On the statistical stability of the optimal solution, found from the regression equation

The results of any experiments are accompanied by errors due to measurement inaccuracy and the influence of uncontrollable factors. This means that when conducting and using the results of experiments, it is necessary to be able to establish the accuracy of the solutions and conclusions obtained. This is especially important when searching for optimal conditions, since optimization problems are poorly conditioned and are very sensitive to measurement and calculation errors. This work is devoted to the study of the sensitivity of statistical optimization models obtained on the basis of the regression equation and used in the study of food technology processes. For an abstract optimization problem, the coordinate of the extremum point was considered as some random variable, the value of which varies under the influence of experimental errors. As a result of the research, formulas for the function and density of distribution of this quantity were obtained. They allow you to calculate the confidence interval of the optimum position. Using data from a literary source as an example, it is shown that even with satisfactory statistical characteristics of the constructed regression equation, the coordinate of the extremum point can vary within a very wide range - more than 100% of the found estimate. Measures are proposed to increase the statistical stability of the solution to the optimization problem by shifting the planning area to the expected vicinity of the optimal point. Using the constructed distribution laws, numerical estimates of the degree of narrowing of the confidence interval of the coordinates of the extremum point after such a shift were obtained. The achieved effect is demonstrated using an example of an optimization problem from a literary source. In addition, it was also found that when constructing a quadratic regression equation in the optimal area, regression significance indicators may deteriorate compared to models built for a remote planning area. Therefore, when experimenting in the optimal region, it is especially important to reduce the influence of experimental errors, for example, by increasing the number of parallel experiments