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On the extension of pseudo-Boolean functions for the
aggregation of interacting criteria
Michel Grabisch, Christophe Labreuche, Jean-Claude Vansnick
To cite this version:
Michel Grabisch, Christophe Labreuche, Jean-Claude Vansnick. On the extension of pseudo-Boolean
functions for the aggregation of interacting criteria. [Research Report] lip6.2000.032, LIP6. 2001.
�hal-02548394�
Fun tions for the Aggregation of Intera ting Criteria Mi hel GRABISCH LIP6 University of Paris VI
4, Pla e Jussieu, 75252 Paris, Fran e email Mi hel.Grabis hlip6.fr
Christophe LABREUCHE
Thomson-CSF, Corporate Resear h Laboratory Domaine de Corbeville, 91404 Orsay Cedex, Fran e
emailflabreu hegl r.thomson- sf. om
Jean-Claude VANSNICK University of Mons-Hainaut
Pla e du Par , 20, B-7000 Mons, Belgium email Jean-Claude.Vansni kumh .a .be
Abstra t
The paperpresents an analysis on theuse of integrals de ned for non-additive measures (or apa ities) as the Choquet and the
Sipos integral, and the multilinearmodel, all seen asextensions of pseudo-Boolean fun tions,and usedasameanstomodelintera tionbetween riteria in a multi riteriade ision making problem. The emphasis is put on the use, besides lassi al omparative information, of infor-mation about di eren e of attra tiveness between a ts, and on the existen e, forea h point of view, of a \neutral level", allowing to in-trodu etheabsolutenotionofattra tive orrepulsivea t. It isshown
Correspondingauthor. OnleavefromThomson-CSF,CorporateResear hLab,91404 OrsayCedex,Fran e
thatinthis ase,theSiposintegralisasuitablesolution,althoughnot unique. Properties of the
Sipos integralas a new wayof aggregating riteria areshown,withemphasis ontheintera tionamong riteria.
Keywords: multi riteriade isionmaking,Choquet integral, apa ity, inter-a tive riteria,negative s ores
1 Introdu tion
Let us onsider a de ision making problem, of whi h the stru turing phase has led tothe identi ationof a family C = fC
1
;::: ;C n
g of n fundamental points of view ( riteria), whi h permitstomeet the on erns of the de ision maker (DM) in harge of the above mentioned (de ision making) problem. Wesuppose hereafter that,during the stru turing phase,one has asso iated to ea h point of view C
i
, i = 1;:::;n, a des riptor (attribute), that is, a set X
i
of referen e levels intended to serve as a basis to des ribe plausible impa ts of potentiala tions withrespe t toC
i .
Wemakealsothe assumptionthat,foralli=1;::: ;n, thereexists inX i twoparti ular elements whi h we all\Neutral
i " and \Good i ", and denoted 0 i and 1 i
respe tively, whi hhave anabsolutesigni ation: 0 i
isanelement whi histhoughtbytheDMtobeneithergoodnorbad,neitherattra tivenor repulsive, relativelyto his on erns with respe t to C
i
,and 1 i
is anelement whi htheDM onsiders asgoodand ompletelysatisfyingif he ouldobtain it on C
i
, even if more attra tive elements ould exist on this point of view. The pra ti alidenti ationoftheseabsoluteelementshas been performedin many real appli ations, see for example [6,8, 9℄.
In multi riteriade ision aid, after the stru turing phase omes the eval-uation phase,inwhi hfor ea hpoint of viewC
i
, intra- riterion information isgathered (i.e. attra tivenessforthe DMofthe elementsofX
i
withrespe t to point of view C
i
), and also, a ording to an aggregation model hosen in agreement with the DM, inter- riteria information. This information, whi haims atdeterminingthe parametersof the hosenaggregation model, generally onsists in some information on the attra tiveness for the DM of some parti ularelements ofX =X
1
X n
. Theseelementsare sele ted so as to enable the resolution of some equation system, whose variables are pre isely the unknown parametersof the aggregationmodel.
Inthispaper, ofwhi haimisprimarilytheoreti al,weadoptwithrespe t tothe lassi alapproa hdes ribed above,arather onverse attitude. Spe if-i ally,wedonot suppose tohavebeforehanda given aggregationmodel, but rathertohavesome information on erningthe attra tiveness fortheDMof a parti ular olle tionof elements of X. Then we study howto extend this
of problem an be alled an identi ation of an aggregation model whi h is ompatible with availableinformation.
The paper is organized as follows. In se tion 2, we introdu e the basi assumptions we make on erning the knowledge on the attra tiveness for the DM of parti ular elements of X. Se tion 3 shows that this kind of informationis ompatiblewith the existen eof someintera tionphenomena betweenpointsofview,andintrodu essomede nitionsrelatedtothe on ept of intera tion. The problem of extending the information on preferen es assumed to be known on a subpart of X, to the whole set X, is addressed in se tion 4, and appears to be the problem of identifying an aggregation model ompatible with given intra- riterion and inter- riteria information. In se tion5,weshowthat this problemamountstode nethe extensionofa given pseudo-Boolean fun tion, and we introdu e some possible extensions, whi hwerelatetoalreadyknownmodelsintheliterature(se tion6). Se tion 7 brie y studies the properties of these models, and on ludes about their usefulness inthis ontext. In se tion8, weshow anequivalent set of axioms for our onstru tion, and in se tion 9, we address the question of uni ity of the solution.
This paper does not deal with the pra ti alaspe ts of the methodology we are proposing, i.e. how to obtain the ne essary informationfor building theaggregationmodel. However,theMACBETHapproa h[7℄ ouldbemost useful forextra ting the informationfrom the DM.
