CERTAIN RESULTS ON MULTI DIMENSIONAL SAMPLING PLANS. The mixed sampling plans are two stage sampling plans in which variable

Transcription

1 CHAPTER III CERTAIN RESULTS ON MULTI DIMENSIONAL SAMPLING PLANS Focus The mixed sampling plans are two stage sampling plans in which variable and attribute quality characteristics are used in deciding the acceptance or rejection of the lot. Due to modern quality control systems, mixed sampling plans are widely applied in various stages of production. In industries, if different sampling plans are used for different quality characteristics, then it would result loss in Economy, Time and Labour. Therefore, an attempt has been made to design, Multi Dimensional Mixed sampling Plans (MDMSP). Based on multi dimensional quality characteristics, MDMSP aims at controlling overall quality of a lot or process. The designing aspect of MDMSP is given in detail which is based on Normal distribution and Poisson model in the st and nd stage respectively. Tables and illustrations are also been provided. This Chapter comprises of three sections that deal with Designing of Multidimensional Mixed Sampling plans for dependent case, Bayesian Sampling Plans for Multidimensional Quality Characteristics and Multidimensional Mixed Sampling plans with Variance criterion. Section 3. Designing and Selection of Multi Dimensional Mixed Sampling Plans for Dependent Case. Section 3. Bayesian Sampling Plans for Multi Dimensional Quality Characteristics. Section 3.3 Multi Dimensional Mixed Single Sampling plans for Maximum Allowable Variance. 77

2 * SECTION 3. DESIGNING AND SELECTION OF MULTI DIMENSIONAL MIXED SAMPLING 3.. INTRODUCTION PLANS FOR DEPENDENT CASE A multidimensional mixed sampling scheme consists of two stages in which several variables and attribute quality characteristics are considered in deciding the acceptance or rejection of the lot. The main advantage of Multi- Dimensional Mixed Sampling Plans (MDMSP) over any other plan is the reduction in the sample size for the same amount of protection. The first stage sample results are used in the second stage. 3.. FORMULATION OF MDMSP The developments of MDMSP and the subsequent discussions are limited only to the upper specification limit. The plans in case of single sided specification (U), standard deviation known can be formulated using the parameters, Where, (n, n, k, k,, k m ; c, c,, c n ) n =First stage sample size with respect variable quality characteristics. n =Second stage sample size with respect attribute quality characteristics. k i = variable factor for i th variable quality characteristics, such that, the lot is accepted if A = U i K i taken for making decision. σ, otherwise a second stage sample is c j = Acceptance number for th j attribute quality characteristics. 78

3 3.. OPERATING PROCEDURE OF MIXED DEPENDENT PLANS: A dependent plan would require the following fundamental steps.. Obtain the first sample.. Test the first sample against a predetermined variables acceptance criterion and: (a) Accept if the test meets the criterion. (b) If the first test fails to meet the criterion, (i) Reject, if the number of defectives in the first sample exceeds a predetermined attributes criterion. (ii) Otherwise resample, call it as the second sample 3. Obtain a second sample, if necessary, according to (b) criterion. 4. Test the first and second sample taken together against a predetermined attributes criterion and accept or reject the lot as indicated by the sampling plan. * A part of this section is published in International Journal of Mathematics and Computation (00), vol 8, No.S0 PP

4 3..3 DESIGNING AND CONSTRUCTION OF MDMSP INDEXED THROUGH AQL WHEN THE FIRST STAGE SAMPLE SIZE n i IS KNOWN Let the two stages of the MDMSP and all the quality characteristics considered are assumed to be dependent. A MDMSP (n, n, k, k km, c, c.c n ) should satisfy the requirement P a (p ) β (3..) Where, p is the AQL. Equation (3..) has to be satisfied for all quality characteristics. Procedure: Let the i variable and j attribute quality characteristics be considered. Step (): Assume that the plan is dependent. Step (): Split the probability of acceptance that will be assigned to the first stage. Let it be β respective to p such that β β Step (3): Decide the sample size n i to be used. Step (4): Calculate the acceptance limit A i = U K iσ,( i =,... m) Where, K i Z( β ) = Z( p ) + n i 80

