Analytic Hierarchy Process (AHP)

How do you set priorities by comparing the importance of several criteria? If you are asked how much weight to put on price, quality, brand and service when choosing a product, the more criteria there are, the harder it becomes to assign importance directly as numbers. The Analytic Hierarchy Process (AHP) compares the criteria two at a time and, based on those judgments, calculates weights, the relative importance of each criterion. It is used for supplier selection, evaluating policy alternatives, setting business priorities, choosing products and more.

AHP breaks a complex decision problem into levels such as the goal, criteria, sub-criteria and alternatives, then compares the elements on the same level two at a time.

The key idea of AHP is that instead of entering the importance of every criterion directly all at once, you judge which of two criteria is more important and by how much. Answering "which matters more, price or performance?" is easier than splitting 100 points across four criteria at once.

Pairwise comparison puts two elements side by side as a pair and rates their relative importance. When comparing price and performance, you ask which is more important, whether they are about equal, or how much more important one is than the other. AHP usually uses a scale from 1 to 9.

A comparison in the opposite direction is expressed as the reciprocal. If performance is rated 3 compared with price, then the importance of price relative to performance is recorded as 1/3. This score is not an objective measurement that the importance is exactly three times as large; it puts the respondent's relative judgment into numbers.

Suppose the criteria are price, performance and battery, and the judgments were made as follows (hypothetical judgments for illustration).

The diagonal values are all 1 because each criterion is compared with itself, and if price vs. performance is 1/3, then performance vs. price is 3. A matrix with these reciprocal relationships is called a reciprocal matrix. AHP finds the eigenvector corresponding to this matrix's largest eigenvalue λ_max, then normalizes it to sum to 1 to obtain the weights.

With these judgments, performance has the highest weight at about 62.5%, price about 23.8%, and battery about 13.6%. When calculating by hand, a common approximation is to take the geometric mean of each row and normalize; for this matrix it gives the same values as the eigenvector. In practice the weights vary with the decision maker's judgments, and the decision is not over once the weights are set. You also need to compare how the alternatives perform before you can set the final ranking.

One important step in AHP is the consistency check. If the judgments are perfectly consistent (e.g., if price is 2× battery and performance is 3× price, then performance is exactly 6× battery), λ_max equals the number of criteria n, and the more the judgments conflict, the further λ_max rises above n.

The matrix above is not perfectly consistent, since performance vs. battery was judged 4 rather than 6, but with λ_max = 3.018 and CR = 0.016 it is quite consistent. In general, CR ≤ 0.10 is often regarded as acceptable consistency. This threshold is not an absolute rule, however, and should be reviewed in light of the purpose of the study and the nature of the judgments.

Conversely, if you judge as in Figure 3 that "price is more important than performance (3), performance is more important than battery (4), yet battery is much more important than price (5)," you get a cycle. λ_max grows to 5.17 and CR, at 1.87, far exceeds the threshold, and the weights spread almost evenly across the three criteria, so no meaningful priority emerges. If the consistency ratio is high, it is a good idea to revisit the comparisons. Checking that the actual judgments make sense matters more than simply getting past a numeric threshold.

The ultimate aim of AHP does not end with the importance of each criterion. Under each criterion, the alternatives are also compared pairwise to get their local priorities, and these are combined using the criterion weights.

Figure 4 sets the criterion weights at price 0.30, performance 0.50 and battery 0.20, and shows the result of hypothetical pairwise comparisons of laptops A, B and C under each criterion. A is highest on price (0.540), B on performance (0.625), and C on battery (0.582). The weighted final priorities are B 0.423, C 0.325, A 0.252, so B, which leads on performance, the criterion with the largest weight, ranks first. Depending on the size of the problem and the data, the alternatives can also be compared using actual performance data instead of pairwise comparisons, or combined with other evaluation methods (Ch. 06, TOPSIS).

With n criteria, the number of distinct pairwise comparisons needed at one level is n(n − 1)/2. Four criteria need 6 comparisons, eight need 28, and ten need 45, so too many criteria put a heavy burden on respondents and tend to hurt the consistency of their judgments. If there are many criteria, it is better to split them into a hierarchy of higher-level criteria and sub-criteria.

• AHP: Break the problem into a hierarchy of goal · criteria · alternatives and compare elements on the same level pairwise.
• Pairwise comparison matrix: The principal eigenvector of the reciprocal matrix, filled in with the 1–9 scale and reciprocals, gives the weights.
• Consistency: CI = (λmax − n)/(n − 1), CR = CI/RI, and CR ≤ 0.10 is commonly used as the acceptance threshold.
• Synthesis: Multiply the criterion weights by the alternatives' priorities under each criterion and add them up to get the final priorities.
• Caution: The number of comparisons grows as n(n − 1)/2, and even with high consistency the validity of the judgments must be checked separately.

Next is 06 Ranking alternatives: TOPSIS. We will look at a method that compares each alternative's distance from an ideal alternative, based on the weights and each alternative's performance.