The Analytic Hierarchy Process
Danny Hahn
The Analytic Hierarchy Process (AHP)
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A Decision Support Tool developed in the 1970s by Thomas L. Saaty,
an American mathematician, currently University Chair, Quantitative
Group, Katz Graduate School of Business, University of Pittsburgh.
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A theory and methodology for modeling problems in the economic,
social and management sciences.
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A problem solving framework used for:
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Determining the best of several alternatives
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Setting Priorities
‰
Allocating Resources
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Requires a “pair-wise” determination of the relative importance of each
of the criteria.
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“Expert Choice” is one commercial software tool based on the AHP.
Slide 2
The Process
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Break down an unstructured situation into its
component parts.
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Arrange the parts or variables into a hierarchic
order.
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Assign numerical values to subjective judgments on
the relative importance of each variable.
„
Synthesize the judgments to determine which
variables have the highest priority and should be
acted upon to influence the outcome of the situation.
Slide 3
The Hierarchy
Goal
Factor 1
Factor 2
Option 1
Factor 3
Option 2
Slide 4
Scale of Relative Importance
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1
Two factors are Equally Important
„
3
One factor is Slightly more Important
than the other
„
5
One is Strongly more Important
„
7
One is Very strongly more Important
„
9
One is Absolutely more Important
„
2, 4, 6, 8 Intermediate Values of one criteria
over the other
Saaty’s book, “The Analytic Hierarchy Process”, provides background and
theory on why he chose the 1 – 9 scale and his validation of this judgment with
measurable science.
Slide 5
Step by Step Example – Buy the Right Car
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Determine the Criteria (factors)
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‰
‰
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Price (lower price is better)
Body Style
Miles per Gallon (more MPG is better)
Interior Quality
Engine Size
Design the Hierarchy
Use an analytic process to help make a
decision
Slide 6
The Car Decision Hierarchy
Buy the
Right Car
Price
Body
Style
X-Treem
MPG
Yaawhee
Interior
Quality
Engine
Size
Zoomer
Slide 7
My Preferences (My Judgments)
„
Body style is more important than Price.
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Price is more important than MPG.
‰
„
I would pay more for the Body Style I want
I would not pay extra for more MPG
Interior Quality is more important than Price.
‰
I would pay more for better Interior Quality
„
Engine Size is more important than Price.
„
Body Style is more important than MPG.
„
Etc. (see next slide)
Slide 8
Pair-wise Comparison of Criteria
Price
Price
Price
Price
Body Style
Body Style
Body Style
MPG
MPG
Interior Quality
More Important
9 8 7 6 5
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More Important
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Body Style
MPG
Interior Quality
Engine Size
MPG
Interior Quality
Engine Size
Interior Quality
Engine Size
Engine Size
Slide 9
Matrix Review
Price
Body Style
MPG
Interior Quality
Engine Size
Price
Body Style
MPG
Interior Quality
Engine Size
1
1/4
3
1/5
1/5
4
1
5
3
1/3
1/3
1/5
1
1/5
1/3
5
1/3
5
1
5
5
3
3
1/5
1
„
An n x n matrix is a square matrix where n is the number of rows and
columns. In this case n = 5.
„
An element is equally important when compared to itself therefore the main
diagonal must be a 1.
„
By convention, the comparison of strength is always of an activity appearing
in the column on the left against an activity appearing in the row on top.
‰
„
Body Style is 5 times more important than MPG
The reverse comparisons (B to A) produce the reciprocal of the basic
comparison. This is called a reciprocal matrix.
‰
MPG is 1/5 as important as Body Style
Slide 10
Convert Criteria Comparisons to a Matrix
Convert the pair-wise comparisons from Slide 9 to a matrix.
Prioritizing the 5 Criteria
Price
Body Style
MPG
Interior Quality
Engine Size
Column Sum
Price
Body Style
MPG
Interior Quality
Engine Size
1
1/4
3
1/5
1/5
4
1
5
3
1/3
1/3
1/5
1
1/5
1/3
5
1/3
5
1
5
5
3
3
1/5
1
15.333
4.783
17.000
4.600
6.867
Normalize the matrix by dividing each value by the column sum (e.g. 1 / 15.33 = 0.065).
Then compute the average value for each row.
