Sustainable Construction A Life Cycle Approach in Engineering

The results of pairwise comparisons are used to define judgment matrices, also called Pairwise
matrices or P matrices.
Each entry p ij of these matrices corresponds to the number that estimates according to selected
scale the relative performance (or weights of importance) of entity E i when it is compared
with E j . Entities can be represented by alternatives A i or criteria C i that have to be compared
in terms of selected criterion or decision goal.
E E E E E
1 2 3 4 n
E p p p
... ... ...
E
2
p21
P
nxn
1 11 12 1n
E
p p
n m1 mn
(5)
The matrix of pairwise comparisons is reciprocal, being p ij =1/p ij and p ii =1. In case it is possible
to have exact value of p ij = w i /w j , with w i actual weigth of importance of entity E i , the matrix
[P] is consistent, that is p ij =p ik·p kj . It can be shown that for real values of p ij the rank of [P] is 1
and that λ=n is the principal eigenvalue. When the values of p ij are close to actual ones but different,
which is the most common case in applications, the [P] matrix is not consistent, the rank
is greater than 1 and the maximum eigenvalue λ max is greater than or equal to n.
It can be demonstrated that the normalized principal eigenvector w of P matrix that corresponds
to the maximum eigenvalue λ max represents the vector (a i1 , a i2 ,…., a in ) with relative performance,
or weights, of compared entities, that is the i-th colum of matrix [A] or the vector of
relative weights of criteria.
In particular, it results:
P w w with max n
(6)
max
w=(a
i1, a
i2,..., a
in)
(7)
A simplified calculation of eigenvector w can be performed by multiplying the entries in
each row of matrix P together and taking the n-th root. In order to have values that add up to 1,
a normalization by the sum of obtained values has to be carried out. A simplified procedure to
obtain λ max is to add the columns of matrix P and the multiplying the resulting vector with w.
To estimate the consistency of pairwise comparison matrix, Saaty proposed the Consistency
Ratio CR, obtained dividing the consistency index by the Random Consistency Index RCI given
in table 2.
CI is defined as follows:
CI ( n)/(n 1)
(8)
max
RCI is an average random consistency index derived from randomly generated reciprocal matrices.
According to Saaty, the CR should be less than 10% in order to have consistency in pairwise
judgments.
Table 2. RCI values of sets of different order n.
n 1 2 3 4 5 6 7 8 9
RCI 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45
180