=Paper=
{{Paper
|id=Vol-3702/paper2
|storemode=property
|title=Simulation of Intelligent Air Transportation Management System Based upon Entropy Approach
|pdfUrl=https://ceur-ws.org/Vol-3702/paper2.pdf
|volume=Vol-3702
|authors=Andriy Goncharenko
|dblpUrl=https://dblp.org/rec/conf/cmis/Goncharenko24
}}
==Simulation of Intelligent Air Transportation Management System Based upon Entropy Approach==
https://ceur-ws.org/Vol-3702/paper2.pdf
Simulation of Intelligent Air Transportation Management
System Based upon Entropy Approach
Andriy V. Goncharenko
National Aviation University, 1, Liubomyra Huzara Avenue, Kyiv, 03058, Ukraine
Abstract
The paper is devoted to the computer modeling for the optimal governing of the intelligent
air transportation management system. The performed computer simulation is based upon the
initial postulate of the subjective entropy maximum principle developed for the active
systems control. The research and calculation experimentations are conducted for both losses
(“harmfulness”) and utilities (“usefulness”) functions. It is discovered the important
phenomenon unknown before, which is the shape of the phase diagrams for the preferences
over the losses functions. Another novelty and significance of the findings related to the
preference functions and entropy are that the conditional optimization of the subjective
individuals’ preferences functions entropy in conjunction with the proposed hybrid combined
relative pseudo-entropy function helps determine the relative certainty/uncertainty degree
concerning prevailing/dominating subjective preferences functions.
Keywords 1
Entropy, preferences, uncertainty, air transport, optimization, intelligence, management,
simulation, objective functional.
1. Introduction
Computer modeling of intelligent systems is an actual and important task for scientific research
and investigations. The urgency for such kinds of study is dictated by the contemporary informative
world and intensifying development of intelligent technologies. Since air transportation management
systems are functioning in some complex operational situations, there must be some corresponding
scientific approaches allowing assessing the circumstances of the occurred multi-alternativeness.
One part of the influential factors, requiring computer modeling, is the issues of the aircraft as
whole (including its powerplants) proper maintenance and repair [1, 2] in order to keep the reliability
of the aeronautical engineering and the flight safety of the entire aircraft itself [3, 4].
Elements of the intelligence, either natural or artificial, are present at the air transportation
management system’s making governing decisions and operational alternatives choice. Such
processes might be considered from the point of view of the utility theory [5, 6].
Anyway, the entropy paradigm formulated in [7 – 9], and used widely in science nowadays [10],
with taking into account economic aspects [11], realized in the theory of active systems and subjective
preferences [12], should be rather effectively implemented to the problems of the intelligent aviation
radio equipment reliability parameters monitoring [13], revealing needed properties of new materials
[14], neural networks of different kinds modeling [15, 16].
The essential feature is the application of such type of the entropy paradigm as used in [7 – 10, 12,
17 – 20] in order to model the properties of the intelligent air transportation management system’s
optimal behavior in conditions of the available operational multi-alternativeness causing the
uncertainty of the individuals’ subjective functions of their preferences.
CMIS-2024: Seventh International Workshop on Computer Modeling and Intelligent Systems, May 3, 2024, Zaporizhzhia, Ukraine
andygoncharenco@yahoo.com (A. V. Goncharenko)
0000-0002-6846-9660 (A. V. Goncharenko)
© 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
2. Development of theoretical models with the elements of simulation
The modeling of the intelligent air transportation management system’s optimal behavior requires
a closer look into theoretical provisions. The uncertainty of the individuals’ subjective preferences
functions should be assessed with the subjective entropy [12]. The well-known Jaynes’ principle of
the entropy maximum [7 – 9] implemented into the field of aviation will help cope with the problems
connected with the notorious human factor. First, let us consider a portion of the developments
dedicated to the subjective preferences theory provisions.
