Sampling Plans for Inspection by Attributes and by Count of Defects

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.

(iii) 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 ame 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, singlesampling plan will be selected in such a case ; (iii) 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 I)-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, IV 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 I)-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 singlesampling plan, from Table 19.4 of IS : 2500 (Part I)-1973, one gets the sample size (n) 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 I)-1973, on the basis of the AQL value chosen and the sample size code letter selected, one gets the values of two samples of the same size (n₁) and acceptance and rejection numbers (a₁ and r₁ respectively) corresponding to the first sample size and the I acceptance and rejection numbers a₂ and r₂) corresponding to the cumulative sample size. Thus first a sample of n₁ items is taken and if the number of' defectives (defects) is less than or equal to a₁  the lot is accepted. If the number of defectives (defects) is equal to or more than n then the lot is rejected ; and it is in between a₁ and r1, then a second sample of m items is taken. The lot is accepted if the number of defectives (defects) in the combined sample of 2n₁ 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 I)-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 (n₁) and the relevant acceptance and rejection numbers corresponding to the given stages?. Thus first sample of size (n₁) 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 n₁ 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.

 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 lighten 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 lightening 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

Table 19.14. Formulae for Constructing Single Sampling AQL,LTPD  plans

LTPDIAQL 0)	Sample size x AQL (2)	Acceptance Number (3)	Sample Sizes x AOQL (4)	LTPD/AOQL (5)
44.890	0.052	0	0.368	12514
10.946	0.355	1	0.840	9.262
6.509	0.818	2	1.371	7.764
4.890	1.366	3	1.942	6.879
4.057	1.970	4	2.544	6.285
3.549	2.613	5	3.168	5.855
3.206	3.286	6	3.812	5.526
2.957	3.981	7	4.472	.5.264
2.768	4.695	8	5.146	5.051
2.618	1426	9	5.831	4.872
2.497	6.169	10	6.528	4.720
2.397	6.924	U	7.233	4.589
2.312	7.690	12	7.948	4.475
2.240	8.464	! 13	8.670	4.373
2.177	9.246	14	9.398	4.283
2.122	10.035	15	10.134	4.204
2.073	10.831	16	10.875	4.129
2.029	11.633	17	11.622	4.061
1.990	12442	18	12374	4.000
1.954	13.254	19	13.131	3.945
1.922	14.072	20	12892	3.894
1.892	14.894	21	14.657	3.848
1.865	15.719	22	15.427	3.799
1.840	16.548	23	16.200	3.759
1.816	17.382	24	16,976	3.723
1.795	18.218	25	17.756	3.683
1.775	19.058	26	18.540	3.652
1.757	19.900	27	19.326	3.617
1.739	20.746	28	20.115	3.589
1.723	21.594	29	20.907	3.559
1.707	22.444	30	21.702	3.530
1.692	23.298	31	22499	3.507
1.679	24.152	32	23.298	3.481
1.665	25.010	33	24.100	3.456
1.653	25.870	34	24.904	3.433
1.641	26.731	35	25.711	3.411
1.630	27.594	36	26.519	3.394
1.619	28.460	37	27.330	3.374
1.609	29.327	38	28.142	3.354
1.599	30.196	39	28.956	3.336
1.590	31.066	40	29.773	3.318
1.581	31.938	41	30.590	3.302
1.572	32812	42	31.410	3.286
1.564	33.686	43	32.231	3.270
1.556	34.563	44	33.054	2.255
1.548	35.441	45	33.878	3.241
1.541	36.320	46	34.704	3.224
1.534	37.200	47	35.532	3.211
1.527	38.082	48	36.360	3.199
1.521	38.965	49	37.191	3.186

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

selecting lower sample size code latter 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 apian with a lower sample size code letter than i he 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 l>y less than 10 lots accepted on reduced inspection, change over to normal inspection.

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.

Example 19.36. Suppose lots containing 100 boiler rivets are submitted for inspection for shank diameter. It is desired to construct a single sampling plan with AQL of 1 per cent and LTPD value of 5 per cent. (The producer's risk and consumer's risk are taken to be 5% and 10% respectively).

Solution. From values of AQL and LTPD calculate LTPD ∕ AQL=0.05 ∕ 0.01 = 6.

Referring to Table 19.14 above, the nearest tabulated value of LTPD/AQL is obtained as 4.890 and the corresponding value of sample size × AQL and acceptance number are 1.366 and 3 respectively. So dividing the value of   n × AQL by the AQL value, the sample size is obtained as  1.366 ∕ 0.01= 137   approximately. The sampling plan would then work as follows :

From each lot of 1000 rivets pick out a sample of 137 rivets at random. Inspect the shank diameter of the rivets in the sample with the help of prescribed gauges and note down the number of defective rivets. The lot shall be accepted if the number of defective rivets in the sample is 3 or less otherwise the lot shall be rejected.

