=Paper=
{{Paper
|id=Vol-2864/paper25
|storemode=property
|title=Synthesis of Spherical Object Configurations: Models and Information Technologies
|pdfUrl=https://ceur-ws.org/Vol-2864/paper25.pdf
|volume=Vol-2864
|authors=Oleksii Kartashov,Kyryl Korobchynskyi,Iryna Yakovleva,Georgiy Yaskov,Olga Yarovaya
|dblpUrl=https://dblp.org/rec/conf/cmis/KartashovKYYY21
}}
==Synthesis of Spherical Object Configurations: Models and Information Technologies==
https://ceur-ws.org/Vol-2864/paper25.pdf
Synthesis of Spherical Object Configurations:
Models and Information Technologies
Oleksii Kartashova, Kyryl Korobchynskyia, Iryna Yakovlevab, Olga Yarovayaa
and Georgiy Yaskovc
a
National Aerospace University "Kharkiv Aviation Institute", 17 Chkalova Street, Kharkiv, 61070, Ukraine
b
National University of Urban Economy in Kharkiv, 17 Marshal Bazhanov Street, Kharkiv, 61002, Ukraine
c
A.N. Podgorny Institute for Mechanical Engineering Problems of the National Academy of Sciences of
Ukraine
Abstract
The problem of synthesis of optimal configurations of spherical objects is considered. The
basis is a general approach to the formalization of the configuration space of geometric
objects. The analysis of models is carried out, their typology and information technologies
for structural synthesis of complex spherical objects are offered. Approaches to the
development of new information technologies for solving problems based on their
generalized information-analytical model are proposed. Data is transformed into an object-
oriented structure for visualization of spherical configurations.
Keywords 1
Spherical object, configuration, analytical model, information technology
1. Introduction
The term configuration is understood as appearance, outline and also the relative positioning of
objects and their parts. In mathematics, as a rule, the configuration means some placement of points in
space. The problem of synthesis of spatial configurations will be considered as the definition of such a
property of geometric objects that would satisfy the given relations (constraints, properties) and would
deliver an extremum to a certain quality criterion.
In this article, we confine ourselves to a class of geometric objects of spherical form. However, the
results obtained can be generalized to broader classes of spatial objects. A significant number of
publications testify to the importance of the problems of spatial synthesis of spherical objects. The
article [1] presents an overview of the works devoted to this topic, containing more than 60
references. In [2] - [10] a number of different approaches are considered for the search for optimal
spherical configurations. Papers [11] - [16] describe practical applications of problems of placing
spherical objects with additional constraints. It should be noted that when considering this class of
problems, much attention is paid to mathematical modeling of systems with spherical objects and the
development of special methods for nonlinear optimization to solve them. Issues related to
information support, analysis of the data structure and ways to transform geometric information in the
optimization process are not fully covered.
The purpose of this paper is to describe the possibilities of using modern information technologies
when considering the investigated class of problems based on existing mathematical models and
analysis of optimization methods.
CMIS-2021: The Fourth International Workshop on Computer Modeling and Intelligent Systems, April 27, 2021, Zaporizhzhia, Ukraine
EMAIL: alexeykartashov@gmail.com (O. Kartashov); kirill.korobchinskiy@gmail.com (K. Korobchynskyi); yakovlevairine@gmail.com
(I. Yakovleva); helga.yarovaya@gmail.com (O. Yarovaya); yaskov@ukr.net (G. Yaskov)
ORCID: 0000-0002-6282-553X (O. Kartashov); 0000-0002-3676-6070 (K. Korobchynskyi); 0000-0001-5204-9779 (I. Yakovleva),
0000-0003-4969-1677 (O. Yarovaya), 0000-0002-1476-1818 (G. Yaskov)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
This paper proposes an information technology that connects into a single system modules for
solving an optimization problem and solution visualizing. The decision maker is included in the
system. Using the results of the decision, especially in its visual representation, the decision-maker
can change the parameters of the decision, relying on his experience and intuition. Such transformed
solutions are used in the next step as a starting point for finding a local extremum. This approach is
because the problems under consideration are multi-extreme and have a large number of local
extrema. The suggested method is one of the ways to improve the local solution.
