Probabilistic model of harmonization of the production process with the regional market

The market can be considered the brightest representative of stochastic processes. But the production of products also cannot be considered deterministic. Therefore, when managing an enterprise, it is necessary to strive for the harmonization of two stochastic processes. The model of market behavior and the production process was chosen that obeys the Kolmogorov equation. Then, the question was raised - how to manage production: to accelerate or freeze production volumes smoothly, or to apply the quickest possible braking and accelerating of production, following the classical rule - time is money. Using the Euler-Lagrange method, it is shown that the matching of stochastic processes can be reduced in this formulation to a third-order differential equation from the control action in the event of a smooth change. In the case of stepwise application of control actions, the Pontryagin maximum principle is applied to solve the optimal control problem. On the basis of the virtual work concept of control actions, as the work is performed in a larger volume with the most frequently encountered values of market demand, these two methods of control are given to the possibility of comparing the results obtained. As a result, it is shown that for a product with a long production cycle, optimal control, based on the Euler-Lagrange method, has the greatest efficiency, and if the production cycle is short, then control is needed with maximum application of control actions that accelerate or slow down production on the Pontryagin maximum principle. The technique proposed distinguished these two cases even with a rough estimate, which proves its efficiency.