MULTIPLE-CRITERIA MODEL FOR OPTIMAL ANTI TANK GROUND MISSILE WEAPON SYSTEM PROCUREMENT

Introduction: Anti Tank Ground Missiles (ATGMs) are one of the most efficient weapon systems to counter armored and mechanized units. Procurement of these weapons is imperative for armed forces of any country. Adequate evaluation and choice of an efficient ATGM system is a very important factor which affects operational capabilities of armed forces. The purpose of this paper is to present that multi criteria methods can be a useful tool for optimal procurement of ATGMs for armed forces. Methods: The implementation of the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and the MABAC (Multi Attributive Border Approximation Area Comparison) multi criteria methods will be proposed for decision makers to solve the problem of ATGM procurement. In this paper, four models of ATGMs will be proposed as possible solutions. These ATGMs will be presented as alternatives A1, A2, A3 and A4. Results: The implementation of the methods in this paper has led to the conclusion that the alternative A2 has the highest value and therefore the best course of action for decision makers is to chose procurement of this model of ATGMs. Conclusion: The conclusion is that the mathematical models of multi-criteria decision making presented in this paper unequivocally point to the alternative with the best characteristics, thus presenting extremely useful tools for decision makers.


Introduction
Anti Tank Ground Missiles (ATGMs) represent one of the most efficient weapon systems present in armed forces globally. The primary purpose of these systems is anti armor defense, as well as partial anti air defense against low flying targets (helicopters and unmanned aerial vehicles). High hit probability and long range of engagement place these systems among the most deadly battlefield threats to armor units.
Regarding the fact that procurement of ATGMs represents one of the imperatives for equipping armed forces, the most optimal selection of ATGMs is one of the most challenging tasks for state decision makers, i.e. officials of the Defense Ministry.
Operational research as a basic management discipline based on mathematical formalism, with multi criteria methods as fundamentals for a decision making process, represents a solution for the ATGM procurement problem.
The objective of this paper is to present the capabilities of the multi criteria model as a comprehensive method for a decision making process where there is a need to select the most optimal solution.
In order to represent the complexity of the ATGM procurement problem, the first part of this paper will give a short history of ATGM development as well as the ATGM basic tactical and technical characteristics. The tactical and technical characteristics of ATGMs represent just a small portion of criteria included in the decision making process. All criteria which are in accordance with the adopted national defense doctrine must be included. Also, one of the crucial criteria is the cost-effectiveness criterion.
The ATGMs which will be described in this paper are fictional. Their specifications are approximate to real weapon systems in usage. The data for ATGMs are collected by accessing available reference documents primarily by the examination of official manufacturers' weapons specifications. As we already mentioned, the presented data will be partially changed in order to avoid similarity with actual weapon systems.
In the second part of this paper, the TOPSIS & MABAC multi-criteria methods will be explained. These methods will be employed as a decision making tool for the ATGM procurement.
The motivation for selecting the TOPSIS and MABAC methods is justified by fact that these methods are comprehensive and represent one of the latest and most advanced scientific methods in the decision-making process.  , (Military-Today, 2021) Рис. 1 -9M14 «Малютка» (Military -Today, 2021) Слика 1 -9M14 "Маљутка" (Military -Today, 2021) to the missile and therefore guides the missile to the target. The communication with the guidance block is conducted by wire or by radio waves. Typical examples of this generation ATGMs are: MILAN, 9K111 "FAGOT"(NATO reporting: AT-4 Spigot), BGM-71 TOW, and "BUMBAR"developed and produced in Serbia. The examples are given in Figure 2. The two-point guidance method is also employed on ATGMs guided by a focused beam (usually a laser beam) where a missile itself has an installed system which provides "riding a missile on a beam" and therefore there is no need for wire communication with the launching platform. The typical examples of this generation ATGMs are the 9M133 Kornet and the 9M119 Svir/Refleks (booth manufactured in Russia). The hit probability of the second generation of ATGMs is about 90%.
The third generation of ATGMs completely employs an automated guidance system. In Western literature, this system is designated as a "Fire and Forget" system. The operator selects a target and after launching the automatic system tracks the target and guides the missile to the target. The typical example of the third generation of ATGMs is the 9M133M Kornet-M. The automated system conducts the target tracking and focuses the laser beam to a target, which is used to guide a missile. This ATGM presents an improved version of the second generation of the 9M133 ATGM. The launching platform is capable of tracking two targets simultaneously. Besides anti-tank engagement, the system is intended for low altitude anti-aircraft defense (e.g. against unmanned aerial vehicles). The 9M133M Kornet-M ATGM is presented in Figure 3. The photo represents a Kornet D variant which is capable of carrying 8 missiles (with additional 8 in reserve) on the GAZ 2975 all-terrain vehicle.
