Rational Air Transportation Technologies Resources Recombination on Condition of the Generalized Values

Abstract

The presented paper is dedicated to the attempt of a theoretical description of the aviation transport technologies elements resources rational use. The maximal value of the generalized “positive” effect is chosen in the view of the optimization criterion. For two competing resources their optimal recombination is found at the segment of the mathematically linearly described solution obtained on condition of the fixed total amount of the production of the aviation transport technologies services. Such fixed total production amount acts like iso-perimeter constraint condition in the stationary problem with the constant capacities/productivities of the competing resources. On the other hand, the generalized “positive” effect parameter is also considered having two competing components, which values ensure yielding the optimal resources recombination. Illustrative example simulation is performed. Necessary diagrams are plotted.

Keywords

• eol>Aviation
• transport
• technology
• resource
• recombination
• entropy
• preferences
• optimization

1. Introduction

Computer simulation of the aviation transport technologies elements is an urgent task. Airlines’ and aviation business enterprises’ resources rational use is important in regards with both aircraft technical operation [1] and the aircraft engines reliable functioning [2]. Based upon the ideas of [1, 2] the aircraft and its powerplant elements reliability and maintenance require decent resources distributions and combinations. A significant amount of the resources are spent upon keeping the predictable level of the aviation risks in the acceptable margins [3, 4].

Analyzing the objectives of the aviation transport technologies elements functioning, it is possible to conclude that, in any case of the airlines operation, the management of the air companies pursues the goals that have some “positive” effects in the view of the managerial endeavors leading to the expected utilities of the individuals’ choice [5, 6].

The required resources recombination, when running an aviation business, also happens at the circumstances of the more or less uncertainties of the different kinds, such as e.g., the aircraft types used, their number, applicable aviation transport technologies elements resources etc.. Those kinds of uncertainties can be evaluated with the use of the entropy approaches, likewise in the references of [7 – 9], having become very popular recently [10].

In conjunction with the economical models of [11], the entropy methods of [7 – 10] resulted in the subjective analysis theory [12], which allows solving various types of the applicable problems, similar to the stated in the references of [13 – 16].

Air transportation management and aviation transport technologies elements resources recombination basic problems have already been set in [17 – 20] implementing the provisions of the previous researches [1 – 12].

According with the described concepts of [1 – 12, 17 – 20] the scientific gap, which needs its resolution, is an elaboration of a plausible theoretical approach to the actual and important problems of the rational aviation transport technologies resources recombination setting, 0000-0002-6846-9660 (A. V. Goncharenko); 0009-0006-2923-1752 (V. V. Iliushyn) © 2024 Copyright for this paper by its authors.

Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). especially in regards with the analytical decision making process supporting based upon the computer modeling advantages.

The problem setting of the presented study is about the theoretical research centered upon the rational aviation transport technologies resources recombination, following the reference of [18].

Thus, the goals of the paper are generally describe possible optimization of the resources recombination and computationally illustrate the theoretically obtained solutions. 2. Resources recombination possible optimization Optimization approach of the presented research is close, in the ideological considerations, to the conditional optimization, analogously to the problem settings implying the problem statements subject to the Lagrange uncertainty multipliers constraints. Nevertheless, herewith, the study uses acceptable simplifications that differ from variational problem of [18], yielding the same theoretical result, though, for the potentially possible resources distributions and combinations.

2.1. Basic concept

Amongst the multiple number of the aviation transport technologies elements is proposed to select and generalize such two competing ones that allow theoretically accumulate resources in considerations of the resources rational recombination.

In assumption that the mentioned above aviation transport technologies elements are determined in terms of the key resources, one can construct the following model, [18]: 1. The total amount of the aviation transport technologies services production: 2. The capacity/productivity of the first competing aviation transport technologies elements resource is ( 1 ) ( 2 ) ( 4 ) ( 5 ) N . p1 .

