223 Validation of the Component Manufacturing Variability Risks Analysis

Validation of the Component Manufacturing Variability Risk, qm, is essential to CA in determining Cpk estimates for component characteristics at the design stage. Collecting component parts from various industrial sources with known statistical histories was central to this. The components were taken from a number of collaborating companies which had produced the components and had measured a critical characteristic using SPC, therefore the process capability indices Cpk and Cp could be determined.

The critical characteristic on each component was analysed, qm calculated from the analysis and the value obtained was plotted against the process capability indices, Cpk and Cp, for the characteristic in question. See Appendix V for descriptions of the 21 components analysed, including the values of Cpk and Cp from the SPC data supplied. Note that some components studied have a zero process capability index. This is a default value given if the process capability index calculated from the SPC data had a mean outside either one of the tolerance limits, which was the case for some of the components submitted. Although it is recognized that negative process capability indices are used for the aim of process improvement, they have little use in the analyses here. A correlation between positive values (or values which are at least within the tolerance limits) will yield a more deterministic relationship between design capability and estimated process capability.

Figure 2.15(a) shows the relationship between qm and Cpk for the component characteristics analysed. Note, there are six points at qm = 9, Cpk = 0. The correlation coefficient, r, between two sets of variables is a measure of the degree of (linear) association. A correlation coefficient of ±1 indicates that the association is deterministic. A negative value indicates an inverse relationship. The data points have a correlation coefficient, r = —0.984. It is evident that the component manufacturing variability risks analysis is satisfactorily modelling the occurrence of manufacturing variability for the components tested.

Confidence limits are also drawn on Figure 2.15(a) to give boundaries of Cpk for a given qm determined from the analysis, which are within 95%. The relationship between qm and Cpk is described by a power law after linear regression giving:

This can be approximated to:

Note that the 'squared' relationship which was initially used to model the degree of difficulty in obtaining more capable tolerances for a given manufacturing route and product design is being returned by the power law. Similarly, a relationship between the process capability index Cp and qm for the components analysed is shown in Figure 2.15(b). The data points have a correlation coefficient, r = —0.956, and the corresponding power law is given by:

This can be approximated by the following equation:

It is possible, therefore, to determine an estimate for Cpk from the formulation given above in equation 2.6, within some confidence. It is assumed that the Cpk values given

Figure 2.15 Empirical relationships between (a) qm and Cp^ and (b) qm and Cp (with 95% confidence limits)

Component manufacturing variability risk {<jmJ

Figure 2.15 Empirical relationships between (a) qm and Cp^ and (b) qm and Cp (with 95% confidence limits)

for the component characteristics were based on the mean values for a number of batches produced and that a confidence limit of 95% is adequate for the outcome for the prediction of Cpk. These values will be later transferred to the quality-cost model for the prediction of the failure costs for given levels of process capability and failure severity (see Section 2.5).

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