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© 2006 Prentice Hall, Inc. S6 – 1
Operations ManagementOperations ManagementSupplement 6 – Statistical Process ControlSupplement 6 – Statistical Process Control
© 2006 Prentice Hall, Inc.
PowerPoint presentation to accompanyPowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 6ePrinciples of Operations Management, 6eOperations Management, 8e Operations Management, 8e
© 2006 Prentice Hall, Inc. S6 – 2
Variability is inherent in every processVariability is inherent in every process Natural or common causesNatural or common causes
Special or assignable causesSpecial or assignable causes
Provides a statistical signal when Provides a statistical signal when assignable causes are presentassignable causes are present
Detect and eliminate assignable Detect and eliminate assignable causes of variationcauses of variation
Statistical Process Control Statistical Process Control (SPC)(SPC)
© 2006 Prentice Hall, Inc. S6 – 3
Natural VariationsNatural Variations Natural variations in the production Natural variations in the production
processprocess
These are to be expectedThese are to be expected
Output measures follow a probability Output measures follow a probability distributiondistribution
For any distribution there is a measure For any distribution there is a measure of central tendency and dispersionof central tendency and dispersion
© 2006 Prentice Hall, Inc. S6 – 4
Assignable VariationsAssignable Variations
Variations that can be traced to a specific Variations that can be traced to a specific reason (machine wear, misadjusted reason (machine wear, misadjusted equipment, fatigued or untrained workers)equipment, fatigued or untrained workers)
The objective is to discover when The objective is to discover when assignable causes are present and assignable causes are present and eliminate themeliminate them
© 2006 Prentice Hall, Inc. S6 – 5
SamplesSamples
To measure the process, we take samples To measure the process, we take samples and analyze the sample statistics following and analyze the sample statistics following these stepsthese steps
(a)(a) Samples of the Samples of the product, say five product, say five boxes of cereal boxes of cereal taken off the filling taken off the filling machine line, vary machine line, vary from each other in from each other in weightweight
Fre
qu
ency
Fre
qu
ency
WeightWeight
##
#### ##
####
####
##
## ## #### ## ####
## ## #### ## #### ## ####
Each of these Each of these represents one represents one sample of five sample of five
boxes of cerealboxes of cereal
Figure S6.1Figure S6.1
© 2006 Prentice Hall, Inc. S6 – 6
SamplesSamples
(b)(b) After enough After enough samples are samples are taken from a taken from a stable process, stable process, they form a they form a pattern called a pattern called a distributiondistribution
The solid line The solid line represents the represents the
distributiondistribution
Fre
qu
ency
Fre
qu
ency
WeightWeightFigure S6.1Figure S6.1
© 2006 Prentice Hall, Inc. S6 – 7
SamplesSamples
(c)(c) There are many types of distributions, including There are many types of distributions, including the normal (bell-shaped) distribution, but the normal (bell-shaped) distribution, but distributions do differ in terms of central distributions do differ in terms of central tendency (mean), standard deviation or tendency (mean), standard deviation or variance, and shapevariance, and shape
WeightWeight
Central tendencyCentral tendency
WeightWeight
VariationVariation
WeightWeight
ShapeShape
Fre
qu
ency
Fre
qu
ency
Figure S6.1Figure S6.1
© 2006 Prentice Hall, Inc. S6 – 8
SamplesSamples
(d)(d) If only natural If only natural causes of causes of variation are variation are present, the present, the output of a output of a process forms a process forms a distribution that distribution that is stable over is stable over time and is time and is predictablepredictable
WeightWeightTimeTimeF
req
uen
cyF
req
uen
cy PredictionPrediction
Figure S6.1Figure S6.1
© 2006 Prentice Hall, Inc. S6 – 9
SamplesSamples
(e)(e) If assignable If assignable causes are causes are present, the present, the process output is process output is not stable over not stable over time and is not time and is not predicablepredicable
WeightWeightTimeTimeF
req
uen
cyF
req
uen
cy PredictionPrediction
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??????
??????
????????????
??????
