Designing Six Sigma Repetitive Group Sampling Variables Plan

Designing Six Sigma Repetitive Group Sampling Variables Plan Indexed by Six Sigma Quality Levels : 

Dr.D.Senthilkumar Associate Professor Department of Statistics PSG College of Arts & Science Coimbatore – 641 014 . email : alamsen@rediffmail.com B. Esha Raffie Research Scholar Department of Statistics PSG College of Arts & Science Coimbatore – 641 014. email : esharaffie@gmail.com Designing Six Sigma Repetitive Group Sampling Variables Plan Indexed by Six Sigma Quality Levels

Overview : 

Overview Six Sigma is the most popular tool to convert management problem to a statistical problem and to find a statistical solution, then convert it to a management solution. This plan proposes the Six Sigma Repetitive Group Sampling Variables plan where the quality characteristic follows normal distribution. Tables are also constructed for the selection of parameters of known standard deviation Six Sigma Repetitive Group Sampling Variables plan indexed by six sigma acceptable quality level and six sigma limiting quality level. Wednesday, November 16, 2011 2

Introduction: 

Introduction Repetitive Group Sampling (RGS) plan is one of the attribute sampling plans developed by Sherman (1965), he pointed out that the RGS plan will give an intermediate in sample size efficiency between the single sampling plan and the sequential sampling plan. Variables sampling plans involve comparing a statistic (mean), with an acceptance limit as the same way that the number of nonconforming items are compared to an acceptance number in attributes plans. Whenever the quality characteristic of interest is measurable, a variables sampling plan can be applied. The main advantage of the variables sampling plan over an attributes sampling plan is that the same operating characteristic (OC) curve can be attained with small sample size. Since variable sampling plan is well suited for variables inspection. Wednesday, November 16, 2011 3

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SSRGSV plan, designated as SSRGSVP (n;k 1 ,k 2 ) is introduced and the probability of acceptance of the lot is 1-3.4 x 10 -6 . This method designing the plan, based on the given Six Sigma AQL, α 1 (producer’s risk), Six Sigma LQL, β 1 (Consumer’s risk) as indicated. The proportion defective corresponding to this probability in the OC curve is termed as Six Sigma Acceptable Quality Level (SSAQL = p ssv1 ) and Six Sigma Limited Quality Level (SSLQL = p ssv2 ). This new variable sampling plan is constructed with a point on the OC curve (p ssv1, 1-  1 ) and (p ssv2 , β 1 ), where  1 =3.4 x 10 -6 and β 1 ≥ 2  1 is similar to (SSAQL, 1-  ) and (SSLQL, β) as suggested by Dodge and Romig (1942). where the quality characteristic follows normal distribution. Wednesday, November 16, 2011 4

Six Sigma Repetitive Group Sampling Variables Plan of Type SSRGSV(n; k1, k2): 

Six Sigma Repetitive Group Sampling Variables Plan of Type SSRGSV(n; k 1, k 2 ) Condition of Applications The following assumptions should be valid for the application of the variables RGS plan. ( i ) Lots are submitted for inspection serially, in the order of production from a process that turns out a constant proportion of non-conforming items. (ii) The consumer has confidence in the supplier and there should be no reason to believe that a particular lot is poorer than the preceding lots. In addition, the usual conditions for the application of single sampling variables plans, with known or unknown standard deviation, should also be valid. Wednesday, November 16, 2011 5

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Assumptions The Quality Characteristic x has a normal distribution with a known standard deviation. A Unit is Defective if x > U or x < L, where U and L are the upper and lower specification   limits respectively. The purpose is to control the fraction defective p in large lots submitted for inspection. Operating Procedure The operating procedure of Six Sigma Repetitive Group Sampling Variables Plan is described below: Step 1 Take a random sample of size n σ from the population. Inspect each unit and record the measurement of the quality characteristics of the sample. Compute sample mean . Step 2 Accept the lot if and reject the lot if . (k 2 < k 1 ) Step 3 If then repeat the steps 1, 2. Thus, the SSRGSV plan has the parameters of the sample size n, and the acceptable criterion k 1 and k 2. Wednesday, November 16, 2011 6

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Operating Characteristic Function According Sherman (1965), the OC function of SSRGVS Plan, which gives the proportion of lots that are expected to be accepted for given product quality, p is given by (1) where is the probability of accepting a lot based on a single sample with parameters (n, k 1 ) and is the probability of rejecting a lot based on a single sample with parameters (n, k 2 ). The fraction non-conforming in a lot will be determined as (2) where Φ(y) is given by (3) Wednesday, November 16, 2011 7

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provided that the quality characteristic of interest is normally distributed with mean µ and standard deviation σ, and the unit is classified as non-conforming if it exceeds the upper specification limit U. Then its probability of acceptance is written as If SSAQL, SSLQL, the producer’s risk 1-α and the consumer’s risk β are prescribed then we have Wednesday, November 16, 2011 8

