ahp lesson 2

L e s s o n The Analytic Hierarchy Process (AHP)

2 In this Exercise • Construct hierarchical representation of a problem. • Construct Pairwise Matrices. • Normalize the Pairwise Matrices. • Calculate Priority Vectors. • Synthesize Weights and calculate overall rankings of Alternatives.

• Use specialized software DecisionMaker.

Duration : 100 minutes

The Problem In the first exercise about AHP method, a simple multicriteria decision making problem is examined

and solved step by step. The problem to be solved is the selection of the cell phone that best covers

the needs of Alexandros, a student of International Hellenic University.

Alexandros some weeks now is thinking to buy a new cell phone. After an extensive market research

he has finally concluded in three different models that more or less fulfill his needs. It is difficult for

him thought to make the final decision as all of them are top models, so he decides to come to you

and ask your help. He knows you are an expert in decision making as recently you have learned that

AHP method. Now you will have to apply your knowledge and guide your friend to the best decision.

Part A

Doing all the calculations

The solution

1. An important part of the AHP methodology is to model the problem correctly. So you asked

Alexandros to tell you some more details about his preferences. The most important feature for

Alexandros is the Camera. He likes to take pictures so there is a need not just for high resolution

(8 megapixels and beyond), but also there is demand for the same image processing capabilities

found in current digital still and video cameras. Another important aspect is the battery timing,

as it is essential for a busy guy like him. Charging of the cell phone device is a liability and he

cannot perform this job often. At the end he told you how much fascinated he is by touch screen

cell phones. Touch screen cell phones have usually large screen with high resolution, and give a

wonderful feeling that you are able to connect to the world literally at a touch of a finger.

The Analytic Hierarchy Process (AHP)

Alexandros also told you that he has concluded among three different models, that all of them

have those features more or less.

Now that you know a little bit more things about the problem you can start applying the AHP

methodology to solve it. First step as mentioned above is to model the problem into a

hierarchical structure. Quickly you construct the following hierarchy that shows clearly, what is

the Goal, the Criteria and the Alternatives of Alexandros’ problem.

Picture 1: The AHP hierarchy for Alexandros’ buying decision. The Goal is blue, the Criteria are

yellow and the Alternatives are green. All the Alternatives three different models of cell phones

are shown below the lowest level of each criterion.

2. The second step in AHP is to define the relative importance of the Criteria. This is done by

comparing each possible pair of Criteria and ranking them on the following scale : Comparing

Criterion I and Criterion j, give a value aij with the following meaning:

Description

1 Element and are equally important

3 Element is weakly more important than

5 Element is strongly more important than

7 Element is very strongly more important than

9 Element is absolutely more important than

Table 1: Pairwise comparison values.

2 International Hellenic University Paraskevopoulos Konstantinos


is assumed to be at least as important as , and is set to 1, and if is m, then .

Now it is time to ask Alexandros to compare in pairs and rate all the Criteria with respect to the

Goal of the problem. Alexandros after some thinking, comes up with some ratings that you

organized in the following matrix :

Battery

Camera

Touch screen

Battery 1 1/4 3

Camera 4 1 7

Touch screen 1/3 1/7 1

Table 2 : Preferences of the Alternatives

in favor of node Buy a new cell phone.

Next due to AHP method you will have to make some calculations to determine the overall

weight that Alexandros is assgnining to each Criterion. There are two methods to do that. The

first one is to find the Normalized Ratings Matrix and the other one a little bit more time

consuming is to find the eigenvector. Here we will work with the first method so we will

calculate the Normalized Ratings Matrix.

3. Normalize the priority matrix by dividing each element by the sum of the column it appears in.

For instance entry would end up as :

11 4 0.333

0.187

The other entries become:

Battery

Camera

Touch screen

Average

Battery 0.188 0.179 0.273 0.213

Camera 0.750 0.718 0.636 0.701

Touch screen 0.063 0.103 0.091 0.085

Sum=1



Table 3 : Weights on Criteria.

That means that 70% of the Criterion weight is on Camera, about 20% is on Battery and 8% is on

Touch screen. It is clearly that Alexandros prefers the Camera over the Battery timing and touch

screen.

4. The next step is to evaluate all the cell phone models on each of the cell phones features. For

example Alexandros told you that the Battery efficiency of Model A is much bigger than that of

Model B and the Battery of Model C is slightly bigger than Model B. You value those

characteristics with the following matrix :

Mod

el A

Mod

el B

Mod

el C

Model A 1 4 3

Model B 1/4 1 1/2

Model C 1/3 2 1

Table 4: Battery scores from each model.

Again you normalize the scoring table dividing each element by the sums of the columns, and

average across rows to get the relative weights of each model regards to Battery. You end up

with the following table :

Mod

el A

Mod

el B

Mod

el C

Average

Model A 0.632 0.571 0.667 0.623

Model B 0.158 0.143 0.111 0.137

Model C 0.210 0.286 0.222 0.239

Sum=1

Table 5: Relative Battery scores from each model.

This means that Model A has the strongest battery as the weight is 62%, followed by 24% of

Model C and 13% of Model B.

Going through a similar process with Camera and Touch screen we get the following matrices :



Mod

el A

Mod

el B

Mod

el C

Model A 1 5 3

Model B 1/5 1 1/4

Model C 1/3 4 1

Table 6: Camera scores from each model.