Figure S6.1Figure S6.1
© 2006 Prentice Hall, Inc. S6 – 10
Control ChartsControl Charts
Constructed from historical data, the Constructed from historical data, the purpose of control charts is to help purpose of control charts is to help distinguish between natural variations distinguish between natural variations and variations due to assignable and variations due to assignable causescauses
© 2006 Prentice Hall, Inc. S6 – 11
Types of DataTypes of Data
Characteristics that Characteristics that can take any real can take any real valuevalue
May be in whole or May be in whole or in fractional in fractional numbersnumbers
Continuous random Continuous random variablesvariables
VariablesVariables AttributesAttributes Defect-related Defect-related
characteristics characteristics
Classify products Classify products as either good or as either good or bad or count bad or count defectsdefects
Categorical or Categorical or discrete random discrete random variablesvariables
© 2006 Prentice Hall, Inc. S6 – 12
Control Charts for VariablesControl Charts for Variables
For variables that have continuous For variables that have continuous dimensionsdimensions Weight, speed, length, strength, etc.Weight, speed, length, strength, etc.
x-charts are to control the central x-charts are to control the central tendency of the processtendency of the process
R-charts are to control the dispersion of R-charts are to control the dispersion of the processthe process
© 2006 Prentice Hall, Inc. S6 – 13
Setting Chart LimitsSetting Chart Limits
For x-Charts when we know For x-Charts when we know
Upper control limit Upper control limit (UCL)(UCL) = x + z = x + zxx
Lower control limit Lower control limit (LCL)(LCL) = x - z = x - zxx
wherewhere xx ==mean of the sample means or mean of the sample means or a target value set for the processa target value set for the process
zz ==number of normal standard number of normal standard deviationsdeviations
xx ==standard deviation of the standard deviation of the sample meanssample means
==/ n/ n
==population standard population standard deviationdeviation
nn ==sample sizesample size
© 2006 Prentice Hall, Inc. S6 – 14
Setting Control LimitsSetting Control LimitsHour 1Hour 1
SampleSample Weight ofWeight ofNumberNumber Oat FlakesOat Flakes
11 1717
22 1313
33 1616
44 1818
55 1717
66 1616
77 1515
88 1717
99 1616
MeanMean 16.116.1
== 11
HourHour MeanMean HourHour MeanMean
11 16.116.1 77 15.215.2
22 16.816.8 88 16.416.4
33 15.515.5 99 16.316.3
44 16.516.5 1010 14.814.8
55 16.516.5 1111 14.214.2
66 16.416.4 1212 17.317.3n = 9n = 9
LCLLCLxx = x - z = x - zxx = = 16 - 3(1/3) = 15 ozs16 - 3(1/3) = 15 ozs
For For 99.73%99.73% control limits, z control limits, z = 3= 3
UCLUCLxx = x + z = x + zxx = 16 + 3(1/3) = 17 ozs= 16 + 3(1/3) = 17 ozs
© 2006 Prentice Hall, Inc. S6 – 15
17 = UCL17 = UCL
15 = LCL15 = LCL
16 = Mean16 = Mean
Setting Control LimitsSetting Control Limits
Control Chart Control Chart for sample of for sample of 9 boxes9 boxes
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
Variation due Variation due to assignable to assignable
causescauses
Variation due Variation due to assignable to assignable
causescauses
Variation due to Variation due to natural causesnatural causes
Out of Out of controlcontrol
Out of Out of controlcontrol
© 2006 Prentice Hall, Inc. S6 – 16
Setting Chart LimitsSetting Chart Limits
For x-Charts when we don’t know For x-Charts when we don’t know
Lower control limit Lower control limit (LCL)(LCL) = x - A = x - A22RR
Upper control limit Upper control limit (UCL)(UCL) = x + A = x + A22RR
wherewhere RR ==average range of the samplesaverage range of the samples
AA22 ==control chart factor found in control chart factor found in Table S6.1 Table S6.1