2.1. Basic concept
It is proposed to apply the generalized model, taking into account the operational uncertainty with
the help of the objective functional of the intelligent air transportation management system, in the
view of [12]:
N N
N (1)
i 1
i 1
i ln i i Li i 1 ,
i 1
where i – corresponding subjective individual’s preference function of the responsible decision
making person distributed with respect to the negative qualities of the achievable for the person’s
goals alternatives i ; and – the values, introduced in the objective intelligent air transportation
management system functional (1), in some respect, could be defined as endogenous parameters
reflecting certain features and properties of psych, or internal parameters of the intelligent air
transportation management system, the uncertain Lagrange multipliers, some coefficients or weight
coefficients at the specific problem settings [12]; L i – the corresponding function of the personally
estimated losses (“harmfulness”) related with the available alternatives; the individual distinguishes a
certain one-sided attitude to the managerial process.
The first member in the intelligent air transportation management system objective functional (1)
having the expression of
N (2)
i ln i ,
i 1
is the subjective entropy H , that is the measure of the operational multi-alternativeness uncertainty
of the individuals’ preferences functions of the effectiveness (in the given case study: losses L i
related to the alternatives).
The last member in the intelligent air transportation management system objective functional (1)
having the expression of
N (3)
i 1 ,
i 1
is the normalizing condition.
On the contrary, formulating the objective functional as, [12]:
N N
N (4)
i 1
i 1
i ln i i U i i 1 ,
i 1
where i – corresponding positive subjective individual’s preference function of the responsible
decision making person; U i – the corresponding function of the personally estimated utility
related with the available alternatives; the individual distinguishes a certain second-sided attitude to
the managerial process.
Both formulation of (1) and (4) that is both the negative (losses, harmfulness) and positive (utility,
usefulness) estimations are possible.
And the both-sided estimated managerial process of (1) and (4) is extremized with the use of the
necessary conditions for the subjective preferences functions entropy conditional optimization in the
view of
i
0 ,
i
0,
i 1, N .
(5)
For both senses of (1) and (4) the conditions of (5) yield the so-called canonical distributions of the
subjective preferences functions [12].
In cases of the object (item, thing) subjective preferences these canonical distributions of the
preferences functions are conventionally called: the subjective preferences functions distributions of
the first kind [12].
The optimal distributions are as follows, [12]:
e L L i eU U i (6)
i N , i N ,
j 1
e L L i j 1
eU U i
where L and L – corresponding coefficients with the subscript for the objective intelligent air
transportation management system functional of (1) having related with the negative sense
alternatives i effectiveness estimations; and U and U – corresponding coefficients if the
subjective attention is drawn to the positive features, in generally speaking terms to the same set of
the alternatives.
Illustration of the theoretical speculations, described with the expressions of (1) – (6), is in [12]
too; and it is shown in the Figure 1.
-(i), +(i)
1
+ (а)
- (б)
0
L(i), U(i)
Figure 1: The shape of the preferences functions, [12]
Now, there arises a big conceptual question. The curve denoted as i is monotonously
decreasing as the corresponding harmfulness function increases, but the point is that, that this is not
always so, because, if it, the preference function for some of the alternatives, decreases, it means that
for the other alternatives it symmetrically increases (absolutely like i preference shown in the
Figure 1 as well).
Let us demonstrate this ideological collision with the simplest modeling.
2.2. Modeling
For the simplest modeling let us consider the linear increase of harmfulness (losses) functions with
respect to a distinguishing parameter for a three-alternative case.
The computational data are as follow:
100100 , L 0.009 , L1 10 , L2 15 , L3 20 . (7)
The results of the simulation are shown in the Figures 2 – 5.
The diagrams plotted in the Figure 2 are for the losses (“harmfulness”) functions values computed
by the last three equations of (7) correspondingly.
3 2000
210
L1( )
L2( ) 0
L3( )
3
210 2000
100 0 100
100 100
Figure 2: Harmfulness functions
The next up is the illustration for the subjective individuals’ preferences functions represented in
the Figure 3.
