# Binary Decision Support Using AHP: A Model for Alternative Analysis
## 2 The AHP Method
#### Analyze the Consistency of Opinions  
In decision-making processes based on the AHP, evaluators may introduce inconsistencies in pairwise comparisons because of uncertainties, cognitive biases, or subjective judgment variations. Decision-makers need to assess the Consistency Ratio (CR) to ensure the reliability and coherence of expert evaluations. [PERSON] defines this ratio as a measure of logically consistent comparisons [5; 6]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Intensity of** & **Definition** & **Explanation** \\
**Importance** & **Definition** & **Explanation** \\ \hline
1 & Equal importance & Both elements contribute equally, \\
3 & Moderate importance & Experience and opinion favor one \\  & & element over the other. \\
5 & Strong importance & One element is strongly favored. \\
7 & Very strong importance & One element is very strongly favored \\  & & over the other. \\
9 & Extreme importance & One element is overwhelmingly \\  & & more important. \\
2, 4, 6, 8 & Intermediate values & Used when a compromise between \\  & & two judgments is needed. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: [PERSON]’s pairwise comparison scale. Numerical scale used for pairwise comparisons in the AHP, providing a structured way to assign importance levels [25]. Figure 1: A visual representation of the AHP method showing the hierarchical relationship between the decision goal, criteria, and alternatives (adapted from [24]). Inconsistency occurs when comparisons contradict each other. For example, if an evaluator assigns A > B and B > C, but then assigns C > A, this sequence introduces a logical inconsistency. Decision-makers must review and adjust their responses when the calculated CR exceeds 0.10 to mitigate such inconsistencies [5; 6]. Decision-makers calculate the CR by multiplying each row of the comparison matrix by the corresponding priority vector to generate a weighted sum vector. 1. \(\mathbf{\lambda}_{max}\)**calculation--According to [PERSON], one can estimate \(\lambda_{\max}\), the principal eigenvalue of the matrix, by dividing the values in the weighted sum vector by the respective priority vector values [28]. 2. **Consistency Index (CI) calculation--[PERSON] computes the Consistency Index using the following formula [28]:** \[CI=\frac{\lambda_{\max}-n}{n-1},\] **where** \(n\) **is the number of criteria or subcriteria in the comparison matrix. **
3. **CR calculation--[PERSON] defines the Consistency Ratio (CR) as the result of dividing the Consistency Index (CI) by the Average Random Index (ARI), a predefined constant that varies depending on the matrix size [27]:** \[\text{CR}=\frac{\text{CI}}{\text{ARI}}\] **If the resulting CR is less than or equal to 0.10, the evaluators consider the judgments as consistent and make no adjustments. However, if the CR exceeds 0.10, the evaluators reassess their pairwise comparisons to improve the logical coherence [29]. **  
Table 2 shows an example of a pairwise comparison matrix that evaluates the consistency of expert opinions, while Table 3 illustrates the corresponding weighted sum vector calculations. The evaluators calculate the Consistency Index (CI) using the \(\lambda_{\max}\) estimate, which they obtain by averaging the weighted sum values divided by their respective priority vector values (see Table 4). \begin{table}
\begin{tabular}{c c c c c} \hline
**Criteria** & **1** & **2** & **3** & **Priority Vector** \\ \hline
1 & 1 & 1/4 & 1/2 & 14.29\% \\
2 & 4 & 1 & 2 & 57.14\% \\
3 & 2 & 1/2 & 1 & 28.57\% \\ CR: 0.0000 & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Pairwise comparison matrix (between criteria) with a consistency check. An example of a pairwise comparison matrix used in a consistency analysis. \begin{table}
\begin{tabular}{c} \hline
**Calculation Method** \\ \hline (1 \(\times\) 0.14) + (1/4 \(\times\) 0.58)+(1/2 \(\times\) 0.29) = 0.43 \\ (4 \(\times\) 0.14) + (1 \(\times\) 0.58)+(2 \(\times\) 0.29) = 1.71 \\ (2 \(\times\) 0.14) + (1/2 \(\times\) 0.58)+(1 \(\times\) 0.29) = 0.86 \\ \hline \end{tabular}
\end{table}