6: II 7: III
3.6 Summary
The graph produced by theplot(gA)statement can identify any outliers that may skew the results. From the boxplots in the upper right and middle left, it can be seen that variability of measurements on each part are reasonably consistent and operators are consistent with no apparent outliers.
From the plots on the top left and middle right, it can be seen thatσgauge2 is small relative to the σ2p. Generally, the gauge or measuring instrument is considered to be suitable if the process to tolerance P/T = U SL−LSL6×σgauge ≤ 0.10 where σgauge =q
σ2gauge and U SL, and LSL are the upper and lower specification limits for the part being measured.
If the P/T ratio is greater than 0.1, looking at the σrepeatability2 and σreproducibility2 gives some information about how to improve the measuring process. Ifσrepeatability2 is the largest portion of measurement error it would indicate that the gauge or measuring device is inadequate since this variance represents the variance of repeat measurements of the same part by the same operator with the same gauge. Ifσ2reproducibility is the largest portion of mea- surement error, and if the plots on the middle and bottom left showed large variability in operator averages or inconsistent trends of measurements across parts for each operator, then perhaps better training of operators could reduce σo2 andσpo2 thereby reducingσgauge2 .
by Schilling and Neubauer[84] shows the average sample numbers for various plans that are matched to a single sampling plan for attributes withn= 50, c= 2. In addition to reduced sample sizes, variable plans provide information like the mean and estimated proportion defective below the lower specifica- tion limit and above the upper specification limit. This information can be valuable to the producer in correcting the cause of rejected lots and improving the process to produce at the AQL level or better.
TABLE 3.2: Average Sample Numbers for Various Plans
Plan Average Sample Number
Single Attributes 50
Double Attributes 43
Multiple Attributes 35
Variables (σunknown) 27
Variables (σknown 12
When a continuous stream of lots is being sampled, the published schemes with switching rules are more appropriate. They provide better protection for producer and consumer at a reduced average sample number. The vari- ables plans and published tables described in this chapter are based on the assumption that the measured characteristic is normally distributed.
That being said, the need for any kind of acceptance sampling is depen- dent on the consistency of the supplier’s process. If the supplier’s process is consistent (or in a state of statistical control) and is producing defects or non- conformities at a level that is acceptable to the customer, Deming[23] pointed out that no inspection is necessary or cost effective. On the other hand, if the supplier’s process is consistent but producing defects or nonconformities at a level that is too high for the customer to tolerate, 100% inspection should al- ways be required. This is because the number (or proportion) nonconforming in a random sample from the lot is uncorrelated with the proportion non- conforming in the remainder of the lot. This can be demonstrated with the following simple example.
If the producer’s process is stable and producing 3% nonconforming and delivering lots of 200 items to the customer, then the sampling results using an attribute single sampling plan a sample size ofn= 46 and an acceptance number ofc= 3 can be simulated with the following R code.psrepresents the proportion defective in the sample, andprrepresents the proportion defective in the remainder of the lot.
R># Lot size N=200 with an average 3% defective R>p<-rbinom(50,200,.03)
R>r<-seq(1:50)
R># This loop simulates the number non-conforming in a sample R># of 46 items from each of the simulated lots
R>for (i in 1:length(p)) {
R> r[i]<-rhyper(1,p[i],200-p[i],46) R> }
R># this statement calculates the proportion non-conforming R># in each lot
R>ps<-r/46
R>#This statement calculates the proportion non-conforming R>pr<-(p-r)/154
R># in the unsampled portion of each lot
R>plot(pr~jitter(ps,1),xlab=’Proportion nonconforming in Sample of 46’, ylab=’Proportion nonconforming in remainder of Lot’)
R>abline(h=(.03*46)/46,lty=2) R>cor(ps,pr)
Figure 3.8 is a plot of the simulated proportion nonconforming in the sample of 46 versus the proportion nonconforming in the remainder of the lot of 200−46. It can be seen that there is no correlation. When the process is stable, rejecting lots with more than 3 defectives in the sample and returning them to the producer will not change the overall proportion of defects the customer is keeping.
FIGURE 3.8: Simulated between the proportion nonconforming in a sample and the proportion nonconforming in the remainder of the lot
The same relationship will be true for attribute or variable sampling. This is exactly the reason that in 1980 Ford Motor Company demanded that their suppliers demonstrate their processes were in a state of statistical control with a proportion of nonconforming items at a level they could tolerate. Under those circumstances, no incoming inspection was necessary. Ford purchased a large enough share of their supplier’s output to make this demand. Following the same philosophy, the U.S. Department of Defense issued MIL-STD-1916 in 1996. This document stated that sampling inspection itself was an inefficient way of demonstrating conformance to the requirements of a contract, and that defense contractors should instead use process controls and statistical control methods. Again the large volume of supplies procured by the Department of Defense allowed them to make this demand of their suppliers. For smaller companies, who do not have that much influence on their suppliers, accep- tance sampling of incoming lots may be the only way to assure themselves of adequate quality of their incoming components.
Chapter 4 will consider statistical process control and process improve- ment techniques that can help a company achieve stable process performance at an acceptable level of nonconformance. These methods should always be used when the producer is in-house. In cases where the producer is external to the customer, the use of sampling schemes like ANSI/ASQ-Z1.9 can encour- age producers to work on making their processes more consistent with a low proportion of non-conformance. This is the case because a high or inconsistent level of nonconformance from lot to lot will result in a switch to tightened in- spection using a sampling scheme. This increases the producer’s risk and the number of returned lots.