## 2 Materials and Methods
In this case, it is important to assess the probability of the process parameters going beyond the permissible limits because this assessment affects the capacity to meet the production targets, the change in the power consumption behavior, and the probability of level-crossings of random processes. The ability to fulfill the production plan is associated with several random factors determined by both internal and external circumstances. One of them is the amount of power consumed (\(P\), kW) at the current time. The second is the power consumed (\(W\), kWh) over the period \(t_{pl}\) required for the timely fulfillment of the production plan, provided the power supply conditions are met. In [63], the study reveals that the relationship between the output volume \(B\) and the electricity consumption \(W\) at the surveyed industrial enterprises is not functional \(B=f(W)\) but probabilistic and is determined based on correlation coefficients changing in the range \(r_{B,W}=0.43-0.89\). This is mainly characteristic of enterprises that produce small-scale products while simultaneously manufacturing a large range of products in various stages of the technological process. Large industrial enterprises, where production processes vary in duration and complexity, are also influenced by the indicated factors. Here are examples of how to calculate the correlation coefficient for two industrial enterprises with different technological processes and different power consumption. **Example 1**. : _During 4 days (\(n=4\)), one of the enterprises of the chemical industry has the volume of output production \(B\) (kg) with daily power consumption \(W\) (MWh), which are given in Table 1._Estimates of the mathematical expectation, dispersion, standard deviations, and correlation coefficient are calculated using known equations:  
\[M^{*}(B)=\frac{\sum_{i=1}^{4}B_{i}}{n}=1167.5\;\text{kg};\;M^{*}(W)=\frac{\sum_{i =1}^{4}W_{i}}{n}=6.75\;\text{MWh};\]  
\[D^{*}(B)=\frac{\sum_{i=1}^{4}[B_{i}-M^{*}(B)]^{2}}{n-1}=22922\;\text{kg}^{2};\]  
\[\sigma_{B}^{*}=\sqrt{D^{*}(B)}=151.4\;\text{kg};\]  
\[D^{*}(W)=\frac{\sum_{i=1}^{4}[W_{i}-M^{*}(W)]^{2}}{n-1}=0.837\;\text{MWh}^{2};\]  
\[\sigma_{W}^{*}=\sqrt{D^{*}(W)}=0.915\;\text{MWh};\]  
\[r_{B,W}^{*}=\frac{\sum_{i=1}^{4}\left[B_{i}-M^{*}(B)\right]\cdot\left[W_{i}-M^{ *}(W)\right]}{(n-1)\sigma_{B}^{*}\sigma_{W}^{*}}=0.98.\]  
The obtained value of the correlation coefficient (0.98) indicates a fairly close (close to functional) dependence between the volume of output of an industrial enterprise and its electricity consumption. The value \(n-1\) in the denominator of the equations for calculating the dispersion and correlation coefficient eliminates the bias of their estimates at a small number of observations. **Example 2**. : _At one of the enterprises of the metalworking industry, the study of the dependence of the volume of output production \(B\) (ton) on the value of electricity consumption \(W\) (MWh) was carried out. The results of measurements for 15 days (\(n=15\)) are shown in Table 2._  
To reveal the dependence between \(B\) and \(W\), their ordering is required. For the purpose of visualization, the paired values for 15 days (\(w_{1}\); \(n_{1}\)), (\(w_{2}\); \(n_{2}\)),..., (\(w_{15}\); \(n_{15}\)) are graphically represented in the Cartesian coordinate system in Figure 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\boldsymbol{B}\), \(\boldsymbol{\text{kg}}\) & 1000 & 1100 & 1220 & 1350 \\ \hline \(\boldsymbol{W}\), \(\boldsymbol{\text{MWh}}\) & 5.6 & 6.6 & 7.0 & 7.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Data of the volume of output production and power consumption of the chemical industry enterprise.