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Here, The value of w 1 at p = p 1 , the value of w 2 at p = p 2 . That is, By fixing the probability of acceptance of the lot, Pa (p) as 1-3.4 x 10 -6 with normal distribution, where ssv 1 is the value of v 1 at SSAQL and ssv 2 is the value of v 2 at SSLQL. For example, if p 1 and p 2 are prescribed, then the corresponding value of ssv 1 and ssv 2 will be fixed and if P a ( p 1 ) and P a ( p 2 ) are fixed at 99.99966% and more than 0 .00068% respectively, Then we have For given SSAQL and SSLQL, the parametric values of the SSRGVS plan namely k 1 , k 2 and the sample size n are determined by using a computer search. Figure 1 shows the OC Curves of SSRGVSP with n=189, k 1 =4.099, k 2 =3.639, 3.5 sigma level and n=298, k 1 =4.199, k 2 =3.939, 3.7 sigma level. It can be observed that the plan OC curves at a good quality, i.e., for very smaller values of fraction defective with more sigma level. Wednesday, November 16, 2011 9

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Figure 1 . OC Curves of SSRGVSP with n=189, k 1 =4.099, k 2 =3.639, 3.5 sigma level and n=298, k 1 =4.199, k 2 =3.939, 3.7 sigma level. Designing SSRGSV Plan Selection of known standard deviation SSRGSV Plan Indexed by SSAQL and SSLQL Table 1 can be used to determine SSRGSVP (n, k 1, k 2 ) for specified values of SSAQL and SSLQL. For example, if it is desired to have a SSRGSVP (n, k 1, k 2 ) for given p ssv1 = 0.00004 and p ssv2 = 0.0003, α 1 = 3.4x10 -6 , β 1 ≥2α 1 , Table 1 gives n = 114, k 1 = 3.839, k 2 = 3.519 which is associated with 3.3 sigma level. Hence the parameters of SSRGSVP are n = 114, k 1 = 3.839, k 2 = 3.519. Wednesday, November 16, 2011 10

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Construction of Table The OC function of SSRGSVP (n, k 1 , k 2 ) is given by the equation (1) for as a specified (p ssv1 , α 1 ) and (p ssv2 , β 1 ). The equation (1) results in Equation (8) and (9) are solved for n, k T and k N (known standard deviation) for as specified pair of points, say, p ssv1 =α 1 and p ssv2 =β 1 on the OC Curve. To identified Six Sigma Repetitive Group Sampling Variables Plan SSRGSVP (n, k 1, k 2 ), a computer search routine was used for given set of (p ssv1 , α 1 ) and (p ssv2 , β 1 ). The plan is identified are tabulated in Table 1. The sigma (σ) value is calculated using the process sigma calculator (http://www.isixsigma.com/) for given n, k 1 and k 2 for known Standard deviation methods. Conclusion: A separate procedure for designing SSRGSV plan with six sigma quality levels are presented and tables are also constructed for the easy selection of the plans. The RGS plans provided in this article will be of much useful to the engineers who are working on the floor of the company, which adopts Six Sigma quality initiatives in their organization. Wednesday, November 16, 2011 11

