MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS
The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problems that involves reducing computing costs, accelerating and effectively finding solutions to such problems....
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General Energy Institute of the National Academy of Sciences of Ukraine
2025
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System Research in Energy| _version_ | 1871104418459418624 |
|---|---|
| author | Khaidurov, Vladyslav Zinchenko, Artem Tsiupii, Tamara Yarovoy, Roman |
| author_facet | Khaidurov, Vladyslav Zinchenko, Artem Tsiupii, Tamara Yarovoy, Roman |
| author_institution_txt_mv | [
{
"author": "Vladyslav Khaidurov",
"institution": null
},
{
"author": "Artem Zinchenko",
"institution": null
},
{
"author": "Tamara Tsiupii",
"institution": null
},
{
"author": "Roman Yarovoy",
"institution": null
}
] |
| author_sort | Khaidurov, Vladyslav |
| baseUrl_str | https://systemre.org/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T12:57:49Z |
| description | The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problems that involves reducing computing costs, accelerating and effectively finding solutions to such problems. Due to the fact that most serious problems involve the search for tens, hundreds or thousands of optimal parameters of mathematical models, the search space for these parameters grows non-linearly. Modern swarm intelligence has significant potential for application in the energy industry due to its ability to optimize and solve complex problems. With its help, it is possible to solve scientific and applied problems of optimizing energy consumption in buildings, industrial complexes and urban systems, reducing energy losses and increasing the efficiency of resource use, as well as for the construction of various elements of energy systems in general. Well-known methods and algorithms of swarm intelligence are also actively used to forecast energy production from renewable sources, such as solar and wind energy. This allows better management of energy sources and planning of their use. The relevance of modifications of methods and algorithms is due to the issues of speeding up their work when solving machine learning problems, in particular, in nonlinear regression models, classification, clustering problems, where the number of observed data can reach tens and hundreds of thousands or more. The work considers and modifies well-known effective methods and algorithms of swarm intelligence (particle swarm optimization algorithm, bee optimization algorithm, differential evolution method) for finding solutions to multidimensional extremal problems with and without restrictions, as well as problems of nonlinear regression analysis. The effectiveness of the modified methods on various classical and applied problems, which are used in the design of elements of complex objects and their systems, is demonstrated. A comparative analysis of the results of these methods was carried out. |
| doi_str_mv | 10.15407/srenergy2025.03.065 |
| first_indexed | 2026-03-24T02:03:36Z |
| format | Article |
| fulltext |
Системні дослідження в енергетиці. 2025. 3(83) 65
ІНФОРМАЦІЙНО-ВИМІРЮВАЛЬНІ ТЕХНОЛОГІЇ,
МОНІТОРИНГ ТА ДІАГНОСТИКА В ЕНЕРГЕТИЦІ
_____________________________________________________________________________
ISSN 2786-7102 (Online), ISSN 2786-7633 (Print)
https://doi.org/10.15407/srenergy2025.03.065
UDC 517.9:519.6
Vladyslav Khaidurov1,2*, PhD (Engin.), Senior Researcher, https://orcid.org/0000-0002-4805-8880
Artem Zinchenko1, PhD (Engin.), Associate Professor, https://orcid.org/0000-0003-1586-3645
Tamara Tsiupii3, PhD (Phys. & Math.), Associate Professor, https://orcid.org/0000-0003-2206-2897
Roman Yarovoy4, PhD (Engin.), https://orcid.org/0000-0001-8978-8137
1National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 37, Beresteiskyi
Prosp., Kyiv, 03056, Ukraine;
2General Energy Institute of NAS of Ukraine, 172, Antonovycha St., Kyiv, 03150, Ukraine;
3National University of Life and Environmental Sciences of Ukraine, 15, Heroiv Oborony St., Kyiv, 03041,
Ukraine;
4European University, 16 V, Academician Vernadsky Blvd., Kyiv, 03115, Ukraine
*Corresponding author: allif0111@gmail.com
_______________________________________________________________________________________
MODIFICATION OF SWARM INTELLIGENCE METHODS FOR
OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS
Abstract. The development of high-speed methods and algorithms for global multidimensional
optimization and their modifications in various fields of science, technology, and economics is an urgent
problems that involves reducing computing costs, accelerating and effectively finding solutions to such
problems. Due to the fact that most serious problems involve the search for tens, hundreds or thousands of
optimal parameters of mathematical models, the search space for these parameters grows non-linearly.
Modern swarm intelligence has significant potential for application in the energy industry due to its ability
to optimize and solve complex problems. With its help, it is possible to solve scientific and applied problems
of optimizing energy consumption in buildings, industrial complexes and urban systems, reducing energy
losses and increasing the efficiency of resource use, as well as for the construction of various elements of
energy systems in general. Well-known methods and algorithms of swarm intelligence are also actively
used to forecast energy production from renewable sources, such as solar and wind energy. This allows
better management of energy sources and planning of their use. The relevance of modifications of methods
and algorithms is due to the issues of speeding up their work when solving machine learning problems, in
particular, in nonlinear regression models, classification, clustering problems, where the number of
observed data can reach tens and hundreds of thousands or more. The work considers and modifies well-
known effective methods and algorithms of swarm intelligence (particle swarm optimization algorithm, bee
optimization algorithm, differential evolution method) for finding solutions to multidimensional extremal
problems with and without restrictions, as well as problems of nonlinear regression analysis. The
effectiveness of the modified methods on various classical and applied problems, which are used in the
design of elements of complex objects and their systems, is demonstrated. A comparative analysis of the
results of these methods was carried out.
Keywords: optimization, swarm intelligence, mathematical modelling, nonlinear regression, complex
objects and systems.
1. Introduction
With the development of modern computing systems, its application for solving optimization problems,
there is a need for the design of complex systems in thermal power engineering [1–3], mathematical modeling.
The modern theory of optimization methods, taking into account its new applications, has undergone
significant changes, which consist in the development of new and modification of existing methods and
algorithms for finding solutions to optimization, inverse problems and problems in incorrect formulation [4].
Considerable interest is now focused on algorithms and methods of swarm intelligence, which are
finding more and more applied applications in modern science and technology. Swarm intelligence is a concept
https://orcid.org/0000-0002-4805-8880
https://orcid.org/0000-0003-1586-3645
https://orcid.org/0000-0003-2206-2897
https://orcid.org/0000-0001-8978-8137
mailto:allif0111@gmail.com
Системні дослідження в енергетиці. 2025. 3(83) 66
that is inspired by observations of the behavior of animal colonies in nature, such as ants, bees, and swarms of
birds. This approach to artificial intelligence and optimization is based on modeling the behavior of individual
agents that interact with each other and with the environment. The collective behavior of these agents is
actively used to solve complex problems and find optimal solutions without centralized management.
