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The paper shows the inevitability of technological mode shift driven by the Industry 4.0, which implies the ubiquitous implementation of information technology, total automation of various processes and creation of cyber-physical systems with artificial intelligence. This requires a complete restructuring of manufacturing systems and production relations, especially in the economies of those countries that want to take a decent place in the new international division of labour of the digital future. An analysis of the world experience of such changes connected with smart industrialization, digital transformations of the economy, the emergence of the industrial Internet of Things and big data processing made it possible to draw the conclusion that it is necessary to apply economic and mathematical methods to justify the expediency of such transformations: economic validity, as well as physical viability of newly created systems. The use of the apparatus of economic and mathematical modeling allows studying properties of the smart system that is being designed, evaluating its effectiveness and risks, anticipating the emergence of problems and errors - without the risk of incurring significant losses which is inevitable when making direct changes in the object of research. Therefore, the purpose of this paper is to study the world experience in the economic and mathematical modeling of smart enterprises and to substantiate its use in the conditions of Ukraine. The review of publications, reflecting the aspects of economic and mathematical modeling in these areas, allowed to conclude that the methodical and methodological apparatus for modeling these processes is unsystematic and inefficient, as well as to formulate recommendations on the economic and mathematical modeling of smart enterprises in Ukraine. In order to take into account the specific features of Ukraine's technological and institutional development, a number of economic and mathematical modeling tools based on the use of production functions, models of inter-branch balance, network optimization models and simulation models based on stochastic dependencies were offered to support the creation of smart enterprises. [ABSTRACT FROM AUTHOR]

The article identifies the factor of production, associated with smart industrialization, and provides modeling results of the corresponding production function on the example of manufacturing industry in Germany, as a country where the "Industry 4.0" development program has been announced and is being implemented at a nationwide level. It is argued that the existing approaches that takes into account scientific and technological progress within the design of production functions are not suitable for modeling the transformation of the impact of production factors in the process of smart industry formation, since scientific and technological progress in most papers is represented not by a specific measurable indicator, but simply by a natural series of numbers, reflecting the part of the change in production that is not explained by changes in the factors considered. It has been found that in German manufacturing industry output is growing while labor and capital expenditure decreases, which indicates the influence of at least one more factor related to the transition to the new technological mode - the smart industry. The difficulties of assessing the impact of the "smart factor" on production have been identified: both objective (the interdependence of computerization factors and the difficulty of distinguishing the contribution of each of them) and subjective (complete absence or fragmented statistical information). Based on the analysis of statistics, it has been found that the costs of software and databases are the most accurate indicator, reflecting the impact of the computerization factor on the output. A model, that is a modification of the Cobb-Douglas production function, has been designed, in which the added value in the processing industry is used as the endogenous variable, and the number of hours worked (labor factor), the cost of machinery and equipment with a 1 year lag (capital factor) and the cost of software and databases (computerization factor) are the exogenous factors. When analyzing the modeling results, authors found that computerization has turned into an important production factor and demonstrates the potential to replace other factors of production - labor and capital. The model can be used to substantiate the directions of smart industry development at the macro level, as well as the basis for developing criteria for assessing the level of enterprise "smartness" at the micro level. [ABSTRACT FROM AUTHOR]