The eigenvectors for each row were calculated using geometric
principles (multiplying the value for each criterion in each column in the same
row of the original pair-wise comparison matrix and then applying this to each
row) as follows:
Where, Egi = eigenvalue for the row i; n = number of elements
in row i. The priority vector is determined by normalizing the eigenvalue to
1(divided by their sum) as follows:
The lambda max (?max) was obtained from the summation of
products between each element of priority vector and the sum of columns of the
reciprocal matrix as shown in the following formula:
Where, aij = the sum of criteria in each column in the
matrix; Wi = the value of weight for each criterion
which is corresponding to the priority vector in the matrix
of decision, where the values (i=1, 2,…., m) and (j= 1, 2,…., n). So, the
lambda max (?max) in this study is equal to 16.852.
The CI (consistency index) was estimated using the following
= (?max ? n)/(n ? 1) (3.5)
CI represents the equivalent to the mean deviation of each comparison element
and the standard deviation of the evaluation error from the true ones (Sólnes,
2003), , and n is size or order of the matrix.
this study, CI = 0.1323.
The consistency ratio (CR) was obtained according to Saaty,
1980, by dividing the value of consistency index (CI) by the Random index
value (RI = 1.59) for n=15 (Table 3.4), where this table displays mean Random
index value RI for matrices with different sizes according to (Saaty, 1980).
CR = (CI/RI) (3.6)
If CR less than 0.1, the ratio indicates a reasonable consistency
level in the pairwise comparison. CR should, therefore, be less than 0.1.
In this study, CR = 0.0832< 0.1 and RI15= 1.59. For any matrix, the judgments are completely consistent if a CR is equal to zero (Coyle, 2004). The pairwise comparison matrices were prepared for fifteen criteria.