=Paper=
{{Paper
|id=Vol-3668/paper14
|storemode=property
|title=Using Probabilistic Dynamics of Innovations to Manage the Recovery and Modernization of Ukrainian Industries
|pdfUrl=https://ceur-ws.org/Vol-3668/paper14.pdf
|volume=Vol-3668
|authors=Olga Gonchar,Marharyta Sharko,Natalia Petrushenko,Oksana Korniienko,Andrii Bitiy,Anton Berdychevskyi
|dblpUrl=https://dblp.org/rec/conf/colins/GoncharSPKBB24
}}
==Using Probabilistic Dynamics of Innovations to Manage the Recovery and Modernization of Ukrainian Industries==
https://ceur-ws.org/Vol-3668/paper14.pdf
Using Probabilistic Dynamics of Innovations to Manage
the Recovery and Modernization of Ukrainian Industries
Olga Gonchar1, Marharyta Sharko2, Natalia Petrushenko3, Oksana Korniienko4,
Andrii Bitiy1 and Anton Berdychevskyi1
1 Khmelnytskyi National University, 11, Instytuts’ka str., 29016, Khmelnytsky, Ukraine
2 Pryazovskyi State Technical University, vul. Universytets’ka 7, 87555, Mariupol (Dnipro), Ukraine
3 Ukrainian Academy of Printing, Pidholosko st., 19, 79020, Lviv, Ukraine
4 Admiral Makarov National University of Shipbuilding, Geroiv Ukrainy ave., 9 , 54050, Mykolayiv, Ukraine
Abstract
The results of using the apparatus of probabilistic dynamics to manage the development of innovation
activity during the recovery and modernization of the Ukrainian economy are presented. Quantitative
estimates of the response to innovation proposals depending on the size of enterprises in various
industries served as input information. It was established that innovation development in conditions
of instability of Ukrainian business relations to new innovative technologies and economically
unbalanced ties has a probabilistic-deterministic character. The conceptual model of probabilities of
innovation dynamics has been developed, in which simulation and event generation reflect the
external environment's influence. Transition matrices and conditional probabilities with different
priorities of resource allocation by sectors of the economy have been calculated. Visualization of the
system of interrelations of Ukrainian business sectors in graphs of innovative development
management during economic recovery is carried out.
Keywords
Markov chains, innovation activity, probabilistic dynamics of innovations, innovation management,
dynamic changes in the external environment, innovation technologies1
1. Introduction
Innovation activity has always been the basis for the sustainable economic development of any
business, as its dynamic development constantly requires the introduction of new technologies
and improvements to consumer demands. Only the one that succeeds in developing its
production constantly monitors new developments and competitors' experience in achieving
exclusive advantages and preferences. Successful enterprises have always had positive
dynamics in the development of innovation activity, which requires constant updating of
production, personnel training, and improvement of personnel skills in mastering new
techniques and technologies. The prospects of success and achieving competitive advantages
have always justified the costs of innovation activities.
The legitimate and justified attention to new modern innovative technologies of all links and
branches of social production was sharply reduced during the COVID-19 pandemic and the
onset of the world economic crisis. The subsequent years have intensified the ambiguous and
jumpy nature of the development of innovation activity.
The propensity for innovation growth depends on the size of the business. As the size of
firms increases, the likelihood of prioritizing the development and use of innovation as a tool in
competition increases. Military actions in Ukraine increase this gap and practically level out the
COLINS-2024: 8th International Conference on Computational Linguistics and Intelligent Systems, April 12–13, 2024,
Lviv, Ukraine
o.i.gonchar@i.ua (O. Gonchar); sharko_m_v@pstu.edu (M. Sharko); natalia.velikaya@gmail.com (N. Petrushenko);
kornienko.oksana@nuos.edu.ua (O. Korniienko); a.bityy@ukr.net (A. Bitiy); tonyberd@ukr.net (A. Berdychevskyi)
0000-0003-3917-7586 (O. Gonchar); 0000-0003-2321-459X (M. Sharko); 0000-0001-7383-
8558 (N. Petrushenko); 0000-0002-9269-6900 (O. Korniienko); 0000-0001-5075-2807 (A. Bitiy); 0009-0008-9487-
1988 (A. Berdychevskyi)
© 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
possibility of rapid recovery and modernization for micro businesses. Still, at the macro level, it
is pretty evident that it is only possible to plan the recovery of Ukrainian business by activating
innovation activity. Quantitative estimates of the probabilistic dynamics of innovation to
manage the recovery and modernization of Ukrainian industries are presented in Table 1.
