Fuzzy integral analysis: The main equations to calculate the
fuzzy measures (ยต ) are shown in the section A of the
Supplementary Material. Once the fuzzy measures for each SA (i in
a set of n ) are calculated, the importance and interaction
indices could be estimated using the Shapley and the Murofushi and
Soneda indices, respectively. The importance index (ฯ ) is based
on the definition proposed by Shapley in the game
theory:41
\(\sigma\left(\mu,i\right)=\ \frac{1}{n}\ \sum_{t=0}^{n=1}{\frac{1}{\left(\frac{n-1}{t}\right)_{\left|\left.\ T\right|=i\right.\ }}\sum_{T\mathrm{\subseteq}\text{Xi}}\left[\mu\left(T\cup i\right)-\mu\left(T\right)\right]}\) (1)
Once normalized, the Shapley index can be interpreted as a weighted
average value of the marginal contribution of each criterion in all
combinations, so the sum of the index of all SA is equal to
1.45
On the other hand, the Murofushi and Soneda indices (I ) represent
the degree of interaction between two SA (i,jโฆ
n ):42
\(I\left(\mu,ij\right)=\ \sum_{T\mathrm{\subseteq}\text{Xij}}{\frac{\left(n-t-2\right)!t!}{n-1}\left[\mu\left(T\cup ij\right)+\mu\left(T\right)-\mu\left(T\cup i\right)-\mu\left(T\cup j\right)\right]}\) (2)
These indices and their interactions (positive or negative) validate the relevance of the selected SA. A positive interaction index for two SA means that the importance of one SA is reinforced by the second SA. A negative interaction index indicates that the SAs are antagonists, and their combined use impairs the final decision. Total interactions (TI) for each attribute are calculated as the sum of
positive values of all its interactions. After normalization (by mean) of Shapley index (๐ฬ
) and of
total interactions (ฬ
๐ฬ
ฬ
๐ผ), a composite index can be calculated, and it serves to guide the selection of the alternative formulations in the next phase.