2. Geometric information and configuration space of spatial objects
Investigations of configurations as mathematical objects are naturally associated with the notion of
configuration space. The configuration space determines the configuration of the system, that is, the
set of values of geometric variables called generalized coordinates and sets the location in the space of
some system and its parts both relative to one another and with respect to a given fixed point.
In the papers [17] - [19] the concepts of geometric information and geometric objects
configuration space are introduced. Geometric information g s , ,p about an object
S R 3 includes a spatial form s as an equivalence class on a set of point sets; metric parameters of
the form ( 1 ,..., k ) , specifying the dimensions of the object; placement parameters
p ( p1 ,..., pt ) , that determine the position of an object S in the space R3 .
In accordance with the general concept of constructing such spaces, we define their structure in the
following way. The basis for specifying the components s and of geometric information g is
the equation of its boundary
f ( , ) 0, R 3 , (1)
where the variables ( , ) have a domain of admissible values D R k , and the function f ( , ) is
defined and continuous everywhere on R 3 D .
Depending on the analytic type of function f ( , ) various classes of objects can be formed. At
the same time, from a practical point of view it is natural to distinguish classes of models of the most
widespread material objects. These objects are called basic.
Different constraints may be imposed on the values of generalized variables in the configuration
space Ξ( SB ) . These restrictions will form the domain of admissible configurations in the
configuration space for the class of tasks under consideration. On the domain of admissible values of
generalized variables may be imposed other additional restrictions that arise when solving specific
classes of tasks, for example, when searching for optimal configurations in a certain sense. We divide
the following restrictions into the following groups: restrictions associated with the fixation of some
generalized variables, restrictions that arise on the mutual location of geometric objects, and the
restrictions imposed by certain physical or mechanical properties of the synthesized system. If to give
on a set of geometric objects some relations, binary in particular, we obtain different classes of
placement configurations.
If for any pair of objects a non-overlapping binary relation must be performed, that is, these
objects do not have common internal points, then the corresponding configuration of the placement is
called the packing configuration of geometric objects. Let us note that in most practical packaging
tasks an additional object is called a container. All geometric objects that are considered, must belong
to the container. Note that the generalized variables in the configuration space may be subject to
additional restrictions that generate special classes of configurations [19]. A distinctive feature of such
configurations is the presence of restrictions on the minimum and the maximum permissible distances
between objects.
In the case when geometric objects are solids with given masses, a balanced system of such bodies
gives the balanced packing configuration. Note that the configuration spaces both layout and balanced
packing configurations are the same.
We indicate also the covering configurations [20], [21] and partitioning configurations [22], [23].
A special class of configurations are Euclidean combinatorial configurations that are formed in the
case when the placement parameters and metric parameters of the objects take discrete values. Such
configurations have a number of important properties, the implementation of which allows us to
propose new approaches to modeling and solving optimization problems [24], [25].
3. Synthesis of spherical object configurations
In this paper, we will examine the class of geometric objects of a spherical shape. Three groups of
spherical objects will be consider: two-dimensional (2D), three-dimensional (3D) and
multidimensional (nD) objects. Each of the basic objects is given by the equation of its boundary.
Then the generalized variables of the configuration space of spherical objects are their radii and the
coordinates of the centers.
Consider the spherical object S R n , n 2 , 3, ... of the radius r 0 . We associate with S the
moving coordinate system by choosing its beginning (the pole of the sphera) in the center of its
symmetry. We define the spatial shape s of the sphere by the equation of its boundary
f ( ,r ) 0, R n , (2)
which, in accordance with the dimension of space, has the form:
f ( ,r ) x 2 y 2 r 2 , ( x, y ) , (3)
f ( ,r ) x 2 y 2 z 2 r 2 , ( x, y, z ) , (4)
n
f ( ,r ) i2 r 2 , ( 1 ,..., n ) . (5)
i 1
If S R 3 , then r is a metric parameter, and we denote p ( v 1 , v 2 , v 3 ) location parameter.
