Generalized mixture operators using weighting functions: a comparative study with WA and OWA

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1 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Geeralzed mxture operators usg weghtg fuctos: a comparatve study wth WA ad OWA Rta Almeda Rbero *, Rcardo Alberto Marques Perera ** * Uversdade Lusíada, Departameto Cêcas Empresaras, Rua da Juquera 192, Lsboa, Portugal, rar@uova.pt ** Uverstà d Treto, Dpartmeto d Iformatca e Stud Azedal, Va Iama 5, TN Treto, Itala, mp@cs.ut.t Abstract. I the cotext of multple attrbute decso makg, we preset a aggregato scheme based o geeralzed mxture operators usg weghtg fuctos ad we compare t wth two stadard aggregato methods: weghted averagg (WA) ad ordered weghted averagg (OWA). Specfcally, we cosder lear ad quadratc weght geeratg fuctos that pealze bad attrbute performaces ad reward good attrbute performaces. A llustratve example, borrowed from the lterature, s used to perform the operators comparso. We beleve that ths comparatve study wll hghlght the potetal ad flexblty of geeralzed mxture operators usg weghtg fuctos that deped o attrbute performaces. Keywords: fuzzy sets, weghted aggregato, lear ad quadratc weght geeratg fuctos, geeralzed mxture operators, multple attrbute decso makg. 1. Itroducto I multple attrbute decso makg problems for a gve umber of alteratves or courses of acto, oe has to detfy a set of relevat attrbutes characterzg the alteratves, elct ther relatve mportace ad express these some form of weght factors. The usual uderlyg assumpto s that attrbutes are depedet,.e. the cotrbuto of ay dvdual attrbute to the total ratg of a alteratve s depedet of the other attrbute values. I ths paper, we assume that a umercal characterzato of all relevat attrbutes s possble ad les wth the famlar ut scale [0,1]. Moreover, as most of the research o weghted aggregato, from the classcal weghted averagg (WA) ad the ordered weghted averagg (OWA) (Yager 1988) (Yager 1992; Yager 1993) to the more recet Choquet tegrato (over a fte doma) 1

2 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) (Grabsch 1995; Grabsch 1996), we assume commesurablty of the varous attrbutes wth the ut scale. Ths meas that the way attrbute satsfacto values are umercally expressed the ut terval s desged so that commesurablty holds to a acceptable degree. We also assume that the relatve mportace of attrbutes ca tself be expressed umercally wth the ut terval. I solvg multple attrbute decso makg problems may types of soluto methods have bee developed (see for stace (Yoo ad Hwag 1995)): scorg ad outrakg methods, trade-off schemes, dstace based methods, value ad utlty fuctos, teractve methods. The frst exteso of multple attrbute decso models to the fuzzy set cotext was proposed by Bellma ad Zadeh (Bellma ad Zadeh 1970). Sce the most of the techques to solve multple attrbute problems have bee exteded to the fuzzy doma (for good overvews see (Che ad Hwag 1992) (Zmmerma 1987) (Kckert 1978) (Rbero 1996)). Scorg techques are wdely used, partcularly the weghted aggregato operators based o multple attrbute value theory (Keeey ad Raffa 1976). The classcal weghted aggregato s usually kow the lterature by weghted averagg (WA) or smple addtve weghtg method (Yoo ad Hwag 1995). Aother mportat aggregato operator, wth the class of weghted aggregato operators, s the ordered weghted averagg OWA (Yager 1988). I ths paper we wll focus o two famles of geeralzed mxture operators (Marques Perera ad Pas 1999; Marques Perera 2000) usg weghtg fuctos that pealze bad attrbute performaces ad reward good attrbute performaces: oe famly s assocated wth lear weght geeratg fuctos ad the other famly s assocated wth quadratc weght geeratg fuctos. Ths aggregato method exteds weghted averagg, whch s a partcular case of geeralzed mxture operator. The questo of defg weghts (.e. weghtg fuctos) that deped o attrbute satsfacto values as (Rbero ad Marques Perera 2001) s at the kerel of ths work ad we beleve t ca be relevat some decso problems. For stace, suppose that the parets of a 5 th grade studet wsh to select a school for ther so. There are two caddate schools ad the parets wat to select the best oe accordg to the followg two crtera. The most mportat crtero for school selecto s dstace ; small dstace meas that school s wth walkg dstace from home, whereas large 2

3 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) dstace meas that the school ca be reached oly by car. The secod (less mportat) crtero s the teachg qualty of the school. The decsoal scheme of the parets s as follows: f both schools are close from home (wth walkg dstace) tha domace w.r.t. dstace s much stroger tha domace w.r.t. qualty, sce the dfferece dstace s expereced walkg. O the other had, f both schools are far from home (ca be reached oly by car) tha domace w.r.t. dstace s comparable wth domace w.r.t. qualty, sce the dfferece dstace s expereced by drvg. We model ths decsoal scheme by cosderg lear weght geeratg fuctos of the followg types: very mportat (wth terval [0.6, 1]) for dstace ad mportat (terval [0.4, 0.8]) for qualty. For the weghted average (WA) we cosder the mddle pot of the weght rages, respectvely 0.8 ad 0.6. Now, Table 1 we preset two possble selecto cases (Case I ad Case II) wth attrbute satsfacto values ad aggregated values for each of the schools uder cosderato, usg the two lear weght geeratg fuctos dcated before. CASE I School 1 School 2 Dstac e Qualt y RESULT S CASE School School II 1 2 Dstac e Qualt y RESULT S Table 1. Example of school selecto wth attrbutes dstace ad qualty To clarfy the use of lear weghtg fuctos computg the aggregated values, cosder that are the weght geeratg fuctos for attrbutes dstace ad qualty, respectvely. The vector x = ( x D, f ( x ) = (1 0.6) x the aggregated value of the school s gve by D x Q ) D D ad f ( x ) = ( ) x represets the attrbute satsfacto values of oe school ad Q Q Q ( x ) = = D, Q w ( x x W ) = j = D Q wth weghtg fuctos w (x) f ( x ) /, f ( x ). (1) For stace, Case I the aggregated value of school 1 s computed as follows, j j 3

