JOURNAL OF MULTI-CRITERIA DECISION ANALYSIS, VOL 5 , 9 9 - 1 1 1 (1996)

Outranking Relations: DoThey Have Special Properties?

DENIS BOUYSSOUESSEC. BP105. F-95021 Cergy-Pontoise Cedex. France

ABSTRACTOutranking relations are most often built using a concordance-discordance principle. Such relations are in generalneither transitive nor complete. This is not to say that the concordance-discordance principle does not impose some'structural' restrictions on these relations. We show why this question may be of some importance for analysing thevarious techniques designed to build a recommendation on the basis of such relations. These restrictions are studiedfor the ELECTRE and PROMETHEE methods.

KEY WORDS: MCDM; outranking relations; social choice theory; binary choice probabilities

1. INTRODUCTION

This paper is concerned with the outrankingapproach to multiple-criteria decision aid (usefulreferences concerning these methods includeScharUg (1985), Roy (1991), Vincke (1992a) andRoy and Bouyssou (1993)). Methods related tothis approach, including the well-known family ofELECTRE methods, are often presented as acombination of two steps:

(1) a 'construction step' in which one or severaloutranking relations are built;

(2) an 'exploitation step' in which outrankingrelations are used to derive a recommendation.

The construction step consists of comparingalternatives taking all criteria into account. Thisleads to a preference model taking the form of oneor several binary relations—the so-called'outranking relations'—which may be crisp orvalued (our definitions and notation concerningbinary relations are introduced at the end of thissection). Outranking relations in most methods arebuilt using a concordance-discordance principle.This principle leads to declaring that an alternativeis 'at least as good as' another when

(1) a 'sufficient' majority of criteria support thisproposition (concordance principle) and

(2) the opposition of the minority is not 'toostrong' (non-discordance principle).

It is well-known that this principle does not ingeneral lead to binary relations possessing'remarkable properties' such as transitivity andcompleteness (this being true for valued relationsindependently of how we interpret these properties

in the valued case; see e.g. Pemy (1992) or Pemyand Roy (1992)).

The exploitation step aims at building arecommendation on the basis of such preferencemodels. Depending on the problem, this recom-mendation may take the form of the selection of asubset of alternatives, the ranking of all alter-natives or the sorting of alternatives into differentcategories. Since the preference models that areused do not in general possess 'remarkable proper-ties' this is not an obvious task. It calls for theapplication of specific techniques that depend onthe type of recommendation that is looked for (seeVanderpooten (1990) or Roy and Bouyssou (1993)).

Outranking methods have often been criticizedfor their lack of axiomatic foundations. Recentlyseveral papers have attempted a theoretical analy-sis of the construction step (see e.g. Bouyssou(1986, 1992a), Bouyssou and Vansnick (1986),Vansnick (1986) or Pemy (1992)) and exploitationstep (see e.g. Bouyssou (1992b, 1995), Bouyssouand Pemy (1992), Vincke (1992b) or Pirlot (1995))of various outranking methods. In these papersoutranking methods are not considered as a whole.They analyse their two steps in a separate way.The absence of 'remarkable properties' of out-ranking relations has often been seen as therationale for separating the analysis of theexploitation step from that of the constructionstep. In particular, papers dealing with theexploitation step have attempted to analyse theproperties of several exploitation techniquesassuming that these techniques are to be appliedto any preference structure. This is a weak line ofreasoning, however. Even if we know thatoutranking relations do not possess 'remarkable

CCC 1057-9214/96/020099-13© 1996 by John Wiley & Sons, Ltd Accepted 12 June 1995

100 D. BOUYSSOU

properties', they may well possess some 'structuralproperties' when they are built using a concor-dance-discordance principle, i.e. it may well beimpossible to obtain any preference structure usinga construction technique based on concordance-discordance. The existence of such 'structuralproperties' would render difficult the interpreta-tion of the above-mentioned analyses ofexploitation techniques. Exploitation techniquescould be envisaged that would not be appealingwhen applied to any relation but that wouldbehave more nicely when applied to relationspossessing some 'structural properties'.

The aim of this paper is to investigate theexistence of such 'structural properties'. Wesummarize our main conclusions below.

(1) With the construction techniques used inELECTRE I and III, outranking relationsdo not possess 'structural properties'. WithELECTRE I it is possible to obtain any crispreflexive binary relation as an outrankingrelation. The same is true for ELECTRE IIIin the valued case. In these methods thisallows one to some extent to separate theanalysis of the exploitation step from that ofthe construction step.

(2) The situation is different when turning tomethods such as PROMETHEE that do notmake use of the discordance concept. Theylead to outranking relations having 'structuralproperties'.

(3) The characterization of these 'structural prop-erties' turns out to be a difficult task. Thisproblem is deeply linked with classic problemsin the theory of choice.

(4) In spite of the absence of a completecharacterization of the 'structural properties'of outranking relations in PROMETHEE-likemethods, we outline how an axiomatic analy-sis of exploitation techniques can be con-ducted taking into account these properties.

The paper is organized as follows. The case ofELECTRE I and III is examined in Section 2. InSection 3 we tum to methods such asPROMETHEE that do not make use of thediscordance concept. The final section presentssome directions for future research. In the rest ofthis section we introduce our basic notation anddefinitions.

A valued (binary) relation T on a set A" is afunction from Xx X into [0; 1]. It is said to bereflexive (resp. irreflexive) if T{x, x) - 1 (resp. 0)

for all X e X A valued relation T on X such thatT{x,y) e {0, l}for all x,y^X\s said to be crisp. Asusual we write x Ty instead of T(x,y) = 1 when Tis a crisp relation. Concerning crisp relations, wewill make use of the following classic definitions(see Fishbum (1970) or Roubens and Vincke(1985)). Let r be a crisp relation on X. Thisrelation is said to be

(1) reflexive ii [xTx],(2) complete if [xTy or yTx],(3) weakly complete if [jc ̂ it;/ =>• xTy or yTx],(4) transitive if [xTy and yTz =• xTz],(5) antisymmetric if [xTy and yTx =^ x = y],(6) asymmetric if [xTy =• Not(y7jc)],(7) Ferrers if [{xTy and zTw) => {xTw or zTy)],(8) semitransitive if [{xTy and yTz) =>• {xTw or

wTz)]

ior 2i\\x,y,z,we X.We say that a crisp relation is

(1) a linear order if it is complete, antisymmetricand transitive,

(2) a strict linear order if it is weakly complete,asymmetric and negatively transitive,

(3) a weak order (sometimes called a preorder) if itis complete and transitive,

(4) a strict weak order if it is asymmetric andnegatively transitive,

(5) a semiorder if it is complete, Ferrers andsemitransitive,

(6) a strict semiorder if it is irreflexive, Ferrers andsemitransitive,

(7) an interval order if it is complete and Ferrers,(8) a strict interval order if it is irrefiexive and

Ferrers,(9) a strict partial order if it is irreflexive and

transitive.

