Using multiple reference levels in Multi-Criteria Decision aid: The Generalized-Additive Independence model and the Choquet integral approaches

In many Multi-Criteria Decision problems, one can construct with the decision maker several reference levels on the attributes such that some decision strategies are conditional on the comparison with these reference levels. The classical models (such as the Choquet integral) cannot represent these preferences. We are then interested in two models. The first one is the Choquet with respect to a p-ary capacity combined with utility functions, where the p-ary capacity is obtained from the reference levels. The second one is a specialization of the Generalized-Additive Independence (GAI) model, which is discretized to fit with the presence of reference levels. These two models share common properties (monotonicity, continuity, properly weighted,…), but differ on the interpolation means (Lovász extension for the Choquet integral, and multi-linear extension for the GAI model). A drawback of the use of the Choquet integral with respect to a p-ary capacity is that it cannot satisfy decision strategies in each domain bounded by two successive reference levels that are completely independent of one another. We show that this is not the case with the GAI model.