statistical process control · statistical process control ;variability is inherent in every...

© 2008 Prentice Hall, Inc. S6 – 1

Statistical Process ControlStatistical Process Control(Session 6, 7)(Session 6, 7)


OutlineOutline

Statistical Process Control (SPC)Statistical Process Control (SPC)Control Charts for VariablesControl Charts for VariablesThe Central Limit TheoremThe Central Limit TheoremSetting Mean Chart Limits (xSetting Mean Chart Limits (x--Charts)Charts)Setting Range Chart Limits (RSetting Range Chart Limits (R--Charts)Charts)Using Mean and Range ChartsUsing Mean and Range ChartsControl Charts for AttributesControl Charts for AttributesManagerial Issues and Control ChartsManagerial Issues and Control Charts


Outline Outline –– ContinuedContinued

Process CapabilityProcess CapabilityProcess Capability Ratio Process Capability Ratio (C(Cpp))Process Capability Index Process Capability Index (C(Cpkpk ))

Acceptance SamplingAcceptance SamplingOperating Characteristic CurveOperating Characteristic CurveAverage Outgoing QualityAverage Outgoing Quality


Statistical Process Control Statistical Process Control

Variability is inherent Variability is inherent in every processin every process

Natural or common causesNatural or common causes(As long as the distribution remains within specified limits, th(As long as the distribution remains within specified limits, the e

process is said to be process is said to be ““in controlin control”” & natural variations can be & natural variations can be tolerated)tolerated)

Special or assignable causesSpecial or assignable causes(These can be traced to a specific reason.)(These can be traced to a specific reason.)

Provides a statistical signal when assignable Provides a statistical signal when assignable causes are presentcauses are presentDetect and eliminate assignable causes of Detect and eliminate assignable causes of variationvariation


Natural VariationsNatural Variations

Also called common causesAlso called common causesAffect virtually all production processesAffect virtually all production processesExpected amount of variationExpected amount of variationOutput measures follow a probability Output measures follow a probability distributiondistributionFor any distribution there is a measure For any distribution there is a measure of central tendency and dispersionof central tendency and dispersionIf 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 be ““in controlin control””


Assignable VariationsAssignable Variations

Also called special causes of variationAlso called special causes of variationGenerally 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 reasonreasonThe objective is to discover when The objective is to discover when assignable causes are presentassignable causes are present

Eliminate the bad causesEliminate the bad causesIncorporate the good causesIncorporate the good causes


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

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WeightWeight

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## ## #### ## #### ## ####

Each of these Each of these represents one represents one sample of five sample of five

boxes of cerealboxes of cereal


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

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WeightWeight


SamplesSamplesTo 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 (bellthe normal (bell--shaped) distribution, but 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

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SamplesSamplesTo 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

WeightWeightTimeTimeFr

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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

WeightWeightTimeTimeFr

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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


Process ControlProcess Control

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


Types of DataTypes of Data

Characteristics that Characteristics that can take any real can take any real valuevalueMay be in whole or May be in whole or in fractional in fractional numbersnumbersContinuous random Continuous random variablesvariables

VariablesVariables AttributesAttributesDefectDefect--related related characteristics characteristics Classify products Classify products as either good or as either good or bad or count bad or count defectsdefectsCategorical or Categorical or discrete random discrete random variablesvariables


Central Limit TheoremCentral Limit TheoremRegardless 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 µµ

x = x = µµ

σσnnσσxx ==

2.2. The standard deviation of the The standard deviation of the sampling distribution sampling distribution ((σσxx)) will will equal the population standard equal the population standard deviation deviation ((σσ)) divided by the divided by the square root of the sample size, nsquare root of the sample size, n


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

= = σσxx ==σσnn

Mean of sample means = xMean of sample means = x

| | | | | | |

--33σσxx --22σσxx --11σσxx xx ++11σσxx ++22σσxx ++33σσxx

99.73%99.73% of all xof all xfall within fall within ±± 33σσxx

95.45%95.45% fall within fall within ±± 22σσxx


Sampling DistributionSampling Distribution

x = x = µµ(mean)(mean)

Sampling Sampling distribution distribution of meansof means

Process Process distribution distribution of meansof means


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.

xx--charts are to control charts are to control the central tendency of the processthe central tendency of the processRR--charts are to control the dispersion of charts are to control the dispersion of the processthe processThese two charts must be used togetherThese two charts must be used together


