1. 1 1 Slide | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | UCL CL LCL Chapter 13 Statistical Methods for Quality Control n Statistical Process Control n Acceptance Sampling

2. 2 2 Slide Quality Terminology n Quality is “the totality of features and characteristics of a product or service that bears on its ability to satisfy given needs.” n Quality assurance refers to the entire system of policies, procedures, and guidelines established by an organization to achieve and maintain quality. n The objective of quality engineering is to include quality in the design of products and processes and to identify potential quality problems prior to production. n Quality control consists of making a series of inspections and measurements to determine whether quality standards are being met.

3. 3 3 Slide Statistical Process Control (SPC) n The goal of SPC is to determine whether the process can be continued or whether it should be adjusted to achieve a desired quality level. n If the variation in the quality of the production output is due to assignable causes (operator error, worn-out tooling, bad raw material,... ) the process should be adjusted or corrected as soon as possible. n If the variation in output is due to common causes (variation in materials, humidity, temperature,... ) which the manager cannot control, the process does not need to be adjusted.

4. 4 4 Slide SPC Hypotheses n SPC procedures are based on hypothesis-testing methodology. n The null hypothesis H 0 is formulated in terms of the production process being in control. n The alternative hypothesis H a is formulated in terms of the process being out of control. n As with other hypothesis-testing procedures, both a Type I error (adjusting an in-control process) and a Type II error (allowing an out-of-control process to continue) are possible.

5. 5 5 Slide The Outcomes of SPC n Type I and Type II Errors State of Production Process State of Production Process H 0 True H a True H 0 True H a True Decision In Control  Out of Control Decision In Control  Out of Control Accept H 0 Correct Type II Accept H 0 Correct Type II Continue Process  Decision  Error Continue Process  Decision  Error Reject H 0 Type I Correct Reject H 0 Type I Correct Adjust Process  Error  Decision Adjust Process  Error  Decision

6. 6 6 Slide Control Charts n SPC uses graphical displays known as control charts to monitor a production process. n Control charts provide a basis for deciding whether the variation in the output is due to common causes (in control) or assignable causes (out of control). n Two important lines on a control chart are the upper control limit (UCL) and lower control limit (LCL). n These lines are chosen so that when the process is in control there will be a high probability that the sample finding will be between the two lines. n Values outside of the control limits provide strong evidence that the process is out of control.

7. 7 7 Slide Types of Control Charts n An x chart is used if the quality of the output is measured in terms of a variable such as length, weight, temperature, and so on. n x represents the mean value found in a sample of the output. n An R chart is used to monitor the range of the measurements in the sample. n A p chart is used to monitor the proportion defective in the sample. n An np chart is used to monitor the number of defective items in the sample.

8. 8 8 Slide Interpretation of Control Charts n The location and pattern of points in a control chart enable us to determine, with a small probability of error, whether a process is in statistical control. n A primary indication that a process may be out of control is a data point outside the control limits. n Certain patterns of points within the control limits can be warning signals of quality problems: Large number of points on one side of center line.Large number of points on one side of center line. Six or seven points in a row that indicate either an increasing or decreasing trend.Six or seven points in a row that indicate either an increasing or decreasing trend.... and other patterns.... and other patterns.

9. 9 9 Slide Control Limits for an x Chart n Process Mean and Standard Deviation Known n Process Mean and Standard Deviation Unknown where: x = overall sample mean R = average range R = average range A 2 = a constant that depends on n ; taken from A 2 = a constant that depends on n ; taken from “Factors for Control Charts” table “Factors for Control Charts” table = _

10. 10 Slide Control Limits for an R Chart UCL = RD 4 LCL = RD 3 where: R = average range R = average range D3, D4 = constants that depend on n ; found in “Factors for Control Charts” table D3, D4 = constants that depend on n ; found in “Factors for Control Charts” table _ _ _

11. 11 Slide Factors for x and R Control Charts n Factors Table (Partial)

12. 12 Slide Example: Granite Rock Co. n Control Limits for an x Chart: Process Mean and Standard Deviation Known The weight of bags of cement filled by Granite’s packaging process is normally distributed with a mean of 50 pounds and a standard deviation of 1.5 pounds. What should be the control limits for samples of 9 bags?

13. 13 Slide Example: Granite Rock Co. n Control Limits for an x Chart: Process Mean and Standard Deviation Known  = 50,  = 1.5, n = 9 UCL = 50 + 3(.5) = 51.5 UCL = 50 + 3(.5) = 51.5 LCL = 50 - 3(.5) = 48.5 LCL = 50 - 3(.5) = 48.5

14. 14 Slide Example: Granite Rock Co. n Control Limits for x and R Charts: Process Mean and Standard Deviation Unknown Suppose Granite does not know the true mean and standard deviation for its bag filling process. It wants to develop x and R charts based on forty samples of 9 bags each. The average of the sample means is 50.1 pounds and the average of the sample ranges is 3.25 pounds.