N = a , p1 N = b , p2 3. The capacity/productivity pertaining with the second competing aviation transport technologies elements resource is p2 . ( 3 )

Obviously, using the introduced parameters of ( 1 ) – ( 3 ), one can obtain the following equations of the stationary case:

where a symbolizes the number of the resources of the first nature required for the aviation transport technologies services production in order to realize the air transport technologies services total amount of N : parameter ( 1 ), on condition of the first nature resources capacity of p1 : parameter ( 2 ).

Analogously to ( 4 ), there is a possibility to describe the stationary process as where b portraits correspondingly the number of the resources of the second kind required, on the competing bases, for the aviation transport technologies services production in order to realize the air transport technologies services total amount of N : parameter ( 1 ), on the condition that the second type resources capacity is of p2 : parameter ( 3 ).

The discussed generalized case of ( 1 ) – ( 5 ) has a well-known linear solution [18] for a possible combination of the resources of both types.

Indeed, let us say the total amount of N : ( 1 ), is realized/produced with/by the resources of the two types together (simultaneously). That means N = N1 + N2 , ( 6 ) ( 7 ) ( 8 ) where N1 is the portion of the total amount realized/produced with the help, or by the use, of the first competing kind of the resources; and N2 is the rest of the total amount realized by the second type resources correspondingly.

Then, according to ( 2 ) and ( 3 ), where n1 is the number of the units of the first competing kind of the resources that took part in the realization/production of the corresponding portion of N1 : ( 6 ), of the total amount of N on the stationary condition of their production/capacity abilities of p1 : parameter ( 2 ).

On the other hand, likewise in ( 7 ),

At the fixed values of ( 1 ) – ( 3 ), the relation of ( 9 ) allows researching dependencies between two variables of n1 and n2 .

Considering one of those variables, for example n1 , as the independent variable, the other n2 can be expressed from ( 9 ) as a function of it.

Namely, n2 (n1 ) = N - p1n1 . ( 10 ) p2 In terms of ( 4 ) and ( 5 ) the equation of ( 10 ) can be rewritten as [18, ( 5 )]:

b ( 11 ) n2 (n1 ) = b - n .

a 1 2.2. Optimization technique

The simplest solution of ( 1 ) – ( 11 ) opens a wide range of conditional optimization problems for the segment of [b, a].

Rational air transportation technologies resources recombination on condition of pertaining to the segment of [b, a] can be conducted based upon the options effectiveness assessments.

Using the options effectiveness assessments parameter in the view of P = D - R , ( 12 ) where P is the “positive” effect parameter value; D is the “developmental” component; R is the “regressive” component.

The models implemented for the equation ( 12 ) components can of different nature and content; similar to, for instance, the proposed in [18, ( 6 ) – ( 11 )] for the activities splitting.

As to the presented study Di = si pini , ( 13 ) where Di is the ith resource kind component, this refers to the corresponding feature of the additive factor for both resources, i.e.

D = D1 + D2 ; ( 14 ) si is the ith resource type component coefficient/function influencing the “developmental” components of the resources; pi and ni are the parameters of ( 1 ) – ( 5 ) and ( 7 ) – ( 11 ) in respect.

As to the values entering ( 13 ), those can be functions of different interdependencies. For the developed herewith computer simulation it is proposed the following where ksi is an augmentation factor/index (also possibly a coefficient/function), for the ith resource type component, scaling the effect of the parameter of ni change/variation; nij is the parameter of ni value symbolizing the “zero” effect for the “developmental” components.

Concerning the “regressive” component R of the dependence ( 12 ), there are a few contemplations of the next sort.

2 ( 16 ) R = е Ri ,

i=1 where Ri is, similarly to ( 14 ), the “regressive” component of R in respect to the resources taken into consideration.

However, in regards with the “regressive” components constructions, the assumed models emphasize their features as Ri = kRi ni , ( 17 ) where kRi is a growth factor/index (also possibly a coefficient/function), for the ith resource kind component, scaling the effect of the parameter of ni change/variation.

