Methodical Approach to the Development of a Mathematical Model for Calculating the Combat Potentials of Strike Unmanned Aircraft Dmytro Ikaiev1, Yaroslav Yaroshenko2, Andrii Shalyhin3, Volodymyr Nerubatskyi4, Valerii Bondar5, Volodymyr Herasymenko6 1,2,5,6 The National Defence University of Ukraine named after Ivan Cherniakhovskiy, 28 Povitroflotskyi avenue, Kyiv, 03049, Ukraine 3,4 Ivan Kozhedub Kharkiv National Air Force University, 77/79 Sumska street, Kharkiv, 61023, Ukraine Abstract The article considers the issue of assessing the combat potential of a strike unmanned aerial vehicle. An analysis of existing methods for assessing the combat potential of manned aircraft and found that they often use methods of expert assessment, which require a significant number of experienced experts and are quite time consuming. The scientific and methodical apparatus for calculations is given: probabilities of unmanned aerial vehicle damage for one departure and for a certain number of departures; mathematical expectation (average number) of combat sorties; mathematical expectation (average value) of the number of single targets hit in one combat flight by an unmanned aerial vehicle; the maximum value of the mathematical expectation (average value) of the relative number of single targets hit in one combat flight by an unmanned aerial vehicle; mathematical expectation of the number of single targets hit by an unmanned aerial vehicle; maximum mathematical value expectations of the number of single targets hit by an unmanned aerial vehicle for the entire period of life; the coefficient of combat potential of the unmanned aerial vehicle; the average combat potential of an unmanned aerial vehicle. The construction of a mathematical model of combat potentials of strike unmanned aerial vehicles based on a block-hierarchical approach is carried out. In this approach, the mathematical model of the modeling object is not represented as a function of many variables, but as a hierarchy of models of much smaller dimension. The basis for building a hierarchy of models is the physical content of the modeling object and the patterns it reflects. Keywords 1 group of manned and unmanned aerial vehicles, combat potentials of unmanned aerial vehicles, indicators of combat effectiveness, unmanned aerial vehicle, methods of calculating combat potential. 1. Introduction cooperation with manned aircraft. In addition to reconnaissance tasks and individual tasks of combat support of manned aircraft, strike tasks of In the field of unmanned aerial vehicles, there UAVs are becoming increasingly important [1-6]. is a transition from the single use of unmanned This raises the questions of the assessment of the aerial vehicles (UAVs) to the group (mass) use in combat potential (CP) of strike UAVs. III International Scientific And Practical Conference “Information Security And Information Technologies”, September 13–19, 2021, Odesa, Ukraine EMAIL: chvtau@gmail.com (A. 1); yar_yaroshenko@ukr.net (A. 2); shalyhin_andrii@ukr.net (A. 3); ner1976@ukr.net (A. 4); valerii.bondar.ua@gmail.com (A. 5); gera410@ukr.net (A.6) ORCID: 0000-0003-4161-7579 (A. 1); 0000-0002-8651-4920 (A. 2); 0000-0002-1828-2443 (A. 3); 0000-0003-3090-1865(A. 4); 0000-0001-8843-680X (A. 5); 0000-0003-2014-7408 (А. 6) © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) The problem is that the methods of estimating other types of maintenance and by other of the CP of UAVs, which are similar to the operational factors. methods of estimating of the CP of manned The construction of mathematical models of aircraft, have not found appropriate distribution. aircraft CP, which take into account a fairly The known methods of estimating of the combat complete list of aircraft characteristics and potential of the manned aircraft are based mainly conditions of their use, proved to be quite a on the methods of expert assessments [7-10,24]. problematic task. An alternative solution of this This is a quite time-consuming process and problem remains the methods of expert requires the involvement of a significant number assessments. of experts with experience in combat use of The purpose of the article is to determine the aircraft. The application of such methods for a approach to construction of a mathematical model large number of existing and perspective UAVs is for calculating the combat potential of a strike problematic and practically impossible to UAV without the involvement of expert or other implement. Therefore, there is a need to develop “fuzzy” information. approaches to the assessment of the CP of UAVs, which do not require the use of expert assessment 2. Presentation of the main material methods. Analysis of the recent researches and publications The difficulty of constructing analytical In [7] it was noted that in the late 50s of the last mathematical models for calculating the CP of aircraft functionally related to the characteristics century the military authorities had the need for a of aircraft, their weapons and parameters of simple and clear way to compare different types combat conditions, can be attributed to the non- of weapons to solve combat tasks and correctly integrability of the vast majority of systems of calculate the balance of forces of the parties in operations. This was due to the fact that with the differential equations [11,23]. Including