MULTI-CRITERIA DECISION-MAKING IN A DEFENSIVE OPERATION OF THE GUIDED ANTI-TANK MISSILE BATTERY : AN EXAMPLE OF THE HYBRID MODEL FUZZY AHP-MABAC

* Corresponding author. E-mail addresses: dragan.bojanic@mod.gov.rs (D. Bojanic), marbojanic@gmail.com (M. Bojanic), vladimir.ristic@mod.gov.rs (V. Ristic) ** An earlier version of this paper was presented at the 1st International Conference on Management, Engineering and Environment – “ICMNEE 2017” (Bojanić et al., 2017). MULTI-CRITERIA DECISION-MAKING IN A DEFENSIVE OPERATION OF THE GUIDED ANTI-TANK MISSILE BATTERY: AN EXAMPLE OF THE HYBRID MODEL FUZZY AHP MABAC


Introduction
MCDM is a systematic way of problem solving for any scientific research area.The military decision-making process is a proven analytical process for designing operations, troops' movements, logistics or air defense planning.
In most cases, the decision-making process in the military organization implies not having relevant information, which is characterized by a high degree of uncertainty, subjectivity and ambiguity.Generally, in the decision-making process for these problems, a large number of key and different criteria are involved.Therefore, it is advisable to use tools for their resolution such as MCDM processes that are nowadays widely used in the military (Kewley & Embrechts, 2002;De Leeneer & Pastijn, 2002;Zanjirani & Asgarib, 2007;Kose et al., 2013;Gyarmati, 2015;Goztepe & Kahraman, 2015;Andersson et al., 2015;Boccia et al., 2017), as well as in other research disciplines (Sánchez-Lozano et al., 2013).Fuzzy logic proves to be an appropriate tool (Sánchez-Lozano et al., 2015) to address the described uncertainties and ambiguities.In that way, fuzzy logic enables the exploitation of tolerance that exists in imprecision, ambiguity and partial truth of the results obtained through given research.
Anti-tank is defined as a combat against enemy's tanks and other armored equipment and vehicles.Its associated tasks include the closing of endangered directions, flanks, and junctions.Anti-tank is a part of the combined combat arms directed against the tanks grouped for an attack or already directly attacking, as well as the armored equipment detached to their battle formation.For a successful combat, the organization of a solid anti-armor system is required, which includes anti-tank units as the basic element.The aim of anti-tank operations in a defensive operation is to prevent accidental penetration of the enemy's armored forces into our elements of combat disposition (Jotić & Slavković,2016).
The new operational environment imposes the need to upgrade the defense science in the field of the preparation and conduct of combat operations (Knežević & Slavković, 2012).In the Serbian Army many decisions are made in the processes of planning, organization and preparation for the execution of missions and tasks.The useful tools which support the decision-making process are the methods of multi criteria decision-making.
The fundamental issue of multi-criteria group decision-making is finding procedures for choosing decisions that correspond to the desired solution, with the option of selecting and allocating the most acceptable alternative.The complexity of group decision-making is reflected in the fact that there are a number of criteria and alternatives with different levels of significance for the decision-makers/experts involved in decision-making.
This paper presents a hybrid model, using the fuzzificated Saaty's scale and the MABAC method (Multi-Attributive Border Approximation Area Comparison) (Božanić et al., 2015;Božanić et al., 2016a;Božanić et al., 2016b;Pamučar et al., 2016b).The paper is focused on the demonstration of a new way of fuzzification of the Saaty's scale used for comparison in pairs while varying a confidence interval depending on the comparison.The scale is used for obtaining criteria weight coefficients, while the MABAC method is used for the final ranking.This model is illustrated by an example of decision-making during the selection of firing position (POP) of the GAMB in the Army's defensive operation.The example presents only one segment from a series of decisions that decision-makers face in the preparation and execution of (military) operations.The sensitivity analysis is used to determine how sensitive a decision model is.

