MULTICRITERIA DECISION MAKING MODEL WITH Z-NUMBERS BASED ON FUCOM AND MABAC MODEL

. In the paper is presented a model for selecting a location for a brigade command post during combat operations. Considering that this is a very complex model, which can be approached from several aspects, this paper is limited only to the criteria related to the construction or arrangement of the command post, respectively, the engineering aspect. The selection process is conducted using hybrid FUCOM – Z-number – MABAC model. The FUCOM method is used to define the weight coefficients of criteria based on which the selection is made. The MABAC method, modified by applying Z-number, is used to rank alternatives. The end results indicate that the application of Z-number in decision making includes broader set of uncertainties than standard fuzzy numbers, which is very important for deciding in combat situations.


Introduction -problem description
The Serbian Army performs various combat and non-combat operations.Through the implementation of these, commanders and leaders (from the highest to the lowest level of command) are often in situations in which they have to make more decisions.Most often the end result is a decision made on the basis of previously acquired, mostly theoretical knowledge and on the experience gained by officers during their military service.One example of such an issue is the selection of a brigade command post in a defense operation.Like others, this problem can be solved on the basis of experience and knowledge, but it is much better when the decision is followed by adequate mathematical decision-making model, used as an aid or tool for decisionmakers.
"A command post presents an area, premises or technical means (ship, tank, conveyor, aircraft) in the area of operation of a unit, in which the command is placed with appropriate means during the preparation and conduct of a combat."(Military Lexicon, 1981).There are numerous factors influencing the selection of a command post.These factors (criteria or conditions) can in principle be divided into two groups: 1) the criteria related to the construction of a command post, respectively, performance of works, and 2) the criteria related to the functionality of a command post.The first set of criteria defines the criteria primarily related to fortifying and partly to masking, while in the second set of criteria would be included the criteria related to successful command during combat operations.
Throughout this paper, the authors focused on the first set of criteria, namely, the development of a model to support decision-making when selecting a brigade command post from the perspective of the ability to perform works, respectively, fortifying and masking.The decision-making support model is based on two methods: 1) the FUCOM method -for defining criteria weights, and 2) the MABAC method, which is fuzzified using standard fuzzy numbers and Z numbers -for ranking alternatives.
The FUCOM method was first presented in 2018 (Pamučar et al., 2018).Due to its simple application and reliable results, this method has quickly begun to be applied in other papers (Prentkovskis et al., 2018;Badi & Abdulshahed, 2019;Puška et al., 2019;Cao et al., 2019;Durmić et al., 2019;Stević et al., 2019;Ibrahimović et al., 2019).The most common application of the FUCOM method is found in the process of defining weight coefficients of criteria.

Methods applied in the paper
In the following part of the paper, the description of the methods used in the paper is provided.

FUCOM Method
Considering that basic version of the FUCOM method is used, which is presented in Pamučar et al. (2018), in the further part of the paper, only the steps of the method are listed.More detailed review with the examples is available at Pamučar et al (2018).The FUCOM method consists of three steps: Step 1.In the first step, the criteria from the predefined set of the evaluation criteria The ranking is performed according to the significance of the criteria, i.e. starting from the criterion which is expected to have the highest weight coefficient to the criterion of the least significance.
Step 2. In the second step, a comparison of the ranked criteria is carried out and the comparative priority should satisfy the two conditions.After the verification of the fulfillment of conditions, the weight coefficients of criteria are defined by using the expression (1):

