The Choquet integral for the aggregation of interval scales in multicriteria decision making

The Cho quet in tegral for the aggregation of in terv al scales in m ulticriteria de cision making ∗ Christ o phe Labreuche Thales Resea rc h & T ec hnolo g y Domaine de Corb evill e 9140 4 Orsay cedex, F ra nce Mic hel Gr abisch † Universit y of Paris VI 4 pla ce Jussieu 7525 2 Paris, F rance Octob er 22, 2018 Abstract This pap er addresses the question of whic h mo d els fit with infor- mation concerning the preferences of the decision make r ov er eac h at- tribute, and h is preferences ab out aggregation of criteria (in teracting criteria). W e sh o w that the conditions in duced b y these information plus some intuitiv e cond itions lead to a u nique p ossible aggregat ion op erator: the Cho quet integ ral. Keyw ords: m ulticriteria decision m aking, Cho qu et integral, ax- iomatic app roac h. 1 In tro duction Let us consider a decision problem that dep ends on n p oints of views de- scrib ed by the attributes X 1 , . . . , X n . The att r ibutes can b e an y set. They can b e o f cardinal nature (f or instance, t he maximum sp eed of a car) or o f ordinal na ture (fo r instance, the colo r { r ed,blue, . . . } ). W e wish to mo del the preferences  o f the decision maker (DM) o v er acts, that is to sa y ov er ele- men ts of X = X 1 × · · · × X n . F o r an y x ∈ X , w e use the follo wing notation: ∗ This pap er is an extended and revised version of a pap er presented in the EUROFUSE 2001 co nference [12]. † On leave from Thales Resea rch & T echnology , do maine de Co rb eville, 91404 Orsay cedex, F rance 1 x = ( x 1 , . . . , x n ) where x 1 ∈ X 1 , . . . , x n ∈ X n . A classical w ay is to mo del  with the help of a n ov erall utility function u : X → I R [11 ]: ∀ x, y ∈ X , x  y ⇔ u ( x ) ≥ u ( y ) . (1) The o v erall utility function u is often tak en in the f ollo wing w a y [11] u ( x ) = F ( u 1 ( x 1 ) , . . . , u n ( x n )) , where the u i ’s a re the so called utility functions , and F is the ag gregation function. The main question is then how to determine the u i ’s and F . The utilit y functions and the aggr egation function a re generally constructed in t w o separate steps. Eac h criterion is considered separately in the first step. The utility function u i represen ts the prefere nces of the DM ov e r the attribute X i . The Macbeth approac h [1 , 2] provide s a metho do logy to construct the u i ’s as scales of difference (see section 6.1). In the second step, all criteria are considered together. As example of aggregatio n functions, one can find the w eigh ted sum, the Cho quet in tegral or the Sugeno in tegral. When F is a Cho quet integral, sev eral metho ds fo r the determination of the fuzzy measure are a v ailable. F or instance, linear metho ds [13], quadratic metho ds [4, 6] and heuristic-based metho ds [5] are a v ailable in the lit era t ur e. F or the Sugeno in tegral, heuristics [7] and metho ds ba sed on fuzzy relations [16, 17] can b e found. Unfortunately , these metho ds address only t he problem of constructing the ag gregation function and refer more to learning pro cedures than to true decision making appro a c hes. The wa y the first step is dealt with is generally not explained, esp ecially with elab orat e aggregators suc h a s fuzzy in tegrals. Y et, the t w o steps are in timately relat ed. F or instance, the notion of util- it y function has no absolute meaning. One cannot construct the u i ’s without an y a priori kno wledge of what kind of F will b e considered. How eve r, when constructing t he u i ’s, F is not already kno wn. As a conseque nce, the con- struction of the utility f unctions is more complicated than what it seems. One metho d addresses b oth p oints in a w a y that is satisfactory in the mea- suremen t theory standp oint: this is the Macb eth approac h. How ev e r, it is restricted to the weigh ted sum. In suc h mo del, there is no in teraction b e- t w een criteria. The aim of t his pap er is t o extend the Macb eth approach in suc h a w a y that in teraction b et w een criteria is allow ed. The u i ’s and F shall b e determined from some information obta ined from the DM ab out his preference o v er acts, e.g. elemen ts o f X . The information w e will consider in this pap er can b e seen as a generalization of the informa- tion neede d in t he M acb eth tec hnology [1]. As in Macb eth, the information is 2 based on the introduction of t w o absolute reference lev e ls ov e r each a ttribute, and the determination of scales of difference, whic h, put t o gether, ensures commensuratenes s. W e will consider more general information than in the Macb eth approac h in suc h a w ay that interaction b etw ee n criteria will b e allo w ed. W e address then the question of whic h mo dels fit with this informa- tion. The information w e hav e and t he measuremen t conditions imply some conditions on the aggregation function F . As a n example, one condition is that F shall enable the construction of the u i ’s. Some conditions give n on sp ecific acts are naturally extended to wider