4 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Case1 school 1 = 0.4 * (0.6 + (1 0.6) * 0.4) * (0.4 + ( ) * 0.8) = (0.6 + (1 0.6) * 0.4) + (0.4 + ( ) * 0.8 Observg Table 1 we ca see that for Case I the selected school s 2, whle Case II the selected school s 1. Ths smple example shows clearly the depedecy of weghts o the satsfacto values of attrbutes, rewardg hgher values of attrbute satsfacto ad pealzg lower values. I both Cases I ad II, there s a domace of 0.2 oe attrbute agast a detcal but reverse domace of 0.2 the other attrbute. I Case II the domace the attrbute dstace ws, whch s just what oe would expect gve that dstace s the most mportat attrbute. I Case I, o the other had, t s the domace the attrbute qualty that ws, because ths case the dstace satsfacto values are low ad therefore the effect of the domace that attrbute s pealzed. Cosderg the chage from Case I to Case II we see that, sce dstace s the most mportat attrbute, whe ts attrbute satsfacto values crease from low to hgh the school choce chages. I other words, attrbutes wth bad performaces are pealzed ad attrbutes wth good performaces are rewarded, wth the correspodg effect o attrbute domace. O the other had, f we apply ether the classcal weghted averagg (WA) or the ordered weghted averagg (OWA) to the school example, the selected school s always 2. Ths s clear the WA case; as far as the OWA case s cocered, ths s because the weght assocated wth attrbute dstace s the same for both schools ad does ot chage from Case I to Case II (otce that the attrbute satsfacto value for dstace s always less or equal tha that of the attrbute qualty ). I summary, the ma focus of ths work s to llustrate the potetal ad flexblty of the aggregato wth geeralzed mxture operators usg weghtg fuctos, by comparso wth two other aggregato methods, the weghted averagg WA ad the OWA (Yager 1988). The paper s orgazed as follows. After ths troductory secto, secto 2 provdes a bref troducto to the three aggregato methods that wll be compared. Secto 3 presets the ma cocepts volved weghted aggregato ad secto 4 descrbes the lear ad quadratc weght geeratg fuctos that wll be used wth the geeralzed mxture operators. Secto 5 compares ad dscusses the three approaches 4

5 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) usg a llustratve example, borrowed from the lterature. Fally, secto 6 presets the coclusos of ths comparatve study. 2. Weghts ad weghtg fuctos The specfcato of a preferece rakg over a set of attrbutes reflects the relatve mportace that each attrbute has the composte. Procedures for elctg ad determg relatve mportace (weghts) have bee the focus of extesve research ad dscusso (Al-Kloub, Al-Shemmer et al. 1997), (Edwards ad Newma 1982) (Hobbs 1980), (Rbero 1996), (Stllwell, Seaver et al. 1981), (Weber ad Borcherdg 1993), (Yoo 1989). However, a mportat problem that has oly bee partally addressed by the research o elctg ad determg weghts s the fact that weghts should sometmes deped o the correspodg attrbute satsfacto values. From a decso maker perspectve, whe cosderg oe attrbute wth a hgh relatve mportace, whch has a bad performace (low satsfacto value), the decsoal weght of ths attrbute should be pealzed, order to reder the gve attrbute less sgfcat the overall evaluato of the alteratves. As a result, whe cosderg two alteratves, the domace effect oe mportat attrbute becomes less sgfcat whe the attrbute satsfacto values are low. I the same sprt, the decsoal weght of a attrbute wth low relatve mportace (weght) that has hgher satsfacto values should be rewarded, thereby rederg more sgfcat the domace effect that attrbute. Summarzg, some cases t may be approprate to have weghts depedg o the attrbute satsfacto performace values. The troducto of weghtg fuctos depedg cotuously o attrbute satsfacto values produces aggregato operators wth complex umercal behavour. The mootocty of the aggregato operator s a crucal ssue (Marques Perera ad Pas 1999; Marques Perera 2000) whch volves costrats o the dervatves of the weghted aggregato operator wth respect to the varous attrbute satsfacto values. Naturally, the questo of mootocty s also relevat the classcal framework of umercal weghts, partcularly whe the weghted aggregato operators have complex x depedeces. I ths case, moreover, there s also the questo of sestvty, volvg costrats o the dervatves of the aggregato operator wth respect to the varous weght values. Roughly speakg, whle mootocty requres that the operator 5

