VOL. 2, NO. 2, March 2012 ISSN ARPN Journal of Science and Technology All rights reserved.

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1 All rights reserved. Construction of Mixed Sampling Plans Indexed Through Six Sigma Quality Levels with TNT - (n; c, c2) Plan as Attribute Plan R. Radhakrishnan and 2 J. Glorypersial P.S.G College of Arts and Science, Coimbatore, Tamil Nadu, India 2 G.R.D College of Science, Coimbatore, Tamil Nadu. India rkrishnan_cbe@yahoo.com glorypersial@yahoo.co.in ABSTRACT Six Sigma is a concept, a process, a measurement, a tool, a quality philosophy, a culture and a management strategy for the improvement in the system of an organization, in order to reduce wastages and increase the profit to the management and satisfaction to the customers. Motorola [] first adopted the concept of six sigma in their organization and established that it can produce less than 3.4 defects per million opportunities. Focusing on reduction of defects will result in more profit to the producer and enhanced satisfaction for the consumer. The concept of Six Sigma can be applied in the process of quality control in general and Acceptance sampling in particular. In this paper a procedure for the construction and selection of Mixed Sampling Plan indexed through Six Sigma Quality level having Tightened - Normal - Tightened plan of the type TNT - (n; c, c 2 ) plan as attribute plan is presented. The plans are constructed using SSQL- and SSQL-2 as indexing parameters. Tables are constructed for easy selection of the plan. Keywords: six sigma quality levels, poisson distribution, operating characteristic curve, mixed sampling plan, tightened-normaltightened plan.. INTRODUCTION Mixed sampling plan is a two stage sampling procedure involving variables inspection in the first stage and attributes inspection in the second stage if the variables inspection of the first sample does not lead to acceptance. Mixed sampling plans are of two types, namely independent and dependent plans. Independent mixed sampling plans do not incorporate first sample results in the assessment of the second sample. Dependent mixed plans combine the results of the first and second samples in making a decision if a second sample is necessary. The mixed sampling has been designed under two cases of significant interest. In the first case the sample size n is fixed and a point on the OC curve is given. In the second case plans are designed when two points on the OC curve are given. The mixed sampling plans are initially introduced by Dodge [2] and later developed by Bowker and Goode [3]. Schilling [4] has given a method for determining the operating characteristics for mixed variables-attributes sampling plans. Tightened-Normal-Tightened (TNT) sampling scheme was first developed by Calvin [5]. Schilling [6] designing the sampling inspection plans, particularly in the area of compliance testing and especially for safety related items, a smaller acceptance number of non-conforming units are particularly desirable. The OC curves of single sampling plans with c = 0 and c = lead to conflicting interests between the producer and consumer. The plan with zero acceptance number behaves favorably for the consumer, while the plan with c = favors the producer. Such a conflict can be overcome if one is able to design a plan with acceptance numbers of zero and one. Hence, in this paper, c = 0 and c 2 = are considered for the selection of the schemes. Radhakrishnan and Sivakumaran [7] introduced SSQL in the construction of sampling plans. Radhakrishnan and Sivakumaran [8] constructed Tightened-Normal-Tightened schemes indexed through Six Sigma Quality Levels. Radhakrishnan [9] constructed Six Sigma based sampling plan using Weighted Poisson Distribution and Intervened Random Effect Poisson Distribution as the base line distribution. Radhakrishnan and Saravanan [0, ] constructed dependent mixed sampling plan with single sampling and chain sampling plan as attribute plan. Radhakrishnan and Glorypersial [2-4] constructed mixed sampling plans indexed through six sigma quality levels with Double Sampling Plan, Conditional Double Sampling Plan and Chain Sampling Plan - (0, ) as Attribute Plans. This paper deals with the construction of mixed variables - attributes sampling plan (independent case) using TNT - (n; c, c 2 ) plan as attribute plan indexed through Six Sigma Quality levels. Tables are constructed for easy selection of the plan and illustrations are also provided. 2. GLOSSARY OF SYMBOLS The symbols used in this paper are as follows: P: submitted quality of lot or process P a (p): probability of acceptance for given quality p n : sample size for variable sampling plan n, 2 : first sample size for attribute sampling plan n 2, 2 : second sample size for attribute sampling plan β j : probability of acceptance for lot quality p j β j ': probability of acceptance assigned to first stage for percent defective p j β j " : probability of acceptance assigned to second stage for percent defective pj k: variable factor such that a lot is accepted if X A = U- kσ 3. OPERATING PROCEDURE OF MIXED SAMPLING PLAN WITH TNT - (N; C, C 2 ) PLAN AS ATTRIBUTE PLAN In this paper only independent mixed sampling plans are considered. The development of mixed sampling plans and the subsequent discussions are limited only to 7

