LOT–BY-LOT ACCEPTANCE SAMPLING BY …fkm.utm.my/~shari/download/qc09 acceptance sampling.pdf ·...
Transcript of LOT–BY-LOT ACCEPTANCE SAMPLING BY …fkm.utm.my/~shari/download/qc09 acceptance sampling.pdf ·...
Acceptance Sampling
acceptancesamplingshari.fkm
LOT–BY-LOT ACCEPTANCE SAMPLING
BY ATTRIBUTES Acceptance Sampling is the process of evaluating a
portion of the product in a lot for the purpose of
accepting or rejecting the entire lot as either
conforming or nonconforming to a quality specification.
• a predetermined no. of units inspected from
each lot.
• If no. of nonconforming < minimum no. lot
accepted else rejected.
• plans established according to severity (major,
minor, critical).
• performed when there is consumer-producer
relationship –2 different depts. within company,
manufacturer to vendor. Want to decide OK or
NG.
Acceptance Sampling
acceptancesamplingshari.fkm
Main advantage of A.S. is econo my/cost involved Situations to use A.S.
1. When test is destructive
2. when cost of 100% insp. is high with respect to
passing a nonconforming unit.
3. when many similar units need to be inspected.
Sampling vs 100% Better or as good as 100%.
Manual inspection boredom, fatigue
monotonous, tend to miss.
4. when info. concerning producers quality not
known/available.
5. when automated insp. not available
Advantages & Disadvantages.
Acceptance Sampling
acceptancesamplingshari.fkm
4 types of Sampling Plans
1. Single
2. Double
3. Multiple
4. Sequential
Single S.P.
Defined by lot size N,
sample size n,
acceptance number c.
If SSP N = 9000 n = 300 c = 2 Intrepretation of Plan Inspect 300 pieces from the lot of 9000 units If nonconforming (nc) found ≤ (c) 2 then ACCEPT LOT or If nc (defectives) > 2 then REJECT LOT
Acceptance Sampling
acceptancesamplingshari.fkm
Doub le Sampling Plan (slightly more complicated) DSP Initial sample, decision based on 1st insp.
(a) accept lot – quality good (b) reject lot – quality bad, no 2nd round. (c) take another sample – neither good
nor bad. ∴ 2nd chance
N = Lot size N1 = Sample size 1st sample (Ac) C1 = Acceptance number on 1st sample (Re) R1 = Rejection number on 1st sample N2 = Sample size 2nd sample C2 = Acc. no. fax both samples R2 = Rej. no. for both samples Ex. N = 9000 N2 = 150 N1 = 60 C2 = 6 C1 = 1 R2 = 7 R1 = 5
Acceptance Sampling
acceptancesamplingshari.fkm
OC Curve
Operating characteristics Curve (OC)
- An impt. Measure of the performance of an
acceptance sampling plan.
- Curve plots the probability of accepting the
lot versus the lot percent non-conf. or lot
frac. def.
- Show probability that a lot with certain
fraction defective will be either accepted or
rejected.
Low % Nonconf will have prob. lot acceptance high. vice-versa
Acceptance Sampling
acceptancesamplingshari.fkm
Construction of an OC Curve (SS)
Ex N = 3000
n = 89
C = 2
Pa = prob. of acceptance
Assume 100po = 2% ∴ po = 0.02
Since n = 89
∴ npo = 1.8
If c ≤ 2, Pa = P(0) + P(1) + P(2)
Refer Poisson Table. Look at npo = 1.8 and c≤ 2
Find P(2 or less) = ________
Assuming lots are coming from steady stream. Use Poisson as an approx. to Binomial to determine Pa.
po = 0.01 npo = (89) x 0.01 = 8.9 = 9
Acceptance Sampling
acceptancesamplingshari.fkm
Now, you can assume for different process quality
levels, i.e. % nonconforming/frac. defective.
Steps. 1) Assume po value
2) Calculate npo value
3) Find Pa values from Poisson Table using
c, npo
4) Plot point (Pa) (100po, 100pa)
You can use a table
Assumed Process Quality
Sample Size (n)
npo Prob. accept
Percent of lots acc
po 100po Pa 100Pa 0.01 0.02 0.03 0.04 0.05 0.06 0.07
1.0 2.0 3.0 4.0 5.0 6.0 7.0
89 89 89 89 89 89 89
0.9 1.8 2.7 3.6 4.5 5.3 6.2
0.938 0.731 0.494 0.302 0.174 0.106 0.055
93.8 73.1 49.4 30.2 17.4 10.6 5.5
In this case 7 points enough to plot the curve.
