weghts w 1,...,w are assgned to the crtera. Weght w reflects the relatve portance of crtera C to the decson, and s assued to be postve. The weghts of the crtera are usually deterned on subjectve bass. They represent the opnon of a sngle decson aker or synthesze the opnons of a group of experts usng a group decson technque, as well. The values x 1,...,x n assocated wth the alternatves n the decson table are used n the MAUT ethods (see below) and are the fnal rankng values of the alternatves. Usually, hgher rankng value eans a better perforance of the alternatve, so the alternatve wth the hghest rankng value s the best of the alternatves. 3

4 x 1 x n A 1 A n w 1 C 1 a 11 a 1 w C a 1 a n Table 1. The decson table Mult-attrbute decson akng technques can partally or copletely rank the alternatves: a sngle ost preferred alternatve can be dentfed or a short lst of a lted nuber of alternatves can be selected for subsequent detaled apprasal. Besdes soe onetary based and eleentary ethods, the two an fales n the ult-attrbute decson akng ethods are those based on the Mult-attrbute Utlty Theory (MAUT) and Outrankng ethods. The faly of MAUT ethods conssts of aggregatng the dfferent crtera nto a functon, whch has to be axzed. Thereby the atheatcal condtons of aggregatons are exaned. Ths theory allows coplete copensaton between crtera,.e. the gan on one crteron can copensate the lost on another (Keeney and Raffa (1976)). The concept of outrankng was proposed by Roy (1968). The basc dea s as follows. Alternatve A outranks A j f on a great part of the crtera A perfors at least as good as A j (concordance condton), whle ts worse perforance s stll acceptable on the other crtera (non-dscordance condton). After havng deterned for each par of alternatves whether one alternatve outranks another, these parwse outrankng assessents can be cobned nto a partal or coplete rankng. Contrary to the MAUT ethods, where the alternatve wth the best value of the aggregated functon can be obtaned and consdered as the best one, a partal rankng of an outrankng ethod ay not render the best alternatve drectly. A subset of alternatves can be deterned such that any alternatve not n the subset be outranked by at least one eber of the subset. The a s to ake ths subset as sall as possble. Ths subset of alternatves can be consdered as a shortlst, wthn whch a good coprose alternatve should be found by further consderatons or ethods. 3.1 Cost-beneft analyss Cost-beneft analyss (CBA) s a worldwde used technque n decson akng. CBA evaluates the costs and benefts of the alternatves on onetary bass. Recently, attepts have been ade to ncorporate the envronental pacts wthn CBA to prove the qualty of envronental decson akng. Although advances have been ade, probles persst n applyng CBA to envronental ssues, ncludng the onetary valuaton of envronental pacts (UK DTLR (2001)). On the other hand, CBA has great attractons as a tool for gudng publc polcy: t consders the gans and losses to all ebers of the socety on whose behalf the CBA s beng undertaken; t values pacts n ters of a sngle, falar easureent scale - oney - and can therefore n prncple show that pleentng an alternatve s worthwhle relatve to dong nothng; the oney values used to weght the relatve portance of the dfferent pacts are based on people's preferences generally usng establshed ethods of easureent. 4

5 Despte ts ltatons, CBA can be effcently ntegrated nto coplex ethods of envronental decson akng. See Munda (1996) how CBA can be ntegrated nto envronental assessent and US EPA (2000) for gudelnes on econoc analyss ncludng cost and beneft analyss. 3.2 Eleentary ethods These eleentary approaches are sple and no coputatonal support s needed to perfor the analyss These ethods are best suted for probles wth a sngle decson aker, few alternatves and crtera that s rarely characterstc n envronental decson akng (Lnkov et al. (2004)) Pros and cons analyss Pros and cons analyss s a qualtatve coparson ethod n whch good thngs (pros) and bad thngs (cons) are dentfed about each alternatve. Lsts of the pros and cons are copared one to another for each alternatve. The alternatve wth the strongest pros and weakest cons s preferred. It requres no atheatcal skll and s easy to pleent. (Baker et al. (2001)) Maxn