18 Incomplete Pairwise Comparisons Method for Estimating …
263
Let C =
c i j
be the combined results of survey formed from set
C
1
, C
2
, . . . , C
M
. Values are presented as intervals c i j =
b i j , t i j
, where b i j are
the values of lower bound and t i j are the values of upper bound. Bounds are defined
according to Eq. 18.3.
b i j =
M
k=1
bottom
k
i j
1
M
t i j =
M
k=1
top
k
i j
1
M
(18.3)
The solution to incomplete pairwise comparisons problem with interval alternatives’ ratings is the solution to incomplete pairwise comparisons problem, which
corresponds to Eq. 18.4.
W : b i j ≤ w i /w j ≤ t i j
(18.4)
It is easy to see that in most cases there is infinite number of solutions. That is
why we determine two supporting solutions: upper (or top) solution and lower (or
bottom) solution. Top solution is the solution to the incomplete pairwise comparisons
problem for matrix T =
t i j
. Bottom solution is built by moving top solution to
lower bound. Middle solution builds based on top solution and bottom solution.
Algorithm of getting this solution will be described in the following sections.
18.3.3 Processing Expert Data
Using expert estimates in pairwise comparisons matrices form (like was described
in Eq. 18.1) we got two matrices of coefficients: upper bound T and lower bound B.
Values of these matrices are presented in Tables 18.2 and 18.3, respectively.
Table 18.2 Upper bound coefficients
ODC ACC QQRC ADIC APIC LRC ITC
CTC ISC
CLC OSC
ODC 1.0000 3.9360 1.4142 1.8612 2.2134 1.8612 1.3161 7.4833 9.0000 9.0000 9.0000
ACC 0.3433 1.0000 0.4083 0.4518 0.5000 0.4083 0.3689 2.2134 2.9130 3.7606 4.3559
QQRC 1.0000 3.4641 1.0000 1.1892 1.6818 1.4142 1.1892 6.7354 7.4539 9.0000 9.0000
ADIC 0.8409 2.7108 1.0000 1.0000 1.1892 1.0000 1.0000 5.4772 7.9686 8.2067 9.0000
APIC 0.7071 2.2134 0.8409 1.0000 1.0000 1.0000 0.7071 4.7287 6.0000 7.4833 9.0000
LRC 1.0000 3.0000 1.1892 1.1892 2.2134 1.0000 1.0000 6.2357 8.4519 7.7373 9.0000
ITC 1.1892 3.4641 1.3161 1.4142 2.0000 1.1892 1.0000 7.2376 9.0000 9.0000 9.0000
CTC 0.1543 0.5000 0.1757 0.2115 0.2541 0.1931 0.1615 1.0000 1.5651 2.0000 2.2795
ISC 0.1179 0.4518 0.1543 0.1964 0.1931 0.1679 0.1361 1.1892 1.0000 1.4142 1.8612
CLC 0.1111 0.4205 0.1183 0.1436 0.1708 0.1382 0.1144 1.0000 1.0000 1.0000 1.1892
OSC 0.1111 0.3433 0.1111 0.1309 0.1183 0.1179 0.1111 0.9036 1.0000 1.1892 1.0000
263
Let C =
c i j
be the combined results of survey formed from set
C
1
, C
2
, . . . , C
M
. Values are presented as intervals c i j =
b i j , t i j
, where b i j are
the values of lower bound and t i j are the values of upper bound. Bounds are defined
according to Eq. 18.3.
b i j =
M
k=1
bottom
k
i j
1
M
t i j =
M
k=1
top
k
i j
1
M
(18.3)
The solution to incomplete pairwise comparisons problem with interval alternatives’ ratings is the solution to incomplete pairwise comparisons problem, which
corresponds to Eq. 18.4.
W : b i j ≤ w i /w j ≤ t i j
(18.4)
It is easy to see that in most cases there is infinite number of solutions. That is
why we determine two supporting solutions: upper (or top) solution and lower (or
bottom) solution. Top solution is the solution to the incomplete pairwise comparisons
problem for matrix T =
t i j
. Bottom solution is built by moving top solution to
lower bound. Middle solution builds based on top solution and bottom solution.
Algorithm of getting this solution will be described in the following sections.
18.3.3 Processing Expert Data
Using expert estimates in pairwise comparisons matrices form (like was described
in Eq. 18.1) we got two matrices of coefficients: upper bound T and lower bound B.
Values of these matrices are presented in Tables 18.2 and 18.3, respectively.
Table 18.2 Upper bound coefficients
ODC ACC QQRC ADIC APIC LRC ITC
CTC ISC
CLC OSC
ODC 1.0000 3.9360 1.4142 1.8612 2.2134 1.8612 1.3161 7.4833 9.0000 9.0000 9.0000
ACC 0.3433 1.0000 0.4083 0.4518 0.5000 0.4083 0.3689 2.2134 2.9130 3.7606 4.3559
QQRC 1.0000 3.4641 1.0000 1.1892 1.6818 1.4142 1.1892 6.7354 7.4539 9.0000 9.0000
ADIC 0.8409 2.7108 1.0000 1.0000 1.1892 1.0000 1.0000 5.4772 7.9686 8.2067 9.0000
APIC 0.7071 2.2134 0.8409 1.0000 1.0000 1.0000 0.7071 4.7287 6.0000 7.4833 9.0000
LRC 1.0000 3.0000 1.1892 1.1892 2.2134 1.0000 1.0000 6.2357 8.4519 7.7373 9.0000
ITC 1.1892 3.4641 1.3161 1.4142 2.0000 1.1892 1.0000 7.2376 9.0000 9.0000 9.0000
CTC 0.1543 0.5000 0.1757 0.2115 0.2541 0.1931 0.1615 1.0000 1.5651 2.0000 2.2795
ISC 0.1179 0.4518 0.1543 0.1964 0.1931 0.1679 0.1361 1.1892 1.0000 1.4142 1.8612
CLC 0.1111 0.4205 0.1183 0.1436 0.1708 0.1382 0.1144 1.0000 1.0000 1.0000 1.1892
OSC 0.1111 0.3433 0.1111 0.1309 0.1183 0.1179 0.1111 0.9036 1.0000 1.1892 1.0000