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© 2008 Prentice Hall, Inc. S6 – 1
Operations ManagementOperations ManagementSupplement 6 – Supplement 6 – Statistical Process Statistical Process ControlControl
PowerPoint presentation to accompany PowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 7ePrinciples of Operations Management, 7eOperations Management, 9e Operations Management, 9e
© 2008 Prentice Hall, Inc. S6 – 2
OutlineOutline
Statistical Process Control (SPC)Statistical Process Control (SPC) Control Charts for VariablesControl Charts for Variables
The Central Limit TheoremThe Central Limit Theorem
Setting Mean Chart Limits (x-Charts)Setting Mean Chart Limits (x-Charts)
Setting Range Chart Limits (R-Charts)Setting Range Chart Limits (R-Charts)
Using Mean and Range ChartsUsing Mean and Range Charts
Control Charts for AttributesControl Charts for Attributes
Managerial Issues and Control ChartsManagerial Issues and Control Charts
© 2008 Prentice Hall, Inc. S6 – 3
Outline – ContinuedOutline – Continued
Process CapabilityProcess Capability Process Capability Ratio Process Capability Ratio (C(Cpp))
Process Capability Index Process Capability Index (C(Cpkpk ))
Acceptance SamplingAcceptance Sampling Operating Characteristic CurveOperating Characteristic Curve
Average Outgoing QualityAverage Outgoing Quality
© 2008 Prentice Hall, Inc. S6 – 4
Learning ObjectivesLearning Objectives
When you complete this supplement When you complete this supplement you should be able to:you should be able to:
1.1. Explain the use of a control chartExplain the use of a control chart
2.2. Explain the role of the central limit Explain the role of the central limit theorem in SPCtheorem in SPC
3.3. Build x-charts and R-chartsBuild x-charts and R-charts
4.4. List the five steps involved in List the five steps involved in building control chartsbuilding control charts
© 2008 Prentice Hall, Inc. S6 – 5
Learning ObjectivesLearning Objectives
When you complete this supplement When you complete this supplement you should be able to:you should be able to:
5.5. Build p-charts and c-chartsBuild p-charts and c-charts
6.6. Explain process capability and Explain process capability and compute compute CCpp and and CCpkpk
7.7. Explain acceptance samplingExplain acceptance sampling
8.8. Compute the AOQCompute the AOQ
© 2008 Prentice Hall, Inc. S6 – 6
Variability is inherent Variability is inherent in every processin every process Natural or common Natural or common
causescauses 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)
© 2008 Prentice Hall, Inc. S6 – 7
Natural VariationsNatural Variations Also called common causesAlso called common causes
Affect virtually all production processesAffect virtually all production processes
Expected amount of variationExpected amount of variation
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
If the distribution of outputs falls within If the distribution of outputs falls within acceptable limits, the process is said to acceptable limits, the process is said to be “in control”be “in control”
© 2008 Prentice Hall, Inc. S6 – 8
Assignable VariationsAssignable Variations
Also called special causes of variationAlso called special causes of variation Generally this is some change in the processGenerally this is some change in the process
Variations that can be traced to a specific Variations that can be traced to a specific reasonreason
The objective is to discover when The objective is to discover when assignable causes are presentassignable causes are present Eliminate the bad causesEliminate the bad causes
Incorporate the good causesIncorporate the good causes
© 2008 Prentice Hall, Inc. S6 – 9
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
© 2008 Prentice Hall, Inc. S6 – 10
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
(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
© 2008 Prentice Hall, Inc. S6 – 11
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
(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
© 2008 Prentice Hall, Inc. S6 – 12
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
(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
© 2008 Prentice Hall, Inc. S6 – 13
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
(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
????????
??????
??????
????????????
??????
