1 acceptance sampling and statistical process control

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Acceptance Sampling and Statistical Process Control

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Probability Review

Permutations (order matters)

• Number of permutations of n objects taken x at a time

x)!(n

n!Pxn

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Probability Review

Permutations (cont.)• Example: Number of permutations of 3

letters (A, B, C) taken 2 at a time

(A,B,C) AB, BA, AC, CA, BC, CB

6!1

!3

)!2(3

3!P23

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Probability Review

Combinations (order does not matter)

• Number of combinations of n objects taken x at a time

x)!(nx!

n!

x

nCxn

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Probability Review (cont.)

Combinations• Example: Number of combinations of 3

letters (A, B, C) taken 2 at a time

(A, B, C) = AB, AC, BC

32

6

!1!2

!3

)!2(32!

3!

2

3

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Probability Review

Binomial Distribution• Sum of a series of independent, identically distributed

Bernoulli random variables• Probability of x defectives in n items• p = probability of “success” (usually determined from

long-term process average)• 1-p = probability of “failure”

n0,1,2,...,x;p)(1px

nP(x) xnx

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Probability Review

Binomial Distribution (cont.)

• Example: What is the probability of 2 defectives in 4 items if p = 0.20?

0.1536(0.80)(0.20)2

4P(2) 22

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Probability Review

Binomial Distribution Exercises

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Probability Review

Hypergeometric Distribution• Random sample of size n selected from N items,

where D of the N items are defective• Probability of finding r defectives in a sample of size n

from a lot of size N

n

N

rn

DN

r

D

P(r)

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Probability Review

Hypergeometric Distribution (cont.)

• Example: What is the probability of finding 0 defects in 10 items taken from a lot of size 100 containing 4 defects?

0.652

10

100

10

96

0

4

P(0)

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Probability Review

Hypergeometric Distribution Exercises

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Probability Review• Binomial Approximation of Hypergeometric

• Use when n/N 0.10• P ~ D/N• Example: What is the probability of finding 0 defects

in 10 items selected from a lot of size 100 containing 40 defects?

• We got 0.652 from the straight hypergeometric.

0.665(0.60)(0.40)0

10P(0) 100

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Probability Review

• Poisson Distribution (to approximate Binomial)• Use when n20 and p0.05 =np (average number of defects)

• Example: What is the probability of finding 2 defects in 4 items if p = 0.20?

• We got 0.1536 with the straight Binomial.

0,1,2,...x;x!

λeP(x)

xλ

0.14382!

(0.8)eP(2)

20.8

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Acceptance Sampling• Definition: process of accepting or rejecting a

lot by inspecting a sample selected according to a predetermined sampling plan

• Notation: N = batch sizen = sample sizec = acceptance numberp = proportion defective (known or

long-term average)

Pa = probability that a batch will be accepted

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and Risks• Type I Error: rejecting an acceptable lot (a.k.a.

producer’s risk)

P(Type I Error) =

• Type II Error: accepting an unacceptable lot (a.k.a. consumer’s risk)

P(Type II Error) =

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Operating Characteristic (OC) Curves

• OC Curves characterize acceptance sampling plans.

• OC Curves are complete plotting of Pa for a lot at all possible values of p.

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OC Curves

• Steepness of OC curves indicates the power of the acceptance sampling plans to distinguish “good” lots from “bad” lots.

Power = 1-P(accepting lot | p)= P(rejecting lot | p)

• For large values of p (i.e., large number of defects), we want the Power to be large (i.e., close to 1).

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OC Curves

• Type A OC Curve• Uses known value for p (i.e., lot composition is

known)• Can use hypergeometric distribution

• Type B OC Curve• Uses process average for p• Can use binomial distribution

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Acceptance Sampling Plans

• Considerations:

-risk

-risk

• Acceptable Quality Level (AQL)

• Lot Tolerance Percent Defective (LTPD)

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Acceptable Quality Level (AQL)• Maximum percent defective that is acceptable

= P(rejecting lot | p = AQL)

• Corresponds to higher Pa (left-hand side of OC Curve)

Lot Tolerance Percent Defective (LTPD)• Worst quality that is acceptable (accepted with low

probability) = P(accepting lot | p = LTPD)

• Corresponds to lower Pa (right-hand side of OC Curve)

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• Single Sampling• Take a single sample from the lot for inspection• Quality of sampled work determines lot decision

• Double Sampling• One small sample from the lot for inspection• If quality of first sample is acceptable, that sample

determines lot decision• If quality of first sample is unacceptable or not clear,

select a second small sample to make lot decision

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Single-Sampling vs. Double-Sampling Plans

• Either plan can satisfy AQL and LTPD requirements

• Double sampling often results in smaller total sample sizes

• Double sampling can increase cost if second sample is required often

• Psychological advantage to double sampling (second chance)

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Inspection• Inspection involves verifying the quality of a

work unit.