6 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) creases whe ay of the attrbute satsfacto values creases, sestvty requres that the relatve cotrbuto of the -th attrbute to the value of the operator creases whe the correspodg weght creases. A terestg study of mootocty ad sestvty the classcal framework of umercal weghts has bee doe (Kaymac ad va Nauta-Lemke 1998). A alteratve approach to modulate the relatve mportace of attrbutes depedg o attrbute satsfacto values was proposed by Rbero (Rbero 2000). The author troduced a lear terpolato fucto, called WIS (Weghtg Importace ad Satsfacto), whch combes lgustc mportace weghts of attrbutes wth attrbute satsfacto values. The lgustc relatve mportace values are represeted by fuzzy sets (Zadeh 1987) o a cotuous terval support, as for stace very_mportat=[0.6, 1], ad attrbute satsfacto values are represeted the terval [0.1,1]. Recetly, wth the geeralzed mxture approach, two weghtg fuctos depedg o attrbute satsfacto values were proposed (Rbero ad Marques- Perera 2001). Oe s obtaed from a lear weght geeratg fucto ad the other s obtaed from a sgmod weght geeratg fucto. I both cases the relatve mportace (weghts) of the attrbutes s provded lgustc format, as for example {very mportat, mportat, ot mportat}, ad each lgustc weght has a lower ad upper lmt gve by the decso maker. The lower ad upper lmts are the tegrated the weght geeratg fuctos to defe ther rage ad slope. I ths paper we develop further the geeralzed mxture approach by cosderg a quadratc exteso of the lear weght geeratg fucto. The ext two sectos descrbe the ma characterstcs of the two weght geeratg fuctos (lear ad quadratc) cosdered. 3. Weghted aggregato operators: WA, OWA, ad geeralzed mxtures I ths secto we brefly recall the defto ad ma propertes of three types of weghted aggregato operators: weghted averagg (WA) operators, ordered weghted averagg (OWA) operators, ad geeralzed mxture operators. The latter exted the classcal weghted averagg operators (whch rema a specal case) the sese that the classcal costat weghts become weghtg fuctos defed o the 6

7 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) aggregato doma. I other words, the weghts are o loger costat but deped o the values of the aggregato varables. The presetato of geeralzed mxtures follows the orgal oes (Marques Perera ad Pas 1999; Marques Perera 2000), where mxture operators ad geeralzed mxture operators have bee troduced. Roughly speakg, mxture operators are cotuous ad compesatve aggregato operators. I other words, they are mea operators wthout the mootocty requremet. Geeralzed mxture operators costtute a partcular class of mxture operators, the same way as geeralzed mea operators (also called quas-arthmetc meas) costtute a partcular class of mea operators. There s fact a strog ad terestg relato betwee the two geeralzed classes, as has bee poted out (Marques Perera ad Pas 1999) Weghted averagg (WA) operators Ths aggregato model, whose relevace stems from the axomatc bass of multple attrbute value theory (Keeey ad Raffa 1976) together wth the cetral role of weght determato multple value aalyss (Weber ad Borcherdg 1993), s also kow the lterature by smple addtve weghtg method (Yoo ad Hwag 1995). Usually, the WA model assumes that weghts are proportoal to the relatve value of a ut chage each attrbute (Hobbs 1980). Cosder varables the ut terval [0,1], wth = 1,...,, ad a equal umber of weghts 0 satsfyg the ormalzato codto = 1. The w vectors cotag the varables ad the weghts are deoted x = [ x ] ad w = [ w ], respectvely. where x j = w 1 j The classcal weghted averagg operator s defed as WA (x) = =1 w x, x = [ x ] ecodes the attrbute satsfacto values of a alteratve. The operator WA s clearly cotuous the aggregato doma [ 0,1]. The weghted averagg operator WA (x) s also compesatve ad mootoc, meag that m( x) WA ( x) max(x) ad x y WA( x) WA( y), 7

8 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) respectvely. The relato x y meas that x y for = 1,...,. Moreover, the WA(x) operator s lear ad the weghts are costat the aggregato doma Ordered weghted averagg (OWA) operators The ordered weghted averagg (OWA) operators have bee troduced (Yager 1988; Yager 1992; Yager 1993). Although the orgal OWA scheme dd ot clude quatfers, ths paper we use quatfed OWA operators (Yager 1996) for our comparatve study. The llustratve example dscussed the fal secto was borrowed from the work (Yager ad Kelma 1999). The ordered weghted averagg operator s defed as OWA ( x) = = 1wx( ), where x ( ) s the th largest elemet of the vector operator WA s cotuous the aggregato doma [ 0,1]. x. Despte the reorderg mechasm, the It s useful to rewrte the weghted averagg operator as OWA (x) = =1 w[ ] x, where w [] s the weght assocated wth the th elemet of the vector x, referrg to the tradtoal OWA operator defto gve above. I ths formulato of the reorderg mechasm t s evdet that the weght assocated wth a elemet of the vector x chages depedg o the order statstcs of that elemet. The ordered weghted averagg operator OWA (x) s also compesatve ad mootoc, that s m( x) OWA( x) max( x) ad x y OWA( x) OWA( y), respectvely. However, the OWA (x) operator s o lear ad the weghts are oly pecewse costat the aggregato doma: they are costat wth comootocty domas but chage dscotuously from oe comootocty doma to aother. We recall that comootocty domas are coes dmesoal aggregato space wth whch all vectors x have the same orderg sgature. For stace, for = 3 the two vectors x = (0.2,0. 7,0.5) ad x'= (0.3,0.9,0.8) have the same orderg sgature ((1),(2),(3))=(2,3,1) ad thus belog to the same comootocty doma. Istead, the vector x"= (0.6,0.1,0.2) wth orderg sgature ((1),(2),(3))=(1,3,2) belogs to a dfferet comootocty doma. 8

