The document discusses the calculations for process capability indices Cp and Cpk for specific product characteristics, assessing their acceptability based on industry standards. It also outlines the principles and methodologies of acceptance sampling, including its importance in quality control and the reasons for utilizing sampling inspection. Various types of sampling plans and relevant definitions, such as AQL (acceptance quality level) and LTPD (lot tolerance percentage defective), are explained to ensure producers meet customer quality expectations.

1. cIC
1. The specification for a certain product characteristic
are 0.005±0.0002in. The standard deviation & Mean are
0.0000429 & 0.0051 respetively. Calculate Cp and Cpk for this
characteristic. Is this process capability acceptable?
2. The specification for a certain product characteristic are
50.05±0.05mm. Mean of the data is 50.04mm. The standard
deviation is 0.01. Calculate Cp and Cpk for this characteristic. Is
this process capability acceptable?
Numerical on Process Capability Index

2. Solution 1:
Usl= 0.005+0.0002= 0.0052
Lsl= 0.005- 0.0002= 0.0048
Mean= 0.0051, Standard deviation = 0.0000429 Find Cp & Cpk
ϭ
Cp= {Usl-Lsl)/6 (sigma)
ϭ
Cp= (0.0052-0.0048)/6* 0.0000429
Cp= 0.0004/0.0002574
Cp = 1.55
Cpk= min { (Usl- Mean)/3 (sigma), (Mean- Lsl)/
ϭ 3 (sigma)
ϭ
Cpk= min { (0.0052-0.0051)/3*0.0000429, (0051-0.0048)/3* 0.0000429}
Cpk= min { 0.0001/0.0001287, 0.0003/0.0001287}
Cpk = min { 0.77, 2.33}
Cpk = 0.77, and since it is less than 1.33 the process is not capable.

3. Solution 2:
Usl= 50.05+0.05= 50.1
Lsl= 50.05-0.05= 50.0
Mean= 50.04, Standard deviation =0.01 Find Cp & Cpk
ϭ
Cp= {Usl-Lsl)/6 (sigma)
ϭ
Cp= (50.1-50.00)/6* 0.01
Cp=0.1 /0.06
Cp = 1.67
Cpk= min { (Usl- Mean)/3 (sigma), (Mean- Lsl)/
ϭ 3 (sigma)
ϭ
Cpk= min { (50.1-50.04)/3*0.01, (50.04-50)/3* 0.01}
Cpk= min { 0.06/0.03, 0.0004/0.03}
Cpk = min { 2.0, 1.33}
Cpk = 1.33, and since it is equal to 1.33 the process is capable.

4. 4
ACCEPTANCE SAMPLING
Contents
Sunjjoy Uvach
2. Reasons for Sampling Inspection
cIC
1. Acceptance Sampling
3. Probability and the Operating Characteristic Curve
4. Lost Acceptance Sampling Plan (LASP)
5. Definitions of Basic Acceptance sampling terms
6. MIL STD 105D for Sampling

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Sunjjoy Uvach
Producer
cIC
QUALITY
Customer
- If a lot is rejected during first sample
then he has to take second sample or
100% inspection of the full lot which
consumes time and money and delay in
dispatches. Vehicle has to wait.
- Customer should get consistent quality of
material as per the terms and condition
discussed earlier.
- Producer should provide customer
consistent quality of material as per the
terms and condition discussed
- If a lot passes the producer site and is
rejected during inspection at their end
the production line can stop in need of
material and producer has to send
person for inspection at customer’s end
- A balance in Quality level to be ensured to satisfy the customer at all times by the
producer.
- A good sampling plan (which is agreed by both) helps in reducing the efforts in inspection
both at producer and customer end.

6. 6
Quality Terms:
1. Quality Control: Day to day activity of
Inspection & Control
(Shop Floor)
2. Quality Assurance: To be designed
in the product or service
(Middle Management)
3. Quality Management: It is the mixture of Quality and Management
for Sustenance & Delighted Customer.
(Top Management)

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Sunjjoy Uvach
cIC 1. Acceptance Quality Control
- Individual sampling plans are used to protect against irregular degradation of
levels of quality in submitted lots below that considered permissible by the
customer.
- A good sampling plan will also protect the producer in the sense that lots
produced at permissible level of quality will have a good chance to be
accepted by the plan.

