The experimental results were analysed by means of the ANOVA test, which provides the statistical significance of the control factors (
P,
E and
C), for the ultimate tensile strength (UTS) as response variable. The results consist of a table containing the degrees of freedom (DoF), the sequential sums of squares (Seq.SS), the contribution percentage (Π%) the adjusted sum of squares (Adj.SS), the adjusted mean squares (Adj.MS), the F-value and the
p-value of each parameter or parameter combination. In general, the term Seq.SS provides a measure of the variation of each parameter with respect to the response variables. This information is quantified by the Π% term, which is the ratio between the Seq.SS term of the analysed parameter and the total one. Unlike the Adj.SS term, the Seq.SS depends on the order the terms are entered into the model. The F-value is used to determine whether a term is associated with the response, comparing the result with the corresponding tabulated value (4.06 for 1-DoF, 3.21 for 2-DoF and 2.59 for 4-DoF): the greater the F-value the greater the influence on the response variable. In this case, the F-value is defined as the ratio between the Adj.MS value of the response variable investigated and the Adj.MS of the error. Finally, the
p-value is used to determine the significance of the factors (the analysis was carried out at a 95% confidence level; thus, a process parameter or their combination is considered significant if the
p-value is lower than 0.05).
Table 5 reports the ANOVA results, in which the significant parameters are highlighted by the bold text. While,
Figure 7 and
Figure 8 show the main effects plot and the significant interaction plot. In
Figure 7 the significant terms (
E and
C) are highlighted by the continuous line.
As reported in
Table 5, the results show that, the laser energy per scan line, the laser cleaning treatment and the interaction between them are statistically significant for the UTS. Among them, the single terms have a greater influence on the response variable if compared to their interaction, as highlighted by the contribution percentage which is greater than 22% against the 8% of the
E*
C term. This is also confirmed by the Fisher value, which is much greater than the tabulated ones for both
E and
C, i.e., ~12 and ~32 against 3.21 and 4.06 respectively. This is sufficient evidence to indicate that the single terms affect the ultimate tensile strength. In the main effects plot (see
Figure 7), the significant parameters are highlighted using continuous lines. The figure shows that increasing the laser energy per scan line from 3 J/mm to 5 J/mm there is a decrease of the UTS to values of almost the half. The same trend is observed using or not the laser cleaning, highlighting in this way the efficacy of the treatment. In fact, by laser cleaning the surface of the sample, a rougher surface is obtained, therefore improving the joining between the materials. This is in accordance with the pertinent literature [
17,
42], in which it is stated that the combination of the surface activation without affection of the integrity of the laminate due to the cleaning of the surface leads to the strengthening of the joint. However, it is worth to note that using the highest power level, i.e., 200 W, involves the adoption of a higher scan speed in order to keep constant the laser energy. As a consequence, there is a shorter interaction time between the laser beam and the substrate, and therefore the heat does not diffuse into the inner layers and tends to be limited on the surface of the CFRP laminate. The result is a larger joined area that exceeds the threshold temperature required for joining. Moreover, in correspondence of the beam axis polycarbonate degradation takes place, while peripheral regions do not reach the melting temperature. In this way, the contribution because of these parts of the sample to the strength of the joint is very weak.
Optimal Fuzzy Regression Model
The first step of the procedures was to find the best regression model able to fit the experimental data among the infinite possible combination of the input parameters and their powers. In particular, the number of terms were fixed at 6, according to the number of terms used in the ANOVA test.
The general model which can be drawn is reported in the following:
In the latter equation, UTS is the response variable, k1 to k6 represent the empirical coefficients, evaluated by standard linear regression, while e1,1 to e6,3 the possible powers. Finally, P, E, C are the control factors. In particular, the first term k1 is the constant term, for which e1,1, e1,2, and e1,3 are equal to 0. While, the other powers were let to assume only few possible values, i.e., between −2 and 2 with a step of 0.5, with a total of nine possible values. In this way, the explored space is discrete and contains 95 × 3 possible models, where 5 is the number of terms and 3 the number of variables constituting each term. In general, convergence was reached in less than 150 generations, in each of which 2000 individuals, i.e., models are evaluated. In practice, the genetic algorithm (GA) explores a space of cardinality C ≅ 2 × 1014 solving only 3 × 105 models. Further 50 generations were computed to verify if mutation can move the optimum from a local minimum toward a better solution. The optimisation was run several times always obtaining the same result, ensuring in this way that the GA reached a global minimum.
It is worth noting that, despite the ANOVA does not indicate the power as statistically significant, in the Equation (5) the same has been maintained. This choice was made for two reasons: the first is that, in any case, the ANOVA shows a non-negligible error (about 30%). The second is because we wanted to test and stress the new procedure.
