The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations
The paper proposes the practical task of choosing a set of programs to protect information at the enterprise. An optimization combinatorial model is built on the configuration of combinations to solve the task. An algorithm for finding a solution to this optimization problem is presented. A practica...
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
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| Cite this: | The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations / L.M. Koliechkina, A.M. Nahirna // Control systems & computers. — 2019. — № 5. — С. 23-29. — Бібліогр.: 18 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1810462025-02-23T18:27:47Z The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations Практичний аспект застосування комбінаторної моделі на конфігурації сполучень Практический аспект применения комбинаторной модели на конфигурации сочетаний Koliechkina, L.M. Nahirna, A.M. Fundamental Problems in Computer Science The paper proposes the practical task of choosing a set of programs to protect information at the enterprise. An optimization combinatorial model is built on the configuration of combinations to solve the task. An algorithm for finding a solution to this optimization problem is presented. A practical example of the use of a optimization combinatorial model and the search for the best choice of a set of programs for data protection at the enterprise is given. Мета статті — демонстрація використання комбінаторної оптимізаційної моделі на множині сполучень і представлення методу розв’язання задач цього типу. Методи. Метод розв’язання задачі умовної оптимізації на комбінаторній множині сполучень. Результати. Сформульовано проблему вибору програмного забезпечення із захисту інформації та запропоновано спосіб її розв’язання. Наразі задача моделюється комбінаторною оптимізаційною моделлю на множині сполучень. Запропоновано метод розв’язання задач цього типу. Наведено практичний приклад використання комбінаторної оптимізаційної моделі на множині сполучень. Целью данной статьи является демонстрация использования комбинаторной оптимизационной модели на множестве сочетаний и представления метода решения задач данного типа. Методы. Метод решения задачи условной оптимизации на комбинаторном множестве сочетаний. Результаты. Сформулирована проблема выбора программного обеспечения по защите информации и предложен подход к ее решению. В данном случае задача моделируется комбинаторной оптимизационной моделью на множестве сочетаний. Предложен метод решения задач данного типа. Рассмотрен практический пример применения комбинаторной оптимизационной модели на множестве сочетаний. 2019 Article The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations / L.M. Koliechkina, A.M. Nahirna // Control systems & computers. — 2019. — № 5. — С. 23-29. — Бібліогр.: 18 назв. — англ. 2706-8145 DOI: doi.org/10.15407/usim.2019.05.023 https://nasplib.isofts.kiev.ua/handle/123456789/181046 364.2:331; 681.513 en Control systems & computers application/pdf Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України |
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Fundamental Problems in Computer Science Fundamental Problems in Computer Science |
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Fundamental Problems in Computer Science Fundamental Problems in Computer Science Koliechkina, L.M. Nahirna, A.M. The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations Control systems & computers |
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The paper proposes the practical task of choosing a set of programs to protect information at the enterprise. An optimization combinatorial model is built on the configuration of combinations to solve the task. An algorithm for finding a solution to this optimization problem is presented. A practical example of the use of a optimization combinatorial model and the search for the best choice of a set of programs for data protection at the enterprise is given. |
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| author |
Koliechkina, L.M. Nahirna, A.M. |
| author_facet |
Koliechkina, L.M. Nahirna, A.M. |
| author_sort |
Koliechkina, L.M. |
| title |
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations |
| title_short |
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations |
| title_full |
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations |
| title_fullStr |
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations |
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The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations |
| title_sort |
practical aspect of using a combinatorial model on configuration of combinations |
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України |
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2019 |
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Fundamental Problems in Computer Science |
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https://nasplib.isofts.kiev.ua/handle/123456789/181046 |
| citation_txt |
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations / L.M. Koliechkina, A.M. Nahirna // Control systems & computers. — 2019. — № 5. — С. 23-29. — Бібліогр.: 18 назв. — англ. |
| series |
Control systems & computers |
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2025-11-24T10:36:25Z |
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| fulltext |
iSSN 2706-8145, control systems and computers, 2019, № 5 23
doi.org/10.15407/usim.2019.05.023
удк 364.2:331; 681.513
l.n. KolIeChKInA, doctor of physical and mathematical sciences, professor,
University of lodz, 22 banaha st.,
lodz, 90-238, poland,
ludapl@ukr.net
A.n. nAhIrnA, phd, physical and mathematical, associate professor,
National University of “kyiv-mohyla academy”,
2 Skovoroda st. , kyiv, 04070, Ukraine
naghirnaalla@ukr.net
the prACtICAl AspeCt
oF usInG A ComBInAtorIAl model
on ConFIGurAtIon oF ComBInAtIons
The paper proposes the practical task of choosing a set of programs to protect information at the enterprise. An optimization combi-
natorial model is built on the configuration of combinations to solve the task. An algorithm for finding a solution to this optimization
problem is presented. A practical example of the use of a optimization combinatorial model and the search for the best choice of a
set of programs for data protection at the enterprise is given.
