Statistical Process Control 5 Part 5 ppt

Process control using variables 127 They pointed out that, regardless of the ease or difficulty of grouping the data from a particular process, the forming of subgroups is an essential step in the investigation of stability and in the setting up of control charts. Furthermore, the use of group ranges to estimate process variability is so widely accepted that ‘the mean of subgroup ranges’ R may be regarded as the central pillar of a standard procedure. Many people follow the standard procedure given on page 116 and achieve great success with their SPC charts. The short-term benefits of the method include fast reliable detection of change which enables early corrective action to be taken. Even greater gains may be achieved in the longer term, however, if charting is carried out within the context of the process itself, to facilitate greater process understanding and reduction in variability. In many processes there is a tendency for observations that are made over a relatively short time period to be more alike than those taken over a longer period. In such instances the additional ‘between group’ or ‘medium-term’ variability may be comparable with or greater than the ‘within group’ or ‘short-term’ variability. If this extra component of variability is random there may be no obvious way that it can be eliminated and the within group variability will be a poor estimate of the natural random longer term variation of the process. It should not then be used to control the process. Caulcutt and Porter observed many cases in which sampling schemes based on the order of output or production gave unrepresentative estimates of the random variation of the process, if R/d n was used to calculate ␴. Use of the standard practice in these cases gave control lines for the mean chart which were too ‘narrow’, and resulted in the process being over-controlled. Unfortunately, not only do many people use bad estimates of the process variability, but in many instances sampling regimes are chosen on an arbitrary basis. It was not uncommon for them to find very different sampling regimes being used in the preliminary process investigation/chart design phase and the subsequent process monitoring phase. Caulcutt and Porter showed an example of this (Figure 6.12) in which mean and range charts were used to control can heights on a can-making production line. (The measurements are expressed as the difference from a nominal value and are in units of 0.001 cm.) It can be seen that 13 of the 50 points lie outside the action lines and the fluctuations in the mean can height result in the process appearing to be ‘out-of-statistical control’. There is, however, no simple pattern to these changes, such as a trend or a step change, and the additional variability appears to be random. This is indeed the case for the process contains random within group variability, and an additional source of random between group variability. This type of additional variability is frequently found in can-making, filling and many other processes. A control chart design based solely on the within group variability is inappropriate in this case. In the example given, the control chart would 128 Process control using variables mislead its user into seeking an assignable cause on 22 occasions out of the 50 samples taken, if a range of decision criteria based on action lines, repeat points in the warning zone and runs and trends are used (page 118). As this additional variation is actually random, operators would soon become frustrated with the search for special causes and corresponding corrective actions. To overcome this problem Caulcutt and Porter suggested calculating the standard error of the means directly from the sample means to obtain, in this case, a value of 2.45. This takes account of within and between group variability. The corresponding control chart is shown in Figure 6.13. The process appears to be in statistical control and the chart provides a basis for effective control of the process. Stages in assessing additional variability 1 Test for additional variability As we have seen, the standard practice yields a value of R from k small samples of size n. This is used to obtain an estimate of the within sample standard deviation ␴: ␴ = R/d n . Figure 6.12 Mean and range chart based on standard practice Process control using variables 129 The standard error calculated from this estimate (␴/ ͱ ස n) will be appropriate if ␴ describes all the natural random variation of the process. A different estimate of the standard error, ␴ e , can be obtained directly from the sample means, X i : ␴ e = ͱ සසසසසසසස ⌺ k i=1 (X i – X ) 2 /(k – 1) X is the overall mean or grand mean of the process. Alternatively, all the sample means may be entered into a statistical calculator and the ␴ n – 1 key gives the value of ␴ e directly. The two estimates are compared. If ␴ e and ␴/ ͱස n are approximately equal there is no extra component of variability and the standard practice for control chart design may be used. If ␴ e is appreciably greater than ␴/ ͱස n there is additional variability. In the can-making example previously considered, the two estimates are: ␴/ ͱස n = 0.94 ␴ e = 2.45 This is a clear indication that additional medium-term variation is present. Figure 6.13 Mean and range chart designed to take account of additional