**6. Example**

To show the potentiality of the RFV approach, a simple example is here reported, where the RFV approach is compared with the GUM approach [1] and the Monte Carlo approach, as suggested by [18].

The GUM approach consists of the application of the law of propagation of the uncertainty [1], while random and systematic contributions to uncertainty are combined applying the quadratic law. The results given by the GUM approach are provided in terms of two specific confidence intervals: the ones at coverage probabilities 95.45% and 68.27%. These intervals are compared with the corresponding α-cuts at the same level of confidence of the RFVs obtained with the RFV approach.

The Monte Carlo approach consists of taking extractions from the given pdfs (in a way to agree with the available information) and combining the extractions to obtain a final histogram. Then, the histogram is converted in a pdf, and the pdf is converted to a PD (through the probability–possibility transformation mentioned above) for an immediate comparison with the RFVs given by the RFV approach.

Let us come to the example. A teacher measures the length and width of her desk with a wooden ruler and evaluates the area of the desk. She/he also asks her/his pupils to take the same measurements (and the area evaluation) with measuring tapes that they have built with some white cloth and a pencil to mark the cloth every half centimeter. The measurements are taken under different assumptions about both the measurement procedure and the uncertainty contributions, as shown in Table 8.


**Table 8.** The considered case studies.

As far as the procedures are concerned, "Known measuring tape" means that the measuring tapes are somehow characterized, and therefore, the systematic error introduced by each of them is known; since the pupil uses their own tape, the systematic error is known and can be compensated. "1 unknown measuring tape" means that both length and width are measured with the same tape taken randomly among the tapes; the systematic error introduced by the tape is not known and it cannot be compensated but, since the same tape is used for the two measurements, the two measurements are correlated with each other. "2 unknown measuring tapes" means that length and width are measured with two different tapes taken randomly among the tapes; the systematic errors introduced by the tapes are not known and they cannot be compensated and since two different tapes are used for the two measurements, and therefore, the two measurements are uncorrelated with each other.

As far as the uncertainty contributions are concerned, the random contributions are supposed to be uniformly distributed; the systematic contributions are compensated (case A), uniformly distributed (case B and C) or without any other knowledge rather than the given interval (case D and E), as in Shafer's total ignorance situation.

The uncertainty contributions reported in Table 8 are related to the pupils' measuring tapes, while no uncertainty is assumed to affect the teacher's measurements, realized with the wooden ruler, so that the teacher's measured values are considered to be the reference values *lref* = 90 cm for the length and *wref* = 60 cm for the width, while *Aref* = 5400 cm<sup>2</sup> is the reference area.

This means that, when the Monte Carlo approach is followed, extractions from the given pdfs in Table 8 are considered; when the RFV method is applied, the given pdfs in Table 8 are transformed into the corresponding PDs by applying the probability–possibility transformation; when the GUM approach is followed, the standard uncertainties are derived from the given pdfs in Table 8, that is, since the pdfs are uniform, the standard uncertainties are equal to the semi-width of the support of the pdfs divided by a factor <sup>√</sup>3 [1,2].

Without entering the details, for which the readers are referred to [10], the obtained results are shown in the following Figures 11–13. When only random contributions to uncertainty are present because the systematic ones are compensated for, the three approaches provide exactly the same results, showing the validity of the RFV method in simulating the presence of the random contributions. When both random and systematic contributions are present and their associated pdfs are known, the GUM approach underestimates the final measuring uncertainty, while the RFV and the Monte Carlo approaches provide very similar results. In this case, the RFV approach has the advantages of being faster and distinguishing, in the final measurement result, the effects due to the two different kinds of contributions. Finally, in the case of total ignorance, neither the GUM or the Monte Carlo approach can represent it in a different way with respect to cases B and C; therefore, they provide incorrect results.

**Figure 11.** Obtained results when case study A is considered: GUM approach (red lines), Monte Carlo approach (blue lines), RFV approach (cyan lines). On the left: the length (**upper plot**) and the width (**lower plot**). On the right: the evaluated area of the desk.

**Figure 12.** Obtained results when case studies B and C are considered: GUM approach (red lines), Monte Carlo approach (blue lines), RFV approach (cyan lines). On the left: the length (**upper plot**) and the width (**lower plot**). On the right: the evaluated area of the desk in case studies B (**upper plot**) and C (**lower plot**).

**Figure 13.** Obtained results when case studies D and E are considered: GUM approach (red lines), Monte Carlo approach (blue lines), RFV approach (cyan lines). On the left: the length (**upper plot**) and the width (**lower plot**). On the right: the evaluated area of the desk in case studies D (**upper plot**) and E (**lower plot**).
