Sampling Plans for Inspection by Attributes and by Count of Defects (Metrology)

19.24.
In industry, very often we come across situations when the quality of a lot of manufactured products has to be evaluated with a view to determining its acceptability or not. This is achieved by carrying out sampling inspection according to certain plans. The sampling plans discussed in this article can be used in those situations where either the items could be classified as defectives or non-defectives (inspection by attributes), or the number of defects of an item could be counted (inspection by count of defects).
A necessary condition for using any sampling plan is that the lot quality should be specified in terms of fraction (or percentage) defective in case of inspection by attributes and in terms of number of defects per item (or per 100 items) in case of inspection by count of defects. It may be mentioned that if the lot quality is uniform and lot comprises of homogeneous items, then a small sample can be taken to represent the lot and the inspection cost will be low. The requirement of good sampling plans is that these should be sufficiently flexible to meet a wide variety of quality requirements and administrative and technical conditions.
The various types of sampling are :
(a) AQL Plans
(i) Single sampling plans. (ii) Double sampling plans.
(Hi) Multiple sampling plans.
(b) One’s own Plans.
(i) Single sampling plans for any desired values of AQL and LTPD
(ii) Sequential plans.
The selection of any particular plan depends upon the following factors :
(a) Cost of inspection.
(b) Protection desired by the producer and the consumer.
As regards the quality protection, all the plans are capable of giving the same equally, but it is mainly due to other factors that one has to choose one particular plan. The other factors are ; (i) the sampling inspection cost i.e., the size of the sample. If inspection cost depends only on the sample size then the single, double, multiple and sequential plans stand in the decreasing order of cost; (ii) amount of information provided on the quality of each lot (in other words on the process producing these lots). If the process average (i.e., average per cent defective of the products) is to be estimated then only first sample is considered, and as single sampling plan gives biggest first sample, single sampling plan will be selected in such a case ; (Hi) the administrative cost involved in using these plans e.g., sequential plans require more computation and recording whereas single plans are cheapest in this respect. However, the double, multiple and sequential sampling plans have the psychological advantage in the sense that the lots are given ‘more than one chance’ for acceptance.
It may once again be stressed that the protection provided by any particular sampling plan to the producer and the consumer can be judged at a glance by its operating characteristic curve ; and steeper is the O.C. curve, (which is closely related to the amount of inspection) better it discriminates between good and bad lots.
It is obvious that for choosing any sampling plan, the choice of AQL (acceptable quality level) is very important. The decision about AQL value is arrived at by striking of compromise between the quality desired and the quality attainable e.g., if AQL is specified more than the quality which a production process (from which lots are coming) is capable of giving then the amount of rejection will be very high. On the other hand if the AQL is not exacting enough then an excessive amount of inferior product are likely to be accepted. The appropriate value of the AQL may be arrived at on the basis of the past performance of the supplier of the product; or else it may be stipulated in accordance with the agreement between the parties concerned.
Indian Standard Specification No. IS : 2500 (Part D-1973 [Sampling Inspection Tables for Inspection by attributes and by Count of Defects], while preparing the Sampling Tables has taken into consideration the inspection level also. By inspection level is meant the relative amount of inspection one is required to do. Obviously if the inspection level is high, then there will be less risk of accepting lots of quality worse than the chosen AQL, and vice-versa. In actual practice one has to strike a compromise between a large sample which gives a reliable estimate of the lot quality (but inspection cost is high) and a small sample which reduces the inspection cost.
The Indian Standard Specification referred above specifies five inspection levels ; namely I, II, III, rV and V. Inspection levels I and II are applicable in the selection of’special small sample inspection plans’. These are intended to be used in cases where relatively small sample sizes are appropriate due to consistent supply of good material or as agreed to between
the producer and the consumer. Specification recommends inspection level IV to be most suitable for majority of cases and under normal conditions of acceptance inspection, it gives a reasonable compromise between high inspection costs and the risks involved. Under the cases where the cost of inspection does not matter much but acceptance of defective item is considered as a serious matter then inspection level V may be used and in the reverse case level III.
For using the sampling table given in this specification, first the ‘sample size code letter’ has to be chosen from Table 1 of IS : 2500 (Part D-1973 corresponding to the lot size and the inspection level chosen. Then from relevant tables, the sample plan can be chosen corresponding to this sample size code letter and the AQL value. For single sampling plan, from Table 19.4 of IS : 2500 (Part I) 1973, one gets the sample size in) and the acceptance number (a). Thus if out of this sample inspected the number of defectives are more than the acceptance number, the lot is rejected otherwise accepted. For double sampling plan, from Table 19.4 of IS : 2500 (Part D-1973, on the basis of the AQL value chosen and the sample size code latter selected, one gets the values of two samples of the same size (n{) and acceptance and rejection numbers {ax and rx respectively) corresponding to the first sample size and the acceptance and rejection numbers a2 and r2) corresponding to the cumulative sample size. Thus first a sample of /ii items is taken and if the number of defectives (defects) is less than or equal to ax the lot is accepted. If the number of defectives (defects) is equal to or more than rx then the lot is rejected ; and if it is in between ax and r1; then a second sample of nx items is taken. The lot is accepted if the number of defectives (defects) in the combined sample of 2nx items is less than or equal to a2 and rejected if it is more than or equal to r2.
For multiple sample plans, from table 4 of IS : 2500 (Part D-1973, corresponding to the chosen value of AQL and the sample size code letter chosen from table 1, one finds the value of seven samples of the same size (n{) and the relevant acceptance and rejection numbers corresponding to the given stages. Thus first sample of size (ni) is selected in a random order and examined for defectives (defects). If this sample is found to contain the defectives (defects) less than or equal to the acceptance number given corresponding to the first sample then the lot is accepted. It will be rejected if the number of defectives (defects) is equal to or more than the rejection number given against first sample. If, however, the number of defectives (defects) in the first sample lies in between the acceptance and rejection numbers given for the first sample, then a second sample of size nx is selected and examined for defectives (defects). The number of defectives (defects) in the combined sample (i.e., the first and the second sample taken together) is then compared against the acceptance and rejection number corresponding to the second stage of sampling for taking decision with regard to the acceptance or rejection of the lot. If no decision is reached, then a third sample and so on are drawn upto the seventh stage till a decision finally reached so as to accept or reject the lot.
19.24.1