We denote COy (resp. SCOx, WOy, SWOx,SOx,SSOx,IOx,SIOx,SVOx) the set of alllinear orders (resp. strict linear orders, weakorders, strict weak orders, semiorders, strictsemiorders, interval orders, strict interval orders,strict partial orders) on a set X, dropping thesubscript when there is no risk of confusionabout the underlying set. It is well-known (seeFishbum (1970) or Roubens and Vincke (1985))that COx^WOx^SOxQIOx and SCOxQSWOx ̂ SSOx £ SJOx £ SPOx, all inclusionsbeing strict as soon as X is large enough.

We denote i{T) (resp. a(7)) the symmetric (resp.asymmetric) part of the crisp relation T, i.e. the

OUTRANKING RELATIONS 101

crisp relations defined respectively by ixL(T)y <^ where[xTy and yTx]a.ndx(x(T)y o [xTy and NotCvTx)].

It is easy to see (see e.g. Roubens andVincke (1985)) that for any crisp relation T on aset X, T € COx => ai(T) e SCOx, TeWOx=>a.{T) e SWOx, T e SOx =)• a(7^ e SSOx andT e IOx => oiiT) e SIOx-

Throughout this paper A = {a,b,c, . . .) willdenote & finite set with \A\ = m'^ 2 elements. Weinterpret the elements of A as 'altematives' to becompared using an outranking method.

aCbj - gji") > ,

1=1

and

aVb •O' [3/ € {1,2, . . ., n} such that

2. ELECTRE I AND III

2.1. A basic resultELECTRE I (Roy, 1968) and III (Roy, 1978) areamong the most popular outranking methods.ELECTRE I aims at building a crisp outrankingrelation starting with a set of altematives eval-uated on several 'true criteria'. ELECTRE IIIbuilds a valued outranking relation starting with aset of altematives evaluated on several 'pseudocriteria' (on the notions of true criterion andpseudo criterion see Roy (1985)). We briefly recallhere, from a purely algorithmic point of view, theprinciples of the construction techniques that areused in these two methods (for a thoroughdiscussion of these methods see Roy andBouyssou (1993)). As indicated in Section 1, wedenote by y4 a finite set (of 'alternatives') with melements.

Consider an 'ELECTRE I situation on A"consisting of

(1) a strictly positive integer n (the 'number ofcriteria'),

(2) a real number (the 'concordance threshold')s e [0.5;l],

(3) n functions (the 'criteria') g\,g2, . . ., ga fromA into R,

(4) n functions (the 'veto thresholds')vi,V2, . . . , Vn from R into M+ such thatVz e {1,2, . . ., n] and Va, b e A,gi{a) ^gi(b) =^ gi{a) + Viigiia)) > gib) + Viigiib)),

(5) n strictly positive real numbers (the 'weights')

Starting with an 'ELECTRE I situation on A", withthe constmction technique of ELECTRE I buildsa crisp relation S on A (i.e. a subset oi Ax A)letting, for all a,b e A, di

aSb ^ [aCi> and

The crisp relation C (resp. V) is called theconcordance (resp. discordance) relation ofELECTRE I.

Consider an 'ELECTRE III situation on Aconsisting of

(1) a strictly positive integer n (the 'number ofcriteria'),

(2) n functions (the 'criteria') gi,g2, • • •, gn fromA into R,

(3) In functions (the 'indifference, preferenceand veto thresholds') q\,p\, vi,q2,P2, V2, . . .,qn,Pn,Vn from R into R.̂ . such thatVie {1,2, . . ., n} and 'ia,b&A, q

qiiSiia)) > giib) + qiigiib)), gi(a) + pi(gi(a))gi{b) + Piigiib)) and g,(a) H-v,(g,(a

(4) n strictly positive real numbers (the 'weights')ki,k2, . . ., kn.

Starting with an 'ELECTRE III situation on A\the construction technique of ELECTRE IIIbuilds a valued relation 5 on ̂ (i.e. a functionfrom Ax A into [0; 1]) letting, for d\\ a,b & A,

= C(a,b){l-D(a,b))

where

1=1

,b) =otherwise

102 D. BOUYSSOU

and

D(a,b) =

if Z)^ = {/e {1,2, . . . , « } :Djia,b)>C(a,b)] = 0

\-Di{a,b)• - n l̂-C(a,b)otherwise

with

between {2,3, . . ., w-f-1} and the ordered pairs(a, b) G A^ such that Not(a Tb). Let us build a'situation' such that

(1) n = l + M,(2) j = i

(3) gi{c) = 0, Vc G A, and vi(;c) = 1, V;c G R,(4) for I = 2, 3, . . . , « + 1 suppose that / corre-

sponds to the ordered pair (a,b) such thatNot(fl Tb) and let us choose the functions g,and V, so that gi(a) = 0, gi(b) = 1, g,(c) = 0.5,Vc G A\[a, b), Vi(x) = 0.6, Vx G R,

. . =A: i+«= 1/2M.otherwise. (5) fc, = i,A:2 = A:3 =

The valued relation C is called the concordancerelation of ELECTRE IIL

It is easily seen that an outranking relation Sbuilt with either method is necessarily reflexive (i.e.Va G ^, aSa in the crisp case and S{a, a) = 1 in thevalued case). Apart from reflexivity, do theserelations possess any 'structural property'? Thefollowing simple proposition shows that this is notthe case.