The configuration space of a sphere is given by generalized coordinates v 1 , v 2 , v 3 , r , and the equation
of the general position of the sphere has the form
F ( x, y , z, v 1 , v 2 , v 3 , r) r 2 ( x v 1 ) 2 (y v 2 ) 2 (z v 3 ) 2 0. (6)
Suppose = S 1 , ..., S n is a set of basic spherical objects. We assign the structure of a complex
object in the form
n
S B = B S 1 , ..., S n S i . (7)
i=1
An object S B whose structure is given by the expression (7), we call a composite object.
Based on the general approach and the proposed general classification, we propose the typology of
spherical objects configuration.
In the configuration spaces of spherical objects S i and S j with corresponding generalized
variables g i ( x i , y i , z i , ri ) and g j ( x j , y j , z j ,r j ) the constrains of non-overlap have the form
( ri r j ) 2 ( xi x j ) 2 ( y i y j ) 2 ( z i z j ) 2 0 . (8)
The condition for the contain of a sphere S i ( g i ) into a spherical container S 0 ( g 0 ) with
generalized variables g 0 ( x 0 , y 0 , z 0 , r0 ) will be written as
( x 0 x i ) 2 ( y 0 y i ) 2 ( z 0 z i ) 2 ( r0 ri ) 2 0 , (9)
where x 0 y 0 z 0 0 .
Note that the generalized variables g 0 , g 1 ,..., g n in the configuration space Ξ( S 0 ) Ξ( S B ) may
be subject to additional constraints that generate special classes of packing configurations. First of all,
it is about layout configuration and balanced packing configuration.
A distinctive feature of such configurations is the constraints on the minimum and the maximum
admissible distances between objects. In this case, the conditions will be
d ijmin ( xi x j ) 2 ( y i y j ) 2 ( z i z j ) 2 ( ri r j ) 2 d ijmax , (10)
where d ijmin ,d ijmax are corresponding distances between objects.
In the case when geometric objects S i ,i J n are solids and have masses mi ,i J n , a balanced
system of such bodies defines the balanced packing configuration. If the poles of the objects
S i ,i J n coincide with the centers of their masses, balanced packing takes place under condition
n n n
xi mi 0 , yi mi 0, zi mi 0 . (11)
i 1 i 1 i 1
Note that the configuration spaces Ξ( S 0 ) Ξ( S B ) both layout and balanced packing configuration
are the same.
The generalization of the spherical objects placement problem is the layout and packing of
composite spheres. For each component of composite sphere (object of the first order) whose
structure has the form (1), we fix the distances between the poles of the base spheres
S i , S j , i, j J n ,i j (objects of zero order). This will create an additional rigid system of
restrictions on the placement parameters p i ( x i , y i , z i ), i J n of the base spheres
( xi x j ) 2 ( y i y j ) 2 ( z i z j ) 2 ij2 . (12)
Depending on the possible affine transformations of complex objects (congruence, translation,
rotation, etc.) the placement options may be subject to additional restrictions.
To formalize of non-overlap conditions of the components of spherical objects, the decomposition
of such conditions is naturally performed on non-overlap of base spheres for various composite
objects.
Let a symmetric matrices B = bij nn and B 0 = bij0 be given with elements
n n
1, if Si S j ; 1, if Si S j
bij bij0 , (13)
0, otherwize, 0, otherwize
where Si S j means non-overlapping Si and S j .
The configuration of the spheres S i ,i J n is said to be admissible if the inequalities (8) holds for
any i, j J n ,i j such that bij 1 . The configuration of the spheres S i ,i J n is said to be valid if it
is included, if the inequality (9) holds for any i, j J n ,i j such that bij0 1 .