Figure S6.1Figure S6.1
© 2008 Prentice Hall, Inc. S6 – 14
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
© 2008 Prentice Hall, Inc. S6 – 15
Process ControlProcess Control
Figure S6.2Figure S6.2
FrequencyFrequency
(weight, length, speed, etc.)(weight, length, speed, etc.)SizeSize
Lower control limitLower control limit Upper control limitUpper control limit
(a) In statistical (a) In statistical control and capable control and capable of producing within of producing within control limitscontrol limits
(b) In statistical (b) In statistical control but not control but not capable of producing capable of producing within control limitswithin control limits
(c) Out of control(c) Out of control
© 2008 Prentice Hall, Inc. S6 – 16
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
© 2008 Prentice Hall, Inc. S6 – 17
Central Limit TheoremCentral Limit Theorem
Regardless of the distribution of the Regardless of the distribution of the population, the distribution of sample means population, the distribution of sample means drawn from the population will tend to follow drawn from the population will tend to follow a normal curvea normal curve
1.1. The mean of the sampling The mean of the sampling distribution distribution ((xx)) will be the same will be the same as the population mean as the population mean
x = x =
nn
xx = =
2.2. The standard deviation of the The standard deviation of the sampling distribution sampling distribution ((xx)) will will equal the population standard equal the population standard deviation deviation (()) divided by the divided by the square root of the sample size, nsquare root of the sample size, n
© 2008 Prentice Hall, Inc. S6 – 18
Population and Sampling Population and Sampling DistributionsDistributions
Three population Three population distributionsdistributions
Beta
Normal
Uniform
Distribution of Distribution of sample meanssample means
Standard Standard deviation of deviation of the sample the sample meansmeans
= = xx = = nn
Mean of sample means = xMean of sample means = x
| | | | | | |
--33xx --22xx --11xx xx ++11xx ++22xx ++33xx
99.73%99.73% of all x of all xfall within fall within ± 3± 3xx
95.45%95.45% fall within fall within ± 2± 2xx
Figure S6.3Figure S6.3
© 2008 Prentice Hall, Inc. S6 – 19
Sampling DistributionSampling Distribution
x = x = (mean)(mean)
Sampling Sampling distribution distribution of meansof means
Process Process distribution distribution of meansof means
Figure S6.4Figure S6.4
© 2008 Prentice Hall, Inc. S6 – 20
Control Charts for VariablesControl Charts for Variables
For variables that have For variables that have continuous dimensionscontinuous dimensions Weight, speed, length, Weight, speed, length,
strength, etc.strength, etc.
x-charts are to control x-charts are to control the central tendency of the processthe central tendency of the process
R-charts are to control the dispersion of R-charts are to control the dispersion of the processthe process
These two charts must be used togetherThese two charts must be used together
© 2008 Prentice Hall, Inc. S6 – 21
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
© 2008 Prentice Hall, Inc. S6 – 22
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
© 2008 Prentice Hall, Inc. S6 – 23
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
© 2008 Prentice Hall, Inc. S6 – 24
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
© 2008 Prentice Hall, Inc. S6 – 25
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
© 2008 Prentice Hall, Inc. S6 – 26
Setting Control LimitsSetting Control Limits
Process average x Process average x = 12= 12 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
© 2008 Prentice Hall, Inc. S6 – 27
Setting Control LimitsSetting Control Limits
UCLUCLxx = x + A= x + A22RR
= 12 + (.577)(.25)= 12 + (.577)(.25)= 12 + .144= 12 + .144= 12.144 = 12.144 ouncesounces
Process average x Process average x = 12= 12 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
From From Table S6.1Table S6.1
© 2008 Prentice Hall, Inc. S6 – 28
Setting Control LimitsSetting Control Limits
UCLUCLxx = x + A= x + A22RR
= 12 + (.577)(.25)= 12 + (.577)(.25)= 12 + .144= 12 + .144= 12.144 = 12.144 ouncesounces
LCLLCLxx = x - A= x - A22RR
= 12 - .144= 12 - .144= 11.857 = 11.857 ouncesounces
Process average x Process average x = 12= 12 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
UCL = 12.144UCL = 12.144
Mean = 12Mean = 12
LCL = 11.857LCL = 11.857
© 2008 Prentice Hall, Inc. S6 – 29
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
© 2008 Prentice Hall, Inc. S6 – 30
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
© 2008 Prentice Hall, Inc. S6 – 31
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
© 2008 Prentice Hall, Inc. S6 – 32
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
