AHP guide: pairwise comparisons and priorities

The Analytic Hierarchy Process (AHP), developed by Thomas Saaty, structures a decision into a goal, criteria and alternatives. You compare items in pairs, calculate their relative weights and combine those weights into a ranking.

Open the AHP calculator

1. Define the goal and feasible alternatives

Write a specific question, such as “Which laptop best fits my work and budget?” First exclude alternatives that fail essential requirements. AHP ranks the alternatives you include; it does not replace checking whether they are feasible.

2. Choose clear criteria

Use criteria that match the goal, such as cost, performance and portability. Avoid overlapping criteria that count the same benefit twice. For each criterion, decide what “better” means: lower cost may be preferred, while higher performance may be preferred.

3. Compare criteria in pairs

Ask which of two criteria matters more for this decision and how strongly. In the calculator, add each criterion and press Enter. Choose Slider or Matrix, then record each comparison on the 1–9 scale.

Intensity	Meaning
1	Equal importance
3	Moderate importance
5	Strong importance
7	Very strong importance
9	Extreme importance
2, 4, 6, 8	Intermediate judgments

If A is preferred to B with intensity 3, the reverse matrix entry is 1/3. This expresses a judgment of preference, not necessarily a physical measurement that is exactly three times larger.

4. Compare alternatives under every criterion

Select a criterion tab and compare each pair of alternatives specifically for that criterion. Repeat for every tab. In Slider mode, move toward the preferred item; press an arrow key to record a comparison, including an equal judgment by returning to the center. In Matrix mode, fill the cells above the diagonal; reciprocal entries are filled automatically.

While comparisons are incomplete, the live projection assumes equal importance for unanswered pairs. Complete every pair before calculating a final ranking.

5. Review consistency

For matrices with three or more items, the consistency ratio (CR) checks how well the comparisons agree with one another. A CR of 0.10 or less is a commonly used acceptance threshold. If it is higher, revisit the reasoning behind your comparisons instead of changing numbers just to pass the check.

A two-item matrix has only one independent comparison, so this consistency check is not applicable. A low CR does not prove that your information is accurate, your criteria are complete or the decision will succeed.

6. Interpret the ranking

The calculator uses the principal eigenvector method to calculate weights. Each alternative’s local weight under a criterion is multiplied by that criterion’s weight, and these contributions are added to obtain a final score. Score shares are not probabilities of success.

Use “Calculate final ranking” to confirm the result, then open the calculation details to inspect matrices, weights and consistency metrics. Download CSV to save the results. Test reasonable changes to your judgments to see whether the top alternative is robust.

Worked example: two laptops

This is a hypothetical teaching example. Suppose cost is moderately more important than performance: their weights are 0.75 and 0.25. Under cost, you prefer laptop A to B with intensity 3, giving A a local weight of 0.75. Under performance, you prefer B with the same intensity, giving A a local weight of 0.25.

Laptop A: 0.75 × 0.75 + 0.25 × 0.25 = 0.625
Laptop B: 0.75 × 0.25 + 0.25 × 0.75 = 0.375

A ranks first in this model because cost has greater weight. If performance becomes more important instead, the ranking reverses. Neither result is a product recommendation or a measure of the laptops’ absolute quality.

Uses and limitations

AHP can help compare suppliers, prioritize projects, evaluate locations and structure expert judgments. It makes preferences explicit, but the result is only as useful as the model and evidence. Overlapping criteria, weak information and unmodelled dependencies can mislead.

A matrix with n items requires n × (n − 1) / 2 independent comparisons. More criteria and alternatives mean more judgments, so begin with a focused model. Reassess the model if you change the available alternatives.