Also, the normalizing condition (3) check is realized with the “one” value made visible in the
Figure 3.
1
1
_L1( )
_L2( )
0.5
_L3( )
_L1( ) _L2( ) _L3( )
4
1.2210 0
100 0 100
100 100
Figure 3: Subjective preferences functions
In order to get the phase portrait equivalent to the one shown in the Figure 1, the diagrams in the
Figure 4 are plotted. The normalizing conditions (3) check is presented there as well.
1
1
_L1( )
_L2( )
0.5
_L3( )
_L1( ) _L2( ) _L3( )
4
1.2210 0
2000 0 2000
3 L1( ) L2( ) L3( ) 3
210 210
Figure 4: Subjective preferences functions with respect to losses functions
Thus, the diagram of i shown in the Figure 1 deals with just a very particular case.
The subjective entropy of H , computed with the help of the expression (2), of the subjective
individuals’ preferences functions i , computed with the use of the first equation of the
expressions of (6), is illustrated in the Figure 5.
1.5
1.099
H__L( ) 1
ln( 3)
0.5
0.062 0
100 0 100
100 100
Figure 5: Subjective preferences functions with respect to losses functions
The maximal subjective entropy value, which is
ln3 , (8)
is also shown in the Figure 5.
Here, one more important theoretical question arises, whether the traditional view entropy H ,
having the expression of (2), is able to distinguish what the preferences distributional uncertainty the
entropy H determines.
In order to clarify this issue, the relative combined pseudo-entropy function is proposed to be used
in addition to the entropy H of (2).
2.3. Modeling based upon pseudo‐entropy function
As it is seen from the Figure 5, the traditional view subjective entropy of H , computed with the
help of the expression (2), is incapable to clarify the entropy value for the specific distribution of the
subjective individuals’ preferences functions i .
Therefore, it is proposed to make use of the measure of the certainty/uncertainty in the view of the
hybrid model of the combined pseudo-entropy function of the subjective functions [17]:
M (9)
N L
H max i ln i
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 – maximal subjective entropy value, in discrete alternatives problem settings this value
constitutes:
H max ln N ; (10)
– factor/index of the preferences functions prevailing/dominance:
M L (11)
j k ,
j 1 k 1
where j – positive and k – negative alternatives correspondingly; M – the number of the positive
alternatives; L – the number of the negative alternatives in respect:
M LN. (12)
Thus, computer modeling allows, designating some subjective individuals’ preferences functions
as “positive” or “negative”, making it visible which alternative preference dominates in the
certainty/uncertainty degree. Moreover, it is important that the value of (9) is relative which is also
more convenient.
2.4. Simulation
For the case considered above, it is possible to distinguish alternatives subjective preferences
functions one by one. For the first alternative the preferences prevailing factor (11) gives:
1 2 3 . (13)
Computation results for (6) – (12), with (13) for (11), are shown in the Figure 6.
1
0.944
RH1( ) 0
0.944 1
100 0 100
100 100
Figure 6: Relative pseudo‐entropy function for the first alternative function dominance
The curve in the diagram in the Figure 6 is plotted by the calculations with formula (9) where (13)
stands for (11). Similar procedures for the other two alternatives:
2 1 3 and 3 1 2 . (14)
bring the results shown in the Figures 7 and 8.
0
0
RH2( ) 0.5
0.944 1
100 0 100
100 100
Figure 7: Relative pseudo‐entropy function for the second alternative function dominance
1
0.944
RH3( ) 0
0.944 1
100 0 100
100 100
Figure 8: Relative pseudo‐entropy function for the third alternative function dominance
All the three pseudo-entropy functions are shown in the Figure 9.
1
0.944
RH1( )
RH2( )
0
RH3( )
0.944 1
100 0 100
100 100
Figure 9: Relative pseudo‐entropy functions for the three alternative functions dominance
For the visibility and comparison analysis the entropies and preferences functions are represented
in the Figure 10.
