1. Multicriteria Decision Making
Analytical Hierarchy Processes

2. Overview of AHP
GP answers “how much?”, whereas AHP answers “which one?”
AHP developed by Saati
Method for ranking decision alternatives and selecting the best one when the decision maker has multiple objectives, or criteria

3. Examples Buying a house Buying a car Going to a college
Cost, proximity of schools, trees, neighborhood, public transportation
Buying a car
Price, interior comfort, mpg, appearance, etc.
Going to a college

4. Demonstrating AHP Technique
Identified three potential location alternatives: A,B, and C
Identified four criteria: Market, labor, Income level, and Transportation,
1st level: Goal (select the best location)
2nd level: How each of the 4 criteria contributes to achieving objective
3rd level: How each of the locations contributes to each of the 4 criteria

5. General Mathematical Process
Establish preferences at each of the levels
Determine our preferences for each location for each criteria
A might have a better labor over the other two
Determine our preferences for the criteria
which one is the most important
Combine these two sets of preferences to mathematically derive a score for each location

6. Pairwise Comparisons Used to score each alternative on a criterion
Preference Level
Numerical Value
Equally preferred 1
Equally to moderately preferred 2
Moderately preferred 3
Moderately to strongly preferred 4
Strongly preferred 5
Strongly to very strongly preferred 6
Very strongly preferred 7
Very strongly to extremely preferred 8
Extremely preferred 9
Used to score each alternative on a criterion
Compare two alternatives according to a criterion and indicate the preference using a preference scale
Standard scale used in AHP

7. Pairwise Comparison Market
If A is compared with B for a criterion and preference value is 3, then the preference value of comparing B with A is 1/3
Pairwise comparison ratings for the market criterion
Any location compared to itself, must equally preferred
Market
location A B C 1 3 2
1/3
1/5
1/2 5

8. Other Pairwise Comparison
Market
location A B C 1 3 2
1/3
1/5
1/2 5
Income level
location A B C 1 6
1/3
1/6
1/9 3 9
labor
location A B C 1
1/3 3 7
1/7
Transportation
location A B C 1
1/3
1/2 3 4 2
1/4

9. Developing Preferences within Criteria
Prioritize the decision alternatives within each criterion
Referred to synthesization
Sum the values in each column of the pairwise comparison matrices
Divide each value in a column by its corresponding column sum to normalize preference values
Values in each column sum to 1
Average the values in each row
Provides the most preferred alternative (A, C, B)
Last column is called preference vector
Market
location A B C 1 3 2
1/3
1/5
1/2 5
11/6 9
16/5
Market
location A B C
6/11
3/9
5/8
2/11
1/9
1/16
3/11
5/9
5/16
Market
location A B C
Average
0.5455
0.333
0.6250
0.5012
0.1818
0.1111
0.0625
0.1185
0.2727
0.5556
0.3125
0.3803

10. Other Preference Vectors
Location
Market
Income Level
labor
Transportation A
0.5012
0.2819
0.1780
0.1561 B
0.1185
0.0598
0.6850
0.6196 C
0.3803
0.6583
0.1360
0.2243

11. Ranking the Criteria
Determine the relative importance or weight of the criteria
which one is the most important and which one is the least important one
Accomplished the same way we ranked the locations within each criterion, using pairwise comparison
Criteria
Market
Income
labor
Transportation 1
1/5 3 4 5 9 7
1/3
1/9 2
1/4
1/7
1/2

12. Normalizing Criteria Market Income labor Transportation Average 0.1519
0.1375
0.2222
0.2857
0.1993
0.7595
0.6878
0.6667
0.5000
0.6535
0.0506
0.0764
0.0741
0.1429
0.0860
0.0380
0.0983
0.0370
0.0714
0.0612
Income level is the highest priority criterion followed by market

13. Developing Overall Ranking
Location
Market
Income Level
labor
Transportation A
0.5012
0.2819
0.1780
0.1561 B
0.1185
0.0598
0.6850
0.6196 C
0.3803
0.6583
0.1360
0.2243
Criteria
Average
Market
0.1993
Income
0.6535
labor
0.0860
Transportation
0.0612
Overall Score A= (0.1993)(0.5012)+(0.6535)(0.2819)+
(0.1780)(0.0860)+(0.1561)(0.0612)
=0.3091
Overall Score B =0.1595
Overall Score C =0.5314
Preference Vector

