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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.22 n.1 San José Jan./Jun. 2015


Optimal production–sales strategies for a company at changing market price

Estrategias óptimas de ventas y producción para una compañía en un mercado con precios cambiantes

Ellina V. Grigorieva*+, Evgenii N. Khailov*


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 productionsales 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 problema 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 máximum 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 bangbang controls can be optimal. Economic analysis of the optimal solutions is discussed.

Keywords: nonlinear microeconomic control model; production-sales strategy; Pontryagin maximum principle; Hamiltonian system.


En este artículo consideramos un monopolio produciendo un producto de consumo de gran demanda. Su precio de mercado depende del volumen de producción descrito por la función de producción de Cobb-Douglas. Una actividad de producción y ventas de la firma es modelada por una ecuación diferencial no lineal con dos controles de frontera: la participación en el resultado de las ventas que la compañía reinvierte para expandir su propia producción, y el monto de los préstamos a corto plazo adquiridos del sistema bancario con el mismo propósito. Se plantea y resuelve el problema de maximizar la ganancia total descontada en un intervalo de tiempo dado. Para encontrar las estrategias óptimas de producción y ventas para la compañía, se usa el principio del máximo de Pontryagin. Para investigar el problema de valores de dos puntos de frontera que aparece para el principio del máximo, se aplica un análisis del sistema hamiltoniano correspondiente. Basado en un análisis cualitativo del sistema, encontramos que dependiendo de las condiciones iniciales y los parámetros del modelo, tanto el control singular como el bang-bang pueden ser óptimos. Se discute un análisis económico de las soluciones óptimas.

Palabras clave: modelo de control microeconómico no lineal; estrategia de producción y ventas; principio del máximo de Pontryagin; sistema hamiltoniano.

Mathematics Subject Classification: 49J15, 58E25, 90A16, 93B03.

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*Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, USA. E-Mail:
Department of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia. E-Mail:

Received: 23/Feb/2014; Revised: 28/Aug/2014; Accepted: 3/Oct/2014

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