1 / 40

SMU EMIS 7364

SMU EMIS 7364. NTU TO-570-N. Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow. Measures of Process Capability Process Capability Ratios Updated: 2/4/04. Process Capability Refers to the uniformity of the process. Variability in the process is a measure of the

jason
Download Presentation

SMU EMIS 7364

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SMU EMIS 7364 NTU TO-570-N Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow Measures of Process Capability Process Capability Ratios Updated: 2/4/04

  2. Process Capability Refers to the uniformity of the process. Variability in the process is a measure of the uniformity of the output. - Instantaneous variability is the natural or inherent variability at a specified time - Variability over time

  3. Process Capability A critical performance measure that addresses process results relative to process/product specifications. A capable process is one for which the process outputs meet or exceed expectation.

  4. Process Capability Measures or Indices • Process capability indices are used to measure the • process variability due to common causes present • in the process • The Cp index • Inherent or potential measure of capability • specification spread • process spread • The CpK index • Realized or actual measure of capability • Other indices • CpM, CpMK Cp =

  5. Measures of Process Capability Customary to use the six sigma spread in the distribution of the product quality characteristic

  6. Key Points • The proportion of the process output that will fall • outside the natural tolerance limits. • Is 0.27% (or 2700 nonconforming parts per million) • if the distribution is normal • May differ considerably from 0.27% if the • distribution is not normal

  7. Measure of Potential Process Capability, Cp • Cp measures potential or inherent capability of the • process, given that the process is stable • Cp is defined as • , for two-sided • specifications • and , for lower • specifications only • , for upper • specifications only

  8. Interpretation of Cp is the percentage of the specification band used up by the process

  9. Measure of Potential Process Capability, CpK • CpK measures realized process capability relative to • actual production, given a stable process • CpK is defined as

  10. Interpretation of CpK < 1, then conclude that the process is stable If CpK = 1, then conclude that the process is marginally capable > 1, then conclude that the process is capable

  11. Recommended Minimum Values of the Process Capability Ratio Two-sided One-sided specifications specifications Existing process 1.33 1.25 New processes 1.50 1.45 Safety, strength, or critical parameter existing process 1.50 1.45 Safety, strength, or critical parameter new process 1.67 1.60

  12. Process Fallout (in defective ppm) PCR One-sided specs Two-sided specs 0.25 226628 453255 0.50 66807 133614 0.60 35931 71861 0.70 17865 35729 0.80 8198 16395 0.90 3467 6934 1.00 1350 2700 1.10 1484 967 1.20 159 318 1.30 48 96 1.40 14 27 1.50 4 7 1.60 1 2 1.70 0.17 0.34 1.80 0.03 0.06 2.00 0.0009 0.0018

  13. Estimation of Process Capability Ratios

  14. Estimation of Cp A point estimate of Cp is: where

  15. Estimation of Cp - example If the specification limits are LSL = 73.95 and USL = 74.05 and

  16. Estimation of CP - example and the process uses of the specification band. ^

  17. Example To illustrate the use of the one sided process capability ratios, suppose that the lower specification limit on bursting strength is 200 psi. We will use x = 264 and S = 32 as estimates of  and , respectively, and the resulting estimate of the one sided lower process capability ratio is

  18. Example The fraction of defective bottles produced by this process is estimated by finding the area to the left of Z = (LSL - )/ = (200 - 264)/32 = -2 under the standard normal distribution. The estimated fallout is about 2.28% defective, or about 22,800 nonconforming bottles per million. Note that if the normal distribution were an inappropriate model for strength, then this last calculation would have to be performed using the appropriate probability distribution. ^ ^

  19. Estimation of Cp A (1 - )  100% confidence interval for Cp is where = = where are the lower /2 and upper /2 percentage points of the chi-squared distribution with n - 1 degrees of freedom.

  20. The Chi-Squared Distribution • Definition - A random variable X is said to have the Chi-Squared • distribution with parameter , called degrees of freedom, if the • probability density function of X is: • , for x >0 • , elsewhere where  is a positive integer.

  21. The Chi-Squared Model Remarks: The Chi-Squared distribution plays a vital role in statistical inference. It has considerable application in both methodology and theory. It is an important component of statistical hypothesis testing and estimation. The Chi-Squared distribution is a special case of the Gamma distribution, i.e., when  = /2 and  = 2.

  22. Properties of the Chi-Squared Model • Mean or Expected Value • Standard Deviation

  23. Properties of the Chi-Squared Model It is customary to let 2a represent the value above which we find an area of p. This is illustrated by the shaded region below. f(x) a x For tabulated values of the Chi-Squared distribution see the Chi-Squared table, which gives values of 2a for various values of p and . The areas, p, are the column headings; the degrees of freedom, , are given in the left column, and the table entries are the 2 values.

