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

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.