xx ==mean of the sample meansmean of the sample means
© 2006 Prentice Hall, Inc. S6 – 17
Control Chart FactorsControl Chart Factors
Table S6.1Table S6.1
Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.8801.880 3.2683.268 00
33 1.0231.023 2.5742.574 00
44 .729.729 2.2822.282 00
55 .577.577 2.1152.115 00
66 .483.483 2.0042.004 00
77 .419.419 1.9241.924 0.0760.076
88 .373.373 1.8641.864 0.1360.136
99 .337.337 1.8161.816 0.1840.184
1010 .308.308 1.7771.777 0.2230.223
1212 .266.266 1.7161.716 0.2840.284
© 2006 Prentice Hall, Inc. S6 – 18
Setting Control LimitsSetting Control Limits
Process average x Process average x = 16.01= 16.01 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
© 2006 Prentice Hall, Inc. S6 – 19
Setting Control LimitsSetting Control Limits
UCLUCLxx = x + A= x + A22RR
= 16.01 + (.577)(.25)= 16.01 + (.577)(.25)= 16.01 + .144= 16.01 + .144= 16.154 = 16.154 ouncesounces
Process average x Process average x = 16.01= 16.01 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
From From Table S6.1Table S6.1
© 2006 Prentice Hall, Inc. S6 – 20
Setting Control LimitsSetting Control Limits
UCLUCLxx = x + A= x + A22RR
= 16.01 + (.577)(.25)= 16.01 + (.577)(.25)= 16.01 + .144= 16.01 + .144= 16.154 = 16.154 ouncesounces
LCLLCLxx = x - A= x - A22RR
= 16.01 - .144= 16.01 - .144= 15.866 = 15.866 ouncesounces
Process average x Process average x = 16.01= 16.01 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
UCL = 16.154UCL = 16.154
Mean = 16.01Mean = 16.01
LCL = 15.866LCL = 15.866
© 2006 Prentice Hall, Inc. S6 – 21
R – ChartR – Chart
Type of variables control chartType of variables control chart Shows sample ranges over timeShows sample ranges over time
Difference between smallest and Difference between smallest and largest values in samplelargest values in sample
Monitors process variabilityMonitors process variability Independent from process meanIndependent from process mean
© 2006 Prentice Hall, Inc. S6 – 22
Setting Chart LimitsSetting Chart Limits
For R-ChartsFor R-Charts
Lower control limit Lower control limit (LCL(LCLRR)) = D = D33RR
Upper control limit Upper control limit (UCL(UCLRR)) = D = D44RR
wherewhere
RR ==average range of the samplesaverage range of the samples
DD33 and D and D44==control chart factors from control chart factors from Table S6.1 Table S6.1
© 2006 Prentice Hall, Inc. S6 – 23
Setting Control LimitsSetting Control Limits
UCLUCLRR = D= D44RR
= (2.115)(5.3)= (2.115)(5.3)= 11.2 = 11.2 poundspounds
LCLLCLRR = D= D33RR
= (0)(5.3)= (0)(5.3)= 0 = 0 poundspounds
Average range R Average range R = 5.3 = 5.3 poundspoundsSample size n Sample size n = 5= 5From From Table S6.1Table S6.1 D D44 = 2.115, = 2.115, DD33 = 0 = 0
UCL = 11.2UCL = 11.2
Mean = 5.3Mean = 5.3
LCL = 0LCL = 0
© 2006 Prentice Hall, Inc. S6 – 24
Mean and Range ChartsMean and Range Charts
(a)(a)
These These sampling sampling distributions distributions result in the result in the charts belowcharts below
(Sampling mean is (Sampling mean is shifting upward but shifting upward but range is consistent)range is consistent)
R-chartR-chart(R-chart does not (R-chart does not detect change in detect change in mean)mean)
UCLUCL
LCLLCL
Figure S6.5Figure S6.5
x-chartx-chart(x-chart detects (x-chart detects shift in central shift in central tendency)tendency)
UCLUCL
LCLLCL
© 2006 Prentice Hall, Inc. S6 – 25
Mean and Range ChartsMean and Range Charts
R-chartR-chart(R-chart detects (R-chart detects increase in increase in dispersion)dispersion)
UCLUCL
LCLLCL
Figure S6.5Figure S6.5
(b)(b)
These These sampling sampling distributions distributions result in the result in the charts belowcharts below
