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1. ENGM 720 - Lecture 08 P, NP, C, & U Control Charts ENGM 720: Statistical Process Control

2. Outline • Assignment • Discrete Distributions and Probability of Outcomes • Examples of discrete distributions • Hypothesis Testing to Control Charts • P- & NP-Charts • C- & U-Charts • Summary of Control Chart Options • Using the Control Chart Decision Chart ENGM 720: Statistical Process Control

3. Assignment: • Reading: • Chapter 6 • Finish reading • Chapter 7 • Sections 7.1 and 7.2 throughp.313 • Sections 7.3 through p.325 • Sections 7.3.2 and 7.5 • Assignments: • Obtain the Control Chart Factors table from Materials Page • Access Excel Template for X-bar, R, & S Control Charts: • Download Assignment 5 for practice • Use the data on the HW5 Excel sheet to do the charting, verify the control limits by hand calculations • Access Excel Template for P, NP, C, & U Control Charts ENGM 720: Statistical Process Control

4. Statistical Quality Control and Improvement Improving Process Capability and Performance Continually Improve the System Characterize Stable Process Capability Head Off Shifts in Location, Spread Time Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability Center the Process LSL 0 USL Process for Statistical Control Of Quality • Removing special causes of variation • Hypothesis Tests • Ishikawa’s Tools • Managing the process with control charts • Process Improvement • Process Stabilization • Confidence in “When to Act” ENGM 720: Statistical Process Control

5. Review • Shewhart Control charts • Are like a sideways hypothesis test (2-sided!) from a Normal distribution • UCL is like the right / upper critical region • CL is like the central location • LCL is like the left / lower critical region • When working with continuous variables, we use two charts: • X-bar for testing for change in location • R or s-chart for testing for change in spread • We check the charts using 4 Western Electric rules ENGM 720: Statistical Process Control

6. Continuous Probability of a range of outcomes is area under PDF (integration) Discrete Probability of a range of outcomes is area under PDF (sum of discrete outcomes) 35.0  2.5 35.0  2.5 30.4 (-3) 34.8 (-) 39.2 (+) 43.6 (+3) 30 34 38 42 32.6 (-2) 37 () 41.4 (+2) 32 36 () 40 Continuous & Discrete Distributions ENGM 720: Statistical Process Control

7. Discrete Distribution Example • Sum of two six-sided dice: • Outcomes range from 2 to 12. • Count the possible ways to obtain each individual sum - forms a histogram • What is the most frequently occurring sum that you could roll? • Most likely outcome is a sum of 7 (there are 6 ways to obtain it) • What is the probability of obtaining the most likely sum in a single roll of the dice? • 6  36 = .167 • What is the probability of obtaining a sum greater than 2 and less than 11? • 32  36 = .889 ENGM 720: Statistical Process Control

8. Continuous & Attribute Variables • Continuous Variables: • Take on a continuum of values. • Ex.: length, diameter, thickness • Modeled by the Normal Distribution • Attribute Variables: • Take on discrete values • Ex.: present/absent, conforming/non-conforming • Modeled by Binomial Distribution if classifying inspection units into defectives • (defective inspection unit can have multiple defects) • Modeled by Poisson Distribution if counting defects occurring within an inspection unit ENGM 720: Statistical Process Control

9. Discrete Variables Classes • Defectives • The presence of a non-conformity ruins the entire unit – the unit is defective • Example – fuses with disconnects • Defects • The presence of one or more non-conformities may lower the value of the unit, but does NOT render the entire unit defective • Example – paneling with scratches ENGM 720: Statistical Process Control

10. Binomial Distribution • Sequence of n trials • Outcome of each trial is “success” or “failure” • Probability of success = p • r.v. X - number of successes in n trials • So: where • Mean: Variance: ENGM 720: Statistical Process Control

11. Binomial Distribution Example • A lot of size 30 contains three defective fuses. • What is the probability that a sample of five fuses selected at random contains exactly one defective fuse? • What is the probability that it contains one or more defectives? ENGM 720: Statistical Process Control

