Visualisation with treemaps and sunbursts in many-objective optimisation

Genetic Programming and Evolvable Machines (2018) 19:421–452 https://doi.org/10.1007/s10710-018-9329-0 Visualisation with treemaps and sunbursts in many‑objective optimisation David J. Walker1 Received: 5 February 2018 / Revised: 6 July 2018 / Published online: 7 August 2018 © The Author(s) 2018 Abstract Visualisation is an important aspect of evolutionary computation, enabling practitioners to explore the operation of their algorithms in an intuitive way and providing a better means for displaying their results to problem owners. The presentation of the complex data arising in many-objective evolutionary algorithms remains a challenge, and this work examines the use of treemaps and sunbursts for visualising such data. We present a novel algorithm for arranging a treemap so that it explicitly displays the dominance relations that characterise many-objective populations, as well as considering approaches for creating trees with which to represent multiand many-objective solutions. We show that treemaps and sunbursts can be used to display important aspects of evolutionary computation, such as the diversity and convergence of a search population, and demonstrate the approaches on a range of test problems and a real-world problem from the literature. Keywords Many-objective optimisation · Visualisation · Evolutionary computation 1 Introduction Visualisation remains an important topic within evolutionary computation and, as many-objective evolutionary algorithms (MaOEAs) continue to mature, the visualisation of solutions to many-objective problems is an important aspect of this [31]. A many-objective optimisation problem comprises four or more competing objectives, such that a solution 𝐱 is quantified by an objective vector 𝐲 with four or more elements: 𝐲 = (f1 (𝐱), … , fM (𝐱)), (1) where M ≥ 4. At various stages during the process of solving a many-objective problem with a MaOEA it is desirable to visualise objective vectors. Visualising * David J. Walker 1 University of Exeter, Exeter, UK 13 Vol.:(0123456789) 422 Genetic Programming and Evolvable Machines (2018) 19:421–452 the objective vectors to such problems is a non-trivial problem because humans are not able to comprehend more than three spatial dimensions. The main motivation behind this work is to facilitate the decision maker’s selection of a final operating solution. In this work the decision maker is considered to be the problem owner— the person who wishes to solve the optimisation problem. They are likely from an industrial or scientific background, and do not necessarily have a background in evolutionary computation. Thus, visualisation is a vital part of the optimisation process as it enables the non-expert user to better understand the results they are presented with. In the case of any MaOEA the task of a decision maker is an important one as the result of executing the algorithm is a set of solutions, which are usually incomparable according to measures such as Pareto dominance. Presented with the solution set, the decision maker must select a single solution that can be implemented to solve the problem. The visualisation methods proposed herein are intended to aid the decision maker in this task. In the past decade, much work has been focussed on the development of methods that can visualise many-objectives. The information that can be extracted from such methods varies depending on the type of visualisation. For example, in some methods the number of dimensions to be visualised is reduced so that a conventional visualisation can be employed (e.g., [7, 46]). Other methods avoid this loss of information by presenting the objective vectors in terms of the full set of objectives (e.g., [11, 12, 15, 20, 25, 36, 36, 46]) or visualising relationships between solutions (for example, conveying which solutions are superior to others) and are constructed in terms of the full objective set (e.g., [48]). Hierarchies are a convenient structure within evolutionary computation. Examples include the use of trees to represent populations of solutions, such as the dominated trees and non-dominated tree structures proposed by [19] and the nondominated tree structure proposed in [33]. The example used later in this work builds on the notion of a quad tree [41]. Solutions are often represented as trees in genetic programming, and a tree-based solution representation was used within NSGA-II in [42]. A study [10, 11] used a tree structure to represent the objectives comprising a many-objective problem in order to reduce the dimensionality of the problem. In this paper we explore the use of treemaps [26] and sunbursts [40]. A pilot study [47] illustrated the potential of using treemaps to visualise manyobjective populations, however it identified two problems. First, the treemaps presented therein were based on trees constructed in terms of dominance. It is well known [16] that the dominance relation is poorly suited to comparing manyobjective solutions since, assuming an uniformly distributed objective space, the solutions are likely to be mutually non-dominating and thus incomparable. The result of constructing a treemap or sunburst with such a tree is that there is little structure to present in the visualisation and the user does not obtain any significant insight. The second problem involved the layout algorithm selected for the visualisations. The treemaps presented in [47] used a standard square layout, and the dominance relations that were present in the tree (for multi-objective examples) were difficult to observe. In this work, the initial study is extended to make the following novel contributions: 13 Genetic Programming and Evolvable Machines (2018) 19:421–452 423 • A new treemap layout algorithm is presented, specifically designed to visu- alise many-objective populations with dominated solutions, and compared to an existing approach proposed by [26]. • A quad tree from the literature [41] is used as the basis for a many-objective visualisation. • The well-known sunburst visualisation [40] is used to visualise many-objective populations; demonstrations show that they can be used to convey information about the optimisation characteristics (e.g., convergence and diversity) as well as the solution quality of a mutually non-dominating set. Therefore, the principal contribution of this paper lies in the application of treebased visualisations to many-objective populations using datastructures already used within MaOEAs. This work is the first to have considered the visualisation of many-objective populations using treemaps and sunbursts. Throughout the paper we present results for a selection of optimisation problems, including well known benchmark problems from the DTLZ problem suite [14], benchmark approximation sets proposed in [43] and solutions to a real-world test problem [24]. The remainder of this paper is organised as follows: Sect. 2 presents some relevant background material, describing (...truncated)