Table 1
Quantitative estimates of the response of firm size to innovation proposals of different levels of
relevance
Size of enterprises Types of innovations, %
deemed irrelevant topical with limited applicability very relevant
Large enterprises 20 42 38
Medium-sized enterprises 27 47 27
Small enterprises 41 46 13
Microenterprises 43 50 8
Source: "Ukrainian Business in a Time of War". Institute for Economic Research and Policy
Consulting 2023
Today, such problems of innovation activity development are dominated by an unfavorable
political situation and low demand at the level of outdated technologies. Attitudes towards
innovation activity for different business sectors are also different (Table 2).
Table 2
Quantitative estimates of response to innovation proposals for different economic sectors
Types of innovations, %
Industries
deemed irrelevant topical with limited applicability very relevant
Metallurgy 36 27 36
Chemical 35 48 17
Machine building 22 50 28
Woodworking 30 48 22
Building materials production 50 46 4
Food processing 31 53 16
Light industry 44 35 21
Printing 64 27 9
Source: "Ukrainian Business in a Time of War". Institute for Economic Research and Policy
Consulting 2023
At the sectoral level, the attitude towards Ukrainian business from the side of new improving
innovative solutions is ambiguous. The industry that suffered the most during the war is related
to the production of construction materials. Despite the high demand for its products, the
industry reveals a shallow and even hostile attitude to new technologies and innovations that
are relevant for foreign similar problem companies and that improve the quality of products.
As it follows from Table 2, the gradation of negative attitudes to innovations is headed by the
printing industry, which visually represents the state of military operations and operational
information about the ongoing changes. The following budget-forming industries also consider
innovation activity during the war irrelevant, among them light industry (44%), food industry
(33%), metallurgy (36%), and machine building (22%).
Current trends in the development of business under military operations also confirm the
instability of business attitudes to new technologies and the imbalance of innovation activity.
2. Problem statement
Innovative activity and modernization of business sectors, in addition to own financing to
eliminate stagnation, destruction of production, requires state support in the form of long-term
programs, fiscal incentives, support for the training of specialists, external consulting support
and establishment of communications with relevant innovators. With limited investment
resources, considering which business sectors should be prioritised is particularly acute in the
face of uncertainty caused by dynamic changes in the external environment.
Under these conditions, the innovative development of Ukrainian enterprises has a random
probabilistic-dynamic character. Markov chains are a convenient tool for describing such
phenomena. Since the process of evaluating innovation activity in the conditions of dynamic
changes in the external environment starts without taking into account corrections for the
current situation, it consists of the sequence of defining states. In this case, only current and
future forecast values are considered without considering experience.
The relevance of the work is to develop a methodology for assessing and forecasting the
development of innovation activity of individual sectors of Ukrainian business in the context of
dynamic changes in the external environment.
The work aims to create an instrumental tool for probabilistic dynamics of management of
the development of innovation activity in Ukraine during unpredictable dynamic changes in the
environment caused by military actions in Ukraine and the recovery of their consequences.
3. Relative works
Under the conditions of uncertainty of the external environment influence and unpredictability
of forthcoming changes, the attitude to innovation activity is different for different industries
and sizes of enterprises. The specificity of reactions to innovation proposals in their various
manifestations (Table 1, Table 2) requires generalizing the presented information, its ordering
and ranking in the form of conditional probabilities for each industry. The experience of using
Markov chains to solve various scientific and technical problems is helpful in this case.
