Double sampling plan for Truncated Life test based on Kumaraswamy-Log-Logistic Distribution

Transcription

1 IOSR Journl of Mthemtics (IOSR-JM) e-issn: ,-ISSN: X, Volume 7, Issue 4 (Jul. - Aug. 203), PP Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution Dr. S. Muthulkshmi, B. Geeth Gnn Selvi Professor in Sttistics, M.Phil Scholr Dertment of Mthemtics Avinshilingm Institute for Home Science nd High Eduction for Women, Coimbtore Abstrct: A double ccetnce smling ln for the life test truncted t the re-ssigned time to decide on ccetnce or rejection of the submitted lots is considered. This double ccetnce smling ln uses only zero nd one filure scheme. The robbility model of the lifetime of the roduct is secified s Kumrswmy-log-logistic distribution which contins severl sub models. The minimum smle sizes of the first nd second smles necessry to ensure the secified medin life re obtined t the given consumer s confidence level. The oerting chrcteristic vlues re nlyzed with vrious rtios of the true medin lifetime to the secified lifetime of the roduct. The minimum rtios of the medin life to the secified life re lso obtined so s to lower the roducer s risk t the secified level. Numericl exmles re rovided to exlin the liction of smling ln. Keywords: Consumer s confidence level; double smling ln; Kumrswmy-log-logistic distribution; Medin life; Poisson distribution. I. Introduction Products or items hve vritions in their lifetimes even though they re roduced by the sme roducer, mchine nd under the sme mnufcturing conditions. The roducer nd the consumer re subject to risks due to the decision on the ccetnce or rejection of roducts bsed on n ccetnce smling ln. Consumer s risk is due to the ccetnce of lots with severl roducts hving the medin lifetime less thn the secified lifetime nd the roducer s risk is due to the rejection of the lots with severl roducts hving medin lifetime greter thn the secified lifetime. Incresing the smle size my minimize both risks to certin level but this will obviously increse the cost of insection. Thus n efficient ccetnce smling scheme nd truncted life tests re the only otion to reduce these risks. The design rmeters of smling ln deend on the underlying sttisticl distribution. Severl studies hve been done for designing single smling lns bsed on the truncted life tests under vrious sttisticl distributions. Estein [4] first introduced single ccetnce smling lns for the truncted life test bsed on the exonentil distribution. Goode nd Ko [5] develoed n ccetnce smling ln using the Weibull distribution s lifetime distribution. Gut nd Groll [7] derived the ccetnce smling ln for the gmm distribution nd Gut [6] designed the ln for the lifetime of the roduct hving log-norml distribution. Duncn [3] ointed out tht double smling lns reduce the smle size or roducer s risk in the re of qulity control. Aslm nd Jun [] introduced double ccetnce smling for the truncted life test bsed on the generlized log-logistic distribution. The urose of this er is to roose the double ccetnce smling ln for the truncted life test ssuming tht the lifetime of roduct follows the Kumrswmy-log-logistic distribution. The beuty nd imortnce of this distribution lies in ccommodting severl imortnce distributions s sub-models. The minimum smle sizes of the first nd second smles for zero-one double smling ln re determined t the secified consumer s confidence level. The oerting chrcteristics re nlyzed s function of the rtio of the true medin life to the secified life. The Kumrswmy-log-logistic distribution is introduced in Section 2.The roosed double smling ln is given in Section 3. The oerting chrcteristics re nlyzed in Section 4. The minimum medin rtios to the secified life re resented in Section 5. Numericl exmles for selection of lns for given condition re resented in Section 6. II. Kumrswmy-Log-Logistic Distribution The lifetime of roduct is ssumed to hve the Kumrswmy-log-logistic distribution (KumLL), whose robbility density function (df) nd cumultive distribution function (cdf) re given resectively s b, t > 0,, b, α nd γ > 0 () b f ( t) t t ( ) t 29 Pge