The applications of swarm intelligence in science and technology include various fields such as
optimization, robotics, data networks, and even machine learning. For example, optimization algorithms based
on swarm intelligence can be used to solve routing problems, find optimal solutions in complex parameter
spaces, or manage distributed systems [1, 5–8].
In the field of robotics and the management of groups of robots, swarm intelligence allows for the
creation of efficient algorithms to coordinate the actions of many robots without centralized control. This is
especially useful in scenarios where adaptability to changing environmental conditions is required. In data
networks, swarm algorithms can be used to optimize routing and traffic management, taking into account
dynamic network conditions and changes in load. In the field of machine learning, swarm methods and
algorithms can be applied to create effective deep learning methods, in particular, in the problems of optimizing
loss functions and finding hyperparameters [1].
2. Methods and materials
2.1. Problem statement
Most of the applied optimization problems are reduced to finding the global extrema of functions in the
classical formulation, which is presented as follows [1, 8]:
( ) ( )min max ,y F= →X , ,nG G X
where ( )1 2 ,, , , nX X X=X G is the search space for the values of the arguments X, n is the dimension of the
search space of the global extremum of the function. , 1;jX j n= – factor (independent) variables that establish
a causal relationship with the dependent (resultant) variable y. The function F(X) is often subject to functional
constraints, which are presented in the form of the following functional dependencies:
( ) , 1; ,i ig S i K =X ( ) , 1; .i ig S i K R= = +X
The complexity of the objective function F(X) and the corresponding constraints gi(X) is determined by
a specific technical problems [1, 9]. The appearance of F(X) and gi(X) determines the choice of the
optimization method that will be used to search for ( )* * * *
1 2 ,, , , nX X X=X F(X*).
It should be noted that the dependence of the species: ( ) min.y F= →X
Such functional dependencies arise in regression (linear and nonlinear) analysis, in the problems of
classification, data clustering, in the construction of effective mathematical models of process control,
forecasting of time series, modernization of objects and systems, as well as in the development of complex
energy complexes in today's conditions.
In this paper, we will consider the problems of searching for a global minimum of problems without
functional constraints on the objective function / functional, problems with constraints in the form of functions,
as well as problems of searching for nonlinear regression parameters, as well as methods for their
search [1, 2, 9]. In this case, further research will focus on methods and algorithms that have a stochastic
component. Three well-known methods and algorithms were taken as a basis: the particle swarm optimization
algorithm, the bee optimization algorithm, and the differential evolution method. The criteria for selecting
basic methods and algorithms are based on the following well-known facts:
– the chosen methods and algorithms are relatively easy to understand, making them accessible to a wide
range of software developers and engineers;
– the chosen methods and algorithms are able to adapt to changes in the environment or optimization
problems by making changes to the parameters or search strategy.
– the selected methods and algorithms are easily parallelized to perform simultaneous data processing;
Системні дослідження в енергетиці. 2025. 3(83) 67
– the chosen methods and algorithms do not use any information about the derivatives of the objective
function and the corresponding constraints on it;
– there are comparatively few parameters in the selected methods and algorithms;
– the chosen methods and algorithms are very effective for finding the global extremum of a function.
2.2. Problem solving methods
2.2.1. PSO algorithm
PSO uses a swarm of particles, where each particle represents a potential solution to a problem. Initially,
all the particles of the swarm occupy a random position in the space of the search for the solution of the
problem and have small random velocities. In the final iterations, the set of particles converges to one or more
optimums that are global (if there are several rather than one). The behavior of a particle in the solution-seeking
hyperspace is constantly adjusting to its experience and that of its neighbors. In addition, each particle
remembers its best position with the achieved local best value of the objective (fitness) function and knows
the best position of the particles of its neighbors, where the global optimum of the function was reached at the
moment. In the search process, the swarm particles exchange information about the best results achieved and
change their positions and speeds according to certain rules based on the currently available information about
local and global achievements. In this case, the global best result is known to all particles and it is corrected in
the case when some particle of the swarm finds a better position with a result that exceeds the current global
optimum. Each i-th particle is characterized by an iteration n of its position ( )ix n in hyperspace and its
velocity of motion ( ).iv n The velocity of the i-th particle is calculated as
( ) ( ) ( ) ( )( ) ( ) ( )( )*
1 1 2 21 ,
best
i i i i iv n v n x n x n r x n x n r + = + − + − (1)
The position of the i-th particle is calculated as
( ) ( ) ( )1 1 ,i i ix n x n v n+ = + + (2)
where ( ) ( ) ( ) ( )( )1 2; ; ;i i i iMx n x n x n x n= – the position of the i-th particle in the iteration n ;
( ) ( ) ( ) ( )( )1 2; ; ;
best best best best
i i i iMx n x n x n x n= – the best position of the i-th particle (personal best position);
( ) ( ) ( ) ( )( )* * * *
1 2; ; ; Mx n x n x n x n= – the best position for the entire population (global best position);
( ) ( ) ( ) ( )( )1 2; , ;i i i iMv n v n v n v n= is the velocity vector of the i-th particle in the iteration ;n 1 2, – positive
acceleration coefficients that regulate the contribution of the cognitive and social components;
( )1 11 12 1; ; ; Mr r r r= , ( )1 21 22 2; ; ; Mr r r r= are random number vectors that introduce an element of randomness
into the search process.
Let us consider the influence of various constituents in calculating the velocity of a particle according
to (1). The first term in (1) ( )iv n preserves the previous direction of the velocity of the i-th parts and can be
considered as the moment that prevents a sharp change in the direction of the velocity and acts as an inertia
velocity. Cognitive velocity ( ) ( )( )1 1
best
i ix n x n r − determines the characteristics of a particle with respect to
its prehistory, which maintains a better position for a given particle. The effect of this term is that it tries to
bring the particle back to a better achieved position. The third term ( ) ( )( )*
2 2ix n x n r − defines the social
velocity, which characterizes the particle in relation to its neighbors. The effect of the social component is that
it tries to direct each particle towards the global optimum achieved by the swarm (or some of its immediate
environment). The displacement of the particle's position is carried out on the basis of (2).
Algorithm for Optimizing Numerical Functions
1. Initializing
1.1. Specifying Parameters 1 2, , and ( )1 2, 0;4 .
Системні дослідження в енергетиці. 2025. 3(83) 68
1.2. Set the maximum number of iterations ,N population size ,K the length of the particle position
vector M minimum and maximum values of the position vector , , 1, ,
min max
j jx x j M minimum
and maximum values for the velocity vector , , 1, ,
min max
j jv v j M And 0.
max
jv
1.3. Defining the Cost Function (Goal Function) ( ) ( )1, , , ,MF min x x→ = x x
where x is the position vector of the particle.
