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Basic Forms of Statistical Basic Forms of Statistical Sampling for Quality ControlSampling for Quality Control

Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling).

Sampling to determine if the process is within acceptable limits (Statistical Process Control)

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ACCEPTANCE SAMPLING (AS)ACCEPTANCE SAMPLING (AS)

• INSPECTION AFTER PRODUCTION.

• HOWEVER - SHOULD NOT TRY TO INSPECT QUALITY INTO PRODUCT.

• ASAS IS AN AUDITING TOOL.

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Acceptance SamplingAcceptance Sampling• Purposes

– Determine quality level– Ensure quality is within predetermined level

• Advantages– Economy– Less handling damage– Fewer inspectors– Upgrading of the inspection job– Applicability to destructive testing– Entire lot rejection (motivation for improvement)

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DESIGNING THE PLAN• Acceptable Quality Level (AQL) = Max.

acceptable percentage of defectives defined by producer.

• (Producer’s risk)The probability of rejecting a good lot.

• Limiting Quality Level (LQL) = Lot Tolerance Percent Defective (LTPD) = Percentage of defectives that defines consumer’s rejection point.

• (Consumer’s risk) =The probability of accepting a bad lot.

Typical Application of Acceptance Sampling

• The decision to accept or reject the shipment is based on the following set standards:– Lot size = N– Sample size = n– Acceptance number = c– Defective items = d

• If d <= c, accept lot• If d > c, reject lot

OC Curve Calculation

• Two Ways of Calculating OC Curves– Binomial Distribution– Poisson formula

• Binomial Distribution– Cannot use because:

• Binomials are based on constant probabilities.• N is not infinite• p changes

OC Curve Calculation• A Poisson formula can be used

• Poisson is a limit – Limitations of using Poisson

• n<= 1/10 total batch N• Little faith in probability calculation when n is quite

small and p quite large.

.

Calculation of OC Curve

• Find your sample size, n

• Find your fraction defect p

• Multiply n*p

• A = d

• From a Poisson table find your PA

Calculation of an OC Curve

• N = 1000• n = 60• p = .01• A = 3

• Find PA for p = .01, .02, .05, .07, .1, and .12?

Np d= 3

.6 99.8

1.2 87.9

3 64.7

4.2 39.5

6 151

7.2 072

Properties of OC Curves• Ideal curve would be

perfectly perpendicular from 0 to 100% for a given fraction defective.

Properties of OC Curves• The acceptance number and sample

size are most important factors.

• Decreasing the acceptance number is preferred over increasing sample size.

• The larger the sample size the steeper the curve.

Properties of OC Curves

Properties of OC Curves• By changing the

acceptance level, the shape of the curve will change. All curves permit the same fraction of sample to be nonconforming.

Operating Characteristics (OC) Curves

• OC curves are graphs which show the probability of accepting a lot given various proportions of defects in the lot

• X-axis shows % of items that are defective in a lot- “lot quality”

• Y-axis shows the probability or chance of accepting a lot

• As proportion of defects increases, the chance of accepting lot decreases

• Example: 90% chance of accepting a lot with 5% defectives; 10% chance of accepting a lot with 24% defectives

AQL, LTPD, Consumer’s Risk (α) & Producer’s Risk (β)

• AQL is the small % of defects that consumers are willing to accept; order of 1-2%

• LTPD is the upper limit of the percentage of defective items consumers are willing to tolerate

• Consumer’s Risk (α) is the chance of accepting a lot that contains a greater number of defects than the LTPD limit; Type II error

• Producer’s risk (β) is the chance a lot containing an acceptable quality level will be rejected; Type I error

Developing OC Curves

• OC curves graphically depict the discriminating power of a sampling plan

• Cumulative binomial tables like partial table below are used to obtain probabilities of accepting a lot given varying levels of lot defectives

• Top of the table shows value of p (proportion of defective items in lot), Left hand column shows values of n (sample size) and x represents the cumulative number of defects found

Constructing an OC Curve

• Lets develop an OC curve for a sampling plan in which a sample of 5 items is drawn from lots of N=1000 items

• The accept /reject criteria are set up in such a way that we accept a lot if no more that one defect (c=1) is found

• Note that we have a 99.74% chance of accepting a lot with 5% defects and a 73.73% chance with 20% defects

Average Outgoing Quality (AOQ)

• With OC curves, the higher the quality of the lot, the higher is the chance that it will be accepted

• Conversely, the lower the quality of the lot, the greater is the chance that it will be rejected

