USE OF THE FUZZY AHP-MABAC HYBRID MODEL IN RANKING POTENTIAL LOCATIONS FOR PREPARING LAYING-UP POSITIONS

The paper proposes a hybrid model, FAHP-MABAC. The fuzzyficated Saaty's scale with the AHP method (FAHP) is used for defining the weight coefficients of criteria. The MABAC method is applied in evaluating and ranking the alternatives. This hybrid model is developed in order to support decision making while selecting locations for the preparation of laying-up positions (facilities intended for concealment, protection and maneuvering of military ships).


Introduction
The decision making process is usually followed by a large number of uncertainties.This is particularly characteristic of combat operations involving different units and different types of modern military equipment.Therefore, the development of a decision making model has an important role in the operational planning process.
This paper presents a hybrid model using the fuzzyficated Saaty's scale, the analytic hierarchy process method (FAHP) and the MABAC (Multi-Attributive Border Approximation area Comparison) method -the FAHP-MABAC model.Herein are briefly explained the methods used, with a more detailed elaboration of the fuzzyfication of Saaty's scale and the MABAC method.The hybrid FAHP-MABAC model is made in order to support decision making while choosing the location for the development of laying-up positions.

Problem description
During the preparation of combat operations, various kinds of masking are carried out in order to protect forces and resources from enemy's surveillance and effects.This term is defined in the literature in different ways, but essentially it refers to the same or similar activities.The most general definition is provided in the Military Lexicon (Vojni leksikon, 1981, p.275), where the term masking refers to "a set of measures and procedures used to conceal the intentions of one's own forces, movements and placements of combat and other means and the facilities from the enemy's surveillance from land, sea, air and cosmic space, achieving a surprise effect and deceiving the enemy."In other words, by means of masking, the enemy is misled into erroneous conclusions, decisions and actions (Milosavljević, 1985, p.129).This is accomplished in several waysby hiding, concealment or deception (Rkman, 1984, pp.21-26).
A large number of war experiences speaks in favor of the importance of including masking in combat operations.In this regard, the Vietnam War  is extremely impressive.Also, the experience from the wars in the former Yugoslavia, and especially the NATO bombing of Yugoslavia in 1999, demonstrated how even small-scale masking protection techniques in the era of electronic means for surveillance can result in the reduction or even avoidance of losses.
One of the elements of masking is the development of laying-up positions.These are "arranged parts of the coast that provide concealment, a partial degree of protection and quick maneuver of river flotilla forces" (Vojni leksikon, 1981, p.276), i.e. military ships.In accordance with the general purpose of masking, the main objective of construction of laying-up positions is leading the enemy to erroneous conclusions, decisions and actions related to the deployment of our military ships (Bozanić et al., 2015a).
The literature generally describes a part of criteria that a location of laying-up positions should meet.However, their precise definition, weight values and relationships are missing.Therefore, the decisions made on this problem rely on the knowledge and experience of decision makers (DM) and their associates.Such a situation represents a field for the application of multiple criteria methods.