differential equations that adequately describe the increase in the variety of weapons and their fighting. Regressive mathematical models, which increasingly narrow specialization, it became can be built on the basis of mathematical almost impossible to determine the ratio of forces modeling data (numerical experiments) or expert by the ratio of individual types of weapons. It was assumed that different types of weapons could be survey data, have significant shortcomings and compared in terms of contribution to the end result limitations [12]. They do not provide a full- fledged replacement for analytical models. In this of hostilities, and therefore each of them could be article, the construction of a mathematical model assigned a weight efficiency factor. Over time, of CP of UAVs is based on a block-hierarchical this factor has been defined as the combat (decomposition) approach. In this approach, the potential (CP) of the sample of armaments. CP of the sample of armaments could be considered as a mathematical model of the modeling object (CP of UAV) is not represented as a function of many criterion of the totality of samples of armaments variables, but as a hierarchy of models of much for its contribution to the achievement of the smaller dimensions. The basis for building a objectives of an operation (hostilities). hierarchy of models, as will be noted below, is the Initially, the СP of armaments samples were determined either empirically, based on statistics physical content of the modeling object and the patterns it reflects [13]. obtained from past wars (armed conflicts) [7], or The concept of “combat potential of a sample by methods of expert evaluation. of weapons”, judging by its various definitions In the 80s years of the last century the methods [14,21], still remains controversial. Below, for of mathematical modeling of hostilities [8] began example, there are two different definitions of to be used to determine the CP. The special studies conducted on mathematical models of combat “the combat potential of a sample of weapons”. Combat potential of a sample of weapons is an operations have revealed that there is no constant integral indicator that characterizes the maximum uniform measure of comparison of different types set of tasks performed by the sample weapons and of weapons. CP of a sample of armaments – is a military equipment (WME) for the intended variable value and it is determined not only by its characteristics, but also by its quantity, structure purpose in the implementation of the limit tactical and technical characteristics (TTC) for the typical of armaments of confronting groups, type of operating time in typical design conditions [15, operation, quality of management, combat and 16,22]. The combat potential of a sample of weapons The number of combat sorties that a UAV can is an integral indicator that characterizes the perform before its defeat is a random variable. maximum amount of combat tasks that can When performing n combat sorties UAV will be perform a sample of weapons for its functional struck with a probability of : purpose in the given (calculated) conditions of use 1 1 (2) during its existence [17,20]. Mathematical expectation (the average In the definition [17,20], the most significant number) of combat sorties is determined by the difference from the definition [15] is that the key ratio of this probability to the probability of defeat feature is not a vague feature – the characteristic in one combat sortie: time, but the time of existence of the sample of 1 1 weapons before its defeat. In particular, if the (3) characteristic time is taken as a time interval that For multiple UAVs n>>1 і n and is reduced is less than the lifetime of the sample, the combat to the inverse probability : potential will be determined essentially by the fire 1 performance or firepower of the sample. But the (4) concept of “fire performance” reflects the meaning of a different indicator than “combat The mathematical expectation (average value) potential”, because it does not take into account of the number of single targets hit in one UAV the ability of the weapon to survive in the face of combat flight is determined by the number of the enemy and continue to function. successful target attacks during the combat flight. As shown in [13,14], the overall purpose of the It is limited by the number of means of destruction operation of weapons at the highest level is in combat charge. It is assumed that the divided into two partial tasks – the failure of ammunition of the aircraft consists of the same enemy targets and maintaining the functioning of type of means of destruction, and launches on one their own means. This follows from the basic law target is carried out by only one means of of armed fight. The indicator that describes the destruction. This simplification is not first task can be “fire performance” or “firepower” fundamental and allows you to reduce the of the weapon. The indicator that describes the recording of basic expressions in the article. second task – “survivability”. The other properties Expression for mathematical expectation (average of the weapon are the means to achieve the main value) of the relative number of single targets hit properties. in one UAV combat flight: It is the indicator “combat potential of the sample of weapons”, which combines the ∙ ∙ (5) indicators of “firepower” and “survivability” should be used in conceptual research on the where – the number of single potential formation of basic requirements for UAVs on a targets of the j-type, hit in the k- complex criterion of “combat potential – cost”. combat flight of the UAV by The model of combat operations of reusable means of the i-type; UAVs can be thought of as a series of repetitive – mathematical expectation of the combat sorties in each of which it hits a number number of single targets of the j- of targets. Each subsequent flight can be type, struck in the k-combat performed provided that the previous ones were flight of the UAV by means of performed. the i-type; Suppose that in the process of performing a – the probability of fulfilling the combat sortie UAV is exposed to fire from the conditions preceding the launch enemy with an intensity of λ, which leads to its (reset, etc.) of the means of defeat in one sortie with probability defeat of the i-type on the target [18,19]. Assuming the Poisson nature of fire of the j-type: detection and effects, this probability is determined by: recognition of the target by 1 (1) external means, long-range where – conditional probability of the guidance, target detection by sample damage under one own means of UAVs, target exposure; attack. Depending on the – duration of the combat flight problem to be solved and the method of using the UAV, the composition of the stages of – the average number of single preparation for the launch of the potential targets of the j-type, means of destruction may differ affected by means of the i-type from the above. For example, in on departures. the case of an autonomous If the CP of UAV is already known, which can method of UAV application, the be taken as a reference, it is more convenient to stages of external targeting may use the CP coefficient instead of the CP. It is be absent, and target detection determined by the ratio of the CP of UAV to the and recognition may be carried reference CP of UAV. It is assumed that UAVs out by its own means; are used in similar conditions. In this case, – the probability of defeat by one expression (9) is simplified because the variables means of defeat of the i-type of are reduced: the j-type target; – the number of means of defeat of ∙ ∙ ∙ (10) the i-type AV used in the k- combat flight; The CP coefficient has a clear physical The maximum value of the mathematical meaning. It is reduced to the product of the ratios expectation (average value) of the relative number of the indicators of effectiveness of the of single targets hit in one combat flight of UAVs destruction means, the number of means in the is determined by the expression: ammunition and the inverse ratio of survivability. Model (10) can be applied during researches at ∙ ∙ (6) the initial stages of UAV creation (external design) when searching for a design compromise where – the total number of means between the combat potential and the cost of of defeat of the i-type UAVs. aircraft, used in the k- To optimize (select) the options of technical combat flight. and design solutions of UAVs, it is necessary to For the entire period of the UAV's life, ie use the dependences of the generalized indicators during n combat sorties, the mathematical , , , of the TTC of expectation of the number of single targets hit by UAVs. Such dependences should be considered as the UAV: components of mathematical models of CP. in (10) is an element of the matrix, (7) where the types of means by which the UAV will hit the enemy's targets are indicated in the lines, The maximum value of the mathematical and the types of targets - in the columns. expectation of the number of single targets hit The matrix (10) is similar to the matrix of during the entire life of the UAV, ie during n efficiency of application of different models of combat sorties by definition is the UAV CP as to weapons in different conditions. The use of the destruction by i-type means of the j-type targets: data contained in the matrix (10) depends on the objectives of research and the method of decision- (8) making based on them. For example, in comparing of = ∙ ∙ ∙ ∙ or different UAVs, several different approaches can be used to select the best option. The simplest approach is to collapse the matrix (9) (10) into a scalar quantity. Then the average CP of UAV is determined: = ∙ ∙ ∙ 1 ∙ ∙ ∙ (11) where – the average value of the number ∙ of means of aircraft destruction (combat kits) of i-type on where – the number of types of UAVs departures; N destruction; M – number of types of targets. – weight factor that determines should be considered as components of the relative frequency mathematical models of CP. (probability) of use of weapons The considered methodical approach to of the i-type, development of mathematical model of combat ∑ 1; potential of strike UAVs allows to build – weighting factor that mathematical models for conducting researches of determines the relative part military and economic efficiency of strike (probability) of the targets of unmanned aerial vehicles without involvement of the j-type that will be expert or other “fuzzy” information. affected, ∑ 1. Weight multipliers , are determined by 4. References the typical composition of enemy targets and weapons of strike UAVs [10]. 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