=Paper=
{{Paper
|id=Vol-3702/paper13
|storemode=property
|title=Computer Modeling of Discrete Systems in the Case of Linear and Non-linear Restrictions on the Optimal Speed of the Aircraft Based on the Lagrange Method
|pdfUrl=https://ceur-ws.org/Vol-3702/paper13.pdf
|volume=Vol-3702
|authors=Andriy Goncharenko,Serhii Teterin
|dblpUrl=https://dblp.org/rec/conf/cmis/GoncharenkoT24
}}
==Computer Modeling of Discrete Systems in the Case of Linear and Non-linear Restrictions on the Optimal Speed of the Aircraft Based on the Lagrange Method==
https://ceur-ws.org/Vol-3702/paper13.pdf
Computer Modeling of Discrete Systems in the Case of
Linear and non‐Linear Restrictions on the Optimal Speed
of the Aircraft Based on the Lagrange Method
Andriy Goncharenko and Serhii Teterin
National Aviation University, 1, Liubomyra Huzara Avenue, Kyiv, 03058, Ukraine
Abstract
The paper of the report represents the study dedicated to the main link of the aircraft flight speed to
the minimal time of the air transportation process. The simplest problem of the aviation
transportation technologies fundamental factors optimization models the considered process of the
delivery and it makes an attempt to the process theoretical description. Two segment air traffic
elementary chain of supply is implied at the presented research. The algorithm, which is used for
calculating the objective parameters of the aircraft motion, is developed. Approaches to aircraft speed
optimization are used. The speed of the delivery by the aircraft at each of the segments have been
conditionally optimized. The objective value is the time of delivery. The aircraft speeds are subject to
both linear and nonlinear constraints. The influence of the speeds’ variations upon the conditionally
minimal objective delivery time values are studied. Theoretical contemplations are conducted in the
framework of the Lagrange uncertainty multipliers implementation. The hypothetical provisions of the
derived mathematical models are illustrated with the help numerical simulation. The part of the
computer modeling is conducted on the Mathcad platform at the educational and scientific laboratory
"Modeling of transport systems and processes" of National Aviation University. The necessary
diagrams are plotted. The obtained results of both theoretical study and computer simulation allows
construction of optimal delivery chains with a better determination of the exchange point location.
Keywords
Computer modeling, optimization, simulation, aviation transport technologies, aircraft flight speed,
speed optimization, delivery time minimization1
1. Introduction
Computer and numerical simulation for searching optimal aircraft speed by the criterion of the
minimal time of the delivery is an urgent task. That requires a proper maintenance of aircraft
itself [1], as well of the airplane engines and powerplants [2], in order to support the
aeronautical components’ reliability [3] and risk [4] at the due level.
However, the unsolved part to the general problem there, at references [1 – 4], is the lack of
the conditional optimization.
In such context, expected operational efficiency and utility [5] is combined with the choice
problems [6].
This means that the transport technologies theoretical constraints, like in reference [7],
strengthening learning for intelligent applications [8, 9] are important.
These variants of uncertainty can be estimated using entropy approaches [10].
In conjunction with the economic models [11], the entropy methods resulted in the
subjective analysis theory [12] allows solving various types of the applicable problems, similar
to the stated in the references of [13 – 16]. Some problems relevant to the air transport
management and aviation transport technologies have already been posed in [17 – 20].
According to the presented concepts, a scientific gap that needs to be solved is the
development of a reliable mathematical approach to the actual and important problems related
to the formulation of the optimal combination of aviation resources, especially in relation to
CMIS-2024: Seventh International Workshop on Computer Modeling and Intelligent Systems, May 3, 2024,
Zaporizhzhia, Ukraine
andygoncharenco@yahoo.com (A. V. Goncharenko); sergiyteterin@gmail.com (S. O. Teterin)
0000-0002-6846-9660 (A. V. Goncharenko); 0009-0004-2662-6252 (S. O. Teterin)
© 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
supporting the process of analytical decision-making based upon the advantages of the
computer modeling.