Lastly,wewant tomentionthat oneof thereasonswhi hhave motivated this resear h is the re ent development of multi riteria methods based on apa ities and the Choquet integral [2℄, whi h seems to open new horizons [12, 18,20℄. Ina sense,this paper aimsat givingatheoreti al foundationof this type of approa h inthe framework of multi riteriade isionmaking.
2 Basi assumptions
We present two basi assumptions, whi h are the starting point of our on-stru tion. We denotethe index set of riteria by N =f1;::: ;ng. Consider-ing two a ts x;y 2 X, and A N, we will oftenuse the notation (x
A ;y
A ) to denote the ompounda t z where z
i =x i if i 2A and y i otherwise. ^;_ denote respe tively minand maxoperators.
We onsidertheparti ularsubsetsX i
,i=1;::: ;n,ofX,whi hare de ned by: X i =f(0 1 ;::: ;0 i 1 ;x i ;0 i+1 ;:::;0 n )jx i 2X i g:
Using our onvention, a ts inX i
are denoted more simplyby (x i
;0 fig
). Weassumetohaveanintervals aledenoted v
i
onea hX i
,whi h quan-ti es the attra tivenessfor the DMofthe elements of X
i
(assumptionA1). In order to simplify the notation, we denote for all i 2 N, u
i : X i ! R, x i 7! u i (x i )= v i (x i ;0 fig
). Thus, assumption A1 means exa tly the follow-ing: (A1.1) 8x i ;y i 2 X i , u i (x i ) u i (y i
) if and only if for the de ision maker (x i ;0 fig )is atleast as attra tive as (y i ;0 fig ). (A1.2) 8x i ;y i ;z i ;w i 2 X i , su h that u i (x i ) > u i (y i ) and u i (w i ) > u i (z i ), we have u i (x i ) u i (y i ) u i (w i ) u i (z i ) =k; k 2R +
ifand onlyif thedi eren eofattra tivenessthatthe DMfeelsbetween (x i ;0 fig )and (y i ;0 fig
)is equaltok times the di eren eof attra tive-ness between (w i ;0 fig ) and (z i ;0 fig ).
We re ognize here information on erning the intra- riterion preferen es (i.e. the attra tiveness of elements of X
i
relatively to C i
), hen e the name of the assumption, whi h is a lassi al type of information in multi riteria de ision aid. Observe however that our presentationavoids the introdu tion of any independen e assumption (preferential or ardinal). This is possible sin e we have introdu ed in every set X
i
an element 0 i
with an absolute meaning in terms of attra tiveness. This strong meaning allows us to x naturallyu i (0 i )=0, 1
i=1;::: ;n, andthusto onsideru i
asaratio s aleon X
i
. We analsotakeadvantageofthe remainingdegreeoffreedomto xthe valueofu
i (1
i
). Contrarilytothe ase ofu i
(0 i
),noparti ularvalue, provided itispositive,ismandatoryhere. However, sin e allelements1
i
,i=1;::: ;n have all the same absolute meaning, we have to hoose for u
i (1
i
) the same numeri alvalueforalli2f1;::: ;ng, whi himpliesthattheonlyadmissible
1
whi histe hni allyalwayspossible,sin eanintervals aleisde ned uptoapositive aÆne transformation (z) = z+ , > 0, whi h means that wehave two degreesof freedom.
i i i >0doesnot depend oni. Thanks tothe elements 0
i and 1 i , the interval s ales u i
be ome thus ommensurable ratio s ales. In the sequel, we take as a onvention u i (1 i )=1, for i=1;::: ;n. 2.2 Inter- riteria assumption
We onsider now another subset of X, denoted Xe f0;1g
, ontaining the fol-lowing elements: Xe f0;1g :=f(1 A ;0 A )jANg; where(1 A ;0 A ) denotes ana t (x 1 ;:::;x n )withx i =1 i if i2A andx i =0 i otherwise, followingour onvention.
We assume to have an interval s ale u f0;1g
on Xe f0;1g
, quantifying the attra tiveness for the DM of all elements in this set (assumption A2). This means that:
(A2.1) for allA;B N, u f0;1g (1 A ;0 A )u f0;1g (1 B ;0 B
)if and only if for the DM (1 A ;0 A )is atleast asattra tiveas (1 B ;0 B ).
(A2.2) for all A;B;C;D N su h that u f0;1g (1 A ;0 A ) > u f0;1g (1 B ;0 B ) and u f0;1g (1 C ;0 C )>u f0;1g (1 D ;0 D ),we have u f0;1g (1 A ;0 A ) u f0;1g (1 B ;0 B ) u f0;1g (1 C ;0 C ) u f0;1g (1 D ;0 D ) =k; k 2R +
if and only if the di eren e of attra tiveness felt by the DM between (1 A ;0 A ) and (1 B ;0 B
) is k times the di eren e of attra tiveness be-tween (1 C ;0 C ) and (1 D ;0 D ).
