## 5 Criteria Significance Research Methods
### Analytic Hierarchy Process (AHP) Method
Quantitatively, the degree of compatibility of an inverse symmetric matrix can be determined by comparing the calculated compatibility index \(C.I.\) of the evaluated matrix with the random index of the inverse symmetric matrix of the same order randomly generated according to the scale 1-3-5-7-9 \(R.I.\)[49], presented in Table 3.  
The AHP method is best suited for researching an object that consists of seven criteria. Their number is considered optimal when there are two criteria less or more than 7, i.e., from 5 to 9 [50,51]. The method cannot be applied when there are less than three criteria because there are no _R.I._ meanings. However, it does not limit the number of criteria beyond \(m=9\), even though the nine-point Saaty scale is used. The greater the number of criteria that make up the research object, the more complex their pairwise comparison matrix is (when \(m>9\)), the more identical elements \(a_{ij}\) are in it. When the matrix consists of more than 15 criteria, _R.I._ value is calculated according to the formula [50]:  
\[R.I.=1.98\frac{m-2}{m}. \tag{17}\]  
To increase the reliability of the research result, the arithmetic averages of the weights assigned to the criteria of the expert group are taken, but only if the opinions of individual experts are harmonized. The consistency of expert opinions is determined by the calculated values of \(W\), \(W_{min}\), \(\chi^{2}\), and \(\chi^{2}_{\alpha,p}\) (Formulas (4)-(7)). The arithmetic mean of the weights of criteria calculated by four MCDM methods is calculated from the formula, without giving preference to any of them:  
\[\overline{\omega}_{i}=\frac{\sum_{k=1}^{r}\omega_{ik}}{r}=\frac{\omega_{i}^{ \rm ARTW-L}+\omega_{i}^{\rm ARTW-N}+\omega_{i}^{\rm DPW}+\overline{\omega}_{ i}^{\rm AHP}}{4}, \tag{18}\]  
where \(\omega_{ik}\)--the weight of the \(i\)-th criterion calculated by the \(k\)-th MCDM method (\(k=1,2\), \(\ldots\), \(r\)) and \(r\) is the number of methods used in the study (\(r=4\)). It is likely that \(\overline{\omega}_{i}\) is closer to the population mean that would be obtained by surveying a large number of experts for the values of the criterion weights calculated by any of the four MCDM methods used in the study. The width of the sample or the width of the data set (range) \(\Delta\omega\)i of the weights of each criterion determined by different methods is calculated from the formula:  
\[\Delta\omega_{i}=\omega_{\rm{imax}}-\omega_{\rm{imin}}, \tag{19}\]  
where \(\omega_{\rm{imax}}\) and \(\omega_{\rm{imin}}\) are, respectively, the maximum and minimum weight of the \(i\)-th criterion determined by the (four) different methods used in the study. The width of the data set \(\Delta_{\omega i}\) is the simplest characteristic of the variation. It can be used for an approximate but quick estimation of the standard deviation \(s_{\Delta\omega i}\)[52]:  
\[s_{\Delta\omega_{i}}=\frac{\Delta\omega_{i}}{\beta}=\frac{\omega_{\rm{imax}}- \omega_{\rm{imin}}}{\beta}, \tag{20}\]  
where \(\Delta\omega_{i}\) is the range determined according to the Formula (19) and \(\beta\)--coefficient depending on the number of data \(r\), i.e., number of MCDM methods used in the study. When \(r=4\), then \(\beta=2.06\). More precisely, the amount of data dispersion can be determined directly by calculating the standard deviation from all data (not from the difference between the maximum \(\omega_{\rm{imax}}\) and minimum \(\omega_{\rm{imin}}\) values):  
\[s_{\omega_{i}}=\sqrt{\frac{\sum_{k=1}^{r}\left(\omega_{ik}-\overline{\omega}_ {i}\right)^{2}}{r-1}}, \tag{21}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline
**Matrix Row \(m\)** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** \\ \hline Random index _R.I._ & 0.58 & 0.90 & 1.12 & 1.24 & 1.32 & 1.41 & 1.45 & 1.49 & 1.51 & 1.48 & 1.56 & 1.57 & 1.59 \\ \hline \end{tabular}
\end{table} 
Table 3: Random Index _R.I._ dependence of values on the order of the matrix (number of compared criteria) \(m\).