Mathematical Modeling of the Management of the Processes of
Training Specialists of Transport Universities
Natalia Valeryevna Kalganova
a
and Nikolay Pavlovich Chuev
b
Ural State University of Railway Transport, Ekaterinburg, Russia
Keywords: Educational process, modeling, mathematical model, control matrix; management of the pro-cesses of training
specialists.
Abstract: The article presents scientific research in the field of mathematical modeling, namely the construction of a
mathematical model for managing the processes of training specialists. When formalizing this complex
process, problems arise that are explained by the presence of a large number of factors that ensure a high-
quality educational process. Therefore, great importance is attached to the use of the mathematical apparatus
in the implementation of quality management of the educational process in higher educational institutions.
The article presents the construction of mathematical models describing the most important links between the
characteristics of the educational process at the university. The considered mathematical models contribute to
the development and adoption of managerial decisions on the strategy of managing the processes of training
specialists.
1 INTRODUCTION
At the present stage of the socio-economic
development of Russia, the development of the higher
education system, the training of highly qualified
specialists depends on the achievements in all areas
of activity of each university in the country. This
requires: improving the system of effective
management of the processes of the university, the
introduction into the practice of developing
universities of new scientific and pedagogical
achievements that correspond to the subject of
management and the requirements of modern
education (Ansoff, 1989). To do this, it is necessary
to solve the following tasks:
1. Development and implementation of methods
and means of planning and management as a
complex of components of effective
management of high-quality training of
specialists.
2. Provision of qualified personnel: managerial,
managerial, scientific, pedagogical; including
their training, systematic professional
development and the effectiveness of forms of
work organization.
a
https://orcid.org/0000-0002-4117-8329
b
https://orcid.org/0000-0003-1549-3533
3. Development of methods and means for the
effective use of material and technical
resources in the organ-ization of the
educational process, ensuring the completeness
and reliability of resource support.
4. The use of modern methods of systems
analysis, strategic forecasting, effective
management and mathematical modeling.
Modeling as a universal method of cognition of
social and pedagogical problems, the tasks of
didactics today is an integral part of the study of the
educational process. The beginning of modeling is a
goal setting that takes into account modern
educational problems, the complexity of which
determines their complex nature. The key task of
building and applying a mathematical model is to
prepare a competent, qualified specialist.
The constructed mathematical model will allow
choosing the optimal control actions, as well as
making an objective forecast of the state of the control
system (Samarsky, Mikhailov, 2005; Neymark, Yu.
I., 2010).
The article (Solodova, 2005) considers a rather
simple from a mathematical point of view and a more
82
Kalganova, N. and Chuev, N.
Mathematical Modeling of the Management of the Processes of Training Specialists of Transport Universities.
DOI: 10.5220/0011579300003527
In Proceedings of the 1st International Scientific and Practical Conference on Transport: Logistics, Construction, Maintenance, Management (TLC2M 2022), pages 82-88
ISBN: 978-989-758-606-4
Copyright
c
2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
general conceptual model developed by the doctor of
philosophical sciences A.S. Panarin.
dt
dy
dt
dy
отраслмежотр
>
,
dt
dy
dt
dy
прикладфундам
>
,
dt
dy
dt
dy
работыучебы
>
(1)
These inequalities (1) can be interpreted as
follows: the first inequality indicates a higher growth
rate of cross-sectoral knowledge compared to sectoral
knowledge. The second inequality indicates an excess
of the growth rate of fundamental knowledge over the
growth rate of applied knowledge. In the third
inequality: the growth rate of study time must be
greater than the growth rate of working time
(Gianetto, Wheeler, 2005).
Projecting system (1) onto the educational process
of the university, the following conclusions can be
drawn: inequalities 1 and 2 determine the content of
education, thus, these inequalities organize the
activities of the university to develop qualification
requirements, curriculum, curriculum and thematic
plan. university plans; Inequality 3 expresses the
principle of lifelong education, the principle of self-
development, formalizes the system of additional
education.
The basis for the development of these
management models is the state standard, which not
only fixes the subject area, but also formulates the
learning objectives. Gosstandart sets the legal and
substantive basis for more detailed knowledge
models designed for a specific training course or part
of it (module), or sets the search vector for the optimal
knowledge acquisition process.