sets of acts. These conditions rule out a wide range of families o f aggregators. W e see finally which ag- gregation functions comes up. With these new prop erties, we show that the only p ossible mo del is the Cho quet integral. W e obta in an axiomatic repre- sen tation of the Cho quet in tegral that is a weak v ersion of a result shown b y J.L. Maric hal [15]. W e already addressed the same kind of problem for ratio scales in [10]. F o r ratio sc ales, the ˇ Sip o ˇ s in tegral s eems to b e the righ t aggregation function. The ot her difference b et w een this pa p er and [10] is that w e restricted ourself only to t he necessary information to construct the mo del ( preferences o v er eac h attribute and the aggregation of criteria) and do not add an y a priori assumption as in [10]. 2 Av ailable information The set of all criteria is denoted by N = { 1 , . . . , n } . Considering t w o acts x, y ∈ X and A ⊂ N , w e use the notatio n ( x A , y − A ) to denote the a ct z ∈ X suc h that z i = x i if i ∈ A a nd z i = y i otherwise. It is w ell-know n in MCDM that the preference relation  can b e mo d- eled b y an ov erall utility function u (see (1) ) if and only if all a ttributes are set commensurate in some w a y [11]. The notion of commensurateness hinges on the idea that w e shall not try to aggregate directly attributes but rather a g gregate v alues tha t represen t the same kind of quan tit y . This is the commensurate scale. The mo deling of the preference relat io n of the DM dep ends en tirely on what ty p e of scale the DM can work o n. Here, w e assume that the D M can handle a scale of difference. This means that , on top of b eing able to ra nk t w o acts x, y ∈ X , the D M can also giv e an assessme n t o f the difference u ( x ) − u ( y ) b etw ee n the ov erall ut ility of x and y . In o ther w ords, if x  y , the DM can give the in tensit y with whic h x is preferred to y . W e assume furthermore tha t the underlying scale is a b ounded unip olar scale. It is b ounded f r om ab ov e and b elo w. In this case, the commensurate scale depicts 3 the satisfaction degree o f the DM ov er att r ibutes. The satisfaction degree is t ypically a n um b er b elonging to the in terv al [0 , 1 ]. The t w o b ounds 0 and 1 hav e a sp ecial meaning. The 0 satisfaction v alue is the v a lue that is considered completely unacceptable b y the DM. The 1 satisfaction v alue is the v alue that is considered p erfectly satisfactor y by the DM. W e assume that these t w o v alues 0 and 1 of the satisfaction scale can b e identified b y t w o particular elemen ts 0 i et 1 i for eac h attribute X i . These tw o particular elemen ts hav e an absolute meaning throughout the at tributes. W e assume that 0 i is the w orst elemen t of X i , that is to sa y ∀ x i ∈ X i , ( x i , 0 − i )  ( 0 i , 0 − i ) . (2) Similarly , 1 i is the b est elemen t of X i : ∀ x i ∈ X i , ( x i , 0 − i )  ( 1 i , 0 − i ) . (3) The in tro duction o f 0 i and 1 i enables us to construct in tra-criterion and in ter-criteria infor ma t ion. 2.1 In tra-criterion information F o r each criterion i , the mapping (denoted b y u i ) from the attribute X i to the satisfaction scale [0 , 1] m ust b e explicited. This corresp o nds to t he pref- erences of the D M ov er eac h at t ribute. Commens urateness implies that the elemen ts of o ne attribute shall b e compared to the elemen ts of a n y other scale. T aking a simple example in v olving tw o criteria (for instance consump- tion and maximal sp eed), this amoun ts to kno w whether the D M prefers a consumption of 5 liters/10 0km to a maxim um sp eed of 200 km/h. This do es not generally mak e sense to the D M, so that he is not generally able to mak e this comparison directly . In order to solv e this problem out, t he Macb eth approac h is based on the idea that a scale of difference is constructed sepa- rately o n each attribute. A scale o f difference is g iven up to tw o degrees of freedom. Fixing t w o p oin ts on the scale determines en tirely the scale. These t w o p oints are c hosen in order to enforce the o v erall commensurateness. As a consequence, these tw o p oin ts are the only elemen ts of the attribute that ha v e to b e compared to the elemen ts of the other attributes. These ele- men ts are actually the 0 i ’s and 1 i ’s. All the 0 i ’s hav e the same meaning : u 1 ( 0 1 ) = · · · = u n ( 0 n ). Similarly , u 1 ( 1 1 ) = · · · = u n ( 1 n ). It is natural for a D M to g ive his preferences ov e r acts. On the ot her hand, t he D M shall not b e ask ed info r ma t ion directly on the parameters of the mo del. Therefore, it is not assumed that the D M can isolate attributes and giv e information regarding directly t he u i ’s. T o this end, as in the 4 Macb eth metho dology , w e consider t he subset X ⌋ i (for i ∈ N ) of X defined b y X ⌋ i := { ( x i , 0 − i ) , x i ∈ X i } . W e ask the DM not only the ra nking of the elemen ts of X ⌋ i but also the difference of satisfaction degree b et w een pairs of elemen ts of X ⌋ i . F r o m this, u i can b e defined by : (In tra a ) ∀ x i , y i ∈ X i , u i ( x i ) ≥ u i ( y i ) ⇔ ( x i , 0 − i )  ( y i , 0 − i ). (In tra b ) ∀ x i , y i , w i , z i ∈ X i suc h that u i ( x i ) > u i ( y i ) and u i ( w i ) > u i ( z i ), w e ha v e u i ( x i ) − u i ( y i ) u i ( w i ) − u i ( z i ) = k ( x i , y i , w i , z i ) , k ( x i , y i , w i , z i ) ∈ I R + if and only if the difference of satisfaction degree that the DM feels b et w een ( x i , 0 − i ) and ( y i , 0 − i ) is k ( x i , y i , w i , z i ) times a s big as the dif - ference of satisfaction b et w een ( w i , 0 − i ) and ( z i , 0 − i ). (In tra c ) u i ( 0 i ) = 0 a nd u i ( 1 i ) = 1. Condition (Intra b ) means t hat k ( x i , y i , w i , z i ) = u ( x i , 0 − i ) − u ( y i , 0 − i ) u ( w i , 0 − i ) − u ( z i , 0 − i ) . The u i corresp ond to a scale of difference. Such a scale is alw a ys g iven up to a shift and a dilatio n. Condition (Intra c ) fixes these tw o degrees of freedom. In order to b e able to construct a unique scale fro m (In tra a ) , (In tra b ) and (In tra c ) , some consistency assumptions shall b e made : (In tra d ) ∀ x i , y i , w i , z i , r i , s i ∈ X i suc h that u i ( x i ) > u i ( y i ), u i ( w i ) > u i ( z i ) and u i ( r i ) > u i ( s i ), k ( x i , y i , w i , z i ) × k ( w i , z i , r i , s i ) = k ( x i , y i , r i , s i ) . 2.2 In ter-criteria information W e are concerned here in the determination of the aggrega tion function F . When F is tak en a s a w eigh ted s um, it is enough to obtain f rom t he DM some information regarding the imp orta nce o f eac h criterion. This is in particular the case of the Macb eth metho dology [1]. In the Macb eth approac h, the follo wing set of acts is in tro duced ˜ X := { ( 0 N ) } [ { ( 1 i , 0 − i ) , i ∈ N } , 5 where ( 0 N ) denotes t he alternative s that is unacceptable on ev ery criteria. On top of giving his preferences on the elemen ts of ˜ X , the DM pro vides some infor ma t io n on the difference of satisfaction b et w een any t w o elemen ts of ˜ X . Thanks to that info rmation, an in terv al scale that represen ts the satisfaction degree of the elemen ts of ˜ X can b e constructed, pro vided the information is consisten t. Then, the imp ort ance of criterion i is defined as b eing prop ortional to the difference of the satisfaction degrees b etw e en ( 1 i , 0 − i ) and ( 0 N ). The constan t of prop ortionality is fixed from the prop ert y that all the imp ortances shall sum up to one. In o rder to generalize the Macb eth approac h, w e in tro duce the following set X ⌉ { 0 , 1 } of acts : X ⌉ { 0 , 1 } := { ( 1 A , 0 − A ) , A ⊂ N } . As b efore, we a sk informat ion from whic h one can o bta in a satisfaction scale defined o n X ⌉ { 0 , 1 } . Define P ( N ) as the set of all subsets o f N . Let µ : P ( N ) → [0 , 1] defined b y (In ter a ) ∀ A, B ⊂ N , µ ( A ) ≥ µ ( B ) ⇔ ( 1 A , 0 − A )  ( 1 B , 0 − B ). (In ter b ) ∀ A, B , C , D ⊂ N suc h tha t µ ( A ) > µ ( B ) and µ ( C ) > µ ( D ) , w e ha v e µ ( A ) − µ ( B ) µ ( C ) − µ ( D ) = k ( A, B , C , D ) , k ( A, B , C , D ) ∈ I R + if and only if the difference of satisfaction degree that the DM feels b et w een ( 1 A , 0 − A ) and ( 1 B , 0 − B ) is k ( A, B , C , D ) times as big as the difference of satisfaction b etw een ( 1 C , 0 − C ) and ( 1 D , 0 − D ). (In ter c ) µ ( ∅ ) = 0 and µ ( N ) = 1. The last condition is rather natural since µ ( ∅ ) = 0 means that the a ct whic h is completely unacceptable on all attributes is also completely unac- ceptable a s a whole, and µ ( N ) = 1 means that the act which is p erfectly satisfactory on a ll attributes is also p erfectly satisfactory as a whole. This depicts the idea o f commensurateness . In or der to b e able to construct a unique scale from (Inter a ) , (In ter b ) and (In ter c ) , some consistency assumptions shall b e made : (In ter d ) ∀ A, B , C , D , E , F ⊂ N suc h that µ ( A ) > µ ( B ), µ ( C ) > µ ( D ) and µ ( E ) > µ ( F ) k ( A, B , C , D ) × k ( C , D , E , F ) = k ( A, B , E , F ) . 6 2.3 Measuremen t cond itions The u i ’s and µ corresp ond to scales of difference. Considering for instance u i , this means that the only type of quantit y that mak es sense is ratio s of the form u i ( x i ) − u i ( y i ) u i ( w i ) − u i ( z i ) . Henceforth, it w ould ha v e b een p ossible to replace condition ( I n tra c ) b y u i ( 0 i ) = β and u i ( 1 i ) = α + β , with α > 0 a nd β ∈ I R. In ot her w ords, w e could replace u i giv en b y conditions (In tra a ) , (In tra b ) and ( In tra c ) by α u i + β . Since the u i ’s map the attributes on to a single commensurate scale, one must change all u i ’s in αu i + β with the same α and β (one cannot c hange only one scale). The measuremen t conditions for scales of difference imply tha t the preference relation  and the ra tios u ( x ) − u ( y ) u ( z ) − u ( t ) for all x, y , z , t ∈ X shall not b e c hanged if a ll the u i ’s are changed in to αu i + β with α > 0 and β ∈ I R. W e assume in this pa p er that this prop erty holds only for the a cts that w e a r e considering in our construction, that is to sa y for x, y , z , t ∈ X ⌋ i and x, y , z , t ∈ X ⌉ { 0 , 1 } . This