7/2/2015tm 720: statistical process control1 tm 720 - lecture 11 acceptance sampling plans

04/19/23 TM 720: Statistical Process Control 1

TM 720 - Lecture 11

Acceptance Sampling Plans


Assignment:

Reading:• Finish Chapter 14

• Sections 14.1 – 14.2

• Sections 14.4

• Start Chapter 12

Assignment:• Download and complete Assign 08: Acceptance Sampling

• Requires MS Word for Nomograph

• Requires MS Excel for AOQ

• Solutions for 8 will post on Thursday


Acceptance Sampling

Company receives shipment fromvendor

Sample taken from lot,Quality characteristic inspected

Lot Sentencing:Accept lot?

YES

Return lotto vendor

NO

Use lot inproduction


Three Important Aspects of Acceptance Sampling

1. Purpose is to sentence lots, not to estimate lot quality

2. Acceptance sampling does not provide any direct form of quality control. It simply rejects or accepts lots. Process controls are used to control and systematically improve quality, but acceptance sampling is not.

3. Most effective use of acceptance sampling is not to “inspect quality into the product,” but rather as audit tool to insure that output of process conforms to requirements.


Three Approaches to Lot Sentencing

1. Accept with no inspection

2. 100% inspection – inspect every item in the lot, remove all defectives

Defectives – returned to vendor, reworked, replaced or discarded

3. Acceptance sampling – sample is taken from lot, a quality characteristic is inspected; then on the basis of information in sample, a decision is made regarding lot disposition.


Acceptance Sampling Used When: Testing is destructive 100% inspection is not technologically feasible 100% inspection error rate results in higher percentage

of defectives being passed than is inherent to product Cost of 100% inspection extremely high Vender has excellent quality history so reduction from

100% is desired but not high enough to eliminate inspection altogether

Potential for serious product liability risks; program for continuously monitoring product required


Advantages of Acceptance Sampling over 100% Inspection

Less expensive because there is less sampling Less handling of product hence reduced damage Applicable to destructive testing Fewer personnel are involved in inspection activities Greatly reduces amount of inspection error Rejection of entire lots as opposed to return of

defectives provides stronger motivation to vendor for quality improvements


Disadvantages of Acceptance Sampling (vs 100% Inspection)

Always a risk of accepting “bad” lots and rejecting “good” lots• Producer’s Risk: chance of rejecting a “good” lot –

• Consumer’s Risk: chance of accepting a “bad” lot –

Less information is generated about the product or the process that manufactured the product

Requires planning and documentation of the procedure – 100% inspection does not


Lot Formation Lots should be homogeneous

• Units in a lot should be produced by the same: • machines, • operators, • from common raw materials, • approximately same time

• If lots are not homogeneous – acceptance-sampling scheme may not function effectively and make it difficult to eliminate the source of defective products.

Larger lots preferred to smaller ones – more economically efficient

Lots should conform to the materials-handling systems in both the vendor and consumer facilities • Lots should be packaged to minimized shipping risks and make

selection of sample units easy


Random Sampling IMPORTANT:

• Units selected for inspection from lot must be chosen at random

• Should be representative of all units in a lot

Watch for Salting:• Vendor may put “good” units on top layer of lot knowing a lax

inspector might only sample from the top layer

Suggested technique:1. Assign a number to each unit, or use location of unit in lot2. Generate/pick a random number for each unit/location in lot3. Sort on the random number – reordering the lot/location pairs4. Select first (or last) n items to make sample


Single Sampling Plans for Attributes Quality characteristic is an attribute, i.e., conforming or

nonconforming• N - Lot size

• n - sample size

• c - acceptance number

Ex. Consider N = 10,000 with sampling plan n = 89 and c = 2• From lot of size N = 10,000

• Draw sample of size n = 89

• If # of defectives c = 2 • Accept lot

• If # of defectives > c = 2 • Reject lot


How to Compute the OC Curve Probabilities

Assume that the lot size N is large (infinite)

d - # defectives ~ Binomial()where • p - fraction defective items in lot

• n - sample size

Probability of acceptance:

0

P 1c

n iia

i

nP d c p p

i


Example Lot fraction defective is p = 0.01,

n = 89 and c = 2. Find probability of accepting lot.


