CHAPTER III CONSTRUCTION AND SELECTION OF SINGLE, DOUBLE AND MULTIPLE SAMPLING PLANS

Transcription

1 CHAPTER III CONSTRUCTION AND SELECTION OF SINGLE, DOUBLE AND MULTIPLE SAMPLING PLANS 3.0 INTRODUCTION When a lot is received by the customer (consumer), he has to decide whether to accept or reject the lot based on its quality. This is called lot sentencing. The consumer will check for defects in the products, if the defects are less than the pre-decided level, the lot is accepted, otherwise rejected. This can be done by 100% inspection. But, in some cases where the testing is destructive or 100% inspection requires more cost or time, then we go for sampling procedure which is known as Acceptance Sampling. A Lot Acceptance Sampling Plan (LASP) is a sampling scheme and a set of rules for making decisions. The decision, based on counting the number of defects in a sample, can be to accept the lot, reject the lot or even, for multiple or sequential sampling schemes, to take another sample and then repeat the decision process. These types of Lot Acceptance Sampling Plans are discussed below SINGLE SAMPLING PLAN The single sampling plan is the most widely used sampling plan in the area of acceptance sampling. A single sampling plan which has acceptance number zero, with a small sample size is often employed in a situations involving costly or destructive testing by attributes. The small sample size is warranted because of costly nature of testing and a zero acceptance number 63

2 arises in practice. A single-sampling plan is a lot-sentencing procedure in which one sample of n units is selected at random from the lot, and the disposition of the lot is determined based on the information contained in that sample. For example, a single-sampling plan would consist of a sample size n and an acceptance number c. The procedure would operate as follows: (i) Select n items at random from the lot. (ii) If there are c or fewer defectives in the sample, accept the lot, (iii) If there are more than c defective items in the sample, reject the lot. These are the most common (and easiest) plans to use although not the most efficient in terms of average number of samples needed. Usually, Binomial, Poisson and Hyper Geometric Distributions are useful to apply the probability of accepting a lot in any produced data in sampling plans. In particular, we are using Binomial Distribution applicable to construct OC function for accepting good lots. The function is as follows: n B (n, p, x) = p x = 0 otherwise x n x ( 1 p), x=0,1,...,n The following assumptions should hold good while applying Binomial Distribution for evaluating the OC function [Guenther (1977)] 1. Each item can be classified into one of two categories, say, defective and non-defective 2. The probability p of obtaining a defective is the same for each item selected 3. Each item is selected independently of any other 64

3 4. A fixed number of items, say n, are selected. The probability of acceptance according to Binomial distribution is L(p) = n c x n x p (1 p) x= 0 x fraction defective. This is called the Binomial OC function where p is the incoming The approximation of hyper geometric to the Binomial holds good n when p lies between 0.1 and 0.9, n>10 and < It is also shown that the N Binomial OC function can be approximated by the Beta distribution with the relationship L (p, n, c, Binomial) = 1 - β(p, c+1, n-c) where β(.) denotes the type I Beta distribution. This method is found to involve less rounding errors than the Binomial terms OC function of Poisson Distribution Suppose the inspection procedure counts the number of defects instead of defectives in the sample, the number of defects in the i th unit of the lot is a random variable X i, i=1,2,...,n. These follows Poisson distribution with parameter λ>0. It means each unit on an average has λ defects. Taking λ=np the following poisson approximation can be used to evaluate the OC function. c e L(p, n, c, Poisson) = np ( np) x= 0 x! x 65

4 The use of Poisson approximation works well when p 0.1, n 30 and N n < 0.1. Very often Poisson OC function is used by practitioners because cumulative binomial tables are easy to adopt Rectifying Inspection Procedure One of the problems in acceptance sampling is regarding the disposal of rejected lots. This is called lot disposal action. There are three options as shown below: 1. Neither the conforming nor the defective units of the test sample are put back into the lot after the inspection and no replacement is made. 2. All defective units in the sample are sorted out and not replaced. 3. All defective units in the sample are taken aside and replaced by good units (sample size n is preserved). If the rejected lot is 100% inspected and if defective units are replaced by good units, the method is called rectifying inspection. Since defectives are replaced in every rejected lot, such lots will not contain any defectives. However, the expected fraction p of defectives will reach the customer through accepted lots. A measure of performance of such a system is called Average Outgoing Quality (AOQ) is given by L ( p) p( N n) AOQ = N n where L(p) is the probability of accepting the lot. When the ratio N becomes very small AOQ can be approximated by L(p).p When the lot quality p is very small, many lots get accepted and naturally fewer defectives reach the customer. It means AOQ is small when p is 66

