Nonmodularity index for capacity identifying with multiple criteria preference information

Nonmodularity is a more general concept than nonadditivity which provides a more inclusive reflection of the relationship between subsets of criteria in multicriteria decision making. In this paper, we construct the nonmodularity index to describe the kind and intensity of interaction within a subset of criteria associated with the nonmodularity, and introduce the total nonmodularity and nonadditivity (amount) index to reflect the interactions within a capacity as a whole. We prove some fundamental properties of the nonmodularity index, develop its expression in Möbius representation and prove that both nonmodularity and nonadditivity indices can serve as alternative representations of a capacity in terms of matrix form. We also discuss in detail the explicit reciprocal transform representations of the nonmodularity index, as well as the nonadditivity index with the capacity and the Möbius representation. In view of these explicit interaction indices and their intuitive merits, we develop specialized capacity identification methods in terms of nonmodularity and nonadditivity indices and formulate the corresponding linear programming problems to represent and aggregate the decision makers’ preference information.