9 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) w (x) As we shall see below, geeralzed mxture operators have weghtg fuctos that deped cotuously o the aggregato varables x. Ths meas cosderg a large class of weghtg fuctos cotag, partcular, arbtrarly good approxmatos of the pecewse costat weghtg fuctos used the OWA operator. Apart from cotuty, t s coveet to assume also some degree of dfferetablty, so that the powerful techques of dfferetal calculus ca be appled. These techques are fact crucal characterzg the geeral mootocty costrats uder whch weghtg fuctos effectvely produce mootoc weghted aggregato operators (Marques Perera ad Pas 1999; Marques Perera 2000). We ote that a partcular class of geeralzed mxture operators had already bee cosdered (Yager ad Flev 1994) but volatos of mootocty had led the authors to abado ths proposal. Now we eed to troduce a lk betwee the OWA operator defto ad ts relatoshp wth lgustc quatfers Q, such as Most. Gve a lgustc quatfer Q, meag for stace that Q of the attrbutes should be satsfed by a acceptable soluto, the weghts ca be determed as (Yager 1996), 1 w ( x ) = Q( u ) Q( u ) (2) j= 1 ( j) j= 1 ( j) j =1 u j = 1 where the u are the ormalzed mportaces of attrbutes, wth 2 quatfer Most s gve by ( r ) r for r 0, 1. Q = [ ], ad the 3.3. Geeralzed mxture operators Let us cosder ow fuctos f (x) defed the ut terval x [0,1], wth = 1,...,. We assume that these fuctos are of class C,.e. each fucto 1 f (x) [ s cotuous ad has a cotuous dervatve f '( x) the ut terval x 0,1], for = 1,...,. We also assume that the fuctos (x) f are postve the ut terval,.e. f ( x) > 0 for x [0,1] ad = 1,...,, ad that ther dervatves '( x) are oegatve the ut terval,.e. '( x) 0 for x [0,1] ad = 1,...,. f f 9

10 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) (x) f 2000), The geeralzed mxture operator W (x) o [ 0,1] geerated by the fuctos, wth = 1,...,, s defed as (Marques Perera ad Pas 1999; Marques Perera W f ( x ) x = 1 (x) = (3) j= 1 f j ( x j ) 1 It s straghtforward to show that W (x) s also C the aggregato doma [ 0,1],.e. t s cotuous ad has cotuous partal dervatves [ 0,1]. Moreover, the geeralzed mxture operator W (x) ca be wrtte as a weghted average of the varables x, wth weghtg fuctos w (x) stead of the classcal w costat weghts, W ( x ) = = 1w ( x) x wth weghtg fuctos w (x) f ( x ) = j = 1 f j ( x j ) (4) Due to the propertes of the geeratg fuctos (x), for = 1,...,, the weghtg f fuctos are strctly postve, w ( x) > 0, ad satsfy the ormalzato codto j=1 w j ( x) = 1. Geeralzed mxture operators are compesatve operators the usual sese, but they are ot ecessarly mootoc. Nevertheless, terestg suffcet codtos for mootocty ca be derved, as show below. Moreover, geeralzed mxture operators are o lear (eve wth comootocty domas) ad ther weghtg fuctos vary cotuously ad smoothly the aggregato doma. I (Marques Perera ad Pas 1999; Marques Perera 2000), the authors have derved a geeral suffcet codto for the strct mootocty of the geeralzed mxture operator W (x), where by strct mootocty we mea ( x) > 0 for = 1,...,. I the ut terval, for postve geeratg fuctos wth o-egatve W 10

11 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) dervatves, the suffcet codto takes the smple form ad = 1,..., f '( x) f ( x), for. I the paper, we call t smply the mootocty codto. x [0,1] From a tutve pot of vew, the mootocty codto establshes a upper boud for the varablty of the weght geeratg fucto, f '( x) / f ( x) 1. I tur, ths costrat lmts the degree to whch the weghtg fuctos (x) w deped o the aggregato varables. I other words, the weght depedece o the aggregato varables must be some sese small ( the classcal weghted averagg case t was ull) so that mootocty s ot volated. Notce that the codtos 0 f '( x) f ( x) > 0 hold depedetly for each = 1,...,. We shall make use of ths fact what follows by applyg those codtos to a geeral lear or quadratc geeratg fucto f (x), wth oly o-classcal parameters regardg the weght depedece o the attrbute satsfacto degrees. Later we cosder the assocated effectve geeratg fucto F( x) = α f ( x) / f ( 1), whose value for ut attrbute satsfacto F(1) = α s gve by the classcal mportace parameter α. I secto 4 we descrbe two dfferet ways whch geeralzed mxture operators ca model multple attrbute aggregato wth weghtg fuctos: oe volves lear weght geeratg fuctos ad the other volves quadratc weght geeratg fuctos. 4. Lear ad quadratc weght geeratg fuctos I ths secto we descrbe weght geeratg fuctos of the lear ad quadratc types. A prelmary study of the lear case has appeared (Rbero ad Marques Perera 2001). I geeral, as dscussed the prevous secto we assume that the weght 1 geeratg fuctos, f (x) for x [0,1] ad = 1,...,, are of class C ad dvdually satsfy three geeral codtos: (I) f ( x) > 0, (II) '( x) 0, ad (III) f '( x) f ( x). The latter s called the mootocty codto ad plays a cetral role the costructo. f 11