Proposition 1(a) Let T be any reflexive crisp relation on a finiteset A. There is an 'ELECTRE I situation on A'such that applying the construction technique ofELECTRE I to this situation leads to an out-ranking relation identical with T.

(b) Let T be any reflexive valued relation on afmite set A. There is an 'ELECTRE III situationon A' such that applying the construction techni-que of ELECTRE III to this situation leads to anoutranking relation identical with T.

ProofThis consists of exhibiting the appropriate'situation'. In both cases the 'situation' consists of

(1) one criterion having 'much weight' on whichall alternatives have identical evaluations and

(2) several criteria having 'little weight' on each ofwhich we introduce a 'discordance effect' for aselected pair of alternatives.

(a) Let r be a reflexive crisp relation on a finiteset A. Let u = m^- \T\, where m = \A\. If M = 0,ELECTRE I will lead to T using a 'situation'consisting of a unique criterion on which allalternatives have an identical evaluation. WhenM / 0, consider a one-to-one correspondence

It is easily seen that applying ELECTRE I tothis 'situation' will lead to an outranking relationidentical with T. Since all alternatives have anidentical evaluation on g, and k]=s = \, theconcordance relation C is complete. For eachordered pair (a,b) such that Not{aTb) we haveintroduced a criterion for which aVb which allowsus to recover T.

(b) Let r be a reflexive valued relation on afmite set A. If T(a,b) = 1, Wa,b€ A, ELECTREIII will lead to T using a 'situation' consisting of asingle criterion on which all alternatives have anidentical evaluation. When this is not the case, let

,Max^ Max Tia,b)(,a,b)sU

By hypothesis we have r^*" e [0; 1) andMG {1,2, . . ., w(m-1)}. Let t he such that,Max<-,< j We denote

• s i - '

It is clear that t^^ < 7 < 1. Consider a one-to-onecorrespondence between U and {2,3, . . ., MH- 1).Let us build a 'situation' such that

(1) « = 1 + M,(2) gj(c) — O,WceA, and g\(x) = p\(x) = 0,

vi(x)= 1 VxeM,(3) for i = 2,2, . . ., u+l suppose that / corre-

sponds to the ordered pair (a, b) € U and let uschoose the functions gi,qi,Pi and v, so thatgib) = 1, gia) = 0, gic) = 0.5 Vce A\{a,b),

= C;, Vx G R, where

C- = 0.5 +0.5y

y-{\-y)na,b)

OUTRANKING RELATIONS 103

(4) it, = t, ki = (1 - 0 /" for I = 2,3, . . ., M -h 1.

Observe that C, ̂ 1, i = 2,3, . . ., M -I-1, so thatthe functions v, are admissible for ELECTRE III.The application of ELECTRE III to this'situation' leads to an outranking relation S.

Consider an ordered pair {a,b) such thatT{a,b) — \. It is easy to see that we haveCi{a,b) = 1 and Di{a,b) = 0, Vi G {1,2, . . .,u+\],so that S(a,fe) = 1.

Consider now an ordered pair (a, b) G U, i.e.such that T{a, b)<\. Let us denote by / the uniqueelement in {2,3, . . ., M-|- 1} corresponding to thisordered pair. We have for the concordance partCj{a, b) = \,'^j^ i, and C^a, b)^0 so thatCia, b) = y and T{a, b) < C{a, b)<\. For thediscordance part we have Dj{a, b) — 0, Vy ^ i, and

= Min

— Min

= Min

l;Max

l;Max

1;0.5 + 0.5y/[y - (1 - y)T(a, b)] - 0.5

Since [y - (1 - y)T(a, b)]/y > y, we have

l-C(a,b)

D

1-y ^ '

This completes the proof

Proposition 1 implies that for the axiomaticinvestigation of exploitation techniques to becoupled with ELECTRE I and III, it makessense to suppose that they will be confrontedwith any (reflexive) preference structure. Suchinvestigations were conducted in the crisp case byVincke (1992b) and in the valued case byBouyssou (1992b, 1995), Bouyssou and Perny(1992) and Pirlot (1995). However, one shouldnot conclude from Proposition 1 that it islegitimate for ELECTRE I and III to completelyseparate the construction and exploitation steps.In particular, in the valued case this propositionsays nothing about the nature and interpretation

of the 'valuations' used to model preferences andconsequently about the operations that we canlegitimately use on them so as to stay consistentwith the way they have been built. This difficultproblem concerning valued relations is still widelyopen (Pemy, 1992).

2.2. Remarks and extensionsIn this subsection we add some comments to thevery simple Proposition 1.

(i) The construction step of ELECTRE I being aparticular case of that of ELECTRE IS (see Royand Skalka (1984) or Roy and Bouyssou (1993)),part (a) of Proposition 1 applies directly toELECTRE IS. A simple modification of thisproof shows that any two nested crisp reflexiverelations on a fmite set can be obtained as theresult of the ELECTRE II construction technique(Roy and Bertier, 1973).

(ii) The proof of Proposition 1 uses 'situations'involving up to m(m — 1) -H 1 criteria (the nextremark shows that m{m — 1) criteria are alwayssufficient for ELECTRE I). In most cases it ispossible to use more 'realistic' 'situations', i.e.'situations' using a more reasonable number ofcriteria. For each method it would be interesting toknow what is the minimimi number of criteria thathas to be used in order to be able to recover anyreflexive relation. This raises interesting combina-torial problems that will not be dealt with here(similar problems arise with the method ofmajority decisions; a basic reference on the subjectis Steams (1959)).

(iii) The proof of part (a) of Proposition 1 makesgreat use of the possibility of introducing dis-cordance effects 'at will' with ELECTRE I.Building upon a famous result of McGarvey(1953) concerning the method of majority deci-sion, let us show that part (a) of the propositionremains true even when there are no discordanceeffects, i.e. when the veto thresholds are largeenough to imply V = 0. Let T be a reflexive crisprelation on A. For any pair of distinct altematives{a, b] we have one and only one of the followingsituations:

(1) [aTb and bTa],(2) [a Tb and Not(bTa)l(3) [Not(a Tb) and Not{b Ta)].

