262
N. M. Kuzmina and A. N. Ridley
18.3 Evaluation of the Contribution of Criteria Based
on the Pairwise Comparisons Method
The definitions of the incomplete pairwise comparison solutions without and with
interval alternative preference ratings are provided in Sects. 18.3.1 and 18.3.2, respectively. Section 18.3.3 contains a method of processing expert data for increasing
matrix consistency. Section 18.3.4 is the main subsection. It contains the description
of the algorithm to obtain a solution to the problem and results of its execution—
solution of the criteria ranking problem. Section 18.3.5 contains a basic analysis of
the results.
18.3.1 Incomplete Pairwise Comparisons Method
Let {O 1 , O 2 , . . . , O N } be a set of alternatives, where N is the object count. S =
s i j
,1 ≤ i, j ≤ N is the pairwise comparisons matrix, where alternatives preference
relations using Saaty’s scale. Some pairs may not be rated and it will be marked
“NA” in pairwise comparisons matrix. Note that all relations are inverse symmetric
(s i j = s
−1
ji ) or both marked NA [7–9].
Let’s construct a directed weighted graph G s := (V, E), where V =
{O 1 , O 2 , . . . , O N } are the vertices of this graph, E =
i, j : s i j = N A, s i j > s ji
are the edges and weights correspond to values from s i j .
Select a connected subgraph G
∗
s from graph G s : delete edges from G s , which
mostly disrupt transitivity s ik = s i j s jk . Thus, for all edges i, j in G
∗
s will be
s i j = w i /w j . This will uniquely identify the alternative weights vector W = (w i ) is
the solution of the incomplete pairwise comparisons problem.
18.3.2 Incomplete Pairwise Comparisons Method
with Interval Alternative Preference Ratings
Let
C
1
, C
2
, . . . , C
M
be the survey results obtained from M experts. They are
presented as incomplete pairwise comparisons matrices with interval estimates of N
alternatives. That is all elements are determined by Eq. 18.1 or as NA.
C
m
=
c
m
i j
, c
m
i j =
bottom
m
i j , top
m
i j
1 ≤ m ≤ M 1 ≤ i, j ≤ N
(18.1)
Due to the principle of inverse symmetry of pairwise comparisons matrices, the
following equality will be true:
bottom
m
i j · top
m
ji = 1.
(18.2)
N. M. Kuzmina and A. N. Ridley
18.3 Evaluation of the Contribution of Criteria Based
on the Pairwise Comparisons Method
The definitions of the incomplete pairwise comparison solutions without and with
interval alternative preference ratings are provided in Sects. 18.3.1 and 18.3.2, respectively. Section 18.3.3 contains a method of processing expert data for increasing
matrix consistency. Section 18.3.4 is the main subsection. It contains the description
of the algorithm to obtain a solution to the problem and results of its execution—
solution of the criteria ranking problem. Section 18.3.5 contains a basic analysis of
the results.
18.3.1 Incomplete Pairwise Comparisons Method
Let {O 1 , O 2 , . . . , O N } be a set of alternatives, where N is the object count. S =
s i j
,1 ≤ i, j ≤ N is the pairwise comparisons matrix, where alternatives preference
relations using Saaty’s scale. Some pairs may not be rated and it will be marked
“NA” in pairwise comparisons matrix. Note that all relations are inverse symmetric
(s i j = s
−1
ji ) or both marked NA [7–9].
Let’s construct a directed weighted graph G s := (V, E), where V =
{O 1 , O 2 , . . . , O N } are the vertices of this graph, E =
i, j : s i j = N A, s i j > s ji
are the edges and weights correspond to values from s i j .
Select a connected subgraph G
∗
s from graph G s : delete edges from G s , which
mostly disrupt transitivity s ik = s i j s jk . Thus, for all edges i, j in G
∗
s will be
s i j = w i /w j . This will uniquely identify the alternative weights vector W = (w i ) is
the solution of the incomplete pairwise comparisons problem.
18.3.2 Incomplete Pairwise Comparisons Method
with Interval Alternative Preference Ratings
Let
C
1
, C
2
, . . . , C
M
be the survey results obtained from M experts. They are
presented as incomplete pairwise comparisons matrices with interval estimates of N
alternatives. That is all elements are determined by Eq. 18.1 or as NA.
C
m
=
c
m
i j
, c
m
i j =
bottom
m
i j , top
m
i j
1 ≤ m ≤ M 1 ≤ i, j ≤ N
(18.1)
Due to the principle of inverse symmetry of pairwise comparisons matrices, the
following equality will be true:
bottom
m
i j · top
m
ji = 1.
(18.2)