=Paper=
{{Paper
|id=Vol-3137/paper17
|storemode=property
|title=Entropy Modeling of Optimal Intelligence Development in Regards with the Air Transport Operation
|pdfUrl=https://ceur-ws.org/Vol-3137/paper17.pdf
|volume=Vol-3137
|authors=Andriy Goncharenko
|dblpUrl=https://dblp.org/rec/conf/cmis/Goncharenko22
}}
==Entropy Modeling of Optimal Intelligence Development in Regards with the Air Transport Operation==
https://ceur-ws.org/Vol-3137/paper17.pdf
Entropy Modeling of Optimal Intelligence Development in
Regards with the Air Transport Operation
Andriy V. Goncharenko 1,2
1
Xi’an Jiaotong University, No.28, Xianning West Road, Xi’an Shaanxi, 710049, P. R. China
2
National Aviation University, 1, Liubomyra Huzara Avenue, Kyiv, 03058, Ukraine
Abstract
The paper is devoted to the entropy computer modeling of optimal intelligence development
in regards with the air transport operation. The subjective analysis theory of the active
systems is used as a framework for theoretical elaborations. The contemplations are based
upon the subjective entropy paradigm. Several solutions are obtained for a few special cases
considered. Conditional optimization of the subjective individuals’ preferences functions
entropy in conjunction with the proposed hybrid combined relative pseudo-entropy function
happened to be helpful in determining the relative certainty/uncertainty degree concerning
prevailing/dominating subjective preferences functions. Illustrative examples simulations are
performed. Necessary diagrams are plotted.
Keywords 1
Entropy, preferences, operation, air transport, optimization, intelligence, management,
simulation, objective functional.
1. Introduction
Rational methods of the air transportation management systems functioning in the operation are
important. Aircraft maintenance and repair [1, 2] procedures are aimed at the aeronautical engineering
reliability support, as well as at the decrease of the risks and negative consequences of the aircraft
systems, systems’ components, and systems’ elements failures [3, 4].
The models of intelligence used at the air transportation management systems functioning
simulation should take into account the elements of the aviation transportation systems activities
usefulness [5, 6].
The uncertainty degree could be evaluated in the framework of the entropy maximum theory terms
as in [7 – 9]. The trend of the entropy research in modern science is very popular altogether [10].
Assessing economical parameters and terms [11], it is logically to combine the issues of the
publications of [7 – 11] into the subjective analysis theory [12] that has been successfully developed
by Professor Kasianov V. A. at the National Aviation University, Kyiv, Ukraine for about the last
three decades.
In fact, the intelligence learning aspects might be and should be implemented to the modeling of
the different nature system and processes, as for example, for the applicable features of the [13 – 17]
publications.
Therefore, the goal of the presented herewith study is to demonstrate the advantages of the entropy
paradigm [7 – 9, 12, 17 – 20] applicably to the generalized value of the air transportation management
systems functioning learning potential.
CMIS-2022: The Fifth International Workshop on Computer Modeling and Intelligent Systems, Zaporizhzhia, Ukraine, May 12, 2022
EMAIL: andygoncharenco@yahoo.com (A.V. Goncharenko)
ORCID: 0000-0002-6846-9660 (A.V. Goncharenko)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. General approach
Intelligent air transportation management systems’ functioning requires that the corresponding
information flow be processed quite effectively and adequately. Participants of an economic process
should have a possibility, taking into account available resources needed for running their own
business, to choose that or another attainable (achievable, reachable) alternative in the problem-
resource situation having been formed.
At this, proceeding from some theoretical speculations and [12, 17 – 20], the subject (the active
element of the intelligent air transportation management system) distributes the preferences functions
in accordance with the postulated optimality [12].
The problem is formulated as to discover the magnitude (possibly some relative value) and
direction of the intelligent conflict situation certainty or uncertainty.
For that purpose, it is proposed to apply the entropy by Shannon, which has been developed for the
probabilities. It is transformed for the intelligent conflict preferences similar to the references of [12,
17 – 20]:
N (1)
H i ln i ,
i 1
where i – the subscript that refers to the corresponding conflicting alternative that can be attained;
N – the total number of the taken into consideration alternatives deemed to be conflicting; i –
intelligent preferences functions that are to be found.
For the intelligent conflict management preferences i of conflicting alternatives it is going be
used canonical expressions of [12, 17 – 20]:
exp i Fi (2)
i t N ,
j 1
exp j F j
where i and j – the structure parameters, their corresponding values relate to the problem setting’s
objective functional of [12, 17 – 20], Fi and Fj – corresponding intelligent conflict management
functions expressing the effectiveness of the i -th and j -th conflicting alternatives.
