# 3 Steps to Reduce the Number of Comparisons in Analytic Hierarchy Process

Written by Dawid Opydo

Analytic Hierarchy Process (AHP) is so time consuming! I don’t have time to make all these comparisons. I can’t expect my colleagues to spend so much time making judgments. Is there a way to make it quicker, please?

No worries. Help is on the way.

Usually, when we hear this plea for help, it’s because people have misunderstood how to get the most from AHP. This article will give you 3 easy steps to make your problem disappear. Oh, and they will probably also improve the quality of your decision.

So read on…

### 1. Replace pairwise comparisons with scales

This is the first step, where you can usually eliminate most of comparisons. Follow the best practices on where to (not) use pairwise comparisons. The main rule is to not use pairwise comparisons to score alternatives, especially if their number is bigger than 9. Read more here

### 2. Organize your criteria into hierarchy

Beginners tend to use flat lists of criteria instead of hierarchies. If you have lots of criteria, you will have a lot of pairwise comparisons to make. Think how to group your criteria together. For example, you might have “Machine Cost”, “Maintenance Cost” and “Production Cost” as criteria, but these should really be sub-criteria under an overall “Costs” criterion.

This not only reduces the number of comparisons you need to make, but it also improves the quality of the decision by focusing discussions on “big tradeoffs” (like Costs vs. Quality) before you work out “how to calculate score for Costs”.

The following example shows the dramatic reduction in comparisons you can achieve simply by organizing your criteria into a hierarchy.

### 3. Use AHP software that can complete your comparisons

Okay, so you’ve implemented scales and you’ve organized your criteria into a hierarchy and you’ve still got 10 sub-criteria on the same level meaning you need make 45 comparisons. This doesn’t happen often, by the way, but it could… so what do you do?

Well, TransparentChoice has the power to read your mind.

Not really, but it **can** extrapolate all those judgments from a much small number of comparisons. So, if you have 10 sub-criteria, you *could* just input 9 judgments and let TransparentChoice extrapolate the rest. The only problem is that the quality of the data will be low. The more comparisons you do, the more TransparentChoice can check for consistency between your answers and improve the quality of the result.

As with so many things in life, it’s a tradeoff. In the number of things you are comparing is small, say up to 6 or 7, then it’s probably worth going through all of the judgments. If, however, you are comparing a larger number of judgments, you may want to trade quality for speed.

Find out more: 5 Must-Have Features for Effective and Intuitive AHP Software

**GEEK CORNER.** If you’re interested in how the trade-off works, let’s look at how the software extrapolates judgments based on your inputs.

TransparentChoice analyzes the comparisons you provide and completes the missing comparisons with values that are most **consistent** with your input. For example, you enter comparisons A = 2xB; B = 4xC; so the algorithm can easily predict that A = 8xC.

Now this is fine if you really believe that A = 8xC, but let’s pretend for a moment that your first judgment was a mistake – you really meant that B = 2xA (your preference is the opposite of what you accidentally entered). If you stop entering judgments before the software asks you to compare A and C, you will never know you made a mistake and your results will be based on this mistake.

If, however, you **do** go on and compare A and C, you’d probably enter something like A = 2xC and TransparentChoice will flag up a whopping inconsistency for you. This gives you a chance to check your answers and increase the quality of your input.

## Example

Warning! You are leaving the safe zone. This part of the blog post includes some math. If you are allergic to it please skip this section and move right along to the summary.

We will use a “car selection” example to show you the process of reducing the number of comparisons you need to make.

Imagine that you are trying to choose a car. There are 6 alternatives and 12 criteria. Let’s assume you start, as beginners often do, with a flat list of criteria (all criteria on the same level of hierarchy) and use pairwise comparisons everywhere in your project (to prioritize criteria and to score alternatives).

Let’s count how many comparisons you would need to make to complete the process. The number of comparisons in given context (e.g. comparing criteria at the same level of a branch in a hierarchy, or comparing alternatives against a criterion) can be calculated as n*(n-1)/2, where n is number of items to compare. You need to:

- Prioritize 12 criteria in context of the overall goal = 66 comparisons
- Score 6 alternatives in context of each of the 12 criteria = 180 comparisons

So, in this project, the **total number of comparisons would be 66 + 180 = 246 **

Wow, that’s a lot of time spent inputting judgments.

And if this is a collaborative decision, where multiple people need to provide input, the time spent banging numbers into TransparentChoice gets silly.

But 6 alternatives and 12 criteria is still quite a small project… so let’s apply our three steps.

**Step 1. Replace pairwise comparisons with custom scales at the bottom of the hierarchy**

Effect: 180 comparisons can be replaced by 72 scores.

**Step 2. Organize the criteria into a hierarchy **

Let’s say that our 12 criteria are: price, fuel efficiency, insurance, airbags, traction control, ABS, noise, speed, fuel tank capacity, torque, audio system, air conditioning.

We will organize them into hierarchy by introducing 4 higher level criteria: convenience, economic aspects, performance, safety.

Let’s count comparisons again:

Prioritize 4 higher level criteria in context of goal = 6 comparisons

Prioritize 4 sets of 3 sub-criteria = 12 comparisons

Effect: Ok, so now, we can prioritize criteria with 18 comparisons instead of 66.

**Step 3. Use software to complete comparisons**

If we are desperate, we could even let TransparentChoice extrapolate some of the results. The minimum number of comparisons you could make is:

Prioritize 4 higher level criteria = 3 comparisons

Prioritize 4 sets of 3 sub-criteria = 8 comparisons

Effect: we reduced number of comparisons needed to prioritize criteria from 18 to 11. Given the small saving here, we’d recommend not bothering with step 3: live with 7 extra judgments and be sure your data are consistent.

In summary we replaced 246 comparisons with 11 comparisons and 72 scores on scale.

## Summary

Beginners often lose enthusiasm because they are faced with so many pairwise comparisons. Usually, this is because they’ve made a couple of basic mistakes; not using scales to score alternatives and having a flat hierarchy.

Usually, fixing these two errors will solve the problem, but if not, you can simply enter less data and let TransparentChoice do the rest. But there’s a tradeoff in decision quality.

I hope this helps!