Description of the methods used in the hybrid AHP-MABAC model
The described MCDM model is based on the knowledge of several decisionmaking methods (areas), fuzzy logic, the AHP method (Saaty's scale) and the MABAC method.Fuzzy logic successfully covers vagueness and uncertainty that are often present in decision-supporting models.The Saaty's scale, which is an indispensable part of the AHP method, shows good results in defining the criteria weight coefficients, and it is increasingly applied with other methods (Knežević et al., 2015;Zhu et al., 1999).The MABAC method provides stable (consistent) solutions and it represents a reliable tool for rational decision-making (Pamučar & Ćirović, 2015).

Fuzzy logic and fuzzy sets
In fuzzy logic, an element's belonging to the specific set is not precisely definedthe element can be more or less part of the set; therefore, it is closer to human perception than conventional logic (Pamučar et al., 2016b).Fuzzy logic allows quantification of seemingly imprecise information, which is a very common situation when describing social phenomena.Fuzzy logic uses the experience of human expert in the form of linguistic if-then rules, while approximate reasoning mechanism uses managing action for the individual case.In this paper, approximate reasoning algorithm is used to display the influence of the entry criteria on decision preference in choosing the most appropriate firing GAMB position.
The first step in designing fuzzy sets is to define the degree of the membership of an element x (xX) in the set A. This is described with membership function A (x), which in the classic theory has a value of 0 (does not belong) or 1 (belongs), while in a fuzzy set the membership function can have any value between 0 and 1.So, it can be said that the closer А (x) is to 1, the greater belonging of x to A is, and vice versa.A fuzzy set A is defined as a set of ordered pairs where:  X is a universal set or a set of considerations based on which fuzzy set A is defined;  A (x) function of element x belonging to set A. In this paper, triangular fuzzy numbers will be used.They will be presented in the form T = (t1, t2, t3), where Fig. 1:  t2 is where the membership function of a fuzzy number has a value of 1;  t1 is the left distribution of the confidence interval of fuzzy number T, and,  t3 is the right distribution of the confidence interval of fuzzy number T.
Triangular fuzzy number T (Pamučar et al., 2016a) The membership function of fuzzy number T is defined in the following way: For its final purpose, fuzzy number T= (t1, t2, t3) is converted into a real number.

Fuzzification of the Saaty's scale
One of the key phases in the application of this method is the development of the comparison matrix by pairs, corresponding to every level of the hierarchy.A pairwise comparison is performed according to the data collected by measuring them on the basis of beliefs, estimates or experiences of those who carry out the assessment (Čupić & Suknović 2010).The shown fuzzification of the Saaty's scale is presented in (Božanić et al., 2015;Božanić et al., 2016a;Božanić et al., 2016b;Pamučar et al., 2016b;Božanić, 2017).
The definition of this new fuzzificated Saaty's scale (Table 1) starts from the assumption that the decision-makers and analysts have a different degree of certainty ji, concerning the accuracy of comparisons in pairs (Božanić et al., 2016a;Pamučar et al., 2016b).This degree of certainty differs from one comparative pair to another.The value of the degree of certainty belongs to interval ji0,1.In the cases when ji=0, it is considered that the decision-maker/analyst has no data about this relationship; hence it should not be used in the decision-making process because it points to absolute ignorance of the decision-making subject.The value of the degree of certainty where ji=1 describes the absolute certainty of the decisionmakers/analysts in the given comparison.The lower the certainty in the performed comparison is, the lower the element ji.
By defining different values of parameter ji, the left and the right distribution of fuzzy numbers change from one comparison to another, according to the expression: the value of t2 represents the value of linguistic expressions from the classic Saaty's scale, which in a fuzzy number has a maximum membership t2 = 1 Fuzzy number The defined scale is further used in standard steps of the AHP method, which is described in a number of papers (Čupić & Suknović2010), (Inđić et al., 2014) and others.
Based on the pre-defined scale, the decision-makers and analysts fill in the new, modified matrix: As can be seen, the matrix is extended with the degree of certainty in the comparison made, whereby ji=ij, where ji0,1.After the calculation is finished, the defuzzification can be performed using one of well-known methods.Some of the well-known expressions for defuzzification (Seiford, 1996) are the following ones: Where  represents the degree of optimism.
The defined scale is also suitable for group decision-making, which is nowadays becoming more and more popular.Involving experts greatly improves the quality of decisions made because knowledge and experience are collected and consolidated into a single unit.