Normal reduced and tightened inspection.

When the quality of the submitted lots shows significant shift, it is desirable to make appropriate changes in the sampling plans. If the quality deteriorates, it is necessary to tighten the inspection ; if the quality improves, it may be desirable to relax the inspection.

Normal Inspection.

Inspection under a sampling plan that is in force for a particular product by producer is called ‘normal inspection’. It may be continued as long as the quality of the product submitted is better than or equal to the chosen AQL. The consistency in maintaining the level of quality by the producer can be ascertained either from a continuous record of inspection data which can be used to estimate the process average of the producer or from a knowledge of the production of the lots that are not accepted. In case the quality becomes consistently better than the stipulated AQL, ‘reduced inspection’ may be undertaken. If however, the quality becomes consistently worse than the chosen AQL, ‘tightened inspection’ is to be restored to.

Tightened Inspection.

Inspection shall be tightened either by raising the inspection level, i.e., by selecting a sample size code letter higher than the one adopted for normal inspection or by employing a smaller AQL. Since the former approach leads to an increased amount of inspection, tightening is done by using a sampling plan with an AQL smaller than that used previously. When tightening of inspection is done, the smaller AQL shall be chosen always with respect to the actual sample size code letter used for normal inspection and not with respect to the intended letter which may not have any sampling plan.
The following criteria is applied for changing from normal to tightened inspection and vice-versa.
(a) If 2 out of 5 (or less) consecutive lots have been rejected while on normal inspection, change over to tightened inspection.
(b) If, while on tightened inspection, 5 consecutive lots have been accepted, change over to normal inspection.
Reduced Inspection. If the quality of the submitted lots is considerably better than the AQL chosen, reduced inspection may be restored to either by selecting lower sample size code letter than the one used for normal inspection or by relation to the AQL. Since the former approach leads to economies in inspection it is preferable unless there is an agreement to the contrary, to reduce inspection by changing over to a plan with a lower sample size code letter than the one adopted for normal inspection.
The following criteria may be applied for changing over from normal to reduced inspection and vice-versa.
(a) If none out of 10 consecutive lots has been rejected while on normal inspection ; change over to reduced inspection.
(b) If a lot is rejected, and if at the same time the rejected lot is preceded by less than 10 lots accepted on reduced inspection, change over to normal inspection.
Table 19.14. Formulae for Constructing Single Sampling AQL, LTPD plans

19.24.2.