random variation 130 Process control using variables (A formal significance test for the additional variability can be carried out by comparing n␴ e 2 /␴ 2 with a required or critical value from tables of the F distribution with (k– 1) and k(n–1) degrees of freedom. A 5 per cent level of significance is usually used. See Appendix G.) 2 Calculate the control lines If stage 1 has identified additional between group variation, then the mean chart action and warning lines are calculated from ␴ e : Action lines X ± 3␴ e ; Warning lines X ± 2␴ e . These formulae can be safely used as an alternative to the standard practice even if there is no additional medium-term variability, i.e. even when ␴ = R/d n is a good estimate of the natural random variation of the process. (The standard procedure is used for the range chart as the range is unaffected by the additional variability. The range chart monitors the within sample variability only.) In the can-making example the alternative procedure gives the following control lines for the mean chart: Upper Action Line 7.39 Lower Action Line –7.31 Upper Warning Line 4.94 Lower Warning Line –4.86. These values provide a sound basis for detecting any systematic variation without over-reacting to the inherent medium-term variation of the process. The use of ␴ e to calculate action and warning lines has important implications for the sampling regime used. Clearly a fixed sample size, n, is required but the sampling frequency must also remain fixed as ␴ e takes account of any random variation over time. It would not be correct to use different sampling frequencies in the control chart design phase and subsequent process monitoring phase. 6.6 Summary of SPC for variables using X and R charts If data is recorded on a regular basis, SPC for variables proceeds in three main stages: Process control using variables 131 1 An examination of the ‘State of Control’ of the process (Are we in control?). A series of measurements are carried out and the results plotted on X and R control charts to discover whether the process is changing due to assignable causes. Once any such causes have been found and removed, the process is said to be ‘in statistical control’ and the variations then result only from the random or common causes. 2A ‘Process Capability’ Study (Are we capable?). It is never possible to remove all random or common causes – some variations will remain. A process capability study shows whether the remaining variations are acceptable and whether the process will generate products or services which match the specified requirements. 3 Process Control Using Charts (Do we continue to be in control?). The X and R charts carry ‘control limits’ which form traffic light signals or decision rules and give operators information about the process and its state of control. Control charts are an essential tool of continuous improvement and great improvements in quality can be gained if well-designed control charts are used by those who operate processes. Badly designed control charts lead to confusion and disillusionment amongst process operators and management. They can impede the improvement process as process workers and management rapidly lose faith in SPC techniques. Unfortunately, the author and his colleagues have observed too many examples of this across a range of industries, when SPC charting can rapidly degenerate into a paper or computer exercise. A well-designed control chart can result only if the nature of the process variation is thoroughly investigated. In this chapter an attempt has been made to address the setting up of mean and range control charts and procedures for designing the charts have been outlined. For mean charts the standard error estimate ␴ e calculated directly from the sample means, rather than the estimate based on R/d n , provides a sound basis for designing charts that take account of complex patterns of random variation as well as simpler short-term or inter-group random variation. It is always sound practice to use pictorial evidence to test the validity of summary statistics used. Chapter highlights ᭹ Control charts are used to monitor processes which are in control, using means (X ) and ranges (R). ᭹ There is a recommended method of collecting data for a process capability study and prescribed layouts for X and R control charts which include warning and action lines (limits). The control limits on the mean and range charts are based on simple calculations from the data. 132 Process control using variables ᭹ Mean chart limits are derived using the process mean X, the mean range R, and either A 2 constants or by calculating the standard error (SE) from R. The range chart limits are derived from R and D 1 constants. ᭹ The interpretation of the plots are based on rules for action, warning and trend signals. Mean and range charts are used together to control the process. ᭹ A set of detailed rules is required to assess the stability of a process and to establish the state of statistical control. The capability of the process can be measured in terms of ␴, and its spread compared with the specified tolerances. ᭹ Mean and range charts may be used to monitor the performance of a process. There are three zones on the charts which are associated with rules for determining what action, if any, is to be taken. ᭹ There are various forms of the charts originally proposed by Shewhart. These include charts without warning limits, which require slightly more complex guidance in use. ᭹ Caulcutt and Porter’s procedure is recommended when short- and medium-term random variation is suspected, in which case the standard procedure leads to over-control of the process. ᭹ SPC for variables is in three stages: 1 Examination of the ‘state of control’ of the process using X and R charts, 2 A process capability study, comparing spread with specifications, 3 Process control using the charts. References Bissell, A.F. (1991) ‘Getting more from Control Chart Data – Part 1’, Total Quality Management, Vol. 2, No. 1, pp. 45–55. Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978) Statistics for Experimenters, John Wiley & Sons, New York, USA. Caulcutt, R. (1995) ‘The Rights and Wrongs of Control Charts’, Applied Statistics, Vol. 44, No. 3, pp. 279–88. Caulcutt, R. and Coates, J. (1991) ‘Statistical Process Control with Chemical Batch Processes’, Total Quality Management, Vol. 2, No. 2, pp. 191–200. Caulcutt, R. and Porter, L.J. (1992) ‘Control Chart Design – A review of standard practice’, Quality and Reliability Engineering International, Vol. 8, pp. 113–122. Duncan, A.J. (1974) Quality Control and Industrial Statistics, 4th Edn, Richard D. Irwin IL, USA. Grant, E.L. and Leavenworth, R.W. (1996) Statistical Quality Control, 7th Edn, McGraw-Hill, New York, USA. Owen, M. (1993) SPC and Business Improvement, IFS Publications, Bedford, UK. Pyzdek, T. (1990) Pyzdek’s Guide to SPC, Vol. 1 – Fundamentals, ASQC Quality Press, Milwaukee WI, USA. Process control using variables 133 Shewhart, W.A. (1931) Economic Control of Quality of Manufactured Product, Van Nostrand, New York, USA. Wheeler, D.J. and Chambers, D.S. (1992) Understanding Statistical Process Control, 2nd Edn, SPC Press, Knoxville TN, USA. Discussion questions 1 (a) Explain the principles of Shewhart control charts for sample mean and sample range. (b) State the Central Limit Theorem and explain its importance in statistical process control. 2 A machine is operated so as to produce ball bearings having a mean diameter of 0.55 cm and with a standard deviation of 0.01 cm. To determine whether the machine is in proper working order a sample of six ball bearings is taken every half-hour and the mean diameter of the six is computed. (a) Design a decision rule whereby one can be fairly certain that the ball bearings constantly meet the requirements. (b) Show how to represent the decision rule graphically. (c) How could even better control of the process be maintained? 134 Process control using variables 3 The following are measures of the impurity, iron, in a fine chemical which is to be used in pharmaceutical products. The data is given in parts per million (ppm). Sample X 1 X 2 X 3 X 4 X 5 1 15 11 8 15 6 2 14161114 7 31369510 4 15 15 9 15 7 5 9 12 9 8 8 6 11141112 5 713129610 810151246 9 8 12 14 9 10 10 10 10 9 14 14 11 13 16 12 15 18 12 7 10 9 11 16 13 11 7 16 10 14 14 11 7 10 10 7 15 13 9 12 13 17 16 17 10 11 9 8 17 4 14 5 11 11 18 8 9 6 13 9 19 9 10 7 10 13 20 15 10 10 12 16 Set up mean and range charts and comment on the possibility of using them for future control of the iron content. Process control using variables 135 4 You are responsible for a small plant which manufactures and packs jollytots, a children’s sweet. The average contents of each packet should be 35 sugar-coated balls of candy which melt in your mouth. Every half-hour a random sample of five packets is taken, and the contents counted. These figures are shown below: Sample Packet contents 12345 1 3336373836 2 3535323735 3 3138353638 4 3735363634 5 3435363637 6 3433383538 7 3436373534 8 3637353231 9 3434323436 10 34 35 37 34 32 11 34 34 35 36 32 12 35 35 41 38 35 13 36 36 37 31 34 14 35 35 32 32 39 15 35 35 34 34 34 16 33 33 35 35 34 17 34 40 36 32 37 18 33 35 33 34 40 19 34 33 37 34 34 20 37 32 34 35 34 Use the data to set up mean and range charts, and briefly outline their usage. 136 Process control using variables 5 Plot the following data on mean and range charts and interpret the results. The sample size is four and the specification is 60.0 ± 2.0. Sample number Mean Range Sample number Mean Range 1 60.0 5 26 59.6 3 2 60.0 3 27 60.0 4 3 61.8 4 28 61.2 3 4 59.2 3 29 60.8 5 5 60.4 4 30 60.8 5 6 59.6 4 31 60.6 4 7 60.0 2 32 60.6 3 8 60.2 1 33 63.6 3 9 60.6 2 34 61.2 2 10 59.6 5 35 61.0 7 11 59.0 2 36 61.0 3 12 61.0 1 37 61.4 5 13 60.4 5 38 60.2 4 14 59.8 2 39 60.2 4 15 60.8 2 40 60.0 7 16 60.4 2 41 61.2 4 17 59.6 1 42 60.6 5 18 59.6 5 43 61.4 5 19 59.4 3 44 60.4 5 20 61.8 4 45 62.4 6 21 60.0 4 46 63.2 5 22 60.0 5 47 63.6 7 23 60.4 7 48 63.8 5 24 60.0 5 49 62.0 6 25 61.2 2 50 64.6 4 (See also Chapter 10, Discussion question 2) . Range 1 55 0.8 4.2 11 55 3.1 3.8 2 55 2.7 4.2 12 55 1.7 3.1 3 55 3.8 6.7 13 56 1.2 3 .5 4 55 5.8 4.7 14 55 4.2 3.4 5 553 .8 3.2 15 552 .3 5. 8 6 54 7 .5 5.8 16 55 2.9 1.6 7 55 0.9 0.7 17 56 2.9 2.7 8 55 2.0 5. 9 18 55 9.4. 60.32 0.6 4 59 .91 2.2 60. 05 0.2 5 60.01 2.4 58 . 95 0.3 6 60.18 2.7 59 .12 0.7 7 59 .67 1.7 58 .80 0 .5 8 60 .57 3.4 59 .68 0.4 9 59 .68 1.7 60.14 0.6 10 59 .55 1 .5 60.96 0.3 11 59 .98 2.3 61. 05 0.2 12 60.22. contents 123 45 1 3336373836 2 353 53237 35 3 3138 353 638 4 37 353 63634 5 34 353 63637 6 343338 353 8 7 343637 353 4 8 3637 353 231 9 3434323436 10 34 35 37 34 32 11 34 34 35 36 32 12 35 35 41 38 35 13 36 36