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2006 Prentice Hall, Inc. S6 1
OperationsManagement
Supplement 6
Statistical Process Control
2006 Prentice Hall, Inc.
PowerPoint presentation to accompanyHeizer/RenderPrinciples of Operations Management, 6eOperations Management, 8e
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Variability is inherent in every process
Natural or common causesSpecial or assignable causes
Provides a statistical signal when
assignable causes are present Detect and eliminate assignable
causes of variation
Statistical Process Control
(SPC)
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Natural Variations
Natural variations in the productionprocess
These are to be expected
Output measures follow a probabilitydistribution
For any distribution there is a measure
of central tendency and dispersion
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Assignable Variations
Variations that can be traced to a specificreason (machine wear, misadjustedequipment, fatigued or untrained workers)
The objective is to discover whenassignable causes are present andeliminate them
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Samples
To measure the process, we take samplesand analyze the sample statistics followingthese steps
(a) Samples of theproduct, say fiveboxes of cerealtaken off the filling
machine line, varyfrom each other inweight
Freque
ncy
Weight
#
#
# #
##
#
#
#
# # ## #
##
# # ## #
## # ##
Each of theserepresents onesample of five
boxes of cereal
Figure S6.1
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Samples
(b) After enoughsamples aretaken from astable process,
they form apattern called adistribution
The solid linerepresents the
distribution
Freque
ncy
WeightFigure S6.1
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Samples
(c) There are many types of distributions, includingthe normal (bell-shaped) distribution, butdistributions do differ in terms of central
tendency (mean), standard deviation orvariance, and shape
Weight
Central tendency
Weight
Variation
Weight
Shape
Frequ
ency
Figure S6.1
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Samples
(d) If only naturalcauses ofvariation arepresent, theoutput of aprocess forms adistribution that
is stable overtime and ispredictable
Weight
Frequency
Prediction
Figure S6.1
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Samples
(e) If assignablecauses arepresent, theprocess output isnot stable overtime and is notpredicable
Weight
Freq
uency
Prediction
????
??
????
?????????
Figure S6.1
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Control Charts
Constructed from historical data, thepurpose of control charts is to helpdistinguish between natural variationsand variations due to assignablecauses
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Types of Data
Characteristics thatcan take any real
value
May be in whole orin fractional
numbers Continuous random
variables
Variables Attributes
Defect-relatedcharacteristics
Classify productsas either good orbad or count
defects Categorical or
discrete randomvariables
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Control Charts for Variables
For variables that have continuousdimensions
Weight, speed, length, strength, etc.
x-charts are to control the centraltendency of the process
R-charts are to control the dispersion ofthe process
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Setting Chart Limits
For x-Charts when we know
Upper control limit(UCL) = x + z x
Lower control limit(LCL) = x - z x
where x = mean of the sample means or a targetvalue set for the process
z = number of normal standard deviationsx = standard deviation of the sample means
= / n
= population standard deviation
n = sample size
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Setting Control Limits
Hour 1
Sample Weight ofNumber Oat Flakes
1 17
2 133 16
4 18
5 17
6 16
7 15
8 17
9 16
Mean 16.1
= 1
Hour Mean Hour Mean
1 16.1 7 15.2
2 16.8 8 16.4
3 15.5 9 16.3
4 16.5 10 14.8
5 16.5 11 14.2
6 16.4 12 17.3n = 9
LCLx= x - z x=16 - 3(1/3) = 15 ozs
For99.73% control limits, z= 3
UCLx= x + z x= 16 + 3(1/3) = 17 ozs
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17 = UCL
15 = LCL
16 = Mean
Setting Control Limits
Control Chartfor sample of9 boxes
Sample number
| | | | | | | | | | | |1 2 3 4 5 6 7 8 9 10 11 12
Variation dueto assignable
causes
Variation dueto assignable
causes
Variation due tonatural causes
Out ofcontrol
Out ofcontrol
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Setting Chart Limits
For x-Charts when we dont know
Lower control limit(LCL) = x - A2R
Upper control limit(UCL) = x + A2R
where R = average range of the samples
A2 = control chart factor found in Table S6.1
x = mean of the sample means
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Control Chart Factors
Table S6.1
Sample Size Mean Factor Upper Range Lower Rangen A2 D4 D3
2 1.880 3.268 0
3 1.023 2.574 0
4 .729 2.282 0
5 .577 2.115 0
6 .483 2.004 0
7 .419 1.924 0.076
8 .373 1.864 0.1369 .337 1.816 0.184
10 .308 1.777 0.223
12 .266 1.716 0.284
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Setting Control Limits
Process average x= 16.01 ouncesAverage range R= .25Sample size n= 5
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Setting Control Limits
UCLx = x + A2R= 16.01 + (.577)(.25)= 16.01 + .144= 16.154 ounces
Process average x= 16.01 ouncesAverage range R= .25Sample size n= 5
FromTable S6.1
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Setting Control Limits
UCLx = x + A2R= 16.01 + (.577)(.25)= 16.01 + .144= 16.154 ounces
LCLx = x - A2R= 16.01 - .144= 15.866 ounces
Process average x= 16.01 ouncesAverage range R= .25Sample size n= 5
UCL = 16.154
Mean = 16.01
LCL = 15.866
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R Chart
Type of variables control chart
