Time Truncated Two Stage Group Sampling Plan For Various Distributions

Transcription

1 Time Truncted Two Stge Group Smpling Pln For Vrious Distributions Dr. A. R. Sudmni Rmswmy, S.Jysri Associte Professor, Deprtment of Mthemtics, Avinshilingm University, Coimbtore Assistnt professor, Deprtment of Mthemtics, CIT, Coimbtore Abstrct: In this pper, two stge group smpling pln is developed for truncted life test when the life time of n item follows different distributions. The prmeters of the proposed pln such s minimum number of testers nd probbility of lot cceptnce re determined when the consumer s risk nd test termintion time re specified. The results re discussed with the help of tbles nd exmples. Keywords: Truncted life test, Generlized Exponentil distribution, Mrshll Olkin extended Lomx distribution, Mrshll Olkin extended exponentil distribution, Weibull Distribution, Generlised Ryleigh Distribution, Inverse Ryleigh Distribution, Operting chrcteristics, Consumer s risk.. INTRODUCTION Acceptnce smpling pln is n inspecting procedure in sttisticl qulity control or relibility tests, widely used in the field of industry nd business to mnge the product relibility nd to mke the decision of ccepting or rejecting the product by the consumer.tody it is very difficult to observe the complete life time of the product tht re designed with high relibility. The cceptnce smpling plns for truncted life test re frequently used to determine the smple size from lot under considertion. For usul smpling pln it is ssumed tht only single item is put in tester. However, in prctice more thn one tester cn be used to test the multiple numbers of items t time becuse testing time nd cost cn be sved by testing items simultneously. For this type of testers the number of items to be equipped in tester is given by the specifiction. The cceptnce smpling pln under this type of testers will be clled group cceptnce smpling pln. The items in tester re referred s group nd the number of items in group is clled the group size. This type of testers is frequently used in sudden deth testing. The sudden deth tests re discussed by Pscul nd Meeker (998) nd Vlcek et l (003). In time truncted cceptnce smpling pln, rndom smple is chosen from submitted lot of items nd put them on test where the number of filures is observed until the pre-specified time. If the number of filures is greter thn the specified cceptnce number, then the submitted lot will be rejected. An ordinry time truncted cceptnce smpling pln hve been discussed by mny uthors, Goode nd Ko (96), Gupt nd Groll(96), Bklizi nd EI Msri(004), Rosih nd Kntm (005) nd Tsi, Tzong nd Shou(006), Blkrishnn, Victor Leiv & Lopez (007). Rdhkrishnn nd Algirismy ws the first to ttempt the ttribute group cceptnce smpling pln using weighted binomil distribution for Preto distribution to determine the minimum number of groups. Sudmni rmswmy nd Priyh nburjn, (0), discussed the two stge group cceptnce smpling plns on truncted life tests for mrshll olkin extended distributions. In this pper, n pproch of designing two stge group cceptnce smpling pln for truncted life test is proposed, when the lifetime of the items follows different life time distributions. The distributions considered in this pper re Mrshll Olkin extended Lomx distribution, Mrshll Olkin extended exponentil distribution, Weibull distribution, Generlised exponentil distribution, Generlised Ryleigh distribution, Inverse Ryleigh distribution. The test termintion time nd men rtios re specified. The minimum number of testers is obtined such tht it stisfies the consumer s risk. The probbility of cceptnce is lso determined. The tbles of the design prmeter re provided for esy selection of the pln. The results re nlysed with the help of tbles nd exmples. Pge 88

2 n - Size of the smple d - Number of defectives c - Acceptnce number of the first smple c - Acceptnce number of the second smple. GLOSSARY OF SYMBOLS P (p) - Probbility of cceptnce of lot submitted for inspection α - Producer s risk β - Consumer s risk σ - Scle prmeter T - Prefixed time µ - Men life µ 0 - Specified life p - Filure probbility m, λ, γ - Shpe prmeter - Test termintion time multiplier g - Number of groups in first stge g - Number of groups in second stge r - Number of items in group. The following re the distributions used in this pper: (i) Generlized exponentil distribution: 3. DISTRIBUTIONS The cumultive distribution function (cdf) of the generlized exponentil distribution is given by F( t, ) e t Where σ is scle prmeter nd λ is the shpe prmeter nd it is fixed s. (ii) Mrshll Olkin extended Lomx distribution:, t > 0, σ > 0 () The cumultive distribution function (cdf) of the Mrshll Olkin extended Lomx distribution is given by ( t ) Ft (, ), ( t ) Where σ is scle prmeter nd θ nd γ re the shpe prmeters nd they re fixed s. (iii) Mrshll Olkin extended exponentil distribution:, t > 0, σ > 0 () The cumultive distribution function (cdf) of the Mrshll Olkin extended exponentil distribution is given by Pge 89