1.4. Creating the Original Population P
1.4.1. Particle Number 1, k P= =
1.4.2. Randomly create a vector position kx
( ) ( ) ( )1, , , ,
min max min
k kM kj j j jx x x x x x rand= = + −kx
where ()rand is a function that returns a uniformly distributed random number in the
range [0; 1]
1.4.3. Creating a Better Position Vector : best best
k k kx x x=
1.4.4. Randomly Generate a Velocity Vector kv
( ) ( ) ( )1, , ,
min max min
k k kM kj j j jv v v v v v v rand= = + −
or
0kjv =
1.4.5. If ( ), , ,
best
k k kx x v P then ( ) , , , 1
best
k k kP P x x v k k= = +
1.4.6. If ,k K then go to step 1.4.2
1.4.7. Define a particle of the current population with the best position
( ) *
* *
argmin , k k
k
k F x x x= =
2. Iteration Number 1n =
3. Particle Number 1k =
4. Velocity vector modification
4.1. ( )1 r rand= , ( )2 r rand=
4.2. ( ) ( )*
1 1 2 2
best
k k k k kv v x x r x x r = + − + −
4.3. Speed limits kv present, i.e.
max , , min , , 1,
min max
kj j kj kj j kjv v v v v v j M= =
or absent
5. Position Modification
5.1. = +
k k k
x x v
5.2. 1j =
5.3. If ,
min
kj jx x then ,
min min
kj j kj j kj kjx x x x v v= + − = −
5.4. If ,
max
kj jx x then ,
max max
kj j kj j kj kjx x x x v v= − − = −
5.5. If ,j M then 1,j j= + Go to step 5.3
6. Definition personal (local) best Position:
if ( ) ( ),best
k kF x F x then best
k kx x=
7. If ,k K then 1k k= +
8. Determine the particle of the current population that is best in terms of the function of the target
( )*
argmin k
k
k F x=
Системні дослідження в енергетиці. 2025. 3(83) 69
9. Determining the Global Best Position:
if ( ) ( )*
*
,
k
F x F x then *
*
k
x x=
10. Stop condition:
If ,n N then 1,n n= + go to step 3
The result is x*.
2.2.2. Bees algorithm
The bee algorithm is based on the behavior of honey bees. It is based on the behavior of foraging bees
and is an extension of the bee system. There is a phase of the worker bee (busy foraging) and the scout bee.
The purpose of the algorithm is to determine the location of good areas in the search space. Scout bees perform
a random search. The found good (in terms of the value of the objective function / functionality) areas are
investigated with the help of local search. The solution corresponds to the position of the bee located in a
certain area.
Algorithm for Optimizing Numerical Functions
1. Initializing
1.1. Specifying Parameters , ,max to create a circle, and ( )0;1 , ( )0;1 ,max ( )0;1 .
1.2. Set the maximum number of iterations ,N population size ,K the number of plots (and the bees
of these plots) ,sL number of elite plots ,esL The number of bee departures in an elite area ,eZ
The number of departures of the worker bee in a regular area ,oZ Length of Bee Position Vector
M (dimension of the search space), minimum and maximum values for the position vector
, , 1, .
min max
j jx x j M
1.3. Defining the Cost Function (Goal Function) ( ) ( )1, , , ,MF min x x→ = x x
where x is the vector of the bee's position.
1.4. Randomly create a vector of a better position
( ) ( )* * * *
1 2, , , , (),
min max min
M kj j j jx x x x x x x x rand= = + −
where ()rand is a function that returns a uniformly distributed random number in the range 0;1
1.5. Creating the Original Population
1.5.1. Bee Number 1, k P= =
1.5.2. Randomly create a vector position kx
( ) ( )1, , , ()
min max min
k kM kj j j jx x x x x x rand= = + −kx
1.5.3. If ,kx P then , 1kP P x k k= = +
1.5.4. If ,k K then go to step 1.5.2
2. Iteration Number 1n =
3. Identify the bee with the best position ( )*
argmin k
k
k F x=
3. Particle Number 1k =
4. If ( ) ( )*
*
,
k
F x F x then *
*
k
x x=
5. Arrange P by the value of the objective function, i.e. ( ) ( )1F F +
k k
x x
6. Worker Bee Phase (Local Search)
6.1. Plot number 1l =
6.2. Determine the size of the circle
Системні дослідження в енергетиці. 2025. 3(83) 70
,1
,
e es
o es s
Z l L
Z
Z L l L
=
6.3. Create a Circle for a Position l-(a) To the extent permitted
6.3.1. ( ) n
maxn =
6.3.2. ( ) ( ) ( )( )1 2 ,
max min
zj lj j jx x n x x rand = + − − + 1; , 1;j M z Z
6.3.3. max ; , min ; ,
min max
zj j zj zj j zjx x x x x x= = 1; , 1;j M z Z
6.4. ( )* argmin z
z
z F x=
6.5. ( ) ( )
*
z
lx
Якщо f x f x then
*
.
z
l x
x x=
6.6. If ,sl L then 1,l l= + Go to step 6.2
7. Scout Bee Phase (Random Search): ( ) ( ) ,
min max min
lj j j jx x x x rand= + − 1; , 1;sj M l L K +
8. Stop Condition
If ,n N then 1,n n= + go to step 3
9. Identify the bee with the best position ( )*
argmin , k
k
k F x= = *
*
k
x x
The result is x*.
2.2.3. Differential evolution method
The main idea of the method of differential evolution is the combination of mutation and crossing to
efficiently find optimal solutions in multidimensional parameter spaces. The method uses some ideas of
genetic algorithms, but, unlike the latter, does not involve working with binary code.
Problem formulation: ( ) ( )1 2min, , , , .MF X X X X X→ =
The main stages of the method are as follows:
1) For each chromosome in a population ,ix ( )1, , , 1;i i iMx x x i N= = (N is the dimension of the
population), three other random members of that population are selected
1 2 3, , ,x x x
1 2 3 1 2 3, , , i i ix x x x x x x x x .
2) A mutant vector is generated: ( ) ( )1 2 3 , 0;2v x F x x F= + −
3) The vector difference 2 3x x− is scaled by a user-defined hyperparameter ( ), 0;2 .F F
A visual illustration of the method is shown in Figure 1.
4) We form a test vector based on crossing
ix with a mutant vector v : for each coordinate of the
chromosome, a number is generated ( )0;1r according to the normal distribution.
➢ If r P , P is a given constant (another parameter of the algorithm ( )0;1P ), then the
corresponding coordinate of the chromosome is replaced by , 1; ,j ijv x j M= M – the
dimension of the search space (the number of independent variables of the fitness function).
Системні дослідження в енергетиці. 2025. 3(83) 71
Fig. 1. Illustrative illustration of the method of differential evolution
5) If ( ) ( ),i if x f v x v = .