For an adequate choice of a mathematical
analytical model, it is necessary to formalize its
parameters:
− to consider educational texts of a textbook,
lecture notes, records of problem solutions, etc.
as information flows of a certain finite amount
of knowledge, taking into account the
sequential dynamics of their development and
memorization;
− to build and investigate the information flow as
a model of knowledge transfer "teacher-
learners" in order to determine the main
characteristics of their interactions,
connections, to identify the effectiveness of this
model in terms of adequate management of the
individual process of mastering professional
knowledge by each student, group, course;
− the mathematical model should form a finite set
of control parameters (components), with the
help of which control decisions can be made for
the further improvement of the educational
process at the university. The managerial
function of the regularity of the process,
contained in mathematical models, can help
management departments to make
scientifically sound decisions to improve it.
The problems of constructing mathematical
models of optimal control of the processes of training
qualified specialists have been considered by many
authors, for example (Vasiliev, 1997; Avetisov,
1998).
As you know, one of the management concepts
that emerged in the 80s of the last century is the
process approach. In accordance with this concept,
the entire activity of a higher educational institution
is a set of sequential and interrelated processes. The
process approach is one of the key elements of
improving the quality of training, therefore, effective
quality management is impossible without replacing
subjective descriptions with objective assessments of
the learning process by building appropriate
mathematical models (Kalganova, 2021; Golubeva,
2016).
1.1 Building a Mathematical Model for
Managing the Processes of Training
Specialists
Studying the content of the educational process,
analyzing statistical data, establishing cause-and-
effect relationships between the elements of the
process being studied, which can be described
quantitatively. This made it possible to formulate the
relationship between the parameters and build a
mathematical model in the form of equations between
the main objects of the model (Stepanov, 2006).
Let x(t) be the amount of knowledge accumulated
by the student at a certain point in time t, including:
the ability to reason, solve problems, understand the
material presented by the teacher. The unit is
important here. As a measure for x(t), you can enter
the sum of the exam grades, the number of
successfully passed credits, etc.
In this study, it is logical to consider the functions
introduced to build a mathematical model as
dimensionless quantities. Thus, the conventional
units (AU) act as a fixed number of credit units. A set
of conventional units is a system of theoretical
knowledge and practical skills formed in the process
Mathematical Modeling of the Management of the Processes of Training Specialists of Transport Universities
83
of implementing the educational process at a
university. Based on the results of examinations for
the differentiated assessment of student performance
in the learning process, it is possible to determine x1
– the minimum value of CU, but sufficient to
complete the training, x2 – the maximum value of
CU, for graduates who have received honors. degree
after graduation.
The task of constructing a mathematical model,
thus, acquires an additional condition: to find the law
of assimilation of knowledge during the period of
study by each student who has the initial amount of
knowledge upon entering the university x0 and who
completed training with the amount of knowledge x ≥
x2.
As a result, the inequality holds for the function
x(t):
x1 ≤ x(t) ≤ x2+d.
where d – is additional knowledge to the main
program (electives, courses, self-education according
to interests, etc.).
The model in this case will have the form:
()
=𝛼
𝑥
(
𝑡
)
𝑥
𝑥
(
𝑡
)
, (2)
where dx(t)/dt – the speed of mastering conventional
units;
α1 – proportionality coefficient;
x(t) – the number of conventional units mastered
by students at a time t;
x2 – x(t) – the amount of knowledge on the
program required for successful completion of the
training.
The coefficient α1>0 depends on the individual
abilities of the student, his attitude to educational
work and the level of the previous one, up to the
moment t, of the qualitative development of the
educational material. The right side of equation (1)
depends only on previously acquired knowledge.
Please note that the value of x(t) with proper self-
organization of the student, tends to only increase;
therefore, it makes sense to apply model (1) on time
intervals: one semester or an academic year, a four-
year study for bachelors or a five-year period for
specialists..
Equation (1), known as the Verhulst equation,
originally arose from the study of population change.
This mathematical model is widely used not only in
various fields of natural science, but also plays an
important role in understanding the mechanisms of
applying nonlinear dynamics to socio-economic
models, in the tasks of introducing technological
innovations and developing science (Zhirkov, 2016;
Bagrinovsky, 1980).