Z number -МАВАС method
The MABAC method is developed by (Pamučar & Ćirović, 2015).It is developed as the method providing crisp values.In this paper is used fuzzified MABAC method by applying Z-numbers.The fuzzification is performed using triangular fuzzy numbers.A general form of triangular fuzzy number is given in the Figure 1.Triangular fuzzy numbers have the form -t1 -the left distribution of the confidence interval of fuzzy number T, t2 -fuzzy number membership function has the maximum value -equal to 1, and t3 -the right distribution of the confidence interval of fuzzy number T (Pamučar et al., 2012).Z-number presents an extension of classic fuzzy number and provides wider opportunities for considering additional uncertainties following decision making.The concept of Z-number was proposed by Zadeh (2011).In 2012 already Kang et al. (2012aKang et al. ( , 2012b) ) shown in detail the application of Z-numbers in uncertain environment.Later authors consider the application of Z-numbers with different methods of multi-criteria decision making.Sahrom & Dom (2015) present the use of Z-numbers in the hybrid AHP-Z-number-DEA method.Azadeh & Kokabi (2016) use Znumbers with the DEA method.Azadeh et al. (2013) with the AHP, Yaakob & Gegov (2015) with the TOPSIS method, Aboutorab et al. (2018) with the Best Worst method.Salari et al. (2014) elaborate a novel earned value management model using Znumber.
Z-number represents an ordered pair of fuzzy numbers that appear as Z=( A , B ) (Zadeh, 2011).The first component, fuzzy number A , represents the fuzzy limit of a particular variable X, while the second component fuzzy number B represents, the reliability of the first component ( A ).The appearance of the Z-number with triangular fuzzy numbers is shown in Figure 2 (Zadeh, 2011).(Kang et al., 2012a) The general record of triangular Z-numbers can be displayed as where the values w represent weight factors of fuzzy numbers A referring to B , which for the initial Z-number the majority of authors defines as is the height of the generalized fuzzy number and A 0 w 1  ) (Chutia et al., 2013).The transformation of the Z-number into the classical fuzzy number, with the presented evidence, is shown in Kang et al. (2012b).This transformation consists of three steps: 1) Convert the second part ( B ) into a crisp number using the centered method (Kang et al., 2012b): 2) Add the weight of the second part ( B ) to the first part ( A ).The weighted Z-number can be denoted as Kang et al. (2012b)   which can be denoted by the figure 3a.This can be written as (Azadeh et al., 2013): ) and it can be present as figure 3b (Kang et al., 2012b).After describing Z-numbers it is necessary to explain their application in a particular model.These numbers present more comprehensive treatment of uncertainty because when the value of an alternative by a criterion in the form of a standard fuzzy number ( A ) is shown, the degree of certainty of the decision maker or expert ( B ) is also presented.By the above expressions (2-7) is made a transformation of the above fuzzy numbers into a unique fuzzy number.Standard fuzzy MABAC method is further applied.The degree of certainty of the decision makers in the values provided for the evaluation of alternatives by criteria is defined by the expressions presented on the fuzzy linguistic scale, as in the Figure 4. Hybrid model Z number -MABAC is taken from Bobar et al. (2020).The Fuzzy MABAC method consists of 7 steps (Božanić et al., 2018, Bobar et al., 2020): Step 1. Forming of the initial decision matrix ( X Step 2. Converting Z-numbers to regular fuzzy number.This process is performed by applying the expressions (2) to ( 7).The output provides new initial fuzzy decisionmaking matrix ( P ) Step 3. Normalization of new initial decision-making matrix ( P ).
The calculation of the elements of normalized matrix ( N ) depends on the type of criteria.For beneficial criteria this calculation is executed according to the expression: For detriment criteria the calculation is executed according to the expression: x  represent elements of the initial matrix of decision-making ( X Step 5. Determination of the approximate border area matrix ( G ).The border approximate area for each criteria is determined based on the expression: The matrix of approximate area () G has a format n x 1, where n presents overall sum of criteria number and is represented in the following way Step 6. Calculation of the matrix elements of alternatives distance from the border approximate area ( Q ).The distance of alternatives from the border approximate area ( ij q ) is defined with the expression: Afterwards the matrix is calculated Q n n m m mn q q q q q q Q q q q (17) Step 7. Ranking of alternatives.The value estimation of criteria functions of alternatives is gained from the sum of the distance of alternatives from the border approximate areas ( i q ).The ultimate values of criteria functions of alternatives are gained from the sum of elements of the matrix Q in rows: By defuzzification of the values obtained, final rank of the alternatives is obtained.Defuzzification can be performed by applying the expressions (Seiford, 1996;Liou and Wang, 1992):

