## 2 The General Plan of the Experiment  
In order to find the optimal flotation parameters, a number of empirical measurements were carried out. Each measurement consisted in establishing the conditions, i.e., the values constituting the process parameters: type of copper ore (\(x_{1}\)), particle size fraction (\(x_{2}\)), flotation time (\(x_{3}\)), collector type (\(x_{4}\)), collector dose (\(x_{5}\)), and in measuring the values resulting from the adopted parameters, i.e., the results of enrichment, i.e., copper content in the concentrate (\(\beta_{w}\)), copper content in tailings (\(\theta_{w}\)) and copper recovery in the concentrate (\(\epsilon_{w}\)). However, it turned out that finding the optimal flotation parameters on the basis of the measurements carried out earlier is not a trivial problem. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Lithological Type** &
\begin{tabular}{c} **Content of Selected Metals** \\ **in Lithological Types** \\ \end{tabular} & \multicolumn{2}{c}{**Povalent Copper-Bearing Minerals**} \\ \hline \multirow{2}{*}{ carbonates} & Cu (\%) & 1.69 \\  & Ag (g/t) & 54 \\ \hline \multirow{2}{*}{shales} & Cu (\%) & 6.02 \\  & Ag (g/t) & 188 \\ \hline \multirow{2}{*}{sandstones} & Cu (\%) & 1.29 \\  & Ag (g/t) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mineralogical composition of lithological types of Polish copper ores. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Lithological Fraction**} & \multicolumn{4}{c}{**Branches of KGHM Polska Miedz S.A.**} & \multirow{2}{*}{**Total**} \\ \cline{2-2} \cline{4-5}  & **Lubin Branch** & & **Polkowice-Sieroszowice Branch** & **Rudna Branch** \\ \hline \multirow{2}{*}{ carbonates} & carbonates & 9.2 & 61.8 & 14.7 & 30.2 \\  & shales & 11.8 & 9.7 & 5.9 & 8.7 \\ \multicolumn{2}{c}{sandstones} & 79.0 & 28.4 & 79.4 & 61.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Proportion of lithological types in the material mined in the KGHM Polska Miedz S.A. Mining Plants. Of course, the simplest solution would be to find one of the measurements carried out that would meet the assumed criterion to the greatest extent. To this end, it would be sufficient to check all samples in a loop and find the one for which the properly formulated criterion assumes the lowest value. However, in such a situation, the result would be limited to the measurements performed. In order to find optimal parameters not only from among the measurement spaces, a model describing the flotation process had to be constructed on the basis of the collected data. It was therefore necessary to create, on the basis of actual measurements, mapping which, for each sample, after providing flotation parameters at the mapping inputs, will result in the values at the mapping outputs that are closest to the enrichment effects present in the sample. In addition, such mapping should also work outside the samples, i.e., in the entire parameter space. There were 5 parameters and 3 enrichment values in each sample, so the mapping should transform the 5-dimensional space into the 3-dimensional space. Finding such mapping is not an easy issue; therefore, a decision to use the neural network to implement such a model was made. A neural network learned with uncertainty back-propagation was used. The network had 5 inputs and 3 outputs. It consisted of 5 layers--4 initial layers consisted of 50 neurons, and the last one consisted of 3 neurons. A decision to use such a number of neurons due to the conducted experiments was made. With fewer neurons, the network could not learn all the samples correctly, and with too many neurons, the process of network learning took too long. The initial stage of the work consisted in the appropriate teaching of the neural network on the basis of samples from measurements, so that it would implement mapping describing the flotation process. Then, evolutionary algorithms were used in order to find the optimal flotation parameters. One of the basic concepts in evolutionary algorithms is an individual.