Development of a Method for Deciding on the Distribution of Efforts to Destroy the Air Enemy Between Anti-Aircraft Missile Forces and Fighter Aircraft Maksim Pavlenko 1, Dmytro Kalinovskiy 2, Nataliya Garmash 3 1,2,3 Ivan Kozhedub Kharkiv National Air Force University, Sumska st. 77/79, Kharkiv, 61023, Ukraine Abstract The publication shows the possibility of using the mathematical apparatus of the multicriteria method of analysis of hierarchies in conjunction with the theory of fuzzy sets, at the stage of assessing the significance of selected parameters that affect the distribution of forces and means in accordance with the conditions of the task, during the task. Binary operations of symmetric difference and drastic sum are used to determine the set of alternatives for the joint use of fighter aircraft and anti-aircraft missile forces, taking into account the projected loss estimate. The conducted mathematical apparatus makes it possible to form a set of rules on the basis of which it is possible to make a decision on the distribution of efforts to destroy the air enemy. Using the method of analysis of hierarchies at one of the stages uses fuzzy logic to form a measure of evaluation of the parameters being evaluated, but the end result has an exact value that requires recalculation when changing at least one of the parameters. Instead, using set theory, namely the operation of symmetric difference and drastic sum, it is possible to determine the set of values that forms alternative decision-making depending on the result obtained. That is, the target that will be in a certain range will be immediately distributed relative to the fighter aircraft and anti-aircraft missile forces, without recalculating the parameters and their coefficients relative to each other. Keywords1 Method of analysis of hierarchies, fuzzy sets, evaluation criterion, symmetric difference, drastic sum. 1. Introduction number of tasks performed by air defense, namely [1, 2]: destruction of enemy air attack means; The experience of wars and armed conflicts of gaining and maintaining an advantage in the the twentieth and twenty-first centuries shows a air; significant increase in the use of air strikes [4]. destruction of unmanned aerial vehicles, The capabilities of air strikes are evolving along which are used in the conditions of hybrid with scientific and technological progress, which warfare, and represent an element of uncertainty allows them to solve not only tactical but also for a set of means of automation or automated strategic tasks. Instead, the use of air defense is control system and require a clear definition not only a defense system against air strikes but (shock or reconnaissance); also a deterrent, the impact of which is assessed in cover of important state and military facilities, the first place. The increase in the number of tasks their transformation into the national defense before the means of air attack correlates with the system of Ukraine; ISIT 2021: II International Scientific and Practical Conference «Intellectual Systems and Information Technologies», September 13–19, 2021, Odesa, Ukraine EMAIL: bpgpma@ukr.net (A. 1); kalinovskyidmytro@gmail.com (A. 2); garmash.n@ukr.net (A. 3) ORCID: 0000-0003-3216-1864 (A. 1); 0000-0003-3184-6458 (A. 2); 0000-0002-8840-2797 (A. 3) ©️ 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) increasing the level of combat capability of the (forces) but also an assessment of the relative Armed Forces of Ukraine and other components losses that can be achieved on the one hand and of the defense forces with the achievement and on the other, resource indicators and time maintenance of certain capabilities for fire defeat parameters from analysis of the situation to of the enemy. completion of the combat mission. The effectiveness of joint combat use of troops Determine the sequence of calculations using (forces) in repelling air strikes is achieved by the method of analysis of hierarchies (MAH) [5, realizing their combat capabilities under the 6, 7], which includes the steps: condition of coordinated interaction and 1. Selection of troops (forces) of fighter centralized management, a single automated aircraft and anti-aircraft missile forces in control system using the decision support system accordance with the existing ones; (DSS). It is also necessary to take into account the 2. Determination of sets of parameters that scale, nature of the actions of the means of air characterize certain troops (forces) (tactical and attack, combat composition, combat capabilities, technical capabilities; time of readiness to weather and geographical conditions. perform the task; means of destruction used; The decision on the distribution of efforts in training of personnel; preparation for re-use; the destruction of air between the anti-aircraft others); missile forces and fighter aircraft is made by the 3. The choice of quality indicators that are decision-maker on the basis of proposed compared (quality indicators in turn can be both alternative solutions to DSS. However, the local and global, which directly affects the priority decision-maker will use the received of the parameter); recommendations only if he trusts this system, 4. Calculation of the generalized criterion and his trust can be built on the degree of validity (global assessment for each fighter or anti-aircraft of the recommendations. missile system, fighter aircraft and anti-aircraft Therefore, at present, the question of choosing missile forces in general relative to others). a mathematical decision-making apparatus Since MAН is based on estimates of the degree remains