Modifications on the entropy of the view of (1), required for the stated problem solution, are going
to be presented and described below herein.
3. Main Content
The traditional view entropy of (1) has some improperness to a certain degree. Let us say in a two
conflicting alternatives situation the distribution of the intelligent conflict preferences are as follows:
1 0.351 and 2 1 1 0.649 . (3)
This means that entropy will be the same for the opposite situation situation:
1 0.649 and 2 1 1 0.351 . (4)
Thus, the entropy of the view (1) shows no difference with respect to the directions of the certainty
or uncertainty of the intelligent conflict management alternating preferences. And that circumstance,
as in the case of (3) and (4) with the mirror reflection preferences distribution change, pertains to any
distribution of conflicting preferences. Hence, it is impossible to realize which uncertainty or certainty
is a “good” one and which is a “bad”. This attitude can be as “right” versus “wrong”.
3.1. General provisions
The relative function [17] as hybrid pseudo-entropy fits the problem solution requirements:
M (5)
N L
i ln i
H max j k
H max H i 1 j 1 k 1 ,
H
H max H max
max M L
j k
j 1 k 1
where H max – the maximally possible value of the entropy, in problems formulated in references of
[12, 17 – 20] it is
H max ln N , (6)
– the factor of the intelligent conflict management alternatives preferences functions domination,
proposed in [17]:
M L (7)
j k ,
j 1 k 1
where – alternatives that are considered to be positive and k – alternatives with the negative
j
content; M – the quantity of the positive alternatives; L – the quantity of the negative subset of the
alternatives, [17]:
M LN. (8)
3.2. Specific cases models construction
Let us consider a basic two-alternative situation:
4 bm 2 g 2 bm 2 g 2 (9)
vT m 4 , vL m 4 4 ,
3 C x0 2 S 2 C x0 2 S 2
where vT m – the speed of an aircraft horizontal flight that is optimal for the maximal duration, it is
obtained as a function of the changeable mass of the aircraft m ; b – the aircraft special aerodynamic
coefficient; g – the acceleration of the force of gravity; C x0 – one more aerodynamic coefficient of
the aircraft drag when the force of the aircraft wing lift equals “zero”; – the air density at the flight
conditions; S – the area that characterizes the aircraft aerodynamics properties; vL m – one more
optimal speed, this time for the aircraft flying in a horizontal path and intended for the maximal
distance, it is also expressed as a function in the terms of the aircraft mass m when it changes.
The aircraft optimal speeds of (9) are obtained as extremum solutions delivering maximum values
to the objective functionals of the aircraft horizontal flights; and these functions are considered in the
presented study as the intelligent cyber conflict management effectiveness functions of the aircraft
horizontal flight effectiveness.
Another two-alternative situation model is with taking into account the intelligent conflict
management effectiveness functions in the view of the solution of the ordinary differential equation
systems of the first order:
dy0 y (10)
1 0 k00 y0 k10 y0 y1
dt Y0
,
dy1 y1
1 k11 y1 k01 y0 y1
dt Y1
where y0 – the first of the two alternative intelligent conflict management effectiveness functions; t
– time; Y0 – marginal value for the first of the two alternative intelligent conflict management
effectiveness functions y0 ; k00 – coefficient of the first function value supposed exponential growth;
k10 – coefficient of the impact of the second of the two alternative intelligent cyber conflict
management effectiveness functions upon the first one, which, by assumption, decreases the rate of
the first conflicting function growth; y1 – the second of the two alternative intelligent conflict
management effectiveness functions; Y1 – marginal value for the second of the two alternative
intelligent conflict management effectiveness functions y1 ; k11 – coefficient of the second function
value supposed exponential decrease; k01 – coefficient of the impact of the first of the two alternative
intelligent conflict management effectiveness functions upon the second one, which, by assumption,
increases the rate of the second conflicting function growth.
The three-alternative case is like the previous but extended:
dy0 y (11)
k 00 y0 1 0 k10 y1 k 20 y2
dt Y0
dy1 y
k 01 y0 k11 y1 1 1 k 01k 21 y0 y2 ,
dt Y1
dy 2 y
k 02 k12 y0 y1 k 22 y2 1 2
dt Y2
where designations and interpretations of functions, coefficients, and values are analogous to the
previous case (10), but it was customized and extended to the intelligent cyber conflict management
situation with some three alternatives.
One more special case to be studied is when there are generalized parameters of the intelligent
system learning.
