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Appendix I 285 From the original data, the number of yield strength values falling between these class limits is recorded to give the frequency and a histogram can be generated, as shown in Figure 5. The mid-class values are determined by taking the mid-point between each pair of class limits. The mean and standard deviation are given by equations 10 and 12 respectively: k i 1 f i x i N 4 Â 4169 Â 43417 Â45210 Â4707 Â4883 Â506 50 457:76 MPa k i 1 f i x i ÿ 2 N s 4 Â416 ÿ457:76 2 9 Â434 ÿ457:76 2 17 Â452 ÿ457:76 2 10 Â470 ÿ457:76 2 7 Â488 ÿ457:76 2 3 Â506 ÿ457:76 2 50 v u u u t 23:45 MPa We can now plot the Normal frequency distribution superimposed over the histogram bars for comparison. The curve is generated using equation 15, where the variables of interest, x, are values in steps of 10 on the x-axis from, say, 380 to 540. The Normal frequency equation is given below, and Figure 6 shows the histogram and the Normal 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Yield strength mid-class (MPa) 416 434 452 470 488 506 Frequency ( f ) Figure 5 Histogram for yield strength data 286 Appendix I distribution for comparison: y Nw 2 p exp ÿ x ÿ 2 2 2 50 Â 18 23:45 2 p exp ÿ x ÿ 457:76 223:45 2 2 To ®nd the strength at ÿ3 from the mean simply requires that we take three standard deviations away from the mean. Therefore, the strength at this point on the distribu- tion is: 457:76323:45387:41 MPa The proportion of individuals that could be expected to have a strength greater than 500 MPa requires using SND theory. The variable of interest is 500 MPa, and so from Equation 16: z x ÿ 500 ÿ 457:76 23:45 1:80 The area under the cumulative SND curve at z 1:80 is equivalent to the probability that the yield strength is less than 500 MPa. Referring to Table 1 gives: P È SND zÈ SND 1:800:964070 We require the proportion that has a strength greater than 500 MPa, therefore: 1 ÿ 0:964070 0:035930 or, in other words, approximately 3.6% of the population can be expected to have a yield strength greater than 500 MPa. 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Yield strength mid-class (MPa) 416 434 452 470 488 506 Frequency ( f ) Superimposed Normal distribution Figure 6 Histogram for yield strength data with superimposed Normal distribution Appendix I 287 Appendix II Process capability studies Process capability concepts A capability study is a statistical tool which measures the variations within a manu- facturing process. Samples of the product are taken, measured and the variation is compared with a tolerance. This comparison is used to establish how `capable' the process is in producing the product. Process capability is attributable to a combina- tion of the variability in all of the inputs. Machine capability is calculated when the rest of the inputs are ®xed. This means that the process capability is not the same as machine capability. A capability study can be carried out on any of the inputs by ®xing all the others. All processes can be described by Figure 1, where the distribu- tion curve for a process shows the variability due to its particular elements. There are ®ve occasions when capability studies should be carried out, these are: . Before the machine/process is bought (to see if it is capable of producing the components you require it to) . When it is installed . At regular intervals to check that the process is given the performance required . If the operating conditions change (for example, materials, lubrication) . As part of a process capability improvement. The aim is to have a process where the product variability is suciently small so that all the products produced are within tolerance. Since variation can never be eliminated, the control of variation is the key to product quality and capability studies give us one tool to achieve this. There are two main kinds of variability: . Common-cause or inherent variability is due to the set of factors that are inherent in a machine/process by virtue of its design, construction and the nature of its operation, for example, positional repeatability, machine rigidity, which cannot be removed without undue expense and/or process redesign. . Assignable-cause or special-cause variability is due to identi®able sources which can be systematically identi®ed and eliminated. When only common-cause variability is present, the process is performing at its best possible level under the current process design. For a process to be capable of produ- cing components to the speci®cation, the sum of the common-cause and assignable- cause variability must be less than the tolerance. The way of measuring capability is to carry out a capability study and calculate a capability index. There are two commonly used process capability indices, C p and C pk . In both cases, it is assumed the data is adequately represented by the Normal distribution (see Appendix I). Process capability index, C p The process capability index is a means of quantifying a process to produce compo- nents within the tolerances of the speci®cation. Its formulation is shown below: C p U ÿ L 6 1 where U upper tolerance limit, L lower tolerance limit, standard deviation. U ÿ L also equals the unilateral tolerance, T. Where a bilateral tolerance, t, is used (for example, Æ0.2 mm), equation (1) simpli®es to: C p t 3 2 where t bilateral tolerance. A value of C p 1:33 would indicate that the distribution of the product character- istics covers 75% of the tolerance. This would be sucient to assume that the process is capable of producing an adequate proportion to speci®cation. The numbers of failures falling out of speci®cation for various values of C p and C pk can be determined from Standard Normal Distribution (SND) theory (see an example later for how to determine the failure in `parts-per-million' or ppm). For example, at C p 1:33, the expected number of failures is 64 ppm in total. The minimum level of capability set by Motorola in their `Six Sigma' quality philosophy is C p 2 which equates to 0.002 ppm failures. Some companies set a Environment ORGANIZATIONAL TECHNOLOGICAL Figure 1 Factors affecting process capability Appendix II 289 capability level of C p 1 which relates to 2700 ppm failures in total. This may be adequate for some manufactured products, say a nail manufacturer, but not for safety critical products and applications where the characteristic controlled has been determined as critical. In general, the more severe the potential failure, the more capable the requirement must be to avoid it. General capability target values are given below. Interpretation of process capability index, C p : Less than 1.33 3 Process not capable. Between 1.33 and 2.5 3 Process capable. Where a process is producing a characteristic with a capability index greater than 2.5 it should be noted that the unnecessary precision may be expensive. Figure 2 shows process capability in terms of the tolerance on a component. The area under each distribution is equal to unity representing the total probability, hence the varying heights and widths. The variability or spread of the data does not always take the form of the true Normal distribution of course. There can be `skewness' in the shape of the distribu- tion curve, this means the distribution is not symmetrical, leading to the distribution appearing `lopsided'. However, the approach is adequate for distributions which are fairly symmetrical about the tolerance limits. But what about when the distribution mean is not symmetrical about the tolerance limits? A second index, C pk ,isusedto accommodate this `shift' or `drift' in the process. It has been estimated that over a very large number of lots produced, the mean could expect to drift about Æ1:5 (standard deviations) from the target value or the centre of the tolerance limits and is caused by some problem in the process, for example tooling settings have been altered or a new supplier for the material being processed. Figure 2 Process capability in terms of tolerance 290 Appendix II Process capability index, C pk By calculating where the process is centred (the mean value) and taking this, rather than the target value, it is possible to account for the shift of a distribution which would render C p inaccurate (see Figure 3). C pk is calculated using the following equation: C pk j ÿ L n j 3 3 where L n nearest tolerance limit and mean. Note, that the j ÿ L n j part of the equation means that the value is always positive. By using the nearest tolerance limit, L n , which is the tolerance limit physically closest to the distribution mean, the worst case scenario is being used ensuring that overopti- mistic values of process capability are not employed. In Figure 3, a ÿ1:5 shift is shown from the target value for a C pk 1:5. C pk is a much more valuable tool than C p because it can be applied accurately to shifted distributions. As a large percentage of distribu- tions are shifted, C p is limited in its usefulness. If C pk is applied to a non-shifted Normal distribution, by the nature of its formula it reverts to C p . Again, the minimum level of capability at Motorola using C pk 1:5 (or $3.4 ppm) at the nearest limit, where it is assumed the sample distribution is Æ1:5 shifted from the target value. From Figure 3, it is evident that at Æ1:5, the number of items falling out of speci®cation on the opposite limit is negligible. However, more typical values are shown below. C pk 1:33 is regarded to be the absolute minimum by industry. This relates to 32 ppm failures, although it is commonly rounded down to 30 ppm. C pk is interpreted in the same way as C p : Less than 1.33 3 Process not capable. Between 1.33 and 2.5 3 Process capable. Figure 3 Process capability for a shifted distribution (C pk 1:5) Appendix II 291 Again, for C pk greater than 2.5, it should be noted that the unnecessary precision may be expensive. Also note that C pk C p ÿ 0:5 4 when the distribution is Æ1:5 shifted from the target value. For a sample set of data, a C p and C pk value can be determined at the same time; however, the selection of which one best models the data is determined by the degree of shift. From equation (3), if C p ÿ 0:5 approaches the value of C pk calculated, then a Æ1:5 shift is evident in the sample distribution, and C pk is a more suitable model. Example ± process capability and failure prediction The component shown in Figure 4 is a spacer from a transmission system. The component is manufactured by turning/boring at the rate of 25 000 per annum and the component characteristic to be controlled, X, is an internal diameter. From the statistical data in the form of a histogram for 40 components manufactured, shown in Figure 5, we can calculate the process capability indices, C p and C pk . It is assumed that a Normal distribution adequately models the sample data. The solution is as follows: k i 1 f i x i N 2 Â 49:95 4 Â 49: 96 7 Â 49:7 10 Â 49:8 8 Â 49:9 6 Â 50 2 Â 50:01 1 Â 50:02 40 49:98 mm k i 1 f i x i ÿ 2 N v u u u u t Figure 4 Spacer component showing a critical characteristic, X 292 Appendix II Figure 5 Measurement data for the characteristic, X, in histogram form ppm 1 000 000 100 000 10 000 1000 100 10 1 0.1 0.01 0.001 0.0001 0.00001 0 0.5 1 1.5 2 2.5 Figure 6 Relationship between C p , C pk and parts-per-million (ppm) failure Appendix II 293 2 Â49:95 ÿ49:98 2 4 Â49:96 ÿ49:98 2 7 Â49:97 ÿ49:98 2 10 Â49:98 ÿ49:98 2 8 Â49:99 ÿ 49:98 2 6 Â50 ÿ49:98 2 2 Â50:01 ÿ49:98 2 1 Â50:02 ÿ49:98 