Aswedidforthe aseofintra- riterioninformation,weusethe twoavailable degrees of freedom of aninterval s ale to x:
u f0;1g (1 ; ;0 N )=u f0;1g (0 1 ;::: ;0 n ):=0 u f0;1g (1 N ;0 ; )=u f0;1g (1 1 ;::: ;1 n ):=1:
Having in mind the meaning of 0 i , i = 1;:::;n, it is natural to impose u f0;1g (0 1 ;::: ;0 n ) = 0. The s ale u f0;1g
is then a ratio s ale. Let us point out that any stri tlypositivevalue ouldhavebeen usedinsteadof 1for the valueofu f0;1g (1 1 ;:::;1 n
). However,itis onvenienttoimposethatthevalue of u f0;1g (1 1 ;::: ;1 n
) isequalto the ommon value hosen for the u i
(1 i
At thispoint,letusremark thatboth u i (1 i ) andu f0;1g (1 i ;0 fig )quantify the attra tiveness of a t (1
i ;0
fig
) for the DM, however their values are on di erentratio s ales,but with thesame 0sin e u
i (0 i )=u f0;1g (0 1 ;:::;0 n )= 0. This means that there exists K
i >0su h that u f0;1g (x i ;0 fig )=K i u i (x i ) for all x i 2 X i
. An important onsequen e of this fa t is that, in order to have ompatibilitybetween theses ales(and hen ebetween assumptionsA1 and A2), we must have
u f0;1g (1 i ;0 fig )>u f0;1g (0 1 ;::: ;0 n )=0; 8i; otherwise no onstant K i
ould exist. This is not restri tive on a pra ti al point of view as soon as ea h point of view really orresponds to a on ern of the DM.
We suppose in addition that whenever A B, the a t (1 B ;0 B ) is at least as attra tive as(1 A ;0 A
),whi his alsoa naturalrequirement.
Under these onditions, and introdu ing the set fun tion : P(N) ! [0;1℄by (A):=u f0;1g (1 A ;0 A ) (1)
wehavede nedanon-additivemeasure,orfuzzymeasure,[36℄or apa ity [2℄, with the additionalrequirement that (fig) >0. Indeed, a apa ity is any non negative set fun tion su h that (;) = 0, (N) =1, and (A) (B) whenever AB.
3 Intera tion among riteria
Ex ept the natural assumptions above for (monotoni ity and (i) > 0 for all i 2 N), no restri tion exists on . Let us take 2 riteria to show the rangeof de ision behaviours we an obtain with apa ities. We suppose in addition that (f1g) = (f2g), whi h means that the DM is indi erent between (1 1 ;0 2 ) and (0 1 ;1 2
) (i.e. equal importan e of riteria, see se tion 4), and onsider 4 a ts x;y;z;t su h that (see gure 1):
x=(0 1 ;0 2 ) y=(0 1 ;1 2 ) z =(1 1 ;1 2 ) t =(1 1 ;0 2 )
other pairs may depend on the de ision maker. Due to the de nition of apa ities, we an range from the two extremal following situations (re all that (f1;2g)=1is xed):
extremal situation 1 (lower bound): weput(f1g)=(f2g)=0,whi h is equivalentto the preferen es xy t, where means indi eren e ( gure 1,left).
extremal situation 2 (upper bound): weput(f1g)=(f2g)=1,whi h is equivalentto the preferen es y z t ( gure 1, middle).
Note that the rst bound annot be rea hed due to the ondition (i) >0. The exa t intermediate situation is (f1g) = (f2g) = 1=2, meaning that z y t x ( gure 1, right), and the di eren e of attra tiveness between x and y,t respe tively is the same than between z and y, t respe tively.
The rst ase orrespondstoasituationwherethe riteriaare omplemen-tary, sin ebothhavetobesatisfa toryinordertogetasatisfa torya t. Oth-erwise said,the DMmakesa onjun tiveaggregation. Wesay thatinsu ha ase,whi h anbe hara terizedbythefa tthat(f1;2g)>(f1g)+(f2g), there is apositive intera tion between riteria.
The se ond ase orresponds to a situationwhere the riteriaare substi-tutive, sin eonlyone has tobesatisfa toryinordertoget asatisfa torya t. Here, the DM aggregates disjun tively. We say that in su h a ase, whi h an be hara terizedby the fa t that(f1;2g)<(f1g)+(f2g),there isa negative intera tion between riteria.
Inthethird ase,wherewehave(f1;2g)=(f1g)+(f2g),wesaythat there is no intera tion among riteria, they are non intera tive.
(b)
(c)
(a)
criterion 1
criterion 2
x
y
z
t
0
1
0
1
1
1
2
2
criterion 1
criterion 2
x
y
z
t
0
1
1
1
2
2
criterion 1
criterion 2
x
y
z
t
0
1
0
1
1
1
2
2
0
1
ofa tsfortheDMisthusperfe tly ompatiblewiththeintera tionsituations between riteria,situationswhi hare worth to onsider on apra ti alpoint of view, but up to nowvery littlestudied.
Inthe above simpleexample, wehad only2 riteria. In the general ase, we use the following de nition proposed by Murofushi and Soneda [28℄.
De nition 1 The intera tion index between riteria i and j is given by:
I ij := X KNnfi;jg (n jKj 2)!jKj! (n 1)! [(K[fi;jg) (K [fig) (K [fjg)+(K)℄: (2)
The de nition of this index has been extended to any oalition A N of riteria by Grabis h[14℄: I(A):= X BNnA (n jBj jAj)!jBj! (n jAj+1)! X KA ( 1) jAj jKj (K[B);8AN: (3) We have I ij
= I(fi;jg). When A = fig, I(fig) is nothing else than the Shapley value of game theory [34℄. Properties of this set fun tion has been studied and related to the Mobius transform [5℄. Also, I has been hara -terized axiomati allyby Grabis hand Roubens [19℄, ina way similarto the Shapley index. Notethat I
ij
>0 (resp. <0;=0) for omplementary (resp. substitutive, non intera tive) riteria.
