MULTI-CRITERIA FUCOM – FUZZY MABAC MODEL FOR THE SELECTION OF LOCATION FOR CONSTRUCTION OF SINGLE-SPAN BAILEY BRIDGE 1

: Selecting the most favorable location for construction of single-span Bailey bridge is ideal for applying multi-criteria decision making. In that regard, it has been developed a model for selecting the most favorable location. The first part of the model is based on the full consistency method (FUCOM), and it is used for the evaluation of weight coefficients of criteria. The second part of the model presents the fuzzification of the Multi-Attributive Border Approximation Area Comparison (MABAC) method, which is used in the evaluation of alternatives. Additionally, in the paper are presented basic criteria, based on which the selection is to be made


Introduction -problem description
The set for launching Bailey bridge consists of a number of elements used to make single-span and multi-span bridges (bridges on standing supports) which are designed for overcoming dry and water barriers.These bridges are mounted on the banks, and after mounting their construction are launched over dry or water barrier.They can be easily adapted to different length or capacity requirements.Their main disadvantage is large mass of the parts of the set, which can significantly slow down the mounting of the bridge itself.These sets are included in the engineering units of the Serbian Army.The bridges made of this material can be found throughout Serbia, and in some places they represent significant link between the two banks.
The selection of location for mounting a single-span Bailey bridge is ideal field for the application of multi-criteria decision making methods.Potential locations where such bridges could be placed usually have significant differences that more or less affect the speed of assembly and human and material resources necessary during the construction process.By correct selection of location for such bridge can be prevented potential problems in the process of its construction and later use.
In this paper, the selection of location for the construction of a single-span Bailey bridge is carried out using the FUCOM -fuzzy MABAC method.Weight coefficients of criteria are calculated using the FUCOM method, while for ranking alternatives is used fuzzy MABAC method.
Both methods are very young and have not been largely applied so far.The FUCOM method was developed in 2018 by Pamučar et al. (2018).In the same year Prentkovskis et al. (2018) used this method as a part of the model for Improving Service Quality Measurement.Crisp MABAC method was announced for the first time in 2015 by Pamučar and Ćirović (2015).As a new method, it has been noted by the researchers quickly, and now there are many papers using this method in problem consideration, independently or as a part of a hybrid model (Božanić et al., 2016a;Peng & Yang, 2016;Chatterjee et al., 2017;Hondro, 2018;Majchrzycka & Poniszewska, 2018;Ji et al., 2018;Peng & Dai, 2018).In some papers, the method is used in fuzzy environment (Roy et al., 2016;Xue et al., 2016;Sun et al., 2017;Hu et al., 2019;Yu et al., 2017), and it has also appeared combined with rough numbers (Sharma et al., 2018;Roy et al. 2017).

Methods
Considering that the hybrid FUCOMfuzzy MABAC model consists of two methods, in the following section of the paper these two methods will be described in detail.

FUCOM
This method is a new MCDM method proposed in (Pamučar et al., 2018).In the following section, the procedure for obtaining the weight coefficients of criteria by using FUCOM is presented.
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.Thus, the criteria ranked according to the expected values of the weight coefficients are obtained: where k represents the rank of the observed criterion.If there is a judgment of the existence of two or more criteria with the same significance, the sign of equality is placed instead of ">" between these criteria in the expression (1) Step 2. In the second step, a comparison of the ranked criteria is carried out and the comparative priority


, 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   represents the significance (priority) that the criterion of the C  rank.The comparative priority of the criteria is defined in one of the two ways defined in the following part: a) Pursuant to their preferences, decision-makers define the comparative priority   among the observed criteria.b) Based on a predefined scale for the comparison of criteria, decision-makers compare the criteria and thus determine the significance of each individual criterion in the expression (1).The comparison is made with respect to the first-ranked (the most significant) criterion.Thus, the significance of the criteria ( ) for all of the criteria ranked in Step 1 is obtained.Since the first-ranked criterion is compared with itself (its significance is , a conclusion can be drawn that the n-1 comparison of the criteria should be performed.
As we can see from the example shown in Step 2b, the FUCOM model allows the pairwise comparison of the criteria by means of using integer, decimal values or the values from the predefined scale for the pairwise comparison of the criteria.
Step 3. In the third step, the final values of the weight coefficients of the evaluation criteria     ) defined in Step 2, i.e. that the following condition is met: (2) In addition to the condition (3), the final values of the weight coefficients should satisfy the condition of mathematical transitivity, i.e. that is obtained.Thus, yet another condition that the final values of the weight coefficients of the evaluation criteria need to meet is obtained, namely: Full consistency i.e. minimum DFC (  ) is satisfied only if transitivity is fully respected, i.e. when the conditions of


