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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.
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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
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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
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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.
.
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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
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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
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Properties of OC Curves• Ideal curve would be
perfectly perpendicular from 0 to 100% for a given fraction defective.
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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.
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Properties of OC Curves
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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AVERAGE TOTAL INSPECTION
• Rectifying plans have greater inspection requirements
• The Average Total Inspections:
)nN)(P1(nATI a
240)10010000)(9859.01(100ATI
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Types of Acceptance sampling Plans
Single-sampling plan
Double-sampling plan
Multiple-sampling plan
Sequential-sampling plan
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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.
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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
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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
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Sequential Sampling Plan
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)(
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
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n ,c’s effect on OC-Curve
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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.
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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.”
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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.
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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)
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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
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SWITCHING RULESSWITCHING RULES
• N > T (NORMAL TO TIGHTENED)
• T > N
• N > R
• R > N
• DISCONTINUANCE BECAUSE OF POOR QUALITY
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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
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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
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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
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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
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USING ISO 2859
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USING ISO 2859
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USING ISO 2859
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USING ISO 2859
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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%,
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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
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DODGE-ROMIG PLANS
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DODGE-ROMIG PLANS