How to Design Attribute The best sample plan minimizes your risk of rejecting by William A. Levinson
Die lot sizes can exceed 10,000, and 100percent die testing is expensive and timeconsuming. If the lot is mostly (e.g., greater than 98percent) good, it's usually cheaper to ship it and accept the extra losses after assembly. Otherwise, it is far more economical to rectify it and cull the bad units in Mountaintop. Sequential sampling plans often allow the decision of whether to pass or reject the lot with a very small sample size. Sequential sampling plans reject very bad lots and pass very good ones quickly. Because the testing equipment can't be programmed to perform sequential sampling, Harris must rely on a singlesample acceptance sampling plan. Singlesample acceptance plans Two numbers define singlesample attribute sampling plans: the sample size, n, and the acceptance number, c. The inspector (or test) examines n pieces, and the lot passes if c or fewer are nonconforming. More than c nonconformances results in rejection of the lot. Such lots are usually rectified or detailed: All the pieces are tested or inspected, and the bad ones are discarded (Figure 1). p is the lot's actual nonconforming fraction. The sampling plan must meet two specifications: The chance of failing a lot of acceptable quality (acceptable quality level or AQL) must be no greater than a, which is the producer's risk, that is, the manufacturer's risk of having to reject an acceptable lot. The chance of accepting a lot of rejectable quality (lot tolerance percent defective or LTPD) must be no greater than b, which is the consumer's risk, that is, the customer's risk of getting an unacceptable lot. Many plans will meet or exceed the requirements P_{A}(AQL) ³ 1  a and P_{A}(LTPD) £ b, where P_{A} is the probability of acceptance. Since inspection and testing cost money, the best plan meets these requirements with the smallest possible sample size. An algorithm for finding the most economical sampling plan follows. Sampling plan economics To determine the plan's economic aspects, the planner must know the perunit inspection cost and the perunit failure cost. The inspection cost can include the cost of purchasing new inspection or test equipmentexisting equipment is a sunk cost even if the accounting system applies its cost to the product under consideration. The failure cost, or penalty cost, is the cost of shipping bad pieces to the next operation. In Harris' application, this is the cost of assembling these units into complete transistor packages before testing and discarding them. The inspection cost is proportional to the plan's average total inspection (ATI), and its penalty cost is proportional to the average outgoing quality (AOQ). Procedure The procedure assumes that the sample is a small portion of the lot, i.e., the binomial approximation to the hypergeometric distribution applies. (The procedure could presumably be modified to use the hypergeometric distribution.) Under the binomial distribution,
n! causes overflow for large sample sizes even in doubleprecision computer variables. The following recursion formula works for such samples:
The algorithm's goal is to find the smallest n for which P_{A}(n_{optimal},c_{optimal},AQL) ³ 1  a P_{A}(n_{optimal},c_{optimal},LTPD) £ b.
Example A singlesample acceptance sampling plan is to have a producer's risk of 6 percent for an acceptable quality level of 0.5 percent nonconforming, and a consumer's risk of 10 percent for a rejectable quality level of 5 percent nonconforming (Juran, 1988, 25.14, Table 25.3). Harris Semiconductor has developed a short Visual Basic program for designing singlesample acceptance sampling plans. Figure 2 shows the user inputs for this problem in the second column under "Plan Design." The results are in the far right column. Juran's example uses a sample plan with n = 78, c = 1 to develop an operating characteristic curve, and this example works the problem in reverse. For n = 78, the acceptance probability at p = 0.005 is P_{A} = 0.995^{78} (the chance of getting no bad pieces) = 0.9415 So a = 0.0585 £ 0.06. The acceptance probability at p = 0.05 is 0.0934 £ 0.10, so n = 78, c = 1 meets the plan requirements. The alpha risk goes down as the sample gets smaller, so a sample of 77 (which yields a consumer's risk of 0.097 £ 0.10) is still capable of meeting the consumer's risk requirement. But for n = 76, b rises to 0.101 > 0.10. The program has therefore found the smallest sample size that fulfills the plan specifications (Table 1).
A c = 0 plan cannot fulfill the plan specifications. A sample of 45 is necessary to meet the consumer's risk requirement of b £ 0.10, but at the acceptable quality level, 0.995^{45} = 0.798 and a = 0.202. Figure 3 shows the operating characteristic curve and average outgoing quality (AOQ) for this plan when the lot size, N, is 1,000. (Enter N, n and c into the Plan Specifications table; click Plot OC Curve; click OC Curve tab.) AOQ is for rectification without replacement; that is, bad units are discarded but are not replaced with good ones.
The numerator is the expected number of bad pieces shipped: (Nn)P_{A} is the number shipped, without inspection or testing, times the nonconforming portion. The denominator is the total pieces shipped: bad units shipped plus N(1p), since all good units are shipped. ATI is the average total inspection. The plot shows that the average outgoing quality limit (AOQL) is about 1 percent when p » 2 percent, and the scrollable OC Curve table shows that the exact value (for p to the nearest 0.1%) is 1.024 percent when the nonconforming fraction is 2.1 percent. Bibliography Juran, J. and Frank Gryna. Juran's Quality Control Handbook, 4th edition. McGrawHill, 1988. McWilliams, Thomas P. How to Use Sequential Statistical Methods. ASQ Quality Press, 1989. Schilling, Edward G. Acceptance Sampling in Quality Control. Marcel Dekker, 1982.
About the author William A. Levinson, P.E., is a staff engineer at Harris Semiconductor's plant in Mountaintop, Pennsylvania. He holds ASQ certifications in quality and reliability engineering, quality auditing, and quality management. He cowrote SPC Essentials and Productivity Improvement: A Manufacturing Approach and edited Leading the Way to Competitive Excellence: The Harris Mountaintop Case Study (ASQ Quality Press). 
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