Evaluation of complex systems: multicriteria approach
The quality of an object's functioning is assessed by a set of properties that are quantitatively expressed (quality criteria). The selection of properties, carried out by the decision-maker (DM), represents a decomposition, resulting in a hierarchical structure. In decision theory, the case wh...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2022
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| Cite this: | Evaluation of complex systems: multicriteria approach / A. Voronin, A. Savchenko // Проблеми керування та інформатики. — 2022. — № 6. — С. 83–89. — Бібліогр.: 2 назв. — англ. |
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| citation_txt | Evaluation of complex systems: multicriteria approach / A. Voronin, A. Savchenko // Проблеми керування та інформатики. — 2022. — № 6. — С. 83–89. — Бібліогр.: 2 назв. — англ. |
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| description | The quality of an object's functioning is assessed by a set of properties that are quantitatively expressed (quality criteria). The selection of properties, carried out by the decision-maker (DM), represents a decomposition, resulting in a hierarchical structure. In decision theory, the case where the multi-criteria problem is represented by a two-level hierarchical system is the most elaborated. In this case, the task of evaluating (combining the criteria of) a simple object is usually solved using a single scalar folding mechanism of the vector criterion. The numerical value of the folding is the overall evaluation of the quality of the object's functioning. However, even with a three-level hierarchy, the object is considered complex, and its evaluation requires other approaches. It is shown that any vector evaluation task of an object can be represented by a hierarchical system of criteria obtained as a result of decomposing the object's properties.
Якість функціонування обʼєкта оцінюється сукупністю властивостей, що мають кількісне вираження (критерії якості). Вибір властивостей обʼєкта, який здійснюється особою, що приймає рішення (ОПР), являє собою декомпозицію, в результаті якої утворюється ієрархічна структура. У теорії прийняття рішень найбільш детально розроблено випадок, коли багатокритеріальна задача представлена дворівневою ієрархічною системою. Тут задача оцінки (композиції критеріїв) простого обʼєкта зазвичай розвʼязується за допомогою механізму одиничної скалярної згортки векторного критерію. Числове значення згортки є оцінкою якості функціонування даного обʼєкта в цілому. Але навіть при трирівневій ієрархії обʼєкт розглядається як складний і його оцінка вимагає інших підходів. Показано, що будь-яка задача векторної оцінки об’єкта може бути представлена ієрархічною системою критеріїв, отриманих у результаті декомпозиції властивостей об’єкта.
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© A. VORONIN, A. SAVCHENKO, 2022
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2022, № 6 83
UDC 519.9
A. Voronin, A. Savchenko
EVALUATION OF COMPLEX SYSTEMS:
MULTICRITERIA APPROACH
Albert Voronin
National Aviation University, Kyiv,
alnv@ukr.net
Alina Savchenko
National Aviation University, Kyiv,
a.s.savchenko@ukr.net
The quality of object functioning is evaluated by a set of properties that have a
quantitative expression (quality criteria). The selection of object properties is
carried out by the decision maker (DM). This selected object properties is a
decomposition, resulting in a hierarchical structure. In decision theory the case
when a multicriteria task is represented by a two-level hierarchical system is
developed most thoroughly in great detail. Here, the problem of evaluation
(composition of criteria) of a simple object is usually solved using the
mechanism of a single scalar convolution of a vector criterion. The numerical
value of the convolution is an assessment of the quality of the functioning of this
object as a whole. But even with a three-level hierarchy, the object is considered
as complex and its evaluation requires other approaches. It is shown that any
problem of vector evaluation of an object can be represented by a hierarchical
system of criteria obtained as a result of decomposition of the objectʼs
properties. With hierarchical decomposition of object properties, the number of
levels depends on the required depth of decomposition. Usually they brought to
such properties that have a quantitative expression (called criteria). The
difficulty lies in the fact that for each initial property, the depth of
decomposition can be different. At the lower level of the hierarchy, the object
(alternative) is evaluated by individual properties using the initial vector of
criteria, and at the upper level, the object as a whole is evaluated by means of the
composition mechanism. The central problem here is the problem of composing
criteria by hierarchy levels, which is solved by the method of nested scalar
convolutions. The necessary and sufficient conditions for the vector estimation
of an object are considered.
Keywords: object properties, decomposition and composition of properties,
hierarchical structure, nested scalar convolutions, multicriteria, necessary and
sufficient conditions.
For a better understanding of abstract concepts, letʼs look at an example. The
problem of assessing the quality of a passenger aircraft project is considered. At the first
level, such properties of this object as comfort, reliability, load capacity, altitude are
distinguished. In turn (the second level), the comfort property is characterized by the
distance between the seats, the noise level in the cabin, and the amplitude of the floor
vibration. The property of reliability consists in the strength of the structure, the
probability of engine failures, etc.