Setting Chart LimitsSetting Chart Limits

For xFor x--Charts when we know Charts when we know σσ

Upper control limit Upper control limit (UCL)(UCL) = x + z= x + zσσxx

Lower control limit Lower control limit (LCL)(LCL) = x = x -- zzσσxx

wherewhere xx == mean of the sample means or a target mean of the sample means or a target value set for the processvalue set for the process

zz == number of normal standard deviationsnumber of normal standard deviationsσσxx == standard deviation of the sample meansstandard deviation of the sample means

== σσ/ n/ nσσ == population standard deviationpopulation standard deviationnn == sample sizesample size


Setting Control LimitsSetting Control LimitsHourHour MeanMean HourHour MeanMean

11 16.116.1 77 15.215.222 16.816.8 88 16.416.433 15.515.5 99 16.316.344 16.516.5 1010 14.814.855 16.516.5 1111 14.214.266 16.416.4 1212 17.317.3

Hour 1Hour 1SampleSample Weight ofWeight ofNumberNumber Oat FlakesOat Flakes

11 171722 131333 161644 181855 171766 161677 151588 171799 1616

MeanMean 16.116.1σσ == 11

n = 9n = 9For For 99.73%99.73% control limits, z control limits, z = 3= 3

UCLUCLxx = x + z= x + zσσxx = 16 + 3(1/3) = 17 ozs= 16 + 3(1/3) = 17 ozs

LCLLCLxx = x = x -- zzσσxx = = 16 16 -- 3(1/3) = 15 ozs3(1/3) = 15 ozs


Setting Control LimitsSetting Control Limits

17 = UCL17 = UCL

15 = LCL15 = LCL

16 = Mean16 = Mean

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



For xFor x--Charts when we donCharts when we don’’t know t know σσ

Upper control limit Upper control limit (UCL)(UCL) = x + A= x + A22RR

Lower control limit Lower control limit (LCL)(LCL) = x = x -- AA22RR

wherewhere RR == average range of the samplesaverage range of the samplesAA22 == control chart factor given in statistical table control chart factor given in statistical table xx == mean of the sample meansmean of the sample means


Control Chart FactorsControl Chart FactorsSample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower RangeLower Range

n n AA22 DD44 DD33

22 1.8801.880 3.2683.268 0033 1.0231.023 2.5742.574 0044 .729.729 2.2822.282 0055 .577.577 2.1152.115 0066 .483.483 2.0042.004 0077 .419.419 1.9241.924 0.0760.07688 .373.373 1.8641.864 0.1360.13699 .337.337 1.8161.816 0.1840.184

1010 .308.308 1.7771.777 0.2230.2231212 .266.266 1.7161.716 0.2840.284



Process average x Process average x = 12= 12 ouncesouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5



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 ouncesouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5

From From Statistical Statistical

TableTable


Setting Control LimitsSetting Control LimitsProcess average x Process average x = 12= 12 ouncesouncesAverage 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

UCLUCLxx = x + A= x + A22RR= 12 + (.577)(.25)= 12 + (.577)(.25)= 12 + .144= 12 + .144= 12.144 = 12.144 ouncesounces

LCLLCLxx = x = x -- AA22RR= 12 = 12 -- .144.144= 11.857 = 11.857 ouncesounces


R R –– ChartChart

Type of variables control chartType of variables control chartShows 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 variabilityIndependent from process meanIndependent from process mean



For RFor R--ChartsCharts

Upper control limit Upper control limit (UCL(UCLRR)) = D= D44RR

Lower control limit Lower control limit (LCL(LCLRR)) = D= D33RR

wherewhereRR == average range of the samplesaverage range of the samples

DD33 and Dand D44 == control chart factors from statistical control chart factors from statistical table table



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 DD44 = 2.115, = 2.115, DD33 = 0= 0

UCL = 11.2UCL = 11.2

Mean = 5.3Mean = 5.3

LCL = 0LCL = 0


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)

RR--chartchart(R(R--chart does not chart does not detect change in detect change in mean)mean)

UCLUCL

LCLLCL

xx--chartchart(x(x--chart detects chart detects shift in central shift in central tendency)tendency)

