Hybrid Group Acceptance Sampling Plan Based on Size Biased Lomax Model

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Randall Leonard
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1 Mthemtics nd Sttistics 2(3): , 2014 DOI: /ms Hybrid Group Acceptnce Smpling Pln Bsed on Size Bised Lomx Model R. Subb Ro 1,*, A. Ng Durgmmb 2, R.R.L. Kntm 3 1 Shri Vishnu Engineering College for Women, Bhimvrm, , Andhr Prdesh, Indi 2 Rghu Institute of Technology, Dkmrri, Viskhptnm, , Andhr Prdesh, Indi 3 Achry Ngrjun University, Guntur, , Andhr Prdesh, Indi *Corresponding Author: rsr_vishnu@rediffmil.com Copyright 2014 Horizon Reserch Publishing All rights reserved. Abstrct In this pper, hybrid group cceptnce smpling pln is introduced for truncted life test if life times of the items follow size bised Lomx model. The minimum number of testers nd cceptnce number re obtined when the consumer s risk nd the test termintion time nd group size re pre-specified. The operting chrcteristic vlues, minimum rtios of the true men life to the specified men life for the given producer s risk re lso derived. The results re discussed through n exmple, comprtive study of proposed smpling pln with existing smpling pln re elborted. Keywords Size Bised Lomx Model (SBLM), Group Acceptnce Smpling Pln (GASP), Consumer s Risk, Producer s Risk, Operting Chrcteristic (O.C), Truncted Life Test 1. Introduction A decision on ccepting or rejecting product depends on its fitness for use. The qulity checking process in qulity control is of different types. One such process is cceptnce smpling plns. In most cceptnce smpling plns, the mjor problem is to determine the smple size from lot under considertion. In usul smpling pln the decision of ccepting or rejecting lot on the bsis of single item. The cceptnce smpling pln tht ccommodtes multiple number of items t time put in tester will be clled group cceptnce smpling pln, which reduces the testing time s well s cost cn be sved by testing those items simultneously. While designing group cceptnce smpling pln determining the smple size is equivlent to determine the number of groups s the number of testers is known. The experiment is truncted if more thn the number of filures occurred ny group during the experiment time. The recent technique tht is incorporte in this pper is hybrid group cceptnce smpling pln, in which the minimum number of testers will be obtin for pre-determined number of groups. Hybrid group cceptnce smpling pln is bsed on truncted life test ssuming tht the life time of product follows specific probbility model. The ttention mde by different reserchers on the development of Hybrid Group Acceptnce Smpling Pln (HGASP) nd its models re: G. Srinivs Ro[7,8] developed hybrid group cceptnce smpling plns for lifetimes bsed on log-logistic distribution nd hybrid group cceptnce smpling pln for lifetimes bsed on generlized exponentil distribution, A. Bklize[1] studied cceptnce smpling bsed on truncted life tests in the Preto distribution of the second kind, Muhmmd Aslm et l.[3] presented group cceptnce pln bsed on truncted life test for Gmm distribution, K. Rosih et l.[6,7] given n economic Relibility test pln with Preto distribution nd Preto distribution in cceptnce smpling bsed on truncted life test, A.R. Sudmni Rmswmy[10] determined hybrid group cceptnce smpling plns for lifetimes bsed on exponentited weibull distribution, Jffer Hussin et l.[2] derived hybrid group cceptnce smpling plns for lifetimes hving generlized Preto distribution, K. Rosih nd R.R.L. Kntm[4] who derived cceptnce smpling bsed on the inverse Ryleigh distribution, K. Rosih et l.[5] discussed Relibility of test plns for exponentited log-logistic distribution. In this pper new hybrid group cceptnce smpling pln (HGASP) is proposed by considering size bised Lomx model with known shpe prmeter. The probbility density function(p.d.f) f(t) nd cumultive distribution function(c.d.f) F(t) of size bised Lomx model re given below α( α 1) t t ( α 1) + f(t) = 1 + ; t 0, α 1, σ 0 (1) σ σ σ α t t F(t) = ; t 0, α 1 σ σ σ 0 Where α is shpe nd σ is scle prmeter. Men nd vrince of size bised Lomx model re given by (2)