Let us consider in general terms a possible approach to the multicriteria evaluation
of hierarchical systems. The methodological basis for analyzing and decomposing the
properties of an object and obtaining criteria at each level of the hierarchy is the
complementarity principle of N. Bohr. According to this principle, for a complete
mailto:alnv@ukr.net
84 ISSN 2786-6491
description of an object, all criteria are equally necessary and therefore do not
contradict, but complement each other. It is the principle of complementarity that makes
it possible to single out and then link these criteria in a multicriteria assessment.
Only a complete set of partial criteria (vector criterion) makes it possible to assess
adequately the functioning of a complex system as a manifestation of the
contradictory unity of all its properties.
However, this possibility is only a necessary but not sufficient condition for the
vector evaluation of the entire complex system as a whole. Indeed, let the lower level of
the hierarchy of criteria define the numerical values of such partial criteria of the aircraft
comfort property as the distance between the seats, the noise level in the cabin, the floor
vibration amplitude, etc. Does this mean that, knowing these values, we can evaluate the
comfort property in general? No we can not.
These separate «models» of comfort reflect various properties of the object, but do
not give a complete picture. For a holistic assessment, it is necessary to leave the lower
level of the hierarchy and climb to the next tier, i.e. carry out the act of composition of
criteria. Such a transition corresponds to Kurt Gödelʼs incompleteness theorem. We can
say that Gödelʼs theorem is the methodological basis for the synthesis in the study of
hierarchical structures.
In relation to our task, this means that in order to assess adequately the object as a
whole, we must solve the problem of composing criteria by hierarchy levels,
successively moving from the lower level to the upper one. A scalar convolution of
criteria can serve as an instrument of the act of composition. Scalar convolution is a
mathematical technique for compressing information and quantifying its integral
properties with a single number.
Historically, the concept of scalar convolution as an optimality principle was
introduced by Blasé Pascal, who is considered to be the founder of decision theory. He
proposed two key concepts of the theory: 1) partial criteria, each of which evaluates one
side of the decision efficiency, and 2) a rule that allows one to calculate a certain
numerical measure of the decision efficiency from the values of the criteria, namely, the
scalar convolution of the criteria vector. By the way, from the whole variety of types of
scalar convolutions, Pascal chose the multiplicative convolution.
Formulation of the problem
The quality of an object is determined by a hierarchical system of vectors
{ }, [2, ],y y j m
where
( 1)j
y
is the vector of criteria at the ( j ‒1)-th level of the hierarchy, the
components of which evaluate the quality of the objectʼs properties at the j-th level; m is
the number of hierarchy levels;
( 1)j
n
— the number of evaluated properties of the
( j ‒1)-th level of the hierarchy. Numerical values of n criteria (1)y y of the first
(lower) hierarchy level for the given object are given. It is clear that (1)n n and
( ) 1.mn
The significance of each of the components of the criterion of the ( j ‒1)-th level in
assessing the k-th property of the j-th level is characterized by a priority coefficient, the
totality of which constitutes a system of priority vectors
( )( 1) ( 1)
1{ } , [2, ].
jj j n
kik ik
p p j m
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2022, № 6 85
It is required to find an analytical estimate *y and a qualitative estimate of this object.
Solution method
For analytical evaluation of the efficiency of hierarchical structures, the
method of nested scalar convolutions is used. The composition is carried out
according to the «matryoshka principle»: scalar convolutions of the weighted
components of the lower level vector criteria serve as components of the higher
level vector criteria. The scalar convolution of criteria obtained at the highest level
automatically becomes an expression for evaluating the effectiveness of the entire
hierarchical system as a whole.
The algorithm for solving the problem by the method of nested scalar convolutions
is represented by a sequence of operations of weighted scalar convolution of vector
criteria of each level of the hierarchy based on the chosen trade-off scheme, taking into
account the priority vectors:
1 1 ( )
[2, ]{( , ) }
j j j
j my p y
and the search for an estimate of the effectiveness of the entire hierarchical system as a
whole is expressed by the problem of determining the scalar convolution of the upper
level of the hierarchy:
( )* .my y
The complementarity principle of N. Bohr is methodological basis for the
decomposition of the properties of an object until the initial vector of criteria is
obtained. This is a necessary condition for the vector evaluation of an object.
The methodology for composing criteria by hierarchy levels is based on the
incompleteness theorem of K. Gödel. This is a sufficient condition for the vector
estimation of an object.
