Preference learning within non-additive aggregation MCDA methods
Résumé
When using Multiple Criteria Decision Analysis (MCDA) methods in real worldapplications, determining the method parameters remains a delicate issue. Indeed,in MCDA methods alternatives are assessedw.r.ta set of criteria, alternative(s)bestsatisfyingthe decision maker’smost importantcriteria is/are selected. To assess theDM’ssatisfactionand criteriaimportance, the DM has to provide some informationabout his/herpreferences. This information is necessary for determining the param-eters defining the MCDA methods. Methods based on a sophisticated models bettermatch the decision maker’s (DM) preferences but require numerous parameters tobe determined. In this case, asking directly to provide parameter values requires aconsiderable cognitive effort from the DM. The challenge is to limit the amount ofinformation requested from the decision maker. For this reason indirect preferenceprocedures are introduced. We focus in this work on multi-attribute utility theory(MAVT) based methods.More formally, in the framework of MAVT, alternatives are assessed using a setofnattributes indexed by the setN={1, ..., n}. Fori∈N, theithattribute takesits values in a spaceXi. The aim of MAVT is to represent the preferences of the DMoverX=X1×...×Xn, i.e., (X,%) by the mean of a value functionV:X→R+suchthat:∀x, y∈X,x%y⇔V(x)≥V(y).(1)To simplify,Vis built using the partial value functionsvion each attribute byassuming theweak preference independenceand theweak difference independenceofthe attributes. In this case, the assessment ofVis simplified by assessing eachvifora single attribute independently (the other attributes are fixed at arbitrarily selectedvalues). As stated by (Krantz et al, 1971) if attributes areweak preference independent,then:∀x∈X, V(x) =F(v1(x1), ..., vn(xn))(2)whereF:Rn→Rincreases in all its arguments andvi:Xi→Ris a value functionthat represents the DM’s preferences for attributei,∀i∈N.In the case of aggregation methods based on additive models ofFsuch as MAC-BETH, a questioning procedure and a disaggregation procedure are implemented todetermineFandvi, i∈N. For non-additive models most of the existing proposalsassume a Choquet integral based on a 2-additive capacity even if it does not match thedecision makers’ preferences. Based on recent advances works concerning the identifi-cation of k-additive capacity resulting from the questioning procedure (Labreuche andGrabisch, 2003), this paper proposes a discussion about the disaggregation procedurewhen a 2-additive model do not match the DM preferences. Considering a sophisti-cated model have to be confronted to the problem of the understanding of parameters.Indeed, more than pair attribute interactions may complicate the understanding of theobtained parameters (Roy, 2009).