# Planning with Optimization Models

Optimization models are based on the selection of the optimal option from a variety of possible ones by comparing them according to the optimization criterion (s). The optimization economic-mathematical model consists of a target function and a system of constraints. The objective function describes the purpose of optimization and reflects the dependence of the indicator by which optimization is carried out on independent variables (constraints). The system of constraints reflects objective economic relations and dependencies and is a system of equalities and inequalities between independent variables.

Examples of optimization models in planning and forecasting: models for optimizing the development and location of production; models for optimizing the structure of production of industrial products, models of the agro-industrial complex, transport tasks, with the help of which the rational attachment of suppliers to consumers is carried out and the minimum transport costs are determined; and others.

Examples of macroeconomic models are static and dynamic models of intersectoral balance.

The static model is as follows:

n

∑  aij xij + yi = xi  (i = 1, n),

j=1

where aij is the direct cost ratio (industry average consumption standard of industry i used as a means of production for the output of a unit of output of industry j);

chij is the gross production of the j-th consumer industry (j = 1, n);

xi — gross production of the i-th industry-supplier (i = 1, n);

yi is the volume of final products of the i-th industry.

At the same time, the ∑ aij xij is an intermediate product (the number of products of the i-th industry used in the j-th industry in the production process).

The statistical model of the intersectoral balance can also be expressed in the following way:

n

chi = ∑  vij ui  (i = 1, n),

j=1

where vij is the coefficient of total material costs, which reflects the value of the products of the i-th industry, necessary at all stages of production to obtain a unit of the final product of the j-th industry.

The coefficients of direct and total costs differ in that the former are determined per unit of gross output of the industry and are industry average, while the latter are calculated per unit of final output and are national economic. Total cost ratios exceed direct cost ratios by indirect costs.

The dynamic model of the intersectoral balance characterizes the production relations of the national economy depending on the volume of investments (i.e., reflects the process of reproduction in dynamics) and provides a link between the plan-forecast of production and the plan-forecast of capital investments. The simplified model looks like

n  n

xit =  ∑  aijt ujt + ∑  Δ Fijt + Zit  (i = 1,n),

j=1  j=1

where t is the index of the year; Δ Fij — products of the i-th industry, directed as production capital investments to expand production in the j-th industry; Zi is the sum of the final product of  the i-th industry, with the exception of products aimed at expanding production.

With the help of intersectoral balances, many macroeconomic problems are solved, among them the analysis of the impact of effective demand, export and import volumes on the gross output of industries and final consumption, the impact of structural changes in individual industries on other industries and the economy as a whole, the impact of technological innovations on the structure of the economy, calculate the volume of investments necessary to achieve the required macroeconomic indicators, etc.