Shows sample ranges over time
Difference between smallest andlargest values in sample
Monitors process variability
Independent from process mean
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Setting Chart Limits
For R-Charts
Lower control limit(LCLR) = D3R
Upper control limit(UCLR) = D4R
where
R = average range of the samples
D3 and D4 = control chart factors from Table S6.1
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Setting Control Limits
UCLR = D4R= (2.115)(5.3)= 11.2 pounds
LCLR = D3R= (0)(5.3)= 0 pounds
Average range R= 5.3 poundsSample size n= 5FromTable S6.1 D4= 2.115, D3 = 0
UCL = 11.2
Mean = 5.3
LCL = 0
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Mean and Range Charts
(a)
Thesesamplingdistributionsresult in the
charts below
(Sampling mean isshifting upward butrange is consistent)
R-chart(R-chart does notdetect change inmean)
UCL
LCL
Figure S6.5
x-chart(x-chart detectsshift in centraltendency)
UCL
LCL
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Mean and Range Charts
R-chart(R-chart detectsincrease indispersion)
UCL
LCL
Figure S6.5
(b)
Thesesamplingdistributionsresult in the
charts below
(Sampling meanis constant butdispersion isincreasing)
x-chart(x-chart does notdetect the increasein dispersion)
UCL
LCL
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Automated Control Charts
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Control Charts for Attributes
For variables that are categorical
Good/bad, yes/no,
acceptable/unacceptable Measurement is typically counting
defectives
Charts may measurePercent defective (p-chart)
Number of defects (c-chart)
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Control Limits for p-Charts
Population will be a binomial distribution,but applying the Central Limit Theorem
allows us to assume a normal distribution
for the sample statistics
UCLp= p + z p^
LCLp= p - z p^
where p = mean fraction defective in the sample
z = number of standard deviations
p = standard deviation of the sampling distribution
n = sample size
^
p(1 - p)np
=^
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p-Chart for Data EntrySample Number Fraction Sample Number FractionNumber of Errors Defective Number of Errors Defective
1 6 .06 11 6 .062 5 .05 12 1 .013 0 .00 13 8 .08
4 1 .01 14 7 .075 4 .04 15 5 .056 2 .02 16 4 .047 5 .05 17 11 .118 3 .03 18 3 .03
9 3 .03 19 0 .0010 2 .02 20 4 .04
Total= 80
(.04)(1 - .04)
100p= = .02^
p= = .0480
(100)(20)
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.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
Sample number
Fractiondefective
| | | | | | | | | |
2 4 6 8 10 12 14 16 18 20
p-Chart for Data Entry
UCLp= p + z p= .04 + 3(.02) = .10^
LCLp= p - z p= .04 - 3(.02) = 0^
UCLp= 0.10
LCLp= 0.00
p= 0.04
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.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
Sample number
Fractiondefective
| | | | | | | | | |
2 4 6 8 10 12 14 16 18 20
UCLp= p + z p= .04 + 3(.02) = .10^
LCLp= p - z p= .04 - 3(.02) = 0^
UCLp= 0.10
LCLp= 0.00
p= 0.04
p-Chart for Data Entry
Possibleassignable
causes present
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Control Limits for c-Charts
Population will be a Poisson distribution,but applying the Central Limit Theorem
allows us to assume a normal distribution
for the sample statistics
where c = mean number defective in the sample
UCLc= c +3 c LCLc= c -3 c
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c-Chart for Cab Company
c= 54 complaints/9 days= 6 complaints/day
|1
|2
|3
|4
|5
|6
|7
|8
|9
Day
Numberdefective14
12
10
8
6
42
0
UCLc = c +3 c
= 6 + 3 6= 13.35
LCLc = c -3 c= 3 - 3 6= 0
UCLc= 13.35
LCLc= 0
c= 6
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Patterns in Control Charts
Normal behavior.Process is in control.
Upper control limit
Target
Lower control limit
Figure S6.7
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Upper control limit
Target
Lower control limit
Patterns in Control Charts
One plot out above (orbelow). Investigate forcause. Process is out
of control.
Figure S6.7
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Upper control limit
Target
Lower control limit
Patterns in Control Charts
Trends in eitherdirection, 5 plots.Investigate for cause of
progressive change.
Figure S6.7
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Upper control limit
Target
Lower control limit
Patterns in Control Charts
Two plots very nearlower (or upper)control. Investigate for
cause.
Figure S6.7
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Upper control limit
Target
Lower control limit
Patterns in Control Charts
Run of 5 above (orbelow) central line.Investigate for cause.Figure S6.7
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Upper control limit
Target
Lower control limit
Patterns in Control Charts
Erratic behavior.Investigate.
Figure S6.7
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Which Control Chart to Use
Using an x-chart and R-chart:
Observations are variables
Collect20 - 25 samples of n= 4, or n=5, or more, each from a stable processand compute the mean for the x-chart
and range for the R-chartTrack samples of n observations each
Variables Data
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Which Control Chart to Use
Using the p-chart:
Observations are attributes that canbe categorized in two states
We deal with fraction, proportion, orpercent defectives
Have several samples, each withmany observations
Attribute Data
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Which Control Chart to Use
Using a c-Chart:
Observations are attributes whosedefects per unit of output can becounted
The number counted is often a smallpart of the possible occurrences
Defects such as number of blemisheson a desk, number of typos in a pageof text, flaws in a bolt of cloth
Attribute Data