Keywords

1 Introduction

The development of Industry 4.0 [1] is associated with the industrial revolution characterized by the following features: new energy technologies are not used, new raw materials are massively used, a significant increase in the energy efficiency of production, the ability to predict the future, a significant acceleration of modernization and the emergence of an exponential increase in labor productivity. In the industrialized countries of Europe and North America, states are the sources of the actors of the ongoing changes. In Russia, a national project for increasing labor productivity has been proposed, which presents the goal of ensuring by 2024 the rate of increase in labor productivity in organizations belonging to the basic, non-resource sectors of at least 5% per year. By 2024, it is planned to increase labor productivity by more than 20%. This will allow the Russian Federation to move to the level of the world's leading economies. The modern fourth industrial revolution has features similar to the previous ones: the modernization of production technology depends on the improvement of the management system. At the same time, for more complex technologies it is required to increase the relative volume of automated control systems. Experts say that more than 40% of enterprises use digital technologies at a critical level, then an exponential growth of industry digitalization is expected [2, 3]. Such digitalization requires the massive implementation of cyber-physical systems capable of self-tuning and adapting to the new needs of manufacturers and consumers.

The structure of these changes includes the following distinctive features: Internet of Things (IoT for short) [4, 5]; digital twins (Digital Twin) [6]; 3-D printing [7,8,‒9]; cloud computing and big data [10,11,‒12]; cybersecurity [13, 14]; digital platforms [15].

The fourth industrial revolution has begun, its distinctive features are manifested quite clearly and extensively. At the same time, it can be argued with confidence that, as in previous revolutions, the one who is able to create, develop and use technologies of distinctive features earlier than others will gain an advantage over other enterprises and countries, and the gap will grow exponentially.

According to expert estimates, the benefits will be given to organizations that are actively working with innovations and have their own competencies in the aspect of digitalization.

One of the possible obstacles is the lack of innovative technologies and models. Fundamentally new technology for constructing and using mathematical models and automated control systems can be based on mathematical models using the mathematical apparatus of status functions [16].

Moreover, in this regard, the problem is the need to develop possible mathematical models of digital twins of socio-economic structures that differ from the known models in specific features.

Let us present a model for the formation of digital twins of socio-economic structures based on status functions developed and proposed in [17].

2 Statement of the Problem of Creating New Mathematical Models for Assessing the State of Socio-Economic Systems

The process of integrating digital twins (CDs) into the classical control system is actively developing in each of the following functions.

  1. 1.

    Observation function. There is a change in the technological order. Industry 4.0 deals with completely new phenomena that are digital in nature. Digital Twin in our case is a software analogue of the socio-economic system (SES).

  2. 2.

    Control function. synthesis and improvement of CD.

A brief description of the subject area of SES digitalization is given in Table 1.

Table 1 Classification of approaches to the digitalization of SES
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The analysis performed in Table 1 shows that the development of models of the CSD of socio-economic structures finds application in the analysis of big data and predictive analytics for the formation of forecasts based on new mathematical models. At the same time, communication between socio-economic structures can be carried out using the CD. Then the communication in the system governing body—controlled socio-economic object can be reduced to the model of meta-communications: governing body—CD—managed socio-economic object, managed socio-economic object—CD—the governing body. The new data analysis model will create a virtual integrated environment for big data analysis and predictive analytics. In the new integrated complex, in the implementation of management actions using big data based on the central data center using problem-oriented software tools, the corresponding technological and technical platform is created by cyber-physical systems. For the synthesis of this organizational and technical platform, the technology of building a CD is of decisive importance.

  1. 3.

    Functions of analysis, management. These functions are required to be provided in the architecture of a functioning integrated platform of a cyber-physical system. When implementing these functions, significant difficulties arise: limited resources, possible lack of feedback in the subsystem: the governing body is a controlled socio-economic system.

A cyber-physical system includes a set of interacting components to achieve a single goal in the control process [17]. Mathematical models of such complex complexes generally correspond to the main provisions of synergetic [18] and are created on the basis of methods developed in physics, chemistry, and biology [19].

The difficulty of using mathematical models and methods is associated with the lack of measurement scales that are used in natural sciences (measurement of speed in m / s, mass in kg, etc.). To overcome it, it is advisable to use linguistic variables to describe, for example, psychophysiological indicators in ergatic systems based on the theory of fuzzy sets [20], as well as to interpret data obtained in the course of field experiments using the so-called status functions (SF).

The SF-based algorithm allows you to enter linguistic assessments and represent them with complex-valued functions, called status functions. Mathematical models based on SF are devoid of many of the shortcomings of classical models of socio-economic processes, such as static nature and the impossibility of taking into account the intersection and mutual influence of the studied components of the processes.

The main trends in the synthesis of CSD of socio-economic systems (SES) are as follows:

  1. 1.

    an increase in the number of states that are evaluated, as well as the components of the data structure that are subject to analysis;

  2. 2.

    increasing the productivity and complexity of information platforms for the implementation of problem-oriented software tools;

  3. 3.

    the complication of requirements for the structure and quality of the estimated values of the SES parameters, the multicomponent nature of the nested structures of the estimates, the overlap and complexity of the processes that are implemented in socio-economic systems;

  4. 4.

    activation of subjects and processes of interaction in organizing management to increase the degree of its automation.

This raises the problem of creating mathematical models that allow increasing the number of controlled variables and parameters to automate the used decision support process (DSS). This initiates the use of modern technologies, techniques and corresponding mathematical methods [19, 20]. So, when developing new methods of managing the educational process, various models are actively used, fuzzy, boolean, etc.