2 nd Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution b, t 0,, b, α nd γ >0 (2) Ft;, b,, t ( ) where α is scle rmeter nd, b nd γ re the she rmeters. Four rmeter Kumrswmy-log-logistic distribution is the result of the work done by Kumrswmy [8] nd Cordeiro nd de Cstro [2]. If T is rndom vrible relting to lifetime of the roduct with density function () is denoted s T~KumLL(,b,α,γ). The medin of the Kumrswmy-log-logistic distribution is derived by 0.5 b m (3) 0.5 b Exression (3) shows tht the medin is roortionl to the scle rmeter, α when the two she rmeters nd b re fixed. It is lso seen tht the medin reduces to m = α regrdless of γ with =b= for the Kumrswmy-log-logistic distribution. Kumrswmy-log-logistic distribution reduces to Burr distribution when =, exonentited log-logistic distribution when b= nd log-logistic distribution when =b=. III. Design Of The Double Smling Pln Assume tht the qulity of roduct cn be reresented by its medin lifetime, m. The lot will be cceted if the submitted lot hs good qulity when the exerimentl dt suorts the null hyothesis, H 0 : m m 0 ginst the lterntive hyothesis, H : m<m 0. where m 0 is secified lifetime. The significnce level for the test is used through -P *, where P * is the consumer s confidence level. The double smling ln for the truncted life test consists of (i) the number of units of first nd second smles (ii) ccetnce numbers (iii) the rtio of true medin life to the secified medin life m/m 0. The consumer risk, the robbility of cceting bd lot which hs the true medin life below the secified life m 0, is fixed s not to exceed -P *. The oerting rocedure of zero-one double smling ln for the truncted life test hs the following stes. Drw first rndom smle of size n from the submitted lot nd ut on the test for re-ssigned exerimentl time t 0. If there re c or less filures occur within the exerimentl time ccet the lot. If there re c 2 + filures occur by t 0 reject the lot. 2. If the number of filures by t 0 is between c + nd c 2 then drw second smle of size n 2 nd ut them on the test for time t 0. If the number of filures in combined smle is t most c 2 ccet the lot. Otherwise, reject the lot. Double smling ln with c =0 nd c 2 = is the zero-one double smling ln. Often it is convenient to set the termintion time s multile of the secified lifetime m 0, in which cse t 0 = ηm 0 for secified multilier η. Then the roosed smling ln is chrcterized by five rmeters (n, n 2, c, c 2, η), where c < c 2. It is ssumed tht the lot size is lrge enough to use the Poisson distribution to clculte the robbility of ccetnce of the lot. Under the roosed double ccetnce smling ln the robbility of ccetnce of the lot is c n i c2 n x c2 x n2 i e n e n e n2 P (4) i0 i! xc x! i0 i! where, the robbility tht n item fils before t 0 is given by t0 b 30 Pge