In turn, the escalation factor/index of kRi , of formula ( 17 ), implies the peculiarities of its own possibilities to change depending upon the ni change/variation.

The proposed model is kRi = k R(0i) + DkRi krnii , where k R(0) i is a boundary/basic value of the growth factor/index; DkRi is the range/diapason n of the growth factor/index possible change/variation; krii is the variated component of kri raised to the power of the parameter of ni value.

In order to investigate the uncertainty of the optimal recombination of the aviation transport technologies resources choice, it is proposed to apply the entropy approach developed and implemented in the works of [17, 18] based upon the entropy paradigm of [7 – 9, 12].

It implies the use of the relative hybrid pseudo-entropy function, i.e.

Hr (n1) = Hmax - H (n1) p1(n1)- p2(n1) , (19)

Hmax p1(n1)- p2(n1) where Hmax is the maximal value of the entropy, in the considered case it equals ln( 2 ) ; H (n1) m si = ksi Х(ni - nij ), j=1 ( 15 ) (18) (20) (21) (22) pi (n1) = е2ebPj (n1 )

, j=1 is the current value of the entropy defined by the expression of

2 H (n1) = -е pi (n1)ln[pi (n1)],

i=1 where pi (n1) are the preferences functions:

ebPi (n1 ) as ( 12 ).

2.3. Simulation

The initial data for computer modeling are as follows: N = 1000 , p1 = 10 , p2 = 20 .

where b is the cognitive parameter essential to forming the objective functional with the preferences functions [7 – 9, 12, 17, 18]; Pi (n1) are the resource-centered effectiveness, defined

Then, implementing the procedure of ( 1 ) – ( 11 ) one can obtain the linear solution results presented in the Figure 1.

n2(n1) 58 67 33 21 correspondingly depicted in the abscise n1 and ordinate n2 axes.

Now, the problem is to determine the optimal recombination of the resources along the solution segment of [b, a].

In order to find the desired optimal air transport technologies resources recombination one can use the models of ( 12 ) – (18).

Concerning the “developmental” components of D , the accepted and used data are following: ks1 = -1,

n11 = 0 , n12 = 100 , ks2 = -2 , n21 = 0 , n22 = 50 . (24) The curves plotted for the augmentation functions of si , calculated by the formulae of ( 15 ), are shown in the Figure 2.

2.5ґ10

33000 s1(n1)2000 s2(n1) s2(n1)1000 0 0

The third curve of s2(n1) , presented in the Figure 2, is plotted, as a phase portrait/curve (phase diagram) in the phase coordinates/space of s2(n1)- n2(n1) , depending upon the phase coordinate of n2(n1) .

In such circumstances as (22) – (24), the computer modeling with the help of ( 13 ) and ( 14 ) gives the results illustrated in the Figure 3.

33 67

The fourth curve of D2(n1) , presented in the Figure 3, is plotted, as a phase portrait/curve (phase diagram) in the phase coordinates/space of D2(n1) - n2(n1) , depending upon the phase coordinate of n2(n1) .

It is also shown in the Figure 3, the extremal values of D1 = 1.4814Ч106 and D2 = 7.4069 Ч105 which are ensured by the values of n1 = 67 and n2 = 33 correspondingly.

Those values are noticeable in the Figure 1 as well.

Next up are the “regressive” components of R .

The accepted data for the calculation experimentations with the parameters of the equations of ( 16 ) – (18) are following: k R(01) = 2 Ч103 , DkR1 = 6 Ч103 , kr1 = 0.954 , k R(02) = 3Ч103 , DkR2 = 8Ч103 , kr2 = 0.9 . (25)

Thus, the results of the conducted computer modeling with the equations of (18) and data of (25) are presented with the diagrams in the Figure 4.

