A conjoint measurement approach to the discrete Sugeno integral

In a recent paper (European Journal of Operational Research, 158, 271-
292, 2004), S. Greco, B. Matarazzo and R. Slowinski have presented conditions
characterizing binary relations on product sets that can be represented using a dis-
crete Sugeno integral. To our knowledge, this is the ?rst result about a 'fuzzy in-
tegral' 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 multiple criteria
decision making.
In this paper, we study a non-numerical model called the noncompensatory model.
We show that, under some conditions, the noncompensatory model and the discrete
Sugeno integral model are equivalent. Our analysis of the noncompensatory model
hence helps us better understand the discrete Sugeno integral model. We 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. We also study the
uniqueness of the representation obtained and the commensurateness implied by
the discrete Sugeno integral model.