In this post, we explore the consistency indexes (CI) for a analytics hierarchy process (AHP).

Context

The AHP is a structured approach to organizing and comprehending decision-making in a multifactor scenario. It was created by Thomas L. Saaty in the 1970s and is used in a variety of domains to help in decision-making, resource allocation and risk analysis.

It consist on the following steps

Breaking down a decision problem into a hierarchical structure of:
- Criteria, i.e., the variables to consider and evaluate
- Alternative, i.e., the options being evaluated by the criteria
Making a pairwise comparison between the criterias. where each variable is compared to another, usually with a scale from 1 to 9, where 1 means equality between the criterias and 9 a higher relevance from criteria to another
After the comparison we can measure the consistency of the evaluation, and review if a evidence of inconsistency appears
Then with the priority weights for each criteria and we can rank the alternatives

Pairwise comparison

Let’s say we have \(n\) criterias then making a pairwise comparison we have \(\frac{n(n-1)}{2}\) comparisons between the criterias.

\[ \begin{bmatrix} c_{11} & c_{12} & ... & c_{1n} \\ c_{21} & c_{22} & ... & c_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ c_{n1} & c_{n2} & ... & c_{nn} \\ \end{bmatrix} \tag{1}\]

where \(c_{11}\) would be the comparison between the criteria \(c_1\) and \(c_1\). \(c_{12}\) would be the comparison between the criteria \(c_1\) and \(c_2\). and so on.

For this matrix we have some established results, such as:

\[ c_{ii} = 1: i = j. \tag{2}\]

The comparison of the same criterias (matrix diagonal) imply in no relevance between them. We also have that:

\[ c_{ij} = \frac{1}{c_{ji}}. \tag{3}\]

Since we compare a criteria in comparison to another, such as \(c_1\) in relation to \(c_2\) we do not need to evalutate \(c_2\) compared to \(c_1\), so we consider the inverse of the original evaluation.

Consistency indexes

Approximate eigenvector

After establishing the comparison matrix we compute the sum for each column:

\[ c_{.j} = \sum\limits_{i=1}^{n} c_{ij}. \tag{4}\]

And then we normalize the matrix by dividing each value by the respective column sum (Equation 4):

\[ \begin{bmatrix} c_{11}/c_{.1} & c_{12}/c_{.2} & ... & c_{1n}/c_{.n} \\ c_{21}/c_{.1} & c_{22}/c_{.2} & ... & c_{2n}/c_{.n} \\ \vdots & \vdots & \ddots & \vdots \\ c_{n1}/c_{.1} & c_{n2}/c_{.2} & ... & c_{nn}/c_{.n} \\ \end{bmatrix} \tag{5}\]

Considering the normalized value as

\[w_{ij} = c_{ij}/c_{.j}, \tag{6}\]

we can compute the mean for each row:

\[ p_{i} = \frac{1}{n}\sum\limits_{j=1}^{n} w_{ij}, \tag{7}\]

where \(p_i\) is the priority value of the respective criteria. Then we can multiply the values of each original criteria by their respective priority:

\[ \lambda_{\mathrm{max}} = \frac{1}{n}\sum\limits_{i=1}^{n} \left[\frac{1}{p_i}\sum\limits_{j=1}^{n} c_{ij}\right]. \tag{8}\]

At last the consistency index (\(CI\)) is computed as

\[ CI = \frac{\lambda_{\mathrm{max}} - n}{n-1}. \tag{9}\]

It is also possible to consider a consistency rate (\(CR\))

\[ CR = \frac{CI}{RI}. \tag{10}\]

where the \(RI\) is the random index, a fixed number based on the calculation from random matrices of different sizes (Saaty, 1980):

\(n\)	\(RI\)
1	0
2	0
3	0.58
4	0.90
5	1.12
6	1.24
7	1.32
8	1.41
9	1.45
10	1.49

Final considerations

The \(CR\) can be used to measure in numeric terms how consistency is the evalutation between the criterias, for example let’s say we evaluate that:

\(c_1\) is 8x more relevant than \(c_2\)
\(c_1\) is 2x more relevant than \(c_3\)
\(c_2\) is equaly relevant in comparison to \(c_3\)

The \(CR\) is computed as 22,7%, giving an evidence of inconsistency, since \(c_2\) and \(c_3\) are considered equal, but \(c_1\) is much more relevant to \(c_2\) than \(c_3\).

But how much can be considered a consistent index? In the literature a \(CR < 10\%\) is considered consistent (Saaty, 1980).

Want to give a try? Checkout this free web application of an AHP priority calculator.