3 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution m m 0 b 0.5 nd b 0.5 b The roosed double smling ln hving c =0 nd c 2 = is clled s zero nd one filure double smling scheme. Therefore, the robbility of ccetnce of the lot is n n2 P e e n (6) The minimum smle sizes n nd n 2 ensuring m m 0 t the consumer s confidence level P * cn be found s the solution of e n n 0 * 2 e n 0 P 0 where 0 is the vlue of evluted t m = m 0 s b 0 (8) Eqution (7) rovides multile solutions for smle sizes n nd n 2 stisfying the secified confidence level. In order to find the otiml smle sizes the minimum of ASN is incororted long with the robbility of the ccetnce of the lot -P * nd n 2 n. The ASN for double smling ln is ASN= np n n2 P (9) where P is the robbility of ccetnce or rejection bsed on the first smle tht is given by P c 2 ic e n n i! i For c =0 nd c 2 =, we hve n ASN = n nn 2 e (0) Therefore, evlution of minimum smle sizes reduces to the following otimiztion roblem. n 0 Minimize ASN = n n n e () 2 0 n 0 n 0 2 e n 0 P * subject to P e where n nd n 2 re integers with n 2 n Minimum smle sizes my be obtined from () by simle serch by vrying the vlues of n nd n 2. Tble resents the minimum smle sizes for the first nd second smles when the lifetime of the roduct follows the Kumrswmy-log-logistic distribution for vrious vlues of P *, η nd different combintions of (, b, γ). Numericl results in Tble revel tht i. increse in the she rmeters (, b, γ) increses the smle sizes ridly for given η t the secified confidence level. ii. increse in η decreses the minimum smle sizes t given confidence level irresective of (, b, γ) nd iii. increse in confidence level increses the minimum smle sizes for secified η irresective of (, b, γ). Tble. Minimum smle sizes to ssure the medin life to exceed given vlue m 0 for vrious (, b, γ) (, b, γ) P* (,, 2) , 8 9, 7 6, 3 4, 3 4, 2 3, 2 3, 2 3, 3, , 27 3, 8, 6 6, 4 5, 3 4, 4 4, 3 4, 2 4, , 37 6, 4 0, 7 7, 7 6, 5 5, 5 5, 3 5, 2 4, 4 η (5) (7) 3 Pge

4 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution , 46 24, 6 5, 8, 6 9, 5 8, 4 7, 5 7, 3 7, 2 (,, 4) , , 26 9, 8 5, 3 3, 3 3, 2, 2 2, 2 2, , 30 43, 40 3, 3 7, 4 5, 2 4, 2 3, 3 3, 2 3, , 38 54, 48 7, 2 8, 8 6, 3 5, 2 4, 3 4, 2 4, , , 65 25, 6 2, 9 8, 6 7, 3 6, 3 6, 2 6, 2 (, 2, 2) , 22 0, 8 6, 4 4, 4 3, 3 3, 2 3, 3, 2, , 32 4, 4 9, 5 6, 5 5, 3 4, 3 4, 2 4, 2 3, , 40 8, 5, 7 8, 4 6, 4 5, 4 5, 2 4, 4 4, , 63 26, 26 6, 9, 7 9, 5 8, 3 7, 3 6, 5 6, 3 (, 2, 4) , , 32 0, 9 5, 3 3, 2 3, 2, 2 2, 2 2, , , 50 5, 2 7, 5 5, 2 4, 2 3, 2 3, 2 3, , , 6 9, 4 9, 5 6, 3 4, 4 4, 2 4, 2 4, , , 82 27, 26 3, 7 8, 5 6, 5 6, 2 5, 4 5, 3 (2, 2, 4) , , 75 8, 6 5, 5 3, 2 2, 2 2, 2 2, 2, , , , 23 8, 5 4, 4 3, 3 3, 2 3, 2 3, , , , 32 0, 6 5, 5 4, 3 4, 2 4, 2 4, , , , 47 4, 9 8, 4 6, 3 6, 2 5, 3 5, 3 (2, 3, 4) , , 224 2, 9 6, 3 3, 2 2, 2 2, 2 2, 2, , , 338 3, 30 8, 6 4, 3 3, 3 3, 2 3, 2 3, , , 42 39, 36 0, 7 5, 4 4, 2 4, 2 4, 4, , , , 49 5, 8 8, 3 6, 3 5, 4 5, 3 5, 3 The effect of the chnge in the she rmeter when b=2 nd γ=4 with reference to the minimum first smle size nd the exeriment time when the confidence level is fixed t 0.90 for Kumrswmy-log-logistic distribution is given in Fig.. Fig.. The first smle size vs. Exeriment time t confidence level, P * = 0.90 for (, 2, 4) n = =2 The figure indictes tht increse in the she rmeter increses the first smle size ridly for shorter exerimenter time comred to the increse of the smle size for longer exerimentl time. Fig 2 shows the trend of first smle sizes ginst the test time for the effect of the chnge in the she rmeter b when = nd γ=4 t the confidence level, P * =0.90. This figure indictes tht t the secified confidence level exerimentl time of longer durtion requires lesser smle sizes comred to exerimentl time of shorter durtion. Fig.2. The first smle size vs. Exeriment time t confidence level, P * = 0.90 for (, b, 4) 32 Pge

