LOT–BY-LOT ACCEPTANCE SAMPLING BY …fkm.utm.my/~shari/download/qc09 acceptance sampling.pdf · Acceptance Sampling acceptancesampling shari.fkm LOT–BY-LOT ACCEPTANCE SAMPLING

Acceptance Sampling

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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


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


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


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


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


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


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


• 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


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


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


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


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


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


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


CONSUMER-PRODUCER RELATIONSHIP

Acceptance Sampling


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


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


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


• 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


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


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


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


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

��

��

Acceptance Sampling


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


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


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

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LOT–BY-LOT ACCEPTANCE SAMPLING BY …fkm.utm.my/~shari/download/qc09 acceptance sampling.pdf · Acceptance Sampling acceptancesampling shari.fkm LOT–BY-LOT ACCEPTANCE SAMPLING