What is the Analytic Hierarchy Process (AHP)?

1. Structure the hierarchy

The first step of AHP is to organize the components of your decision-making problem into a hierarchy, where the hierarchy’s first level amounts to being clear about your over-arching decision-making goal – e.g. “I want to rank smartphones so I can choose one to buy”.

The second level involves identifying the criteria that matter to you when evaluating the alternatives you are considering for achieving your goal, and the third level is those alternatives – e.g. criteria for choosing a smartphone might include “camera”, “storage”, “looks” and “price”, and the alternatives are smartphone models.

It can sometimes be useful to further decompose your criteria into various sub-criteria which would form another level of the hierarchy – e.g. “camera” could be broken down into “for taking photos” and “for taking videos”.

The simple three-level hierarchical structure outlined above can be represented by a tree diagram like this:

2. Pairwise compare your criteria

The second step of the Analytic Hierarchy Process is for you to assess the relative importance of your criteria. This assessment is performed via a series of pairwise comparisons, often in the form of a pairwise comparison chart, also known as a pairwise comparison table or matrix.

To get started, create a pairwise comparison chart (matrix) by entering your criteria both down the left-hand side and across the top:

Next, go through each cell in the chart, one-by-one, and ask yourself: “How much more important is the criterion in the row than the criterion in the column?” – e.g. “How much more important is “camera” than “storage”?

With respect to your answers, fundamental to AHP are the following ratio scale values and corresponding verbal descriptions that are intended to capture the intensity of your preferences – e.g. where “3” means the row criterion is “3-times as important” as the column criterion, corresponding to “moderately more important”.

1: Equally important

2: Equally to moderately more important

3: Moderately more important

4: Moderately to strongly more important

5: Strongly more important

6: Strongly to very strongly more important

7: Very strongly more important

8: Very strongly to extremely more important

9: Extremely more important

Alternatively, if you think the row criterion is less important than the column criterion, use reciprocals of the numbers above to quantify the relative importance: 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8 or 1/9 – e.g. where “1/9” means the row criterion is “1/9th as important” as the column criterion, corresponding to “extremely unimportant”.

Thus, AHP requires you to pairwise rate the criteria on, in effect, a 17-point scale, resulting in the pairwise comparison table being filled in like this (where the 1s down the diagonal are criterion self-comparisons, such as “camera” vs “camera”, and are, in effect, filled in by default):

Notice that matching entries in the chart are reciprocal: e.g. if you say “camera” is 4-times as important as “price”, then “price” should be 1/4 as important as “camera”.

3. Calculate criterion weights

The weights on the criteria, representing their relative importance, are derived from your preference ratios in the pairwise comparison chart (matrix) via some intricate calculations usually performed using Excel or specialized software.

After summarizing the steps for performing the calculations, we’ll take you through each step in turn to give you confidence by demystifying them.

The calculations, which are based on an eigenvalue and power method, involve the following five steps. Other methods are available and produce similar results; e.g. see the comparative study by Ishizaka and Lusti (2006).

1. Square the pairwise comparison matrix.

2. Sum each row of the squared matrix to get a “row total” for each row.

3. Sum the “row totals” and then divide each “row total” by the sum. The resulting “priority vector” corresponds to the criterion weights (summing to 1).

4. Using the resulting matrix from step 1, repeat steps 1-3.

5. Repeat step 4 until there are no further significant changes to the priority vector (weights), e.g. to three or four decimal places of accuracy.

If you have any sub-criteria in your decision hierarchy, these steps are performed with your sub-criteria to find their weights too.

Let’s run through an example!

Here is the earlier pairwise comparison table with the fractions now expressed in decimal form, from which the criterion weights will be derived by following the five steps:

Step 1 is to square this matrix – i.e. multiply the matrix by itself – which involves calculating its “dot products”.

Start by multiplying each number in the “camera” row by the corresponding number in the “camera” column and sum the products: (1 × 1) + (0.2 × 5) + (3 × 0.33̇̇) + (4 × 0.25) = 4. This first “dot product” appears in the top-left cell of the squared matrix below. The other 15 dot products are calculated analogously.

Thus, next multiply each number in the “camera” row as above but this time by the corresponding number in the “storage” column, and again sum the products (= 1.305); and then multiply the “camera” row by the “looks” column and sum the products (9.800), and, finally, multiply the “camera” row by the “price” column and sum the products (15.400). Then, move down to the “storage” row and repeat the multiplications/summations outlined above for each of the columns; and then for the “looks” row; and, finally, the “price” row.

Thus, after squaring the matrix above, we have:

Step 2 is much easier, just requiring us to sum each row of the squared matrix to get a “row total” for each row:

Step 3 is to sum the “row totals and then divide each “row total” by sum. The resulting “priority vector” (final column) corresponds to the criterion weights (summing to 1).

Step 4 is to square the squared pairwise comparison matrix above, and repeat the earlier steps 1-3, resulting in this table and, most importantly, a slightly different set of weights:

Step 5 is to repeat step 4 (i.e. again, square the squared, squared pairwise comparison matrix) until there are no further significant changes to the priority vector (weights), e.g. to three or four decimal places of accuracy. In our example, this constancy (convergence) of the final weights is achieved after just one repeat, as above.

Thus, the most important criterion is “storage” (weight = 0.667), followed by “camera” (0.193) and then “looks” (0.081) and – of least importance – “price” (0.058).

