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S6-1 Operations Management Statistical Process Control Supplement 6

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S6-2 Outline Statistical Process Control (SPC). Mean charts or X -Charts. Range chart or R -Charts. Control charts for attributes. Managerial issues and control charts. Acceptance Sampling.

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S6-3 Statistical technique to identify when non- random variation is present in a process. All processes are subject to variability. Natural causes: Random variations. Assignable causes: Correctable problems. Machine wear, unskilled workers, poor materials. Uses process control charts. Statistical Process Control (SPC)

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S6-4 Produce Good Provide Service Stop Process No Yes Is process in control? Take Samples Find Out Why Create Control Chart Start Statistical Process Control Steps Take Sample Inspect Sample

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S6-5 Process Control Charts Plot of Sample Data Over Time 0 20 40 60 80 159131721 Time Sample Value Upper control limit Lower control limit

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S6-6 Process is not in control if: Sample is not between upper and lower control limits. A non-random pattern is present, even when between upper and lower control limits. Based on sample being normally distributed. Control Charts

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S6-7 Distribution of Sample Means Standard deviation of the sample means (mean)

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S6-8 As sample size gets large enough, distribution of mean values becomes approximately normal for any population distribution. Central Limit Theorem

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S6-9 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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S6-10 Characteristics for which you focus on defects. Categorical or discrete values. ‘Good’ or ‘Bad’. # of defects. AttributesVariables Quality Characteristics Characteristics that you measure, e.g., weight, length. Continuous values.

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S6-11 Shows sample means over time. Monitors process average. Example: Weigh samples of coffee. Collect many samples, each of n bags. Sample size = n. Compute mean and range for each sample. Compute upper and lower control limits (UCL, LCL). Plot sample means and control limits. X Chart

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S6-12 X Chart Control Limits - Std. Dev. of Process Is Known sample mean at time i = known process standard deviation

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S6-13 Each sample is 4 measurements. Process mean is 5 lbs. Process standard deviation is 0.1 lbs. Determine 3 control limits. X Chart - Example 1

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S6-14 X Chart Control Limits - Std. Dev. of Process is Not Known sample range at time i A 2 is from Table S6.1 sample mean at time i

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S6-15 Factors for Computing Control Chart Limits Sample Size, n Mean Factor, A 2 Upper Range, D 4 Lower Range, D 3 21.8803.2680 31.0232.5740 40.7292.2820 50.5772.1150 60.4832.0040 70.4191.9240.076 80.3731.8640.136 90.3371.8160.184 100.3081.7770.223

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S6-16 Each sample is 4 measurements. Determine 3 control limits. sample mean range. 1 5.02.124.96, 5.03, 5.01, 5.08 2 4.99.08. 3 4.97.13. 4 5.03.18. 5 4.99.14. X Chart - Example 2

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S6-17 X Chart - Example 2 4.9 5.0 5.1 Time Sample Mean Upper control limit Lower control limit

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S6-18 sample valuesmean range 6 5.05, 5.00, 4.80, 4.95 4.95 0.25 7 5.00, 5.10, 5.10, 5.00 5.05 0.10 8 4.80, 5.20, 5.10, 5.00 5.025 0.40 4.9 5.0 5.1 Time Sample Mean Upper control limit Lower control limit Example 2 – New Samples

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S6-19 Shows sample ranges over time. Sample range = largest - smallest value in sample. Monitors process variability. Example: Weigh samples of coffee. Collect many samples, each of n bags. Sample size = n. Compute range for each sample & average range. Compute upper and lower control limits (UCL, LCL). Plot sample ranges and control limits. R Chart

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S6-20 sample range at time i From Table S6.1 R Chart Control Limits

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S6-21 Each sample is 4 measurements. Determine 3 control limits. sample mean range 1 5.02.12 2 4.99.08 3 4.97.13 4 5.03.18 5 4.99.14 R Chart - Example 2 4.96, 5.03, 5.01, 5.08

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S6-22 R Chart - Example 2 0 0.2 0.3 Time Sample Range Upper control limit Lower control limit 0.1

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S6-23 sample valuesmean range 6 5.05, 5.00, 4.80, 4.95 4.95 0.25 7 5.00, 5.10, 5.10, 5.00 5.05 0.10 8 4.80, 5.20, 5.10, 5.00 5.025 0.40 Example 2 – New Samples 0 0.2 0.3 Time Sample Range Upper control limit Lower control limit 0.1

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S6-24 Control Chart Steps Collect 20 to 25 samples of n=4 or n=5 from a stable process & compute the mean and range. Compute the overall mean and average range. Calculate upper and lower control limits. Collect new samples, and plot the means and ranges on their respective control charts.

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S6-25 Control Chart Steps - Continued Investigate points or patterns that indicate the process is out of control. Assign causes for the variations. Collect additional samples and revalidate the control limits.

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S6-26 Use of Control Charts

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S6-27 sample valuesmean range 1 4.9, 5.0, 5.1 5.0 0.2 2 5.2, 5.3, 5.4 5.3 0.2 3 5.5, 5.6, 5.7 5.6 0.2 4 5.8, 5.9, 6.0 5.9 0.2 Example 3

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S6-28 Example 3 – Control Charts 5.0 5.5 6.0 Time Sample Mean Upper control limit = 5.6546 Lower control limit = 5.2454 0.0 0.5 1.0 Time Sample Range Upper control limit = 0.5148 Lower control limit = 0

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S6-29 sample valuesmean range 1 5.0, 5.0, 5.0 5.0 0.0 2 4.5, 5.0, 5.5 5.0 1.0 3 4.0, 5.0, 6.0 5.0 2.0 4 3.0, 5.0, 7.0 5.0 4.0 Example 4

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S6-30 Example 4 – Control Charts 3.0 5.0 7.0 Time Sample Mean Upper control limit = 6.79025 Lower control limit = 3.20975 0.0 3.0 6.0 Time Sample Range Upper control limit = 4.5045 Lower control limit = 0

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S6-31 Attributes control chart. Shows % of nonconforming items. Example: Count # defective chairs & divide by total chairs inspected. Chair is either defective or not defective. p Chart

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S6-32 Attributes control chart. Shows number of defects in a unit. Unit may be chair, steel sheet, car, etc. Size of unit must be constant. Example: Count # defects (scratches, chips etc.) in each chair of a sample of 100 chairs. c Chart

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S6-33 Quality testing for incoming materials or finished goods. Procedure: Take one or more samples at random from a lot (shipment) of items. Inspect each of the items in the sample. Decide whether to reject the whole lot based on the inspection results. Acceptance Sampling

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S6-34 Inspecting all items is too expensive. The larger the sample inspected: The greater the cost for inspection. The less likely you are to accept a “bad” lot or to reject a “good” lot. Key questions: How many should be inspected in each lot? How confident are you in the accept/reject decision? Acceptance Sampling

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