Critical Assessment 3: The unique contributions of multi-objective evolutionary and genetic algorithms in materials research

Abstract

The current state of the art of materials research using multi-objective genetic and evolutionary algorithms is briefly presented with critical analyses. The basic concepts of multi-objective optimisation and Pareto optimality are explained in simple terms and the advantages of an evolutionary approach are emphasised. Current materials related research in this area is summarised, focusing on the achievements to date and the specific needs for further improvement.

Keywords:

• Genetic algorithms
• Multi-objective optimisation
• Materials design and processing
• Reviews
• Critical assessments

Preamble

To elaborate the scope of multi-objective optimisation we begin with a simple example. Suppose that for a specific alloy of known composition it is necessary to develop thermo-mechanical processing parameters consistent with the maximum achievable strength and ductility. Balancing strength and ductility is a ubiquitous requirement, but far from being trivial or simple. This is because those two targeted properties are often conflicting requirements: an improvement in one generally leads to a deterioration in the other. It is not therefore possible to discover a processing schedule that would set both the properties at their individual best: all that can be achieved is a tradeoff, a compromise. Is there some optimum solution to this problem? Yes, and for that we need to enter the domain of Pareto-optimality, conceived and convincingly established by the legendary Italian mathematician and sociologist Vilfredo Pareto (1843–1923). Some glimpses of its rigour are readily available in recent texts.^1–3

The balance between strength and ductility can be additionally complex. It may be that we know the alloy system, but not the composition leading to the optimum strength and ductility, which itself remains a subject of investigation. Taking the example of Ni-based superalloys,^4–6 direct experimentation might be prohibitively resource consuming. The optimisation, as its first requirement, would involve modelling both objectives in terms of the process variables, where the conventional strategy of thermodynamic, kinetic or transport phenomena based approaches might have limited applicability, and a data-driven empirical modelling strategy^7–9 be the only viable option. Furthermore, it may not be clear whether all the apparent process variables actually influence the objectives in question: do all the potential solutes have significant impact on the strength or ductility of a superalloy, or are some redundant? In what follows, the state of the art in such perplexing problem is set out in the context of materials, beginning with Pareto-optimality in a nutshell.

The essence of Pareto-optimality

Avoiding the complex mathematical treatment that is now well documented,¹ one can attempt to establish the notion of Pareto-optimality in plain English.^10,11 When it is necessary simultaneously to optimise conflicting objectives, such as strength, ductility, and the cost of production as well; it is necessary to define dominance, which has a precise mathematical description. While comparing two possible solutions, say two distinct sets of alloy composition and heat treatment schedules, leading to different values of the objectives such as strength, ductility and cost, it can be concluded that one solution dominates if (i) it produces objective values which are at least as good as those produced by the other and (ii) it produces at least one objective value which is better than the corresponding value obtained using the alternate with which it is being compared.

This is known as the weak dominance and is generally used. A strong dominance would require the dominating solution to be better in all the objectives. If the dominance condition is not satisfied, then the two solutions are taken as non-dominating. In multi-objective optimisation, the locus of the non-dominated solutions is known as the Pareto-front, and its member solutions constitute the Pareto-set. When no additional feasible and non-dominating solution can be added to the Pareto-set, its locus is called the Pareto-frontier, which will constitute the optimum solutions in a multi-objective scenario, as no feasible solution can dominate any of its members.^1–3

Thus, in a multi-objective situation, the optimum is not just one particular solution; rather it is constituted by a set of solutions, where each member represents an optimum tradeoff between the objectives. This is of immense advantage to a decision maker (DM), since it offers several alternates, from which a suitable one can be picked using even some additional criteria, if needed. This flexibility is not available in the conventional single objective scenario.^12–14 Take, for example, two members of an optimum frontier; suppose one points to a longer annealing time than the other, the DM can prefer the one with the shorter annealing cycle, citing the need for faster production at the industrial level, without losing the optimality. The advantage of such flexibility is simply overwhelming.

Why genetic algorithms?

Numerous gradient based methods are available¹ for calculating the Pareto-optimality. They however require calculating one optimum solution at a time. Thus, calculation of the entire Pareto-front might become prohibitively time consuming and often impossible. Also, every objective may not be differentiable and gradient calculation may not be possible at all times. Recently the genetic and evolutionary algorithms^2,3 based approaches have made some very significant contributions in this area.^15–19 These algorithms are non-traditional, in that they are of non-calculus type, and do not use gradient or derivative information, but are nature inspired, thus tend to mimic biological processes like crossover and mutation as well as the group behaviour of a flock of birds, a colony of ants, or the working principles of the immune systems in the developed species. In addition, these algorithms are robust and are easily portable for one type of problem to another with no significant loss of accuracy. Traditionally, these types of algorithms have been widely used for solving the single objective optimisation problems in the materials domain,^15–19 and now their usage in the multi-objective problems is quite prominent.²⁰ Some of the significant materials related applications are now described.

