## 3 The Methods of the Average Rank Transformation into Weight (ARTIW-L and ARTIW-N)
The average of the weights \(\overline{\omega}_{j}\) calculated for each criterion by these two methods can be considered as the result of the task calculation:  
\[\overline{\omega}_{j}=\frac{\omega_{j}+\omega^{\prime}_{j}}{2}. \tag{5}\]  
The consistency of expert group estimates is determined by the concordance coefficient \(W\). The concordance coefficient \(W\) in the absence of tied ranks is expressed in terms of the relationship between the obtained sum \(S\) and the respective largest sum \(S_{\text{max}}\)[58]:  
\[W=\frac{12S}{m^{2}n(n^{2}-1)}=\frac{12S}{m^{2}(n^{3}-n)}. \tag{6}\]  
When the estimates provided by the experts or respondents are in agreement, the Kendall coefficient of concordance, \(W\), is about one. When the estimates differ considerably, the value of \(W\) is close to zero. The deviations of the ranks \(R_{\bar{w}}\) of each criterion from the sum of squares of the average rank can be calculated as follows:  
\[S=\sum_{j=1}^{n}\left[\sum_{i=1}^{m}R_{ij}-\frac{1}{2}m(n+1)\right]^{2}, \tag{7}\]  
where \(n\) is the number of criteria (\(j=1,2,...,n\)), \(m\) is the number of experts (respondents) (\(i=1,2,...,m\)). The random value for \(S\) was calculated by Equation (7), adding the squared values of all the criteria given in parentheses. The concordance coefficient may be used in practice when its ultimate value, showing when expert estimates can be considered consistent, has been found. [PERSON] [57] has shown that if the number of criteria is \(n>7\), the significance of the concordance coefficient \(W\) can be determined using Pearson's chi-squared test statistic \(\chi^{2}\). The random value  
\[\chi^{2}=Wm(n-1)=\frac{12S}{mn(n+1)}, \tag{8}\]  
is distributed according to \(\chi^{2}\) distribution, with the degree of freedom \(\
u=n-1\). When the number of the compared criteria \(n\) ranges from 3 to 7, the distribution \(\chi^{2}\) cannot be used in all cases because sometimes the critical value of \(\chi^{2}_{\
u,\alpha}\) may be larger than the calculated value (even though the consistency of the estimates is still sufficiently high). In this case, the probability tables of the concordance coefficient or the tables of critical values \(S\) (with \(3\leq n\leq 7\)) can be used [64]. The smallest value of the concordance coefficient \(W_{\text{min}}\) allowing the authors to consider that the estimates of \(m\) experts of the quality of the research object based on \(n\) criteria, with the assigned (required) significance level \(\alpha\) and degree of freedom \(\
u=n-1\), are consistent, can be calculated as follows [63]:  
\[W_{\text{min}}=\frac{\chi^{2}_{\
u,\alpha}}{m(n-1)}, \tag{9}\]  
where \(\chi^{2}_{\
u,\alpha}\) is the critical Pearson's statistic found in the table [65], assuming the degree of freedom \(\
u=n-1\) and the significance level \(\alpha\). The quality of the research object is evaluated by the additive mathematical model used for calculating its comprehensive quality index, which allows for describing the quality of the object by a single number. It also allows for comparing it with the quality of other similar objects, and the coefficients \(\overline{\omega}_{j}\) of the normalized criteria weights (rather than the average criteria ranks \(\overline{R}_{j}\), which cannot show how one criterion is more important that another) are used. The weights of the criteria describing the research object (the selection of rail transport rather than road transport by passengers) can be calculated by using a very popular but complicated approach referred to as the analytic hierarchy process (AHP) offered by [PERSON] [54; 55; 56; 66]. Passengers are not highly qualified experts and, therefore, can hardly fill in a pairwise comparison matrix properly, particularly if the number of the criteria compared is large. This number may be more than nine (e.g., fifteen) criteria. In the study [67], passengers completed pairwise comparison matrices with 32 criteria, 22 of which were rejected because their C.R. (consistency ratio) was greater than 0.1.