The use of information technology to assess the readiness of enterprises for innovative
transformation with the help of Markov chains is presented in [1,2]. Models of Markov
processes of logical transitions of probabilistic assessments of transitions are presented in [3,4].
In [5], the main methodological constructions of a mixed number of states of Markov processes
in discrete time are described. Markov chains in state sequence estimation are given in [6,7],
and an information system for water treatment quality assessment is considered in [8]. The
relationship between control and human factors of complex control models of Markov
processes using Monte Carlo methods is described in [9]. Parametric identification of stochastic
state uncertainty in asymptotic choice of alternatives using Markov chains is considered in [10].
In [11], stochastic estimates of transport materials are given. Features of the scenario-
optimization model of stochastic processes are described in [12], and intelligent information
management [13] and management of resources under an unstable external environment is
presented in [14-15]. Mathematical support for eliminating the human factor in navigation
equipment systems under uncertainty and risk is given in [16-17], management of development
of enterprises in [18], hidden Markov model for lymphatic turor progression in the head and
neck in [19]. Practical applications of Markov chains and probabilistic dynamics are vast and
diverse in different works: modeling economic impact of COVID-19 epidemic in Kenya using
Markov Chains [20], parallel probabilistic swarm guidance by exploiting Kronecker product
structures in discrete-time Markov chains [21], innovative activity in Ukraine and other
countries [22, 26], Stern assessing the accuracy of record linkages with Markov chain [23],
transition distributions of a Markov sequence [24], scarcity exchange model for different
processes [25], detecting IoT malware by Markov chains [27], and analysis of changing
uncertainty of metro tunnel's long-term settlement via hierarchy Bayesian network [28]. Some
fragments and peculiarities of the Markov process methodology have been used in this paper.
4. Materials and Methods
The current state of qualitative assessments of innovation proposals for different industries,
obtained based on a large number of statistical data and their forecast expert assessments on of
the state and modernization of industries, obtained based on a large number of statistical data,
due to innovation activities of technological processes, expressed through conditional
probabilities under conditions of uncertainty and risk, were used as research materials. Since
the state budget funds are considered a single source of financing innovation activities, the sum
of conditional probabilities of different industries in a particular time interval should equal 1.
Markov chains and the peculiarities of their application to the management of recovery and
modernization of industries of the Ukrainian economy were used as research methods.
Input information on the probabilistic dynamics of the current state of the industries of
Ukraine is presented in Table 3.
Table 3
Current state of probabilities of industry dynamics
№ Industries Weight Parameter
ratio designation
1 Metallurgy 0.12 v1
2 Chemical 0.11 v2
3 Machine building 0.2 v3
4 Woodworking 0.05 v4
5 Building materials production 0.17 v5
6 Food processing 0.13 v6
7 Light industry 0.07 v7
8 Printing 0.15 v8
Source: "Ukrainian Business in a Time of War". Institute for Economic Research and Policy
Consulting 2023
5. Methodology
The dynamics of innovations required to manage the recovery and modernization of Ukraine's
economic sectors can be described using the following equation:
𝑑𝑥 (1)
= 𝑓(𝑥, 𝑢, 𝑣 , 𝑣 , … , 𝑣 ), x(t0) = x0
1 2 𝑛
𝑑𝑡
where xEm – vector of phase variables, and Em - state space at each moment of time t.
Let us denote the general change in the dynamics of the management system for the
restoration and modernization of the country's economic sectors as follows u(t) U, and
changes in the dynamics of the control subsystems v1(t), v2(t), …, vn(t), where vi(t) Vi.
Assuming a continuous variation of the control parameter u varying with time, the resulting
function will have the form (t), t [t0, t]. u(t) U, will change with the change t.
Sets U, v1, v2, … , vn represent sets of admissible variants of management.
Every managerial step u(t), t [t0, t] allows to define the path of motion of the control
system x(t), t[t0, t]. The set of endpoints of the trajectories of equation (1) forms the set of
reachability, originating from the initial state under all possible program management of the
restoration and modernization of the economy u(t) U, t [t0, t]. Each subsequent event
depends solely on the previous one and has no dependence on other events. Accordingly, the
final control trajectory ends at the point of x(t), achieved by the system at the moment of time t.
