## 2 Materials and Methods
### Fuzzy AHP  
Since the weight values for factors in the aforementioned forest fire susceptibility formula were pre-assigned, there is a justified need for a more objective distribution of the weighting values for each factor that is used for forest fire susceptibility determination. In order to obtain such distribution, the fuzzy AHP method was used. It should be emphasized that that in this case, same parameters with the same class values as for the Forest Fire Susceptibility Index calculation were used. The difference is in the weighted values of each factor used. Fuzzy sets were introduced for the first time by [PERSON] [38] in the form of object classes with a continuum of membership grades. Fuzzy AHP is a technique of incorporating vagueness or fuzziness of human thoughts in decision making [16]. In the fuzzy AHP model, a combination of AHP and fuzzy sets is used to weight the contributing factors in forest fire occurrence [10] and to model forest fire susceptibility. This modelling method uses expert ideas to express the importance and priority of each factor that contributes to forest fire occurrence. The fuzzy sets enter into the modelling process to express uncertainty and to obtain more accurate results than the ordinary AHP method [39]. The constituent of a fuzzy system is a fuzzy number. In this paper, for forest fire susceptibility assessment, a triangular fuzzy number was used, which is characterized by the form of the affiliation function \(\mu\)(x). The triangular fuzzy number is defined by the values \(\mathrm{l_{ij}}\), \(\mathrm{m_{ij}}\), and \(\mathrm{u_{ij}}\) (provided that \(\mathrm{l_{ij}}\leq\mathrm{m_{ij}}\leq\mathrm{u_{ij}}\)). The given values determine the boundaries of the fuzzy set, where the x-axis shows the size of the fuzzy number (left and right distribution), and the y-axis shows the degree of affiliation of the fuzzy number (between 0 and 1) [9; 40]. The fuzzy AHP method involves the process of fuzzification of [PERSON]'s scale [41]. The conversion of [PERSON]'s numerical to the fuzzy AHP scale was performed using a triangular fuzzy number with a predefined confidence interval [40; 42; 43]. The first phase of fuzzy AHP method involves creating a fuzzy comparison matrix A at each level of the hierarchical structure using a triangular fuzzy number and fuzzy Saaty's scale [10; 44; 45; 46; 47]. After creating the fuzzy comparison matrix, it is necessary to calculate the value of the fuzzy synthetic extent (Si) for each element [9; 10; 44; 48; 49; 50]. In the next phase, calculation of the degree of possibility that \(\mathrm{M_{2}}=(\mathrm{l_{2}}\), \(\mathrm{m_{2}}\), \(\mathrm{u_{2}})\geq\mathrm{M_{1}}=(\mathrm{l_{1}}\), \(\mathrm{m_{1}}\), \(\mathrm{u_{1}})\) should be carried out [9; 45; 49]. In the following, it is needed to analyze the degree of possibility that the convex fuzzy number is greater than the k convex number \(\mathrm{M_{i}}\) (i = 1,2,..., k) and is defined by the expression [9; 48; 50; 49]. \[\mathrm{V(M\geq M_{1}}\mathrm{,\,M_{2},\,\ldots,\,M_{k})=V[(M\geq M_{1})\text { and (M\geq M_{2}) and }\ldots\text{ and (M\geq M_{k})] =}} \tag{5}\] \[=\min\,V(M\geq M_{i})\text{, i = 1,2,...,\,k}\]  
Assume that  
\[d^{\prime}\,(\mathrm{A_{i}})=\min\,V(\mathrm{S_{i}\geq S_{k});\,k=1,2,\, \ldots,\,n;\,k\
eq i} \tag{6}\]  
From the previous step arises the weight vector \(W^{\prime}\), and through normalization, the vector is reduced to the form \(W\)[9; 48; 50; 49],  
\[W^{\prime}=(d^{\prime}\,(\mathrm{A_{1}})^{\prime},\,d^{\prime}\,(\mathrm{A_{2} })^{\prime},\,\ldots,\,d^{\prime}\,(\mathrm{A_{n}}))^{T} \tag{7}\]  
\[W=(d(A_{1}),\,d(A_{2}),\,\ldots,\,d(A_{n}))^{T} \tag{8}\]  
where W is a non-fuzzy number [9; 48]. The final phase of the procedure represents the calculation of comparison matrix consistency. The comparison matrix, created using a triangular fuzzy number, needs a process of defuzzification.