© 2008 Prentice Hall, Inc. S6 – 33
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
© 2008 Prentice Hall, Inc. S6 – 34
Steps In Creating Control Steps In Creating Control ChartsCharts
1.1. Take samples from the population and Take samples from the population and compute the appropriate sample statisticcompute the appropriate sample statistic
2.2. Use the sample statistic to calculate control Use the sample statistic to calculate control limits and draw the control chartlimits and draw the control chart
3.3. Plot sample results on the control chart and Plot sample results on the control chart and determine the state of the process (in or out of determine the state of the process (in or out of control)control)
4.4. Investigate possible assignable causes and Investigate possible assignable causes and take any indicated actionstake any indicated actions
5.5. Continue sampling from the process and reset Continue sampling from the process and reset the control limits when necessarythe control limits when necessary
© 2008 Prentice Hall, Inc. S6 – 35
Manual and AutomatedManual and AutomatedControl ChartsControl Charts
© 2008 Prentice Hall, Inc. S6 – 36
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)
© 2008 Prentice Hall, Inc. S6 – 37
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 = =^̂
© 2008 Prentice Hall, Inc. S6 – 38
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)
© 2008 Prentice Hall, Inc. S6 – 39
.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
© 2008 Prentice Hall, Inc. S6 – 40
.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
© 2008 Prentice Hall, Inc. S6 – 41
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
© 2008 Prentice Hall, Inc. S6 – 42
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
= 6 - 3 6= 6 - 3 6= 0= 0
UCLUCLcc = 13.35= 13.35
LCLLCLcc = 0= 0
c c = 6= 6
© 2008 Prentice Hall, Inc. S6 – 43
Managerial Issues andManagerial Issues andControl ChartsControl Charts
Select points in the processes that Select points in the processes that need SPCneed SPC
Determine the appropriate charting Determine the appropriate charting techniquetechnique
Set clear policies and proceduresSet clear policies and procedures
Three major management decisions:Three major management decisions:
© 2008 Prentice Hall, Inc. S6 – 44
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
© 2008 Prentice Hall, Inc. S6 – 45
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
© 2008 Prentice Hall, Inc. S6 – 46
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 a small part of The number counted is a small part of the possible occurrencesthe 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
© 2008 Prentice Hall, Inc. S6 – 47
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
© 2008 Prentice Hall, Inc. S6 – 48
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
© 2008 Prentice Hall, Inc. S6 – 49
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
© 2008 Prentice Hall, Inc. S6 – 50
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
© 2008 Prentice Hall, Inc. S6 – 51
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
© 2008 Prentice Hall, Inc. S6 – 52
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
© 2008 Prentice Hall, Inc. S6 – 53
Process CapabilityProcess Capability
The natural variation of a process The natural variation of a process should be small enough to produce should be small enough to produce products that meet the standards products that meet the standards requiredrequired
A process in statistical control does not A process in statistical control does not necessarily meet the design necessarily meet the design specificationsspecifications
Process capability is a measure of the Process capability is a measure of the relationship between the natural relationship between the natural variation of the process and the design variation of the process and the design specificationsspecifications
© 2008 Prentice Hall, Inc. S6 – 54
Process Capability RatioProcess Capability Ratio
CCpp = = Upper Specification - Lower SpecificationUpper Specification - Lower Specification
66
A capable process must have a A capable process must have a CCpp of at of at least least 1.01.0
Does not look at how well the process Does not look at how well the process is centered in the specification range is centered in the specification range
Often a target value of Often a target value of CCpp = 1.33 = 1.33 is used is used to allow for off-center processesto allow for off-center processes
Six Sigma quality requires aSix Sigma quality requires a C Cpp = 2.0 = 2.0
© 2008 Prentice Hall, Inc. S6 – 55
Process Capability RatioProcess Capability Ratio
CCpp = = Upper Specification - Lower SpecificationUpper Specification - Lower Specification
66