4 Constru ting the model
We will only onsider in this paper the general type of aggregation model introdu edby Krantzet al. [25, Chap. 7℄:
A tx=(x 1
;::: ;x n
)isatleastasattra tiveasa ty=(y 1 ;::: ;y n ) if and only if F(u 1 (x 1 );::: ;u n (x n ))F(u 1 (y 1 );:::;u n (y n ));
where the aggregation fun tion F : R n
!R is stri tly in reasing in all its arguments.
Indeed,thistypeofmodelislargelyused,and hasthe advantage ofbeing rather general, and to lead to a omplete and transitive preferen e relation on X.
aggregation fun tion F whi h is ompatible with intra- riterion and inter- riteriainformationde nedbyassumptionsA1and A2,and satis esnatural onditions. Spe i ally, we are lookingfor a mapping F : R
n ! R of the form F(u 1 (x 1 );::: ;u n (x n ))=u(x 1 ;::: ;x n )
satisfying the following requirements (in whi h the presen e of is due to the fa tthat the u
i
are ommensurable ratio s ales):
(i) ompatibilitywith intra- riteria information (assumptionA1)
8i2N and 8x i ;y i 2X i , u i (x i )u i (y i ),u(x i ;0 fig )u(y i ;0 fig )
whi hbe omes,intermsofF (duetothe onsequen esof assump-tion A1 onthe s ale):
u i (x i )u i (y i ), F(0;::: ;0; u i (x i );0;::: ;0)F(0;::: ;0; u i (y i );0;:::;0) (4)
for all >0. In fa t,the onstant here is useless,sin e for any >0,u i (x i )u i (y i ), u i (x i ) u i (y i ). 8i 2 N and 8w i ;x i ;y i ;z i su h that u i (w i ) > u i (x i ) and u i (y i ) > u i (z i ), u(w i ;0 fig ) u(x i ;0 fig ) u(y i ;0 fig ) u(z i ;0 fig ) = u i (w i ) u i (x i ) u i (y i ) u i (z i )
whi h be omes interms of F:
F(0;::: ;0; u i (w i );0;::: ;0) F(0;::: ;0; u i (x i );0;::: ;0) F(0;:::;0; u i (y i );0;::: ;0) F(0;::: ;0; u i (z i );0;::: ;0) = u i (w i ) u i (x i ) u i (y i ) u i (z i ) (5) for all >0.
u f0;1g (1 A ;0 A ) u f0;1g (1 B ;0 B ),u(1 A ;0 A )u(1 B ;0 B )
whi h be omes, interms of F:
u f0;1g (1 A ;0 A ) u f0;1g (1 B ;0 B ) ,F( 1 A ;0 A )F( 1 B ;0 B )
for all >0,where for any AN, (1 A
;0 A
) is the ve tor whose omponent x
i
is1 whenever i2A, and 0otherwise.
8A;B;C;D N, with u f0;1g (1 A ;0 A ) > u f0;1g (1 B ;0 B ) and u f0;1g (1 C ;0 C ) >u f0;1g (1 D ;0 D ), we have: u(1 A ;0 A ) u(1 B ;0 B ) u(1 C ;0 C ) u(1 D ;0 D ) = u f0;1g (1 A ;0 A ) u f0;1g (1 B ;0 B ) u f0;1g (1 C ;0 C ) u f0;1g (1 D ;0 D )
whi h be omes, interms of F:
F( 1 A ;0 A ) F( 1 B ;0 B ) F( 1 C ;0 C ) F( 1 D ;0 D ) = u f0;1g (1 A ;0 A ) u f0;1g (1 B ;0 B ) u f0;1g (1 C ;0 C ) u f0;1g (1 D ;0 D ) (6) for all >0.
(iii) onditions related to absolute information We impose that s ales u and u
f0;1g
oin ide on parti ular a ts orre-sponding toabsolute information,namely:
u(0 1 ;:::;0 n ) = u f0;1g (0 1 ;:::;0 n ) := 0, whi h leads to F(0;::: ;0)=0. u(1 1 ;:::;1 n ) = u f0;1g (1 1 ;:::;1 n ) := 1, whi h leads to F(1;::: ;1) = 1. However, remember that the hoi e of value \1"wasarbitrarywhenbuildings alesu
i and u
f0;1g
, andany pos-itive onstant an do. Hen e, we should satisfy more generally F( ;:::; )= , 8 >0.
(iv) monotoni ity of F. This property is a fundamental requirement for any aggregation fun tion:
8(t 1 ;::: ;t n );8(t 0 1 ;::: ;t 0 n )2R n ; t 0 i t i ;i=1;::: ;n)F(t 0 1 ;:::;t 0 n )F(t n ;::: ;t n ):
The monotoni ity is stri t if all inequalities are stri t. Remark that monotoni ity entails the rst ondition of (i), namely formula(4).
an aggregation fun tion, and thus our problemamounts tothe sear h of an aggregation model whi h is ompatible with intra- and inter- riteria infor-mation de ned by assumptions A1and A2.
At this point,letus make two remarks.
the readermaywonderabout theveryspe i formofinter- riteria in-formationaskedfor,thatis,attra tivenessofa tsoftheform(1
A ;0
A ). These a ts present the double advantage to be non related with real a ts, whi h permits to avoid any emotional answer from the DM, and to have, taking into a ount the de nition of 0
i
and 1 i
, a very lear meaning, and onsequently, tobevery wellper eived and understood.
They are urrently used in real world appli ations of the MACBETH approa h [6, 8, 9℄ . Until now, these appli ations were done in the framework of anadditiveaggregation model. In su h a ase, only a ts of the form (1 i ;0 fig ) have tobe introdu ed.
