Visualizing Exact and Approximated 3D Empirical Attainment Functions

Most real-world engineering optimization problems are inherently multiobjective, for example, searching for trade-off solutions of high quality and low cost. As no single optimal solution exists for such problems, multiobjective optimizers provide sets of optimal (or near-optimal) trade-off solutions to choose from. The empirical attainment function (EAF) is a powerful tool that can be used to analyze and compare the performance of these optimizers. While the visualization of EAFs is rather straightforward in two objectives, the three-objective case presents a great challenge as we need to visualize a large number of 3D cuboids. This paper addresses the visualization of exact as well as approximated 3D EAF values and differences in these values provided by two competing multiobjective optimizers. We show that the exact EAFs can be visualized using slicing and maximum intensity projection (MIP), while the approximated EAFs can be visualized using slicing, MIP, and direct volume rendering. In addition, the paper demonstrates the use of the proposed visualization techniques on a steel casting optimization problem.