1

Chapter 10

Quality Control

2

Phases of Quality Assurance

Acceptancesampling

Processcontrol

Continuousimprovement

Inspectionbefore/afterproduction

Correctiveaction duringproduction

Quality builtinto theprocess

The leastprogressive

The mostprogressive

3

Inspection: Appraisal of good/service quality

• How Much (sample size) /How Often (hourly, daily)

Co

st

OptimalAmount of Inspection

Cost of inspection(appraisal and Prevention cost)

Cost of passingdefectives(failure cost)

Total Cost

4

Inspection

• Where/When • Raw materials• Finished products

• Before a costly operation, PhD comp. exam before candidacy

• Before an irreversible process, firing pottery

• Before a covering process, painting, assembly

• Centralized vs. On-Site, my friend checks quality at cruise lines

Inputs Transformation Outputs

Acceptancesampling

Processcontrol

Acceptancesampling

5

Examples of Inspection Points

Type ofbusiness

Inspectionpoints

Characteristics

Fast Food CashierCounter areaEating areaBuildingKitchen

AccuracyAppearance, productivityCleanlinessAppearanceHealth regulations

Hotel/motel Parking lotAccountingBuildingMain desk

Safe, well lightedAccuracy, timelinessAppearance, safetyWaiting times

Supermarket CashiersDeliveries

Accuracy, courtesyQuality, quantity

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Statistical Process Control (SPC)

• SPC: Statistical evaluation of the output of a process during production

• The Control Process– Define– Measure– Compare to a standard– Evaluate– Take corrective action– Evaluate corrective action

7

Statistical Process Control

• Shewhart’s classification of variability: common cause vs. assignable cause

• Variations and Control– Random variation: Natural variations in the

output of process, created by countless minor factors, e.g. temperature, humidity variations.

– Assignable variation: A variation whose source can be identified. This source is generally a major factor, e.g. tool failure.

8

Mean and Variance

• Given a population of numbers, how to compute the mean and the variance?

deviation Standard

)(Variance

Mean

},...,,{Population

1

2

2

1

21

N

x

N

x

xxx

N

ii

N

ii

N

9

Statistical Process Control

• From a large population of goods or services (random if possible) a sample is drawn. – Example sample: Midterm grades of BA3352

students whose last name starts with letter R {60, 64, 72, 86}, with letter S {54, 60}

• Sample size= n• Sample average or sample mean= • Sample range= R• Standard deviation of sample means=

x

population theofdeviation Standard: where n

x

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Sampling Distribution

Sampling distribution

Variability of the average scores of people with last name R and S

Process distribution

Variability of the scores for the entire class

Mean

Sampling distribution is the distribution of sample means.

Grouping reduces the variability.

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Normal Distribution

Mean

95.44%

99.74%

x

at x. cdf normal )1,_,,(normdist:functions lstatistica Excel

at x. pdf normal )0,_,,(normdist:functions lstatistica Excel

devstmeanx

devstmeanx

normdist(x,.,.,0)

Probab

normdist(x,.,.,1)

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Cumulative Normal Density

)_,,(norminv :prob""at cdf offunction Inverse

)1,_,,(normdist:at x (cdf)function Cumulative

:functions lstatistica Excel

devstmeanprob

devstmeanx

0

1

x

normdist(x,mean,st_dev,1)

prob

norminv(prob,mean,st_dev)

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Normal Probabilities: Example

• If temperature inside a firing oven has a normal distribution with mean 200 oC and standard deviation of 40 oC, what is the probability that– The temperature is lower than 220 oC

=normdist(220,200,40,1)

– The temperature is between 190 oC and 220oC=normdist(220,200,40,1)-normdist(190,200,40,1)

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Control Limits

Samplingdistribution

Processdistribution

Mean

LCLLowercontrol

limit

UCLUppercontrol

limit

Process is in control if sample mean is between control limits. These limits have nothing to do with product specifications!

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Setting Control Limits: Hypothesis Testing Framework

• Null hypothesis: Process is in control• Alternative hypothesis: Process is out of control• Alpha=P(Type I error)=P(reject the null when it is true)=

P(out of control when in control)• Beta=P(Type II error)=P(accept the null when it is false)

P(in control when out of control)

• If LCL decreases and UCL increases what happens to – Alpha ?– Beta?

• Not possible to target alpha and beta simultaneously, control charts target a desired level of Alpha.

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Type I Error=Alpha

Mean

LCL UCL

/2 /2

Probabilityof Type I error

st_dev)mean,/2,-norminv(1UCL

st_dev)mean,/2,norminv(LCL

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Control Chart

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

Sample number

Mean

Out ofcontrol

Normal variationdue to chance

Abnormal variationdue to assignable sources

Abnormal variationdue to assignable sources

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Observations from Sample Distribution

Sample number

UCL

LCL

1 2 3 4

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Control Charts

• Control charts for variables (measurable quantities), e.g. length, temperature– Mean control charts

• To check mean

– Range control charts• To check variability

• Control charts for attributes, e.g. fit, defective – p-charts

• To check proportion of defectives (occurrences)

– c-charts• To check the number of defectives (occurrences)

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Mean control chart

deviation standard of multiple a minusmean grand

deviation standard of multiple a plusmean grand

x

x

zxLCL

zxUCL

xofaveragexmeanGrand

x

x

x

),x/2,-norminv(1z

xxUCL

Most often z is set to 2 or 3.

If the standard deviation of the sample means is not known, use the average of sample ranges to get the limits:

ranges sample of average theof multiple a minusmean grand

ranges sample of average theof multiple a plusmean grand

R ranges sample of average

2

2

RAxLCL

RAxUCL

R

Multiplier A_2 depends on n and is available in Table 10-2.