5 small. When as p increases, the AOQ tend to increase, reach as a maximum and starts to decrease as p increases further. The maximum value of AOQ is called Average Outgoing Quality limit or AOQL. It is the average level of quality across a large stream of lots. The Average Total Inspection per lot is denoted by ATI and it is given by ATI = n + (1 L(p))(N-n) where L(p) is the probability of accepting the lot. The expression for ATI shows that in every rejected lot, all the N units will be compulsorily inspected. Hence when the incoming lot quality is poor (p is high), the ATI increases rapidly when the lot size increases. ATI is also one criterion for designing a sampling plan Procedure for Construction and selection of Single Sampling Plan In a Single Sampling Plan, we take only one sample to justify the lot to make lot sentencing. The Procedure used for construction of the single sampling plan is as follows: (i) Select a sample of n = 100 units (ii) If the number of defects is less than or equal to c (= 5,4,3,2,1 or 0) then accept the lot (iii) If the number of defects is greater than c (= 5,4,3,2,1 or 0) as specified in the above step, then reject the lot (iv) The Operating characteristic function P a (p) is calculated by using e the formula P a (p) = np ( np) C! c for different values of c (i.e., c=5,4,3,2,1 and 0) separately. For some varying values of p, the proportion of defectives, taken from 0.2 to 5.0 (values of np), the probability of acceptance P a (p) for various 67

6 values of acceptance number c i.e., 0,1,2,3,4 and 5is calculated in the following table. TABLE 3.1 OC FUNCTION VALUES FOR SINGLE SAMPLING PLAN Pa(p) (x)np c=5 c=4 c=3 c=2 c=1 c= In the above table, the Operating Characteristic function P a (p) values are calculated. For those values, the Average Outgoing Quality (AOQ) values are also calculated in the following table by using the following formula: 68

7 AOQ(p) = p.p a (p) and the maximum value for each c value is highlighted in each column as we call them Average Outgoing Quality Limit (AOQL). TABLE 3.2 AOQ VALUES FOR SINGLE SAMPLING PLAN (y)aoq (x)np c=5 c=4 c=3 c=2 c=1 c=

8 3.1.4 Maximum Allowable Average Outgoing Quality Around Linear Trend (MAAOQ) LT As the AOQL is the worse quality level that the consumer will receive in the long run, we are calculating the Maximum Allowable Average Outgoing Quality over Linear Trend, (MAAOQ) LT. The principle of Least Squares is the most popular and widely used technique for fitting mathematical functions to the any given set of observations. Let us consider the straight line equation y = a+bx The above equation represents a family of straight lines for different values of the arbitrary constants a and b. Now, we are able to obtain a, b values using the legender s principle of least squares. From the principle of maxima and minima, the partial derivatives of E (the residual sum of squares) with respect to a, b and equating to zero, we get the following normal equations n i= 1 y i = na + b n i= 1 x i n i= 1 y i = a n i= 1 x + b i n i= 1 x 2 i These are the normal equations for estimating a and b. We apply the Trend procedure for AOQ values in the above table 3.2. It is also a procedure for construction and selection of sampling plans for variable inspection. The fitted Linear Trend to the AOQ (y) values after estimating the values of constants a and b by using Principle of Least Squares from the data is presented in the following table. The values of (MAAOQ) LT are highlighted in the following table. 70

