## Operating Characteristic Curves

For large shipments consisting of many units, say 5,000, we must determine a sample size n and an acceptance number c such that we are sufficiently assured that our accept/reject decision, based on the sample, is correct. The choices for n and c determine the characteristics of our sampling plan. Standard procedures are available for determining the sam

4. This section has been adapted from E. E. Adam and R. J. Ebert, Production and Operations Management, 5th ed. (New York: Prentice-Hall, 1992), pp. 653-655. Reproduced by permission of Everett Adam.

pling plan parameters, n and c, that will meet the performance requirements specified by the user. The performance requirements include the following four items of information: AQL, a conventional notation standing for "acceptable quality level" or "good quality"; LTPD, standing for "lot tolerance percent defective" or "poor quality level"; a, the producer's risk; and p, the consumer's risk. Assigning numeric values to these four parameters is largely a matter of managerial judgment. As soon as their numeric values have been assigned, values for n and c can be determined:

Example: A large medical clinic purchases shipments of pregnancy test kits (PTKs). A shipment contains 10,000 PTKs. It is important that the chemical composition of the PTK shipment be evaluated so that prescribing physicians are assured of valid tests.

Physicians have agreed that a shipment has acceptable quality if no more than 2 percent of the PTKs in the shipment have an incorrect chemical composition. They consider shipments having 5 percent or more defective PTKs to be an extremely bad-quality shipment. We want a sampling plan that affords a 0.95 probability of accepting good shipments but only a 0.10 probability of accepting extremely bad shipments. These performance specifications for the sampling plan are summarized on the left side of Table 20-6. A sampling plan was derived to meet these performance requirements. The plan calls for 308 PTKs to be sampled from each shipment (right side of Table 20-6). If more than 10 of these PTKs are defective, the entire shipment is rejected. If 10 or fewer PTKs are defective, the shipment is accepted. In this way, a shipment having 2 percent defective PTKs has only 5 chances out of 100 of being rejected, whereas a shipment having 5 percent defective has only 10 chances out of 100 of being accepted. This sampling plan includes procedures for determining the probability of accepting the shipment if its percent defective is between 2 and 5. These probabilities are shown in Figure 20-32.

The curves in Figure 20-32, called the operating characteristic curves or OC curves, reveal how sampling plans discriminate among shipments. If a shipment is of high quality (low percent defective), a good sampling plan yields a high probability of accepting the shipment. If a shipment is of poor quality (high percent defective), the plan yields a low probability of accepting the shipment.

You can see from the OC curve in Figure 20-32 that the desired probabilities of accepting good- and bad-quality PTK shipments have been obtained. The second OC curve represents a different sampling plan, n = 154 and c = 5, that does not meet desired performance specifications: It offers only a 0.88 probability of accepting a good-quality shipment, and a 0.22 probability of accepting a bad shipment.

TABLE 20-6. SAMPLING PLAN AND SPECIFICATIONS FOR PTKs

Performance Specifications Parameters of Sampling Plan

Good quality (AQL) = .02 or few defectives

Desired probability of accepting a good quality shipment = .95 n = 308

Bad quality (LTPD) = .05 or more defectives

Desired probability of accepting a bad quality shipment = .10 Risk: probability of p errors = .10

QUALITY OF SHIPMENT (IN PERCENT DEFECTIVE) FIGURE 20-32. Probabilities of accepting a PTK shipment; OC curves.

A sampling plan specifying a unique pair of n and c has a unique OC curve. Sampling plans calling for a large sample size are more discriminating than plans calling for a small sample size. Figure 20-32 shows OC curves for two sampling plans with different sample sizes and acceptance numbers. Comparing plans, the ratio of the acceptance number c and the sample size n is constant. For plans with larger ns, the probability of accepting good-quality lots is higher than for plans with smaller ns; similarly, for plans with larger ns, the probability of accepting bad-quality lots is lower than for plans with smaller ns. Of course, these benefits are not obtained without incurring the higher inspection costs associated with large sample sizes.

The effect of increasing the acceptance number c (for a given value of n) is to increase the probability of accepting the shipment for all levels of percent defective other than zero (Figure 20-33). By increasing c, more defective units are allowed to pass inspection. By decreasing c, inspection is tightened.

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