Additive and decomposable conjoint measurement with ordered categories

Conjoint measurement studies binary relations defined on product sets and investigates the existence and uniqueness of numerical representations of such relations. It has proved to be quite a powerful tool to analyze and compare MCDM techniques designed to build a preference relation between multiattributed alternatives and has been an inspiring guide to many assessment protocols. These MCDM techniques lead to a relative evaluation model of the alternatives through a preference relation. Such models are not always appropriate to build meaningful recommendations. This has recently lead to the development of MCDM techniques aiming at building evaluation models having a more absolute character. In such techniques, the output of the analysis is, most often, a partition of the set of alternatives into several ordered categories defined with respect to outside norms, e.g., separating "Attractive" and "Unattractive" alternatives. In spite of their interest, the theoretical foundations of such MCDM techniques have not been much investigated. The purpose of this paper is to contribute to this analysis. More precisely, we show how to adapt classic conjoint measurement results to make them applicable for the study of such MCDM techniques. We concentrate on additive models. Our results may be seen as an attempt to provide an axiomatic basis to the well-known UTADIS technique that sorts alternatives using an additive value function model.