Consistency of pairwise comparisons

The consistency of your pairwise comparisons, as summarized in the pairwise comparison chart, can be considered with respect to how well your comparisons satisfy the logical property of transitivity: e.g. if you say “camera” is 2-times as important as “storage”, and “storage” is 3-times as important as “looks”, then – by transitivity – “camera” should be 6-times as important as “looks”.

As well as transitivity, consistency also requires, as discussed earlier, that matching entries in the chart are reciprocal: e.g. if you say “camera” is 4-times as important as “price”, then “price” should be 1/4 as important as “camera”.

Consistency is measured by calculating Saaty’s “consistency ratio” (Saaty 1977, 1980); however, explaining its calculation is outside the scope of this article. A review of consistency measures for AHP is available from Pant et al (2022).

In general terms, and returned to later in the article, the more consistent you are, the higher the “quality” of your judgments in terms of their validity and reliability, and accordingly the more accurate are the weights derived from them.

From a practical perspective, the more consistent your pairwise comparisons, the faster your weights converge at step 5 of the calculations above.

4. Evaluate alternatives

Evaluating the alternatives on the criteria is like assessing the relative importance of your criteria at the previous step. This assessment is also done via a series of pairwise comparisons, again in the form of a pairwise comparison chart (matrix) but this time with a chart for each criterion.

As before, enter the alternatives both down the left-hand side and across the top of each pairwise comparison chart, beginning here with the “camera” criterion:

Next, go through each cell in the chart, one-by-one, and ask yourself: “With respect to this particular criterion, how much more do you prefer the alternative in the row to the alternative in the column?” – e.g. “With respect to their cameras, how much more do you prefer Phone X to Phone Y?

Ask analogous questions for every pair of alternatives on every criterion.

As at step 3, answers to these pairwise comparison questions are chosen from a nine-point ratio scale:

1: Equally preferred

2: Equally to moderately preferred

3: Moderately preferred

4: Moderately to strongly preferred

5: Strongly preferred

6: Strongly to very strongly preferred

7: Very strongly preferred

8: Very strongly to extremely preferred

9: Extremely preferred

If you think the row alternative is less preferred than the column alternative, use reciprocals of the nine-point scale, i.e. from 1 (“equally unpreferred”) to 1/9 (“extremely unpreferred”).

Thus, AHP requires you to pairwise rate the alternatives on a 17-point ratio scale for each criterion, where, again, matching entries in the chart need to be consistent; e.g. if you say Phone X is 6-times more preferred than Phone Z, then Z should be 1/6 as preferred as X:

Because there are four criteria for this example, we also need to create a pairwise comparison matrix of the alternatives for the other three criteria (notice, the ratios are all reciprocal, as well as being transitive):

Finally, in the same way as you did for the criterion weights earlier, calculate the priority vectors for the alternatives – one priority vector per criterion – to obtain scores for each alternative for each criterion by performing these five steps:

1. Square the pairwise comparison matrix.

2. Sum each row of the squared matrix to get a “row total” for each row.

3. Sum the “row totals” and then divide each “row total” by the sum. The resulting “priority vector” corresponds to the alternatives’ scores (summing to 1).

4. Using the resulting matrix from step 1, repeat steps 1-3.

5. Repeat step 4 until there are no further significant changes to the priority vector (scores), e.g. to three or four decimal places of accuracy.

Because the pairwise comparison matrixes above are consistent – both reciprocal and transitive (discussed earlier) – there is no need to perform steps 4 and 5 as the scores will not change, i.e. steps 1-3 are sufficient for consistent matrixes.

The results for each criterion are summarized in the four tables below. In each table, columns 2-4 comprise the squared pairwise comparison matrix (step 1), culminating in the “priority vector” (step 3) in the final column, corresponding to the alternatives’ scores on each criterion.

You might like to check your understanding by downloading our free Excel workbook and having a go at reproducing the four tables yourself (start with decimalized versions of the pairwise comparison charts above).

As for your criteria, but again outside the scope of this article – and unnecessary for this example – the consistency of your pairwise comparisons of alternatives for each criterion can be evaluated by calculating Saaty’s “consistency ratio”.

5. Combining weights and scores to rank alternatives

The final step of AHP is to combine the criterion weights from step 3 with the alternatives’ scores from step 5.

Weighted-sum model

For each alternative, multiply its score on each criterion by the criterion’s weight and sum the products to get the alternative’s total score, by which the alternatives are ranked.

The total scores for the three phones in our example are calculated from this simple linear equation:

(W_camera × S_camera) + (W_storage × S_storage) + (W_looks × S_looks) + (W_price × S_price),

where the W and S variables are the weights and scores respectively.

Thus, applying the weights and scores from the tables above:

• Phone X: (0.193 × 0.6) + (0.667 × 0.077) + (0.081 × 0.692) + (0.058 × 0.714) = 0.265

• Phone Y: (0.193 × 0.3) + (0.667 × 0.308) + (0.081 × 0.231) + (0.058 × 0.143) = 0.290

• Phone Z: (0.193 × 0.1) + (0.667 × 0.615) + (0.081 × 0.077) + (0.058 × 0.143) = 0.444

Now that we’ve gone through the Analytic Hierarchy Process, we find that Phone Z has the highest total score and so is the best (first), followed by Phone Y in second place and Phone X in third.

References