Reliable Optimal Production Control with Cobb-Douglas Model

Production is the most fundamental activity in our economy. In this paper, a Cobb-Douglas production function is used as the mathematical model to describe the relationship among production, labor and capital. Two reliable production optimal control problems are studied. Algorithms to find dynamic optimal control intervals are provided with interval parameter presentations and interval computations. 1. Optimal Production Control with Cobb-Douglas Model: Traditional (Non-Interval) Case Production. Every day, various products are produced to meet different demands in our society. Production is truly one of the most fundamental activities in our economy. Every producing firm wants to maximize its profits: • In an equilibrium market, where the amount produced is more or less fixed (by the demand and by the firm’s market share), in order to maximize profits, the firm needs to minimize production costs. • In a seller’s market, in which the supply of a product is smaller than the demand for it, maximizing profits means producing the maximum amount within the available production costs. Production function. In both cases, to optimize production, we must know how the output Q depends on the production costs. The dependence of Q on controllable parameters is called a production function. One of the most widely used production functions was proposed by Cobb and Douglas and has the following form (see, e.g., [3]): Q = A ⋅ Lα ⋅ Kβ , (1) where L is labor (measured in certain units), K is capital, and A, α, and β are (constant) parameters; these parameters depend on the firm, on the produced unit, etc., and have to be determined experimentally. Comment. In many real-life situations, α + β = 1. This equality has a simple economic interpretation: if we increase both labor and capital twofold, we will thus