Justification of the Optimal Composition of the Air Defense Troops Grouping

The article deals with the problem of determining the number of antiaircraft missile systems of various types in grouping of Air Defense troops for the purpose of providing the required effectiveness of repelling the enemy air strikes on the objects and troops. The methodology of justification of ratio of different types of antiaircraft missile systems in the grouping of Air Defense troops using Lagrange’s method of undetermined multipliers has been developed. This technique allows justifying a set of types of antiaircraft missile systems which will provide maximum effectiveness of the employment of grouping of Air Defense troops. This approach can also be used to justify the composition of mixed military formations of Air Defense.


Introduction
Typically, antiaircraft missile systems (AAMs) of different types which differ in tactical and technical characteristics [1][2][3] are usually used to form a grouping of Air Defense (AD) troops. It is well known that the effectiveness of the employment of the grouping of AD troops to repel air strikes on the objects and troops, as well as any system designed to engage the objects (targets), nonlinearly depends on the number and types of antiaircraft missile systems in it. It differs in different types of AAMs [2].
There is a well-known regularity: the more effective the functioning of the system, the more means are necessary for its further increase [2]. That is, it can be assumed that there is such a ratio of different types of AAMs that can provide the maximum effectiveness of employment of the grouping of AD troops. Therefore, the

Preliminaries and Related Works
It is evident that the main task of the grouping of AD troops is to provide the required effectiveness of repelling air strikes on the objects and troops [1][2][3][4][5][6][7][8][9][10][11][12]. Thus, in order to provide such effectiveness in creation of the grouping of AD troops, the task of determining the required number of forces and means, in particular the AAMs, is relevant. The required amount of AAMs to repel enemy air strikes is usually determined by the results of evaluation of the required effectiveness of AD troops.
In the papers [1,[3][4][5][6][7][8] an analysis of existing and perspective air defense systems was carried out. However, the methodological theses for estimating the correlation of different types of AAMs in the grouping of AD troops are not considered. In [2] the task is to determine the rational number of target channels of various types of AAMs in the grouping of AD troops, depending on the effectiveness of their combat employment. However, the methodological guidelines for determining the number of different types of AAMs are not given in the work.
In [9,10] the theoretical bases of synthesis of adaptive structures of the antiaircraft missile artillery system support are described. In addition, questions of determination of composition and operational structure of AD grouping, structure of combat orders of units (forces) of antiaircraft missile troops (AAMt) are examined. In [11], the methodology of rational structure of the antiaircraft missile and artillery support of the AD grouping of the Land forces in the Operational zone (area) is given. This method provides prediction of the effectiveness of the antiaircraft missile and artillery support, as well as taxonomic evaluation and direction of the formation of the most effective vector of parameters of the antiaircraft missile and artillery support of the AD grouping of the Land forces in the Operational zone (area), which defines a rational version of the structure of the antiaircraft missile and artillery support of AD grouping of the Land forces in the Operational zone (area). In [12] a methodology for pre-justification of the required composition of the AAMs of grouping of Air Defense troops, based on the compliance of the combat capabilities of the AAMs with the number and capabilities of the means of air attack expected in the strike is suggested. However, these techniques do not take into account the possibility of increasing the required efficiency of AD due to the ratio of AAMs of different types in the grouping of AD troops and, as a consequence, the possibility of reducing their number. Thus, the task of determining the ratio of different types of AAMs in the grouping of AD troops to provide the required effectiveness of the repelling the enemy air strikes on the objects and troops requires further research.
The purpose of the article is to develop the main provisions of the methodology for justification of the ratio of various types of AAMs in the grouping of AD troops.