The FGM-148 Javelin ATGM, manufactured by Raytheon Technologies, USA, represents one of the most sophisticated combat systems. One of the distinctive characteristics featured in this combat system is ability to attack the target from above where the upper armor is much thinner than the frontal armor, thus significantly improving target destruction probability.
This missile employs a section designated as a Seeker section. The seeker section consists of an imaging infrared system (I2R) and a contact fuse for warhead detonation. The I2R system provides the Fire and Forget capability. During the missile flight, the I2R system tracks a target and provides data to the electronic unit installed in the missile. Figures 4 and 5 show the target acquisition. After locating a target, the operator turns on the narrow field of view which engages track gates and crosshairs projected in intervals, indicating that the seeker is detecting target thermal reflection, or so called "target lock". When the strongest thermal reflection is detected, the track gates and the crosshairs become solid, which is the indication that the target is "locked", i.e. the firing conditions are fulfilled and the system is ready to launch a missile ( Figure 5). VOJNOTEHNIČKI GLASNIK / MILITARY TECHNICAL COURIER, 2021, Vol. 69, Issue 4 . Figure 5 -Target "Locked" firing conditions fulfilled; system ready to launch a missile Рис. 5 -Условия стрельбы по «Закрытой мишени» выполнены Слика 5 -Циљ "закључан" -услови за гађање испуњени ATGMs are equipped with a warhead which destroys a target with an explosion cumulative effect 1 . A cumulative explosion effect represents focusing energy to a surface smaller than the outer surface of the explosive charge. Energy accumulation is created by a proper shape of the explosive charge. If the outer surface is shaped with a cone cavity (so called cumulative cavity), detonation products are focused in the center of the cavity. In that manner, energy accumulation is created on a smaller surface and therefore the effect of a larger magnitude is created.
The coating of the cumulative cavity also increases penetration capability. For the coating, the most used metal is copper, but steel can also be used as well as sintered metals. Figure 6 shows the formation of a cumulative beam while a cumulative beam formation radiograph is presented in Figure 7. Modern ATGMs employ a so-called tandem-cumulative warhead 2 which is capable of forming two cumulative beams following each other. These beams are formed by two separated cumulative cavities. The purpose of the primary beam is to activate Explosive Reactive Armor (ERA) 3 , and then the secondary beam penetrates the main armor. Modern third generation ATGMs penetrate more than 1,300 mm of homogenous armor behind ERA.
One of important characteristics of ATGMs is missile velocity. ATGM velocity is usually measured in meters per second (m/s) and it is ranged from 120 m/s (First generation ATGM e.g. 9M14 Malyutka) up to 300 m/s (Third generation ATGM e.g. ATGM 9M133M Kornet-M). It is evident that missile velocity is an important characteristic considering battlefield multiple targets engagement. From the previous example, it can be noticed that the third generation of ATGMs has the velocity almost twice as high as that of the first generation. Let us have an example when the operator locates two targets at a range of 2400 meters. For the guidance of the first generation of ATGMs, the operator needs approximately 20 seconds. For the guidance of the third generation of ATGMs, the operator needs approximately 8 seconds. This means that the operator armed with a third generation ATGM has enough time to destroy two targets, unlike the operator armed with a first generation ATGM.
The ATGM range of engagement is also an important characteristic. The ATGM range of engagement is form 3000 meters for the first generation ATGMs (e.g. 9M14 Malyutka) up to 8000 meters for the third generation ATGMs (e.g. 9M133M Kornet-M).
Regarding ATGM procurement for armed forces, the price of ATGMs is a factor which must be considered in conjunction with economic capabilities of the armed forces in question. Also, the ATGM weight must be considered in conjunction with armed forces transportation capabilities, as well as with the mobility of units armed with ATGMs.
In modern armed conflicts, electronic warfare (EW) is employed through system jamming, so electronic counter-countermeasures (ECCM) must be implemented, which for ATGMs can be expressed as jamming resistance. Besides jamming resistance, reliability is demanded for weapon systems.
Based on the previously mentioned ATGM development history and the presented tactical & technical characteristics, the following criteria should be taken into account when procuring ATGMs:  Engagement range,  Hit probability,  Weight,  Price,  Training simplicity,  Reliability,  Jamming resistance, and  Missile velocity.