The objective functional is
2
2 (12)
Pri lnPri Pr1 R Pr2 R0 Pri 1 ,
i 1 i 1
where Pr1 and Pr2 are the generalized perception functions of the two alternatives for resources; R
and R0 are the input learning variable and threshold value resources correspondingly; is the
coefficient for the subjective preferences normalized assessment; it is likewise , i , and j ; they
are the corresponding weight coefficients or they might be the structure parameters that are internal,
also, these coefficients can be considered as the uncertainty Lagrange multipliers [12]. For the
presented study these parameters are interpreted as the internal intelligent control parameters which
have some properties of the intelligent object “attitude” to the alternatives [12].
The generalized perceptions functions of Pr1 and Pr2 are analogous to the preferences functions of
(1) – (4).
Extremizing the objective functional (12) under conditions of
(13)
0,
i
one can get the expressions similar to (2):
exp R expR0 (14)
Pr1 , Pr2 .
exp R exp R0 expR exp R0
The corresponding potential for the intelligence growth could be expressed as
V R V0 Pr2 R , (15)
where V0 is the amount of the intelligence potential available for the intelligence growth.
The intelligence growth output could be represented with one more generalized function:
Output R V R R . (16)
3.3. Solutions to the specific cases
In first situation (9) the solution is shown in the Figure 1.
30
26.215
26.215
102 150
vL( M )
25
21.618
vT ( M )
17.1 20
16.426 15
100 120 140 160
90 M 150
Figure 1: Intelligent conflict of optimal aircraft speeds
The designations in the Figure 1 are as follows: vLM is for vL m , obtained by the second
equation of (9); vTM is for vT m , obtained by the first equation of (9) respectively.
The results of the system (10) solution are presented in the Figures 2 and 3.
550 Y0l
500
400
y0
300
294.629 200
0 50 100 150 200
t0 t t1
Figure 2: Intelligent cyber conflict system internal self‐management for the self‐growing
effectiveness function
In the Figures 2 and 3 it is designated: y0 is for y0 , Y0l is for Y0 ; and y1 is for y1 , Y1l is for
Y1 correspondingly.
The third case of solution, with the system of equations (11), is demonstrated with the diagrams
plotted in the Figures 4 – 6.
60
49.973 Y1l
40
y1
20
7.936
0 50 100 150 200
t0 t t1
Figure 3: Intelligent conflict system internal self‐management for the self‐decreasing effectiveness
function
The designations in the Figures 4 – 6 are analogous to those for the Figures 2 and 3, and simply
extended to the considered three-alternative intelligent conflict situation.
The solutions expressed with the formulae of (14) – (16) to the special case of (12) under
conditions of (13) has already been described above.
15
11.575
10
y0 5
0 20 40 60 80 100
0.5 5
t0 t t1
Figure 4: Intelligent conflict system internal self‐management for the self‐decreasing effectiveness
function in case of the three‐alternative situation
5
4.128
y1
0 20 40 60 80 100
4.347 5
t0 t t1
Figure 5: Intelligent conflict system internal self‐management for the self‐growing effectiveness
function in case of the three‐alternative situation
5
4.487
y2
0 20 40 60 80 100
4.041 5
t0 t t1
Figure 6: Intelligent conflict system internal self‐management for the other self‐decreasing
effectiveness function in case of the three‐alternative situation
3.4. Computer simulation
When the system of equation (11) is adapted to the composition that combines the complete self-
management in the intelligent cyber conflict, it gives the results of computer simulation shown in the
Figures 7 – 9.
40
35.261
20
y0
0 5 10 15 20 25
20
20.056 40
t0 t t1
Figure 7: Intelligent conflict system complete internal self‐management for the self‐decreasing
effectiveness function in case of the three‐alternative situation
50
35.361
y1
0 5 10 15 20 25
38.639 50
t0 t t1
Figure 8: Intelligent conflict system complete internal self‐management for the self‐growing
effectiveness function in case of the three‐alternative situation
60
48.715
40
y2
20
1.669
0 5 10 15 20 25
t0 t t1
Figure 9: Intelligent conflict system complete internal self‐management for the other self‐decreasing
effectiveness function in case of the three‐alternative situation
The intelligent cyber conflict management preferences computed by (2) are illustrated in the
Figure 10.
In the Figure 10: p0 stands for 0 , p1 stands for 1 , and p2 stands for 2 , correspondingly.
The traditional entropy of the intelligent conflict management preferences calculated by (1) is
plotted in the Figure 11.
1
0.858
p0
p1 0.5
p2
0.03 0
0 10 20 30
0 t 25
Figure 10: Intelligent conflict system complete internal self‐management effectiveness functions
preferences in case of the three‐alternative situation
1.5
1.093
ln( 3)
2.275 10.825
1
H
0.5
0.488 0
0 5 10 15 20 25
0 t 25
Figure 11: Entropy of intelligent conflict system complete internal self‐management effectiveness
functions preferences in case of the three‐alternative situation
The pointed above hybrid pseudo-entropy relative function (5) of [17], when 2 is a “positive”
preference and 0 and 1 are not, that is they are considered as “negative” preferences, is shown in
the Figure 12.
