Operating characteristics curves11th April 2021
O.C. curves quantifies manufacturer’s (producer’s) risk and consumer’s (purchaser’s) risk. This is a graph of the percentage defective in a lot versus the probability that the sampling plan will accept a lot.
An O.C. Curve drawn for sampling plan of n = 300 and C = 10 at Fig. 60.1 indicates the following:
AQL = 0.02 or 2%
Manufacturer’s risk = 0.05
Consumer’s risk = 0.10
LTPD = 0.05 or more defectives.
All practical sampling plans have an operating characteristics curve, briefly called O.C. curve.
Following points need emphasis regarding O.C. curves:
(i) There is some chance that good lots will be rejected.
(ii) There is some chance that bad lots will be rejected.
(iii) These risks can be calculated by the theory of probability and depends on the number of samples inspected, the acceptance number, and the percent defectives in the lot offered for sample inspection. Given the amount of risks which can be tolerated, a sampling plan can be devised to meet these requirements.
(iv) The larger the sample used for inspection, the nearer the O.C. curve approaches the ideal. However beyond a certain point, the added cost in inspecting a large number of parts far exceeds the benefit derived.
In any acceptance sampling plan, three parameters are specified. The first parameter is number of articles N in the lot from which sample is to be drawn. The second parameter is the number of articles n in the random sample drawn from the lot, and the third is the acceptance number C.
This acceptance number C is the maximum allowable number of defective articles in the sample. If more than C defectives are found in a sample the lot is liable to be rejected. Since the lot size has little affect on the probability of acceptance, therefore lot size is generally ignored in deriving a sampling plan.
O.C. curve of an acceptance sampling plan (i.e. for a particular combination of n and C) shows how well the sampling plan discriminates between good and bad lots. In order to examine the suitability of an acceptance sampling plan, it is necessary to compare their performance over a range of possible quality levels of the product.
The graph of this performance is known as operating characteristic curve. Fig. 60.2 below shows an ideal O.C. curve where it is desired to accept all lots having 3% or less defectives, and to reject all lots having more than 3% defectives.
In this curve, all lots with less than 3% defectives have a probability of acceptance of 100%, while all lots with more than 3% defectives have a probability of acceptance as 0%. However, such a plan does not exist in reality.
Zones of O.C. Curve:
O.C. curve can be divided into following 3 zones:
(a) Acceptance Zone:
In this zone all the batches are accepted, therefore, the O.C. curve should be so selected that its acceptance zone accepts what is considered to be satisfactory lot.
(b) Rejection Zone:
In this zone, all the batches are rejected. Hence the O.C. curve selected should be such that it rejects what is considered to be an unsatisfactory lot.
(c) Zone of Indecision:
This is the zone where there is no purity that whether any particular batch or lot will be accepted or rejected. This problem can be solved either by adopting 100% inspection or by taking larger sample, but these will increase the inspection costs.
A batch or lot in this zone is worse than acceptable lot, and better than those what is considered as unacceptable. Thus its quality is border-line, and practically does not matter much whether a lot is finally accepted or rejected from this zone.