Construction of ones own sampling plan. (Single sampling AQL, LTPD plans for inspection by attributes and by count of defects):

Single sampling plans for inspection by attributes and by count of defects corresponding to any specified combination of AQL and LTPD values can be constructed by making use of Table 19.14. The plans so obtained are virtually independent of the lot size. They assume the producer’s risk to be 5% and the consumer’s risk to be 10%. Since the acceptance of a bad lot is more serious when the lot is large than when it is small while selecting a sampling plan for inspection of a large lot such as AQL and LTPD combination should be preferred which lead to large sample sizes.
All values of AQL, LTPD and AOQL are to be expressed as fraction defective or number of defects per item in Table 19.14.
19.24.3.

Sequential Analysis.

It has already been stressed that in the case of double sampling for attribute inspection, the average amount of sampling required is less for the same protection, if the lot quality is not appreciably different. Theoretically the amount of sampling (in other words inspection cost) could be reduced further by designing triple or quadruple sampling plans, but in actual practice it becomes very difficult to construct and administer such plans. However sequential sampling plans offer both of these advantages. Due to least inspection cost, these are best suited in cases where inspection is of destructive type or samples are difficult to obtain.
Two types of sequential sampling plans are available for inspection by attributes and by count of defects, namely, (a) item by item sequential plans, and (b) group-by-group sequential plans.
In sequential analysis (which is the ultimate in multiple sampling), the number of items to be inspected is not predetermined. The sample items are inspected one at a time in the case of item-by-item sequential plans (or in small groups in the case of group-by-group sequential
sampling plans). After inspecting it, a decision is made using appropriate criteria based on the inspection data available upto that stage, either (i) to accept the lot
or (ii) to reject the lot
or (iii) to continue sampling.
The procedure is continued until the lot is either accepted or rejected.
For the operation of this type of sampling plan either the graphical procedure shown in Fig. 19.31, or a tabular procedure is followed. The formulae for the acceptance and rejection criteria are given below.

Fig. 19.31. Sequential analysis plan for attribute insnection.

In preparing a sampling table, d2 is taken to be the next higher whole number to the computed value, and dx is taken to be the next smaller.
If d2 is equalled or exceeded after n items have been inspected the lot is rejected.
If di is equalled or a value less than d\ is obtained after n items have been inspected, the lot is accepted.
If a value between d\ and d2 is obtained after inspecting n items then inspection is continued. For facilitating the use of this plan, the values of dx and d2 for different values of n can be calculated before hand and kept ready either in the form of a graph or a table.
If it is not convenient to take samples one by one (i.e. item-by-item sequential sampling), then sample may be taken in groups (group-by-group sequential plans) of say 10, 20, 50 or any other convenient number at a time and the decision based upon the total inspected. (In many instances it is found more practical to inspect a sequence of groups of items rather than a sequence of items themselves).
The above formulae for computing the acceptance and rejection criteria for item-by-item sequential sampling plans for inspection by count of defects get modified as below : Let cx = AQL expressed as number of defects per item c2 = LTPD expressed as number of defects per item a = Producer’s risk P = Consumer’s risk

and the maximum amount of sampling (on the average) required to make a decision will generally occur when the quality of lots submitted lies between pi and p2.
It may sometimes so happen in certain border line cases that the sample size required to reach a decision could reach infinity. (Though in practice sample size for taking a decision will never exceed 2-3 times npX or np2).
For practical reasons it is desirable to stop at a point and decide arbitrarily in one of the following ways:
(i) Reject the lot, if not accepted so far (In this case producer’s risk is high).
(ii) Accept the lot, if not rejected so far (In this case, the consumer’s risk is increased). (iii) Take decision to accept on the basis that number of defectives are closer to di or d2.
Smallest sample number at which lot may be accepted (n{) or rejected (n2) can be determined by the formula

If a comparison was to be made between the amount of sampling required for the three types of sampling, the curves for these will look like as shown in Fig. 19.32, provided other things remain same, i.e. OC characteristic is same. It may be noted that sequential sampling

Fig. 19.32
plan has marked advantage over single sampling plan regardless of the quality of lot submitted. The sequential sampling plan is better than double sampling plan only if lot contains more than Pa% of defectives and after Pb% it is definitely advantageous.