The Evolution of the Water–Energy–Food Nexus …
51
index value according to Saaty’s AHP pairwise comparison matrix (PCM), and then
normalised to have the indicator [70, 71].
The AHP was used to numerically relate indicators through the pairwise comparison matrix (PCM) (Eq. 1). This theory (AHP) has a comparison matrix that compares
two factors/indicators at a time, applying a scale ratio of between 1/9 and 9 (Table 3)
[71]. An array between 1 and 9 represents a significant relationship, and a range
between 1/3 and 1/9 symbolises a less important relationship. These ranges are well
illustrated and described in Table 3.
A ranking of 9 shows that in relation to the column factor, the row factor is 9 times
more important. On the contrary, a rating of 1/9 indicates that relative to the column
indicator, the row indicator is 1/9 less important. But in instances where the column
and row indicators are equally important, they have a rating of 1. In WEF nexus, the
indices are dependent on the impact of the indicator on its overall scoring.
The PCM was used to determine indices, taking the eigenvector (a vector with the
same direction even if a linear transformation is applied) conforming to the largest
eigenvalue (the rank of the eigenvector) of the pattern (matrix), and thereafter normalising the total of the factors or indices [76]. Figure 8 shows a graphical presentation
of the processes followed when applying the AHP.
Table 3 Fundamental scale for pairwise comparisons
Intensity of importance
Definition
Explanation
1
Equal importance
Elements a and b contribute
equally to the objective
3
Moderate/weak importance
of one over another
Experience and judgement
slightly favour element a over b
5
Essential or strong
importance
Experience and judgement
strongly favour element a over b
7
Demonstrated importance
Element a is favoured very
strongly over b; its dominance
is demonstrated in practice
9
Absolute importance
The evidence favouring element
a over b is of the highest
possible order of affirmation
2, 4, 6, 8, 1/2, 1/4, 1/6, 1/8
Intermediate values between
the two adjacent judgements
When compromise is needed.
For example, 4 can be used for
the intermediate value between
3 and 5
1/3
Moderately less important
1/5
Strongly less important
1/7
Very strongly less important
1/9
Extremely less important
Reciprocals of above nonzero If a has one of the above numbers assigned to it when compared
with b, then b has the reciprocal value when compared with a
Source Saaty [71]
51
index value according to Saaty’s AHP pairwise comparison matrix (PCM), and then
normalised to have the indicator [70, 71].
The AHP was used to numerically relate indicators through the pairwise comparison matrix (PCM) (Eq. 1). This theory (AHP) has a comparison matrix that compares
two factors/indicators at a time, applying a scale ratio of between 1/9 and 9 (Table 3)
[71]. An array between 1 and 9 represents a significant relationship, and a range
between 1/3 and 1/9 symbolises a less important relationship. These ranges are well
illustrated and described in Table 3.
A ranking of 9 shows that in relation to the column factor, the row factor is 9 times
more important. On the contrary, a rating of 1/9 indicates that relative to the column
indicator, the row indicator is 1/9 less important. But in instances where the column
and row indicators are equally important, they have a rating of 1. In WEF nexus, the
indices are dependent on the impact of the indicator on its overall scoring.
The PCM was used to determine indices, taking the eigenvector (a vector with the
same direction even if a linear transformation is applied) conforming to the largest
eigenvalue (the rank of the eigenvector) of the pattern (matrix), and thereafter normalising the total of the factors or indices [76]. Figure 8 shows a graphical presentation
of the processes followed when applying the AHP.
Table 3 Fundamental scale for pairwise comparisons
Intensity of importance
Definition
Explanation
1
Equal importance
Elements a and b contribute
equally to the objective
3
Moderate/weak importance
of one over another
Experience and judgement
slightly favour element a over b
5
Essential or strong
importance
Experience and judgement
strongly favour element a over b
7
Demonstrated importance
Element a is favoured very
strongly over b; its dominance
is demonstrated in practice
9
Absolute importance
The evidence favouring element
a over b is of the highest
possible order of affirmation
2, 4, 6, 8, 1/2, 1/4, 1/6, 1/8
Intermediate values between
the two adjacent judgements
When compromise is needed.
For example, 4 can be used for
the intermediate value between
3 and 5
1/3
Moderately less important
1/5
Strongly less important
1/7
Very strongly less important
1/9
Extremely less important
Reciprocals of above nonzero If a has one of the above numbers assigned to it when compared
with b, then b has the reciprocal value when compared with a
Source Saaty [71]