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Operations Management Statistical Process Control Supplement 6

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## Operations Management Statistical Process Control Supplement 6

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**Operations ManagementStatistical Process ControlSupplement 6****Outline**• Statistical Process Control (SPC). • Mean chartsor X-Charts. • Range chart or R-Charts. • Control charts for attributes. • Managerial issues and control charts. • Acceptance Sampling.**Statistical Process Control (SPC)**• 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.**Start**Take Sample Produce Good Inspect Sample Provide Service Take Samples No Is process in control? Create Stop Process Control Chart Yes Find Out Why Statistical Process Control Steps**Plot of Sample Data Over Time**80 Upper control limit 60 Sample Value 40 20 Lower control limit 0 1 5 9 13 17 21 Time Process Control Charts**Control Charts**• 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.**Distribution of Sample Means**Standard deviation of the sample means (mean)**Central Limit Theorem**As sample size gets large enough, distribution of meanvalues becomes approximately normal for any population distribution. Central Limit Theorem**Control Chart Types**Control Categorical or Discrete Numerical Data Charts Continuous Numerical Data Variables Attributes Charts Charts R P C X Chart Chart Chart Chart**Quality Characteristics**Variables Attributes • Characteristics for which you focus on defects. • Categorical or discrete values. • ‘Good’ or ‘Bad’. • # of defects. • Characteristics that you measure, e.g., weight, length. • Continuous values.**X Chart**• 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 Control Limits**A2 is from Table S6.1 sample range at time i sample mean at time i**Sample**Mean Upper Lower Size, n Factor, A Range, D Range, D 2 4 3 2 1.880 3.268 0 3 1.023 2.574 0 4 0.729 2.282 0 5 0.577 2.115 0 6 0.483 2.004 0 7 0.419 1.924 0.076 8 0.373 1.864 0.136 9 0.337 1.816 0.184 10 0.308 1.777 0.223 Factors for Computing Control Chart Limits**X Chart - Example 2**Each sample is 4 measurements. Determine 3 control limits. sample mean range. 1 5.02 .12 4.96, 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**5.1 Upper control limit=5.095 Sample Mean 5.0 Lower control limit=4.905 4.9 Time**Example 2 – New Samples**sample values mean 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 5.1 Upper control limit=5.095 Sample Mean 5.0 Lower control limit=4.905 4.9 Time**R Chart**• 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 Control Limits**From Table S6.1 sample range at time i**R Chart - Example 2**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 4.96, 5.03, 5.01, 5.08**R Chart - Example 2**0.3 Upper control limit=0.297 Sample Range 0.2 0.1 Lower control limit=0 0 Time**Example 2 – New Samples**sample values mean 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 0.3 Upper control limit=0.297 Sample Range 0.2 0.1 Lower control limit=0 0 Time**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.**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.**Example 3**sample values mean 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**6.0**Upper control limit = 5.6546 Sample Mean 5.5 Lower control limit = 5.2454 5.0 Time 1.0 Upper control limit = 0.5148 Sample Range 0.5 Lower control limit = 0 0.0 Time Example 3 – Control Charts**Example 4**sample values mean 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**7.0**Upper control limit = 6.79025 Sample Mean 5.0 Lower control limit = 3.20975 3.0 Time 6.0 Upper control limit = 4.5045 Sample Range 3.0 Lower control limit = 0 0.0 Time Example 4 – Control Charts**p Chart**• Attributes control chart. • Shows % of nonconforming items. • Example: Count # defective chairs & divide by total chairs inspected. • Chair is either defective or not defective.**c Chart**• 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.**Acceptance Sampling**• 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**• 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?