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p 1 p 2 n k 1 k 2 σ - level 0.00001 0.00003 349 4.259 3.959 3.8 0.00004 298 4.199 3.939 3.7 0.00005 265 4.159 3.939 3.7 0.00006 210 4.139 3.919 3.6 0.00007 198 4.109 3.909 3.5 0.00008 190 4.089 3.899 3.5 0.00009 156 4.089 3.849 3.4 0.0001 135 4.095 3.815 3.4 0.0002 107 3.959 3.769 3.3 0.0003 82 3.909 3.689 3.2 0.0004 60 3.909 3.589 3.0 0.0005 45 3.929 3.589 2.9 0.0006 36 3.949 3.509 2.7 0.0007 27 4.029 3.389 2.5 0.0008 20 4.139 3.239 2.3 0.0009 18 4.169 3.179 2.2 0.001 13 4.349 2.959 1.9 0.002 12 4.149 2.939 1.9 0.003 10 4.149 2.809 1.7 0.004 7 4.359 2.489 1.2 0.00002 0.00005 271 4.195 3.779 3.7 0.00006 235 4.169 3.749 3.6 0.00007 226 4.125 3.739 3.6 0.00008 208 4.095 3.735 3.6 0.00009 188 4.085 3.715 3.5 0.0001 180 4.055 3.715 3.5 0.0002 115 3.939 3.679 3.3 0.0003 109 3.839 3.669 3.3 0.0004 92 3.799 3.629 3.2 0.0005 84 3.759 3.609 3.2 0.0006 69 3.759 3.559 3.1 0.0007 62 3.739 3.519 3.1 0.0008 56 3.729 3.499 3.0 0.0009 51 3.719 3.469 3.0 0.001 40 3.769 3.389 2.8 0.002 34 3.619 3.329 2.7 0.003 16 3.839 2.969 2.2 0.004 11 3.979 2.719 1.9 0.005 9 4.069 2.559 1.6 Table 1 SSRGSVP(n, k 1 ,k 2 , σ – level) with known standard deviation indexed by AQL and LQL (α 1 =3.4 x 10 -6 and β 1 ≥2α 1 ). p 1 p 2 n k 1 k 2 σ - level 0.00003 0.00009 189 4.099 3.639 3.5 0.0001 183 4.069 3.639 3.5 0.0002 127 3.929 3.609 3.4 0.0003 113 3.839 3.589 3.3 0.0004 99 3.779 3.559 3.3 0.0005 87 3.749 3.529 3.2 0.0006 80 3.719 3.509 3.2 0.0007 74 3.689 3.479 3.1 0.0008 66 3.689 3.459 3.1 0.0009 64 3.659 3.449 3.1 0.001 54 3.679 3.399 3.0 0.002 49 3.499 3.369 3.0 0.003 22 3.679 3.039 2.5 0.004 19 3.649 2.979 2.4 0.005 11 3.919 2.629 1.9 0.00004 0.0001 186 4.099 3.559 3.5 0.0002 130 3.929 3.539 3.4 0.0003 114 3.839 3.519 3.3 0.0004 100 3.789 3.469 3.3 0.0005 89 3.749 3.459 3.2 0.0006 85 3.709 3.429 3.2 0.0007 76 3.689 3.429 3.2 0.0008 69 3.679 3.399 3.1 0.0009 68 3.639 3.389 3.1 0.001 59 3.649 3.359 3.0 0.002 52 3.479 3.319 3.0 0.003 29 3.549 3.109 2.7 0.004 22 3.569 2.969 2.5 0.005 16 3.669 2.809 2.2 0.00005 0.0002 134 3.939 3.479 3.4 0.0003 115 3.839 3.459 3.3 0.0004 101 3.779 3.439 3.3 0.0005 90 3.749 3.409 3.2 0.0006 87 3.699 3.399 3.2 0.0007 78 3.679 3.379 3.2 0.0008 75 3.619 3.369 3.2 0.0009 72 3.629 3.359 3.1 0.001 69 3.609 3.349 3.1 0.002 54 3.469 3.279 3.0 p 1 p 2 n k 1 k 2 σ - level 0.003 35 3.479 3.129 2.8 0.004 26 3.499 3.009 2.6 0.00007 0.0002 137 3.979 3.359 3.4 0.0003 115 3.859 3.359 3.3 0.0004 102 3.789 3.339 3.3 0.0005 93 3.749 3.329 3.2 0.0006 88 3.709 3.309 3.2 0.0007 82 3.669 3.309 3.2 0.0008 79 3.639 3.299 3.2 0.0009 75 3.619 3.279 3.2 0.001 71 3.599 3.269 3.1 0.002 57 3.449 3.209 3.1 0.003 48 3.369 3.159 3.0 0.004 32 3.409 3.009 2.7 0.005 28 3.389 2.959 2.7 0.00008 0.0003 115 3.889 3.289 3.3 0.0004 102 3.799 3.309 3.3 0.0005 94 3.749 3.299 3.3 0.0006 88 3.709 3.289 3.2 0.0007 82 3.669 3.269 3.2 0.0008 79 3.649 3.259 3.2 0.0009 75 3.619 3.249 3.2 0.001 72 3.599 3.239 3.1 0.002 57 3.449 3.179 3.1 0.003 50 3.359 3.319 3.0 0.004 34 3.389 2.999 2.8 0.005 30 3.359 3.939 2.7 0.0001 0.0002 616 3.72 3.53 4.0 0.0003 568 3.61 3.53 4.0 0.0004 511 3.54 3.52 4.0 0.0005 389 3.51 3.49 3.9 0.0006 302 3.49 3.46 3.8 0.0007 98 3.63 3.26 3.3 0.0008 61 3.74 3.12 3.0 0.0009 58 3.71 3.11 3.0 0.001 49 3.74 3.05 2.9 0.002 47 3.51 3.06 2.9 0.003 44 3.4 3.04 2.9 Wednesday, November 16, 2011 12

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References: Dodge, H.F. and Roming , H.G.(1942): Army service forces tables, Bell telephone laboratories, United States. Hamaker , H. C. (1979) Acceptance sampling for percent defective by variables and by attributes, Journal of Quality Technology, 11, pp. 139–148. Lieberman, G. J., Resnikoff , G. J. (1955). Sampling plans for inspection by variables. J. Amer. Stat. Assoc. 50:457–516. Sherman, R. E. (1965). Design and evaluation of repetitive group sampling plan. Technometrics 7:11–21. Soundrarajan , V., and Ramasamy , V. (1986): Procedures and tables for construction and selection of Repetitive Group Sampling (RGS)Plan, The QR Journal, 13, 13, pp.14-21. Process Sigma Calculation: http://www.isixsigma.com/ Wednesday, November 16, 2011 13

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