2 All rights reserved. the upper specification limit U. By symmetry a parallel discussion can be made use for lower specification limits. Also it is suggested that the mixed sampling plan with TNT - (n; c, c 2 ) in the case of single sided specification U, S.D (σ ) known can be formulated by the parameters n, 2, n 2, 2, s and t. By giving the values for the parameters an independent plan for single sided specification, σ known would be carried out as follows: a) Determine the parameters with reference to OC curves b) Take a random sample of size n, from the lot assumed to be large c) If a sample average X A = U - kσ, accept the lot. If the sample average X > A = U - kσ, take another sample of size n,2 (i) inspect using tightened inspection with a larger sample size n,2 and acceptance number c. (ii) switch to normal inspection when t lots in a row are accepted under tightened inspection. (iii) inspect using normal inspection with smaller sample size n 2,2 and acceptance number c 2 (>c ) (iv) switch to tightened inspection after a rejection if an additional lot is rejected in the next s lots. When σ is not known, simply substitute the sample standard deviation (s ) Where s = n i= ( X X) i 2 for σ in the known n standard deviation procedure by choosing an appropriate value of m and sample size n for the unknown standard deviation case. 4. CONDITIONS FOR APPLICATIONS (i) Production process should be steady and continuous (ii) Lots are submitted sequentially in the order of their production (iii) Inspection is by variable in the first stage and attribute in the second stage with quality defined as the fraction defective (iv) Human involvement should be less in the process 5. DEFINITION OF SSQL- AND SSQL-2 The proportion defective corresponding to the probability of acceptance of the lot as -3.4 x 0-6, (the concept of six sigma quality suggested by Motorola [4] in the OC curve is termed as Six Sigma Quality Level- (SSQL-). This new sampling plan is constructed with a point on the OC curve (SSQL-, -α ), where α =3.4 x 0-6 suggested by Radhakrishnan and Sivakumaran [7]. Further the proportion defective corresponding to the probability 2α in the OC curve is termed as Six Sigma Quality Level-2 (SSQL-2). This new sampling plan is constructed with a point on the OC curve (SSQL-2, β 2 ), where β 2 =2α suggested by Radhakrishnan and Sivakumaran [7]. The Tables for the selection of the TNT - (n; c, c 2 ) scheme when c = 0 and c 2 = are also provided for the easy selection of the schemes using the above concepts. 6. DESIGNING THE MIXED SAMPLING PLAN WHEN A SINGLE POINT ON THE OC CURVE IS KNOWN The procedure for the construction of mixed variables - attributes sampling plans is provided by Schilling [4] for a given n and a point p on the OC curve. A modified procedure for the construction of independent mixed variables - attributes sampling plan for a given SSQL-, SSQL-2 and n is given below. Split the probability of acceptance (β j ) determining the probability of acceptance that will be assigned to the first stage. Let it be β j '. Decide the sample size n (for variable sampling plan) to be used Calculate the acceptance limit for the variable sampling plan as A = U - kσ = U [z (p j ) + {z (β j ')/ n }]σ, where z (t) is the standard normal variate corresponding to t such u that t = 2 /2 e du zt () 2π Determine the sample average X. If a sample average X > A = U - kσ, take a second stage sample of size n 2 using attribute sampling plan. Determine β j ", the probability of acceptance assigned to the attributes plan associated with the second stage sample as β j " = (β j - β j ') / (-β j ') n 2, 2 from Pa (p) = β j " for p=p j. Now determine β ", the probability of acceptance assigned to the attributes plan associated with the second stage sample as β " = (β - β ') / (-β '). n 2 and c from P a (p) = β " for p=ssql-. Determine β 2 ", the probability of acceptance assigned to the attributes plan associated with the second stage sample as β 2 " = (β 2 - β 2 ') / (-β 2 '). n 2 and c from P a (p) = β 2 " for p=ssql-2. Using the above procedure Tables can be constructed to facilitate easy selection of mixed sampling plan with TNT - (n; c, c 2 ) plan as attribute plan indexed through SSQL- and SSQL OPERATING CHARACTERISTIC FUNCTION Under the assumption of Poisson model, the OC function of the independent mixed sampling plan having TNT - (n; c, c2 ) Plan is given by Pa(p)=P( X A)+[- P( X A)] P( P )( P )( P) + PP ( P)(2 P ) s t t s s t t s ( P2 )( P )( P2) + P ( P)(2 P2 ) 8