Consider a one-to-one correspondence between{1,3, . . ., m(m - 1) - 1} and the m(m - l)/2 pairs

104 D. BOUYSSOU

of distinct alternatives in A. Consider a 'situation'such that

(1) « = m(m - 1),(2) l < . r < i + l / / « ( / M - l ) ,(3) for / = 1,3, . . ., m[m- 1 ) - 1 suppose that /

corresponds to the pair [a,b} of distinctalternatives, consider a one-to-one correspon-dence/between A\[a, b) and {1,2, . . ., m - 2}and define gt and g,+i lettinggi{a) = gi(b) = m-l, gi(c) =fic)

Vc € ^\{fl, b], andgi+i{a) = gi+\(b) = l,gi+\{c) = m -fie),

Vc € A\{a, b], if [aTb and bTa],gi{a) = m,gi{b) = m- \,gi{c) =f{c),

'ic€A\{a,b], andgi+\{a) = l,gi+\ib) = O,g,+i(c) = m -f[c),

Vc e A\{a, b], if [aTb and Not(bTa)],gi(a) = m,gi{b) = w - l,gi{c) =f[c),

Vc e A\{a, b], and+i) ,g,Hi() ,f.+iW f{),Vc e A\{a, b], if [Not(a7Y)) and Not(Z>ra)],

(4) ki=k2= . . . =fc« = l/n.

Let us show that C=T, C being the concor-dance relation obtained by applying ELECTRE Ito this 'situation'. Let {a, b] be a pair of distinctalternatives in A. We denote

gj(b) J- gj(b) > gj(a)

It is easy to see that

(1) rab = rta = mint - l ) /2 +lif aTb and b Ta,(2) Tab = m{m- l ) /2 -I-1 and rba = m{m- l ) / 2 - 1

if a Tb and Not(Z> Ta),(3) Tab = rba = m(m - l ) /2 if Not(a Tb) and

Not(b Ta).

Thus for all s e {^; ^+l/m{m-I)) we haveT=C.

Extending in a similar way a previous result byDeb (1976), it is not difficult to show that thefollowing stronger result holds.

Let r be a reflexive crisp relation on a finite setA and X&{^;1). There is an 'ELECTRE Isituation on A' with s = X such that applying theconstruction technique of ELECTRE I to thissituation leads to a concordance relation Cidentical with T.

This result does not hold for A = 1 (resp. 0.5),since in that case C is necessarily transitive (resp.complete).

Let us finally mention that these techniques can

easily be transposed to other ways of building crispconcordance relations. For instance, it can beshown that every asymmetric crisp relation T canbe obtained as a concordance relation in theTACTIC method (see Vansnick (1986); let usrecall that concordance relations in TACTIC arenecessarily asymmetric).

(iv) Contrary to the situation with ELECTRE I,it is not possible to obtain any reflexive valuedrelation as the result of ELECTRE III if thediscordance part of the method is not used, i.e.when S{a, b) = C{a, b), "ia, b e A, which amountsto choosing 'very large' veto thresholds v/.Consider the following valued relation T (valuedrelations in matrix form are always read from rowto column):

T

abc

a

101

b

110

c

011

Suppose that there is an 'ELECTRE III situationon A' such that C=T (from part (b) ofProposition 1 we know that there is a 'situation'such that 5 = 7 ) . In such a 'situation' we have,V/ e {1,2, . . ., n], giia) > giib) + pi(gi(b)), gi(b) >gi(c) +Pi(gi{c)), giic) > gi{a) +Pi(gi(a)), which iscontradictory since /?, is always non-negative.Thus concordance relations in ELECTRE IIIpossess 'structural properties'. We study them inthe next section.

3. VALUED CONCORDANCE RELATIONSAND 'BINARY CHOICE PROBABILITIES'

3.1, Valued concordance relations, steppedrelations and semiordersIn order to analyse the 'structural properties' ofvalued concordance relations, we use the followinggeneral framework. Consider a 'generalized strictconcordance situation on A' consisting of

(1) a strictly positive integer n,(2) n functions g\,g2, . . ., gn from A into R,(3) n functions t],t2, . . . , / „ from R^ into [0; 1]

such that V/ e {1,2, . . ., n}, /, is non-decreas-ing (resp. non-increasing) in its first (resp.second) argument and ti(x, x) = 0, Vx G R,

(4) n strictly positive real numbers k\,k2, . . ., *:„.

OUTRANKING RELATIONS 105

On the basis of such a 'situation' the'generalized strict concordance' (GSC) method(inspired by Pemy (1992) and Pemy and Roy(1992)) leads to a valued relation P on A letting,Va, be A,

Monjardet (1984) (after Fishbum (1973)), theonly difference being the addition here of arestriction on t{x,x). By construction, relationsin QSC are 'convex mixtures' of t-g relations.

1=1

1=1

We denote by QSCA (or simply QSC when there isno risk of confusion) the set of all valued relationson A that can be obtained with the GSC methodon the basis of a 'generalized strict concordancesituation on A'. Since it was supposed thattjix, x) — 0, all relations in QSC are obviouslyirrefiexive (i.e. T(a, a) = 0, Va G A).

The interest of the GSC method lies in its linkswith the PROMETHEE method (see Brans et al.(1984) or Brans and Vincke (1985)) and theconcordance part of ELECTRE III. First, it iseasy to show that the PROMETHEE method is aparticular case of the GSC method with'ifc(^). gi(.l>)) = ^iiSii^) ~ gii^)), the functions A,used in PROMETHEE being non-decreasing andsuch that A,(0) = 0. Second, consider an'ELECTRE III situation' and let'i(̂ i(̂ )> f i(6)) — 1 ~ Q(*> '')• Such functions /, areclearly admissible in the GSC method. Thus toeach concordance relation C obtained withELECTRE III corresponds a valued relation Pobtained with the GSC method such thatP(a, b) = l - C(b, a), Wa,beA.

In what follows, we study the structural proper-ties of relations in QSC. The following definitionswill prove useful for this purpose.