14. Summary
Develop a pairwise comparison matrix for each decision alternative for each criterion
Synthesization
Sum values in each column
Divide each value in each column by the corresponding column sum
Average the values in each row (provides preference vector for decision alternatives)
Combine the preference vectors
Develop the preference vector for criteria in the same way
Compute an overall score for each decision alternative
Rank the decision alternatives

15. AHP Consistency
Decision maker uses pairwise comparison to establish the preferences using the preference scale
In case of many comparisons, the decision maker may lose track of previous responses
Responses have to be valid and consistent from a set of comparisons to another set
Consistency Index (CI) measures the degree of inconsistency in the pairwise comparisons

16. CI Computation Consider the pairwise comparisons for the 4 criteria
Pairwise Comparison Matrix
Consider the pairwise comparisons for the 4 criteria
Multiply the Pairwise Comparison Matrix by the Preference Vector
Divide each value by the corresponding weights from the preference vector
If the decision maker was a perfectly consistent decision maker, then each of these ratios would be exactly 4
CI=( n)/(n-1), where n is the number of being compared
Criteria
Market
Income
labor
Transportation 1
1/5 3 4 5 9 7
1/3
1/9 2
1/4
1/7
1/2
Preference
Vector
.1993
.6535
.0860
.0612 *
(1)(0.1993)+ (1/5)(0.6535)+…+(4)(0.0612)=0.8328
(5)(0.1993)+ (1)(0.6535)+…+(9)(0.0612)=2.8524
(1/3)(0.1993)+ (1/9)(0.6535)+…+(2)(0.0612)=0.3474
(1/4)(0.1993)+ (1/7)(0.6535)+…+(1)(0.0612)=0.2473
0.8328/0.1993=4.1786
2.8524/06535=4.3648
0.3474/.0760=4.0401
0.2473/0.0612=4.0422
Ave =4.1564

17. Degree of Consistency CI=(4.1564-4)/(4-1)=0.0521
If CI=0, there would a perfectly consistent decision maker
Determine the inconsistency degree
Determined by comparing CI to a Random Index (RI)
RI values depend on n
Degree of consistency =CI/RI
IF CI/RI <0.1, the degree of consistency is acceptable
Otherwise AHP is not meaningful
CI/RI=0.0521/0.90=0.0580<0.1 n 2 3 4 5 6 7 8 9 10 RI
0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.51

18. Scoring Model Similar to AHP, but mathematically simpler
Decision criteria are weighted in terms of their relative importance
Each decision alternative is graded in terms of how well it satisfies the criteria using Si=Σgijwj, where
Wj=a weight between 0 and 1.00 assigned to criterion j indicating its relative importance
gij=a grade between 0 and 100 indicating how well the decision alternative i satisfies criterion j
Si=the total score for decision alternative i

19. Decision Alternatives
Example
Decision Alternatives
Decision Criteria
Weight
Alt.1
Alt.2
Alt.3
Alt.4
Criterion 1
0.30 40 60 90
Criterion 2
0.25 75 80 65
Criterion 3 79 85
Criterion 4
0.10
100
Criterion 5 30 50 70
Weight assigned to each criterion indicates its relative importance
Grades assigned to each alternative indicate how well it satisfies each criterion
(0.3)(40)+ (0.25)(75)+…+(0.10)(80)=62.75
(0.3)(60)+ (0.25)(80)+…+(0.10)(30)=73.50
(0.3)(90)+ (0.25)(65)+…+(0.10)(50)=76.00
(0.3)(60)+ (0.25)(90)+…+(0.10)(70)=77.75
Si=Σgijwj=

20. III-Weight/Durability
Example
II-Gear Action
Bike A B C 1
1/3
1/7 3
1/4 7 4
Purchasing a mountain bike
Three criteria: price, gear action, weight/durability
Three types of bikes: A,B,C
Developed pairwise comparison matrices I,II,III
Ranked the decision criteria based on the pairwise comparison
Select the best bike using AHP
III-Weight/Durability
Bike A B C 1 3
1/3
1/2 2
I-Price
Bike A B C 1 3 6
1/3 2
1/6
Criteria
Price
Gear
Weight 1 3 5
1/3 2
1/5
1/2