• Most inspection includes rectification of errors found.

• Rectification:• 100% of defective work units are repaired or replaced

• Rejected lots are 100% verified and rectified

• Verifiers should have higher level of understanding to establish confidence in the inspection/rectification process.

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Average Outgoing Quality

Average Outgoing Quality (AOQ) = proportion defective after inspection and rectification• AOQ curve relates outgoing quality to incoming

quality

• Average Outgoing Quality Level (AOQL) = point where outgoing quality is worst

• Maximum AOQ over all possible values of p

N

Ppn)(NAOQ a

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Average Outgoing Quality Level

(Table 1 provides values of y for c = 0,1,2,…,40)

For small sampling fractions,

yn

)Nn

(1AOQL

n

yAOQL

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Determining sample size for desired outgoing quality (AOQL):• Given desired AOQL• Given desired acceptable number of defects, c• Given y-value from Table 1 for desired c

AOQL

yn

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Determining AOQL for known sample size and desired acceptable number of rejects, c:• Given sample size, n• Given desired c-value• Given y-value from Table 1 for desired c-value

• Adjust c-value and n to manipulate the AOQL

n

yAOQL

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Additional Acceptance Sampling Terminology

Average Sample Number (ASN)

• Average number of sample units inspected to reach lot decision

• In single-sampling plan: ASN = n• In double-sampling plan: n1 ASN n1 + n2

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• Average Total Inspection (ATI):

• Average number of units inspected per lot• In single-sampling plan: n ATI N• ATI = n + (1-Pa)(N-n)

• Better incoming quality less inspection

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• Average Fraction Inspected (AFI):

• Average fraction of units inspected per lot• AFI = ATI/N• In single-sampling plan: n/N AFI 1

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Acceptance Sampling

Acceptance Sampling Plan Exercise

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Other Sampling Plans• Continuous Sampling

• Not possible to use lots/batches• Level of inspection depends on perceived quality

level

• Continuous Sampling Plan 1 (CSP1)• Start at 100% inspection• After i consecutive non-defectives, go to sample

inspection• Go back to 100% inspection when a defective is

found

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Other Sampling Plans

• Chain Sampling• Like standard sampling plans with c=0• However, allow 1 defect in a lot if previous i lots were

defective-free• Useful for small lots where c=0 is required

• Skip-Lot Sampling• Lot-based continuous sampling• Lots inspected 100% until i lots are defective-free,

then go to sample inspection of lots

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Acceptance Sampling Trade-Off

Sampling Fraction vs. Acceptance Criteria

• Assuming a set quality level (AOQL), we can make choices regarding inspection rates and acceptance criteria

• To decrease inspection rate, we must tighten acceptance criteria

• To allow more defects, we must increase sample size

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Acceptance Sampling Trade-Off

• Example: Desired AOQL = 5%, Batch Size 1-800 items

• 25% inspection accept if 6 or fewer defects in a sample of size 60

• 10% inspection accept if 5 or fewer defects in a sample of size 60

• See Tables 1 and 2

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Creating Work Units

• Homogeneity• Alike items within a batch• Helps to keep samples representative

• Batch Size• Rule-of-thumb: ½ to 1 day of work• Very small batch frequent QC, more paperwork• Very large batch delayed feedback, higher rework

risk

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Sample Selection

Important for sample selection to be completely random

• Random Number Table

• Number batch items 1 through N

• Select random point in random number table

• Use next n numbers in the table as the batch items to select

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Sample Selection

Systematic Sampling

• Select a random start between 1 and 10

• Select every xth item until n items are selected

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U.S. Census Bureau Applications

Address Canvassing• Three random starts within listing pages• One random start provides start for check of

total listings• Two random starts provide start for check of

added housing units and deleted housing units• No errors allowed• QA form example from upcoming 2004 Census