The vast majority of modern publications are devoted to the tasks of the layout and packaging of
objects, the metric parameters are fixed, with the exception of the container. Introduction of the
configuration spaces of geometric objects allows us to investigate fundamentally new, practically
important problems from the general positions, one of the important classes of which is the problem
of synthesis of optimal configurations of spherical objects.
Suppose there is a certain function on the set of permissible configurations of the spheres, which
we call the quality criterion of the configuration. Then the problem of synthesis of optimal
configurations of spherical objects is to determine the generalized variables of the configuration
space, which has a lot of geometric interpretations and practical applications.
Select a set of numbers L J n . Spheres S i ,i L are called prohibition zones and fix their
position in R 3 , putting
xi ˆxi , y i ˆy i , z i ˆz i , i L . (14)
We will form a matrix B = bij nn with elements
0, i , j L;
bij (15)
1, otherwize.
A set of admissible configurations will be given by a system of inequalities
( ri r j ) 2 ( xi x j ) 2 ( y i y j ) 2 ( z i z j ) 2 0, (16)
where i J n \ L, j J n .
Let a quality criterion , which need to be minimum, is
1 x 1 , y 1 , z 1 , r1 , ..., x n , y n , z n , rn max xi ri , y i ri , z i ri . (17)
iJ n \ L
If L = , we have a classic problem of placing unequal spheres in a sphere of minimal radius. In
this case, we can consider a container S 0 as a sphere of variable radius r0 with the placement
parameters x0 y 0 z 0 0 . Then
1 x 0 , y 0 , z 0 , r0 , x 1 , y 1 , z 1 , r1 , ..., x n , y n , z n , rn r0 . (18)
New classes of problems of synthesis of optimal configurations of spherical objects are
considered, generalizing the classical problems of packing and arrangement in the case of variables of
metric parameters of objects (radii of spheres).
We fix the generalized variables
x0 y 0 z 0 0, r0 rˆ0 , xi ˆxi , y i ˆy i , z i ˆz i , i L , (19)
in the configuration space Ξ( S B ) Ξ( S 0 ) Ξ( S 1 ) ... Ξ( S n ) .
Then, depending on the choice of specific constraints, we describe the appropriate set of
admissible configurations. In turn, the variety of quality criteria of the configurations expands. For
example, we will consider such a criterion as the maximum of occupied part of a container with
variable radii of spheres:
r1 ,...,rn 4
3 ri3 . (20)
iJ n \ L
In the configuration space Ξ( S B ) Ξ( S 1 ) ... Ξ( S n ) we fix the placement
parameters x x i , y y , z z i of spheres S ,i J .
i i i i i n
Let r1 , r2 ,..., rn - arbitrary convex function on Rn . Then we have the problem of convex
programming
r1 , r2 , ..., rn min (21)
with linear constraints
ri r j ij 0, i, j J n , i j , (22)
where
ij ( ˆxi ˆx j ) 2 ( ˆy i ˆy j ) 2 ( ˆz i ˆz j ) 2 , i, j J n , i j. (23)
Let a given matrix C = cij whose elements determine the number of connections between
nn
the spheres S i and S j , i, j J n , i j . Then the total length of the network connecting the
objects is given by the function
x1 , y 1 , z 1 , r1 ,..., x n , y n , z n , rn cij (S i , S j ) , (24)
iJ n jJ n
j i
where
(S i ,S j ) (xi x j ) 2 ( y i y j ) 2 (z i z j ) 2 ri r j . (25)
Let the spheres have fixed radii and represent a solid density body. We minimize the function that
characterizes the deviation of the center of the masses of the combination of bullets from a given
point. We have
2
mi xi
2
mi y i mi z i
2
x 1 , y 1 , z 1 , r1 ,..., x n , y n , z n , rn x0 y0 z0 , (26)
iJ n m m m
where mi 4 / 3 ri3 , i J n , m mi .
iJ n
Thus, the formalization of problems of synthesis of optimal configurations of spherical objects
allows us to conclude that depending on the quality criteria and the choice of specific constraints that
form the set of admissible configurations, such tasks can be attributed to the corresponding class of
mathematical programming problems.