5 Step (5): Now the sample size n of the Multi Dimensional mixed dependent plan is fixed as n = n i and variable factor k = max(k i ) Step (6): Obtain the values of P (y, n j ) by successive approximation to satisfy the relation β c j c j x β = Pn ( x, x > A ) P( y, n ) forp p i i j = x= 0 y= 0 (3..) if such values exist. This will determine the second stage sample size n j and the acceptance number (j;, m) for each attribute characteristics. Step (7): Now the second stage sample size n of MD mixed dependent sampling plan is fixed as the maximum sample size of the individual quality characteristic sizes. n = max { n j }. Step(8): The acceptance number for multi characteristics quality is re-estimated after the second stage sample size has been fixed satisfying (3..). 8

6 Illustration 3.. Let the items in the lot shall be tested with respect to seven variables and seven attribute quality characteristics. Let the AQL s of quality characteristics be.00 (.00).007 and the probability of acceptance at AQL s be 95%. Based on these, determine the Multi-Dimensional Mixed Dependent Sampling Plans. Solution: Initially the Multi-Dimensional Mixed Dependent Sampling Plans are determined for each quality characteristics and given in table (3..). The first stage sample size of Multi-Dimensional Mixed Dependent Sampling Plan is fixed as 0. The variable factors are given in table (3..). The second stage sample size is found and fixed as n =58, the acceptance numbers of MDMDSP are calculated and provided in the same table. Thus, the MDMDSP parameters are determined and given by (0, 58,.9598,.9598,.9598,.9598,.9598,.9598,.9598,,,,,,, ) i.e.,

7 Table 3.. Shows the values of the parameters for each quality characteristics at AQL for given β =0.95, β =0.65 and n i = 0 Quality AQL Variable factor Acceptance nd stage Characteristics p number sample size n j

8 Table 3..: Shows the values of the parameters for each quality characteristics at AQL for given β =0.65, β =0.65 and the first stage sample size n = 0 after fixing the nd stage sample size. Quality AQL Variable Acceptance nd stage Characteristics p factor number sample size k n DESIGNING AND CONSTRUCTION OF MDMDSP INDEXED THROUGH AQL AND LQL Consider a lot of size very large is delivered for the inspection with i variable and j attribute quality characteristics. Let the th i quality characteristics have an 84

9 AQL & LQL with corresponding producer risk β and consumer risk β. By providing values for the parameters, an dependent plan for single sided specification with known S.D would be carried out as follows: Let the two stages of the MDMSP and all the quality characteristics considered are assumed to be dependent. A MDMSP (n, n, k, k. km; c, c.. cn) should satisfy the following requirements P ( β β a p ) & Pa ( p) (3..3) Where, p is AQL and p is LQL. The equation (3..) has to be satisfied for all quality characteristics. Procedure: Let the i variable and j attribute quality characteristics be considered. Step (): Split the probability of acceptance that will be assigned to the first stage. Call it β & β respective to p and p such that β & β Step (): Using the standard variable procedure determine the st stage sample sizes n i for each variable quality characteristics as n i ' ' Z ( β ) Z ( β ) = Z ( p) Z ( p ) (3..4) 85

10 Step (3): Calculate the acceptance limit for each variable quality characteristics as A i = U K i σ (i =,.. m) (3..5) Z ( β ) Where, K i = Z( p) + ( ) (3..6) n i Step (4): Now the sample size n of the MDMSP is fixed as the maximum sample size of the individual variable characteristic plan n = ma x ( n ) (3..7) i Step (5): The variable factor (k = k i ) is re-estimated after the sample size n has been fixed satisfying eqn (3..). Step (6): Now determine the appropriate second stage sample of size n and the acceptance number ( j :,,3..n) for each attribute characteristics such that β c j c j x β = Pn ( x, x > A ) P( y, n ) forp p i i j = x= 0 y= 0 β c j c j x β = Pn ( x, x > A ) P( y, n ) forp p i i j = x= 0 y= 0 (3..8) 86

11 Step (7): Now the nd stage sample size n of MDMSP is fixed as the maximum sample size of the individual quality characteristic sizes. n = max { n } (3..9) j Step (8): The acceptance numbers cj for multi characteristics quality are reestimated after the second stage sample size n has been fixed satisfying (3..8) Illustration 3.. Let the items in the lot shall be tested with respect to six variables and six attribute quality characteristics. The AQL s quality characteristics be.005 (.00).0 and the LQL s be.06,.08,.0,.09,.09,.0. Let the probability of acceptance at AQL s be 95 % and the probability of acceptance at LQL s be 5 %. Based on these, determine the MDMDSP. Solution: Initially the Multi-Dimensional Mixed Dependent Sampling Plans are determined for each quality characteristics and given in table (3..3). The first stage sample size of MDMSP dependent case is fixed as 5 which is the maximum sample size of individual variable characteristic plans. By fixing the sample size as n =5, the variable factors are re-estimated and given in table (3..4). 87

