3 Analytical Hierarchy Process (AHP)
23
Table 3.2 Corresponding
random index values for
different matrix dimensions
Matrix dimensions (n)
Random Index value (RI)
1
0.0
2
0.00
3
0.58
4
0.90
5
1.12
6
1.24
7
1.32
8
1.41
9
1.45
10
1.49
Otherwise, if the consistency ratio is found to be greater than the standard value of 0.1,
then the problem is insufficiently consistent and requires re-evaluation. In the case
where the problem is denoted as inconsistent, the decision-makers must re-assess the
preferential reasoning and identify the source of the variance or inconsistencies to
then rectify and refine them so that consistency holds true for the problem at hand.
Notice that as the number of criteria increases or as the matrix dimension gets
larger, a subsequent incremental effect is imposed onto the corresponding index
value as well. This indicates that as the problem becomes more complex, meaning
it encompasses a greater number of criteria and sub-criteria (dimensions), then the
possibility of inconsistencies in judgments also increases. Relating this ideology to
logic helps enhance the fundamental understanding of the concept. In this sense, if
we increase the instances where the decision-maker has to provide an input based on
personal preference, i.e. criteria, then a higher chance of variations will be induced.
The work done by Saaty proved that for the consistency problem, the pairwise
matrix is reciprocal and positive and holds a maximum Eigen value equivalent to the
dimensions of the comparison matrix (n). In other words:
λmax = n
In addition to this, he introduced the consistency index as a measure of consistency
and this is found using the formula,
C.I =
λmax − n
n − 1
where n is the dimension of the reciprocal positive comparison matrix and λ max
represents the maximum Eigen value, which in the ideal case (no inconsistencies)
would hold the identical value n (dimension of the matrix). The closer the maximum
Eigen value is to the dimension of the matrix, the more the consistency index
decreases, and the consistency ratio decreases accordingly.
23
Table 3.2 Corresponding
random index values for
different matrix dimensions
Matrix dimensions (n)
Random Index value (RI)
1
0.0
2
0.00
3
0.58
4
0.90
5
1.12
6
1.24
7
1.32
8
1.41
9
1.45
10
1.49
Otherwise, if the consistency ratio is found to be greater than the standard value of 0.1,
then the problem is insufficiently consistent and requires re-evaluation. In the case
where the problem is denoted as inconsistent, the decision-makers must re-assess the
preferential reasoning and identify the source of the variance or inconsistencies to
then rectify and refine them so that consistency holds true for the problem at hand.
Notice that as the number of criteria increases or as the matrix dimension gets
larger, a subsequent incremental effect is imposed onto the corresponding index
value as well. This indicates that as the problem becomes more complex, meaning
it encompasses a greater number of criteria and sub-criteria (dimensions), then the
possibility of inconsistencies in judgments also increases. Relating this ideology to
logic helps enhance the fundamental understanding of the concept. In this sense, if
we increase the instances where the decision-maker has to provide an input based on
personal preference, i.e. criteria, then a higher chance of variations will be induced.
The work done by Saaty proved that for the consistency problem, the pairwise
matrix is reciprocal and positive and holds a maximum Eigen value equivalent to the
dimensions of the comparison matrix (n). In other words:
λmax = n
In addition to this, he introduced the consistency index as a measure of consistency
and this is found using the formula,
C.I =
λmax − n
n − 1
where n is the dimension of the reciprocal positive comparison matrix and λ max
represents the maximum Eigen value, which in the ideal case (no inconsistencies)
would hold the identical value n (dimension of the matrix). The closer the maximum
Eigen value is to the dimension of the matrix, the more the consistency index
decreases, and the consistency ratio decreases accordingly.