MATHEMATICAL MODEL OF THE OPTIMAL MARKET OF MANY GOODS
Abstract
A nonlinear dynamic mathematical model of the free market for many goods is considered. In the considered model of the inertial market of both one and many goods with a fixed demand line, it is assumed that the sales volume of each product at each step (interval) of discrete time is determined by the minimum of two quantities: the volume of goods supplied to the market (supply of goods) and the volume of demand. At the same time, there are 3 zones of supply volumes: a zone of shortage of goods, a zone of market overstocking and a zone of dynamic market equilibrium. In the first zone, demand exceeds supply, so that the seller receives less profit. For each of the zones, a detailed analysis of the behavior of the supply and demand functions was carried out. Since the sales volumes depend on the price of the goods and the ratio of supply and demand, each zone has its own conditionally optimal prices of the goods, which depend on the level of supply in each zone, and ensure the maximum profit of the seller for each fixed volume of supply of the goods. It is shown that the state of the market for many (n> 1) goods is characterized by the indicated three possible zones for each of the goods, which leads to 3n possible zones of the market state. Due to the constraints of the type of inequalities inherent in the considered mathematical model of the market, the target function of the market (the total profit of the seller) is a piecewise differentiable function of the vectors of prices and offers of goods with discontinuities in the gradient of this functions on the lines of equality of supply and demand, that is, on the division lines of zones 1 and 2 (in zone 3), which makes it difficult to solve the problem of vector optimization of this function. It is not possible to construct an analytical solution to the problem for each of the 3n zones. In this regard, an original approach to a unified representation of an objective piecewise differentiable function through a system of indicator matrix predicate functions is proposed, which made it possible to represent the objective function of many variables as conditionally smooth, everywhere differentiable for hypothetical values of predicate functions, and obtain an analytical solution to the problem of multidimensional optimization.
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