Keywords: information security, combinatorial optimization, mathematical model, a configuration of combinations, objective
function, restrictions.
Introduction
Currently, the information security of an enterprise
is one of the leading factors in its effective deve-
lopment . The information has a real value weight,
which is clearly determined by the profit received
during its use, or the damage that may be caused
to an enterprise if it is used by other persons [1–2] .
The share of organizations’ expenses on ensuring
the integrity of information and protecting it from
possible external threats is constantly growing .
However, the profitability of the enterprise depends
on the profit and costs spent on making a profit .
Therefore, enterprises are trying to optimize costs,
and they want to buy only what is really necessary
to build a reliable information protection system
with minimal costs [3] .
Various aspects of data protection and the use
of various methods based on cryptography are de-
scribed in [4–6] .
An integrated information security system at an
enterprise provides for the solution of a number of
tasks that give a solution to security problems at
the software, hardware and organizational levels
[7, 8] . From a practical point of view, the software
level of protection is the most strategically impor-
tant . The market for data security software is very
diverse . For reliable protection of information at
the enterprise, it is necessary to have several pro-
grams in a package that should provide needed
protection in terms of identifying destabilizing
factors and successfully preventing various types of
threats . At the same time, it should be noted that
comprehensive software packages for data protec-
24 iSSN 2706-8145, системи керування та комп'ютери, 2019, № 5
L.N. Koliechkina, A.N. Nahirna
tion have a high cost, therefore it is necessary to
be able to make the best choice from the provided
software, taking into account the necessary condi-
tions that form the basic principles of information
security in an enterprise .
When modelling various processes and pheno-
mena in various fields of activity, models are often
used, which are presented as tasks of conditional
optimization . Particularly noteworthy are the mo-
dels of such problems considered on combinatorial
sets . To date, a significant amount of work has been
devoted to the study and investigation of models of
combinatorial nature [9–10] .
Despite the simplicity of constructing combi-
natorial sets, combinatorial optimization prob-
lems are complex and time-consuming from a
computational point of view [11–12], so there is a
need to develop new methods and algorithms for
solving them .
Formulation of the problem
In the software market, there are many software
products to protect information . There is no need
to purchase large quantities, but you need to make
an effective choice that ensures the prevention of
threats and minimal losses during unauthorized in-
trusions . It should be noted that the price policy in
the software market plays the same important role
when choosing .
Therefore, coming out of the main tasks of data
protection and financial capabilities, as a rule, an
enterprise needs to make a choice from the existing
availability of programs of this type . It should be
noted that data protection programs have a fairly
wide range of overlapping functionalities . At the
same time, vary significantly in price and imple-
mentation requirements .
Each program has a rating in a general database
of programs for certain functions . For example,
ratings of antivirus programs can be found on cer-
tain sites taking into account the assessment of the
main characteristics, such as threat detection, per-
formance, false positives and others .
For ease of presentation, we assume that 1 is
the program with the lowest rating, respectively,
n – with the highest rating . The main task is to
choose a set of programs with the highest ra-
ting taking into account the conditions of an en-
terprise . We assume that the average statistical
prices for programs of the corresponding rating
are known, then the choice should provide for
minimal costs for acquiring the programs with
the highest rating .