The dynamics described by the Ferhulst equation
is a logistic curve in Figure 1.
Figure 1: Logistic curve for x_2 = 1, x_0 = 0,5, α_1= 1.
The completeness and target volume of
knowledge necessary for the successful assimilation
of educational material during the period of study at
a university is a key component of the quality of
education, therefore, to study their dynamics, it is
advisable to use a logistic equation. However, other
parameters (directions) of the university's activity
also affect the successful assimilation of educational-
theoretical and practical material when studying at a
university.
Two important components are considered that
directly affect the quality of training of specialists.
This is the qualitative composition of the teaching
staff of the university and the presence of a modern
and fully material and technical base (Kalganova,
2020).
Let's consider the first component of the
pedagogical process and its influence on the quality
of training of specialists. The pedagogical activity of
the teacher is united by the main components of this
activity:
− transfer of knowledge through direct
interaction «teacher - student»;
− methodical work;
− scientific work;
− educational work.
In order to formalize a rather complex,
multifaceted activity of a teacher and reflect it in the
form of quantitative characteristics of numerical
assessments, it is necessary to find the criteria of
pedagogical work and the value of such assessments;
the methodology for calculating rating indicators is
partially presented in Table 1.
Thus, using the methods of organizing ratings at
the university using quantitative and qualitative
indicators of teachers' academic performance in
points, it is possible to introduce a numerical,
functional relationship with other parameters of the
mathematical model being developed.
TLC2M 2022 - INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE TLC2M TRANSPORT: LOGISTICS,
CONSTRUCTION, MAINTENANCE, MANAGEMENT
84
Table 1: Methodology for calculating the rating indicators
of teachers.
Indicators The value
of the
indicator in
p
oints
Educational activities
Innovative pedagogical activity (online
courses; development of practice-
oriented technolo
g
ies; tests
)
0-3
Students' assessment of the teacher's
activities by factors: the availability of
material, the development of work
programs by disciplines, practices, - the
quality of classes
2-5
Professional development over the last
y
ea
r
0-2
Availability of developed educational
and methodological documentation for
training courses
0-3
Preparation of students for international
and (or) republican Olympiads,
competitions
0-2
Research wor
k
Participation in research, research and
development work, in obtaining grants,
fulfillment of business contracts
0-3
Obtaining patents/certificates issued to
the universit
y
0-3
Scientific
p
ublications 0-2
References to the author's works
received in the reporting year
0-2
Reports at conferences 0-2
Defense of dissertations 0-2
Let us introduce the function y(t) – the rating
indicators of the teacher, as a measure of the
effectiveness of the influence of the teaching staff on
the quality of teaching. The numerical value of the
overall grade for each teacher will satisfy the
inequality:
y1 ≤ y(t)≤ y2,
where y1 – the lowest rating value in conventional
units (points), y2 – maximum rating value.
The function y(t) will be considered
dimensionless, which characterizes the direct effect
on the increase in the growth rate of the volume of
students' knowledge with the coefficient β1.
Coefficient β1 there is a function depending on
β11 – staffing with teaching staff, β12 – optimal ratio
of teachers with academic degrees of candidate and
doctor of sciences, β13 – the level of scientific and
scientific-methodical work of each teacher, β14 –
regular completion of refresher courses and other
ranking indicators.
Let us introduce the function z(t) – the relative
total cost of all scientific and educational equipment
The function z(t) will be considered
dimensionless in conventional units and will have a
direct impact on the increase in the growth rate of
students' knowledge with a coefficient of y1. The
effectiveness of the influence of educational and
material resources of the university on the quality of
training of specialists can be expressed using the
coefficient y1. This coefficient represents a certain
function depending on y11 – the full staffing of the
educational and material base of the university, y12 –
the optimal use of scientific and educational
laboratories in the educational process, y13 –
compliance with the modern level of scientific and
educational laboratories, y14 – the availability of
testing and training grounds, etc. Taking into account
the above, we finally come to the construction of the
following mathematical model for managing the
processes of training specialists.
)()(γ)(β))()((
)(
11121
tftztytxxtx
dt
tdx
+++−=
α
)()(γ)(β
)(
222
tftzty
d
t
tdy
++=
)()(γ)(β
)(
222
tftzty
dt
tdy
++=
(2)
where the functions fi(t) 𝑖 = 1,2,3, can act as
additional conditions due to additional sources of
knowledge, lectures by foreign scientists, the use of
research and testing laboratories of research
institutes, enterprises.