15. 15 Slide Example: Granite Rock Co. n Control Limits for x and R Charts: Process Mean and Standard Deviation Unknown x = 50.1, R = 3.25, n = 9 x = 50.1, R = 3.25, n = 9 R Chart R Chart UCL = RD 4 = 3.25(1.816) = 5.9 LCL = RD 3 = 3.25(0.184) = 0.6 LCL = RD 3 = 3.25(0.184) = 0.6 x Chart x Chart UCL = x + A 2 R = 50.1 +.337(3.25) = 51.2 UCL = x + A 2 R = 50.1 +.337(3.25) = 51.2 LCL = x - A 2 R = 50.1 -.337(3.25) = 49.0 LCL = x - A 2 R = 50.1 -.337(3.25) = 49.0 = = _ _ _ _ = _

16. 16 Slide Control Limits for a p Chart where:assuming: np > 5 np > 5 n (1- p ) > 5 n (1- p ) > 5

17. 17 Slide Control Limits for an np Chart assuming: np > 5 np > 5 n (1- p ) > 5 n (1- p ) > 5 Note: If computed LCL is negative, set LCL = 0

18. 18 Slide Acceptance Sampling n Acceptance sampling is a statistical method that enables us to base the accept-reject decision on the inspection of a sample of items from the lot. n Acceptance sampling has advantages over 100% inspection including: less expensive, less product damage, fewer people involved,... and more.

19. 19 Slide Acceptance Sampling Procedure Lot received Sample selected Sampled items inspected for quality Results compared with specified quality characteristics Accept the lot Reject the lot Send to production or customer Decide on disposition of the lot Quality is not satisfactory satisfactory Quality is Quality issatisfactory

20. 20 Slide Acceptance Sampling n Acceptance sampling is based on hypothesis-testing methodology. n The hypothesis are: H 0 : Good-quality lot H a : Poor-quality lot

21. 21 Slide The Outcomes of Acceptance Sampling n Type I and Type II Errors State of the Lot State of the Lot H 0 True H a True H 0 True H a True Decision Good-Quality Lot  Poor-Quality Lot Decision Good-Quality Lot  Poor-Quality Lot Accept H 0 Correct Type II Error Accept H 0 Correct Type II Error Accept the Lot  Decision Consumer’s Risk Reject H 0 Type I Error Correct Reject H 0 Type I Error Correct Reject the Lot Producer’s Risk  Decision Reject the Lot Producer’s Risk  Decision

22. 22 Slide Probability of Accepting a Lot n Binomial Probability Function for Acceptance Sampling where: n = sample size p = proportion of defective items in lot x = number of defective items in sample f ( x ) = probability of x defective items in sample f ( x ) = probability of x defective items in sample

23. 23 Slide Example: Acceptance Sampling An inspector takes a sample of 20 items from a lot. Her policy is to accept a lot if no more than 2 defective items are found in the sample. Assuming that 5 percent of a lot is defective, what is the probability that she will accept a lot? Reject a lot? n = 20, c = 2, and p =.05 n = 20, c = 2, and p =.05 P (Accept Lot) = f (0) + f (1) + f (2) P (Accept Lot) = f (0) + f (1) + f (2) =.3585 +.3774 +.1887 =.3585 +.3774 +.1887 =.9246 =.9246 P (Reject Lot) = 1 -.9246 =.0754 P (Reject Lot) = 1 -.9246 =.0754

24. 24 Slide Example: Acceptance Sampling n Using the Tables of Binomial Probabilities

25. 25 Slide Operating Characteristic Curve.10.20.30.40.50.60.70.80.90 Probability of Accepting the Lot 0 5 10 15 20 25 1.00 Percent Defective in the Lot p0p0 p1p1  (1 -  )  n = 15, c = 0 p 0 =.03, p 1 =.15  =.3667,  =.0874

26. 26 Slide Multiple Sampling Plans n A multiple sampling plan uses two or more stages of sampling. n At each stage the decision possibilities are: stop sampling and accept the lot,stop sampling and accept the lot, stop sampling and reject the lot, orstop sampling and reject the lot, or continue sampling.continue sampling. n Multiple sampling plans often result in a smaller total sample size than single-sample plans with the same Type I error and Type II error probabilities.

27. 27 Slide A Two-Stage Acceptance Sampling Plan Inspect n 1 items Find x 1 defective items in this sample Is x 1 < c 1 ? Is x 1 > c 2 ? Inspect n 2 additional items Accept the lot Reject Is x 1 + x 2 < c 3 ? Find x 2 defective items in this sample Yes Yes No No No Yes

28. 28 Slide The End of Chapter 13