12 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) We recall that the geeralzed mxture operator costructed from the geeratg fuctos s defed as W ( x) = = 1w ( x) x, where the weghtg fuctos j = w (x) = f ( x ) / 1 f ( x j j ) satsfy the usual ormalzato codto j=1 w j ( x) = 1 for all x [0,1] Lear weght geeratg fuctos Cosder the lear geeratg fucto l ( x) = 1+ βx, wth doma x [0,1] ad parametrc rage 0 β = 0. β 1. The classcal case, wth costat weghts, correspods to It s straghtforward to check that the above codtos (I-III) hold the parametrc rage 0 β 1. The frst two codtos, l ( x) > 0 ad l' ( x) 0, are clearly verfed. The mootocty codto ca be wrtte as β ( 1 x) 1. Accordgly, sce ( 1 x) [0,1] for x [0,1], the mootocty codto s also verfed. l' ( x) l(x) The effectve geeratg fucto L (x) assocated wth l (x) s gve by l( x) 1+ βx L ( x) = α = α (5) l(1) 1+ β wth parameters 0 < α 1 ad 0 β 1. The mmum ad maxmum values L (0) ad L(1) of the effectve geeratg fucto, wth 0 < L(0) L(1) 1, are gve by L( 0) = α /( 1+ β ) ad L(1) = α, respectvely. Parameter 0 < α 1 cotrols the maxmum value L(1) = α of the effectve geeratg fucto, whe the attrbute satsfacto value s oe. I tur, parameter 0 β 1 cotrols the rato L( 1) / L(0) = 1+ β betwee the maxmum ad mmum values of the effectve geeratg fucto. I ths respect, the parametrc codto 0 β 1 ca be wrtte the form 1 L( 1) / L(0) 2, meag that the rato L( 1) / L(0) s at most 2 the lear case. 12

13 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) I the classcal weghted averagg case, the effectve geeratg fuctos have ull parameter β ( ths way they do ot deped o attrbute satsfacto) ad the dfferet costat weghts are obtaed by dfferet choces of the parameter α. Lear wth alpha=0.8 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 beta=0 beta=0.5 beta=1 Fgure 1. Lear weght geeratg fuctos Fgure 1 above shows three effectve geeratg fuctos of the lear type, assocated wth α = 0.8 ad three dfferet choces of parameter β. Parameter α cotrols the maxmum value of the effectve geeratg fucto, whe the attrbute satsfacto value s 1, whle parameter β cotrols the rato betwee the maxmum ad mmum values of the geeratg fucto Quadratc weght geeratg fuctos I complete aalogy wth the lear case descrbed prevously, oe ca study geeratg fuctos depededg quadratcally o attrbute satsfacto values. Cosder thus the quadratc geeratg fucto q( x) = 1+ γ x 2, wth doma x [0,1] ad parametrc rage 0 γ 1. The classcal case, wth costat weghts, correspods to γ = 0. As before, t s straghtforward to check that codtos (I-III) hold the parametrc rage 0 γ 1. The frst two codtos, q( x) > 0 ad q' ( x) 0, are clearly verfed. I tur, the mootocty codto q '( x) q( x) ca be wrtte as 13

14 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) γ (2x x ) 1. Accordgly, sce (2x x ) [0,1] for x [0,1], the mootocty codto s also verfed. The effectve geeratg fucto Q (x) assocated wth q (x) s gve by q( x) Q(x) = α q(1) 1+ γ x = α 1 + γ 2 (6) wth parameters 0 < α 1 ad 0 γ 1. As the prevous case, the mmum ad maxmum values Q (0) ad Q (1) of the effectve geeratg fucto, wth 0 < Q(0) Q( 1) 1, are gve by Q( 0) = α /(1 + γ ) ad Q (1) = α, respectvely. Parameter 0 < α 1 cotrols the maxmum value Q(1) = α of the effectve geeratg fucto, whe the attrbute satsfacto value s oe. I tur, parameter 0 γ 1 cotrols the rato Q (1) / Q(0) = 1+ γ betwee the maxmum ad mmum values of the effectve geeratg fucto. I ths respect, the parametrc codto 0 γ 1 ca be wrtte the form 1 Q(1) / Q(0) 2, meag that the rato Q ( 1) / Q(0) s at most 2 also the quadratc case. As before, the classcal weghted averagg case the effectve geeratg fuctos have ull parameter γ ( ths way they do ot deped o attrbute satsfacto) ad the dfferet costat weghts are obtaed by dfferet choces of the parameter α. The two cases, lear ad quadratc, are equvalet most respects, partcularly the aalogous role of the parameters β,γ ad the detcal rage (from 1 to 2) of the rato betwee the maxmum ad mmum values of the effectve geeratg fuctos. The real dfferece betwee the lear ad quadratc cases les the fact that the lear depedecy o attrbute satsfacto values s homogeeous the doma x [0,1], whereas the quadratc depedecy s weak for low attrbute satsfacto values x [ 0, 1 2] ad strog for hgh attrbute satsfacto values x [ 1 2,1]. Ths homogeeous depedecy o attrbute satsfacto values has the effect of 14

15 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) emphaszg the o classcal behavour of the assocated geeralzed mxture operators, as wll be llustrated the example preseted the last secto of the paper. Regardg the mportat questo of the rato betwee the maxmum ad mmum values of the effectve geeratg fuctos, t s possble to sgfcatly exted the preset 1 to 2 rage by cosderg a more geeral famly of quadratc 2 geeratg fuctos, cotag also a lear term: q( x) = 1+ ( β γ ) x + γ x. Roughly speakg, ths exteded geeratg fucto the presece of parameter β ehaces the effect of parameter γ ad (wth approprate parameter tug) the rato Q( 1) / Q(0) ca reach up to 2.6, approxmately. The theoretcal dscusso of ths possblty, however, s beyod the scope of ths paper ad wll be preseted elsewhere. Lear & Quadratc wth alpha=0.8 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Lbeta=0 Lbeta=0.5 Lbeta=1 Qgamma=0 Qgamma=0.5 Qgamma=1 Fgure 2. Lear ad quadratc weght geeratg fuctos Fgure 2 above shows three effectve geeratg fuctos of the quadratc type, assocated wth α = 0.8 lear geeratg fuctos wth ad three dfferet choces of parameter β. The correspodg β = γ are also depcted. As before, parameter α cotrols the maxmum value of the effectve geeratg fucto, whe the attrbute satsfacto value s 1, whle parameter γ cotrols the rato betwee the maxmum ad mmum values of the geeratg fucto. 15