A preliminary analysis of the relationship
between the parameters included in the construction
of a mathematical model allows us to conclude that it
is possible to build a mathematical model for
managing the processes of training specialists using
systems of three linear differential equations.
Such a system, composed according to the same
principle as system (2), will have the following form.
)()(γ)(β)(
)(
1111
tftztytx
dt
tdx
+++=
α
)()(γ)(β
)(
222
tftzty
dt
tdy
++=
)()(γ
)(
23
tftz
dt
tdz
+=
(3)
We introduce the matrix А of coefficients of the
corresponding system of linear differential equations
(3):
A =
𝛼
𝛽
𝛾
0𝛽
𝛾
00𝛾
. (4)
Mathematical Modeling of the Management of the Processes of Training Specialists of Transport Universities
85
Additionally, we introduce two matrices:
− matrix – a column of unknown functions
X =
𝑥(𝑡)
𝑦(𝑡)
𝑧(𝑡)
; (5)
− matrix – a column of pivot members
F =
𝑓
(𝑡)
𝑓
(
t
)
𝑓
(
𝑡
)
. (6)
Using the introduced matrices, system (3) can be
briefly written in matrix form:
.
d
XAXF
dt
=+
(7)
Please note that system (2) is nonlinear, and an
analytical solution in the form of a solution formula
is not possible. To study the solutions of this system,
the use of approximate or numerical methods is
required.
Therefore, let us consider in more detail the
solution of the linear system (3).
For systems (3), (7), we formulate the Cauchy
problem or the problem with initial conditions for t =
0:
x(0) = x
0
, y(0) = y
0
, z(0) = z
0
(8)
1.2 Study of the Mathematical Model and
Solution of the Cauchy Problem for
Systems of Differential Equations (3)
Let us prove the existence and uniqueness theorem
for the solution of the Cauchy problem for system (3)
of differential equations with the initial conditions
(8).
Theorem.
Let the functions f1(t), i = 1, 2, 3 be continuous
on some interval t∈[0,T] then the Cauchy problem for
the inhomogeneous system of first-order differential
equations (3), (7), (8) with constants coefficients will
have the only solution.
Proof.
The third differential equation of system (3) is a
first-order linear equation with a solution (Stepanov,
2006):
𝑧
(
𝑡
)
= 𝑒
𝑧
+
𝑓
(
𝜏
)
𝑒
𝑑𝜏
=φ
(
𝑡
)
, (9)
where the function φ
1
(t) denotes the solution to the
third equation of the system (3).
Let us write the second equation taking into
account (9) as follows:
)()(γ)(β
)(
2122
tftty
dt
ty
++=
ϕ
(10)
the resulting differential equation is also a first-
order linear equation, and its solution φ
2
(t) will have
the form:
𝑦
(
𝑡
)
= 𝑒
𝑦
+
(
𝛾
φ
(
𝜏
)
+𝑓
(𝜏)
)
𝑒
𝑑𝜏
=
φ
(𝑡), (11)
We continue to consistently perform similar
actions, we find the solution to the first equation:
𝑥
(
𝑡
)
= 𝑒
𝑥
+
(
β
φ
(
𝜏
)
+ 𝑓
(𝜏)
)
𝑒
𝑑𝜏
=
φ
(𝑡). (12)
Suppose that the functions f (t), i=1, 2, 3 are
bounded,
|
𝑓
(
𝑡
)
|
<В, 𝑖=1,2,3.
Then for all 𝑡∈[0,𝑇] we obtain the inequalities:
].0[,
γ
B
γ
B
|
γ
B
||||])([||)(|
3
γ
0
3
0
γ
γ
3
0
γγ
0
30
γ
33
3333
TtconstDxx
zdfztz
Tt
tt
t
t
∈=<+≤+=
=+≤+=
−−
ττ
τ
Finally:
|z(t)| ˂ D
Similar inequalities can be obtained for all
subsequent solutions of the systems of equations (3)
and (7).