Description of the methods used in the hybrid FAHP-MABAC model
The described 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 covers successfully 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 the criteria weight coefficients defining, and it is increasingly applied with other methods (Ertuğrul, Karakaşoğlu, 2009), (Das, et al., 2012), (Beikkhakhian, et al. , 2015), (Zhu, et al., 2015), (Sara, et al., 2015), (Kazan, et al. 2015), (Knežević, et al., 2015).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, unlike in conventional sets, belonging of one element to the given set is not precisely defined; the element can be more or less a part of the set (Pamučar, et al., 2011a, p.594).Therefore, fuzzy logic is closer to human perception than conventional logic (Pamučar, et al., 2011a, p.594).This feature allows fuzzy logic to quantify the information which, in classic logic, is considered imprecise.The existence of apparently imprecise information, which in fuzzy logic is very well handled, frequently occurs in social sciences, including decision-making processes.
The creator of fuzzy logic is Lotfi Zadeh.In a series of papers, he presented the basics of fuzzy logic (Zadeh, 1965), (Zadeh, 1972), (Zadeh, 1973) and others.These basics were enough to empower fuzzy logic and now cause it to constantly evolve and increasingly be applied in practice.For the purposes of this paper, several segments of fuzzy logic are significant: definition of fuzzy set, selection of a membership function form and confidence intervals.
The fuzzy set A is defined as a set of arranged pairs where: -X -is a universal set or a set of considerations based on which a fuzzy set A is defined; -μ A (x) is a membership function of the element x (x∈X) to the set A; A membership function can have any value between 1 and 0; so, as the value of the function is closer to 1, the membership of the element x to the set A is greater, and vice versa.
Each fuzzy set is completely and uniquely defined by its membership function (Zadeh, 1965).Membership functions may have different forms, but the most commonly used are triangular, trapezoidal and Gaussian.The selection of the membership function is carried out so that it best describes the phenomenon it represents, for which there are no certain rules.In this paper, triangular fuzzy numbers T = (t 1 , t 2 , t 3 ), will be used (Figure 1), where t 1 represents left and t 3 right distribution of the confidence interval of the fuzzy number T and t 2 represents the point where the membership function of the fuzzy number has its maximum value, or value 1.The elements of fuzzy sets are taken from the confidence interval.The confidence interval contains all the elements that can be considered.Therefore, a fuzzy variable can have only the values from the confidence interval.Determination of the confidence interval of each fuzzy variable is the task of the designer and the most natural solution is to adopt the confidence interval so that it matches the physical limits of the variable (Pamučar, et al., 2016).If the variable is not of physical origin, one of the standard ones is adopted or an abstract confidence interval is defined (Božanić, Pamučar, 2010), (Pamučar, et al., 2011a).
For the purpose of end use, the fuzzy number T = (t 1 , t 2 , t 3 ) is converted into a real number.For this operation, a number of methods are used (Herrera Martínez, 2000).Some of the well-known expressions for the defuzzyfication are the following ones (Seiford, 1996) where λ is the optimism index, which can be described as the belief/attitude of DM considering the risk in decision making (Milićević, 2014, p.186).The most common optimism index is 0, 0.5 or 1, corresponding to the pessimistic, average and optimistic view of the decision-maker (Milićević, 2014, p.186).

Analytical Hierarchy Process and the Saaty's scale
The AHP method was developed by Thomas L. Saaty.It is based on the decomposition of a complex problem through the hierarchy approach, with the goal at the top, criteria, sub-criteria and alternatives at the levels and sub-levels of the hierarchy (Saaty, 1980), Figure 2. The key element of the AHP method is the development of a comparison matrix in pairs.It is made at each level of the hierarchy.

