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What is Operating Characteristic Curves

Operating Characteristic Curves — The OC curve is a means of quantifying the producer’s and consumer’s risk. The OC curve for an attribute plan is a graph of the percent defective in a lot versus the probability that the sampling plan will accept the lot. The probability must be stated for all values of “P” (percent defective) since “P” is unknown. An assumption is made that an infinite number of lots will be produced.

It is characteristic of sampling plans that the probability of acceptance is high as long as product quality is good, but decreases as product quality becomes poorer. An example of an optimum OC curve is shown in Figure 8.3.1. Let us assume that we desire to accept all lots less than 2% defective and reject all lots greater than 2% defective. All lots less than 2% defective have a probability of acceptance of 1.0 (certainty), and all lots greater than 2% defective have a probability of acceptance of 0%. However, in reality, there are no sampling plans that are perfect. There will always be some chance that a good lot will be rejected or a bad lot will be accepted. The one major goal in developing a sampling plan should be to make the acceptance of good lots more likely than the acceptance of bad lots.

Optimum sampling plan performance.

The shape of the OC curve

The shape of the OC curve can be affected greatly by the parameters of the sampling plan. Figure 8.3.2 illustrates this by showing the curve for perfect discrimination as well as the curves for the other sampling plans. As the sample size approaches the lot size and an appropriate accept number (c) is used, the OC curve approaches the perfect curve (P1). With an accept number of 0, the resultant OC curve will be exponential in shape, or concave upward as in curves 2 and 3. Increasing the acceptance number tends to push up the OC curve for low values of P (percent defective), as in curve 1. Increasing both the accept number and sample size at the same time (curve 1) gives a curve which most closely resembles the perfect discriminator curve of P1.

How OC curves are affected by the parameters of a sampling plan. Reproduced with permission from J.M. Juran, “Quality Control Handbook,” 3d ed., 1979, McGraw-Hill Book Company.

Constructing OC Curves

Construct an OC curve by determining the probability of acceptance for various values of P (percent defective in a lot). The probabilities of all values of P must be determined since P is unknown. There are three distributions which may be used to find the probability of acceptance. They are the Poisson, hypergeometric, and the binomial. Probably the easiest one to use is the Poisson distribution, if all assumptions are met for its use. 

Some assumptions which must be met are as follows: 

  • Sample size must be 16 or greater. 
  • Lot size must be at least 10 times greater than the sample size. 
  • Percent defective is less than 0.01.

Many tools exist which can aid in calculating and plotting the OC curves. One such tool is shown in Figure 8.3.3. The following steps are required to plot the OC curve for a single sampling plan. 

  • Set up a table like the one shown in Table 8.3.1 for various values of P (percent defective). Express P as a decimal. The P value range should cover both good and bad product. 
  • Complete the second column of the table by multiplying each of the P values in column 1 by n (sample size).
  • Find the probability of acceptance (Pa) by using the table of curves in Figure 8.3.3.
  • Plot the probability of acceptance (Pa) for each corresponding value of P as shown in Figure 8.3.2. When comparing a number of OC curves, make sure the same vertical and horizontal scales are used.

Calculations for plotting an OC curve.

Inspection Lot Formation

Lot formation is one of the most important factors in acceptance sampling. It is imperative that we know the pertinent details about a lot (i.e., who, when, and what) before we can make intelligent decisions with the inspection data.

The following guidelines are given to ensure the validity of inspection data. Others may exist, but are more related to individual processes.

  • Do not mix products from different sources (process or machines, production shifts, raw materials, etc.), unless you can prove that variation is small enough to be ignored. 
  • Do not mix products from various time periods. Require suppliers to date code their products to allow you greater flexibility in your lot information. 
  • Keep the lots as large as possible to take advantage of the fact that lot size has very little effect on the OC curve. Large lots may create some problems such as storage, production, and delivery problems, when rejected. 
  • Make use of additional information, such as capability studies and prior inspection results in lot formation. This information can prove to be very helpful when the lots are few and far between.

 Cumulative probability curves of the Poisson distribution. Reproduced with permission from Harold F. Dodge and Harry C. Romig, “Sampling Inspection Tables, Single and Double Sampling,” 2d ed., 1959, John Wiley & Sons, Inc. (A modification of a chart given by F. Thorndike in “The Bell System Technical Journal,” Oct. 1926.) These curves serve as a generalized set of OC curves for single sampling plans when the Poisson distribution is applicable.

Sampling Justification

Sampling may or may not be the most effective solution for a given situation. Each situation must be evaluated individually in deciding whether to sample. Three alternatives exist for a product. One can choose to do (1) 100% inspection, (2) sample, or (3) no inspection. A thorough cost analysis of all three alternatives should be completed prior to adopting any one method. Many excellent references exist that may aid in this process. One such reference is Quality Planning and Analysis, by Joseph Juran and Frank Gryna.

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