Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 243
P(Xi)=
F(Xi)
� n
j=1 F(Xj)
where P(Xi) is the probability of selection for the i-th individual with
fitness F(Xi) ofthen-sized population.
• Mutation. Mutation operator consists of the random substitution of some
bits (nucleotides) in the numeric string representing an individual. The role
of mutation is to enhance the probability of exploring untouched areas
of the design space avoiding premature convergence. Mutation generally
involves less than 10% of the individuals.
• Crossover. Crossover is a genetic recombination between two individuals,
whose strings are randomly cut and re-assembled.
The version of GA in modeFRONTIER c○ implements a fourth operator called
directional crossover. It assumes that a direction of improvement can be detected
comparing the fitness values of two reference individuals. This operator
usually speeds up the convergence process, though it reduces robustness.
Directional cross over works as follows:
1. Select an individual i;
2. Select reference individuals i1 and i2;
3. Create the new individual as:
Xinew = Xi + s · sign(Fi − Fi1)(Xi − Xi1)+t · sign(Fi − Fi2)(Xi − Xi2)
where s and t are two random parameters. The genetic operator behavior is
illustrated in Fig. 8.10. Each operator can be applied with a certain probability.
Different combinations of operator probabilities may lead to different
levels of robustness, accuracy and convergence rate.
8.7.2 Multi-objective Approaches
When there is a single-objective function, the definition of a metric for evaluating
design fitness is straightforward. Pareto optimal set reduces to a single
point and the best design is a global extreme. On the other hand, the introduction
of multiple objectives tangles things a bit. It has been already
stressed that in multi-objective optimization, there exists a set of solutions
that present equal quality or effectiveness. These are the non-dominated designs
as defined in Eq. (8.41). The problem arises as to how to compare and
judge designs in order to get a scalar evaluation scale for their fitness.
MOEA approaches can be roughly divided into three categories: Aggregating
Functions, Population-based Approaches, andPareto-based Approaches.