leads to the follo wing requiremen t : (In tra e ) The preference relation  and the u ( x ) − u ( y ) u ( z ) − u ( t ) for x, y , z , t ∈ X ⌋ i and for x, y , z , t ∈ X ⌉ { 0 , 1 } shall not b e c hanged if all the u i ’s are c hanged in to αu i + β with α > 0 and β ∈ I R. This condition is no more than the requiremen t that the u i ’s are commensu- rate scales of differences. Lo oking at X ⌉ { 0 , 1 } as a generalization of the set ˜ X used in the Macb eth approac h, µ ( { i } ) represen ts some kind of imp ortanc e of criterion i . More precisely , the term µ ( { i } ) corresp onds to the difference o f the satisfaction degrees b et w een ( 1 i , 0 − i ) a nd ( 0 N ). G eneralizing this to coalitions of any cardinalit y , µ ( A ) corresp onds in f act to the difference of the satisfaction de- grees b et w een the alternativ es ( 1 A , 0 − A ) and ( 0 N ). Applying this to A = ∅ , the v alue µ ( ∅ ) shall alw a ys b e equal to zero, whatev er the interv al scale attac hed to ˜ X ma y b e. If the in terv al scale atta ched to ˜ X is changed to another in terv al scale, the v alues of µ ( A ) could v ary , except µ ( ∅ ) that will alw a ys v anish. More precisely , µ will b e replaced by γ µ , with γ ∈ I R. Hence- forth, µ corresp onds to a rat io scale. As a consequence, it would b e p o ssible to replace condition (In ter c ) b y µ ( ∅ ) = 0 and µ ( N ) = γ , with γ ∈ I R . As previously , the follo wing requiremen t is imp o sed : (In ter e ) The preference relat io n  a nd the u ( x ) − u ( y ) u ( z ) − u ( t ) for x, y , z , t ∈ X ⌋ i and for x, y , z , t ∈ X ⌉ { 0 , 1 } shall not b e changed if µ is c hanged in to γ µ with γ ∈ I R. 7 3 Condit i o ns on the mo del In measuremen t theory , it is classical t o split the o v erall ev aluation mo del u in to tw o part s [11]: the utility functions (that map the attributes on to a sin- gle satisfaction degree scale), and the ag gregation function (that aggregat es commensurate scales). The function u i corresp onds to satisfaction degrees o v er criterion i . Thanks to assumptions (Intra a ) and (Intra b ) , u i are scales of difference. These scales are commensurate b y condition (I n tra c ) . The term µ ( A ) represen ts the imp ortance that the DM giv es to the coalit io n A in the D M pro cess. Consequen tly , it is natural to write u as fo llo ws: u ( x ) = F µ ( u 1 ( x 1 ) , . . . , u n ( x n )) , (4) where F µ is the ag gregation op erator. F µ dep ends on µ in a w a y that is not kno wn for the momen t. F rom now on, w e assume that the preferences o f the DM can b e mo deled b y u giv en b y (4). As example of agg r ega tion op erators, F µ could b e a w eigh ted sum F µ ( u 1 , . . . , u n ) = n X i =1 α i u i . F o r a normalized weigh ted sum ( i.e. when the we ights α i sum up to one), one clearly has F µ ( β , . . . , β ) = β for any β . This prop ert y is explained b y the fact that F µ aggregates commensurate scales. It can b e naturally generalized to a n y F µ . The condition that the weigh ted sum is normalized b ecomes t ha t µ is normalized, that is to sa y µ ( N ) = 1. Henceforth, it is nat ural to assume that (whenev er µ satisfies µ ( N ) = 1) F µ ( β , . . . , β ) = β , ∀ β ∈ [0 , 1] . Due to condition ( In tra c ) and relations (2) and (3), u i ( x i ) ∈ [0 , 1] for an y x i ∈ X i . This is why β ∈ [0 , 1 ] in previous relation. Ho w ev er, condition (In tra e ) implies tha t u i ( x i ) could tak e virtually an y real v alue. Hence , F µ shall b e defined on I R n . This show s that previous relation on F µ shall hold for a n y β ∈ I R : F µ ( β , . . . , β ) = β , ∀ β ∈ I R . Consider now the case when µ ( N ) 6 = 1. Since µ corresp onds to a ratio scale, it is reasonable to assume that µ acts as a dilatio n of the ov e rall ev aluation scale of u . In other w ords, when µ ( N ) 6 = 1, we should hav e F µ ( β , . . . , β ) = β µ ( N ) , ∀ β ∈ I R . (5) 8 3.1 In tra-criterion information Lemma 1 If u satisfies (4), a nd i f c onditions (Intra a ) , (Intra b ) , ( In tra c ) , (In tra d ) , (Intra e ) , (Inter a ) , (Inter b ) , (Inter c ) , (In ter d ) and (Inter e ) ar e fulfil le d, then for al l a i , b i , c i , d i ∈ [0 , 1] , and for al l α > 0 , γ , β ∈ I R , F γ µ ( αa i + β , β − i ) − F γ µ ( αb i + β , β − i ) F γ µ ( αc i + β , β − i ) − F γ µ ( αd i + β , β − i ) = a i − b i c i − d i . (6) Pro of : u ( x i , 0 − i ) and u i ( x i ) corresp ond t o tw o p ossible scales of difference related to the same act ( x i , 0 − i ) ∈ X ⌋ i . Henceforth u ( x i , 0 − i ) − u ( y i , 0 − i ) u ( w i , 0 − i ) − u ( z i , 0 − i ) = u i ( x i ) − u i ( y i ) u i ( w i ) − u i ( z i ) , whic h giv es F µ ( u i ( x i ) , u − i ( 0 − i )) − F µ ( u i ( y i ) , u − i ( 0 − i )) F µ ( u i ( w i ) , u − i ( 0 − i )) − F µ ( u i ( z i ) , u − i ( 0 − i )) = u i ( x i ) − u i ( y i ) u i ( w i ) − u i ( z i ) . (7) F r om (Intra e ) , o ne might change all u j in αu j + β at the same time without an y change in (7). In this case, the utility functions will tak e v a lues in the in terv a l [ β , α + β ]. F rom ( In ter e ) , one might change µ into γ µ ( γ ∈ I R) without