Page 367 - Biofuels for a More Sustainable Future

Fuzzy MCDM for ranking the biofuels production pathways 323

(11.3) (11.4) (11.5) (11.6) (11.7) (11.8) (11.9)

λ2R

λ>0,
and
numbers,

fuzzy + b 3 Þ a 3 a 3 b 1 Þ

triangular + b 2 , a 2 a 2 b 2 , a 3 b 3 Þ a 3 = b 1 a 2 = b 2, a 3 b 3 Þ 2 i 1=2

numbers two are + b 1 , a 1 a 1 b 3 , a 2 b 2 , a 1 b 1 , a 1 = b 2, + ð

fuzzy b 1 , b 2 , b 3 Þ b 3 Þ ¼ ð b 3 Þ ¼ ð b 2 , b 3 Þ ¼ ð λa 3 Þ λa 2 , b 3 Þ ¼ b 2 , a 2 b 2 Þ 2 0<b 1 b 2 b 3

triangular e ¼ ð b 2 , b 1 , b 1 , b 2 , b 1 , λa 1 , b 1 , 1 a 1 a 2 , + ð and

the B and a 3 Þ + ð a 2 , a 3 Þ ð a 2 , a 3 Þ
ð a 3 Þ ¼ ð a 3 Þ ð a 2 , 1 1 a 3 , a 1 b 1 Þ 2 0<a 1 a 2 a 3
between a 1 , a 2 , a 3 Þ e a 1 , e a 1 , a 2 , a 1 , a 2 , a 1 , e a 1 , 1 , a 3 Þ ¼ a 1 , a 2 h e ¼ ð B

operations e ¼ ð A e + B ¼ ð A e B ¼ ð A e A
e ¼ ð B λ e ¼ λð A e B ¼ ð A 1 e ¼ ð A e A, d where

arithmetic

The distance
11.1 Subtraction Multiplication
Table Addition Scalar Division Reciprocal Euclidean