Definition 1Let r be a valued relation on a finite set A. Therelation T is said to be a t-g relation if there are

(1) a real-valued function g on A and(2) a function t from g[A] x ̂ A] into [0; 1] being

non-decreasing (resp. non-increasing) in itsfirst (resp. second) argiunent and such thatt{x,x) = O,Vxeg[A],

such that for a.lla,beA, 1\a, b) = t{g{a),g{b)).

The notion of t-g relation is very closely relatedto that of 'monotone scalability' used in

Definition 2Let r be a valued relation on a finite set A. We saythat T is upper diagonal stepped if there is a linearorder (i.e. a complete, antisymmetric and transitivecrisp relation) V on A such that for d\\ a,b & A,

T{c, b).(1) aVb^ 7X6, a) = 0 and(2) aVb => T{a, c) ̂ T{b, c) and T{c, a)

'ice A.

Apart from the restriction that aVb =^T\b, a) = 0, an upper diagonal stepped relation isidentical with a relation having a 'monotoneboard' as defined in Monjardet (1984).

Definition 3Let rbe a valued relation on a finite set A. We saythat T is linear if for all a,b,c,deA, [T{a, c) >T(b,c) or nc,a)<nc,b)]^[na,d)^nb,d)and T{d,a)^ T{d,b)].

The following lemma is a direct consequence ofTheorem 13 in Monjardet (1984). For the sake ofcompleteness we outline its proof.

Lemma 1Let r be a valued relation on a finite set A. Thefollowing statements are equivalent.

(i) r is a t-g relation.(ii) T is irreflexive and linear.(iii) T is upper diagonal stepped.

Proof(i) =^ (ii). Obvious.

(ii) zd^ (iii). Define the crisp relation Sj-, the stricttrace of T, letting, for all a,b e A,aSTb <^[T{a, c) > Jib, c) or T(c, a) < T{c, b) for somec e A]. The relation Sj- is clearly negativelytransitive. It is easy to see that it is also asymmetricwhen T is linear, so that ST e SWO^. Considernow any linear order V such that Sj^ttiV) (sucha linear order exists by Szpilrajn's lemma). Usingthe irreflexivity and linearity of T, it is easy toprove that T is upper diagonal stepped using sucha linear order.

106 D. BOUYSSOU

(iii)=>(i). Since 4̂ is a finite set and K is a linearorder, there is a real-valued function g on A suchthat for all a,b e A, gia) ^ g{b) <> aVb (see e.g.Roubens and Vincke (1985)). Given such afunction g, define t(g(a),g(b))= T(a,b). Thevalued relation T being upper diagonal stepped,it is easy to prove that / is a well-defined real-valued function on g[A] x g[A], has the requiredmonotonicity property and is such that'(sic), g{a)) = 0 for all a e ^ (for details seeTheorems 1 and A in Fishbum (1973)). •

Following the terminology used in Doignon etal. (1986), a valued relation satisfying the condi-tions of Lemma 1 will be called a strict linearsemiordered valued relation or, more briefly, avalued strict semiorder. More general forms oflinearity and closely related definitions and resultscan be found in Doignon et al. (1986). This paperalso gives many references and a thoroughhistorical background on the subject.

Definition 4Let /C be a set of crisp relations on a finite set A.The valued relation T on A is said to berepresentable in K. if there is a function ip from /Cinto [0; 1] such that

KeK

for which

, b),Keic

From Lemma 1 we know that relations in QSCare 'convex mixtures' of upper diagonal steppedrelations, i.e. valued strict semiorders. We proceedby showing that valued strict semiorders areparticular convex mixtures of elements of SSO,i.e. crisp strict semiorders.

Lemma 3Let r be a valued relation on a finite set A.U T\supper diagonal stepped, then it is representable inSSO

( ) /m(m-\)/2, not

positive values arem-i)/2 < 1. Let

any / = 1,2, . . .,

ProofSince A is finite and T is upper diagonal stepped,the latter takes at most m{m - \)/2 strictlypositives values. Thesenecessarily distinct, strictlysuch that 0 < 9i ^ 2̂ ^ •^* = 1 - 9m(m-l)/2- Form{m - l)/2 define the crisp relation T, on Aletting, for z.\\a,beA, aTjb ^ T(a, b) ^ 9,. SinceTis upper diagonal stepped, it is easy to see that Tjis upper diagonal stepped for i = l ,2, . . . ,m{m - l)/2. Thus we know from Lemma 2 thatTj 6 SSO A. Let T* be the (crisp) empty relation onA, i.e. the relation such that Not(aT*b) for alla,beA. It is clear that T*e SSO A.

Define a function ip from SSO^ into [0; 1]letting, for all We SSO A ,

r* if W=T*

if Pr=rm(m -

otherwiseO

It is easy to see that

and that with this function T is representable inSSO A. •

Simple examples show that the converse ofLemma 3 not true. Combining Lemmas 1-3 allowsfor a first characterization of the elements of QSC,showing that a valued relation is a convex mixtureof valued strict semiorders if and only if it is aconvex mixture of crisp strict semiorders.

Lemma 2Let r be a crisp relation on a finite set A. Therelation T is upper diagonal stepped if and only ifTSSO

ProofThis results immediately from the classical proper-ties of strict semiorders (see e.g. Fishbum (1970) orRoubens and Vincke (1985)). •

Proposition 2Let /• be a valued relation on a finite set A. ThenP € QSC4 if and only if it representable in the setSSOx of all strict semiorders on A.

Proof(a) [P is representable in SSO^ => P e QSC^]. LetfC = {Ti,T2, . . .,T,] be the set of all strict

OUTRANKING RELATIONS 107

semiorders J i n SSOA such that (p(,T)>0 (since Ais finite, so is SSO and hence K.). The set A beingfinite, for any T,, = 1,2, . . . , / , there is (see e.g.Roubens and Vincke (1985)) a function «, from Ainto R such that Va, b e A, [aTib] -O- [M,(a) >Ui(b) + 1]. Consider a 'situation' involving /criteriaand let A:, = i/j(r,), g, = w,-, f,(x,j) = 1 if x > y -I-1and 0 otherwise. It is obvious that applying theGSC method to this 'situation' leads to P.