Methodology Description and Basic Mathematical Equations
The mathematical expectation of the number of targets that can be destroyed by the grouping of AD troops is determined by the amount (1) It is necessary to define a set of AAMs of each i-type i x ( n і , 1 = ), which provides maximum mathematical expectation M with the total number of AAMs in the grouping of AD troops ( In this case, the following condition must be met: . This problem can be solved by enumerating of possibilities of xi, where a large number of variants of the grouping of AD troop, as well as considerable time, are required to be examined. Taking into consideration the nonlinearity of dependencies Mi(xi) = f(xi), Lagrange's method of undetermined multipliers can be used to solve such a problem [13][14].
The block diagram of the methodology for justifying the ratio of different types of AAMs in the grouping of AD troops is shown in Fig. 1.
The process that takes place in the air defense system during the repulsion of an air strike is usually considered as a continuous Markov chain (the Markov process with discrete states of the system and continuous time). So determination of the mathematical expectation of the number of targets that can be destroyed by the AAMs of i -type Mi(xi) when changing their number in the grouping of AD troops may be accomplished by using mass service theory methods, in particular the method used to evaluate the mass service system with refusals. In such case, according to [15][16], the mathematical expectation Mi(xi) for AAMs with a small defeat zone is determined by the formula 58 DOI 10.3849/aimt.01303   To use Lagrange's method of undetermined multipliers in order to determine the optimal ratio of the different types of AAMs in the grouping of AD troops, it is necessary to have the functions Mi(xi) = f(xi). It is known that in natural science many regularities are expressed by means of power functions y = ax b . The power function can be approximated by the mathematical expectations of the number of targets that are destroyed by the AAMs of i-type (Tab. 1). Then the mathematical expectation is Function Eq. (4) for 1 > bi > 0 is convexity in the top. Lines that correspond to the power function are sometimes called polytropic lines.
The selection of the power function can be done in the form of equal sums [17]. To do this, all values (points) xi are divided into two equal groups (in our case, in two groups by N/2 = 5 points). A system of equations is formed ( ) ( ) According to the Tab. 1 for the first type of AAMs which has tfir,1 = 1 min the system of equations has the form ( ) The indicator of the degree of the selected function bi is determined by dividing the equations of the system Eq. (5). This excludes the unknown ai. So we get the function For our case we obtain the equation   It should be noted that the dependence of the mathematical expectation on the number of targets that may be destroyed on the number of AAMs of i-type in the grouping of Air Defense troops can be approximated by functions of another type.
Applying Lagrange's method of undetermined multipliers to find the extremum of a function ( ) The system of differential equations for finding the conditional extremum of a function This system is supplemented by the equation In this way n + 1 equations are obtained, which allows solving the problem with n + 1 unknown xi, λ.
Lagrange's method of undetermined multipliers gives only the necessary conditions for the existence of a conditional extremum for a continuous function. Therefore, the value of the obtained functions must be checked using the sufficient conditions of the extremum of the function of many variables. For this purpose, the second derivative functions are analyzed, on the basis of which the fulfillment of the Sylvester condition [13,18] is checked.
Adequate conditions for the function extremum existence are determined by the results of an analysis of the matrix A = (aij) whose elements are coefficients aij, that are determined by the relation The values of ∆i are determined from the matrix A = (aij): 11 By the Sylvester conditions [17], if all values are ∆i > 0, the acquired values of the variables respond the function's minimum. If ∆i have an odd order, ∆1, ∆3, ∆5, … < 0, and a pair order ∆2, ∆4, ∆6, … > 0, then the values of the obtained variables respond to the function's maximum. If the conditions above are not fulfilled, then the function has neither a maximum nor a minimum at the point that is considered.
As an example, let's examine the grouping of Air Defense troops, in which there can be three types of AAMs which are 2, 3, 4 (Tab. 1).
Thus, in the grouping of AD troops, it is advisable to have three second-class AAMs, four AAMS of the third type, and three AAMs of the fourth type. Such a ratio of the AAMs types provides the maximum potential effectiveness of the grouping of AD troops. According to Tab. 1, the value of the extremum of the function ( ) The effectiveness of the grouping of AD troops, which can be realized during combat operations, is determined taking into account the coefficients of participation Kpart.i Eq. (1), which refers to the proportions of the AAMs of i -type, which take part in repelling the enemy air strikes. It is believed that the magnitude of the coefficient Kpart.i is proportional to the ratio of the total area of the defeat zones of the AAMs of i -type to the area of responsibility of the grouping of AD troops [9]. The graphical representation of the coefficient Kpart.i is shown in Fig. 3.

Fig. 3 Graphical representation of the coefficient Kpart.i
With an equal distribution of the means of air attack on the objects in the area of responsibility of the grouping of AD troops, Kpart.i can be determined by the formula 2 tch, part.
As an example that is under consideration, when R2 = 10 km, R3 = 15 km, R4 = 20 km, and Lw × Ld = 50 × 50 km. By the Eq. ). This approach can also be used to justify the composition of mixed military formations of Air Defense.

Conclusions
The method of justification of the ratio of different types of AAMs in the grouping of Air Defense troops is proposed, taking into account the nonlinear nature of the change in the effectiveness of their number employment. The nonlinearity of changing the mathematical expectation of the number of targets that is destroyed by a particular type of AAMs is determined by the known dependence which is used to evaluate the efficiency of functioning of the multi-channel mass-service system with refusals. The optimal ratio of different types of AAMs in the grouping of Air Defense troops is determined using Lagrange's method of undetermined multipliers. The developed methodology should be used to justify the composition of the grouping of Air Defense troops, which may include different types of AAMs. In the future, it is advisable to improve the given method of obtaining dependencies of the effectiveness of the employment of different types of AAMs taking into account their number in the grouping of Air Defense troops.