(Module 1)
Industrial Systems Engineering Dept.- IU
Office: Room 508

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.

M1.1
Introduction
M1.2
Multifactor Evaluation Process
M1.3
Analytic Hierarchy Process


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.

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
Salary
Importance
(weight)
0.3
AA
Co.
0.7
EDS,
LTD.
0.8
PW,
Inc.
0.9
Career
Advancement
Location
Weights should sum to 1
0.6
0.1
0.9
0.7
0.6
0.8
Score Table
0.6
0.9

=
Weighted
Weight Evaluation Evaluation
Factor Factor Factor Weighted
Name Weight Evaluation Evaluation
Salary
Career
Location
0.3 0.7
0.6 0.9
0.1 0.6
0.21
0.54
0.06
Total 0.81

Factor
Salary
Career
Location
Weighted
Evaluation
AA Co.
0.21
0.54
0.06
0.81
EDS,LTD.
0.24
0.42
0.08
0.74
Decision is AA Co: Highest weighted evaluation
PW,Inc.
0.27
0.36
0.09
0.72





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 quantitative criteria in evaluation.
Wide range of applications exists:

Selecting a car for purchasing



Dr. Thomas L. Saaty
Distinguished Prof. at U. of Pittsburgh
Deciding upon a place to visit for vacation
Deciding upon an MBA program after graduation.
…
8

 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.
9

 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 Goal
More General
Level 1 (factors)
Level 2 (criteria)
Sub-criteria at the lowest level
C
1
C
2
C
3
C
11
C
12
C
13
C
21
C
22
C
31
C
32
C
33
More Specific
Level ..
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
Size
Size
Comparison
Apple A Apple B Apple C
Apple A Apple B Apple C
Apple A 1 2 6
Resulting
Priority
Eigenvector
6/10
Relative Size of Apple
A
Apple B 1/2 1 3 3/10 B
Apple C
Criteria #1
1/6 1/3 1
Criteria #2
1/10 C
9 8 7 6 5 4 3 2
1
Intensity of
Importance
2 3 4 5 6 7 8 9

Pairwise Comparison Matrix to A1 A2 A3
Pairwise Comparison Matrix A = ( a ij
)
A1 a
11 a
12 a
13
A2 a
21 a
22 a
32
(a) a ii
= 1 A comparison of criterion i with itself: equally important
(b) a ij
= 1/ a ji a ji are reverse comparisons and must be the reciprocals of a ij
A3 a
31 a
32 a
33
Ranking Scale for Criteria and Alternatives
Values for a ij
:
Numerical values
5
7
1
3
9
Verbal judgment of preferences equally important weakly more important strongly more important very strongly more important absolutely more important
2,4,6,8 => intermediate values reciprocals => reverse comparisons

 Objective

Selecting a car
 Criteria

Style, Reliability, Fuel-economy Cost?
 Alternatives

Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata
14

Hierarchy tree
Selecting a New Car
Style Reliability Fuel Economy
Civic Saturn Escort Miata
15

Ranking of Criteria
Style
Reliability
Fuel Economy
Style
1/1
2/1
1/3
Reliability Fuel Economy
1/2 3/1
1/1 4/1
1/4 1/1
16

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
Norm. Pairwise Comp. Matrix
Normalized
Column Sums
0.30 0.29 0.38
0.60 0.57 0.50
0.10 0.14 0.13
Priority vector
Row
Averages
X=
0.3196
0.5584
0.1220
Column sums 3.33 1.75 8.00
1.00 1.00 1.00
17

Ranking of Priorities (cont.)



Style
Criteria weights
.3196 ≈ .3
Second most important criterion
Reliability .5584 ≈ .6
Fuel Economy .1220 ≈ .1
First important criterion
The least important criterion
Here is the tree of criteria with the criteria weights
Selecting a New Car
1.0
Style
.3196
Reliability
.5584
Fuel Economy
.1220
18

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.
19

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.
1 0.5 3
2 1 4
0.333 0.25
1.0
0.30
0.60
0.10
0.90
1.60
0.35
=

0.30
0.60
0.10
0.90/0.30
 Consistency Vector = 1.60/0.60
0.35/0.10
=
3.00
2.67
3.50
 
3 .
0

2 .
67

3 .
5
3
 Consistency index (CI) is found by
CI

  n n

1

3 .
06

3
3

1

0 .
03
 Note: This is just an approximate method to determine value of λ

3 .
06
20

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)
5
6
7
8
3
4 n RI
2 0.0
0.58
0.90
9
10
1.12
1.24
1.32
1.41
1.45
1.51

