Analytic Hierarchy Process

By: Aman Kumar Gupta
(2012MB31)

1
Table of Content
•
•
•
•
•

What is AHP
General Idea
Example: Car Selection
Ranking Scale for Criteria and Alternatives
Pr...
Analytic Hierarchy Process(AHP)
• The Analytic Hierarchy Process (AHP) is a structured
technique for organizing and analyz...
Analytic Hierarchy Process (Cont.)
Wide range of applications exists:
•
Selecting a car for purchasing
•
Deciding upon a p...
General Idea
AHP algorithm is basically composed of two steps:
1. Determine the relative weights of the decision criteria
...
Example: Car Selection
•

Objective
•

•

Criteria
•

•

Selecting a car
Style, Reliability, Fuel-economy

Cost?

Alternat...
Hierarchy tree
Selecting
a New Car

Style

Civic

Reliability

Saturn

Fuel Economy

Escort

Alternative courses of action...
Ranking Scale for Criteria and
Alternatives

http://en.wikipedia.org/wiki/Talk:Analytic_Hier
archy_Process/Example_Car

8
Ranking of criteria
Style

Reliability

Fuel Economy

Style

1

1/2

3

Reliability

2

1

4

1/3

1/4

1

Fuel Economy

9
Ranking of priorities
• Consider [Ax = maxx] where
• A is the comparison matrix of size n n, for n criteria, also called t...
Criteria weights
• Style
.32
• Reliability
.56
• Fuel Economy .12

Selecting a New Car
1.00

Style
0.32

Reliability
0.56
...
Checking for Consistency
• The next stage is to calculate a Consistency Ratio (CR) to
measure how consistent the judgments...
Calculation of Consistency Ratio
• The next stage is to calculate max so as to lead to the
Consistency Index and the Consi...
C.R. = C.I./R.I. where R.I. is the random index
n 1
2
3
4
5
6
R.I.
0
0
.52
.88
1.11 1.25

7
1.35

C.I. = 0.02
n=3
R.I. = 0...
Ranking alternatives
Style
Civic

Civic
1

Saturn

4

1

4

1/4

Escort
Miata

1/4
6

1/4
4

1
5

1/5
1

Reliability Civic...
Ranking alternatives
Miles/gallon

Priority Vector

Civic

34

.30

Saturn
Escort
Miata

Fuel Economy

27
24
28
113

.24
....
Selecting a New Car
1.00

Style
0.32
Civic
Saturn
Escort
Miata

0.13
0.24
0.07
0.56

Reliability
0.56
Civic
Saturn
Escort
...
Fuel
Economy

Reliability

Style

Ranking of alternatives

Civic

.13 .38 .30

Saturn
Escort

.24 .29 .24

Miata

.56 .26 ...
Including Cost as a Decision Criteria
Adding “cost” as a a new criterion is very difficult in AHP. A new column and a
new ...
• The “ESCORT” Is the winner with the highest benefit to
COST RATIO and we rank it 1st ,
• Then at 2nd position Saturn,
• ...
More about AHP: Pros and Cons
Pros

• It allows multi criteria decision making.
• It is applicable when it is difficult to...
22
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Analytic hierarchy process

