1. Acceptance Sampling Plans by Variables
CH 16

2. Contents
Advantages and Disadvantages of Acceptance Sampling by Variables.
Types of Acceptance Sampling by Variables.
Chain Sampling.
Continuous Sampling.

3. Advantages of Variables Sampling
The same OC curve can be obtained with a smaller sample size than would be required by an attributes sampling plan.
Measurement data provide more information about the manufacturing process.
When AQLs are very small, the sample sizes required by attributes sampling plans are very large.

4. Disadvantages of Variables Sampling
The distribution of the quality characteristic must be known.
A separate sampling plan must be employed for each quality characteristic that is being inspected.

5. Types of Variables Sampling Plans
Plans that control the lot or process fraction defective.
Plans that control a lot or process mean.

6. Plans to Control Process Fraction Defective
Since the quality characteristic is a variable, there will exist either LSL, USL, or both, that define the acceptable values of this parameter.
Fig. 1 illustrates the situation in which the quality characteristic x is normally distributed and there is LSL on this parameter.
Fig. (1)

7. Plans to Control Process Fraction Defective
Procedure 1 (k-Method)
Take a random sample of n items from the lot and
compute
If there is a critical value of p of interest that should not be exceeded with stated probability, we can translate this value of p into critical distance k.
If ZLSL ≥ k, we would accept the lot because the sample data imply that the lot mean is sufficiently far above LSL to insure that p is satisfactory.

8. Plans to Control Process Fraction Defective
Procedure 2 (M-Method)
Compute ZLSL .
Use ZLSL to estimate the fraction defective of the lot or
process .
Determine the max. allowable fraction defective M (using specific values of n, k).
If exceeds M, reject the lot; otherwise, accept it.

9. Plans to Control Process Fraction Defective
Notes
In the case of an USL, we compute
If is unknown, it is estimated by s.
When there is only a single specification limit (LSL or USL), either procedure may be used.
When there are both LSL and USL, M-method should be used by computing ZLSL and ZUSL, finding the corresponding fraction defective estimates and
Then, if ≤ M, the lot will be
accepted.
pLSL ^ ^
pUSL
pLSL ^
pUSL ^

10. Designing a variables sampling plan with a specified OC curve
Let be the two points on the OC curve of interest.
are the levels of lot or process fraction nonconforming that correspond to acceptable and rejectable levels of quality, respectively.
p1 and p2

11. Designing a variables sampling plan with a specified OC curve
Example 1

12. Designing a variables sampling plan with a specified OC curve

13. Designing a variables sampling plan with a specified OC curve
Example 2 :Design a sampling plan using M-method

14. Designing a variables sampling plan with a specified OC curve

15. MIL STD 414
There are five general levels of inspection, and level IV is designated as “normal”.

16. MIL STD 414
As MIL STD 105E, sample size code letters are used, but the same code letter does not imply the same sample size in both standards.
Sample sizes are a function of the lot size and the inspection level.
All the sampling plans in the standards assume that the quality characteristic is normally distributed.

17. MIL STD 414
Organization of MIL STD 414

18. MIL STD 414 Example 3: Using MIL STD 414 Solution
From table, if we use IV level, the sample size code letter is O.
From a second table, we find n=100.
For AQL of 1%, on normal inspection, k=2.
For AQL of 1%, on tightened inspection, k=2.14

19. MIL STD 414

20. Plans to Control A Process Mean
Example 4
Solution
Let XA be the value of the sample average below witch the lot will be accepted.
If lots have 0.95 probability of acceptance, then
P (X ≤ XA ) = 0.95 - -

21. Plans to Control A Process Mean
P (Z ≤ ) = 0.95
=1.64
If lots have 0.1 probability of acceptance, then
P (X ≤ XA ) = 0.1
p (Z ≤ ) = 0.1
= -1.28
These two equations can be solved for n and XA , giving n=9 and
XA=0.356 - - -

22. Chain Sampling Plan
It is used for small sample size plans that have acceptance number of zero.
Lots should be produced under the same conditions, and be expected to be of the same quality.
There should be a good record of quality performance on the part of supplier.
The general procedure for ChSP-1:
For each lot, select the sample of size n and observe the defectives.
If sample has zero defectives, accept the lot .
If sample has two or more defectives, reject the lot.
If sample has 1 defective, accept the lot provided that there have been no defectives in the previous i lots.

23. Chain Sampling Plan The effect of chain sampling on OC curve:
It is more difficult to reject lots
with very small p with ChSp-1
Plan than it is with ordinary single
Plan.
In practice, values of i vary between
3 and 5, since OC curves of such plans
approximate OC curve for S-S plan.
The points on the OC curve of
a ChSP-1 plan are given by:

24. Continuous Sampling Plan
Used when production is continuous. That is, the manufacturing operation do not result in the natural formation of lots.
It consists of altering sequences of sampling inspection and screening (100% inspection).

25. Continuous Sampling Plan
Procedure for CSP-1 plans

26. Continuous Sampling Plan
The average number of units inspected in a 100% screening is
equal to , where q=1-p
The average number of units passed under the sampling inspection is
The average fraction of total manufactured units inspected in the
long run is
The average fraction of manufactured units passed under the
sampling procedure is

27. Continuous Sampling Plan
OC curves for various continuous sampling plans, CSP-1.