1.5
1.099
H__L( )
1
ln( 3)
_L1( )
_L2( ) 0.5
_L3( )
RH1( )
0
RH2( )
RH3( )
_( ) 0.5
0.944 1
100 0 100
100 100
Figure 10: Relative pseudo‐entropy functions for the three alternative functions dominance
2.5. Modeling a more general case
Now, it is possible to consider some more generalized models when some of the functions of
losses are nonlinear.
The data that are different from the above case are as follows:
L2 0.002253 , L3 0.2 2 . (15)
The results of the computer modeling calculation simulations in the style of (1) – (14) with the
corresponding functions of (15) are represented in the Figures 11 – 19.
The diagrams for functions of losses (harmfulness) are shown in the Figure 11.
3 4000
2.2510
L1( ) 2000
L2( ) 0
L3( )
2000
3
2.2510 4000
100 0 100
100 100
Figure 11: Functions of losses for the three alternatives
The subjective preferences functions are portrayed in the Figure 12.
1
1
_L1( )
_L2( )
0.5
_L3( )
_L1( ) _L2( ) _L3( )
0 0
100 0 100
100 100
Figure 12: Functions of subjective preferences for the three alternatives
The normalizing conditions are also presented in the Figure 12.
Phase diagrams of the subjective individuals’ preferences functions with regards to the
corresponding harmfulness functions are illustrated in the Figure 13.
1
1
_L1( )
_L2( )
0.5
_L3( )
_L1( ) _L2( ) _L3( )
0 0
3000 2000 1000 0 1000 2000 3000
2.2510
3 L1( ) L2( ) L3( ) L2( ) 2.2510
3
Figure 13: Phase diagrams of subjective preferences functions with respect to the losses functions
for the three alternatives
The phase portrays as well as the check line for the normalizing conditions are, in fact, the
corresponding projections of the curves within the three coordinates reference system (see and
compare the Figures 11 – 13). The traditional view subjective entropy of the individuals’ preferences
functions is shown in the Figure 14.
1.5
1.099
59.9 6.8
H__L( ) 1
ln( 3)
0.5
3
6.11810 0
100 0 100
100 100
Figure 14: Traditional view subjective entropy of the individuals’ preferences functions
The relative pseudo-entropy function in the case of the first alternative preference function
domination is shown in the Figure 15.
1
0.546
0.36898
59.9 6.8
0.054348
RH1( ) 0
0.994 1
100 0 100
100 100
Figure 15: Relative pseudo‐entropy function for the first alternative subjective individuals’
preferences function prevailing
As for the second alternative preference function prevailing, the relative pseudo-entropy function
is shown in the Figure 16.
1
0.714
RH2( ) 0
0.994 1
100 0 100
100 100
Figure 16: Relative pseudo‐entropy function for the second alternative subjective individuals’
preferences function dominance
The third alternative preference function domination impact upon the relative pseudo-entropy
function is represented in the Figure 17.
The relative pseudo-entropy functions for all three cases of the three alternative preference
functions domination are shown in the Figure 18.
1
0.994
RH3( ) 0
0.714 1
100 0 100
100 100
Figure 17: Relative pseudo‐entropy function for the third alternative subjective individuals’
preferences function prevalence
1
0.994
RH1( )
RH2( )
0
RH3( )
0.994 1
100 0 100
100 100
Figure 18: Relative pseudo‐entropy functions for the three alternatives subjective individuals’
preferences functions prevalence
The diagrams showing all curves of the case study are plotted the Figure 19.
1.5
1.099
H__L( )
1
ln( 3)
_L1( )
_L2( ) 0.5
_L3( )
RH1( )
0
RH2( )
RH3( )
_( ) 0.5
0.994 1
100 0 100
100 100
Figure 19: All curves of the case study
3. Discussion
Computer modeling of intelligent systems should be based upon the laws and regularities of the
natural intellect functioning. One of such regularities is the available for an individual’s alternatives
choice subjective preferences optimal distribution in accordance with the postulated in subjective
analysis theory of preferences principle which is called the subjective entropy maximum principle.
