Helsinki University Of Technology, Systems Analysis Laboratory Mat Independent research projects in applied mathematics (3 cr)

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1 Helsnk Unversty Of Technology, Systems Analyss Laboratory Mat-2.08 Independent research projects n appled mathematcs (3 cr) "! #$&% Antt Laukkanen 506 R ajlaukka@cc.hut.f

2 2 Introducton Multattrbute decson-makng Multattrbute decson-makng problems Lower boundng of weghts Rank orderng weght approxmaton methods Decson rules Central Values Decson rule Smulaton Evaluaton measures Dstrbuton of scores Smulaton steps Results Conclusons References...7

3 3 Introducton In a multattrbute decson-makng problem,.e. problem wth several alternatves and attrbutes, the decson maker (DM) s often assumed to be able to provde nformaton of rank orderng of attrbutes nstead of specfc, or true, weghts of mportance of the attrbutes, as s done n []. Such assumpton, complete rank orderng beng provded by DM, s also made n ths study. The true weghts of attrbutes reman unknown n practce, because accuracy that s needed for elctaton of exact weghts may be actually absent n the mnd of the DM []. Even f the elctaton of precse weghts was possble, t would probably be tme-consumng and dffcult and therefore mpractcal. Rank orderng weghtng methods provde approxmatons of true weghts of attrbutes when rank orderng nformaton s known, for best outcome of the decson-makng process. Rank orderng weghtng methods take nto account DM s nformaton about rank orderng of attrbute weghts, from whch approxmatons for attrbute weghts are calculated by usng correspondng formula. Rank sum [2], rank order centrod [3] and rank recprocal [3] are the rank orderng weghtng methods benchmarked n ths study. In addton to the rank orderng methods, usage of a decson rule s benchmarked n ths study. Decson rules are heurstcs that recommend one alternatve of the alternatves avalable. In a multattrbute decson-makng problem the avalable alternatves are evaluated by decson rules, whch are used by takng nto account the rank orderng nformaton. Usage of Central Values [4] decson-rule s benchmarked n ths study. Smulaton of 36 dfferent runs, 5000 rounds each, was done n ths study. Dfferent decson-makng problems were smulated. Amount of alternatves and attrbutes n the problem was vared. Trangularand unform dstrbutons for alternatves scores were appled and effect of lower bounded weghts for attrbutes studed. Ths paper s arranged as follows: Multattrbute decson-makng problem s dscussed and llustrated va example n Secton 2. Used weghtng methods and the decson rule are ntroduced n Secton 3. Secton 4 descrbes the smulaton procedure and Secton 5 the results of the smulaton. Conclusons are dscussed n Secton 6.

4 2 Multattrbute decson-makng 4 2. Multattrbute decson-makng problems Many mplementatons for multattrbute decson-makng problems exst n practce. There are many challenges consderng the solvng of such problems n practce. To make a defntely rght decson, attrbutes' weghts should be exactly known and scores for alternatves evaluated. As dscussed n the prevous secton, defnng of exact weghts for attrbutes s not easy for DM, or perhaps even possble, n practce. Followng notatons are used throughout ths study: n = number of attrbutes m = number of alternatves W = true weght of :th attrbute In ths study, only possble weghts are assumed to exst for alternatve. True weghts for attrbutes: W 0, =,..., n, where W stands for the "real" weght for th attrbute. In ths study, all weghts are assumed to be normalzed,.e. ther sum equals one: m W = =. () Model for alternatves evaluaton s assumed to be addtve multattrbute value (MAV) of form, []: v ( j) = n = W v j, j =,..., m (2) where v(j) s the overall multattrbute value of alternatve j, whereas total number of alternatves s m. v j stands for the value, score, assocated to jth attrbute of th alternatve. Complete rank orderng, that DM s n ths study assumed to able to provde, defnes the rank orderng of the attrbute weghts, r = (r, r 2,, r n ), (3)