2 138 Hybrid Group Acceptnce Smpling Pln Bsed on Size Bised Lomx Model Construction of hybrid group cceptnce smpling pln for size bised Lomx model is presented in Section 2. The operting chrcteristic vlues re given in Section 3. Description of tbles nd exmples re discussed in Section 4. Concluding remrks re given in Section Design of the Proposed Smpling Pln Let µ be the true men life of product nd µ 0 is the specified men life of n item, under the ssumption tht the lifetime of n item follows size bised Lomx model. If H 0 : µ µ 0, the lot of the product is ccepted, otherwise the lot of the product is rejected. In cceptnce smpling schemes, this hypothesis is tested on the bsis of number of filures in smple with pre-fixed time. HGASP follows the following steps: Select the number of testers, r nd ssign the r items to ech predefined groups, g, the required smple size for lot is n = rg. Pre-fix the cceptnce number, c for ech group nd the experiment time t 0. Accept the lot if t most c filures occurs in ech of ll groups. Terminte the experiment if more thn c filures occur in ny group nd reject the lot. Determine the number of tester s r, for size bised Lomx 2σ 2ασ 2 E ( t) = ; α 2 nd V ( t) = ; α 3 α 2 α 2 2 α 3 ( ) ( ) model nd vrious vlues of cceptnce number c, wheres the number of groups g, nd the termintion time t 0 re ssumed to be specified. It is convenient to consider tht termintion time s multiple of the specified men life µ 0, consider t 0 =. µ 0, where is constnt termintion rtio. The probbility of rejecting good lot is clled the producer s risk nd the probbility of ccepting bd lot is clled the consumer s risk nd re respectively γ nd β. The prmeter vlue of r of the proposed smpling pln is derived by consumer s risk β. If the confidence level is p *, then β = 1-p *. If the lot size is lrge enough, we cn use binomil distribution to develop the HGASP. The lot of the product is ccepted only if every group g hving t most c filures. The HGASP is chrcterized by the three prmeters ( n,c,t/σ 0 ). The lot cceptnce probbility is g c L( p) = r p i ( 1 p r i c i ) i= o where p is the probbility tht n item in tester fils before the termintion time t 0 =.µ 0 nd is given by 2α 2 p = F(t) = µ µ ( α 2) ( α 2) µ 0 µ 0 (3) (4) (5) The minimum number of testers r cn be derived by considering the consumer s risk when the true men life equls to specified men life (µ = µ 0 ) through the following inequlity where p 0 is the filure probbility t µ = µ 0, nd it is given by 2α 2 p = F(t) = 1 (7) 1+ 1 ( α 2) + ( α 2) Tble 1 show for the pre-fix consumer s risk (β), number of groups (g), cceptnce number (c) nd termintion time () to obtin the minimum testers (r). The minimum number of testers required for the size bised Lomx model t α = 3 re clculted nd re given in Tble 2. Tble 1. The pre-fix vlues of β, g, c nd β g c Tble 2. Number of testers required for the proposed pln SBLM t α = 3 β g c L( p0 ) β (6)

3 Mthemtics nd Sttistics 2(3): , Operting Chrcteristic The Operting chrcteristic function of smpling pln is the probbility of cceptnce i.e., function of the devition of specified men µ 0 from its true men µ. Once the minimum smple size is obtined, one my be interested to find the probbility of cceptnce of lot when the qulity of the product is good enough. The product is considered to be good if µ µ 0. The probbilities bsed on (4) for vrious men life times (µ/µ 0 = 2, 4, 6, 8, 10, 12) under β =,,, with the termintion rtio = 0.7, 0.8, 1.0, 1.2, 1.5, 2.0 choosing cceptnce number c = 2 re presented in Tble 3. For given producer s risk the miniml rtios of true men life to the specified men life cn be obtined by using the inequlity g c r p i ( 1 p r i c ) 1 i i= 0 where p is given in (5) nd for vrious combintions of consumer s risk β, nd γ =, the minimum men rtio s re found for fixed vlues of c nd re given in Tble 4. (8) Tble 3. O.C vlues of the HGASP with g = 4 nd c = 2 for SBLM with α = 3. µ/µ 0 β r

4 140 Hybrid Group Acceptnce Smpling Pln Bsed on Size Bised Lomx Model Tble 4. Minimum rtio vlues of true men life to specified men life for γ = for SBLM t α = 3. β g c Description of Tbles nd Exmples The require prmeters of HGASP re evluted t vrious vlues of the consumer s risk nd the termintion time in Tble 2. The minimum smple size is clculted by using the reltion n = rg. Tble 2 indictes tht, s the test termintion time increses, the number of testers r either decrese or constnt, i.e., smller number of testers is needed, if the termintion time increses t fixed number of groups. Suppose, from Tble w, if β =, g = 4, c = 2 nd chnges from 0.7 to 0.8, the required vlues of design prmeters of HGASP chnges from r = 4 to r = 3. The probbility of cceptnce for the lot t the men rtio corresponding to the producer s risk is presented in Tble 3. For n exmple, the lifetime of product follows the size bised Lomx distribution with = 3. It is desired to design HGASP to test if the men life is greter thn 1000 hrs bsed on testing time of 700 hrs nd using 4 groups. Thus, we will drw rndom smple of size 16 (n = r. g) items nd llocte 4 items to ech of 4 groups to put on test for 700 hrs. We will ccept the lot if no more thn 2 filures occurs before 700 hrs in ech of 4 groups. We truncte the experiment s soon s the 3 rd filure occurs before the 700 th hr.

