Marginal contribution representation of capacity-based multicriteria decision making

Integrals defined with respect to fuzzy measures (capacities) are powerful tools in multicriteria decision making. Monotonicity is a basic property of capacity, which means that the marginal contribution of any single criterion to any subset of criteria is always nonnegative. In this paper, we present the capacity-based decision making theory in terms of marginal contributions, which provides an alternative perspective to this widely used decision making strategy. We construct the marginal contribution representations of the equivalent transformations of capacities, some particular capacities, three types of nonlinear integrals, and discuss the capacity identification methods. We also introduce some new concepts and representations, such as nonadditivity and nonmodularity indices, 0 to 1 variables-based linear constraints of k-maxitive capacity, a special representation of the Choquet integral and pan integral. We discuss constraints on marginal contributions which ensure supermodularity of capacities. Finally, an illustrative example is given to show the use of marginal contribution presentation in capacity-based decision making methods.