A Technique Based on Trade-off Maps to Visualise and Analyse Relationships Between Objectives in Optimisation Problems

TL;DR

This paper proposes a four-step technique to analyse relationships between objectives in many-objective optimisation problems, combining Kendall correlation for global relationships, objective-range analysis, Gray-code region maps for trade-off regions, and multiobjective scatter plots for local relationships. It demonstrates the approach on three combinatorial problems (MOMKP, MONSP, MOVRPTW), revealing how global, local, and composite relationships vary across instances and informing algorithm design and benchmarking. The region maps and threshold analyses highlight where trade-offs concentrate and where single-objective approaches may suffice or fail, while scatter plots expose local conflicts and harmonies. The work offers a practical framework to understand and compare fitness landscapes across problems, with potential extensions to heatmaps and adaptive integration into optimization workflows.

Abstract

Understanding the relationships between objectives in a multiobjective optimisation problem is important for developing tailored and efficient solving techniques. In particular, when tackling combinatorial optimisation problems with many objectives, that arise in real-world logistic scenarios, better support for the decision maker can be achieved through better understanding of the often complex fitness landscape. This paper makes a contribution in this direction by presenting a technique that allows a visualisation and analysis of the local and global relationships between objectives in optimisation problems with many objectives. The proposed technique uses four steps: First, the global pairwise relationships are analysed using the Kendall correlation method; then, the ranges of the values found on the given Pareto front are estimated and assessed; next, these ranges are used to plot a map using Gray code, similar to Karnaugh maps, that has the ability to highlight the trade-offs between multiple objectives; and finally, local relationships are identified using scatter plots. Experiments are presented for three combinatorial optimisation problems: multiobjective multidimensional knapsack problem, multiobjective nurse scheduling problem, and multiobjective vehicle routing problem with time windows . Results show that the proposed technique helps in the gaining of insights into the problem difficulty arising from the relationships between objectives.

Figures (23)

• Figure 1: Example of complex relationship between two objectives $Z_1$ and $Z_2$ in a 3-objectives optimisation problem, $Z_3$ is omitted for simplicity.
• Figure 2: Three-way conflicting objectives
• Figure 3: Region map schematics, illustrating the use for 3, 4 or 5 objectives.
• Figure 4: Importance of choosing the right pivot on a three-objective approximation set: in a) $Z_1$--$Z_3$ the conflicting relationship is easily spotted, while in b) the same relationship can not be visualised.
• Figure 5: Pairwise correlations for each set of instances.