Thus, in the case with the aircraft transportation speeds, it is necessary to formulate the
scientific hypothesis of the conducted research as the speeds’ variations, subject to both linear
and nonlinear constraints, impact upon the conditional minimal value of the delivery time.
The problem statement of the presented research concerns with the theoretical studies
focused on the calculations of the optimal speed of the aircraft as the key element of the aviation
transport technologies.
Thus, the goals of the article are a general description of the possible optimization of the
aircraft speed for the theoretical and mathematical obtaining of rational solutions.
2. Possibilities of optimization
The simplest problem of the aviation transportation technologies fundamental factors
optimization stated here is based upon an elementary two segment supply chain consideration.
The speed of the delivery by the aircraft at each of the segments could be conditionally
optimized.
This necessitates the further development of the optimization methods of [18] and [19].
2.1. Basic concept
It is going to be considered the theoretical background for calculating aircraft movement
parameters and approaches to their optimization.
The simulation of the aircraft motion was conducted with use of the software capabilities of
the educational and scientific laboratory "Modeling of transport systems and processes".
2.1.1. A case of a linearly dependable constraint
Taking into account the speed of the aviation transportation delivery
AB (1)
T v1 , v2 ,
v1 v2
where T v1 , v2 is the time of the delivery; v 1 and v 2 are correspondingly the speeds of the
first and the second aircraft that fly towards each other the same time T v1 , v2 ; AB is the
distance covered by both aircraft in the time of T v1 , v2 and at the speeds of v 1 and v 2 in
respect.
Model (1) imply, for instance, a case when allows you to find an initial reference solution,
and then, improving it, get the optimal solution.
Considering the condition of
P f v v
1 1 f v v
1min
cost idem ,
2 2 2 min (2)
where P is some idempotent (independent upon the parameters of the considered problem
model, stable, steady, unchanged, constant) value; f 1 and f 2 are the corresponding speeds
coefficients; v1min and v2min are the minimal values in respect for the aircraft speeds of v 1 and
v 2 possible range of variation. The idea of (2) is close to [18].
The linear dependence between the aircraft speeds of v 1 and v 2 coefficients values of f 1
and f 2 could be derived supposing as from (2).
P f v v 1
f v v1 . 1min 2 2
(3)
2 min
And from (3)
v 2 v1
P
f2
f
1 v1 v1min v 2 min .
f2
(4)
Also assuming
v2 max v2 min (5)
v2 v1 v2 max
v1max v1min
v v ,
1 1min
where v1max and v2max are the maximal values in respect for the aircraft speeds of v 1 and v 2
possible range of variation.
Comparing the corresponding members of (4) and (5)
P
v2min v2max ;
f2
(6)
f1 v2max v2min
.
f 2 v1max v1min
System (6) yields
P
f2 ;
v2 max v2 min
(7)
P
f1 .
v1max v1min
Having determined the coefficients of f 1 and f 2 from (2) – (7), it is possible to consider now
the condition of (2) as a constraint to the objective function of (1):
v1 v1min v2 v2min
v1 , v2 1 0 . (8)
v v 1maxv v 1min 2 max 2 min
Therefore, the problem is becoming a problem of a conditional optimization.
Namely, find the optimal aircraft speeds: v 1 and v 2 , (1) – (8), extremizing the time of the
delivery by the aviation transportation: T v1 , v2 , (1), subject to the only constraint of (2) as (8).
Thus, the extended Lagrange function is
AB v1 v1min v 2 v 2 min
L v1 , v 2 T v1 , v 2 v1 , v 2 1 , (9)
v1 v 2
v1max v1min v 2 max v 2 min
where is the Lagrange uncertain multiplier.
The necessary conditions for a possible extremum of (9) existence are
L v1 , v 2
0;
v1
L v1 , v 2
0; (10)
v 2
L v1 , v 2
0 .