Price
Body Style
MPG
Interior Quality
Engine Size
0.065
0.052
0.176
0.043
0.029
Price
0.261
0.209
0.294
0.652
0.049
Body Style
0.022
0.042
0.059
0.043
0.049
MPG
0.326
0.070
0.294
0.217
0.728
Interior Quality
0.326
0.627
0.176
0.043
0.146
Engine Size
Looking at the average value of each row, notice that 33% of my objective weight is on Interior Quality,
29% on Body Style, 26% on Engine Size. These are the weights of the criteria.
Average
0.073
0.293
0.043
0.327
0.264
Slide 11
The Decision Candidates
The cars under consideration
Price
Body Style
MPG
Interior Quality
Engine Size
X-Treem
Yaawhee
Zoomer
$25,000
4-door Mid-size
$27,000
5-door SportWagon
$29,000
4-door Full-size
19
Standard
22
Deluxe
17
Above Average
3.8 Liter V-6
2.8 Liter 4 Cyl.
5.0 Liter V-8
The next step is to evaluate all three cars on each of the five criteria as
shown in the Hierarchy Chart on Slide 7
Slide 12
Evaluate Price for Each Car
X-Treem = $25,000, Yaawhee = $27,000, Zoomer = $29,000
Price of three cars. Lower Cost is better.
X is slightly more important than Y. X strongly more important than Z.
Y is slightly more important than Z.
X-Treem
Yaawhee
3
Zoomer
5
X-treem
Yaawhee
Zoomer
1/3
1
1/5
1/3
1
Column Sum
1.533
4.333
9.000
1
3
Normalize the Price matrix by dividing each value by the column sum
(e.g. 1 / 1.533 = 0.652). Then compute the average value for each row.
X-treem
Yaawhee
Zoomer
X-Treem
Yaawhee
Zoomer
Average
0.652
0.217
0.130
0.692
0.231
0.077
0.556
0.333
0.111
0.633
0.260
0.106
Slide 13
Evaluate Body Style for Each Car
X-Treem = Mid-sized, Yaawhee = Wagon, Zoomer = Full-sized
Body Style of three cars. I prefer Full-size, then Wagon, then Mid-size.
Z is slightly more important than Y. Z is strongly more important than X.
Y is slightly more important than X.
X-Treem
Yaawhee
Zoomer
1
3
1/3
1/5
1/3
X-treem
Yaawhee
Zoomer
5
3
Column Sum
9.000
4.333
1.533
X-Treem
Yaawhee
Zoomer
Average
0.111
0.333
0.556
0.077
0.231
0.692
0.130
0.217
0.652
0.106
0.260
0.633
1
1
Normalized Body Style Matrix
X-treem
Yaawhee
Zoomer
Slide 14
Evaluate MPG for Each Car
X-Treem = 19 MPG, Yaawhee = 22 MPG, Zoomer = 17 MPG
MPG of three cars. Higher MPG is better.
Y is slightly better than X. Y is strongly better than Z.
X is slightly better than Z.
X-Treem
Yaawhee
Zoomer
1
1/3
3
X-treem
Yaawhee
Zoomer
3
1
5
1/3
1/5
1
Column Sum
4.333
1.533
9.000
X-Treem
Yaawhee
Zoomer
Average
0.231
0.217
0.333
0.260
0.692
0.077
0.652
0.130
0.556
0.111
0.633
0.106
Normalized MPG matrix
X-treem
Yaawhee
Zoomer
Slide 15
Evaluate Interior Quality
X-Treem = Standard, Yaawhee = Deluxe, Zoomer = Above Average
Interior Quality of three cars. I prefer Deluxe, then Above Avg, then Standard.
Y is slightly better than Z. Y is strongly better than X.
Z is slightly better than X.
X-Treem
Yaawhee
Zoomer
1
1/5
1/3
5
1
1/3
3
1
1.533
4.333
X-Treem
Yaawhee
Zoomer
Average
0.111
0.130
0.077
0.106
0.556
0.333
0.652
0.217
0.692
0.231
0.633
0.260
X-treem
Yaawhee
Zoomer
3
Column Sum
9.000
Normalized Interior Quality Matrix
X-treem
Yaawhee
Zoomer
Slide 16
Evaluate Engine Size
X-Treem = 3.8 liter V-6, Yaawhee = 2.8 liter 4 Cyl, Zoomer = 5.0 liter V-8
Engine Size of three cars. I prefer V6, then V8, then 4-Cylinder
X is slightly better than Z. X is strongly better than Y.
