Analytical Hierarchy Process-Detailed guide?

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Analytical Hierarchy Process-Detailed guide?

Don’t you think that our “tomorrow” lies in the decisions we are making today? Throughout our entire life, we make choices, sometimes based on 2 criteria and sometimes based on more. The process of decision making based on multiple criteria called MCDA or Multi-Criteria Decision Analysis. So, The Analytical Hierarchy Process is one of the most famous MCDM methods. But What is the Analytical Hierarchy Process? Let’s dive in without further introduction.


What is Analytical Hierarchy Process 

When you combine individual performance indicators to one key performance indicator you can give each one a different weight. Now the question is how to derive the weights, For this, a mathematical method is available which is called AHP, i.e. the Analytical Hierarchy Process.

Also, The AHP or Analytical Hierarchy Process is an organized strategy for arranging and breaking down complex choices, based on arithmetic and brain research.

As said in Wikipedia, in the 1970s Thomas L. Saaty suggested the AHP for the first time and revolutionized the studies of MCDM. Thomas L. Saaty, later partnering with Ernest Forman developed the Expert Choice in 1983.

Calculation of the AHP

The method is to derive ratio scales from paired comparisons. It also allows some small inconsistency in judgment. As an import, you can use actual measurements like price, weights, and so on or subjective opinions like satisfactory feelings or preferences, and as an output, you will get ratio scales and a consistency index. So, The method is based on the solution often an eigenvalue problem. The ratio scales result from eigenvectors and the consistency index from the eigenvalue. 

The process is done in several steps. First, you have to define your objective, Then you have to structure the elements in groups of criteria, sub-criteria, and alternatives. In each group, you make a pairwise comparison of elements and calculate the weighing and consistency ratio. Then you can evaluate the alternatives according to the weighing and get a ranking. 

Let us take an example. Your objective is to buy a gadget like a smartphone or MP3 player. The criteria are the colors of the model, the memory, and the delivery time. The colors pink, blue, green, and black are available. The memory space ranges from 8GB to 64GB and the delivery time is immediate or 5 days, or 4 weeks. 

4 models are available as shown here. 2 models with a price of $120 and 2 models with a price of $150. These are your alternatives.

Mathematical Representation of the AHP

Now in the first step, you structure the elements in groups of criteria, sub-criteria, and alternatives. The objective is to buy the gadget and your criteria are color, memory, and delivery time. 

To each criterion, you have sub-criteria, the color, the memory space, and the delivery times. You then have to compare all elements pairwise concerning the objective. So in the first step, you have to compare the criteria, color, memory, and delivery. 

You start to compare the color with memory and you are using a scale ranging from 9 to 1/9. 1 means both criteria have the same importance and they are equal. 9 means criteria color is 9 times more important than memory. 1/9 means memory is 9 times more important than color.

Analytical Hierarchy Process

So let’s make the comparison. You compare the color with memory and in your opinion memory is 3 times more important than color so on the scale you will have 1/3. Then you compare the color with delivery you say delivery is 2 times more important than color then you get the result as 1/2. Memory and delivery are equally important in your opinion, so you put in a 1. For 3 criteria you have 3 comparisons. 

Analytical Hierarchy Process

In the next step, you’ll arrange your comparisons into a matrix as shown in this image. From this matrix, you compute the normalized principal eigenvector. 

As a result, you get the following weighting. Colour- 17 percent, Memory- 43 percent ,and Delivery- 40 percent. 

Analytical Hierarchy Process


So the most important criteria are memory followed by delivery followed by color. Now you make the same pairwise comparison for the sub-criteria, in this case, the different colors. 

Analytical Hierarchy Process

For example, you compare the pink color and the blue color, you think pink is 2 times nicer than blue so you put a 2. Pink compared to green, in your opinion pink is being 3 times better than green and so on. 

In total, because you have 5 criteria, you need to do 10 comparisons. Again you arrange the result of that comparison in the matrix and compute the normalized principal eigenvector of the matrix. As a result, you will get the percentages as shown here. 