0.5
0.402
2.275 10.825 0
0
H1
0.5
0.556 1
0 5 10 15 20 25
0 t 25
Figure 12: Hybrid pseudo‐entropy relative function of intelligent conflict system complete internal
self‐management effectiveness functions preferences in case of the three‐alternative situation
The results of computer simulation in case of (12) – (16) are represented in the Figures 13 – 15.
The generalized intelligence perception function curve, plotted in the Figure 13, is for the
calculations by the second equation of (14) with the value of the threshold generalized resource of
R0 100 .
1
0.996
Pr( R) 0.5
3
4.49610 0
0 100 200
0 R 200
Figure 13: Generalized intelligence perception function
100
99.55
V( R) 50
0.45 0
0 100 200
0 R 200
Figure 14: Generalized intelligence potential for the intelligence growth function
The generalized intelligence potential function (see the Figure 14) is used for the intelligence
growth function shown in the Figure 15.
It is calculated with the amount of the intelligence potential available for the intelligence growth
V0 100 .
4
1 10
3
5.97810
Output ( R) 5000
0 0
0 100 200
0 R 200
Figure 15: Generalized intelligence growth output function
4. Discussion
As can be seen from the results of the computer simulation (see the Figures 1 – 12) of the
intelligent conflict management models involving entropy tools of (1) – (8) in application to (9) –
(11), the hybrid pseudo-entropy relative function (5) of [17], of the intelligent conflict management
effectiveness functions preferences has advantages over the traditional entropy (1).
The optimal generalized resource value for the intelligence growth output (see the Figures 13 – 15)
is also obtainable with the help of the entropy paradigm (12) – (16).
The optimal value, which can be seen in the Figure 15, is lower than the threshold value accepted
in the calculation simulations.
4.1. Comparison analysis to the known results
In the case of conflicting alternative speeds of the aircraft horizontal flight (see the curves
calculated by the equations of (9) and plotted in the Figure 9), the intelligent cybernetic conflict
management function with respect to the available and achievable conflicting alternatives (options)
preferences functions entropy in the traditional view of (1) will not show the relative certainty or
uncertainty degree for the alternatives, and its direction either. The proposed hybrid pseudo-entropy
relative function (5) will show those required qualities and quantities.
Such effects are noticeable when comparing the entropies plotted in the Figures 11 and 12.
The entropy illustrated in the Figure 11 does not represent when, how much, and to which
alternative or group of alternatives, that is to “good” or “bad”, “correct” or “wrong”, the intelligent
conflict management has its inclination.
Whereas, the proposed combined hybrid pseudo-entropy relative function (5), that takes into
account the relative value of the intelligent conflict management situation uncertainty, together with
the composition with the intelligent conflict management effectiveness preferences functions index of
domination, shows that there are periods of time when the certainty of the considered intelligent cyber
conflict management has some definitely “positive” values. These values of time: t 2.275 and
t 10.825 are represented in the Figure 12. Also, there is such effect at the time about t 19.325 .
The mentioned above points in time portrait practically not that much referred back to the
traditional measure of uncertainty pictured in the Figure 11.
In the special case described with the equations of (12) – (16), illustrated in the Figures 13 – 15,
the optimal input value of the generalized resources R delivers maximum value to the generalized
intelligence growth output.
4.2. Evidently promising investigations
The calculation experimentations with the procedures described with the mathematical expressions
of (1) – (11) and their adaptations have been conducted with several abstracted supposed data values
and some initial conditions voluntary selected to a certain degree. Therefore, there is no need in their
indication herewith.
Though the main ideas and provisions of the presented study are quite comprehensively performed
in the models, there is a potential for the further research in the areas of the coefficient estimations.
However, some other representations are also possible, as well as some other models of the similar
interpretation could be elaborated.
The input value in the special case described with the equations of (12) – (16) is the generalized
resources and it is necessary to investigate more complex models of the resource-output relations.
5. Conclusion
The entropy theory of conflicts in general sense can be successfully implemented to the solutions
of the intelligent conflict management in the terms of the conflicts genesis, development, transitions,
and exodus. The intelligent conflict being a primary reason for the intelligent cyber system motion
can be properly managed with the relative function based upon the hybrid pseudo-entropy.
The optimal generalized input intelligent resources value found with the use of the resources
generalized perceptions functions entropy conditional extremization ensures the maximum value to
the generalized intelligence growth output. Further endeavors in the intelligent conflict management
have prospects in the used values estimations and methodological modifications.
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