(Sampling mean (Sampling mean is constant but is constant but dispersion is dispersion is increasing)increasing)
x-chartx-chart(x-chart does not (x-chart does not detect the increase detect the increase in dispersion)in dispersion)
UCLUCL
LCLLCL
© 2006 Prentice Hall, Inc. S6 – 27
Control Charts for AttributesControl Charts for Attributes
For variables that are categoricalFor variables that are categorical Good/bad, yes/no, Good/bad, yes/no,
acceptable/unacceptableacceptable/unacceptable
Measurement is typically counting Measurement is typically counting defectivesdefectives
Charts may measureCharts may measure Percent defective (p-chart)Percent defective (p-chart)
Number of defects (c-chart)Number of defects (c-chart)
© 2006 Prentice Hall, Inc. S6 – 28
Control Limits for p-ChartsControl Limits for p-Charts
Population will be a binomial distribution, Population will be a binomial distribution, but applying the Central Limit Theorem but applying the Central Limit Theorem
allows us to assume a normal distribution allows us to assume a normal distribution for the sample statisticsfor the sample statistics
UCLUCLpp = p + z = p + zpp^̂
LCLLCLpp = p - z = p - zpp^̂
wherewhere pp ==mean fraction defective in the samplemean fraction defective in the samplezz ==number of standard deviationsnumber of standard deviationspp ==standard deviation of the sampling distributionstandard deviation of the sampling distribution
nn ==sample sizesample size
^̂
pp(1 -(1 - p p))nn
pp = =^̂
© 2006 Prentice Hall, Inc. S6 – 29
p-Chart for Data Entryp-Chart for Data EntrySampleSample NumberNumber FractionFraction SampleSample NumberNumber FractionFractionNumberNumber of Errorsof Errors DefectiveDefective NumberNumber of Errorsof Errors DefectiveDefective
11 66 .06.06 1111 66 .06.0622 55 .05.05 1212 11 .01.0133 00 .00.00 1313 88 .08.0844 11 .01.01 1414 77 .07.0755 44 .04.04 1515 55 .05.0566 22 .02.02 1616 44 .04.0477 55 .05.05 1717 1111 .11.1188 33 .03.03 1818 33 .03.0399 33 .03.03 1919 00 .00.00
1010 22 .02.02 2020 44 .04.04
Total Total = 80= 80
(.04)(1 - .04)(.04)(1 - .04)
100100pp = = = .02= .02^̂p p = = .04= = .04
8080
(100)(20)(100)(20)
© 2006 Prentice Hall, Inc. S6 – 30
.11 .11 –
.10 .10 –
.09 .09 –
.08 .08 –
.07 .07 –
.06 .06 –
.05 .05 –
.04 .04 –
.03 .03 –
.02 .02 –
.01 .01 –
.00 .00 –
Sample numberSample number
Fra
ctio
n d
efec
tive
Fra
ctio
n d
efec
tive
| | | | | | | | | |
22 44 66 88 1010 1212 1414 1616 1818 2020
p-Chart for Data Entryp-Chart for Data Entry
UCLUCLpp = p + z = p + zpp = .04 + 3(.02) = .10= .04 + 3(.02) = .10^̂
LCLLCLpp = p - z = p - zpp = .04 - 3(.02) = 0 = .04 - 3(.02) = 0^̂
UCLUCLpp = 0.10= 0.10
LCLLCLpp = 0.00= 0.00
p p = 0.04= 0.04
© 2006 Prentice Hall, Inc. S6 – 31
.11 .11 –
.10 .10 –
.09 .09 –
.08 .08 –
.07 .07 –
.06 .06 –
.05 .05 –
.04 .04 –
.03 .03 –
.02 .02 –
.01 .01 –
.00 .00 –
Sample numberSample number
Fra
ctio
n d
efec
tive
Fra
ctio
n d
efec
tive
| | | | | | | | | |
22 44 66 88 1010 1212 1414 1616 1818 2020
UCLUCLpp = p + z = p + zpp = .04 + 3(.02) = .10= .04 + 3(.02) = .10^̂
LCLLCLpp = p - z = p - zpp = .04 - 3(.02) = 0 = .04 - 3(.02) = 0^̂
UCLUCLpp = 0.10= 0.10
LCLLCLpp = 0.00= 0.00
p p = 0.04= 0.04
p-Chart for Data Entryp-Chart for Data Entry
Possible assignable
causes present
© 2006 Prentice Hall, Inc. S6 – 32
Control Limits for c-ChartsControl Limits for c-Charts
Population will be a Poisson distribution, Population will be a Poisson distribution, but applying the Central Limit Theorem but applying the Central Limit Theorem
allows us to assume a normal distribution allows us to assume a normal distribution for the sample statisticsfor the sample statistics
wherewhere cc ==mean number defective in the samplemean number defective in the sample
