Derive the weights, then check yourself
Compare criteria against each other, compare options on each criterion, and find out whether your judgements actually hang together before you trust the ranking.
Analytic Hierarchy Process
Weight the criteria by comparison, score the options by comparison, and check whether your judgements were self-consistent before trusting any of it.
Step 1 — How do the criteria compare against each other?
Step 2 — How do the options compare, on each criterion?
Result
Consistency
The mathematics
Each comparison grid is a reciprocal matrix: if A is 5× B, then B is 1/5 of A. Only the upper triangle is entered.
Priorities are the principal eigenvector of that matrix, obtained by power iteration. The widely-taught shortcut — normalise the columns and average the rows — agrees closely when judgements are consistent and drifts when they are not, which is precisely the case the consistency check is about.
Consistency ratio CR = CI ÷ RI, where CI = (λmax − n) ÷ (n − 1) and RI is Saaty's published random index for a matrix of that size. Below 0.10 is the accepted convention.
Final scores are each option's priority under a criterion, weighted by that criterion's priority, summed. All scores sum to 1, so they are shares rather than points.
What AHP adds over a scoring matrix
An ordinary weighted matrix asks you to invent the weights. AHP derives them from comparisons, which people make far more reliably than they assign numbers — and then it checks whether your comparisons were mutually consistent, which no scoring matrix can do at all.
That check is the reason to use it. 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 satisfies all three. A scoring matrix accepts the contradiction silently and produces a confident answer built on it.
Reading the consistency ratio
Below 0.10 your judgements hang together well enough to rely on. Above it, something is wrong — and it is usually 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 rather than thinking. Or two items are genuinely so close that the direction flipped between comparisons, which is worth knowing and easy to fix.
This is why the page names the specific judgements most out of line. On five criteria and four options that is fifty comparisons, and "your matrix is inconsistent" without a location is not actionable.
The honest criticisms
AHP has real critics and they have a point. The 1-to-9 scale is arbitrary. Adding an irrelevant option can in principle reorder the others — rank reversal — which offends against a property most people assume decision methods have. And the arithmetic gives the whole exercise a precision that the underlying judgements do not possess.
What it is genuinely good for is structuring a disagreement. When a group cannot agree on a decision, comparing pairs surfaces exactly where they differ far faster than arguing about the conclusion. The number at the end matters less than the conversation that produced it.
Keep it small
Comparisons grow as n(n−1)/2 per matrix, and there is one matrix per criterion plus one for the criteria. Four criteria and four options is 6 + 24 = thirty judgements, which is about the practical ceiling before fatigue starts showing up as inconsistency.