6: II 7: III
3.3 Sampling Schemes
ˆ
pL=Bx(a, b) = 0.000095, (3.18) where
a=b= 63
2 −1 = 30.5, x= max
0, .5−.5QL √
63 63−1
= 0.2733, and
QL=97.006−90
1.9783 = 3.541. M =BBM
63−2 2 ,63−2
2
= 0.02284, (3.19) whereBM =.5
1−1.9740363√−631
= 0.37364.
Therefore, ˆp= (ˆpL+ˆpU) = 0.06416>0.02284 =M, and again the decision would be to reject the lot.
ˆ
pandM can again be calculated using theEPn()andMPn()functions as shown below.
R>library(AQLSchemes) R># sigma unknown
R>P<-EPn(sided="two",stype="unknown",LSL=90,USL=100, xbar=97.006,s=1.9783,n=63)
R>P
[1] 0.06416326
R>M<-MPn(k=1.97403,stype="unknown",n=63) R>M
[1] 0.02284391
of MIL-STD-414. The Range method (shown in the grey boxes) simplified hand-calculations by using the range (R) rather than the sample standard deviation (s). However, with modern computers and calculating devices these methods are no longer necessary. The M-method can be used for either single or double specification limits and eliminates the need for the k-Method.
When MIL-STD-105 was updated to version D and E, it destroyed the match between MIL-STD-414 and MIL-STD-105. Commander Gascoigne of the British Navy showed how to restore the match. His ideas were incorpo- rated into the civilian standard ANSI/ASQ Z1.9 in 1980. It now matches OC performance of the plans with the same AQL between the variable plans in ANSI/ASQ Z1.9 and the attribute plans in ANSI/ASQ Z1.4. Therefore it is possible to switch back and forth between an attribute plan in ANSI/ASQ Z1.4 and a variables plan from ANSI/ASQ Z1.9, for the same lot size inspec- tion level and AQL, and keep the same operating characteristic. These plans are recommended for in-house or U.S. domestic trade partners.
These variables sampling schemes are meant to be used for sampling a stream of lots from a supplier. They include normal, tightened, and re- duced sampling plans and the same switching rules used by MIL-STD- 105E(ANSI/ASQ-Z1.4) Schilling and Neubauer[84]. To use the standard, the supplier and customer companies should agree on an AQL level and adhere to the switching rules. The switching rules must be followed to gain the full benefit of the scheme. Following the rules results in a steep scheme OC curve (approaching the ideal as shown in Figure 2.2 for MIL-STD-105E) with a higher protection level for both the producer and supplier than could be ob- tained with a single sampling plan with similar sample requirement.
The international derivative of MIL-STD-414 is ISO 3951-1. This set of plans and switching rules is recommended for international trade. The ISO scheme has dropped the plans that use the range (R) as an estimate of pro- cess variability (grey boxes inFigure 3.6), and it uses a graphical acceptance criterion for double specification limits in place of the M-method. Using this graphical criterion, a user calculates xand the sample standard deviation s from a sample of data, then plots the coordinates (x,s) on a curve to see if it falls in the acceptance region.
The functionAAZ19()in the R packageAQLSchemescan retrieve the nor- mal, tightened or reduced sampling plans for the variability known or unknown cases from the ANSI/ASQ Z1.9 standard. This function eliminates the need to reference the tables, and provides the required sample size, the acceptabil- ity constant (k) and the maximum allowable proportion nonconforming (M).
When given the sample data and the specification limits, the function EPn in the same package can calculate the estimated proportion non-conforming from sample data as illustrated in section 3.2.
To illustrate the use of these functions, consider the following example of the use of ANSI/ASQ Z1.9. The minimum operating temperature for op- eration of a device is specified to be 180◦F and the maximum operating
FIGURE 3.6: Content of MIL-STD 414
temperature is 209◦F. A lot size N = 40 is submitted for inspection with the variability unknown. The AQL is 1%, and the inspection level is II.
The function AAZ19() has two required arguments, the first argument type, that can take on the values ’Normal’,’Tightened’ or ’Reduced’. It must be supplied to override the default value’Normal’. A second argument stypecan take on the values ’unknown’or’known’. It must be supplied to override the default value ’unknown’. The function AAZ19()is called in the same the way the functions AASingle() and AADouble() were called. They were illustrated in the last chapter.
The section of R code below illustrates the function call, the interactive queries and answers and the resulting plan. The second argument to the func- tionAAZ19()was left out to get the default value.
R>library(AQLSchemes) R>AAZ19(’Normal’)
MIL-STD-414 ANSI/ASQ Z1.9 What is the Inspection Level?
1: S-3 2: S-4 3: I 4: II 5: III
Selection: 4
What is the Lot Size?
1: 2-8 2: 9-15
3: 16-25 4: 26-50
5: 51-90 6: 91-150
7: 151-280 8: 281-400
9: 401-500 10: 501-1200
11: 1201-3200 12: 3201-10,000 13: 10,001-35,000 14: 35,001-150,000 15: 150,001-500,000 16: 500,001 and over
Selection: 4
What is the AQL in percent nonconforming per 100 items?
1: 0.10 2: 0.15 3: 0.25 4: 0.40 5: 0.65 6: 1.0 7: 1.5 8: 2.5 9: 4.0 10: 6.5 11: 10
Selection: 6
n k M
5.000000 1.524668 0.033300
The result shows that the sampling plan consists of taking a sample of 5 devices from the lot of 40 and comparing the estimated proportion non- conforming to 0.0333.
If the operating temperatures of the 5 sampled devices were: (197,188,184,205, and 201), theEPn()function can be called to calculate the estimated propor- tion non-conforming as shown in the R code below.
R>library(AQLSchemes)
R>sample<-c(197,188,184,205,201)
R>EPn(sample,sided="two",LSL=180,USL=209) [1] 0.02799209
The argument sample in the function call is the vector of sample data values. The argumentsided can be equal to"two" or "one", depending on whether there are double specification limits or a single specification limit.
Finally, the arguments LSLand USL give the specification limits. If there is only a lower specification limit, change sided="two" to sided="one" and leave out USL. If there is only an upper specification limit leave out LSLin addition to changing the value ofsided.
The results of the function call indicate that the estimated proportion non-conforming is 0.02799209, which is less than the maximum tolerable pro- portion non-conforming = 0.0333; therefore the lot should be accepted. If the sample mean, sample standard deviation, and the sample size have already been calculated and stored in the variablesxb,sd, andns, then the function call can also be given as:
R>EPn(sided="two",LSL=180,USL=209,xbar=xb,s=sd,n=ns)
The tightened sampling plan for the same inspection level, lot size, and AQL is found with the call:
R>library(AQLSchemes) R>AAZ19(’Tightened’)
Answering the queries the same way as shown above results in the plan:
n k M
5.000000 1.645868 0.013400
Thus the lot would be rejected under tightened inspection since 0.02799209 > 0.0134.
The reduced sampling plan for the same inspection level, lot size and AQL is found with the call:
R>library(AQLSchemes) R>AAZ19(’Reduced’)
Answering the queries the same way as shown above results in the plan:
n k M
4.0000 1.3362 0.0546