Influence of Input factors on Responses MRR and SR
Experimental results obtained based on CCD were investigated to find the influence of various input factors on the responses (MRR and SR) using ANOVA at 0.05 level significant. ANOVA for MRR and SR (refer to Table 3 and 4) with the small value of probability indicates a larger value of correlation coefficient. According to the RSM investigation, the quadratic model used is statistically significant for both MRR and SR. The “coefficient of determination” R2 attaining 1 indicates the output characteristics fitting the real data, and this even helps in checking predicted and adjacent R2 (i.e., Pred-R2 and Adj-R2) reached unity. Above 90% for R2 as well as Adj-R2 of both machining characteristics (MRR and SR) shows that the mathematical model developed by regression has a good relationship between process factors and output results. The inputs like P, square terms - CXC, and interaction terms-PxC, are found to be most significant. V, Q, and C are found to an insignificant process parameter for MRR. Similarly, the ANOVA table for SR shows that C, P, and Q, and square term-CxC are essential to process parameters. V is found to be insignificant input factors for SR.
Table 3.
Table 4.
Figure 1.
Pareto charts also indicate most significant and significant factors at individual, square and interaction levels of factors as shown in Figure 1 (a) for MRR and (b) for SR.
Figure 2.
Figure 2 shows the 3D surface plot of output MRR varying to the level of input factors, and each Figure 2a to 2f shows the role of interaction of A, B, C, and D on MRR. The MRR decreases more with increased C and less with Pulse on time. That means an increase in C (from 2 to 4A) may severely decrease the MRR without even increasing the Pulse on time as it encourages swift melting along with alloy vaporization. But the mid-side value of C might improve MRR up to A=37.5 µs and remains same up to 45 µs and then MRR increase further irrespective of A and C change after 45 µs because high value of C directs a large amount of energy hooked on the targeted area to remove the higher amount of metal along with impulsive forces in the dielectric fluid for taking away molten metal of targeted area. The MRR also gets increased due to change of shape as well as the size of surface pits, pores, etc., with an increase in C., whereas factors B and D have a counter effect on MRR, i.e., they contribute to MRR value only if A and C are also changing. The interactions of the input factors on SR are depicted in Figures 3 (a) to (f). At a low value of C, SR reduces with increasing A because of low impulsive forces retaining longer time. But at higher C, SR increases along with the A because high impulsive forces along with sparks and maintaining a more extended time would damage the surface, and hence SR increases. The pits, pores increase due to the high rate of erosion of an alloy. It is also noticed that low thermal conductivity and melting point gives a higher SR . The decisive empirical formed equations of machining characteristics (MRR and SR) are characterized in Eq. (4) and (5).
MRR
|
=
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6.30 - 0.0971 P + 0.426 Q - 1.923 C - 0.033 V + 0.000144 P2 - 0.01338 Q2+ 0.4116 C2 + 0.00067 V2+0.000548 P*Q+ 0.01060 P*C+ 0.000523 P*V- 0.0390 Q*C- 0.00094 Q*V - 0.01224 C*V
|
(4)
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SR
|
=
|
2.58+ 0.0067 P+ 0.0283 Q+ 1.195 C- 0.1198 V- 0.000240 P2- 0.00189 Q2- 0.1595 C2+ 0.00122 V2- 0.000377 P*Q+ 0.002193 P*C + 0.000023 P*V + 0.00271 Q*C + 0.000136 Q*D + 0.00146 C*V
|
(5)
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Multi response optimization using RSM
To increase the WEDM performance, optimum factors set have to be selected by considering the responses such as high MRR and low SR, which are very difficult to get for achieving high-quality surface. For this, composite desirability (cd) approach is a most suitable technique for optimizing input factors to satisfy the conditions of responses by using a response function which determines the scale-free value (di) of the responses called desirability (lies between 0 to 1) expresses the zero and absolute issues for the responses which are at outer allowable margins. Whereas cd is the weighted statistical average of the distinct desirability of responses.
Figure 3.