3 F( t, ) e e t t,, t > 0, σ > 0 (3) Where σ is scle prmeter nd γ is the shpe prmeter nd it is fixed s. (iv) Weibull distribution: The cumultive distribution function (cdf) of the Weibull distribution is given by t m F( t, ) e, t > 0, σ > 0 (5) Where σ is scle prmeter. (v) Generlised Ryleigh distribution (vi) The cumultive distribution function (cdf) of the Ryleigh distribution is given by e t F ( t, ), t > 0, σ > 0 (4) (vii) where σ is scle prmeter Inverse Ryleigh distribution The cumultive distribution function (cdf) of the Inverse Ryleigh distribution is given by Where σ is scle prmeter. t F( t, ) e, t > 0, σ > 0 (6) If some other prmeters re involved, then they re ssumed to be known. The filure probbility of n item by time t 0 is given by p = F(t 0 : σ) (7) The qulity of n item is usully represented by its true men lifetime. Let us ssume tht the true men μ cn be represented by the scle prmeter. In ddition, it is convenient to specify the test time s multiple of the specified life so tht μ 0 nd the qulity of n item s rtio of the true men to the specified life (μ/ μ 0 ). Then we cn rewrite (6) s function of (termintion time) nd the rtio μ/μ 0. p = F( μ 0 : μ/ μ 0 ) (8) Here when the underlying distribution is the inverse Ryleigh distribution p exp 0 (9) Pge 90

4 When the underlying distribution is Generlised Ryleigh distribution p Where m = Γ(k+/)/ Γ(k+) k j0 m / 0 j e j! m 0 (0) When the underlying distribution is the Mrshll Olkin extended exponentil distribution p e e ,.5708 When the underlying distribution is the Mrshll Olkin extended Lomx distribution 0 () p 0,.5708 / ( ).5708 / ( ) 0 When the underlying distribution is the Generlised exponentil distribution p e.79 When the underlying distribution is the Weibull distribution 0 () (3) p m b 0 e (4) 4. OPERATING PROCEDURE FOR TWO STAGE GROUP ACCEPTANCE SAMPLING PLAN FOR TRUNCATED LIFE TEST According to Abbur Rzzque Mughl, Muhmmd Hnif, Azhr Ali Imrn, Muhmmd Rfi nd Munir Ahmd [3] for the two stge group cceptnce smpling pln for truncted life test.. (First stge) Drw the first rndom smple size n from lot, llocte r items to ech of g groups (or testers) so tht n = rg nd put them on test for the durtion of t 0. Accept the lot if the number of filures from ech group is c or less. Truncte the test nd reject the lot s soon s the number of filures in ny group is lrger thn c before t 0. Otherwise, go to the second stge.. (Second stge) Drw the second rndom smple of size n from lot, llocte r items to ech of g groups so tht n = rg nd put them on test for t 0. Accept the lot if the number of filures in ech group is c or less. Truncte the test nd reject the lot if the number of filures in ny group is lrger thn c before t 0. The following is the operting procedure for two stge group cceptnce smpling pln for life test in the form of flow chrt. Pge 9

5 5. FLOW CHART Drw the first smple of size n Assign r to ech group g nd put into test for t 0 Clculte d for ech group g Accept the lot if d c for ech group g If c <d< c Reject the lot if d > c in ny group g Drw the second smple of size n Assign r to ech group g nd put into test for t 0 Clculte d for ech group g Accept the lot if d c in ech group g Reject the lot if d>c in ny group g The two stge group cceptnce smpling pln constitute the design prmeters of g, g, c nd c when the number of testers r re specified. Pge 9