4. Informational technology of synthesis of spherical object configurations
The paper proposes a unified approach for solving all types of problems of synthesis of spherical
objects under consideration. This approach is based on information technology, which includes the
interaction of several interconnected units. Each of these units solves its own specific tasks. In our
work, 4 such units can be distinguished:
• optimization;
• visualization;
• consolidated data warehouse;
• modeling and control.
The optimization unit as a basis contains a solver that allows you to effectively solve nonlinear
optimization problems to obtain local extrema. In our work, we use the IpOpt software package for
this purpose. It uses interior point techniques for high-dimensional NLP problems. Freely distributed,
has open source code and does not require obligations when creating commercial software.
The visualization unit is an interactive graphical interface for working with 3D graphics. For this
purpose, the work uses software from Autodesk 3ds MAX, which is provided under a free license. To
automate the display of data, programs are used - scripts written in the built-in 3ds MAX Scripts
language. They are executed in the environment of the internal Listener code interpreter, which
renders the objects. Thanks to this unit, you can display the obtained solutions, interactively change
their parameters or create a new problem using a graphical interface.
The need for a consolidated data warehouse is caused by the following. When using traditional
tools, there is redundancy in data volumes, and when processing significant resources, they are
converted. Thus, the data obtained from a variety of sources and different types of information
resources integrated into the system must be consolidated into an adequate information storage model.
For this we used ORM (Object Relational Mapping), and to transform ADO.NET Entity Framework
Code First into a repository strategy, and to save the results, we used a portable SQLite database.
The modeling and control unit is the core of the entire system. It is implemented in C ++ language.
In it, using polymorphic classes, information models of both spherical objects of varying degrees of
complexity and problems of synthesis of configurations of all considered types are built, stored and
transformed. In addition, this unit provides interaction between all units of the system and
automatically converts the internal representation of the problem into the form required for an
optimization, visualization or data store units.
The model of information technology for synthesis of spatial configurations of spherical objects is
shown in Fig. 1.
Figure 1: The model of information technology for synthesis of spatial configurations
The expediency of this approach is due to the following. All the problems of the considered classes
of synthesis of spatial configurations are are NP-complex complicated ones, which have a high
dimension (in real systems with more than 1000 variables), are multiextreme (with a non-linear
number of local extrema). For many of them it is possible to find the local extremum using standard
classical methods. One of the ways to improve a local solution, based on the proposed information
model, consists in a dialogue between a solver and a decision maker. The dialog is based on a
graphical display of the solver result. The decision maker interactively modifies the received locally
optimal configuration and sends it as an initial placement to the solver. This dialogue can continue
until the decision-maker stops the process. All obtained locally optimal solutions are stored in the
database and are the basis for the final selection. This process does not preclude the use of automatic
global search methods, for example, an genetic algorithms [26, 27].
Process of synthesis of optimal spherical configurations including the participation of a decision
maker is shown on Fig. 2.
The inclusion of a decision maker in the process will also allow solving problems of the types
under consideration with additional constraints that are difficult to formalize. In such cases, only the
decision maker can choose the final outcome. Consider as an example the spherical objects placement
problem. It is necessary to find the minimum spherical shell for a set of spheres of different radii.
Starting point is shown on Fig. 3 and a locally optimal solution obtained by the solver given on Fig. 4.
Figure 2: UML diagram of a process of synthesis of optimal spherical configurations including a
decision maker
Figure 3: Starting point
Figure 4: Locally optimal solution obtained by the Solver
The decision maker changed the received decision and sent it back to the Solver (Fig. 5). A new
locally optimal solution is shown on Fig. 6.
