Visualisation of Pareto Front Approximation: A Short Survey and Empirical Comparisons

Abstract

Visualisation is an effective way to facilitate the analysis and understanding of multivariate data. In the context of multi-objective optimisation, comparing to quantitative performance metrics, visualisation is, in principle, able to provide a decision maker better insights about Pareto front approximation sets (e.g. the distribution of solutions, the geometric characteristics of Pareto front approximation) thus to facilitate the decision-making (e.g. the exploration of trade-off relationship, the knee region or region of interest). In this paper, we overview some currently prevalent visualisation techniques according to the way how data is represented. To have a better understanding of the pros and cons of different visualisation techniques, we empirically compare six representative visualisation techniques for the exploratory analysis of different Pareto front approximation sets obtained by four state-of-the-art evolutionary multi-objective optimisation algorithms on the classic DTLZ benchmark test problems. From the empirical results, we find that visual comparisons also follow the No-Free-Lunch theorem where no single visualisation technique is able to provide a comprehensive understanding of the characteristics of a Pareto front approximation set. In other words, a specific type of visualisation technique is only good at exploring a particular aspect of the data.

Decomposition Multi-Objective Evolutionary Optimization: From State-of-the-Art to Future Opportunities

“…To be self-contained, we started with a gentle tutorial about the basic working mechanism of MOEA/D. Thereafter, our survey is conducted according to the four core design components, i.e., weight vector settings, subproblem formulations, selection mechanisms and reproduction operators, along with some selected advanced topics including constraint handling, expensive optimization and preference incorporation [37,76,229,231,[306][307][308][309][310][311][312][313][314][315][316][317][318][319][320][321][322][323]. At the end, we outlined some emerging directions for future developments of MOEA/D that have not yet been broadly studied so far.…”

Optimal emergency shutdown valve configuration for pressurised pipelines

“…This technique is the most frequently used in the case of two or three objectives. Although a scatter plot matrix is able to visualise more than three objectives by creating an array of the plot projecting all the pairwise combinations of coordinates, the complexity of the information provided to the decision maker increases significantly with the number of objectives (Gao et al, 2019).…”

“…It should be noted that each polyline denotes for a solution for all objectives which the corresponding value can be obtained from the intersection at each parallel axis. Parallel coordinates plot is a very advantageous technique to present dependencies between objectives; however, it becomes difficult to interpret and analyse large sets of data due to the cluttering effect with crossing lines (Gao et al, 2019).…”

“…Another popular technique for high-dimensional data is heatmap which projects the objective value into a colour map. Similar to parallel coordinates plot, it has advantages on presenting dependencies between objectives but has the same cluttering problem (Gao et al, 2019).…”

“…The authors claimed that this technique is capable to investigate many typical features of Pareto Front including closeness to ideal point, discontinuities and ranges of attainable values. However, it becomes difficult to analyse as the increasing number of objectives can result in significant number of subplots(Gao et al, 2019).…”