  24. Estimation of CpK An approximate (1 - )  100% confidence interval for CpK is where = =

  25. Example A sample of size n = 20 is taken from a stable process is used to estimate CpK, with the result that = 1.33. An approximate 95% confidence interval on CpK is = =

  26. Example = = 0.99  CpK  1.67 This is an extremely wide confidence interval. Based on the sample data, the ratio CpK could be less than one (a very bad situation), or it could be as large as 1.67 (a very good situation). Thus, we have learned very little about the actual process capability, because CpK is very imprecisely estimated. The reason for this is that a very small sample (n = 20) has been used.

  27. Testing Hypotheses About PCR’s A practice that is becoming increasingly common in industry is to require a supplier to demonstrate process capability as part of the contractual agreement. Thus, it is frequently necessary to demonstrate that the process capability ratio Cp meets or exceeds some particular target value - say, Cp0. This problem may be formulated as a hypothesis testing problem. H0: Cp = Cp0 (or the process is not capable) H1: Cp> Cp0 (or the process is capable)

  28. Testing Hypotheses About PCR’s  =  = 0.10  =  = 0.05 Sample Cp(high)/ Cp(high)/ Size Cp(low) C/Cp(low) Cp(low) C/Cp(low) 10 1.88 1.27 2.26 1.37 20 1.53 1.20 1.73 1.26 30 1.41 1.16 1.55 1.21 40 1.34 1.14 1.46 1.18 50 1.30 1.13 1.40 1.16 60 1.27 1.11 1.36 1.15 70 1.25 1.10 1.33 1.14 80 1.23 1.10 1.30 1.13 90 1.21 1.10 1.28 1.12 100 1.20 1.09 1.26 1.11

  29. Testing Hypotheses About PCR’s We would like to reject H0, thereby demonstrating that the process is capable. We can formulate the statistical test in terms of Cp, so that we will reject H0, if Cp exceeds a critical value C. ^ ^

  30. Example A customer has told his supplier that, in order to qualify for business with his company, the supplier must demonstrate that his process capability exceeds Cp = 1.33. Thus his supplier is interested in establishing a procedure to test the hypothesis. H0: Cp = 1.33 H1: Cp > 1.33 The supplier wants to be sure that if the process capability is below 1.33 there will be a high probability of detecting this (say, 0.90), whereas if the process capability exceeds 1.66 there will be a high probability of judging the process capable

  31. Example again (say, 0.90). This would imply that Cp(low) = 1.33, Cp(high) = 1.66 and  =  = 0.10. To find the sample size and critical value C from the table, compute and enter the table value where  =  = 0.10. This yields n = 70 and

  32. Example from which we calculate Thus, to demonstrate capability, the supplier must take a sample of n = 70 parts and the sample process ratio Cp must exceed C = 1.46. ^

  33. Testing Hypotheses In many situations the reason for gathering and analyzing data is to provide a basis for deciding on a course of action. Let us assume that either of two courses of action is possible: A1 or A2, and that we would be clear whether one or the other is the better action, if only we knew the nature of a certain population - that is, if we knew the probability distribution of a certain random variable.

  34. Testing Hypotheses The whole population or the distribution of probability is usually unattainable, therefore, we are forced to settle for such information as can be gleaned from a sample and to make our choice between the two actions on the basis of the sample. 1. Obtain random sample of size n 2. Apply decision rule to data

  35. Testing Hypotheses Statistical Hypothesis - is a statement about a probability distribution and is usually a statement about the values of one or more parameters of the distribution. For example, a company may want to test the hypothesis that the true average lifetime of a certain type of TV is at least 500 hours, i.e., that   500.

  36. Testing Hypotheses The hypothesis to be tested is called the null hypothesis and is denoted by H0. To construct a criterion for testing a given null hypothesis, analternate hypothesis must be formulated, denoted by H1. Remark: To test the validity of a statistical hypothesis the test is conducted, and according to the test plan the hypothesis is rejected if the results are improbable under the hypothesis. If not, the hypothesis is accepted. The test leads to one of two possible actions: accept H0 or reject H0 (accept H1)

  37. Testing Hypotheses Test Statistic - The statistic upon which a test of a statistical hypothesis is based. Critical Region - The range of values of a test statistic which, for a given test, requires the rejection of H0. Remark: Acceptance or rejection of a statistical hypothesis does not prove nor disprove the hypothesis! Whenever a statistical hypothesis is accepted or rejected on the basis of a sample, there is always the risk of making a wrong decision. The uncertainty with which a decision is made is measured in terms of probability.

  38. Testing Hypotheses There are two possible decision errors associated with testing a statistical hypothesis: A Type I error is made when a true hypothesis is rejected. A Type II error is made when a false hypothesis is accepted. True Situation Decision H0 trueH0 false Accept H0 correct Type II error Reject H0 Type I error correct (Accept H1)

  39. Uses of Results from a Process Capability Analysis 1. Predicting how well the process will hold the tolerances. 2. Assisting product developers/designers in selecting or modifying a process. 3. Assisting in establishing an interval between sampling for process monitoring. 4. Specifying performance requirements for new equipment 5. Selecting between competing vendors. 6. Planning the sequence of production processes when there is an interactive effect on processes or tolerances. 7. Reducing the variability in a manufacturing process.

More Related