The Analytic Hierarchy Process (AHP)

Choosing by Weighing Up Many Subjective Factors

The Analytic Hierarchy Process (AHP) - Choosing By Weighing Up Many Subjective Factors

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Tread carefully, and be systematic in your approach.

How do you make a choice in a complex, subjective situation with more than a few realistic options?

You could sit and think over each option, hoping for divine inspiration – but you may end up more confused than when you started. Or you could leave it to fate – draw straws or pick a number. But, of course, this won't win you the Decision Maker of the Year award!

An all-too-common strategy is to simply wait out the problem, doing nothing proactively, until a solution is somehow chosen for you by circumstances.

None of these approaches are very effective. What you need is a systematic, organized way to evaluate your choices and figure out which one offers the best solution to your problem. So what do you do when you're faced with a decision that needs significant personal judgment and subjective evaluation? How do you avoid getting caught in the "thinking it over" stage? And how can you be more objective?

Combining Qualitative and Quantitative

As rational beings, we usually like to quantify variables and options to make objective decisions. However, the problem is that not all criteria are easy to measure.

To address this problem, Thomas Saaty created the Analytic Hierarchy Process (AHP) in the 1970s. This system is useful because it combines two approaches – the "black and white" of mathematics, and the subjectivity and intuitiveness of psychology – to evaluate information and make decisions that are easy to defend.

Let's look at an (admittedly slightly trivial) example. If you want to determine the best route to work in the morning, and travel time is your deciding factor, the decision making process is very straightforward and simple. You would use each alternative route for a week, time the commute, and choose the one that's fastest on average.

If, however, you carpool with other riders, and you have to consider everyone's priorities, the decision becomes much more complex. Larry is concerned about his personal safety, because one route goes through a dangerous part of town. Joanne wants to factor in a stop at a drive-through coffee shop, so that everyone can get coffee. Richard points out that Java Jolt is better than Cuppa Jo. There are several branches of each in the city, with both types accessible from all routes, although at different distances.

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Now you've got tangible and intangible, and quantitative and qualitative, factors to think about. And you have to consider the different perspectives and priorities of the various people.

AHP can combine these different types of factors and turn them into a standardized numerical scale. You can use this to make your choice objectively, while including all the decision criteria.

AHP Snapshot

Here's a quick overview of the Analytic Hierarchy Process.

  1. Build your "hierarchy"
    • Define your goal or objective.
    • Identify the choices you're considering.
    • Outline the major factors you'll use to evaluate each option.
    • Identify criteria (and any subcriteria of these) that you need to consider for each of these major factors. Link these to the major factors (see figure 1 below for an example of this.)
    • Continue to build a hierarchy of decision criteria until all factors are identified and linked.
  2. Establish your priorities
    • Using paired comparison, determine your criteria preferences (perhaps A is a little more preferred than B, B is much more preferred than C, and so on).
    • Rate these preferences from 1-9.
    • Repeat this for each level in your hierarchy. (The example below will make this clearer.)
  3. Synthesize, or combine, the ratings
    • Calculate weighted criteria scores that combine all of the ranking data.
  4. Compare the alternatives
    • Using those combined scores, calculate a final score for each alternative.

Tip 1:

If you're familiar with Paired Comparison Analysis, then this approach might sound familiar. The power of AHP is that it uses paired comparison to determine the relative weights of various criteria, and then it transfers them across each level of criteria to calculate overall weightings. From this you can calculate an objective score for each alternative.

Tip 2:

This is a complicated approach, and one that needs careful thought and calculation. Only use it if Paired Comparison Analysis isn't giving you the answer you want, or where the problem to be solved is complex and significant, and involves many different subjective factors.

Let's work through our example to determine which route – A, B, or C – is best for the group of carpoolers.

Step 1: Build Your Hierarchy

Define your goal at the top of the hierarchy:

  • Find the best route to work.

Identify your first level of evaluation criteria:

  • Commute time.
  • Safety.
  • Drive-through access (to coffee).

Decide if there are second level criteria, or subcriteria of these, related to any of your Level 1 criteria:

  • Drive-through access has two subcriteria:
    1. Java Jolt.
    2. Cuppa Jo.