Since this set of programs should ensure the
protection of information of the enterprise as a
whole, it is natural to assume that each structural
unit of the enterprise has its requirements for data
protection software . Programs may coincide or
differ, which mathematically can be formulated as
a system of inequalities .
It is necessary to choose a set of programs that
should have the highest rating possible and satisfy the
conditions of the structural units of the enterprise .
The choice of such programs should ensure the mi-
nimization of the costs of their acquisition . When de-
tailing the choice, it is possible to perform additional
calculations of the objective function in the range of
maximum and minimum average prices of data pro-
tection programs, taking into account their rating .
Combination optimization model
Let suppose that in the software market for an en-
terprise in a certain industry, information protec-
tion n programs that have their own specific rating
A = (a
1
, a
2
, . . ., a
n
), (n ∈ N) . Having conducted a
comprehensive assessment to identify destabilizing
factors and possible losses from the invasion, taking
into account the financial activities of the enter-
prise, it was found that there is a need to purchase k
from n programs for the comprehensive protection
of information .
Rating of
Programms
Price, $
p
1
p
2
a
1
= 1 c
11
c
12
a
2
= 2 c
21
c
22
a
3
= 3 c
31
c
32
... ... ...
a
n
= n c
n1
c
n2
Тable 1. Price range of programs depending on rating
iSSN 2706-8145, control systems and computers, 2019, № 5 25
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations
The price of each program, depending on the ra-
ting, can fluctuate in the following range (таb . 1) .
The company has m departments involved in the
processing of information, which is a trade secret .
The main signs of the selection of programs is
their ability to prevent threats of the following type:
virus attacks, unauthorized intrusions into the net-
work, leakage of confidential information .
Each department has formed its necessary condi-
tions for the selection of data protection programs,
which are illustrated by the following constraints:
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
... ( ) ,
... ( ) ,
....................................,
... ( ) .
n n
n n
m m mn n m
a x a x a x b
a x a x a x b
a x a x a x b
+ + + ≥ ≤
+ + + ≥ ≤
+ + + ≥ ≤
(1)
It is necessary to make a choice of k out of n
possible programs, those .to find such an element of
a multitude of combinations that, taking into ac-
count the fulfillment of restrictions (1), would pro-
vide the minimum costs for purchasing programs
of this rating .
Then, the objective function will be:
min F = c
1
x
1
+ c
2
x
2
+ . . . + c
n
x
n
, (2)
where c
1
= c
12
- c
11
, . . ., c
n
= c
n2
- c
n1
.
Since it is necessary to choose k programs from
n (n ≥ k), taking into account their rating, we can
find a solution on the combinatorial set of combi-
nations without repetitions .knC
Then, taking into account constraints and ra-
ting of programs, the mathematical model of the
problem of choosing the best set of data protection
programs for an enterprise is presented as:
( , ( )) :min{ ( ) | }k k
n nZ C A a a CΦ Φ ∈
(3)
{ | ( ) }nD x R Gx b= ∈ ≤ ≥ , G ∈ Rm×m, b ∈ Rm, (4)
where
1
(a)
n
j j
j
c x
=
Φ =∑ – the objective function of the
combinatorial set of combinations .knC
Definition 1 . Let a set A = (a
1
, a
2
, . . ., a
n
), (n ∈ N) .
A combination without repetitions of n elements by
k (n ≥ k) is called k-elemental subset C of the set
A and is denoted by .knC Since the order of writing
elements of the set is irrelevant, therefore, as a rule,
the elements in each combination are written in
ascending order [13] .
We realize the bijective mapping of the set
.knC into space R n by assigning a relevant vector
x ∈ R n to each element a ∈ .knC The image of a
set .knC is denoted by
n n
kE R⊂ . As a result, we have
the problem of combinatorial optimization in the
Euclidean formulation (the problem of Euclidean
combinatorial optimization)
(5)
additional constraints
{ | ( ) }k n
nD x C R Gx b= ∈ ⊂ ≤ ≥ , (6)
where G - m × n matrix, b ∈ R m , while Φ(a) = F(x),
a ∈ .knC , k
nx E∈ .
