Page 145 - Materials Chemistry, Second Edition

7.2 Methods 141
This study develops a more efficient method to the multicriteria decision analysis of
heating, ventilating, and air conditioning systems, and to solve the problems of the uncer-
tainties in criteria PVs and weighting, which were not treated carefully in previous studies.
This paper is organized as following. Firstly, the SMAA models and FWS concept, as well
as the way to handle the uncertainties, are introduced; followed by a case study in China,
where the developed methods are demonstrated with seven candidate DH system options;
finally the conclusion is drawn according to the results and discussion of the study.

7.2 Methods

SMAA is a family of models that encompasses many different variants (Tervonen and
Figueira, 2008). This paper proposes to use SMAA-2 and SMAA-O models to solve the
multicriteria decision making problems that have both quantitative and qualitative criteria
(Lahdelma et al., 2001).

7.2.1 The SMAA-2 model
Let’s take an MCDM problem, which has m alternatives A¼{x 1 ,x 2 ,x 3 , …,x m } and n criteria.
SMAA-2 model assumes that DM’s preference can be expressed by a utility function u(x i ,w);
this function calculates the utility value for alternative x i when using weight vector w.We
introduce a rank acceptability index to evaluate each alternative’s acceptability according
to the utility calculation results. A ranking function is defined to determine the ranking se-
quences from the best (1) to the worst (m), as in Lahdelma and Salminen (2001):
X
Þ ¼ 1+ (7.1)
i k i
rank ξ , wð k ρ u ξ , wð½ Þ > u ξ , wð ފ
where ρ(true)¼and ρ(false)¼0, ξ is used to stand for criteria PVs having a stochastic distri-
bution of f X (ξ); similarly w has a stochastic distribution of f W (w). Then the favorable rank
r
weights, W i (ξ) is defined:
n X n o
r n
i i Þ ¼ rg, where W ¼ w 2 R : w j 0, w j ¼ 1 (7.2)
f
W ξðÞ ¼ w 2 W : rank ξ , wð j¼1
r
If a weight vector w2W i (ξ), then it makes that alternative x i obtains rank r. Based on this,
r
the rank acceptability index, b i , can be defined as:
ð ð
r f W wðÞdwdξ (7.3)
i
b ¼ f X ξðÞ
X r
i
W ξðÞ
r
In fact, b i indicates all the different valuations that make alternative x i rank r. It is not pos-
r
sible to calculate b i directly from the integral formula, but it can be calculated by using the
Monte Carlo simulation. From this point of view, rank acceptability also can be explained
as the share (%) of Monte Carlo simulations that make alternative x i rank r. SMAA-2 uses
a holistic acceptability index shown in Eq. (7.4) to consider contributions of all ranks; this
is an improvement compared to the original SMAA model (Lahdelma et al., 1998).
m
h r
X
i
a ¼ r¼1 αb i (7.4)