Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions
In the article the task of finding the optimal combination of objective functions given in the view of a logical conditional system of equations is considered. It is modeled the behavior of an active system controlled by the intellectual (active) element. The principle of the subjective entropy ma...
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irk-123456789-851872015-07-22T03:02:19Z Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions Goncharenko, A.V. Концептуальные проблемы создания систем искусственного интеллекта In the article the task of finding the optimal combination of objective functions given in the view of a logical conditional system of equations is considered. It is modeled the behavior of an active system controlled by the intellectual (active) element. The principle of the subjective entropy maximum is applied for obtaining the canonical distribution of the individual’s preferences as the solution to the problem for the conditional extremum. У статті розглядається задача відшукання оптимальної комбінації цільових функцій, заданих у вигляді логічної умовної системи рівнянь. Моделюється поведінка активної системи, керованої інтелектуальним (активним) елементом. Принцип максимуму суб’єктивної ентропії застосовується для отримання канонічного розподілу індивідуальних переваг у якості розв’язка задачі на умовний екстремум. В статье рассматривается задача отыскания оптимальной комбинации целевых функций, заданных в виде логической условной системы уравнений. Моделируется поведение активной системы, управляемой интеллектуальным (активным) элементом. Принцип максимума субъективной энтропии применяется для получения канонического распределения индивидуальных предпочтений в качестве решения задачи на условный экстремум. 2013 Article Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions / A.V. Goncharenko // Искусственный интеллект. — 2013. — № 4. — С. 4–9. — Бібліогр.: 9 назв. — англ. 1561-5359 http://dspace.nbuv.gov.ua/handle/123456789/85187 303. 725.36:159.9.015:159.964.21:519.86(02) en Искусственный интеллект Інститут проблем штучного інтелекту МОН України та НАН України |
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Концептуальные проблемы создания систем искусственного интеллекта Концептуальные проблемы создания систем искусственного интеллекта |
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Концептуальные проблемы создания систем искусственного интеллекта Концептуальные проблемы создания систем искусственного интеллекта Goncharenko, A.V. Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions Искусственный интеллект |
| description |
In the article the task of finding the optimal combination of objective functions given in the view of a logical
conditional system of equations is considered. It is modeled the behavior of an active system controlled by the
intellectual (active) element. The principle of the subjective entropy maximum is applied for obtaining the canonical
distribution of the individual’s preferences as the solution to the problem for the conditional extremum. |
| format |
Article |
| author |
Goncharenko, A.V. |
| author_facet |
Goncharenko, A.V. |
| author_sort |
Goncharenko, A.V. |
| title |
Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions |
| title_short |
Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions |
| title_full |
Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions |
| title_fullStr |
Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions |
| title_full_unstemmed |
Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions |
| title_sort |
subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions |
| publisher |
Інститут проблем штучного інтелекту МОН України та НАН України |
| publishDate |
2013 |
| topic_facet |
Концептуальные проблемы создания систем искусственного интеллекта |
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http://dspace.nbuv.gov.ua/handle/123456789/85187 |
| citation_txt |
Subjective entropy maximum principle for preferences functions of alternatives given in the view of logical conditions / A.V. Goncharenko // Искусственный интеллект. — 2013. — № 4. — С. 4–9. — Бібліогр.: 9 назв. — англ. |
| series |
Искусственный интеллект |
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AT goncharenkoav subjectiveentropymaximumprincipleforpreferencesfunctionsofalternativesgivenintheviewoflogicalconditions |
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2025-07-06T12:21:03Z |
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2025-07-06T12:21:03Z |
| _version_ |
1836900117978808320 |
| fulltext |
ISSN 1561-5359 «Искусственный интеллект» 2013 № 4 4
1G
УДК 303. 725.36:159.9.015:159.964.21:519.86(02)
A.V. Goncharenko
National Aviation University, Kyiv, Ukraine
Ukraine, 03680, c. Kyiv, Kosmonavta Komarova av., 1
Subjective Entropy Maximum Principle
for Preferences Functions of Alternatives Given
in the View of Logical Conditions
А.В. Гончаренко
Национальный авиационный университет, г. Киев, Украина
Украина, 03680, г. Киев, пр. Космонавта Комарова, 1
Принцип максимума субъективной энтропии
для функций предпочтений альтернатив,
заданных в виде логических условий
А.В. Гончаренко
Національний авіаційний університет, м. Київ, Україна
Україна, 03680, м. Київ, пр. Космонавта Комарова, 1
Принцип максимуму суб’єктивної ентропії
для функцій переваг альтернатив, заданих
у вигляді логічних умов
In the article the task of finding the optimal combination of objective functions given in the view of a logical
conditional system of equations is considered. It is modeled the behavior of an active system controlled by the
intellectual (active) element. The principle of the subjective entropy maximum is applied for obtaining the canonical
distribution of the individual’s preferences as the solution to the problem for the conditional extremum.
