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IENG 486 - Lecture 16

IENG 486 - Lecture 16. P, NP, C, & U Control Charts (Attributes Charts). Assignment:. Reading: Chapter 3.5 Chapter 7 Sections 7.1 – 7.2.2: pp. 288 – 304 Sections 7.3 – 7.3.2: pp. 308 - 321 Chapter 6.4: pp. 259 - 265 Chapter 9 Sections 9.1 – 9.1.5: pp. 399 - 410

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IENG 486 - Lecture 16

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  1. IENG 486 - Lecture 16 P, NP, C, & U Control Charts (Attributes Charts) IENG 486: Statistical Quality & Process Control

  2. Assignment: • Reading: • Chapter 3.5 • Chapter 7 • Sections 7.1 – 7.2.2: pp. 288 – 304 • Sections 7.3 – 7.3.2: pp. 308 - 321 • Chapter 6.4: pp. 259 - 265 • Chapter 9 • Sections 9.1 – 9.1.5: pp. 399 - 410 • Sections 9.2 – 9.2.4: pp. 419 - 425 • Sections 9.3: pp. 428 - 430 • Assignment: • CH7 # 6; 11; 27a,b; 31; 47 • Access Excel Template for P, NP, C, & U Control Charts IENG 486: Statistical Quality & Process Control

  3. 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” IENG 486: Statistical Quality & Process Control

  4. 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 IENG 486: Statistical Quality & Process Control

  5. 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 Continuous & Discrete Distributions 30.4 (-3) 34.8 (-) 39.2 (+) 43.6 (+3) 30 34 38 42 32.6 (-2) 37 () 41.4 (+2) 32 36 () 40 IENG 486: Statistical Quality & Process Control

  6. 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 IENG 486: Statistical Quality & Process Control

  7. 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 IENG 486: Statistical Quality & Process Control

  8. 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: IENG 486: Statistical Quality & Process Control

  9. 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? IENG 486: Statistical Quality & Process Control

  10. 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: IENG 486: Statistical Quality & Process Control

  11. 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? IENG 486: Statistical Quality & Process Control

  12. 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 IENG 486: Statistical Quality & Process Control

  13. 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 IENG 486: Statistical Quality & Process Control

  14. 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 IENG 486: Statistical Quality & Process Control

  15. 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 ! IENG 486: Statistical Quality & Process Control

  16. 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 ! IENG 486: Statistical Quality & Process Control

  17. 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! IENG 486: Statistical Quality & Process Control

  18. Continuous Variable Charts Smaller changes detected faster Require smaller sample sizes Can be applied to attributes data as well (by CLT)* Attribute Charts Can cover several defects with one chart Less costly inspection Summary of Control Charts • Use of the control chart decision tree… IENG 486: Statistical Quality & Process Control

  19. Control Chart Decision Tree 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 What is the inspection basis? Is the size of the inspection unit fixed? 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 IENG 486: Statistical Quality & Process Control

  20. 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 IENG 486: Statistical Quality & Process Control

  21. Questions & Issues IENG 486: Statistical Quality & Process Control

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