MABAC method
The MABAC method was developed by Pamučar and Ćirović (Pamučar & Ćirović, 2015).The basic setting of the MABAC method is reflected in the definition of the distance of the criterion function of each of the observed alternatives from the approximate border area.The text that follows shows the procedure of implementation of the MABAC method in six steps, its mathematical formulation being: Step 1. Creation of initial decision matrix (X).In the first step, the evaluation of m alternatives by n criteria is carried out.The alternatives are presented with vectors Ai = (xi1,xi2, ……xin), where xij.is the value of alternative i according to criteria j (i = 1,2,..., m; j = 1,2,..., n).where m indicates the number of alternatives, and n indicates the total number of criteria.
Step 2. Normalization of initial matrix (X) elements.
Multi-criteria decision-making in A defensive operation of the Guided anti-tank.The elements of normalized matrix (N) are obtained using the following expressions: a) For the "benefit" type criteria b) For the "cost" type criteria Step 3. Calculation of weighted matrix elements (V).The elements of weighted matrix (V) are calculated on the basis of expression (17): where tij are the elements of normalized matrix (N), and w represents the weight coefficient of criteria.By applying expression (17), we get weighted matrix V that otherwise can be written as: where n is the total number of criteria, and m is the total number of alternatives.
Step 4. Determination of approximate border area (G) matrix.The border approximate area (BAA) for each criterion is determined by expression ( 19) where vij are weighted matrix elements (V) and m represents the total number of alternatives.
After determining value gi according to the criteria, we form the matrix of approximate border areas G (20) size nx1 (n is the total number of criteria by which the election of the offered alternatives is made).
  Step 5. Calculation of the matrix elements distance from border approximate area (Q) 11 12 1n 21 22 2n m1 m2 mn q q ... q q q q Q ... ... ... ... q q ... q The distance of the alternatives from border approximate area (qij) is defined as the difference between weighted matrix elements (V) and the values of border approximate areas (G).
which otherwise can be written as: where gi represents the border approximate area for criterion Ci, vij. is weighted matrix elements (V), n represents the number of criteria, and m represents the number of alternatives.
Alternative Ai may belong to border approximate area (G), upper approximate area (G + ) or lower approximate area (G¯), respectively Ai⋲{G¯˅G + V G¯}.Upper approximate area (G + ) is an area in which the ideal alternative is found (A + ), while lower approximate area (G¯) is an area in which the anti-ideal alternative (A¯) is found (Fig. 2).approximation areas (Pamučar, Ćirović, 2015) Belonging of alternative Ai to approximate area (G ,G + or G¯) is determined on the basis of expression ( 24 In order for alternative At to be chosen as the best from the set, it is necessary that, according to as many criteria as possible, it belongs to upper approximate area (G + ).
Step 6. Ranking alternatives.Calculation of the criteria function values by alternatives is obtained as the sum of the distances of the alternatives from border approximate areas (qi).Summing the elements of matrix Q by rows gives the final values of the criteria function alternatives where n represents the number of criteria, and m represents the number of alternatives.

Identification of the criteria and calculation of the weight coefficients of the criteria
A guided anti-tank missile battery performs its tasks from the firing position (POP).It is composed of self-propelled launchers LRSPM83.The (POP) is a part of the land in the area of the operation, prepared and occupied or intended to be occupied by the artillery units for the execution of firing support (The military lexicon, 1981).
"In order to ensure the effectiveness of fire control and fire coordination, the distance in length is 100-300m between self-propelled launchers, and 300-400m between platoons.The dimensions of deployment areas can extend to 1,4-2 km in width and 1 km in length for batteries, 1km in width and 500m in length for platoons depending on the combat situation, terrain, and the number of assets involved."(Fig. 3) (Rule book self-propelled anti-tank battery -platoon, 2016).