Test

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Review of Map Improvement FilesSelect map “features” for QC reviewAcceptance Sampling plan

Global AQL requirementSample size dependent on “batch” sizeAcceptance number dependent on sample size

Uses stratified sampling

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Sample = 200 HIDs

Acceptance Number = 14

0 to 10,000 HIDs

AQL = 4%

Matched Unmatched Added

Road

Water

Other

Sample = 315 HIDs

Acceptance Number = 21

10,001 to 500,000 HIDs

AQL = 4%

Matched Unmatched Added

Road

Water

Other

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Process Control• Allows us to plan quality into our

processes

• Spend fewer resources on inspection and rework

• Observe processes, collect samples, measure quality, determine if the process produces acceptable results

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Process Control

Stresses prevention over inspection• Traditional approach to QC:

• Most resources spent on inspection and rework

• Process Control approach to QC:• Most resources spent on prevention with

relatively little spent on inspection and rework• Lower overall cost

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Sources of Variation

Random Sources

• Cause of variation is common or unassignable

• Process is still “in control”

• Difficult or impossible to eliminate

• Requires modification to the process itself

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Non-Random Sources

• Cause of variation is special and assignable

• Could be difficult to eliminate

• Causes process to be “out of control”

• Can address the specific cause of the variation

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Diagnosing non-random variation

• “2-out-of-3” – if two out of three consecutive points are out of control

• “4-out-of-5” – if four out of five consecutive points are out of control

• “7 successive” – if seven consecutive points are on one side of the process average

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Control Charts

• Graphical device for assessing statistical control

• Plots data from a process in time order

• Three reference lines:

• Upper Limit

• Center Line (CL)

• Lower Limit

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Control Charts

• CL represents the process average

• Upper and lower limits represent the region the process moves in under random variation

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Control Charts

Control chart limits can be Control Limits or Specification Limits• Control Limits: based on quality capability of

the process• Control limits traditionally set at 3 standard

deviations away from the CL

• Specification Limits: based on quality goal

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Control Charts

• Example: Chemical process with assumed concentration of 3%, desired concentration 3.2% and 2.8%

• Specification Limits: LSL = 2.8, USL = 3.2

• Control Limits: LCL = 2.5, UCL = 3.5 (based on true capability of process)

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Control Charts

A process is considered to be “in control” if the process data move randomly between the upper and lower limits

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Control ChartsTime for Enumerating Blocks

0

2

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1 3 5 7 9 11 13 15 17 19 21 23 25

Day

Ho

urs

to

En

um

era

te B

lock

UCL

CL

LCL

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Control Charts

• Examples:

• Process: enumerator canvassing a block

• Measure: hours to canvass the block

• Is the process in statistical control?

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Control Charts

• The process data in time order move randomly inside the control limits

• Therefore, the process IS in statistical control

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Control ChartsTime for Enumerating Blocks

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1 3 5 7 9 11 13 15 17 19 21 23 25

Day

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UCL

CL

LCL

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Control Charts

• Some points above the UCL• Therefore, process IS NOT in statistical

control• Might be hard-to-enumerate blocks

(special cause)• Field supervisor might want to send a

more experienced lister to canvass those blocks

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Control ChartsTime to Enumerate Blocks

0

5

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15

20

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Day

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lock

UCL

CL

LCL

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Control Charts

• Process has definite upward trend

• Therefore, process IS NOT in statistical control

• Perhaps lister has forgotten proper techniques (special cause)

• Field supervisor might want to consider re-training the lister

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1 3 5 7 9 11 13 15 17 19 21 23 25

Day

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to

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era

te B

locks

UCL

CL

LCL

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Control Charts

• Sometimes, patterns are hard to see

• This example seems to show a random distribution of points

• However, look what happens when we connect the points:

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0

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1 3 5 7 9 11 13 15 17 19 21 23 25

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UCL

CL

LCL

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Control Charts

• The points are all inside the control limits

• So, the process IS in statistical control

• However, there is a definite cyclical pattern

• These types of patterns are not random and should be investigated

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Control Charts

Control Chart Exercises

1 acceptance sampling and statistical process control

Documents

paccepting lot p

lot composition

unacceptable lot

probability of success

n objects

oc curvestype

n itemsp

large number of defects