6 6. CONSTRUCTION OF TABLES The probbility of cceptnce cn be regrded s function of the devition of specified verge from the true verge. This function is clled operting chrcteristic (oc) function of the smpling pln. Once the minimum number of testers is obtined one my be interested to find the probbility of cceptnce of lot when the qulity of the product is good enough. The probbility of lot cceptnce t the first stge cn be evluted s, P c i0 r i p ( i p) ri g (5) The probbility of lot rejection t the first stge is given by, P r c i0 r i p ( i p) ri g (6) where the probbility of lot cceptnce t the second stge is P ( P P r ) c i0 r i p ( i p) ri g Therefore, the probbility of lot cceptnce for the proposed two stge group cceptnce smpling pln is given by (7) L ( p) P P g g 0 c, c (8) The minimum number of testers required cn be determined by considering the consumer s risk when the true men life equls the specified men life (μ = μ 0 ) (worst cse) by mens of the following inequlity: Lp ( 0) Where p 0 is the filure probbility t μ = μ 0. Here minimum number of testers r is obtined using (8) nd (9).The filure probbilities re obtined by fixing the time multiplier s 0.7, 0.8,.0,.,.5 nd.0 nd the men rtios μ/μ 0 s, 4, 6, 8, 0 nd. The minimum number of testers is determined by fixing the number of groups nd the consumer s risk β s 0.5, 0.0,, nd 0.0 in the equtions (8) nd (9). The minimum numbers of testers is determined for the bove cited distributions nd re presented in the Tble to Tble 6 respectively. The minimum smple size for the bove distributions cn be obtined by using n = rg, from Tble to Tble 6. The tbles indictes tht, s the test termintion time multiplier increses, the number of testers r decreses, i.e., smller number of testers is needed, if the test termintion time multiplier increses for fixed number of groups. The probbility of cceptnce re lso clculted nd re presented in the Tble 7 to Tble when the life time of the item follows different distributions. (9) Pge 93

7 Probbility of cceptnce Interntionl Journl of Mthemtics nd Physicl Sciences Reserch ISSN (Online). Operting chrcteristic σ/σₒ Gen Exp Mr Lmx Mr Exp Weibull Ry Inv Ry Figure : OC curve for Probbility of cceptnce ginst μ/μ 0 of two stge group smpling pln when the life time of the item follows different distributions. Tble: Minimum number of testers for the two- stge group smpling pln with c = 0 nd c = when the life time of the item follows Generlized exponentil distribution β g g Pge 94

8 Tble : Minimum number of testers for the two- stge group smpling pln with c = 0 nd c = when the life time of the item follows Mrshll Olkin extended Lomx distribution β g g Tble 3: Minimum number of testers for the two- stge group smpling pln with c = 0 nd c = when the life time of the item follows Mrshll Olkin extended exponentil distribution β g g Pge 95

9 Tble 4: Minimum number of testers for the two- stge group smpling pln with c = 0 nd c = when the life time of the item follows Weibull distribution β g g Tble 5: Minimum number of testers for the two- stge group smpling pln with c = 0 nd c = when the life time of the item follows Generlised Ryleigh distribution β g g Pge 96

10 Tble 6: Minimum number of testers for the two- stge group smpling pln with c = 0 nd c = when the life time of the item follows Inverse Ryleigh distribution β g g Tble 7: Probbility of cceptnce for the two- stge group smpling pln with g = & g =, when the life time of the item follows Generlized exponentil distribution μ/μ 0 β r Pge 97

11 Tble 8: Probbility of cceptnce for the two- stge group smpling pln with g = & g = when the life time of the item follows Mrshll Olkin extended Lomx distribution r μ/μ Tble 9: Probbility of cceptnce for the two- stge group smpling pln with g = & g = when the life time of the item follows Mrshll Olkin extended exponentil distribution μ/μ 0 β r Pge 98

12 Tble 0: Probbility of cceptnce for the two- stge group smpling pln with g = & g = when the life time of the item follows Weibull distribution μ/μ 0 β r Tble : Probbility of cceptnce for the two- stge group smpling pln with g = & g = when the life time of the item follows Ryleigh distribution β r μ/μ Pge 99

13 Tble : Probbility of cceptnce for the two- stge group smpling pln with g = & g = when the life time of the item follows inverse Ryleigh distribution β r μ/μ Pge 00