and axax ethods The axn ethod s based upon a strategy that tres to avod the worst possble perforance, axzng the nal perforng crteron. The alternatve for whch the score of ts weakest crteron s the hghest s preferred. The axn ethod can be used only when all crtera are coparable so that they can be easured on a coon scale, whch s a ltaton (Lnkov et al. (2004)) Conjunctve and dsjunctve ethods These ethods requre satsfactory rather than best perforance n each crteron. The conjunctve ethod requres that an alternatve ust eet a nal perforance threshold for all crtera. The dsjunctve ethod requres that the alternatve should exceed the gven threshold for at least one crteron. Any alternatve that does not eet the conjunctve or dsjunctve rules s deleted fro the further consderaton. These screenng rules can be used to select a subset of alternatves for analyss by other, ore coplex decson akng tools (Lnkov et al. (2004)). Screenng by conjunctve and dsjunctve rules can also be appled n Step 2 (Deterne requreents) of the decson akng process (see Secton 1) Lexcographc ethod In the lexcographc ethod crtera are ranked n the order of ther portance. The alternatve wth the best perforance score on the ost portant crteron s chosen. If there are tes wth respect to ths crteron, the perforance of the ted alternatves on the next ost portant crteron wll be copared, and so on, tll a unque alternatve s found (Lnkov et al. (2004)). 3.3 MAUT ethods In ost of the approaches based on the Mult-attrbute Utlty Theory (MAUT), the weghts assocated wth the crtera can properly reflect the relatve portance of the crtera only f the scores a j are fro a coon, densonless scale. The bass of MAUT s the use of utlty functons. Utlty functons can be appled to transfor the raw perforance values of the alternatves aganst dverse crtera, both factual (objectve, quanttatve) and judgental (subjectve, qualtatve), to a coon, densonless scale. In the practce, the ntervals [0,1] or [0,100] are used for ths purpose. Utlty functons play another very portant role: they convert the raw perforance values so that a ore preferred perforance obtans a hgher utlty value. A 5

6 good exaple s a crteron reflectng the goal of cost nzaton. The assocated utlty functon ust result n hgher utlty values for lower cost values. It s coon that soe noralzaton s perfored on a nonnegatve row n the atrx of the a j entres. The entres n a row can be dvded by the su of the entres n the row, by the axal eleent n the row, or by a desred value greater than any entry n the row. These noralzatons can be also foralzed as applyng utlty functons Sple ultattrbute ratng technque (SMART) SMART s the splest for of the MAUT ethods. The rankng value x j of alternatve A j s obtaned sply as the weghted algebrac ean of the utlty values assocated wth t,.e. x j = = 1 w a j / = 1 w, j = 1,..., n. Besdes the above sple addtve odel, Edwards (1977) also proposed a sple ethod to assess weghts for each of the crtera to reflect ts relatve portance to the decson. Frst, the crtera are ranked n order of portance and 10 ponts are assgned to the least portant crteron. Then, the next-least-portant crteron s chosen, ore ponts are assgned to t, and so on, to reflect ther relatve portance. The fnal weghts are obtaned by noralzng the su of the ponts to one. However, as Edwards and Barron (1994) ponted out, the coparson of the portance of attrbutes s eanngless f t does not reflect the range of the utlty values of the alternatves as well. They proposed a varant naed SMARTS (SMART usng Swngs) that n the course of the coparson of the portance of the crtera also consders the apltude of the utlty values,.e. the changes fro the worst utlty value level to the best level aong the alternatves. See also Barron and Barrett (1996) for further technques Generalzed eans In a decson proble the vector x=(x 1,...,x n ) plays a role of aggregaton takng the perforance scores for every crteron wth the gven weght nto account. Ths eans that the vector x should ft nto the rows of the decson atrx as well as possble. Mészáros and Rapcsák (1996) ntroduced an entropy optzaton proble to fnd the vector x of best ft. They ponted out that the optal soluton s a postve ultple of the vector of the weghted geoetrc eans of the coluns, consequently, wth the values x j w = w = 1 w / w = aj, = 1,..., n = 1, consttute a reasonable and theoretcally establshed syste of rankng values. By ntroducng another entropy optzaton proble, based on another easure of fttng, the weghted algebrac eans (used also n SMART and addtve lnear odels) were obtaned as best fttng rankng values. Mészáros and Rapcsák (1996) also proposed to deterne the ratng values n the for of generalzed ean: 6

