Analytical Hierarchy Process
A systematic method for comparing a list of
objectives or alternatives
When used in the systems engineering
process,AHP can be a powerful tool for
comparing alternative design concepts
ReferenceReferenceReference,Ernest H,Forman,Decision by
Objectives,
http://mdm.gwu.edu/Forman/DBO.pdf
ing
in i epts
,
AHP
Assume that a set of objectives has been established
(VSD,OH),and that we are trying to establish a
normalized set of weights to be used when
comparing alternatives using these objectives,
For simplicity,we assume that there are 4
objectives,O
1
,O
2
,O
3
,and O
4
,
ssume that a set of objectives has been established
(VSD,OH),and that we are trying to establish a
set of weights to be used when
comparing alternatives using these objectives,
or simplicity,we assume that there are 4
objectives:
1
,O
2
,O
3 4
,
AHP
Form a pairwise comparison matrix A,where
the number in the i
th
row and j
th
column gives
the relative importance of O
i
as compared with
O
j
Use a 1–9 scale,with
– a
ij
= 1 if the two objectives are equal in importance
– a
ij
= 3 if O
i
is weakly more important than O
j
– a
ij
= 5 if O
i
is strongly more important than O
j
– a
ij
= 7 if O
i
is very strongly more important than O
j
– a
ij
= 9 if O
i
is absolutely more important than O
j
– a
ij
= 1/3 if O
j
is weakly more important than O
i
i
th
j
th
i
j
se a 1 9 scale,with
ij
= 1 if the two objectives are equal in imp rtance
ij
= 3 if O
i
is weakly more important tha
j
ij
= 5 if O
i
is strongly more important tha
j
ij
= 7 if O
i
is very strongly more i
j
ij
= 9 if O
i
is absolutel important than O
j
ij
/3 if O
j
is weakly more important than O
i
AHP
Thus we might arrive at the following matrix:
To normalize the weights,compute the sum of each
column and then divide each column by the
corresponding sum
Using an overbar to denote normalization,we get:
hus we might arrive at the following matrix:
o normalize the weights,compute the sum of each
column and then divide each col mn by the
sing an o erbar to denote normalization,we get:
=
=
000.1333.0200.0000.7
000.3000.1333.0000.3
000.5000.3000.1000.5
143.0333.0200.0000.1
13/15/17
313/13
5315
7/13/15/11
A
=
109.0071.0115.0438.0
328.0214.0192.0188.0
547.0643.0577.0313.0
016.0071.0115.0063.0
A
AHP
The numbers in the second row are generally larger
than the rest of the numbers,except for the case of
column 1
This indicates some inconsistency in the comparisons
used in the original matrix
Ideally,the 4 normalized columns would all be
identical if the pairwise comparisons were consistent
In practice,one can compute a consistency measure
using the eigenvalues of the normalized comparison
matrix.
he numbers in the second row are generally larger
than the rest of the n mbers,except for the case of
his indicates some inconsistency in the co s
used in the original m trix
Ideally,the 4 normalized columns would all be
identical if the pairwise comparisons were consistent
In practice,one can compute a consistency measure
using the eigenvalues of the nor alize co
matrix.
=
109.0071.0115.0438.0
328.0214.0192.0188.0
547.0643.0577.0313.0
016.0071.0115.0063.0
A
AHP
The next step is to compute the average
values of each row and use these as the
weights in the Objective Hierarchy
For this example,the weights would be:
Note that by construction,
These weights would be used in summing
the measures as required in the evaluation
of the Objective Hierarchy.
s,
[]
T
w 183.0231.0520.0066.0=
∑
=
=
4
1
.1
i
i
w
A systematic method for comparing a list of
objectives or alternatives
When used in the systems engineering
process,AHP can be a powerful tool for
comparing alternative design concepts
ReferenceReferenceReference,Ernest H,Forman,Decision by
Objectives,
http://mdm.gwu.edu/Forman/DBO.pdf
ing
in i epts
,
AHP
Assume that a set of objectives has been established
(VSD,OH),and that we are trying to establish a
normalized set of weights to be used when
comparing alternatives using these objectives,
For simplicity,we assume that there are 4
objectives,O
1
,O
2
,O
3
,and O
4
,
ssume that a set of objectives has been established
(VSD,OH),and that we are trying to establish a
set of weights to be used when
comparing alternatives using these objectives,
or simplicity,we assume that there are 4
objectives:
1
,O
2
,O
3 4
,
AHP
Form a pairwise comparison matrix A,where
the number in the i
th
row and j
th
column gives
the relative importance of O
i
as compared with
O
j
Use a 1–9 scale,with
– a
ij
= 1 if the two objectives are equal in importance
– a
ij
= 3 if O
i
is weakly more important than O
j
– a
ij
= 5 if O
i
is strongly more important than O
j
– a
ij
= 7 if O
i
is very strongly more important than O
j
– a
ij
= 9 if O
i
is absolutely more important than O
j
– a
ij
= 1/3 if O
j
is weakly more important than O
i
i
th
j
th
i
j
se a 1 9 scale,with
ij
= 1 if the two objectives are equal in imp rtance
ij
= 3 if O
i
is weakly more important tha
j
ij
= 5 if O
i
is strongly more important tha
j
ij
= 7 if O
i
is very strongly more i
j
ij
= 9 if O
i
is absolutel important than O
j
ij
/3 if O
j
is weakly more important than O
i
AHP
Thus we might arrive at the following matrix:
To normalize the weights,compute the sum of each
column and then divide each column by the
corresponding sum
Using an overbar to denote normalization,we get:
hus we might arrive at the following matrix:
o normalize the weights,compute the sum of each
column and then divide each col mn by the
sing an o erbar to denote normalization,we get:
=
=
000.1333.0200.0000.7
000.3000.1333.0000.3
000.5000.3000.1000.5
143.0333.0200.0000.1
13/15/17
313/13
5315
7/13/15/11
A
=
109.0071.0115.0438.0
328.0214.0192.0188.0
547.0643.0577.0313.0
016.0071.0115.0063.0
A
AHP
The numbers in the second row are generally larger
than the rest of the numbers,except for the case of
column 1
This indicates some inconsistency in the comparisons
used in the original matrix
Ideally,the 4 normalized columns would all be
identical if the pairwise comparisons were consistent
In practice,one can compute a consistency measure
using the eigenvalues of the normalized comparison
matrix.
he numbers in the second row are generally larger
than the rest of the n mbers,except for the case of
his indicates some inconsistency in the co s
used in the original m trix
Ideally,the 4 normalized columns would all be
identical if the pairwise comparisons were consistent
In practice,one can compute a consistency measure
using the eigenvalues of the nor alize co
matrix.
=
109.0071.0115.0438.0
328.0214.0192.0188.0
547.0643.0577.0313.0
016.0071.0115.0063.0
A
AHP
The next step is to compute the average
values of each row and use these as the
weights in the Objective Hierarchy
For this example,the weights would be:
Note that by construction,
These weights would be used in summing
the measures as required in the evaluation
of the Objective Hierarchy.
s,
[]
T
w 183.0231.0520.0066.0=
∑
=
=
4
1
.1
i
i
w