are met.In that way, the requirement for maximum consistency is fulfilled, i.e.DFC is 0   for the obtained values of the weight coefficients.In order for the conditions to be met, it is necessary that the values of the weight coefficients   T 1 2 n w , w ,..., w meet the condition of , with the minimization of the value  .In that manner the requirement for maximum consistency is satisfied.
Based on the defined settings, the final model for determining the final values of the weight coefficients of the evaluation criteria can be defined.

Fuzzy МАВАС method
The MABAC method is developed by (Pamučar & Ćirović, 2015).It is developed as the method providing crisp values.In this paper is carried out its fuzzification.The fuzzyfication is performed using triangular fuzzy numbers.A general form of triangular fuzzy number is given in the Figure 1.The fuzzyfication of the MABAC method is taken from (Božanić et al., 2018), and its mathematical formulation is presented in seven steps.
Step 1. Forming of the initial decision matrix ( X ).In the first step the evaluation of m alternatives by n criteria is performed.The alternatives are shown by vectors , where xij is the value of the i alternative by j criterion (i = 1,2, ... m; j = 1,2, ..., n).where m denotes the number of the alternatives, and n denotes total number of criteria.
Step 2. Normalization of the initial matrix elements ( X ).The elements of the normalized matrix ( N ) are obtained by using the expressions: For benefit-type criteria The elements of the weighted matrix ( V ) are calculated on the basis of the expression (11 where ij t represent the elements of the normalized matrix ( N ), i w represents the weighted coefficients of the criterion.
Step 4. Determination of the approximate border area matrix ( G ).The border approximate area for every criterion is determined by the expression ( 12): where ij v represent the elements of the weighted matrix ( V ), m represents total number of alternatives.
After calculating the value of i g by criteria, a matrix of border approximate areas G is developed in the form n x 1 (n represents total number of criteria by which the selection of the offered alternatives is performed).
  Step 5. Calculation of the matrix elements of alternatives distance from the border approximate area ( Q ) 21 22 2n m1 m 2 mn q q ... q q q q Q ... ... ... ... q q ... 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 ).
The values of alternative i A may belong to the border approximate area ( G ), to the upper approximate area ( G  ), or to the lower approximate area ( G  ), i.e., The upper approximate area ( G  ) represents the area in which the ideal alternative is found ( A  ), while the lower approximate area ( G  ) represents the area where the anti-ideal alternative is found ( A  ), as presented in the Figure 2. The membership of alternative For alternative i A to be chosen as the best from the set, it is necessary for it to belong, by as many as possible criteria, to the upper approximate area ( G  ).The higher the value i qG   indicates that the alternative is closer to the ideal alternative, while the lower the value i qG   indicates that the alternative is closer to the anti-ideal alternative.
Step 6. Ranking of alternatives.The calculation of the values of the criteria functions by alternatives is obtained as the sum of the distance of alternatives from the border approximate areas ( i q ).By summing up the matrix Q elements per rows, the final values of the criteria function of alternatives are obtained where n represents the number of criteria, and m is the number of alternatives.
Step 7. Final ranking of alternatives.By defuzzification of the obtained values i S , the final rank of alternatives is obtained.The defuzzification can be performed with the next expressions (Seiford, 1996):