Description of criteria and calculation of weight coefficients
The selection of a location for a command post is made on the basis of five criteria, obtained by analyzing available literature.Basic criteria for selecting a location of a brigade command post are shown from the most significant (C1) to the least significant (C5), respectively, C1> C2> C3> C4> C5.The criteria on which depends the location of a command post are as follows:  C1 -Time required for engineering works.This criterion implies the total time required for preparatory, main and final works on the engineering arrangement of a command post.(Hristov, 1978).Through this criterion, various elements such as the influence of land to the selection of the type of object to be constructed, geological composition of the soil, etc., are indirectly evaluated. C2 -Deposits of building materials.Various materials are used in the construction of fortification structures, such as: timber, steel and concrete elements and stone.Through this criterion, the existence of material deposits in the vicinity of the area of works, the quantities and types of materials, as well as the possibility of its incorporation into facilities in its existing form or after processing are evaluated. C3 -Masking conditions.Masking conditions include the possibility of concealing preparations for the execution of works, centralized processing of certain elements (timber, reinforced concrete elements, etc.) and direct works on the fortification. C4 -Influence of the enemy.This criterion implies the ability of the enemy to detect the preparation and execution of works and the possibility of direct action from the ground and from the air.(Šećković, 1972). C5 -Possibilities of use of workshops, technical means and tools.In the areas of a potential command posts, it is desirable to have the possibility of using local plants (workshops, quarries, sawmills, etc.), tools (pickaxes, shovels, crowbars, etc.) and technical means suitable for fortification (dozers, loaders, diggers, etc.), in order to economize the forces, resources and time required to perform the works.The set of criteria from C1 to C5 consists of two subsets:  the "C +" is a set of criteria of the benefit type, which means that the higher value of criteria is more favorable (the criteria C2, C3 and C5), and  the"C -" is a set of criteria of the cost type, which means that the lower value of criteria is more favorable (the criteria C1 and C4).The criterion C1 is presented as numerical, while the other criteria are presented as linguistic.
The weight coefficients of the criteria are obtained using the FUCOM method.Criteria ranks are calculated based on the data on their mutual comparison, as in the Table 1.
The values of the calculated weight coefficients are provided in the Table 2.

Model testing
Ten alternatives were defined to test the model.Prior to the process of selecting the best alternative from the set of offered ones, a scale for evaluating linguistic criteria had been defined, as in the Figure 5  Linguistic criterion can be described with five values: very small (VS), small (S), medium (M), large (L), very large (VL).
The initial decision-making matrix is shown in the Table 3.

Table 3. Initial decision making matrix
Alternative index In the next step, the quantification of linguistic descriptors is performed, as shown in the Table 4.In the second step, Z-numbers were converted to regular fuzzy numbers.After converting into a regular fuzzy number, new initial decision-making matrix was obtained, as in the Table 5.In the third step, the normalization of the new initial decision-making matrix was performed, using the expressions 9 and 10 respectively, as in the Table 6.By applying the expression (12) in the following step is obtained the weighted matrix ( V ), as in the Table 7.In the fifth step is obtained the approximate border area matrix ( G ), by applying the expression (14), as in the Table 8.In the sixth step, using the expression ( 16), the distance of the alternatives from the border approximate area was obtained, as in the Table 9.The final values of the criteria functions with the rank of alternatives are provided in the Table 10. 2 (-0.26,0.11,0.45) 0.1 3 0.19 3 A3 (-0.28,0.01,0.22)-0.01 5 (-0.28,0.17,0.49) 0.13 2 0.2 1 A4 (-0.17,0.03,0.27)0.04 3 (-0.37,-0.01,0.44) 0.02 5 -0.01 7 A5 (-0.41,-0.24,0.14)-0.17 10 (-0.4,-0.12,0.36) -0.05 8 -0.17 9 A6 (0.07,0.3,0.5)0.29 1 (-0.23,0.17,0.54) 0.16 1 0.19 2 A7 (-0.24,-0.06,0.15)-0.05 8 (-0.33,0,0.38)0.02 6 0 6 A8 (-0.22,0,0.26)0.01 4 (-0.5,-0.13,0.34) -0.09 9 -0.18 10 A9 (-0.32,0.01,0.22)-0.03 6 (-0.5,-0.04,0.24) -0.1 10 0.04 8 A10 (-0.36,0,0.24) -0.04 7 (-0.34,0.16,0.47) 0.1 4 0.16 4 In addition to the rank of alternatives obtained by applying Z-number MABAC model, in the Table 10 are also provided the ranks of alternatives obtained by applying classic MABAC method and by applying fuzzy MABAC method (excluding Znumber).Comparative ranking of alternatives provides significant differences in ranking.The alternative A6, the second-ranked in the application of classic MABAC method, appears as the first-ranked in the rest of the cases.The alternative A3, which is the first-ranked when applied the MABAC method, is the second-ranked when fuzzy MABAC method is applied, and even the fifth-ranked when Z -number MABAC model is applied.This difference clearly indicates the need to mathematically examine rank correlation.Considering that these are different models, rank differences can be expected, but they should not be significantly different.In this sense, rank correlation control is performed using the Spearman's coefficient where is:  S -the value of the Spearman's coefficient,  Di -the difference in the rank of the given element in the vector w and the rank of the correspondent element in the reference vector,  n -number of ranked elements.Table 13 shows different ranks of alternatives for different scenarios, indicating that the model produced is sensitive to changes in the criteria weights.Regardless of the different ranks, it is noted that the alternatives A6 and A2 are at the top in all scenarios, while the alternatives A1 and A5 are ranked as the worst in most scenarios.The next step in sensitivity analysis is the application of the Spearman's coefficient, to establish and analyze rank correlations when applying different scenarios, as in the Table 14.As observed from the Table 14, the values of the Spearman's coefficient are extremely high and very close to the ideal positive correlation.This indicates a stable and sensitive enough model

