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Statistical Process Control

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Overview Variation Control charts – R charts – X-bar charts – P charts

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Measures performance of a process Primary tool - statistics Involves collecting, organizing, & interpreting data Used to: – Control the process as products are produced – Inspect samples of finished products Statistical Quality Control (SPC)

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Bottling Company Machine automatically fills a 20 oz bottle. Problem with filling too much? Problems with filling to little? So Monday the average is 20.2 ounces. Tuesday the average is 19.6 ounces. Is this normal? Do we need to be concerned? Wed is 19.4 ounces.

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Natural Variation Machine can not fill every bottle exactly the same amount – close but not exactly.

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Assignable variation A cause for part of the variation

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SPC Objective: provide statistical signal when assignable causes of variation are present

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Control Charts R Chart Variables Charts Attributes Charts X Chart P C Continuous Numerical Data Categorical or Discrete Numerical Data Control Chart Types

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Characteristics for which you focus on defects Classify products as either ‘good’ or ‘bad’, or count # defects – e.g., radio works or not Categorical or discrete random variables Attributes Measuring quality Characteristics that you measure, e.g., weight, length May be in whole or in fractional numbers Continuous random variables Variables

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Show changes in data pattern – e.g., trends Make corrections before process is out of control Show causes of changes in data – Assignable causes Data outside control limits or trend in data – Natural causes Random variations around average Control Chart Purposes

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Figure S6.7

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Steps to Follow When Using Control Charts TO SET CONTROL CHART LIMITS 1.Collect 20-25 samples of n=4 or n=5 a stable process compute the mean of each sample. 2.Calculate control limits Compute the overall means Calculate the upper and lower control limits.

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Steps to Follow When Using Control Charts - continued TO MONITOR PROCESS USING THE CONTROL CHARTS: 1.Collect and graph data Graph the sample means and ranges on their respective control charts Determine whether they fall outside the acceptable limits. 2.Investigate points or patterns that indicate the process is out of control. Assign causes for the variations. 3.Collect additional samples and revalidate the control limits.

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Control Charts for Variables Glacier Bottling Manage at Glacier Bottling is concerned about their filling process. In particular, they want to know whether or not the machines are really filling the bottles with 16 ounces. Create an Xbar chart that will be used to monitor the process. Collected data for 25 days. Each day, pulled 4 bottles from the filling line and measured the amount in the bottle.

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4 115.8516.0215.8315.93 216.1216.0015.8516.01 316.0015.9115.9415.83 416.2015.8515.7415.93 515.7415.8616.2116.10 615.9416.0116.1416.03 715.7516.2116.0115.86 815.8215.9416.0215.94 916.0415.9815.8315.98 1015.6415.8615.9415.89 1116.1116.0016.0115.82 1215.7215.8516.1216.15 1315.8515.7615.7415.98 1415.7315.8415.9616.10 1516.2016.0116.1015.89 1616.1216.0815.8315.94 1716.0115.9315.8115.68 1815.7816.0416.1116.12 1915.8415.9216.0516.12 2015.9216.0916.1215.93 2116.1116.0216.0015.88 2215.9815.8215.89 2316.0515.73 15.93 2416.01 15.8915.86 2516.0815.7815.9215.98

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Glacier Bottling Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4 115.8516.0215.8315.93 216.1216.0015.8516.01 316.0015.9115.9415.83 416.2015.8515.7415.93 515.7415.8616.2116.10 Remember: There are 25 samples of size 4 to calculate the control limits. We are doing the first 5 right now…

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4R 115.8516.0215.8315.93 216.1216.0015.8516.01 316.0015.9115.9415.83 416.2015.8515.7415.93 515.7415.8616.2116.10 Glacier Bottling 16.02 – 15.83=0.19

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4R 115.8516.0215.8315.930.19 216.1216.0015.8516.010.27 316.0015.9115.9415.830.17 416.2015.8515.7415.930.46 515.7415.8616.2116.100.47 615.9416.0116.1416.030.20 715.7516.2116.0115.860.46 815.8215.9416.0215.940.20 916.0415.9815.8315.980.21 1015.6415.8615.9415.890.30 1116.1116.0016.0115.820.29 1215.7215.8516.1216.150.43 1315.8515.7615.7415.980.24 1415.7315.8415.9616.100.37 1516.2016.0116.1015.890.31 1616.1216.0815.8315.940.29 1716.0115.9315.8115.680.33 1815.7816.0416.1116.120.34 1915.8415.9216.0516.120.28 2015.9216.0916.1215.930.20 2116.1116.0216.0015.880.23 2215.9815.8215.89 0.16 2316.0515.73 15.930.32 2416.01 15.8915.860.15 2516.0815.7815.9215.980.30 Rbar =0.29 ounce

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Glacier Bottling R -Charts UCL R = D 4 R LCL R = D 3 R R = 0.29

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Control Chart Factors Control Chart Factors Factor for UCLFactor forFactor Size ofand LCL forLCL forUCL for Samplex-ChartsR-ChartsR-Charts (n)(A 2 )(D 3 )(D 4 ) 21.880 03.267 31.023 02.575 40.729 02.282 50.577 02.115 60.483 02.004 70.419 0.0761.924 This chart is in your text and will be provided for exams if needed.