  1. 1. Analytic Hierarchy Process By: Aman Kumar Gupta (2012MB31) 1
  2. 2. Table of Content • • • • • What is AHP General Idea Example: Car Selection Ranking Scale for Criteria and Alternatives Pros and Cons of AHP 2
  3. 3. Analytic Hierarchy Process(AHP) • The Analytic Hierarchy Process (AHP) is a structured technique for organizing and analyzing complex decisions. • It was developed by Thomas L. Saaty in the 1970s. • Application in group decision making. 3
  4. 4. Analytic Hierarchy Process (Cont.) Wide range of applications exists: • Selecting a car for purchasing • Deciding upon a place to visit for vacation • Deciding upon an MBA program after graduation. 4
  5. 5. General Idea AHP algorithm is basically composed of two steps: 1. Determine the relative weights of the decision criteria 2. Determine the relative rankings (priorities) of alternatives Both qualitative and quantitative information can be compared by using informed judgments to derive weights and priorities. 5
  6. 6. Example: Car Selection • Objective • • Criteria • • Selecting a car Style, Reliability, Fuel-economy Cost? Alternatives • Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata 6
  7. 7. Hierarchy tree Selecting a New Car Style Civic Reliability Saturn Fuel Economy Escort Alternative courses of action Miata 7
  8. 8. Ranking Scale for Criteria and Alternatives http://en.wikipedia.org/wiki/Talk:Analytic_Hier archy_Process/Example_Car 8
  9. 9. Ranking of criteria Style Reliability Fuel Economy Style 1 1/2 3 Reliability 2 1 4 1/3 1/4 1 Fuel Economy 9
  10. 10. Ranking of priorities • Consider [Ax = maxx] where • A is the comparison matrix of size n n, for n criteria, also called the priority matrix. • x is the Eigenvector of size n 1, also called the priority vector. • max is the Eigenvalue. • To find the ranking of priorities, namely the Eigen Vector X: 1) Normalize the column entries by dividing each entry by the sum of the column. 2) Take the overall row averages. A= 1 0.5 3 2 1 4 0.33 0.25 1.0 Column sums 3.33 1.75 8.00 Normalized Column Sums 0.30 0.60 0.10 0.28 0.57 0.15 0.37 0.51 0.12 1.00 1.00 1.00 Row averages X= 0.32 0.56 0.12 Priority vector 10
  11. 11. Criteria weights • Style .32 • Reliability .56 • Fuel Economy .12 Selecting a New Car 1.00 Style 0.32 Reliability 0.56 Fuel Economy 0.12 11
  12. 12. Checking for Consistency • The next stage is to calculate a Consistency Ratio (CR) to measure how consistent the judgments have been relative to large samples of purely random judgments. • AHP evaluations are based on the aasumption that the decision maker is rational, i.e., if A is preferred to B and B is preferred to C, then A is preferred to C. • If the CR is greater than 0.1 the judgments are untrustworthy because they are too close for comfort to randomness and the exercise is valueless or must be repeated. 12
  13. 13. Calculation of Consistency Ratio • The next stage is to calculate max so as to lead to the Consistency Index and the Consistency Ratio. • Consider [Ax = max x] where x is the Eigenvector. A 1 0.5 2 1 0.333 0.25 x 3 4 1.0 Ax x 0.32 0.56 0.12 0.98 1.68 0.36 0.32 0.56 0.12 = = max λmax=average{0.98/0.32, 1.68/0.56, 0.36/0.12}=3.04 Consistency index , CI is found by CI=(λmax-n)/(n-1)=(3.04-3)/(3-1)= 0.02 13
  14. 14. C.R. = C.I./R.I. where R.I. is the random index n 1 2 3 4 5 6 R.I. 0 0 .52 .88 1.11 1.25 7 1.35 C.I. = 0.02 n=3 R.I. = 0.50(from table) So, C.R. = C.I./R.I. = 0.02/0.52 = 0.04 C.R. ≤ 0.1 indicates sufficient consistency for decision. 14
  15. 15. Ranking alternatives Style Civic Civic 1 Saturn 4 1 4 1/4 Escort Miata 1/4 6 1/4 4 1 5 1/5 1 Reliability Civic Saturn 1/4 Saturn 2 Escort Miata 4 1/6 Escort Miata 5 1 Civic 1 Saturn Escort 1/2 1/5 1 1/3 3 1 2 1/4 Miata 1 1/2 4 1 Priority vector 0.13 0.24 0.07 0.56 0.38 0.29 0.07 0.26 15
  16. 16. Ranking alternatives Miles/gallon Priority Vector Civic 34 .30 Saturn Escort Miata Fuel Economy 27 24 28 113 .24 .21 .25 1.0 Since fuel economy is a quantitative measure, fuel consumption ratios can be used to determine the relative ranking of alternatives. 16
  17. 17. Selecting a New Car 1.00 Style 0.32 Civic Saturn Escort Miata 0.13 0.24 0.07 0.56 Reliability 0.56 Civic Saturn Escort Miata 0.38 0.29 0.07 0.26 Fuel Economy 0.12 Civic Saturn Escort Miata 0.30 0.24 0.21 0.25 17
  18. 18. Fuel Economy Reliability Style Ranking of alternatives Civic .13 .38 .30 Saturn Escort .24 .29 .24 Miata .56 .26 .25 .07 .07 .21 Priority matrix .32 x .56 .12 .28 .25 = .07 .34 Criteria Weights 18
  19. 19. Including Cost as a Decision Criteria Adding “cost” as a a new criterion is very difficult in AHP. A new column and a new row will be added in the evaluation matrix. However, whole evaluation should be repeated since addition of a new criterion might affect the relative importance of other criteria as well! Instead one may think of normalizing the costs directly and calculate the cost/benefit ratio for comparing alternatives! Cost • • • • CIVIC SATURN ESCORT MIATA Normalized Cost Benefits Cost/Benefits Ratio $ 12k $15K $9K $18K .22 .28 .17 .33 .28 .25 .07 .34 0.78 1.12 2.42 0.97 19
  20. 20. • The “ESCORT” Is the winner with the highest benefit to COST RATIO and we rank it 1st , • Then at 2nd position Saturn, • At 3rd Miata, • At 4th Civic. 20
  21. 21. More about AHP: Pros and Cons Pros • It allows multi criteria decision making. • It is applicable when it is difficult to formulate criteria evaluations, i.e., it allows qualitative evaluation as well as quantitative evaluation. • It is applicable for group decision making environments Cons •There are hidden assumptions like consistency. Repeating evaluations is cumbersome. •Difficult to use when the number of criteria or alternatives is high, i.e., more than 7. •Difficult to add a new criterion or alternative •Difficult to take out an existing criterion or alternative, since the best alternative might differ if the worst one is excluded. 21
  22. 22. 22
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