Under each bottom level criterion, write down the alternatives you're considering:

  • Route A.
  • Route B.
  • Route C.

Figure 1: An Example Hierarchy

Example Hierarchy

Step 2: Establish Your Priorities

Have each decision maker rate the relative importance or preference for each criterion, at each level. Use a paired comparison approach.

Set up a matrix to compare each criterion to the others. We have three Level 1 criteria, therefore we need a 3x3 matrix.

Criterion Commute Time Safety Drive-through Access
Commute Time      
Drive-through Access      

Rank the importance of each criterion relative to the others, using the scale below.

Relative Importance Value
Equal importance/quality 1
Somewhat more important/better 3
Definitely more important/better 5
Much more important/better 7
Very much more important/better 9

Note 1:

The numbers 2, 4, 6, and 8 are half way positions between the values above.

Note 2:

The scale assumes that the ROW (first) criterion being ranked is of equal or greater importance than the COLUMN (second) criterion. If you have a pairing where the row criterion is less important than the column, use the reciprocal value (1/3, 1/5, 1/7, or 1/9).

Compare the criteria in the columns to the criteria in the rows.

  • Commute time is [much MORE important] than safety
  • Commute time is [somewhat MORE important] than drive-through access
  • Drive-through access is [extremely MORE important] than safety
  • Safety is [extremely LESS important] than drive-through access
  • Safety is [much LESS important] than commute time
  • Drive-through access is [somewhat LESS important] than commute time
Criterion Commute Time Safety Drive-through Access
Commute Time 1 7 3
Safety 1/7 1 1/9
Drive-through Access 1/3 9 1

Step 3: Calculate the Ratings

Now you have to calculate the overall weighting for each criterion. This is called a "priority vector" (PV) (don't worry about this term – it's not very helpful.)

Add each column in your ratings matrix.

Criterion Commute Time Safety Drive-through Access
Commute Time 1 7 3
Safety 0.14 1 0.11
Drive-through Access 0.33 9 1
TOTAL 1.47 17 4.11

Divide each entry by the total of its column.
Commute time ÷ Commute time total = 1/1.47
Safety ÷ Commute time total = 0.14/1.47

Criterion Commute Time Safety Drive-through Access
Commute Time 0.68 0.41 0.73
Safety 0.10 0.06 0.03
Drive-through Access 0.22 0.53 0.24

Notice how the columns add up to approximately 1.0. This is because the weights have now been standardized.

Because the ratings are subjective, we sometimes see inconsistencies. To "smooth" these out, calculate the average of each row. This is the final weight (priority vector) for each criterion.

Criterion Commute Time Safety Drive-through Access Priority Vector
Commute Time 0.68 0.41 0.73 0.61
Safety 0.10 0.06 0.03 0.06
Drive-through Access 0.22 0.53 0.24 0.33

This weighted score suggests the following:

  • Commute time represents about 61% of the final decision
  • Nearness to a coffee drive-through represents about 33% of the decision
  • Safety represents about 6% of the decision

Level 2 Criteria – If you had no subcriteria, you could move onto the next step and calculate final scores for each alternative route. In our example, drive-through access has another variable to consider – whether the coffee shop on the route is Java Jolt or Cuppa Jo.

Following the same steps as the Level 1 criteria, start with a 2x2 matrix.

Then add the ratings. For our example, we'll assume that Java Jolt is somewhat better (3) than Cuppa Jo.

Brand Java Jolt Cuppa Jo
Java Jolt 1 3
Cuppa Jo 0.33 1

Calculate the overall weighting.
Add each column.

Brand Java Jolt Cuppa Jo
Java Jolt 1 3
Cuppa Jo 0.33 1
Total 1.33 4

Divide each entry by the column total. Then average each row.

Brand Java Jolt Cuppa Jo Average/Priority Vector
Java Jolt 0.75 0.75 0.75
Cuppa Jo 0.25 0.25 0.25

Access to a Java Jolt is three times more preferred than access to Cuppa Jo.

Step 4: Compare the Alternatives

Compare each alternative based on the lowest level in your hierarchy of decision criteria. In our example (see figure 1), the lowest-level criteria are as follows:

  • Commute time.
  • Safety.
  • Java Jolt.
  • Cuppa Jo.