Fuzzyfication of the Saaty's scale
The Saaty's scale, although it is a standard of the AHP, has its drawbacks.It often happens that DM/experts are not entirely sure of the accuracy of comparisons by pairs.Therefore, in the literature, there are an increasing number of papers which in various ways elaborate fuzzyfication of the Saaty's scale (Zhu, et al., 1999), (Srđević, et al., 2008), (Gardaševic-Filipović, Šaletić, 2010), (Pamučar , et al., 2011b), (Božanić, et al., 2013), (Janacković, et al., 2013), (Rezaei, et al. 2014), (Janjić, et al., 2014), (Pamučar, et al., 2015), (Božanić, et al., 2015c) and others.A part of presented fuzzyfications, or scales, is fuzzyficated so that the confidence interval of the membership function is determined prior to research.The second part of papers leaves the possibility that the confidence interval depends on the specific parameters determined in the course of research.
In order to define the weight of criteria, in this paper the fuzzyficated Saaty's scale shown in (Božanić, Pamučar), (Božanić, et al. 2015b) is applied.With this scale, it is defined that DM/experts have a different degree of certainty γ ij in the accuracy of comparisons in pairs they perform.This degree of certainty differs from one to another pair of comparison.The value of the degree of certainty belongs to the interval γ ij ∈[0,1].In cases where γ ij =0, it is considered that DM/experts have no data on this relationship, so they should not use it in the decision-making process, because it points to the absolute ignorance of the decisionmaking matter.The value of the degree of certainty where γ ij =1 describes the absolute DM/expert certainty in the defined comparison, so, in such cases, a fuzzy number is not used but the standard values of the Saaty's scale.As the certainty in the comparisons lowers, so does γ ij .The same importance By defining different values of the parameter γ ji , the left and right distribution of fuzzy numbers change from a comparison to a comparison, according to the expression: (5) t 2 value represents the value of the linguistic expression from the classic Saaty's scale, in which a fuzzy number has its maximum membership t 2 = 1.
A fuzzy number ( ) ( ) ( ) x , is defined with experessions: The inverse fuzzy number ( ) ( ) ( ) ) On the basis of the previously defined scale, DM/experts fill a new, modified matrix: A new scale can be applied in group decision-making, which has significantly improved the quality of decision-making.In specific cases, the data collected by the Delphi technique are analyzed separately for each expert.The data analysis is performed by the AIJ synthesis (Aggregating Individual Judgments), where numerical ratings of element preferences are aggregated at the local level (for each matrix separately), in order to obtain a synthetic set of matrices for a fictitious ("group") decision-maker representing the group, and then the standard AHP synthesis could be executed (Zoranović, Srdjević, 2003).After all members of the group have performed the necessary comparisons in pairs of elements of the hierarchy, the filled matrices of type A(e)={a ij (e)} (e = 1, 2, ..., E, where e represents the number of group members), are aggregated into the correspondent unique matrices for the group by applying, at each position (i,j), a micro aggregation by geometric averaging (Zoranović, Srdjević, 2003) using the formula: ( ) The MABAC method The MABAC method was developed by Pamučar and 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, i.e. its mathematical formulation: Step1.Creating the initial decision matrix (X).In the first step, the evaluation of the m alternatives by the n criteria is carried out.The alternatives are presented with vectors , ,..., , where ij x is the value of the i alternative according to the j criteria ( 1, 2,..., ; 1, 2,..., i m j n = = ).
1 2 where m indicates the number of alternatives, and n indicates the total number of criteria.
Step 2. Normalization of the elements of the initial matrix (X).
The elements of the normalized matrix (N) are obtained using the following expressions: a) For the "benefit" type criteria where ij x , i x + and i x − are the components of the initial decision matrix (X), where i x + and i x − are defined as: ( ) and represents the maximum value of the observed criteria by alternatives.
( ) and represents the minimum value of the observed criteria by alternatives.
Step 3. Calculation of the weighted matrix elements (V).
The elements of the weighted matrix (V) are calculated on the basis of expression (19): where ij t are the elements of the normalized matrix (N), and i w represents the weight coefficient of criteria.By applying the expression (19), we get the 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 the approximate border area (G) matrix.The border approximate area (BAA) for each criterion is determined by expression ( 21) where ij v are the weighted matrix elements (V) and m represents the total number of alternatives.
After determining the value i g according to the criteria, we form the matrix of approximate border areas G (22) 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 the border approximate area (Q) The distance of the alternatives from the border approximate area ( ij q ) is defined as the difference between the weighted matrix elements (V) and the values of the border approximate areas (G).
which otherwise can be written as: where i g represents the border approximate area for the criterion i K , ij v is the weighted matrix elements (V), n represents the number of criteria, and m represents the number of alternatives.The alternative i A may belong to the border approximate area (G), the upper approximate area ( G + ) or the lower approximate area ( G − ), respectively The upper approximate area ( G + ) is an area in which the ideal alternative is found ( A + ), while the lower approximate area ( G − ) is an area in which the anti-ideal alternative ( A − ) is found (Figure 3).Belonging of the alternative i A to the approximate area ( G , G + or G − ) is determined on the basis of expression ( 26) In order for the alternative i A to be chosen as the best from the set, it is necessary that, according to as many criteria as possible, it belongs to the upper approximate area ( G + ).If, for example, the alternative i A according to 5 criteria (out of 6) belongs to the upper approximate area, and according to one criterion belongs to the lower approximate area ( G − ), this means that by 5 criteria the alternative is close or equal to the ideal alternative, while by one criterion it is close or equal to the anti-ideal alternative.The higher the value i g G + ∈ , the closer the alternative i A is to the ideal alternative, while the lower the value i g G − ∈ , the closer the alternative i A is to the anti-ideal alternative.
Step 6. Ranking alternatives.Calculation of criteria function values by alternatives is obtained as the sum of the distances of alternatives from the border approximate areas ( i q ).Summing the elements of the matrix Q by rows gives the final values of the criteria function alternatives