The escalation factors/indices (as the functions of the aviation transport technologies resources units numbers) of kRi , of formulae ( 17 ) and (18), for kR2 , depicted as kR2(n1) in the Figure 4, are shown in both dependencies; namely, for both, with respect to the only, so far, independent variable of n1 and n2 (n1 ) . The latter case, the curve number three as kR2(n1) in the Figure 4, is symbolized with the phase portrait/curve (phase diagram) in the phase coordinates/space of kR2(n1)- n2(n1) , depending upon the phase coordinate of n2(n1) .

Then, the diagrams of the “regressive” components of R , computed with the help of the formulae of ( 16 ), and ( 17 ), are shown in the Figure 5.

2.4Ч1052.21.64 ..110055

1.92 .105 R(n1) 1.68 .105

The fourth curve of R2(n1), presented in the Figure 5, is plotted, as a phase portrait/curve (phase diagram) in the phase coordinates/space of R2(n1) - n2(n1), depending upon the phase coordinate of n2(n1) .

At last, the “positive” effect parameter values of P , computed by the equation ( 12 ) components, are illustrated in the Figure 6.

21 58

The last three curves of P(n1) , P1(n1), and P2(n1) , presented in the Figure 6, are plotted, as the phase portraits/curves (phase diagrams) in the phase coordinates/spaces of P(n1) - n2(n1) , P1(n1) - n2(n1), and P2(n1) - n2(n1) , depending upon the phase coordinate of n2(n1) .

It is visible from the plots illustrated in the Figure 6 that at a certain aviation transport technologies resources recombination (circumstances) the “positive” effect parameters values turn to the “negative” zone.

The optimal recombination of the aviation transport technologies resources is also depicted in the Figure 6, i.e. the extremal values of P(n1 ) = 1.7044 Ч106 are ensured by the values of the optimal recombination for n1 = 58 and n2 = 21 correspondingly.

This optimal recombination of the aviation transport technologies resources is shown in the Figure 1 as well.

Computer modeling of the uncertainty situation with the help of (19) – (21) yields the results represented in the Figures 7 – 9.

The data accepted for the simulation are as follows: b = 3.6 Ч10-6 . (26)

The resource-centered effectiveness: Pi (n1), are defined by ( 12 ), ( 14 ), and ( 16 ) as P(n1 ) = D(n1) - R(n1) = D1(n1) -R1(n1 ) + D2 (n1 ) -R2 (n1 ) = P1(n1) + P2 (n1 ) . (27) Preferences by (21) and entropy by (20) are shown in the Figures 7 and 8 respectively.

0.98 Prn1(n1) Prn2(n1) 0.02 0.5 1 0 0 0 33 73

0.5 50 n1 100 100

3. Discussion

The demonstrated approach of ( 1 ) – (21) with the accepted data of (22) – (27) has a number of the generalized simplifying, although plausible, assumptions.

The simplified models make it easier finding the aviation transport technologies elements resources optimal recombination. It is accepted undisputable that the aviation transport technologies services production resources have their own subjectively preferred rational use as that was claimed in the references dealing with the individuals’ choices, like in [5, 6], with respect to the economical issues marked at [11], subjective individual preferences uncertainty emphasized in [12].

One of the significant simplifications of the discussed model presented herein with this paper is the stationary condition. Dynamics of the resources optimal recombination must inevitably have its own consequences. Other important simplifications of the study are the suppositions of just one independent variable n1 and independence of the resources capacities of pi : ( 2 ) and ( 3 ) upon it. Introducing other independent variables, e.g. n2 , as well as implying all other parameters dependences upon ni in more elaborated model constructions, it will predictably change the models relations ( 1 ) – ( 11 ) and results of simulation shown in Figures 1 – 6.

Since it has been supposed the additive properties of ( 6 ), ( 9 ), and ( 12 ), and their components, both in respect to the generalized resources types and kinds of the effect parameter values, the expected solution will be complicated as well if all that have possible multiplicative effects (see and compare Figures 2 and 4).