All of the above steps are performed up to the method stop criterion (classical, as in other algorithms
and swarm intelligence methods).
2.3. Combination of deterministic and stochastic methods
In complex scientific and technical applied problems, the objective function or functional can be
computed for a relatively long time, even on modern computing systems. Stochastic methods and algorithms
of global optimization can find global extremes of objective functions without requiring additional information
about them, in particular, the differentiability of objective functions. In this case, the total number of
calculations increases. This means that the processor time of searching for the global extremum of the objective
function also increases. In this case, methods and algorithms are developed that provide for a deterministic
component, as well as a stochastic component. The deterministic component works quickly and efficiently
refines the solution found, and the stochastic component prevents the method / algorithm from getting stuck
in local extremums.
3. Practical results
3.1. Modification of the described methods and algorithms of swarm intelligence
In computational mathematics, each modification of an optimization method/algorithm involves a robot
in several ways:
a) reducing the total number of iterations to achieve a certain error. The positive effect of reducing the
total number of iterations leads to a decrease in the total processor time for solving a specific problem or
solving a specific problem. This, in turn, reduces the overall load on the system;
b) improving the accuracy of calculations. The positive effect of this is to accelerate the convergence of
the optimization method/algorithm. This, in turn, again leads to a decrease in the number of iterations, which
means a decrease in the processor time required to solve a problem or a specific task.
This paper proposes several modifications of the algorithms described above.
Системні дослідження в енергетиці. 2025. 3(83) 72
1) Modification of the position of the worst element of the population in the algorithms and
methods of swarm intelligence described above. Replace the worst element of t
j
wors
x the population with the
average position of the element throughout the population Average
jx 1;j M = according to the following
formula:
1
1
1; ,
N
i
Average
j
i
jx x j M
N =
= = (3)
where N is the total number of items in the population, and M is the dimension of the search space. This
approach can be performed on a per-iteration or periodically every K iterations.
2) Modification of the position of the elements of the population to the best in increments h. The
renewal of each element of the population is carried out according to the formula:
,
, 1; , 1; ,
-
,
-
Best j ij
ij ij Best i
Best i
x x
x x h x x i M
x x
N j+ == = (4)
where Bestx is the best vector of the population in terms of the value of the objective function. It should be
noted that the step of motion h can be determined proportionally (according to the linear law) to the
circumference of the population, which will coincide to the global optimum over time.
3) Changing the hyperparameters of optimization methods/algorithms. This approach involves
changing one or more hyperparameters during the operation of the program. For example, in a differential
evolution algorithm, the parameter F can be replaced by a random real number in the range from 0 to 2 through
K iterations.
In the bee algorithm, there is a dependence of the parameter change ( )n on the number of iterations
n. In this case, in order to optimize functions of large dimensions, a problem arises, which is to reduce this
parameter in advance. In order to prevent this from happening before finding the global extremum of the
function or functional, such a case is assumed to be
( ) /
,
n K
maxn = (5)
where [⋅] is the integer part of the number, K is the period that determines how many iterations the value of
the η parameter will be updated. In the method of differential evolution, it is proposed to take the
hyperparameter F as a random number from 0 to 2 every K iterations. This is due to the fact that at each
iteration, the method finds values closer to the global optimum of the function or functional.
3.2. The structure of the developed software
In the course of the study, a software package for optimization problems was developed, which includes
a module of the main wreath of transition to the modules for solving the optimization problems considered in
the work, 6 modules (3 modules for solving problems by conventional methods / algorithms and 3 modules
with modifications of the corresponding methods / algorithms).
3.3. Functions for testing methods and algorithms and their modifications
The first test function is the function, which looks like this:
( ) ( )( )2
1
cos 2 ,
n
i i
i
f X An x A x
=
= + − (6)
where 10, 5,12;5,12 , 1, .iA x i n= − = is the Global Low at ( ) ( )* * * *
1 2; ; 0; ;0 .nX x x x= = ( )*
0.f X =
The second test function is the Rosenbrock function with functional limitations). Rosenbrock function
bounded by a cubic curve and a straight line.
( ) ( ) ( )
( )
22 2
3
, 1 100 min,
1 1 0; 2 0, 1,5;1,5 , 0,5;2,5 .
f x y x y x
x y x y x y
= − + − →
− − + + − − −
(7)
Системні дослідження в енергетиці. 2025. 3(83) 73
Optimal value: ( )1;1 0.minf =
The third test function is the Rosenbrock function, bounded by the disk. It looks like this:
( ) ( ) ( )
22 2
2 2
, 1 100 min,
2, 1,5;1,5 , 1,5;2,5 .
f x y x y x
x y x y
= − + − →
+ − −
(8)
Optimal value: ( )1;1 0.minf =
The fourth Mishra-Bird test function with functional limitations, which looks like this:
( ) ( ) ( ) ( )
( ) ( )
2 2 21 cos 1 sin
2 2
, sin cos min,
5 5 25, 10;0 , 6,5;0 .
x y
f x y e y e x x y
x y x y
− −
= + + − →
+ + + − −
(9)
Optimal value: ( )3,1302468; 1,5821422 106,7645367.minf − − = −
The last classical function for testing algorithms and methods of swarm intelligence is the Simionescu
function with functional limitations. It looks like this:
( )
( )( )( )
2
2 2
, 0,1 min,
1 0,2cos 8 1,25;1,25 ; 1,25;1,25, .
f x y xy
x y arctg x yx y
= →
+ + − −
(10)
Optimal value: ( )0,84852813; 0,84852813 0,0720.minf = −
3.4. Results of testing methods and algorithms and their modifications on ordinary functions
The software package for testing methods and algorithms was developed in the MATLAB 2023b
computer mathematics system. Processor on which the calculations were carried out: INTEL Xeon E-2288G
8C/16T/3.7GHz/16MB/FCLGA1151/TRAY (CM8068404224102).
This section of the paper contains practical results that demonstrate the effectiveness of both classical
methods and algorithms of global multidimensional optimization, as well as modifications obtained for them
in the work. Results of comparison of the classical algorithm of global optimization by a swarm of particles
and its modification. Table 1 shows the results of the computational algorithm for the hyperparameters
vmin = –5, vmax = 5. Variable N Table 1 shows the number of elements in the swarm (population dimension), a1,
a2 – the hyperparameters of the algorithm, which were described above in the pseudocode. The calculation
error ε = 10-12.
Table 1. Comparative Analysis of the Classical Algorithm of Global Multivariate Optimization by a Flock of Particles
and Its Modifications
Function, Algorithm Parameters
(Classic/Modified)
Space
Average number of
iterations
Working hours of the
program,
Sec.