12 Also by fixing the second stage sample size n =88, the acceptance numbers of MDMSP(D) are re-calculated and provided in table(3..4) Table 3..3 Shows the values of the parameters for each quality characteristics at AQL and LQL for given β = 0.95, β = 0.05 β = 0.65 & β = 0.0 Quality characteristic AQL p LQL p Variable factor Acceptance number st stage sample size nd stage sample size n i n j

13 Table 3..4 Values of the parameters for each quality characteristics at AQL and LQL for given β = 0.95 β = 0.05 β = 0.65 & β = 0.0 after fixing the st and nd stage sample sizes. Quality characteristic (i) AQL p LQL p Variable factor Acceptance number st stage sample size nd stage sample size n n

14 Tabe Values of the parameters for each quality characteristics at AQL and LQL for given β =0.95, β =0.05, β =0.65 & β =0.0 Quality Characteristics (i) AQL p LQL p Variable factor Acceptance Number st stage sample size n i nd stage sample size n j

15 Table 3..6 Shows the values of the parameters for each quality characteristics at AQL and LQL for given β =0.95, β =0.05, β =0.65 & β =0.0 after fixing the st and nd stage sample sizes. Quality Characteristics (i) AQL p LQL p Variable factor Acceptance Number st stage sample size n nd stage sample size n

16 3..5 DESIGNING AND CONSTRUCTION OF MDMDSP INDEXED THROUGH IQL WHEN THE FIRST STAGE SAMPLE SIZE n i IS KNOWN Let the two stages of the MDMSP and all the quality characteristics considered are assumed to be dependent. A MDMSP (n, n, k, k k m, c, c.c n ) should satisfy the requirement P a ( ) β 0 (3..0) Where, p 0 is the IQL. Equation (3..0) has to be satisfied for all quality characteristics. Procedure: Let the i variable and j attribute quality characteristics be considered. Step (): Assume that the plan is dependent. Step (): Split the probability of acceptance that will be assigned to the first stage. Let it be respective to p 0 such that 0 Step (3): Decide the sample size to be used. Step (4): Calculate the acceptance limit A i = U k iσ,( i =,... m) Where, k i Z( β 0 ) = Z( p0 ) + n i Step (5): Now the sample size n of the MD mixed dependent plan is fixed as 9

17 n = n i and variable factor k = max(k i ) Step (6): Obtain the values of p (y, n j ) by successive approximation to satisfy the relation,, x A i c j x > ) P( y, n j ) for p= (3..) if such values exist. This will determine the second stage sample size n j and the acceptance number (j;, m) for each attribute characteristics. Step (7): Now the second stage sample size n of MDMDSP is fixed as the maximum sample size of the individual quality characteristic sizes. n = max{ n j }. Step (8): The acceptance number for multi characteristics quality is reestimated after the second stage sample size has been fixed satisfying (3..0). Illustration 3..3 Let the items in the lot shall be tested with respect to seven variables and seven attribute quality characteristics. Let the IQL s of quality characteristics be.0 (.005).04 and the probability of acceptance at IQL s be 50%. Based on these, determine the Multi-Dimensional Mixed Dependent Sampling Plans. Solution: Initially the Multi-Dimensional Mixed Dependent Sampling Plans are determined for each quality characteristics and given in table (3..7). The first stage sample size of MDMDSP is fixed as 5. The variable factors are given in table (3..8). The second stage sample size is found and fixed as n =0. The acceptance numbers of MDMDSP are calculated and provided in the same table. 93

18 Thus, the MDMDSP parameters are determined and given by (5,0,.66,.66,.66,.66,.66,.66,.66,,,,,,, ) i.e., Table 3..7 Values of the parameters for each quality characteristics at IQL for given a β 0 =0.50, =0.5 and n i = 5 are shown in the table Quality IQL Variable Acceptance nd stage Characteristics p 0 factor number C j sample size n j

