heizer supp 06

© 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


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


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


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


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


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)


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”


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


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


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


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



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



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

????????

??????

??????

????????????

??????



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 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


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


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



Sampling DistributionSampling Distribution

x = x = (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.

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


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


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


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



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


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



Process average x Process average x = 12= 12 ounces ouncesAverage 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


From From Table S6.1Table S6.1



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


UCL = 12.144UCL = 12.144

Mean = 12Mean = 12

LCL = 11.857LCL = 11.857


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



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



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


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


x-chartx-chart(x-chart detects (x-chart detects shift in central shift in central tendency)tendency)

UCLUCL

LCLLCL


Mean and Range ChartsMean and Range Charts

R-chartR-chart(R-chart detects (R-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)

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

UCLUCL

LCLLCL


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


Manual and AutomatedManual and AutomatedControl ChartsControl Charts


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)


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 = =^̂


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)


.11 .11 –

.10 .10 –

.09 .09 –

.08 .08 –

.07 .07 –

.06 .06 –

.05 .05 –

.04 .04 –

.03 .03 –

.02 .02 –

.01 .01 –

.00 .00 –


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


.11 .11 –

.10 .10 –

.09 .09 –

.08 .08 –

.07 .07 –

.06 .06 –

.05 .05 –

.04 .04 –

.03 .03 –

.02 .02 –

.01 .01 –

.00 .00 –


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


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


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


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:


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



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



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


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




TargetTarget



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.”




TargetTarget



Trends in either Trends in either direction, 5 plots. direction, 5 plots. Investigate for cause of Investigate for cause of progressive change.progressive change.




TargetTarget



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




TargetTarget



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



TargetTarget



Erratic behavior. Erratic behavior. Investigate.Investigate.



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


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




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




66



= = 1.938= = 1.938213 - 207213 - 207

6(.516)6(.516)




66



= = 1.938= = 1.938213 - 207213 - 207

6(.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 1.01.0

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



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





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


Interpreting Interpreting CCpkpk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1



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


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)


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


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


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



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


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


OC Curves for Different OC Curves for Different Sampling PlansSampling Plans

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

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



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



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)


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


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

heizer supp 06

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