Next, we consider linear objective functions of
the form
1
( )
n
j j
j
F x c x
=
= ∑
.
.
(7)
Additional linear constraints form a multifaceted
set D ⊂ R n .
Consider the algorithm of solving the problem
(5)-(7) .
Algorithm for solving
The algorithm for solving the formulated problem
consists of three steps, which ensure that the
minimum of the objective function is found with
additional restrictions on the set of combinations .
1 . Finding the first reference solution .
According to Definition 1, the elements of a set
of combinations are written in ascending order,
so the first element of a set of combinations (x
1
,
x
2
, . . ., x
n-1
, x
n
), where (x
1
< x
2
< . . . < x
n-1
< x
n
) , must
be taken as the starting point . Next, check the
constraints (6) .
If constraints (6) are satisfied, the first support
solution is found and the target function is calcu-
lated . Next, go to step 2 .
Otherwise, select the next point in ascending or-
der . Notably, the number of elements in the set of
combinations is a finite set .
2 . Formation of the initial search conditions for
the optimal solution .
A point of a set of combinations that satisfies all
constraints (6) will be the first reference solution .
For further search for the optimal solution, the ini-
( , ) : min{ ( ) | }k k
n nZ F C F a X D C∈ ⊂
26 iSSN 2706-8145, системи керування та комп'ютери, 2019, № 5
L.N. Koliechkina, A.N. Nahirna
tial conditions for finding the optimal solution are
formed:
f(x
1
, x
2
, . . ., x
n
) = b,
1 1 2 1 1 1 1 1 2
2 1 2 2 2 2 2 1 2
1 2
( , ,..., ) ( ) , ( , ,..., ),
( , ,..., ) ( ) , ( , ,..., ),
...............................................................................,
( , ,..., ) ( )
n n
n n
n n
g x x x b b b g x x x
g x x x b b b g x x x
g x x x
∆ ≤ ≥ ∆ ∆ = −
∆ ≤ ≥ ∆ ∆ = −
∆ ≤ ≥ ∆ 1 2, ( , ,..., ).n n n n nb b b g x x x
∆ = −
(8)
The next point of the set of combinations is
checked according to the constraints (8) . If they are
not fulfilled, then return to p . 1 .1 ., if completed, go
to step 3 . For all subsequent points of calculating
the growth of the constraints, be located behind the
formula:
2 1g g g∆ = ∆ − ∆ = 2 1 2 1( ) ( )g g g g
j i j i j ic x x c x x− + − . (9)
3 . Improving the first reference solution .
The obtained point in step 2 is the optimal so-
lution, it is necessary to check whether it can be
improved . To do this, we find the increase in the
objective function:
2 1 2 1
2 1 ( * ) ( )f f f f
j i j i j if f f c x x c x x∆ = ∆ − ∆ = − + −
.
(10)
Since the minimum value of the objective func-
tion must be find, a necessary condition for im-
proving the first reference solution is to reduce the
growth of the objective function:
∆f ≤ 0 . (11)
If (11) is not satisfied, then the next point of the
set of combinations in ascending order is consi-
dered and we verify it according to (8) .
If (11) is satisfied, then the optimal solution
is found .
It should be noted that condition (8) is sufficient
for the existence of an optimal solution, and condi-
tion (11) is necessary for its determination .
example
In the software market for the «OZONINVEST»
pharmaceutical industry, six information protec-
tion programs have been selected that have their
own specific rating A = (1, 2, 3, 4, 5) . After a
comprehensive assessment at the enterprise, it
was found that there is a need to purchase three of
the six programs for comprehensive information
protection . The price of each program, depen-
ding on the rating, can fluctuate in the following
range (таb . 2) .
The enterprise «OZONINVEST» consists of 3
divisions, which have formed their necessary con-
ditions for choosing information protection pro-
grams, represented by the following constraints:
1 2 3
1 2 3
1 2 3
10 7 4 20,
8 5 2 5,
2 4 6 43.
x x x
x x x
x x x
+ + ≥ − + ≤
+ + ≥
According to how much, it is necessary to form
a set of programs with the highest rating, but at
minimal cost, then you should consider the values
of the coefficients of the objective function, as the
values p
1
, таble 2:
min F = c
1
x
1
+ c
2
x
2
+ c
3
x
3
+ c
4
x
4
+ c
5
x
5
+ c
6
x
6
.
