# Skrgic Selection in GPdotNET

## Introduction

This document presents Skrgic Selection Method, the one of several selection methods in GPdotNET. While I was developing GPdotNET I have had several conversation with my friend Fikret Skrgic (master in computer scinence and math) regarding selection in Evolutionary algorithms. He is an incredible man, and in just a few minutes while I was describing to him what I want from a new way of selection, he came out with the basic idea of the new selection method. After that time in just a few mails I had a new selection method ready for implementation in GPdotNET. I gave the name to the new selection method by his surname. It is a little thankfulness to him.

In the flowing text it will be presented the idea behind Skrgic Selection.

## Liner Skrgic Selection (LSS)

The Idea behind this selection is based on chromosome fitness. The rule is simple: The bigger chromosome fitness gives bigger chance to select better chromosome. In Population, chromosomes are already sorted from best to worst chromosomes.
The process of selection is the following:

1. Find the maximum and minimum fitness value from the population.
2. The best chromosome (maximum fitness) has position 0 in the population (zero based index), and the worst chromosome has index $N-1$.
3. Let’s give them name as $f_{max}$ and $f_{min}$ respectively.
4. Choose random number $r$ between $f_{min}$ and $f_{max}$.
5. Choose random chromosome from the Population $chrom$.
6. Compare the values of $r$ and $f_{chrom}$. If the $f_{chrom}$ is greater than $r$, select $chrom$.
7. Repeat steps 3 and 4 until you select one chromosome.
8. Repeat steps 2,3,4,5 until you select $n$ chromosomes.

The following picture shows graph of linear Skrgic selection:

## Nonlinear Skrgic Selection (NSS)

In LSS we have linear graph of selection probability. This means that if we have chromosome with $f_{chrom}$ fitness value, and another chromosome with $\frac{f_{chrom}}{2}$ fitness value, the probability of selection of two chromosomes is $p$ and $\frac{p}{2}$ respectively. This means that probability of selection is growing linearly.
If we want to change probability between chromosomes we can define a factor $k$ to be like selection pressure on whole chromosomes in the population. So let the k be a real value, and define the fitness of each chromosome as: $f_{nss}=f_{chrom} (1+k \frac{f_{chrom}}{f_{max}})$,

where:

• $f_{chrom}$ – fitness value of the chromosome,
• $f_{max}$ – maximum fitness value(fitness of the best chromosome of the population),
• $k$ – selection pressure,
• $f_{nss}$ – nonlinear fitness value of the chromosome.

We can conclude that the maximum nonlinear fitness value of the best chromosome is given by: $f_{max(nss)}=f_{max}(1+k)$.

For various value of parameter $k$  we can define graph of probability selection like on the following picture:

On the picture above we can see if $k=0$ we get the standard Linear Skrgic Selection (LSS). We can also conclude that there is a fitness value when the probability of selection is constant and not depends of parameter $k$. Very interesting graph is when the $k=-1$. In this case the selection probability of the worst and the best chromosome are equal.