9 TABLE 3.3 TREND VALUES FOR SINGLE SAMPLING PLAN (y)aoq Trend (x)np c=5 c=4 c=3 c=2 c=1 c=0 c=5 c=4 c=3 c=2 c=1 c=

10 From the above two tables, the highlighted values i.e., AOQL and (MAAOQ) LT values are listed in the following table TABLE 3.4 VALUES OF AOQL AND (MAAOQ) LT FOR A SINGLE SAMPLING PLAN AOQL (MAAOQ) LT OC Curves for single sampling plan From the first table, i.e., table 3.1 where the probability of acceptance for various p values and various Acceptance numbers (c values) are presented, the OC curves are constructed to test the performance of this Single Sampling Plan c=5 c=4 c=3 c=2 c=1 c= Pa(p) np FIGURE 3.1 OC CURVES FOR SINGLE SAMPLING PLAN 72

11 3.1.6 AOQ Curves for SSP From the second table i.e., table 3.2 where AOQ values are calculated, the AOQ curves are drawn by taking np values on X-axis and AOQ values on Y-axis AOQ(p) = p.pa(p) AOQ c=0 c=1 c=2 c=3 c=4 c= np FIGURE 3.2 AOQ CURVES FOR SINGLE SAMPLING PLAN Trend lines for AOQ of Single Sampling Plan For the fitted values of AOQ, i.e., for y^ values in the above table, the trend lines are drawn for various c values which is shown below 73

12 Trend c=5 c=4 c=3 c=2 c=1 c= np FIGURE 3.3 TREND LINES FOR AOQ OF SINGLE SAMPLING PLAN AOQ curves along with Trend lines for Single Sampling Plan Again, for each value of the acceptance number c, AOQ curve and its trend line are drawn separately presented from the above two figures as follows AOQ Trend y ŷ np FIGURE 3.4 AOQ ALONG WITH ITS TREND LINE FOR SINGLE SAMPLING PLAN WITH c=5 74

13 AOQ Trend y ŷ np FIGURE 3.5 AOQ ALONG WITH ITS TREND LINE FOR SINGLE SAMPLING PLAN WITH c= y ŷ AOQ Trend np FIGURE 3.6 AOQ ALONG WITH ITS TREND LINE FOR SINGLE SAMPLING PLAN WITH c=3 75

14 FIGURE 3.7 AOQ ALONG WITH ITS TREND LINE FOR SINGLE SAMPLING PLAN WITH C=2 y ŷ AOQ Trend np FIGURE 3.8 AOQ ALONG WITH ITS TREND LINE FOR SINGLE SAMPLING PLAN WITH c=1 76

15 AOQ Trend y ŷ np FIGURE 3.9 AOQ ALONG WITH ITS TREND LINE FOR SINGLE SAMPLING PLAN WITH C=0 From the values of the AOQL column we take maximum for AOQL, theoretically it is good, but it is worst quality level that the consumer gets in the long run. So, we are taking the (MAAOQ) LT as the function of all observations in AOQ column. We observed that (MAAOQ) LT is the better performance measure instead of AOQL. From the above tables and OC curves, we conclude that the curves indexed with (MAAOQ) LT gives the better performance compared with the plans indexed with AOQL. 3.2 DOUBLE SAMPLING PLANS When we can t arrive at a decision by using a single sample, we introduce a second sample to decide whether to accept or reject the lot. This is called Double Sampling Plan as we are taking two samples to arrive at a decision of lot sentencing. In a Double Sampling Plan, we take two samples of sizes n 1 and n 2. Here we need two acceptance numbers c 1 and c 2. First, we select a first sample of size n 1 and check for defects in the products. If the number of defects is less than or equal to c 1, the lot is accepted and if it is 77

16 greater than c 2, the lot is rejected. If this number of defects is in between c 1 and c 2, then take a second sample of size n 2. Now, check the combined sample of size n 1 +n 2 for defects. If this number of defects is less than or equal to c 2, the lot is accepted and if the number of defects is greater than c 2, the lot is rejected Construction of a Double Sampling Plan with n value fixed (i.e. n 1 =n 2 =100) First we take a Double Sampling Plan with equal n 1 and n 2 values i.e., n 1 = n 2 = 100 for various c 1 and c 2 values i.e., c 1 =0, c 2 =3; c 1 =1, c 2 =4; c 1 =2, c 2 =5 separately. For these values probability of acceptance and hence OC curves are constructed to check the performance of the plans. The stepwise procedure for construction of a Double Sampling Plan is as follows: (i) First take a sample of n 1 = 100 units (ii) If the number of defects d 1 in these 100 units are less than c 1, accept the lot and if the number of defects are greater than c 2, reject the lot. (iii) If the number of defects lies between c 1 and c 2, then take a second sample of size n 2 = 100 units. (iv) If the number of defects in the combined sample of n 1 + n 2 = 200 units d 2 is less than or equal to c 2, accept the lot, otherwise reject the lot. (v) The operating characteristic function is calculated by using the formula e P a (p) = np ( np) c! c at each stage. 78