Description of criteria and calculation of weight coefficients
The criteria for selecting the most favorable location for a single-span Bailey bridge are defined based on the analysis of the available literature.The analysis sets out seven key criteria that have the greatest influence on the selection, and they are the following (Kočić, 2017): C1-Access roads C2-Scope of work on site arrangement C3-Properties of banks C4-Width of water barrier C5-Masking conditions C6-Scope of works on joining access roads with the crossing point C7-Protection of units The concept of access roads (C1) refers to the number and quality of the roads by which the resources are brought to the location for construction and launching of the bridge over the water barrier, or close to it.These are the roads with adequate surface which does not require significant repairs and reconstructions.Through this criterion several elements are considered: capacity, number and width of access roads, as well as the position of roads in relation to the barrier (administrative or lateral) (Pamučar et al., 2011).
The scope of work on site arrangement (C2) represents the workload required for the site arrangement.In other words, it refers to the works necessary for arranging a place of work, where the space for storage of the parts of the set is arranged, parking of motor vehicles, place for stuff operation, space for rest, material disposal, and space for assembly and launching of the bridge (Božanić, 2017).
Properties of the banks (C3) refer to the soil composition of the bank, height of the bank, slope of the bank, forestation, artificial barriers, and the like.
The width of water barrier (C4) is defined as the distance from one bank to the other, measured by the surface of water (Pifat, 1980).
Masking conditions (C5) include measures and procedures undertaken to hide the activities and arrangement of the forces, assets and objects from the enemy, in order to lead the enemy to wrong conclusions, to make wrong decisions and apply wrong actions (Rkman, 1984).
Scope of works on joining access roads with the crossing point (C6) refers to the roads that ensure moving the unit from the nearest access road to the crossing point over the water barrier.
Unit protection (C7) is an integral and essential part of every operation.This criterion includes the assessment of the measures that must be taken to ensure required level of unit protection.
The set of criteria from C1 to C7 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 C1, C3, C5 and C7), 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 C2, C4 and C6).
The criterion C4 is presented as numerical, while the other criteria are presented as linguistic.
The weight coefficients of criteria are obtained by applying the FUCOM method.The evaluation of the weight coefficients is performed by 9 decision makers (DM)experts in the field of the subject matter.For all decison makers is carried out the evaluation of competence.
In the second step, the decision makers compared in pairs the ranked criteria from the step 1.The comparison is made according to the first-ranked criterion, based on the scale   1, 7 .This is how the importance of the criteria is obtained ( for all the criteria ranked in the step 1 (Table 1).
Finally, in the third step and based on the comparison performed by DM, applying the expressions 3-5 are obtained the values presented in the Table 2. Having been obtained the weight coefficients of criteria by every DM, it is performed the calculation of the aggregated weight coefficient.Such calculation was carried out by subsequent synthesis of individual decisions by the method of averaging using geometric mean (Geometric Mean Method -GMM) applying the expression (Zoranović & Srđević, 2003): where: Final, aggregated values of the weight coefficients are presented in the Table 3.

Model testing
The testing of the model, respectively, fuzzy MABAC method is performed with six alternatives.Before the very beginning of the testing, fuzzy linguistic descriptors had been defined which were used to describe linguistic criteria C2 and C6: a=very small (VS), b=small (S), c=medium (M), d=large (L), e=very large (VL).
The initial decision making matrix is shown in the Table 4.

Sensitivity analysis
In this section is presented sensitivity analysis, as a logical sequence of the development of the multi-criteria decision-making model.The sensitivity assessment was done by changing the weight coefficients of the criteria, using seven different scenarios, where in each scenario the second criterion was favorable (Pamučar et. al. 2017).The display of weight coefficients according to the scenarios is given in Table 7.Based on sensitivity analysis of the results from the Table 8, it can be observed that the model in the midst of change of weight coefficients provides also the change of ranks of the given alternatives.It is interesting to note, though, that the firstranked alternative A6, no matter the scenario, not once was ranked as the fifth or the sixth, and the alternative A3 which was ranked as the last, not in one scenario appeared as the first one.
For the mathematical determination of the correlation of ranks, the values of Spirman's coefficient were used: where is:  S -the value of the Spirman 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, final values of the weight coefficients should satisfy the two conditions: (1) that the ratio of the weight coefficients is equal to the comparative priority among the observed criteria ( Figure 1.Triangular fuzzy number

Figure 2 .
Figure 2. Display of upper ( G  ), lower ( G  ) and border ( G ) approximatearea(Pamučar & Ćirović, 2015) aggregated value of the weight coefficient,   i ak-value of the weight coefficient for every k-th DM where k=1,...K, k b -additionally normalized competence coefficient of the k-th DM;

Figure 3 .
Figure 3. Graphic display of fuzzy linguistic descriptors(Božanić et al., 2016b) Every criterion can be described with five values: Multi-criteria FUCOM -Fuzzy MABAC model for the selection of location for construction of … i x  and i x  represent the elements of the initial decision matrix ( X ), whereby i x  and i x  are defined as follows:

Table 2 .
Weight coefficient of criteria by every DM individually

Table 3 .
Final weight coefficient of criteria

Table 5 .
Quantification of linguistic descriptors

Table 6 .
Ranking of alternatives

Table 7 .
Weight coefficient in different scenario The values obtained by applying different scenarios are given in Table8.

Table 8 .
Ranking of alternatives by applying different scenarios