an y c hange in (7). Consequen tly , o ne must ha v e F γ µ ( αu i ( x i ) + β , α u − i ( 0 − i ) + β ) − F γ µ ( αu i ( y i ) + β , α u − i ( 0 − i ) + β ) F γ µ ( αu i ( w i ) + β , α u − i ( 0 − i ) + β ) − F γ µ ( αu i ( z i ) + β , α u − i ( 0 − i ) + β ) = ( αu i ( x i ) + β ) − ( αu i ( y i ) + β ) ( αu i ( w i ) + β ) − ( α u i ( z i ) + β ) . No w, setting a i = u i ( x i ), b i = u i ( y i ), c i = u i ( w i ) and d i = u i ( z i ), w e hav e a i , b i , c i , d i ∈ [0 , 1] b y (2), (3) and (Intra c ) . Hence, the lemma is prov ed. 3.2 In ter-criteria information Lemma 2 If u satisfies (4), a nd i f c onditions (Intra a ) , (Intra b ) , ( In tra c ) , (In tra d ) , (Intra e ) , (Inter a ) , (Inter b ) , (Inter c ) , (In ter d ) and (Inter e ) ar e fulfil le d, then for any α > 0 an d γ , β ∈ I R , it holds that F γ µ (( α + β ) A , β − A ) − F γ µ (( α + β ) B , β − B ) F γ µ (( α + β ) C , β − C ) − F γ µ (( α + β ) D , β − D ) = µ ( A ) − µ ( B ) µ ( C ) − µ ( D ) . (8) 9 Pro of : By conditions (I n ter a ) and (In ter b ) , u ( 1 A , 0 − A ) and µ ( A ) corre- sp ond to tw o p ossible in terv al scales related to the act ( 1 A , 0 − A ) ∈ X ⌉ { 0 , 1 } . Hence u ( 1 A , 0 − A ) − u ( 1 B , 0 − B ) u ( 1 C , 0 − C ) − u ( 1 D , 0 − D ) = µ ( A ) − µ ( B ) µ ( C ) − µ ( D ) , whic h giv es F µ ( u A ( 1 A ) , u − A ( 0 − A )) − F µ ( u B ( 1 B ) , u − B ( 0 − B )) F µ ( u C ( 1 C ) , u − C ( 0 − C )) − F µ ( u D ( 1 D ) , u − D ( 0 − D )) = µ ( A ) − µ ( B ) µ ( C ) − µ ( D ) . (9) As previously , from (Intra e ) and (In ter e ) , one can c hange u j in α u j + β , and µ in γ µ without an y change in (9) : F γ µ ( αu A ( 1 A ) + β , α u − A ( 0 − A ) + β ) − F γ µ ( αu B ( 1 B ) + β , α u − B ( 0 − B ) + β ) F γ µ ( αu C ( 1 C ) + β , α u − C ( 0 − C ) + β ) − F γ µ ( αu D ( 1 D ) + β , α u − D ( 0 − D ) + β ) = γ µ ( A ) − γ µ ( B ) γ µ ( C ) − γ µ ( D ) . Hence t he lemma is pro v ed. 4 Generalize d conditi o ns on the mo del Lemma 3 If u satisfies (4), and if c onditions (5) , (Intra a ) , (In tra b ) , (In tra c ) , (In tra d ) , (In ter a ) , (I n ter b ) , (In ter c ) and (Inter d ) ar e fulfil le d, then we have for a ny η , β ∈ I R F γ µ + δ µ ′ ( η A , β − A ) = γ F µ ( η A , β − A ) + δ F µ ′ ( η A , β − A ) . (10) Pro of : T aking (8) with γ = 1, B = D = ∅ and C = N , it holds t ha t F µ (( α + β ) A , β − A ) = α µ ( A ) + β µ ( N ) . (11) This relation holds for an y µ satisfying µ ( ∅ ) = 0 . Conse quen tly , replacing µ b y γ µ + δ µ ′ , w e obta in for any β ∈ I R, α > 0 F γ µ + δ µ ′ (( α + β ) A , β − A ) = γ F µ (( α + β ) A , β − A ) + δ F µ ′ (( α + β ) A , β − A ) . Hence, setting η = α + β , we obtain for an y β ∈ I R and η > β , F γ µ + δ µ ′ ( η A , β − A ) = γ F µ ( η A , β − A ) + δ F µ ′ ( η A , β − A ) . 10 Replacing A b y N \ A in previous relation, w e obtain for any β ∈ I R and η > β , F γ µ + δ µ ′ ( β A , η − A ) = γ F µ ( β A , η − A ) + δ F µ ′ ( β A , η − A ) . Putting together last t w o results, w e get for any η , β ∈ I R with β 6 = η F γ µ + δ µ ′ ( η A , β − A ) = γ F µ ( η A , β − A ) + δ F µ ′ ( η A , β − A ) . Previous relation holds also when β = η by (5). Hence, w e hav e for any η , β ∈ I R F γ µ + δ µ ′ ( η A , β − A ) = γ F µ ( η A , β − A ) + δ F µ ′ ( η A , β − A ) . W e wish to generalize (10) in the follo wing w a y F γ µ + δ µ ′ ( x ) = γ F µ ( x ) + δ F µ ′ ( x ) , ∀ x ∈ I R n , ∀ γ , δ ∈ I R . W e should ha v e F γ µ ( x ) = γ F µ ( x ) ( γ ∈ I R) since µ corr esp o nds to a ratio scale. One should hav e F µ + µ ′ 2 ( x ) = 1 2 ( F µ ( x ) + F µ ′ ( x )) for the following r ea- son. If t w o decision ma kers giv e the imp ortances of the coa lit ions ( µ and µ ′ resp ectiv ely), the consensus o f these t w o decision mak ers can b e p erformed b y ta king the mean v alue of these info r mation (that is to say µ + µ ′ 2 ). Then it is reasonable t ha t the ov erall agg r ega tion functions equals the mean v alue of the ag gregation for t he t w o D M. Based on these considerations, we prop ose the f ollo wing axiom : Linearit y wrt the Measure (LM) : F or all x ∈ I R n and γ , δ ∈ I R, F γ µ + δ µ ′ ( x ) = γ F µ ( x ) + δ F µ ′ ( x ) . (12) This axiom cannot b e completely deduced from the construction of the u i ’s and µ . How eve r, Lemma 3 prov es that formula (12) is obtained from the construction of the u i ’s a nd µ for particular alternat ives x . Since F µ aggregates satisfaction scales, it is natural to assume that x 7→ F µ ( x ) is increasing. Increasingness (In) : ∀ x, x ′ ∈ I R n , x i ≤ x ′ i ∀ i ∈ N ⇒ F µ ( x ) ≤ F µ ( x ′ ) 11 This axiom is not deduced from the construction of the u i ’s and µ . It is a necessary requiremen t for F µ to b e a n agg regation f unction [3]. Let us give no w the last t w o axioms. Prop erly W eigh ted (PW) : If µ satisfies condition ( In ter c ) , then F µ (1 A , 0 − A ) = µ ( A ), ∀ A ⊂ N . Stabilit y for the admissible P ositiv e Linear transforma- tions (weak SPL) : If µ satisfies condition ( In ter c ) , t hen for a ll A ⊂ N , α > 0, and β ∈ I R, F µ (( α + β ) A , β − A ) = α F µ (1 A , 0 − A ) + β This axiom is