constraint’’ of this kind of partial preferential information. It suggests a number of structures as potential models being less demanding
than the classical one in which differences in utilities can be used to represent the comparison of differences in attractiveness. The models
are more attractive than d, and that a and d are incomparable, as well as b and c. From a theoretical point of view, such an
information can be encoded in what Roubens and Vincke (1985) call a preference (relational) structure.
that between z and w]. For example, imagine that he says there is a weak difference between a and b ((a,b) 2 C
1
), a
*
Tel.: +32 65373252; fax: +32 65373054.
E-mail address: mousset@umh.ac.be
www.elsevier.com/locate/ejor
Available online at www.sciencedirect.com
European Journal of Operational Research 192 (2009) 538–548Definition 1. A preference (relational) structure on a set X is a triple (P, I,J) of disjoint binary relations on X, where P
(preference) is asymmetric (xPy) y(notP)x), where I (indifference) is symmetric (xIy () yIx) and reflexive ("x xIx),
where J (incomparability) is symmetric, and where P [ I [ J is strongly complete ("x,yxP[ I [ Jy or yP [ I [ Jx).
For example, the judgements given by our DM correspond to a preference structure (P, I,J) where
P ={(a,b), (a,c), (b,d), (c,d)}, I ={(a,a), (b,b), (c,c), (d,d)} and J ={(a,d), (d,a), (b,c), (c,b)} (Fig. 1 offers a graph represen-
tation of the binary relation P).
Now suppose that the DM is able to characterize the difference in attractiveness he feels between x and y when he said
that x is more attractive than y. More precisely, let us introduce some categories C
1
,C
2
,...,C
n
(n > 1) of difference in
attractiveness, and suppose that the DM assigns each ordered pair (x,y)ofP to one (and only one) of the categories in
such a way that if (x,y) 2 C
i
and (z,w) 2 C
j
, then [i > j () the difference in attractiveness between x and y is greater thanare characterized in the more general context of families of non-complete preference structures, according to two different perspectives
(called ‘‘semantico-numerical’’ and ‘‘matrix’’). Both perspectives open the door to further practical applications connected with elicita-
tion of the preferences of a decision maker.
 2007 Elsevier B.V. All rights reserved.
Keywords: Multiple criteria analysis; Preference modelling; Partial relation; Family of relations; Preferential information
1. Introduction
Let us suppose that we are trying to elicit the preferences of a decision maker (DM) in a decision aiding context. For
example, imagine that, given four alternatives a, b, c and d, the DM says that a is more attractive than b and c, that b and cDecision Support
Families of relations modelling preferences under
incomplete information
Ce´line Mousset
*
Warocque Research Center, University of Mons-Hainaut, Place Warocque´ 17, 7000 Mons, Belgium
Received 18 August 2005; accepted 14 September 2007
Available online 7 November 2007
Abstract
Let us consider a preferential information of type preference–indifference–incomparability (P, I,J), with additional information about
differences in attractiveness between pairs of alternatives. The present paper offers a theoretical framework for the study of the ‘‘level of0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2007.09.030

Fig. 1. Graph representation of P.
C. Mousset / European Journal of Operational Research 192 (2009) 538–548 539moderate difference between b and d as well as between c and d ((b,d) 2 C
2
,(c,d) 2 C
2
), and a strong difference between a
and c ((a,c) 2 C
3
).
Such an information can be encoded in three nested preference structures (P
1
, I
1
,J
1
), (P
2
, I
2
,J
2
) and (P
3
, I
3
,J
3
) where
• P
1
= P, I
1
= I and J
1
= J,
• P
2
= C
2
[ C
3
, I
2
¼ I [ C
1
[ C

1
and J
2
= J,
• P
3
= C
3
, I
3
¼ I
2
[ C
2
[ C

2
and J
3
= J,
(with xC

j
y () yC
j
x). Let us note that xP
1
y (resp. xP
2
y, resp. xP
3
y) if and only if the difference in attractiveness between x
and y is at least weak (resp. at least moderate, resp. at least strong). Fig. 2 offers a graph representation of the relations P
i
–
arrows– and I
i
–double lines–.
More generally, we introduce the following definition.
Definition 2. A family P of preference structures on a set X is a collection of n (n is a strictly positive integer) preference
structures on X : P¼½ðP
1
; I
1
; J
1
Þ; ðP
2
; I
2
; J
2
Þ; ...; ðP
n
; I
n
; J
n
Þ.
This paper investigates families of non-complete (J
i
5 ;) preference structures. Such structures are very interesting from
a practical point of view because, in the field of preference elicitation, complete preferential information is rarely available
immediately. Indeed, preference elicitation is a dynamic process. The preference structures of the DM are built step by step,
adding more and more information. So, before the very end of the questioning procedure, we always handle partial infor-
mation, more precisely non-complete preference structures (as shown in the previous example). For a profound compre-
hension of what is happening at every moment of the process, families of non-complete preference structures have to be
better known. That was precisely the starting point of the present paper.
In particular, given a family of partial preference structures P (as defined by Definition 2) which encodes the informa-
tion obtained from the DM at one point in the learning process, we may be interested in knowing which families of com-
plete preference structures are compatible with the current information encoded in P. And more precisely what are the
‘‘more constrained’’ of these families (we say that a type of family is more constrained than a second one if the former
is a particular case of the latter) that one can expect to obtain when the information is completed. To answer that question,
we need a theoretical characterization of the various families of non-complete structures.
From a practical point of view, this type of question has already been treated in a very particular case. Indeed, the M-
MACBETH software (see Bana e Costa et al., 2003) offers a computational tool to find a real-valued function f such that
(1) xPy ) f(x)>f(y).
(2) xIy ) f(x)=f(y).
(3) [(x,y) 2 C
i
and (z,w) 2 C
j
and i > j]) f(x)  f(y)>f(z)  f(w).
Moreover, being a progressive learning process, M-MACBETH points out the possible incompatibilities between thejudgements of the DM and the existence of a mapping f satisfying the three previous conditions (see Bana e Costa
Fig. 2. Graph representation of P
1
and I
1
, P
2
and I
2
, P
3
and I
3
.