Optimal management of efficiency gains manufacturing enterprises

Сформулированы две ключевые задачи, связанные с повышением эффективности производственных предприятий на основе оптимального управления качеством и снижением себестоимости производимой продукции. Решение первой задачи сводится к допустимому росту качества производимой предприятием продукции путем ее модернизации с учетом покупательской способности основной массы потребителей на различных сегментах рынка. Приведен критерий позволяющий определить оптимальные объемы производства продукции с различным уровнем качества для различных сегментов рынка, обеспечивающие производственному предприятию получение максимально возможной прибыли за счет роста потребительских свойств производимой продукции и повышения на этой основе ее рыночной стоимости. Решение второй задачи связано с условной минимизацией переменных издержек производства без потери требуемого уровня качества различных видов производимой предприятием продукции, которая обеспечивается путем сбалансированного ввода факторов производства в производственный процесс. Такой ввод факторов производства сопровождается снижением себестоимости производимой предприятием продукции и получением на этой основе дополнительной прибыли. Для проведения условной оптимизации, когда решение задачи оптимального управления получением дополнительной прибыли находится на границе области допустимых значений вводимых в него факторов, приводится критерий определяющий условие сбалансированного их ввода в производственный процесс и снижения на этой основе переменных издержек производства различных видов продукции в краткосрочном периоде. Two key tasks have been formulated related to improving the efficiency of manufacturing enterprises based on optimal quality management and reducing the cost of production. The solution to the first problem is reduced to an acceptable increase in the quality of the products produced by the enterprise by means of its modernization, taking into account the purchasing power of the bulk of consumers in various market segments. A criterion is given that allows you to determine the optimal production volumes of products with different levels of quality for different market segments, providing a manufacturing enterprise to obtain the maximum possible profit due to the growth of consumer properties of the products produced and on this basis increase its market value. The solution of the second problem is associated with the conditional minimization of variable production costs without losing the required level of quality of various types of products produced by the enterprise, which is ensured by balanced input of production factors into the production process. This input of factors of production is accompanied by a decrease in the cost of products manufactured by the enterprise and the receipt of additional profit on this basis. To carry out conditional optimization, when the solution to the problem of optimal control for obtaining additional profit is on the border of the region of admissible values ​​of the factors introduced into it, a criterion is given that determines the condition for their balanced input into the production process and, on this basis, reduce the variable production costs of various types of products in the short term.

SYNTHESIS OF AN ADAPTIVE AUTOMATIC SYSTEM OF AIRCRAFT CONTROLS WITH MULTIDIMENSIONAL PI-REGULATOR

The article presents the synthesis of a functional diagram of an adaptive automatic control system (ACS) for controlling an aircraft with an automatically reconfigurable multidimensional PI controller, which provides the minimum static and minimum mean square error of control with minimal energy consumption for the formation of the control exposure. The synthesis of ACS algorithms is performed as a result of solving the problem of conditionally minimizing the quadratic functional of the generalized work (taking into account restrictions on state variables and control actions given by differential equations of the control object (CO) and inequalities). The mathematical description of the multidimensional CO is carried out using the CO model in the state space, which automatically takes into account the mutual influence of individual control loops on each other. As the state variables of the aircraft, linear displacements, speeds and accelerations of the center of mass of the aircraft, and angular displacements, speeds and accelerations of the rotational movement of the aircraft relative to the center of mass are used. The matrix equation of dynamics of the aircraft is formed by a system of nonlinear differential equations of the first order of forces and moments of forces acting on the aircraft. To ensure the minimum static control error, integrators are included in the ACS (for each control action). The algorithm for the formation of control actions of the extended CO, providing the declared properties of the ACS, is obtained as a result of solving the problem of conditional minimization of the generalized work functional. The task of conditional minimization of a functional with constraints is performed by the maximum principle. The resulting two-point boundary value problem is transformed by the invariant immersion method into a Cauchy problem for optimal values of state variables. The evaluation of the characteristics of a specific adaptive ACS for the spacecraft is expected to be obtained as a result of further research by mathematical modeling.

OPTIMIZATION OF UNIMODAL FUNCTIONS USING PARABOLIC PREDICTOR search

A combined parabolic predictor search is proposed for the conditional minimization of the unimodal function using the predictive-based selective application of phases of extremum search by golden section search and parabolic search. The formula for calculating the value of parabolic predictor function is given, with its help it is possible to work out the forecast and tactics of extremum search of the minimized function. Predictor includes forecasting extremeness, monotony and constancy of function on a segment of uncertainty. Identification forecast for a direct function is described, using which allows to find a solution in three calculations. The assertion is made that if three successive computations of a function give points with similar ordinates, then abscissa of each point can be a solution of the problem. The procedure of identifying non-direct monotonic functions is described. It is shown that the reliability of monotonicity forecast can be determined by five calculations of the function. There has been described the procedure of using phases of parabolic method, which can be performed at favorable prediction of detecting the internal extremum of function. It has been stated that carrying out these phases, even with favorable forecast, can be considered inexpedient for cases when it is recognized that the problem is weakly sensitive or insensitive to the parabolic forecast. Block diagrams of algorithms implementing the method are given. It is shown that, compared to golden section search, the predictor has 3-5 times faster response for smooth functions and is comparable by this criterion to Brent method. The predictor achieves the greatest speed when minimizing monotonic functions. The method works somewhat slower than golden section search, however, it is much faster than Brent method when searching for the minimum of piecewise, flat, planar and other functions of a similar nature for which approximation of parabola does not give the expected effect. In comparison with Brent method, parabolic predictor has 1.5-4 times more speed in solving problems of such type.

About the estimation of the convergence rate of projection-iteration processes of conditional minimization of a functional

We study projection-iterative processes based on the conditional gradient method to solve the problem of minimizing a functional in a real separable Hilbert space. To solve extremal problems, methods of approximate (projection) type are often used, which make it possible to replace the initial problem by a sequence of auxiliary approximating extremal problems. The work of many authors is devoted to the problems of approximating various classes of extremal problems. Investigations of projection and projection-iteration methods for solving extremal problems with constraints in Hilbert and reflexive Banach spaces were carried out, in particular, in the works of S.D. Balashova, in which the general conditions for approximation and convergence of sequences of exact and approximate solutions of approximating extremal problems considered both in subspaces of the original space and in certain spaces isomorphic to them were proposed. The projection-iterative approach to the approximate solution of an extremal problem is based on the possibility of applying iterative methods to the solution of approximating problems. Moreover, for each of the "approximate" extremal problems, only a few approximations are obtained with the help of a certain iteration method and the last of them as the initial approximation for the next "approximate" problem is used. This paper, in continuation of the author's past work to solve the problem of minimizing a functional on a convex set of Hilbert space, is devoted to obtaining theoretical estimates of the rate of convergence of the projection-iteration method based on the conditional gradient method (for different ways of specifying a step multiplier) of minimization of approximating functionals in certain spaces isomorphic to subspaces of the original space. We prove theorems on the convergence of a projection-iteration method and obtain estimates of error and convergence degree

Application of methods of conditional multidimensional minimization to the ballistic trajectory calculation problem

The paper focuses on the problem of calculating the approximate ballistic missile trajectory, the calculation ensuring that the missile travels from a given launch point to the finish point and covering the entire range rate for the missiles of the type considered. The missile trajectory is defined by a system of nonlinear differential equations. A different range is achieved by changing the initial values of the flight-path angle and the operating time of the missile stages. Due to the physical significance, these variables are constrained. The problem of multidimensional conditional minimization by the method of barrier functions with minimization of Nelder - Meed method