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Supplement to Chapter 17 Statistical process control (SPC) 523 Figure S17.3 Process capability compares the natural variation of the process with the specification range which is required Worked example In the case of the process filling boxes of rice, described previously, process capability can be calculated as follows: Specification range = 214 − 198 = 16 g Natural variation of process = 6 × standard deviation = 6 × 2 = 12 g C p = process capability UTL − LTL = 6s 214 − 198 = = 6 × 2 = 1.333 If the natural variation of the filling process changed to have a process average of 210 grams but the standard deviation of the process remained at 2 grams: 214 − 210 4 C pu = = = 0.666 3 × 2 6 210 − 198 12 C pl = = = 2.0 3 × 2 6 C pk = min(0.666, 2.0) = 0.666 16 12
524 Part Three Planning and control Figure S17.4 Control chart for the impact resistance of door panels, together with control limits Control limits Assignable causes of variation Not all variation in processes is the result of common causes. There may be something wrong with the process which is assignable to a particular and preventable cause. Machinery may have worn or been set up badly. An untrained person may not be following prescribed procedures. The causes of such variation are called assignable causes. The question is whether the results from any particular sample, when plotted on the control chart, simply represent the variation due to common causes or due to some specific and correctable assignable cause. Figure S17.4, for example, shows the control chart for the average impact resistance of samples of door panels taken over time. Like any process the results vary, but the last three points seem to be lower than usual. So, is this natural (common cause) variation, or the symptom of some more serious (assignable) cause? To help make this decision, control limits can be added to the control chart (the red dashed lines) which indicates the expected extent of ‘common-cause’ variation. If any points lie outside these control limits (the shaded zone) then the process can be deemed out of control in the sense that variation is likely to be due to assignable causes. These control limits could be set intuitively by examining past variation during a period when the process was thought to be free of any variation which could be due to assignable causes. But control limits can also be set in a more statistically revealing manner. For example, if the process which tests door panels had been measured to determine the normal distribution which represents its common-cause variation, then control limits can be based on this distribution. Figure S17.4 also shows how control limits can be added; here they are put at ±3 standard deviations (of the population of sample means) away from the mean of sample averages. It shows that the probability of the final point on the chart being influenced by an assignable cause is very high indeed. When the process is exhibiting behaviour which is outside its normal ‘common-cause’ range, it is said to be ‘out of control’. Yet there is a small but finite chance that the (seemingly out of limits) point is just one of the rare but natural results at the tail of the distribution which describes perfectly normal behaviour. Stopping the process under these circumstances would represent a type I error because the process is actually in control. Alternatively, ignoring a result which in reality is due to an assignable cause is a type II error (see Table S17.1). Control limits are usually set at three standard deviations either side of the population mean. This would mean that there is only a 0.3 per cent chance of any sample mean falling outside these limits by chance causes (that is, a chance of a type I error of 0.3 per cent). The control limits may be set at any distance from the population mean, but the closer the limits
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sixth edition OperatiOns ManageMent
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Further reading in Operations Manag
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Pearson Education Limited Edinburgh
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Contents Guide to ‘operations in
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Contents ix Chapter 12 Inventory pl
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Guide to ‘operations in practice
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Guide to ‘operations in practice
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Making the most of this book and My
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Making the most of this book and My
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Preface xix Illustrations-based Ope
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To the Student . . . Making the mos
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About the authors Nigel Slack is th
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Publisher’s acknowledgements We a
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Part One INTRODUCTION This part of
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