Chapter 13 Statistical Process Control
Assuring Quality • Acceptance Sampling • Accepting (or rejecting) a lot (batch) after it is produced • Based on a previously determined criteria for acceptance (AQL) • Done by “inspectors” (QA/QC personnel) • Still in practice by many organizations today • Statistical Process Control
Statistical Process Control (SPC) • A methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when appropriate • When special causes are present, the process is deemed to be out of control. • If the variation in the process is due to common causes alone, the process is said to be in statistical control. • SPC relies on control charts
Histograms vs. Control Charts • Histograms do not take into account changes over time. • Control charts can tell us when a process changes
Key Idea Process capability calculations make little sense if the process is not in statistical control because the data are confounded by special causes that do not represent the inherent capability of the process.
Control In Control Out of Control Capability Capable Not Capable IDEAL Improving Capability and Control
SPC Metrics • An attribute is a performance characteristic that is either present or absent in the product or service under consideration (e.g., in or out of tolerance, error/defect present or absent). • Expressed as proportions or rates • Variable data are continuous (e.g., length, weight, and time). • Expressed with such statistics as averages and standard deviations
Key Idea Collecting attribute data is usually easier than collecting variable data because the assessment can usually be done more quickly by a simple inspection or count, whereas variable data require the use of some type of measuring instrument.
Control Charts for Variables Data • x-bar and R-charts – used when sample size is small and insufficient to reliably estimate the standard deviation` • x-bar and s-charts – used with large sample sizes • Charts for individuals (x-charts; also called I-charts); usually used in conjunction with a Moving Range (MR) chart
Constructing x-bar and R-Charts • Collect k = 25-30 samples, with sample sizes generally between n = 3 and 10 • Compute the mean and range of each sample • Compute the overall mean and average range: • Compute control limits:
Estimating Process Capability • Estimate of standard deviation using control chart data: • Use this estimate in process capability index calculations • Not as accurate as calculating the standard deviation using the complete set of data.
Key Idea Control limits relate to averages of samples, whereas specification limits relate to individual measurements. Control limits are not the same as specification limits!
Process Monitoring and Control • After a process is determined to be in control, the charts should be used on a daily basis to monitor performance, identify any special causes that might arise, and make corrections only as necessary. • Workers who run a process should use control charts and need to be trained to use them properly.
Key Idea Control charts indicate when to take action, and more importantly, when to leave a process alone.
Case Study: La Ventana Window Company • La Ventana received some complaints about narrow, misfitting gaps between the upper and lower window sashes • The plant manager wants to evaluate the capability of a critical cutting operation that he suspects might be the source of the gap problem. • The nominal specification for this cutting operation is 25.500 inches with a tolerance of 0.030 inch. • Inspect five consecutive window panels in the middle of each shift over a 15-day period and recording the dimension of the cut
Control Limit Calculations • n = 5; A2 = 0.577 and D4 = 2.114
Key Idea When a process is in statistical control, the points on a control chart fluctuate randomly between the control limits with no recognizable pattern.
Interpretation • Sample 24 out of control in R-chart • Samples 9, 21, and 24 out of control in x-bar chart • Common characteristic: Shane was process operator • Attribute results to special cause variation and delete these samples • New control limits
Controlled Process • No points are outside control limits. • The number of points above and below the center line is about the same. • The points seem to fall randomly above and below the center line. • Most points, but not all, are near the center line, and only a few are close to the control limits.
Investigation Needed • Point outside control limits (out of control!) • Sudden shift in process average • Cycles • Trends • Hugging the center line • Hugging the control limits • Instability
Charts for Individuals • Applications • Sampling from a homogeneous mixture • Low volume operations • Control limits for x-chart • Control limits for moving range-chart
Key Idea Control charts for individuals offer the advantage of being able to draw specifications on the chart for direct comparison with the control limits.
Charts for Attributes • A nonconformance (defect, error) is a single nonconforming quality characteristic of a unit of work. • If a unit of work has one or more nonconformances, we term the entire unit nonconforming. • Attribute charts are used formonitoring nonconformances as well as the number nonconforming
Fraction Nonconforming (p) Chart • Collect k samples, each of size n • Compute the fraction nonconforming in each sample, pi • Average fraction nonconforming • Compute standard deviation • Control limits
Variable Sample Size • If each sample has a different sample size, the the calculation of the average fraction nonconforming is • The p-chart will have control limits that vary with the sample size
p-Chart With Average Sample Size • Use the average sample size (n-bar) to compute approximate control limits • Use the average sample size method when the sample sizes fall within 25 percent of the average.
u-Chart for Nonconformances Per Unit • Applies if the samples are of unequal (or equal) size • Collect samples and count the number of nonconformances • Compute the average number of nonconformances per unit, u-bar • Control limits
Key Idea Confusion often exists over which chart is appropriate for a specific application, because the c- and u-charts apply to situations in which the quality characteristics inspected do not necessarily come from discrete units.
Control Chart Selection Quality Characteristic variable attribute defective defect no n>1? x and MR constant sampling unit? yes constant sample size? yes p or np no n>=10 or computer? x and R yes no no yes p-chart with variable sample size c u x and s