relevant, with the possibility of of influence of lower hierarchy factors on the explaining the recommendations in DSS, which criteria and indicators of higher levels of the provides proposals for the distribution of efforts hierarchy, we assume that the more factors and in the destruction of air strikes between anti- indicators will be taken into account, the more aircraft missiles and fighter aircraft. accurate the final result will be. On the other hand, the assessment of the significance of some 2. The main part parameters may be insignificant in comparison with other estimates of parameters, which implies the effect of insignificance of the selected The distribution of efforts is carried out in parameters, which will give a negligible order to achieve the maximum effect in the advantage to the parameters that should not be destruction of air attack means by joint efforts, as considered at all. well as to ensure the safety of their aircraft. Given To select the optimal grouping of troops the fact that anti-aircraft missile forces can be (forces) that will be tasked, we use the algorithm used for ground (surface) purposes, it follows that of expert assessment of the impact of the the scope is common. characteristics of complex technical systems on Depending on the combat capabilities and the quality indicators, proposed by T. Saaty using a nature of the tasks to be solved, fighter aircraft and nine-point scale for comparing alternatives anti-aircraft missile forces share efforts: Tab. 1. However, this measure of comparison is  in space (in directions, boundaries, not a dogma and, if necessary, it is possible to use areas, strips, sectors, zones, altitudes); your own.  by time; If several experts take part in the assessment,  method of performing the task [3]. the assessment is agreed by consensus, or each The distribution of goals according to the expert builds his own table, and the estimates are forces that must be allocated for their destruction presented as geometric averages. is carried out by the decision-maker from a It is important to emphasize the importance of particular point of control. To make such choosing the right priorities, because in the case decisions requires not only a preliminary of a gross mistake, the priority may not be assessment of the conditions of use of troops important, but important will be ignored, which will lead to the accumulation of error or even error reliability, as well as take into account possible and making the wrong decision. Therefore, for a errors. more accurate assessment, you need to specify a more accurate set of input data, assess their Table 1 Comparison of alternatives Relative Definition Explanation importance Equal contribution of two indicators to the 1 Equal importance of indicators assessment A slight advantage of one over Experience and research give a slight advantage of 3 the other one indicator over another Experience and research give a great advantage of 5 Great advantage one indicator over another The advantage is so strong that it becomes 7 A significant advantage significant 9 Full advantage The obvious advantage is most fully confirmed 2,4,6,8 Intermediate values Used as an alternative After calculations, we obtain a matrix of unless they operate jointly in one area but priorities in accordance with certain conditions separated by sometimes. (known). This will allow you to make a decision Therefore, it would be rational to apply a on the distribution of effort on the basis of the symmetrical difference (drastic product A  B ) average value, or using special data analysis [9, 10, 11] of two sets where A is the set of targets software [8] to determine at what level of the that will be attributed to the targets of fighter hierarchy the defined indicators have the greatest aircraft and B is the set of targets that will be impact. attributed to the targets of anti-aircraft missile For example, using a color gradation of the forces. appropriate range. However, changing the Also ( A  B ) based on the definition conditions of the task, taking into account the ( A B) \ (A B) can be described as Fig.1., that risks or making a decision in conditions of is, a set that includes all elements of both sets that complete uncertainty requires a recalculation of are not common to the two sets or the priorities of the parameters on the basis of new ( A \ B) (В\ А) , a set that includes all elements input data. of the first set that are not included in the second Although the set of comparative alternatives is set, as well as elements of the second set that are used at the stage of expert evaluation in MAH, it not included in the first set. affects only the initial level of evaluation of coefficients, which does not allow to determine the set of alternatives of the final result. Using instead of the specified exact value, a fuzzy unlimited multivalued estimate, we can increase the set of values, all values of which satisfy the task and are within acceptable limits. However, when considering the distribution of efforts in one area, the security of their troops (forces) remains essential, so the set of targets set for fighter Figure 1: Image of the symmetric difference of aircraft should not intersect with multiple targets sets A and B using Euler circles that will apply to anti-aircraft missile forces, Using a system of algebraic equations can be Symmetrical difference, respectively. represented as follows: Given the safety of joint use, we define the set mB ( x), if mA ( x)  1 for x  X C - the estimated estimate of the loss of fighter  aircraft and D - the estimated estimate of the loss mAB ( x)  mA ( x), if mB ( x)  