*2.3. Procedure 3—Comparison of Sensor Placement Solutions*

Sensor placement solutions were evaluated using the following four contamination impact indicators. The first is function *f4* in Equation (5), followed by functions *f5*, *f6* and *f7* reported in the following Equations (6)–(8), respectively.

The function *f5* is the detection likelihood (i.e., the probability of detection):

$$f\_5 = P\_s = \frac{1}{S} \sum\_{r=1}^{S} d\_r \tag{6}$$

where *dr* = 1 if contamination scenario *r* is detected, and zero otherwise; and *S* denotes the total number of the contamination scenarios considered.

The function *f6* is the detection time. For each detected contamination scenario, the sensor detection time corresponds to the elapsed time from the start of the contamination event, to the first identified presence of a nonzero contaminant concentration. If *tj* is the time of the first detection (referred to the *j*-th sensor location), the detection time (*td*) for the solution for each contamination event, is the minimum among all present sensors *td* = *min*(*tj*); the characteristic detection time of the solution is defined as the mean of all *td* for the contamination scenarios detected by at least one sensor:

$$f\_6 = mean(t\_d) \tag{7}$$

Finally, the function *f7* is the sensor redundancy. In a generic scenario, the variable *Red* corresponds to the number of sensors (including the first) that detect the contamination within 30 minutes from the first detection; the redundancy *Red* of the solution is defined as the mean of all the values of redundancy *Red* for all the considered contamination scenarios:

$$f\_7 = \text{Red} = \text{mean}(\text{Red})\tag{8}$$

which contributes to the safety of the monitoring systems, especially in the case of sensor failures or false positive/negative detection, conferring a higher reliability.

As for the choice of the objective functions, it must be remarked that theoretically more than two of them could be inserted in the same optimization framework. However, to prevent this framework from becoming overly complex, we preferred to keep only two objective functions (number of sensors and exposed population) in the optimization framework, while other assessment criteria (e.g., detection likelihood, detection time, and sensor redundancy) will be considered in the postprocessing of the optimal solutions.