What we are doing here is merely a generalization, onsidering not only single riteria,but any oalition of riteria. This natural general-izationfromsingletons tosubsetsis indeedthe key tothe modellingof intera tion, as explained in se tion 3. In this sense, the global utility u(1
A ;0
A
), whi h is a apa ity (see se tion 2.2), ould represent the importan eof oalition A tomake de ision.
it an be observed that onditions (ii) and (iii)above entail that the fun tion F : R
n
! R to be determined must oin ide with on f0;1g n , i.e.: F(1 A ;0 A )=(A); 8AN:
Indeed, just onsider equation (6) with B = D = ;;C = N, and use (iii),and de nition of (eq. (1)).
Thus, F must be anextension of on R n
. In other words, the assign-ment of importan e to oalitions is tightly linked with the evaluation fun tion. This fa t is well known in the MCDM ommunity (see e.g. Mousseau [27℄), but the argument above puts it more pre isely. The nextse tionaddressesinfulldetailtheproblemofextending apa ities.
5 Extension of pseudo-Boolean fun tions
The problem of extending a apa ity an be ni ely formalized through the use of pseudo-Boolean fun tions (see e.g. [21℄).
Any fun tionf :f0;1g !R isasaid tobeapseudo-Booleanfun tion. By making the usual bije tion between f0;1g
n
and P(N), it is lear that pseudo-Boolean fun tions on f0;1g
n
oin ide with real-valued set fun tions on N (of whi h apa ities are a parti ular ase). More spe i ally, if we de ne for any subset A N the ve tor Æ
A = [Æ A (1)Æ A (n)℄ in f0;1g n by Æ A
(i)=1ifi2A, and0 otherwise,then forany setfun tion v we ande ne its asso iatedpseudo-Boolean fun tionf by
f(Æ A
):=v(A); 8A N;
and re ipro ally. It has been shown by Hammer and Rudeanu[22℄ that any pseudo-Booleanfun tion an bewritten ina multilinearform:
f(t)= X AN m(A) Y i2A t i ; 8t 2f0;1g n : (7)
m(A) orresponds tothe Mobius transform (see e.g. Rota [31℄) of v, asso i-ated to f, whi his de ned by:
m(A)= X BA ( 1) jAnBj v(B): (8)
Re ipro ally,v an be re overed from the Mobius transformby
v(A)= X BA m(B): (9) If ne essary, we write m v
for the Mobius transform of v. Note that (7) an beput in anequivalent form,whi his
f(t)= X AN m(A) ^ i2A t i ; 8t2f0;1g n : (10)
More generally, the produ t an be repla ed by any operator on [0;1℄ n
oin iding with the produ t on f0;1g n
, su h as t-norms [32℄ (see e.g. [10℄ for a survey on this topi , and [24℄ for a omplete treatment). We re all that a t-norm is a binary operator T on [0;1℄ whi h is ommutative, as-so iative, non de reasing in ea h pla e, and su h that T(x;1) = x, for all x 2 [0;1℄. Asso iativity permits to unambiguously de ne t-norms for more than 2 arguments.
Theseare not the onlyways to writepseudo-Boolean fun tions. When v is a apa ity, itispossibleto repla ethe sum by _,as the following formula shows [15℄: f(t)= _ AN m _ (A)^ ^ i2A t i ! : (11)
_ by m
_
(A) =v(A) whenever v(A) >v(Ani) for all i 2 A, and 0 otherwise. Note that onversely we have ( ompare with (9)):
v(A)= _ BA m _ (B);8AN: (12)
In the sequel, we fo us on formulas (7) and (10). We will ome ba k on alternativesto these formulas inse tion 8.
Inordertoextend f toR n
,whi hisne essary inourframeworksin e the DM an judge that anelement(x
i ;0
fig
) is less attra tivethan (0 1
;::: ;0 n
) (inthat ase u
i (x
i
)<0),two immediate extensions ome from(7)and (10), where we simplyuse any t2R
n
instead of f0;1g n
. Wewill denotethem
f (t):= X AN m(A) Y i2A t i ; 8t2R n ; (13) f ^ (t):= X AN m(A) ^ i2A t i ; 8t2R n : (14)
However ase ond way an beobtained by onsidering the fa tthat any real number t an be writtenunder the form t= t
+
t , where t +
=t_0, and t = t _ 0. If,byanalogywiththisremark,werepla e
Q i t i by Q i t + i Q i t i , and similarlywith
V
, we obtaintwo new extensions:
f (t):= X AN m(A) " Y i2A t + i Y i2A t i # ; 8t2R n ; (15) f ^ (t):= X AN m(A) " ^ i2A t + i ^ i2A t i # ; 8t2R n : (16)
These are not the only possible extensions. In fa t, nothing prevents us to introdu e for the negative part another apa ity, e.g. equation (16) ould be ome: f ^ 12 (t):= X AN m 1 (A) ^ i2A t + i X AN m 2 (A) ^ i2A t i ; 8t2R n : (17)
However, wewillnot onsiderthispossibilityinthesubsequentdevelopment, ex ept in se tion 9 where the question of uni ity is addressed. In the next se tions weinvestigatewhether extensions (13) to(16) are relatedtoknown models of aggregation, and whi h one satisfy the requirements (i) to (iv) introdu edin se tion 4, and an be thus used as an aggregationfun tion in our ase.
We introdu e the Choquet integral with respe t to a apa ity, whi h has beenintrodu edasanaggregationoperatorby Grabis h[11, 12℄. Letbea apa ity onN, and t=(t 1 ;:::;t n )2(R + ) n
. The Choquet integral of t with respe t to is de ned by [29℄: C (t)= n X i=1 (t (i) t (i 1) )(f(i);::: ;(n)g) (18) where (i)
indi ates a permutationon N so that t (1) t (2) t (n) , and t (0)
:= 0 by onvention. It an be shown that the Choquet integral an be written asfollows: C (t)= X AN m(A) ^ i2A t i ; 8t2(R + ) n (19)
where m denotes the Mobius transform of . This result has been shown by Chateauneuf and Ja ray [1℄ (also by Walley [40℄), extendingDempster's result [3℄.
