Diversity

The convergence of the genetic algorithm is estimated through its population diversity. Initially, the population has a high diversity since all the individuals are randomly selected. As the algorithm converges, the individuals in the population converge towards the best solution, thus decreasing the diversity. In our case, the individuals are points in a heterogeneous dimension space, with $\alpha$, $\beta$, $\gamma_{A}$ and $\gamma_{B}$ $\in \Re^{+}$ while the other parameters ranging between 0 and 1. Hence we use the Mahalanobis distance measure to determine the diversity of a population [22].

The Mahalanobis distance takes into account the heterogeneity in dimensions and correspondingly scales each dimension while estimating the distance between two points. Given a set of data points {$z_{i}$} with each data point $z_{i}$ being an n-tuple $\langle z_{ij} \vert 1 \leq j \leq n \rangle$, the Mahalanobis distance $d_{m}$ between two points $z_{k}$ and $z_{l}$ is given as

$$d_{m}(z_{k}, z_{l}) = (z_{k} - z_{l})^{T}¥ \Sigma^{-1} (z_{k}- z_{l})$$

Here $\Sigma$ is the $n \times n$ variance-covariance matrix for the given data points. To compare the diversity of populations across generations, the covariance matrix is computed taking into account all the chromosomes over all generations. The diversity of a population is then calculated as the average Mahalanobis distance of each chromosome from the mean chromosome.