Deviation from the exact
value of the objective
function
Classical
algorithm
Modified
algorithm
Classical
algorithm
Modified
algorithm
Classical
algorithm
Modified
algorithm
Problem (6), N = 200; a1=2,
a2 – rand(0; 4) / a1=2, a2 = 3
2 112 59 2,700E-07 9,998E-08 4,969E-13 3,040E-13
Problem (6), N = 600, a1=2,
a2 – rand(0; 4) / a1=2, a2 = 3
5 48 29 1,880E-06 1,160E-06 8,319E-13 7,199E-13
Problem (6), N = 350, a1=2,
a2 – rand(0; 4) / a1=2, a2 = 3
20 629 373 2,688E-05 1,590E-05 8,869E-13 8,949E-13
Problem (7), N = 350; a1=2,
a2 – rand(0; 4) / a1=2, a2 = 2
2 85 63 2,800E-07 1,940E-07 7,199E-13 6,409E-13
Problem (8), N = 350; a1=2,
a2 – rand(0; 4) / a1=2, a2 = 2
2 56 35 3,719E-07 1,260E-07 5,879E-13 4,859E-13
Problem (9), N = 350; a1=2,
a2 – rand(0; 4) / a1=2, a2 = 2
2 212 69 6,803E-06 2,232E-06 5,659E-13 5,969E-13
Problem (10), N = 350; a1=2,
a2 – rand(0; 4) / a1=2, a2 = 2
2 198 58 6,564E-06 2,544E-06 5,081E-13 5,001E-13
Системні дослідження в енергетиці. 2025. 3(83) 74
As can be seen from the results obtained, modifications of the algorithm give an advantage in the total
number of iterations, which reduces the total load on the processor and reduces the total number of calculations
to obtain the optimum of a function with a given accuracy (computational error that is the result of deviation
from the exact value for test functions / functionals). Similarly, a study was conducted with a modification of
the bee algorithm. Table 2 shows the results of the computational algorithm for the hyperparameters Zo = 150,
Ze = 100, ηmax = 1. The calculation error ε = 10-12. The studies take into account the dependence (5), the value
of the parameter K = 5.
Table 2. Comparative Analysis of the Classical Bee Algorithm of Global Multivariate Optimization
and its Modifications
Function, Algorithm
Parameters
(Classic/Modified)
Space
Average number of iterations
Working hours of the
program,
Sec.
Deviation from the exact
value of the objective
function
Classical
algorithm
Modified
algorithm
Classical
algorithm
Modified
algorithm
Classical
algorithm
Modified
algorithm
Problem (6) 2 37 26 2,700E-07 1,998E-08 4,969E-13 3,039E-13
Problem (6) 5 138 84 1,880E-02 1,160E-02 8,319E-13 7,199E-13
Problem (6) 20 597 354 2,688E-05 1,590E-05 8,868E-13 8,948E-13
Problem (7) 2 48 33 2,800E-06 1,940E-06 7,199E-13 6,409E-13
Problem (8) 2 84 53 3,719E-07 1,260E-07 5,879E-13 4,859E-13
Problem (9) 2 182 121 6,803E-06 2,232E-06 5,659E-13 5,969E-13
Problem (10) 2 171 100 7,131E-06 2,541E-06 5,217E-13 5,021E-13
The results of modifying the classical algorithm give an acceleration of about 1.5–2 times compared to
the classical algorithm. Table 3 presents the results of the study with a modification of the method of
differential evolution. The table shows the results of the computational algorithm, taking into account (3)
and (4). Calculation error ε = 10-12. The studies take into account the update of the value of the parameter F
= 2 rand( ) every K iterations, with K = 5.
Table 3. Comparative Analysis of the Classical Method of Differential Evolution for Global Multivariate Optimization
and Its Modification
Function, Method
Parameters
(Classic/Modified)
Space
Average number of iterations
Working hours of the
program,
Sec.
The resulting value of the
function
Classical
method
Modified
Method
Classical
method
Modified
Method
Classical
method
Modified
Method
Problem (6) 2 8 5 1,500E-07 5,555E-08 2,761E-13 1,689E-13
Problem (6) 5 28 17 1,044E-06 6,443E-07 4,621E-13 3,999E-13
Problem (6) 20 371 220 1,493E-05 8,832E-06 4,927E-13 4,971E-13
Problem (7) 2 89 37 1,555E-07 1,078E-07 3,999E-13 3,561E-13
Problem (8) 2 64 41 2,066E-07 6,999E-08 3,266E-13 2,700E-13
Problem (9) 2 125 72 3,779E-06 1,240E-06 3,144E-13 3,316E-13
Problem (10) 2 97 43 3,251E-06 1,014E-06 3,544E-13 3,154E-13
3.5. Non-linear regression model
Let's consider the basic principles of finding a solution to a problem using a specific model mathematical
example. Let the mathematical model that describes the process look like this:
2 3
1 2 3 4
2 3
5 6 7
.
1
b b x b x b x
y
b x b x b x
+ + +
=
+ + +
Task: to find such values of parameters , 1;7,ib i = that the mathematical model describes statistical data
as accurately as possible.
Loss Function is a target function that has one of the following forms:
( ) ( )( )
2
1 7 1 7
1
1
, , , , ;
S
Model
i i
i
E b b y y b b
S =
= − ( ) ( )1 7 1 7
1
1
, , , , ;
S
Model
i i
i
E b b y y b b
S =
= −
Системні дослідження в енергетиці. 2025. 3(83) 75
( ) ( )1 7 1 7, , max , , ,
Model
i i
i
E b b y y b b = −
Model
iy is the value of the functional dependence for the set ( )1 7, ,b b in the i-th observation, S is the number
of selected observations. Dataset: 36 observations – 1 independent variable (x), 1 dependent variable (y).
The data looks like Table 4:
Table 4. Input data for building a nonlinear regression model and their visualization
y x y x y x Graphical representation of data
80,574 -3,067 401,672 -1,501 1273,514 -0,103
84,248 -2,981 390,724 -1,460 1288,339 0,010
87,264 -2,921 567,534 -1,274 1327,543 0,119
87,195 -2,912 635,316 -1,212 1353,863 0,377
89,076 -2,840 733,054 -1,100 1414,509 0,790
89,608 -2,797 759,087 -1,046 1425,208 0,963
89,868 -2,702 894,206 -0,915 1421,384 1,006
90,101 -2,699 990,785 -0,714 1442,962 1,115
92,405 -2,633 1090,109 -0,566 1464,350 1,572
95,854 -2,481 1080,914 -0,545 1468,705 1,841
100,696 -2,363 1122,643 -0,400 1447,894 2,047
101,060 -2,322 1178,351 -0,309 1457,628 2,200
The search for a solution to the problem of finding unknown coefficients of the nonlinear regression
model was carried out by steps in the general methodology of machine learning.