7 1 w x j = f ( f ( aj )), = 1,..., n, w = 1 where f s a strctly onotone real functon. Ths wde class of eans also ncludes the weghted algebrac and geoetrc eans wth f(t)=t and f(t)=log(t), respectvely The Analytc Herarchy Process The Analytc Herarchy Process (AHP) was proposed by Saaty (1980). The basc dea of the approach s to convert subjectve assessents of relatve portance to a set of overall scores or weghts. AHP s one of the ore wdely appled ultattrbute decson akng ethods. We follow here the suary of UK DTRL (2000) on the AHP. The ethodology of AHP s based on parwse coparsons of the followng type 'How portant s crteron C relatve to crteron C j?' Questons of ths type are used to establsh the weghts for crtera and slar questons are to be answered to assess the perforance scores for alternatves on the subjectve (judgental) crtera. Consder how to derve the weghts of the crtera. Assue frst that the crtera are not arranged n a tree-structure. For each par of crtera, the decson aker s requred to respond to a parwse coparson queston askng the relatve portance of the two. The responses can use the followng nne-pont scale expressng the ntensty of the preference for one crteron versus another 1= Equal portance or preference. 3= Moderate portance or preference of one over another. 5= Strong or essental portance or preference. 7= Very strong or deonstrated portance or preference. 9= Extree portance or preference. If the judgeent s that crteron C j s ore portant than crteron C, then the recprocal of the relevant ndex value s assgned. Let c j denote the value obtaned by coparng crteron C relatve to crteron C j. Because the decson aker s assued to be consstent n akng judgeents about any one par of crtera and snce all crtera wll always rank equally when copared to theselves, we have c j =1/c j and c =1. Ths eans that t s only necessary to ake 1 / 2 ( - 1) coparsons to establsh the full set of parwse judgeents for crtera. The entres c j,,j=1,, can be arranged n a parwse coparson atrx C of sze x. The next step s to estate the set of weghts that are ost consstent wth the relatvtes expressed n the coparson atrx. Note that whle there s coplete consstency n the (recprocal) judgeents ade about any one par, consstency of judgeents between pars,.e. c j c kj = c k for all,j,k, s not guaranteed. Thus the task s to search for an -vector of the weghts such that the x atrx W of entres w /w j wll provde the best ft to the judgents recorded n the parwse coparson atrx C. Several of technques were proposed for ths purpose. Saaty's orgnal ethod to copute the weghts s based on atrx algebra and deternes the as the eleents n the egenvector assocated wth the axu egenvalue of the atrx. The egenvalue ethod has been crtczed both fro prortzaton and consstency ponts of vew and several other technques have been developed. A nuber of other ethods are based on the nzaton of the dstance between atrces C and W. Soe of these approaches gve the vector 7

8 w drectly or by sple coputatons, soe other ones requre the soluton of nuercally dffcult optzaton probles. One of these approaches, the logarthc least squares ethod, results n a straghtforward way of coputng vector w: calculate the geoetrc ean of each row n the atrx C, calculate the su of the geoetrc eans, and noralze each of the geoetrc eans by dvdng by the su just coputed (Saaty and Vargas (1984)). See Gass and Rapcsák (2004) for further references on dstance-nzng ethods and a new approach based on sngular value decoposton. In the practce the crtera are often arranged n a tree-structure. Then, AHP perfors a seres of parwse coparsons wthn saller segents of tree and then between sectons at a hgher level n the tree-structure. Slarly to calculaton of the weghts for the crtera, AHP also uses the technque based on parwse coparsons to deterne the relatve perforance scores of the