Multiple-criteria decision process
In the previous chapter, the problem of ATGM procurement for armed forces has been presented. Finding an optimal solution, i.e. the decision making process and the selection of the "best" alternative, is based on multiple criteria and on series of limitations. The concerned criteria can be, by their nature, very diverse, and expressed in different units of measurement, from currency followed by physical units of measurement to subjective assumptions measured by a scale formed for a problem in question. Therefore, a final, unique solution cannot be determined without the participation of a decision maker. A decision maker must decide on a particular solution. The solution adopted by the decision maker is designated as the best or the most preferred solution. The task of multiplecriteria decision process is to provide assistance to the decision maker to select the best solution in the given conditions. Therefore, efforts made for solving multiple-criteria problems are designated as a multiple-criteria decision analysis.
The problems of multi-criteria optimization are characterized by:  Presence of multiple criteria (objective function, criteria function) for decision making,  Presence of multiple alternatives (solutions) for the selection,  Presence of a designing process or solution selection,  Presence of conflicts between the criteria, which is the most common case in real life problems,  Incomparable units of measurement for different criteria, and  Selection process of one final solution, which can be presented as a design of an action (alternative), or a selection of one particular action from the previously defined convolution of final actions.
There are many multiple-criteria decision process methods which have common aspects. The characteristic terms which occur in these methods are:  Alternativesrepresent selections of different actions which are available for a decision maker. The convolution of alternatives implies limited convolution, ranging from a few to a few hundreds. It is assumed that alternatives are verified, prioritized and in some cases ranked.  Multiple attributesdifferent designations of attributes are "objectives" or "decision criteria" and represent different dimensions for which alternatives can be observed.  Criteria conflictdifferent criteria represent different dimensions which can be in conflict.  Incomparable units of measurementdifferent criteria can be linked with different units of measurement.  Decision magnitudemost methods of the multiple-criteria decision process demand assignment of the weight criteria in accordance with their importance. Usually, these weights are normalized so that their sum equals one.  Decision matrixthe multiple-criteria decision process can be represented in the form of a matrix. The decision matrix (mxn) is a matrix in which the element aij represents the characteristics of the alternative Ai where (i=1,2,…, m), when the alternative Ai is valued in accordance with the decision criteria Cj (j=1, 2, …, n). Also, it is presumed that the decision maker has assigned the weight criterion wj (j=1, 2, …, n).
Multiple-criteria decision methods are employed in multiple disciplines and can be used in problem solving for any decision making process with multiple criteria. We well cite only a fraction of areas in which multiple criteria decision methods are employed:   comparison of two fuzzy multi criteria decision methods for selecting a potential airport location (Belbag et al, 2013)  comparative analysis of fuzzy multi-criteria decision making for selecting a location for a textile plant in Turkey (Demirel et al, 2016)  wind farms site selection (Deveci et al, 2020b) and .
Description of multiple-criteria decision methods used in the decision making process for the problem in question The methods of multiple-criteria decision making which will be used for finding the best solution are the TOPSIS и MABAC methods: The TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method was developed by Hwang and Joon. This method implies ranking alternatives by multiple criteria based on the comparison of distances from the ideal solution and the negative ideal solution. The ideal solution minimizes the cost based criteria and maximizes the benefit type criteria, while for the negative ideal solution the opposite rule is implied.
The TOPSIS consists of six steps. Prior to the presentation of these steps, the terms to be used will be defined. In this method, the decision matrix R is used, where every alternative is matched by one row and every column is matched by one criterion. The element represents the alternative performance related to the criterion . For m criteria ( 1 , 2 ..., ) and n alternatives ( 1 , 2 ..., ), the matrix R has the following form: Step 1 Decision matrix values normalization. In the majority of multiple-criteria decision methods, the first step is decision matrix elements normalization in order to get a matrix in which all elements are un-dimensional values. In the TOPSIS method, vector normalization is applied, represented by expressions (2.1) and (2.2). (2.2) Step 2 Multiplication of the X matrix normalized values with the weight criteria coefficients.
With relation (2.3), the elements of the weight normalized matrix = ( ), are obtained, where every is a product of the normalized alternative performance and the appropriate criterion coefficient. where: The best alternatives are those which have the biggest relative to the criteria which are maximized and the smallest relative to the criteria which are minimized. * indicates the best alternative (the ideal solution), and, using the same logic, ′′ indicates the negative ideal solution.
Step 4 Determination of alternative distances from the ideal solution. In this step, using the following relations: dimensional Euclid distances are calculated of all alternative distances from the ideal solution and from the negative ideal solution.