17 73

Models of ( 13 ), and their components can differ in mathematical expressions. Herein it has been studied the “parallel” models, identically mathematically expressed.

Multiplicative effects are probable with the consideration of the individuals’ subjective preferences [12] in combinations with the dynamical issues. The study of the subjective preferences entropy influence upon the desired optimal air transport technologies resources recombination looks like a huge separate problem with the internal applicative incentives.

In this sense, the potential challenges and limitations associated with the implementing the proposed model in the diverse aviation contexts can be applicably demonstrated in the optimal distributions of the aircraft numbers of a certain or specified types over the airlines’ fleets. Such problems are supposedly prospective in the studies that are going to be prolonged in the future considerations dealing with the problem settings of conditional optimization.

Thus, the rational recombination may not belong with the solution presented in the Figure 1.

In conditions of the stated problem the solution with the values of n1 = 67 and n2 = 33 is impossible since it is not in the solution fragment (see Figure 1). It happens because that is not in the compliance with the theoretically developed model dependencies ( 1 ) – (21) and accepted calculation data of (22) – (25) too; although the mentioned above impossible solution delivers maximal values to the “developmental” components of D (see and compare Figures 1 and 3).

As to the uncertainty of the optimal recombination of the aviation transport technologies resources choice, it is worth saying that (19) – (21) used the parameter of (26) on the purpose of the contrast distinguishing between two alternatives of (27); the corresponding values of the preferences functions and entropy are visible in the Figures 7 and 8.

Herewith, it should be noted that the traditional view entropy of (20): [7 – 9, 12], is incapable to distinguish the directions of the certainty or uncertainty between the alternative preferences around the designated points (see Figure 8).

Such a lack of the informative deficiency is supposed to be compensated with the relative hybrid pseudo-entropy, for instance, around the argument values of “17” and ‘73’; “bad”, that is “negative”, vs. “good”, for the represented case study computer modeling “positive”, relative certainty (see Figure 9). In the illustrated interpretation zero value of the pseudo-entropy corresponds to the situation of the complete uncertainty.

4. Conclusion

The developed theoretical approach allowed discovering the rational aviation transport technologies resources recombination as a kind of an optimal solution in regards with the generalized values. The segment of the linear solution has a maximal value of the generalized “positive” effect parameter in the framework of the accepted suppositions for the technologies fixed production. Computational intelligence and modeling are the indispensable aids in the optimization techniques for taking into account the operational alternatives subjective preferences uncertainty.

The dynamics of the desired optimal air transport technologies resources recombination selection process as well as some more developed models implementing operational alternatives subjective preferences entropy are proposed to be investigated in the further studies. https://ieeexplore.ieee.org/abstract/document/8943902.

doi:10.1109/APUAVD47061.2019.8943902. [18] A. V. Goncharenko, Relative pseudo-entropy functions and variation model theoretically adjusted to an activity splitting, in: Proceedings of the IEEE International Conference on Advanced Computer Information Technologies, ACIT’2019, Ceske Budejovice, Czech Republic, 2019, pp. 52–55. https://ieeexplore.ieee.org/abstract/document/8779876. doi:10.1109/ACITT.2019.8779876. [19] A. Goncharenko, Development of a theoretical approach to the conditional optimization of aircraft maintenance preference uncertainty, Aviation 22 ( 2 ) (2018) 40–44. https://journals.vilniustech.lt/index.php/Aviation/article/view/5929.

https://doi.org/10.3846/aviation.2018.5929. [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.

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] A. M. B Pavani et al., "VISIR+ Project Follow-up after four years: Educational and research impact," 2023 IEEE Frontiers in Education Conference (FIE), College Station , TX, USA, 2023 , pp. 1 - 8 , doi: 10.1109/FIE58773. 2023 . 10343298 .

[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 .