decson table for each of the alternatves on each subjectve (judgeental) crteron. Now, the parwse questons to be answered ask about the relatve portance of the perforances of pars of alternatves relatng the consdered crteron. Responses use the sae set of nne ndex assessents as before, and the sae technques can be used as at coputng the weghts of crtera. Wth the weghts and perforance scores deterned by the parwse coparson technque above, and after further possble noralzaton, alternatves are evaluated usng any of the decson table aggregaton technques of the MAUT ethods. The so-called addtve AHP uses the sae weghted algebrac eans as SMART, and the ultplcatve AHP s essentally based on the coputaton of the weghted geoetrc eans. A nuber of specalsts have voced a nuber of concerns about the AHP, ncludng the potental nternal nconsstency and the questonable theoretcal foundaton of the rgd 1-9 scale, as well as the phenoenon of rank reversal possbly arsng when a new alternatve s ntroduced. On the sae te, there have also been attepts to derve slar ethods that retan the strengths of AHP whle avodng soe of the crtcss. See Trantaphyllou, E. (2000) and Fguera et al. (2004) for state-of-art surveys and further references. 3.4 Outrankng ethods The prncpal outrankng ethods assue data avalablty broadly slar to that requred for the MAUT ethods. That s, they requre alternatves and crtera to be specfed, and use the sae data of the decson table, naely the a j s and w s. Vncke (1992) provdes an ntroducton to the best known outrankng ethods; see also Fguera et al. (2004) for state-of-art surveys. Here, the two ost popular fales of the outrankng ethods, the ELECTRE and the PROMETHEE ethods wll be brefly outlned The ELECTRE ethods The splest ethod of the ELECTRE faly s ELECTRE I. The ELECTRE ethodology s based on the concordance and dscordance ndces defned as follows. We start fro the data of the decson atrx, and assue here that the su of the weghts of all crtera equals to 1. For an ordered par of alternatves (A j,a k ), the concordance ndex c jk s the su of all the weghts for those crtera where the perforance score of A j s least as hgh as that of A k,.e. 8

9 c jk = w, : a j a k j, k = 1,.., n, j k. Clearly, the concordance ndex les between 0 and 1. The coputaton of the dscordance ndex d jk s a bt ore coplcated: d jk =0 f a j >a k, =1,...,,.e. the dscordance ndex s zero f A j perfors better than A k on all crtera,. Otherwse, d jk = ax = 1,.., a ax a j= 1,.., n k j a j n a j = 1,.., n j, j, k = 1,.., n, j k,.e. for each crteron where A k outperfors A j, the rato s calculated between the dfference n perforance level between A k and A j and the axu dfference n score on the crteron concerned between any par of alternatves. The axu of these ratos (whch ust le between 0 and 1) s the dscordance ndex. A concordance threshold c* and dscordance threshold d* are then defned such that 0<d*<c*<1. Then, A j outranks A k f the c jk >c* and d jk <d*,.e. the concordance ndex s above and the dscordance ndex s below ts threshold, respectvely. Ths outrankng defnes a partal rankng on the set of alternatves. Consder the set of all alternatves that outrank at least one other alternatve and are theselves not outranked. Ths set contans the prosng alternatves for ths decson proble. Interactvely changng the level thresholds, we also can change the sze of ths set. The ELECTRE I ethod s used to construct a partal rankng and choose a set of prosng alternatves. ELECTRE II s used for rankng the alternatves. In ELECTRE III an outrankng degree s establshed, representng an outrankng credtablty between two alternatves whch akes ths ethod ore sophstcated (and, of course, ore coplcated and dffcult to nterpret). See Fguera et al (2004) for ore detals and further ebers of the ELECTRE faly The PROMETHEE ethods The decson table s the startng pont of the PROMETHEE ethodology ntroduced by Brans and Vncke (1985) and Brans et al. (1986). The scores a j need not necessarly be noralzed or transfored nto a coon densonless scale. We only assue that, for the sake of splcty, a hgher score value eans a better perforance. It s also assued that the weghts w of the crtera have been deterned by an approprate ethod (ths s not a part of the PROMETHEE ethods), furtherore, = w = 1 1. Here, followng Brans and Mareschal (1994), we gve a bref revew of the PROMETHEE ethods. In order to take the devatons and the scales of the crtera nto account, a preference functon s assocated to each crteron. For ths purpose, a preference functon P (A j,a k ) s defned, representng the degree of the preference of alternatve A j over A k for crteron C. We consder a degree n noralzed for, so that 0 P (A j,a k ) 1 and P (A j,a k ) =0 eans no preference or ndfference, P (A j,a k ) 0 eans weak preference, P (A j,a k ) 1 eans strong preference, and P (A j,a k ) =1 eans strct preference. 9

