1. 8/4/2015IENG 486: Statistical Quality & Process Control 1 IENG 486 - Lecture 16 P, NP, C, & U Control Charts (Attributes Charts)

2. 8/4/2015 IENG 486: Statistical Quality & Process Control2 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

3. 8/4/2015 IENG 486: Statistical Quality & Process Control3 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” Reduce Variability Identify Special Causes - Good (Incorporate) Improving Process Capability and Performance Characterize Stable Process Capability Head Off Shifts in Location, Spread Identify Special Causes - Bad (Remove) Continually Improve the System Statistical Quality Control and Improvement Time Center the Process LSL  0 USL

4. 8/4/2015 IENG 486: Statistical Quality & Process Control4 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

5. 8/4/2015 IENG 486: Statistical Quality & Process Control5 Continuous & Discrete Distributions  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 37 (  ) 41.4 (  +2  ) 32.6 (  -2  ) 43.6 (  +3  ) 30.4 (  -3  ) 39.2 (  +  ) 34.8 (  -  ) 35.0  2.5 36 (  ) 4032 42303834

6. 8/4/2015 IENG 486: Statistical Quality & Process Control6 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

7. 8/4/2015 IENG 486: Statistical Quality & Process Control7 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

8. 8/4/2015 IENG 486: Statistical Quality & Process Control8 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:

9. 8/4/2015 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?

10. 8/4/2015 IENG 486: Statistical Quality & Process Control10 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:

11. 8/4/2015 IENG 486: Statistical Quality & Process Control11 Poisson Distribution Example  A sheet of 4’x8’ paneling (= 4608 in 2 ) 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 in 2 ?

12. 8/4/2015 IENG 486: Statistical Quality & Process Control12 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 00 22 22 00 22 22 2-Sided Hypothesis TestShewhart Control ChartSideways Hypothesis Test CLCL LCL UCL Sample Number

13. 8/4/2015 IENG 486: Statistical Quality & Process Control13 P-Charts  Sample Control Limits: Approximate 3σ limits are found from trial samples:  Standard Control Limits: Approximate 3σ limits continue from standard:  Tracks proportion defective in a sample of insp. units  Can have a constant number of inspection units in the sample

14. 8/4/2015 IENG 486: Statistical Quality & Process Control14 P-Charts (continued)  Mean Sample Size Limits: Approximate 3σ limits are found from sample mean:  Variable Width Limits: Approximate 3σ limits vary with individual sample size:  More commonly has variable number of inspection units  Can’t use run rules with variable control limits

15. 8/4/2015 IENG 486: Statistical Quality & Process Control15 NP-Charts  Sample Control Limits: Approximate 3σ limits are found from trial samples:  Standard Control Limits: Approximate 3σ limits continue from standard:  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 !

16. 8/4/2015 IENG 486: Statistical Quality & Process Control16 C-Charts  Sample Control Limits: Approximate 3σ limits are found from trial samples:  Standard Control Limits: Approximate 3σ limits continue from standard:  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 !

17. 8/4/2015 IENG 486: Statistical Quality & Process Control17 U-Charts  Mean Sample Size Limits: Approximate 3σ limits are found from sample mean:  Variable Width Limits: Approximate 3σ limits vary with individual sample size:  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!

18. 8/4/2015 IENG 486: Statistical Quality & Process Control18 Summary of Control Charts  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  Use of the control chart decision tree…

19. 8/4/2015 IENG 486: Statistical Quality & Process Control19 Use p-Chart No, varies Yes, constant Use np-Chart Individual Defects Poisson Distribution Use c-Chart Use u-Chart No, varies Discrete Attribute What is the inspection basis? Is the size of the inspection unit fixed? Yes, constant Is the size of the inspection sample fixed? Continuous Variable Range Standard Deviation Which spread method preferred? Use X-bar and R-Chart Use X-bar and S-Chart Kind of inspection variable? Defective Units (possibly with multiple defects) Binomial Distribution Control Chart Decision Tree

20. 8/4/2015 IENG 486: Statistical Quality & Process Control20 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

21. 8/4/2015 IENG 486: Statistical Quality & Process Control21 Questions & Issues