Then
Lv1 , v2 AB
0;
v1 v1 v2 v1max v1min
2
Lv1 , v2 AB (11)
0;
v2 v1 v2 2 v2 max v2 min
Lv1 , v2 v1 v1min v2 v2 min
1 0.
v1max v1min v2 max v2 min
The systems of (10) and (11) yield
AB
;
v1max v1min v1 v 2 2
AB (12)
;
v2 max v 2 min v1 v 2 2
v1 v1min v2 v2 min
1.
v1max v1min v 2 max v 2 min
From the first two equations of system (12)
v1max v1min v2max v2min C . (13)
Then, using the third equation of system (12)
C v1 v1min v2 v2min . (14)
Applying (13) and (14) to the first or second equation of system (12)
AB
. (15)
v1 v1min v2 v2 min v1 v2 2
From (15)
AB v1 v1min v 2 v 2 min . (16)
v1 v 2 2
But system (12) might have a solution. It is because its first two equations are the same.
Indeed, instead of (12) there is a possibility to write
AB
;
C v1 v2 2
(17)
AB
;
C v1 v2 2
v1 v1min v2 v2 min
1.
C C
The rewritten system (12) means
AB C
;
v1 v2 2
(18)
v1 v1min v2 v2min C.
Then
AB C
;
C v1min v2min
2
(19)
v1 v2 C v1min v2min .
So, v1 v2 has a constant (idempotent) value. One of the speeds can be determined through
the other one. It can be resolved with the help of the expressions of (2) – (5), or conditions of
(8), (9), the third equations of (11), (12), as well as from (14), the third equation of (17), and the
second equations of (18) or (19) too.
Moreover, therefore, the duration of flight has a idempotent (stable, steady, unchanged,
constant) value as well. That follows the model expression (1).
Thus, the system of two equations (18) obtained from/of (12) has happened to be a system
of two equations with three unknowns. And (19) is actually the one equation with the two
unknowns.
The following sections dedicated to simulation and discussion will visualize and dispute
upon the case set as (1) – (19).
2.1.2. A variation upon the constraint
This subsection deals with the time: T v1 , v2 , (1), but subject a nonlinear constraint in the
type of the variation to the equation of (2) or to the equation (4).
The necessary conditions for a possible extremum of (9) existence are
Now, it is going to be
v2 v2 min (20)
v2 v1 v2 max max
v1max v1min
v1 v1min v1 ,
where v1 is the variation to the linear dependence of v 2 , (2), or the equation (4), both v 2
and v1 being dependent upon v 1 .
Suppose a nonlinear variation of v1 , (20). The proposed model is
v k v v v v ,
1 1 1 1 min 1
(21)
1 max
where k1 is a coefficient.
On the other hand, it is possible to model the opposite side, of the equation of (2), or to the
equation (4), dependence of v 2 , (2), upon v 1 variation. That is
v2
P
f2
f
1 v1 v1 min v 2 min v1 ,
f2 (22)
where v1 is the variation to the linear dependence of v 2 , (2) or the equation (4), upon v 1 ,
however this time the variation provides the v 2 values on the contrary to the previous option of
(20) and (21).
The variation itself can have formally the mathematically identical expression though:
v1 k 1 v1 v1min v1 v1max , (23)
where k1 is a coefficient.
Making allowance for the above option of (20) and (21), just for the certainty of the problem
setting, the new constraint will have the view of
P f1
v1 , v 2
f2
f2
v v v
1
. 1 min
(24)
2 min v1 v 2 0
That means
P
Lv1 , v2 T v1 , v2 v1 , v2
v1 v2
fAB
1 v1 v1min v2min v1 v2 . (25)
f2 f2
Or in the view convenient for differentiating
P f
Lv1 , v2
AB
v1 v2
1 v1 v1min v2min k1 v1 v1min v1 v1max v2 . (26)
f2 f2
After applying the conditions of (10) to (25)
Lv1 , v2
AB f
1 k 1 v1 v1min v1 v1max 0;
v1 v1 v2 2
f2
Lv1 , v2 AB (27)
0;
v2 v1 v2 2
Lv1 , v2 P
f
1 v1 v1min v2 min k 1 v1 v1min
f2 f2
v1 v1max v2 0.