Active air transportation systems management elements have to include the corresponding
subjective individual’s preferences functions of the responsible decision making persons distributed
with respect to the negative qualities of the achievable for the person’s goals alternatives modeled
with the objective intelligent air transportation management system functional (1). Subjective entropy
(2) and normalizing constraint (3) are essential components of the objective functionals (1) and (4) for
obtaining solutions (6) on conditions of (5).
However, the generally accepted character of the subjective preferences functions (6) dependence
upon the functions of losses (“harmfulness”) (illustrated in the Figure 1, [12]), is now broadened to
the understandings of the modeled cases (7) – (15) (see the Figures 2 – 19). The point is that the
subjective individuals’ preferences functions (6) presumably measure the relative either harmfulness
(losses) or the utilities (“usefulness”) of the attainable alternatives due to the subjective preferences
entropy conditional optimization principle.
That is why the shape of the phase diagrams for preferences to losses functions (illustrated in the
Figure 1, [12]) is revealed to be correct only for the greatest of the linear case losses functions with
respect to the determining parameter (see and compare the Figures 1 – 4 and Figures 11 – 13
correspondingly).
The proposed relative pseudo-entropy function (9) happened to be helpful in determining the
relative certainty/uncertainty degree concerning prevailing/dominating preference (11). For instance,
the hybrid relative combined pseudo-entropy function (9) (see the Figure 15) in the diapason of the
determining parameter of 59.9 6.8 shows the systems certainty increase with the following
decrease because of the first alternative subjective preference function domination. Although the
traditional view entropy (2) is incapable to distinguish such phenomenon (see and compare the
Figures 15 and 14 in regards). For other diapasons the certainty/uncertainty varies in negative values,
which means that the first alternative subjective preference function has no prevalence (13) (also see
and compare the Figures 15 and 14 in correspondences). The same effectiveness of the proposed
hybrid relative combined pseudo-entropy function (9) comparatively to the traditional view entropy
(2) is visible in the Figures 14 and 16 – 19; as well as in the Figures 5 – 10. The relative value of the
pseudo-entropy function (9), having “positive certainty”: “+1”, “negative certainty”: “-1”, and
“uncertainty”: “0” values, is more convenient rather than the bare value of the traditional view
entropy (2). The same is observed for the utilities (“usefulness”) subjective preferences functions
entropy conditional optimization described with the formulas of (4) and (6).
4. Conclusion
Computer modeling of the intelligent air transportation management system functioning in the
conditions of the operational alternatives subjective preferences uncertainty helps reveal the important
phenomenon unknown before. That is the shape of the phase diagrams for the preferences to losses
functions. 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.
Further studies should investigate more special cases of the intelligent systems multi-
alternativeness.
5. References
[1] M. J. Kroes, W. A. Watkins, F. Delp, R. Sterkenburg, Aircraft Maintenance and Repair, 7th. ed.,
McGraw-Hill, Education, New York, NY, 2013.
[2] T. W. Wild, M. J. Kroes, Aircraft Powerplants, 8th. ed., McGraw-Hill, Education, New York,
NY, 2014.
[3] B. S. Dhillon, Maintainability, Maintenance, and Reliability for Engineers, Taylor & Francis
Group, New York, NY, 2006.
[4] D. J. Smith, Reliability, Maintainability and Risk. Practical Methods for Engineers, Elsevier,
London, 2005.
[5] R. D. Luce, D. H. Krantz, Conditional expected utility, Econometrica 39 (1971) 253–271.
[6] R. D. Luce, Individual Choice Behavior: A theoretical analysis, Dover Publications, Mineola,
NY, 2014.
[7] E. T. Jaynes, Information theory and statistical mechanics, Physical review 106 (4) (1957) 620–
630. doi:10.1103/PhysRev.106.620.