5 5 where r s the rank orderng of the th attrbute, r= represents the most mportant attrbute and r=n the least mportant. For nstance, n a problem of three attrbutes, for a rank orderng r = (r =, r 2 =3, r 3 =2) or n more compact form, r = (,3,2), the real weghts must fulfll W W 3 W 2. In addton, no equally mportant ranks are assumed to exst, and therefore rank orderng s always defned. More generally, r < r j W W j,, j {,2,.., n} j. (4) A multattrbute decson-makng problem s llustrated for nstance by takng a look at a famly buyng a new car. There are plenty of alternatves (dfferent car brands, dfferent models), and varous attrbutes (prce, comfort, looks, speed, economy, space, brand attractveness, etc.) nvolved. Due to lmted testng effort of the famly, number of alternatves s reduced to seven by defnng whch are among the best seven alternatves. Smlarly number of attrbutes to be taken nto account s lmted by namng the ones belongng to top fve. Thus, the famly has ended up to a decson problem wth seven alternatves and fve attrbutes. It s easly realzed that precse weghts for the fve attrbutes of a car are certanly hard for the famly to defne. Wth rank orderng (3) of the attrbutes known, rank orderng weghtng methods can be used to acheve approxmatons w for assumed real attrbute weghts behnd the rank orderng n the mnds of the DMs: w ~ W. (5) Naturally, also approxmatons must fulfll the rank orderng: r < r j w w j,, j {,2 n} j,..,. (6) In addton, scores of alternatves must be evaluated, whch may not be an easy task n practce ether. Verbal evaluaton may gve consderable ad for evaluaton of scores for attrbutes. After evaluatng the scores, famly can calculate values (2) for ther attrbutes, usng the calculated weght approxmatons and scores of alternatves, and defne the best alternatve. Another approach would be usage of decson rules [4],.e. heurstcs to choose one alternatve of the avalable ones. The decson-rule approach s qute dfferent compared to the weght approxmaton. No attrbute weghts are approxmated, but nstead possble rank orderngs for attrbute weghts are

6 6 defned from provded rank orderng nformaton. In the case wth complete rank orderng nformaton provded, approach of decson rules s straghtforward. Scores for alternatves must be evaluated. Based on the possbltes for weghts provded by rank orderng nformaton, possble score ranges for alternatves can be calculated, and therefore decson-rule appled to the problem. Thus, the use of decson rules accepts the uncertanty of attrbute weghts all the way and assumes nothng about the exact weghts. 2.2 Lower boundng of weghts It may be realstc to assume that all the weghts are at some reasonable level, so that there s no extremely small value for any attrbute. Ths assumpton s done by lower boundng the weghts relatvely to the number of attrbutes n, [6]: W, =,..., n. (7) 3n 3 Rank orderng weght approxmaton methods As t was dscussed n Sectons and 2, values for weghts can be approxmated va rank orderng weght approxmaton methods. The dea of the methods s followng; as the complete rank orderng nformaton of attrbute weghts s known, approxmatons for weghts are calculated by usng approprate formula, and by usng the approxmated weghts the values for alternatves are calculated and the alternatve wth the hghest value chosen. Used formulas are n force only when sum of the weghts equals one (). Followng three formulas for approxmated weghts, w, are used n ths study: Rank-order centrod weghts, [2]: w ( ROC) = n n j= j, =,..., n, (8) where w * s the approxmated weght for th mportant attrbute. Approxmatons for attrbute weghts must correspond to the rank orderng. Approxmated weghts w are acheved followng way: w = w j *, r =j, =,..., n. (9)

7 7 Smlarly, weghts correspondng to the ranks, rank sum [3]: ( n + ) n( n ) n + 2 w ( RS) = =, =,..., n (0) j j=... n and an approach based on the recprocal of the ranks, rank recprocal [3]: w ( RR) = j=... n j, =,..., n () Weghts of RS are reduced lnearly lower from the most mportant to the least mportant. RREC weghts descend aggressvely after the most mportant, but n the least mportant end the ROC weghts are the lowest ones. Dfferences of the weghtng methods are llustrated n the followng example of fve attrbutes (Fgure ). 0,50 0,40 weght 0,30 0,20 0,0 0,00 rank order ROC 0,46 0,26 0,6 0,09 0,04 RSUM 0,33 0,27 0,20 0,3 0,07 RREC 0,44 0,22 0,5 0, 0,09 ROC RSUM RREC Fgure Approxmatons for attrbute weghts gven by used formulas, n case of fve attrbutes

