19.23.
Having realised that 100% inspection is not feasible, it was considered desirable to go in for sampling inspection in which the decision about the acceptance or rejection of a complete lot is made on the basis of evidence obtained from a small sample. Due to sampling error there are chances that good lot is likely to be rejected and producer suffers in that case ; also chances are there that bad lots may be accepted and thus the consumer suffers in that case. Also in any case the accepted lots will contain some defectives and they can’t be eliminated altogether. The rejected lots are 100% inspected and defective parts are replaced by non-defectives. The 100% inspection of lots definitely exerts a positive influence on the producer to improve the quality of his products.
It is thus obvious that both the producer and the consumer have to take certain amount of risk (estipulated above) and then the sampling plans can be designed to ensure that neither the producer nor the consumer are going to be affected to an extent more than specified.
If producer is maintaining the control charts for his process, he knows the average quality of his products, and he can’t produce goods better than this quality. In case it is
acceptable to the consumer then the producer desires that all his lots upto this quality should be accepted by the sampling inspection plan. This limit in the control terminology is called acceptable quality level (AQL) and all goods of this quality and better than this are considered as good.
On the other hand consumer, depending, on his requirements, can specify a quality (bad quality, in control terminology known as lot tolerance per cent defective or LTPD) beyond which he does not want to accept any lot. The goods of quality LTPD or worse than this are considered as bad.
As already indicated, both the producer and the consumer are made to take some risks due to sampling errors. The producer’s risk is usually taken as 5% [i.e. 5% of the lots of quality (AQL) are likely to be rejected] and the customer’s risk is taken as 10% (i.e. 10% of lots of quality worse than LTPD are likely to be accepted).
19.23.1.
Operating Characteristic Curve. (O.C. Curve) (Fig. 19.28).
The characteristics of the system (sampling plan) is best shown by plotting a graph between probability
of acceptance against percentage of defectives. Such a curve is called the O.C. Curve. It compares the performance of sampling plans over a range of possible quality levels of submitted product. The two important points of the curve are :
The sampling plan chosen should be such as to pass through these two points. The interest of producer is that most of those components which are less defective should be accepted in full. The consumer’s interest lies in accepting the more defective parts to a minimum extent. Thus saying producer’s risk is represented by Po.95 = 0.25% means producer wants if a lot contains 0.25% defectives, then 95% of the lots must be accepted. Similarly saying consumer’s risk is po.io = 2.5% means that consumer does not want a worse quality
Fig. 19.28
(i) Producer’s risk and (ii) Customer’s risk.
containing more than 2.5% defectives and he would at the most accept 10% of lots containing 2.5% defectives, 2.5% is also called lot tolerance per cent defective (LTPD), i.e. the worst lot acceptable.
Theoretically the O.C. curve should be as APBCD. But this is not possible and curve is usually as shown by A’PCB’. Here, if a large sample and acceptance number are used, the plan becomes more discriminating i.e. a greater number of good lots are accepted and a greater number of bad lots are rejected. At the same time it is desired to have economical plan also in which the total number of articles inspected will be minimum and which will give the required degree of protection to both producer and consumer.
Type A, O.C. Curve. This gives the probabilities of acceptance for various fraction defectives as a function of the lot quality of finite lots. The computation for this curve is based on hypergeometric probabilities ; the binomial or Poisson distributions often give satisfactory approximations. The curve is definitely not a continuous one ; e.g., a lot of 200 items may be 0.5% or 1.0% defective but not 0.8% ; but in practice it is customary to draw it as continuous one.
Type B, O.C. Curve. This gives the probabilities of acceptance of a lot as a function of product quality. The computation for such a curve is done as if the lot size is infinite. For type B curve the binomial is exact and the Poisson often gives a satisfactory approximation. Such a curve is correctly viewed as continuous. When the sample size n is not more than l/10th of the lot size N, Type A and Type B curves may be considered as identical for most practical purposes. It may be noted that for some conditions type B curve always falls above type A curve, meaning thereby that the use of typeS curve tends go give a figure for Consumer’s Risk that is somewhat too high.
