4/13/2020 IENG 486 Statistical Quality & Process Control 1

 Reading:

Chapter 9
 Sections 9.1
– 9.1.5: pp. 399 - 410
 Sections 9.2
– 9.2.4: pp. 419 - 425
 Sections 9.3: pp. 428 - 430
 Homework:

Due 03 DEC
CH 9 Textbook Problems:
 1a, 17, 26 Hint: Use Excel!
 Last Assignment:

Download and complete Last Assign: Acceptance Sampling
 Requires MS Word for Nomograph
 Requires MS Excel for AOQ
4/13/2020 IENG 486 Statistical Quality & Process Control 2

4/13/2020
Company receives shipment from vendor
Sample taken from lot,
Quality characteristic inspected
NO Lot Sentencing:
Accept lot?
YES
Use lot in production
Return lot to vendor
IENG 486 Statistical Quality & Process Control 3

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.
4 4/13/2020 IENG 486 Statistical Quality & Process Control

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.
4/13/2020 IENG 486 Statistical Quality & Process Control 5

 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
4/13/2020 IENG 486 Statistical Quality & Process Control 6

 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
4/13/2020 IENG 486 Statistical Quality & Process Control 7

 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
8 4/13/2020 IENG 486 Statistical Quality & Process Control

 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 minimize shipping risks and make selection of sample units easy
4/13/2020 IENG 486 Statistical Quality & Process Control 9

 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.
2.
3.
4.
Assign a number to each unit, or use location of unit in lot
Generate / pick a random number for each unit / location in lot
Sort on the random number – reordering the lot / location pairs
Select first (or last) n items to make sample
4/13/2020 IENG 486 Statistical Quality & Process Control 10

 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
4/13/2020 IENG 486 Statistical Quality & Process Control 11

 Assume that the lot size N is large (infinite)
 d - # defectives ~ Binomial(p,n) where

 p - fraction defective items in lot n - sample size
 Probability of acceptance:
P a

P
 d
 c

 i c 

0 p i

1
 p

4/13/2020 IENG 486 Statistical Quality & Process Control 12

 Lot fraction defective is p = 0.01,
n = 89 and c = 2. Find probability of accepting lot.
4/13/2020 IENG 486 Statistical Quality & Process Control 13

 Performance measure of acceptance-sampling plan

 displays discriminatory power of sampling plan

Plot of: P a
P a vs. p
= P[Accepting Lot]
 p = lot fraction defective
p = fraction defective in lot P a
= P[Accepting Lot]
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.9897
0.9397
0.8502
0.7366
0.6153
0.4985
0.3936
4/13/2020 IENG 486 Statistical Quality & Process Control 14

Probability of Acceptance, Pa
Pa
1.0
0.8
0.6
0.4
0.2
0.0
0.00
0.02
n=89 c=2
0.04
0.06
0.08
0.10
Lot fraction defective, p
 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
4/13/2020 IENG 486 Statistical Quality & Process Control 15

 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:
Probability of Acceptance, Pa
1.00
4/13/2020
0.01
0.02
0.03
Lot fraction defective, p
0.04
IENG 486 Statistical Quality & Process Control 16

 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.
4/13/2020 IENG 486 Statistical Quality & Process Control 17

n
Probability of Acceptance, Pa
1.00
0.80
Pa
0.60
0.40
0.20
0.00
0.00
n=50, c=1 n=100, c=2 n=200, c=4 n=1000, c=20
0.02
0.04
0.06
0.08
Lot fraction defective, p
0.10
 Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases
4/13/2020 IENG 486 Statistical Quality & Process Control 18

c
Probability of Acceptance, Pa
1.0
0.8
Pa
0.6
0.4
0.2
0.0
0.00
n=89, c=2 n=89, c=1 n=89, c=0
0.02
0.04
0.06
0.08
Lot fraction defective, p
0.10
 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
4/13/2020 IENG 486 Statistical Quality & Process Control 19

 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 –

)
4/13/2020 IENG 486 Statistical Quality & Process Control 20


 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: p
1
= 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
 p
1
 That is,
 
P


Lot is rejected given that process has an acceptable quality level


 
P Lot is rejected p

AQL
4/13/2020 IENG 486 Statistical Quality & Process Control 21


 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 p
2
” p
2
= LTPD = Lot Tolerance Percent Defective
 is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective
 p
2
 That is,




P


Lot accepted given that lot has unacceptable quality level


P Lot accepted p

LTPD
4/13/2020 IENG 486 Statistical Quality & Process Control 22

 Use a chart called a Binomial Nomograph to design plan
 Specify:
 p
1
= AQL (Acceptable Quality Level)
 p
2
= LTPD (Lot Tolerance Percent Defective)

1 – 
= P[Lot is accepted | p = AQL]

β = P[Lot is accepted | p = LTPD]
4/13/2020 IENG 486 Statistical Quality & Process Control 23

 Draw two lines on nomograph


Line 1 connects p
1
Line 2 connects p
2
= AQL to ( 1-

)
= LTPD to


Pick n and c from the intersection of the lines
 Example: Suppose



 p
1
= 0.01,
α = 0.05, p
2
= 0.06,
β = 0.10.
Find the acceptance sampling plan.
4/13/2020 IENG 486 Statistical Quality & Process Control 24

p
1
= AQL = .01
p - Axis p
2
= LTPD = .06
n = 120
Greek - Axis

= .10
4/13/2020
1 – 
= 1 – .05 = .95
c = 3
Take a sample of size 120.
Accept lot if defectives ≤ 3.
Otherwise, reject entire lot!
IENG 486 Statistical Quality & Process Control 25

 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
4/13/2020 IENG 486 Statistical Quality & Process Control 26

4/13/2020
Incoming Lots:
Fraction Defective p
0
Inspection
Activity
Rejected Lots:
100%
Inspected
Fraction
Defective = 0
Accepted
Lots
Fraction
Defective p
0
Outgoing Lots:
Fraction Defective p
1
 p
0
IENG 486 Statistical Quality & Process Control 27

 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
28 4/13/2020 IENG 486 Statistical Quality & Process Control

 Quality that results from application of rectifying inspection

Average value obtained over long sequence of lots from process with fraction defective p a
  n

AOQ

N
 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
4/13/2020 IENG 486 Statistical Quality & Process Control 29

 If lot accepted:
Number defective units in lot:
  
N
 n
 fraction defective

# units remaining in lot




 Expected number of defective units:

P a
 
N
 n
 
Prob


Lot
 
 
# defective accepted
  units in lot


 Average fraction defective,
Average Outgoing Quality, AOQ:
AOQ

P p

N
 n

N
4/13/2020 IENG 486 Statistical Quality & Process Control 30

 Suppose N = 10,000, n = 89, c = 2, and incoming lot quality is
p = 0.01. Find the average outgoing lot quality.
4/13/2020 IENG 486 Statistical Quality & Process Control 31

4/13/2020 IENG 486 Statistical Quality & Process Control 32