Assume the existence of two x
1
(t) and x
2
(t)
solutions of system (3), (7) with initial conditions (8),
then the function X(t)=X
2
(t) - X
1
(t) will satisfy a
homogeneous equation fo f
1
= f
2
= f
3
= 0 and zero
initial conditions.
Then from formula (9) follows z(t)=0, similarly,
formula (11) follows y¯(t)=0 and¯x (t)=0 Hence the
equality x
1
(t) = x
2
(t).
Thus, there is only one solution for problem (9) –
(10).
The theorem of existence and uniqueness of the
solution of the Cauchy problem for systems of
differential equations (3) and (7) with initial
conditions (8) is proved.
Based on the obtained solution of the Cauchy
problem for systems (2), (3), (7). it is possible to
develop proposals for managing the learning process
by choosing the values of the initial valuesx
0
, y
0
, z
0
and positive coefficients α
1
, β
1
, β
2
, y
1
, y
2
, y
3
of the
matrices А and F, the elements of which are a set of
tunable parameters affecting the optimal control of
solutions of systems ordinary differential equations
(2), (3) or (7). By varying the elements of the matrix
А (coefficients of the system) and the function fi(t)
(free terms of the system), one can achieve an
TLC2M 2022 - INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE TLC2M TRANSPORT: LOGISTICS,
CONSTRUCTION, MAINTENANCE, MANAGEMENT
86
increase or decrease in the rate of change of x(t) – the
amount of accumulated knowledge by a student.
Let's introduce a definition. The set of coefficients is
called α
1
, β
1
, β
2
, y
1
, y
2
, y
3
and functions fi(t) of system
(2), matrices (4), (6) of systems (3), (7) control
matrices for solving systems of ordinary differential
equations (2), (4) or (6), i.e. this is a set of control
parameters from a set of matrix elements А and the
time functions of the matrix F, which have a direct
impact on the solutions of systems (2), (4), (6) –
systems that model the management of the future
development of the educational system of the
university and the processes of training specialists.
1.3 Verification of the Mathematical
Model
We will check the verification of mathematical
models, that is, the correspondence of the solutions of
systems of differential equations to the real
educational processes of the university, the optimal
control of the processes of training specialists.
Analysis of the construction of models, for
example, system (2), shows that in the process of
building the target value x
2
– the full amount of
knowledge for the formation of a specialist, is an
integral part of the system to which the function
aspires x(t) при t → ∞.
The members β
1
y(t), y
1
z(t), f
1
z(t), system (3) only
accelerate the achievement of the target value in a
limited time interval. This solution shows the
possibility of organizing training that is optimal for
all students at the same time. In this case, the
achievement of the target results depends on the
initial position and control matrices for solving the
systems of ordinary differential equations.
Let us give examples showing the role of the
control matrix А (4) for achieving target values in
managing the process of training specialists.
Example 1.
Consider a system of three equations and use the
MathCAD 15 software package (Makarov, 2009) to
obtain a numerical-graphic solution.
Given the system: (Given)
Here the steering matrix has the form:
A =
0.1 0.008 0.034
0.027 0.27 0.024
0 0 0.67
Figure: 2: Graphs of functions x(t),y(t), z(t) and x(t)-z(t).
x
(0) = 7.0
x
(5) = 11.698
y(0) = 2.6 y(5) = 7.572
z
(0) = 1
z
(5) = 0.035
Consider the case when some coefficients are zero
x
(0) = 7.0
y(0) = 0.6
z
(0) = 1
Figure: 3: Graphs of functions x(t), y(t) and z(t).
