## 2 Materials and Methods
### Selection of Optimization Methods  
Construction production has its own peculiarities and specifics: the instability of production, its discrete nature, mobility, the number of participants in the construction process, a long production cycle compared to industry, the stationary nature of the use of construction products, and the influence of random factors (weather, geographical conditions, etc.). The management of construction production occurs through optimization. The golden section method emerges as one of the most promising, achieving the highest accuracy with a limited number of target function \(f(x)\) calculations [11,14]. According to the concept of the golden ratio method, the optimization of any management decision is performed by the practical tools of dividing the initial interval \([a\dots b]\) of possible alternative construction options of length (\(\tau=1\)) into two parts in such a way that the ratio of the whole to the larger part is equal to the ratio of the larger part to the smaller part. This unusual division principle creates two points, \(x_{\tau 1}\) and \(x_{\tau 2}\), on the research interval, which realize the optimization principle of the golden section. To ensure the symmetry of the search optimal solution, the distance \((1-\tau)\) should be \(\tau\)-th of the length of the total interval. Under such conditions of choosing the value of \(\tau\), the next test point is placed at a distance equal to the \(\tau\)-th part of the length of the interval from the right limit point of the segment boundary [19]. When choosing \(\tau\) in accordance with the condition \(1-\tau=\tau^{2}\), the search solution is preserved when moving to a reduced interval (see Figures 1-4). Next, the following quadratic equation is solved:  
\[\tau=\frac{-1\pm\sqrt{5}}{2} \tag{1}\]  
The positive value of the equation is obtained: \(\tau=0.61803\approx 0.62\). Figure 1: Geometric scheme of the golden section method in optimization problems of construction production (source: own findings based on [1,19]). Figure 2: The search for the symmetry of the golden section in the optimization tasks of construction production in production systems (source: own findings based on [1,19]). The mathematical model of the proportion is formed as follows:  
\[\frac{\tau}{y_{2}}=\frac{y_{2}}{y_{1}} \tag{2}\]  
or  
\[\frac{\tau}{y-y_{1}}=\frac{\tau-y_{1}}{y_{1}} \tag{3}\]  
The following notation is introduced into the mathematical calculations:  
\[\frac{y_{1}}{\tau}=z \tag{4}\]  
the formalization of which obtains the following:  
\[\frac{1}{1-z}=\frac{1-z}{1} \tag{5}\]  
and the following result was obtained: \(z=\frac{3-\sqrt{5}}{2}\approx 0.382\). We performed a further search for the harmonic accompaniment of the optimal solution using the method of the golden section, which is based on dividing the segment of a set of possible options by trial points into intervals in the golden ratio. The management decision optimization function \(f(x)\) is unimodal on the interval [\(a\)... \(b\)]. Under these conditions, the position of the optimal point of the function (min) is found in the two inner points of the golden ratio (see Table 1). The mutual location of the optimal solution points should be considered in the following two cases:  
\[f(x_{\tau 1})<f(x_{\tau 2}) \tag{6}\]  
\[f(x_{\tau 1})\geq f(x_{\tau 2}) \tag{7}\]  
Figure 4: Symmetry of the golden section in optimization tasks of construction production (source own based on [1; 19]). Figure 1:0 The formation of “golden intervals” in the work of the golden section method of optimizing construction production in production systems (source: own findings based on [1; 19]). The first case (6) realizes the minimum of the optimality function in the segment [\(a\ldots x_{\tau 2}\)]. The second condition (7) provides the implementation of the optimal function in the segment [\(x_{\tau 1}\ldots b\)].