Development of an Algorithm for Evaluating the Dominant Factors that have the Greatest Impact on the Energy Intensity of Products

The article deals with the assessment of the dominant factors that have the greatest impact on the energy intensity of products. A method is proposed that makes it possible to single out the most significant factors from all the variety of factors affecting energy indicators and to give an appropriate assessment to each of them. Methods and mathematical models are proposed that can be used for current and medium-term forecasting of electricity consumption. The analysis is carried out and the energy characteristics of production units are developed, depending on the reduced and total volume of manufactured products. Using linear programming methods, the energy intensity coefficients of the entire range of products manufactured by individual production units are obtained.


Introduction
Development of an algorithm for evaluating the dominant factors that have the greatest impact on the energy intensity of products In addition to a study of the impact of each factor individually, it is also necessary to assess their cumulative impact, since in the production environment all factors are interconnected and operate simultaneously.

Main part
Dispersion, correlation and regression analysis methods are mainly used to solve multivariate statistical problems. Dispersion and correlation methods ensure the determination of the degree of influence of various factors on each other and on dependent variables, while regression methods make it possible to analytically present the nature of the regularities of the influence of these factors and to evaluate and forecast the results for the future period. The multiple regression equation is generally as follows: 12  Where, B0 -free member of the equation; B1coefficient i=1÷n; Х1 -factors i=1÷n; n -number of factors. If the class of functions describing the phenomenon under study is not known, the type of relationship is determined empirically by selection, building a number of functions and statistical reliability using the multiple correlation coefficients, F -criterion, residual dispersion and relative approximation error [1][2][3][4][5].
In empirical function selection, the regression equation is initially represented by a Taylor segment. For practical tasks, it is usually limited to a second-degree polynomial: (2) 0 1 1 1 1 1 The second-order terms Xk, Xi for i ≠ k are included in the regression equation (2) in order to take into account the so-called effects of the combined action of factors of arguments on the dependent variable (Y), and the terms Xk, Xi for i = k (i.e. X2) take into account the nonlinearity of the change in the dependence of the variable (Y) when changing the i-th argument [6][7][8].
The process of finding a significant segment of the Taylor series equation is carried out as follows. First, all unknowns in the first degrees are included in the model and the resulting modules are assessed by the Fcriterion with a relative approximation error. If the assessment of the F -criterion turns out to be insignificant, then the model includes the values of the unknowns in the second powers (paired products). The process of increasing the degree of the polynomial continues until the estimate of the equation becomes significant. However, this method has significant disadvantages: -the number of coefficients of the regression equation when using even a second degree polynomial (2) grows very quickly with an increase in the number of arguments; -with a not very large sample size, the random arrangement of points in space can lead to the fact that some random (false) connections characterizing the effects of the relationship will also be significant; -when studying the regression equation given in the form (2), a circumstance arises that complicates the application of the least squares method. In this regard, the equation (2) is preferred to be set in the following form: -average values i-th and и k -th parameters in the studied statistical population. In addition, the complexity in integrating the results obtained causes computational difficulties, since the matrix of the system of normal equations becomes close to degenerate [9][10][11].
Research has established that of all the existing methods for solving the problem, it is advisable to use the regression step method of Efroimson. This method is economical from a computational point of view and includes the advantageous aspects of the method of all possible regressions, the method of elimination, and also allows you to obtain an adequate mathematical model of the object under study, suitable for practical use. The main idea of this method is to find regression with several variables in the form of a series of linear regression dependencies and to rebuild the correlation matrix step by step to the end. The method starts with a simple correlation matrix. The variables (factors) are included in the equation in turn. The order of inclusion is determined using the partial correlation coefficient, as a measure of the importance of variables not yet included in the equation. The variable Xi is selected (let's say it is xi), which is most correlated with Yi, and a linear (first-order) regression equation is found [12][13][14].
Then the partial correlation coefficient xi (i ≠ 1) is determined (taking into account the correction for xi). Mathematically, this is equivalent to determining the correlation between: Then such a value xk (let's say x2) is selected, which possesses the above properties, and as a result of the calculation, the second regression equation is obtained where , m -the number of factors; n -the number of measurements by factors; Yi -investigated energy parameter (e, P, W). 1. The average value is determined by the following expression: The adjusted sum of squares is calculated:: The unadjusted sum of the mixed works is: A matrix of uncorrected sums of mixed products X / X is constructed, where X / is the transported matrix. 7. The correlation coefficients between each factor and the response are calculated:

12
(11) , 12 12 ( We expand the correlation matrix R as follows: negative identity matrix. The elements of this matrix will be denoted ij A .
10. To introduce a factor into the regression model. We calculate Vi which is determined by the formula: , 11. The Vi with the maximum value is chosen, let's say it is Vi, then XL is a factor to be considered [17][18][19] .. ,  -the number of residual degrees of freedom after the inclusion of the factor. 16. Since the XL factor is included in the regression, the correlation matrix must be transformed, for which the Lth row of the matrix is divided by and a second table is compiled, the elements of which are denoted by letters and they are determined as follows:  Determine the value of F -to include the factor X by the formula (3.30). 26. If F> Fij, then the factor Xj is not included in the regression equation and the step method ends. 27. The free term of the equation at each step is calculated by the following formula: 28. When F <Fn -th factor is included in the regression equation and repeated calculations are made starting from the 15th stage. 29. Since the factors Xj and XL are included in the regression equation, particular F -criteria are determined. Partial F -criterion for XL, in the presence of Xj in the model is determined by the formula: , 30. For the factors L and j, the coefficients of variation, the deviation of the residuals, the standardized bcoefficients for Xj and XL are determined according to the formulas specified in steps 17, 18, 19. 31. If F is a criterion for either the j-th or L-th factor is less than the critical one, that is, Fxj <Fcr, then the j-th factor is excluded from the regression equation. 32. Go to step 16. 33. When calculating Vmax, the value of F is determined to include the factor. If some k-th factor has just been excluded from the regression dependence at the previous step and F for inclusion is more than F -critical, that is, Fк> Fкр, then the process of including and excluding factors in the model stops and the free term of the equation is determined by the formula (3.40 ). 34. The mathematical model of specific power consumption, depending on the factors obtained, can be expressed by the following equation: where knumbers of factors included. In (33), we introduce the following transformation: When the i-th factor changes by β percentage, which is determined by the following formula: Specific electricity consumption will change by β percent, which is determined by the following formula: The given method makes it possible to single out from the whole variety of factors influencing energy indicators, the most significant ones and to give an appropriate assessment to each of them. The above algorithm is unified for all the above calculation options and is included in the corresponding resulting calculations of the specific power consumption rates.

Conclusion
-The proposed methods and mathematical models can be used for current and medium-term forecasting of electricity consumption.
-The analysis has been performed and energy characteristics of production units have been developed depending on the given and total volume of products manufactured. Energy intensity coefficients for the entire range of products manufactured by individual production units were obtained using linear programming methods; -the possibility of applying a mathematical method of experiment planning to calculate energy characteristics and identify technological factors that have a significant impact on energy indicators has been justified; -a method has been proposed to assess the impact of the volume of yarn and spindle downtime on the energy performance of coiled, twisted, spinning and weaving production units