3 All rights reserved. where P = Probability of acceptance under tightened inspection P 2 = Probability of acceptance under normal inspection s = Criteria for switching to tightened inspection and t = Criteria for switching to normal inspection Under the conditions of the application of the Poisson model, when c = 0 the probability of acceptance for both,2 e n p tightened inspection becomes P = When c 2 =, the probability of acceptance under normal 2,2 e n p inspection becomes P 2 = (+np) Since n, 2 > n 2, 2, we set n, 2 equal to some multiple of n 2, 2 say, mn 2, 2. The Tables furnished in this paper are for the case when m=2. 8. CONSTRUCTION OF MSP WITH TNT - (N; C, C 2 ) PLAN AS ATTRIBUTE PLAN INDEXED THROUGH SSQL- In this section the mixed sampling plan indexed through SSQL- is constructed. A point on the OC curve can be fixed such that the probability of acceptance of fraction defective SSQL- is β. The general procedure given by Schilling [4] is used for constructing the mixed sampling plan as attribute plan indexed through SSQL- [for β " = (β - β ') / (-β ')] with β = and β '=0.50, the n,2 SSQL- values are calculated for different values of s and t using visual basic program and is presented in Table-. The sample size n 2, 2 is obtained using n 2, 2 = n 2, 2 SSQL-/SSQL-and and n, 2 = (n 2, 2 ) (m). Hence the parameters of the TNT - (n; c, c 2 ) plan n, 2, n 2, 2, s and t are obtained for various values of SSQL-. The sigma level of the process [5] is calculated using the Process Sigma Calculator by providing the sample size and acceptance number. Selection of the plan Table- is used to construct the plans when SSQL-, s and t are given. For any given values of SSQL-, s and t one can determine n 2 value using n 2 = n 2 SSQL-/SSQL-. Example : Given SSQL-= , s = 2, t = 3 and β '=0.50, the value of SSQL- is selected from Table- as and the corresponding second stage sample size n 2, 2 is computed as n 2, 2 = n 2, 2 SSQL-/SSQL-= / = 40 and n, 2 = (n 2, 2 ) (m) = 820 which are associated with 4.3 and 4.5 sigma levels respectively. For a fixed β '=0.50, the Mixed Sampling Plan with TNT - (n; c, c 2 ) plan as attribute plan are n, 2 = 820, n 2, 2 = 40, s = 2 and t = 3 for a specified SSQL- = Practical Application Suppose the plan with n = 0, k =.5, s = 2, and t = 5 is to be applied to the lot-by-lot acceptance inspection of soccer balls. The characteristic to be inspected is the weight of the soccer balls is gm for which there is a specified upper limit (U) of 450 gm with a known standard deviation (σ ) of gm. In this example, U = 450 gm, σ = gm and k =.5. Now, in applying the variable inspection first, take a random sample of size n =0 from the lot. Record the sample results and find X. If X A = U - kσ = mm, accept the lot otherwise take a random sample of size 820 and apply attribute inspection. Under attribute inspection, the TNT - (n; 0, ) plan as attribute plan, if the manufacturer fixes the values β ' = 0.50, SSQL-= (9 non-conformity soccer balls out of million balls ), then inspect under tightened inspection with sample of size 820 soccer balls from the manufactured lot of a particular month with acceptance number c = 0. If 3 lots in a row are accepted under tightened inspection, then switch to normal inspection and inspect under normal inspection with sample of size 40 soccer balls from the manufactured lot of a particular month with acceptance number c 2 =. If a lot is rejected then switch to tightened inspection, if a further lot is also rejected in the next 2 lots and inform the management for quality improvement. The OC curve of the schemes in Example is presented in the Figure-. pa(p) , 4.5 sigma levels p Figure-. OC curve for the schemes n, 2 = 820, n 2, 2 = 40, s = 2 and t = CONSTRUCTION OF MSP WITH TNT - (N; C, C 2 ) PLAN AS ATTRIBUTE PLAN INDEXED THROUGH SSQL-2 In this section the mixed sampling plan indexed through SSQL-2 is constructed. A point on the OC curve can be fixed such that the probability of acceptance of fraction defective SSQL-2 is β 2. The general procedure given by Schilling [4] is used for constructing the mixed sampling plan as attribute plan indexed through SSQL-2 [for β 2 " = (β 2 - β 2 ') / (-β 2 ')] with β 2 = and β 2 ' = , the n 2,2 SSQL-2 values are calculated for different values of s and t using visual basic program and is presented in Table-2. Example 2: Given SSQL-2=0.008, s = 3, t = 5 and β '=0.50, the value of SSQL-2 is selected from Table-2 as and the corresponding second stage sample size n 2, 2 is computed as n 2, 2 = n 2, 2 SSQL-2/SSQL-2= /0.008 = 924 and n, 2 = (n 2, 2 ) (m) = 3848 which are associated with 4.8 and 5.0 sigma levels respectively. For a fixed β '=0.50, the Mixed Sampling Plan with TNT - (n; c, c 2 ) plan as attribute plan are n, 2 = 3848, n 2, 2 = 924, s = 3 and t = 5 for a specified SSQL- 2=