8 8 4 Decson rules The use of decson-rules n a multattrbute decson-makng problem s based on defnng of a feasble regon for attrbute weghts, and therefore for alternatves scores as well. Use of decson rules under ncomplete nformaton s dscussed n [4], and ther topology and related defntons n more detal n [6]. Extreme ponts are the boundary ponts for feasble regon, whch fulfls the rank orderng. When complete rank orderng of attrbute weghts s known, there are n extreme ponts and ther defnng s qute straghtforward. If the weghts are not lower bounded, for kth extreme pont: w =, r() k w = 0, r() > k,, k =,..., n (2) where w s the value of attrbute weght n the extreme pont. For nstance f there are three attrbutes and ther rank orderng s r = (,2,3), correspondng extreme ponts are (,0,0), (0.5, 0.5, 0) and (0.33, 0.33, 0.33). Feasble regon for weghts les between the extreme ponts, whch can be seen from llustraton of the example s extreme ponts and resultng feasble regon n Fgure 2 below: fgure 2 llustraton of extreme ponts n case of three attrbutes

9 9 If the weghts are lower bounded, the pont calculaton changes somewhat as every attrbute must have at least the weght of the lower bound. Therefore () must be modfed, as there s lesser amount of free weght as n- tmes the lower bound s reserved for the other weghts. Extreme value calculaton n case of lower bounded weghts becomes therefore: w nb =, r() k r w = l, r() > k,, k =,..., n (3) where b s the lower bound of the weght, / 3n n ths study (7). For nstance, extreme ponts for smlar three-attrbute problem, n case of lower-bounded weghts are (0.77, 0., 0.), (0.44, 0.44, 0.) and (0.33, 0.33, 0.33), Fgure 3. Fgure 3 Illustraton of extreme ponts n case of lower bounded weghts and three attrbutes

10 0 4. Central Values Decson rule Central values [4] decson rule uses followng heurstc: Choose the alternatve x for whch the mdpont of the feasble value nterval s the greatest,.e. [ v x) + v( x) ] [ v( x') + v( x' )], x, x' X (, (4) where v(x) s the best value of the alternatve x n feasble regon and v(x) s the worst. Consderng the calculaton, worst and best values are found n extreme ponts, as the valuaton functon (2) s lnear. In a case wth complete rank orderng of attrbute weghts and scores of alternatves known, as t s the case n ths study, t can be sad that usage of decson rules s qute straghtforward. Frst the extreme ponts are dentfed and therefore feasble values for alternatves calculated. By applyng ths nformaton to the chosen decson rule, mnmum and maxmum values n case of central values (4), best alternatve s chosen. 5 Smulaton As the scope of ths study s to compare the three weghtng methods (8),(0),(), and also benchmark the central values decson rule (4), exact scores for alternatves attrbutes are assumed to be known as well as complete attrbute rank orderng nformaton. The true weghts of attrbutes are assumed to exst n smulaton, so that real correct choces can be defned and therefore success of the used methods evaluated and benchmarked. Smulaton s done by smulatng 5000 cases wth randomzed attrbute weghts and scores for alternatves' attrbutes. Smulatons were run by varyng number of alternatves (5, 7 or 0) and attrbutes (5, 7 or 0). Lower boundng of weghts (7) was appled to half of the runs. Unform and trangular dstrbutons for scores were used. All combnatons were run, and 3 * 3 * 2 * 2 = 36 smulatons were therefore done. Smulaton was carred out usng Matlab ( n a UNIX envronment. The smulatons took approxmately an hour.

11 5. Evaluaton measures Success of the dfferent methods s evaluated by two measures, expected loss of value and percentage of correct choces. The latter one s calculated smply by countng the number of correctly chosen cases and dvdng t by total number of cases. Expected loss of value (ELV) s the average loss n overall value when the alternatve that has been chosen s compared to the correct choce [5]. ELV s calculated as follows: ELV = n = w * ' ( v ( x ) v ( x ), (5) where w s the real weght of attrbute, x * s the correct choce,.e. best alternatve, and x s the alternatve chosen by usng decson rule. 5.2 Dstrbuton of scores Scores are generated ether from an unform dstrbuton or a trangular dstrbuton [5], to have values over [0,]. Unformly dstrbuted scores are gven smply by n smulaton generated random number u over [0,],.e. s j = u (2). Therefore t s assumemed that alternatves do not have any common level of mportance as levels of attrbutes can vary very much. Trangular dstrbuton, n contrast, reflects a stuaton where the alternatves score qute smlar values as the densty of the dstrbuton s at hghest level around value of 0.5, whch s also dstrbuton s mean. The densty of trangular dstrbuton s n other hand dmnshngly small n ts extremes, 0 and. Tr-dstrbuted scores s are acheved from followng functon of u, [5]: u s =, when u 0.5 and 2 ( u) s =, when u > 0.5. (5) 2