UCLUCLcc = c + = c + 33 c c LCLLCLcc = c - = c - 33 c c
© 2006 Prentice Hall, Inc. S6 – 33
c-Chart for Cab Companyc-Chart for Cab Company
c c = 54= 54 complaints complaints/9/9 days days = 6 = 6 complaintscomplaints//dayday
|1
|2
|3
|4
|5
|6
|7
|8
|9
DayDay
Nu
mb
er d
efec
tive
Nu
mb
er d
efec
tive14 14 –
12 12 –
10 10 –
8 8 –
6 6 –
4 –
2 –
0 0 –
UCLUCLcc = c + = c + 33 c c
= 6 + 3 6= 6 + 3 6= 13.35= 13.35
LCLLCLcc = c - = c - 33 c c
= 3 - 3 6= 3 - 3 6= 0= 0
UCLUCLcc = 13.35= 13.35
LCLLCLcc = 0= 0
c c = 6= 6
© 2006 Prentice Hall, Inc. S6 – 34
Patterns in Control ChartsPatterns in Control Charts
Normal behavior. Normal behavior. Process is “in control.”Process is “in control.”
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Figure S6.7Figure S6.7
© 2006 Prentice Hall, Inc. S6 – 35
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
One plot out above (or One plot out above (or below). Investigate for below). Investigate for cause. Process is “out cause. Process is “out of control.”of control.”
Figure S6.7Figure S6.7
© 2006 Prentice Hall, Inc. S6 – 36
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Trends in either Trends in either direction, 5 plots. direction, 5 plots. Investigate for cause of Investigate for cause of progressive change.progressive change.
Figure S6.7Figure S6.7
© 2006 Prentice Hall, Inc. S6 – 37
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Two plots very near Two plots very near lower (or upper) lower (or upper) control. Investigate for control. Investigate for cause.cause.
Figure S6.7Figure S6.7
© 2006 Prentice Hall, Inc. S6 – 38
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Run of 5 above (or Run of 5 above (or below) central line. below) central line. Investigate for cause. Investigate for cause. Figure S6.7Figure S6.7
© 2006 Prentice Hall, Inc. S6 – 39
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Erratic behavior. Erratic behavior. Investigate.Investigate.
Figure S6.7Figure S6.7
© 2006 Prentice Hall, Inc. S6 – 40
Which Control Chart to UseWhich Control Chart to Use
Using an x-chart and R-chart:Using an x-chart and R-chart: Observations are variablesObservations are variables
Collect Collect 20 - 2520 - 25 samples of n samples of n = 4= 4, or n , or n = = 55, or more, each from a stable process , or more, each from a stable process and compute the mean for the x-chart and compute the mean for the x-chart and range for the R-chartand range for the R-chart
Track samples of n observations eachTrack samples of n observations each
Variables DataVariables Data
© 2006 Prentice Hall, Inc. S6 – 41
Which Control Chart to UseWhich Control Chart to Use
Using the p-chart:Using the p-chart: Observations are attributes that can Observations are attributes that can
be categorized in two states be categorized in two states We deal with fraction, proportion, or We deal with fraction, proportion, or
percent defectivespercent defectives Have several samples, each with Have several samples, each with
many observationsmany observations
Attribute DataAttribute Data
© 2006 Prentice Hall, Inc. S6 – 42
Which Control Chart to UseWhich Control Chart to Use
Using a c-Chart:Using a c-Chart: Observations are attributes whose Observations are attributes whose
defects per unit of output can be defects per unit of output can be countedcounted
The number counted is often a small The number counted is often a small part of the possible occurrencespart of the possible occurrences
Defects such as number of blemishes Defects such as number of blemishes on a desk, number of typos in a page on a desk, number of typos in a page of text, flaws in a bolt of clothof text, flaws in a bolt of cloth
Attribute DataAttribute Data