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Statistical Process Control

Statistical Process Control. Processes that are  not  in a state of  statistical control show excessive variations or exhibit variations that change with time Control charts are used to detect whether a process is statistically stable .

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Statistical Process Control

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  1. Statistical Process Control • Processes that are not in a state of statistical control show excessive variations or exhibit variations that change with time • Control charts are used to detect whether a process is statistically stable. • Control charts differentiates between variations that is normally expected of the process due chance or common causes that change over time due to assignable or special causes

  2. Common cause variation Variations due to common causes • are inherent to the process because of: • the nature of the system • the way the system is managed • the way the process is organized and operated • can only be removed by • making modifications to the process • changing the process • are the responsibility of higher management

  3. Special Cause Variation Variations due to special causes are • localized in nature • exceptions to the system • considered abnormalities • often specific to a • certain operator • certain machine • certain batch of material, etc. Investigation and removal of variations due to special causes are key to process improvement

  4. Causes of Variation • Two basic categories of variation in output include common causes and assignable causes. • Commoncauses are the purely random, unidentifiable sources of variation that are unavoidable with the current process. • If processvariability results solely from common causes of variation, a typical assumption is that the distribution is symmetric, with most observations near the center. • Assignablecauses of variation are any variation-causing factors that can be identified and eliminated, such as a machine needing repair.

  5. Statistical Process Control (SPC) Charts • Statistical process control (SPC) charts are used to help us distinguish between common and assignable causes of variation. We will cover 2 types: • Variables Control Charts: Service or product characteristics that is continuous and can be measured, such as weight, length, volume, or time. • Attributes Control Charts: Service or product characteristics that can be counted...pass/fail, good/bad, rating scale. Color, inspection, test results (pass/fail), number of defects, types of defects SPC charts Explained

  6. Control Charts • Control Charts are run charts with superimposed normal distributions

  7. Purpose of Control Charts • Control charts provide a graphical means for testing hypotheses about the data being monitored.

  8. Control and warning limits • The probability of a sample having a particular value is given by its location on the chart. Assuming that the plotted statistic is normally distributed, the probability of a value lying beyond the: • warning limits is approximately 0.025 or 2.5% chance (plus or minus 2-sigma from the mean) • control limits is approximately 0.001 or 0.1% chance (plus or minus 3 sigma from the mean), this is rare and indicates that • the variation is due to an assignable cause • the process is out-of-statistical control

  9. Out of Control Processes Run rules are rules that are used to indicate out-of-statistical control situations. Typical run rules for Shewhart X-charts with control and warning limits are: • a point lying beyond the control limits • 2 consecutive points lying beyond the warning limits (0.025x0.025x100 = 0.06% chance of occurring) • 7 or more consecutive points lying on one side of the mean ( 0.57x100 = 0.8% chance of occurring and indicates a shift in the mean of the process) • 5 or 6 consecutive points going in the same direction (indicates a trend) • Other run rules can be formulated using similar principles

  10. UCL Nominal Variations LCL Sample number Control Charts

  11. UCL Nominal Variations LCL Sample number Control Charts

  12. UCL Nominal Variations LCL Sample number Control Charts

  13. UCL Nominal Variations LCL Sample number Control Charts

  14. Sampling • Sampling plan: A plan that specifies a sample size, the time between successive samples, and decision rules that determine when action should be taken. • Sample size: A quantity of randomly selected observations of process outputs. Why do I need to sample? The case of potato chips

  15. Means and Ranges The mean of the sample is the sum of all observations in a sample divided by the number of observations in the sample. The range of the sampleis the difference between the largest observation in a sample and the smallest. The mean of the process is the sum of all sample means divided by the number of samples The mean range for the process is the sum of all ranges divided by the number of samples

  16. Statistical ProcessControl Methods • Control Charts for variables are used to monitor the mean and variability of the process distribution. • R-chart (Range Chart) is used to monitor process variability. - x-chart is used to see whether the process is generating output, on average, consistent with a target value set by management for the process or whether its current performance, with respect to the average of the performance measure, is consistent with past performance. • If the standard deviation of the process is known, we can place UCL and LCL at “z” standard deviations from the mean at the desired confidence level.

  17. – The control limits for the R-chartare UCLR = D4R and LCLR = D3Rwhere R = average of several past R values and the central line of the chart.D3,D4 = constants that provide 3 standard deviations (three-sigma) limits for a given sample size. Another way to develop control charts so for small sample sizes The control limits for the x-chart are: UCLx= x + A2R and LCLx = x - A2R Where X = central line of the chart, which can be either the average of past sample means or a target value set for the process. A2 = constant to provide three-sigma limits for the sample mean. = = – =

  18. Control Chart Factors

  19. Over the past 2 years, Professor Matta has been asked to teach one section of Process Analytics in each quarter (4 Quarters/ Year). Each time he taught, he would give 3 exams. The class average grade on these exams over the last 8 quarters have been as follows:

  20. Exam Grades in Prof Matta’s PA class during the past 2 years with Averages:

  21. Exam Grades in Prof Matta’s PA class during the past 2 years with Averages and Ranges:

  22. For Prof. Matta’s Case:

  23. Control Charts for Prof Matta’s Exams:

  24. Control Charts Example: • At Quikie Car Wash, the wash process is advertised to take less than 7 minutes. Consequently, management has set a target average of 390 seconds for the wash process. Suppose that the average range for a sample of 9 cars is 10 seconds. Establish the means and ranges control limits using this data.

  25. Solved Example • X= 390 sec, n = 9, R= 10 sec From Table 5.1 in your book, • A2 = 0.337, D3 = 0.184, D4 = 1.816 • UCLR= D4 R= 1.816(10 sec) = 18.16 sec • LCLR= D3 R= 0.184(10 sec) = 1.84 sec • UCLx = x + A2 R= 390 sec + 0.337(10 sec) = 393.37 sec • LCLx = x - A2 R= 390 sec – 0.337(10 sec) = 386.63 sec

  26. Marlin Bottling Company Problem 5 in the book chapter 2

  27. Marlin Co. Bottling – In class Calculations Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.00

  28. p = p(1 – p)/n Where n = sample size p = central line on the chart, which can be either the historical average population proportion defective or a target value. – – Control limits are: UCLp = p+zp and LCLp= p−zp Control Charts for Attributes • p-chart: A chart used for controlling the proportion of defective services or products generated by the process. z = normal deviate (number of standard deviations from the average)

  29. Hometown Bank Example 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 results for the past 12 weeks are shown in the following table. Is the booking process out of statistical control? Use three-sigma control limits.

  30. Sample Wrong Proportion Number Account # Defective 1 15 0.006 2 12 0.0048 3 19 0.0076 4 2 0.0008 5 19 0.0076 6 4 0.0016 7 24 0.0096 8 7 0.0028 9 10 0.004 10 17 0.0068 11 15 0.006 12 3 0.0012 Total 147 = 0.0049 147 12(2500) p = p = p(1 – p)/n p = 0.0049(1 – 0.0049)/2500 p = 0.0014 UCLp = 0.0049 + 3(0.0014) = 0.0091 LCLp = 0.0049 – 3(0.0014) = 0.0007 Hometown Bank Using a p-Chart to monitor a process n = 2500

  31. Hometown Bank Using a p-Chart to monitor a process Example

  32. In class Problem

  33. In Class Problem

  34. Control Charts Two types of error are possible with control charts • A type I error occurs when a process is thought to be out of control when in fact it is not • A type II error occurs when a process is thought to be in control when it is actually out of statistical control These errors can be controlled by the choice of control limits

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