Figure 5: New point proposed by the Decision maker
Figure 6: Locally optimal solution obtained by the Solver
5. Conclusions
The paper provides a general formulation of problems for the synthesis of spatial configurations
and the classification of these problems. Analytical models of various types are given for the case of
spherical objects. To solve the problems of synthesis of spherical configurations, an information
technology has been proposed, including modules for optimization, visualization and the participation
of a decision maker. This approach made it possible to use the experience and intuition of a specialist
for a global search for a solution, i.e. attempts to improve local extrema. An example of using the
proposed technology is given.
6. References
[1] M. Hifi, R. M’Hallah, A literature review on circle and sphere packing problems: models and
methodologies, Advances in Operation Research, no. 7, pp. 1–22, 2009.
doi:10.1155/2009/150624.
[2] J. Liu, Y. Yao, Yu. Zheng, et al. An effective hybrid algorithm for the circles and spheres
packing problems, Combinatorial Optimization and Applications, Lecture Notes in Computer
Science, vol. 5573, pp. 135–144, 2009. doi:10.1007/978-3-642-02026-1_12.
[3] A. Sutou, Y. Day, Global optimization approach to unequal sphere packing problems in 3D,
Journal of Optimization Theory and Applications, vol. 114, no. 3, pp. 671–694, 2002.
doi:10.1023/A:1016083231326.
[4] E.G. Birgin, F.N.C Sobral, Minimizing the object dimensions in circle and sphere packing
problems, Operations Research, no. 35, pp. 2357–2375, 2008. doi:10.1016/j.cor.2006.11.002.
[5] Yu. G. Stoyan, G. Scheithauer, G. N. Yaskov, Packing unequal spheres into various containers,
Cybernetics and Systems Analysis, vol. 52, no.3, pp. 419–426, 2016.
doi:10.1007/s10559-016-9842-1.
[6] I. Castillo, F.J. Kampas, J.D. Pinter, Solving circle packing problems by global optimization:
numerical results and industrial applications, Eur. J. Oper. Res., vol. 191, no. 3, pp. 786–802,
2008. doi:10.1016 / j.ejor.2007.01.054.
[7] Y. Stoyan, G. Yaskov, T. Romanova et al., Optimized packing multidimensional hyperspheres:
a unified approach, Mathematical Biosciences and Engineering, vol. 17, no.6, pp. 6601–6630,
2020. doi: 10.3934 / mbe.2020344.
[8] C.O. López, J.E. Beasley, Packing unequal circles using formulation space search, Comput.
Oper. Res., vol. 40, no. 5, pp. 1276– 1288, 2013. doi:10.1016/j.cor.2012.11.022.
[9] S. Yakovlev, O. Kartashov, K. Korobchynskyi, B. Skripka, Numerical results of variable radii
method in the unequal circles packing problem, in: Proceedings of the 15th International
Conference on the Experience of Designing and Application of CAD Systems (CADSM 2019),
pp. 1-4, 2019. doi:10.1109/CADSM.2019.8779288.
[10] Z.Z. Zeng, W.Q. Huang, R.C. Xu, Z.H. Fu, An algorithm to packing unequal spheres in a larger
sphere, Advanced Materials Research, vol. 546, pp. 1464–1469, 2012.
doi: 10.4028/www.scientific.net/AMR.
[11] J. Wang, Packing of unequal spheres and automated radiosurgical treatment planning, Journal of
Combinatorial Optimization, vol. 3, pp. 453–463, 1999. doi:10.1023/A:1022951027498.
[12] J. Zhou, Y. Zhang, J. K. Chen, Numerical simulation of random packing of spherical particles for
powder- based additive manufacturing, ASME. J. Manuf. Sci. Eng., vol. 131, pp.1-8, 2009.
doi:o rg/10.l l 15/1.3123324.
[13] N. Yu. Ilyasova, A. S. Shirokanev, A. V. Kupriyanov, R. A. Paringer, Technology of intellectual
feature selection for a system of automatic formation of a coagulate plan on retina, Computer
Optics, vol. 43, no 2, 2019. doi: 10.18287 / 2412-6179-2019-43-2-304-315.