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Sunjjoy Uvach
cIC 2. Reasons for Sampling Inspection
1. If tests are destructive, necessitating sampling. It is obvious that 100%
sampling will remove the defective parts but at the same time leaving no
material for dispatch.
2. Process not in control, necessitating sampling to evaluate the product. An
out of control condition implies erratic behavior that cannot be predicted.
Therefore, to evaluate the product it is necessary to take a random sample
of production after fixed time.
3. 100% inspection is inefficient; 0% is risky. The efficiency of 100%
inspection is estimated at around 80% in screening the product. Under
sampling, rejected pieces are returned, getting managements attention.

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Sunjjoy Uvach
cIC
4. Special causes may occur after process inspection. Process control ends
when the control chart is plotted, but the product moves on and is affected
by subsequent causes on its way to the customer. Sampling the final or
incoming product provides assurance against problems occurring after the
process is complete.
5. Need for assurance while instituting process control. The process must
operate for some time to implement control charts and achieve control. The
product produced in this period of unknown control must be evaluated.
Sampling is a way to evaluate this product and provide information useful in
the start-up of process control.

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Sunjjoy Uvach
cIC
6. Deliberate submission of defective material. A real world experience has
shown that pressures of production or profit may lead to fraud. Sampling
can help prevent this by detecting.
7. Process control may be impractical because of cost or lack of
sophistication of personnel. It is sometimes not cost-effective to institute
process control , yet the product needs to be evaluated. Sampling is also
easier to implement.
8. 100% inspection does not promote process/product improvement.
Sampling with feedback of information often leads to process improvement.
9. Customer mandates sampling plan. Customer may insist on mandatory
sampling procedures which are to be met.

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Sunjjoy Uvach
cIC 3. Probability and the Operating Characteristic Curve
Undoubtedly the most important tool in acceptance quality control is
probability theory. It is vital that the quality engineers are acquainted with
probability theory.
3.1 Probability:
Definition: “If there be a number of events of which one must happen and
all are equally likely, and if one of a (smaller) number of these events will
produce a certain results which cannot otherwise happen, the probability of
this result is expressed by the ratio of this smaller to the whole number of
events” (Whiteworth 1965, rule IV). Here the probability is defined as the
ratio of favorable to total possible equally likely and mutually exclusive
cases.
Example: Chances of getting one ace from a pack of cards is 4/52=1/13
Total aces is 4 nos. & Pack of cards has 52 nos.

12. Sunjjoy Uvach
14
3.2. Operating Characteristic Curve
cIC
This curve shows the relation between probability of accepting a lot versus
the per cent defectives. A fundamental use of probability with regard to
acceptance sampling comes in describing the chances of a lot passing
sampling inspection if it is composed of a given proportion defective. The
very simplest sampling plan is, of course, as follows:
1. Sample one piece from the lot.
2. If it is good , accept the lot,
3. If it is defective, reject the lot.
- If the sample has ¾ th ok pieces then chances of lot getting ok is 75% and
getting rejected is 25%

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Sunjjoy Uvach
cIC OC Curve n=1 & c=0
Here, we see that for any proposition
defective p, the probability of acceptance Pa
is just complimentary of p,
i.e. Pa=1-p This is only true of the plan
n (sample)=1,
c (acceptance no.)=0
Here, OC curve gives at a glance a
characterization of potential performance of
the plan.
Pa
p
0.10.20.30.40.50.60.70.80.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

14. Sunjjoy Uvach 16
cIC
For n=5 (samples), c=0 & p=0.005,0.01, 0.02,0.03.0.04 & 0.05
Probability of acceptance Pa for any
proportion defective p can be computed
as,
Pa= (1-p) (1-p) (1-p) (1-p) (1-p)= (1-p)⁵
p (1-p) Pd=(1-p)⁵
0.005 0.995 0.975
0.01 0.99 0.951
0.05 0.95 0.774
0.10 0.90 0.590
0.20 0.80 0.328
0.30 0.70 0.168
0.40 0.60 0.078
0.50 0.50 0.031
Pa
p
0.005 0.010 0.050 0.100 0.200 0.300 0.400 0.500
0.000
0.200
0.400
0.600
0.800
1.000
1.200
Make O.C curve for following situation

15. Sunjjoy Uvach
p
cIC
OC curve n=5, c=0
From fig it is clear that if the producer can
maintain a defective less than 0.01 the product
will be accepted 95% of time or more by the
plan.
0.005 0.010 0.050 0.100 0.200 0.300 0.400 0.500
0.000
0.200
0.400
0.600
0.800
1.000
1.200
If the product is 13% defective then it will have a
50-50 % chance of acceptance.
Pa