4 All rights reserved. Practical Application Suppose the plan with n = 0, k =.5 s = 3 and t = 5 is to be applied to the lot-by-lot acceptance inspection of hand balls. The characteristic to be inspected is the weight of the hand balls is gm for which there is a specified upper limit (U) of 275 gm with a known standard deviation (σ ) of gm. In this example, U = 275 gm, σ = gm and k =.5. Now, in applying the variable inspection first, take a random sample of size n =0 from the lot. Record the sample results and find X. If X A = U - kσ = mm, accept the lot otherwise take a random sample of size 3848 and apply attribute inspection. Under attribute inspection, the TNT - (n; 0, ) plan as attribute plan, if the distributer fixes the values β ' = 0.50, SSQL-=0.008 (8 non-conformity hand balls out of thousand balls), then inspect under tightened inspection with sample of size 3848 hand balls from the manufactured lot of a particular month with acceptance number c = 0. If 5 lots in a row are accepted under tightened inspection, then switch to normal inspection and inspect under normal inspection with sample of size 924 hand balls from the manufactured lot of a particular month with acceptance number c 2 =. If a lot is rejected then switch to tightened inspection, if a further lot is also rejected in the next 3 lots and inform the management for quality improvement. The OC curve of the schemes in Example 2 is presented in the Figure-2. [2] Dodge H.F Statistical control in sampling Inspection. American Mechanist, Oct , Nov 932, [3] Bowker A. H and Goode H.P Sampling Inspection by Variables. McGraw Hill, New York. [4] Schilling E.G A general method for determining the operating characteristics of mixed variables. Attribute sampling Plans single side specifications, S.D known. Ph. D thesis Rutgers, The State University, New Brunswick, New Jersy. [5] Calvin T.W TNT zero acceptance number sampling. American Society for Quality Control, Annual Technical Conference Transactions, Philadelphia, PA. pp [6] Schilling E.G Acceptance Sampling in Quality Control. Marcel Dekker, New York. [7] Radhakrishnan R. and Sivakumaran P.K Construction and Selection of Six Sigma Sampling Plan indexed through Six Sigma Quality Level. International Journal of Statistics and Systems. 3(2): [8] Radhakrishnan R. and Sivakumaran P. K Construction of Tightened-Normal-Tightened schemes indexed through Six Sigma Quality Levels. International Journal of Advanced Operations Management (IJAOM). 2(/2): pa(p) p 4.8, 5.0 sigma levels Figure-2. OC curve for the schemes n, 2 = 3848, n 2, 2 = 924, s = 3 and t = CONCLUSIONS This paper provides a procedure to engineers for the selection of Mixed Sampling Plan through Six Sigma Quality Levels having TNT - (n; c, c 2 ) Plan as attribute plan. These plans are very effective in place of classical plans indexed through SSQL- and SSQL-2 and these plans are useful for the companies in developed and developing countries which are practicing six sigma quality initiatives in the process. The procedure outlined in this paper can be used for other plans also. REFERENCES [] Motorola [9] Radhakrishnan R Construction of Six Sigma based sampling plans. A D.Sc. Thesis submitted to Bharathiar University, Coimbatore, India. [0] Radhakrishnan R. and Sampath Kumar R. and Saravanan P.G Construction of Dependent Mixed Sampling Plan using Single Sampling Plan as attribute plan. Accepted for publication and to appear in The International Journal of Statistics and Systems. 4(, ISSN ): [] Radhakrishanan R. and Saravanan P.G Construction of dependent Mixed Sampling Plans using Chain Sampling Plan of type ChSP- indexed through AQL. National Journal of Technology. 6(4, ISSN ): [2] Radhakrishnan R. and Glorypersial J. 20a. Construction of Mixed SamplingPlans Indexed through Six Sigma Quality Levels with Double Sampling Plan as Attribute Plan as Attribute Plan. International Journal of Statistics and Analysis. (): -0. [3] Radhakrishnan R. and Glorypersial J. 20b. Construction of Mixed Sampling Plans Indexed through Six Sigma Quality Levels with Conditional Double Sampling Plan as Attribute Plan. International Journal of Recent Scientific Research. 2(7):

5 All rights reserved. [4] Radhakrishnan R. and Glorypersial J. 20c. Construction of Mixed Sampling Plans Indexed through Six Sigma Quality Levels with Chain Sampling Plan - (0, ) as Attribute Plan. International Journals of Multi Disciplinary Research Academy, ISSN (6): [5] Process Sigma Calculation - Table-. Various characteristics of the MSP when (SSQL-, β ) is known with β = , β ' = s t n 2,2 SSQL Table-2. Various characteristics of the MSP when (SSQL-2, β 2 ) isknown with β 2 = and β 2 ' = s t n 2,2 SSQL