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Supplement to Chapter 17 Statistical process control (SPC) 529 Figure S17.8 The process mean or the process range (or both) can change over time Control limits for variables control chart As with attributes control charts, a statistical description of how the process operates under normal conditions (when there are no assignable causes) can be used to calculate control limits. The first task in calculating the control limits is to estimate the grand average or population mean (D) and average range (A) using m samples each of sample size n. The population mean is estimated from the average of a large number (m) of sample means: D = The average range is estimated from the ranges of the large number of samples: A = The control limits for the sample means chart are: The control limits for the range charts are: B 1 + B 2 + ...+ B m m R 1 + R 2 + ...+ R m m Upper control limit (UCL) = D + A 2 A Lower control limit (LCL) = D − A 2 A Upper control limit (UCL) = D 4 A Lower control limit (LCL) = D 3 A The factors A 2 , D 3 and D 4 vary with sample size and are shown in Table S17.2. The LCL for the means chart may be negative (for example, temperature or profit may be less than zero) but it may not be negative for a range chart (or the smallest measurement in the sample would be larger than the largest). If the calculation indicates a negative LCL for a range chart then the LCL should be set to zero.
530 Part Three Planning and control Table S17.2 Factors for the calculation of control limits Sample size n A 2 D 3 D 4 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 12 0.266 0.284 1.716 14 0.235 0.329 1.671 16 0.212 0.364 1.636 18 0.194 0.392 1.608 20 0.180 0.414 1.586 22 0.167 0.434 1.566 24 0.157 0.452 1.548 Worked example GAM (Groupe As Maquillage) is a contract cosmetics company, based in France but with plants around Europe, which manufactures and packs cosmetics and perfumes for other companies. One of its plants, in Ireland, operates a filling line which automatically fills plastic bottles with skin cream and seals the bottles with a screw-top cap. The tightness with which the screw-top cap is fixed is an important part of the quality of the filling line process. If the cap is screwed on too tightly, there is a danger that it will crack; if screwed on too loosely it might come loose when packed. Either outcome could cause leakage of the product during its journey between the factory and the customer. The Irish plant had received some complaints of product leakage which it suspected was caused by inconsistent fixing of the screw-top caps on its filling line. The ‘tightness’ of the screw tops could be measured by a simple test device which recorded the amount of turning force (torque) that was required to unfasten the tops. The company decided to take samples of the bottles coming out of the filling-line process, test them for their unfastening torque and plot the results on a control chart. Several samples of four bottles were taken during a period when the process was regarded as being in control. The following data were calculated from this exercise: The grand average of all samples D = 812 g/cm 3 The average range of the sample A = 6 g/cm 3 Control limits for the means (B) chart were calculated as follows: UCL = D + A 2 A = 812 + (A 2 × 6) From Table 17.6, we know, for a sample size of four, A 2 = 0.729. Thus: UCL = 812 + (0.729 × 6) = 816.37 LCL = D − (A 2 A) = 812 – (0.729 × 6) = 807.63
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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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