Let's make calculations
x
(0) = 7
x
(0) = 0,6
z
(0) = 1
x(5) = 12,712 y(1) =1,091
z(5) =
0,035
y
(1)
–
0,6 = 0,491
y
(2) = 1,788
y
(2)
–
y
(1) = 0,697
y
(3) = 2,776
y
(3)
–
y
(2) = 0,989
y
(4) = 4,179
y
(4)
–
y
(3) = 1,403
y
(5) = 6,17
y
(5)
–
y
(4) = 1,991
The graphs in Figures 2, 3 show an increase in the
volume of students' knowledge while improving the
t
xt()
d
d
0.1 x t()⋅ 0.008 y t()⋅+ 0.034 z t()⋅−
t
yt()
d
d
0.27 y t()⋅ 0.028 x t()⋅− 0.024 z t()⋅+
t
zt()
d
d
0.67− zt()⋅
x0( ) 7.0
y0( ) 2.6
z0( ) 1.0
0 1 2 3 4 5
0
5
10
xt()
yt()
zt()
xt() zt()−
t
t
xt()
d
d
0.1 x t()⋅ 0.08 y t()⋅+ 0.034 z t()⋅−
t
yt()
d
d
0.35 y t()⋅ 0.2+
t
zt()
d
d
0.67− zt()⋅
0 1 2 3 4 5
0
5
10
xt()
yt()
zt()
xt() zt()−
t
Mathematical Modeling of the Management of the Processes of Training Specialists of Transport Universities
87
quality of the educational process and educational and
material support.
Indeed, the results of analytical studies, the
examples given show that it is possible to develop
proposals for managing the learning process based on
the solutions of systems (3), (7), choosing for each t
the values of the initial values
x
0
, y
0
, z
0
the values of
the coefficients as a set of adjustable parameters
affecting the optimal control of the solution of the
systems of ordinary differential equations (3) or (7),
as well as the achievement of target values.
Thus, the models of the educational process
management process (2), (3) and (7) made it possible
to obtain a number of practical recommendations
expressed in numerical form. At the same time, there
was no need to clarify methods for measuring the
amount of knowledge available to students. It is
sufficient that these quantities satisfy the qualitative
relations leading to the systems of equations (2), (3),
and (7).
2 CONCLUSIONS
The article discusses mathematical models that
contribute to making managerial decisions about an
alternative choice of strategy for managing the
processes of training specialists.
The first mathematical model (2) allows you to
build an optimal acquisition strategy, taking into
account the initial state and the formulated target
value at the end of the training period (bachelor's,
specialist, master's degree) based on the logistic
function. obtained using a differential equation. This
model makes it possible to predict and implement
strategic programs of the educational process of the
university for a longer period.
Based on the analysis of the relationship between
the parameters included in the construction of a
mathematical model, a mathematical model for
managing the processes of training specialists using
systems of three linear differential equations with
constant coefficients has been built.
The matrices А and F of the coefficients of the
corresponding system of linear differential equations
and their free terms determine the control matrices for
the solution of the systems of ordinary differential
equations (3) and (7), which have a direct impact on
the process of managing the training of specialists.
The article conducts research on the verification
of mathematical models based on the consideration of
control matrices and the multivariance of parameters
included in the solution of differential equations that
form the basis of the mathematical models under
consideration.
REFERENCES
Ansoff, I., 1989. Strategic management. p. 358.
Samarsky, A. A., Mikhailov, A. P., 2005. Mathematical
modeling. p. 320.
Neymark, Yu. I., 2010. Mathematical modeling as a
science and art. p. 420.
Solodova, E. A., Antonov, Yu. P., 2005. Mathematical
modeling of pedagogical systems. p. 332.
Vasiliev, V. N., et al. 1997. On mathematical models of
optimal control of the system of training specialists. p.
136.
Avetisov, A. A., Kamyshnikova, T. V., 1998. Optimization
model for assessing and managing the quality of
student training at a university. pp. 105-109.
Kalganova, N. V., 2021. Mathematical modeling of the
process of sustainable development of the personnel
potential of industry universities. E3S Web of
Conferences. p. 296.
Golubeva, N. V., 2016. Mathematical modeling of systems
and processes. p. 192.
Zhirkov, A. M., 2016. Mathematical modeling of systems
and processes. p. 192.
Bagrinovsky, K. A., Busygin, V. P., 1980. Mathematics of
planning decisions. p. 224.
Kalganova, N. V., 2020. Mathematical models of optimal
management of the material support of educational and
scientific activities of industry universities. Bulletin of
USTU. 4, pp. 120-126.
Gianetto, K., Wheeler, E., 2005. Knowledge Management.
p. 192.
Stepanov, V. V., 2006. Course of differential equations. p.
472.
Makarov, E. G., 2009. MathCAD: training course. p. 384.
TLC2M 2022 - INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE TLC2M TRANSPORT: LOGISTICS,
CONSTRUCTION, MAINTENANCE, MANAGEMENT
88
