A conjoint measurement approach to the discrete Sugeno integral: a note on a result of Greco, Matarazzo and Slowinski

In a recent paper (European Journal of Operational Research, 158, 271-292, 2004), S. Greco, B. Matarazzo and R. Slowinski have stated without proof a result characterizing binary relations on product sets that can be represented using a discrete Sugeno integral. To our knowledge, this is the first result about a fuzzy integral that applies to non-necessarily homogeneous product sets and only uses a binary relation on this set as a primitive. This is of direct interest to MCDM. The main purpose of this note is to propose a proof of this important result. Thereby, we study the connections between the discrete Sugeno integral and a non-numerical model called the noncompensatory model. We also show that the main condition used in the result of S. Greco, B. Matarazzo and R. Slowinski can be factorized in such a way that the discrete Sugeno integral model can be viewed as a particular case of a general decomposable representation.