A Group Acceptance Sampling Plans Based on Truncated Life Tests for Type-II Generalized Log-Logistic Distribution

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1 ProbSa Forum, Volume 09, July 2016, Pages ISSN ProbSa Forum is an e-journal. For deails please visi A Group Accepance Sampling Plans Based on Truncaed Life Tess for Type-II Generalized Log-Logisic Disribuion K Rosaiah a, G S Rao b, S V S V S V Prasad c a Deparmen of Saisics, Acharya Nagarjuna Universiy, Gunur , India. b Deparmen of Saisics, The Universiy of Dodoma, Dodoma, PO. Box: 259, Tanzania. c Deparmen of Saisics, Acharya Nagarjuna Universiy, Gunur , India. Absrac. In his paper, we have developed a group accepance sampling plan for runcaed life ess when he lifeime of iems follows he Type-II generalized log logisic disribuion (TGLLD). A a specified level of consumer risk and a he predefined es erminaion ime, derived he minimum number of groups reuired for a given group size and he accepance number. The values of operaing characerisic (OC) funcion according o various ualiy levels are derived and he minimum raios of rue average life o he specified life a given producers risk are obained. The resuls are illusraed wih example. 1. Inroducion Saisical ualiy conrol refers o he user of saisical mehods in monioring and mainaining of he ualiy of producs and services. One mehod, referred o as accepance sampling, can be used when a decision mus be made o accep or rejec a group of pars or iems based on he ualiy found in a sample. Accepance Sampling is an imporan field of saisical ualiy conrol ha popularized by Dodge and Roming and originally applied by he U.S. miliary o he esing of bulles during World War II. Accepance sampling is a saisical mehod of inspecing a sample of goods and deciding wheher o accep he enire lo based on he resuls. The accepance sampling plans for runcaed life ess are freuenly used o deermine he sample size fromalo under consideraion. Forusualsamplingplan, i is assumeha onlyasingleiem is pu in aeser. Tesers accommodaing muliple number of iems a a ime in order o save he ime and cos. For his ype of esing he number of iems o be euipped in a eser is given by he specificaion. The accepance sampling plan under his ype of esers will be called group accepance sampling plan (GASP). If GASP is used in conjuncion wih runcaed life ess, i is called a GASP based on runcaed life es assuming ha he lifeime of produc follows a cerain probabiliy disribuion. For such a es he deerminaion of sample size is euivalen o deermine he number of groups. Many auhors have discussed accepance sampling plans based on runcaed life ess of single iem groups for various disribuions. Kanam and Rosaiah (1998), Kanam e al. (2001), Rosaiah e al. (2006), Aslam and Jun (2009), Rao (2009, 2010, 2011), Aslam and Jun (2009a), Aslam e al. (2011), Ramaswamy 2010 Mahemaics Subjec Classificaion. 62N05, 90B25, 62P30 Keywords. Type-II generalized log-logisic disribuion, group accepance sampling, Consumer risk, producer risk, operaing characerisic, runcaed life es. Received: 26 Sepember 2015; Acceped: 28 July addresses: rosaiah1959@gmail.com (K Rosaiah), gaddesrao@gmail.com (G S Rao), svsvsvprasad@gmail.com (S V S V S V Prasad)

2 Rosaiah, Rao and Prasad / ProbSa Forum, Volume 09, July 2016, Pages and Anburajan (2012) and Rao e al. (2014). The main aim of his sudy is o develop a GASP based on runcaed life ess assuming he life ime disribuion of he producs follows TGLLD. Rao e al. (2012, 2013) have discussed reliabiliy es plans for TGLLD and also developed Economic reliabiliy es plans for TGLLD. The res of he paper is arranged in he following way, inroducion of TGLLD is given in Secion 2 and he proposed GASP is discussed in Secion 3. Descripion of he proposed mehodology wih real life daa is given in Secion 4 and finally, some conclusions are esablished in Secion Type-II Generalized Log Logisic Disribuion Log logisic disribuion(lld) has proven is imporance in ualiy conrol. Differen ypes of accepance sampling plans are developed for LLD. The cumulaive disribuion funcion (cdf) of he log-logisic disribuion (LLD) is ( λ F() = ) 1+ ( ) λ ]; > 0, > 0, λ > 1 (1) Rosaiah e al. (2008) inroduced a generalizaion of LLD and called as Type-II generalized log logisic disribuion (TGLLD), is cumulaive disribuion funcion (cdf) is F(;λ,,θ) = 1 1+ ( ) ] λ θ ; > 0,,θ > 0, λ > 1 (2) I can be seen ha his is he failure probabiliy funcion of series sysem of θ componens, where life imes of he componens are independenly and idenically disribued wih he log logisic disribuion given by (1) as a common model. I may be noed ha he disribuion given in (2) is defined hrough he reliabiliy oriened generalizaion of log-logisic disribuion. In shor we call his as he Type-II generalized log logisic disribuion Type-I generalized (exponeniaed) log logisic disribuion is deal wih by Rosaiah e al. (2006)]. The corresponding probabiliy densiy funcion (pdf) is given by f(t;θ,λ,) = λθ ( ) λ 1 1+ ( ) ] λ θ+1 ; > 0,,θ > 0, λ > 1 (3) where is he scale parameer, λ and θ are shape parameers and θ akes an ineger values. The 100 h percenile of he TGLLD is given by ] = (1 p) 1 1 λ θ 1 = η (4) where η = ] (1 p) 1 1 λ θ 1 Hence, for fixed values of θ = θ 0 andλ = λ 0, he uanile in euaion (4) is he funcion of scale parameer = 0 i.e., 0 0 where 0 = ( 0 η ) (5) I may be noed ha 0 depends on θ 0 and λ 0 o draw he accepance sampling plans for TGLLD.