Z is slightly better than Y.
X-Treem
X-Treem
Yaawhee
Zoomer
1
Yaawhee
5
1/5
1/3
1.533
Zoomer
3
1
3
9.000
1/3
1
4.333
Normalize the Engine Size matrix and compute the average of each row
X-Treem
Yaawhee
Zoomer
X-Treem
Yaawhee
Zoomer
Average
0.652
0.130
0.217
0.556
0.111
0.333
0.692
0.077
0.231
0.633
0.106
0.260
Slide 17
Compute the Final Result
Relative Scores for each Objective. Collect all the computed average values
from each normalized matrix and multiply the original criteria weights.
Price
Body Style
MPG
Interior Quality
Engine Size
X-Treem
Yaawhee
Zoomer
Criteria Weight
0.633
0.260
0.106
0.073
0.106
0.260
0.260
0.633
0.633
0.106
0.293
0.043
0.106
0.633
0.633
0.106
0.260
0.260
0.327
0.264
Use these relative scores for each objective and multiply by the
original weights of the criteria:
X-Treem = 0.633(.073) +.106(.293) +.260(.043) +.106(.327) +.633(.264) =
0.290
Yaawhee = 0.260(.073) + .260(.293) + .633(.043) + .633(.327) + .106(.264) =
Zoomer = 0.106(.073) + .633(.293) + .106(.043) + .260(.327) + .260(.264) =
0.357
0.351
The winner is the Yaawhee.
Slide 18
What Happened??
„
This is not the result I expected from my original preferences!
‰ I would not buy a 4-cylinder sport wagon. Should I have given Engine
Size a wider separation in importance?
„
I forgot to consider consistency.
‰ If A is bigger than B, and B is bigger than C, then A must be bigger
than C.
‰
‰
„
Perfect consistency would be if A is 2 times bigger than B, and B is 3
times bigger than C, then A must be 6 times bigger than C.
Or if Body Style is 4 times more important than Price, and Price is 3
times more important than MPG, then Body Style must be 12 times
more important than MPG.
Determining the Consistency Index and the Consistency Ratio should
have been done on the initial “pair-wise” comparisons. This was the very
first matrix that defined the relative priorities of the criteria (Slide 11).
Slide 19
Consistency Index and Consistency Ratio
„
„
There are at least 2 methods to evaluate the weights for errors in judgment:
logarithmic least squares, and Saaty’s eigenvector method. Additionally
there are several techniques available to estimate Saaty’s eigenvector
method.
Important terms needed to understand Saaty’s method:
‰
‰
‰
‰
Lambda max. = the maximum eigenvalue (Perron root) of the matrix =
Lmax = λmax
C.I. = Consistency Index = (λmax – n) / (n – 1)
R.I. = Random Index. For each matrix of size n, Saaty’s team
generated random matrices and computed their mean C.I. value and
called it the Random Index. These values are shown in the next slide.
C.R. = Consistency Ratio = (C.I.) / (R.I.). A value less than or equal to
0.1 is acceptable. Larger values require the decision maker to reduce
the inconsistencies by revising judgments.
Slide 20
Random Index
Random Consistency Index Table
n
1
2
3
4
5
6
7
8
9
10
Random Index
0
0
0.58
0.90
1.12
1.24
1.32
1.41
1.45
1.49
This table represents a composite of two different experiments performed
by Saaty and his colleagues at the Oak Ridge National Laboratory and at
the Wharton School of the University of Pennsylvania.
500 random reciprocal n x n matrices were generated for n = 3 to n = 15
using the 1 to 9 scale.
The maximum eigenvalue was determined by raising each random matrix
to increasing powers and normalizing the result until the process
converged.
The consistency index was then computed on each matrix for n = 1
through n = 15. Only n = 1 through n = 10 is presented here.
Slide 21
Step by Step on Original Matrix
Prioritizing the 5 Criteria
Price
Body Style
MPG
Interior Quality
Engine Size
Column Sum
1.
2.
3.
4.
5.
6.
7.
Price
Body Style
MPG
Interior Quality
Engine Size
1
1/4
3
1/5
1/5
4
1
5
3
1/3
1/3
1/5
1
1/5
1/3
5
1/3
5
1
5
5
3
3
1/5
1
15.333
4.783
17.000
4.600
6.867
Add a new column (5th Root) and compute the 5th root of the product of the
values in each row.