10 In ost practcal cases P (A j,a k ) s functon of the devaton d = a j ak,.e. P (A j,a k )= p ( aj ak ), where p s a nondecreasng functon, p (d)=0 for d 0, and 0 p ( d ) 1 for d > 0. A set of sx typcal preference functons was proposed by Brans and Vncke (1985) and Brans et al. (1986). The splcty s the an advantage of these preferences functons: no ore than two paraeters n each case, each havng a clear econocal sgnfcance. A ultcrtera preference ndex π (A j,a k ) of A j over A k can then be defned consderng all the crtera: π (A j,a k ) = w P =1 (A j,a k ). Ths ndex also takes values between 0 and 1, and represents the global ntensty of preference between the couples of alternatves. In order to rank the alternatves, the followng precedence flows are defned: Postve outrankng flow: Negatve outrankng flow: + φ (A j ) = 1 n n 1 k = 1 1 n φ (A j ) = n 1 k = 1 π (A j,a k ). π (A k,a j ). The postve outrankng flow expresses how uch each alternatve s outrankng all the others. The + hgher φ (A j ), the better the alternatve. φ + (A j ) represents the power of A j, ts outrankng character. The negatve outrankng flow expresses how uch each alternatve s outranked by all the others. The saller φ (A j ), the better the alternatve. φ (A j ) represents the weakness of A j, ts outranked character. The PROMETHEE I partal rankng A j s preferred to A k when holds as a strct nequalty. A j and A k are ndfferent when + φ (A j ) φ + (A k ), A j and A k are ncoparable otherwse. φ (A j ) + φ (A j )= φ + (A k ) and φ (A j )= φ (A k ). φ (A k ), and at least one of the nequaltes In ths partal rankng soe couples of alternatves are coparable, soe others are not. Ths nforaton can be useful n concrete applcatons for decson akng. The PROMETHEE II coplete partal rankng If a coplete rankng of the alternatves s requested by the decson aker, avodng any ncoparabltes, the net outrankng flow can be consdered: φ (A j ) = φ + (A j ) -φ (A j ). 10

11 The PROMETHEE II coplete rankng s then defned: A j s preferred to A k when φ (A j )>φ (A k ), and A j and A k are ndfferent when φ (A j )=φ (A k ). All alternatves are now coparable, the alternatve wth the hghest φ (A j ) can be consdered as best one. Of course, a consderable part of nforaton gets lost by takng the dfference of the postve and negatve outrankng flows. PROMETHEE V: Optzaton under constrants Optzaton under constrants s a typcal proble of operatons research. The proble of fndng an optal selecton of several alternatves, gven a set of constrants, belongs to ths feld. PROMETHEE V extends PROMEHEE II to ths selecton proble. The objectve s to axze the total net outrankng flow value of the selected alternatves eanwhle they are feasble to the constrants. Bnary varables are ntroduced to represent whether an alternatve s selected or not, and nteger prograng technques are appled to solve the optzaton proble. The GAIA vsual odellng ethod The set of alternatves can be represented by n ponts n the -densonal space, where s the nuber of crtera. As the nuber of crtera s usually greater than two, t s possble to have a clear vson of these ponts. GAIA offers a vsualzaton technque by projectng the ponts on a two-densonal plane, where the plane s defned so that as few nforaton as possble gets lost by the projecton. The GAIA plane provdes the decson aker wth a powerful tool for the analyss of the dfferentaton power of the crtera and ther conflctng aspects. Clusters of slar alternatves as well as ncoparablty between two alternatves are clearly represented. The projecton of the vector of the weghts of crtera suggests the drecton, where the ost prosng alternatves can be found on the plane. The ethodology appled n GAIA appeared earler n statstcs as a vsualzaton tool under the nae of prncpal coponents analyss. See Rapcsák (2004) for the atheatcal background of the ethodology. Strengthenng PROMETHEE wth deas of AHP Soe deas of AHP can also be appled n the PROMEETHE ethodology. Recently, Machars et al. (2004) proposed to use the parwse coparson technque of AHP to deterne the weghts of the crtera. Slarly, the use of the tree-structure to decopose the decson proble nto saller parts can also be benefcal. 