## Introduction¶

In this section, we're going to take a look at how to calculate the contributing hypervolume indicator values for a population of solutions.

The Contributing Hypervolume (CHV) indicator is a population sorting mechanism based on an adaptation of the hypervolume indicator. The hypervolume indicator works by calculating the size of the objective space that has been dominated by an entire approximation set with respect to a specified reference point, whereas the CHV indicator assigns each solution in an approximation set with the size of the space that has been dominated by each solution exclusively. With this information, the population can be sorted by the most dominant and diverse solutions. This has been illustrated below in two-dimensional space with a population of three solutions.

Calculating the exact CHV indicator is attractively simple. The method begins by first calculating the hypervolume indicator quality of a population $\mathrm{X}$, and then for each solution in the population, the solution is removed and the hypervolume indicator quality is again calculated for the new population. The new hypervolume indicator value is then subtracted from the hypervolume indicator value of the whole population, which results in the CHV indicator value of the solution which was removed. It is then possible to calculate the CHV indicator values of all the solutions in the population, order them by descending value so that they are ordered by the greatest explicit hypervolume indicator contribution, and select the better solutions to form the next parent population. This approach has been listed in Algorithm below.

$$
\begin{aligned}
&\textbf{CHVIndicator}(f^{ref}, X)\\
&\;\;\;\;X_{HV} \leftarrow HV(f^{ref},X)\\
&\;\;\;\;for\;\;{n = 1 : \lambda}\;\;do\\
&\;\;\;\;X_t \leftarrow X \backslash {X_n}\\
&HV_n \leftarrow HV(f^{ref},X_t)\\
&CHV_n \leftarrow X_{HV} - HV_n\\
&\textbf{return}\;\;CHV
\end{aligned}
$$

Although many optimisation algorithms use the CHV as a sorting criterion for selection, its calculation becomes computationally infeasible as the number of objectives being considered increase. Monte Carlo approximations have been used to speed up the calculation of the CHV in which through empirical experiments has shown that the method does not impair the quality of the approximation set. However, the speed increase provided by the Monte Carlo approximation method still results in an infeasibility of the CHV indicator on problems consisting of five objectives or more.

This particular measure of diversity preservation can also be used to reduce the size of a final approximation set produced by an optimiser, to a size that will not overwhelm and confuse a decision-maker.