Analytic Hierarchy Process - Calculation Methods

After you enter pairwise comparisons for criteria or options, AHP Software calculates a relative weight for each factor. Those weights are then used to calculate the final weighted sum or weighted product for each option.

SpiceLogic AHP Software gives you several calculation methods. Most users can safely keep the defaults, but the options are available when you want to compare methods or reproduce a specific academic approach.

Try it live. Run a quick AHP calculation now: enter your criteria, make the pairwise comparisons, and see the priority weights and the consistency ratio instantly.

Open the AHP Calculator

AHP Software panel for selecting the priority calculation method (Approximate Eigen Vector, Largest Eigenvector, Geometric Mean, Fuzzy Geometric Mean) used to derive weights from the pairwise comparison matrix.

If you are in a hurry and unsure which method to choose, keep the default settings. The default methods are standard and recommended for normal use. The other methods are available for experimentation or for cases where your study requires a specific calculation method.

In practice, these methods usually produce very similar rankings. In rare cases, a different method may change the final recommendation, so it is useful to know what each method does.

Approximate Eigenvector Method

By default, AHP Software uses the Approximate Eigenvector method unless you choose another method.

This method is easy to understand. It provides a useful approximation of the weights, especially when the comparison matrix has low inconsistency.

Step 1: Normalize the Columns

Suppose we have the pairwise comparison matrix below.

Original pairwise comparison matrix for Cost, Comfort, and Safety used as the worked example for explaining the AHP Approximate Eigen Vector calculation.

Normalize each column so the values in that column add up to 1. First calculate the sum of the column, then divide each cell in that column by the column sum. The following figure shows the process.

Step 1 of the Approximate Eigen Vector method: each column of the pairwise comparison matrix is normalized so its values sum to 1, by dividing every cell by the column sum.

Step 2: Take the Arithmetic Mean of Each Row

After the matrix is normalized, calculate the arithmetic mean of each row. Those row averages become the priority weights.

Step 2 of the Approximate Eigen Vector method: take the arithmetic mean across each row of the normalized matrix to obtain the final criterion priorities.

Pie chart visualization of the criteria priorities obtained by the Approximate Eigen Vector method, with each slice sized by the criterion's calculated weight.

This is how SpiceLogic AHP Software calculates priorities when the Approximate Eigenvector method is selected.

AHP Software pairwise comparison panel showing the final calculated priorities for Cost, Comfort, and Safety when the Approximate Eigen Vector method is selected.

Largest Eigenvector Method

You can learn more about this method from this article: https://medium.com/dlprodteam/the-ahp-pairwise-process-c639eadcbd0e

Geometric Mean Method

With the geometric mean method, the software first calculates the geometric mean of each row in the pairwise comparison matrix. That value represents the raw priority for the factor in that row.

Because priority weights should add up to 1, the raw priorities are then normalized. Each raw priority is divided by the sum of all raw priorities. The figure below shows the process.

Worked illustration of the Geometric Mean method for AHP: compute the nth root of the row product for each row, then normalize the resulting vector so its values sum to 1.

The left side of the figure shows the original pairwise comparison matrix. For the first row, "Cost", multiply the row values: 1 x 5 x 4 = 20. The geometric mean is the third root of 20, written as 20 ^ (1/3.0). In Excel, you can calculate it with the formula shown below.

Excel POWER formula used to take the cube root of 20 (the product of the first row 1 x 5 x 4) when calculating a geometric mean by hand for the AHP example.

The geometric mean of the Cost row is 20 ^ (1/3.0) = 2.71. In the same way, the geometric means for Comfort and Safety are 0.405 and 0.908.

Next, add the raw priority values: 2.7144 + 0.405 + 0.908 = 4.028. Then divide each raw priority by 4.028. The final normalized priorities are Cost = 0.67, Comfort = 0.10, and Safety = 0.22.

When you choose the geometric mean method in SpiceLogic AHP Software, the diagram and table match this hand calculation.

AHP Software pairwise comparison panel set to the Geometric Mean method, displaying priorities (Cost 0.67, Comfort 0.10, Safety 0.22) that match the hand-calculated values.

Fuzzy Geometric Mean Method

For fuzzy AHP, we use the fuzzy geometric mean approach. The following video explains the method we adopted for the software: https://vimeo.com/1201933899

Calculating the Consistency Ratio

Every pairwise comparison panel also shows a "Consistency Ratio". To see how it is calculated, read the dedicated consistency-ratio page.

Calculating the Final Option Value

After the software calculates priorities from a pairwise comparison matrix, it uses those priorities to score the options.

Suppose the relative weight for Cost is 0.665. Also suppose there are two options, "Car 1" and "Car 2". The software compares Car 1 and Car 2 against Cost and produces a priority vector for that criterion.

AHP Software pairwise comparison of Car 1 versus Car 2 against the Cost criterion, used in the worked example for calculating the final option value.

The same process is repeated for each criterion. Each criterion gets its own option-priority vector.

AHP Software pairwise comparison of Car 1 versus Car 2 against the Comfort criterion in the worked option-value calculation.

AHP Software pairwise comparison of Car 1 versus Car 2 against the Safety criterion, the third comparison feeding into the weighted-sum option value calculation.

Then the software combines those values with either the Weighted Sum method or the Weighted Product method to calculate the final priority for each option.

After the pairwise comparisons, suppose the attributes for Car 1 are:

AHP Software panel listing Car 1's attribute values (Cost 0.75, Comfort 0.333, Safety 0.75) derived from the pairwise comparisons against each criterion.

And suppose the attributes for Car 2 are:

AHP Software panel listing Car 2's attribute values (Cost 0.25, Comfort 0.667, Safety 0.25), the complementary scores to Car 1 from the same option-vs-option comparisons.

The software now combines the option attributes to calculate the final value. The default method is "Weighted Sum", which is the regular recommended method. "Weighted Product" is also available if you want to experiment or compare methods.

AHP Software panel with the Weighted Sum method radio selected and the formula displayed for combining criterion weights with per-option attribute values.

If you switch to "Weighted Product", the displayed formula changes as shown below.

AHP Software panel with the Weighted Product method radio selected; the displayed formula switches from sum-of-products to product-of-powers for combining option attributes.

If you keep the Weighted Sum method, the following formula is used to calculate the option value.

= [Weight of Cost factor] * [Cost attribute for Car 1] + [Weight of Comfort factor] * [Comfort attribute for Car 1] + [Weight of Safety factor] * [Safety attribute for Car 1]

Calculate the value for Car 1:

= (0.67 x 0.75) + (0.1 x 0.333) + (0.23 x 0.75)

= 0.71

Calculate the value for Car 2:

= (0.67 x 0.25) + (0.1 x 0.667) + (0.23 x 0.25)

= 0.29

Car 1 scores 0.71, which is higher than Car 2 at 0.29. So AHP Software recommends "Car 1".

AHP Software final option value display with Car 1 scoring 0.71 versus Car 2's 0.29, so the application recommends Car 1 as the winning choice.