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

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

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 Tv1,v2  

  v1  v2 , where T v1, v2  is the time of the delivery; v1 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 v1 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  f1v1  v1min  f2 v2  v2min   cost  idem ,  where P is some idempotent (independent upon the parameters of the considered problem model, stable, steady, unchanged, constant) value; f1 and f 2 are the corresponding speeds coefficients; v1min and v2min are the minimal values in respect for the aircraft speeds of v1 and v 2 possible range of variation. The idea of ( 2 ) is close to [18].

The linear dependence between the aircraft speeds of v1 and v 2 coefficients values of f1 and f 2 could be derived supposing as from ( 2 ).

And from (3)

Also assuming

P  f1v1  v1min   f2 v2  v2min  . v2 v1   P  f1 v1  v1min  v2min .

f 2 f 2 ( 1 )  ( 2 )  ( 3 ) ( 4 ) where v1max and v2max are the maximal values in respect for the aircraft speeds of v 1 and v 2 ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10 ) ( 11 ) possible range of variation.

Comparing the corresponding members of (4) and (5)

P  v2min  v2max ;  f2

System (6) yields

 f1  v2max  v2min . f2 v1max  v1min  f2 

P ; v2max  v2min 

P f1  v1max  v1min . 

Having determined the coefficients of f1 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 1  0 . v1, v2  

v1max  v1min v2max  v2min

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 Lv1, v2   T v1, v2    v1, v2   v1  v2   v1  v1min   v1max  v1min The systems of ( 10 ) and ( 11 ) yield

Then, using the third equation of system (12)

C  v1 v1min v2 v2min .

AB v1  v1min  v2  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).

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     AB ;  C v1  v2 2   v1  v1min  v2  v2min  1. C C 

The rewritten system (12) means

Then

AB C    ;  v1  v2 2 

 v1  v1min  v2  v2min  C.

AB C    C  v1min  v2min 2 ;  v1  v2  C  v1min  v2min . ( 12 ) (13) (14) (15) (16) (17) (18) (19)

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

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 v1 .

Suppose a nonlinear variation of v1, (20). The proposed model is

v1   k1 v1  v1min v1  v1max  , where k1 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 v1 variation. That is v2  P  f1 v1  v1min  v2min  v1  ,

f 2 f 2 2.1.2. A variation upon the constraint (20) (21) (22) (23) (24) (25) (26) (27) (28) where v1 is the variation to the linear dependence of v 2 , ( 2 ) or the equation ( 4 ), upon v1 , 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  , where k1 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  v1, v2   P  f1 v1  v1min  v2min   v1   v2  0 .

f 2 f 2

That means

Lv1,v2   T v1,v2   v1,v2  

AB v1  v2   P  f1 v1  v1min  v2min  v1   v2  .   f2 f2 

Or in the view convenient for differentiating

AB  P Lv1,v2     v1  v2  f2   ff12 v1  v1min  v2min  k1 v1  v1min v1  v1max  v2  .

After applying the conditions of (10) to (25)

AB   ff12  k1 v1  v1min  v1  v1max   0; Lv1,v2      v1 v1  v2 2   Lv1,v2    AB    0;  v2 v1  v2 2   Lv1,v2   Pf2  ff12 v1  v1min  v2min  k1 v1  v1min v1  v1max  v2  0.

The second equation of (27) immediately meant that

AB   

Then, substituting (28) for its value into the first equation of (27) it yields And And

Which means



AB  f   v  v2 2  f1  k1 v1  v1min  v1  v1max   0 .

1 2 2.2. Simulation v2opt  Pf2  ff12 v1opt  v1min  v2min  k1 v1opt  v1min v1opt  v1max . (29) (30) (31) (32) (33)

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  1104 , v1  6001103 and v2  6001103 .

The results for T v1, v2  obtained by ( 1 ) with the use of data (28) are shown in the Figure 1 (34) 8.333 T( v1 600) T( v1 650) T( v1 700) T( v1 750) T( v1 800) T( v1 850) T( v1 900) T( v1 950) T( v1 1000) 9 8 7 6 5 The three-dimensional plot of T v1, v2  by ( 1 ) is shown in the Figure 2.

Applying the constraint in the view of ( 2 ), or v2 v1  : ( 4 ), or ( 5 ), to the duration of the aircraft transportation delivery (flight time), i.e.

Tv1,v2v1 

AB v1  v2v1  Tv1 .

(35)

In such case, T v1, v2 v1 : ( 1 ), modified to (35), it proves to have no extremum shown in the Figure 3.

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.

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. 1000 1103

T ( X Y Z)

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 k1 : entering the expressions (21) and (23), and used throughout the modeling (20) – (33). Those data were as the following:

k1  5103 and k1  5103 .

The results are presented in the Figure 6. (36) 200 200 ( v1) ( v1)

0 1000 1103

The variated speeds with the basic one are shown in the Figure 7.

6.5

6 8.

Computer modeling for the time of the air transportation delivery is illustrated in the Figure

The extreme values are shown in the Figures 10 and 11 as well.

The phase portraits in the Figures 10 and 11 demonstrate the minimal time and the optimal speeds combination.

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 theoreticalmathematical 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.

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