The second equation of (27) immediately meant that
AB (28)
.
v1 v2 2
Then, substituting (28) for its value into the first equation of (27) it yields
AB
AB f1
k 1 v1 v1min v1 v1max 0 .
2
(29)
v1 v2 2
v1 v2 f 2
And
AB f
1 1 k 1 2v1 v1max v1min 0 .
2
(30)
v1 v2 f 2
Which means
1
f1
f2
k 1 2 v1 v1max v1min 0 .
(31)
And
1 f1 f 2
v1opt v1max v1min . (32)
2 k 1 f 2
The solution (32) ensures
v 2 opt
P
f2
f
1 v1opt v1min v 2 min k 1 v1opt v1min v1opt v1max .
f2
(33)
2.2. Simulation
Let's consider the results obtained with the help of the theoretical considerations mentioned
above using formulas (1) - (20) and calculation procedures. In the interests of achieving the goal
of the study, computer modeling of the process of objectivity of the criteria for evaluating the
optimization of transport work in the implementation of air transportation was carried out.
2.2.1. Computer modeling with the linearly dependable constraint
In case of (1) – (19), the accepted calculation data are as follows:
AB 110 4 , v1 6001 103 and v2 6001 103 . (34)
The results for T v1 , v2 obtained by (1) with the use of data (28) are shown in the Figure 1
9
8.333
T ( v1 600)
T ( v1 650)
8
T ( v1 700)
T ( v1 750)
T ( v1 800) 7
T ( v1 850)
T ( v1 900)
T ( v1 950) 6
T ( v1 1000)
5 5
600 700 800 900 1000
600 v1 3
110
Figure 1: Time of the air transportation delivery
The three-dimensional plot of T v1 , v2 by (1) is shown in the Figure 2.
T
Figure 2: Time of the air transportation delivery
Applying the constraint in the view of (2), or v 2 v1 : (4), or (5), to the duration of the aircraft
transportation delivery (flight time), i.e.
AB (35)
T v1, v2 v1 T v1 .
v1 v2 v1
In such case, T v1 , v2 v1 : (1), modified to (35), it proves to have no extremum shown in the
Figure 3.
6.26
6.256
6.255
T ( v1) 6.25
6.245
6.244 6.24
600 800 1000
600 v1 3
110
Figure 3: Absence of the extremum of the time of the delivery
The constant (idempotent) value of T v1 , v2 v1 : (1), modified to (35), visible in the Figure 3
relates to the equations of (2) – (5), represented in the Figure 4.
31000
110
v2( v1) 800
600 600
600 800 1000
600 v1 3
110
Figure 4: Linearly constrained speeds dependence
Additional data used for plotting diagrams in the Figures 3 and 4 may be relevant to the P
value, (2) – (4) and (6), (7), however, it can be canceled or substituted, (8) – (19).
The absence of the extremum is noticeable in the three-dimensional plots of T v1 , v2 v1 : (1),
and the equations of v2 v1 : (2) – (5), illustrated for the perceptional ease in the Figure 5.
T ( X Y Z)
Figure 5: Absence of the extremum of the time of the air transport delivery
X,Y,Z shown in the Figure 5 are the parametric equations of the plain symbolizing the linear
constraints (2) – (5).
1. 1. The results of additional experimental studies made it possible to obtain new data
regarding the values of the weighting coefficients of the component indicators of the
integral indicator and to reveal a significant deviation from the values obtained
according to experts' assessments.
2. Local extrema are more inherent in the solution of optimization tasks of the parameters
of specific technologies and devices, since, as a rule, they have technical limitations of
independent variable objective functions.
2.2.2. Computer modeling with the nonlinearly dependable constraint
In the case with the aircraft transportation speeds variations of (20) – (33), in addition to the
data of (34), there is a need to have data for the computer simulations of the variations: v1
and v1 .
In fact, it was necessary to accept the values for the coefficients of k 1 and k1 : entering the
expressions (21) and (23), and used throughout the modeling (20) – (33). Those data were as
the following:
k1 5 103 and k1 5 103 . (36)
The results are presented in the Figure 6.
