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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

IENG 486 - Lecture 16

P, NP, C, & U Control Charts

(Attributes Charts)

IENG 486: Statistical Quality & Process Control

assignment
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

process for statistical control of quality

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

review
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

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

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

continuous attribute variables
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

discrete variables classes
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

binomial distribution
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

binomial distribution example
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

poisson distribution
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

poisson distribution example
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

moving from hypothesis testing to control charts

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

p charts
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

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:

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

np charts
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

c charts
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

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:

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

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

Summary of Control Charts
  • Use of the control chart decision tree…

IENG 486: Statistical Quality & Process Control

control chart decision tree
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

attribute chart applications
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

questions issues
Questions & Issues

IENG 486: Statistical Quality & Process Control