16 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Illustratve example Ths example s borrowed from Yager ad Kelma (Yager ad Kelma 1999) ad cossts the problem of selectg the best caddate for a computer developer job. There are three caddates ad ther evaluato s made o the bass of sx attrbutes. We dscuss ths decsoal problem wth the followg techques: Yager s quatfed OWA operator (Yager 1988); the classcal weghted averagg operator; ad the geeralzed mxture operators assocated wth our lear ad quadratc weght geeratg fuctos. We compare the results obtaed wth the four techques order to llustrate the flexblty of our weght geeratg fuctos the cotext of decsoal problems whch depedece o attrbute satsfacto s a relevat ssue. As metoed before, there are three caddates for the computer developer job ad the decso maker evaluates them wth respect to sx attrbutes. The sx attrbutes are the computer laguage kowledge of: Basc, Pascal, Lsp, Prolog, C++, ad Vsual Basc. Moreover, the decso maker thks that some laguages are more mportat tha others ad assgg dfferet mportace values to dfferet attrbutes wll represet ths. For what cocers the quatfed OWA method, the oly dfferece betwee our dscusso of the computer developer problem ad the orgal oe descrbed (Yager ad Kelma 1999) s that we wat to select the caddate wth the best kowledge of the most mportat laguages, stead of selectg the caddate wth the best kowledge of most laguages. For ths reaso, we use the lgustc quatfer Few stead of the lgustc quatfer Most orgally used (Yager ad Kelma 1999). Mathematcally, we represet the lgustc quatfer Few by the ut terval [ ] trasformato Q ( r ) = r for r 0, 1. Ths substtuto of the lgustc quatfer Most wth Few eables us to make a more meagful comparso betwee our geeralzed mxtures ad the quatfed OWA operators sce, ths paper, we oly study the case of creasg weght geeratg fuctos. Ths class of geeratg fuctos produces larger weghts for larger attrbute satsfacto values, whch s roughly speakg the average behavour of quatfed OWA operators based o the lgustc quatfer Few. If o the other had we were studyg decreasg weght geeratg fuctos we could have kept the lgustc quatfer Most, as the orgal verso of the decsoal problem. 16

17 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) I Table 2 we show the orgal data for the decsoal problem questo (see also (Yager ad Kelma 1999)). It cludes the satsfacto values of each attrbute, for the three caddates, as well as the mportace values of the sx attrbutes. Importace Attrbutes Basc Pascal Lsp Prolog C++ Vsual Basc Caddate A Caddate B Caddate C Table 2. Data for the computer developer problem The frst step to solve ths decsoal problem s to trasform the mportace values to a [0,1] scale, sce our effectve weght geeratg fuctos are o that scale. Hece, we dvde the mportace of each attrbute by the maxmum mportace value. These scaled mportace values wll be used throughout the dscusso. Table 3 shows the orgal mportace values ad the scaled oes. Attrbutes Basc Pascal Lsp Prolog C++ Vsual Basc Importace Scaled Importace Table 3. Scaled mportace values The scaled mportace values are used our lear ad quadratc weght geeratg fuctos to defe the parameter α, whch correspods to the upper lmt of the fuctos. Both parameters β,γ for the lear ad quadratc geeratg fuctos (respectvely) are set to 1, so as to have the maxmal slope both cases. Furthermore, lgustc terms we ca terpret the attrbute mportace values as {very_low, low, hgh, very_hgh} ad these terms correspod to the upper lmts α ={0.2, 0.4, 0.6, 1}, respectvely. I Fgure 3 we show the lear ad quadratc geeratg fuctos for the weghts whose mportace s hgh ad very_hgh (Pascal, C++, ad Vsual Basc). 17

18 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Weghts Hgh ad Very-Hgh 0,9 1 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Lear(0.6, 1) Lear(1, 1) Quadratc(0.6,1,1) Quadratc(1,1,1) Fgure 3. hgh: Lear(0.6,1), Quadratc(0.6,1); very_hgh: Lear(1,1), Quadratc(1,1). The remag weght geeratg fuctos used our dscusso have upper lmts α = 0.2 ad α =0.4, correspodg to very_low ad low mportace values. For what cocers the WA ad OWA operators used ths example, the costructo s as follows. The weghts the WA operator are gve by the ormalzed (ut sum) mportace values computed from Table 3,.e. the weght for Pascal s 0.2/3.6 ad so o. The same values of ormalzed mportace are also used the OWA operator costructo. I ths case they play the role of the u, for = 1,...,, whch eter the defto of the OWA weghts through the quatfer scheme descrbed at the ed of secto 3.2. We are ow the posto to preset the results usg the four methods: the classcal weghted averagg operator WA; Yager s quatfed OWA aggregato operator; ad the geeralzed mxture operators assocated wth our lear ad quadratc weght geeratg fuctos. Table 4 shows the aggregato results obtaed wth the four methods. Lear Quadratc WA OWA Caddate A Caddate B Caddate C Table 4. Aggregato results Observg Table 4 we ca see that all techques select caddate A as the best oe, ad that the weghted averagg operator WA fals to detect the secod best 18