Thus, the development of new methods for assessing the state of socio-economic systems, taking into account the complexity, subjectivity and multidimensionality is a very urgent problem. It is advisable to apply the proposed mathematical method in the construction of an intelligent monitoring system, as well as PM in the analysis of indicators of the functioning of the SES, such as, for example, individual enterprises or the structure of indicators of the activities of the regions of the Russian Federation to increase the effectiveness and efficiency of management according to the characteristics of the SES and the successful solution of problems caused by the goals of functioning. management systems for individual enterprises or regions.

3 The Concept of Development and Integration of New Mathematical Models

The concept of mathematical modelling based on effective mathematical methods and building a model that describes ergatic interactions in the form of corresponding communications of digital twins based on an integrated platform for various cyber-physical systems, it is advisable to display as a sequence of five stages (see Fig. 1).

Fig. 1
figure 1

Architecture of an integrated cyber-physical system platform using mathematical modelling to apply effective mathematical methods based on status functions

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Stage 1. Interpretation of data obtained as a result of field experiments. At this stage, an analysis of the data of the controlled variables of the SES is required. So, for example, the analysis of indicators of socio-economic activity of the regions of the Russian Federation and models was carried out in works [22, 23].

Stage 2. Mathematical modelling of socio-economic processes and objects. The best basis that is advisable to use for the synthesis of mathematical models of digital twins is SF. From their combination, it is possible to form the final mathematical models, which are proposed and presented in the article [14]. The new mathematical model allows building an integrated platform for cyber-physical systems. On this platform, it is advisable to implement training for the synthesis of hybrid ergatic systems.

Stage 3. Mathematical model of communications in the environment. We will focus on meta-communication in a cyber-physical system. Moreover, the direct communication and interaction of the governing body—the managed socio-economic object are replaced by meta-communication of digital twins: the governing body—the central office—the managed socio-economic object, the managed socio-economic object—the central office—the governing body.

Stage 4. Checking the adequacy of the proposed mathematical models using data from field experiments. It is advisable to interpret the information obtained as a result of experiments by the SF method, as well as on the basis of the Kolmogorov-Chapman equations [25, 26] based on the data obtained in the FIRA PRO system.

Stage 5. The complex of problem-oriented software. Below is a description of the architecture for assessing the parameters for assessing the activities of SES on the example of assessing the competitiveness of the regions of the Russian Federation.

4 Example of Implementation of the Concept

Let us describe the 5th stage of building a complex of problem-oriented programs. First, based on the collection of data, a linguistic description of the assessment results is formed. To assess the state of the variables “Degree of depreciation of fixed assets”, “The number of populations with incomes below the subsistence level”, “The number of official unemployed”, “Dilapidated and dilapidated housing”, “Debt on taxes and fees”, Table 2 with the corresponding linguistic description is used. For the generated assessment tables, SFs are put into correspondence, which are presented in Table 3.

Table 2 Linguistic description of variables for data from the FIRA PRO system
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Table 3 Expressions of basic SFs for substitution in accordance with the table of linguistic description of variable values for data from the FIRA PRO system
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Basic SF values consist of two parts: amplitude and phase. The amplitude parts are real and are shown in Table 4. They are the orthonormal basis of the system and represent a set of basis functions that can describe all possible states of the system. The phase part of the SF is complex and indicates the direction of the trend—or in which direction the shift of the mathematical value of the SF is expected.

Table 4 Examples of linguistic estimates, mathematical expressions and the form of the amplitude part of the basic SF
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As a result, the corresponding complex-valued status functions (SF) are formed.

Secondly, the formation of hierarchical interconnected structures is carried out. The result is a causal graph (see Fig. 2).

Fig. 2
figure 2

A causal graph for assessing the risks of regional competitiveness

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In the diagram shown in Fig. 2 structure of the following indicators: the cost of fixed assets (E29); depreciation rate of fixed assets (E30); density of railway tracks (E31); road density (E32); per capita money income (E33); population with incomes below the subsistence level (E34); the number of officially unemployed people (E35); life expectancy (E36); population growth rate (E37); migration growth rate (E38); dilapidated and dilapidated housing (E39); the number of doctors per 10,000 population (E40); the number of reported crimes (E41); emissions of polluting products (E42); discharge into water bodies of polluted wastewater (E43) was obtained as a result of data collection and analysis [28, 29]. The rest of the indicators uniting leaf vertices into a causal graph were introduced as a result of logical operations presented in [16].

Third, a system of differential operators is formed for the structure of groups of higher indicators. As a result, the system of differential equations based on SF becomes an analogue of the described structure. The system is obtained similarly to Forrester's equations of world dynamics.

Note that recently, in general, interest in the use of modern mathematical models, digital technologies for mathematical modelling of socio-economic systems using the ideas of digital twins have increased significantly [29, 30].

5 Conclusion

Thus, based on the estimates presented by the SF method, it is possible to construct dynamic models of digital twins for use in cyber-physical systems. Structures that can be built on the basis of status functions can be simply formed from models for evaluating indicators based on linguistic terms. Status functions are a “bridge” between a person's simple ideas about the world around him and phenomena to the strict uncertainty of quantum mechanics. The advantage of the model is that it takes into account the complexity, high degree of uncertainty, and the intersection of socio-economic processes. Models allow building-level structures of possible states. The main advantage is the predictivity of the model and its suitability both for predictions and for describing meta-interactions in an integrated environment of cyber-physical systems.

The proposed generalized scheme of mathematical modeling based on the use of effective numerical methods and a complex of problem-oriented programs is advisable to use to assess communications in an integrated environment of cyber-physical systems based on status functions.