Therefore, unselected programs are rated 0 in
the objective function .
Since the elements of many combinations are
ordered in increasing order, it is natural to assume
that with an increase in the rating of programs, the
value of the objective function increases . Therefore,
the search for a solution should begin with the lo-
west rating of programs, taking into account the
growth of the objective function .
Consider the point of combination set (1, 2, 3) .
Find the values of the constraints: g
1
= 16 < 20,
constraint is not satisfied .
Аnalogically, for the point (1, 2, 4): g
1
= 20 ≥ 20,
g
2
= 6 > 5 constraint is not satisfied .
For the point (1, 2, 5): g
1
= 24 ≥ 20, g
2
= 8 > 5
constraint is not satisfied .
Rating of
Programms
Price, $
p
1
p
2
a
1
= 1 1500 2700
a
2
= 2 2300 3800
a
3
= 3 5300 7000
a
4
= 4 6000 8700
a
5
= 5 9000 10500
a
6
= 6 11200 14300
Тable 2. The price range of programs depending on the rating
for enterprise «OZONINVEST»
iSSN 2706-8145, control systems and computers, 2019, № 5 27
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations
For the point (1, 2, 6): g
1
= 28 ≥ 20, g
2
= 10 > 5
constraint is not satisfied .
For the point (1, 3, 4): g
1
= 27 ≥ 20, g
2
= 1 ≤ 5,
g
1
= 38 < 43 constraint is not satisfied .
For the point (1, 3, 5): g
1
= 31 ≥ 20, g
2
= 3 ≤ 5,
g
1
= 44 ≥ 43 constraints are satisfied, F(135) =
= 62400 . Accordingly, the first supporting solution
is found . Then the initial search conditions for the
next supporting solution:
1
2
3
11,
2,
1.
g
g
g
∆ ≥ −∆ ≤
∆ ≥ −
Consider the next point (1, 3, 6): ∆g
1
= 4 ≥ -11,
∆g
2
= 2 ≤ 2, ∆g
3
= 6 ≥ -1 constraints are satisfied,
not ∆F(136) = 22200 . Therefore, this solution is
not better than the previous one .
Therefore, point (1, 3, 5) is optimal solution .
The minimum values of the objective function
minF(136) = {62400 – 76200} .
Answer: for comprehensive data protection, the
enterprise «OZONINVEST» needs to purchase
programs that have a 1st, 3th and 5th rating . The
minimum cost of acquiring them will range from
$ 62400 to $ 7600 .
Conclusion
The analysis of the problem of choosing a set of
programs for protecting information at an enter-
prise of any industry is carried out . An optimiza-
tion combinatorial model is constructed, taking
into account the emerging conditions for the
selection of data protection programs, as well as
the financial activities of the enterprise . Using the
bijective mapping of multiple combinations into
Euclidean space, the mathematical model was
considered on the configuration of combinations .
An algorithm for solving the problem, consis-
ting of three steps . At the first and second steps,
the search for the supporting solution was carried
out, and the third step ensures the finding of the
optimal solution . A numerical example of the
implementation of the considered optimization
combinatorial model on the set of configuration
of combinations is given .
Further research is aimed at constructing ma-
thematical models on other combinatorial con-
figurations with nonlinear objective functions .
