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 (ฯƒ ) 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,jโ€ฆ 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 (๐œŽฬ…) and of total interactions (ฬ…๐‘‡ฬ…ฬ…๐ผ), a composite index can be calculated, and it serves to guide the selection of the alternative formulations in the next phase.