## 3 The Methods of the Average Rank Transformation into Weight (ARTIW-L and ARTIW-N)
This method of determining the criteria weights is logical; however, its accuracy is low. Therefore, it can be used only at the initial stage of analysis. Using more accurate and complicated methods still requires preliminary ranking of the criteria. The weights \(\omega_{j}\) of the ranked indicators (criteria) can be determined by applying different methods (algorithms) that do not have theoretical advantages over one another. However, the general principle of all algorithms is the same: the most important criterion is assigned the highest weight. The values of the weights \(\omega_{j}\) must correspond to the criteria ranks (lower rank-higher weight). The sum of the weights \(\omega_{j}\) of all the criteria describing the research object must be equal to one, i.e., weights must be normalized. It is convenient to transform the ranks assigned to the criteria by a group of experts into weights by using the new ARTIW-L and ARTIW-N methods, whose sequence of operations and calculations is given below (see Figure 1).The average rank \(\overline{R}_{j}\), representing the ranks assigned by all \(i\)-th experts (\(i=1,\,2,\,\ldots,\,m\)) is calculated for each \(j\)-th criterion (\(j=1,\,2,\,\ldots,\,n\)) by the formula:  
\[\overline{R}_{j}=\frac{\sum\limits_{i=1}^{m}R_{ij}}{m}. \tag{1}\]  
The more important the criterion, the smaller its average rank \(\overline{R}_{j}\). In practice, it is more convenient to use the estimates of the criteria significance, whose numerical values show higher importance. For this purpose, the normalized weights of criteria \(j\), expressing relative importance, are used. Significances (weights) of the evaluation criteria of an object can be determined in the process of their normalization (setting their sum equal to one) by transforming the average ranks into weights (the ARTIW method). This method was first proposed in 2011 [63]; however, at that time it was not called ARTIW. A relative weight \(\omega_{j}\) of the criterion is calculated as follows:  
\[\omega_{j}=\frac{(n+1)-\overline{R}_{j}}{\sum\limits_{j=1}^{n}\overline{R}_{ j}}, \tag{2}\]  
where \(n\) is the number of criteria describing the quality of the considered object, \(\overline{R}_{j}\) is the average \(j\)-th criterion rank calculated by Equation (1). The normalized weights \(\omega_{j}\) of the \(j\)-th criteria calculated according to formula (2) have a linear inverse correlation (functional) relationship with the rank averages (average ranks) \(\overline{R}_{j}\) of these criteria calculated according to Formula (1). Therefore, this method is called the average rank transformation into weight-linear (ARTIW-L). The normalized weights \(\omega^{\prime}_{j}\) of the criteria can be calculated by using another method of transforming rank averages into weights. The criterion weights calculated according to Formulas (3) and (4) are related to criteria rank averages \(\overline{R}_{j}\) by a non-linear inverse correlation (functional) dependence. Therefore, this method is called the average rank transformation into weight-non-linear (ARTIW-N). Using the ARTIW-N method, the ratio of the min \(\overline{R}_{j}\) of the most important criterion (with the lowest average ranks \(\overline{R}_{j}\)) to the average of the ranks \(\overline{R}_{j}\) of all other \(j\)-th criteria is initially calculated:  
\[u_{j}=\frac{\min\limits_{j}\overline{R}_{j}}{\overline{R}_{j}}, \tag{3}\]  
After normalizing the values \(u_{j}\) for each criterion, their subjective significances \(\omega^{\prime}_{j}\) are calculated:  
\[\omega^{\prime}_{j}=\frac{u_{j}}{\sum\limits_{j=1}^{n}u_{j}}. \tag{4}\]  
Neither of these two methods (ARTIW-L and ARTIW-N) can be considered more accurate than the other, and neither of them can be looked at as the reference method. Figure 1: The algorithm of ARTIW-L and ARTIW-N methods of the average rank transformation into weight used for calculating normalized subjective weights of the criteria describing the research object.