Page 148 - Materials Chemistry, Second Edition

144 7. MCDM for sustainability ranking of district heating systems considering uncertainties

1.0 1
0.9 0.9
0.8 0.8
0.7 0.7
Cardinal values 0.6 Cardinal values 0.6

0.5
0.5
0.4
0.3 0.4
0.3
0.2
0.2
0.1 0.1
0.0 0
1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11
(A) Ordinal scales (B) Ordinal scales
1 1
0.9 0.9
0.8 0.8
0.7 0.7
Cardinal values 0.6 Cardinal values 0.5
0.6
0.5
0.4
0.4
0.3 0.3
0.2 0.2

0.1 0.1
0 0
1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11
(C) Ordinal scales (D) Ordinal scales
FIG. 7.2 Variation of Γ j mapping ordinal scales onto cardinal values with simulation iterations K from 1 to 1000 in
SMAA-O when j max ¼ m ¼11 (the straight lines stand for linear mapping) (Wang, 2013). (A) K ¼1, (B) K ¼100,
(C) K ¼500, and (D) K ¼1000.

7.2.4 Handling the uncertainties
A certain probability distribution around the expected values of the criteria PVs is used to
express the uncertainties. The most popular distributions are uniform and normal distribu-
tions (Lahdelma et al., 1998) and the former one is used in this study. The SMAA-O model
already takes into account the uncertainties when simulating the mapping processes; there-
fore, we only focus on how to treat the uncertainties in weighting by taking a 3-criterion ex-
ample. However, the same technique can be used in higher dimensions.