If the sampling plan is operated on the acceptance-rectification basis, then I lie resulting AOQL of the plan may also be obtained with the help of above Table. The value of   n× AOQL corresponding to the figure of 4.890 of  LTPD/AQL is read as 1.942. Dividing this figure by the sample size n which is 137, the AOQL is obtained as 1.4%.

Note. The above table can also be used for construction of ones own plan when the stipulations are made in terms of AOQL and LTPD in the same fashion as above entering the table through column 5 instead of column 1.

 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.

For designing a sequential analysis plan for inspection by attributes, first we should decide about the values of the following parameters which should be based on the practical and economic aspects of the product involved:

(i)   Good quality (AQL) = p₁ (expressed as fraction defective)

(ii)   Bad quality (LTPD) = P₂               (—do—)

(iii)  Producer's risk = α (expressed as a fraction)

(iv)  Consumer's risk = β           (—do—)

Then the values of m₁, m₂ and S in the equation of acceptance and rejection lines (i.e. d₁=Sn− m₁  and d₂=Sn + m₂ respectively) are com­puted as follows:

m₁=B  ∕  g_{1 +} g₂

m₂=A ∕  g_{1 +} g₂

S= g₂ ∕  g_{1 +} g₂

where          g₁= log p₂ – log p₁

g₂=log (1 –P₁) −log (1 –p₂)

A = log (1 – β) – log α

B = log (1 – α) – log β.

In preparing a sampling table, d₂ is taken to be the next higher whole number to the computed value, and d₁ is taken to be the next smaller.

If  d₂  is equalled or exceeded after n items have been inspected the lot is rejected.

If  d₁  is equalled or a value less than d₁ is obtained after n items have been inspected, the lot is accepted. .

If a value between  d₁and d₂  is obtained after inspecting n items then inspection is continued. For facilitating the use of this plan, the values of d₁ 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          C₁ = AQL expressed as number of defects per item

C₂ = LTPD expressed as number of defects per item

α = Producer's risk

β = Consumer's risk

Then   A = log (1 – β) – log α   and B = log (1 – α) – log β

g =log C₂ − log C₁

m₁=B ∕ g      and         m₂=A ∕ g

S=C₂ − C₁ ∕g

The acceptance limit     d₁=S_n − m₁

and rejection limit        d₂=S_n + m₂

The average amount of sampling required at quality levels p₁ and p₂ will respectively be

np₁=(1−α) m₁₋α m₂ ∕ s – P₁

and          np₂=(1−β) m₂−β m₁ ∕ P₂ − S

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 p₁ 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 np₁or 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 d₁ or d₂.

Smallest sample number at which lot may be accepted (n₁) or rejected (n₂) can be determined by the formula

n₁ —> next larger integer to m_{1 ∕}  S

n₂  —>   next larger integer to m2 ∕ S .

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 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 P_A% of defectives and after P_B% it is definitely advantageous.

Example 19.37. (a) What is the difference between sequential sampling plan and the multiple sampling plan ?

(b) Prepare a unit sequential sampling plan to meet the following specifications :

α= 0.05                       β = 0.10

p₁ =0.02                    p₂ = 0.05

Determine the probability of acceptance of a lot with 0.03 fraction defective.

Solution, (a) In multiple sampling plans, three or more samples of a stated size are inspected and decision on acceptance or rejection is reached after inspecting the stated number of samples.

In sequential sampling plan, decision is possible after each item has been inspected and there is no specified limit on the total number of units to be inspected.

(b) Symbol α represents 1 – P_a for a lot of quality p₁, symbol β  is P_a for a lot of a quality p₂

The sequential sampling plan is fully defined by the equation of the rejection line, d₂= Sn + m₂ and the acceptance line, d₁ = Sn−m₁(Refer Fig. 19.31). To compute S, the slope of these lines, and m₁ and ,m₂ the intercepts, certain auxiliary symbols, g₁,  g_{2 ,} A andB are used.

g_{1 =} log P₂ ∕ P₁ = log 0.05∕ 0.02=log 2.5=0.3979

g₂ = log 1− P₁  ∕ 1− P₂=log0.98 ∕0.95=log1.031=0.0132

A=log 1−β ∕ α=log 0.90 ∕ 0.05=log 18=1.2553

B=log 1  −α  ∕ β=log0.95∕ 0.10=log 9.5=0.9777

m₁=B ∕ g_{1 +} g₂=0.9777 ∕ 0.3979 +0.0132==0.9777 ∕ 0.4111=2.38

m₂=A ∕  g_{1 +} g₂=1.2553 ∕0.4111=3.05

S=g₂ ∕  g_{1 +} g₂=0.0132∕ 0.0132 +0.3979 =0.0132 0.4111

=132∕ 4111=0.032

.'.    d₂=0.032× n +3.05

and   d₁=0.032× n− 2.38

With the help of these equations, the acceptance and rejection lines can' be converted into an item-by-item table of acceptance and rejection numbers.

It will be noted that the computed values of d₁ and d₂ are generally not whole numbers. Usually the rejection number is taken as the next whole number above , d₂  and the acceptance number is taken as the next whole number below d_1..

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