1) Shuffle the sample with the rows in the database that contains the observations (database rows are
swapped randomly).
2) Selection from the beginning of most of the data for training (search for parameters * *
1 7, , ,b b that best
describe the statistics). This is an educational (training sample). Typically, this subsample contains 75 %
to 80 % of all observations in the sample.
3) Finding * *
1 7, ,b b parameters using the optimization method / algorithm of the objective error function.
4) Finding an error on the test training and test subsamples for the found * *
1 7, , .b b ones They are compared
with each other, conclusions are formulated.
5) For the found, * *
1 7, ,b b dependency graphs are constructed: real data on the test sample and model data
on the training sample; real data on the test sample and model data on the test sample.
6) The results of the dependence of real data on the test sample and model data on the training sample and
real data on the test sample and model data on the test sample are compared with the dependence y = x.
7) Graphical derivation of the statistical dependence of the independent variable (x) and the dependent
variable (y) on the entire sample. Imposition of functional dependence on a given coordinate system
* * * 2 * 3
1 2 3 4
* * 2 * 3
5 6 7
.
1
b b x b x b x
y
b x b x b x
+ + +
=
+ + +
If a functional dependency does describe a certain statistical process correctly, then the Loss Function
will have relatively small values. The main results of the simulation are shown in Figure 2 and Figure 3.
Системні дослідження в енергетиці. 2025. 3(83) 76
Fig. 2. Convergence process when searching for nonlinear regression parameters for test and training samples
It took more iterations to solve the nonlinear regression problem than to find the global extremum of the
classical test functions, which are given above together with the results of the application of optimization
methods and algorithms. It should also be noted that the adaptation of algorithms and methods of swarm
intelligence to the tasks of nonlinear regression analysis can be an interesting area of research.
Fig. 3. Forecast graphs and real data for training and test samples
Most technical problems involve building optimization models on statistical data, which are mostly
reduced to the form where the least squares method can be effectively applied. Even if the least-squares method
is difficult to apply due to the poor conditionality of matrix structures, here regularization is performed and
again algorithms and methods of swarm intelligence are used to minimize the generalized functionality. For
most tasks of regression analysis, the results of machine learning are built in the space "Model data – real
data". This result is shown in Figure 3, which shows the results of the "forecast" on the training sample (left),
as well as the results of the "forecast" on the test sample (on the right).
3.6. Applied problems with functional limitations
In this section of the work, several applied problems are considered, which involve the use of swarm
intelligence methods. Other well-known methods and algorithms of global multivariate optimization have
worse results than those considered on similar applied problems.
3.6.1. The problem of minimizing the weight of the reducer [10]
The problem minimizes the weight of the gearbox (Figure 4) taking into account the constraints on
wheel tooth bending, surface stress, transverse deflections of the shafts and stress in the shafts.
Системні дослідження в енергетиці. 2025. 3(83) 77
Fig. 4. Gearbox geometers
Problem parameters:
b – Plating width (
1x );
m – Wheel Module ( 2x );
z – Number of wheel teeth ( 3x – integer variable);
1l – The length of the first shaft between the bearings ( 4x );
2l – The length of the second shaft between the bearings ( 5x );
1d – The diameter of the first shaft ( 6x );
2d – The diameter of the second shaft ( 7x ).
The objective function looks like this:
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
2 2
1 2 3 3
2 2 3 3 2 2
1 6 7 6 7 4 6 5 7
3
4
1 2 32 2 2 4
1 2 3 1 2 3 2 3 6
4 54 3
2 3 7 6
0.7854 3.3333 14.9334 43.0934
1.508 7.4777 0.7854 ,
1.9327 397.5
1 0, 1 0, 1 0,
1.93 1.0
1 0,
110
f x x x x x
x x x x x x x x x min
x
g x g x g x
x x x x x x x x x
g x g x
x x x x
= + − −
− + + + + + →
= − = − = −
= − =
( ) ( )
( ) ( ) ( ) ( )
2
64
2 3
2
65 2 3
6 73
7 2 3
6 72 1
8 9 10 11
1 2 4 5
1 2
745.0
16.9 10 1 0,
745.01.0
157.5 10 1 0, 1 0,
85 40
1.5 1.9 1.1 1.95
1 0, 1 0, 1 0, 1 0,
12
2,6 3,6; 0,7 0,8
x
x x
x x x
g x g x
x x x
x xx x
g x g x g x g x
x x x x
x x
+ −
= + − = −
+ +
= − = − = − = −
3 4 3
5 6 7
; 17 28; 7,3 8,3; ,
7,8 8,3; 2,9 3,9; 5,0 5,5.
x x x
x x x
(11)
Table 5 shows the results of the method of differential evolution and its modification, which are
described in the paper for 40 different runs of the program for searching for the global extremum. To generate
the results in Table 5, the best of the three tools for finding the global extremum of test multivariate functions
was chosen. The best tool is the method of differential evolution. Table 5 takes into account the corresponding
modifications of the method that were used to generate the results in the previous tables.
Table 5. Computational Experiments for the Problem (11)
Computational
Experiment
Number
Number of iterations
Working hours of the program,
Sec.
Deviations from the exact value
of the objective function [10]
Classical
method
Modified
Method
Classical method Modified Method
Classical
method
Modified Method
1 356 211 1,28070E+00 7,59531E-01 9,89873E-13 4,26860E-13
2 366 193 1,35873E-02 8,11026E-03 2,50646E-13 3,07247E-13
3 390 167 1,85683E-02 8,61480E-03 2,60083E-13 2,34069E-13
… … … … … … …
38 464 140 9,22273E-04 7,14465E-04 2,31168E-13 1,31756E-13
39 444 159 2,23053E-04 5,71327E-04 7,10985E-13 6,39946E-13
40 432 138 3,47420E-04 2,24475E-04 4,80957E-13 6,19187E-13
3.6.2. The problem of optimizing the spring design [10]
The problem minimizes the tension / compression of the spring taking into account the limitations of
minimum deviation, shear stress, overvoltage frequency, diameter constraints and design variables (Figure 5).
Системні дослідження в енергетиці. 2025. 3(83) 78
Fig. 5. Structural diagram of the spring and its parameters
Parameters of the problem:
d – wire diameter (
1x ),
D – average coil diameter ( 2x ),
N – number of active coils ( 3x ).
The mathematical model of the optimization applied problem is as follows:
( ) ( )
( ) ( ) ( )
( )
( )
2
3 2 1
3
2 3 1 2 1
1 3 44 2
1 2 3
2
2 1 2
2 23 4
12 1 1
1 2 3
2 .
