The world is at the threshold of the fourth industrial revolution, which has already begun. It requires enterprises and even sectors to move toward Industry 4.0. Innovative systems (IS) play an important role in this process. In the article, the innovative systems in Industry 4.0 are considered to be complex systems whose components have a lot of tasks; in particular, to produce innovative policy; to provide the subjects of innovative activity with the necessary resources; to participate directly in the process of creation, commercialization, and the practical use of new knowledge; to implement integration approaches between these processes, etc. The complexity of the innovation system leads to the development of modern approaches to their modeling, as a tool for further designing, creating, and modifying real innovative systems of different levels of organization under the conditions of Industry 4.0. In the simulation of IS under the conditions of Industry 4.0, the description of the subsystems by a number of sets is proposed. The model is described by the graph of relationships, including the abstract level of the hierarchical model of IS, its elements, indicators and their values, functions, actions and operations, their states and efficiency, and the tree of goals. In order to make optimal solutions, using the mathematical apparatus of the theory of Markov chains to study the dynamic and static characteristics of the states of the IS is proposed. This approach can be widely used in the simulation, designing, development, and rebuilding of IS at different levels of an organization.

In order to revive innovative processes, which in the post-war period should be directed to ensuring the revitalization of the economic system on an innovative basis, restoring infrastructure and ensuring the primacy of promising technologies in Ukraine, the article examines the theoretical foundations of the development of innovative activity. The structural and logical model of the study consisted in identifying approaches to the definition of the concept of "innovative development of the enterprise" and the keywords inherent in them, graphical visualization of the obtained results using the Sankey diagram and their use as basic components that outline the meaning of innovative development for enterprises. The article uses the methods of scientific research, theoretical generalization, systematization, and statistical analysis, which were carried out to construct the Sankey diagram. The basic components outlining the meaning of innovative development for enterprises are defined as: ensuring the enterprise's ability to use innovative ideas and realizing its innovative potential (23,8%); formation of conditions that accompany a change in the state of the enterprise based on innovations with the aim of achieving positive changes (30,9%); renewal and implementation of reforms based on innovations (7,1%); guaranteeing changes in the enterprise based on innovation and realization of growth-oriented potential (11,9%); activities implemented for comprehensive development of the enterprise and management of innovative potential with the aim of achieving a state of effective functioning (2,3%); ensuring the company's response to the threats of the innovative environment and responding to changes in the market situation (2,3%); achieving a balanced state and an appropriate level of implementation of innovative processes in the market environment (2,3%); updating the system components of enterprise management by implementing reforms on an innovative basis (7,1%); increasing the level of competitiveness based on the growth of the economic efficiency of economic activity based on innovations, etc (2,3%). З метою пожвавлення інноваційних процесів, які у післявоєнний період слід скерувати на забезпечення ревіталізації економічної системи на інноваційній основі, відновлення інфраструктури та укріплення першості перспективних технологій в Україні, у статті вивчено теоретичні засади розвитку інноваційної діяльності. Структурно-логічна модель дослідження полягала у виявленні підходів до дефініції поняття «інноваційний розвиток підприємства» та притаманних їм ключовим словам, графічній візуалізації отриманих результатів та їх використання як базових складових, які окреслюють значення інноваційного розвитку для підприємств (передбачає використання інноваційних ідей та реалізацію інноваційного потенціалу; зміну стану підприємства на основі інновацій із метою досягнення позитивних змін тощо). У статті використано теоретичне узагальнення, систематизацію та статистичний аналіз, діаграму Санкей.

The article examines the theoretical foundations of mentoring as a system of relationships in which one person (mentor) provides support for new knowledge, development and progress of another person (student, mentee). A study of legislative and institutional support, as well as the results of practical implementation of mentoring as a form of professional socialization at enterprises (MTS Ukraine, Coca-Cola Beverages Ukraine, 3M Ukraine, KPMG in Ukraine, JSC Ukrzaliznytsia, etc.) was conducted. Positive results of mentoring determined the increase in the level of internal system of training of employees, the development of own know-how of enterprises, increasing the level of satisfaction of employees with work, the formation of high-quality intellectual and human resources potential to ensure digital transformations at enterprises, the elimination of conflicts, increasing the level of motivation, the introduction of new rules and traditions of the company, increasing the level of relevance and reliability of Management Information, expanding the network of international contacts between employees (exchange of experience, solutions and ideas), improving the skills of employees and their performance, etc. The disadvantages of implementing mentoring systems of enterprises include resistance from mentors and their unwillingness to share experience with young colleagues, as well as the spread of mentoring practices only for new employees of enterprises, young specialists, beginners in a particular profession, undeveloped feedback between mentor and mentee (mutual mentoring). Using the method of theoretical generalization, analysis and synthesis, the article examines the analytical tools of mentoring activities, which are summarized in four groups: a targeted questionnaire survey of respondents, an online survey in the internet environment, a critical analysis of literature sources and the regulatory framework for the problem, a combination of various methods (questionnaire surveys, interviews, demonstration projects, etc.). In the context of deepening the practice of combining various methods of analysis, in order to increase the effectiveness of mentoring activities, when implementing analytical work, it is recommended to use methods of personnel assessment: 365 degrees method, Hay method, point-factor assessment, assessment center. The article outlines the essence of the recommended methods and points out the advantages of using them when analyzing mentoring systems in enterprises.

The article investigates the processes of strategic planning of innovative activity of the enterprise as an activity to determine the directions of enterprise innovative activity (strategy), which would be consistent with the available resource provision, innovative potential of the enterprise and meet the needs of the market and have a reasonable use of the mathematical apparatus of Markov chains for its modeling. The article gives examples of practical use of Markov chains in the management of innovative projects, forecasting the structure of the existing land fund of Ukraine, in the tasks of buying and selling shares, allocation of financial resources between executors of investment projects in a closed system, etc. Markov chain is used to model strategic planning of innovative activity of the enterprise.

In order to increase the level of efficiency of strategic management of the enterprise as a result of strategy implementation by means of various tactical approaches, the article investigates methods and models for selecting tactical approaches to strategy implementation: enterprise strategy and tactics tree (S&T tree), marginal approach to the choice of tactics, hidden Markov model, method of morphological analysis, etc. Critical analysis of the methods has proved the necessity of combining them in order to increase the effectiveness of the mathematical apparatus, which is designed to reduce the subjectivity, which is inherent in heuristic methods (provided by most of the studied methods). In order to achieve the goals of the article, it is recommended the procedure for choosing tactical approaches to implementing the strategy: definition of the purposes and tasks, information maintenance of process of a choice of the tactical approach; formation of a set of strategic approaches for realization of strategy of the enterprise for what it is possible to use elements of a suffix tree; selection of criteria for acceptability of the tactical approach (Boolean variables that characterize the costs of the enterprise and the implementation of the tactical approach, the level of prices for finished products and services, market capacity in which the company operates, the level of market competition, international economic relations) and their expert evaluation; formation of a table of values of the Boolean function of an acceptable tactical approach to implementing enterprise strategy and evaluating the Boolean vector number; formation of an evaluation scale depending on the set of tactical approaches of the enterprise; selection tactical approach on the recommended scale.