19 Table 3..8 Values of the parameters for each quality characteristics at IQL for given β 0 =0.50, =0.5 and the first stage sample size n = 5 after fixing the nd stage sample size are shown in the table. Quality IQL Variable factor Acceptance nd stage Characteristics p 0 k number Cj sample size n

20 Table 3..9 Shows the values of the parameters for each quality characteristics at IQL for given β 0 =0.50, =0.5 and n i = 4 Quality IQL Variable factor Acceptance nd stage Characteristics p 0 k i number sample size n j

21 Table 3..0 Shows the values of the parameters for each quality characteristics at IQL for given β 0 =0.50, =0.5 and the first stage sample size n = 4after fixing the nd stage sample size. Quality IQL Variable factor Acceptance nd stage Characteristics p 0 k number sample size n Interpretation: The Mixed Sampling Plans are developed and designed through various Standard Quality levels such as AQL, LQL and IQL which will facilitate easy application in various industrial circumstances. Multi-Dimensional Quality characteristics converge to unique sample size and variable factor k. 97

22 *SECTION 3. BAYESIAN SAMPLING PLANS FOR MULTI DIMENSIONAL QUALITY CHARACTERISTICS 3.. INTRODUCTION Multi-dimensional sampling plans are widely applied in various stages of production. In industries, if different quality characteristics occur, then it would result in loss in economy, time and labors. Therefore, an attempt has been made to design Multi Dimensional Bayesian single sampling plans (MDBSSP) based on polya distribution. Based on multidimensional quality characteristics, MDBSSP aims at controlling the overall quality of a lot of process. The designing aspect of MDBSSP based on polya distribution is given in detail. Tables and illustrations are also been provided. The main advantage of MDBSSP over any other plan is reduction in the sample size for the same account of protection. The Probability mass function and the Probability of acceptance of SSP based on polya distribution can be seen in Loganathan.A, Rajagopal.K and Vijayaraghavan (007). * A part of this section is published in International journal of Emerging Technologies in Science and Engineering, Vol 4 No July 0, pp 73-85,Canada. 98

23 The Probability mass function of polya distribution is,!!!!!!!!!, (3..) for x = 0,,,3,.n This can be evaluated by using the moment estimate of and for each n. For instance, let,,. denote the observed fraction non conforming of independent lots, then the moment estimate of and are given by, Where,, n = Sample Size x = The random variable specifying the number of non-confirming units. p = Fraction Defective = Average Process Fraction Defective = Probability of Acceptance. 99

24 It can be noted that depend on, and also on. The probabilities of (3..) can be computed for each triplet (,, ) instead of (n,, ).The OC function of MDB SSP based on polya distribution is given by, (3..) which gives the probability of acceptance at each value of for given, and. As is the average lot quality for individual lot, is replaced by for each lot. Now, having obtained an estimate of, the probability of acceptance for the submitted lot can be calculated using (3..) for given values of, and. 3.. CLASSICAL ALGORITHM FOR SENTENCING A LOT Step (): Determine the parameters with reference to ASN or OC curves. Step (): Take a random sample of size n from the lot assumed to be large. Step (3): Count the number of defectives in each attribute quality characteristics. Step (4): If the number of defectives, accept the lot, otherwise reject the lot. This algorithm is useful in constructing initial tables. 00

25 3..3 MODIFIED CLASSICAL ALGORITHM Step(): Determine the parameters (,c) with reference to ASN or OC curves. Step(): Take a random sample of size,i=,,.n from the lot assumed to be large. Step(3): Count the number of defectives x in each attribute quality characteristics. Step(4): If the number of defectives x accept the lot, otherwise reject the lot. This modified algorithm is useful for quality control engineers after fixing the sample size DESIGNING AND CONSTRUCTION OF MDBSSP INDEXED THROUGH AQL AND LQL Let the th quality characteristics have an AQL of and LQL of with corresponding producer risk and consumer risk. A MDBSSP,, should satisfy the following requirements. (3..3) The equation (3..3) has to be satisfied for all quality characteristics. 0

26 PROCEDURE Step (): Let the quality characteristics be determined. Step (): Now determine the appropriate sample size and the acceptance Number,,., for each attribute characteristics such that, for fraction defective, for fraction defective (3..4) Step (3): Now the sample size of MDBSSP is fixed as the maximum sample size of the individual characteristic sizes is max Step (4): The acceptance numbers for multicharacteristics quality are reestimated after the sample size n has been fixed satisfying (3..3). 0