(b) [P e QSCA ^ P is representable in SSOAITo prove that P is representable in SSO A, it issufficient to prove that the relations defined byPi(a,b) = ti(gi(a),gi(b)), 'ia,beA, are representa-ble in SSOA since /* is a convex mixture of therelations P,. From Lemma 1 we know that therelations P, are upper diagonal stepped and the useof Lemma 3 completes the proof. D

We already know that PROMETHEE is aparticular case of the GSC method. The aboveproof shows that if a valued relation is represen-table in SSO, it can be obtained as the result ofPROMETHEE, since the functions /,(x, j ) used inthe proof of Proposition 2 only depend on x-y.Thus a valued relation can be obtained with theGSC method if any only if it can be obtained withPROMETHEE.*

Consider now the 'generalized large concor-dance' (GLC) method, which is identical with GSCexcept that it uses functions t, such thatti{x,x) = 1. The concordance part of ELECTREIII is obviously a particular case of the GLCmethod. The immediate transposition ofProposition 2 and the preceding remark to thiscase shows the equivalence of the following threepropositions.

(1) R can be obtained with the GLC method.(2) R can be obtained as a concordance relation in

ELECTRE III.(3) R is representable in SO A, the set of all

semiorders on A.

3.2. 'Generalized concordance' and binary choiceprobabilitiesProposition 2 leads us to study the conditions for avalued relation to be representable in the set SSO

*This does not imply that any <-g relation can beobtained with PROMETHEE with a single criterion, i.e.with a function t depending only on the differenceg{a) - g{b). On this point see Marchant (1995).

of Strict semiorders. Observe that irreflexivity isclearly a necessary condition for the representationin any set of irreflexive crisp relations and thus inSSO. In the rest of this subsection we thereforeconcentrate on conditions between 'off-diagonal'elements.

Before tuming to the representation problem inSSO, let us recall that the characterization ofvalued relations representable in the set SCO ofstrict linear orders (notice that, apart fromreflexivity conditions, this problem is equivalentto that of the representation in CO) has receivedmuch attention in the literature, where it is knownas the 'binary choice probabilities' problem(among the numerous papers dealing with thisproblem, dating back to the early 1950s, let usmention the more recent contributions of Dridi(1980), Fishbum (1987, 1990), Fishbum andFahnagne (1989), Cohen and Falmagne (1990),Gilboa (1990), Gilboa and Monderer (1992), Suck(1992) and Koppen (1995); Fishbum (1992) offersan excellent survey of the available results). This isa difficult problem. Apart from the trivial irre-flexivity requirement, it amounts to characterizingthe set of all facets of a polyhedron in W"^-'"-^having \SCO\ — m\ vertices. We will not try here togive an exhaustive survey of the important andcomplex literature on the subject. We will justmention a few points that are important for ourpurposes.

Let P be a valued relation on A. Consider thefollowing conditions for all distinct a,b,ceA:

and

(1)

(2)

Since a strict linear order is weakly complete,asymmetric and transitive, the necessity of (1) and(2) for the representability of P in SCO follows.Let us notice that (2) is a necessary conditionfor the representation in any set of transitivecrisp relations (e.g. WO, SWO, SIO or SVO).Conditions (2) is more often presented under theform of the 'triangle inequality'

P(a, b) + P(b, c) ^ P(a, c) (20

which, together with (1), is equivalent to (2).Together with irreflexivity, conditions (1) and

(2) are known to be sufficient for the representa-tion in SCO when w ^ 5 (see Dridi (1980) and

108 D. BOUYSSOU

Fishbum (1987)). They are no longer sufficient assoon as m ^ 6, as shown by the following well-known example (see e.g. Dridi (1980) or Gilboa(1990); Dridi (1980) offers a whole family of suchexamples):

p

a

bcd

e

f

a

0

4

0

01

b

1

0

2

01

0

c

51

01

0

0

d

1

1

{0

e

1\1

50

\

f

\

11

\

0

Routine verification shows that P is irreflexive andsatisfies (1) and (2). It is not representable in SCOhowever. This is shown by observing that strictlinear orders T can be used to represent P only ifthey are such that

aTd, aTe, bTd, bTf, cTe and cTf (3)

Any strict linear order T satisfying (3) can includeat most one of the following relations:fTa, eTb and dTc. Thus it is impossible to haveat the same time P{f,a) = \ P(e, 6) = i andP(d) \( , ) \

It is known that no finite set of necessary andsufficient conditions can guarantee the representa-tion of a relation in SCO for all m (see Fishbum(1990) or Fishbum and Falmagne (1989)). Suchconditions exist for each value of m, liowever. Thediscovery of such conditions form = 6,7, . . . is anopen—and difficult—problem (Fishbum (1992)mentions a recent result, obtained by enumeration,of G. Reinelt giving such conditions for m — 6).

The preceding remarks lead us for the repre-sentation problem in SSO to consider thefollowing conditions for all distinct a,b,c e A:

and

P{a, b) + P(b, P(a, c)

(1')

(2)

Together with irreflexivity, their necessity for therepresentation in SSO is obvious (notice that inthe presence of (1'), (2') is no longer equivalent to(2)). Since they are also necessary for therepresentation in any set of asymmetric and

transitive relations (e.g. SIO or SPO), we havegood reasons to believe that they are not sufficientas soon as m is large enough (i.e. when w ^ 4,since in that case there are asymmetric andtransitive relations which are not strict semior-ders). The following example shows that this isindeed the case:

p

a

b

c

d

a

0

0

0X

b

0

01

0

c

5X

0

0

d

1

01

0

It is obvious that P is irreflexive and satisfies (T)and (2). Suppose that P is representable in SSO.The sum of the 'weights' i^T) of the strictsemiorders T for which bTc has to be \. Forthese strict semiorders it is impossible to have cTd,since transitivity would imply bTd. The sum of theweights of the strict semiorders for which cTb hasto be J. For these strict semiorders it is impossibleto have a aTc, since transitivity would imply aTb.If P is representable in SSO, it has to berepresentable using only two families of strictsemiorders that are respectively such that

(1) bTc, aTc and(2) cTb,cTd.

No strict semiorder of the first family can havedTa, because transitivity would imply dTc.However, dTa is also impossible in the secondfamily, since transitivity would lead to cTa. Wehave thus shown that P cannot be represented inSSO.