[14] S. Bereg, J.M. Diaz-Banez, M.A. Lopez et al., A decentralized geometric approach for the
formation keeping in unmanned aircraft navigation, in: 2015 International Conference on
Unmanned Aircraft Systems (ICUAS), pp. 989–997, 2015.
[15] O. Blyuss, L. Koriashkina, E. Kiseleva, R. Molchanov, Optimal placement of irradiation sources
in the planning of radiotherapy: mathematical models and methods of solving, Computational
and Mathematical Methods in Medicine, vol. 2015, pp. 1-8, 2015. doi: 10.1155/2015/142987.
[16] Lin Lu, Y. Guo, R. Fan, et al., Content-aware photo collage using circle packing, IEEE
Transactions on Visualization and Computer Graphics, vol. 20, no.2, pp. 182–195, 2014.
doi:10.1109/TVCG.2013.106.
[17] Y.G. Stoyan, S.V. Yakovlev, Configuration space of geometric objects, Cybernetics and Systems
Analysis, vol. 54, no. 5, pp. 716-726, 2018. doi:10.1007/s10559-018-0073-5.
[18] S.V. Yakovlev, Formalizing spatial configuration optimization problems with the use of a special
function class, Cybernetics and Systems Analysis, vol. 55, no. 4, pp. 581-589, 2019.
doi:10.1007/s10559-019-00167-y.
[19] S.V. Yakovlev, On some classes of spatial configurations of geometric objects and their
formalization, Journal of Automation and Information Science, vol. 50, no. 9, pp. 38–50, 2018.
doi:10.1615/JAutomatInfScien.v50.i9.30.
[20] S.V. Yakovlev, On a class of problems on covering of a bounded set, Acta Mathematica
Hungarica, vol. 53, no. 3, pp. 253-262, 1999. doi:10.1007/BF01953365.
[21] S.B. Shekhovtsov, S.V. Yakovlev, Formalization and solution of one class of covering problem
in design of control and monitoring systems, Avtomatika i Telemekhanika, vol. 5, pp. 160–168,
1989. http://mi.mathnet.ru/eng/at6296.
[22] E.M. Kiseleva, The emergence and formation of the theory of optimal set partitioning for sets of
the n-dimensional Euclidean space: theory and application, Journal of Automation and
Information Science, vol. 50, no. 9, pp. 1-24, 2018. doi: 10.1615/JAutomatInfScien.v50.i9.10.
[23] O.Blyuss, A. Zaikin, V. Cherepanova et al., Development of PancRISK, a urine biomarker-based
risk score for stratified screening of pancreatic cancer patients, British Journal of Cancer, vol.
122, no. 5, pp. 692–696, 2020. doi:10.1038/s41416-019-0694-0.
[24] Y.G. Stoyan, S.V. Yakovlev, Theory and methods of Euclidian combinatorial optimization:
current status and prospects, Cybernetics and System Analysis, vol. 56, no. 3, pp. 366–379, 2020.
doi:10.1007/s10559-020-00253-6.
[25] O. Pichugina, S. Yakovlev, Euclidean combinatorial configurations: typology and applications,
in: 2019 IEEE 2nd Ukraine Conference on Electrical and Computer Engineering (UKRCON),
pp. 1065–1070, 2019. doi:10.1109/UKRCON.2019.8879912.
[26] S. Yakovlev, O. Kartashov, O. Pichugina, Optimization on combinatorial configurations using
genetic algorithms, in: CEUR Workshop Proceedings, vol. 2353, pp. 28–40, 2019.
http://ceur-ws.org/Vol-2353/paper3.pdf.
[27] S. Yakovlev, O. Kartashov, O. Yarovaya, On class of genetic algorithms in optimization
problems on combinatorial configurations, in: Proceedings of the 2018 IEEE 13th International
Scientific and Technical Conference on Computer Sciences and Information Technologies
(CSIT 2018), vol. 1, pp. 374–377, 2018. http://toc.proceedings.com/41726webtoc.pdf.