Transcript Analytic Hierarchy Process

Lecture 08
Analytic Hierarchy Process
(Module 1)
Industrial Systems Engineering Dept.- IU
Office: Room 508
Learning Objectives
Students will be able to:
1.
2.
3.
Use the multifactor evaluation process in making
decisions that involve a number of factors, where
importance weights can be assigned.
Understand the use of the analytic hierarchy process
in decision making.
Contrast multifactor evaluation with the analytic
hierarchy process.
Module Outline
M1.1 Introduction
M1.2 Multifactor Evaluation Process
M1.3 Analytic Hierarchy Process
Introduction
 Multifactor decision making involves individuals
subjectively and intuitively considering various
factors prior to making a decision.
 Multifactor evaluation process (MFEP) is a
quantitative approach that gives weights to each
factor and scores to each alternative.
 Analytic hierarchy process (AHP) is an approach
designed to quantify the preferences for various
factors and alternatives.
Multifactor Evaluation Process
Example: Steve. M.: considering employment with 3 companies.
determined 3 factors important to him, assigned each factor a weight.
Steve evaluated the various factors on a 0 to 1 scale for each
of these jobs.
Factor
Importance
(weight)
AA
Co.
EDS,
LTD.
PW,
Inc.
Salary
0.3
0.7
0.8
0.9
Career
Advancement
0.6
0.9
0.7
0.6
Location
0.1
0.6
0.8
0.9
Weights should sum to 1
Score Table
Evaluation of AA Co.
Factor X Factor
Weight
Evaluation
Factor
Name
Salary
Career
Location
Factor
Weight
0.3
0.6
0.1
=
Factor
Evaluation
0.7
0.9
0.6
Total
Weighted
Evaluation
Weighted
Evaluation
0.21
0.54
0.06
0.81
Comparison of Results
Factor
AA Co.
EDS,LTD.
PW,Inc.
Salary
0.21
0.24
0.27
Career
0.54
0.42
0.36
Location
0.06
0.08
0.09
Weighted
Evaluation
0.81
0.74
0.72
Decision is AA Co: Highest weighted evaluation
The Analytic Hierarchy Process (AHP)




Founded by Saaty in 1980.
It is a popular and widely used method
for multi-criteria decision making.
Allows the use of qualitative, as well as
Dr. Thomas L. Saaty
Distinguished Prof. at U. of Pittsburgh
quantitative criteria in evaluation.
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.
 …
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AHP-General Idea

Develop an hierarchy of decision criteria and define the
alternative courses of actions.

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.
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Steps

Step 0: Construction of Hierarchy Structure
(including: Goal, Factors, Criteria, and Alternatives)

Step 1: Calculation of Factor Weight



Step 1-1: Pairwise Comparison Matrix
Step 1-2: Eigenvalue and Eigenvector (Priority vector)
Step 1-3:Consistency Test
Consistency Index
 Consistency Ratio
Step 2:Calculation of Level Weight



Step 3: Calculation of Overall Ranking
Hierarchy Tree
Level 0
More General
Goal
C1
Level 1 (factors)
Level 2 (criteria)
C3
C11 C12 C13 C21 C22 C31 C32 C33
Sub-criteria at the
lowest level
Level ..
C2
More Specific
Alternatives
Tom Saaty suggests that hierarchies be limited to six levels and nine items per
level.
This is based on the psychological result that people can consider 7 +/- 2
items simultaneously (Miller, 1956).
Pairwise Comparisons
Apple A
Size
Apple B
Apple A
Size
Comparison
Apple B
Apple C
Apple C
Relative Size
of Apple
Apple A
1
2
6
6/10
A
Apple B
1/2
1
3
3/10
B
Apple C
1/6
1/3
1
1/10
C
Criteria #1
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Resulting
Priority
Eigenvector
8
7
6
5
Criteria #2
4
3
2
2
1
Intensity of
Importance
3
4
5
6
7
8
9
Ranking of Criteria and Alternatives
Pairwise Comparison Matrix
to
A1
A2
A3
A1 a11
a12
a13
A2 a21
a22
a32
(a) aii = 1
A3 a31
a32
a33
(b) aij = 1/ aji aji are reverse comparisons and must be the reciprocals of aij
Pairwise Comparison Matrix A = ( aij )
A comparison of criterion i with itself: equally important
Ranking Scale for Criteria and Alternatives
Values for aij :
Numerical values
Verbal judgment of preferences
1
equally important
3
weakly more important
5
strongly more important
7
very strongly more important
9
absolutely more important
intermediate
2,4,6,8 => values
reciprocals => reverse
comparisons
Example 1: Car Selection (1/15)

Objective


Criteria


Selecting a car
Style, Reliability, Fuel-economy
Cost?
Alternatives

Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata
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Example 1: Car Selection (2/15)
Hierarchy tree
S e lecting
a N e w C ar
S tyle
Civic
R e lia b ility
Saturn
Escort
F u e l E con o m y
Miata
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Example 1: Car Selection (3/15)
Ranking of Criteria
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
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Example 1: Car Selection (4/15)
Ranking of Priorities

Consider [Ax = x] 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.
  is the Eigenvalue,   > n.
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.