Acceptance Sampling
acceptancesamplingshari.fkm
• It shows the chance/prob. of a lot being accepted
for particular incoming quality level.
• If 2.3 % process quality
Pa = 0.66
If 100 lots insp. from that lot
66 lots will be accepted
34 lots will be rejected.
• Curve is unique to each sampling plan (N, n, c)
• When % NC low, Pa is large
% NC high, Pa is small.
Acceptance Sampling
acceptancesamplingshari.fkm
OC Curve Properties
4 main properties giving diff. OC Curves
1. Sample size as a fixed percentage of lot size.
Before A.S. concepts, insp. done on fixed % of lot
size and with zero accept. no.
Say, 10% for lot size 900,300,90
∴ 1. N = 900 n = 90 c = 0
2. N = 300 n = 90 c = 0
3. N = 90 n = 90 c = 0
OC Curves for 10% of lot
They offer different levels of protection if 100p0 = 5% n = 900 Pa = 0.02 N = 300 Pa = 0.22 N = 90 Pa = 0.63 (Better protection)
Acceptance Sampling
acceptancesamplingshari.fkm
2. Fixed sample size
• Fixed sample size, OC curves are similar
Type B n ≥ 10% N – Poisson/Binomial Type A n < 10% N - Hypergeometric
True for both cases When n ≥ 10% N or when n < 10% N Shape of curve changes with sample size more than N itself.
Acceptance Sampling
acceptancesamplingshari.fkm
3. As sample size increases, curve becomes steeper
• also approaches a straight vertical line
• Sampling plans with large sample sizes better
able to discriminate acceptable and
unacceptable quality
• Greater the slope the better the discriminating
power – good & bad lots.
Acceptance Sampling
acceptancesamplingshari.fkm
4. Acceptance number and shape of curve
• As c �
, curve becomes steeper , justify use
of c = 0
• notice that slope steeper with higher n = 300
as compared with n = 50
OC Curves for different acceptance numbers (c)
Acceptance Sampling
acceptancesamplingshari.fkm
Consumer-Produ cer Relationship
• When A.S. used � conflicting interest between
consumer – producer.
• Producer - all acceptable lots to be accepted.
• Consumer - all UNACCEPTABLE lots rejected.
• An ideal sampling plan can satisfy this
- ‘ideal’ OC Curve thru 100% insp.
∴ When you do sampling, it will carry risks of
rejecting good lots and accepting bad. lots
Runs horiz at Pa = 1.00 until lot qual. Level considered bad Pa = 0
Acceptance Sampling
acceptancesamplingshari.fkm
2 TYPES OF RISKS
1. PRODUCER’S RISK
• represented by α
• α = prob. of rejection of good lot
α usually 0.05; Ranges 0.0001 – 0.10
• since α - prob. of rejection on the OC curve
α = 1 – Pa OR Pa = 1 -α
• Related to α is numerical definition of
acceptable lot called ACCEPTABLE
QUALITY LEVEL (AQL).
• AQL = max. % nonconforming considered
satisfactory for PURPOSE of A.S. It is ref.
point on OC curve NOT meant to show that
any % nonconforming is acceptable.
Eg. N = 4000 n = 300 c = 4 AQL = 0.7% α = 0.05
Means: Product with 0.7% non-conf will have rejection prob. of 0.05 or 5% of time
Acceptance Sampling
acceptancesamplingshari.fkm
CONSUMER-PRODUCER RELATIONSHIP
Acceptance Sampling
acceptancesamplingshari.fkm
II. CONSUMER’S RISK
• Represented by β
• β is prob. of accepting a bad lot
• usually β = 0.10
• since shown as Pa, no change
• related with β is numerical def. Of
nonconforming lot, LIMITING QUALITY
LEVEL (LQL).
• LQL = % non conforming in a lot which
consumers wants Pa to be low.
• LQL = 2.6%, β = 0.10 means lots with 2.6%
nc will have chance of acceptance equals
0.10 or 10% i.e. 1 out of 10 with 2.6% nc will
be accepted by this S.P.
Acceptance Sampling
acceptancesamplingshari.fkm
AVERAGE OUTGOING QUALITY (AOQ)
• AOQ another way to evaluate a sampling
plan
• Provides answer to :
‘What is the average quality in all the lots
after rejected lots have been sorted 100%
and defectives removed?’
• Because rejected lots are rectified AOQ is
always better than incoming quality level.
