Abstract
Over the years, Taguchi method for process optimisation has become very popular among the engineers. However, Taguchi method focuses on the optimisation of a single-response variable only, whereas most of the modern manufacturing processes demand for simultaneous optimisation of multiple response variables, and some of these responses are often correlated. Several methods have been proposed in literature which aims at making the Taguchi method useful for solving multi-response optimisation problems too. However, only few of these methods take into account the possible correlations that may exist among the response variables. Among these, principal component analysis (PCA)-based approaches are quite popular among the practitioners. However, we find that the PCA-based approaches suffer from some weaknesses, e.g. problem due to using signal-to-noise ratios as input data, problem due to scaling of the input data, problem due to difference in PCA results given by different software. This article aims at drawing attention of the researchers/practitioners to these problem areas of the PCA-based approaches so that appropriate research initiatives can be taken up by the researchers/practitioners to overcome those weaknesses.
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Appendix: sample calculation for implementation of WPC method
Appendix: sample calculation for implementation of WPC method
The sample calculations are given here with respect to the case study 1. The computed and scaled SN ratios for three response variables are available in Table 7. It can be observed from Table 7 that the scaled SN ratio values for the three response variables are 0.9723, 1.0000 and 0.5935, respectively, i.e. Y 11 = 0.9723, Y 12 = 1.0000 and Y 13 = 0.5935.
Principal component analysis (PCA) of actual SN ratios or scaled SN ratios using particular software (S1) results in the same values of eigenvalues and eigenvectors. These values are shown in Table 8. It is observed from Table 8 that the eigenvector of the first principal component is [0.146, 0.717, 0.681]. This implies that a 11 = 0.146, a 12 = 0.717, a 13 = 0.681. Therefore, the value of the first PC corresponding to first trial (Z 11) is computed as follows:
Similarly, the values of the second and third PCs corresponding to the first trial (Z 12 and Z 13) are obtained as 0.8366 and −0.0003, respectively. It can be noted from Table 8 that the proportion of variation explained by the first, second and third PCs are 0.6062, 0.3583 and 0.0335, respectively, i.e. the relative weights of the first PC (W 1), second PC (W 2) and third PC (W 3) are 0.6062, 0.3583 and 0.0335, respectively. Therefore, the MPI value corresponding to the first trial is computed as
Similarly, the MPI values corresponding to all the nine trials are computed. When the WPC method is implemented based on the actual SN ratios, the Y values are replaced by η values in the equations for the computation of principal component values. The computed MPI values based on actual SN ratios as well as scaled SN ratios for all the nine trials are shown in Table 9.
It can be observed from Table 9 that factor A was set in the first level in the first three trials, and the computed MPI values (based on scaled SN ratios) in these trials are 1.0685, 1.1706 and 0.8953. So the average MPI value at the first level of factor A is computed as
Similarly, the level averages for all the factors are estimated, and these values are shown in Table 10.
Examining the level averages, it is found that the optimal combination with respect to computed MPI values (based on scaled SN ratios) is A1B2C2D1. The expected SN ratio value of SR at this optimal solution is estimated using additive model as follows:
where μ is the overall average SN ratio value for SR, and \( {\overline{A}}_1,{\overline{B}}_2,{\overline{C}}_2 \) and \( {\overline{D}}_1 \) are average SN ratio values for SR at level 1 of factor A, level 2 of factor B, level 2 of factor C and level 1 of factor D, respectively. These values are computed separately from the actual SN ratio values of SR, corresponding to different trial s shown in Table 7. It is found that \( \begin{array}{llllll}\mu =5.56,\hfill & {\overline{A}}_1=7.26,\hfill & {\overline{B}}_2=5.71,\hfill & {\overline{C}}_2=5.68\hfill & \mathrm{and}\hfill & {\overline{D}}_1=5.55\hfill \end{array} \). This implies SR E = 7.53 ⋅ dB. Similarly, the expected values of all the response variables at the optimal solutions (derived based on scaled SN ratios and actual SN ratios) are estimated.
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Gauri, S.K., Pal, S. The principal component analysis (PCA)-based approaches for multi-response optimization: some areas of concerns. Int J Adv Manuf Technol 70, 1875–1887 (2014). https://doi.org/10.1007/s00170-013-5389-8
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DOI: https://doi.org/10.1007/s00170-013-5389-8