OC Curve Performance measure of acceptance-sampling plan

• displays discriminatory power of sampling plan Plot of: Pa vs. p

• Pa = P[Accepting Lot]

• p = lot fraction defective

p = fraction defective in lot Pa = P[Accepting Lot]

0.005 0.9897

0.010 0.9397

0.015 0.8502

0.020 0.7366

0.025 0.6153

0.030 0.4985

0.035 0.3936


OC curve displays the probability that a lot submitted with a certain fraction defective will be either accepted or rejected given the current sampling plan

Probability of Acceptance, Pa

0.00.20.40.60.81.0

0.00 0.02 0.04 0.06 0.08 0.10

Lot fraction defective, p

Pa

n=89c=2

OC Curve


Ideal OC Curve Suppose the lot quality is considered bad if p = 0.01 or more A sampling plan that discriminated perfectly between good and

bad lots would have an OC curve like:

1.00

0.040.01 0.02 0.03




Ideal OC Curve

In theory it is obtainable by 100% inspection IF inspection were error free.

Obviously, ideal OC curve is unobtainable in practice

But, ideal OC curve can be approached by increasing sample size, n.


Effect of n on OC Curve

The precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases


0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.02 0.04 0.06 0.08 0.10


Pan=50, c=1

n=100, c=2

n=200, c=4

n=1000, c=20


Effect of c on OC Curve

Changing acceptance number, c, does not dramatically change slope of OC curve.

Plans with smaller values of c provide discrimination at lower levels of lot fraction defective


0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.06 0.08 0.10


Pa

n=89, c=2

n=89, c=1

n=89, c=0


Producer and Consumer Risks in Acceptance Sampling

Because we take only a sub-sample from a lot, there is a risk that: • a good lot will be rejected

(Producer’s Risk – )

and• a bad lot will be accepted

(Consumer’s Risk – )


Producer’s Risk - Producer wants as many lots accepted by consumer as possible so

• Producer “makes sure” the process produces a level of fraction defective equal to or less than:

p1 = AQL = Acceptable Quality Level

is the probability that a good lot will be rejected by the consumer even though the lot really has a fraction defective p1

That is,

Lot is rejected given that process

has an acceptable quality levelP

Lot is rejectedP p AQL


Consumer’s Risk - Consumer wants to make sure that no bad lots are accepted

• Consumer says, “I will not accept a lot if percent defective is greater than or equal to p2”

p2 = LPTD = Lot Tolerance Percent Defective

probability bad lot is accepted by the consumer when lot really has a fraction defective p2

That is,

Lot accepted given that lot

has unacceptable quality levelP

Lot acceptedP p LTPD


Designing a Single-Sampling Plan with a Specified OC Curve

Use a chart called a Binomial Nomograph to design plan

Specify:• p1 = AQL (Acceptable Quality Level)

• p2 = LTPD (Lot Tolerance Percent Defective)

• 1 – = P[Lot is accepted | p = AQL]

• β = P[Lot is accepted | p = LTPD]


Use a Binomial Nomograph to Find Sampling Plan (Figure 14-9, p. 658)

Draw two lines on nomograph• Line 1 connects p1 = AQL to (1- )

• Line 2 connects p2 = LTPD to • Pick n and c from intersection of lines

Example: Suppose • p1 = 0.01,

• α = 0.05,

• p2 = 0.06,

• β = 0.10.

Find the acceptance sampling plan.


