Monotonic influence diagrams: Foundations and application to optimal design

Monotonic influence diagrams (MID) are proposed for qualitative and mathematical functional reasoning about the relationships between the variables of a design problem. The theory of MID’s is based on a graph-theoretic representation of an optimization problem which can be topologically transformed as a means of solving the problem and exploring variable-objective-constraint relationships. Formally, a monotonic influence diagram is a directed graph consisting of nodes and arcs. The nodes represent design variables and the arcs reveal their relationships. A deterministic qualitative relation between two variables is given by the sign of the partial derivative of the function defining one of the variables with respect to the other variable. Topological transformations such as arc reversal and node removal allow qualitative relations to be determined between constrained design variables and the objective function to be minimized or maximized. In this sense, MID’s provide a reasoning mechanism about constraint activity, so candidates for active constraints or flaws in the problem formulation can be detected.