Matrix representation of capacity-based multicriteria decision analysis

Matrix algebra is an efficient way to represent the relationship between different concepts in the multiple criteria decision analysis theory. The main purpose of this paper is to provide a compact and systematic representation of the capacities and nonlinear integrals based decision theory and models from the perspective of matrices or vectors. We first investigate the forms and properties of the matrix representation of some equivalent transformations of capacities and introduce some new types of equivalent representations. Then we represent respectively the specified conditions corresponding to several particular families of capacities as well as the calculation formulas of three types of nonlinear integrals in terms of vectors and matrices without the process of presorting the integrand values. Finally, we discuss the applications of the matrix representations in the capacity identification models and illustrate them through a multiple criteria decision making problem. The use of matrix-vector formalism facilitates formulation of the relevant optimization problems and their solutions in capacity based decision analysis can be easily programmed and implemented by means of standard software tools and libraries in the environments such as R and Matlab.