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Range Control Chart

ranges sample of average theof multipleA

ranges sample of average theof multipleA

3

4

RDLCL

RDUCL

Multipliers D_4 and D_3 depend on n and are available in Table 10-2.

EX: In the last five years, the range of GMAT scores of incoming PhD class is 88, 64, 102, 70, 74. If each class has 6 students, what are UCL and LCL for GMAT ranges?

079.6*0 159.279.6*2

.0D ,2D 6,nFor .6.795/)74701026488(

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34

RDLCLRDUCL

R

Are the GMAT ranges in control?

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Mean and Range Charts: Which?

UCL

LCL

UCL

LCL

R-chart

x-Chart Detects shift

Does notdetect shift

(process mean is shifting upward)

SamplingDistribution

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Mean and Range Charts: Which?

UCL

LCL

LCL

R-chart Reveals increase

x-Chart

UCL

Does notreveal increase

(process variability is increasing)SamplingDistribution

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Use of p-Charts

• p=proportion defective, assumed to be known

• When observations can be placed into two categories.– Good or bad

– Pass or fail

– Operate or don’t operate

– Go or no-go gauge

before as , )1(

LCL

zn

ppwhere

zpzpUCL

p

pp

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Use of c-Charts

• c=number of occurrences per unit• Use only when the number of occurrences per unit can be counted.

• Scratches, chips, dents, or errors per item• Cracks or faults per unit of distance• Breaks or Tears per unit of area• Bacteria or pollutants per unit of volume• Calls, complaints, failures per unit of time

c average theuse known,not is c if

LCL czcczcUCL

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C-chart Example

• While the nuclear submarine Kursk was being raised in the Barents sea (between Svalbard, No and Novaya Zemlya, Ru), which took 15 hours, engineers took a reading of number of Geiger counts per hour to detect any increase in radiation levels. Should they have stopped before 5th or 10th hour given 3-sigma control and the readings data: 42, 48, 50, 45, 52, 66, 64, 84, 92, 76.At the 5th hour, average number of counts=47.4, stdev of counts=6.88,

UCL=47.4+3*6.88=68.05, LCL=47.4-3*6.88=26.75. Do not stop.

At the 10th hour, average number of counts=61.9, stdev of counts=7.87, UCL=61.9+3*7.87=85.51, LCL=61.9-3*7.87=38.29. Stop, 9th reading is out of control.

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Up and Down Run Charts

• If all readings are in control, is the process really in control?

• There could be trends in readings even when they are in control.

Counting Up/Down Runs (r=8 runs)

U U D U D U D U U D

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Up and Down Run Charts

samples ofNumber K90

2916 and

3

1-2KE(r)

runs of stdev of multiple a minus runs Expected)(

runs of stdev of multiple a plus runs Expected)(

K

zrELCL

zrEUCL

r

r

r

EX: What are 3-sigma UCL and LCL for the number of runs in 50 samples?

2.92*3-33)(

2.92*333)(

92.290

2916 and 33

3

1-2KE(r) 50,K

r

r

r

zrELCL

zrEUCL

K

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• Tolerances/Specifications– Requirements of the design or customers

• Process variability– Natural variability in a process– Variance of the measurements coming from the process

• Process capability– Process variability relative to specification– Capability=Process specifications / Process variability

Process Capability

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Process Capability: Specification limits are not control chart limits

LowerSpecification

UpperSpecification

Process variability matches specifications

LowerSpecification

UpperSpecification

Process variability well within specifications

LowerSpecification

UpperSpecification

Process variability exceeds specifications

Sampling Distribution

is used

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Process Capability Ratio

When the process is centered, process capability ratio

Upper specification – lower specification6

Cp =

A capable process has large Cp.

Example: The standard deviation, of sample averages of the midterm 1scores obtained by students whose last names start with R, has been 7. The SOM management requires the scores not to differ by more than 50% in an exam. That is the highest score can be at most 50 points above the lowest score. Suppose that the scores are centered, what is the process capability ratio?Answer: 50/42

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Process Capability Ratio

When the process is not centered, process capability ratio

Min{Process mean - lower spec , Upper spec - Process mean} 3

Cpk=

When the process is not centered, the closest spec to mean determinesthe capability of the process because that spec is likely to be more of a limiting factor than the other.

Example: Suppose that the process is not centered in the previous example and the SOM wants all the scores to fall within 50% and 100%. What is the Capability ratio if the average score was 70?

Answer: From the lower limit, we have (70-50)/21 From the upper limit, we have (100-70)/21 Then the ratio is 20/21

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Processmean

Lowerspecification

Upperspecification

+/- 3 Sigma

+/- 6 Sigma

3 Sigma and 6 Sigma Quality

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Chapter 10 Supplement

Acceptance Sampling

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Acceptance Sampling

• Acceptance sampling: Is a lot of N products good if a random sample of n (n<N) products contain only c defects?– For example take a sample of 10(=n) milk bottles out

of every 100(=N). If 1(=c) or more bottles do not fit specifications, reject the entire lot of 100 bottles.

• c is determined to balance type I and type II errors.

• This is a smart compromise between 100% inspection and no inspection.

• Generally used for input/output inspection.

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Why not to emphasize Acceptance Sampling (AS)

• AS plans have no clearly stated economic objective. They target some levels of type I and II errors.

• AS incorporate an attitude of punishment by rejecting entire lots after examining small samples. This feeds the mistrust between supplier and the customer.

• AS does not attempt to find the root cause of defectives. It merely detects defectives. Real problem is actually finding the root cause. Some people say that:– “AS provides elegant solutions to balance type I and II

errors by making a type III error: solving the wrong problem”.