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 different 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.
Highlights
The development of solution techniques for multiobjective optimisation problems (MOPs) has witnessed many improvements in recent years, partly prompted by real-world applications, and due to the increasing computational power available; affordable computers are much more capable of performing the complex computations which are required to solve such problems
Dataset A does not present relevant global or local pairwise relationships according to the global pairwise analysis and the multiobjective scatter plot analysis
The information from the trade-off region maps can be used to interact with the decision-maker to identify which regions are of more interest and use single-objective optimisation algorithms to find solutions in that region
Summary
The development of solution techniques for multiobjective optimisation problems (MOPs) has witnessed many improvements in recent years, partly prompted by real-world applications, and due to the increasing computational power available; affordable computers are much more capable of performing the complex computations which are required to solve such problems. The analysis technique is performed in four steps: first the global pairwise relationships are analysed using the Kendall correlation method; the ranges of the values found on the given Pareto front are estimated and assessed; these ranges are used to plot a map using Gray code, similar to Karnaugh maps, that have the ability to highlight the trade-offs between multiple objectives; and local relationships are identified using scatter-plots It was suggested in (Pinheiro et al, 2015) that the technique could be used to compare the multiobjective natures of different problems and that the information obtained could be used in the design of algorithms.
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