Simon Huband

A review of multiobjective test problems and a scalable test problem toolkit

When attempting to better understand the strengths and weaknesses of an algorithm, it is important to have a strong understanding of the problem at hand. This is true for the field of multiobjective evolutionary algorithms (EAs) as it is for any other field. Many of the multiobjective test problems employed in the EA literature have not been rigorously analyzed, which makes it difficult to draw accurate conclusions about the strengths and weaknesses of the algorithms tested on them. In this paper, we systematically review and analyze many problems from the EA literature, each belonging to the important class of real-valued, unconstrained, multiobjective test problems. To support this, we first introduce a set of test problem criteria, which are in turn supported by a set of definitions. Our analysis of test problems highlights a number of areas requiring attention. Not only are many test problems poorly constructed but also the important class of nonseparable problems, particularly nonseparable multimodal problems, is poorly represented. Motivated by these findings, we present a flexible toolkit for constructing well-designed test problems. We also present empirical results demonstrating how the toolkit can be used to test an optimizer in ways that existing test suites do not

A faster algorithm for calculating hypervolume

We present an algorithm for calculating hypervolume exactly, the Hypervolume by Slicing Objectives (HSO) algorithm, that is faster than any that has previously been published. HSO processes objectives instead of points, an idea that has been considered before but that has never been properly evaluated in the literature. We show that both previously studied exact hypervolume algorithms are exponential in at least the number of objectives and that although HSO is also exponential in the number of objectives in the worst case, it runs in significantly less time, i.e., two to three orders of magnitude less for randomly generated and benchmark data in three to eight objectives. Thus, HSO increases the utility of hypervolume, both as a metric for general optimization algorithms and as a diversity mechanism for evolutionary algorithms.

A scalable multi-objective test problem toolkit

This paper presents a new toolkit for creating scalable multi-objective test problems. The WFG Toolkit is flexible, allowing characteristics such as bias, multi-modality, and non-separability to be incorporated and combined as desired. A wide variety of Pareto optimal geometries are also supported, including convex, concave, mixed convex/concave, linear, degenerate, and disconnected geometries. All problems created by the WFG Toolkit are well defined, are scalable with respect to both the number of objectives and the number of parameters, and have known Pareto optimal sets. Nine benchmark multi-objective problems are suggested, including one that is both multi-modal and non-separable, an important combination of characteristics that is lacking among existing (scalable) multi-objective problems.

An evolution strategy with probabilistic mutation for multi-objective optimisation

Evolutionary algorithms have been applied with great success to the difficult field of multiobjective optimisation. Nevertheless, the need for improvements in this field is still strong. We present a new evolutionary algorithm, ESP (the Evolution Strategy with Probabilistic mutation). ESP extends traditional evolution strategies in two principal ways: it applies mutation probabilistically in a GA-like fashion, and it uses a new hyper-volume based, parameterless, scaling independent measure for resolving ties during the selection process. ESP outperforms the state-of-the-art algorithms on a suite of benchmark multiobjective test functions using a range of popular metrics.

Debugging parallel programs using incomplete information

Many parallel programs employ regular topological structures to support their computation. This topological information is exploitable in the debugging process. Communications not normally part of a topology ones that are either missing or unexpected, are immediately recognisable. Furthermore, animations used to assist the debugging may be enhanced by arranging representations of the executing tasks with reference to the programs topology. However direct topology support is lacking in many environments, including workstation clusters, where popular language extensions such as the Parallel Virtual Machine (PVM) and the Message Passing Interface (MPI) are common. Programmers are required to implement topology support themselves. Moreover debugger support that exploits topological information is lacking; without explicit knowledge, determining a programs topology is difficult. This paper presents a methodology to identify program topologies using only standard trace facilities. This methodology uses the concept of distance between graphs. To demonstrate the feasibility of the approach, several generic algorithms are implemented, and results on five different types of topologies reported.

A multi-objective evolutionary algorithm approach for crusher optimisation and flowsheet design

The performance of crushing equipment in mineral processing circuits is often critical to the generation of final product. A multiobjective evolutionary algorithm has been developed that allows the crusher internal geometry to be created and evaluated against multiple performance objectives. The multiple-objective approach is particularly important in mineral processing, as the optimum performance of single machines is often a trade-off between competing process drivers. A case study is presented that demonstrates the application of the technique to the design of cone crusher liners. New crusher liner profiles resulting from the application of the evolutionary algorithm suggest that significant improvements in the generation of lump product can be obtained. The extension of the approach to wider process plant design is discussed in terms of the objectives and issues to be addressed. 2004 Elsevier Ltd. All rights reserved.

Multi-level ranking for constrained multi-objective evolutionary optimisation

In real-world optimisation problems, feasibility of solutions is invariably an essential requirement. A natural way to deal with feasibility is to cast it as an additional objective in a multi-objective optimisation setting. In this paper, we consider two possible ways to do this, using a multi-level scheme for ranking solutions. One strategy considers feasibility first, before considering objective values, while the other reverses this ordering. The first strategy has been explored before, while the second has not. Experiments show that the second strategy can be much more successful on some difficult problems.

Designing comminution circuits with a multi-objective evolutionary algorithm

Mining is an important industry in Australia, contributing billions of dollars to the economy. The performance of a processing plant has a large impact on the profitability of a mining operation, yet plant design decisions are typically guided more by intuition and experience than by analysis. In this paper, we motivate the use of an evolutionary algorithm to aid in the design of such plants. We formalise plant design in terms suitable for application in a multi-objective evolutionary algorithm and create a simulation to assess the performance of candidate solutions. Results show the effectiveness of this approach with our algorithm producing designs superior to those used in practice today, promising significant financial benefits.

Economic optimisation of an ore processing plant with a constrained multi-objective evolutionary algorithm

Existing ore processing plant designs are often conservative and so the opportunity to achieve full value is lost. Even for well-designed plants, the usage and profitability of mineral processing circuits can change over time, due to a variety of factors from geological variation through processing characteristics to changing market forces. Consequently, existing plant designs often require optimisation in relation to numerous objectives. To facilitate this task, a multi-objective evolutionary algorithm has been developed to optimise existing plants, as evaluated by simulation, against multiple competing process drivers. A case study involving primary through to quaternary crushing is presented, in which the evolutionary algorithm explores a selection of flowsheet configurations, in addition to local machine setting optimisations. Results suggest that significant improvements can be achieved over the existing design, promising substantial financial benefits.