Some recent evolutionary studies

To date, there is only one review that deals exclusively with the applications of multi-objective genetic and evolutionary algorithms in the materials area.²⁰ Several subsequent publications have dealt with diverse materials including iron and steel,^21,22 titanium alloys,^23,24 bulk metallic glasses,^25,26 composites,^27,28 industrial chemicals,²⁹ polymers,³⁰ and so on. Diverse processes such as friction stir welding,^31,32 complex sheet metal forming,³³ residual stress minimisation during machining,³⁴ semi-conductor solar cell design,³⁵ iron ore sintering,³⁶ etc. have come under its purview, in addition to materials studies of more fundamental nature,³⁷ providing vital insights to materials design, fabrication and processing. The success of the optimisation scheme depends often on efficient modelling of the objective functions and, given the limitations of the existing theory, data-driven empirical modelling turns out to be the most practical and efficient option. Genetic and evolutionary algorithms are population based strategies. In other words, instead of one starting point and its iterative update, as common in the classical methods, they start with a large random collection of initial guesses and process them over many generations. This, however, adds to the computing burden, and for efficient execution, this optimisation strategy requires models that themselves are not computing intensive. It is essential then to employ accurate meta-models or surrogate models, constructed either from the experimental data, or from a limited amount of simulation using strategies like molecular dynamics,^38–40 the finite element method,³³ or computational fluid dynamics⁴¹ among others. Such meta-models need to be sufficiently accurate and should not inherit random noise from the original data source. In recent years newer algorithms have been proposed^7,42,43 that can efficiently use the multi-objective genetic algorithms themselves to come up with meta-models of right accuracy and complexity, which are increasingly being used in engineering studies.^42–44 Such meta-models enabled prediction of newer steels with superior and optimised properties, which were also verified experimentally.²² It made it possible accurately to predict the yield point in steel, through calculation of the Hill’s coefficients,⁴⁵ without making customary ad hoc assumptions regarding the strain offset level.³⁷ Such calculations were not possible previously but are feasible now, simply because in recent years the evolutionary multi-objective optimisation algorithms have become mature and advanced.^2,3 Recently, Pareto-optimality has been applied to friction stir welding,⁴⁶ coupling it with a heat transfer model.³² For an industrial casting process it was coupled with a commercial simulator.⁴⁷ Evolutionary multi-objective procedures were also utilised to optimise the residual stresses during machining, using an elaborate finite element scheme.³⁴ Evolutionary meta-models can further enhance the scopes of such studies. Dealing with the uncertainties is another major aspect of data-driven models that would ultimately lead to Pareto-optimality; there the fuzzy rule based systems have a major role to play, as demonstrated in a recent study of friction stir welding of aluminium alloys³¹ and another on epoxy polymerisation.³⁰ It is also crucial to analyse the interaction between the input variables and to identify the significant variables while constructing the meta-models for the purpose of computing Pareto-optimality. Some simple intuitive approaches are now available to study the individual variable responses in a meta-model⁴⁸ and a recently proposed pruning algorithm⁴⁹ appeared to be very successful in that, as demonstrated in a number of recent studies.¹⁹

Concluding remarks: where do we go from here?

Most of the evolutionary multi-objective algorithms are actually quite recent.^2,3 Their impressive success opens avenues for further improvements that will require quite extensive research in the coming years. Some important issues are emphasised below:

1.	Implementing Pareto-optimality for a problem having more than three objectives is still very difficult with most of these algorithms, and more and more materials problems now require a large dimensional capability in the objective space. Practically, the Pareto-optimality condition is quite difficult to implement when a large number of objectives are present; as it does not allow a dominating solution to be inferior in terms of any of them. For such problems, alternates to Pareto-optimality are now available, where this strict requirement is ingeniously relaxed without losing the essence of optimality.50 Serious implementation of such alternate conditions is yet to be taken up in the materials domain.
2.	Visualisation of the results with several objectives is a cumbersome task;51 more efforts will be required in this direction, as we move towards increasingly complex problems.
3.	Data driven modelling can now be coupled with evolutionary multi-objective optimisation through some commercial software;52,53 however, for highly non-linear and noisy problems the results often vary considerably between one module and another,21 which simply indicates that tough challenges still lie ahead in terms of reliable software development. It is quite usual to carry out some analysis of variance (ANOVA) for this type of problem,54 and reliable software is needed to undertake an in depth statistical analysis of the noisy data in hand, before actually constructing a meta-model or optimising it. Often this is quite difficult, particularly when only a small amount of data is available with significant random noise. Nonetheless, the future algorithms must learn how to convincingly accomplish this.
4.	The flexibility and advantage of having a non-calculus type of approach in the evolutionary algorithms, where no gradient computation is required, also comes with a price that attainment of exact Pareto-optimality cannot be mathematically established in many cases. It is possible to use Karush–Kuhn–Tucker (KKT) optimality conditions55 for Pareto-optimality. Future algorithms should try to implement it more vigorously in the evolutionary domain as well.
5.	Finally, the Pareto-frontier being the boundary between the feasible and infeasible solutions, the decision maker is often faced with a tough choice of picking up a robust solution,56 which will not become infeasible due to real-life fluctuations of the decision variables. A thorough stability and error analysis of the Pareto solutions is therefore needed. The theoretical basis for asserting the robustness to a particular solution is still at a preliminary stage and needs to be pursued at a high priority.

It can be expected that all these issues will be successfully resolved in the near future and that the materials world will be able to take the advantage of multi-objective optimisation at the fullest extent. Optimisation plays a crucially important role in current industrial scenarios: for example, a newly designed aircraft or a novel semiconductor chip for a specific application is actually routinely optimised for many attributes. In this context, Pareto-optimality has a distinct advantage over single objective optimisation, as it provides multiple optimum options, instead of a single rigid solution. In the metallurgical world, full scale industrial problems have been tackled using such strategies; for example, in blast furnace ironmaking⁵⁷ and hot rolling.⁵⁸ More such studies will be forthcoming in the near future.