14 7. EXAMPLES Bsed on the consumer s risk vlues nd the test termintion rtios, the minimum number of testers is determined nd hence the minimum smple size is obtined. Suppose tht the experimenter is interested in estblishing tht the true unknown verge life is t lest 000 hours with β = 0.5. It is desired to stop the experiment t t = 68 hours. It is ssumed tht c = 0 nd c =. Following re the results obtined when the lifetime of the test items follows the Generlized Exponentil distribution, Mrshll Olkin extended Lomx distribution, Mrshll Olkin extended exponentil distribution, Weibull Distribution, Inverse Ryleigh Distribution respectively. 7. Generlized Exponentil Distribution: Let the distribution followed be Generlized Exponentil, it is ssumed tht g =, g =, c = 0,c = nd β = 0.5. From Tble the minimum number of testers required is r = 4.Thus we will drw first smple of size n = 8 items nd llocte 4 items to ech of groups to put on tester for 700 hours. (n = rg ), if no filure occur during 700 hours we ccept the lot. The test is terminted nd the lot is rejected if more thn filures occurs otherwise if or filures occurs, then we move on to the second stge, where the second smple of size n = 4 (n = rg ) is chosen nd tested. For the bove conditions the probbility of cceptnce from Tble 7 is when the rtio of the unknown verge life is. 7. Mrshll Lomx extended exponentil distribution: Let the distribution followed be Mrshll Olkin Extended Lomx, it is ssumed tht g =, g =, c = 0,c = nd β = 0.5. From Tble the minimum number of testers required is r = 3.Thus we will drw first smple of size n = 6 items nd llocte 3 items to ech of groups to put on tester for 700 hours. (n = rg ), if no filure occur during 700 hours we ccept the lot. The test is terminted nd the lot is rejected if more thn filures occurs otherwise if or filures occurs, then we move on to the second stge, where the second smple of size n = 3 (n = rg ) is chosen nd tested. For the bove conditions the probbility of cceptnce from Tble 8 is when the rtio of the unknown verge life is. 7.3 Mrshll Olkin Extended Exponentil Distribution : Let the distribution followed be Mrshll Olkin Extended Exponentil, it is ssumed tht g =, g =, c = 0,c = nd β = 0.5. From Tble 3 the minimum number of testers required is r =.Thus we will drw first smple of size n = 4 items nd llocte items to ech of groups to put on tester for 700 hours. (n = rg ), if no filure occur during 700 hours we ccept the lot. The test is terminted nd the lot is rejected if more thn filures occurs otherwise if or filures occurs, then we move on to the second stge, where the second smple of size n = (n = rg ) is chosen nd tested. For the bove conditions the probbility of cceptnce from Tble 9 is when the rtio of the unknown verge life is. 7.4 Weibull Distribution: Let the distribution followed be Weibull,, it is ssumed tht g =, g =, c = 0,c = nd β = 0.5. From Tble 4 the minimum number of testers required is r =.Thus we will drw first smple of size n = 4 items nd llocte items to ech of groups to put on tester for 700 hours. (n = rg ), if no filure occur during 700 hours we ccept the lot. The test is terminted nd the lot is rejected if more thn filures occurs otherwise if or filures occurs, then we move on to the second stge, where the second smple of size n = (n = rg ) is chosen nd tested. For the bove conditions the probbility of cceptnce from Tble 0 is when the rtio of the unknown verge life is. 7.5 Generlised Ryleigh Distribution: Let the distribution followed be Ryleigh,, it is ssumed tht g =, g =, c = 0,c = nd β = 0.5. From Tble 5 the minimum number of testers required is r = 3.Thus we will drw first smple of size n = 6 items nd llocte 3 items to ech of groups to put on tester for 700 hours. (n = rg ), if no filure occur during 700 hours we ccept the lot. The test is terminted nd the lot is rejected if more thn filures occurs otherwise if or filures occurs, then we move on to the second stge, where the second smple of size n = 3 (n = rg ) is chosen nd tested. For the bove conditions the probbility of cceptnce from Tble is when the rtio of the unknown verge life is. 7.6 Inverse-Ryleigh Distribution: Let the distribution followed be Inverse-Ryleigh,, it is ssumed tht g =, g =, c = 0,c = nd β = 0.5. From Tble 6 the minimum number of testers required is r = 3.Thus we will drw first smple of size n = 6 items nd llocte 3 items to ech of groups to put on tester for 700 hours. (n = rg ), if no filure occur during 700 hours we ccept the lot. The test is terminted nd the lot is rejected if more thn filures occurs otherwise if or filures occurs, then we move on to the second stge, where the second smple of size n = 3 (n = rg ) is chosen nd tested. For the bove conditions the probbility of cceptnce from Tble is when the rtio of the unknown verge life is. Pge 0