140,45
1 0, 1 0, 1 0,
7,178 1.5
4 1
1 0,
5,10812,566
0,005 2,0, 0,25 1,3, 2,0 15,0.
f x x x x min
x x x x x
g x g x g x
x x x
x x x
g x
xx x x
x x x
= + →
+
= − = − = −
−
= + −
−
(12)
Table 5 shows the results of the method of differential evolution and its modification, which are
described in the paper for 30 different runs of the program for searching for the global extremum.
Table 6. Computational Experiments for the Problem (12)
Computational
Experiment
Number
Number of iterations
Working hours of the program,
Sec.
Deviations from the exact value of
the objective function [10]
Classical
algorithm
Modified
algorithm
Classical
algorithm
Modified
algorithm
Classical
algorithm
Modified algorithm
1 356 211 9,32504E-01 4,08713E-01 9,13226E-13 2,54923E-13
2 386 203 8,68428E-01 5,38674E-01 7,05207E-13 1,71713E-13
3 400 224 2,17573E-01 1,22818E-01 3,92392E-13 1,35203E-13
… … … … … … …
28 371 196 5,64807E-01 3,94011E-01 1,66983E-13 1,83788E-13
29 395 213 6,04579E-01 4,29167E-01 1,45964E-13 2,54636E-13
30 402 231 2,17123E-01 1,32962E-01 3,66866E-13 1,89860E-13
As we can see from the results of testing the method, it has proven to be very effective on applied
problems. A modification of the classical method showed an acceleration of about 1.5–2 times.
4. Discussion
The main advantage of modern methods and algorithms of swarm intelligence is the intuitive structure
of the natural origin of the swarm, which means that it is a relatively simple software implementation.
Therefore, it makes swarm intelligence tools popular in various fields of research.
Most processes are structurally nonlinear. Swarm intelligence demonstrates high efficiency in finding
global optima for a wide range of functions, including nonlinear and non-convex functions, and works
effectively with nonlinear functions through the appropriate mechanism of evolution, which allows you to
move in the parameter space to find the global optimum. Another of the main advantages of the methods and
algorithms under consideration is the small number of hyperparameters that need to be configured. This
simplifies the use of methods and algorithms and customization compared to other algorithms.
Системні дослідження в енергетиці. 2025. 3(83) 79
5. Conclusions
The development of computing technologies makes it possible to simulate the behavior of complex
objects and systems. The development of new and modification of existing systems involves the construction
of mathematical models in the form of optimization in order to study various parameters of such systems. Such
models make it possible to have an idea of the modes of operation of various objects and systems, their optimal
parameters (settings), geometric properties of such systems. The use of swarm intelligence contributes to the
study and development of such objects and systems, since the usual approximate methods of global
multivariate optimization do not have such an opportunity due to the complexity and nonlinearity of functional
dependencies that describe systems and their operation.
In this work, three relevant methods and algorithms of swarm intelligence are studied: the algorithm of
global optimization by a swarm of particles, the bee algorithm for finding global solutions to problems that are
presented in extreme statements, and the method of differential evolution. These methods have been tested on
various scientific and applied problems, including problems that boil down to the search for global extremes
of multivariate functions with and without constraints, as well as on problems that reduce to the use of
nonlinear multivariate regression models and forecasting time series that arise during the study of the work of
various system complexes. Three modifications of the considered methods and algorithms have been obtained.
The first modification consists in replacing the worst element of the population with the average position of
the element throughout the population. This approach works effectively when applied at each iteration or
periodically at every K iteration. The second modification of these methods and algorithms is to apply the
principle of deterministic movement towards the best element of the population. Movement step h it is
determined in proportion (according to the linear law) to the circumference of the population, which will
coincide to the global optimum over time. The third modification consists in changing the hyperparameters of
optimization methods/algorithms. This involves changing one or more hyperparameters in the course of the
program's operation. The practical results of the article are that the complex application of three modifications
for each method/algorithm gives advantages in increasing the dimension of search in an extreme problem. For
multivariate problems, modifications affect the elements of the population and allow for a more accurate
definition of the solution. This reduces the total number of iterations of the method/algorithm, which means
that it reduces the time to find the optimum of the problem with a given accuracy.
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МОДИФІКАЦІЯ МЕТОДІВ РОЙОВОГО ІНТЕЛЕКТУ ДЛЯ
ЗАДАЧ ОПТИМІЗАЦІЇ СКЛАДНИХ ПРОЦЕСІВ, ОБ’ЄКТІВ І
СИСТЕМ
Владислав Хайдуров1,2*, канд. техн. наук, ст. досл., https://orcid.org/0000-0002-4805-8880
Артем Зінченко1, канд. техн. наук, доцент, https://orcid.org/0000-0003-1586-3645
Тамара Цюпій3, канд. фіз.-мат. наук, доцент, https://orcid.org/0000-0003-2206-2897
Роман Яровий4, канд. техн. наук, https://orcid.org/0000-0001-8978-8137
1Національний технічний університет України «Київський політехнічний інститут імені Ігоря
Сікорського», Берестейський просп., 37, Київ, 03056, Україна;
2Інститут загальної енергетики НАН України, вул. Антоновича, 172, Київ, 03150, Україна;
3Національний університет біоресурсів і природокористування України, вул. Героїв Оборони, 15,
Київ, 03041, Україна;
4ПЗВО «Європейський університет», бульвар Академіка Вернадського, 16 В, Київ, 03041, Україна
*Автор-кореспондент: allif0111@gmail.com
Анотація. Розробка швидкісних методів й алгоритмів глобальної багатовимірної оптимізації і їх
модифікацій у різних сферах науки, техніки, економіки є актуальним завданням, яке передбачає
зменшення обчислювальних затрат, прискорення і ефективний пошук розв’язків такого роду задач.
У зв’язку з тим, що більшість серйозних завдань передбачають пошук десятків, сотень або тисяч
оптимальних параметрів математичних моделей, простір пошуку цих параметрів зростає
нелінійно. Сучасний ройовий інтелект має значний потенціал для застосування в енергетичній
галузі через свою здатність до оптимізації та розв’язання складних проблем. За допомогою нього
можна вирішувати науково-прикладні задачі оптимізації споживання енергії в будівлях,
промислових комплексах та міських системах, зменшуючи втрати енергії та підвищуючи
ефективність використання ресурсів, а також для побудови різних елементів енергетичних
систем загалом. Відомі методи й алгоритми ройового інтелекту також активно застосовують
для прогнозування виробництва енергії від відновлюваних джерел, таких як сонячна та вітрова
енергія. Це дозволяє краще управляти джерелами енергії та планувати їхнє використання.