Key words: active system, individual’s preferences, subjective entropy, multi-alternativeness.
В статье рассматривается задача отыскания оптимальной комбинации целевых функций, заданных в
виде логической условной системы уравнений. Моделируется поведение активной системы, управляемой
интеллектуальным (активным) элементом. Принцип максимума субъективной энтропии применяется
для получения канонического распределения индивидуальных предпочтений в качестве решения
задачи на условный экстремум.
Ключевые слова: активная система, индивидуальные предпочтения,
субъективная энтропия, многоальтернативность.
У статті розглядається задача відшукання оптимальної комбінації цільових функцій, заданих у вигляді
логічної умовної системи рівнянь. Моделюється поведінка активної системи, керованої інтелектуальним
(активним) елементом. Принцип максимуму суб’єктивної ентропії застосовується для отримання канонічного
розподілу індивідуальних переваг у якості розв’язка задачі на умовний екстремум.
Ключові слова: активна система, індивідуальні переваги, суб’єктивна ентропія,
багатоальтернативність.
In subjective analysis we encounter a problem of evaluation of an individual-subject
participation in systems and processes of control in the vast majority of scientific re-
searches that consider the functioning and control of the so-called active systems [1-9].
Subjective Entropy Maximum Principle for Preferences Functions...
«Штучний інтелект» 2013 № 4 5
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Because an active system includes the active element (a person who is responsible for ma-
king managing decisions), therefore, in our opinion the active system has the following
peculiarities [4, P. 58, § 3.1]:
1. Being closed, is able to decrease its own entropy
2. Requires the presence (existence) of the active element, the bearer of individual
preferences
3. The extremized functional includes the subjective entropy
4. The behavior of the active element is dictated by the postulated variational
principle and directed upon the optimization of a certain problem-resource situation with
extremizing the entropy
5. Has the ability to aggregate preferences
Thus, an active system is guided by the intellect of its active element and the active
element acts on the basis of his own preferences distributed at each moment in time on the
achievable for his objectives set of alternatives.
Let us consider theoretically a logical conditional system of, for instance, some three
equations [9, P. 20, (1)]:
.,
;,
;,
3
32122
31211
max
otherwisexy
xyxyxyxyifxy
xyxyxyxyifxy
xY (1)
The concept (1) implies xyi – dependence, for example, of the control of safety of
an active system, an effectiveness of the ith operator (alternative) of an engineering system,
the ith alternative function of safety, the ith alternative strategy, role of the ith political party
(alternative) in sociology/politics, usefulness or utility function of the ith alternative in
economics etc. depending upon some parameter of x – for instance, a certain indicator of
the mode of operation, parameter that exerts an influence upon the safety of the active
system, the value of an income taxation rate, index of policy, profitableness, light and
shadow economic proportion and so on and so forth. Here, in the system of equations (1),
the intellectual system is seeking after the positive value. Naturally it must be maximal.
Without doubt, if the functions of xyi represent some negative qualities, modeling
should be made with respect to the corresponding minimization [9, P. 20, (1)]:
.,
;,
;,
3
32122
31211
otherwisexy
xyxyxyxyifxy
xyxyxyxyifxy
xYmix (2)
Thus, the systems of (1, 2) look like natural intellect application for decision making,
nevertheless they do not represent their system’s active element’s intentions (individual’s
preferences), they do not show how much and to which alternative the responsible for
making decisions person is inclined to [9, P. 20].
The conceptual framework of subjective analysis [1-9] allows creating a model of an
artificial intellect of active systems with the use of the postulated functional in the view of
[2, P. 119, (3.38)], [9, P. 20, (2)]:
1ln
111
N
i
i
N
i
ii
N
i
ii F , (3)
where i – function of the individual’s subjective preferences of the ith achievable
alternative; N – number of the achievable alternatives; – structural parameter; iF – fun-
Goncharenko A.V.