Let us represent the initial probability distribution in the form of the following equation:
P( x0 S ) q0 ( S ) SE (2)
where ∀ – universality quantum, S – discrete states, q0 – probability distribution at time
t0 = 0.
The elements of the set E are restricted to a finite number of possible states.
E e1 , e1 ,..., en (3)
Range of values of the random parameter {xn}, which defines the characteristics of
innovation dynamics for managing the recovery and modernization of Ukrainian economic
sectors, is a state space, where the variable n denotes the step number and characterizes the
evolution of this parameter in the control system. Probabilities of transition from one state to
another are represented by square matrices.
Pij n P xn1 j xn i (4)
s1 s2 ... sn
s1 p11 p12 ... p1n
P s2 p p2 n
21 p22 ... (5)
. . . ... .
sn pn1 pn 2 ... pnn
Elements, pij represent transition probabilities from the current state si in the following sj.
Transition probabilities, represented as a matrix, express the probability that the state of the
control system at step n + 1 is the next state for the current state of the initial system.
P xn1 Sn1 xn Sn P Sn , Sn 1 Sn 1 , Sn E E (6)
A Markov chain can be considered homogeneous when the transition probability matrix is
independent of a particular step number. This can be expressed by the following equation:
Pij ( n ) Pij (7)
In accordance with the Kolmogorov-Chapman equation, the matrix of transition probabilities
for the number of steps in a homogeneous Markov chain is expressed as a matrix of states of
degree n of transition for one step.
The key characteristics of the Markov chain at any time period are the vector-string of
transition probabilities P.
The conceptual model developed in accordance with this approach is shown in Fig. 1.
Defining the field of
knowledge
Construction of
the transition Construction of stochastic
Search for a priori input probability models
information matrix
Determining the starting
point of likely dynamics
Determining the focus of Simulation and event
the process generation
Establishing the state Calculation of quantitative
space Determining the number indicators
of sampling steps
Determination of current
state probabilities Visualisation
of links
Figure 1: Conceptual model of probabilistic dynamics of innovation
According to the above definitions, the probability distribution in identifying the state of
management of rehabilitation and modernization of economic sectors in Ukraine does not
depend on time, but is conditioned only by the transitions from the current state to the relevant
management operations. By further developing the proposed approach, it is possible to
establish a sequence of transitions from the current initial state, forming the necessary basis for
management decisions.
The novelty of the presented conceptual model is the transition from discrete time
observation of the evolution of probabilistic representations to a continuous sequence of states
characterized by intervals of the system being in the equilibrium position of an uncertain
situation with varying discretization intervals.
6. Experiment
A specific example of calculating innovation dynamics indicators for managing the recovery and
modernization of Ukrainian industries based on Markov chains is considered. The parameters
of expert assessment of managerial activity in various country sectors are used as input data.
The probability distributions of the current state are presented in Table 3.
Each set of parameters of innovation dynamics indicators for managing the recovery and
modernization of industries is assigned a specific probability, which is recorded in a row of the
state matrix. The total sum of probabilities in each matrix row always equals one.
The values of conditional probabilities of the parameters of innovation dynamics at different
stages of management are presented in Table 4.
Table 4
Conditional probabilities of innovation dynamics indicators for managing the recovery and
modernization of Ukrainian industries
Current Metal- Chem. Mech. Wood- Production Food Light- Print.
State lurgy eng. working of building weight
Subsequent materials
condition
Metallurgy 0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
Chemical 0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
Machine building 0.23 0.07 0.2 0.12 0.06 0.17 0.11 0.04
Woodworking 0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11
processing
Construction 0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
materials production
Food processing 0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14
Light industry 0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
Printing 0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
7. Result and Discussion
The initial vector of states, according to Table 3, is written in the form:
p(0)=(0.12, 0.11, 0.20, 0.05, 0.17, 0.13, 0.07, 0.15) (8)
The matrix displaying the current state of probabilities of transitional probabilities of
industry development dynamics has the form:
0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
|0.23 0.07 0.20 0.12 0.06 0.17 0.11 0.04|
0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11 (9)
A= | |
0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
|0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14|
0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
Each of the rows in the presented matrix is characterized by its own probability distribution.
