## 5 Criteria Significance Research Methods
### Average Rank Transformation into Weight-Linear (ARTIW-L) Method  
This method calculates the averages of the ranks given by the experts (\(j=1,2,\ldots,n\)) participating in the research for each criterion (\(i=1,2,\ldots,m\)):  
\[\overline{R}_{i}=\frac{\sum_{j=1}^{n}R_{ij}}{n}, \tag{2}\]  
where \(R_{ij}\)--the rank assigned to the \(i\)-th criterion of the \(j\)-th expert and \(n\)--number of experts.  
The subjective normalized weight of each criterion is calculated using the linear method of transforming rank averages into weights (ARTIW-L):  
\[\omega_{i}=\frac{(m+1)-\overline{R}_{i}}{\sum_{i=1}^{m}\overline{R}_{i}}, \tag{3}\]  
where \(\overline{R}_{i}\)--\(i\)-th criterion average rank and \(m\)--the number of criteria describing the research object.  
The criteria weights \(\omega_{i}\) calculated by the ARTIW-L method are linearly correlated with the average ranks of these criteria \(\overline{R}_{i}\).  
To be able to take the experts group opinions' average expressed in ranks, scores, or weights, as the result of solving the problem, it is necessary to check whether the opinions of all experts are not contradictory. Only when the assessments of all experts are similar (non-contradictory), the average is a reasonable decision. For this reason, Kendall's variance concordance coefficient \(W\)[46], which shows the compatibility of expert opinions, is calculated and its numerical value can vary from 0 to 1:  
\[W=\frac{12S}{n^{2}(m^{3}-m)}=\frac{12S}{n^{2}m(m^{2}-1)}. \tag{4}\]  
The sum of the squares \(S\) of the deviations of the sum of ranks \(\sum_{j=1}^{n}R_{ij}\) of each \(m\)-th criterion from the average rank \(\overline{R}\) is calculated from the formula:  
\[S=\sum_{i=1}^{m}\left[\sum_{j=1}^{n}R_{ij}-\frac{1}{2}n(m+1)\right]^{2}, \tag{5}\]  
where \(m\)--criterions number (\(i=1,2,\ldots,m\)) and \(n\)--expert number (\(j=1,2,\ldots,n\)).  
The calculated value of \(W\) is compared with its minimum value \(W_{min}\), depending on the chosen significance level \(\alpha\) (take \(\alpha=0.05\) or stricter \(\alpha=0.01\)) and the number of degrees of freedom \(\
u=m-1\):  
\[W_{\min}=\frac{\chi_{\alpha,\
u}^{2}}{n(m-1)}, \tag{6}\]  
where \(\chi_{\alpha,\
u}^{2}\)--[PERSON]'s criterion statistic is found in the of Mathematical Statistics Appendix Table [46].  
Whether the opinions of the experts are consistent can be checked by calculating a random variable:  
\[\chi^{2}=Wn(m-1)=\frac{12S}{nm(m+1)}, \tag{7}\]  
which is distributed according to the \(\chi^{2}\) (chi-square) distribution with \(\
u=m-1\) degree of freedom. According to the selected significance level \(\alpha\) (0.05 or 0.01) from the \(\chi^{2}\) distribution table [47] with \(\
u=m-1\) degree of freedom, the critical value \(\chi_{\alpha,\
u}^{2}\) is found. If the value of \(\chi^{2}\) calculated according to Formula (7) is greater than \(\chi_{\alpha,\
u}^{2}\), it is considered that the experts' assessments are consistent (uncontradictory, similar).  
How many times the calculated concordance coefficient \(W\) is greater than its minimum (critical) value \(W_{\min}\) and how many times \(\chi^{2}\) is greater than its critical value \(\chi^{2}_{a,v}\) is shown by the new consistency coefficient \(k_{c}\), which is calculated from the formula:  
\[k_{c}=\frac{W}{W_{\min}}=\frac{\chi^{2}}{\chi^{2}_{a,v}}. \tag{8}\]  
When the experts' opinions are aligned, then \(k_{c}>1\). Otherwise (when \(k_{c}<1\)), the opinions of the expert group differ significantly, i.e., they are not aligned.