[8] E. T. Jaynes, Information theory and statistical mechanics. II, Physical review 108 (2) (1957)
171–190. doi:10.1103/PhysRev.108.171.
[9] E. T. Jaynes, On the rationale of maximum-entropy methods, Proceedings of the IEEE 70 (1982)
939–952. https://ieeexplore.ieee.org/document/1456693. doi:10.1109/PROC.1982.12425
[10] F. C. Ma, P. H. Lv, M. Ye, Study on global science and social science entropy research trend, in:
Proceedings of the IEEE International Conference on Advanced Computational Intelligence,
ICACI, Nanjing, Jiangsu, China, 2012, pp. 238–242.
https://ieeexplore.ieee.org/document/6463159. doi:10.1109/ICACI.2012.6463159
[11] E. Silberberg, W. Suen, The Structure of Economics. A Mathematical Analysis, McGraw-Hill
Higher Education, New York, NY, 2001.
[12] V. Kasianov, Subjective Entropy of Preferences. Subjective Analysis, Institute of Aviation
Scientific Publications, Warsaw, Poland, 2013.
[13] O. Solomentsev, M. Zaliskyi, T. Herasymenko, O. Kozhokhina, Yu. Petrova, Data processing in
case of radio equipment reliability parameters monitoring, in: Proceedings of the International
Conference on Advances in Wireless and Optical Communications, RTUWO, Riga, Latvia,
2018, pp. 219–222. https://ieeexplore.ieee.org/abstract/document/8587882.
doi:10.1109/RTUWO.2018.8587882.
[14] V. Marchuk, M. Kindrachuk, Ya. Krysak, O. Tisov, O. Dukhota, Y. Gradiskiy, The mathematical
model of motion trajectory of wear particle between textured surfaces, Tribology in Industry
(Tribol. Ind.) 43 (2) (2021) 241–246. https://doi.org/10.24874/ti.1001.11.20.03.
[15] D. Shevchuk, O. Yakushenko, L. Pomytkina, D. Medynskyi, Y. Shevchenko, Neural network
model for predicting the performance of a transport task, Lecture Notes in Civil Engineering. 130
LNCE (2021) 271–278. doi:10.1007/978-981-33-6208-6_27.
[16] S. Subbotin, The neuro-fuzzy network synthesis and simplification on precedents in problems of
diagnosis and pattern recognition, Opt. Mem. Neural Networks 22 (2013) 97–103.
https://doi.org/10.3103/S1060992X13020082.
[17] A. V. Goncharenko, Multi-optional hybridization for UAV maintenance purposes, in:
Proceedings of the IEEE International Conference on Actual Problems of UAV Developments,
APUAVD, Kyiv, Ukraine, 2019, pp. 48–51.
https://ieeexplore.ieee.org/abstract/document/8943902.
doi:10.1109/APUAVD47061.2019.8943902.
[18] A. V. Goncharenko, Active systems communicational control assessment in multi-alternative
navigational situations, in: Proceedings of the IEEE International Conference on Methods and
Systems of Navigation and Motion Control, MSNMC, Kyiv, Ukraine, 2018, pp. 254–257.
https://ieeexplore.ieee.org/abstract/document/8576285. doi:10.1109/MSNMC.2018.8576285.
[19] A. Goncharenko, A multi-optional hybrid functions entropy as a tool for transportation means
repair optimal periodicity determination, Aviation 22 (2) (2018) 60–66.
https://journals.vgtu.lt/index.php/Aviation/article/view/5930.
https://doi.org/10.3846/aviation.2018.5930.
[20] A. V. Goncharenko, Airworthiness support measures analogy to the prospective roundabouts
alternatives: theoretical aspects, Journal of Advanced Transportation Article ID 9370597 2018
(2018) 1–7. https://www.hindawi.com/journals/jat/2018/9370597/.
https://doi.org/10.1155/2018/9370597.