4. Group decson akng Group decson s usually understood as aggregatng dfferent ndvdual preferences on a gven set of alternatves to a sngle collectve preference. It s assued that the ndvduals partcpatng n akng a group decson face the sae coon proble and are all nterested n fndng a soluton. A group decson stuaton nvolves ultple actors (decson akers), each wth dfferent sklls, experence and knowledge relatng to dfferent aspects (crtera) of the proble. In a correct ethod for syntheszng group decsons, the copetence of the dfferent actors to the dfferent professonal felds has also to be taken nto account. 11

12 We assue that each actor consders the sae sets of alternatves and crtera. It s also assued that there s a specal actor wth authorty for establshng consensus rules and deternng votng powers to the group ebers on the dfferent crtera. Keeney and Raffa (1976) call ths entty the Supra Decson Maker (SDM). The fnal decson s derved by aggregatng (syntheszng) the opnons of the group ebers accordng to the rules and prortes defned by the SDM. There are several approaches to extend the basc ultattrbute decson akng technques for the case of group decson. Soe earler MAUT ethods of group decson are revewed by Bose et al. (1997). Here we present the ethod appled n the WINGDSS software (Csák et al. (1995)). Consder a decson proble wth l group ebers (decson akers) D 1,,D l, n alternatves A 1,..,A n and crtera C 1,,C. In case of a factual crteron the evaluaton scores ust be dentcal for any alternatve and any decson aker, whle subjectve (judgental) crtera can be evaluated dfferently by each decson aker. Denote the result of the evaluaton of decson aker D k for alternatve A j on the crteron C by a. Assue that the possble proble arsng fro the k j dfferent densons of the crtera has already been settled, and the a k j values are the result of proper transforatons. The ndvdual preferences on the crtera are expressed as weghts: let the weghts of portance w 0 be assgned at crteron C by decson aker D k, =1,...,; k=1,...,l. k The dfferent knowledge and prorty of the group ebers are expressed by votng powers both for weghng the crtera and qualfyng (scorng) the alternatves aganst the crtera. For factual crtera only the preference weghts gven by the decson akers wll be revsed at each crteron by the votng powers for weghng. However, n case of subjectve crtera, not only the weghts but also the a values wll be odfed by the votng powers for qualfyng. k j k Let V ( w) denote the votng power assgned to D k for weghng on crteron C, and votng power assgned to D k for qualfyng (scorng) on crteron C, =1,...,; k=1,...,l. V ( q) the k The ethod of calculatng the group utlty (group rankng value) of alternatve A j s as follows: For each crteron C, the ndvdual weghts of portance of the crtera wll be aggregated nto the group weghts W : l = k k V ( w) w k 1 W, = 1,..., l k V ( w) =. k = 1 The group qualfcaton Q j of alternatve A j aganst crteron C s: l = k k V ( q) a k 1 j Qj, = 1,...,, j = 1,..., n l k V ( q) k = 1 =. The group utlty U j of A j s deterned as the weghted algebrac ean of the aggregated qualfcaton values wth the aggregated weghts: W Q 1 j U j, j = 1,..., n W = =. = 1 12