Characteristics ofO. C. Curve. Following are a few characteristics associated with every O.C.curve:
(a) With the fixed value of acceptance number c and N, the larger the value of n, the steeper the O.C. curve and thus it gives better discrimination between good and bad lots.
(6) With the fixed value of acceptance number and sample size, the O.C. curves for different values of N are not appreciably different and thus indicating that it is the absolute size of the sample that is more important rather than its relative size compared to the size of the lot.
(c) With the fixed value of N, if a number of O.C. curves all of which necessarily satisfy only one set of co-ordinates, then the O.C. curve with larger n and c gives a better discrimination between good and bad lots.
The most economical plans for LTPD and consumer’s risks are published by Messrs Dodge and Roming, popularly known as Dodge and Roming tables. Sampling Inspection Tables for inspection by attributes and by count of defects, and for inspection by variables for per cent defective have been prepared by Bureau of Indian Standard and we shall be discussing the practice recommended by BIS here.
In acceptance sampling (Dodge and Roming, tables as well as BIS tables), the rejected lots are 100% inspected. The average outgoing quality therefore first increases with increasing %age defectives in incoming products. As it increases too much, the rejected parts are replaced by good parts and outgoing quality again improves. Thus average outgoing quality level iAOQL) attains maximum value and then falls down.
As operating characteristic curve effectively summarises the nature of quality protection obtained by operating a particular plan on a number of lots (whose quality may vary within any limits), many of the published plans furnish the O.C. curves of the plans for selection purposes to enable choice from the point of view of quality protection.
Many a time, in addition to the producer’s risk and consumer’s risk being used as a measure of protection offered by a plan, the lot quality which has a 50% chance of being accepted or rejected by the plan (normally referred to as indifference quality) is also sometimes considered.
The O.C. curve does not give one useful information i.e., the average outgoing quality limit iAOQL). The consumer is very much concerned with iAOQL) value as it ensures him that the submitted lots may be of any quality but he will not get quality worse than AOQL value.
It may be noted that if the incoming quality is very poor, then many of the lots will be rejected and they have to be 100% inspected and defective items are to be rectified. As a result of this after certain value of the incoming quality the outgoing quality will start improving as the rejected lots do not contain any defective. This is shown in Fig. 19.29.
It may be stressed here that the adoption of a certain particular sampling inspection plan will ensure its AOQL value only in the long run and it does not guarantee this value for short periods.
Incoming Quality Fig. 19.29
19.23.2.
Definitions.
We shall now define the various terms commonly used in connection with the acceptance sampling.
{i)AQL. Acceptable quality level ; this is the fraction defective that can be tolerated without serious effect upon further processing operation or customer reaction. In other words AQL is the maximum per cent defective that, for the purpose of sampling inspection, can be considered satisfactory as a process average. Probably the first step in deciding upon a satisfactory sampling plan will be to decide upon a quality level that is acceptable. The level (AQL) is commonly designated by the symbol Pi.
(ii) RQL. Rejectable quality level; it is also commonly known as lot tolerance per cent defective (LTPD) and designated by either pi or p2. The only way to get real protection against unsatisfactory material is to use a sampling plan that will reject most of the lots offered that would seriously interfere with further processing operations or cause too much unfavourable customer reaction. LTPD represents the percentage defectives in a lot that can be tolerated in only a specified proportion of lots.
(iii) Producer’s risk (a). This is the risk which the producer runs of having lots of quality Pi (specified AQL) rejected. Saying a = 0.05 means that in the long run about one lot in 20 will be rejected, provided the lots are coming from a process controlled at quality level pi. This risk should be kept as low as possible. Reducing this risk too much would require an increased sampling that would generally be uneconomical. Of course, the producer can decrease his risk by producing material at a better quality level than pi depending on the other economical considerations involved.
(iv) Consumer’s risk (P). This is the risk which the consumer runs of accepting lots of qualityp2 (specified LTPD). Saying P = 0.01 means that in the long run, only about one lot in ten of quality p2 will be accepted. The risk of accepting lots of quality worse than the LTPD will be smaller than the designated consumer’s risk.
(v) N. Number of pieces in given lot (lot size).
(vi) n. Number of pieces in sample (sample size).
