## 2 Materials and Methods
### Possibilities of Using Bayesian Networks  
A harmonious production system is a complex of component parts (subsystems) interconnected according to the rule of the golden ratio. The management and organization of such a system should be based on the concepts of harmonious management [11; 18]. Under these circumstances, the application of Bayesian Networks can be seen as a natural extension of the principle of harmonious management. By modeling the probabilistic dependencies between different variables in the system, Bayesian Networks can help to \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Level of Symmetry** & \multicolumn{3}{c}{**Location of the Optimal Solution in the Golden Ratio**} \\ \cline{2-4}
**of the Golden Section** & \(f(y_{\mathbf{0}})<f(x_{\tau 2})\) & \(f(y_{\mathbf{0}})\geq f(x_{\mathbf{0}})\) \\ \hline I & 0.62 (\(b_{0}-a_{0}\)) & 0.38 (\(b_{0}-a_{0}\)) & 0.38 (\(b_{0}-a_{0}\)) & 0.62 (\(b_{0}-a_{0}\)) \\ II & 0.38 (\(b_{1}-a_{1}\)) & 0.24 (\(b_{1}-a_{1}\)) & 0.24 (\(b_{1}-a_{1}\)) & 0.38 (\(b_{1}-a_{1}\)) \\ III & 0.24 (\(b_{2}-a_{2}\)) & 0.14 (\(b_{2}-a_{2}\)) & 0.14 (\(b_{2}-a_{2}\)) & 0.24 (\(b_{2}-a_{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operation of the optimization procedure by the method of the golden section. Figure 5: Graph of the optimization function of the golden section method with the mutual location (**a**) of points \(f(x_{\tau 1})<f(x_{\tau 2})\); (**b**) of points \(f(x_{\tau 1})\geq f(x_{\tau 2})\) (source: own findings based on [1; 19]). capture the complex dynamics of the system and thus contribute to the achievement of a more harmonious state [23]. Bayesian Networks (BNs) are powerful tools in the field of construction, offering a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph [24; 25; 26]. The nodes in this graph represent the variables, and the edges between nodes represent the probabilistic dependencies between the variables. By integrating Bayesian Networks into the management process, it is possible to make more informed decisions and thus further enhance the harmony of the system. Bayesian Networks are a type of probabilistic graphical model that uses Bayesian inference for probability computations. Bayesian Networks aim to model conditional dependence, and therefore causation, by representing conditional dependence by edges in a directed graph. Through these relationships, one can efficiently conduct the inference of the random variables in the graph using the Bayesian Network [27]. Mathematically, a Bayesian Network is defined by two components: a directed acyclic graph (DAG), G, where each node represents a variable, and the absence of an edge represents a conditional independence statement, and a set of conditional probability distributions (CPDs). Each node, \(X_{i}\), in the graph has an associated CPD, \(P(X_{i}|\textit{Parents}(X_{i}))\), where \(\textit{Parents}(X_{i})\) denotes the parents of \(X_{i}\) in G.  
The joint probability distribution over all variables \(X_{1}\), \(X_{2}\),..., \(X_{n}\) in the Bayesian Network can be expressed as a product of the CPDs in the network [28; 29]:  
\[P\left(X_{1},\ X_{2},\ \ldots,\ X_{n}\prod_{i=1}^{n}P(X_{1}\left|\textit{Parents }\left(X_{1}\right)\right.\right) \tag{9}\]  
This factorization allows for the efficient computation of the joint distribution and is the key feature of Bayesian Networks that makes them useful for modeling complex systems. The structure of the graph can also provide valuable insight into the conditional independence relationships in the modeled system. In the context of construction, these variables could represent different aspects of a construction project, such as cost, time, quality, and safety. The dependencies between these variables can help in understanding how changes in one aspect of the project can affect others. For instance, an increase in cost might lead to an increase in quality but might also lead to an extension in the project timeline.