called (weak SP L) since it is a w eak v ersion of the axiom (SPL) in tro duced by J.L. Maric hal [14, 15] : (SPL) : F o r all x ∈ I R n , α > 0, and β ∈ I R, F µ ( αx + β ) = αF µ ( x ) + β Lemma 4 Axioms (PW) and (weak SPL) c an b e de duc e d fr om the c o n- struction of the u i ’s and µ . Pro of : Axiom (P W ) is obtained from (11) with β = 0 and α = 1. Finally , using (11) and ( PW) , ( w eak SPL) ho lds. Next lemma shows that axioms (LM) , (PW) and (weak SPL) contain the tw o relations (6) and (8). Lemma 5 Conditions (LM) , (PW) and ( w eak SP L) imply r elations (6) and (8). Pro of : Consider F µ satisfying (LM) , (PW ) and (w eak SPL) . By (LM) and (w eak SPL) , w e ha v e for any α > 0 and a i ∈ I R + , F γ µ ( αa i + β , β − i ) = γ F µ ( αa i + β , β − i ) = γ α a i F µ (1 i , 0 − i ) + γ β . Consequen tly , (6) is fulfilled. Condition (8) is clearly satisfied from (LM) , (PW) a nd (w eak SPL) . T o end up this section, let us recall that the construction of the u i ’s (con- ditions (I ntra a ) , (Intra b ) , (Intra c ) , (Intra d ) ), (In tra e ) ) and µ (conditions (In ter a ) , (Inter b ) , (Inter c ) , (Inter d ) ) and ( In ter e ) , plus a ssumptions (4) and ( 5 ) lead to axioms (PW) and (w eak SPL) , and partially to axiom (LM) (see Lemma 3). Moreov er, w e saw that axiom (I n) is a v ery natural requiremen t. 12 5 Expressio n of t he aggre gation funct i on In this section, w e wish to find whic h a ggregation f unctions F µ are c haracter- ized b y conditions ( LM) , (I n) , (PW) a nd (w eak SPL) . It can b e noticed that the Cho quet in tegral satisfies these conditions. Our main result sho ws that this is t he o nly a g gregator satisfying these conditions. 5.1 Bac kground on the Cho quet in tegral Thanks t o condition (Inter c ) and axiom (In) , the set function µ satisfies (i) µ ( ∅ ) = 0, µ ( N ) = 1. (ii) A ⊂ B ⊂ N implies µ ( A ) ≤ µ ( B ). These t w o prop erties define the so-called fuzzy me asur es . As a consequence, F µ could b e a Cho quet integral. The Cho quet integral [3, 8] is a g eneralization o f t he commonly used w eigh ted s um. Whereas the w eigh ted s um is the d iscrete for m of the Leb esgue in tegral with an a dditiv e measure, w e consider as an aggrega tor the discrete form of the g eneralization o f Leb esgue’s in tegral to the case of non-additive measures. This generalization is precisely the Cho quet integral. Definition 1 L et µ b e a fuzzy me asur e on N . The discrete Cho quet in tegral of an ele ment x ∈ I R n with r esp e ct to µ is defi ne d b y C µ ( x ) = n X i =1 x τ ( i ) [ µ ( { τ ( i ) , . . . , τ ( n ) } ) − µ ( { τ ( i + 1) , . . . , τ ( n ) } )] wher e τ is a p ermutation satisfying x τ (1) ≤ · · · ≤ x τ ( n ) . Besides, the Cho quet in tegral can mo del ty pical h uman b eha vior suc h as the ve to. This op erator is also a ble to mo del the imp ortance of criteria and the interaction b et w een criteria. Con v erse ly , the Cho quet inte gral can b e interpreted in term of the imp ortance of criteria, the interaction b et w een criteria, and v eto [6, 8, 3]. 13 5.2 Expression of F µ The result shown here is close to a result pro v ed in [15 ]. Before giving the final result, let us giv e a n in termediate lemma. Lemma 6 If F : I R n → I R satisfies (In) and (w eak SPL) , F (0 N ) = 0 , F (1 N ) = 1 and that F (1 A , 0 − A ) ∈ { 0 , 1 } for al l A ⊂ N , then F ≡ C µ , wher e the f uzzy me asur e µ is given by µ ( A ) = F (1 A , 0 − A ) for al l A ⊂ N . Pro of : Let σ b e a p erm utation of N . F or i ∈ N , let θ σ ( i ) := F  1 { σ ( i ) ,... ,σ ( n ) } , 0 { σ (1) ,..., σ ( i − 1) }  . W e hav e θ σ ( i ) ∈ { 0 , 1 } . F rom ( I n) , there exists k σ ∈ N suc h that θ σ ( i ) = 0 f o r i ∈ { k σ + 1 , . . . , n } and θ σ ( i ) = 1 f o r i ∈ { 1 , . . . , k σ } . W e ha v e k σ ∈ { 1 , . . . , n − 1 } since F (0 N ) = 0 and F (1 N ) = 1. F or x ∈ I R n , the following nota tion [ x 1 , . . . , x n ] σ − 1 denotes the elemen t y ∈ I R n suc h that y i = x σ − 1 ( i ) . With this notation, x ∈ I R n reads x =  x σ (1) , . . . , x σ ( n )  σ − 1 . Let B σ :=  x ∈ I R n , x σ (1) ≤ . . . ≤ x σ ( n )  and x ∈ B σ . F rom (In) and (w eak SPL) , and since x σ ( k σ ) ≤ x σ ( n ) , w e ha v e F ( x ) = F  x σ (1) , . . . , x σ ( n )  σ − 1  ≤ F ([ x σ ( k σ ) , . . . , x σ ( k σ ) | {z } k σ times , x σ ( n ) , . . . , x σ ( n ) | {z } ( n − k σ ) times ] σ − 1 ) = x σ ( k σ ) +  x σ ( n ) − x σ ( k σ )  F ([0 , . . . , 0 | {z } k σ times , 1 , . . . , 1 | {z } ( n − k σ ) times ] σ − 1 ) = x σ ( k σ ) +  x σ ( n ) − x σ ( k σ )  F  1 { σ ( k σ +1) ,...,σ ( n ) } , 0 { σ (1) ,..., σ ( k σ ) }  = x σ ( k σ ) +  x σ ( n ) − x σ ( k σ )  θ σ ( k σ + 1) = x σ ( k σ ) since θ σ ( k σ + 1) = 0. O n the other hand, by ( In) , F ( x ) = F  x σ (1) , . . . , x σ ( n )  σ − 1  ≥ F ([ x σ (1) , . . . , x σ (1) | {z } ( k σ − 1) times , x σ ( k σ ) , . . . , x σ ( k σ ) | {z } ( n − k σ +1) times ] σ − 1 ) = x σ (1) +  x σ ( k σ ) − x σ (1)  F  1 { σ ( k σ ) ,...,σ ( n ) } , 0 { σ (1) ,..., σ ( k σ − 1) }  = x σ (1) +  x σ ( k σ ) − x σ (1)  θ σ ( k σ ) = x σ ( k σ ) since θ σ ( k σ ) = 1. Hence we ha v e F ( x ) = x σ ( k σ ) . 