Definition

️️️️️️️In this selection method we define the notion of temperature $T$ as a tuneable parameter for determining selection pressure. This notion of selection pressure is the extent to which high-fitness individuals are preferred over low-fitness ones. We can implement it similarly to the two previous ones:

Roulette-Wheel Selection

The probability $p_{i}$ of selecting individual $i$ is $$p_{i}=\frac{e^{\frac{F_{i}}{T}}}{\sum\limits_{j=1}^{N}e^{\frac{F_{i}}{T}}}$$Individuals are selected with approximately equal probabilities if $T$ is large and vice versa for small $T$ (high-fitness preferred).

Tournament Selection

Pick two individuals $k$ and $j$ randomly from the population. During selection a random number $r$ is generated and the selected individual $i$ is determined according to $$i=\begin{cases}j&\mbox{if }b(F_{j},F_{k})>r \\ k&\mbox{otherwise} \end{cases} \ \ b(F_{j},F_{k})=\frac{1}{1+e^{\frac{1}{T}(\frac{1}{F_{j}}- \frac{1}{F_{k}})}}$$ with $F_{j}$ and $F_{k}$ as the fitness values of the two individuals. For large $T$ the probability of either getting chosen is equal, if $T$ is small, the better of the two is selected more often.