Discovering design principles from dominated solutions

Important progress has been made by many researchers in extracting fundamental design principles from patterns in design parameters along the non-dominated front generated by evolutionary algorithms in biobjective optimization problems. However, to our knowledge, no attention has been given to discovering design principles from the wealth of additional information available from patterns in dominated solutions. To explore that space, we use heatmaps of dominated solutions to visualize how relevant variables self-organize with respect to the objectives throughout the feasible region. We overlay ceteris paribus lines on these heatmaps to show how the objective values change when a given design variable is varied while all others are held constant. We use three biobjective optimization problems to demonstrate various ways in which these visualization techniques can provide useful information beyond that which can be determined from the non-dominated front. Specifically, we investigate a simple two-member truss design problem, a simple welded beam design problem, and a real-world watershed management design problem to illustrate (a) how principles derived from the non-dominated front alone can be misleading, (b) how new principles can be derived from the dominated solutions, and (c) how non-dominated solutions can often be fragile with respect to assumptions about uncertain external forcing conditions, whereas solutions a short distance inside the front are often much more robust.