Acceptance Sampling by Variables

Transcription

1 Acceptance Sampling by Variables Advantages of Variables Sampling o Smaller sample sizes are required o Measurement data usually provide more information about the manufacturing process o When AQLs are very small, the sample sizes required by attributes sampling plans are very large. Disadvantages of Variables Sampling o The distribution of the quality characteristic must be known o A separate sampling plan must be employed for each quality characteristic that is being inspected. o It is possible that the use of a variables sampling plan will lead to rejection of a lot even though the actual sample inspected does not contain any defective items. Two types of variables sampling procedures 1. Plans that control the lot or process fraction defective (or nonconforming). [Procedure 1] Take a random sample of n units. Calculate: Z LSL x LSL If Z LSL k, accept the lot, otherwise, reject it. The value of k, a critical distance can be found from the requirement of the plan. 2. Plans that control a lot or process parameter (usually the mean). 140

2 [Procedure 2] Take a random sample of n units. Calculate: Z LSL x LSL Use Z or Q Z n( n 1) to estimate pˆ of the process. If LSL LSL LSL pˆ M, accept the lot, otherwise, reject the lot. M is a specified value. Both procedures can be used when an upper specification limit is given. The variable then is Z USL USL x When both upper and lower specifications are given, the second procedure should be used. If the standard deviation is not given, then sample standard deviations will be used to replace the in the above equations. Caution in the use of variables sampling o The distribution of the quality characteristic must be known o The usual assumption is that the parameter of interest follows the normal distribution. This is a critical assumption. o If the normality assumption is not satisfied, then estimates of the fraction defective based on the sample mean and standard 141

3 deviation will not be the same as if the parameter were normally distributed o It is possible to use variables sampling plans when the parameter of interest does not have a normal distribution. We then need to know the distribution the parameter follows and develop a corresponding procedure. Design a variable sampling plan for Procedure 1. o Similar to the plan for attribute sampling, based on the OC curve with 1 for lots with fraction defective p 1, for lots with fraction defective p 2. o We can use the nomograph in Figure 16.2 to find the required values of the sampling plan. Example For a given process, the given lower specification is 225. If p 1 =0.01, 1 =0.95 and p 2 =0.06, =0.10, we can find that, when the process is unknown, n=40 and k=1.9 from the nomograph,. When the process is known, then the sample size would be 15, from the same nomograph. Assume that a sample with n=40 is taken and we have x 255 and s 15, then, Z LSL x LSL s Since 2 k 1. 9, we accept the lot. Z LSL

4 We can also use the same nomograph for designing a variables sampling plan using Procedure 2. An additional chart is needed to determine the big M used in this procedure. After the n and k are determined using the nomograph, the chart in Fig 16-3 should be used to find the big M. It is 143

5 also necessary to convert Z LSL or Z USL into an estimated fraction defective. A chart shown in Fig can be used for this purpose. 144

6 Example 16-2 Continue from Example If the process is unknown, we have n=40, k=1.9. From Fig.16-3, we use the abscissa equation to get the abscissa value abscissa = k n 1 n 1 2 k n n

7 Then from the nomograph in Figure 16.3, we read that M=0.030 from the intersection of the curve for sample size 40 and the 0.35 vertical line. If we use the same sample of 40 units (n=40) to find that x 255 and s 15, then as calculated before, we have, Z LSL x LSL s From Figure 16-4, we read pˆ M and the lot will be accepted. 146

8 When we have double-specification limits, Procedure 2 can also be used to decide if a lot can be accepted. We can start in the same way to get sample size n and the critical value k as we did in a single limit plan to have the same values of p 1, p 2 and as the desired doublespecification-limit plan. Then we can find the value of M from Fig We then compute the values of Z LSL and Z USL. We then use Fig to find the corresponding fraction defective estimates pˆ LSL and pˆ USL. If pˆ LSL pˆ USL M, the lot is accepted; otherwise, it will be rejected. 147

9 Example For a plan with process unknown and with p 1 =0.014, =0.95, p 2 =0.09 and =0.10, we can see from the nomograph in Fig that k=1.7 and n=29. Use the abscissa calculation in Fig.16-3, we can calculate the abscissa value: k n n 1 28 abscissa = From the nomograph shown in page 150 (same but more precise than that in Fig.16-3), we can find that M= Assume for this example, that LSL=550 and USL=700. Also assume that we take a sample of 29 and find that the mean of this sample is 570 and the sample standard deviation s is 40, we then calculate: Z LSL Z USL x LSL s USL x s From the chart in page 151 (same but more precise than that in Fig. 16-4), we can read that pˆ and p ˆ Since LSL USL pˆ LSL pˆ USL M 0.040, we accept the lot. 148

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12 Military Standard MIL STD 414 (ANSI/ASQC Z1.9) MIL STD 414 is a lot-by-lot acceptance-sampling plan for variables introduced in Sample size code letters are used as in MIL STD 105E, but the same code letter does not imply the same sample size in both standards. Sample sizes are a function of the lot size and the inspection level. All sampling plans assume the quality characteristic of interest is normally distributed. MIL STD 414 is divided into four sections: 151

13 o A: general description of the sampling plans including definitions, sample size code letters, and OC curves for the plans. o B: variables sampling plans based on the sample standard deviation for the case in which the process or lot variability is unknown. o C: variables sampling plans based on the sample range method o D: variables sampling plans for the case where the process standard deviation is known. ANSI/ASQC Z1.9 is the civilian counterpart of MIL STD 414. Differences and revisions 1. Lot size ranges were adjusted to correspond to MIL STD 105D 2. Code letters assigned to the various lot size ranges were arranged to make protection equal to that of MIL STD 105E 3. AQLs of 0.04, 0.065, and 15 were deleted 4. Original inspection levels I, II, III, IV, and V were relabeled S3, S4, I, II, III, respectively. 5. Original switching rules were replaced by those of MIL STD 105E, with slight revisions. 152

14 Table for the sample size code of the standard 153

15 Example.16-3 Assume that for a quality characteristic of interest, the lower specification limit is 225 psi. The AQL at this specification limit is 1%. Assume that the lot size is 100,000. We use Procedure 1 from MIL STD414 for the testing. Lot standard deviation is unknown. From Table 16-1, if inspection level is IV, the sample size letter is O. Then from Table 16-2, we read that n=100. For the acceptable quality level of 1%, on normal inspection, the k value is On tightened inspection, k is The standard has rules to shift to tightened or reduced inspection based on the process average. MIL STD414 and ANSI/ASQC Z1.9 The standard becomes similar to that in MIL STD105E such as the rule of switching from tightened to normal and so on. Please refer to the text book in Section

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