Mathematical Model of Optimal Planning of the Production Process

In the first half of the last century, due to the increasing complexity of production processes, there was a need for their more efficient organization. During this period, the foundations of mathematical modeling of economic processes were laid. Modern mathematical models of optimal planning integrate artificial intelligence, machine learning methods and large databases, take into account uncertainty and risks in production processes using stochastic dependencies and probability theory methods in the models. This makes it possible to model even more complex systems and take into account more factors, such as fluctuations in demand, changes in production supply chains, etc. Today, optimal planning models are the basis of enterprise management systems (ERP) and are used in various industries: from manufacturing to logistics and energy. This article considers an economic-mathematical model for planning the optimal production process under certain assumptions about the economic state. That is, the profit of the enterprise, which can produce different types of products, for each type of these products depends on the economic state of the country. It is determined what share in the total production of the enterprise will be occupied by a certain type of product in order to obtain maximum profit. The profit of the enterprise depends on the state of the economy, therefore the expected profit is characterized by the mathematical expectation of profit. For an optimal production plan, you need to strive for the best ratio between expected profit and risk (mean square deviation), that is, it is necessary to find the maximum of the function that characterizes this ratio and is a function of n unknowns. To find the extremum of this function, we find partial derivatives and obtain n nonlinear equations with n unknowns. Performing some transformations, we reduce this system to a system of n linear equations. If the non-negativity conditions are not imposed on the variables, then when solving the system, it may happen that some variables will take negative values. This means that in order to obtain optimal profit, it is not recommended to manufacture the corresponding type of products