Next, it is necessary to determine the probability of influence of innovation dynamics indicators
to manage the recovery and modernization of industries of the Ukrainian economy at different
stages of their use.
Initial state S0 is characterized by the fact that the parameters of functioning are stable and
do not depend on the influence of environmental factors. The indicators at the initial stage will
be determined by the parameters vi presented in Table 2.
The state of innovation dynamics indicators for managing the recovery and modernization of
the country's industries at the first stage of management S1 is defined by the first vector row of
the matrix A. Probability of influence of this parameter p(1), describing the dynamics of
industry vi innovation in accordance with the Markov chain methodology is as follows:
p(1)= (0.12, 0.11, 0.20, 0.05, 0.17, 0.13, 0.07, 0.15)х
0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
|0.23 0.07 0.20 0.12 0.06 0.17 0.11 0.04|
0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11
х| |= (10)
0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
|0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14|
0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
= (0.1525, 0.1101, 0.1443, 0.1645, 0.0988, 0.1249, 0.0991, 0.1058)
The probability that from the state S1, the indicators of innovation dynamics to manage the
recovery of industries of the Ukrainian economy will move to the state of S2, characterized by
unstable, poorly predictable changes in environmental parameters is equal to p(2).
p(2)= (0.1525, 0.1101, 0.1443, 0.1645, 0.0988, 0.1249, 0.0991, 0.1058)х
0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
|0.23 0.07 0.20 0.12 0.06 0.17 0.11 0.04|
0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11
х| |= (11)
0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
|0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14|
0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
=(0.1374, 0.11594, 0.14789, 0.1485, 0.11202, 0.12721, 0.10263, 0.10842)
Comparison of values of probability indicators described by vector-string p(0), with
corresponding probability distributions of innovation dynamics to manage the recovery and
modernization of Ukrainian industries at the first stage of the economy p(1), led to the
elaboration of several proposals applicable in practice. Comparison of the same values of the
indicators available in equations (8) and (10) showed their increase at the first stage of
managing the recovery and modernization of the country's industries, which can be considered
satisfactory, except for two parameters "metallurgy" (v1) and "woodworking industry" (v4), that
have declined. This fact requires a transition to the next stage of calculation of innovation
dynamics to manage the recovery and modernization of economic sectors with a probability
of p(2).
In the context of unstable functioning due to the influence of external environment impacts,
the probability of transition of innovation dynamics indicators for managing the recovery and
modernization of Ukrainian industries from the state of S2 in S3 denoted as p(3).
At this stage of management, the transition probability of innovation dynamics indicators is
determined by the parameter v3. The probability of impact of this parameter on the transition
probability of innovation dynamics indicators, reflecting their controllability, i.e. the ability to
correct control actions, is as follows:
p(3)= (0.1374, 0.11594, 0.14789, 0.1485, 0.11202, 0.12721, 0.10263, 0.10842)х
0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
|0.23 0.07 0.20 0.12 0.06 0.17 0.11 0.04|
0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11
х| |= (12)
0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
|0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14|
0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
=(0.1393, 0.11439, 0.1462, 0.15218, 0.11004, 0.12682, 0.10156, 0.10954)
Comparison of the same parameters presented in equations (11) and (12) showed an
increase for parameters v2 and v8 while all other parameters decreased. This was the basis for
proceeding to the next stage of calculations. Note that the state p(0) describes only the initial
state of the system before control. The first stage starts with the step p(1). The transition
probability of innovation dynamics indicators at each stage of management should decrease.
