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Risk Management

Published: Monday, October 15, 2018 - 12:03

All articles in this series:

In my first article, the merits and cautions of AS9138 c=0 sampling plans were discussed and a simple formula was provided to determine the required sample size to detect nonconforming units. In the second article, the process control properties of MIL-STD-105 c>0 sampling plans were demonstrated, and the connectivity to other process control techniques was discussed. Here, a third alternative will be explored that applies the procedures of MIL-STD-105 to “imaginary limits” which are set proportionally inside the real “engineering specification.” This imaginary limit procedure thereby does not allow nonconforming units in the sample and has superior detection capabilities.

Following World War II, at the 105th annual meeting of the American Statistical Association, a discussion ensued regarding an existing procedure called “The test of increased severity.” In this procedure, a paper bag was designed to ship clothing overseas, and its engineering requirement was that the filled bag must withstand a drop to a concrete floor from a height of 6 ft. An attribute test was developed for a more severe “pseudo” specification of 12 feet.^{1} It was concluded that “if not more than 20 percent of the bags failed after a drop of 12 ft, then not more than 1 percent would fail after a drop of 6 ft.” The discussion continued, and finally, W. L. Gore discussed the possibility of applying this method, using a two-step snap gauge to the production problem of machining parts to an engineering specification. He threw the challenge to the group, stating “if anyone is interested in doing some good for us simple engineers, here is a problem.” During the decades that followed, Bernard Dudding, William Jennett, W. L. Stevens, Ellis Ott, Edward Schilling, Shaul Ladany, August Mundel, and others developed these techniques further and published several papers.

Instead of thinking about a specification that is on the engineering drawing, they proposed a second, more severe set of limits—one that is set inside the engineering specification. Sometimes these limits are referred to as a “pseudo specification,” “compressed specification,” “imaginary limits,” or “narrow limits.” The idea is simple: In order to detect nonconforming units from a marginal process, using conventional acceptance sampling, the sample size required is prohibitive. This fundamental problem is remedied with “imaginary limits,” which provide a counting mechanism and proportion. Like the “Test of increased severity,” the proportion of product exceeding the imaginary limits can be used to determine the likely proportion of units exceeding the engineering specification.

A sketch of the relationship between imaginary limits and the engineering specification is shown in figure 1. The imaginary limits are set inside the engineering specification by distance tσ. In this example t=1.5σ, so that the imaginary specification is equal to 50 percent of the engineering specification. For the illustrated process, the imaginary limits facilitate the observation of 6.7 percent of the inspected items that exceed each limit. The same sample compared to the engineering specification could likely indicate zero nonconforming units. Referring to figure 1, if during inspection, 10 of 20 sampled units exceed the upper imaginary limit (p=0.50), we could conclude that the process is roughly centered on the imaginary limit. Given that the upper engineering specification is only 1.5σ from the imaginary limit (now process center), we conclude, using standard z-values, that 6.7 percent of the population in the lot will exceed the engineering specification, even though there may have been zero nonconforming units actually observed in the sample.

The utilization of imaginary limits provides a middle ground between variables and attribute methods. The discussion that follows will demonstrate that imaginary limit techniques require smaller sample sizes than other attribute methods, the calculations are simpler than variables methods, and the sensitivity is superior to attribute methods and roughly equal to variables methods.

Figure 1: |

The following discussion will illustrate how to correlate the proportions obtained from imaginary limits with the acceptance quality limit (AQL) levels of MIL-STD-105. For the following discussion, we refer to back to figure 1, with the imaginary limits set t=1.5σ inside the engineering specification. For simplicity the discussion will focus only on a single-sided upper imaginary limit, which has 6.7 percent of the area under the curve exceeding this upper imaginary limit.

Since the sampling plans of MIL-STD-105 are based primarily on the binomial distribution, the related probabilities are sufficiently defined by the proportion ‘p,’ and the sample size ‘n.’ Let us assume for the moment that we have a marginal process, and we are sampling with sample of size n=13. During inspection, each sample unit is evaluated against the imaginary limit, which is set t=1.5σ inside the engineering specification. A graph of the related binomial distribution for n=13, p=0.067 is shown in figure 2 (left), and it can be seen that it is possible to observe either 0, 1, 2, 3, 4, 5, or even 6 observations exceeding the upper imaginary limit. In fact, the probability associated with three or greater parts exceeding the upper imaginary limit is only 0.05. This 5-percent producer’s risk is aligned with the intention of MIL-STD-105. This is the risk of claiming this process has shifted or increased in variability, when in fact it did not. The parameters of the sampling plan would be n=13, t=1.5, accept (Ac) = 2, and reject (Re) = 3. In a similar manner, the binomial distribution for n=20, p=0.067 is shown in figure 2 (right) and provides Ac = 3 and Re = 4 units exceeding the upper imaginary limit with a producer’s risk of 4 percent.

If the process has less variability than the one illustrated in figure 1 but is centered, the Re number should not be reached, and the lot of material would be accepted. If, on the other hand, the process is worse than the process illustrated in figure 1, either because of increased variability or process shift, the lot would be rejected. The lot would be screened for nonconforming units even if none were observed in the sample. Stevens^{2} provides additional theory about the use of imaginary limits for process and product control. He further states that “an assumption of normality is not essential,” and, “we can assume that the distribution has a single ‘hump’ near the mean and is not violently asymmetrical.”

