OPTIMAL PRODUCTION–SALES STRATEGIES FOR A COMPANY AT CHANGING MARKET PRICE - ELLINA V. GRIGORIEVA; EVGENII N. KHAILOV (Revista de Matemática: Teoría y Aplicaciones, 2015)
In this paper we consider a monopoly producing a consumer good of high demand. Its market price depends on the volume of the produced goods described by the Cobb-Douglas production function. A production-sales activity of the firm is modeled by a nonlinear differential equation with two bounded controls: the share of the profit obtained from sales that the company reinvests into expanding own production, and the amount of short-term loans taken from a bank for the same purpose. The problem of maximizing discounted total profit on a given time interval is stated and solved. In order to find the optimal production and sales strategies for the company, the Pontryagin maximum principle is used. In order to investigate the arising two-point boundary value problem for the maximum principle, an analysis of the corresponding Hamiltonian system is applied. Based on a qualitative analysis of this system, we found that depending on the initial conditions and parameters of the model, both, singular and bang- bang controls can be optimal. Economic analysis of the optimal solutions is discussed.