19 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) caddate B. Both the OWA ad our lear ad quadratc weght geeratg fuctos select caddate B as the secod best ad caddate C as the worst caddate. I order to llustrate the flexblty of our weght geeratg fuctos, whch deped o the satsfacto values of the attrbutes, we made a smulato cosderg that caddate A had a lower attrbute satsfacto value for laguage C++. More specfcally, we wated to see what would happe f caddate A had a satsfacto value of 0.2 stead of 0.5 for a attrbute wth very_hgh mportace. The correspodg aggregato results are show table 5. Lear Quadratc WA OWA Caddate A Caddate B Caddate C Table 5. Aggregato results usg attrbute C++ wth 0.2 (stead of 0.5) for caddate A. Observg the results obtaed Table 5 we ca see that, ow, the best caddate s B for our lear ad quadratc weght geeratg fuctos, whereas the weghted averagg operator WA does ot dstgush betwee caddate B ad C, ad the quatfed OWA operator stll thks that caddate A s the best oe. However, f we observe the orgal attrbute satsfacto values (Table 2), wth the chage of 0.2 for attrbute C++ o caddate A, t s clear that caddate B performs much better (o average) attrbutes wth hgh ad very_hgh mportace, respectvely Pascal, C++, ad Vsual Basc: wth the orgal data (Table 2), the weghted average values of attrbute satsfacto restrcted to the three mportat attrbutes are A > B > C 0.615; o the other had, whe the attrbute satsfacto value of caddate A for C++ s chaged from 0.5 to 0.2, the weghted average values become B > C > A 0.515, wth A as the worst caddate ow. However, ether the WA or the OWA are able to take to accout ths fact. Ths smulato clearly shows that our weght geeratg fuctos ca pealze (or reward) alteratves that have lower (or hgher) satsfacto values for the attrbutes, partcularly whe the attrbute has a hgh or very_hgh mportace. Aother mportat aspect to study s the comparso betwee the actual weghtg fuctos of our geeralzed mxture operators ad the OWA weghts. Hece, we made 19

20 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) a smulato that llustrates the behavour of our weghtg fuctos ad the OWA weghts whe the satsfacto value of caddate A for attrbute C++ creases the terval [0.2, 0.5]. The umercal results are show Fgure 4. C++ weght comparso Lear Quadratc OWA WA Fgure 4. Comparso betwee our weghtg fuctos ad the OWA weghts Observg Fgure 4, otce that the OWA weght s costat for the terval cosdered whle our weghtg fuctos reward the crease the satsfacto value of the attrbute C++. The lear weghtg fucto has a faster crease the the quadratc oe, as could also be see Fgure 4. The choce betwee the lear ad quadratc weght geeratg fuctos should be left to the decso maker because t depeds o the problem, partcularly o how much the decso maker wshes to pealze or reward chages attrbute satsfacto. A fal smulato was made to study volatos of preferetal depedece (Keeey ad Raffa 1976) beyod those that occur the OWA aggregato scheme. Ths meas lookg for preferetal depedece volatos wth comootocty domas, whch would ecessarly mply that our geeralzed mxtures are able to represet preferece structures that ca ot be represeted by OWA operators. Cosder a vector x = x ] of attrbute satsfacto values, wth x [0,1]. As [ the OWA operator defto, we deote by () the dex value of the th largest elemet of the vector x. More precsely, we say that (( 1), (2),..., ( ) ) s a decreasg x ) orderg sgature assocated wth f x ( 1) x(2... x( ). I what follows the decreasg specfcato wll always be mplct ad s therefore omtted. The orderg 20

21 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) sgature s clearly ot uque whe the vector x has equal elemets. I geeral, we deote by π (x) the set of orderg sgatures assocated wth x. We say that two-attrbute satsfacto vectors x = [ x ] ad y = [ y ] [ 0,1] are comootoc f the assocated orderg sgatures π (x) ad π (y) cocde (as sets). Comootocty s thus a equvalece relato, whch parttos the aggregato doma [0,1] to mutually dsjot comootocty domas. Geometrcally, each comootocty doma s a coe from the org bouded by the lmts of the aggregato doma [ 0,1]. Now, cosder two vectors x = ] ad [ x 1,..., x 1, t, x + 1,..., x y = [ y y, 1,..., 1 t, y + 1,..., y ] [ 0,1] wth the same th elemet. We say that a aggregato operator W o [ 0,1] satsfes preferetal depedece wth respect to the th attrbute f, gve ay two reduced vectors x \ = [ x 1,..., x 1,x+ 1,..., x ] ad y\ = [ y 1,..., y 1, y+ 1,..., y ], we obta the same relato W ( x ) W( y ) or W( x ) W( y ) for all values of t [0,1]. We say that the aggregato operator W satsfes preferetal depedece f t satsfes preferetal depedece wth respect to all attrbutes. It s well kow that the weghted averagg WA operator satsfes preferetal depedece ad that the weghted averagg operator OWA oly satsfes preferetal depedece wth comootocty domas. We wll show below that our aggregato operators, the geeralzed mxtures wth lear ad quadratc weght geeratg fuctos, do ot satsfy preferetal depedece eve wth comootocty domas. For ths reaso they are able to represet preferece structures ot represetable by OWA operators. The example s as follows. Suppose that we have two ew caddates ad that they oly kow Pascal, Lsp ad Prolog. Table 6 shows the attrbute satsfacto values assocated wth the two ew caddates, usg the same attrbute mportace values as the oes gve Table 4. The smulato cossts creasg the satsfacto value of attrbute Pascal from 0.5 to 0.7. Pascal Lsp Prolog 21

22 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Caddate D Caddate E Table 6. Ital data for the three attrbutes Pascal, Lsp ad Prolog. The aggregato results are show Table 7. Ideed, a volato of preferetal depedece occurs, ad wth the sgle comootocty doma x < x < x Lsp Pascal Prolog better tha caddate E. : caddate D s tally worse tha caddate E, but eds up Lear OWA Satsfacto Pascal Cad. D Cad. E Cad. D Cad. E Table 7. Aggregato results wth creasg satsfacto of attrbute Pascal ( ) Fgure 5 shows the plot of the aggregato results, llustratg the volato of preferetal depedece produced by the lear weght geeratg fucto. As metoed before, ths meas that our geeralzed mxture operators ca represet preferece structures that are ot possble to represet wth OWA operators, or for that matter wth ay other aggregato operator satsfyg preferetal depedece wth comootocty domas (as, for stace, Choquet tegrals). Dferece D-E Dferece D-E Fgure 5. Preferetal depedece volato wth lear weght geeratg fuctos 22