Phase 1 -criteria identification and the calculation of criteria weight coefficients
For the full implementation of the model, it is necessary to define two types of criteria -the limit criteria and the evaluation criteria.In the set of the limit criteria, the criteria are classified on the basis of which the acceptance of specific locations as an alternative is made, or its rejection.After passing through the limit criteria assesment, the alternatives are further ranked using the evaluation criteria.The limit criteria are difficult to define as universal, because they depend on a number of circumstances based on which the decision is made, such as: type of operation, balance of powers, time it takes to create laying-up positions, etc.
In the most general sense, a certain location will not be considered as an alternative in the following situations: when it cannot provide even the minimum conditions for the protection and masking, when it cannot provide even the minimum conditions for the stay of military ships in all meteorological and hydrological conditions, when it is located next to visible objects, when it is not at a sufficient distance from the existing laying-up positions or other protection and masking facilities on the coast and along the coast, when the volume of work is so large that the production can be considered unrewarding, etc.
The evaluation criteria in the selection of the most suitable location for the development of laying-up positions are defined based on the analysis of the available literature.The most detailed description of the conditions which a location for making laying up positions should meet is given in (Milovanović, 1971, pp.169-170).Through a detailed analysis of all conditions, five key evaluation criteria stand out (Milovanović, 1971, pp.169-170), (Božanić et al., 2015a, pp. 695-696): • C 1 -masking conditions -the place where the laying up position is going to be placed should provide a good concealment of military ships, and during masking, as simple mask base as possible should be used (one that can be quickly set and removed); • C 2 -scope of work -a place where production of laying up positions is carried out must ensure the most favorable conditions for its development, ie. as small as possible volume of work, especially when performing underwater works; • C 3 -degree of protection -laying up position should provide the best possible protection from enemy effects from the air, by land or by river; • C 4 -benefits for sailing in and out-approaches to laying-up positions should be safe for quick sailing in and out; • C 5 -terrain conditions for the immediate security organization -the surrounding terrain should provide favorable conditions for possible defense against an imminent attack of a military ship, primarily from sabotage and terrorist groups.
The set of criteria C i (i = 1, .. 5) consists of two subsets: • C + -a subset of the "benefit" type criteria, which means that the higher value of the criteria is preferable or better (criteria: C 1 , C 3 , C 4 , C 5 ), and • C -a subset of the "cost" type criteria, which means that the lower value is preferable or better (criterion C 2 ).

Phase 2 -evaluation and selection of the optimal alternative
Through the second phase, the evaluation and ranking of alternatives is performed by the application of the MABAC method.The first step in the implementation of the method is to define the initial decision matrix.Since the evaluation criteria have a qualitative character, for the evaluation of alternatives by all criteria, the fuzzyficated Likert scale was used (Campari, 2013), Table 8.In the specific case, the fuzzyficated Likert scale is applied to provide the assessment of ten illustrative alternatives, whose values are defuzzyficated by using expression (2) and shown in Table 9.The next step is the normalization of the initial matrix elements by the application of expressions ( 16) and ( 17).The normalized matrix (N) is presented in Table 10.The weighted matrix (V) is calculated using expression (19), Table 11.Then the approximate border areas are calculated using expression (21), Table 12.The penultimate step is the calculation of the distance of alternatives from the border approximate area by the application of expression (24), Table 13.In the end, expression ( 27) is applied for ranking the alternatives, Table 14.Based on the obtained values Si, the alternatives are ranked from the most suitable alternative (A 1 ) to the most unfavorable one (A 9 ).

Sensitivity analysis of the output results
A logical sequence in most multi criteria decision-making processes is to analyze sensitivity.It is recommended as a means of checking the stability of the results against the subjectivity of decision-makers (Meszaros, Rapcsak, 1996).The sensitivity analysis of the results is carried out by changing the initial weight coefficients of the evaluation criteria.The situations of changing weight coefficients, from A to F, are provided in Table 15.Ranks of alternatives using different situations are given in Table 16.Table 16 shows a significant result stability compared to situations A, B, C and slightly less D (especially when it comes to the first five alternatives).The stability of the results compared to situations E and F is much smaller, which could be expected if we bear in mind that the importance of the favored criteria in the developed model is several times smaller.