12 2 5.3 Dstrbuton of weghts The weghts are normalzed, so that ther sum equals one (). The weghts of the attrbutes are assumed to be randomly dstrbuted from flat-dstrbuton, and have values over [0,] n not lower bounded cases. Lower boundng (7) of weghts s appled to the half of the smulaton runs. Lower boundng of weghts guarantees that the weghts have no extreme devaton as all are above certan level, and therefore reflect the real stuaton better. The weghts n lowerbounded cases have values over [b,-nb]. 5.4 Smulaton steps The smulaton prncpals are llustrated n the fgure 4 below: (0) Intate smulaton parameters rank orderng weghtng methods no yes () Randomze: scores weghts. Defne rank orderng Calculate attrbute weght approx. (2.) (2.3) Calculate extreme ponts (2.2) Choose best alternatve usng RROC RS, RREC (2.4) Choose best alternatve usng CV enough rounds? (3) Wrte down chosen alternatves and correct choce decson rule (4) Evaluate decsons; ELV, CC% Fgure 4 Smulaton flowchart

13 3 The smulaton steps n more detal: 0. Defne smulaton parameters: number of alternatves, k (5/7/0) number of attrbutes, n (5/7/0) lowerbounded (true/false) trdstrbuted (true/false) rounds (5000) Then, repeat rounds tmes steps -3:. Smulate decson problem. Randomze values for attrbutes of alternatves If trdstrbuted case then randomze from tr-dstrbuton, otherwse randomze from unformdstrbuton. Result s (k,n)-matrx..2 Randomze real weghts for attrbutes If lowerbounded case then lmt attrbutes' lower bounds.3 Normalze scores for each attrbute Each score must have value from [0,].4 Calculate true values for alternatves, and choose the correct choce, CC 2. Choose best alternatves gven by the methods Rank orderng attrbute weghtng methods: 2. Calculate weghts for RROC, RSUM, RREC 2.2 Usng approxmated attrbute weghts, calculate values for every alternatve and choose the best, RROC_best, RSUM_best, RREC_best.

14 4 Decson rule: 2.3 Calculate extreme ponts for central values-method 2.4 For each alternatve calculate central values usng extreme ponts, and choose the best, CV_best 3. Calculate and wrte down results 3.. For each method compare the best chosen alternatve and the correct choce CC. If the alternatve chosen s dfferent than the the CC, wrte down ncorrect, otherwse correct choce For each ncorrect choce made, calculate loss of value, e. dfference of values of real and heurstcs best alternatves. 4. After all the rounds have passed, calculate Expected loss of value, ELV (5) Percentage of correct choces

15 5 6 Results Smulaton parameters percentage of correct choces expected loss of value number of number of bounded dstrbuton alternatves attrbutes weghts of scores rroc rsum rrec cv rroc rsum rrec cv 7 5 yes tr 0,829 0,804 0,87 0,787 0, , , , no tr 0,832 0,794 0,820 0,770 0, , , , yes flat 0,822 0,796 0,8 0,775 0,0074 0,074 0, , no flat 0,825 0,790 0,80 0,765 0, ,04 0, , yes tr 0,86 0,80 0,789 0,746 0,0043 0, , , no tr 0,837 0,776 0,8 0,78 0,0040 0, ,0053 0, yes flat 0,826 0,807 0,802 0,762 0, ,0090 0,0073 0, no flat 0,837 0,776 0,807 0,726 0,0053 0,053 0, , yes tr 0,836 0,792 0,792 0,73 0, , , , no tr 0,84 0,763 0,79 0,665 0, , , , yes flat 0,838 0,803 0,794 0,738 0,0043 0, ,0067 0, no flat 0,849 0,759 0,80 0,684 0, ,067 0, , yes tr 0,799 0,77 0,780 0,75 0,006 0, , , no tr 0,809 0,756 0,790 0,739 0, ,0075 0, , yes flat 0,796 0,777 0,774 0,739 0, ,067 0,0028 0, no flat 0,8 0,783 0,79 0,73 0,007 0,066 0, , yes tr 0,800 0,773 0,76 0,723 0, , , , no tr 0,86 0,752 0,788 0,686 0, , , , yes flat 0,8 0,775 0,789 0,732 0, ,0027 0,0077 0, no flat 0,88 0,752 0,787 0,690 0,0057 0,0299 0, , yes tr 0,803 0,766 0,755 0,69 0,0039 0, , , no tr 0,830 0,740 0,780 0,633 0, , ,0052 0, yes flat 0,80 0,776 0,778 0,7 0, , , , no flat 0,827 0,753 0,772 0,634 0, ,082 0, , yes tr 0,840 0,89 0,832 0,85 0, , , , no tr 0,860 0,820 0,845 0,807 0, , , , yes flat 0,840 0,88 0,824 0,798 0, ,0033 0,0083 0, no flat 0,857 0,83 0,840 0,803 0, ,0202 0, , yes tr 0,847 0,822 0,832 0,804 0, , ,0044 0, no tr 0,867 0,8 0,852 0,778 0, , , , yes flat 0,855 0,823 0,829 0,796 0, ,0084 0, , no flat 0,859 0,87 0,835 0,76 0,0042 0, , , yes tr 0,852 0,82 0,83 0,768 0, , , , no tr 0,863 0,80 0,82 0,74 0, , ,0047 0,0 5 0 yes flat 0,856 0,824 0,8 0,760 0, , , , no flat 0,873 0,803 0,834 0,730 0, ,0096 0, ,052 The results are qute smlar n all the cases smulated, however some dfferences and nterestng ssues were found. Common for all the runs s the successfulness of the methods. The RROC s the most successful method n all cases. Decson rule central values gves most wrong decsons. RREC s often better than RSUM, but not n all cases. The order s same n terms of ELV, except when k=7, n=5 and k=n=5, as the rank sum has the worst and central values the second worst ELV.

