control chart u np c p
TRANSCRIPT
![Page 1: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/1.jpg)
Chapter 7. Control Charts for Attributes
![Page 2: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/2.jpg)
Control Chart for Fraction Nonconformingg
Fraction nonconforming is based on the binomial distribution.n: size of populationp pp: probability of nonconformanceD: number of products not conformingSuccessive products are independent.
Mean of D = npVariance of D = np(1-p)
![Page 3: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/3.jpg)
Sample fraction nonconformance
ˆMean of p:Mean of p:
ˆVariance of p:
![Page 4: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/4.jpg)
w: statistics for qualityMean of w: µwMean of w: µwVariance of w: σw
2
L: distance of control limit from center line (in standard deviation units)
If p is the true fraction nonconformance:
![Page 5: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/5.jpg)
If p is not know, we estimate it from samples.m: samples, each with n units (or observations)D : number of nonconforming units in sample iDi: number of nonconforming units in sample i
Average of all observations:
![Page 6: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/6.jpg)
![Page 7: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/7.jpg)
Example 6-1. 6-oz cardboard cans of orange juice
![Page 8: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/8.jpg)
Design of Fraction Nonconforming ChartDesign of Fraction Nonconforming Chart
Three parameters to be specified:p p1. sample size2. frequency of sampling3 width of control limits3. 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.
![Page 9: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/9.jpg)
np Control Chart
![Page 10: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/10.jpg)
Variable Sample Size
Variable-Width Control Limits
( ) ( )( ) ( )1- 1-UCL 3 LCL 3
i i
p p p pp p
n n= + = −
![Page 11: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/11.jpg)
![Page 12: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/12.jpg)
Variable Sample SizeControl Limits Based on an Average Sample Size
U l i F i lUse average sample size. For previous example:
![Page 13: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/13.jpg)
![Page 14: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/14.jpg)
Variable Sample Size
Standard Control Chart- Points are plotted in standard deviation units.
UCL = 3UCL = 3Center line = 0LCL = -3
![Page 15: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/15.jpg)
![Page 16: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/16.jpg)
Operating Characteristic Function andOperating Characteristic Function and Average Run Length Calculations
Probability of type II error
{ } { }{ } { }
ˆ ˆUCL | LCL |
UCL | LCL |
P p p P p p
P D n p P D n p
β = < − ≤
< ≤{ } { } UCL | LCL |P D n p P D n p= < − ≤
![Page 17: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/17.jpg)
C t l Ch t f N f iti ( D f t )Control Charts for Nonconformities (or Defects)
Procedures with Constant Sample Sizex: number of nonconformitiesc > 0: parameter of Poisson distribution
Set to zero if negative
![Page 18: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/18.jpg)
If no standard is given, estimate c then use the following parameters:
Set to zero if negativeg
![Page 19: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/19.jpg)
![Page 20: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/20.jpg)
![Page 21: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/21.jpg)
Choice of Sample Size: µ Chartx: total nonconformities in n inspection unitsu: average number of nonconformities per inspection unit
: obserd average number of nonconformities per inspection unitu: obserd average number of nonconformities per inspection unitu
![Page 22: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/22.jpg)
![Page 23: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/23.jpg)
Control Charts for NonconformitiesProcedure with Variable Sample SizeProcedure with Variable Sample Size
![Page 24: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/24.jpg)
![Page 25: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/25.jpg)
Control Charts for NonconformitiesDemerit Systems: not all defects are of equal importance
![Page 26: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/26.jpg)
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 weightsConstants 100, 50, 10, and 1 are demerit weights.
: inspection unitsb f d it it
n: number of demerits per unit
where
in
i i
uDu D d= =∑
1i i
in =∑
![Page 27: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/27.jpg)
µi: linear combination of independent Poisson variables
is average number of Class A defects per unit, etc.Aµ
![Page 28: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/28.jpg)
Control Charts for Nonconformities
Operating Characteristic Functionx: Poisson random variablec: true mean valueβ: type II error probabilityβ ype e o p obab y
![Page 29: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/29.jpg)
For example 6-3
Number of nonconformities is integer.
![Page 30: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/30.jpg)
![Page 31: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/31.jpg)
Control Charts for Nonconformities
• If defect level is low, <1000 per million, c and u charts become ineffective
Dealing with Low Defect Levels
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 p y pdistribution.
• 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• To use normal approximation, translate exponential distribution to Weibull distribution and then approximate with normal variable
: normal approximation for exponential variable x y1
0.27773.6 x y y= =
![Page 32: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/32.jpg)
![Page 33: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/33.jpg)
![Page 34: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/34.jpg)
![Page 35: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/35.jpg)
Guidelines for Implementing Control Charts
Applicable for both variable and attribute control
![Page 36: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/36.jpg)
Determining Which Characteristics and Where to Put Control ChartsWhere to Put Control Charts
![Page 37: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/37.jpg)
![Page 38: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/38.jpg)
Choosing Proper Type of Control Chart
![Page 39: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/39.jpg)
![Page 40: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/40.jpg)
![Page 41: control Chart u np c p](https://reader034.vdocument.in/reader034/viewer/2022052522/554a08a7b4c9055b7a8b57e4/html5/thumbnails/41.jpg)
Actions Taken to Improve Process