«Искусственный интеллект» 2013 № 4 6
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ction, related to the ith achievable alternative; – structural parameter. The structural
parameters and can be considered in different situations as Lagrange coefficients, weight
coefficients or endogenous parameters that represent some certain properties of the individual’s
psych.
Let us consider, as an example, the problem setting with the logical conditions systems of
(1) and (2). Hence, N = 3. Assume, the functions of xyi are
2
1
1
111
kkk dxcdxbdxaxy ,
2
2
2
1
222 zdxcdxbdxaxy kkk , 33 znxexy nx . (4)
where a , b , c , 1d , 2d , k , 2z , 3z , n – coefficients and parameters of the models.
The necessary conditions for the extremum of functional (3) to exist, in the form of
the system of equations 0
i
, yield the canonical distributions of the individual’s
preferences, likewise in [2, P. 115-135], [5, P. 58, (4)], that is for the systems (1, 2), where
3N , ii yF , it will be, [9, P. 21, (3)]
3
1i
y
y
j
i
j
e
e . (5)
For the system (1) the optimal combination of the preferences
.,
;,
;,
3
32122
31211
max
otherwisex
xxxxifx
xxxxifx
x (6)
Analogously for (2)
.,
;,
;,
3
32122
31211
min
otherwisex
xxxxifx
xxxxifx
x (7)
Let us introduce a function of the intellectual diapason in the view of
xxx minmax (8)
The function (8) is not normalized itself, though being formed as a summation of the
optimized combinations of the normalized functions of preferences comprised by the systems
of (6, 7), has the possible values diapason for its changes from “0” up to “1”.
For the special data given: a = 0.5; b = – 0.632; c = 9.487; 1d = 5; 2d = 0; k = 2;
2z = 70; 3z = 30; n = – 0.4; = – 0.063 the results of modeling by the procedure of (1-8)
are illustrated in fig. 1 [9, P. 21, fig. 1].
From the diagrams plotted in fig. 1 in the corresponding scales it is visible that the
optimal combinations of functions xyi correspond the optimal combinations of the pre-
ferences functions xi . The corner points of the changes of xyi have the same abscissas as
for the xi . The function of the intellectual diapason on one hand has its own minima
corner points with the same abscissas as for the maximal combination of the objective fun-
ctions of xyi and corresponding individual’s preferences xi . On the other hand the
Subjective Entropy Maximum Principle for Preferences Functions...
«Штучний інтелект» 2013 № 4 7
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function of the intellectual diapason has its own maxima corner points with the same ab-
scissas as for the minimal combination of the individual’s preferences and the objective
functions of xyi .
10 5 0 5 10
0
20
40
60
80
100
110
0
90max x( )
90min x( )
90 1 x( )
90 2 x( )
90 3 x( )
90 x( )
Ymax x( )
Ymin x( )
y1 x( )
y2 x( )
y3 x( )
90 x( )
1010 x
Figure 1 – Objective functions, preferences, their optimal combinations,
and function of intellectual diapason
The considered problem means the application of the subjective entropy extremi-
zation principle being developed in subjective analysis [1-9]. For the given problem setting
the entropy of preferences is
3
1
ln
N
i
ii xxxH ; (9)
the entropy of optimal combinations of preferences –
xxxxxH opt minminmaxmax lnln ; (10)
the entropy of normalized optimal combinations of preferences –
x
xxn
max
max_
;
x
xxn
min
min_
; (11)
will be –
xxxxxH nnnnnopt min_min_max_max__ lnln . (12)
The entropies calculated by formula (9-12) are presented in fig. 2.
The entropy paradigm illustrated with the corresponding diagrams plotted in fig. 2
makes it noticeable that the measures of uncertainty are appropriate for the estimation of
Goncharenko A.V.
«Искусственный интеллект» 2013 № 4 8
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the decision making thresholds. Nevertheless, the comparison of the entropies with the
function of the intellectual diapason and the optimal combinations of preferences functions
testifies that the corner points of the function of the intellectual diapason, thus, the optimal
transitions from one objective function to other, do not necessarily coincide with the extre-
mums of the traditional entropy (9) and the entropy of the traditional form (9) has no
corner points as it is the smooth (everywhere differentiated curve).
On the contrary, the curves of the entropies (10, 12) indeed have those corner points
at the same with the function of the intellectual diapason abscissas; therefore they have the
diagnostic value for the optimal transitions.
