Monotonic influence diagrams: Application to optimal and robust design

In this dissertation monotonic influence diagrams (MID's) are proposed for qualitative and mathematical functional reasoning about the relationships between the variables of a design problem. Monotonic influence diagrams are a synthesis of influence diagrams and monotonicity analysis. 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-goal-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. Nodes in a MID can represent either deterministic or uncertain quantities. An arc in a MID is associated with the qualitative relation between the corresponding variables. 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. A probabilistic qualitative relation is defined in terms of a constraint on the joint probability distribution of the variables.; Topological transformations such as arc reversal and node removal allow us to determine qualitative relations between constrained design variables and the objective (utility) 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.; We present algorithms for the removal of nodes associated with unconstrained variables and for the analysis of the problem, by means of the first rule of monotonicity analysis, for different combinations of inactive constraints.; The qualitative nature of the information represented by a monotonic influence diagram might result in loss of qualitative information for some sequences of topological transformations. We derive sufficient conditions for a transformation to avoid loss of information that could result in an undetected active constraint.; An energy storage flywheel and a multiobjective spring design problem are solved in a parametric fashion using monotonic influence diagrams. The optimality conditions of monotonicity analysis are also extended to deal with non-monotonic functions such as the expected value of the absolute deviation of a monotonic characteristic function with respect to its target value. These new conditions are used for the formal solution of Taguchi N-type parameter design problems.