Insurance claims processInsurance claims process
Process mean x Process mean x = 210.0= 210.0 minutes minutesProcess standard deviation Process standard deviation = .516 = .516 minutes minutesDesign specification Design specification = 210 ± 3= 210 ± 3 minutes minutes
© 2008 Prentice Hall, Inc. S6 – 56
Process Capability RatioProcess Capability Ratio
CCpp = = Upper Specification - Lower SpecificationUpper Specification - Lower Specification
66
Insurance claims processInsurance claims process
Process mean x Process mean x = 210.0= 210.0 minutes minutesProcess standard deviation Process standard deviation = .516 = .516 minutes minutesDesign specification Design specification = 210 ± 3= 210 ± 3 minutes minutes
= = 1.938= = 1.938213 - 207213 - 207
6(.516)6(.516)
© 2008 Prentice Hall, Inc. S6 – 57
Process Capability RatioProcess Capability Ratio
CCpp = = Upper Specification - Lower SpecificationUpper Specification - Lower Specification
66
Insurance claims processInsurance claims process
Process mean x Process mean x = 210.0= 210.0 minutes minutesProcess standard deviation Process standard deviation = .516 = .516 minutes minutesDesign specification Design specification = 210 ± 3= 210 ± 3 minutes minutes
= = 1.938= = 1.938213 - 207213 - 207
6(.516)6(.516)Process is
capable
© 2008 Prentice Hall, Inc. S6 – 58
Process Capability IndexProcess Capability Index
A capable process must have a A capable process must have a CCpkpk of at of at least least 1.01.0
A capable process is not necessarily in the A capable process is not necessarily in the center of the specification, but it falls within center of the specification, but it falls within the specification limit at both extremesthe specification limit at both extremes
CCpkpk = minimum of , = minimum of ,
UpperUpperSpecification - xSpecification - xLimitLimit
LowerLowerx -x - SpecificationSpecification
LimitLimit
© 2008 Prentice Hall, Inc. S6 – 59
Process Capability IndexProcess Capability Index
New Cutting MachineNew Cutting Machine
New process mean x New process mean x = .250 inches= .250 inchesProcess standard deviation Process standard deviation = .0005 inches = .0005 inchesUpper Specification Limit Upper Specification Limit = .251 inches= .251 inchesLower Specification LimitLower Specification Limit = .249 inches = .249 inches
© 2008 Prentice Hall, Inc. S6 – 60
Process Capability IndexProcess Capability Index
New Cutting MachineNew Cutting Machine
New process mean x New process mean x = .250 inches= .250 inchesProcess standard deviation Process standard deviation = .0005 inches = .0005 inchesUpper Specification Limit Upper Specification Limit = .251 inches= .251 inchesLower Specification LimitLower Specification Limit = .249 inches = .249 inches
CCpkpk = minimum of , = minimum of ,(.251) - .250(.251) - .250
(3).0005(3).0005
© 2008 Prentice Hall, Inc. S6 – 61
Process Capability IndexProcess Capability Index
New Cutting MachineNew Cutting Machine
New process mean x New process mean x = .250 inches= .250 inchesProcess standard deviation Process standard deviation = .0005 inches = .0005 inchesUpper Specification Limit Upper Specification Limit = .251 inches= .251 inchesLower Specification LimitLower Specification Limit = .249 inches = .249 inches
CCpkpk = = 0.67 = = 0.67.001.001
.0015.0015
New machine is NOT capable
CCpkpk = minimum of , = minimum of ,(.251) - .250(.251) - .250
(3).0005(3).0005.250 - (.249).250 - (.249)
(3).0005(3).0005
Both calculations result inBoth calculations result in
© 2008 Prentice Hall, Inc. S6 – 62
Interpreting Interpreting CCpkpk
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Figure S6.8Figure S6.8
© 2008 Prentice Hall, Inc. S6 – 63
Acceptance SamplingAcceptance Sampling
Form of quality testing used for Form of quality testing used for incoming materials or finished goodsincoming materials or finished goods Take samples at random from a lot Take samples at random from a lot
(shipment) of items(shipment) of items
Inspect each of the items in the sampleInspect each of the items in the sample
Decide whether to reject the whole lot Decide whether to reject the whole lot based on the inspection resultsbased on the inspection results
Only screens lots; does not drive Only screens lots; does not drive quality improvement effortsquality improvement efforts
© 2008 Prentice Hall, Inc. S6 – 64
Acceptance SamplingAcceptance Sampling
Form of quality testing used for Form of quality testing used for incoming materials or finished goodsincoming materials or finished goods Take samples at random from a lot Take samples at random from a lot
(shipment) of items(shipment) of items
Inspect each of the items in the sampleInspect each of the items in the sample
Decide whether to reject the whole lot Decide whether to reject the whole lot based on the inspection resultsbased on the inspection results