Sum this 5th Root column.
Add another column (Priority Vector) and divide each value from Step 1 by
the sum in Step 2.
Add a new row (Priority Row) under Column Sum row and multiply the
Column Sum vector by the Priority Vector.
Lambda Max = the sum of the values computed in Step 4.
C.I. = (Lambda Max – 5) / (4)
C.R. = (C.I.) / (R.I.) = Step 6 divided by 1.12 from the Random Index Table
for n = 5
Slide 22
Determine My Consistency (Inconsistency?)
Back to my original “pair-wise” comparisons
Price
Body Style
MPG
Interior
Quality
Engine Size
1
4
1/4
1
3
5
1/5
3
1/5
1/3
0.496
1.821
0.079
0.288
1/3
1/5
1
1/5
1/3
0.339
0.054
5
1/3
5
1
5
2.108
0.334
5
3
3
1/5
1
1.552
0.246
Sum row
15.333
4.783
17.000
4.600
6.867
6.315
1.000
Priority row
1.204
1.379
0.911
1.536
1.687
Price
Body Style
MPG
Interior Quality
Engine Size
Compute the n-th root of the product of the values in each row.
e.g. 0.496 = the fifth root of (1*1/4*3*1/5*1/5)
This technique is called the geometric mean.
5th Root of Priority
Product
Vector
(n is the number of criteria = 5)
Priority Vector is the nth root divided by the sum of the nth root values.
e.g. 0.079 = (0.496 / 6.315)
Sum row = sum of each column
Priority row = (sum row value)*(priority vector)
LambdaMax = 6.717
= sum of Priority Row
Consistency Index (CI) = 0.429
= (LambdaMax -n) / (n-1) = (6.717 - 5) / (4)
Consistency Ratio (CR) = 0.383
= (CI) / (Random Index) = 0.429 / 1.12
CR should be less than 0.10 (up to 0.20 is tolerable) 38.3% is too inconsistent
Slide 23
Reconsider My Judgments
Price
Price
Price
Price
Body Style
Body Style
Body Style
MPG
MPG
Interior Quality
More Important
9 8 7 6 5
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More Important
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Body Style
MPG
Interior Quality
Engine Size
MPG
Interior Quality
Engine Size
Interior Quality
Engine Size
Engine Size
Engine Size is slightly more important to me than Interior Quality, not vice versa. This
was a mistake on my initial judgment matrix. After making this change my Consistency
Ratio moved from 0.383 to 0.145. Saaty states if this value is more than 10%, the
judgments may be somewhat random and should perhaps be revised.
Engine Size is strongly more important than MPG. I really just changed my mind here to
be more consistent. My Consistency Ratio then became 0.108.
Slide 24
Convert Revised Judgments to Matrix
Convert the pair-wise comparisons to a matrix. Shaded items are changes
from the original.
Prioritizing the 5 Criteria for New Judgements
Price
Body Style
MPG
Interior Quality
Engine Size
Column Sum
Price
Body Style
MPG
Interior Quality
Engine Size
1
1/4
3
1/5
1/5
4
1
5
3
1/3
1/3
1/5
1
1/5
1/5
5
1/3
5
1
1/3
5
3
5
15.333
4.783
19.000
3
7.400
1
2.067
Normalize the matrix by dividing each value by the column sum (e.g. 1 / 15.33 = 0.065).
Then compute the average value for each row.
Price
Body Style
MPG
Interior Quality
Engine Size
0.065
0.052
0.158
0.027
0.097
Price
0.261
0.209
0.263
0.405
0.161
Body Style
0.022
0.042
0.053
0.027
0.097
MPG
0.326
0.070
0.263
0.135
0.161
Interior Quality
0.326
0.627
0.263
0.405
0.484
Engine Size
Looking at the average value of each row, notice that 42% of my objective weight is now on Engine Size,
26% on Body Style, 19% on Interior Quality. These are the new weights of the criteria.