UCLUCL

LCLLCL


Mean and Range ChartsMean and Range Charts

RR--chartchart(R(R--chart detects chart detects increase in increase in dispersion)dispersion)

UCLUCL

LCLLCL

(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)

xx--chartchart(x(x--chart does not chart does not detect the increase detect the increase in dispersion)in dispersion)

UCLUCL

LCLLCL


Steps to Create Control ChartsSteps to Create Control Charts

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


Manual and AutomatedManual and AutomatedControl ChartsControl Charts


Control Charts for AttributesControl Charts for Attributes

For variables that are categoricalFor variables that are categoricalGood/bad, yes/no, Good/bad, yes/no, acceptable/unacceptableacceptable/unacceptable

Measurement is typically counting Measurement is typically counting defectivesdefectivesCharts may measureCharts may measure

Percent defective (pPercent defective (p--chart)chart)Number of defects (cNumber of defects (c--chart)chart)


Control Limits for pControl Limits for p--ChartsChartsPopulation 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 + zσσpp̂̂

LCLLCLpp = p = p -- zzσσpp̂̂

wherewhere pp == mean fraction defective in the samplemean fraction defective in the samplezz == number of standard deviationsnumber of standard deviationsσσpp == standard deviation of the sampling distributionstandard deviation of the sampling distributionnn == sample sizesample size^̂

pp(1 (1 -- pp))nnσσpp ==^̂


pp--Chart for Data EntryChart 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 .04.04Total Total = 80= 80

(.04)(1 (.04)(1 -- .04).04)100100σσpp = = = .02= .02^̂p p = = .04= = .048080

(100)(20)(100)(20)


pp--Chart for Data EntryChart for Data EntryUCLUCLpp = p + z= p + zσσpp = .04 + 3(.02) = .10= .04 + 3(.02) = .10^̂

.11 .11 –

.10 .10 –

.09 .09 –

.08 .08 –

.07 .07 –

.06 .06 –

.05 .05 –

.04 .04 –

.03 .03 –

.02 .02 –

.01 .01 –

.00 .00 –


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22 44 66 88 1010 1212 1414 1616 1818 2020

LCLLCLpp = p = p -- zzσσpp = .04 = .04 -- 3(.02) = 03(.02) = 0^̂

UCLUCLpp = 0.10= 0.10

LCLLCLpp = 0.00= 0.00

p p = 0.04= 0.04


pp--Chart for Data EntryChart for Data Entry

.11 .11 –

.10 .10 –

.09 .09 –

.08 .08 –

.07 .07 –

.06 .06 –

.05 .05 –

.04 .04 –

.03 .03 –

.02 .02 –

.01 .01 –

.00 .00 –


Frac

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22 44 66 88 1010 1212 1414 1616 1818 2020

UCLUCLpp = p + z= p + zσσpp = .04 + 3(.02) = .10= .04 + 3(.02) = .10^̂

LCLLCLpp = p = p -- zzσσpp = .04 = .04 -- 3(.02) = 03(.02) = 0^̂

UCLUCLpp = 0.10= 0.10

LCLLCLpp = 0.00= 0.00

p p = 0.04= 0.04

Possible assignable

causes present


Control Limits for cControl Limits for c--ChartsChartsPopulation 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 cc LCLLCLcc = c = c -- 33 cc


cc--Chart for Cab CompanyChart for Cab Company

c c = 54= 54 complaintscomplaints/9/9 days days = 6 = 6 complaintscomplaints//dayday

|1

|2

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|4

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DayDay

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ber d

efec

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Num

ber d

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tive14 14 –

12 12 –10 10 –8 8 –6 6 –4 –2 –0 0 –

UCLUCLcc = c + = c + 33 cc= 6 + 3 6= 6 + 3 6= 13.35= 13.35

LCLLCLcc = c = c -- 33 cc= 6 = 6 -- 3 63 6= 0= 0

UCLUCLcc = 13.35= 13.35

LCLLCLcc = 0= 0

c c = 6= 6


Managerial Issues &Managerial Issues &Control ChartsControl Charts

Three major management decisions:Three major management decisions:

Select points in the processes that Select points in the processes that need SPCneed SPCDetermine the appropriate charting Determine the appropriate charting techniquetechniqueSet clear policies and proceduresSet clear policies and procedures