Incoming lots ➔ ➔ fraction defective Po
A.S.
Rej. Lots
Acc. Lots
Frac Def = 0
Frac def PO
OUT GOING
P,< Po
Acceptance Sampling
acceptancesamplingshari.fkm
Using same example : N = 3000, n= 89, c=2
Need one more column (for AOQ Table )
AOQ = (100po) x (Pa) – (in % def)
100pO - % defective Pa ia prob of
accepting
Process
Quality
100Po
Sample
size
N
npO
Prob of
acceptance
Pa
AOQ
= 100po. Pa
1.0 89 0.9 0.938 0.938
2.0 89 1.8 0.731 1.146
3.0 89 2.7 0.494
4.0 89 3.6 0.302
5.0 89 4.5 0.174
6.0 89 5.3 0.106
7.0 89 6.2 0.055
Acceptance Sampling
acceptancesamplingshari.fkm
• Analysis – when incoming quality is 2.0% non-conf,
the average outgoing quality is 1.46%
• There is a limit – AOQL- for this sampling plan as
the percentage non-conf changes, the average
outgoing quality never exceeds 1.6%
Acceptance Sampling
acceptancesamplingshari.fkm
To understand better A.S. concept
Suppose 15 lots of 3000 prod. ➔ consumer
Lots are 2% nc and the SSP
is n = 89 c = z
Acceptance Sampling
acceptancesamplingshari.fkm
At 2% nc Pa = 0.731
∴ Out of 15, No of Accepted Lots = 0.731 x 15 =
10.97 ~ 11 lots
∴ 4 lots rejected.
Total number:
11 lots @ 2% nc 11 x 3000 = 33000
4 lots @ 0% nc 4(3000) (0.98) = 11,760
(240 discarded) 44,760 ====== Number non-conf form the11 lots = 33,000x0.02 = 660
AOQ(% nc) = =%100x760,44
660
1.47%
Acceptance Sampling
acceptancesamplingshari.fkm
SAMPLING PLAN DESIGN Specified Producer’s risk When α, AQL specified - a family of sampling plan can be found. Say α = 0.05
AQL = 1.2 %
All of these ensures that prdt. Having 1.2% nc rejected 5% of time or acepted 95% i.e. Pa = 0.95
Acceptance Sampling
acceptancesamplingshari.fkm
ALL GIVE SAME PROT. FOR PRODUCER
How to get the above SSP?
(1) assume C
(2) find np using Table below
Pa = 0.95 p0.95 = 0.012
When c = 1, np0.95 = 0.355 ∴ n = 012.0355.0
Pnp 95.0 =
c = 2, np0.95 = 0.818 ∴ n c = 6, np0.95 = 0.826 ∴ n
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Acceptance Sampling
acceptancesamplingshari.fkm
But if consumer risk β = 0.10 c = 1, n = 30 ➔ LQL = 13% Pa = 0.10
c = 2, n = 68 ➔ LQL = 7.8% Pa = 0.10
c = 1, n = 274 ➔ LQL = 3.8% Pa = 0.10
Which one gives better protection for consumer?
Stipulated α, AQL does not guarantee consumer
protection.
Acceptance Sampling
acceptancesamplingshari.fkm
Comments
• Sampling plan designs involve fixing α, β, AQL &
LQL values
• α = 0.05 and β = 0.10 was used to illustrate
technique.
Usually α set at 0.05 but can be 0.01 to 0.15
β usually at 0.10 and can be 0.01 to 0.20
• Sampling Plans can also be specified using
AOQL
(Avg. Outgoing Quality Level)
if AOQL = 1.5%
for an incoming (process) quality level of 2.0%
than AOQL = 100 PO x Pa
1.5 = 2.0 x Pa
∴ Pa = 0.75
Fig 9.22 AOQL Samp. Plans.
Acceptance Sampling
acceptancesamplingshari.fkm
DESIGN OF SAMPLING PLANS
(i) PRODUCERS RISK
SPECIFIED α = 0.5, Pα = .95 AQL = .012
(ii) CONSUMER’S RISK β � �
α = 0.10
LQ -= P0.10 = 0.06
(i) Get np from Table get np from Table C np C np 1 .35 1 3.89 2 .818 3 6.681 � � : : 6 3.286 7 11.771
(ii) p
npncak =
npncak =
c = 1 30
012.355.
n ∆=
c = 1 65
06.89.3
n Ξ=
c = 6 n = 274 c = 7 n = 196
Need α, AQL, β, LQ