Average
0.080
0.260
0.048
0.191
0.421
Slide 25
Re- Determine My Consistency
Price
Body Style
MPG
Interior
Quality
1
1/4
3
1/5
1/5
0.496
0.073
4
1
5
3
1/3
1.821
0.268
1/3
1/5
1
1/5
1/5
0.306
0.045
5
1/3
5
1
1/3
1.227
0.180
5
3
5
1
2.954
0.434
Column total
15.333
4.783
19.000
7.400
2.067
6.803
1.000
Priority row
1.118
1.280
0.854
1.334
0.897
Price
Body Style
MPG
Interior Quality
Engine Size
Compute the n-th root of the product of the values in each row.
e.g. 0.496 = the fifth root of (1*1/4*3*1/5*1/5)
5th Root of Priority
Engine Size Product
Vector
3
(n is the matrix size = 5)
In Excel this formula is =Power(number,power)
This technique is called the geometric mean.
Priority Vector is the nth root divided by the sum of the nth root values.
e.g. 0.073 = (0.496 / 6.803)
Column Total = sum of each column
Priority row = (sum row value)*(priority vector)
LambdaMax = 5.483
= sum of Priority Row
Consistency Index (CI) = 0.121
Consistency Ratio (CR) = 0.108
= (LambdaMax -n) / (n-1) = (5.483 - 5) / (4)
= (CI) / (Random Index) = 0.121 / 1.12
CR should be less than 0.10 (up to 0.20 is tolerable)
I am now more consistent but still not perfect.
Slide 26
Re-Compute Using New Criteria Weights
Note the average scores comparing the criteria against each car remain
unchanged. Only the Criteria Weights have changed.
Price
Body Style
MPG
Interior Quality
Engine Size
X-Treem
Yaawhee
Zoomer
Criteria Weight
0.633
0.260
0.106
0.080
0.106
0.260
0.633
0.260
0.260
0.633
0.106
0.048
0.106
0.633
0.260
0.191
0.633
0.106
0.260
0.421
Use the same relative scores for each objective and multiply by the revised
weights of the criteria:
X-Treem = 0.633(.080) +.106(.260) +.260(.048) +.106(.191) +.633(.421) =
0.377
Yaawhee = 0.260(.080) + .260(.260) + .633(.048) + .633(.191) + .106(.421) =
0.284
Zoomer = 0.106(.080) + .633(.260) + .106(.048) + .260(.191) + .260(.421) =
0.337
The winner is the X-Treem, the Mid-sized V-6. This is the result I really
expected.
Slide 27
Summary
1. Define the problem and specify the solution desired.
•
•
Lay out the elements of a problem as a hierarchy.
Structure the hierarchy from the top levels to the level at which decisions to
solve the problem is possible.
2. Do paired comparisons among the elements of a level as required by
the criteria of the next higher level.
•
•
•
Give a judgment that indicates the dominance as a whole number.
Enter that number and its reciprocal in the appropriate position in the matrix.
An element on the left is examined regarding its dominance over an element
at the top of the matrix.
3. These comparisons produce priorities and finally, through synthesis, to
overall priorities.
•
Check for consistency.
4. Repeat steps 2 and 3 for all levels in the hierarchy.
Slide 28
New Developments
„
The example just presented used the Geometric Mean technique for approximating
an eigenvector. This technique is described in Saaty’s book, “The Analytic Hierarchy
Process”, written in 1980, and is also the technique presented in Appendix D.9 of the
INCOSE Systems Engineering Handbook, Version 2a, dated 2004.
„
In 2001, Saaty wrote another book, “Decision Making for Leaders”, that in some
respects differs from his original technique.
‰
‰
‰
„
He recomputed the Random Consistency Index. (Comparison on next slide.)
For the approximation procedure to obtain Lambda Max, he states that the geometric mean
method (using the nth root of the products) should only be used for a matrix of size n = 3.
Otherwise the row average method should be used.
The consistency ratio should be 5% or less for n = 3; 9% or less for n = 4; and 10% or less for
n > 4.
Both the Geometric Mean and the Row Average techniques for approximating the
eigenvector of a reciprocal matrix are described in Saaty’s 1980 book and in the
reference sited in the INCOSE SE Handbook (IEEE Transactions on Engineering
Management, August 1983).
‰
The INCOSE SE Handbook only presents the Geometric Mean technique.
Slide 29
Random Consistency Index Changes
Random Consistency Index Table - 1980
n
Random Index
1
0
2
0
3
0.58
4
0.90
5
1.12
6
1.24
7
1.32
8
1.41
9
1.45
10
1.49
6
1.25
7
1.35
8
1.40
9
1.45
10
1.49
Random Consistency Index Table - 2001
n
Random Index
1
0
2
0
3
0.52
4
0.89
5
1.11
In Saaty’s 2001 book he notes these values were recently recalculated.