Актуальність модифікацій методів й алгоритмів зумовлена питаннями прискорення швидкості їх
роботи під час розв’язання задач машинного навчання, зокрема у моделях нелінійної регресії,
задачах класифікації, де кількість спостережуваних даних може сягати десяти і сотні тисяч або
більше. У роботі розглянуті й модифіковані відомі ефективні методи й алгоритми ройового
інтелекту (алгоритм оптимізації роєм частинок, бджолиний алгоритм оптимізації, метод
диференціальної еволюції) для пошуку розв’язків багатовимірних екстремальних задач з
обмеженнями і без обмежень, а також задач нелінійного регресійного аналізу. Продемонстрована
ефективність роботи модифікованих методів на різних класичних і прикладних задачах, які
використовуються в проєктуванні елементів складних об’єктів та їх систем. Проведено
порівняльний аналіз результатів роботи даних методів.
Ключові слова: оптимізація, ройовий інтелект, математичне моделювання, нелінійна регресія,
складні об’єкти та системи.
Надійшла до редколегії: 27.01.2025
https://www.j-ets.net/collection/published-issues/12_3
https://www.j-ets.net/collection/published-issues/12_3
https://orcid.org/0000-0002-4805-8880
https://orcid.org/0000-0003-1586-3645
https://orcid.org/0000-0003-2206-2897
https://orcid.org/0000-0001-8978-8137
mailto:allif0111@gmail.com
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| id | systemreorg-article-907 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:23:47Z |
| publishDate | 2025 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/e3/3f958eafbb1726f2d321d52b6acc99e3.pdf |
| spelling | systemreorg-article-9072026-07-18T12:57:49Z MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS Модифікація методів ройового інтелекту для задач оптимізації складних процесів, об’єктів і систем Khaidurov, Vladyslav Zinchenko, Artem Tsiupii, Tamara Yarovoy, Roman optimization, swarm intelligence, mathematical modelling, nonlinear regression, complex objects and systems. оптимізація, ройовий інтелект, математичне моделювання, нелінійна регресія, складні об’єкти та системи. The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problems that involves reducing computing costs, accelerating and effectively finding solutions to such problems. Due to the fact that most serious problems involve the search for tens, hundreds or thousands of optimal parameters of mathematical models, the search space for these parameters grows non-linearly. Modern swarm intelligence has significant potential for application in the energy industry due to its ability to optimize and solve complex problems. With its help, it is possible to solve scientific and applied problems of optimizing energy consumption in buildings, industrial complexes and urban systems, reducing energy losses and increasing the efficiency of resource use, as well as for the construction of various elements of energy systems in general. Well-known methods and algorithms of swarm intelligence are also actively used to forecast energy production from renewable sources, such as solar and wind energy. This allows better management of energy sources and planning of their use. The relevance of modifications of methods and algorithms is due to the issues of speeding up their work when solving machine learning problems, in particular, in nonlinear regression models, classification, clustering problems, where the number of observed data can reach tens and hundreds of thousands or more. The work considers and modifies well-known effective methods and algorithms of swarm intelligence (particle swarm optimization algorithm, bee optimization algorithm, differential evolution method) for finding solutions to multidimensional extremal problems with and without restrictions, as well as problems of nonlinear regression analysis. The effectiveness of the modified methods on various classical and applied problems, which are used in the design of elements of complex objects and their systems, is demonstrated. A comparative analysis of the results of these methods was carried out. Розробка швидкісних методів й алгоритмів глобальної багатовимірної оптимізації і їх модифікацій у різних сферах науки, техніки, економіки є актуальним завданням, яке передбачає зменшення обчислювальних затрат, прискорення і ефективний пошук розв’язків такого роду задач. У зв’язку з тим, що більшість серйозних завдань передбачають пошук десятків, сотень або тисяч оптимальних параметрів математичних моделей, простір пошуку цих параметрів зростає нелінійно. Сучасний ройовий інтелект має значний потенціал для застосування в енергетичній галузі через свою здатність до оптимізації та розв’язання складних проблем. За допомогою нього можна вирішувати науково-прикладні задачі оптимізації споживання енергії в будівлях, промислових комплексах та міських системах, зменшуючи втрати енергії та підвищуючи ефективність використання ресурсів, а також для побудови різних елементів енергетичних систем загалом. Відомі методи й алгоритми ройового інтелекту також активно застосовують для прогнозування виробництва енергії від відновлюваних джерел, таких як сонячна та вітрова енергія. Це дозволяє краще управляти джерелами енергії та планувати їхнє використання. Актуальність модифікацій методів й алгоритмів зумовлена питаннями прискорення швидкості їх роботи під час розв’язання задач машинного навчання, зокрема у моделях нелінійної регресії, задачах класифікації, де кількість спостережуваних даних може сягати десяти і сотні тисяч або більше. У роботі розглянуті й модифіковані відомі ефективні методи й алгоритми ройового інтелекту (алгоритм оптимізації роєм частинок, бджолиний алгоритм оптимізації, метод диференціальної еволюції) для пошуку розв’язків багатовимірних екстремальних задач з обмеженнями і без обмежень, а також задач нелінійного регресійного аналізу. Продемонстрована ефективність роботи модифікованих методів на різних класичних і прикладних задачах, які використовуються в проєктуванні елементів складних об’єктів та їх систем. Проведено порівняльний аналіз результатів роботи даних методів. General Energy Institute of the National Academy of Sciences of Ukraine 2025-08-26 Article Article application/pdf https://systemre.org/index.php/journal/article/view/907 10.15407/srenergy2025.03.065 System Research in Energy; No. 3 (83) (2025): System Research in Energy; 65-80 Системні дослідження в енергетиці; № 3 (83) (2025): Системні дослідження в енергетиці; 65-80 2786-7102 2786-7633 en https://systemre.org/index.php/journal/article/view/907/817 Copyright (c) 2025 Vladyslav Khaidurov, Artem Zinchenko, Tamara Tsiupii, Roman Yarovoy https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | optimization swarm intelligence mathematical modelling nonlinear regression complex objects and systems. Khaidurov, Vladyslav Zinchenko, Artem Tsiupii, Tamara Yarovoy, Roman MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS |
| title | MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS |
| title_alt | Модифікація методів ройового інтелекту для задач оптимізації складних процесів, об’єктів і систем |
| title_full | MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS |
| title_fullStr | MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS |
| title_full_unstemmed | MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS |
| title_short | MODIFICATION OF SWARM INTELLIGENCE METHODS FOR OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS |
| title_sort | modification of swarm intelligence methods for optimization of complex processes, objects, and systems |
| topic | optimization swarm intelligence mathematical modelling nonlinear regression complex objects and systems. |
| topic_facet | optimization swarm intelligence mathematical modelling nonlinear regression complex objects and systems. оптимізація ройовий інтелект математичне моделювання нелінійна регресія складні об’єкти та системи. |
| url | https://systemre.org/index.php/journal/article/view/907 |
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