17 The table of probability of acceptance P a (p) for varying values of c 1 and c 2 are presented in the following table TABLE 3.5 OC FUNCTION OF DOUBLE SAMPLING PLAN Pa(p) np c 1 =0, c 2 =3 c 1 =1, c 2 =4 c 1 =2, c 2 =5 A C B A C B A C B

18 3.2.2 OC curves for DSP The Operating Characteristic (OC) curves from the above table to check the performance of the Double Sampling Plan for different values of c 1 and c 2 are shown below FIGURE 3.10 OC CURVE FOR DOUBLE SAMPLING PLAN WITHc 1 =0, c 2 =3 FIGURE 3.11 OC CURVE FOR DOUBLE SAMPLING PLAN WITH c 1 =1, c 2 =4 80

19 FIGURE 3.12 OC CURVE FOR DOUBLE SAMPLING PLAN WITH c 1 =2, c 2 =5 From the above Figures we conclude that the performance of the plan is satisfactory Double Sampling Plan with varying n value (n 1 =100 and n 2 =150) Now, we go for a Double Sampling Plan with varying values of n 1 and n 2 with c 1 and c 2 values i.e., c 1 =0, c 2 =3; c 1 =1, c 2 =4; c 1 =2, c 2 =5. Let us take n 1 = 100 and n 2 =150. Now the probability of acceptance P a (p) values are calculated in the following table is as follows: The stepwise procedure for construction of a Double Sampling Plan (i) First take a sample of n 1 = 100 units 81

20 (ii) If the number of defects d 1 in these 100 units are less than c 1, accept the lot and if the number of defects are greater than c 2, reject the lot. (iii) If the number of defects lies between c 1 and c 2, then take a second sample of size n 2 = 150 units. (iv) If the number of defects in the combined sample of n 1 + n 2 = 250 units d 2 is less than or equal to c 2, accept the lot, otherwise reject the lot. (v) The operating characteristic function is calculated by using the formula P a (p) = e np ( np) c! c at each stage Operating Characteristic Values for DSP The table of probability of acceptance P a (p) for varying values of c 1 and c 2 and for different values of np are presented in the following table 82

21 TABLE 3.6 OC FUNCTION VALUES OF DOUBLE SAMPLING PLAN FOR VARYING VALUES OF n 1 AND n 2 Pa(p) np c 1 =0,c 2 =3 c 1 =1, c 2 =4 c 1 =2, c 2 = OC curves for DSP For the values in the above table, the OC curves to check the performance of the Double Sampling Plan are presented as follows 83

22 Pa(p) np FIGURE 3.13 OC CURVE FOR DOUBLE SAMPLING PLAN The above OC curves are separately presented for each (c 1, c 2 ) set of values as shown below Pa(p) np FIGURE 3.14 OC CURVE FOR DOUBLE SAMPLING PLAN WITHc 1 =0, c 2 =3 84

23 Pa(p) np FIGURE 3.15 OC CURVE FOR DOUBLE SAMPLING PLAN WITH c 1 =1, c 2 = Pa(p) np FIGURE 3.16 OC CURVE FOR DOUBLE SAMPLING PLAN WITH c 1 =2, c 2 = AOQ Values for DSP The Average Outgoing Quality (AOQ) values are calculated for the values in the above table by using the formula AOQ(P*) = p.p a (P*). These AOQ values are denoted with y whereas np values are denoted with x. The highest value of AOQ, called AOQL is highlighted/determined in each column. TABLE