Pairwise Comp. Matrix
A=
1
0.5 3
2
1
4
0.33 0.25 1.0
Column sums 3.33 1.75
8.00
Norm. Pairwise Comp. Matrix
Normalized
Column Sums
0.30
0.60
0.10
0.29
0.57
0.14
0.38
0.50
0.13
1.00
1.00
1.00
Priority vector
Row
Averages
0.3196
0.5584
X=
0.1220
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Example 1: Car Selection (5/15)
Ranking of Priorities (cont.)



Style
Reliability
Fuel Economy
Criteria weights
Second most important criterion
.3196 ≈ .3
First important criterion
.5584 ≈ .6
The least important criterion
.1220 ≈ .1
Here is the tree of criteria with the criteria weights
S e lec ting
a N e w C ar
1 .0
S tyle
.3 196
R e lia b ility
.5 584
F u e l E c on o m y
.1 220
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Example 1: Car Selection (6/15)
Checking for Consistency

Consistency Ratio (CR): measure how consistent the judgments have been
relative to large samples of purely random judgments.

AHP evaluations are based on the asumption 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.



Suppose we judge apple A to be twice as large as apple B and apple
B to be three times as large as apple C.
To be perfectly consistent, apple A must be six times as large as
apple 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.
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Example 1: Car Selection (7/15)
Calculation of Consistency Ratio


The next stage is to calculate , Consistency Index (CI) and the
Consistency Ratio (CR).
Consider [Ax = x] where x is the Eigenvector.
A
1
0.5
2
1
0.333 0.25
Consistency

3
4
1.0
Vector =
x
Ax
0.30
0.60
0.10
0.90
1.60
0.35
0.90/0.30
1.60/0.60
0.35/0.10
=
=
3.00
2.67
3.50
Consistency index (CI) is found by
CI 

n
n 1
x
0.30
=  0.60
0.10
 
3 . 0  2 . 67  3 . 5
 3 . 06
3

3 . 06  3
3 1
Note: This is just an approximate method to determine value of λ
 0 . 03
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Example 1: Car Selection (8/15)
Consistency Index

reflects the consistency of
one’s judgement
CI 

n
n 1
Random Index (RI)

the CI of a randomly-generated
pairwise comparison matrix
Tabulated by size of matrix (n):
(given by author)
n
2
3
4
5
6
7
8
9
10
RI
0.0
0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.51
Example 1: Car Selection (9/15)
Consistency Ratio
CR  CI
RI
In practice, a CR of 0.1 or below is considered acceptable.
 Any higher value at any level indicate that the judgements
warrant re-examination.
In the above example:
CR 
CI
RI

0 . 03
 0 . 052  0 . 1
0 . 58
so, the evaluations are consistent
Example 1: Car Selection (10/15)
Ranking Alternatives
Style
Civic
Civic
1
Saturn
Escort
Miata
4
1/4
6
Reliability Civic
Civic
1
Saturn 1/2
Escort 1/5
Miata 1
Saturn
1/4
1
1/4
4
Saturn
2
1
1/3
1/2
Escort Miata
4
1/6
Priority vector
1/4
1/5
1
0.13
0.24
0.07
0.56
Escort Miata
5
1
3
2
1
1/4
4
1
0.38
0.29
0.07
0.26
4
1
5
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Example 1: Car Selection (11/15)
Ranking Alternatives (cont.)
Fuel Economy
Miles/gallon
Normalized
Civic
34
.30
Saturn
Escort
27
24
.24
.21
Miata
28
113
.25
1.0
! Since fuel economy is a quantitative measure, fuel consumption ratios
can be used to determine the relative ranking of alternatives; however
this is not obligatory. Pairwise comparisons may still be used in some
cases.
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Example 1: Car Selection (12/15)
Ranking Alternatives (cont.)
Selecting a New Car
1.00
Style
0.30
Civic
0.13
Saturn 0.24
Escort 0.07
Miata 0.56
Reliability
0.60
Civic 0.38
Saturn 0.29
Escort 0.07
Miata 0.26
Reliability(0.6)
Fuel Economy
0.10
Civic
0.30
Saturn 0.24
Escort 0.21
Miata 0.25
Car
Style(0.3)
Fuel Economy(0.1)
Total
Civic
0.13
0.38
0.30
0.30
Saturn
0.24
0.29
0.24
0.27
Escort
0.07
0.07
0.21
0.08
Miata
0.56
0.26
0.25
0.35 largest
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Example 1: Car Selection (13/15)
Fuel
Economy
Reliability
Style
Ranking of Alternatives (cont.)
Civic
.13
.38 .30
Saturn
Escort
Miata
.24
.29 .24
.07
.07 .21
.56
.26 .25
Priority matrix
.30
x
.60
.10
.30
.27
=
.08
.35
Factor Weights
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Example 1: Car Selection (14/15)
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!