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Glacier Bottling R -Charts UCL R = D 4 R LCL R = D 3 R R = 0.29 D 4 = 2.282 D 3 = 0 UCL R = 2.282 (0.29) = 0.654 ounce LCL R = 0(0.29) = 0 ounce

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Glacier Bottling R -Charts UCL R = D 4 R LCL R = D 3 R R = 0.29 D 4 = 2.282 D 3 = 0 UCL R = 2.282 (0.29) = 0.654 ounce LCL R = 0(0.29) = 0 ounce

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Glacier Bottling

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Figure S6.7

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4 115.8516.0215.8315.93 216.1216.0015.8516.01 316.0015.9115.9415.83 416.2015.8515.7415.93 515.7415.8616.2116.10 615.9416.0116.1416.03 715.7516.2116.0115.86 815.8215.9416.0215.94 916.0415.9815.8315.98 1015.6415.8615.9415.89 1116.1116.0016.0115.82 1215.7215.8516.1216.15 1315.8515.7615.7415.98 1415.7315.8415.9616.10 1516.2016.0116.1015.89 1616.1216.0815.8315.94 1716.0115.9315.8115.68 1815.7816.0416.1116.12 1915.8415.9216.0516.12 2015.9216.0916.1215.93 2116.1116.0216.0015.88 2215.9815.8215.89 2316.0515.73 15.93 2416.01 15.8915.86 2516.0815.7815.9215.98

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4RXbar 115.8516.0215.8315.930.19 216.1216.0015.8516.010.27 316.0015.9115.9415.830.17 416.2015.8515.7415.930.46 515.7415.8616.2116.100.47 Glacier Bottling (15.85+16.02+15.83+15.93)/4 = 15.908

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4RXbar 115.8516.0215.8315.930.19 15.908 216.1216.0015.8516.010.27 316.0015.9115.9415.830.17 416.2015.8515.7415.930.46 515.7415.8616.2116.100.47 Glacier Bottling (16.12+16.00+15.85+16.01)/4 = 15.995

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Bottle Volume in Ounces Sample NumObs 1Obs 2Obs 3Obs 4RXbar 115.8516.0215.8315.930.1915.908 216.1216.0015.8516.010.2715.995 316.0015.9115.9415.830.1715.920 416.2015.8515.7415.930.4615.930 515.7415.8616.2116.100.4715.978 615.9416.0116.1416.030.2016.030 715.7516.2116.0115.860.4615.958 815.8215.9416.0215.940.2015.930 916.0415.9815.8315.980.2115.958 1015.6415.8615.9415.890.3015.833 1116.1116.0016.0115.820.2915.985 1215.7215.8516.1216.150.4315.960 1315.8515.7615.7415.980.2415.833 1415.7315.8415.9616.100.3715.908 1516.2016.0116.1015.890.3116.050 1616.1216.0815.8315.940.2915.993 1716.0115.9315.8115.680.3315.858 1815.7816.0416.1116.120.3416.013 1915.8415.9216.0516.120.2815.983 2015.9216.0916.1215.930.2016.015 2116.1116.0216.0015.880.2316.003 2215.9815.8215.89 0.1615.895 2316.0515.73 15.930.3215.860 2416.01 15.8915.860.1515.943 2516.0815.7815.9215.980.3015.940 RBar = 0.29 ounce XBarBar = 15.9469 ounces

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Shows % of nonconforming items Attributes control chart – Nominally scaled categorical data e.g., good-bad p Chart

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p Chart Control Limits # Defective Items in Sample i Size of sample i z = 2 for 95.5% limits; z = 3 for 99.7% limits

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HOMETOWN BANK Hometown Bank The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel. Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The records for the past 12 weeks are shown in the following table. Is the process out of control? Use 3-sigma control limits.

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Hometown Bank UCL p = p + z p LCL p = p - z p p = p (1 - p )/ n SampleWrong NumberAccount Number 115 212 319 4 2 519 6 4 724 8 7 910 1017 1115 12 3 Total 147 Total defectives Total observations p = n = Control Charts for Attributes

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Control Charts for Attributes Hometown Bank UCL p = p + z p LCL p = p – z p p = p (1 – p )/ n n = 2500 p = 0.0049

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Control Charts for Attributes Hometown Bank UCL p = p + z p LCL p = p – z p p = 0.0049(1 – 0.0049)/2500 n = 2500 p = 0.0049

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Control Charts for Attributes Hometown Bank UCL p = p + z p LCL p = p – z p p = 0.0014 n = 2500 p = 0.0049

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Control Charts for Attributes Hometown Bank p = 0.0014 n = 2500 p = 0.0049 UCL p = 0.0049 + 3(0.0014) LCL p = 0.0049 – 3(0.0014)

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Control Charts for Attributes Hometown Bank p = 0.0014 n = 2500 p = 0.0049 UCL p = 0.0049 + 3(0.0014) LCL p = 0.0049 – 3(0.0014) Why 3? 3-sigma limits Also to within 99.7%

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UCL p = 0.0091 LCL p = 0.0007 Control Charts for Attributes Hometown Bank p = 0.0014 n = 2500 p = 0.0049

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p -Chart Wrong Account Numbers

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Figure S6.7

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Which control chart is appropriate? Webster Chemical Company produces mastics and caulking for the construction industry. The product is blended in large mixers and then pumped into tubes and capped. Webster is concerned whether the filling process for tubes of caulking is in statistical control. The process should be centered on 8 ounces per tube. Several samples of eight tubes are taken and each tube is weighed in ounces.

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Which control chart is appropriate? A sticky scale brings Webster’s attention to whether caulking tubes are being properly capped. If a significant proportion of the tubes aren’t being sealed, Webster is placing their customers in a messy situation. Tubes are packaged in large boxes of 144. Several boxes are inspected. The number of leaking tubes in each box is recorded.

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