24 AOQ VALUES OF DOUBLE SAMPLING PLAN Pa(p) (y)aoq (x)np c 1 =0,c 2 =3 c 1 =1, c 2 =4 c 1 =2, c 2 =5 c 1 =0,c 2 =3 c 1 =1, c 2 =4 c 1 =2, c 2 = AOQ Curves for DSP The AOQ curves for the values in the above table are shown below for various (c 1,c 2 ) set of values 86

25 AOQ np FIGURE 3.17 AOQ CURVES FOR DOUBLE SAMPLING PLAN Maximum Allowable Average Outgoing Quality Around Linear Trend : As the AOQL is the worse quality level that the consumer will receive in the long run, we are calculating the Maximum Allowable Average Outgoing Quality over Linear Trend, (MAAOQ) LT. Let Linear Transformation by taking the principles of Least Squares obtain required constants for y=a+bx and it be a fitted Linear curve. We apply the above procedure for AOQ values in the above table. It is also a procedure for construction and selection of sampling plans for variable inspection. The values of (MAAOQ) LT are highlighted/determined in the following table. The fitted Linear Trend to the AOQ (y) values after estimating the values of constants a and b by using Principle of Least Squares from the data is presented below Now a Linear Trend to the AOQ values by using Principle of Least Squares by taking np as x and AOQ as y. The values of (MAAOQ) LT are also highlighted in the following table 87

26 TABLE 3.8 TREND VALUES FOR AOQ OF DOUBLE SAMPLING PLAN (x)np (y)aoq y^(trend) c 1 =0,c 2 =3 c 1 =1,c 2 =4 c 1 =2,c 2 =5 c 1 =0,c 2 =3 c 1 =1,c 2 =4 c 1 =2,c 2 = TREND LINES for DSP For the trend values (y^) calculated in the above table, trend lines are presented as follows 88

27 Trend np FIGURE 3.18 TREND LINES FOR AOQ OF DOUBLE SAMPLING PLAN AOQ CURVES ALONGWITH TREND LINES FOR DSP Now, the AOQ curves along with their trend lines are presented separately for (c 1,c 2 ) set of values i.e., c 1 =0, c 2 =3; c 1 =1, c 2 =4; c 1 =2, c 2 =5 values in the following figures AOP & Trend np FIGURE 3.19 AOQ ALONG WITH ITS TREND LINE FOR DOUBLE SAMPLING PLAN WITH c 1 =0,c 2 =3 89

28 AOP & Trend np FIGURE 3.20 AOQ ALONG WITH ITS TREND LINE FOR DOUBLE SAMPLING PLAN WITH c 1 =1, c 2 =4 AOP & Trend np FIGURE 3.21 AOQ ALONG WITH ITS TREND LINE FOR DOUBLE SAMPLING PLAN WITH c 1 =2,c 2 =5 is satisfactory From the above Figures we conclude that the performance of the plan 90

29 3.3 MULTIPLE-SAMPLING PLANS A multiple-sampling plan is an extension of double-sampling inspection scheme in that more than two samples can be required to sentence a lot. This plan will operate as follows: If, at the completion of any stage of sampling, the number of defective items is less than or equal to the acceptance number, the lot is accepted. If, during any stage, the number of defective items equals or exceeds the rejection number, the lot is rejected; otherwise the next sample is taken. The multiple-sampling procedure continues until the fifth sample is taken, at which time a lot disposition decision must be made. The first sample is usually inspected 100%, although subsequent samples are usually subject to curtailment. The construction of OC curves for multiple-sampling is a straightforward extension of the approach used in double-sampling. Similarly, it is also possible to compute the average sample number curve of multiplesampling plans. One may also design a multiple-sampling plan for specified values of p 1, c 1, p 2, and c 2. The principal advantage of multiple-sampling plans is that the samples required at each stage are usually smaller than those in single or double sampling; thus, some economic efficiency is connected with the use of this procedure. However, multiple-sampling is much more complex to administer and preserves good lots. A Multiple Sampling Plan with 4 samples and having various Acceptance numbers and Rejection numbers is calculated as follows: 91