We are now ready to relate previous extensions to known aggregation models.
the extension f
is known in multiattributeutility theory asthe mul-tilinear model [23℄, whi h we denote by MLE. Note that our pre-sentation gives a meaning to the oeÆ ients of the polynom, sin e they are the Mobius transform of the underlying apa ity de ned by (A) = u(1
A ;0
A
), for all A N. Up to now, no lear interpretation of these oeÆ ients were given.
on erning f
, toour knowledge,it does not orrespond to anything known intheliterature. Wewilldenoteitby SMLE(symmetri MLE).
onsidering f ^ restri ted to (R + ) n
, it appears due to the above result (19) that f
^
is the Choquet integral of t with respe t to , where orresponds tof. Thisextensionisalsoknown astheLovaszextension off [26,35℄. Atthispoint,letusremarkthattheextensionofthe Cho-quet integraltonegativeargumentshas been onsideredby Denneberg [4℄, who gives two possibilities:
1. the symmetri extension S C de ned by S C (t)=C (t + ) C (t ); 8t2R n : (20)
2. the asymmetri extension C de ned by AS C (t)=C (t + ) C (t ); 8t2R n ; (21)
where is the onjugate apa ity de ned by (A) := (N) (A
).
The rstextensionhas been proposed rstby
Sipos[39℄,whilethe se -ond oneis onsideredasthe lassi alde nition ofthe Choquet integral on real numbers. In the sequel, we will denote the
Sipos integral by S , while we keep C
forthe (usual)Choquet integral.
ThefollowingpropositiongivestheexpressionofChoquet and
Siposintegrals in terms of the Mobius transform,and shows that f
^ C and f ^ S .
Proposition 1 Let be a apa ity. For any t2R n , C (t)= X AN m(A) ^ i2A t i ; (22) S (t)= X AN m(A) " ^ i2A t + i ^ i2A t i # = X AN + m(A) ^ i2A t i + X AN m(A) _ i2A t i ; (23) where N + :=fi2Njt i 0g and N =N nN + .
The proof isbased on the following lemma,shown in[16℄.
Lemma 1 Let v be any set fun tion su h that v(;) = 0, and onsider its o-Mobius transform
2 [13℄, de ned by: m v (A):= X BNnA ( 1) n jBj v(B)= X BA ( 1) jBj v(N nB);8AN:
Then, if v denotes the onjugate set fun tion:
m v (A)=( 1) jAj+1 m v (A); 8AN;A6=; (24)
and for any a2(R + ) n , C v (a)= X AN;A6=; ( 1) jAj+1 m v (A) _ i2A a i : (25) 2
Proof of Prop. 1: The ase of Sipos integral is lear from (14) and (20). Forthe aseof Choquet, the proofisbased onthe abovelemma. Using(14), we have: C (t + )= X AN m(A) ^ i2A t + i = X AN;A\N =; m(A) ^ i2A t i
Also, using (24) and (25) and remarking that m(;)=0, we get:
C (t )= X AN;A6=; ( 1) jAj+1 m (A) _ i2A t i = X AN m(A) _ i2A t i : Now _ i2A t i = V i2A t i ; if A\N 6=; 0; otherwise Thus C (t )= X AN;A\N 6=; m(A) ^ i2A t i so that C (t)=C (t + )+C (t )= X AN m(A) ^ i2A t i :
The next proposition gives the expression of Choquet and
Sipos integral dire tly in terms of the apa ity.
Proposition 2 Let be a apa ity. For any t2R n , C (t)=t (1) + n X i=2 t (i) t (i 1) (f(i);:::;(n)g) (26) S (t)= p 1 X i=1 t (i) t (i+1) (f(1);::: ;(i)g)+t (p) (f(1);:::;(p)g) +t (p+1) (f(p+1);:::;(n)g)+ n X i=p+2 t (i) t (i 1) (f(i);::: ;(n)g) (27)
(i) (1) (2) (p) t (p+1) t (n) .
Proof: fromthe de nition (18), we have:
C (t)=t (1) + n X i=2 t (i) t (i 1) (f(i);:::;(n)g): Let t2R n
. Wesplit t intoits positive and negativeparts t + ;t . Sin e 8 > > > < > > > : (t + ) (1) =(t + ) (2) ==(t + ) (p) =0 (t + ) (p+1) =t (p+1) . . . (t + ) (n) =t (n) we have C (t + )=t (p+1) (f(p+1);::: ;(n)g)+ n X i=p+2 t (i) t (i 1) (f(i);::: ;(n)g) :
In the same way,one has
C (t )= t (p) (f(p);:::;(1)g) p 1 X i=1 t (i) t (i+1) (f(i);::: ;(1)g) :
This gives the desired expression for
Sipos integral. The ase of Choquet integral pro eeds similarly.
Remarking that C (0) = S
(0) for any apa ity, we have from proposi-tion 2: C ( t)= C (t) (28) S ( t)= S (t) (29) for any t in R n
, hen e the terms asymmetri and symmetri .
Insummary,threeamongthefourextensions orrespondtoknownmodels of aggregation, even if ontexts may di er.
