Notes on bipolar outranking

We consider a finite set of alternatives A = {a1, a2, . . . , am} evaluated on a family of n criteria F = {g1, g2, . . . , gn}. Let N = {1, 2, . . . , n}. To each criterion gi ∈ F is assigned a positive weight wi. It is supposed wlog that the weights are normalized so that ∑n i=1wi = 1. Let Xi = {gi(a) : a ∈ A} be the set of evaluations of the alternatives on the ith criterion. It is supposed that a semi-order Pi (i.e., an asymmetric, Ferrers and semitransitive binary relation) is defined on Xi. We denote by Ii the symmetric complement of Pi. Let Si = Pi ∪ Ii. The relations Pi (resp. Ii) models strict preference (resp. indifference) on the ith criterion. In order to model discordance, we introduce a second semiorder Vi on Xi. It is supposed that Vi ⊆ Pi and that there is a weak order compatible with both Pi and Vi (i.e., there is a weak order %i on Xi such that [α %i β and β Pi γ implies α Pi γ] and [α %i β and δ Pi α implies δ Pi β], with similar relations holding with Vi instead of Pi).