Session XVII. Sampling Statistical Aspects

Transcription

1 Session XVII Sampling Statistical Aspects

2 Outline Fundamental Aspects Statistical Aspects Sampling Plan Design Sampling Plan System

3 Statistical Aspects The Operating Characteristic Curve (OCC): Measures the performance of an acceptance sampling plan Plots the probability of accepting the lot versus the lot fraction defective Shows the probability that a lot submitted with a certain fraction defective will be either accepted or rejected

4 Operating Characteristic Curve This curve plots the probability of accepting the lot (Y-axis) versus the lot fraction or percent defectives (X-axis). The OCC curve is the primary tool for displaying and investigating the properties of a Lot Acceptance Sampling Plan.

5 Operating Characteristic Curve OC Curve for the Single Sampling Plan N = 3000, n = 89, and c = 2

6 OC Curve for Double Sampling Plan OC Curve for the Double Sampling Plan N = 2400 n1 = 150, c1 = 1, r1 = 4 N2 = 200, c1 = 5, r1 = 6

7 Double Sampling Plan Inspect a sample of 150 from lot of 2400 If 1 or less Nonconforming units accept lots and stop If 2 or 3 nonconforming units, inspect a second sample of 200 If 4 or more Nonconforming units the lot is not accepted and stop If 5 or less Nonconforming units On both samples, Accept the lot If 6 or more Nonconforming units On both samples The lot is not accepted Graphical description of the double sampling plan: N=2400 n1=150, c1=1, r1=4 n2=200, c2=5, and r2=6

8 OCC for a Multiple Sampling Plan OC Curve for the Multible Sampling Plan N = 3000, n = 89, and c = 2

9 OC Curves Types There are two types of OC curves: Type A Gives the probability of acceptance of an individual lot coming from finite production Type B Gives the probability of acceptance for lots coming from a continuous production

10 OC Curves Types Types A and B OC Curves

11 OC Curve Properties Sample size as a fixed % of lot size. Sample size that is 10% of lot size. Larger the sample size, the curve gets steeper. Decrease the acceptance number, the curve gets steeper.

12 OC Curve Properties OC Curves for Sample Sizes that are 10% of the Lot Size OC Curves for Fixed Sample Size (Type A)

13 OC Curve Properties OC Curves Illustrating Change in Sample Size OC Curves Illustrating Change in Acceptance Number

14 OC Curve Properties Ideal OC Curve

15 Acceptable Quality Level (AQL) The AQL is a percent defective that is the base line requirement for the quality of the producer's product. The producer would like to design a sampling plan such that there is a high probability of accepting a lot that has a defect level less than or equal to the AQL.

16 Producer s Risk (Type I Error) n, c α

17 Limiting Quality (LQ) The LQ is a designated high defect level that would be unacceptable to the consumer. The consumer would like the sampling plan to have a low probability of accepting a lot with a defect level as high as the LQ.

18 Consumer s Risk (Type II Error) This is the probability, for a given (n, c) sampling plan, of accepting a lot with a defect level equal to the LTPD. The consumer suffers when this occurs, because a lot with unacceptable quality was accepted. The symbol β is commonly used for the Type II error and typical values range from 0.2 to 0.01.

19 Consumer-Producer Relations Consumer Producer Relationship

20 Average Outgoing Quality (AOQ) A common procedure, when sampling and testing nondestructive lots, is to 100% inspect rejected lots and replace all defectives with good units. In this case, all rejected lots are made perfect and the only defects left are those in lots that were accepted. AOQ (Est)= 100p0 x Pa (See Table 10-3)

21 Average Outgoing Quality (AOQ) Average Outgoing Quality Curve for the Sampling Plan N = 3000, n = 89, and c = 2

22 AOQ and Acceptance Sampling Producer 15 lots 2% nonconforming N=3000 n=89 c=2 11 lots 2% nonconforming Consumer 4 lots 2% nonconforming 4 lots 0% nonconforming Total Number Number Nonconforming 11 lots- 2% Nonconforming 11(3000)=33,000 33,000(0.02)=660 4 lots- 0% Nonconforming 4(3000)(0.98)=11, , Percent Nonconforming (AOQ) = 660/44,760 X 100 =1.47%

23 Average Outgoing Quality Limit A plot of the AOQ (Y-axis) versus the incoming lot p (Xaxis) will start at 0 for p = 0, and return to 0 for p = 1 (where every lot is 100% inspected and rectified). In between, it will rise to a maximum. This maximum, which is the worst possible long term AOQ, is called the Average Outgoing Quality Limit AOQL.

24 Average Sample Number (ASN) For a single sampling (n, c) we know each and every lot has a sample of size n taken and inspected or tested. For double, multiple and sequential plans, the amount of sampling varies depending on the number of defects observed.

25 Average Sample Number (ASN) For any given double, multiple or sequential plan, a long term ASN can be calculated assuming all lots come in with a defect level of p. A plot of the ASN, versus the incoming defect level p, describes the sampling efficiency of a given plan scheme. for a double sampling plan ASN = n1 + n2 (1 P1)

26 Average Sample Number (ASN) ASN Curves for Single, Double, Multiple, and Sequential Sampling

27 Average Total Inspection (ATI) When rejected lots are 100% inspected, it is easy to calculate the ATI if lots come consistently with a defect level of p. For a single sampling plan (n,c) with a probability Pa of accepting a lot with defect level p, we have: ATI = n + (1 - Pa) (N - n) where N is the lot size.

28 Average Total Inspection (ATI) ATI Curve for N = 3000, n = 89, and c = 2

29 Computer Program EXCEL program files on the website will solve for OC and AOQ curves.