Which Control Chart to UseWhich Control Chart to Use

Variables DataVariables Data

Using an xUsing an x--chart and Rchart and R--chart:chart:Observations are variablesObservations are variablesCollect Collect 20 20 -- 2525 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 xand compute the mean for the x--chart chart and range for the Rand range for the R--chartchartTrack samples of n observations eachTrack samples of n observations each



Attribute DataAttribute Data

Using the pUsing the p--chart: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 defectivesHave several samples, each with Have several samples, each with many observationsmany observations



Attribute DataAttribute Data

Using a cUsing a c--Chart:Chart:Observations are attributes whose Observations are attributes whose defects per unit of output can be defects per unit of output can be countedcountedThe number counted is a small part of The number counted is a small part of the possible occurrencesthe possible occurrencesDefects 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


Patterns in Control ChartsPatterns in Control Charts

Normal behavior. Normal behavior. Process is Process is ““in control.in control.””

Upper control limitUpper control limit

TargetTarget

Lower control limitLower control limit




TargetTarget

Lower control limitLower control limitOne plot out above (or One plot out above (or below). Investigate for below). Investigate for cause. Process is cause. Process is ““out out of control.of control.””




TargetTarget

Lower control limitLower control limitTrends in either Trends in either direction, 5 plots. direction, 5 plots. Investigate for cause of Investigate for cause of progressive change.progressive change.




TargetTarget

Lower control limitLower control limitTwo plots very near Two plots very near lower (or upper) lower (or upper) control. Investigate for control. Investigate for cause.cause.




TargetTarget

Lower control limitLower control limitRun of 5 above (or Run of 5 above (or below) central line. below) central line. Investigate for cause. Investigate for cause.




TargetTarget

Lower control limitLower control limitErratic behavior. Erratic behavior. Investigate.Investigate.


Process CapabilityProcess CapabilityThe 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 requiredrequiredA process in statistical control does not A process in statistical control does not necessarily meet the design necessarily meet the design specificationsspecificationsProcess 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


Process Capability RatioProcess Capability Ratio

CCpp = = Upper Specification Upper Specification -- Lower SpecificationLower Specification66σσ

A capable process must have a A capable process must have a CCpp of at of at least least 1.01.0Does 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 offto allow for off--center processescenter processesSix Sigma quality requires aSix Sigma quality requires a CCpp = 2.0= 2.0




Insurance claims processInsurance claims process

Process mean x Process mean x = 210.0= 210.0 minutesminutesProcess standard deviation Process standard deviation σσ = .516= .516 minutesminutesDesign specification Design specification = 210 = 210 ±± 33 minutesminutes






= = 1.938= = 1.938213 213 -- 2072076(.516)6(.516)






= = 1.938= = 1.938213 213 -- 2072076(.516)6(.516)

Process is capable


Process Capability IndexProcess Capability Index

A capable process must have a A capable process must have a CCpkpk of at of at least least 1.01.0A 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 Specification -- xxLimitLimit

3σ3σ

LowerLowerx x -- SpecificationSpecification

LimitLimit3σ3σ



New Cutting MachineNew Cutting MachineNew process mean x New process mean x = .250 inches= .250 inchesProcess standard deviation Process standard deviation σσ = .0005 inches= .0005 inchesUpper Specification Limit Upper Specification Limit = .251 inches= .251 inchesLower Specification LimitLower Specification Limit = .249 inches= .249 inches



New Cutting MachineNew Cutting MachineNew process mean x New process mean x = .250 inches= .250 inchesProcess standard deviation Process standard deviation σσ = .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) (.251) -- .250.250

(3).0005(3).0005


Process Capability IndexProcess Capability IndexNew Cutting MachineNew Cutting Machine

New process mean x New process mean x = .250 inches= .250 inchesProcess standard deviation Process standard deviation σσ = .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) (.251) -- .250.250

(3).0005(3).0005.250 .250 -- (.249)(.249)

(3).0005(3).0005

Both calculations result inBoth calculations result in


Interpreting Interpreting CCpkpk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1


Acceptance SamplingAcceptance SamplingForm 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 itemsInspect each of the items in the sampleInspect each of the items in the sampleDecide 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


Acceptance SamplingAcceptance SamplingForm 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 itemsInspect each of the items in the sampleInspect each of the items in the sampleDecide 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 supplierCulled for defectives (100% inspection)