200
200
( v1)
0
( v1)
200 200
600 800 1000
600 v1 3
110
Figure 6: Aircraft speeds variations
The variated speeds with the basic one are shown in the Figure 7.
3 1200
1.0510
1000
1000 800
v2( v1)
v2f ( v1) 800
v2F( v1)
600
550 400
600 800 1000
600 v1 3
110
Figure 7: Variated aircraft speeds
Computer modeling for the time of the air transportation delivery is illustrated in the Figure
8.
6.5
6.25
v1opt
T ( v1 v2) 6
5.5556
5.556 5.5
600 800 1000
600 v1 3
110
Figure 8: Time of the air transportation delivery constrained by the nonlinearly dependable
aircraft speeds
The three-dimensional plots of T v1 , v2 by (1) and v2 v1 by (2) – (5), as well as v2 v1 by
(20) are shown in the Figure 9.
T ( X Y Z) ( X Ym Z)
Figure 9: Time of the air transportation delivery subject to constraints upon the aircraft speeds
The extreme values are shown in the Figures 10 and 11 as well.
6.5
6.25
v2opt
T ( v1 v2) 6
5.5556
5.556 5.5
600 800 1000 1200
600 v2F( v1) 3
1.0510
Figure 10: Phase diagram of the time of the air transportation delivery subject to constraints
upon the aircraft speeds
The phase portraits in the Figures 10 and 11 demonstrate the minimal time and the optimal
speeds combination.
6.5
6.25
800 1000
T ( v1 v2)
6
T ( v1 v2)
T ( v1opt v2)
5.556 5.5
600 800 1000 1200
600 v1 v2F ( v1) 1.05103
Figure 11: Combined phase portrait of the time of the air transportation delivery subject to
constraints upon the aircraft speeds
3. Discussion
As it was shown, the creation of similar formalized models, that is, the relationship of target
functions at different levels of the system hierarchy, will allow to maximize adequacy to optimal
conditions of air transport operation.
The results of the experiment and the computational experiment based on the mathematical
model by successive approximation by appropriate iterative methods are compared.
1. Optimization will always end with the search for local extrema of the objective
functions, since the intervals of variation of the independent variables included in the objective
functions are set a priori.
2. Phase portraits and trajectories of oscillation forms in the configuration space of the
system were constructed and analyzed. The conditions for the localization of the forms of
oscillations of the system have been obtained. The stability of the oscillation forms was studied.
The presented system and mathematical model can be a source for new modeling approaches.
At the first stage of the research, a problem was identified that lay in the optimization of the
aircraft's speed. To achieve this goal and systematize our understanding of the problem and
potential ways to solve it, the following was defined:
The problem that required our attention and what goal we want to achieve through the
research.
Possible ways of solving the problem, various alternatives were considered and the
most effective and suitable variant of its mathematical solution was chosen.
The improvement of the criteria for evaluating the transport work in the performance of air
transportation is carried out in the direction of expanding the list of factors that are taken into
account when determining the relevant indicators, successively - the number (mass) of objects
of transportation (passengers and cargo), range (distance between the points of departure and
destination) and speed (time) of their spatial movement (delivery) from the point of departure
to the destination.
If we draw a parallel between the ratio of optimal speed and the theory of individual risk
perception, then this may make it possible to conduct another study regarding the theoretical-
mathematical model of the demand for insurance services based on the conditional
optimization apparatus. When solving this problem by the method of undetermined Lagrange
multipliers, the derivative must equal zero before the necessary extremum condition.
Accordingly, in this case, at the optimal point, the budgetary limitation of the insurance cost
should be beneficial for the policyholder.
4. Conclusion
The obtained values as the implementation of this study allowed to analytically and graphically
determine the region of the optimal solution, taking into account the limitations of the objective
function.
The conditionally optimized aircraft speeds ensure minimal time of the air transport
delivery, which in turn leads to the improvement of the air transportation technologies.
For further research, it is proposed to investigate the dynamics, of the process of choosing
the desired optimal technologies of air transport, and models based on the given calculation
conditions that implement operational alternatives and the subjective entropy.
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