Table 5 shows the results of the coatings characterization process and that were entered with the projection parameters (input variables) in the Eureqa Formulize-Desktop program to obtain the objective functions.
Table 5. Results of the characterization
Experiment N°
|
140MXC-530AS
|
140MXC-560AS
|
Microhardness (Hv)
|
Particle Size
(µm)
|
Microhardness (Hv)
|
Particle Size
(µm)
|
1
|
584.53
|
155
|
186.51
|
87
|
2
|
148.93
|
139
|
218.30
|
119
|
3
|
164.86
|
85
|
422.86
|
133
|
4
|
189.66
|
111
|
487.73
|
89
|
5
|
294.76
|
79
|
121.62
|
137
|
6
|
520.93
|
137
|
314.98
|
74
|
7
|
351.60
|
276
|
324.60
|
89
|
8
|
154.37
|
120
|
298.73
|
176
|
9
|
173.30
|
134
|
202.13
|
101
|
According to the analysis of the results of Table 5, the optimization conditions required for the runs of the objective functions in MATLAB were established for the mixture of the coatings as shown in Table 6.
Table 6. Optimization conditions
Optimization Conditions Required
|
Microhardness (Hv) 140MXC-530AS / 140MXC-560AS
|
Minimum
|
Maximum
|
500
|
800
|
Particle Size (µm) 140MXC-530AS / 140MXC-560AS
|
Minimum
|
Maximum
|
80
|
160
|
For the 140MXC-530AS mixture, 6 target functions were obtained for the microhardness attribute and 6 for the particle size; while for 140MXC-560AS mixture were 7 and 8 respectively. Through eq 1 (calculation of the proportional size of the solution space), the criterion for selection of the functions was established in terms of their quality, which is defined by the best proportion of the search space determined by the user (minimum and maximum value) in relation to the global search space offered by each function. Table 7 shows the ordering of the pairing for multi-objective analysis of the best objective functions for the mixtures of 140MXC-530AS and 140MXC-560AS coatings according to the expected attributes.
Table 7. Matching for multi-objective analysis
140MXC-530AS
|
140MXC-560AS
|
Couple N°
|
Attribute
|
Mathematical Selection
|
Formulated Value
|
Function No.
|
Couple N°
|
Attribute
|
Mathematical Selection
|
Formulated Value
|
Function No.
|
1
|
Hv
|
1
|
5.46
|
2
|
1
|
Hv
|
1
|
4.26
|
7
|
µm
|
1
|
5.62
|
5
|
µm
|
1
|
2.82
|
5
|
2
|
Hv
|
2
|
5.99
|
4
|
2
|
Hv
|
2
|
4.93
|
6
|
µm
|
2
|
5.88
|
6
|
µm
|
2
|
2.87
|
6
|
3
|
Hv
|
3
|
9.58
|
5
|
3
|
Hv
|
3
|
6.74
|
3
|
µm
|
3
|
6.79
|
4
|
µm
|
3
|
3.43
|
7
|
Table 8 shows the best objective functions obtained according to the values formulated by equation 1 for the mixtures of coatings 140MXC-530AS and 140MXC-560AS for the microhardness and particle size attributes respectively.
The objective function No 1 corresponds to the microhardness attribute for the mixture of coatings 140MXC-530AS.
The objective function No 2 corresponds to the particle size attribute for the mixture of coatings 140MXC-530AS.
The objective function No 3 corresponds to the microhardness attribute for the mixture of coatings 140MXC-560AS.
The objective function No 4 corresponds to the particle size attribute for the mixture of coatings 140MXC-560AS.
Table 8. Objective functions of coatings mixtures 140MXC-530AS and 140MXC-560AS
N°
|
Objective Function for mixing 140MXC-530AS / Attribute: Microhardness (Hv)
|
1
|
10025.8717696593 + 45.5559873990753*A + 22.2395466925751*Ps + 21.0496919996887*V + 528.073096469254*Pp^2 + 0.184097466280761*A*Pp^2 + -0.731839755539107*A^2/Pp - 5539.23800112966*Pp
|
N°
|
Objective Function for mixing 140MXC-530AS / Attribute: Particle size (µm)
|
2
|
0.570621174976964 + 2.0297809250124*cos (1.85931009469647 + Ps) + 1.18390307276845*sin (5.37807343097015 + A + Pp) - 0.057557968428169*Pp - 0.038324437107436*Ps*cos (1.37568335042401*A - 0.65229880278333*V)
|
N°
|
Objective Function for mixing 140MXC-560AS / Attribute: Microhardness (Hv)
|
3
|
1574.30406778521*Pp + 298.150342662608*Ps + 0.681797613141099*A + 550.220928526876*sin(V) + -1777.66598085211*sin(V)/Ps - 3540.16559556089 - 10.3537290404493*V - 210.985961933397*Pp^2
|
N°
|
Objective Function for mixing 140MXC-560AS / Attribute: Particle size (µm)
|
4
|
0.0520645141336136*V + 0.0171407127971521*V*Pp*sin(V) + -0.00248390483036672*A*sin(V)/Ps - 0.325491914509165 - 2.20407853879951*sin(V) - 0.402194869142805*sin(0.537136306307155*Pp^2*sin(V)^2) ^2
|
3.1 Validity and Reliability Criteria
The criteria of validity and reliability were based on the verification of the algorithm that was done with the development of routines and subroutines generated by exercises that contained or not restrictions of the similar functions generated by Eureqa. We started by testing a simple genetic algorithm with one-variable exercises; the conditions of the algorithm used for each objective function were maintained in 100 generations and a population size of 100 individuals under a crossover probability of 95% and a mutation probability of 1%. Then test exercises with 2 and 3 variables were performed, thus increasing the complexity of the exercises used; in this case the conditions of the algorithm were varied according to the complexity of each function, but always maintaining the crossover probability of 95% and the probability of mutation in 1%. Figure 8 shows the results of the NSGA-II multi-objective genetic algorithm with respect to the "Dominance Fronts" (figure 8a), "Pareto Optimal Fronts" (Figure 8b) and "Unmanaged Solutions" (Figure 8c) Of the final population; this same behavior can be seen in the results reported by [12].