14 Let us note that fo r the fuzzy measure µ defined b y µ ( S ) = F (1 S , 0 − S ), w e ha v e C µ ( x ) = n X i =1 x σ ( i ) [ µ ( { σ ( i ) , . . . , σ ( n ) } ) − µ ( { σ ( i + 1) , . . . , σ ( n ) } )] . Moreo v er, µ ( { σ ( i ) , . . . , σ ( n ) } ) = F  1 { σ ( i ) ,... ,σ ( n ) } , 0 { σ (1) ,..., σ ( i − 1) }  = θ σ ( i ) =  1 if i ∈ { 1 , . . . , k σ } 0 if i ∈ { k σ + 1 , . . . , n } Hence C µ ( x ) = x σ ( k σ ) = F ( x ) . In Lemma 6, ( In) and (weak SPL) are sufficien t to obtain that F is a Cho quet in tegral o nly b ecause the range of F (1 A , 0 − A ) b elongs to { 0 , 1 } . This is no more true in the general case. Theorem 1 F µ satisfies (LM) , (In) , (PW) and (w eak SPL) if and only if F µ ≡ C µ in I R n . Pro of : Clearly , the Cho quet in tegral satisfies (LM) , (In) , ( PW) and (w eak SPL) . Consider now F µ satisfying (LM) , (In) , ( P W) a nd (weak SP L) . W e write µ ( A ) = P B ⊂ N m ( B ) u B ( A ), where m is the M¨ obius transform m ( B ) = P C ⊂ B ( − 1) | B |−| C | µ ( C ), and u B is the unanimity g ame, t ha t is to sa y u B ( A ) equals 1 if B ⊂ A and 0 otherwise. Since m ( ∅ ) = 0, w e ha v e µ ( A ) = P B ⊂ N , B 6 = ∅ m ( B ) u B ( A ) Thanks to ( LM) , F µ ( x ) = X B ⊂ N , B 6 = ∅ m ( B ) F u B ( x ) , where u B is a { 0 , 1 } v alued fuzzy measure. By lemma 6, for B 6 = ∅ , there exists a fuzzy measure v suc h that F u B ( x ) = C v ( x ) f or all x ∈ I R n . Clearly , b y ( PW) , we hav e for B 6 = ∅ v ( A ) = F u B (1 A , 0 − A ) = u B ( A ) . Hence F u B ( x ) = C u B ( x ) ∀ x ∈ I R n . 15 By linearity o f the Cho quet in tegral with resp ect to the fuzzy measure, ∀ x ∈ I R n , F µ ( x ) = X B ⊂ N , B 6 = ∅ m ( B ) C u B ( x ) = C P B ⊂ N , B 6 = ∅ m ( B ) u B ( x ) = C µ ( x ) . This theorem is a generalization of a result of J.L. Marichal [15]. J.L. Maric hal prov e d in [15 ] t ha t the Cho quet in tegral is the only ag gregation function satisfying ( LM) , ( In) , (PW) a nd (SPL) . Let us sum up the res ult w e ha v e s ho wn. F rom the construction of the u i ’s (conditions (In tra a ) , (Intra b ) , (Intra c ) , (Intra d ) ), (Intra e ) ) and µ (con- ditions (Inter a ) , (Inter b ) , (In ter c ) , (In ter d ) ), (I nter e ) ), and a dding the t w o assumptions (4) and (5 ), we obtained axioms (PW) and (weak SPL) (see L emma 5), and axiom (LM) w as partially o bta ined (see Lemma 3 ). Moreo v er, w e saw that axiom (I n) is a v ery natura l requiremen t. Theorem 1 sho ws that there is o nly o ne mo del that fits with pr evious informatio n. This is the Cho quet in tegral. Therefore, the Cho quet in tegral is the only mo del that is suitable with our construction. 6 Discuss ion on the pract i cal construc tion of the u i ’s and µ Some practical issues concerning the construction of u i and µ are considered in this section. 6.1 In tra-criterion information Let us inv es tigate here ho w to construct in practice the utility function u i . W e follo w the Macb eth metho dology [1, 2]. The goal of this section is not to giv e v ery precise details concerning the Macb eth approa c h. W e refer to references [1, 2 ] for a more detailed explanatio n o f the Macb eth approach . F o r the sak e of simplicit y , assume that attribute X i has a finite set of v a lues : X i =  a i 1 , . . . , a i p i  with a i 1 = 0 i et a i p i = 1 i . W e aim at determining u i ( a i j ) for all j ∈ { 1 , . . . , p i } . F r om a theoretical standp oin t, the construction of u i from ( In tra a ) , (In tra b ) a nd (I n tra c ) is straightforw ard if the data is consisten t. Indeed, applying (I n tra b ) with x i = a i j , y i = z i = 0 i and w i = 1 i , together with (In tra c ) yields u i ( a i j ) = k ( a i j , 0 i , 1 i , 0 i ) 16 where k is given b y (Intra b ) . Pro ceeding in this w a y ensures uniqueness of u i . Of course, it is not reasonable to ask a DM to give directly the v alue o f k as a real num ber. The idea of the Macb eth metho do logy is to a sk an information of an o r dinal nature to the DM. It is we ll-know n fro m psyc hological studies that human b eings can handle at most 5 plus or min us 2 items at the same time. The Macb eth metho dology prop oses to ask to the DM a satisfaction lev el b elonging to a n or dina l scale comp osed of 6 elemen ts: { very smal l, smal l, me an, lar ge, ve ry lar ge, extr eme } =: E . If the D M assesses the v alue of u i ( a i j ) directly in this scale, the utility functions will b e v ery rough. In order to cop e with the finiteness of the satisfaction scale E , w e ask the DM to give muc h more information than just a ssessing the u i ( a i j ) for j ∈ { 1 , . . . , p i } . In fact, the DM is a ske d to assess (giving a v alue in the scale E ) the difference of satisfaction u i ( a i j ) − u i ( a i k ) b et w een t w o v alues a i j and a i k , for any j 6 = k suc h that  0 − i , a i j  ≻ ( 0 − i , a i k ). The information ask ed in practice is th us quite similar to (I n tra b ) . The adv an tage of a sking u i ( a i j ) − u i ( a i k ) is that it leads to a redundant information that will enable to compute accurate