5 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution n b= b=2 The effect of the chnge in the she rmeter γ when = nd b=2 with reference to the minimum first smle size nd the exeriment time when the confidence level is fixed t 0.90 for Kumrswmy-log-logistic distribution is given in Fig.3. Fig.3. The first smle size vs. Exeriment time t confidence level, P * = 0.90 for (, 2, γ) n γ=2 γ=4 The figure indictes tht increse in the she rmeter increses the first smle size ridly for shorter exeriment time comred to the smle size for longer exerimentl time. IV. Oerting Chrcteristic Vlues OC vlues deict the erformnce of the smling ln ccording to the submitted qulity of the roduct. The robbility of the ccetnce will increse if the true medin increses beyond the secified life. Therefore, we need to know the oerting chrcteristic vlues for the roosed ln ccording to the rtio m/m 0 of the true medin life to the secified life. Obviously, ln will be more desirble if its OC vlues increse more shrly to one. The OC vlues s function of the rtio m/m 0 when P * =0.90, =, b=2, γ=4 nd η=0.7 re resented in Tble 2. It is seen tht the OC increses to one more ridly to move from lower vlue to higher vlue of the rtio m/m 0. This rises due to the increse in the smle sizes required t higher she rmeters. It is lso seen tht i. increse in confidence level decreses the oerting chrcteristic vlues for given rtio m/m 0 ii. increse in confidence level decreses the oerting chrcteristic vlues for given η iii. increse in the rtio m/m 0 increses the oerting chrcteristic vlues for ny given η nd ny secified confidence level Tble 2. OC vlues with =, b = 2 nd γ = 4 t secified consumer s confidence level P* n n 2 η m/m Pge

6 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution V. Minimum Medin Rtios To The Secified Life Consider now tht the roducer wnts to know wht will be the minimum roduct qulity level in order to kee the roducer s risk t the secified level. At the roducer s risk of α the minimum rtio m/m 0 cn be obtined by solving P (2) where P is given by Eqution (4) or (6) for the zero nd one filure schemes nd is given by Eqution (5). Tbles 3 nd 4 show the minimum medin rtios to the secified life t the given consumer s confidence level nd test times corresonding to the roducer s risks of 0.05 nd 0.0. Numericl vlues in tbles revel tht i. increse in roducer s risk decreses the minimum medin rtios for ny given combintions of (, b, γ), η nd P * 34 Pge

7 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution ii. increse in P * increses the minimum medin rtios for ny given combintions of (, b, γ) nd η for given roducer s risk iii. increse in η increses the minimum medin rtios for ny given combintions of (, b, γ) nd P * for given roducer s risk Tble 3. Minimum medin rtios to the secified life t roducer s risk of 0.05 η (, b, γ) P* (,, 2) (,, 4) (, 2, 2) (, 2, 4) (2, 2, 4) (2, 3, 4) Tble 4. Minimum medin rtios to the secified life t roducer s risk of 0.0 η (, b, γ) P* (,, 2) (,, 4) Pge