For further presentation, we introduce some particular assumptions. Without
loss of generality, we will assume that the object quality criteria are subject to
minimization, i.e. the smaller they are, the better for the object. The constraints for
the criteria have the form
[0, ], [1, ],k ky A k s
where s is the number of criteria at the j-th level of the hierarchy.
Approach of the criteria to their limit values is dangerous for the functioning of the
object. Under these assumptions, the evaluation function is adequate, i.e. scalar
convolution according to the nonlinear trade-off scheme [1]:
1
( ) .
s
k
k k k
A
Y y
A y
We normalize the criteria according to the rule
0 0, [0;1], [1, ].k
k k
k
y
y y k s
A
Then the algorithm for solving the problem by the method of nested scalar
convolutions using a nonlinear trade-off scheme takes the form of a recursion that
gives an expression for estimating the k-th property of an object at the j-th level of
the hierarchy:
86 ISSN 2786-6491
( 1)
( ) ( 1) ( 1) 1 ( )
0
1
[1 ] , [1, ],
j
kn
j j j j
k ik ik
i
y p y k n
[2, ],j m
where
( 1)
0
j
ik
y
are the components of the normalized vector
( 1)
0
j
y
involved in the eva-
luation of the k-th property of the object at the j-th level of the hierarchy;
( 1)j
k
n
—
their number; ( )jn is the number of evaluated properties at the j-th level.
The determination of the priority coefficients p at each level of the hierarchy can
be performed by the method of expert assessments on a scale of points [2].
In order for this formula to reflect the idea of the method of nested scalar
convolutions, it is necessary to normalize the resulting expression, i.e. obtain a relative
criterion
( )
0
[0;1]
j
k
y such that it is minimizable and its limiting value is unity. The
final expression for the recursion for calculating analytical estimates of object properties
at all levels of the hierarchy takes the form
( 1) 1
( ) ( 1) ( 1) 1 ( )
0 0
1
1 [1 ] , [1, ], [2, ].
j
kn
j j j j
k ik ik
i
y p y k n j m
(1)
Qualitative assessment of the object
A qualitative (linguistic) assessment of an object is obtained by comparing the
analytical assessment with Harringtonʼs verbal-numerical scale. The Harrington
scale is a characteristic of the expressiveness of a criterion property and has a
universal character. The numerical values of the gradations are obtained on the
basis of the analysis and processing of a large array of statistical data. Harrington ʼs
verbal-numerical scale is presented in Table. It shows the relationship between the
qualitative gradations of the properties of objects and the corresponding normalized
quantitative estimates y0.
Table
Description of gradations Numerical value y0
Very high 0,8– 1,0
High 0,64 –0,8
Average 0,37 –0,64
Low 0,2– 0,37
Very low 0,0– 0,2
It can be said that in terms of the theory of fuzzy sets, the verbal-numerical
scale acts as a universal membership function for the transition from a number to
the corresponding qualitative gradation and vice versa. A transition is made from a
linguistic variable (average, high score, etc.) to the corresponding quantitative
scores on a scale of points, i.e. transition from fuzzy qualitative gradations to
numbers and vice versa.
Evaluation of objects according to a single verbal-numeric Harrington scale makes
it possible to solve multicriteria tasks, in addition to traditional formulations, and in the
case when it is required to choose an alternative from a variety of heterogeneous
alternatives for which it is impossible to formulate a single set of quantitative evaluation
criteria, as well as to evaluate a single (unique) object.
In the problem of choosing solutions, the number of options (alternatives) is
1.an Each option is characterized by its hierarchical structure. When 1,an the task
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2022, № 6 87
is transformed into the task of evaluating the given hierarchical structure. If 1,an
then each structure is evaluated as given and the variant which hierarchical structure has
received the best estimate is selected. Therefore, in discrete multicriteria optimization,
the problem of estimating a given hierarchical structure can be considered as a basic
one. However, this can only be done in the case of a relatively small number of
alternatives ,an when the simple enumeration method does not cause significant
computational difficulties. For large volumes of sets of alternatives, other, more
complex optimization methods should be used.
The considered method for solving complex multicriteria estimation and
optimization problems, based on the approach of nested scalar convolutions of vector
criteria, makes it quite easy to solve problems of structural and parametric synthesis of
hierarchical systems. In this case, the hierarchy can be both natural (multilevel systems
with subordination from top to bottom) and arising as a result of the decomposition of
the object properties to the level of criteria (hierarchy of criteria).
