Ahp

Analytic Hierarchy Process
• Multiple-criteria decision-making
• Real world decision problems
– multiple, diverse criteria
– qualitative as well as quantitative information

Comparing apples and oranges?
Spend on defence or agriculture?
Open the refrigerator - apple or orange?

AHP
• Information is decomposed into a hierarchy of
alternatives and criteria
• Information is then synthesized to determine
relative ranking of alternatives
• Both qualitative and quantitative information
can be compared using informed judgements
to derive weights and priorities

Example: Car Selection
• Objective
– Selecting a car

• Criteria
– Style, Reliability, Fuel-economy

Cost?

• Alternatives
– Civic Coupe, Saturn Coupe, Ford Escort,
Mazda Miata

Hierarchical tree
S e le c t in g
a N ew C ar
S t y le
- Civic
- Saturn
- Escort
- Miata

R e lia b ilit y
- Civic
- Saturn
- Escort
- Miata

FuelE conom y
- Civic
- Saturn
- Escort
- Miata

Ranking of criteria
• Weights?
• AHP
– pair-wise relative importance
[1:Equal, 3:Moderate, 5:Strong, 7:Very strong, 9:Extreme]

Style

Reliability

Fuel Economy

Style

1/1

1/2

3/1

Reliability

2/1

1/1

4/1

Fuel Economy

1/3

1/4

1/1

Ranking of priorities
• Eigenvector

[Ax = x]

Iterate
1. Take successive squared powers of matrix
2. Normalize the row sums
Until difference between successive row sums is
less than a pre-specified value

1
0.5
2
1
0.333 0.25
Row sums
12.75
22.3332
4.8333
39.9165

3
4
1.0

squared

3.0
1.75 8.0
5.3332 3.0
14.0
1.1666 0.6667 3.0

Normalized
Row sums
0.3194
0.5595
0.1211
1.0

0.3196
• New iteration gives normalized row sum 0.5584
0.1220
• Difference is:

0.3194
0.5595 0.1211

0.3196
0.5584
0.1220

- 0.0002
= 0.0011
- 0.0009

Preference
• Style
.3196
• Reliability
.5584
• Fuel Economy .1220
S e le c t in g
a N ew C ar
1 .0
S t y le
.3 1 9 6

R e lia b ilit y
.5 5 8 4

FuelE conom y
.1 2 2 0

Ranking alternatives
Style
Civic

Civic
1/1

Saturn
Escort
Miata

4/1
1/4
6/1

Reliability Civic
Civic
1/1
Saturn
Escort
Miata

1/2
1/5
1/1

Saturn
1/4
1/1
1/4
4/1
Saturn
2/1
1/1
1/3
1/2

Escort Miata
4/1
1/6
4/1
1/1
5/1

1/4
1/5
1/1

Escort Miata
5/1
1/1
3/1
1/1
4/1

2/1
1/4
1/1

Eigenvector
.1160
.2470
.0600
.5770

.3790
.2900
.0740
.2570

Miles/gallon

Fuel Economy
(quantitative
information)

Normalized

Civic

34

.3010

Saturn
Escort
Miata

27
24
28

.2390
.2120
.2480

113

1.0

S e le c t in g
a N ew C ar
1 .0
S t y le
.3 1 9 6

- Civic .1160
- Saturn .2470
- Escort .0600
- Miata .5770

R e lia b ilit y
.5 5 8 4

- Civic
- Saturn
- Escort
- Miata

.3790
.2900
.0740
.2570

Fuel E conom y
.1 2 2 0

- Civic
- Saturn
- Escort
- Miata

.3010
.2390
.2120
.2480

Ranking of alternatives
Style Reliability
Civic
Saturn
Escort
Miata

.1160

Fuel
Economy
.3196

.3790 .3010

.2470
.0600

.2900 .2390
.0740 .2120

.5770

.2570

.2480

*

.5584
.1220

=

.3060
.2720
.0940
.3280

Handling Costs
• Dangers of including Cost as another criterion
– political, emotional responses?

• Separate Benefits and Costs hierarchical trees
• Costs vs. Benefits evaluation
– Alternative with best benefits/costs ratio

Cost vs. Benefits
•
•
•
•

MIATA
CIVIC
SATURN
ESCORT

Cost

Normalized
Cost

Cost/Benefits
Ratio

$18K
$12K
$15K
$9K

.333
.222
.2778
.1667

.9840
1.3771
.9791
.5639

Complex decisions
•Many levels of criteria and sub-criteria

• Application areas
–
–
–
–
–

strategic planning
resource allocation
source selection, program selection
business policy
etc., etc., etc..

• AHP software (ExpertChoice)
– computations
– sensitivity analysis
– graphs, tables

• Group AHP