The optimum factors with the greatest desirability will be chosen for the mathematical models of MRR and SR by combining the objectives to satisfy the combined goals of all the responses. The optimality solution evaluates higher MRR and Lower SR and their corresponding predicted optimum input factors values presented in Table 6. The desirability need not always be equal to one, but its value discloses the proximity of the margins set pertaining to the real optimum values. An optimal set of factors using Minitab 19 software was evaluated and presented in Table 5. Once the optimum values are achieved, it becomes necessary to validate to check the confirmation with experiments at these optimal values. Table 6 gives % error for confirming with experimentation results of WEDM and found very small, which is acceptable by the researchers.
Table 5.
Table 6.
Figure 4.
The desirability gradient and line graph of the WEDM process (refer Figure 4) shows the di value obtained for each response generated between low and high values. If the targeted responses di value is closer to 1 show having good desirability, then the desirability is good. The global desirability D = 0.8748 shows closeness of responses to the target set.
Results of MOOPSO
The PSO technique was applied, keeping in mind that both should not be minimized as when SR reduces by minimizing the objective function, the MRR also gets reduced. But we need the maximization of MRR while minimizing the SR. To avoid conflict between objectives of MRR and SR and to achieve higher MRR and lower, the optimum set of input factors are to be located. Thus, the objective function of MRR is altered into minimization form in the following way as below:
Function 1= Minimize (1/MRR);
Function 2 = Minimize (SR).
To achieve MO optimization as per specific goal, a PSO toolbox of MATLAB program was employed to execute the source code of the anticipated algorithm. Distribution of Pareto front containing 100 optimum sets of factors satisfying the condition of both functions. Figure 5 showing the Pareto front distribution of top scored ND solutions out of the 100-optimum set of populations while performing optimizations of two functions. The best solution depends on product requirements or upon the choice of engineer for a specifically designed process. MOOPSO forecasted the finest results within levels of factors. Top scored solutions come from 100 top global solutions, but only the finest 25 solutions are provided in Table 6. After both analysis and comparing with empirical outputs of MRR using Pareto front, the maximum value of 3.5420068 mm3/min is found at pulse-on-duration of 25.1118229 μs, pulse-off-duration of 12.3503394 μs, C of 2.009124 amp and V of 45.120498 volts, with corresponding to experiment no. 27 in Table 2 where the maximum MRR = 3.5663.23mm3/min at pulse-on-duration =25 μs, Q = 14 μs, C = 14 amp, and V = 50 volts. The MRR of the Pareto front optimal solution exhibits their values are slightly lesser than the experiment results. Furthermore, the 3-D MRR output plot (referring Figure 5) reveals that the MRR decreasing as the Pulse on time and C increases. The proper combination of input factor levels may yield a higher value of MRR. It is also found that minimum MRR of 2.508 mm3/min (refer Table 2) in experiments having s.no. 11 at pulse-on-duration =35 μs , Q =14 μs, C =3amp, and V =45volts. while Pareto front provided the minimum value of MRR of 2.66760113mm3/min at pulse-on-duration = 49.83780 μs , Q = 13.8012141 μs, C = 2.00763 amp, and V =45.50042649.614 Volts. A comparison of the minimum MRR provides the Pareto front result is 0.13 mm3/min higher experimental one, which is highly acceptable, and this difference is because of the factor level changes.
Similarly, Pareto front results revealed the minimum SR of 1.4855502 μm at P = 4549.83780 μs, Q = 13.801214 μs, C = 2.00763 amp, and V = 45.500426 volts and experiment output no. 5 (refer Table 2), the minimum SR of 1.501μm at P = 50 μs, Q = 14 μs, C = 2 amp, and V = 50 volts is slightly lower than the Pareto front analysis. Furthermore, the 3D response plots of SR (Fig. 6) show that the SR decreases with the increasing values of P, V and C. Therefore, by proper levels of input, factors yield the minimum SR, which is need of an hour. Further, experimental output no. 9, the maximum SR of 2.615 μm at P = 25 μs, Q = 8 μs, C = 4 amp, and V = 40volts. While Corresponding results of Pareto front seen with the maximum SR of 1.7886815 μm at P = 25.1118229 μs, Q = 12.3503394 μs, C = 2.009124amp, and V = 45.1204989 volts. Comparing the maximum SR of experimental is 0.4 μm higher than Pareto optimal results (refer Table 7). This may be because of factors level differences between these two methods. The five most excellent pareto front results keeping minimum SR and maximum MRR are presented in the Table 8.
Table 7.
Table 8.
Figure 5.