Page 316 - Materials Chemistry, Second Edition

15.3 Methods 315
In a similar way, the others-to-worst (BO) vector can also be determined by comparing the
relative preferences of all the T criteria with C L regarding their influences on the j-th factor by
using the above-mentioned five linguistic phrases. The linguistic terms in BO and OW vectors
can be transformed into triangular fuzzy numbers, denoted by V BO and V OW , respectively. It
e
e
is apparent that e a Bk ¼¼ 1, 1, 1ð Þ when k¼M, and e a kW ¼ 1, 1, 1ð Þ when k¼L.
½
V BO ¼ e a B1 e a B2 ⋯ e a BT Š (15.10)
e
e
V OW ¼ e a 1W e a 2W ⋯ e a TW Š (15.11)
½
where e a Bk k ¼ 1, 2, ⋯, Tð Þ and e a kW k ¼ 1, 2, ⋯, Tð Þ, which are fuzzy numbers, represent the rel-
ative preference of C M comparing with the k-th criterion and that of the k-th criterion compar-
ing with C L regarding their influences on the j-th criterion.
h i
Substep 3: Determining the fuzzy optimum weights of the T elements e ω j e ω j ⋯ e ω j .
1 2 T
The weights of these T factors, which represent the relative influences of these T criteria on
the j-th criterion, can be determined by solving programming (15.12).

minξ ∗
s:t:
j,L
ω

B a ξ ∗
j,L
j,U Bk
ω

k
j,M
ω
B a j,M ξ ∗
j,M Bk
ω
k

ω j,U
B j,U ξ ∗
j,L a Bk
ω
k
j,L
ω

k a j,L ξ ∗
j,U kW
ω

W
j,M
ω
(15.12)
k a j,M ξ ∗
j,M kW
ω

W
j,U
ω
k a j,U ξ ∗
j,L kW
ω
W
n Xω j,L +4ω j,M + ω j,U
n
X j j j j
R e ω ¼ ¼ 1
k 6
k¼1 k¼1

j,L
j
e ω ¼ ω , ω j,M , ω j,U
k k k k
ω j,L ω j,M ω j,U
k k k
ω j,L 0
k
k ¼ 1,2,⋯,T