27 Illustration 3..: Let the items in the lot be tested with respect to nine attribute quality characteristics. The AQL quality characteristics be 0.0(0.005)0.05 and the LQL s be 0.0(0.05)0.50 and the estimated value of S is 3. Find the MDBSSP. Solution: Let the probability of acceptance at AQL s be 95% and the probability of acceptance at LQL s be 5%. Based on these, the multidimensional single sampling plans based on polya distribution for each quality characteristics are given in table (3..). The parameters of MDBSSP are determined and given by (39, 4,, 3, 3, 3, 3, 3, ) and S=3 03

28 Table 3.. Shows the values of the parameters for each quality characteristics for 3, Quality AQL LQL Sample Acceptance Characteristics() p p Size ( ) Number

29 Table 3.. Shows the values of the parameters for each quality characteristics at AQL and LQL for the given 3, after fixing the sample size from table 3.. Quality AQL LQL Sample Acceptance Characteristics() p p Size ( ) Number

30 Table 3..3 Shows the values of the parameters for each quality characteristics at AQL and LQL for the given 3, Quality AQL LQL Sample Acceptance Characteristics() p p Size ( ) Number

31 Table 3..4 Shows the values of the parameters for each quality characteristics at AQL and LQL for the given 3, after fixing the sample size from table 3..3 Quality AQL LQL Sample Acceptance Characteristics() p p Size ( ) Number

32 3..5 DESIGNING AND CONSTRUCTION OF MDPSP INDEXED THROUGH LQL Let the th quality characteristics have an LQL of with corresponding consumer risk. A MDBSSP,, should satisfy the following requirements (3..4) The Equation 3..4 has to be satisfied for all quality characteristics. Designing Procedure Step (): Let the quality characteristics be monitored. Step (): Now determine the appropriate sample size and the acceptance number c i (i=,,..n), for each attribute characteristics such that, for fraction defective,. Step (3): Now the sample size of MDBSSP is fixed as the maximum sample size of the individual characteristic sizes which will satisfy equation (3..4), max Step (4): The acceptance numbers for multi characteristics quality are re- estimated after the sample size n has been fixed. 08

33 Table 3..5 Shows the values of the parameters for each quality characteristics for 4, 0.05 Quality LQL Sample Acceptance Characteristics() Size( ) Number

34 Table 3..6 Shows the values of the parameters for each quality characteristics for S=4, 0.05 after fixing the sample size Quality LQL Sample Acceptance Characteristics() Size( ) Number(

35 Table 3..7 Shows the values of the parameters for each quality characteristics for 5,0.05 Quality LQL Sample Acceptance Characteristics() Size( ) Number

36 Table 3..8 Shows the values of the parameters for each quality characteristics for 5,0.05 after fixing the sample size Quality LQL Sample Size Acceptance Characteristics() () Number

37 3..6 DESIGNING AND CONSTRUCTION OF MDPSP INDEXED THROUGH IQL Let the th quality characteristics have an IQL of. A MDBSSP,, should satisfy the following requirements (3..5) The Equation 3..5 should satisfy for all quality characteristics. Designing Procedure Step (): Let the quality characteristics be monitored. Step (): Now determine the appropriate sample size and the acceptance number,,., for each attribute characteristics such that, for fraction defective,. Step (3): Now the sample size of MDBSSP is fixed as the maximum sample size of the individual characteristic sizes which will satisfy equation (3..5) max Step (4): The acceptance numbers for multi characteristics quality are reestimated after the sample size n has been fixed. Table 3..9 Shows the values of the parameters for each quality characteristics 3

38 for 3, 0.5 Quality IQL Sample Acceptance Characteristics() Size( ) Number

39 Table 3..0 Shows the values of the parameters for each quality characteristics for 3,0.5 after fixing the sample size Quality IQL Sample Acceptance Characteristics() Size() Number

40 Table 3.. Shows the values of the parameters for each quality characteristics for 3, 0.30 Quality IQL Sample Acceptance Characteristics() Size( ) Number

41 Table 3.. Shows the values of the parameters for each quality characteristics for s=3, 0.30 after fixing the sample size Quality IQL Sample Acceptance Characteristics() Size() Number