10 5 0 5 10
0
0.2
0.4
0.6
0.8
1
1.15
2.081 10 7
x( )
H x( )
H x( )
H_n x( )
ln 3( )
ln 2( )
max x( )
min x( )
1010 x
Figure 2 – Entropies of preferences and function of intellectual diapason
For the estimation of the conflictability of the making managing decisions and control
process, as well as conflictability of the individual’s preferences distributions it is ap-
plicable the pseudo entropy hybrid function proposed in [6], [7].
Literatura
1. Kasianov V.A. Elements of subjective analysis: monograph / V.A. Kasianov. – Кyiv : National Aviation
University, 2003. – 224 p.
2. Kasianov V.A. Subjective analysis: monograph / V.A. Kasianov. – Кyiv : National Aviation University,
2007. – 512 p.
3. Kasianov V. Subjective entropy of preferences. Subjective analysis: monograph / V. Kasianov. – Warsaw :
Institute of Aviation, 2013. – 644 p.
4. Kasianov V.A., Goncharenko A.V. Light and Shadow. Proportions of shadow economy. Entropy approach:
monograph. – Кyiv: Kafedra, 2013. – 86 p.
5. Kasyanov V.O. Variational principle in the problem of ship propulsion and power plant operation with
respect to subjective preferences / V.O. Kasyanov, A.V. Goncharenko // Scientific proceedings of Kherson state
maritime academy: Scientific journal. – Kherson: Publishing House of Kherson State Maritime Academy, 2012. –
№ 2(7). – P. 56-61.
Subjective Entropy Maximum Principle for Preferences Functions...
«Штучний інтелект» 2013 № 4 9
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6. Goncharenko A.V. Mathematical modeling of the ship’s main engine random operational process /
A.V. Goncharenko // Internal combustion engines: Scientifically-engineering journal. Kharkov: National
Technical University “Kharkovskiy Polytechnicheskiy Insitut”, 2012. – № 2. – P. 117-125.
7. Goncharenko A.V. Measures for estimating transport vessels operators’ subjective preferences uncertainty
/ A.V. Goncharenko // Scientific proceedings of Kherson state maritime academy: Scientific journal. –
Kherson: Publishing House of Kherson State Maritime Academy, 2012. – № 1(6). – P. 59-69.
8. Goncharenko A.V. Conflictability of operational situations in terms of entropy paradigm /
A.V. Goncharenko // Modern information and innovation technologies in transport (MINTT-2013)
[Collection of materials of the 5th international scientific and practical conference. In 2 volumes (28-30
May 2013 year, Kherson)]. – V. 1. – Kherson: Kherson State Maritime Academy, 2013. – P. 115-118.
9. Goncharenko A.V. Artificial versus natural intellect in control of optimality / A.V. Goncharenko //
Intellectual Systems for Decision Making and Problems of Computational Intelligence: International
Scientific Conference, Yevpatoria, 20-24 May 2013 year: Conference Proceedings. – Kherson: Kherson
National Technical University, 2013. – P. 20-22.
RESUME
A.V. Goncharenko
Subjective Entropy Maximum Principle for Preferences Functions
of Alternatives Given in the View of Logical Conditions
In the given article the models for evaluation of an individual-subject participation in
systems and processes of control in the problems, and the vast majority of scientific
researches, that consider the functioning and control of the so-called active systems are
developed.
The models are developed on the basis of the subjective analysis theory elaborated in
monographs [1], [4] with the application of the principle of the subjective entropy of
individual preferences extremization. The concept of an active system is interpreted in
terms of monograph [4]. The active system is guided by the intellect of its active element
and the active element acts on the basis of his own preferences distributed at each moment
in time on the achievable for his objectives set of alternatives.
Postulated in subjective analysis functional allows obtaining canonical distributions
of individual’s preferences on the basis of the systems of the necessary conditions for
extremums to exist equations solutions. The corner points of the optimal combinations of
the preferences represent the optimal combinations of the objective functions.
The developed in the article entropies and function of intellectual diapason allow
estimating the potential for control in situations of multi-alternativeness and conflicts. The
conflictability of individual preferences is estimated with the application of the pseudo
entropy hybrid function proposed in works [6], [7].
Realization of such approach creates a theoretical background for solutions of the
conceptual problems of the artificial intellect.
The paper is received by the edition 26.06.2013.
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