Chapter 7. Control Charts for Attributes

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hapter 7. Control Charts for Attribute

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Chapter 7. Control Charts for Attributes. Control Chart for Fraction Nonconforming. Fraction nonconforming is based on the binomial distribution. n : size of population p : probability of nonconformance D : number of products not conforming Successive products are independent. - PowerPoint PPT Presentation

Transcript of Chapter 7. Control Charts for Attributes

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Chapter 7. Control Charts for Attributes

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Control Chart for Fraction Nonconforming

Fraction nonconforming is based on the binomial distribution.n: size of populationp: probability of nonconformanceD: number of products not conformingSuccessive products are independent.

Mean of D = npVariance of D = np(1-p)

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Sample fraction nonconformance

ˆMean of p:

ˆVariance of p:

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w: statistics for qualityMean of w: μw

Variance of w: σw2

L: distance of control limit from center line (in standard deviation units)

If p is the true fraction nonconformance:

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If p is not know, we estimate it from samples.m: samples, each with n units (or observations)Di: number of nonconforming units in sample i

Average of all observations:

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Example 6-1. 6-oz cardboard cans of orange juice

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If samples 15 and 23 are eliminated:

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Additional samples collected after adjustment of control chart:

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Control chart variables using only the recent 24 samples:

Set equal to zero for negative value

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Design of Fraction Nonconforming Chart

Three parameters to be specified:1. sample size

2. frequency of sampling

3. width of control limits

Common to base chart on 100% inspection of all process output over time.

Rational subgroups may also play role in determining sampling frequency.

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np Control Chart

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Variable Sample Size

Variable-Width Control Limits

1- 1-UCL 3 LCL 3

i i

p p p pp p

n n

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Variable Sample Size

Control Limits Based on an Average Sample Size

Use average sample size. For previous example:

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Variable Sample Size

Standard Control Chart - Points are plotted in standard deviation units. UCL = 3 Center line = 0 LCL = -3

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Skip Section 6.2.3 pages 284 - 285

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Operating Characteristic Function and Average Run Length Calculations

Probability of type II error

ˆ ˆUCL | process not incontrol LCL | process not incontrol

UCL | process not incontrol LCL | process not incontrol

P p P p

P D n P D n

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Since D is an integer,

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Average run length

If the process is in control:

If the process is out of control

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For Table 6-6: 50, UCL 0.3698, LCL 0.0303, center line 0.20.

If process is in control with , probability of point plotting in control = 0.9973.

1- 0.0027.

n p

p p

If process shifts out of control to 0.3, 0.8594.p

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Control Charts for Nonconformities (or Defects)

Procedures with Constant Sample Size x: number of nonconformities c > 0: parameter of Poisson distribution

Set to zero if negative

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Set to zero if negative

If no standard is given, estimate c then use the following parameters:

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There are 516 defects in total of 26 samples. Thus.

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Sample 6 was due to inspection error.Sample 20 was due to a problem in wave soldering machine. Eliminate these two samples, and recalculate the control parameters.

There are 516 defects in total of 26 samples. Thus.

New control limits:

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Additional samples collected.

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Further Analysis of Nonconformities

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Choice of Sample Size: μ Chart

x: total nonconformities in n inspection unitsu: average number of nonconformities per inspection unit

: observed average number of nonconformities per inspection unitu

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Control Charts for NonconformitiesProcedure with Variable Sample Size

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Control Charts for Nonconformities

Demerit Systems: not all defects are of equal importance

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ciA: number of Class A defects in ith inspection units Similarly for ciB, ciC, and ciD for Classes B, C, and D.di: number of demerits in inspection unit i

Constants 100, 50, 10, and 1 are demerit weights.

1

: inspection units

: number of demerits per unit

where

i

n

i ii

n

u

Du D d

n

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µi: linear combination of independent Poisson variables

is average number of Class A defects per unit, etc.A

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Control Charts for Nonconformities

Operating Characteristic Functionx: Poisson random variablec: true mean valueβ: type II error probability

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For example 6-3

Number of nonconformities is integer.

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• If defect level is low, <1000 per million, c and u charts become ineffective.

• The time-between-events control chart is more effective.• If the defects occur according to a Poisson distribution, the

probability distribution of the time between events is the exponential distribution.

• Constructing a time-between-events control chart is essentially equivalent to control charting an exponentially distributed variable.

• To use normal approximation, translate exponential distribution to Weibull distribution and then approximate with normal variable

Control Charts for Nonconformities

Dealing with Low Defect Levels

10.27773.6

: normal approximation for exponential variable

x y

x y y

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Guidelines for Implementing Control Charts

Applicable for both variable and attribute control

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Determining Which Characteristics and Where to Put Control Charts

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Choosing Proper Type of Control Chart

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Actions Taken to Improve Process