16 6 An nterestng ssue s the effect of lower-boundng of weghts. The RSUM and CV succeed better when the weghts are lower bounded. In contrast, RROC and RREC succeed worse when the weghts are lower bounded. As t can be seen from the Fgure, weghts of RS are more unformly dstrbuted than RREC and RROC. The lower boundng of the weghts reduces ther mutual dfferences, and therefore RS fts closer to the real stuaton. In addton, lower boundng of weghts seems to mprove the success of central values, especally when there are many attrbutes. The dstrbuton of scores does not seem to have a clear mpact to the percentage of correct choces. However, expected loss of value seems to get somewhat bgger for all the methods when the scores come from unform-dstrbuton. Ths s a result of bgger devaton of scores and therefore also dfferences of alternatves values ncrease. 7 Conclusons Usage of three rank orderng attrbute weghtng methods, rank order centrod (8), rank sum (0), rank recprocal (), and central values decson rule (4), were dscussed and smulated n ths study. Smulaton of total 36 dfferent runs was made, 5000 rounds each. Dfferent decson-makng problems were smulated. Amount of alternatves and attrbutes was vared, as well as dstrbuton of scores of alternatves and weghts of attrbutes. The smulaton results were qute smlar n all the cases, whereas some nterestng ssues were found. Rank order centrod was the most successful method n terms of both used crtera, percentage of correct choces and expected loss of value. The central values decson rule was the worst method. Rank recprocal was mostly better than rank sum. Lower boundng of weghts has an mprovng effect to the success of rank sum method and central values, whereas rank order centrod and rank recprocal score worse when weghts are lower bounded. Expected loss of value ncreases when unform dstrbuton of weghts s used nstead of trangular dstrbuton.

17 7 8 References [] Hutton B., Barrett B., 996. Decson Qualty Usng Ranked Attrbute Weghts. Management Scence, Vol 42, No, November 996. [2] Barron, F.H., 992. Selectng a Best Multattrbute Alternatve wth Partal Informaton About Attrbute Weghts. Acta Psychologca, 80 (992), [3] Stllwell, W.G., D. A. Seaver, and W. Edwards, 98. A comparson of Weght Approxmaton Technques n Multattrbute Utlty Decson Makng. Organzatonal Behavor and Human Performance, 28 (98), [4] Salo A., Hämälänen R., 200. Preference ratos n multattrbute evaluaton.(prime) Elctaton and decson procedures under ncomplete nformaton. Transactons n systems, man, and cybernetcs part A: systems and humans, Vol 3, No 6, November 200 [5] Punkka A., 2003: Uses of Ordnal Preference Informaton n Interactve Decson Support. Systems Analyss Laboratory, Helsnk Unversty of Technology, [6] Salo A., Punkka A., Rank Incluson n Crtera Herarches. European Journal of Operatonal Research. (to appear).