Acceptance Sampling İST 252 EMRE KAÇMAZ B4 / 14.00-17.00

Acceptance Sampling Acceptance sampling is usually done when products leave the factory (or in some cases even within the factory). The standard situation in acceptance sampling is that a producer supplies to a consumer (a buyer or wholesaler) a lot of N items (a carton of screws, for instance). The decision to accept or reject the lot is made by determining the number x of defectives (=defective items) in a sample of size n from the lot.

Acceptance Sampling The lot is accepted if x ≤ c, where c is called the acceptance number, giving the allowable number of defectives. If x > c, the consumer rejects the lot. Clearly, producer and consumer must agree on a certain smapling plan giving n and c.

Acceptance Sampling From the hypergeometric distribution we see that the event A: ‘Accept the lot’ has probability where M is the number of defectives in a lot of N items. In terms of the fraction defective θ = M/N we can write (1) as

Acceptance Sampling P(A;θ) can assume n+1 values corresponding to θ=0, 1/N, 2/N,….,N/N here, n and c are fixed. A monotone smooth curve through these points is called the operating characteristic curve (OC curve) of the sampling plan considered.

Example 1 Sampling Plan Suppose that certain tool bits are packaged 20 to a box, and the following sampling plan is used. A sample of two tool bits is drawn, and the corresponding box is accepted if and only if both bits in the sample are good. In this case, N = 20, n = 2, c = 0, and (2) takes the form (a factor 2 drops out)

Example 1 The values of P(A,𝜃) for 𝜃 = 0,1/20,2/20,……..,20/20 and the resulting OC curve are shown if Fig. 538. (Verify!)

Figure 538 – Figure 539

Example 1 In most practical cases θ will be small (less than 10%). Then if we take small samples compared to N, we can approximate (2) by the Poisson distribution; thus

Example 2 Sampling Plan. Poisson Distribution Suppose that for large lots the following sampling plan is used. A sample of size n = 20 is taken. If it contains not more than one defective, the lot is accepted. If the sample contains two or more defectives, the lot is rejected.

Example 2 In this plan, we obtain from (3) The corresponding OC curve is shown in Fig. 539.

Errors in Acceptance Sampling We show how acceptance samplings fits into general test theory and what this means from a practical point of view. The producer wants the probability α of rejecting an acceptable lot (a lot for which θ does not exceed a certain number θ0 on which the two parties agree) to be small. θ0 is called the acceptable quality level (AQL). Similarly, the consumer (the buyer) wants the probability β of accepting an unacceptable lot (a lot for which θ is greater than or equal to some θ1) to be small.

Errors in Acceptance Sampling θ1 is called the lot tolerance percent defective (LTPD) or the rejectable quality level (RQL). α is called producer’s risk. It corresponds to a Type I error in Sec. 25.4 β is called consumer’s risk and corresponds to a Type II error. Figure 540 shows an example.

Errors in Acceptance Sampling We see that the points (θ0, 1-α) and (θ1,β ) lie on the OC curve. It can be shown that for large lots we can choose θ0, θ1(> θ0), α, β and then determine n and c such that the OC curve runs very close to those prescribed points. Table 25.6 shows the analogy between acceptance sampling and hypothesis testing in Sec. 25.4.

Figure 540

Table 25.6

Table in Turkish

Rectification Rectification of a rejected lot means that the lot is inspected item by item and all defectives are removed and replaced by nondefective items. (This may be too expensive is the lot is cheap; in this case the lot may be sold at a cut-rate price or scrapped.) If a production turns out 100θ% defectives, then in K lots of size N each, KNθ of the KN items are defectives.

Rectification Now KP(A; θ) of these lots are accepted. These contain KPNθ defectives, whereas the rejected and rectified lots contain no defectives, because of the rectification. Hence after the rectification the fraction defective in all K lots equals KPNθ / KN. This is called the average outgoing quality (AOQ); thus

Rectification Figure 541 shows an example. Since AOQ(0) = 0 and P(A; 1) = 0, the AOQ curve has a maximum at some θ = θ*, giving the average outgoing quality limit (AOQL). This is the worst average quality that may be expected to be accepted under rectification.

Figure 541