AHP Consistency Check
The priority vector shows relative weights among the things we compare and can be used in the AHP method.
Matrices used in the analytic hierarchy process (AHP) compile expert knowledge as pairwise comparisons among various criteria and alternatives in decision-making problems. Many items are usually considered in the same comparison process, so judgment is not entirely consistent – and sometimes, the level of consistency may be unacceptable.
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Fill every step with “Weighing laptop criteria” and see a result in one click.
Enter at least three criteria
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How AHP Consistency works
In the Analytic Hierarchy Process (AHP) you express preferences as pairwise comparisons: how much more important is one criterion than another? Because people can contradict themselves, AHP measures this with a consistency ratio. This tool calculates the consistency ratio of your pairwise comparison matrix and the priority vector, which you can use as criteria weights.
Step by step
- List 3 to 10 criteria (or any items you want to compare).
- Fill in the comparisons above the diagonal. A value of 1 means two criteria are equally important, 3 means the row criterion is moderately more important than the column criterion, 5 strongly more, 7 very strongly more and 9 extremely more. The calculator fills in the reciprocal values below the diagonal for you.
- Each column is normalized by its sum, and the priority vector is the average of every row of the normalized matrix.
- The principal eigenvalue is estimated by multiplying each column sum by the matching priority vector entry and adding the results.
- The Consistency Index is the distance of that eigenvalue from the matrix size n, and the Consistency Ratio divides it by the Random Index for that size.
- A Consistency Ratio below 0.10 is treated as consistent.
Formulas
Worked example
Four laptop criteria are compared in pairs. Price is judged 2 times as important as Performance, 3 times as Battery and 5 times as Weight; Performance 2 times as Battery and 3 times as Weight; Battery 2 times as Weight.
| Price | Performance | Battery | Weight | |
|---|---|---|---|---|
| Price | 1 | 2 | 3 | 5 |
| Performance | 1/2 | 1 | 2 | 3 |
| Battery | 1/3 | 1/2 | 1 | 2 |
| Weight | 1/5 | 1/3 | 1/2 | 1 |
| Criterion | Priority (weight) |
|---|---|
| Price | 0.482 |
| Performance | 0.272 |
| Battery | 0.158 |
| Weight | 0.088 |
The Consistency Index is about 0.006 and the Consistency Ratio about 0.007, far below 0.10, so the judgments are consistent. The priority vector (0.482, 0.272, 0.158, 0.088) can be typed in as the weights in TOPSIS, ELECTRE, SAW, WPM or Linear Assignment.
Use the "Load example" button at the top of this page to fill in this data and run it yourself.
Frequently asked questions
What is a good consistency ratio?
Below 0.10 (10%) is the usual AHP threshold, and the calculator reports the matrix as consistent when the ratio is under it. A higher ratio means your comparisons contradict each other too much, for example A is much more important than B, B much more than C, but C is judged more important than A.
How do I fill in the pairwise comparison matrix?
Only the cells above the diagonal are editable. Enter how many times more important the row criterion is than the column criterion (typically between 1 and 9, or a fraction such as 0.5 if the column criterion matters more). The cell below the diagonal shows the reciprocal automatically.
What can I do if my matrix is not consistent?
Review your comparisons and adjust the ones that contradict others. The calculator reports the ratio but does not point to the specific inconsistent judgments.
What is the priority vector for?
It is the set of weights implied by your comparisons and adds up to 1. Use it as the criteria weights in the other calculators.
How many criteria can I compare?
Between 3 and 10 criteria. The comparison values must be greater than zero.