## 3 Related Work
### Comparison of Characteristics for the DEA Method  
When choosing a DEA model, the first decision is whether to use a constant scale effect (CRS) or variable scale effect (VRS). Upon closer examination, it can be said that the output parameters change proportionally to the input parameters during the CRS. When the VRS changes the output parameters, the output parameters may change disproportionately [60]. [PERSON], [PERSON], and [PERSON] [63] divide technical efficiency into two related multiplier components. The division is based on the dependence of efficiency on the magnitude of the scale. Considering enterprises for returns on scale, we can speak of the dependence of productivity on an increase in the resources used for production. If the increase in productivity is proportional to the increase in the number of resources used, then the enterprise has a CRS. An enterprise achieves the highest possible level of productivity if it is 100% efficient, with both a constant and a variable return on scale. If an enterprise is 100% efficient with VRS, but fails to achieve full efficiency with CRS, there then arises the efficiency depending on the magnitude of scale--scale efficiency (SE). The formula of this regularity is as follows:  
\[\text{TE}=\text{SE}\times\text{PTE} \tag{7}\]  
where  
TE--technical efficiency with constant return of scale;  
PTE--technical efficiency with variable return of scale, or pure technical efficiency;  
SE--efficiency depending on the magnitude of the scale. When calculating these performance indicators, it is possible to identify the sources of inefficiency. In the case of PTE, inefficiency may be caused by the inefficient operation of the enterprise. Then, in the case of SE, inefficiency can be caused by unfavorable conditions. In the experimental section, we will show this with examples. Figure 2 below demonstrates the difference between the three performance indicators (TE, SE, and PTE) discussed above. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Scale Effect**} & **Range** & **Type of** & \multicolumn{2}{c}{**Data**} \\  & & **Efficiency** & **Efficiency** & **x** & **y** \\ \hline CCR-Output & CRS & [0.1] & TE & S-p & U-s \\ \hline CCR-Input & CRS & [0.1] & TE & S-p & U-s \\ \hline BCC-Output & VRS & [0.1] & TE & U-s & S-p \\ \hline BCC-Input & VRS & [0.1] & TE & S-p & U-s \\ \hline \hline \end{tabular}  
* CRS—Constant Return to Scale, VRS—Variable Return to Scale, TE—Technical Efficiency, U-s—unrestricted in sign, S-p—Semi-p (= semi-positive)—only positive values are allowed.  
\end{table} 
Table 1: Comparison of the characteristics of the described CCR and BCC models. In Figure 2, the input parameters are located along the \(X\)-axis. The \(Y\)-axis shows the output parameters. The direct line is a line on which 100% efficient enterprises with a constant return of scale should be located. On the F curve, there are 100% efficient enterprises with variable returns of scale. Thus, enterprise A, lying on the curve F, is fully efficient with VRS (PTE = 1), but not efficient enough with CRS. Therefore, the efficiency, depending on the magnitude of the scale, is determined by the ratio [64]  
\[SE(A)=\frac{\text{LM}}{LA}<1 \tag{8}\]  
Both enterprise B and enterprise C are 100% effective at both CRS and VRS, so they operate at the highest possible level of productivity. For an inefficient enterprise E, efficiency can be determined using the following formula:  
\[SE(E)=\frac{PQ}{PE}=\frac{PE}{PR} \tag{9}\]  
Figure 2 and the calculations show that the inefficiency of enterprise E is caused by both inefficient production and unfavorable conditions. Materials and Methods of using DEA Models to Calculate the Efficiency of Fuel and Energy Complex Enterprises  
In the previous paragraphs, the basic DEA models were investigated. We now consider the possibility of applying these models in practice in a specific area and build an algorithm for using DEA models to calculate the efficiency of fuel and energy complex enterprises.