Multi-Objective Optimisation Algorithms After proximity to the true Pareto-optimal front, diversity of solutions in the
2.6 Preference Articulation in Evolutionary Multi-Objective Optimisation
When solving real-world multi-objective problems, the ideal optimisation algo- rithm is one which converges to non-dominated Pareto-optimal solutions within the DM’s ROI. This allows for the DM to be presented with a small set of trade- off solutions which are within their ROI (illustrated in Figure 2.14), as opposed to a larger set of trade-off solutions within the entire objective space (illustrated in Figure 2.15). Subsequently, the DM is not overwhelmed with a large set of candidate solutions when using expert knowledge to select their desired solution to the problem. Furthermore, when an ROI is specified by the definition of pref- erences, the algorithm is able to use this information during the search to discard trade-off solutions which do not fall within the desired region, and to skew the search towards the region by influencing the EMO algorithm’s selection operator. This additional preference information ultimately reduces the area of feasible so- lutions within the objective space, thus reducing the computational effort needed to produce a diverse set of pertinent solutions to aid the DM in making a decision. The role of the DM in EMO is usually to choose a single compromise solution from the approximation set presented to them. Although there may be a poten- tially infinite number of Pareto-optimal solutions in the global trade-off surface, in practice the DM will usually only be interested in a small subset of these. Therefore, allowing the DM to focus the optimisation process on relevant areas of the search space both increases the efficiency of the search effort and reduces the amount of irrelevant information the DM has to consider [75].
2.6. Preference Articulation in Evolutionary Multi-Objective Optimisation 41
Figure 2.14: A Pareto-optimal approximation set containing five solutions with ideal pertinence.
Figure 2.15: A Pareto-optimal approximation set containing seven solutions with undesirable pertinence.
A prerequisite for this type of convergence is the articulation of preferences by the DM. The preferences of a DM can be incorporated into the optimisation process in three ways:
• A priori, in which preferences are defined before the search.
• A posteriori, in which the DM selects a solution after completion of a search.
• Progressively, involving interaction with the DM during execution of the search.
A posteriori methods of preference articulation involve the DM selecting a compromise solution from the global approximation set of Pareto-optimal solu- tions found at the end of the optimisation process, whilsta priori and progressive preference articulation methods aim to achieve a good representation of the trade- off surface in the ROI of the DM. The key advantage ofa priori and progressive preference articulation methods is the reduction in the size of the search space explored by the optimiser because the search is focussed on a sub-set of the global trade-off surface.
In a priori articulation of preferences the DM expresses their preferences be- fore the start of the optimisation process. However, often the DM may not be sure of their preferences prior to optimisation, and by stating their preferences a priori, the DM may not investigate some areas of the search space which po- tentially deserve attention. A better method is often progressive articulation of preferences, which enables the DM to alter their preferences during the optimisa-
2.6. Preference Articulation in Evolutionary Multi-Objective Optimisation 43
tion process and thus incorporate information that only becomes available during the search process [76] (such as the exact nature of trade-offs between objectives). One of the first schemes for progressive preference articulation in EMO algo- rithms was introduced by [77], and extended the Pareto-based ranking scheme used in the Multiple Objective Genetic Algorithm (MOGA) [78] to allow pref- erences to be expressed throughout the run of an EMO algorithm. These pref- erences were then used in a modified version of dominance which combines the concept of Pareto-optimality with a preference operator to rank the candidate solutions according to both preference information and Pareto dominance. This progressive preference articulation method has been used in a wide variety of engineering applications such as the optimisation of robust control strategies for gasifier power plants [79], and the design of lateral stability controllers for aircraft [80].
More recently, the Reference-point-based Nondominated Sorting Genetic Al- gorithm II (R-NSGA-II) presented in [81], combines a preference based strategy with an EMO methodology, in order to demonstrate how a preferred set of solu- tions near a number of reference points can be found simultaneously. The paper suggests two approaches for the incorporation of preferences: a modified EMO procedure based on the NSGA-II; and a predator-prey approach based on an original grid based procedure [82]. Both approaches appeared to perform well, with the modified NSGA-II approach performing better overall.
The Preference-Based Evolutionary Algorithm (PBEA) was introduced in [83] in order to address the short-comings of not having preference information in the solution process. The algorithm uses the Indicator-Based Evolutionary Algorithm (IBEA) introduced in [84] as a base, in combination with a binary indicator which
has been modified with an achievement function (based on a reference point) which directly represents the preference information. The experimental results were obtained primarily from bi-objective synthetic test functions from the ZDT synthetic test suite. The authors suggest that the incorporation of preferences results in more relevant approximations throughout the optimisation process.
These preference driven multi-objective optimisers offer promising results, and suggest that the incorporation of DM preferences into multi-objective search can reduce computational cost of the optimisation process and improve the pertinence of the final approximation set presented to the DM.
The approaches introduced appear to lack rigorous benchmarking consisting of many test suites, real-world problems, and test-cases. The approaches introduced also involve the tight integration of the preference method into an existing EMO method. Instead, it would be desirable to have a preference articulation operator which is designed for portability. Such a portable preference articulation operator could be incorporated into any multi-objective optimiser, as different optimisers are more suitable for different problems, and some optimisers become redundant after years of further research.
2.7. State of the art Evolutionary Multi-Objective Optimisation Algorithms 45