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 .
58

0 .
052

0 .
1 so, the evaluations are consistent

Ranking Alternatives
Style Civic Saturn Escort Miata
Priority vector
Civic 1 1/4 4 1/6
0.13
Saturn
Escort
4
1/4
1
1/4
4
1
1/4
1/5
0.24
0.07
0.56
Miata 6 4 5 1
Reliability Civic Saturn Escort Miata
Civic 1 2 5 1
Saturn 1/2 1 3 2
Escort 1/5 1/3
Miata 1 1/2
1 1/4
4 1
0.38
0.29
0.07
0.26
23

Fuel Economy
Civic
Saturn
Escort
Miles/gallon Normalized
34
27
24
28
113
.30
.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; however this is not obligatory. Pairwise comparisons may still be used in some cases.
24

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
Fuel Economy
0.10
Civic 0.30
Saturn 0.24
Escort 0.21
Miata 0.25
Car Style(0.3) Reliability(0.6) Fuel Economy(0.1) Total
Civic
Saturn
Escort
Miata
0.13
0.24
0.07
0.56
0.38
0.29
0.07
0.26
0.30
0.24
0.21
0.25
0.30
0.27
0.08
0.35
largest
25

Ranking of Alternatives (cont.)
Civic
Saturn
Escort
.13 .38 .30
.24 .29 .24
.07
.07 .21
.56
.26 .25
x
.30
.60
.10
=
.30
.27
.08
.35
Priority matrix Factor Weights
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 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
$12K
$15K
$ 9K
$18K
Normalized
Cost
.22
.28
.17
.33
Benefits
.30
.27
.08
.35
Cost/Benefits
Ratio
0.73
1.04
2.13
0.94
27

Methods for including Cost Criterion
 Use graphical representations to make trade-offs.
40
Miata
Civic
30
25
20
Saturn
Escort
10
5
0
0 5 10 15
Cost
20 25 30
Miata
Saturn
35



Calculate cost/benefit ratios
Use linear programming
Use seperate benefit and cost trees and then combine the results
28

•Many levels of criteria and sub-criteria exists for complex problems.
29

*Goal: Buying the best car
*There are three criteria:
 Cost
 Quality
 Maintenance

Insurance

Services
*Three alternatives: Honda, Mercedes, Hyundai

Level 0 Select the
"best" car
Level 1
Criteria
Level 2
Sub-criteria
Alternatives
COST
Honda
Insurance
Maintenance Quality
Service
Mercedes Huyndai
The Hierarchy for problem Buying the best car

Step 1: Criterion comparison
• Criterion comparison
Price
Price Mantenance Quality
1
Maintenance 1/3
3
1
5
2
• Normalize values:
Quality 1/5 1/2 1
Price
Maintenance
Quality
• Find Column vector
Price
0.652
0.217
0.131
Maintenance
0.667
0.222
0.111
Price
Mainternance
Quality
Price
0.648
0.23
0.122
Quality
0.625
0.25
0.125
• The process is repeated for the sub-criteria until the evaluation for all other alternatives. This example will be supported by Expert Choice software

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
1 3 5
1/3 1 2
1/5 1/2 1
Consistency vector
Weighted sum vector
1.948
Consistency vector =
Weighted sum vector/ Column vector
0.690
0.366

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:
Insurance
Service
Insurance Service
1 3
1/3 1
• For Cost
HONDA
MER.
HUYNDAI
25000
60000
15000
• For Insurance:
Honda Mer Huyndai
Honda 1 1/3 1/4
Mer
Huyndai
3
4
1
1/2
2
1
• For Service
Honda Mer Huyndai
Honda
Mer
1
1/3
Huyndai 1/4
3
1
1/2
4
2
1
• For Quality
Honda
Mer
Huyndai
Honda Mer Huyndai
1
4
5
1/4
1
2

And make your final evaluation (students self develop this evaluation)
1/5
1/2
1

Select the
"best" car
COST Maintenance Quality
Honda
Insurance Service
Mercedes Huyndai

1) Weights are defined for each hierarchical level...
0.6
0.4
0.7
0.3
0.2
0.6
0.2
2) ...and multiplied down to get the final lower level weights.
0.6
0.4
Multiply
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

•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
•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.
Users should be trained to use
AHP methodology .
Use cost/benefit ratio if applicable
41


M 1.4, M 1.10, M 1.11