## Research

### Multi-Objective Genetic Algorithms

**Multi-Objective Genetic Algorithms** (**MOGAs**) are an extension of the classical single objective Genetic Algorithms (GAs).
MOGAs are used to solve **multi-objective optimization problems** in which the objectives are in competition among them:
in this case it could be useful to find a collection of solutions with a good trade-off between all the objectives.

The trade-off surface is called **Pareto Front**, and it is composed by non-dominated solutions
(a solution *s _{1}* dominates a solution

*s*if

_{2}*s*is not worse than

_{1}*s*on all the objectives and better than

_{2}*s*in at least one objective). Using a MOGA to find a good approximation of the Pareto Front, a

_{2}**fitness vector**is associated to each element (chromosome) of the actual population, in order to evaluate its level of goodness, instead of a single fitness value such as in the classical GAs.

The research focuses on:

- the analysis of the state-of-art MOGAs
- the study of new MOGAs
- the applications of MOGAs to real problems
- the study and implementation of ad-hoc genetic operators (selection, crossover and mutation) for a specific multi-objective problem.

In the last year, we studied how to use a MOGA to identify the rule base of a **Mamdani type Fuzzy System**, from numerical data,
with application to regression, functions approximation and time series forecasting problems.
Usually, during the identification process of a Fuzzy System, only the maximization of the accuracy of the system is considered,
but we also tried to minimize the complexity. These two objectives are in competition and a good trade-off between
the accuracy and the complexity should be found.

We used the **Pareto Archived Evolutionary Strategy** (**PAES**) to identify a set of non dominated rule bases,
for a Mamdani System in which the fuzzy partitions of input and output variables have pre-determined numbers
of fuzzy sets and uniformly distributed membership functions.