Multiobjective evolutionary algorithms

Like numerous other search techniques, multiobjective evolutionary algorithms were initially designed as a means to find trade-offs between the objective performance measures of candidate solutions. The overall benefits that evolutionary algorithms bring to multiobjective decision making problems are today the reason for the growing interest of researchers in this field, in particular with regard to their ability to generate several elements of the Pareto optimal set in a single run [81]. It is believed that evolutionary algorithms and, in particular, genetic algorithms, are so well suited for these types of problems as a result of their analogous connection to biological processes exhibiting multiobjective features in nature. According to Zitzler et al. [167], two major problems must be addressed when solving a multiobjective optimisation problem by means of a genetic algorithm. One is to configure the fitness assignment and selection processes effectively in order to guide the search towards the Pareto front, and the other is to maintain a diverse population in order to prevent premature convergence and generate a good spread of solutions along the approximate Pareto front.

During the period 1993–1995, researchers began in earnest to investigate the possibility of using evolutionary algorithmic techniques for solving multiobjective optimisation problems. Among these early approaches, the multiobjective genetic algorithm [105], the non-dominated sorting

2.4. Evolutionary algorithms 29

Generate initial population

Evaluate fitness of individuals

Select parent solutions for reproduction

Generate offspring population

Apply mutation operator on offspring solutions

Evaluate fitness of offspring solutions

Select individuals to be carried over to next generation

Update incumbent solution (if necessary)

Stopping criterion met?

False Report

incumbent solution True

genetic algorithm (NSGA-I) [134] and the niched Pareto genetic algorithm [69] received partic- ular attention. These algorithms were designed with the necessary additional components to convert single objective evolutionary algorithms to multiobjective counterparts without the use of weighting techniques, which often induce inconsistencies. Moreover, they are able to solve problems in multidimentional objective space and generate multiple non-dominated solutions near/along the Pareto front in a single run for a wide variety of optimisation problems. In gen- eral, two fundamental features are found in such techniques: (1) the assignment of fitness values to population individuals based on non-dominated sorting and (2) the preservation of diversity among solutions located on the same non-dominated front.

Researchers have, however, realised that, although the above-mentioned approaches are per- haps surprisingly effective in finding non-dominated solutions for many test problems, a wider variety of operators than mere crossover and mutation are required to improve computational performance. For example, elitism was introduced in an attempt to improve the convergence properties of the algorithms and to prevent the extinction of good solutions once they have been uncovered. The suggestion to include elitism was indeed a good one, as it has been shown that multiobjective evolutionary algorithms are subsequently able to achieve better converging prop- erties [167], and are generally able to find non-dominated solutions faster. The strength Pareto evolutionary algorithm [169] and the Pareto archived evolution strategy [79], in particular, are popular multiobjective genetic algorithm techniques which make use of the notion of elitism. The NSGA-I mentioned above was one of the very first multiobjective evolutionary algorithms to function without the use of user-defined objective preference weights (which amounts to solving a multiobjective problem in uni-dimensional objective space). In other words, the non-dominated population of the last generation represents an approximation of the Pareto-optimal set, from which a compromise solution may be chosen at the discretion of the decision maker. During the 1990s, the performance level of this algorithm was very high relative to the alternative, more traditional methods then available. Continual research in the field of multiobjective evolutionary algorithms over time, however, allowed for the discovery of shortcomings in the method. More specifically, three important criticisms were raised. The first had to do with the high computa- tional complexity of the NSGA-I. Implementation of the algorithm is computationally expensive for large population sizes, particularly due to the complexity associated with its non-dominated sorting procedure, which has to be performed at every generation. Secondly, the NSGA-I lacked the incorporation of elitism. As mentioned earlier, not only does elitism improve the convergence rate of the algorithm, it also prevents the loss of good solutions once they have been uncovered. Lastly, it was argued that the NSGA-I requires an additional parameter for a more effective diversity-preservation mechanism. In particular, the NSGA-I incorporated a so-called sharing parameter to this effect, which may be thought of as the distance metric chosen to calculate a proximity measure between two individuals in the current population. However, the addition of such a parameter raises two main complications. First, the performance of the sharing function in terms of maintaining a good spread of solutions relies too heavily on the parameter value chosen (which is user-defined). Secondly, according to this diversity preservation mechanism, every solution in the current population must be compared with every other solution, which may dramatically increase the problem complexity when the size of the population is large (in addition to the complexity associated with the non-dominated sorting procedure pointed out above).

Seven years later, Deb et al. [41] developed an improved version of the original NSGA-I, and called it the NSGA-II. Their method incorporates a better sorting algorithm, allows for elitism, and does not require the user to define a sharing parameter a priori, hence addressing all of the disadvantages raised in respect of the original NSGA-I. Perhaps the main difference between

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the two algorithm versions is that, while the fitness of a candidate solution in the NSGA- I is evaluated using non-dominated sorting only, the NSGA-II makes use of non-dominated sorting as well as diversity preservation in order to assess the fitness of individuals. Moreover, it has been demonstrated that the NSGA-II outperforms two other successful multiobjective evolutionary algorithms (namely the Strength Pareto Evolutionary Algorithm and the Pareto Archived Evolution Strategy) on the grounds of not only finding a more diverse set of candidate solutions but also in achieving convergence in close proximity to the true Pareto-optimal set for a wide variety of problems.

Overall, multiobjective evolutionary algorithms generally share certain fundamental common features. Their differences are not as complex as one might expect; they are often just individ- ually designed incarnations of the same basic principle achieved in a unique, creative manner. According to Coello et al. [29], the generic goals and operator design of all these algorithmic variations are (1) to preserve non-dominated points in objective space and respective solution points in decision space, (2) to continue to make algorithmic progress toward the Pareto front in objective space, (3) to maintain diversity among the points on or near the Pareto front, and (4) to provide the decision maker with a sufficient, but not overwhelming number of non-dominated points for alternative selection.