Table 6 reports the values of the coefficients and the powers of each term of the optimal regression model obtained by using the genetic algorithm presented in section computational, while
Figure 9 shows the comparison among the model and the experimental results, which are characterized by a mean error of about 7%.
Starting from this empirical model, the fuzzy model is built considering the coefficients,
k1 to
k6 as triangular fuzzy numbers, denoted as
, to
. Thus, the resulting fuzzy model is:
The aim was to produce a fuzzy input-output relation, based on the experimental observations, that links the control factors, i.e., laser energy, laser power and laser cleaning, to the achieved ultimate tensile strength. The model can be used to evaluate how much a given experimental sample, characterized by a certain value of E, P and C, and the corresponding UTS, belong to the fuzzy set defined by Equation (6). It is important to notice that the values of the process parameters are measured and deterministic thus remain regular numbers, while the uncertainty is modelled within the fuzzy coefficients.
Generally speaking, it is possible to state that the nominal model (Equation (5)) does not represent any experimental data (i.e., there is no experimental evaluation that can fall over the model surface). As the level of uncertainty is increased, measured by a decrease in the membership function, the model accommodates a larger number of samples with lower membership level. In other words, the fuzzy model is able to describe, as the membership function decreases, an increasing number of experimental data and, thanks to the genetic algorithm, with the highest degree of belonging to the fuzzy set defined by the model itself [
29].
All the fuzzy parameters are described by eight α-cuts and the interval at each α-level is discretized with three points, the upper and the lower bound and a midpoint. For each α-cut, the transformation method requires, in a combinatorial scheme, the evaluation of the points at each α-level to the power of the number of fuzzy parameters, six in this case, leading to 729 evaluations. The transformation method requires that, for each α-cut, all these models are evaluated obtaining for each of them the hypersurface of the output quantity as a function of the process parameters. The fuzzy result for the given α-cut is then obtained by computing the envelope of these hypersurfaces, which are reported in
Figure 10. In particular, along the x-axis are reported the experimental tests ordered for increasing values of the input parameters combinations, while along the y-axis the response variable, i.e., UTS.
From the inspection of the fuzzy results reported in
Figure 9, the uncertainty level related to the fuzzy models appears to be not constant with respect to the parameters’ combination used during the experimental test. It is worth to note that the extent of the input uncertainty in the model, because of the choice of a specific confidence interval, is not only related to the accuracy of the regression model adopted but also to the variability of the process. So, the transformation method, which in this case was used to propagate the uncertainty to the outputs, also provides information about the uncertainty at the input level due to the regression model adopted. This effect can be therefore considered the reason for a non-constant level of uncertainty.
In general, this kind of process map can be used to select operational parameters in order to obtain a desired process output. They provide, as additional information, how much the uncertainty of the model and the process varies by changing the operational parameters. It is important to notice here that the variability of the process is highlighted by the combined fuzzy-genetic algorithm model through bands of uncertainty, represented in the latter figure by grey shaded areas. Moreover, it is worth to note that this information is not available by considering just the nominal regression model, nor directly obtained from the values of the confidence intervals.
The proposed model can also be inverted (see
Figure 11) in order to obtain the most suitable operational parameters’ combinations, in terms of laser energy and laser power, while considering the laser cleaning as a constant, leading to a desired output. For the case study, the fuzzy-genetic algorithm model has been used to assess the optimal parameters in order to satisfy the highest resistance (set over 65% of the maximum value achieved, i.e., at about 0.8 kN). The membership level of the fuzzy model is represented as a grey shaded area (white to black corresponds to µ(x) from 0 to 1), while the experimental data and their occurrences as red dots (the dimension of each dot is proportional to the number of occurrences reported as green numbers).
From
Figure 11, it is evident that there are different solutions that can satisfy the requested requirement of UTS > 0.8 kN, as highlighted by the grey shaded area. In particular, among the various combinations, the one with the lowest degree of uncertainty is given by the darkest area, which is characterized by a laser power lower than 150 W and a laser energy lower than 4 J/mm. This is also supported by the three experimental occurrences over three repetitions, reported as green numbers, which fall exactly in the darkest area. Moreover, analysing the fuzzy inverse map it is recommended to use a combination of
E and
P within 3–3.3 J/mm and 100–160 W, respectively, as highlighted by the darkest area. Among these, wanting to remain in the central part of the optimal working (i.e., the darkest area) area and therefore the farthest from the lighter zones, reasonable values for
E and
P may be 3.15 J/mm and 135 W, respectively. It is worth to highlight that this does not mean that it is not possible to satisfy the requirement with the other combinations, but that those scenarios are characterized by a certain degree of uncertainty, which is higher for a high value of laser power and laser energy because the fuzzy map is lighter.