• The average outgoing quality level of the product (AOQ) can be computed as follows: AOQ=(Pac)p

• AOQ can be calculated for each proportion of defects in a lot by using the above equation

• This graph is for n=5 and x=1 (same as c=1)

• AOQ is highest for lots close to 30% defects

Implications for Managers

• How much and how often to inspect?– Consider product cost and product volume– Consider process stability– Consider lot size

• Where to inspect?– Inbound materials– Finished products– Prior to costly processing

• Which tools to use?– Control charts are best used for in-process production– Acceptance sampling is best used for

inbound/outbound

OC CurvesP

rob

abil

ity

of

Acc

epti

ng

Lo

t

Lot Quality (Fraction Defective)

100%

75%

50%

25%

.03 .06 .09

OC Curves come in various shapes depending on the sample size and risk of and errors

This curve is more discriminating

This curve is less discriminating

OC Definitions on the CurveP

rob

abil

ity

of

Acc

epti

ng

Lo

t

Lot Quality (Fraction Defective)

100%

75%

50%

25%

.03 .06 .09

= 0.1090%

= 0.10

AQ

L

LTP

D

IndifferentGood Bad

DEFINING GOOD AND BAD SHIPMENTS: AQL VERSUS LTPD

• Instead of simply "good" versus "bad", we will define "really good", "really bad", and "ok, but not great" shipments

– A really good shipment has p <= AQL– A really bad shipment has p >= LTPD– Anything in between (AQL < p < LTPD) is ok, but not

great

THE OPERATING-CHARACTERISTIC (OC) CURVE

• For a given a sampling plan and a specified true fraction defective p, we can calculate – Pa -- Probability of accepting lot

• If lot is truly good, 1 - Pa =

• If lot is truly bad, Pa =

• A plot of Pa as a function of p is called the OC curve for a given sampling plan

THE OPERATING-CHARACTERISTIC (OC) CURVE

• The ideal sampling plan discriminates perfectly between good and bad shipments – Both and are zero in this example!– This requires a sample size equal to the population -- not

feasible

p

Pa

1.0

0.0

GOOD BAD

USING AN OC CURVE

• How do we find and using an OC curve?– AQL = 0.01– LTPD = 0.05

• Then 1 – Pa(p=0.01) = 1 - 0.9206 = 0.0794

• And = Pa(p=0.05) = 0.1183

AVERAGE OUTGOING QUALITY • Consider a part with a long-term fraction

nonconforming of p– Samples of size n are taken from a lot of size N and

inspected– Any defectives in the sample of size n are replaced, accept

or reject

• When a lot of is accepted, we expect p(N-n) defectives in the remainder of the lot

• When a lot is rejected, it will be sorted and defective units replaced, leaving N-n good units in the remainder

• This is referred to as "rectifying" inspection

AVERAGE OUTGOING QUALITY

• If Pa is the probability of accepting a lot, then the average outgoing quality is:

N

)nN(pPAOQ a

0049.010000

)10010000)(005.0(9859.0AOQ

• The worst possible AOQ is the AOQ Limit or AOQL

AVERAGE TOTAL INSPECTION

• Rectifying plans have greater inspection requirements

• The Average Total Inspections:

)nN)(P1(nATI a

240)10010000)(9859.01(100ATI

Types of Acceptance sampling Plans

Single-sampling plan

Double-sampling plan

Multiple-sampling plan

Sequential-sampling plan

Single-sampling plan

(n,c)

Acc the lot

Reject the lot

Sn <C

Sn>C

(N, p)

Total number ： N

The proportion of defects :P

Where Sn is the number of the actual defects in the sample.

DOUBLE SAMPLING PLANS

• Define:– n1 -- sample size on first sample– c1 -- acceptance number for first sample– d1 -- defectives in first sample– n2 -- sample size on second sample– c2 -- acceptance number for both samples– d2 -- defectives in second sample

• Take sample of size n1 – Accept if d1 c1; reject if d1 c2; – Take second sample of size n2 if c1 < d1 c2

– Accept if d1+d2 c2; reject if d1+d2 c2

Multiple-sampling plan

……..(n,p)

Acc the lot

Reject the lot

Sn1<c1

Sn1>c1

(n1+n2)c1<Sn1<r1

Acc the lot

Reject the lot

S(n1+n2)<r2

S(n1+n2)>r2

(n1+n2+n3)c2<S(n1+n2)<r2

Sequential Sampling Plan

)(

1)(

'

'

2

1

POC

POC

risksConsumer

risksproducer

Example:AQL=1 %LQL=6 %OC (1%)=0.95OC(6 %)=0.10

(n,c)=(89, 2)

0.05;

0.10

n ,c’s effect on OC-Curve

Example: Acceptance Sampling Problem

Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.

Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel.

Example: Step 1. What is given and what is not?

In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.

What you need to determine your sampling plan is “c” and “n.”

Example: Step 2. Determine “c”

First divide LTPD by AQL.LTPD

AQL =

.03

.01 = 3

Then find the value for “c” by selecting the value in the Table “n(AQL)”column that is equal to or just greater than the ratio above.

Exhibit Exhibit

c LTPD/AQL n AQL c LTPD/AQL n AQL0 44.890 0.052 5 3.549 2.6131 10.946 0.355 6 3.206 3.2862 6.509 0.818 7 2.957 3.9813 4.890 1.366 8 2.768 4.6954 4.057 1.970 9 2.618 5.426

So, c = 6.

Example: Step 3. Determine Sample Size

c = 6, from Tablen (AQL) = 3.286, from TableAQL = .01, given in problem

Sampling Plan:Take a random sample of 329 units from a lot. Reject the lot if more than 6 units are defective.

Now given the information below, compute the sample size in units to generate your sampling plan.

n(AQL/AQL) = 3.286/.01 = 328.6, or 329 (always round up)

TYPES OF INSP. AND TYPES OF INSP. AND SWITCHINGSWITCHING

• STD. PLANS FACILITATE CONDITIONAL INSPECTIONS

• NORMAL - WITH NORMAL LEVEL OF DEFECTS

• TIGHTENED - WITH HIGH LEVEL OF DEFECTS

• REDUCED - WITH OUTPUT REDUCED DEFECTS

SWITCHING RULESSWITCHING RULES

• N > T (NORMAL TO TIGHTENED)

• T > N

• N > R

• R > N

• DISCONTINUANCE BECAUSE OF POOR QUALITY

ISO 2859 (ANSI/ASQC Z1.4) • One of oldest sampling systems

– Covers single, double, & multiple sampling– AQL-based: Type I error ranges 9%-1% as sample size

increases– Minimal control over Type II error – Type II error decreases as general inspection level (I, II,

III) increases– “Special” inspection levels when small samples needed

(and high Type II error probability tolerated)

• Mechanism for reduced or tightened inspection depending on recent vendor performance– Tightened -- more inspection– Reduced -- less inspection

ISO 2859 (ANSI/ASQC Z1.4)

• A vendor begins at a "normal" inspection level– Normal to tightened: 2/5 lots rejected– Normal to reduced:

• Previous 10 lots accepted (NOT ISO 2859)

• Total defectives from 10 lots ok (NOT ISO 2859)

• If a vendor is at a tightened level:– Tightened to normal: 5 previous lots accepted

• If a vendor is at a reduced level:– Reduced to normal: a lot is rejected

ISO 2859

• A vendor begins at a "normal" inspection level– Normal to reduced:

• “Switching score” set to zero• If acceptance number is 0 or 1:

– Add 3 to the score if the lot would still have been accepted with an AQL one step tighter; else reset score to 0

• If acceptance number is 2 or more: – Add 3 to the score if the lot is accepted; else reset score to 0

• If score hits 30, switch to reduced inspection

USING ISO 2859 1. Choose the AQL

2. Choose the general inspection level

3. Determine lot size

4. Find sample size code

5. Choose type of sampling plan

6. Select appropriate plan from table

7. Switch to reduced/tightened inspection as required

USING ISO 2859

USING ISO 2859

USING ISO 2859

USING ISO 2859

DODGE-ROMIG PLANS

• Developed in the 1920's

• Rectifying plans

• Requires knowledge of vendor's long-term process average (fraction non-conforming)

• Choice of LTPD or AOQL orientation– Both minimize ATI for specified process average– Type II error = 10%,

DODGE-ROMIG PLANS

• AOQL plans:– 1) Determine N, p, and AOQL– 2) Use table to find n and c– Finds plan with specified AOQL which minimizes ATI– Calculate resulting LTPD with Type II error = 10%

• LTPD plans:– 1) Determine N, p, and LTPD– 2) Use table to find n and c– Finds plan with specified LTPD which minimizes ATI– Calculate resulting AOQL

DODGE-ROMIG PLANS

DODGE-ROMIG PLANS