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Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Statistical Process Control Operations Management - 5 th Edition Chapter 3 Roberta Russell & Bernard W. Taylor, III

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Copyright 2006 John Wiley & Sons, Inc.3-2 Basics of Statistical Process Control Statistical Process Control (SPC) monitoring production process to detect and prevent poor quality monitoring production process to detect and prevent poor quality Sample subset of items produced to use for inspection subset of items produced to use for inspection Control Charts process is within statistical control limits process is within statistical control limits UCL LCL

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Copyright 2006 John Wiley & Sons, Inc.3-3 Variability Random common causes common causes inherent in a process inherent in a process can be eliminated only through improvements in the system can be eliminated only through improvements in the system Non-Random special causes special causes due to identifiable factors due to identifiable factors can be modified through operator or management action can be modified through operator or management action

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Copyright 2006 John Wiley & Sons, Inc.3-4 Quality Measures Attribute a product characteristic that can be evaluated with a discrete response a product characteristic that can be evaluated with a discrete response good – bad; yes - no good – bad; yes - no Variable a product characteristic that is continuous and can be measured a product characteristic that is continuous and can be measured weight - length weight - length

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Copyright 2006 John Wiley & Sons, Inc.3-5 Where to Use Control Charts Process has a tendency to go out of control Process is particularly harmful and costly if it goes out of control Examples at the beginning of a process because it is a waste of time and money to begin production process with bad supplies at the beginning of a process because it is a waste of time and money to begin production process with bad supplies before a costly or irreversible point, after which product is difficult to rework or correct before a costly or irreversible point, after which product is difficult to rework or correct before and after assembly or painting operations that might cover defects before and after assembly or painting operations that might cover defects before the outgoing final product or service is delivered before the outgoing final product or service is delivered

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Copyright 2006 John Wiley & Sons, Inc.3-6 Control Charts A graph that establishes control limits of a process Control limits upper and lower bands of a control chart upper and lower bands of a control chart Types of charts Attributes Attributes p-chart p-chart c-chart c-chart Variables Variables range (R-chart) range (R-chart) mean (x bar – chart) mean (x bar – chart)

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Copyright 2006 John Wiley & Sons, Inc.3-7 Process Control Chart 12345678910 Sample number Uppercontrollimit Processaverage Lowercontrollimit Out of control

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Copyright 2006 John Wiley & Sons, Inc.3-8 Normal Distribution =0 1111 2222 3333 -1 -2 -3 95% 99.74%

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Copyright 2006 John Wiley & Sons, Inc.3-9 A Process Is in Control If … 1.… no sample points outside limits 2.… most points near process average 3.… about equal number of points above and below centerline 4.… points appear randomly distributed

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Copyright 2006 John Wiley & Sons, Inc.3-10 Control Charts for Attributes p-charts uses portion defective in a sample c-charts uses number of defects in an item

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Copyright 2006 John Wiley & Sons, Inc.3-11 p-Chart UCL = p + z p LCL = p - z p z=number of standard deviations from process average p=sample proportion defective; an estimate of process average p = standard deviation of sample proportion p =p =p =p = p(1 - p) n

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Copyright 2006 John Wiley & Sons, Inc.3-12 p-Chart Example (cont.) UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = 0.10 - 3 p(1 - p) n 0.10(1 - 0.10) 100 = 200 / 20(100) = 0.10 total defectives total sample observations p =

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Copyright 2006 John Wiley & Sons, Inc.3-13 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Proportion defective Sample number 2468101214161820 UCL = 0.190 LCL = 0.010 p = 0.10 p-Chart Example (cont.)

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Copyright 2006 John Wiley & Sons, Inc.3-14 Control Charts for Variables Mean chart ( x -Chart ) uses average of a sample Range chart ( R-Chart ) uses amount of dispersion in a sample

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Copyright 2006 John Wiley & Sons, Inc.3-15 x-bar Chart x =x =x =x = x 1 + x 2 +... x k k = UCL = x + A 2 RLCL = x - A 2 R == where x= average of sample means =

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Copyright 2006 John Wiley & Sons, Inc.3-16 x- bar Chart Example (cont.) UCL = 5.08 LCL = 4.94 Mean Sample number |1|1 |2|2 |3|3 |4|4 |5|5 |6|6 |7|7 |8|8 |9|9 | 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01 =

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Copyright 2006 John Wiley & Sons, Inc.3-17 R-Chart Example (cont.) UCL = 0.243 LCL = 0 Range Sample number R = 0.115 |1|1 |2|2 |3|3 |4|4 |5|5 |6|6 |7|7 |8|8 |9|9 | 10 0.28 – 0.24 – 0.20 – 0.16 – 0.12 – 0.08 – 0.04 – 0 –

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Copyright 2006 John Wiley & Sons, Inc.3-18 Using x- bar and R-Charts Together Process average and process variability must be in control. It is possible for samples to have very narrow ranges, but their averages is beyond control limits. It is possible for sample averages to be in control, but ranges might be very large.

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Copyright 2006 John Wiley & Sons, Inc.3-19 Control Chart Patterns UCL LCL Sample observations consistently above the center line LCL UCL Sample observations consistently below the center line

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Copyright 2006 John Wiley & Sons, Inc.3-20 Control Chart Patterns (cont.) LCL UCL Sample observations consistently increasing UCL LCL Sample observations consistently decreasing

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Copyright 2006 John Wiley & Sons, Inc.3-21 Zones for Pattern Tests UCL LCL Zone A Zone B Zone C Zone B Zone A Process average 3 sigma = x + A 2 R = 3 sigma = x - A 2 R = 2 sigma = x + (A 2 R) = 2323 2 sigma = x - (A 2 R) = 2323 1 sigma = x + (A 2 R) = 1313 1 sigma = x - (A 2 R) = 1313 x = Sample number |1|1 |2|2 |3|3 |4|4 |5|5 |6|6 |7|7 |8|8 |9|9 | 10 | 11 | 12 | 13

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Copyright 2006 John Wiley & Sons, Inc.3-22 Process Capability Tolerances design specifications reflecting product requirements design specifications reflecting product requirements Process capability range of natural variability in a process what we measure with control charts range of natural variability in a process what we measure with control charts

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Copyright 2006 John Wiley & Sons, Inc.3-23 Process Capability Measures Process Capability Ratio Cp==Cp== tolerance range process range upper specification limit - lower specification limit 6

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Copyright 2006 John Wiley & Sons, Inc.3-24 Computing C p Net weight specification = 9.0 oz 0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz C p = = = 1.39 upper specification limit - lower specification limit 6 9.5 - 8.5 6(0.12)

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Copyright 2006 John Wiley & Sons, Inc.3-25 Process Capability Measures Process Capability Index C pk = minimum x - lower specification limit 3 = upper specification limit - x 3 =,

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