Step 5 Determination of relative alternatives proximity to the ideal solution. For each alternative, relative distance is determined: where 0 ≤ * ≤ 1. The alternative is closer to the ideal solution if * is closer to the value of 1, or, which is the same, if * is closer to the value of 0.
Step 6 Ranking of alternatives. The alternatives are ranked by the descending values * . The best alternative is the alternative with the highest * value and vice versa.
The MABAC (Multi-Attributive Border Approximation area Comparison) represents one of recent date methods. The MABAC method was developed by Dragan Pamucar, PhD, University of Defence in Belgrade, Military Academy, Department of Logistics. Up to this date, this method has found wide application and modifications in multiple-criteria problem solving. The MABAC method basic setting is to define every criterion function distance from the border approximate area. The MABAC method consists of 6 steps: Step 1: Base matrix forming(X). In the first step, the evaluation of alternatives per criteria is performed. The alternatives are presented by vectors = ( , 2 , … , ) where the value is the value of the alternative per the criterion( = 1,2, … , ; = 1,2, … , ). (2.10) and where represents an alternative number while is the total number of criteria.
Step 2 Elements normalization of the base matrix . ( 1 , 2 , … , ) and represents the minimum values of a considered criterion per alternatives.
Step 3 Aggravated matrix( ) elements calculation. The elements of the aggravated matrix ( ) are calculated by expression (2.14): where ( ) represents the elements of the normalized matrix( ), , represents weight criteria coefficients. By the application of expression (2.14), the aggravated matrix( )is obtained: where represents the total number of criteria and represents the total number of alternatives.
Step 4 Determination of the border approximate area( ). The border approximate area (BAA) is determinated by expression (2.15): represents the elements of the aggravated matrix ( ) and represents the total number of alternatives. After the calculation of the values per the border approximate area( ) criterion , the matrix is formed with a format 1 ( represents the total number of criteria by which the selection of the offered alternatives is performed). represents the border approximate area for the criterion , represents the elements of the aggravated matrix ( ), represents the number of criteria, and represents the number of alternatives.
The alternative can belong to the border approximate area ( ), the upper approximate area ( + )or the lower approximate area ( − ) or ∈ { ∨ + ∨ − }. The upper approximate area ( + ) represents the area where the alternative( + ) can be found, while the lower approximate area ( − ) represents the area where the anti-ideal alternative ( − ) can be found (Figure 9). In order to select the alternative as the best one from the congregation, it is required that this alternative, per most criteria, belongs to the upper approximate area( + ). If, for example, the alternative per 5 criteria (from the total of 6 criteria) belongs to the upper approximate area, while by one criterion it belongs to the lower approximate area ( − ), it means that this alternative is close or equal to the anti-ideal alternative per 5 criteria. If the values > 0 or ∈ + , then the alternative is close or equal to the ideal alternative. The value < 0 or ∈ − points out that the alternative is close or equal to the anti-ideal alternative.
Step 6 Ranking of alternatives. Calculation of the criteria function values per alternatives (2.21) is obtained as the sum of the alternatives distances from the border approximate areas( ). By summing the matrix by rows, the final criteria function of the alternatives values is obtained: where represents the total number of criteria and represents the number of alternatives.

Best alternative selection for ATGM procurement by the application of the TOPSIS method
In the chapter entitled "ATGM development history and basic characteristics" we have explained the basic ATGM characteristics. These characteristics are always listed in weapon systems manuals because of their tactical significance. That is why these characteristics will be used as the selection criteria.
To repeat, these criteria are as follows:  K1: Engagement range in meters,  K2: Hit probability (0-1),  K3: Weight in kilograms,  K4: Price in USD,  K5: Training simplicity,  K6: Reliability,  K7: Jamming resistance, and  K8: Missile velocity, Under the assumption that we are selecting one ATGM from the proposed four ATGMs of different manufactures, represented as А1, А2, А3 and А4, the decision matrix for a ATGM selection is presented in Table  1.
The characteristics are similar to the characteristics of real weapon systems already in use. In the given table, we can notice that, for the criteria К1, К2, К5, К6, К7 and К8, the most desirable values are the highest ones (max), while for the criteria К3 and К4 (weight and price, respectively), the most desirable are the lowest values (min).
Consider that the criteria К5, К6 and К7 are represented by quality grades; quantification by quality grades will be performed by the application of the scale presented in Table 2. By the application of the quantitative values from Table 2, the quantified matrix is presented. , we will normalize the elements of the base decision and we will obtain the matrix presented in Table 4. In order to obtain the aggravated normalized value matrix, it is necessary to multiply the values form Table 4 with the weight coefficients.
