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This method of decision support is based on a breakdown of the decision process into a hierarchical structure and a pairwise comparison of the different criteria..

## Introduction

The AHP method, Analytic Hierarchy Process, was developed by Saaty in the 1970s 1 . While consulting for the American government, he developed it to optimize the allocation of resources. This method can be used whenever we have to make a decision based on several criteria :

• Manufacturing technology
• Choosing to design a new product

## The principle

This method is based on a breakdown of the decision process into a hierarchical structure. This pyramid structure is always composed at the top of the decision subject, then at a criterion level, then at a sub-criterion level … and at the bottom of the different alternatives that we have. ## 1 – Define the subject

First obvious step, define the subject on which we must make a decision. For example : What is the best technology to ensure the proper crimping of a bottle? What is the best car regarding my my need ?

## 2 – Construct a hierarchical analysis of the subject

We will start by setting up levels of evaluation, from the most general level to the most specific. There can be 1 or X level. More generally, there will be 2 levels of criteria: global criteria and specific sub-criteria.

Example :

We wish to choose a car. We hesitate between 3 models, and we build the hierarchy of choice as follows. ## 3 – Put in place the first level evaluation matrix

For all levels, we will set up a pair analysis matrix to assess the importance of each one of them.

Saaty2 offers a 5-level rating according to the following table :

 Description Numerical value Both elements have equal importance 1 The online element is a bit bigger than the column one 3 The online element is more important than the column element 5 The online element is much more important than the columnar one 7 The online element is absolutely more important than the columnar one 9 The online element is a bit less important than the columnar one 1/3 The online element is less important than the column element 1/5 The online element is much less important than the columnar one 1/7 The online element is absolutely less important than the one in column 1/9 In some cases, if we have difficulties to evaluate, we will refine by adding intermediate values. 2, 4, 6, 8

Exemple

Our subject is therefore the choice of the car. The first level of criteria is Price, Equipment and Space. We build the evaluation matrix and we get the following result :

 Prix Equipment Habitability Price 1 9 3 Equipment 1/9 1 1/5 Habitability 1/3 5 1

The matrix reads as follows :

• The price is absolutely more important than the equipment: it was noted matrix reads as follows 9
• Space is more important than the equipment : the note 5

We notice that :

• The diagonal is always composed of 1, a criterion having an importance equal to itself …
• If the price has a rating of 9 in relation to the equipment, then the equipment has a rating of 1/9 against the price.

## 4 – Calculate the weight of each criterion

In the previous step, we constructed a square matrix that results from a 2 to 2 evaluation of the criteria. By calculating the eigenvalues of this matrix, we will obtain the weight of each of our criteria. The calculation is as follows :

1. The sum is done by column.
2. We divide each of the values of the column by the sum of these.
3. The weight (which corresponds to the eigenvalue of the matrix) is given by calculating the average of each of the lines.

Exemple

Using the previous example, we get the following results :

 Weight Price 67% Equipment 6% Habitability 27%

## 5 – Check the consistency of the result

At this point, we have the “weight” of each of the level 1 criteria. The AHP method then proposes to validate the reliability of the result by calculating a consistency index. Indeed, the potential inconsistency can come from the fact that a criterion may not be judged coherently with respect to one or more others. This index will therefore allow us to detect important defects in our calculation and our evaluation.

For example, if we reverse the judgment of the equipment and the space, the rest being equal. In other words, we now consider that equipment is more important than space. We obtain the following matrix.

 Prix Equipment Habitability Price 1 9 3 Equipment 1/9 1 1/5 Habitability 1/3 5 1

It is clear that we have an inconsistency of judgment since on the one hand we have the price which is absolutely more important than the equipment and slightly more important than the space, whereas the equipment is considered more important than the equipment than space.

The consistency ratio can put this forward. His calculation is as follows :

RC = IC / RI

Avec :

• RI, Random index developed by Saaty : It indicates the level of reliability of the same evaluation carried out several times.
• IC, Coherence Index : it represents the level of reliability of our judgment (details of the calculation below).
• RC, Ratio of Coherence : By dividing IC by RI, we compare our real reliability with a theoretical reliability. At most this ratio is close to 0, the more our evaluation is consistent. Saaty gives a value of more than 10% error in order to accept the evaluation. Otherwise, it will be necessary to redo the evaluation.

The consistency index is calculated as follows :

CI = (λmax – n) / (n- 1)

•  λmax : Maximum eigenvalue. For details of the calculation, download the attached excel file.
• n : Matrix size.

We simply note that for a size 2 matrix, it will not be possible to calculate an incoherence ratio, since incoherence is not possible….

Example

In our case, the IC value is 0.023 and the RC is 0.04. Our matrix is therefore coherent.

In the second example where we deliberately create an inconsistency, we obtain the following values: IC at 0.74 and RC at 1.28, so very largely incoherent. In this case, it will be necessary to investigate to find out where the inconsistency comes from.

## 6 – Iterate steps 4 and 5 for subcriteria groups

For each of the following levels, we will set up a matrix per criterion of the previous level, then we will redo a pairwise evaluation of each alternative with respect to this criterion..

This step is repeated as many times as there are levels.

Example

Let’s continue our example by doing the same work for the sub-criteria groups. We obtain the following matrices :

 Equipment GPS Sunroof Weight GPS 1 7 88% Sunroof 1/7 1 12%

 Habitability Number of places Trunk size Weight Number of places 1 5 83% Trunk size 1/5 1 17%

## 7 – Calculate the final “weight”

Once all the weights of the criteria by level, we will calculate the final weight simply by multiplying the weights of the criteria of the lowest level by the weights of the corresponding criteria of the higher levels..

Exemple

In our case, we get the following result :

 Level 1 criterion Level 2 criterion Final weight Description Weight Description Weight Price 67% – – 67% Equipment 6% GPS 88% 5,3% Sunroof 12% 0,72% Habitability 27% Nb de places 83% 22,4% Taille du coffre 17% 4,6%

For example, for GPS, we multiply 6% by 88%.

## 8 – Evaluate and choice

It remains for us to evaluate our different alternatives with respect to the matrix of weighted criteria that we have calculated previously. By performing the evaluation and multiplying the scores by weighting, we can identify the best choice.

More generally, it will suffice to identify the alternative with the best sum of the weighted scores. However, we can in some “adjust” using the rank of each alternative on each criterion.

Indeed, in some cases, it is possible for an alternative to have the best overall score without being ranked highest in each of the criteria. We may then be able to improve this alternative by using the strengths of the other alternatives.

## Source

1 – T. L. (1980) – The analytic hierarchy process

2 – L. G. Vargas, T. L. Saaty (1991) – Prediction, projection and forecasting