3.2 Pilot test
According to the revised literature and to what was established in the previous numerals, it was decided to maintain the probability of crossing and the probability of mutation in 95% and 1% respectively; therefore, the size of population (T) and the number of generations (G) developed for the algorithm were established in the pilot under systematic approaches (trial and error), depending on the quality of results (Optima’s Solutions Pareto) and computational resource (processing time). Different values and combinations of T and G were run at a suitable decision criterion for these parameters of 200 for each respectively. The result is shown in figure 9 for 140MXC-530AS a) and 140MXC-560AS b) coating mixes.
3.3 Final solutions
Once the number of partial solutions was defined, a new reclassification was carried out in order to obtain a single Optimal Pareto Front for each of the coating mixtures. The best conditions that could be obtained with less standard deviation in the data means that these values do not have much dispersion and that represent a similarity between individuals seen from the condition of parameterization of the technique in the equipment of thermal projection.
Figure 10 shows the graphical results of the Pareto optimal fronts, as a response to the processing of the objective functions for mixing 140MXC-530AS coatings with their 10 possible selections.
Table 9 presents the tabulation of the results of figure 10 for microhardness (Hv) and particle size (µm) attributes as a function of projection parameters (input variables), according to the algorithm used. In this case, the selection that should be taken into account to maximize microhardness (Hv) and minimize particle size (µm) at the same time would be run # 6. Maximum Microhardness =797.48 (Hv) and minimum Particle Size = 83.13 (µm), are obtained with the projection parameters: Current ≈ 109.8 (A), Primary air pressure ≈ 3.2 (bar), Secondary air pressure ≈ 3.6 (bar) and Voltage ≈ 26.5 (V).
Table 9. Final Result - Pareto Front for 140MXC-530AS
Current (A)
|
Primary air pressure (bar)
|
Secondary air pressure (bar)
|
Voltage (V)
|
Microhardness (Hv)
|
Particle Size
(µm)
|
114.8
|
3.3
|
3.7
|
29.7
|
741.83
|
80.00
|
114.8
|
3.3
|
3.7
|
30.7
|
762.88
|
80.01
|
108.7
|
3.3
|
3.6
|
30.9
|
777.09
|
80.01
|
109.5
|
3.3
|
3.7
|
31.1
|
782.86
|
80.06
|
109.4
|
3.3
|
3.7
|
31.3
|
787.17
|
80.09
|
109.8
|
3.2
|
3.6
|
26.5
|
797.48
|
80.13
|
109.7
|
3.3
|
3.6
|
32.1
|
799.38
|
80.75
|
102.8
|
3.2
|
3.8
|
26.1
|
799.66
|
81.06
|
115.1
|
3.3
|
3.8
|
32.4
|
799.86
|
82.66
|
109.2
|
3.3
|
3.7
|
31.9
|
799.99
|
83.65
|
For the mixing 140MXC-560AS coatings, the same selection criteria were considered as for the 140MXC-530AS, Figure 11 shows the graphical results of the Pareto optimal fronts, as a response to the processing of the objective functions for mixing 140MXC-560AS coatings with their 10 possible selections too.
Similarly, table 10 presents the tabulation of the results of figure 11 for microhardness (Hv) and particle size (µm) attributes as a function of projection parameters (input variables), according to the algorithm used, Here, the selection that should be taken into account to maximize microhardness (Hv) and minimize particle size (µm) at the same time would be run # 4, Maximum Microhardness =797,56 (Hv) and minimum Particle Size = 80,19 (µm), are obtained with the projection parameters: Current ≈ 102,8 (A), Primary air pressure ≈ 3,2 (bar), Secondary air pressure ≈ 3,8 (bar) and Voltage ≈ 25,9 (V) [19, 20].
Therefore, tables 9 and 10 indicate the configuration to be made to the thermal projection equipment as projection parameters and the production of the mixtures 140MXC-530As and 140MXC-560AS coatings for the expected attributes.
Table 10. Final Results Parameterization for 140MXC-560AS
Current (A)
|
Primary air pressure (bar)
|
Secondary air pressure (bar)
|
Voltage (V)
|
Microhardness (Hv)
|
Particle Size
(µm)
|
115.7
|
3.2
|
3.7
|
25.5
|
754.28
|
80.00
|
108.7
|
3.3
|
3.6
|
30.9
|
777.09
|
80.01
|
109.1
|
3.2
|
3.8
|
26.1
|
795.58
|
80.02
|
102.8
|
3.2
|
3.8
|
25.9
|
797.56
|
80.19
|
108.9
|
3.2
|
3.8
|
26.2
|
797.93
|
81.04
|
103.2
|
3.3
|
3.7
|
32.1
|
799.35
|
81.06
|
102.8
|
3.2
|
3.8
|
25.9
|
799.66
|
81.06
|
103.6
|
3.2
|
3.7
|
26.1
|
799.79
|
82.61
|
115.1
|
3.3
|
3.8
|
32.4
|
799.86
|
82.66
|
109.2
|
3.3
|
3.7
|
31.9
|
799.99
|
83.65
|