The weighting for drive-through access will be included as 33% of the final decision. It will be split 75/25 between nearness to Java Jolt or Cuppa Jo.

By using the lowest level of your hierarchy, you ensure that all variations of all options are considered.

You can have as many levels of subcriteria as you need to make a final decision. If Java Jolt and Cuppa Jo were relatively equal in their preference (score of 3 or less), then you could further break down the decision into taste, price, and muffin selection. With each level, the total weight always adds to about 1.0, and the overall weight is spread up the hierarchy through each subsequent calculation.

Create a comparison matrix for the first decision criterion you want to evaluate. We'll use commute time.

Commute Time Route A Route B Route C
Route A      
Route B      
Route C      

Then use the same 1-9 rating scale to determine how each route compares to the others, based on that decision criterion. We'll assume the following:

  • Route A is somewhat faster than Route B.
  • Route A is very much faster than Route C.
  • Route B is much faster than Route C.

Fill in the matrix, and calculate the priority vector.

Commute A B C
A 1 3 9
B 0.33 1 7
C 0.11 0.14 1
Total 1.44 4.14 17
Commute A B C Priority Vector
A 0.69 0.72 0.53 0.65
B 0.23 0.24 0.41 0.29
C 0.08 0.03 0.06 0.06

Repeat this comparison process for each of the remaining decision criteria.

  • Which route is safer than the other?
  • How much closer to a Java Jolt is one route than the other?
  • How much closer to a Cuppa Jo is one route than the other?
Safety A B C
A 1 0.14 0.14
B 7.00 1 0.2
C 7.00 5 1
Total 15 6.14 1.34
Safety A B C Priority Vector
A 0.07 0.02 0.11 0.07
B 0.47 0.16 0.15 0.26
C 0.47 0.81 0.74 0.68
Access to Java Jolt A B C
A 1 4 7
B 0.25 1 3
C 0.14 0.33 1
Total 1.39 5.33 11
Access to Java Jolt A B C Priority Vector
A 0.72 0.75 0.64 0.70
B 0.18 0.19 0.27 0.21
C 0.10 0.06 0.09 0.09
Access to Cuppa Jo A B C
A 1 1 0.11
B 1 1 0.2
C 9 5 1
Total 11 7 1.31
Access to Cuppa Jo A B C Priority Vector
A 0.09 0.14 0.08 0.11
B 0.09 0.14 0.15 0.13
C 0.82 0.71 0.76 0.77

Combine the overall weights, and determine a value for each route. (See below for this.)

The value for each route is the weighted sum of all rankings that are associated with it. The priority values (PVs) for each criterion have been added to the hierarchy below.

Figure 2: The Example Hierarchy Developed

Hierarchy Example

Calculate Route A's final score:

Commute Time PV (0.61) x Route A's Commute PV (0.65) +
Safety PV (0.06 ) x Route A's Safety PV (0.07) +
Drive-through Access PV (0.33) x Java Jolt PV (0.75) x Route A's JJ PV (0.7) +
Drive-through Access PV (0.33) x Cuppa Jo PV (0.25) x Route A's CJ PV (0.11) +
= 0.40 + 0 + 0.17 + 0.01
= 0.58

Complete the calculations for Routes B and C.

Final Scores
Criterion Route A Route B Route C
Commute Time 0.40 0.18 0.04
Safety 0.00 0.02 0.04
Java Jolt 0.17 0.05 0.02
Cuppa Jo 0.01 0.01 0.06
Total 0.58 0.26 0.16

Route A is the clear winner! You can interpret this to mean that Route A meets 58% of all the decision criteria considered. Route B meets only 26% of the criteria, and Route C meets 16%.

Key Points

The Analytic Hierarchy Process can help you quantify the judgments you use in decision making. When problems become complex, it's hard to justify and explain all the reasons why one alternative is better, or more preferable, than another. With AHP, you calculate weighted scores for each set of criteria that you consider, and then you use those weights to calculate a final score for each alternative. The result is an "apples to apples," quantitative comparison of your choices. Whether you use this method to make a final choice or as one of many tools in your decision making process, the results can be remarkably clear.