Normal Probability Plot
5. CHAPTER FIVE
5.1.2 Geospatial data preparation and presentation
5.1.2.1 The Analytical Hierarchy Process method: Criteria weights
The pairwise comparison matrix is the key to the AHP method. It serves as the input while the relative weights are produced to serve as outputs. The matrix was computed for pairwise comparison using a scale with values from 1 to 9 to measure the alternatives against each other (Table 5.3). The site selection process included eleven factors. As a result, the AHP method was employed to compute their weights.
Table 5.3: The pairwise comparison scale (Saaty, 1980) Intensity of importance Definition
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1 Equal importance
2 Equal to moderate importance
3 Moderate importance
4 Moderate to strong importance
5 Strong importance
6 Strong to very strong importance
7 Very strong importance
8 Very to extremely strong importance
9 Extreme importance
The matrix is used to determine the importance of one criterion against the other.
Furthermore, by measuring the consistency ratios, the AHP approach helps in evaluating inconsistency in datasets (Sisay et al., 2021). The typical pairwise comparison matrix for n objectives (Eq. 1).
[
]
(1)
Eq. 2 is the pairwise comparison matrix with the relative weights of the objectives. The greater a factor's influencing weight, the more significant it is (Makonyo and Msabi, 2021).
[
]
[ ]
(2)
Table 5.4 present the pairwise comparison matrix of the study criteria. The criteria were compared against each other using the 1-9 significance scale in table 5.3.
97 Table 5.4: Pairwise comparison matrix of the study criteria River Village Landus
e
Land elevation
Slope Soil type
Road Powerl ine
Protect ed site
Aspect
River 1 1 1 3 3 3 5 9 7 7
Village 1 1 1 3 3 1 5 9 5 7
Landuse 1 1 1 5 5 3 1 5 5 9
Land elevation
0.33 0.33 0.2 1 1 1 3 5 7 5
Slope 0.33 0.33 0.2 1 1 1 1 7 3 5
Soil type 0.33 1 0.33 1 1 1 5 5 3 5
Road 0.2 0.2 1 0.33 1 0.2 1 5 3 3
Powerlin e
0.11 0.11 0.2 0.2 0.14 0.2 0.2 1 1 3
Protected site
0.14 0.2 0.2 0.14 0.33 0.33 0.33 1 1 1
Aspect 0.14 0.14 0.11 0.2 0.2 0.2 0.33 0.33 1 1
The consistency ratio is very important as it confirms if the comparison is consistent or not. Therefore, after the weights are given and criteria are compared to one another, a consistency ratio is carried out to validate the comparison. That is acquired through a step by step that involves calculations of several equations as follows:
The Eigenvector calculations
The pairwise comparison matrix's output value is multiplied by the value for each criterion in each column of the same row to determine the eigenvectors (Egi) for each row. The output value is then subtracted from the root for the number of elements in each row and applied to each row in turn. (Eq.3).
The eigenvalue of a row is determined using the following equation:
√ (3)
Where Egi is the eigenvalue for row I and n denote the number of elements in row i.
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The priority vector is computed by normalizing the eigenvalue to 1 (the sum of the eigenvalues).
(∑
) (4)
Where Egk is the sum of the Egi
Table 5.5 represent the normalized pairwise comparison matrix of the study criteria. The normalized matrix is the first step that allows the performance of the consistency analysis to obtain the criteria weights and validate them.
Table 5.5: The normalized pairwise comparison matrix of the criteria River Village Landu
se
Elevati on
Slope Soil type
Road Powerl ines
Protect ed sites
Aspect CW
0.22 0.18 0.19 0.20 0.19 0.27 0.23 0.19 0.19 0.15 0.20 0.22 0.18 0.19 0.20 0.19 0.09 0.23 0.19 0.14 0.15 0.18 0.22 0.18 0.19 0.33 0.32 0.27 0.05 0.11 0.14 0.20 0.20 0.07 0.06 0.03 0.06 0.06 0.09 0.14 0.11 0.19 0.11 0.09 0.07 0.06 0.04 0.07 0.06 0.09 0.05 0.15 0.08 0.11 0.08 0.07 0.19 0.06 0.07 0.06 0.09 0.23 0.11 0.08 0.11 0.11 0.04 0.038 0.19 0.02 0.06 0.02 0.05 0.11 0.08 0.07 0.07 0.02 0.02 0.04 0.01 0.01 0.02 0.01 0.02 0.03 0.07 0.02 0.03 0.04 0.04 0.01 0.02 0.03 0.02 0.02 0.03 0.02 0.03 0.03 0.06 0.02 0.01 0.01 0.02 0.02 0.01 0.03 0.02 0.02
The weighted sum vector is divided by the predetermined criterion weights to get the consistency vector. The first criterion's weight is multiplied by the first column of the original comparison matrix in pairs, the second criterion's weight is multiplied by the second column, the third criterion's weight is multiplied by the third column of the original multiply, and then these values are added to produce the weighted sum vector.
After determining the eigenvalue and the priority vector, the lambda max (max) is produced from the summation of products by multiplying the sum of each matrix column by the corresponding value of the priority vector. By multiplying each priority vector
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component by the sum of the columns of the reciprocal matrix, the lambda max (max) was obtained., as given in the formula below.
∑ ( )
(4)
Where aij is the sum of criteria in each column of the matrix; Wi is the weight value for each criterion that corresponds to the priority vector in the decision matrix, with values (i=1, 2... n).
The right eigenvector, which is obtained from the maximum absolute eigenvalue (max, 1,2), is used to determine the weight coefficients of the ranking criterion and decision sub-criteria. All the criteria's grade values have been normalized to 1. The corresponding eigenvector of max is W, and the weight value for ranking is wi I = 1, 2..., n).
W is the associated primary eigenvector, Wi is the weight of criteria value, and I = 1, 2...
n is the number of criteria involved.
The Consistency Ratio (CR)
In AHP, the consistency index (CR) of a matrix is computed to access the Consistency Index (CI) of the utilised judgment during the weighing of the criteria (Saaty, 1980).
Consistency is measured by the Consistency Index (CI), which is a measure of deviance. The Consistency Index formula (Eq. 7).
( ) ( )
where, n is the size or order of the matrix, and CI is the equivalent of the mean deviation of each comparison element and the standard deviation of the evaluation error from the actual ones.
According to (Saaty, 1980), the consistency ratio (CR) was calculated by dividing the consistency index (CI) value by the Random index value (Eq. 8). The Random index value RI for matrices of various sizes (Saaty, 1980) for this study is displayed in table 3.6.
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The RI is determined by the number of elements being evaluated (see table 5.6). The conditions are: If CR< 0.1, this means there is consistency in the ratio values of the pairwise comparison, however, CR ≥ 0.10, means there is no consistency in the ratio values of the pairwise comparison.
Table 5.6: The "n" numbers of the Random Inconsistency Indices (Saaty, 1980).
N 1 2 3 4 5 6 7 8 9 10
RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49