Applying the Optimal Control Methodology to the Mathematical Model of Sales Volume Distribution in a Company

In an era of change, business management should rely more on rigorous mathematical calculations and forecasts, rather than expert estimates, especially when it comes to distributed indicators of the company. This paper focuses on a mathematical model for the distribution of sales by their value, which is an initial-boundary-value problem for a partial differential equation of parabolic type and taking into account the stochastic nature of the sales process. The purpose of the study is to provide a mathematical justification for the methodology for calculating such a distributed sales plan, which will bring the company as close as possible to the desired distribution of sales. The methods of the theory of optimal control of distributed systems, the theory of partial differential equations, mathematical, and computer modeling are applied. Two problems of optimal control of the distributed system under consideration are formulated and interpreted in terms of the model. In the first task, the share of sales for the minimum amount plays the role of the control function. In the second task, we focus on the distribution of sales to new customers. Optimal laws are given for these functions, as well as optimality systems in strong form. An example of a numerical calculation of the model performed by Comsol Multiphysics is considered in the case when it is necessary to switch from a sales structure with dominant sales in the economy price segment to a structure in which sales in the expensive price segment prevail. Sales calculations based on the methodology proposed by the authors can be used in strategic planning of the company’s development.