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 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.

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].

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".

A case of a linearly dependable constraint

Taking into account the speed of the aviation transportation delivery

  2 1 2 1 , v v AB v v T   , (1)

where   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

    idem cost min min 2 2 2 1 1 1       v v f v v f P , (2)

where P is some idempotent (independent upon the parameters of the considered problem model, stable, steady, unchanged, constant) value; v possible range of variation. The idea of (2) is close to [18]. The linear dependence between the aircraft speeds of 1 v and 2 v coefficients values of 1 f and 2 f could be derived supposing as from (2).

    min min 2 2 2 1 1 1 v v f v v f P     . (3)

And from (3)

    min min 2 1 1 2 1 2 1 2 v v v f f f P v v     . (4)

Also assuming

    min min max min max max 1 1 1 1 2 2 2 1 2 v v v v v v v v v      ,(5)

where max 1 v and max 2 v are the maximal values in respect for the aircraft speeds of 1 v and 2 v possible range of variation.

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

            . ; min max min max max min 1 1 2 2 2 1 2 2 2 v v v v f f v v f P (6) System (6) yields            . ; min max min max 1 1 1 2 2 2 v v P f v v P f (7)

Having determined the coefficients of 1 f and 2 f from ( 2) -( 7), it is possible to consider now the condition of (2) as a constraint to the objective function of (1):

  0 1 , min max min min max min 2 2 2 2 1 1 1 1 2 1          v v v v v v v v v v . (8)

Therefore, the problem is becoming a problem of a conditional optimization.

Namely, find the optimal aircraft speeds: 1 v and 2 v , ( 1) -( 8), extremizing the time of the delivery by the aviation transportation:  

2 1 , v v T ,(1)

, subject to the only constraint of ( 2) as (8). Thus, the extended Lagrange function is

                            1 , , , min max min min max min 2 2 2 2 1 1 1 1 2 1 2 1 2 1 2 1 v v v v v v v v v v AB v v v v T v v L , (9)

where  is the Lagrange uncertain multiplier.

The necessary conditions for a possible extremum of (9) existence are

                         . 0 , ; 0 , ; 0 , 2 1 2 2 1 1 2 1 v v L v v v L v v v L (10) Then                                                   . 0 1 , ; 0 , ; 0 , min max min min max min min max min max 2 2 2 2 1 1 1 1 2 1 2 2 2 2 1 2 2 1 1 1 2 2 1 1 2 1 v v v v v v v v v v L v v v v AB v v v L v v v v AB v v v L (11)

The systems of ( 10) and (11)

yield                              . 1 ; ; min max min min max min min max min max 2 2 2 2 1 1 1 1 2 2 1 2 2 2 2 1 1 1 v v v v v v v v v v AB v v v v AB v v (12)

From the first two equations of system ( 12)

C v v v v     min max min max 2 2 1 1 . (13)

Then, using the third equation of system ( 12)

min min 2 2 1 1 v v v v C     . (14)

Applying ( 13) and ( 14) to the first or second equation of system ( 12)

  2 2 1 2 2 1 1 min min v v AB v v v v       . (15)

From ( 15)

    2 2 1 2 2 1 1 min min v v v v v v AB       . (16)

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

                         . 1 ; ; min min 2 2 1 1 2 2 1 2 2 1 C v v C v v v v AB C v v AB C (17)

The rewritten system (12) means

              . ; min min 2 2 1 1 2 2 1 C v v v v v v C AB (18) Then                . ; min min min min 2 1 2 1 2 2 1 v v C v v v v C C AB (19) So, 2 1 v v 

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).

A variation upon the constraint

This subsection deals with the time:  

2 1 , v v T

, (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

      1 1 1 1 1 2 2 2 1 2 min min max min max max v v v v v v v v v v        ,(20)

where  

1 v 

is the variation to the linear dependence of 2 v , (2), or the equation ( 4), both 2 v and  

1 v  being dependent upon 1 v .

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

1 v  ,     max min 1 1 1 1 1 1 v v v v k v      , (21)

where

1  k 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 2 v , (2), upon 1 v variation. That is

    1 2 1 1 2 1 2 2 min min v v v v f f f P v       , (22)

where  

1 v 

is the variation to the linear dependence of 2 v , (2) or the equation ( 4), upon 1 v , however this time the variation provides the 2 v values on the contrary to the previous option of ( 20) and ( 21).

The variation itself can have formally the mathematically identical expression though:

     max min 1 1 1 1 1 1 v v v v k v      ,(23) where 1



k 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

      0 , 2 1 2 1 1 2 1 2 2 1 min min          v v v v v f f f P v v . (24)

That means

                              2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 min min , , , v v v v v f f f P v v AB v v v v T v v L . (25)

Or in the view convenient for differentiating

                         2 1 1 1 1 2 1 1 2 1 2 2 1 2 1 max min 1 min min , v v v v v k v v v f f f P v v AB v v L . (26)

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

                                                                         . 0 , ; 0 , ; 0 , 2 1 1 1 1 2 1 1 2 1 2 2 1 2 2 1 2 2 1 1 1 1 1 2 1 2 2 1 1 2 1 max min 1 min min max min 1 v v v v v k v v v f f f P v v L v v AB v v v L v v v v k f f v v AB v v v L (27)

The second equation of (27) immediately meant that

  2 2 1 v v AB     . (28)

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

) it yields           0 max min 1 1 1 1 1 2 1 2 2 1 2 2 1                  v v v v k f f v v AB v v AB . (29) And     0 2 1 min max 1 1 1 1 2 1 2 2 1               v v v k f f v v AB . (30)

Which means

  0 2 1 min max 1 1 1 1 2 1       v v v k f f . (31)

And

             min max 1 opt 1 1 2 2 1 1 2 1 v v f k f f v . (32)

The solution (32) ensures

     max opt min opt 1 min min opt opt 1 1 1 1 2 1 1 2 1 2 2 v v v v k v v v f f f P v         .

(33)

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.

Computer modeling with the linearly dependable constraint

In case of ( 1) -( 19), the accepted calculation data are as follows:

4 10 1  AB , 3 1 10 1 600    v and 3 2 10 1 600    v . (34)

The results for  

        1 1 2 1 1 2 1 , v T v v v AB v v v T    . (35)

In such case,

    1 2 1 , v v v T

: (1), modified to (35), it proves to have no extremum shown in the Figure 3. The constant (idempotent) value of

    1 2 1 , v v v T :(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

    1 2 1 , v v v T :(1),

and the equations of  

1 2 v v :(2)

-( 5), illustrated for the perceptional ease in the Figure 5.

T X Y  Z  ( ) 

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:  

  k . (36)

The results are presented in the Figure 6.

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.

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.