13 In addton to the weghted algebrac eans used n the above aggregatons, WINGDSS also offers the weghted geoetrc ean, but generalzed eans can also be appled (see subsecton 3.3.2). Csák et al. (1995) also descrbes the forulas for coputng n the case when the crtera are gven n a tree-structure. The best alternatve of group decson s the one assocated wth the hghest group utlty. A correct group utlty functon for cardnal rankng ust satsfy the axos gven n Keeney (1976). The utlty functon coputed by the WINGDSS ethodology s approprate n ths respect. The approach of the Analytc Herarchy Process can also be extended to group decson support (Dyer and Foran (1992)), see also La et al. (2202) for a recent applcaton and further references. Snce the AHP s based on parwse coparson atrces, the key queston s how to synthesze the ndvdual parwse coparson atrces of the group ebers. Aczél and Saaty (1983) showed that under reasonable assuptons (recprocty and hoogenety) the only syntheszng functon s the geoetrc ean. Another approach was proposed by Gass and Rapcsák (1998) for syntheszng group decsons n AHP. It conssts of the aggregaton of the ndvdual weght vectors deterned by sngular value decoposton, takng the votng powers of the group ebers also nto account. Of course, the extensons of the outrankng ethods for group decson support have also been developed. Machars et al. (1998) presents a PROMETHEE procedure for group decson support. Another ethod, based on ELECTRE ethodology, was proposed by Leyva-López and Fernández- González (2003) for group decson support. 5. Senstvty analyss Soe values of the ultattrbute decson odels are often subjectve. The weghts of the crtera and the scorng values of the alternatves aganst the subjectve (judgental) crtera contan always soe uncertantes. It s therefore an portant queston how the fnal rankng or the rankng values of the alternatves are senstve to the changes of soe nput paraeters of the decson odel. The splest case s when the value of the weght of a sngle crteron s allowed to vary. For addtve ultattrbute odels, the rankng values of the alternatves are sple lnear functons of ths sngle varable and attractve graphcal tools can be appled to present a sple senstvty analyss to a user (Foran and Selly (2001)). For a wde class of ultattrbute decson odels Mareschal (1988) showed how to deterne the stablty ntervals or regons for the weghts of dfferent crtera. These consst of the values that the weghts of one or ore crtera can take wthout alterng the results gven by the ntal set of weghts, all other weghts beng kept constant. Wolters and Mareschal (1995) proposed a lnear prograng odel to fnd the nu odfcaton of the weghts requred to ake a certan alternatve ranked frst. Trantaphyllou and Sanchez (1997) presented an approach of a ore coplex senstvty analyss wth the change of the scores of the alternatves aganst the crtera, as well. A general and coprehensve ethodology was presented by Mészáros and Rapcsák (1996) for a wde class of MAUT odels where the aggregaton s based on generalzed eans, ncludng so the addtve and ultplcatve odels as well. In ths approach the weghts and the scores of the alternatves aganst the crtera can change sultaneously n gven ntervals. The followng questons were addressed: What are the ntervals of the fnal rankng values of the alternatves wth the restrcton that the ntervals of the weghts and scores are gven? 13

14 What are the ntervals of the weghts and scores wth the restrcton that the fnal rankng of the alternatves does not change? Consder a subset of alternatves whose rankng values are allowed to change n an nterval. In what ntervals are the weghts and scores allowed to vary, and how wll these odfcatons effect the rankng values of the entre set of alternatves? Mészáros and Rapcsák (1996) ponted out that these questons lead to the optzaton of lnear fractonal functons over rectangles and proposed an effcent technque to solve these probles. Soe of the results of Mészáros and Rapcsák (1996) were recently extended by Ekárt and Néeth (2005) for ore general decson functons. References Aczél, J. and Saaty, T.L. (1983) Procedures for syntheszng rato judgeents, Journal of Matheatcal Psychology, 27, Baker, D., Brdges, D., Hunter, R., Johnson, G., Krupa, J., Murphy, J. and Sorenson, K. (2002) Gudebook to Decson- Makng Methods, WSRC-IM , Departent of Energy, USA. Barron, F.H. and Barrett, B.E. (1996) The effcacy of SMARTER Sple Mult-Attrbute Ratng Technque Extended to Rankng, Acta Psychologca, 93, Bose, U., Davey, A.M. and Olson, D.L. (1997) Mult-attrbute utlty ethods n group decson akng: Past applcatons and potental for ncluson n GDSS, Oega, 25, Brans, J.P. and Vncke, Ph. (1985) "A preference rankng organzaton ethod", Manageent Scence, 31, Brans, J.P., Vncke, Ph. and Marechal, B. (1986) "How to select and how to rank projects: The PROMETHEEethod", European Journal of Operatonal Research, 24, Brans, J.