Quantifying the shape of a pareto front in support of many-objective trade space exploration

Complex design optimization problems typically include many conflicting objectives, and the resulting trade space is comprised of numerous design solutions. To efficiently explore a many-objective trade space, form preferences, and select a final design, one must identify and negotiate tradeoffs between multiple, often conflicting, objectives. Identifying conflicting objective pairs allows decision-makers to concentrate on these objectives when selecting preferred designs from the non-dominated solution set, i.e., the Pareto front. Techniques exist to identify and visualize tradeoffs between these conflicting objectives to support trade space exploration; however, these techniques donotquantify, or differentiate, the shape of the Pareto front, which might be useful information for a decisionmaker. More specifically, designers could gain insight from the degree of diminishing returns among solutions on the Pareto front, which can be used to understand the extent of the tradeoffs in the problem. Therefore, the shape of the Pareto front could be used to prioritize exploration of conflicting objective pairs. In this paper, we introduce a novel index that quantifies the shape of the Pareto front to provide information about the degree of diminishing returns. The aim of the index is to help designers gain insight into the underlying tradeoffs in a many-objective optimization problem and support trade space exploration by prioritizing the negotiation of conflicting objectives. The proposed Pareto Shape Index is based on analytical geometry and derived from the coordinates of the Pareto solutions in the n objective trade space. The utility of the Pareto Shape Index in differentiating diminishing returns between conflicting objectives is demonstrated by application to an eight-objective benchmark optimization problem.