# A Simplified Algorithm for Dealing with Inconsistencies Using the Analytic Hierarchy Process
## 3 Results
### Circular Triads  
Circular triads, or three-way cycles, are extreme cases of inconsistency where judgements are intransitive, such that \(i>j>k>i\). In cases where circular triads exist, the approach can still provide an indication of how preferences may be determined. If we consider a variant of the matrix presented in Section 3.1, such that \(A=\begin{bmatrix}1&4&1/5\\ 1/4&1&5\\ 5&1/5&1\end{bmatrix}\), where \(i>j\), \(j>k\) but \(k>i\). The scores are subsequently highly inconsistent (GCI = 7.06). The derived weights from the matrix are fairly equal, i.e., 0.31, 0.36 and 0.33 for \(i\), \(j\), and  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Attribute** & **Initial** & **Revised** \\ \hline
1 & 0.673 & 0.706 \\
2 & 0.251 & 0.176 \\
3 & 0.075 & 0.117 \\ GCI & 0.483 & 0.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial and revised relative weights from the example matrix.  
\(k\) respectively. In this case, the estimated expected values are \(\hat{a}_{1,3}=a_{1,2}\cdot a_{2,3}=4\cdot 5\Rightarrow 9\), \(\hat{a}_{2,3}=a_{2,1}\cdot a_{1,3}=(1/4)\cdot(1/5)\Rightarrow 1/9\) and \(\hat{a}_{1,2}=a_{1,3}\cdot a_{3,2}=(1/5)\cdot(1/5)\Rightarrow 1/9\). The normalized deviations in this case suggest that \(\hat{a}_{1,3}\) is the furthest from what may be expected. Replacing the value in the matrix with the expected value gives the new matrix \(\hat{A}=\begin{bmatrix}1&4&9\\ 1/4&1&5\\ 1/9&1/5&1\end{bmatrix}\). This matrix has an acceptable level of consistency (GCI = 0.213), however the derived weights from the matrix have changed considerably, i.e., 0.71, 0.23, and 0.06 for \(i,j\), and \(k\) respectively. Correcting the inconsistency has changed the ranking from \(j>k>i\) to \(i>j>k\) (i.e., a variant on the rank reversal phenomenon but not related to addition or removal of an attribute).  
The approach is less successful dealing with extreme circular triads. Consider the case of an extreme circular triad in the initial preference matrix with three attributes, given by \(A=\begin{bmatrix}1&9&9\\ 1/9&1&9\\ 1/9&1/9&1\end{bmatrix}\). In this case, it might be interpreted that the respondent feels both the first attributes is very much more important than the other two (but the strength of this is limited to a score of 9), and the second is also substantially more important than the third (again with the scores being limited to 9). The weights derived from this combination are 0.78, 0.18, and 0.04, reflecting the substantial importance of the first alternative and the minor importance of the latter. However, the matrix is inconsistent (GCI = 1.609).  
From step 1, the expected value in each case is \(\hat{a}_{1,3}=a_{1,2}\cdot a_{2,3}=9\cdot 9\gg 9\), \(\hat{a}_{2,3}=a_{2,1}\cdot a_{1,3}=9\cdot 1/9=1\) and \(\hat{a}_{1,2}=a_{1,3}\cdot a_{3,2}=9\cdot 1/9=1\). As the normalized deviation is the same for both \(\hat{a}_{1,2}\) and \(\hat{a}_{2,3}\), each case, i.e., \(\big{(}a_{i,j}-a_{i,j}\big{)}^{2}/a_{i,j}=(1-9)^{2}/9\), both values (and their inverse relationships) are replaced by their estimate. The revised matrix results in weights of 0.58, 0.28 and 0.14, maintaining the rank order of the three alternatives but reducing the importance of the first alternative. However, the consistency score with the revised matrix is unchanged (GCI = 1.609). Repeating the process in this case reverts back to the original matrix, and the process cycles without resolution.