30 Sample number TABLE 3.9 EXAMPLE FOR MULTIPLE SAMPLING PLAN Size Combined size Acceptance number Rejection number The calculation of the OC function and OC curve for the Multiple Sampling Plan follows the same pattern as the Double Sampling Plan. The following calculations shows the probability of acceptance and rejection of a 2% defective lot. To simplify the calculations, it is assumed that the unsampled portion of the lot will still be substantially 2% defective regardless of the results of the past samples. Under this assumption, it is not necessary to recalculate µ np based on the number of defectives found in the first sample. Here we take a Multiple Sampling Plan consisting of four samples of 100 units each. Under the above assumption, we calculate µ np as µ np = (100)(0.02) = 2 From the tables of Poisson Distribution, we calculate the necessary probabilities. The calculations for the Multiple Sampling Plan are as follows P 0 =0.135 (P 0 ) P 1 = =0.271 P 2 = =0.271 P 3 or more = 1 - ( ) = =

31 Sample I P 0 = (Continue sampling, entering sample II with zero defectives) P 1 = (Continue sampling, entering sampling II with 1 defective) P 2 or more = = (Reject) Sample II P 0 : P 0-0 = (0.135)(0.135)= 8225 (ACCEPT) P 1 : P 0-1 = (0.135)(0.271)= P 1-0 = (0.271)(0.135)= (Continue Sampling, entering sample III with one defective) P 2 : P 0-1 = (0.135)(0.271)= P 1-1 = (0.271)(0.271)= (Continue Sampling, entering sample III with two defectives) P 3 or more : P 0-3 or more = (0.135)(0.323) = P 1-2 or more = (0.271)(0.594) = (Reject) Check: P 0 II +P 1 II + P 2 II +P II 3 or more = = NOTE: The probability of continuing sampling is 93

32 P 1 II +P 2 II = = The sum of the computed probabilities of the different possible results in sample III must be equal to this figure. Sample III III P 1 III P 2 P 3 III : P 1-0 = ( )(0.135) = 9878 (ACCEPT) : P 1-1 = ( )(0.271) = 9829 P 2-0 = ( )(0.135) = (Continue sampling, entering Sample IV with 2 defectives) : P 1-2 or more = ( )(0.594) = P 2-1 or more = ( )(0.865) = (Reject) (P 1 or more = 1-P 0 = ) Check: P III 1 +P III III 2 + P 3 = = SAMPLE IV IV P 2 IV P 3 : P 2-0 = ( )(0.135) = 4682 (ACCEPT) : P 2-1 = ( )(0.271) = 9399 P 3-0 = ( )(0.135) = P IV 4 or more : P 2-2 or more = ( )(0.729) = P 3-1 or more = ( )(0.865) = (Reject) 94

33 The Values of Probability of Acceptance and the Probability of Rejection are presented in the following Table TABLE 3.10 PROBABILITY OF ACCEPTANCE AND REJECTION OF MSP Sample Number Probability of Acceptance Rejection The OC curve which is drawn for sample number and Probability of Acceptance is given by Pa(p) Saple Number FIGURE 3.21 OC CURVE FOR MULTIPLE SAMPLING PLAN 95

34 Also another curve is drawn for sample number and Probability of Rejection which is given by Pa(p) Sample Number FIGURE 3.22 REJECTION CURVE FOR MULTIPLE SAMPLING PLAN Here we get a single value of AOQ which is given by AOQ = 0.02* = Here we got only a single value for AOQ and hence we cannot get any AOQL value or we can say that that value only becomes AOQL value. Also calculation of Trend is not possible here for a single value. Hence in this case, (MAAOQ) LT is not possible. Variate difference method is the approximation of Time series analysis of different samples for randomness. In this thesis, we took the random data for drawing OC, AOQ curves and for constructing their tables. So, for randomness, applied variate difference method was applied and conclude that 96

35 the procedures for construction of sampling plans indexed with (MAAOQ) LT gives the better performance compared with that of AOQL plans For all the above Sampling plans, the Average Outgoing Quality Limit (AOQL) and the Maximum Allowable Average Outgoing Quality over Linear Trend (MAAOQ) LT values are calculated and by using Variate Difference Method, we conclude that the Sampling Plans indexed with Maximum Allowable Average Outgoing Quality over Linear Trend (MAAOQ) LT are better than those indexed with AOQL. 97