Fitness assignment

When dealing with a multi-objective problem, fitness assignment can not be done straightforwardly, due to there being not only one objective function, but at least two of them which have to be taken into account. In general, one can distinguish the fitness assignment approaches below [264].

2.4.1.1 Aggregation

The most intuitive approach to assign fitness in the presence of multiple objective functions is to combine them into a single function. An example of this approach is

a sum of weights of the form f (x) = F X i=1 wi fi(x) , (2.11)

where coefficients wi ≥ 0 are weighting values representing the relative importance

of the F objective functions. It is usually assumed that PF_i=1wi = 1, however

the coefficients are varied during the optimisation process in order to find a set of non-dominated solutions. Although this technique does not require any changes to the basic mechanism of an EA, the major drawbacks of this approach are the uncertainty of the weighting coefficients [39], and that for some kind of problems it can not generate proper members of the Pareto optimum set [55].

2.4.1.2 Criterion

This approach switches between the objectives during the selection phase. Each time an individual is chosen, potentially a different objective can be used to make this decision [212, 152]. In Figure 2.5(a) we see an example showing that solutions which objectives are contained in the upper-left ellipse have the minimum values for function f1, while the those in the lower-right ellipse have the minimum values

for function f2. In this case, when one individual has to be selected, it will come

from the sets specified by any of the two ellipses, according to the proportion or probability assigned to each objective function.

2.4.1.3 Pareto dominance

There are different methods to assign fitness under this approach [264], from which three are described below.

Dominance rank. This method considers the number of solutions in the population P by which an individual is dominated [90]. To find the rank of solution si ∈ P , we

(a) Criterion (b) Dominance depth

Figure 2.5: Criterion and dominance depth methods for fitness assignment in multi-objective Evolutionary Algorithms.

need to compare it with every other sj ∈ P and count how many solutions dominate

si. That is, the rank r(si) of solution si ∈ P is

r(si) =

 {sj ∈ P | sj ≺ si}

 . (2.12)

Dominance count. This technique takes into consideration the number of solu- tions dominated by a certain individual [262]. In contrast with dominance rank, this method counts the number of individuals sj ∈ P that are dominated by si.

Specifically, the dominance count c(si) of solution si ∈ P is

c(si) =

 {sj ∈ P | si ≺ sj}

 . (2.13)

Dominance depth. This approach groups the population into non-dominated

fronts and their depth indicates the fitness of the individuals belonging to them

[223,67]. This method is represented in Figure 2.5(b). A naive approach to identify the front to which each solution si ∈ P belongs to, is to compare each solution si

with every other sj ∈ P to know if they are non-dominated. However, this operation

will only result in the first non-dominated front. If this procedure is repeated, the second non-dominated front will be found. That is, this process has to be executed

as many times as needed in order to complete the assignment of solutions to fronts. The worst case is when there are popSize fronts and there exists only one solution in each front. In this case, this method requires an overall O(F popSize3_{) comparisons,}

where F is the number of objective functions.

2.4.2 Diversity preservation

Diversity in the Pareto approximation is important because it is desirable that the solutions contained in this set are different. Density information reflects a good estimation of population diversity and could be used to increase it. This means that the probability of a solution being selected decreases as the density of solutions in its neighbourhood increases. The methods used in multi-objective EAs can be classified according to the categories of the techniques used in statistical density estimation [217, 264].

2.4.2.1 Kernel methods

In this kind of method, the distance in the objective space between each solution and all other in the population is calculated, as shown in Figure 2.6(a) for vector xi. Then, a Kernel function is applied over those values. The density estimate for

a solution will be the sum of all evaluations of the Kernel function [67].

2.4.2.2 Nearest neighbour

Methods in this category take into account the distance in the objective space bet- ween a given point and its k-th nearest neighbour to estimate density in its neighbou- rhood [262]. In the example shown in Figure 2.6(b), the xi’s third nearest neighbour

(a) Kernel (b) Nearest neighbour (c) Histogram

Figure 2.6: Kernel, nearest neighbour, and histogram methods for density es- timation which are used to preserve diversity in multi-objective Evolutionary Algorithms.

2.4.2.3 Histogram

Techniques in this category define, as neighbourhoods, grids in the F -dimensional space as shown in Figure 2.6(c). The number of individuals which objectives are in the same grid as those of a given solution is then its density estimate [145, 41]. In the example, we see that xi is sharing the neighbourhood with another vector.