42 Table 3..3 Shows comparision of OC values (SSP,PSSP,MDBSSP) for the given strength 0.0, 0., 0.05, 0.0 PROBABILITY OF ACCEPTANCE Fraction SSP-Single PSSP-Polya Single MDBSSP-Multi defective Sampling Plan Sampling Plan Dimensional Bayesian p n= 5,c= n=54,c=3,s=3 Single Sampling Plan n =39,c=,s=

43 FIGURE 3.. -OC CURVES COMPARISION 0.99 P a (p) Single Sampling Plan Polya Single Sampling Plan Multi Dimensional Baysian Single Sampling Plan p (fraction defective) Interpretation The comparison among SSP, PSSP and MDBSSP shows better discrimination and the probability of acceptance is lower for MDBSSP than the other sampling plans. From figure 3.. it is found that the MDBSSP gives higher probability of acceptance for the same quality level. Also it is found that when process fraction defective increases, probability of acceptance decreases rapidly. This shows that the MDBSSP gives more protection to the producer when standard quality is maintained. It also gives protection to the consumer by fraction by rejecting the lots, when the fraction defective increases. 9

44 SECTION 3.3 MULTI -DIMENSIONAL MIXED SINGLE SAMPLING PLANS FOR MAXIMUM ALLOWABLE VARIANCE 3.3. INTRODUCTION In acceptance sampling by variables, mean is the most commonly used criterion. However, there are occasions where the variance of the quality characteristics is used as the criterion. That is, a lot may be judged to be an acceptable, if the variance of the quality characteristics is less than or equal to a pre-specified maximum (σ o ) value. For example, in case of measuring devices, it may be considered acceptable if the variance of the measurement is less than or equal to σ o - a specified maximum allowable variance value. Similarly, a lot of weapons or detonators may be judged to be satisfactory, if the simultaneity of detonation of these items when ignited at the same time is not larger than σ o. In this mixed plan, if the measured variance is greater than σ o, the lot is not rejected, but another sample is taken and the decision is based on attribute criteria. Thus before rejecting a lot, second sample is taken to ensure more protection for producer. Theorem 3.3. Let n be the first stage sample size, n be the second stage sample size and be the sample variance ratio.the probability of acceptance is =, 0

45 Proof: In mixed sampling plans, the first stage inspection is done with variable inspection and the second stage inspection is done with attribute inspection. If the first stage inspection fails to accept the lot, then the second stage of attribute inspection becomes more important to discriminate the lot. The lot will be accepted either in the first stage or in the second stage. But, it will be rejected only in the second stage due to the sampling procedure of the mixed plans. The possible combinations for the acceptance of the lot in mixed single sampling plans are as follows: (A) The lot will be accepted in the first stage, if the sample variance ratio, (or) (B) The lot is not accepted in the first stage if. Inspect and count the number of defectives d in the second stage. If d c, accept the lot. The above two events are mutually exclusive. Therefore, the probability of acceptance is given as i.e =, Hence, the derivation of OC function.

46 Theorem 3.3. The ASN function (Average Sample number) of mixed sampling plans with variance criterion is Let P(A) be the probability that,in the variable inspection, in a sample of size n The event B is defined as follows; If, in n, take another sample of size n. Inspect and count the number of defectives d in the second stage. (i) (ii) If d c, accept the lot. If d > c, reject the lot. Therefore, P(B) =, For event A, the expected sample size for decision is n P(, For event B, the expected sample size for decision is Since A and B are mutually exclusive, ASN of the entire plan ASN

47 3.3. FORMULATION AND OPERATING PROCEDURE OF MIXED PLAN WITH VARIANCE CRITERION. The development of mixed plans and the subsequent discussions are limited only to the known variance (σ ). The mixed sampling plan with variance criterion can be formulated with four parameters n, n,, c. For predetermined values of the parameters, an independent plan with known variance would be carried out as follows: Step : Determine the four parameters, usually with reference OC curve. Step : Draw a random sample of size n from the lot assumed to be large. Step 3: If the sample ratio, accept the lot. Step 4: If the ratio, take another sample of size n. Let it be the second stage. Step 5: Inspect and count the number of defectives d in the second stage. (i) If d c, accept the lot. (ii) If d > c, reject the lot. Step 6: Replace all the defectives with good ones. 3

