A technique based on trade-off maps to visualise and analyse relationships between objectives in optimisation problems

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. KEYWORDS fitness landscape visualisation, multiobjective combinatorial problems, multiobjective fitness landscape analysis, trade-off region maps 1 throughout the majority of the search space and is often easy to identify. A local relationship exists in a restricted region of the search space and can be difficult to spot. Understanding local relationships is a useful tool to design tailored algorithms (Garrett & Dasgupta, 2008).

Multi-cost Bounded Reachability in MDP

“…However, Pareto set visualisations alone may not provide sufficient information, about, e.g., which objectives are aligned or conflicting (see e.g. [39] for a discussion in the non-probabilistic case). Cost bounds furthermore add an extra dimension for each cost structure.…”

“…Evaluation using a prototypical implementation in Storm [20] shows promising results. In addition, we equip our algorithm with means to visualise (inspired by the recent techniques in [39]) the trade-offs between various objectives that go beyond Pareto curves; we believe that this is key to obtain better insights into multi-objective decision making. An example is given in Fig.…”

Multi-cost Bounded Tradeoff Analysis in MDP

“…In addition, we equip our algorithm with means to visualise (inspired by the recent techniques in [43]) the tradeoffs between various objectives that go beyond Pareto curves. We believe that this is key to obtain better insights into multi-objective decision making.…”

“…which objectives are aligned or conflicting (see e.g. [43] for a discussion in the non-probabilistic case). Cost bounds furthermore add an extra dimension for each cost structure.…”