Let us assume that, in our example, an authority expert team, formed to procure ATGMs for the Armed Forces of one particular state, has reached the decision that some criteria have advantage over the others. In such a case, we can rank the before mentioned criteria as follows: I. Hit probability, II.
Engagement range in meters, III.
Missile velocity, and VIII.
Jamming resistance. Based on this requirement, we can assign numerical values, in the range from the minimum value of 0.062 for the criteria with the least importance to the value of 0.246 for the criteria with the highest importance. Two criteriahit probability and engagement range -have the highest values because of their tactical significance. By the application of the expression = • ; = 1,2 … , the aggravated normalized matrix is obtained and presented in Table 5. Step 3 Table 6. During the calculation of the values ( * ) and ( − ) , the elements of the normalized matrix are examined (Table 5), the ideal solutions * and − are obtained in the previous step.
Step 5 Determining the relative distance of the alternatives to the ideal solution. By the application of the expression: * = − * − − , = 1, … , relative distance is determined for each solution, Table 7. Step 6 Alternative ranking. As it has already been mentioned, the alternatives are ranked by declining values * . The best alternative is the alternative which has the highest value * , i.e. in our example the alternative 2. The alternative rank is presented in Table 7, i.e. 2 → 4 → 3 → 1.

Best alternative selection for ATGM procurement by the application of the MABAC method
The procurement of an ATGM will be examined, i.e. the selection of the best alternative, using the MABAC method of the multiple-criteria decision making process.
Step 1 Base decision matrix forming( ). As in the previous example, we will examine four ATGMs. In Table 8, the characteristics of four ATMGs are presented. Table 8 also gives the base decision matrix. For evaluating the alternatives according to the C5, C6 and C7 criteria, the Likert scale (Table 9) will be applied. By the application of the Likert scale, the base decision matrix with numerical values is obtained. Step 2 Base matrix elements normalization( ). After forming the base decision matrix( ), Table 10, the normalization of the base matrix elements is facilitated.
In our example, the "benefit" type criteria (highest values desired) are C1, C2, C5,C6, C7 and C8; therefore, for calculating the normalized matrix N elements, the following expression will be applied: For the "cost" type criteria (lowest criteria values desired), which in our example are C3 and C4, the following expression is applied for calculation: By the application of the previously given expressions, the normalized matrix N is obtained and presented in Table 11. 3 The calculation of the aggravated matrix (Table 12) elements is performed by multiplying the weight coefficients, presented in the last row of the table. The weight coefficients are identical to those in the decisionmaking process when the TOPSIS method is used. Therefore, by the application of the expression = • + , the elements of the aggravated matrix are obtained and presented in Table 12. Step 4 Determination of the border approximate areas values( ). By the application of geometric averaging for the values of the alternatives, with the expression the border approximate areas matrix ( )is obtained, Table 13. Step 5 Calculation of the distance of alternatives (matrix elements) from the border approximate area( ). The distance od the alternatives from the BAA (Table 14)   Step 6 Alternative ranking. The criteria function values by alternatives are obtained by summing the elements from the matrix by rows, by the application of the expression: = ∑ =1 , = 1,2, … , , = 1,2, … , The values of the criteria functions and the final rank of the alternatives are presented in Table 15. Based on the obtained results, we can conclude that the alternative 2 is first ranked, i.e. the alternatives are ranked as follows: 2 → 3 → 4 → 1.

Conclusion
By analyzing the data presented in Table 7(relative alternative proximity( * ) by the application of the TOPSIS method) and the data presented in Table 15 (alternative ranking by the application of the MABAC method), we can observe that the alternative A2 has the highest value and the alternative A1 has the lowest value. The alternatives A3 and A4 have relatively close values. Based on the stated, we can conclude that both methods of multi-criteria decision making are pointing that the best alternative, therefore a decision for the best ATGM weapon system, is presented in the alternative A2. Both methods show that the selection of the alternative A1 would be a wrong decision for procurement. In the applied methods, the alternatives ranking of A3 and A4 are different, so a decision maker can optionally decide (for example to lower the expenses) to select the alternative A3.
The characteristics of the alternative A2 are similar to the characteristics of the 9M133M Kornet-M ATGM, which represents one of the best, battle proven, weapon systems in world. We can conclude that the mathematical models of multi-criteria decision methodswhich in our example unequivocally point to the alternative A2 -present an extremely helpful tool for decision makers.
The novelty of the method described in this paper is in the fact that a specific scientific method can be used for weapon system procurement for armed forces, therefore for gaining advancement in operational capabilities of armed forces.