Figure 3. Battery deployment on POP
Decision-makers usually have to select an (POP) area relying on the acquired theoretical knowledge, experience and assessment in the specific situation.A number of criteria that influence the ranking and selection of alternatives indicate a possibility of applying multiple criteria methods.Founded on the available literature, the ranking criteria that are the basis for selection-making are also defined: C1-"distance to the targets area"-the distance at which the guided armorpiercing missiles will neutralize the enemy armored vehicle in the direction of the attack.The range of guided armor-piercing missiles is determined by their maximum launching distance and the distance of the fire line -influenced by the terrain in front of their firing positions; C2-"speed of occupancy POP" -represents the estimate of time for which launching devices from the expected region will make an arrival at POP; C3-"maneuver capability " -the maneuver capability of guided anti-tank missile units is determined by their mobility (movements to firing positions, during maneuvers to deployment areas), and the time required for occupation and leaving of the deployment areas; C4 -"fortification conditions" -terrain features that allow successful fortification of artillery to enhance force protection; C5 -"quality of access roads" -road characteristics that appreciate the possibility of fast and successful settling and abandonment of POP; C6 -"masking conditions " -terrain features that enable successful masking of GAMB and movement of GAMB of parts as well as masking of the effects of the rocket launching.

Application of the MABAC method for ranking alternatives
In order to apply the MABAC method, six alternatives have been selected (from A1 to A6) (Table 6).The alternatives represent a land area that the artillery means will be deployed on.The initial decision matrix is presented in Table 6.Since evaluation criteria C3, C4, C5 and C6 have a qualitative character, for the evaluation of alternatives by criteria, the fuzzy linguistic descriptors scale is used.Using the next steps of the MABAC method, the distance values of the alternatives from the border approximate area are obtained and their ranking is shown in Table 8.

Sensitivity analysis of the output results
It is recommended as a means of checking the stability of the results against the subjectivity of decision-makers.The sensitivity analysis of the results is carried out by changing the initial weight coefficients of the evaluation criteria.Table 9 gives the scenarios of change of weight coefficients (seven scenarios), based on which the ranking of the already presented alternatives is performed.The ranking of the alternatives after the application of the scenario is given in Table 10.By analyzing the obtained results, it can be concluded that there is significant stability of the output results in most of the scenarios.This is supported by the fact that A1 and A4 are most often ranked as the first or the second, which is expected for a stable system.

Conclusions
The application of the (MCDM) model is successfully presented in the paper while the sensitivity analysis indicates the potential of the application of the created models to support decision-making during the planning process of the Land Forces operation.In a similar way, the application would be conducted in group decisionmaking.
A practical example of the fuzzy scale demonstrated a possibility of using the hybrid fuzzy AHP-MABAC model, its performance in ranking the offered alternatives.The analysis of the output results sensitivity has shown that the hybrid FAHP-MABAC model provides stable solutions for the problem of choosing an optimal firing position of the Guided anti-tank missile battery.The proposed hybrid approach can solve not only the problems of location choice, but also many other decisionmaking ones.
From all of the above mentioned, it can be concluded that the fuzzificated Saaty's scale improves decision-making by taking into account the degree of certainty of decision-makers in the shown pairwise comparison.
16) where xij, xi + and xi − are the components of initial decision matrix (X), where xi + and xi − are defined as: xi + = max (x1, x2,..., xm) and represents the maximum value of the observed criteria by alternatives, xi + = min(x1, x2,..., xm) and represents the minimum value of the observed criteria by alternatives.

Table 2 .
Comparison matrix in pairs

Table 3 .
Comparison matrix in pairs after fuzzification

Table 4 .
Fuzzy weight vectors of the criteria ) the defuzzification of the weight coefficients of the evaluation criteria is performed.The final values of the weight coefficients of the criteria are given in Table5.

Table 5 .
Final values of the weight coefficients

Table 6 .
Initial decision matrixDefuzzification is performed by applying expression (12).The next step is the normalization of the matrix elements and the final values of the normalized matrix are presented in Table7.

Table 8 .
Ranking of the alternatives Based on the obtained distances from the ideal alternative, it can be concluded that alternative (A1) is the most appropriate alternative while alternative (A3) is the least favorable one.

Table 9 .
Scenarios with different weights of the criteria

Table 10 .
Ranks of alternatives by applying different situations