What if I re-compute my consistency using these new developments?
Slide 30
Re-Compute Consistency
Prioritized New Judgments
Price
Price
Body Style
MPG
Interior Quality
Engine Size
Column Sum
1
4
1/3
5
5
15.33
Body
Style
1/4
1
1/5
1/3
3
4.78
MPG
3
5
1
5
5
19.00
Interior
Quality
Engine
Size
1/5
3
1/5
1
1/5
1/3
1/5
1/3
3
7.40
1
2.07
Normalize the matrix above by dividing each entry by its column sum
Add a column to sum each row and then take the average.
Price
Body Style
MPG
Interior Quality
Engine Size
Column sum
Price
Body
Style
0.07
0.26
0.02
0.33
0.33
1.000
Row
Sum
Priority Vector
(Row sum
average)
MPG
Interior
Quality
Engine
Size
0.05
0.21
0.04
0.07
0.63
0.16
0.26
0.05
0.26
0.26
0.03
0.41
0.03
0.14
0.41
0.10
0.16
0.10
0.16
0.48
0.40
1.30
0.24
0.96
2.11
0.08
0.26
0.05
0.19
0.42
1.000
1.000
1.000
1.000
5.000
1.000
Slide 31
Row Average Technique - Continued
Multiply original non-normalized matrix by Priority Vector
Total each Row
Price
Body Style
MPG
Interior Quality
Engine Size
Price
Body
Style
0.08
0.32
0.03
0.40
0.40
0.07
0.26
0.05
0.09
0.78
MPG
Interior
Quality
Engine
Size
0.15
0.25
0.05
0.25
0.25
0.04
0.57
0.04
0.19
0.57
0.08
0.14
0.08
0.14
0.42
Row
Totals
0.42
1.54
0.25
1.07
2.42
e.g. For the first row of the matrix on Slide 31 --1 * .08, ¼ * .26, 3 * .05, 1/5 * .19, and 1/5 * .42
Slide 32
Estimate the Eigenvector
Take column of Row Totals and divide by the Priority Vector
0.42
1.54
0.25
1.07
2.42
divide by
0.08
0.26
0.05
0.19
0.42
equals
5.21
5.92
5.01
5.61
5.76
Now average the result to obtain Lambda Max (5.21 + 5.92 + 5.01 + 5.61 + 5.76) / 5
Lambda Max
5.50
Consistency Index
0.13 = (LambdaMax -n) / (n-1) = (5.50 - 5) / (4)
Consistency Ratio
0.12 = (CI) / (Random Index) = 0.13 / 1.11 ...note new RI value used here
Techniques compared:
Lambda Max
Consistency Index
Consistency Ratio
Row Average
5.50
.13
.12
Geometric Mean
5.48
.12
.11
The Row Average technique produces a consistency ratio that is slightly
worse than the Geometric Mean technique (0.11).
Slide 33
Summary
„
„
Consistency in the pair-wise comparisons of your criteria in very
important.
‰ My first attempt would have led to an incorrect decision. Revising
my judgments changed my consistency ratio from 38% to 11%,
where the goal is 10% or less. These more consistent judgments
changed the results of my decision.
Using Saaty’s recommendations from his 2001 book instead of
his original 1980 book produced a larger inconsistency (12%) of
my judgments of the pair-wise comparisons.
‰ This implies I should go back to my judgments (pair-wise
comparisons) of the criteria and reconsider their relative
importance to me.
Slide 34
References
„
Golden, Bruce L., Wasil, Edward A, and Harker, Patrick T.
(editors): “The Analytic Hierarchy Process - Applications and
Studies”, Springer-Verlag, Berlin, 1989.
„
INCOSE Systems Engineering Handbook, Version 2a, Appendix
D.9, International Council on Systems Engineering, INCOSE-TP2003-016-02, Version 2a, 1 June 2004.
„
Saaty, Thomas L.: “Decision Making for Leaders”, RWS
Publications, Pittsburgh, 2001.
„
Saaty, Thomas L.: “Priority Setting in Complex Problems”, IEEE
Transactions on Engineering Management, Vol. EM-30, No. 3,
August 1983.
„
Saaty, Thomas L.: “The Analytic Hierarchy Process”, McGraw-Hill,
Inc., New York, 1980.
Slide 35