Let us notice that the preceding argument onlymakes use of the transitivity of the elements ofSSO. Thus this example shows that (T) and (2)together with irreflexivity are not sufficient for therepresentability of P in any set of asymmetric andtransitive crisp relations as soon as w ^ 4 (e.g.SXO or SPO). A tedious but simple proof showsthat (1') and (2) together with irrefiexivity aresufficient for the representation in SSO whenm = 3. We do not reproduce it here.

The preceding remarks lead us to believe thatthe problem of the characterization of the elementsin QSC is not easier than the 'binary choiceprobabilities' problem as soon as m is moderatelylarge. In view of the preceding example with

OUTRANKING RELATIONS 109

m = 4, we will not in this paper proceed anyfurther in that direction. We conclude this sectionwith two remarks.

(i) Not having a simple characterization ofQSC for all values of m is an incentive to try todirectly infer from Proposition 2 a number ofuseful properties of this set. In particular, if wechoose the functions f, and the weights it, so asto be rational, every relation P in QSC can beinterpreted as the summary of an 'election' inwhich each voter would indicate her preferencefor the elements of A through a strict semiorder.The values P(a,b) in this context represent thepercentage of voters having declared that 'a isstrictly preferred to b\ For instance, in such acontext the 'net flow' exploitation technique usedin PROMETHEE II is equivalent to rankingaltematives according to Borda's rule. Thissimple analogy can easily be exploited to transferin our context many useful results in socialchoice theory conceming the characterization ofchoice or ranking procedures (concemingBorda's mle see Young (1974), Hansson andSahlquist (1976) and Debord (1987, 1992)). Suchtranspositions are often self-evident, since mostresults of this type in social choice theory arevalid whenever the 'voters' have (strict)preferences included in any set of relationscontaining the set of strict linear orders SCO.Bouyssou (1993) offers examples of such trans-positions.

(ii) The absence of simple characterization of theelements of QSC does not facilitate the analysis ofexploitation techniques to be coupled with theGSC method when the above-mentioned analogycannot be used. The following very simpleproposition might be helpful to quickly eliminate'bad' exploitation techniques.

Proposition 3Let P be an irreflexive valued relation on A. Wehave

from Lenmia 2 that for all distincta,beA, P^b e SSO A. Since

a,beA

ProofGiven Proposition 2, we only have to show that Pis representable in SSO A. For all distinct a,b e Aconsider the crisp relation P^b on A such that forall c,deA, cP^bd ^[c = a and d = b]. It is clear

a, beA

and P is irreflexive, it is obvious that P isrepresentable in SSOA using the relations Pgb withthe weights P{a,b) and the strict semiordercorresponding to the empty relation on A withthe weight

1 - Da,beA

Since most exploitation techniques give a similarresult when applied to P or when applied to arelation Q such that Q = (xP,a. €(0; 1), this pro-position might prove useful in building 'counter-examples' for exploitation techniques to becoupled with the GSC method.

4. CONCLUSIONS

These few remarks conceming the constructionand exploitation of outranking relations leavemany important questions open. We mention herewhat we consider to be interesting directions forfuture research.

We already mentioned that Propositions 1 and 2were far from exhausting all the hnks betweenconstruction and exploitation techniques thatwould be interesting to investigate in order toobtain a good 'interface' between them. Theproblem of the interpretation of 'valuations' inmethods leading to valued outranking relationsand its corollary in terms of 'admissible opera-tions' on these valuations remains widely open. Aprofitable line of research would consist of tryingto entirely characterize and/or analyse a pairconsisting of a method of construction and amethod of exploitation. In the valued case it seemsthat only such an analysis would settle the already-mentioned questions of the 'interpretation of thevaluations' and the 'admissible operations' onthem.

Concerning the characterization of outrankingrelations that can be obtained with a givenconstruction method, many questions are stillpending. We already noticed some open interestingcombinatorial problems conceming Proposition 1.

110 D. BOUYSSOU

Moreover, a lot of work remains to be done inorder to go beyond Proposition 2. Fishbum (1987)mentions a long list of open problems concemingthe problem of representability. To this list we canadd the ones mentioned in Section 3.2: problemsof the representation in the set of strict semiorders,strict interval orders and strict partial orders whenw > 4 .

ACKNOWLEDGEMENTS

I wish to thank Patrice Perny, Marc Pirlot, BernardRoy, Alexis Tsoukias and Philippe Vincke for helpfuldiscussions on the subject of this paper. BernardMonjardet provided extremely helpful comments on anearlier draft of this text. The usual caveat applies. Part ofthis work was accomplished while the author wasvisiting the Universite Libre de Bruxelles, Service deMathematiques de la Gestion, thanks to a 'Research inBrussels' action grant from the Brussels-Capital Region.

REFERENCES

Bouyssou, D., 'Some remarks on the notion ofcompensation in MCDM'. Eur. J. Oper. Res., 26,150-160(1986).

Bouyssou, D., 'On some properties of outrankingrelations based on a concordance-discordance princi-ple', in Goicoechea, A., Duckstein, L. and Zionts, S.(eds). Multiple Criteria Decision Making, New York:Springer, 1992a, pp. 93-̂ 106.

Bouyssou, D., 'Ranking methods based on valuedpreference relations: a characterization of the net flowmethod', Eur. J. Oper. Res., 60, 61-68 (1992b).

Bouyssou, D., 'Les relations de surclassement ont-ellesdes proprietes remarquablesT, Working Paper CER-ESSEC 93032, ESSEC, Cergy, 1993.

Bouyssou, D., 'A note on the "min in favor" choiceprocedure for fuzzy preference relations', in Pardalos,P. M., Siskos, Y. and Zopounidis, C. (eds). Advancesin Multicriteria Analysis, Dordrecht: Kluwer, 1995,pp. 9-16.

Bouyssou, D. and Perny, P., 'Ranking methods forvalued preference relations: a characterization of amethod based on entering and leaving flows', Eur. J.Oper. Res., 61, 186-194 (1992).

Bouyssou, D. and Vansnick, J. C, 'Noncompensatoryand generalized noncompensatory preference struc-tures'. Theory Decis., 21, 251-266 (1986).