12. Poisson Distribution • Let X be the number of times that a certain event occurs per unit of length, area, volume, or time • So: where x = 0, 1, 2, … • Mean: Variance: ENGM 720: Statistical Process Control

13. Poisson Distribution Example • A sheet of 4’x8’ paneling (= 4608 in2) has 22 scratches. • What is the expected number of scratches if checking only one square inch (randomly selected)? • What is the probability of finding at least two scratches in 25 in2? ENGM 720: Statistical Process Control

14. UCL 0 CL LCL 0 Sample Number 2-Sided Hypothesis Test Sideways Hypothesis Test Shewhart Control Chart  2  2  2  2 Moving from Hypothesis Testing to Control Charts • Attribute control charts are also like a sideways hypothesis test • Detects a shift in the process • Heads-off costly errors by detecting trends – if constant control limits are used ENGM 720: Statistical Process Control

15. Sample Control Limits: Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard: P-Charts • Tracks proportion defective in a sample of insp. units • Can have a constant number of inspection units in the sample ENGM 720: Statistical Process Control

16. Mean Sample Size Limits: Approximate 3σ limits are found from sample mean: Variable Width Limits: Approximate 3σ limits vary with individual sample size: P-Charts (continued) • More commonly has variable number of inspection units • Can’t use run rules with variable control limits ENGM 720: Statistical Process Control

17. Sample Control Limits: Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard: NP-Charts • Tracks number of defectives in a sample of insp. units • Must have a constant number of inspection units in each sample • Use of run rules is allowed if LCL > 0 - adds power ! ENGM 720: Statistical Process Control

18. Sample Control Limits: Approximate 3σ limits are found from trial samples: Standard Control Limits: Approximate 3σ limits continue from standard: C-Charts • Tracks number of defects in a logical inspection unit • Must have a constant size inspection unit containing the defects • Use of run rules is allowed if LCL > 0 - adds power ! ENGM 720: Statistical Process Control

19. Mean Sample Size Limits: Approximate 3σ limits are found from sample mean: Variable Width Limits: Approximate 3σ limits vary with individual sample size: U-Charts • Number of defects occurring in variably sized inspection unit • (Ex. Solder defects per 100 joints - 350 joints in board = 3.5 insp. units) • Can’t use run rules with variable control limits, watch clustering! ENGM 720: Statistical Process Control

20. Steps for Trial Control Limits • Start with 20 to 25 samples • Use all data to calculate initial control limits • Plot each sample in time-order on chart. • Check for out of control sample points • If one (or more) found, then: • Investigate the process; • Remove the special cause; and • Remove the special cause point and recalculate control limits. • If can’t find special cause - drop point & recalculate anyway ENGM 720: Statistical Process Control

21. Continuous Variable Charts Smaller changes detected faster Apply to attributes data as well (by CLT)* Require smaller sample sizes Attribute Charts Can cover several defects with one chart Less costly inspection Summary of Control Charts • Use of the control chart decision table. ENGM 720: Statistical Process Control

22. Control Chart Decision Table Is the size of the inspection sample fixed? Defective Units (possibly with multiple defects) Binomial Distribution Use p-Chart No, varies Use np-Chart Yes, constant Is the size of the inspection unit fixed? What is the inspection basis? Individual Defects Poisson Distribution Use c-Chart Discrete Attribute Yes, constant Kind of inspection variable? Use u-Chart No, varies Which spread method preferred? Range Use X-bar and R-Chart Continuous Variable Standard Deviation Use X-bar and S-Chart ENGM 720: Statistical Process Control

23. Control Chart Sensitizing Rules • Western Electric Rules: • One point plots outside the three-sigma limits; • Eight consecutive points plot on one side of the center line (run rule!); • Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line(zone rule!); • Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line(zone rule!). • If chart shows lack of control, investigate for special cause ENGM 720: Statistical Process Control

24. Attribute Chart Applications • Attribute control charts apply to “service” applications, too. • Number of incorrect invoices per customer • Proportion of incorrect orders taken in a day • Number of return service calls to resolve problem ENGM 720: Statistical Process Control

25. Questions & Issues ENGM 720: Statistical Process Control