7 Properties of the extensions
This se tion is devoted to the study of the four extensions, regarding the properties requested in the onstru tion of the aggregation model (se tion 4).
alling that u i
(0 i
)=08i2N, and noting that m(fig)=(fig), a straight-forward omputationshows that forany >0:
C (0;::: ;0; u i (x i );0;:::;0)= (fig)u i (x i ) if x i i 0 i (fig)u i (x i ) if x i i 0 i (30) S (0;::: ;0; u i (x i );0;:::;0)= (fig)u i (x i ) (31) MLE (0;::: ;0; u i (x i );0;:::;0)= (fig)u i (x i ) (32) SMLE (0;::: ;0; u i (x i );0;:::;0)= (fig)u i (x i ): (33)
In the general ase, we have (fx i
g) 6= (fx i
g). Thus there is an angular point around the origin for the Choquet integral. The onsequen e is that equation (5), and hen e assumption A1, are not satis ed by the Choquet integral in general.
This urious property an be explained as follows. For the
Sipos in-tegral, the zero has a spe ial role, sin e it is the zero of the ratio s ale, and all is symmetri with respe t to this point. For the Choquet integral, the zero has no spe ial meaning, but observe that if x
i 0 i y i , the a ts (0 1 ;:::0 i 1 ;x i ;0 i+1 ;::: ;0 n ) and (0 1 ;:::0 i 1 ;y i ;0 i+1 ;::: ;0 n ) are not omonotoni , i.e. they indu e a di erent orderingof the integrand.
ompatibility with inter- riteria information (assumption A2) It results from the de nitions of C
, S , MLE and SMLE that, 8AN and 8 >0, MLE ( 1 A ;0 A ) =SMLE ( 1 A ;0 A ) = X BA m(B) jBj ; (34) and C ( 1 A ;0 A )= S ( 1 A ;0 A ) = (A):
Consequently, MLEand SMLEare inadequate for our model.
use of absolute information Obviously any extension satis es F(0;:::;0) = 0, and taking into a ount the fa t that (N) = 1, we have C ( ;::: ; )= S
( ;:::; )= ,forall >0. Butfrom(34),thisproperty is not satis ed by MLE and SMLE.
Monotoni ity It an be shown that, for any t;t 0 2R n , t i t 0 i ;i=1;:::;n )C (t 1 ;:::;t n )C (t 0 1 ;::: ;t 0 n ) (35) t i t 0 i ;i=1;:::;n ) S (t 1 ;::: ;t n ) S (t 0 1 ;::: ;t 0 n ): (36)
any t2(R +
) n
, an equivalent formof (18) is:
C (t)= n X i=1 t (i) [(f(i);:::;(n)g) (f(i+1);::: ;(n)g)℄:
Monotoni ity is immediate from the fa t that A B implies (A) (B). Now, foranyt 2R
n
,monotoni ityofthe Choquet and
Siposintegralsfollow from equations (20) and (21). To obtain stri t monotoni ity,we need stri t monotoni ity of the apa ity, i.e. A$B implies (A)<(B).
Itiseasytoseefromde nitionthatMLEandSMLEare monotoni when the oeÆ ients m(A) are allpositive. But in general, the Mobius transform of a apa ity is not always positive. To our knowledge, there is noresult in the general ase. The following an beproven.
Proposition 3 For any t 2 [0;1℄ n
, for any apa ity , MLE
is non de- reasing with respe t to t
i
, i = 1;:::;n. Stri t in reasingness is ensured i is stri tly monotoni .
Proof: We an express easily MLEwith respe t to (see Owen [30℄):
MLE (t)= X AN " Y i2A t i #" Y i62A (1 t i ) # (A):
Then we have,for any t 2[0;1℄ n and any k 2N: MLE(t) t k = X ANnk " Y i2A t i #" Y i62A;i6=k (1 t i ) # (A[k) X ANnk " Y i2A t i #" Y i62A;i6=k (1 t i ) # (A) = X ANnk " Y i2A t i #" Y i62A;i6=k (1 t i ) # ((A[k) (A)):
Clearly, the expression isnon negative(resp. positive) forany k 2N i is monotoni (resp. stri tly monotoni ).
The proof shows learly that MLE ould be non in reasing when t is no more in [0;1℄
n
. Taking for example n = 2, with (f1g) = (f2g) =0:9, we have: MLE (1;1)=0:9+0:9 0:8 =1 MLE (3;3)=(3)(0:9)+(3)(0:9) (9)(0:8)= 1:8 <MLE (1;1):
s ores are limited to [0;1℄, that is, unipolar bounded riteria. Also, SMLE whi hdi ers fromMLE only for negative values, is learly useless.
S ale preservation Although this property is not required by our on-stru tion (but it somehow underlies it in assumptions A1 and A2), it is interesting toinvestigate whether the extensions satisfy it.
The following is easyto prove.
(C.1) invarian eto the same positive aÆne transformation
C ( t 1 + ;:::; t n + )= C (t 1 ;::: ;t n )+ ; 8 0;8 2R: (S.1) homogeneity S ( t 1 ;::: ; t n )= S (t 1 ;::: ;t n );8 2R:
As remarked by Sugeno and Murofushi [37℄, this means that if the s ores t
i
are on ommensurable interval s ales, then the global s ore omputed by the Choquet integral isalsoon anintervals ale (i.e. relativeposition of the zero), and if the s ores are ona ratio s ale, then the global s ore omputed by the
Siposintegralis ona ratio s ale (absolute position of the zero). By ontrast, MLE and SMLE neither preserve the interval nor the ratio s ale,sin etheyare nothomogeneous. Indeed,takingn =2andany 2R
: MLE ( t 1 ; t 2 )=m(f1g) t 1 +m(f2g) t 2 +m(f1;2g) 2 t 1 t 2 6= MLE (t 1 ;t 2 ):
This isthereasonwhyMLE andSMLEfailedtoful llassumptionA2. Note however that MLE satis es(5)but not (6).