8 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution (, 2, 2) (, 2, 4) (2, 2, 4) (2, 3, 4) VI. Selection Of Smling Pln Tbles my be used for the selection of zero-one double smling ln with truncted life test for the secified confidence level when the lifetime of the roduct follows Kumrswmy-log-logistic distribution with different she rmeters. Exmle. Assume tht the lifetime of the roduct under considertion follows Kumrswmy-log-logistic distribution with she rmeters =, b=2 nd γ =4. Suose tht the roducer wnts to estblish tht the true medin life is greter thn or equl to 2000 hours with the confidence level of The exerimenter wnts to sto the exeriment t 000 hours. This leds to the exeriment termintion multilier, η=0.5. For the roblem under considertion Tble gives zero-one double smling ln with the minimum smle sizes n =63 nd n 2 =6. This smling ln is ut into oertion s follows First, selected rndomly 63 items from the lot nd ut on test for 000 hours nd ccet the lot if no filure occurs nd reject if more thn one filure occurs during the exerimentl time. If one filure is observed select second smle of size 6 nd ut on test for 000 hours. Accet the lot if there re no filures in the second smle. otherwise, reject the lot. The roducer my be concerned with the ccetnce robbility s the qulity imroves becuse they wnt to minimize the roducer s risk. Suose tht the roducer is interested in knowing wht qulity level will led to the roducer s risk less thn This cn be nswered from Tble 3. The minimum rtio for =, b=2 nd γ =4, P * =0.95 nd η=0.5 is So, the true medin required of the roduct should be t lest 9869 hours. Exmle.2 Suose tht the lifetime of the roduct follows Kumrswmy-log-logistic distribution with she rmeters =2, b=3 nd γ =4. Assume tht the roducer wishes to estblish tht the true medin life is lrger thn 000 hours with the confidence level of The exerimenter wnts to terminte the exeriment t 700 hours. For this roblem Tble gives zero-one double smling ln with the minimum smle sizes n =3 nd n 2 =30. As the termintion time leds to η=0.7. This smling ln is crried out s follows First, select 3 items from the lot nd ut on test for 700 hours nd ccet the lot if no filure occurs nd reject the lot if more thn one filure occurs during the exerimentl time. If one filure is observed select second smle of size 30 nd ut on test for 700 hours. Accet the lot if there re no filures from the second smle nd reject the lot otherwise. The roducer my be concerned with the ccetnce robbility s the qulity imroves becuse they wnt to minimize the roducer s risk. Suose tht the roducer is interested in knowing wht qulity level will 36 Pge

9 Double smling ln for Truncted Life test bsed on Kumrswmy-Log-Logistic Distribution led to the roducer s risk less thn 0.0. This cn be nswered from Tble 4. The minimum rtio for =2, b=3 nd γ =4, P * =0.90 nd η=0.5 is So, the true medin required of the roduct should be t lest 3083 hours. VII. Conclusion The double smling ln with zero-one filure scheme is considered nd tbles re rered to mke decision of ccetnce or rejection on the submitted lot under the ssumtion tht the lifetime of the roduct following Kumrswmy-log-logistic distribution with different she rmeters, which hs wider licbility due to the inclusion of severl sub-models. A generl double smling ln would be referble rther thn zero-one double smling ln. Therefore, generl double smling ln for truncted life test is under rertion s future work. References [] M. Aslm, nd C.H. Jun, A double ccetnce smling ln for generlized log-logistic distributions with known she rmeters, Journl of Alied Sttistics, 37(3), 200, [2] G. M. Cordeiro, nd M. de Cstro, A new fmily of generlized distributions, Journl of Sttisticl Comuttion nd Simultion, 8, 20, [3] A.J. Duncn, Qulity Control nd Industril Sttistics, 5th ed., Richrd D. Irwin, Homewood, Illinois, 986. [4] B. Estein, Truncted life tests in the exonentil cse, Ann. Mth. Sttist. 25, 954, [5] H.P. Goode, nd J.H.K. Ko, Smling lns bsed on the Weibull distribution, In Proceeding of the Seventh Ntionl Symosium on Relibility nd Qulity Control, Phildelhi, 96, [6] S.S. Gut, Life test smling lns for norml nd lognorml distributions, Technometrics 4, 962, [7] S.S. Gut, nd P.A. Groll, Gmm distribution in ccetnce smling bsed on life tests, J.Am. Sttist. Assoc. 56, 96, [8] P. Kumrswmy, A generlized robbility density function for double-bounded rndom rocesses, Journl of Hydrology 46, 980, Pge