AHP, and checking whether your judgements agree
People compare pairs reliably and assign weights badly. AHP exploits the first and then checks whether your comparisons contradict each other.
Why comparisons beat assigned weights
Asked to distribute 100 points across five criteria, people anchor on the first number, cluster the rest, and produce something different if asked again next week. Asked whether cost matters more than support, and by roughly how much, the same people answer consistently.
AHP asks only the second kind of question and derives the weights from the answers. That is the whole idea, and it is a good one.
The consistency ratio is the real contribution
If you say cost matters three times as much as support, support twice as much as reliability, but reliability more than cost, no set of weights can satisfy all three. You have contradicted yourself.
An ordinary scoring matrix accepts that silently and produces a confident answer built on it. AHP detects it. The consistency ratio compares your inconsistency against what randomly generated judgements of the same size would produce, and Saaty's convention is that 0.10 or below is acceptable.
Below three items there is nothing to contradict, so no ratio applies — and a tool reporting one there is reporting noise.
Inconsistency usually means one of three things
The items may not be comparable on a single dimension, in which case the question is malformed rather than the answer wrong. Someone may have been clicking through without thinking. Or two items are genuinely so close that the direction flipped between comparisons.
All three are worth knowing, and none is visible from a ranking. This is why a good implementation names which judgements are most out of line — on five criteria and four options there are around fifty comparisons, and "inconsistent" without a location is not actionable.
The eigenvector, and why the shortcut is not good enough
Priorities are the principal eigenvector of the comparison matrix. The commonly taught shortcut is to normalise the columns and average the rows.
That approximation is close for consistent matrices and drifts precisely when a matrix is inconsistent — which is the case you most need to be right about, since it is the case you are about to raise a warning over. Power iteration costs nothing and removes the question.
The criticisms are fair
AHP has serious critics. The 1-to-9 scale is arbitrary. Adding an irrelevant option can in principle reorder the others, which is called rank reversal and offends against a property most people assume decision methods have. And the arithmetic lends the exercise a precision the underlying judgements do not possess.
What it is genuinely good at is structuring a disagreement. When a group cannot agree, comparing pairs surfaces exactly where they differ far faster than arguing about the conclusion. The number at the end matters less than the argument it makes visible.
Keep it small
Comparisons grow as n(n−1)/2 per matrix, with one matrix per criterion plus one for the criteria themselves. Four criteria and four options is thirty judgements, which is roughly the ceiling before fatigue starts showing up as inconsistency — at which point you are measuring tiredness.