48 If a dependent plan is desired, then, Step : Determine the four parameters, usually with reference to OC curve. Step : Draw a random sample of size n from the lot assumed to be large. Step 3: If the sample variance ratio, accept the lot. Step 4: If the ratio, examine the first stage sample for number of defectives (d ) therein. Step 5: (i) If d > c, reject the lot (ii) If d c, take second stage sample of size n and count the number of defectives d there from. (iii) (iv) If d + d c, accept the lot If d + d > c, reject the lot. Step 6: Replace all the defectives with good ones MEASURES OF THE INDEPENDENT MIXED SAMPLING PLANS Probability of Acceptance =, (3.3.) Average Sample Number ASN= (3.3.) 4

49 Average Total Inspection ATI=ASN+(N n n ) ( P a (p) ) (3.3.3) Average Outgoing Quality AOQ=p.P a (p) for any lot of large size (3.3.4) Formulation of Multi-Dimensional Sampling Plans (MDMSP) with Variance Criterion The mixed sampling plan with variance criterion can be formulated with using the parameters (n, n,,,,., c, c,. c n ) Where, n = First stage sample size with respect variable quality characteristics n = Second stage sample size with respect variable qualitycharacteristics = Variable factor for i th quality characteristics such that the lot is accepted if, otherwise a second sample is taken for inspection c j= Acceptance number for j th quality characteristics 5

50 3.3.5 Operating procedure of Multi-Dimensional Mixed Sampling plans. A Multi dimensional plan would require the following fundamental steps.. Draw a random sample of size n from the lot. This is known to be the first stage ( ). Test the first sample against a predetermined multidimensional variable acceptance criterion and (a) Accept the lot if the test meets the criterion. (b) If the test fails to meet the criterion, count the number of defectives there in. (i) Reject, if the number of defectives in the first sample exceeds a predetermined attribute criterion. (ii) Otherwise accept the lot DESIGNING AND CONSTRUCTION OF MDMSP FOR MAXIMUM ALLOWABLE VARIANCE, INDEXED THROUGH AQL, WHEN THE FIRST STAGE SAMPLE SIZE n i IS GIVEN. Let the two stages of the MDMSP and all quality characteristics considered be independent. A MDMSP ( n, n,,,,., c, c,. c n ) should satisfy the requirement. P a (p ) β (3.3.5) Where is the AQL and equation (3.3 5 ) has to be satisfied for all quality characteristics. 6

51 Procedure: Let the i variable and j attribute quality characteristics be considered. Step () Step () Assume that the plan is independent. Split the probability of acceptance, that will be assigned to the first stage. Let it be respective to such that Step (3) Decide the sample size n to be used. Let ` for ith variable quality characteristics (i =,,, n). Obtain S,the sum of the squares from the sample observations. Step (4), Calculate the acceptance limit, by using the equation S ' α = P > k = ( ) i f z dz (3.3.6) σ 0 ' ki λ Step (5) Where, and z follows chi-square distribution with n- degrees of freedom for the specified producer s risk. Now the sample size n for the MDMSP is fixed as ` and the variable factor Step (6) Determine the appropriate second stage sample size n j and the acceptance number (j=,,, m) from the relation. c j x= 0 e n p ( n x! p) x = β ' (3.3.7) Step (7) Now the second stage sample size n of MDMSP is fixed as n = max { n j } 7

52 Step (8) The acceptance number for multi characteristics quality are re Illustration 3.3. estimated after the second stage sample size has been fixed satisfying ( 3.3.) In a production process, the fraction defective is given as 0.0(0.0)0.0, β = Find the multi dimensional mixed sampling plan for variance criterion. Solution: Let the first stage sample size be = 0 Given β = 0.99 and the st stage probability of acceptance be, = 0.95, from Table (3.3.), the acceptance criterion is =6.9, the nd stage probability is 0.8 and the nd stage sample size from table (3.3.3) is 50. Hence, the MDMSP for variance criterion is ,, 3, 3, 4, 4, 8

53 Table 3.3. Shows the Values of acceptance criterion sample size n i for known β = 0.99 and = 0.95 n i n i and the first stage Table Shows the second stage sample size and the acceptance number for lot fraction defective, assuming β = 0.95 and β = 0.99 Quality Characteristics

54 Table Shows the values of the parameter for each quality characteristics at AQL for given β = 0.99, β = 0.95 and the first stage sample size n =0 after fixing the second stage sample size. Quality Characteristics AQL Variable factor Second stage Sample size n Acceptance number