CIVIC
SATURN
ESCORT
MIATA
Cost
Normalized
Cost
Benefits
Cost/Benefits
Ratio
$12K
$15K
$ 9K
$18K
.22
.28
.17
.33
.30
.27
.08
.35
0.73
1.04
2.13
0.94
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Example 1: Car Selection (15/15)
Methods for including Cost Criterion

Use graphical representations to make trade-offs.
Miata
40
Civic
35
Miata
Civic
Benefit
30
25
Saturn
20
15
Escort
Saturn
Escort
10
5
0
0
5
10
15
20
25
30
35
Cost



Calculate cost/benefit ratios
Use linear programming
Use seperate benefit and cost trees and then combine the results
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Complex Decisions
•Many levels of criteria and sub-criteria exists for
complex problems.
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Example 2: Buying the best car
*Goal: Buying the best car
*There are three criteria:

Cost

Quality

Maintenance

Insurance

Services
*Three alternatives: Honda, Mercedes, Hyundai
Example 2: Buying the best car
Select the
"best" car
Level 0
Level 1
COST
Maintenance
Quality
Criteria
Level 2
Insurance
Service
Sub-criteria
Alternatives
Honda
Mercedes
Huyndai
The Hierarchy for problem Buying the best car
Example 2: Buying the best car
Step 1: Criterion comparison
• Criterion comparison
Price
Mantenance
Quality
Price
1
3
5
Maintenance
1/3
1
2
Quality
1/5
1/2
1
• Normalize values:
Price
Maintenance
Quality
• Find Column vector
•
Price
0.652
0.217
0.131
Maintenance
0.667
0.222
0.111
Quality
0.625
0.25
0.125
Price
Price
0.648
Mainternance
0.23
Quality
0.122
The process is repeated for the sub-criteria until the evaluation for all other
alternatives. This example will be supported by Expert Choice software
Example 2: Buying the best car
Step 2: Determining the Consistency Ratio - CR
2.1. Determining the Consistency vector
• We begin by determining the weighted sum vector. This is done by
multiplying the column vector times the pairwise comparison matrix.
Column vector:
Pairwise comparison matrix:
Price
Mainternance
Quality
=
0.648
0.230
0.122
X
Consistency vector
1
3
5
1/3
1
2
1/5
1/2
1
Weighted sum vector
1.948
Consistency vector =
Weighted sum vector/ Column vector
0.690
0.366
Example 2: Buying the best car
2.2. Determining  and the Consistency Index-CI
 = (3.006+3.0+3.0) / 3 = 3.002
The CI is:
CI = (3.002 - 3) / (3 - 1) = 0.001
2.3. Determining the Consistency Ratio-CR
with n = 3, we get RI = 0.58
CR = 0.001 / 0.58 = 0.0017
Since 0< CR < 0.1, we accept this result and move to the lower
level. The procedure is repeated till the lowest level.

Continue for other levels:
 For subcriteria Insurance – Service:
• For Cost
Insurance
Service
Insurance
1
3
Service
1/3
1
• For Service
HONDA
Honda
Mer
Huyndai
Honda
1
3
4
Mer
1/3
1
2
Huyndai
1/4
1/2
1
Honda
Mer
Huyndai
25000
MER.
60000
HUYNDAI
15000
• For Insurance:
• For Quality
Honda
Mer
Huyndai
Honda
1
1/3
1/4
Honda
1
1/4
1/5
Mer
3
1
2
Mer
4
1
1/2
Huyndai
4
1/2
1
Huyndai
5
2
1
 And make your final evaluation (students self develop this evaluation)
Select the
"best" car
COST
Maintenance
Insurance
Honda
Quality
Service
Mercedes
Huyndai
1) Weights are defined for
each hierarchical level...
0.6
2) ...and multiplied down to get
the final lower level weights.
0.6
0.4
0.4
Multiply
0.7
0.3
0.2
0.6
0.2
0.7
0.3
0.2
0.6
0.2
0.42
0.18
0.08
0.24
0.08
Notes:
• In general, the evaluation scores are collected from
many experts and the average scores is used in the
pairwise comparison matrix.
•The AHP solving is computer-aided by Expert Choice
(EC) software.
- Building structure of problem !!!
- Enter judgments (Pairwise Comparisons)
- Analysis the weights
- Sensitivity Analysis
- Advantages and disadvantages
- Miscellaneous
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.
Users should be trained to use
AHP methodology.
•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
Use cost/benefit ratio if
applicable
•Difficult to take out an existing criterion or
alternative, since the best alternative might
differ if the worst one is excluded.
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Homework 08
(Due: next class)

M 1.4, M 1.10, M 1.11