Rectifying Inspection Programs

Acceptance sampling programs usually require corrective action when lots are rejected, that is, • Screening rejected lots

• Screening means doing 100% inspection on lot

In screening, defective items are• Removed or

• Reworked or

• Returned to vendor or

• Replaced with known good items


Rectifying Inspection Programs

InspectionActivity

Rejected Lots: 100%

Inspected

AcceptedLots

FractionDefective

Incoming Lots:Fraction Defective

FractionDefective = 0

Outgoing Lots:Fraction Defective

0p

0p

1 0p p


Where to Use Rectifying Inspection Used when manufacturer wishes to know average level

of quality that is likely to result at given stage of manufacturing

Example stages:• Receiving inspection

• In-process inspection of semi-finished goods

• Final inspection of finished goods

Objective: give assurance regarding average quality of material used in next stage of manufacturing operations


Average Outgoing Quality: AOQ Quality that results from application of rectifying

inspection• Average value obtained over long sequence of lots from

process with fraction defective p

N - Lot size, n = # units in sample Assumes all known defective units replaced with good

ones, that is, • If lot rejected, replace all bad units in lot

• If lot accepted, just replace the bad units in sample

aP p N nAOQ

N


Development of AOQ If lot accepted:

Number defective units in lot:

Expected number of defective units:

Average fraction defective,Average Outgoing Quality, AOQ:

# units

fraction remaining

defectivein lot

p N n

Lot # defectiveProb

accepted units in lotaP p N n

aP p N nAOQ

N


Example for AOQ Suppose N = 10,000, n = 89, c = 2, and incoming lot quality is p

= 0.01. Find the average outgoing lot quality.


Military Standard 105E(MIL STD 105E)(ANSI/ASQC Z1.4, ISO 2859)

Most widely used acceptance sampling system for attributes

MIL STD 105E is Acceptance Sampling System• collection of sampling schemes

Can be used with single, double or multiple sampling plans • We will consider single sampling plans for this course


Inspection Types Normal Inspection

• Used at start of inspection activity

Tightened Inspection• Instituted when vendor’s recent quality history has

deteriorated

• Acceptance requirements for lots are more stringent

Reduced Inspection• Instituted when vendor’s recent quality history has been

exceptionally good

• Sample size is usually smaller than under normal inspection


Switching Rules

- Production Steady- 10 consecutive lots accepted- Approved by responsible authority

NormalReduced Tightened

- Lot rejected- Irregular production- Lot meets neither accept nor reject criteria- Other conditions warrant return to normal inspection

2 out of 5 consecutive lotsrejected

5 consecutivelots accepted

10 consecutive lots remainon tightened inspection

Start

DiscontinueInspection

AND conditions

OR conditions


Procedure for MIL STD 105E

STEP 1: Choose AQL• MIL STD 105E designed around Acceptable Quality

Level, AQL• Recall that the Acceptable Quality Level, AQL, is producer's

largest acceptable fraction defective in process

• Typical AQL range: • 0.01% AQL 10%

• Specified by contract or authority responsible for sampling


STEP 2: Choose inspection level• Level II

• Designated as normal

• Level I

• Requires about one-half the amount of inspection as Level II

• Use when less discrimination needed

• Level III

• Requires about twice as much

• Use when more discrimination needed

• Four special inspection levels used if very small samples necessary

• S-1, S-2, S-3, S-4



STEP 3–Determine lot size, N• Lot size most likely dictated by vendor

STEP 4: Find sample size code letter • From Table 14-4, p 675

• Given lot size, N, and Inspection Level, use table to determine sample size code letters

STEP 5: Determine appropriate type sampling plan• Decide if Single, Double or Multiple sampling plan is to be

used



STEP 6: Find Sample Size, n, and Acceptance Level, c

• Given sample size letter code, use Master Tables: 14-5, 14-6, and 14-7 on pp.676-678

• Find n and c for all three inspection types:• Normal Inspection

• Tightened Inspection

• Reduced Inspection



Example Suppose product comes from vendor in lots of size 2000 units.

The acceptable quality level is 0.65%. Determine the MIL STD 105E acceptance-sampling system.


Questions & Issues

7/2/2015tm 720: statistical process control1 tm 720 - lecture 11 acceptance sampling plans

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