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L.N. Koliechkina, A.N. Nahirna
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Колєчкіна Л.М., доктор фіз.-мат. наук, професор,
Лодзький університет, вул. Банаха 22, Лодзь 90-238, Польща,
ludapl@ukr.net,
Нагірна А.М., канд. фіз.-мат. наук, доцент,
Національний університет «Києво-Могилянська академія»,
вул. Г. Сковороди, 2, м. Київ, 04070, Україна,
naghirnaalla@ukr.net
ПРаКТичНий аСПЕКТ заСТоСУВаННя
КоМБіНаТоРНої МодЕЛі На КоНфіГУРації СПоЛУчЕНь
Вступ. На ринку програмного забезпечення є багато програмних продуктів із захисту інформації. Виходячи з
головних задач захисту інформації та фінансових можливостей, підприємству зддебільшого необхідно здійснювати
вибір з поміж наявних програм цього типу. Вибір таких програм має забезпечувати мінімізацію витрат на їхнє
придбання. При деталізації вибору, можна виконати додаткові розрахунки цільової функції в діапазоні максимуму
та мінімуму середньо-статистичних цін програм захисту інформації, враховуючи їхні рейтинги. для розв’язання
цієї проблеми доцільно використовувати комбінаторну оптимізаційну модель на множині сполучень.
Мета статті — демонстрація використання комбінаторної оптимізаційної моделі на множині сполучень і
представлення методу розв’язання задач цього типу.
Методи. Метод розв’язання задачі умовної оптимізації на комбінаторній множині сполучень.
Результати. Сформульовано проблему вибору програмного забезпечення із захисту інформації та запропоновано
спосіб її розв’язання. Наразі задача моделюється комбінаторною оптимізаційною моделлю на множині сполучень.
запропоновано метод розв’язання задач цього типу. Наведено практичний приклад використання комбінаторної
оптимізаційної моделі на множині сполучень.
Висновки. запропоновану модель можна використовувати для моделювання задач, які передбачають сполучення
об’єктів, процесів і т.ін. за умови мінімізації функції мети. Подальші дослідження будуть спрямовані на побудову
оптимізаційних моделей на інших комбінаторних множинах із нелінійними функціями мети.
Ключові слова: інформаційна безпека, комбінаторна оптимізаційна модель, множина сполучень, цільова функція,
обмеження.
iSSN 2706-8145, control systems and computers, 2019, № 5 29
The Practical Aspect of Using a Combinatorial Model on Configuration of Combinations
Колечкина Л.Н., док . физ .-мат . наук, профессор,
Лодзинский университет, ул . Банаха 22, Лодзь 90-238, Польша,
ludapl@ukr .net,
Нагорная А.Н., канд . физ .-мат . наук, доцент,
Национальный университет «Киево-Могилянская академия»,
ул . Г . Сковороды, 2, м . Киев, 04070, Украина,
naghirnaalla@ukr .net,
ПРАКТИЧЕСКИЙ АСПЕКТ ПРИМЕНЕНИЯ
КОМБИНАТОРНОЙ МОДЕЛИ НА КОНФИГУРАЦИИ СОЧЕТАНИЙ
Введение. На рынке программного обеспечения существует множество программных продуктов по защите ин-
формации . Выходя из основных задач о защите информации и финансовых возможностях, как правило, пред-
приятию необходимо осуществлять выбор из имеющегося наличия программ данного типа . Выбор таких про-
грамм должен обеспечивать минимизацию затрат на их приобретение . При детализации выбора, можно произ-
вести дополнительные расчеты целевой функции в диапазоне максимума и минимума среднестатистических цен
программ по защите информации с учетом их рейтинга . При решении данной проблемы можно использовать
комбинаторную оптимизационную модель на множестве сочетаний .
Целью данной статьи является демонстрация использования комбинаторной оптимизационной модели на
множестве сочетаний и представления метода решения задач данного типа .
Методы. Метод решения задачи условной оптимизации на комбинаторном множестве сочетаний .
Результаты. Сформулирована проблема выбора программного обеспечения по защите информации и пред-
ложен подход к ее решению . В данном случае задача моделируется комбинаторной оптимизационной моделью
на множестве сочетаний . Предложен метод решения задач данного типа . Рассмотрен практический пример при-
менения комбинаторной оптимизационной модели на множестве сочетаний .
Выводы. С помощью предложенной модели можно моделировать задачи, которые предусматривают соче-
тание объектов, процессов и т .п . при условии минимизации функции цели . Дальнейшие исследования будут
направлены на построение оптимизационных моделей на других комбинаторных множествах с нелинейными
функциями цели .
Ключевые слова: информационная безопасность, комбинаторная оптимизационная модель, множество сочетаний,
целевая функция, ограничения.
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