Probability of change in the dynamics of innovation to manage the recovery and
modernization of Ukrainian economic sectors from the state of S3 in the state of S4 denoted
as p(4).
p(4)= (0.1393, 0.11439, 0.1462, 0.15218, 0.11004, 0.12682, 0.10156, 0.10954)х
0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
|0.23 0.07 0.20 0.12 0.06 0.17 0.11 0.04|
0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11
х| |= (13)
0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
|0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14|
0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
=(0.13877, 0.1147, 0.14632, 0.15173, 0.11044, 0.12666, 0.10194, 0.10947)
Comparison of the respective economic sectors presented in (12) and (13) showed a
decrease in all parameters except for the v2, v5 and v7. This was the basis for the transition to the
next stage of management.
The probability of changing the dynamics of innovations to manage the recovery and
modernization of the country's economic sectors from the state of S4 in the state of S5 denoted
as p(5).
p(5)= (0.13877, 0.1147, 0.14632, 0.15173, 0.11044, 0.12666, 0.10194, 0.10947)х
0.12 0.17 0.23 0.06 0.13 0.16 0.08 0.05
0.17 0.11 0.07 0.27 0.04 0.10 0.06 0.18
|0.23 0.07 0.20 0.12 0.06 0.17 0.11 0.04|
0.06 0.12 0.17 0.05 0.21 0.11 0.17 0.11
х| |= (14)
0.14 0.08 0.19 0.11 0.17 0.12 0.10 0.09
|0.20 0.14 0.05 0.14 0.15 0.13 0.05 0.14|
0.08 0.09 0.16 0.21 0.07 0.17 0.07 0.15
0.10 0.13 0.07 0.33 0.02 0.04 0.16 0.15
=(0.1386, 0.11433, 0.14629, 0.15155, 0.11037, 0.12664, 0.1019, 0.10941)
Comparison of similar parameters of change in the dynamics of innovations for managing the
recovery and modernization of industries, presented in (13) and (14), showed a decrease in all
parameters. This indicates the quality of application of innovations for the management of
recovery and modernization by industrial sectors in Ukraine.
Fig. 2 shows an oriented graph of Markov chains for the considered example of distribution
of probability indicators of innovation dynamics for managing the restoration and
modernization of industries in Ukraine.
0.12 0.23
0.17
building materials production,
metallurgical industry, v1 v5
0.27
0.19
0.40
0.10
0.26 0.25
0.13
0.11
0.13
chemical industry, v2 0.19 food industry, v6
0.40 0.15
0.37 0.47 0.10
0.18
0.25
0.20 0.07
mechanical engineering, v3 0.18 light industry industry, v7
0.22
0.15
0.23
0.52 0.44
0.23
0.26
0.05 0.15
0.24
wood industry, v4 0.46 printing industry, v8
0.26
Figure 2: Oriented graph of Markov chains of distribution of probabilistic indicators of
innovation dynamics to manage the recovery and modernisation of industries of the Ukrainian
economy
Since in the above oriented graph the sum of output probabilities for each of the industries of
Ukraine is equal to 1, we can say that the graph corresponds to the calculations performed.
8. Conclusion
1. The solution to one of the situational problems of modern recovery and modernization
of industries in Ukraine based on innovation activity is given. There is a sharp decline in
interest in innovative technologies for all sectors of Ukrainian business, associated with
the consequences of the COVID-19 pandemic, the onset of the global economic crisis,
and the military invasion of the Russian Federation in Ukraine.
2. Attitudes toward innovation activity for different sectors of the Ukrainian economy are
randomly jumping ambiguously. With limited investment resources, the question of
which sectors should be prioritized is particularly acute.
3. A conceptual model of probabilities of innovation dynamics based on Markov chains is
developed, in which simulation and generation of events reflect the influence of
environmental factors. The novelty of the model is that the argument is not time but a
sequence of states of resource interaction and step number reflecting discrediting
intervals.
4. Calculations of transition matrices and conditional probabilities of innovation resources
by sectors of Ukrainian business are given.
5. The growth of total probabilities with increasing discretization steps is found.
6. The visualization of the priority use of investment resources by different industries in
the form of a graph reflecting their mutual probabilities in real-time is made.
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