Operations Management Statistical Process Control Supplement 6

Outline Statistical Process Control (SPC). Acceptance Sampling. Mean charts or 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. This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues.

Statistical Process Control Steps Start Take Sample Produce Good Provide Service Inspect Sample Take Samples No Is process in control? This slide introduces the statistical control process. It may be helpful here to walk students through an example or two of the process. The first walk through should probably be for a manufacturing process. The next several slides present information about the various types of process control slides: Create Stop Process Control Chart Yes Find Out Why

Process Control Charts Plot of Sample Data Over Time 20 40 60 80 1 5 9 13 17 21 Time Sample Value Upper control limit Lower control limit An example of a control chart. .

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 Central Limit Theorem As sample size gets large enough, distribution of mean values becomes approximately normal for any population distribution. Central Limit Theorem The next three slides can be used in a discussion of the theoretical basis for statistical process control.

Control Chart Types Control Categorical or Discrete Numerical Data Continuous Numerical Data Charts Variables Attributes Charts Charts This slide simply introduces the various types of control charts. R X P C Chart Chart Chart Chart

Quality Characteristics Variables Attributes Characteristics that you measure, e.g., weight, length. Continuous values. Characteristics for which you focus on defects. Categorical or discrete values. ‘Good’ or ‘Bad’. # of defects. Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.

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 - Std. Dev. of Process Is Known The following slide provides much of the data from Table S4.1. sample mean at time i  = known process standard deviation

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

X Chart Control Limits - Std. Dev. of Process is Not Known A2 is from Table S6.1 The following slide provides much of the data from Table S4.1. sample range at time i sample mean at time i

Factors for Computing Control Chart Limits Sample Size, n Mean Factor, A 2 Upper Range, D 4 Lower 3 1.880 3.268 1.023 2.574 0.729 2.282 5 0.577 2.115 6 0.483 2.004 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

X Chart - Example 2 sample mean range. 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 Sample Mean 5.0 4.9 Time Upper control limit Lower control limit An example of a control chart. . 4.9 Time

Example 2 – New Samples 5.1 Sample Mean 5.0 4.9 Time 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 Sample Mean 5.0 Lower control limit 4.9 Time

R Chart Shows sample ranges over time. Monitors process variability. 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 sample mean range 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 Sample Range 0.2 0.1 Time Upper control limit An example of a control chart. . Lower control limit Time

Example 2 – New Samples 0.3 Sample Range 0.2 0.1 Time 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 Sample Range 0.2 0.1 Lower control limit 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.

Use of Control Charts

Example 3 sample values mean range 1 4.9, 5.0, 5.1 5.0 0.2 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 – 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

Example 4 sample values mean range 1 5.0, 5.0, 5.0 5.0 0.0 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 – 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

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. Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.

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? Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.