## 4 Results  
The determination of the optimal grain size composition of the porous filler was conducted using the mathematical planning method of the full-factor experiment (FFE 2k) using the program \"MathCAD\". The plans were used when the first-order planning did not allow an adequate regression model to be obtained, and the need for its complication became clear. Additionally, it may be used if it is known that the research object has significant nonlinear properties in advance. Compared to plans of the first order, plans of the second order are more complex in structure, have a larger number of points and levels of variation for each factor, and require an increased number of experiments in their implementation. The parameters were taken as the response function: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(\frac{W}{C}\) & **C, kg/m\({}^{3}\)** & **W, L/m\({}^{3}\)** & **CS, kg/m\({}^{3}\)** & **S, kg/m\({}^{3}\)** & \(\rho_{cm}\), kg/m\({}^{3}\)** \\ \hline Value & 0.55 & 373 & 205 & 1139 & 720 & 2437 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters of the composition of the concrete mixture. * \(\rho_{b}\)--bulk density of the slag pumice mixture, kg/m\({}^{3}\);
* \(\rho_{bv}\)--bulk density of vibro-compacted slag pumice, kg/m\({}^{3}\) (vibration compaction time 1.5 min);
* \(V_{sp}\)--the voidness of the slag pumice mixture, %. The results of experimental studies are presented in Table 8.  
As a result of statistic processing of the experimental data, a mathematical model was obtained for the bulk density of the mixture of aggregates in its natural state:  
\[\rho_{b}=968\,x_{1}+1003\,x_{2}+1038\,x_{3}+150\,x_{1}\,x_{2}+448\,x_{1}\,x_{ 3}+298\,x_{2}\,x_{3} \tag{6}\]  
In the case of a vibrationally compacted mixture of grains, the equation takes the form:  
\[\rho_{bv}=1088\,x_{1}+1130\,x_{2}+1155\,x_{3}+4\,x_{1}\,x_{2}+602\,x_{1}\,x_{ 3}+222\,x_{2}\,x_{3} \tag{7}\]  
In this case, the porosity of the vibro-compacted mixture was described by the following regression equation:  
\[V_{sp}=37.8\,x_{1}+32.3\,x_{2}+27\,x_{3}-7.8\,x_{1}\,x_{2}-52\,x_{1}\,x_{3}-6. 6\,x_{2}\,x_{3} \tag{8}\]  
The resulting equations are graphically presented in Figures 4-6, from which it can be seen that:  
* The nature of the change in the bulk density of the mixture in its natural and vibro-compacted state is the same;
* In the vibro-compacted state, the maximum bulk density can reach 1272 kg/m\({}^{3}\) and can reach 1115 kg/m\({}^{3}\) in the natural state;
* The maximum density and maximum porosity are achieved with discontinuous granulometry (no grains with a fraction of 2.5-5 mm);
* When the relative content of coarse aggregates (5-10 mm) in the mixture is more than 40%, the voidness and bulk density of the mixture is influenced by the fraction of 1.25-2.5 mm;
* With a relative content of a mixture of coarse aggregate (5-10 mm) less than 40%, fluctuations in the number of fine fractions have practically no effect on the voidness and bulk density. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Slag pumice Fraction Size, mm**} & **Bulk Density** & **Bulk Density of** & **Voidness of** \\ \cline{2-5}  & **5–10 (x\({}_{1}\))** & **2.5–5 (x\({}_{2}\))** & **1.25–2.5 (x\({}_{3}\))** & **of Slag Pumice** & **Vibro-Compacted** & **Slag Pumice** \\ \cline{2-5}  & \multicolumn{3}{c}{**Volume Content of This Fraction, \%**} & **Mixture, kg/m\({}^{3}\)** & **Slag Pumice, kg/m\({}^{3}\)** & **Mixture,\%** \\ \hline
1 & 1 & 0 & 0 & 968 & 1088 & 37.8 \\
2 & 0 & 1 & 0 & 1003 & 1130 & 32.3 \\
3 & 0 & 0 & 1 & 1038 & 1155 & 27.0 \\
4 & 1/2 & 1/2 & 0 & 1023 & 1110 & 37.0 \\
5 & 1/2 & 0 & 1/2 & 1115 & 1272 & 14.5 \\
6 & 0 & 1/2 & 1/2 & 1095 & 1198 & 28.0 \\
7 & 1/3 & 1/3 & 1/3 & 1096 & 1205 & 26.1 \\
8 & 3/5 & 1/5 & 1/5 & 1076 & 1196 & 29.4 \\
9 & 1/5 & 3/5 & 1/5 & 1076 & 1186 & 31.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of the slag pumice experimental studies.