2.3.1. Fuzzy DEMATEL

As already underlined, the fuzzy version of DEMATEL is more suitable than the crisp version to reduce uncertainty, and get more reliable results. We describe now the steps to implement the method as it exists in the literature. After having highlighted the general objective of the decision-making problem and the elements to be evaluated, and properly chosen the group of experts, the procedure is the following.

1. Defining the fuzzy linguistic scale that will be used to assess the elements belonging to the system. Judgments must be collected by pairwise comparing all the elements to express the influence of one element, *i*, over another, *j*, and vice versa. To such an aim, in [21], it is defined the linguistic variable "influence" through five linguistic terms of evaluation, each one associated to a positive triangular fuzzy number (TFN) (*aij*, *bij*, *cij*). TFNs expressing those evaluations are given in Table 1.


**Table 1.** Fuzzy linguistic scale for the linguistic variable "influence".


$$Z = \mathbf{s} \times \mathbf{D},\tag{9}$$

*s* being a positive number slightly smaller than

$$\min\left(\frac{1}{\max\_{1\le i\le n}\sum\_{j=1}^{n}d\_{ij}},\frac{1}{\max\_{1\le j\le n}\sum\_{i=1}^{n}d\_{ij}}\right).\tag{10}$$

After deriving matrix *Z*, it is possible to proceed to the calculation of the TRM, *T* = (*tij*), as follows:

$$T = Z(I - Z)^{-1},\tag{11}$$

*I* being the identity matrix. This matrix represents the build-up of mutual direct and indirect effects among elements, since *T*, being the sum of all the powers of *Z*, reflects both direct and indirect effects among elements (note that the series of powers of *Z* is convergent (see, for example, [36]), since, because of Equation (10), the spectral radius of *Z* is smaller than 1).

4. Building the relational chart on the plane "prominence" (horizontal axis) versus "relation" (vertical axis). Values of prominence (*A* + *B*) and relation (*A* − *B*) can be derived from matrix *T*, by calculating the sums of the rows, *A*, and the sums of the columns, *B*:

$$A = \sum\_{j=1}^{n} t\_{ij\prime} \tag{12}$$

$$B = \sum\_{i=1}^{n} t\_{ij}.\tag{13}$$

The resulting mapping represents the core of the methodology, since just by observing the positions of the elements in the four quadrants of the plane, it is possible to establish which elements have: (i) high prominence and high relation; (ii) low prominence and low relation; (iii) high prominence and low relation; and (iv) low prominence and high relation. This distinction is very useful to understand how interdependence among the elements is organised, and thus to establish future lines of intervention. Moreover, relations among elements are visually represented by means of arrows: two elements are linked by an arrow if the corresponding value of the TRM overcomes a given threshold, herein calculated by averaging all the values of the TRM [37].

5. Ordering in a decreasing way the elements of the decision-making problem, according to their corresponding values of prominence, to obtain a structured ranking. The "prominence" of an element indicates how much it influences the others, thereby providing a global measurement of its importance. Values of "relation" are instead useful to cluster factors into groups of causes or effects. If the "relation" value corresponding to an element is positive, that means that it has to be considered as a cause, while as an effect otherwise.