Illustration example
It is required to find a quantitative
(3)
0 0y y and qualitative assessment of the
aircraft project according to two main properties: comfort, characterized by the
assessment of the criterion
(2)
01y and reliability, which is compared with the assessment
of the criterion
(2)
02 .y The comfort property, in turn, is evaluated according to three
criteria: the distance between the seats in the passenger compartment 01,y the noise
level in the cabin 02y and the vibration level of the floor in the cabin 03.y Reliability is
assessed by the probability of equipment failures 04y and structural strength 05.y In
addition to these two, the criterion of the floor vibration level takes part in the reliability
assessment 03 ,y i.e. there is one cross-link. All of these criteria are normalized and
reduced to one method of extremization, namely, all of them are subject to
minimization. The lower level criteria are taken into account in the evaluation of the
properties of the higher level with priority coefficients
( 1)
, [2, ].
j
ik
p j m
The following numerical values are given. Criteria of the lower (first) level of the
hierarchy: 01y 0,3; 02y 0,5; 03y 07; 04y 0,2; 05y 0,1. Priority coefficients:
(1)
11p 0,7;
(1)
21p 0,2;
(1)
31p 0,1;
(1)
32p 0,1;
(1)
42p 0,45;
(1)
52p 0,45;
(2)
13p 0,5;
(2)
23p 0,5.
At the first stage of criteria composition, based on the recursive formula (1), we
obtain an expression for the analytical assessment of the comfort property (the second
level of the hierarchy):
(1)
1
(2)
01
(1) (1) 1
1 0 1
1
1
1 ,
(1 )
n
i i
i
y
p y
where
(1)
1n 3 and
(1) (1) (1)
01 02 03011 021 031; ; .y y y y y y Substituting numerical values,
we get
88 ISSN 2786-6491
(2)
01
1
1 0,42.
1 1 1
0,7 0,2 0,1
1 0,3 1 0,5 1 0,7
y
Comparing this analytical assessment with Table, we find that the comfort property
for this aircraft design is qualitatively assessed as satisfactory.
The expression for the analytical evaluation of the reliability property (also the
second level of the hierarchy) has the form
(1)
2
(2)
02
(1) (1) 1
2 0 2
1
1
1 ,
(1 )
n
i i
i
y
p y
where, taking into account the cross-relationship,
(1)
2n 3 and
(1) (1)
03 04012 022; ;y y y y
(1)
05032 .y y Priority coefficients
(1) (1) (1) (1) (1) (1)
12 32 22 42 32 52; ; .p p p p p p Substitute the
numerical values and get
(2)
02
1
1 0,28.
1 1 1
0,1 0,45 0,45
1 0,7 1 0,2 1 0,1
y
According to Table, the quality of the reliability property for this project is rated
as high.
At the final (second) stage of criteria composition, formula (1) takes the form
(2)
3
(3)
0 0
(2) (2) 1
3 0 3
1
1
1 ,
(1 )
n
i i
i
y y
p y
where
(2)
3n 2 and
(2) (2) (2) (2)
013 01 023 02; .y y y y Substituting numerical values, we get
0
1
1 0,36.
1 1
0,5 0,5
1 0,42 1 0,28
y
Table says that according to this analytical assessment, the quality of this aircraft design
is generally assessed as good.
А.М. Воронін, А.С. Савченко
ОЦІНКА СКЛАДНИХ СИСТЕМ:
БАГАТОКРИТЕРІАЛЬНИЙ ПІДХІД
Воронін Альберт Миколайович
Національний авіаційний університет, м. Київ,
alnv@ukr.net
Савченко Аліна Станіславівна
Національний авіаційний університет, м. Київ,
mailto:alnv@ukr.net
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2022, № 6 89
a.s.savchenko@ukr.net
Якість функціонування обʼєкта оцінюється сукупністю властивостей,
що мають кількісне вираження (критерії якості). Вибір властивостей
обʼєкта, який здійснюється особою, що приймає рішення (ОПР), являє
собою декомпозицію, в результаті якої утворюється ієрархічна струк-
тура. У теорії прийняття рішень найбільш детально розроблено випа-
док, коли багатокритеріальна задача представлена дворівневою ієрар-
хічною системою. Тут задача оцінки (композиції критеріїв) простого
обʼєкта зазвичай розвʼязується за допомогою механізму одиничної
скалярної згортки векторного критерію. Числове значення згортки є
оцінкою якості функціонування даного обʼєкта в цілому. Але навіть
при трирівневій ієрархії обʼєкт розглядається як складний і його оцін-
ка вимагає інших підходів. Показано, що будь-яка задача векторної
оцінки об’єкта може бути представлена ієрархічною системою крите-
ріїв, отриманих у результаті декомпозиції властивостей об’єкта. При
ієрархічній декомпозиції властивостей обʼєктів кількість рівнів зале-
жить від необхідної глибини декомпозиції. Зазвичай доходять до таких
властивостей, які мають кількісне вираження (називаються критерія-
ми). Складність полягає в тому, що для кожної початкової властивості
глибина декомпозиції може бути різною. На нижньому рівні ієрархії
об’єкт (альтернатива) оцінюється за окремими властивостями з вико-
ристанням початкового вектора критеріїв, а на верхньому — об’єкт у
цілому за допомогою механізму композиції. Центральною проблемою
тут є композиція критеріїв за рівнями ієрархії, що вирішується мето-
дом вкладених скалярних згорток. Розглянуто необхідні та достатні
умови векторної оцінки обʼєкта.