Brans, J. P., Mareschal, B. and Vincke, Ph., 'PRO-METHEE: a new family of outranking methods inmulticriteria analysis', in Brans, J. P. (ed.), OR'84,Amsterdam: North-Holland, 1984, pp. 477-490.

Brans, J. P. and Vincke, Ph., 'A preference ranking

organization method', Manag. Sci., 31, 647-656(1985).

Cohen, M. and Falmagne, J. C, 'Random utilityrepresentation of binary choice: a new class ofnecessary conditions', J. Math. Psychol, 34, 88-94(1990).

Deb, R., 'On constructing generalized voting para-doxes', .Rev. Econ. Stud., 43, 347-351 (1976).

Debord, B., 'Axiomatisation de procedures d'agregationde preferences'. These, Universite de Grenoble, 1987.

Debord, B., 'An axiomatic characterization of Borda'sfc-choice function'. Social Choice Welf, 9, 337-343(1992).

Doignon, J. P., Monjardet, B., Roubens, M. andVincke, Ph., 'Biorder families, valued relations andpreference modelling', J. Math. Psychol., 30, 435-480(1986).

Dridi, T., 'Sur les distributions binaires associees a desdistributions ordinales'. Math. Sci. Humaines, (69),15-31 (1980).

Fishbum, P. C, Utility Theory for Decision Making,New York: Wiley, 1970.

Fishbum, P. C, 'Binary choice probabilities: on thevariety of stochastic transitivity', J. Math Psychol., 10,327-352 (1973).

Fishbum, P. C, 'Decomposing weighted digraphs intosums of chains'. Discrete Appl. Math., 16, 223-238(1987).

Fishbum, P. C, 'Binary probabilities induced byrankings', SIAM J. Discrete Math., 3, 478-488 (1990).

Fishbum, P. C, 'Induced binary probabilities and thelinear ordering polytope: a status report'. Math.Social Sci., 23, 67-80 (1992).

Fishbum, P. C. and Falmagne, J. C, 'Binary choiceprobabilities and rankings', Econ. Lett., 31, 113-117(1989).

Gilboa, I., 'A necessary but insufficient condition for thestochastic binary choice problem', J. Math. Psychol.,34, 371-392 (1990).

Gilboa, I. and Monderer, D., 'A game-theoreticapproach to the binary stochastic choice problem',J. Math. Psychol, 36, 555-572 (1992).

Hansson, B. and Sahlquist, H., 'A proof technique forsocial choice with variable electorate', J. Econ.Theory, 13, 193-200 (1976).

Koppen, M., 'Random utility representation of binarychoice probabilities: critical graphs yielding criticalnecessary conditions', J. Math. Psychol., 39, 21-39(1995).

McGarvey, D. C, 'A theorem on the construction ofvoting paradoxes', Econometrica, 21, 608-610 (1953).

Marchant, Th., Doctoral Dissertation, Universite Librede Bruxelles, 1995.

Monjardet, B., 'Probabilistic consistency, homogeneousfamilies of relations and linear A-relations', inDegreef, E. and van Buggenhaut, J. (eds). Trends inMathematical Psychology, Amsterdam: North-Hol-land, 1984, pp. 271-281.

OUTRANKING RELATIONS 111

Perny, P., 'Modelisation, agregation et exploitation despreferences floues dans une problematique de range-ment: bases axiomatiques, procedures et logiciels',Doctoral Dissertation, Universite de Paris-Dauphine,1992 (unpublished).

Pemy, P. and Roy B., 'The use of fuzzy outrankingrelations in preference modelling'. Fuzzy Sets Syst.,49, 33-53 (1992).

Pirlot, M., 'A characterization of "min" as a procedurefor exploiting valued preference relations and relatedresults', J. Multi-Crit. Decis. Anal., 4, 37-56 (1995).

Roubens, M. and Vincke, Ph., Preference Modelling,Berlin: Springer, 1985.

Roy B., 'Classement et choix en presence de points devue multiples (la methode ELECTRE)', RIRO, 2eannee, 57-75 (1968).

Roy, B., 'ELECTRE III: un algorithme de classementfonde sur une representation floue des preferences enpresence de criteres multiples', Cah. CERO, 20, 3-24(1978).

Roy, B., Methodologie Multicritere d'Aide a la Decision,Paris: Economica, 1985.

Roy, B., 'The outranking approach and the foundationsof ELECTRE methods'. Theory Decis., 31, 49-73(1991).

Roy B. and Bertier, P., 'La methode ELECTRE II:une application au media-planning', in Ross, M.

(ed.), OR72, Amsterdam: North-Holland, 1973,pp. 291-302.

Roy, B. and Bouyssou, D., Aide Multicritere a laDecision: Methodes et Cas, Paris: Economica, 1993.

Roy, B. and Skalka, J. M., 'ELECTRE IS—aspectsmethodologiques et guide d'utilisation'. Doc. LAM-SAD E 30, Universite de Paris-Dauphine, 1984.

Scharlig, A., Decider sur Plusieurs Criteres—Panoramade I'Aide a la Decision Multicritere, Lausanne: PressesPolytechniques Romandes, 1985.

Stearns, R., 'The voting problem'. Am. Math. Mon., 6,761-763 (1959).

Suck, R., 'Geometric and combinatorial properties ofthe polytope of binary choice probabilities'. Math.Social Sci., 23, 81-102 (1992).

Vanderpooten, D., 'The construction of prescriptions inoutranking methods', in Bana e Costa, C. A. (ed.).Readings in Multiple Criteria Decision Aid, New York:Springer, 1990, pp. 184-215.

Vansnick, J. C, 'On the problem of weights in muhiplecriteria decision-making (the noncompensatory ap-proach)', Eur. J. Oper Res. 24, 288-294 (1986).

Vincke, Ph., Multicriteria Decision-Aid, Chichester:Wiley, 1992a.

Vincke, Ph., 'Exploitation of a crisp relation in aranking problem'. Theory Decis., 32, 221-240 (1992b).

Young, H. P., 'An axiomatization of Borda's rule', J.Econ. Theory, 9, 43-52 (1974).