1 for x  X , (1) of anti-aircraft missile forces in joint use. Based 0 in other cases  on the above use, we must take into account that where X is the set of real numbers, mA ( x) is the the projected loss estimate will be relative to the set of common use, so similarly use the symmetric set of targets of fighter aircraft, mB ( x) is the set of difference and the drastic sum to determine targets of anti-aircraft missile forces. kCD ( x) and kCD ( x) where kC ( x) and kD ( x) - the From the point of view of the use of fighter set of projected losses. aircraft and anti-aircraft missile forces, it can be To determine the set of decision-making described as used jointly in one area but divided alternatives, taking into account the projected into ranges, in the purpose for which fighter estimate of losses when used together, consider aircraft are used, anti-aircraft missile forces are not considered and vice versa. max( AB ) , min( AB ) and max(CD ) , On the other hand, it is necessary to consider min( CD ) , in the figure it will look like a limit, the whole set of purposes for which fighter aircraft and its division is described by determining the and anti-aircraft missile forces can be used based maximum of the maximum allowable value and on their capabilities, given the limit of joint use to the minimum of the minimum allowable. We get: determine this set, we use a drastic sum ( AB ) max( AB )  max(max(mB ( x)) max(max(mA ( x)) (6) which can be written as: min( AB )  min(min(mB ( x)) min(min(mA ( x)) (7) mB ( x), if mA ( x)  0 for x  X Accordingly, for max(CD ) , min( CD ) . Using  mAB ( x)  mA ( x), if mB ( x)  0 for x  X , (2) Euler's circles, the set of decisions takes the form 1 in other cases of Fig.2  In the case of the use of a symmetric sum, a set of purposes became known which can be considered for joint use, and in the case of the use of a drastic sum using the previously described max( AB ) , we define the boundaries of joint use, which can be described as: max(mB ( x)), if min(mA ( x))  0 for x  X  mAB ( x)  max(mA ( x)), if min(mB ( x))  0 for x  X , (3) 1 in other cases  then the symmetric difference can be written as: min(mB ( x)), if max(mA ( x))  1 for x  X Figure 2: A variant of many decision-making  alternatives mAB ( x)  min(mA ( x)), if max(mB ( x))  1 for x  X , (4) 0 in other cases  To describe the knowledge for decision- Using a symmetrical difference and a drastic making based on the given set of values, we use sum to determine the boundaries of alternatives to the production model [12, 13]. In our case we can the decision to use fighter aircraft and anti-aircraft form the following rules: missile forces, there are other cases in which joint 1. If max( AB ) max(CD ) then use fighter use will take place. Let us denote the joint application by  , since aircraft; it is necessary to consider all possible cases, then 2. If min( AB ) min(CD ) then use anti- the system of equations of the drastic sum takes aircraft missile forces; the form: 3. If max( AB ) min( AB ) then joint use. mB ( x), if mA ( x)  0 for x  X Otherwise, it is possible to form a set of  hierarchically dependent rules depending on the mAB ( x)  mA ( x), if mB ( x)  0 for x  X , (5)  level of the production model, or the level of  AB hierarchy of the method of analysis of hierarchies. 3. Conclusions [9] Mathematical apparatus of artificial intelligence in power systems / Kirik V. V. - Kyiv: 2019. - 224p. 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Laboratory of Knowledge, 2013. - minimal costs and maximum planned effect. 798 p.: ill. - (Adaptive and intelligent Using the method of analysis of hierarchies at systems). - ISBN 978-5-9963-1495-9 one of the stages uses fuzzy logic to form a [12] Estimation of reliability of decision-making measure of evaluation of the parameters being methods on the generalized indicator of evaluated, but the end result has an exact value quality of production with use of fuzzy sets / that requires recalculation when changing at least Zubretskaya N. A, Fedin S. S // Information one of the parameters. Instead, using set theory, processing systems. - 2015. - № 5 (130). - P. namely the operation of symmetric difference and 93 - 96. drastic sum, it is possible to determine the set of [13] Construction of a production knowledge base values that forms alternative decision-making using an adaptive neural network / depending on the result obtained. That is, the A. Savelyev // VESTNIK AGTU. - 2007. - target that will be in a certain range will be № 1 (36) - S. 144—149. immediately distributed in relation to fighter aircraft or anti-aircraft missile forces, without recalculating the parameters and their coefficients relative to each other. 4. References [1] Military Security Strategy of Ukraine. URL: https://www.president.gov.ua/documents/12 12021-37661 [2] Strategic Defense Bulletin of Ukraine / NSC of Ukraine 2013. – P. 22–23. [3] Toropchin A. Ya., Romanenko I. O. and Danik Yu. G. (2003), “Dovidnyk z protypovitrianoi oborony” [Air Defense Reference Book], Kyiv, 366 p. [4] Yermoshin M. O. and Feday V. M. (2004), “Borotba v povitri” [Fighting in the air], Kharkiv, 384 p. [5] Saaty T. L. Axiomatic foundation of the analytic hierarchy process//Management Science. 1986, July. – Vol. 32, №7. – P. 841- 855. [6] Saaty T. L. Decision making. “Hierarchy analysis method" М.: 1993.- 320 p. [7] Unmanned aerial vehicles: Methods of comparative assessment of combat capabilities / Mitrahovich M. M., Silkov V. I., Samkov A. V., Burshtynskaya H. V., Stankevich S. A., Semenov V. B. 2012. - 288 p. [8] Perform analysis of complex data. https://support.microsoft.com