Conclusions
By the FUCOM -Z-number -MABAC model presented have successfully been evaluated the locations for a command post selection in military combat operations.With a parallel presentation of the application of classic MABAC method and its modifications by the use of fuzzy numbers, respectively, Z-number, it can be noted that the modification of the MABAC method using Z-number provides broader range of possibilities for considering uncertainty.It is difficult to cover large number of uncertainties following combat operations through conventional multi-criteria decision-making methods, which is why it is important to include at least a part of those uncertainties in the decision-making process.Fuzzy MABAC model includes some uncertainties, while Z-number -MABAC model increases the number of uncertainties treated.This indicates that the application of Z-number is extremely useful in the processes in which is not possible to predict all the elements that influence decision making, because, contrary to uncertainty, it can include other factors that are not fully measurable but can influence the final outcome.
The introduction of a model for the selection of a command post location in combat operations significantly advances this process: it helps decision makers understand the factors that influence the selection more comprehensively, and provides less experienced decision makers with the support in decision making based on their predecessor's experience.Considering that there is a number of decisions made during combat operations and followed by a high degree of uncertainty, undoubtedly, the model presented can significantly facilitate making decision on selection of a command post.
where k represents the rank of the criteria) of the evaluation criteria is determined.The comparative priority of the evaluation criteria ( k / (k 1)   ) is an advantage of the criterion of the In the third step, the final values of the weight coefficients of the The final values of the weight coefficients

Figure 3 .
Figure 3. Z-number after multiplying the reliability (a) and the regular fuzzy number transformed from Z-number (b)3) Convert the weighted Z-number into a regular fuzzy number.The regular fuzzy set can be denoted asKang et al. (2012b)

Figure 4 .
Figure 4. Fuzzy linguistic descriptors for evaluating the degree of conviction of experts(Bobar et al., 2020)

Figure 5 .
Figure5.Graphic display of fuzzy linguistic descriptors(Božanić et al., 2016b) ). Matrix is formed with a grade of alternatives based on criteria ). Calculation of the weighted matrix ( V ) elements.Elements of this matrix are calculated based on the following expression:

Table 2 .
Weight coefficient of criteria

Table 4 .
Quantification of linguistic descriptors

Table 5 .
New initial decision-making matrix

Table 6 .
Normalized initial decision-making matrix

Table 8 .
Approximate border area matrix

Table 9 .
Matrix of the distance of alternatives from border approximate areas

Table 10 .
Ranking of alternatives

Table 13 .
Ranking of alternatives by applying different scenarios

Table 14 .
Spearman's coefficient values obtained using different sensitivity analysis scenarios