**3. Numerical Example**

In this section we apply the proposed procedure, as an example, to a very small WDN. Using this small network means indeed to deal with a small number of elements to be evaluated, and then with a small number of matrices. This enables us to show the calculations of our procedure step-by-step.

The small network used as the numerical example is known as a two-loop network [38] (Figure 2). This network has six junctions, eight pipes, and one reservoir. Classically, the network is used as a benchmark for optimal design in water distribution systems. For this example, the optimally designed network is used, and a demand curve with residential features has been added. This allows the simulation of the network for 24-h. Leaks are created using emitters node by node. An emitter coefficient equal to 1 has been used, in accordance with [39], which investigated the effects of the emitter coefficient on different geometries and hydraulic head loads to simulate leaks. Observe that the authors established an interval to simulate single leaks varying from 0.5 to 8.

**Figure 2.** Topology and nodal demand for the 2-loop network.

Considering the size of the network, certainly only one sensor should be installed. Given the simplicity of the problem we only maximise the sensitivity, thereby using just the sensitivity matrix as the input for the fuzzy DEMATEL.

Table 2 details the linguistic evaluations of influence (associated to the TFNs of Table 1) referring to the two-loop network nodes, each one possibly hosting a sensor. Each element has been codified as *Ni* (*i* = 1, ... , 6). Tables 3 and 4 respectively present the defuzzified DRM and the corresponding TRM, this last one also presenting the final ranking of elements. For ease of replication, we specify that the graded mean integration approach has been applied to get the crisp values *dij* of the DRM matrix:

$$d\_{ij} = \frac{a\_{ij} + 4b\_{ij} + c\_{ij}}{6}.\tag{14}$$

The obtained results shown as the nodes occupying the first positions of the ranking (*N*6, *N*5, *N*4) are more suitable to host sensors because they have higher associated sensitivity. By assuming this condition, the monitoring capability can be enhanced in the considered network. From the hydraulic point of view, these nodes can be identified as those presenting lower pressure during the hydraulic simulations.

Figure 3 shows the final chart graphically showing interdependencies. As it is possible to note, the first three nodes of the ranking are in the first quadrant, being characterised by both high prominence and relation.

**Table 2.** Linguistic evaluations of input.


**Table 3.** Defuzzified direct relation matrix (DRM).


**Table 4.** Total relation matrix (TRM) and final ranking.


**Figure 3.** Relational chart.