Only screens lots; does not drive Only screens lots; does not drive quality improvement effortsquality improvement efforts
Rejected lots can be:
Returned to the supplier
Culled for defectives (100% inspection)
© 2008 Prentice Hall, Inc. S6 – 65
Operating Characteristic Operating Characteristic CurveCurve
Shows how well a sampling plan Shows how well a sampling plan discriminates between good and discriminates between good and bad lots (shipments)bad lots (shipments)
Shows the relationship between Shows the relationship between the probability of accepting a lot the probability of accepting a lot and its quality leveland its quality level
© 2008 Prentice Hall, Inc. S6 – 66
Return whole shipment
The “Perfect” OC CurveThe “Perfect” OC Curve
% Defective in Lot% Defective in Lot
P(A
cc
ept
Wh
ole
Sh
ipm
en
t)P
(Ac
cep
t W
ho
le S
hip
me
nt)
100 100 –
75 75 –
50 50 –
25 25 –
0 0 –| | | | | | | | | | |
00 1010 2020 3030 4040 5050 6060 7070 8080 9090 100100
Cut-Off
Keep whole Keep whole shipmentshipment
© 2008 Prentice Hall, Inc. S6 – 67
An OC CurveAn OC Curve
Probability Probability of of
AcceptanceAcceptance
Percent Percent defectivedefective
| | | | | | | | |
00 11 22 33 44 55 66 77 88
100 100 –95 95 –
75 75 –
50 50 –
25 25 –
10 10 –
0 0 –
= 0.05= 0.05 producer’s risk for AQL producer’s risk for AQL
= 0.10= 0.10
Consumer’s Consumer’s risk for LTPDrisk for LTPD
LTPDLTPDAQLAQL
Bad lotsBad lotsIndifference Indifference zonezone
Good Good lotslots
Figure S6.9Figure S6.9
© 2008 Prentice Hall, Inc. S6 – 68
AQL and LTPDAQL and LTPD
Acceptable Quality Level (AQL)Acceptable Quality Level (AQL) Poorest level of quality we are Poorest level of quality we are
willing to acceptwilling to accept
Lot Tolerance Percent Defective Lot Tolerance Percent Defective (LTPD)(LTPD) Quality level we consider badQuality level we consider bad
Consumer (buyer) does not want to Consumer (buyer) does not want to accept lots with more defects than accept lots with more defects than LTPDLTPD
© 2008 Prentice Hall, Inc. S6 – 69
Producer’s and Consumer’s Producer’s and Consumer’s RisksRisks
Producer's risk Producer's risk (()) Probability of rejecting a good lot Probability of rejecting a good lot Probability of rejecting a lot when the Probability of rejecting a lot when the
fraction defective is at or above the fraction defective is at or above the AQLAQL
Consumer's risk Consumer's risk (()) Probability of accepting a bad lot Probability of accepting a bad lot Probability of accepting a lot when Probability of accepting a lot when
fraction defective is below the LTPDfraction defective is below the LTPD
© 2008 Prentice Hall, Inc. S6 – 70
OC Curves for Different OC Curves for Different Sampling PlansSampling Plans
nn = 50, = 50, cc = 1 = 1
nn = 100, = 100, cc = 2 = 2
© 2008 Prentice Hall, Inc. S6 – 71
Average Outgoing QualityAverage Outgoing Quality
wherewhere
PPdd = true percent defective of the lot= true percent defective of the lot
PPaa = probability of accepting the lot= probability of accepting the lot
NN = number of items in the lot= number of items in the lot
nn = number of items in the sample= number of items in the sample
AOQ = AOQ = ((PPdd)()(PPaa)()(N - nN - n))
NN
© 2008 Prentice Hall, Inc. S6 – 72
Average Outgoing QualityAverage Outgoing Quality
1.1. If a sampling plan replaces all defectivesIf a sampling plan replaces all defectives
2.2. If we know the incoming percent If we know the incoming percent defective for the lotdefective for the lot
We can compute the average outgoing We can compute the average outgoing quality (AOQ) in percent defectivequality (AOQ) in percent defective
The maximum AOQ is the highest percent The maximum AOQ is the highest percent defective or the lowest average quality defective or the lowest average quality and is called the average outgoing quality and is called the average outgoing quality level (AOQL)level (AOQL)
© 2008 Prentice Hall, Inc. S6 – 73
Automated InspectionAutomated Inspection
Modern Modern technologies technologies allow virtually allow virtually 100% 100% inspection at inspection at minimal costsminimal costs
Not suitable Not suitable for all for all situationssituations
© 2008 Prentice Hall, Inc. S6 – 74
SPC and Process VariabilitySPC and Process Variability
(a)(a) Acceptance Acceptance sampling (Some sampling (Some bad units accepted)bad units accepted)
(b)(b) Statistical process Statistical process control (Keep the control (Keep the process in control)process in control)
(c)(c) CCpkpk >1 >1 (Design (Design a process that a process that is in control)is in control)
Lower Lower specification specification
limitlimit
Upper Upper specification specification
limitlimit
Process mean, Process mean, Figure S6.10Figure S6.10