Multicriteria Decision Making techniques

Multicriteria Decision Making techniques

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
–Cost, proximity of schools, trees, nationhood,
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, Infrastructure,
Income level, and Transportation,
•1
st
level: Goal (select the best location)
•2
nd
level: How each of the 4 criteria contributes
to achieving objective
•3
rd
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 infrastructure 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
•Compare two
alternatives according
to a criterion and
indicate the
preference using a
preference scale
•Standard scale used
in AHP
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

7. Pairwise Comparison
•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
locationA B C
A 1 3 2
B 1/3 1 1/5
C 1/2 5 1

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

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
A 1 3 2
B 1/3 1 1/5
C 1/2 5 1
11/6 9 16/5
Market
location A B C
A 6/11 3/9 5/8
B 2/11 1/9 1/16
C 3/11 5/9 5/16
Market
location A B C Average
A 0.5455 0.333 0.62500.5012
B 0.18180.11110.06250.1185
C 0.27270.55560.31250.3803

10. Other Preference Vectors
Location Market Income LevelInfrastructureTransportation
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
M
a
r
k
e
t
I
n
c
o
m
e
in
f
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t
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T
r
a
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s
p
o
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t
a
t
io
n
Market 11/534
Income 5197
infrastructure1/31/912
Transportation1/41/71/21

12. Normalizing
Criteria
M
a
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k
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t
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c
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m
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f
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T
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p
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A
v
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r
a
g
e
Market 0.15190.13750.22220.28570.1993
Income 0.75950.68780.66670.50000.6535
Infrastructure0.05060.07640.07410.14290.0860
Transportation 0.03800.09830.03700.07140.0612
Income level is the highest priority criterion followed by market

13. Developing Overall Ranking
L
o
c
a
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n
M
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k
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t
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c
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m
e

L
e
v
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l
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A 0.50120.28190.17800.1561
B 0.11850.05980.68500.6196
C 0.38030.65830.13600.2243
Criteria
A
v
e
r
a
g
e
Market 0.1993
Income 0.6535
Infrastructure0.0860
Transportation0.0612
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
•Suppose for a criterion
–A is “very strongly preferred” to B and A is “moderately preferred”
to C
–C is “equally preferred” to B
–Not consistent with the previous comparisons
•Consistency Index (CI) measures the degree of
inconsistency in the pairwise comparisons

16. CI Computation
•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=(4.1564-n)/(n-1), where n
is the number of being
compared
Criteria
M
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Market 1 1/5 3 4
Income 5 1 9 7
infrastructure1/3 1/9 1 2
Transportation1/4 1/71/21
.1993
.6535
.0860
.0612
*
Pairwise Comparison Matrix
Preference
Vector
(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 0
.
5
8
0
.
9
0
1
.
1
2
1
.
2
4
1
.
3
2
1
.
4
1
1
.
4
5
1
.
5
1

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 S
i
=Σg
ij
w
j
,
where
–W
j=a weight between 0 and 1.00 assigned to criterion j
indicating its relative importance
–g
ij=a grade between 0 and 100 indicating how well the
decision alternative i satisfies criterion j
–S
i=the total score for decision alternative i

19. Example
Decision Alternatives
Decision CriteriaWeight Alt.1 Alt.2 Alt.3 Alt.4
Criterion 1 0.30 40 60 90 60
Criterion 2 0.25 75 80 65 90
Criterion 3 0.25 60 90 79 85
Criterion 4 0.10 90 100 80 90
Criterion 5 0.10 80 30 50 70
Weight assigned to each criterion indicates its relative importance
Grades assigned to each alternative indicate how well it satisfies each criterion
S
i
=Σg
ij
w
j
=
(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

20. Example
•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
A 1 3 1
B 1/3 1 1/2
C 1 2 1
Criteria Price Gear Weight
Price 1 3 5
Gear 1/3 1 2
Weight 1/5 1/2 1
I-Price
Bike A B C
A 1 3 6
B 1/3 1 2
C 1/6 2 1
II-Gear Action
Bike A B C
A 1 1/3 1/7
B 3 1 1/4
C 7 4 1