Page 146 - Materials Chemistry, Second Edition

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

where α r are the meta-weights, which means the contribution of each rank acceptability index
to the holistic evaluation. In general, first ranks contribute most and the worst ranks contrib-
ute least to the holistic acceptability index.
c
The central weight vector, w i , can be expressed as in Eq. (7.5).
ð ð
f W wðÞwdwdξ
f X ξðÞ
X 1
c W ξðÞ
i
i
w ¼ 1 (7.5)
b
i
The central weight vector can be deemed as the best single representation of the preference
c
from a DM supporting x i . w i is actually the average value of the weight vectors that make
alternative i the best.
c
The confidence factor, p i , is the probability that x i ranks first when its central weight vector is
1
used.Thatistosay,onlythefirst rankacceptabilityb i hastheconfidencefactor.Itisexpressedas:
ð
c f X ξðÞdξ
i (7.6)
p ¼
c
ξ2X:rank ξ, w Þ¼1
i
ð
The confidence factor is used to evaluate whether the criteria PVs are accurate to differen-
tiate alternatives using the central weight vectors.
In addition, we also can calculate the confidence factors for different alternatives using
each other’s’ central weight vectors, which are called cross confidence factors. Better discrim-
ination capability can be observed based on these cross confidence factors. The cross confi-
dence factor for alternative x i with respect to target alternative x k is defined as:
ð
c f X ξðÞdξ
ik
p ¼ (7.7)
1 c 1
k k i
ξ2X,b 6¼0:w 2W ξðÞ
The cross confidence factor measures the probability that x i will obtain the first rank when
the central weight vector of x k is used. Nonzero cross confidence factors means that the alter-
native x i will compete for the first rank with the central weight vector of alternative x k and the
c
competence extent can also be determined. Note that the cross confidence factor p ii is exactly
c
the confidence factor p i . In all, rank acceptability indices, holistic acceptability indices, central
weight vectors, and confidence factors are used to facilitate the evaluation of DH systems.
7.2.2 The SMAA-O model
The SMAA-O model was developed for problems with ordinal criteria (Lahdelma et al.,
2003). It uses rank level numbers, r j ¼1, 2, …, j max , to sort the alternatives in terms of each cri-
terion. It is clear that 1 is the best and j max is the worst rank level. In reality, two or more al-
ternatives may be considered equally good; so that j max m. In SMAA-O, the ordinal
measurements are mapped into the cardinal values. All consistent mappings between the
ordinal scales and cardinal values are considered. Monte Carlo simulations are used to gen-
erate random cardinal values corresponding to the ordinal values. Let γ j be the cardinal
values for rank levels, r j , then the mapping (David and Nagaraja, 2003) is:

(7.8)
γ ¼ v j r j
j