Analytical Hierarchy Process

Now you’ll weight the sub-criteria according to the weights of the main criteria the complete results will look like this. The next step is now is to evaluate, the alternatives. We have available model 1, the color is pink and pink weights of 2.2 percent. The memory space of model one is 32 GB. 32 GB has a weightage of 19 percent. The delivery time is immediate. Immediate, as a result, get 18 percent. So after counting together you will get the first result of the benefit for model 1, i.e. 39 percent. You do the same for all of the models and then you would see that model 3 has the highest ranking.

Analytical Hierarchy Process

Probably you’ll notice that we didn’t put in the price of the models as one of the criteria. The reason is very simple, If you separate the benefits from the course we can do a cost-benefit analysis. 

Graphical representation of the AHP

You can draw a diagram showing the benefits as a function of the relative costs of the models. 

So you’ll put in the models according to the benefits resulting from the analysis over the relative costs as it is shown here for models 1 to 4. Now let’s assume you have only selected model 1 or 3.

As you can see, in the diagram model 3 has similar benefits compared to model 1 but with a higher cost. Probably you would go for model 1 with immediate delivery and a lower price. If only model 2 and model 4 would be available, then you can see that model 4  has significantly higher benefits than model 2, Probably you would go for model 4 accepting the longer delivery and higher price. 

Uses of the AHP

As opposed to recommending a “right” choice, the AHP enables decision-makers to discover one that best suits their objective and their comprehension of the issue. It gives an extensive and reasonable system for organizing a decision problem, measuring its components, relating those components to general objectives, and assessing alternative solutions.

The users of the AHP first disintegrate their decision problem into a chain of command, i.e, the hierarchy of all the more effortlessly comprehended sub-problems, where each one can be analyzed individually. The components of the hierarchy can relate to any part of the decision problem.

When the hierarchy is developed, the decision-makers methodically assess its different elements by contrasting them with one another two at once, as for their effect on a component above them in the hierarchy. 

In conducting the comparison process, the decision-makers can utilize solid information about the components. It is the essence of the AHP that human decides, and not simply the fundamental data can be utilized in playing out the assessments.

The AHP changes over these assessments to numerical values that can be prepared and looked over to the whole scope of the problem. A numerical weight or priority is determined for every element of the hierarchy, permitting assorted and frequently incommensurable elements to be contrasted with each other rationally and consistently. This ability recognizes AHP from other decision-making procedures.

In the final step of the procedure, numerical priorities are determined for every one of the choices. These numbers represent the option’s relative capacity to accomplish the decision goal, so they permit a direct thought of the different courses of action.

Decision circumstances to which the AHP is applied include: 


The choice of one option from a given arrangement of options, for the most part where there are various choice standards or criteria included. 


Putting a lot of choices all together from most to least alluring. 


Determining the overall value of individuals from a set of choices, rather than choosing a solitary one or simply positioning them.


Comparing the procedures in one’s association with those of other best-of-breed associations.


Settling debates between parties with obviously contrary objectives or positions.

The utilizations of AHP to complex choice circumstances have numbered in thousands, and have delivered results. Numerous AHP applications are never answered to the world everywhere because they occur at significant levels of enormous associations where security and protection considerations prohibit their revelation.

Real-life Examples of AHP

Experienced professionals realize that the ideal approach to comprehend the AHP is to work through cases and models. Two nitty-gritty contextual analyses, explicitly structured as inside and out showing examples, are given as informative supplements to this article: 

Basic bit by bit example with four Criteria and three Alternatives: Choosing a pioneer for an association. 

Increasingly intricate bit by bit example with ten Criteria/Subcriteria and six Alternatives: Buying a family vehicle and Machinery Selection Example. 

Some portion of the books on AHP contains practical instances of its utilization. However, they are not normally expected to be step by step learning tools. One of them contains a bunch of extended models, in addition to around 400 AHP hierarchies quickly depicted and represented with figures. 


The AHP i.e. the Analytical Hierarchy Process is defined as an organized strategy for arranging and breaking down complex choices, based on arithmetic or brain research that has revolutionized the studies of Mathematics, Statistics, and Psychology since the 1970s.

Its contribution to the modern studies of MCDM is considerable.

Also You can read our Blog on Paired Comparison Analysis- A tool for Decision Making


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