but non-unique v alues of u i ( a i l ) for l ∈ { 1 , . . . , p i } . The second adv an tage is that it is easier f o r a h uman b eing to give some relative information regarding a difference (for instance u i ( a i j ) − u i ( a i k )) than to give some absolute information (for instance u i ( a i j )). There is no unique utilit y function u i corresp onding to the data comp osed of the v alues u i ( a i j ) − u i ( a i k ) (for j 6 = k ) assessed in the scale E . All p ossible solutions are consisten t with t he giv en information. In practice, one utility function is c hosen among all p ossible ones [1]. The dra wbac k of a sking redundan t information is that some inconsisten- cies ma y b e in tro duced by the DM. F or instance, for a i j 1 , a i j 2 and a i j 3 suc h that  0 − i , a i j 1  ≻  0 − i , a i j 2  ≻  0 − i , a i j 3  , here is an example of inconsistency : the difference o f satisfaction degree b et w een a i j 1 and a i j 2 is judged very s m al l b y the D M, the one b et w een a i j 2 and a i j 3 is also judged very sm a l l , and the one b et w een a i j 1 and a i j 3 is judged extr eme . C .A. Bana e Costa, and J.C. V ansnic k sho w ed that inconsistencies are related to cyclones in the prefer- ence relation structure [1]. This prop erty enables t o detect and explain all p ossible inconsistencies [1]. 6.2 In ter-criteria information W e could imagine pro ceeding as for u i to o btain µ . The D M would b e a ske d ab out the difference o f satisfaction b et w een the alternative s ( 1 A , 0 − A ) and ( 1 B , 0 − B ) (f or A 6 = B ). Actually , ev en if this w a y is p ossible, it is not 17 generally used b ecause of the following tw o reasons. The first one is that it ma y not b e natural for a DM to giv e his preferences on the prototypical acts (1 A , 0 − A ). The second one is that it enforces the DM to construct a ratio scale o v er 2 n alternativ es. This requires roughly 4 n questions to b e a ske d to the D M. This is t o o muc h in practice. Conditions ( In ter a ) , ( In ter b ) a nd (I n ter c ) w ere introduced to sho w the practicalit y of the us e of the Cho quet in tegral to mo del the pre ference relation ≻ o n t he Cartesian pro duct of the attr ibutes. In practice, (In ter a ) , (In ter b ) and (Inter c ) are replaced b y any classical metho d designed to deduce the fuzz y measure from information on acts given in [0 , 1] n . These acts are directly describ ed b y satisfaction degrees in [0 , 1] o v er all criteria. The metho ds men tioned in the introduction are w ell-suited for p erforming this step: we can cite a linear metho d [13, 8], a quadratic metho d [4, 6] and an heuristic-based metho d [5]. 7 Conclus ion W e give in this pap er a result that pro vides some informatio n concerning the preferences o f the DM ov er eac h attribute and the aggrega t ion of criteria, and a priori assumptions leading to the Cho quet mo del. The difficult y of the in tra-criterion step is that one has to determine the utility functions (whic h ha v e a meaning only through the ov e rall mo del and th us through the aggregation function) without a precise kno wledge of the aggregation function (tha t is determined in the next step). Let us recall the main property that mak es this p ossible. It is in fa ct a consequenc e of the (w eak SP L) prop erty : u ( x i , 0 − i ) u ( 1 i , 0 − i ) = F µ ( u i ( x i ) , 0 − i ) F µ (1 i , 0 − i ) = u i ( x i ) . Hence, u i ( x i ) is prop ortional to u ( x i , 0 − i ) whatev er the v a lue o f µ may b e. Some q uestions ab out the practicalit y of the informatio n ask e d to the DM are no w raised. Note t hat regarding the questions to b e ask ed to t he D M, psyc hological asp ects m ust b e ta k en into consideration. F or the construction of u i , w e prop ose here to use the acts ( x i , 0 − i ). Let us recall that 0 cor r esp o nds to the c ompletely unac c eptable absolute lev el. As a consequenc e, since the act ( x i , 0 − i ) is unacceptable a lmost ev erywhere, one could argue that this act can b e considered as almost unacceptable, so t hat the DM will hav e some 18 troubles giving his feeling ab out the differences b etw ee n some ( x i , 0 − i ) and ( y i , 0 − i ). If this happ ens, it w ould b e reasonable to replace the space X ⌋ i b y X ⌋ ′ i := { ( x i , 1 − i ) , x i ∈ X i } , and th us condition (Intra a ) by (In tra a ’) ∀ x i , y i ∈ X i , u i ( x i ) ≥ u i ( y i ) ⇔ ( x i , 1 − i )  ( y i , 1 − i ). Then (6) shall b e replaced b y F γ µ ( αa i + β , ( α + β ) − i ) − F γ µ ( αb i + β , ( α + β ) − i ) F γ µ ( αc i + β , ( α + β ) − i ) − F γ µ ( αd i + β , ( α + β ) − i ) = a i − b i c i − d i for all a i , b i , c i , d i ∈ I R + , and for a ll α, γ > 0, β ∈ I R. It can b e chec k ed that the r est of the pa p er remains unchanged, leading also t o theorem 1. If the same phenomenon o ccurs with X ⌋ ′ i , the DM satisfies strongly to the satur ation e ff e ct . In t his case, it may b e more appropriate to mo del the DM with the help of a purely o r dinal aggr ega tor, lik e the Sugeno op erator. 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