Operating Characteristic CurveOperating Characteristic Curve

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


The The ““PerfectPerfect”” OC CurveOC Curve

Return whole shipment

% Defective in Lot% Defective in Lot

P(A

ccep

t Who

le S

hipm

ent)

P(A

ccep

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hipm

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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


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 –

αα = 0.05= 0.05 producerproducer’’s risk for AQLs risk for AQL

ββ = 0.10= 0.10

ConsumerConsumer’’s s risk for LTPDrisk for LTPD

LTPDLTPDAQLAQLBad lotsBad lotsIndifference Indifference

zonezoneGood Good lotslots


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 badConsumer (buyer) does not want to Consumer (buyer) does not want to accept lots with more defects than accept lots with more defects than LTPDLTPD


ProducerProducer’’s & Consumers & Consumer’’s Riskss Risks

Producer's risk Producer's risk ((αα))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 ((ββ))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


OC Curves for Different OC Curves for Different Sampling PlansSampling Plans

nn = 50, = 50, cc = 1= 1

nn = 100, = 100, cc = 2= 2


Average Outgoing QualityAverage Outgoing Quality

AOQ = AOQ = ((PPdd)()(PPaa)()(N N -- nn))NN

wherewherePPdd = true percent defective of the lot= true percent defective of the lotPPaa = probability of accepting the lot= probability of accepting the lotNN = number of items in the lot= number of items in the lotnn = number of items in the sample= number of items in the sample


Average Outgoing QualityAverage Outgoing Quality

1.1. If a sampling plan replaces all defectivesIf a sampling plan replaces all defectives2.2. 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)


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, µµ


Numerical: Numerical:

Sampling 4 pieces Sampling 4 pieces of precisionof precision--cut cut wire (to be used in wire (to be used in Computer Computer assembly) assembly) every hour for the every hour for the past 24 hours has past 24 hours has produced the produced the following results: following results:

HourHour XX RR HourHour XX RR11 3.253.25 0.710.71 1313 3.113.11 0.850.8522 3.13.1 1.181.18 1414 2.832.83 1.311.3133 3.223.22 1.431.43 1515 3.123.12 1.061.0644 3.393.39 1.261.26 1616 2.842.84 0.50.555 3.073.07 1.171.17 1717 2.862.86 1.431.4366 2.862.86 0.320.32 1818 2.742.74 1.291.2977 3.053.05 0.530.53 1919 3.413.41 1.611.6188 2.652.65 1.131.13 2020 2.892.89 1.091.0999 3.023.02 0.710.71 2121 2.652.65 1.081.08

1010 2.852.85 1.331.33 2222 3.283.28 0.460.461111 2.832.83 1.171.17 2323 2.942.94 1.581.581212 2.972.97 0.400.40 2424 2.652.65 0.970.97


SampleSample Nos. of Nos. of DefectivesDefectives

%defectives%defectives SampleSample Nos. of Nos. of DefectivesDefectives

%defectives%defectives

11 1212 0.240.24 1616 88 0.160.1622 1515 0.30.3 1717 1010 0.20.233 88 0.160.16 1818 55 0.10.144 1010 0.20.2 1919 1313 0.260.2655 44 0.080.08 2020 1111 0.220.2266 77 0.140.14 2121 2020 0.40.477 1616 0.320.32 2222 1818 0.360.3688 99 0.180.18 2323 2424 0.480.4899 1414 0.280.28 2424 1515 0.30.3

1010 1010 0.20.2 2525 99 0.180.181111 55 0.10.1 2626 1212 0.240.241212 66 0.120.12 2727 77 0.140.141313 1717 0.340.34 2828 1313 0.260.261414 1212 0.240.24 2929 99 0.180.181515 2222 0.440.44 3030 66 0.120.12

TotalTotal 347347 P = 0.2313P = 0.2313

Numerical: Sample size (n) is 50Numerical: Sample size (n) is 50


If we know the probability of part produced If we know the probability of part produced being defective is p OR this is a standard then being defective is p OR this is a standard then we donwe don’’t need to calculate %defective.t need to calculate %defective.

If n=50, and p = 0.24 If n=50, and p = 0.24 then then

calculate UCL and LCL for the control charts.calculate UCL and LCL for the control charts.

statistical process control · statistical process control ;variability is inherent in every...

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