-P. and Mareschal, B. (1994) The PROMCALC & GAIA decson support syste for ultcrtera decson ad, Decson Support Systes, 12, Csák, P., Rapcsák, T., Turchány, P. and Veres, M. (1995) Research and developent for group decson ad n Hungary by WINGDSS, a Mcrosoft Wndows based group decson support syste, Decson Support Systes 14, Dyer, R.F. and Foran, E.H. (1992) Group decson support wth the Analytc Herarchy Process, Decson Support Systes, 8, Edwards, W. (1977) How to use ultattrbute utlty easureent for socal decsonakng, IEEE Transactons on Systes, Man, and Cybernetcs, SMC-7, Edwards, W. and Barron, F.H. (1994) SMARTS and SMARTER: Iproved sple ethods for ultattrbute utlty easureents, Organzatonal Behavor and Huan Decson Processes, 60, Ekárt, A. and Néeth, S.Z. (2005) Stablty analyss of tree structured decson functons, European Journal of Operatonal Research, 160, Fguera, J., Greco, S. and Ehrgott, M. (Eds.) (2004) Multple Crtera Decson Analyss: State of the Art Surveys, Sprnger, New York. Foran, E. and Selly, M.A. (2001) Decson by Objectves, World Scentfc. Gass, S. I. and Rapcsák, T. (1998) A note on syntheszng group decsons, Decson Support Systes, 22, Gass, S. I. and Rapcsák, T. (2004) Sngular value decoposton n AHP, European Journal of Operatonal Research, 154, Harrs, R. (1998) Introducton to Decson Makng, VrtualSalt. Keeney, R.L. and Raffa, H. (1976) Decsons wth Multple Objectves: Perforances and Value Trade-Offs, Wley, New York. Keeney, R.L. (1976) A group preference axoatzaton wth cardnal utlty, Manageent Scence, 23,

15 La, V.S., Bo K.W. and Cheung, W. (2002) Group decson akng n a ultple crtera envronent: A case usng the AHP n software selecton, European Journal of Operatonal Research, 137, Leyva-López, J-C. and Fernández-González, E. (2003) A new ethod for group decson support based on ELECTRE III ethodology, European Journal of Operatonal Research, 148, Lnkov, I., Varghese, A., Jal, S., Seager, T.P., Kker, G. and Brdges, T. (2004) Mult-crtera decson analyss: A fraework for structurng reedal decsons at the contanated stes, In: Lnkov, I. and Raadan, A.B. (Eds.) Coparatve Rsk Assessent and Envronental Decson Makng, Sprnger, New York, pp Machars, C., Brans, J.P. and Mareschal, B. (1998). The GDSS PROMETHEE Procedure, Journal of Decson Systes, 7, Machars, C., Sprngael, J., De Brucker, K. and Verbeke, A. (2004) PROMETHEE and AHP: The desgn of operatonal synerges n ultcrtera analyss.: Strengthenng PROMETHEE wth deas of AHP, European Journal of Operatonal Research, 153, Mareschal, B. (1988) Weght stablty ntervals n ultcrtera decson ad, European Journal of Operatonal Research, 33, Mészáros, Cs. and Rapcsák, T. (1996) On senstvty analyss for a class of decson systes, Decson Support Systes 16, Munda, G. (1996) Cost-beneft analyss n ntegrated envronental assessent: soe ethodologcal ssues, Ecologcal Econocs, 19, Nehauser, G.L., Rnnoy Kan, A.H.G. and Todd, M.J. (1989) Handbooks n Operatons Research and Manageent Scence: Volue 1 Optzaton, North-Holland, Asterda. Rapcsák, T. (2004) Multattrbute Decson Makng, Lecture notes, Departent of Decsons n Econocs, Corvnus Unversty, Budapest. (n Hungaran) Roy, B. (1968) "Classeent et chox en présence de ponts de vue ultple (la éthode electre), RAIRO, 2, Saaty, T.L. (1980) The Analytc Herarchy Process, McGraw Hll. Saaty, T.L. and Vargas, L.G. (1984) Coparson of egenvalue, logarthc least squares and least squares ethods n estatng ratos, Matheatcal Modellng, 5, Steuer, R. E. (1986) Multple Crtera Optzaton: Theory, Coputaton and Applcaton, Wley, New York. Trantaphyllou, E. and Sanchez, A. (1997) "A senstvty analyss approach for soe deternstc ult-crtera decson akng ethods", Decson Scences, 28, Trantaphyllou, E. (2000) Mult-Crtera Decson Makng Methods: A Coparatve Study, Kluwer Acadec Publshers, Dordrecht. UK DTLR (2001) Mult Crtera Analyss: A Manual, Departent for Transport, Local Governent and the Regons, UK. US EPA (2000) Gudelnes for Preparng Econoc Analyss, Unted States Envronental Protecton Agency, EPA 240-R Vncke, P. (1992) Mult-crtera Decson-Ad, John Wley, Chchester. Wolters, W.T.M. and Mareschal, B. (1995) Novel types of senstvty analyss for addtve MCDM ethods, European Journal of Operatonal Research, 81, ` 15