23 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Coclusos We have dscussed two geeralzed mxture operators, assocated wth lear ad quadratc weght geeratg fuctos, that pealze poorly satsfed attrbutes ad reward well satsfed attrbutes. We have preseted a example comparg our two geeralzed mxture operators wth the classcal weghted averagg (WA) ad the ordered weghted averagg (OWA) operators. The results obtaed hghlght the potetal ad flexblty of geeralzed mxture operators usg weghtg fuctos that deped o attrbute performaces. Fally, we have show that the geeralzed mxture operators assocated wth (say) lear weght geeratg fuctos do ot satsfy preferetal depedece eve wth comootocty domas ad therefore they ca be used to represet preferece structures whch the OWA operators (ad also the Choquet tegrals) ca ot represet. Refereces Al-Kloub, B., T. Al-Shemmer, et al The role of weghts mult-crtera decso ad, ad the rakg of water projects Jorda. Europea Joural of Operatoal Research 99 : Bellma, R. E. ad L. A. Zadeh Decso-makg a fuzzy evromet. Maagemet Scece 17(4): Che, S.-J. ad C.-L. Hwag Fuzzy multple attrbute decso makg, Sprger-Verlag. Edwards, W. ad J. R. Newma, Eds Multattrbute evaluato. Quattatve applcatos the socal sceces, Sage Publcatos. Grabsch, M Fuzzy tegral multcrtera decso makg. Fuzzy Sets ad Systems 69 : Grabsch, M The applcatos of fuzzy tegrals multcrtera decso makg. Europea Joural of Operatoal Research 89: Hobbs, B. F A comparso of weghtg methods power plat sttg. Decso Sceces 11 : Kaymac, U. ad H. R. va Nauta-Lemke A sestvty aalyss approach to troducg weght factors to decso fucto fuzzy multcrtera decso makg. Fuzzy Sets ad Systems 97: Keeey, R. L. ad H. Raffa Multattrbute prefereces uder ucertaty: more tha two attrbutes. Decsos wth Multple Objectves: Prefereces ad Value Tradeoffs, Joh Wley ad Sos :

24 Draft of paper I: Europea Joural of Operatoal Research 145 (2003) Kckert, W. J. M Fuzzy theores o decso makg. Martus Njhoff Socal Sceces Dvso. Marques Perera, R. A The oress of mxture operators: the expoetal case. Proc. 8th Iteratoal Coferece o Iformato Processg ad Maagemet of Ucertaty Kowledge-Based Systems (IPMU 2000), Madrd, Spa. Marques Perera, R. A. ad G. Pas O o-mootoc aggregato: mxture operators. Proc. 4th Meetg of the EURO Workg Group o Fuzzy Sets (EUROFUSE 99) ad 2d Itl. Coferece o Soft ad Itellget Computg (SIC 99), Budapest, Hugary. Rbero, R. A Fuzzy multple attrbute decso makg: a revew ad ew preferece elctato techques. Fuzzy Sets ad Systems 78: Rbero, R. A Crtera mportace ad satsfacto decso makg. Proc. 8th Iteratoal Coferece o Iformato Processg ad Maagemet of Ucertaty Kowledge-Based Systems (IPMU 2000), Madrd, Spa Rbero, R. A. ad R. A. Marques-Perera Weghts as fuctos of attrbute satsfacto values. Proc. of the Workshop o Preferece Modellg ad Applcatos (EUROFUSE 2001), Graada, Spa. Stllwell, W. G., D. A. Seaver, et al A comparso of weght approxmato techques multattrbute utlty decso makg. Ogazatoal behavour ad Huma Performace 28: Weber, M. ad K. Borcherdg Behavoural flueces o weght judgmets multattrbute decso makg. Europea Joural of Operatoal Research 67: Yager, R. R O ordered weghted averagg aggregato operators multcrtera decso makg. IEEE Trasactos o Systems, Ma, ad Cyberetcs 18(1): Yager, R. R Applcatos ad extesos of OWA aggregatos. It. J. of Ma- Mache Studes 37: Yager, R. R Famles of OWA operators. Fuzzy Sets ad Systems 59: Yager, R. R Quatfer guded aggregato usg OWA operators. It. Joural of Itellget Systems 11: Yager, R. R. ad D. P. Flev Parametrzed AND-lke ad OR-lke OWA operators. It. J. of Geeral Systems 22: Yager, R. R. ad A. Kelma A exteso of the aalytcal herarchy process usg OWA operators. Joural of Itellget & Fuzzy Systems 7(4): Yoo, K The propagato of errors multple-attrbute decso aalyss: a pratcal approach. Joural Operatoal Research Socety 40(7): Yoo, K. P. ad C.-L. Hwag Multple attrbute decso makg, Sage Publcatos. Zadeh, L. A The cocept of a lgustc varable ad ts applcato to approxmate reasog - I. Fuzzy Sets ad Applcatos: Selected Papers by L. A. Zadeh. R. R. Yager, S. Ovchkv, R. Tog ad H. T. Nguye. New York, Joh Wley & Sos. Reprted from: Iformato Sceces, 8 (1975): Zmmerma, H.-J Fuzzy sets, decso makg, ad expert systems. Bosto, Kluwer Academc Publshers. 24