Ключові слова: властивості об’єктів, декомпозиція та композиція власти-
востей, ієрархічна структура, вкладені скалярні згортки, багатокритеріаль-
ність, необхідні та достатні умови.
REFERENCES
1. Voronin A.N., Ziatdinov Yu.K., Kuklinsky M.V. Multicriteria decisions: models and methods.
Kiev : NAU, 2010. 348 p.
2. Voronin A. Multicriteria decision making for the management of complex systems. USA : IGI
Global, 2017. 201 p.
Submitted 31.01.2023
mailto:a.s.savchenko@ukr.net
|
| id | nasplib_isofts_kiev_ua-123456789-210922 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-19T21:29:01Z |
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| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Voronin, A. Savchenko, A. 2025-12-20T22:15:42Z 2022 Evaluation of complex systems: multicriteria approach / A. Voronin, A. Savchenko // Проблеми керування та інформатики. — 2022. — № 6. — С. 83–89. — Бібліогр.: 2 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/210922 519.9 10.34229/2786-6505-2022-6-7 The quality of an object's functioning is assessed by a set of properties that are quantitatively expressed (quality criteria). The selection of properties, carried out by the decision-maker (DM), represents a decomposition, resulting in a hierarchical structure. In decision theory, the case where the multi-criteria problem is represented by a two-level hierarchical system is the most elaborated. In this case, the task of evaluating (combining the criteria of) a simple object is usually solved using a single scalar folding mechanism of the vector criterion. The numerical value of the folding is the overall evaluation of the quality of the object's functioning. However, even with a three-level hierarchy, the object is considered complex, and its evaluation requires other approaches. It is shown that any vector evaluation task of an object can be represented by a hierarchical system of criteria obtained as a result of decomposing the object's properties. Якість функціонування обʼєкта оцінюється сукупністю властивостей, що мають кількісне вираження (критерії якості). Вибір властивостей обʼєкта, який здійснюється особою, що приймає рішення (ОПР), являє собою декомпозицію, в результаті якої утворюється ієрархічна структура. У теорії прийняття рішень найбільш детально розроблено випадок, коли багатокритеріальна задача представлена дворівневою ієрархічною системою. Тут задача оцінки (композиції критеріїв) простого обʼєкта зазвичай розвʼязується за допомогою механізму одиничної скалярної згортки векторного критерію. Числове значення згортки є оцінкою якості функціонування даного обʼєкта в цілому. Але навіть при трирівневій ієрархії обʼєкт розглядається як складний і його оцінка вимагає інших підходів. Показано, що будь-яка задача векторної оцінки об’єкта може бути представлена ієрархічною системою критеріїв, отриманих у результаті декомпозиції властивостей об’єкта. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблемы управления и информатики Дослідження операцій та системний аналіз Evaluation of complex systems: multicriteria approach Оцінка складних систем: багатокритеріальний підхід Article published earlier |
| spellingShingle | Evaluation of complex systems: multicriteria approach Voronin, A. Savchenko, A. Дослідження операцій та системний аналіз |
| title | Evaluation of complex systems: multicriteria approach |
| title_alt | Оцінка складних систем: багатокритеріальний підхід |
| title_full | Evaluation of complex systems: multicriteria approach |
| title_fullStr | Evaluation of complex systems: multicriteria approach |
| title_full_unstemmed | Evaluation of complex systems: multicriteria approach |
| title_short | Evaluation of complex systems: multicriteria approach |
| title_sort | evaluation of complex systems: multicriteria approach |
| topic | Дослідження операцій та системний аналіз |
| topic_facet | Дослідження операцій та системний аналіз |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210922 |
| work_keys_str_mv | AT voronina evaluationofcomplexsystemsmulticriteriaapproach AT savchenkoa evaluationofcomplexsystemsmulticriteriaapproach AT voronina ocínkaskladnihsistembagatokriteríalʹniipídhíd AT savchenkoa ocínkaskladnihsistembagatokriteríalʹniipídhíd |