Asa on lusion, onlythe
Siposintegal amongourfour andidates an t all requirementsof our onstru tion.
8 An equivalent axiomati
Our onstru tionis based on a ertain number of requirements for aggrega-tion fun tion F, whi h we sum up below:
restri ted monotoni ity (M1), omingfrom assumption A1:
8i=1;:::;n;8a i ;a 0 i 2R;a i a 0 i )F(a i ;0 fig )F(a 0 i ;0 fig )
F( a i ;0 fig ) F( b i ;0 fig ) F( i ;0 fig ) F( d i ;0 fig ) = a i b i i d i ;8 >0;8a i ;b i ; i ;d i 2R; i 6=d i
interval s ale forinter- riteriainformation (A2):
F( 1 A ;0 A ) F( 1 B ;0 B ) F( 1 C ;0 C ) F( 1 D ;0 D ) = (A) (B) (C) (D) ; 8 >0 idempoten e(I): F( ;:::; )= ; 8 0;
with restri ted versions (I0)for =0 and (I1) for =1.
monotoni ity (M),whi his non de reasingness of F for ea hpla e.
As already noted,(M) implies(M1). All these requirements ome from on-siderations linked with thepreferen e of the DMands ales ofmeasurement. Itispossibletoshowthattheyareequivalenttoamu hsimplersetofaxioms about F.
Proposition 4 Let F : R n
) R and a apa ity on N. Then the set of axioms (A1), (A2), (I),(M) isequivalent to the following set of axioms:
1. homogeneous extension (HE):
F( 1 A
;0 A
)= (A); 8 0;8A N
2. restri ted aÆnity (A)
F(a i ;0 fig )=a i F(1 i ;0 fig ); 8a i 2R;8i =1;::: ;n 3. monotoni ity (M).
Proof: ()) Letting B = D = ;;C = N in (A2) and using (I) lead to F( 1
A
;0)= (A),whi his(HE).Now, using(A1)with b i =d i =0, i =1, =1 and using (I0) weget F(a
i ;0 fig )=a i F(1 i ;0 fig ), whi h is (A). (()Using(A), we get:
F( a i ;0 fig ) F( b i ;0 fig ) F( i ;0 fig ) F( d i ;0 fig ) = a i F(1 i ;0 fig ) b i F(1 i ;0 fig ) i F(1 i ;0 fig ) d i F(1 i ;0 fig ) = a i b i i d i ;
F( 1 A ;0 A ) F( 1 B ;0 B ) F( 1 C ;0 C ) F( 1 D ;0 D ) = (A) (B) (C) (D)
whi h is (A2). Finally, from (HE) with A = N, we get (I) sin e (N) =1.
Nota: (M) an be dropped from the 2 sets of axioms without hanging the equivalen e.
9 The uni ity issue
Having this simpler set of axioms, we address the question of the uni ity of thesolution,i.e. isthe
Siposintegraltheonlyaggregationfun tionsatisfying the requirements?
First we examine the following extension on [0;1℄ n of pseudo-Boolean fun tions: F(a 1 ;::: ;a n )= X AN m(A)( i2A a i );8a i 2[0;1℄ (37)
as suggested in se tion 5, where is a \pseudo-produ t". Re all that m is the Mobius transform of the underlying apa ity. Letus suppose as a basi requirement that is a ommutative and asso iative operator, otherwise our expression of F would be ill-de ned sin e
i2A a
i
would depend on the order of elements in A ( ommutativity), and on the grouping of elements (asso iativity). Thus, itis suÆ ienttode ne on[0;1℄
2
. The following an beshown.
Proposition 5 Let : [0;1℄ 2
! [0;1℄ be a ommutative and asso iative operator, and F be given by (37). Then:
(i) F satis es(HE) on[0;1℄ n
ifand onlyif oin idewith theprodu ton f0;1g, satis es = for all 2[0;1℄, and 0=0.
(ii) F satis es (M) implies is non de reasing.
Proof: (i) ()) Let us onsider the parti ular apa ity u 1;2
de ned by u
1;2
(A) = 1 if f1;2g A, and 0 otherwise (unanimity game). It is easy tosee thatitsMobius transformis su h that m(f1;2g)=1and 0elsewhere. Let us onsider (HE)with A=;, =1,and the apa ity u
1;2
. We obtain
F(0;::: ;0)=1(00)=u 1;2
F(1;::: ;1)=1(11)=u 1;2
(N)=1;
hen e 11 = 1. Now let us take A = f1g, with any > 0 and we obtain from (HE):
F( ;0;:::;0)=1( 0)= u 1;2
(f1g)=0;
hen e 0 = 0 for any = > 0, in parti ular when = 1. Thus, oin ides with the produ t on f0;1g. Lastly, letus apply (HE) with A=N and again the apa ity u
1;2
. We obtain:
F( ; ;:::; )=1( )=
hen e = .
(()Forany apa ity , any A N, any 2[0;1℄:
F( 1 A ;0 A ) = X BA m(B)( i2B )+ X B6A m(B)[( i2A )( i62A 0)℄ = X BA m(B)+0 = (A):
(ii)Ifis de reasinginsome pla e,and m ispositive,then F annotbe in reasing, a ontradi tion. Thus, isnon de reasingin ea hpla e.
To go further in the analysis, let us assume in the sequel that is non de reasing. Then we obtainthe following result.
Corollary 1 Let:[0;1℄ 2
