Acceptance Sampling

Acceptance Sampling (AS) is used to qualify products in order to verify that the quality requirements are met [4, 5, 7, 15, 22, 23, 24, 25, 26]. It uses scientifically based sampling plans, through which conclusions are made about the quality of whole batches when checking only a small part (sample) of batches. It replaces 100% inspection, which is expensive and imperfect, and as production increases, it becomes difficult to implement.


When to Use
  • If we need to decide if the lot is likely to be acceptable
  • The sample inspection can be done much precise than 100% inspection
  • Then the inspection is destructible, 100% inspection is impossible

Characteristics

The most important characteristic is the operating characteristic (OC). It represents the probability for approvement of a lot.

The OC can be explained by the following example:

Let’s have a that sample consist of one piece of the lot. The rules for acceptance/rejection of the lot are: if the sample is “OK”, we accept the lot. If the sample is “NOT OK”, we reject the lot.

If the sample size is denoted by \( n \), and the acceptance number is denoted by \( c \), this means we have a plan: \( n = 1 \), \( c = 0 \).

Suppose the lot consists of 0 defective items, then the probability of acceptance is 1. If the lot consists of

only defective items then the probability of acceptance is 0. Other possibility is the lot to have equal number of defective and non-defective items. In this case the probability of acceptance is 0.5.

There are a lot of combinations of defective and non-defective items in the lot. All of them can be described wit the OC curve. The curve for the plan \( n = 1 \), \( c = 0 \) is shown on Fig. 9.11:


oc curve c=0 n=1
Fig. 9.11 OC curve for plan n=1, c=0


The OC curve can be represented by formula (9.44):


\( c=1-p \) (9.44)


The curves for other plans can be easily constructed. For example, let’s have a plan \( n=5 \), \( c=0 \). In this case \( P_a \) for any proportion of defective \( p \) can be calculated by formula (9.45):


\( P_a=\left(1-p\right)^5 \) (9.45)


The data for constructing the curve \( n=5 \), \( c=0 \) is given in table 9.13:


Table 9.13
\( p \)
 \( 1-p \)
 \( P_a \)

0

1

1.000

0.1

0.9

0.590

0.2

0.8

0.328

0.3

0.7

0.168

0.4

0.6

0.078

0.5

0.5

0.031

0.6

0.4

0.010

0.7

0.3

0.002

0.8

0.2

0.000


The OC curve for this case is given in Fig. 9.12:


oc curve n5 c0
Fig. 9.12 OC curve for plan n=5, c=0

From the graphic in Fig. 9.12, we can derive the following:

If \( p\approx0.01 \) than \( P_a=0.95 \)

If \( p\approx0.13 \) than \( P_a=0.5 \)

If \( p\approx0.37 \) than \( P_a=0.1 \)


These are some of the particular points used in acceptance sampling

\( P_a=0.95 \) - AQL (Acceptance Quality Level)
\( P_a=0.5 \) - IQ (Indifference Quality)
\( P_a=0.1 \) - LTPD (Lot tolerance percent defective) or RQL (Rejection Quality Level)

 

  1. A lot with acceptable quality can be rejected. The probability in this case is called Producer’s Risk or Type I Error. The corresponding level of quality is called Producers Quality Level (PQL).
  2. A lot with bad quality can be accepted. The probability in this case is called Consumer’s Risk or Type II Error. The corresponding level of quality is called Consumer’s Quality Level (CQL).

 The relation between OC curve, PQL and CQL is given on Fig. 913:


oc alpha and betha
Fig. 9.13 Relation between OC curve, PQL and CQL

The sampling plans are developed by the terms of these two points and the risks associated with them. Usually, the value for \( \alpha \) is 0.5 and the value for \( \beta \) is 0.1.

There are two types of sampling – Type A (sampling from an individual lot) and Type B (sampling from a process). The Type A sampling gives us the probability for acceptance of a lot according to the proportion of defectives in the lot. The Type B sampling gives us the proportion of the accepted lots according to the proportion of defectives in the process.

If during the inspection, we are counting the number of defects in a sample we have Poisson distribution. In case we are evaluating the proportion defective from process or large lots we have Binomial distribution. If we are evaluating the proportion defective from individual lost we have hypergeometric distribution.

 

The effect of changing the values of \( c \) and \( n \) is demonstrated on Fig. 9.14:


oc curves

Fig. 9.14 The effect of changing the values of \( c \) and \( n \)