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Sunjjoy Uvach
cIC
The probability of observing exactly d defectives is given by the following
binomial:
Pd = f(d) =
n!
d!(n-d)!
Pd (1-p)n-d
The probability of acceptance is the probability that d, the no. of defectives,
is less than or equal to c, the accept no. This means that
P(d ≤ c) =∑
n!
d!(n-d)!
Pd (1-p)n-d
Pa =
c
d=0
The probability of acceptance is the probability that d, the no. of defectives,
is less than or equal to c, the accept no. This means that Using this formula
with n=52, c=3 and p=0.01, 0.02,……….0.12, we find the result as in table

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Sunjjoy Uvach
cIC
Pa 0.998 0.980 0.930 0.845 0.739 0.620 0.502 0.394 0.300 0.223 0.162 0.115
p 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
OC Curve n=52,c=3

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Sunjjoy Uvach
cIC 4. Lot Acceptance Sampling Plan (LASP)
LASP is a sampling scheme and a set of rules for making decisions for
accepting/rejecting a lot.
LASP are following types:
1. Single sampling plans
2. Double sampling plans
3. Multiple sampling plans
4. Sequential sampling plans
5. Skip lot sampling plans
1. Single sampling plans: These are usually denoted as (n,c) plans, for a
sample size n, where the lot is rejected if there are more than c defectives.
This is the easiest of plans but not cost effective considering the nos of
lots to be inspected.

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Sunjjoy Uvach
cIC
2. Double sampling plans: After the first sample is tested , there are three
possibilities
a) Accept the lot, b) Reject the lot, c) No decision (Indecisiveness)
If decision is c then another sample is taken and combined result is evaluated
to take the decision.
3. Multiple sampling plans: This is an extension of double sampling plan
where more than two lots are inspected to make a decision. The advantage is
that small sample sizes are taken.
4. Sequential sampling plans: This is the ultimate extension of multiple
sampling where items are selected from a lot one at a time and after it’s
inspection a decision of either accepting/rejecting the lot or take another
sample
5. Skip lot sampling plans: In this only some of the lots in offering are
inspected.

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Sunjjoy Uvach
cIC 5. Definitions of Basic Acceptance sampling terms
1. AQL (Acceptance Quality Level): AQL is a percentage defective that is the
baseline requirement for the quality of the producer’s product. The producer
would like to design a sampling plan such that there is high probability of
accepting a lot that has a defect level ≤ AQL
2. LTPD (Lot Tolerance Percentage Defective): LTPD is a designated high defect
level that would be unacceptable to the consumer. Consumer will want a
sampling plan which will have low probability of accepting a lot having defect
level as high as LTPD.
3. Type 1 error (Producer’s risk): This is the probability that a (n,c) sampling
plan will be rejecting a lot having defect level equal to AQL. It is represented by
‘α’. It ranges from 0.2 to 0.01
4. Type 2 error (Consumer’s risk): This is the probability that a (n,c) sampling
plan will be accepting a lot having defect level equal to LTPD. It is represented
by ‘β’. It ranges from 0.2 to 0.01

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Sunjjoy Uvach
cIC 5. Average Outgoing Quality (AOQ): AOQ represents .’the average
% defective in the outgoing products after inspection.
Example: Calculate sample size and AOQ for single sampling plan using
following data of acceptance sampling:
1) Probability of acceptance of a 0.5% defectives is 0.525
2) Lot size= 10,000 nos
3) Acceptance no. =1
4) np= 1.6 {=sample size (n) X defectives in sample(p)}
5) Defectives in the sample not to be replaced
Rejection in the lot
Total qty. in the lot
(N-n)
N
AOQ = =PaXP’
Pa – Probability of acceptance,
P’-
percentage defective
N= Lot size
n= Sample size

22. Solution:

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Sunjjoy Uvach
cIC
Sr.
No
Point shown in
Fig.
Probability of
Acceptance, Pa
Description
1 A’ 100% Zero Defectives
2 A 95% 5% Defectives
3 B’ 0% 100% Defectives
4 B 10% 90% Defectives
5 AQL Higher Acceptance Quality
Level
6 IQL 50% Indifference Quality
Level
7 LTPD Lower Lot Tolerance
Percentage Defective
8 Producer’s
Risk
---------- Risk of rejecting a good
lot
9 Consumer’s risk ----------- Risk of accepting a bad
lot
6. Actual OC Curve

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Sunjjoy Uvach
cIC
6. MIL STD 105D for Sampling

25. Engineering Engineers
Questions are Welcome
Thanks for Patience