Mod

el A

Mod

el B

Mod

el C

Average

Model A 0.652 0.500 0.706 0.619

Model B 0.130 0.100 0.059 0.096

Model C 0.217 0.400 0.235 0.284

Sum=1

Table 7: Relative Camera scores from each model.

Mod

el A

Mod

el B

Mod

el C

Model A 1 0.333 0.143

Model B 3.000 1 0.500

Model C 7.000 2.000 1

Table 8: Touch screen ratings from each model.



Mod

el A

Mod

el B

Mod

el C

Average

Model A 0.091 0.100 0.087 0.093

Model B 0.273 0.300 0.304 0.292

Model C 0.636 0.600 0.609 0.615

Sum=1

Table 9: Relative Touch screen scores from each model.

5. We can now place all the overall weights on the hierarchy model from picture 1 and get a more

clear picture of how much each element in the model counts in the final decision.

6. We can now get a final value for each cell phone model :

The value for Model A is :

(0.623*0.213)+(0.619*0.701)+(0.093*0.085)=0.133+0.434+0.008=0.575

The value for Model B is:

(0.137*0.213)+(0.096*0.701)+(0.292*0.085)=0.029+0.068+0.025=0.122

The value for Model C is:

(0.239*0.213)+(0.284*0.701)+(0.615*0.085)=0.051+0.199+0.052=0.303



The winner is Model A cell phone, as it fits 57.5% the needs of Alexandros comparing with the

other two models.



Part B

Using specialized software.

In this part the solution to the problem “Select a new cell phone” is given using specialized software

named Decision Maker.

1. Run the program (Java Runtime Environment needed).

2. Since now you are opening the program the tree canvas area should be empty. Select the node

Goal and from the left menu click “Edit node…”. The dialog “Edit node” will show.

3. In the Node name text field type “Buy cell phone”. If you like type a short description of the

node in the field “Node Description”, as it is shown in the following capture:

4. In the following steps we will build

the hierarchy of the problem as

presented in Part A. With the root

node selected click the button

“Insert new node” on the left menu.

The dialog “Add new problem node”

will come up as shown in the

following capture. This dialog allows

to insert a new node in the problem’s hierarchy, as well as to set the properties of this new node



such as node name, node type (Goal, Criteria, Alternative), and the information type that this

node will hold (either Qualitative or Quantitative information).

5. In the text field “New node name”, type “Battery”.

6. In the field “New node type”, select Criteria (only the value Criteria is available for now as it is

forbidden to insert an Alternative node in the root node).

7. In the field “Information type”, select the value Qualitative as the “Battery” criterion holds

Qualitative information.

8. Click Ok to insert the node, and double click the root node to show the new inserted node.

9. Repeat the steps 4‐7 to insert the nodes “Camera” and “Touch screen” in the root node. Now

the hierarchy structure should look like the one in the following capture. Take a moment to see

the comments in that capture and what the differences in the appearance of the nodes mean.

10. Now it is time to insert in the hierarchy the Alternative nodes as shown in Part A of this exercise.

Select a leaf node from the hierarchy then click the button “Insert new node” placed on the left

side of the window. The familiar dialog “Insert new node” appears. Type “Model A” for the

name of the node select Alternative for the type and for the information type leave as is the

value “Parent node type”

11. Click Ok. You just entered in your hierarchy, the first Alternative node with the name “Model

A”. Note that Alternative nodes are colored with orange, and they also have slightly different

graphic.

12. Repeat the previous step to insert the Alternative nodes “Model B” and “Model C”. Your tree

hierarchy should now look like the one in the following capture.



13. Click the button “Next”, located almost at the top of the window. It will open the interface for

the pairwise comparisons of the Criteria nodes as shown in the following picture.

Click the Next button to take you in step 2 of the solution process.

14. Remember what Alexandros told you about his preferences and make the correct ratings as

shown in the following capture:

You can notice that while making the selections for the pairwise comparisons, two numbers are

changing at the top of the selection tool. Those are two indices named “Consistency Index” and



“Consistency Ratio”. We will talk about those two indices in the next exercise. Now keep in

mind that if those indices are colored red then you should rethink your ratings as they are

inconsistent.

15. When finished with the selections click on the “Show Calculations” button at the left of the

window. The dialog “AHP Calculations” will open showing to you all the calculations that you did

in the first part of the exercise when the pairwise comparisons were made. Check to see if you

are correct!

16. Close the dialog “AHP Calculations” and click again on “Next” button. The necessary tools to

make the pair wise comparisons for the Alternative nodes are shown.

17. Rate the Alternatives as you did in the first part of the exercise. When done with the ratings

your screen should look like the following capture:



International Hellenic University Paraskevopoulos Konstantinos

18. Check the calculations if you want by clicking on the button “Show Calculations” on the left side

of the window.

19. When done with the ratings of the

Alternatives click “Next” button once

more. Next step screen will show as

you can see in the following capture.

20. Here are the final rankings of the

Alternatives. Model A is the winner.

You can see also the score of each

Alternative in respect to its last

Criterion.

21. You can click also if you want at the

“Show Graphs” button and see

graphic representation of your

ratings as shown in the picture.

22. Save the problem in a file named Exercise_2.xml.

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ahp lesson 2

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