15 8. CONCLUSIONS In this pper, designing two stge group smpling pln for the truncted life test is presented. The minimum number of testers nd the probbility of cceptnce re clculted, when the consumer s risk β nd other pln prmeters re specified, ssuming tht the lifetime of n item follows different distributions. When ll the bove tbles (Tble 7 to Tble ) re compred, considering both the fctors. i.e. minimum number of testers nd the probbility of cceptnce, the Inverse Ryleigh distribution is comprtively the best. It cn lso be observed tht the minimum number of testers required decreses s the test termintion time multiplier increses in ll the bove cited distributions nd thus it is concluded tht the tbles provided cn be used conveniently in prcticl situtions when multiple number of items t time re dopted for life test to sve the cost nd time of the experiment. REFERENCES [] Aslm, M., Kundu, D. nd Ahmd, M. (00). Time truncted cceptnce smpling pln for generlized exponentil distribution. J. App. Sttist., 37, [] Abdur Rzzque Mughl, Muhmmd Aslm, Jffer Hussin nd Abdur Rehmn (00). Economic relibility group cceptnce smpling plns for lifetimes following Mrshll-Olkin extended distribution, Middle Estern Finnce nd Economics, Issue 7. [3] Abdur Rzzque Mughl, Muhmmd Hnif, Azhr Ali Imrn, Muhmmd Rfi nd Munir Ahmd (0), Economic relibility two stge group cceptnce smpling plns for truncted life test hving Weibull distribution. Europen Journl of Scientific Reserch, Vol. 54(4), [4] Sudmni rmswmy A.R. nd Priyh nburjn, A Two Stge Group Acceptnce Smpling Plns on Truncted Life Tests for Mrshll Olkin Extended Distributions, Interntionl Journl of Mthemtics Reserch. ISSN , Vol. 4(6), 0. [5] Aslm,M., nd C.H. Jun. A group cceptnce smpling plns for truncted life tests bsed on the inverse Ryleigh nd log-logistic distributions. Pkistn Journl of Sttistics 5, -3, 009. [6] Aslm, M., Jun, C.-H., Lee, H., Ahmd, M., nd Rsool, M. (0). Improved group smpling plns bsed on truncted life tests. The Chilen J. Sttist., (), [7] Bklizi,. nd El Msri,A.E.K. (004), Acceptnce smpling plns bsed on truncted life tests in the Birnbum Sunders model, Risk Anlysis, vol.4, [8] Gupt,R.D.nd Kundu,D. (007), Generlized exponentil distribution : existing methods nd recent developments, Journl of Sttisticl plnning nd inference, vol.37, [9] Gupt,S.S. nd Groll,P.A. (96), Gmm distribution in cceptnce smpling bsed on life tests,journl of the Americn Sttisticl Assocition,vol.56, [0] Rdhkrishnn R. nd Algirismy K. (0). Construction of group cceptnce smpling pln using weighted binomil distribution, Interntionl Journl of Recent Scientific Reserch, Vol. (7), 9-3. [] Srinivs Ro (00), Group cceptnce smpling plns bsed on truncted life tests for Mrshll olkin extended Lomx distribution, Electronic journl of Applied Sttisticl Anlysis, Vol.3,Isse (00),8-7. [] Srinivs Ro G. (009), A group cceptnce smpling plns for lifetimes following generlized exponentil distribution, EQC, Vol 4 (009), No., [3] Srinivs Ro, G., (0), A hybrid group cceptnce smpling plns for lifetimes bsed on generlized exponentil distribution, Journl of Applied Sciences, : [4] Srinivs Ro, G., (0), A hybrid group cceptnce smpling plns for lifetimes bsed on Log logistic distribution, Journl of Relibility nd Sttisticl Studies, Vol. 4(), Pge 0