Module 8. Statistical quality control

Lesson 30

OC AND AOQ CURVES

30.1  Introduction

The two main considerations on the basis of which sampling inspection plans may be compared are the Operating Characteristic (OC), Average Outgoing Quality (AOQ) and Average Sampling Number (ASN). Let us consider two equivalents sampling inspection plans viz., single and double sampling plan for which the OC curves are practically the same. The two plans are equivalent in the sense that they give the same amount of protection against rejection (or 100% inspection) of good lots or acceptance of bad lots. The average amount of inspection required per lot is maximum for single sampling plan and relatively less for double sampling plan. The exact amount of saving depends on the lot-quality and the particular plan under consideration. Generally speaking double sampling plan requires 25 to 33 percent less inspection on an average than single sampling plan. In this lesson, the OC and AOQ curves for single sampling plan have been discussed.

30.2  The Operating Curve (OC) Curve

The OC curve of an Acceptance Sampling Plan shows the ability of the plan to distinguish between good and bad lots. In judging various acceptance sampling plans it is desirable to compare their performance over a range of possible quality levels of submitted product. An excellent picture of this performance is given by the OC Curve. For any given fraction defective p in the submitted lot, the OC Curve shows the probability of acceptance Pa that such a lot will be accepted by the given sampling plan or it shows the long run percentage of submitted lots that would be accepted if a great many lots of any stated quality were submitted for Inspection. The OC is the Mathematical Expression L(p) or Pa, stating the probability of accepting a lot as a function of p, the fraction defective of the lot. The curve obtained by plotting the Operating Characteristic known as Probability of Acceptance Pa against p is called the OC curve. The steeper the OC Curve, the greater is the protection to the Consumer. An ideal plan, of course, would be one which rejects all lots which are of worse quality than some predetermined value of the fraction defective p and accepts all lots which are equal to or better than that quality. Such a plan, however, can never be attained. As ‘p’ the fraction defective increases, the probability of acceptance Pa decreases.      

Fig. 30.1 OC Curve

An ideal sampling plan would be one that rejected all lots that were worse than specified quality   and accepted all lots of specified quality or better. The OC curve of such an ideal plan would be

Fig. 30.2 OC Curve for ideal plan

The probability of acceptance (Pa) is calculated by using the following formula:

               

The OC Curve for the incoming quality ‘p’ is given by

                                                                                                                                             ………(Eq. 30.1)         

When p<0.10. a good approximation to above equation (30.1) is given by the first (c+1) terms of the binomial expansion

                                                                                                                                                                                                      ………(Eq. 30.2)

               

When p<0.10 and also n/N<0.10, the equation (30.1) is given by Poisson Distribution.

               

30.3  Average Outgoing Quality (AOQ) Curve

The expected fraction defective remaining in the lot after the application of the sampling plan is called the Average Outgoing Quality (AOQ). This is a function of ‘p’, the actual fraction defective in the lot. The maximum value of the average outgoing Quality, the maximum being taken with respect to p, is known as Average Outgoing Quality Limit (AOQL). Suppose a sample of n items is drawn from a lot of size N and Pa is the probability of acceptance of a lot of average quality level p then

               

If the sampling fraction n/N is negligible, then AOQL = p.Pa

The AOQ Curve is obtained by plotting p. Pa known as Average Outgoing Quality (AOQ) against ‘p’, the fraction defective of the lot.

30.4  Average Sample Number (ASN)

The expected value of the sample size required for coming to a decision i.e., for acceptance or rejection, under the sampling inspection plan of a lot is called the Average Sample Number (ASN). This is a function of p, the actual fraction defective of the lot. The curve obtained by plotting ASN against p is called the ASN curve. Obviously, other factors remaining the same, the lower the ASN Curve the better is the sampling inspection plan.