The MIL-STD-105 (ANSI/ASQC Z1.4) single sampling plan for normal inspection is reproduced in figure 3. Using the AQL column of 6.5 as the closest match for p=0.67, the reader can validate the Ac and Re criterion for the sample sizes 13 and 20 that were calculated in figure 2. The reader can also validate the producer’s risk of figure 2 by observing the MIL-STD-105 operating characteristic (OC) curves shown in figure 4. For code E (n=13), the producer risk is 5 percent, and for code F (n=20), the producer’s risk is 4 percent.

In summary, we have designed our own sampling plan using an upper imaginary limit setting t=1.5σ inside the engineering specification, and we have validated this plan using MIL-STD-105. Going forward, we may use MIL-STD-105 directly after setting imaginary limits that correspond with the given AQL values.

Figure 3: |

We have effectively applied MIL-STD-105 using imaginary limits, and any sampled units that exceed the imaginary limits pose no moral dilemma for acceptance as long as the Re number has not been reached. A lot would be rejected when the imaginary limit Ac criterion are exceeded, even though actual nonconforming units may not have been observed in the sample. A lot would also be rejected if a nonconforming unit is observed within the sample, even though the Re number may not be reached. When a submitted lot is from a process that wanders off target or exhibits excessive variation, it will be detected by the imaginary limits, and therefore the procedure incorporates an element of process control. The imaginary limits can be used to monitor, or control, the likelihood of exceeding the engineering specification.

It is important to note that the OC curves shown in figure 4 relate to the imaginary limits and *not* the engineering specification. The engineering specification OC curve is shown in figure 5 and compares an imaginary limit sampling plan (n=20, t=1.5, Ac=2, Re=3) with a conventional c=0 sampling plan (n=20, Ac=0, Re=1). The superior detection ability of the imaginary limit sampling plan is obvious. Ellis Ott and August Mundel^{3} provide a detailed description for calculating operating characteristic curves for various compression values of n, t, and Ac.

We are not limited to single-sided specifications, and imaginary limit sampling plans can be derived as shown above, or by using MIL-STD-105 directly as long as the “t” value is adjusted to correspond with available AQL plans. The good news is that we are not constrained by MIL-STD-105. We are free to follow the principles illustrated here and make the appropriate trades of producer risk, consumer risk, sample size, and compression factor (t) to suit our needs.

Alternately, if we wanted to hold constant the operating characteristic curves of MIL-STD-105, then design an equivalent imaginary-limit sampling plan, we could make a significant reduction in sample size. Schilling and Sommers^{4} provide equivalency tables that match MIL-STD-105 OC curves to imaginary limit plans having smaller sample sizes.

Shaul P. Ladany^{5} provides a nomograph for determining the optimal imaginary limit parameters n, t, and c for any two-point producer risk pr consumer risk combinations. He notes that the attribute sample sizes for imaginary limit sampling plans are similar to sample sizes for variables sampling plans having similar sensitivity.

It should be noted that not all processes lend themselves neatly to the imaginary limit technique. There is no substitute for human intervention and engineering knowledge. For example, a reaming process naturally runs in the upper 25 percent of the engineering specification. There is no adjustment to “center” this process within the specification because the reamed hole size is determined by standard reamer sizes. When done properly, reaming is a process with very little variation. In this case, the imaginary limit technique will give a false signal, perhaps counting 100 percent of the inspected units as exceeding the imaginary limit. The process in this case is operating correctly, and there may be no risk of exceeding the engineering specification.

Figure 6 illustrates a more complex machining processes where thousands of part characteristics are generated. Narrow-limit methods can be used as a screen to characterize the processes and identify the high risk areas. A high-risk characteristic is defined as one exceeding the imaginary limit Re criterion. Automated measuring machines can reduce volumes of data to a single simple graph. In addition, attribute control charts and Pareto charts that count occurrences exceeding the imaginary limit can supplement the imaginary limit method. Note in this example, conventional attribute acceptance sampling would find no parts exceeding the engineering specification.

Let’s not turn our backs on MIL-STD-105, and let’s start supporting its innovative application. The individuals who conceived and wrote this procedure were visionaries ahead of their time. The talented team of pioneers in the Bell Labs were imaginative and creative enough to give birth to this specification. Let’s not let our own imagination and limitations make MIL-STD-105 irrelevant in today’s environment.

**Sources cited**

1. American Statistical Association, journaled 105th annual meeting, Jan 27, 1946, Cleveland. Special topic, “Acceptance Sampling,” published by ASA, Washington D.C. , 1950, pp. 45–51.

2. W. L. Stevens, “Control by Gauging,” *Journal of the Royal Statistical Society*, Vol. 10, No. 1, 1948, pp. 54–108.

3. Ellis R. Ott, August B. Mundel, “Narrow-Limit Gaging” *Industrial Quality Control,* March 1954, pp. 21–28.

4. Edward G. Schilling, Dan J. Sommers, “Two-Point Optimal Narrow Limit Plans with Applications to MIL-STD105D,” *Journal of Quality Technology,* Vol. 13, No. 2, April 1981, pp. 83–92.

5. Shaul P. Ladany, “Determination of Optimal Compressed Limit Gaging Sampling Plans,” *Journal of Quality Technology,* Vol. 8, No. 4, October 1976, pp. 225–231.