5 Mthemtics nd Sttistics 2(3): , Tble 5. At r=4 nd =0.7, O.C vlues of the HGASP with g = 4 nd c = 2 for SBLM with α = 3. µ/µ p Tble 5 explins (borrowed from tble 3) tht, if the true men life is 4 times of 1000 hrs, the producer s risk is So, lot of submitted items shll be ccepted with probbility if the true men life is 2 times the specified men life. The cceptnce probbility of submitted lot is incresed up to if the true men life of n item in lot is 12 times the specified men life. In order to compre proposed HGASP with tht of G. Srinivs Ro(2012), we consider the bove exmple. The HGASP for the log-logistic distribution re (g, r, c, ) = (4,10,2,0.7) with δ = 3 nd the HGASP for proposed smpling pln re (g, r, c, ) = (4,4,2,0.7). So, our proposed HGASP requires 16 (n = r. g) items where s the existing pln given by G. Srinivs Ro(2012) requires 32 items respectively to rech on sme decision bout submitted items. A comprison of smple sizes for both the plns with β = re exhibited in Tble 6. Tble 6. Comprisons of smple size (n) when g = 4 nd c = β Existing HGASP Proposed HGASP Conclusion In this pper, hybrid group cceptnce smpling pln from the truncted life test ws proposed, the number of testers nd the cceptnce number ws derived for size bised Lomx model with α = 3 when the consumer s risk (β) nd other prmeters specified. It cn be observed tht the minimum number of testers required is decreses s test termintion time increses nd lso the operting chrcteristics vlues increses more rpidly s the qulity improves. Hybrid group cceptnce smpling pln is more preferble thn group cceptnce smpling pln. The evluted vlues of r for given g by hybrid group cceptnce smpling pln is showing stbility resulting n usge tht common tble t some stge for ll vlues of α wheres in group cceptnce smpling pln the derived vlues of g for given r re incresing rpidly for vrious vlues of α. If the dt follows size bised Lomx mode, our proposed smpling pln of HGASP gives more efficient results thn existing smpling pln. REFERENCES [1] A. Bklize. Acceptnce smpling bsed on truncted life tests in the Preto distribution of the second kind. Advnces nd Applictions in Sttistics, 3(1), (2003). [2] Jffer Hussin, Abdur Rzzque Mughl, Muhmmd Khlid Perviz nd Usmn Ali. A hybrid group cceptnce smpling plns for lifetimes hving generlized Preto distribution, Journl of Sttistics, vol 19, 31-42, (2012). [3] Muhmmd Aslm, Chi-Hyuck Jun nd Munir Ahmd. A group cceptnce pln bsed on truncted life test for Gmm distribution, Pk. J. Sttist., vol. 25(3), , (2009). [4] K. Rosih nd R.R.L. Kntm. Acceptnce smpling bsed on the inverse Ryleigh distribution, Economic Qulity Control 20(2), , (2005). [5] K. Rosih, R.R.L. Kntm nd Ch. Sntosh Kumr. Relibility of test plns for exponentited log-logistic distribution, Economic Qulity Control, 21(2), , (2006). [6] K. Rosih, R.R.L. Kntm nd R. Subb Ro. An economic Relibility test pln with Preto distribution, Interntionl Journl of Agriculturl Sttisticl Science, 3(2), , (2007). [7] K. Rosih, R.R.L. Kntm nd R. Subb Ro. Preto distribution in cceptnce smpling bsed on truncted life tests. IAPQR Trnsctions Vol. 34, No. 1, (2009). [8] G. Srinivs Ro. A hybrid group cceptnce smpling plns for lifetimes bsed on log-logistic distribution, Journl of Relibility nd Sttisticl Studies, 4(1), 31-40, (2011). [9] G. Srinivs Ro. A hybrid group cceptnce smpling pln for lifetimes bsed on generlized exponentil distribution, Journl of Applied Sciences, 11(12), , (2011). [10] A.R. Sudmni Rmswmy. A hybrid group cceptnce smpling plns for lifetimes bsed on exponentited weibull distribution, Interntionl Journl of Mthemticl Archives, 3(10), , (2012).

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