2 40 v u u u u u t 0:0162 mm C p t 3 0:05 3 Â 0:0162 1:03 C pk j ÿ L n j 3 49:9 ÿ 49:95 3 Â 0:0162 0:62 Compare to see l if shift C pk C p ÿ 0:5 1:03 ÿ 0:5 0:53 approaches Æ 1:5 It is evident that an approximate ÿ1:5 shift can be determined from the data and so the C pk value is more suitable as a model. Using the graph on Figure 6, which shows the relationship C p , C pk (at Æ1:5 shift) and parts-per-million (ppm) failure at the nearest limit, the likely annual failure rate of the product can be calculated. The ®gure has been constructed using the Standard Normal Distribution (SND) for various limits. The number of components that would fall out of tolerance at the nearest limit, L n , is potentially 30 000 ppm at C pk 0:62, that is, 750 components of the 25 000 manufactured per annum. Of course, action in the form of a process cap- ability study would prevent further out of tolerance components from being produced and avoid this failure rate in the future and a target C pk 1:33 would be aimed for. 294 Appendix II [...]... particularly simple and eective and the approach is outlined in Figure 11 In this ®gure, values from Figure 10(b) are used to generate plots of factors Figure 11 Graphical analysis of the experimental results 311 312 Appendix III and levels The plots are employed to study the eects of both individual results and combinations Note that only four of the six graphs are included in Figure 11 In the case of... maps Index to maps Sheet A Casting processes Sand casting ± Steel and iron Sand casting ± Copper alloys Sand casting ± Aluminium and magnesium alloys Shell moulding ± Steel, iron and copper alloys Shell moulding ± Aluminium and magnesium alloys Centrifugal casting ± All metals Sheet B Pressure die casting ± Zinc alloys Pressure die casting ± Aluminium and magnesium alloys Pressure die casting ± Copper... both the noise and factors to be controlled during the experiment and select the levels to be considered These should be representative of normal operating range and suciently spaced to spot changes Establish an eective measuring system Understand its variance and the likely eects of this apparent variation in output Phase 2 ± Experimentation and analysis Carrying out the planned trials and analytical... from fewer suppliers and possibly fewer quality problems Systematic component costing and process selection Improved yields Lower component and assembly costs Standardize components, assembly sequence and methods across product `families' leading to improved reproducibility Faster product development and reduced time to market Lower level of engineering changes, modi®cations and concessions Typical... considerable time and resources, and good preparation is all important The results from other techniques are important inputs to this phase of the methodology, providing focus and priority selection ®ltering A summary of the steps that should be considered is given below: De®ne the problem to be solved Agree the objectives and prepare a project plan Examine and understand the situation Obtain and study... manufacture and use Optimize the product or process Reduce cost Placement in product development QFD, FMEA and CA are useful in identifying critical characteristics early on in product development and the results from these can be fed into DOE DOE is useful in investigating and validating these critical characteristics with respect to technical requirements and their in¯uence on product and process... development Key issues An overall strategy for its implementation and application should be in place with clear objectives Can be complex at ®rst Begin on small experiments and then expand to larger ones Poor understanding of the concepts and underlying methods can lead to poor results Problems associated with interpretation of the results and assessing their signi®cance are common Team should involve... Key issues Can be used on products, software or services Must be management led and have an overall strategy for implementation and application Training required to use analysis initially Multi-disciplinary team-based application essential Can be subjective and tedious Organizations do not extend the use of QFD past the ®rst phase usually Involvement of customers and suppliers essential Review... design stage using experience or judgement, or integrated with existing data and knowledge on components and products FMEA was ®rst used in the 1960s by the aerospace sector, but has since found applications in the nuclear, electronics, chemical and motor manufacturing sectors FMEA can also apply to oce processes as well as design and manufacturing processes, which are the main application areas Placement... structure and increased assembly design eciency resulted Overall, component and assembly costs were signi®cantly reduced Figure 9(c) summarizes the results of the analysis D Design of Experiments (DOE) (Grove and Davis, 1992; Kapur, 1993; Taguchi et al., 1989) Description DOE encompasses a range of techniques used to enable a business to understand the eects of important variables in product and process . produced and avoid this failure rate in the future and a target C pk 1:33 would be aimed for. 294 Appendix II Appendix III Overview of the key tools and techniques A. Failure Mode and Effects. existing data and knowledge on components and products. FMEA was ®rst used in the 1960s by the aerospace sector, but has since found applications in the nuclear, electronics, chemical and motor manufacturing. Calculation of component manufacturing and assembly costs . Ease of part handling . Ease of assembly . Ability to reproduce identically and without waste products which satisfy customer requirements. 304
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