(vii) Defect. A failure to meet a specified quality standard. (viii) Defective. An item that contains one or more defects.
(ix) Group size. A number of successive observations that are considered together for control chart purposes.
(x) Acceptance number (c). The maximum allowable number of defective pieces in a sample of size n, for acceptance of the lot.
(xi) Rejection number (r). The minimum number of defectives in the sample for the rejection of lot.
(xii) Fraction defective (p). For a sample, this is equal to the ratio of the number of defectives (m) to the number of items in a sample (n) ; and for a lot this is the ratio of the number of defectives (M) to the number of item in lot (AT).
(xiii) Process average. It represents the average per cent defective of the products submitted by the producer for original inspection.
(xiv) Probability of acceptance (Pa). (xv) Probability of rejection (1 -Pa).
(xvi) Operating characteristic (O.C.) curve. It, at a glance, shows the percentage of lots likely to be accepted for varying percentage of defectives.
(xvii) Po.95> ^o.50 etc. Fraction defective having a probability of acceptance 0.95, 0.50 etc. under any given acceptance criteria.
(xviii) Single Sampling Plan. The procedure in acceptance sampling is to consider each submitted lot of product separately and to base the decision on acceptance or rejection of the lot on the evidence of one or more samples chosen at random from the lot. In single sampling plan only one sample is taken out of lot and the decision to accept or reject the lot is based on this sample.
Any systematic plan for simple sampling requires that three numbers be specified.
(a) N. Number of articles in the complete lot from which the sample is to be drawn.
(b) n. Sample size i.e., the number of articles in the random sample drawn from the lot.
(c) c. acceptance number. This is the maximum allowable number of defectives in a sample and more than c defectives will cause the rejection of the lot.
Thus saying a plan N = 100, n = 5, c = 0 would mean “Take a random sample of 5 out of a lot of 100. If the sample contains more than zero (0) defectives, reject the lot, otherwise, accept the lot”.
(xix) Double Sampling Plan. If a lot is very good or very bad, a small sample can detect it. If it be somewhere in between, then second sample can give still better idea. Thus total inspection is reduced in double sampling plan.
In double sampling plans, the attributes are usually specified by five constants, viz., N = lot size.
ni = the first sample size.
ci = acceptance number for the first sample.
n2 = the second sample size.
c2 = maximum number of defectives allowed in the combined first and second samples. (xx) Average Outgoing Quality (AOQ). It represents the average quality (average percentage defectives in the outgoing products (after inspection) including all accepted and all rejected lots which have been 100% inspected and defectives replaced by non-defectives.
In the case of single sampling plan, for lots of quality p, if the probability of acceptance is Pa, then AOQ = (p) (Pa), if defectives found in samples are not to be replaced. This may be proved as follows :
Let the total lots each containing N items submitted be X + Y.
Let the lots accepted be X.
Defectives in X lots are pNX.
Since the rejected lots are fully screened and therefore are free from defectives. Therefore Y”lots will not contain any defective.
(xxi) Average Outgoing Quality Limit (AOQL). It represents the maximum average per cent defective in the outgoing products.
(xxii) Important points about Sampling Inspection. In judging various acceptance sampling plans it is desirable to compare their performance over a range of possible quality levels of submitted product. An excellent picture of this performance is given by the O.C. curve (which shows the ability of the plan to distinguish between good and bad lots). As already explained the O.C. curve shows the probability Pa (for a given fraction defective p, in a submitted lot) that such a lot will be accepted by the given sampling plan.
In connection with various sampling plans i.e. different values ofN, n and c the following comments may be offered as are observed by drawing their O.C. curves.
(i) Sampling acceptance plans with same per cent samples give very different quality protection. Thus it is very wrong notion that the sample size should be in proportion to the lot size.
(ii) Fixed sample size tends towards constant quality protection. In other words, the size of the lot has not much influence on the shape of O.C. curve.
(Hi) In sampling plans developed without benefit of statistical analysis c is often specified as zero under the illusion that if sample is perfect, the lot will be perfect. Actually the acceptance number need not be zero.
(iv) The sampling plan chosen should be such which satisfies both producer and the consumer and requires total minimum inspection.