3 Rosaiah, Rao and Prasad / ProbSa Forum, Volume 09, July 2016, Pages Design of he proposed Sampling Plan We prefer o use he percenile poin as he ualiy parameer insead of mean and i is denoed by. According o he accepance sampling plans he hypohesis H 0 : 0 is esed based on he number of observed failures from a sample in a pre-fixed ime. The number of producs n in he lo o be esed and is divided ino eual sized groups subjec o he availabiliy of he number of esers. I is assumed ha he experimenal ime and number of producs in each group are fixed in advance. Since n = rg deermining n is euivalen o deermining g. The following group accepance sampling plan on he runcaed life es is proposed. i Selec a random sample of n producs and allocae r iems o each of g groups so ha n = rg ii Fix an accepance number c and he pre-specified es erminaion ime 0 iii Accep he lo if he number of failures from all groups is less han or eual o c. iv Terminae he experimen if more han c failures observed from all groups before he erminaion 0. The probabiliy of acceping lo for group sampling plan based on he number of failures from all groups under a runcaed life es before he prefixed ime 0 is given by ( ) rg F(p) = p i (1 p) rg i (6) i where g is number of groups, c is he accepance number, r is he group size and p is he probabiliy of geing a failure wihin he es erminaion ime 0. Since he lifeime of he produc follows TGLLD, we have p = F( 0 ). Since i is convenien o se he erminaion ime as muliple of he argeed 100 h lifeime percenile, 0 and a consan δ. Le be he rue 100 h lifeime percenile. Then p can be rewrien as p = 1 1+ ( ) ] λ θ 0 In order o find he design parameers of he proposed group accepance sampling plan, we prefer he approach based on wo poins on he curve by considering he producers and consumers risk. In his approach he ualiy level is measured hrough he raio of is percenile lifeime o he rue lifeime, 0.To ensure and improve he ualiy of he producs, producer may use he percenile raios. Producer prefers ha he probabiliy of lo accepance shall be a leas 1 α a he accepable reliabiliy level(arl), p 1. Hence he producer wans ha he lo o be acceped a differen values of percenile raios, say for he values of 0 = 2,4,6,8 given in (7). In view of he consumer he lo rejecion probabiliy shall no be exceed a he lo olerance reliabiliy level (LTRL), p 2. Hence, consumer may rejec he lo if 0 =1 according o he euaion (7). c i=0 c i=0 ( ) rg p i 1 (1 p 1 ) rg i 1 α (8) i ( ) rg p i 2 (1 p 2 ) rg i i where p 1 and p 2 are given by p 1 = 1 1+ η δ 0 ( ) 0 λ θ and p 2 = 1 1+ ( η δ) 0 λ ] θ (10) Esimaes of he parameers of he proposed plan for differen values of shape parameers are obained. i.e., a he given values of λ = 2, θ = 2, producers risk α = 0.05 and es erminaion ime 0 = δ 0 wih δ (7) (9)

4 Rosaiah, Rao and Prasad / ProbSa Forum, Volume 09, July 2016, Pages = 0.5 or 1.0, he parameers of he proposed GASP are esimaed for 10 h and 50 h perceniles a differen confidence levels =0.25, 0.10, 0.05, The plan parameers are presened in Tables 1 and 2 for λ=2, θ=2 a 10 h and 50 h perceniles respecively. Tables 3 and 4 are consruced using ML esimaes ˆθ = and ˆλ= a 10 h and 50 h perceniles respecively. We noiced from Tables 1 o 4, as percenile raio increases, he number of groups g decreases. Similarly, as r increases from 5 o 10, he number of groups reduces. 4. Descripion of he Proposed Mehodology wih Real Daa Example In his secion, we use a real daa se o show ha he ype-ii generalized log logisic disribuion can be a suiable model. Folks & Chhikara (1978) presened several ses of daa o describe he Birnbaum-Saunders disribuion.one of he daa se gives he runoff amouns a Jug Bridge, Maryland. For ready reference his daa se is reproduced as follows: 0.17, 0.23, 0.33, 0.39, 0.39, 0.40, 0.45, 0.52, 0.56, 0.59, 0.64, 0.66, 0.70, 0.76, 0.77, 0.78, 0.95, 0.97, 1.02, 1.12, 1.19, 1.24, 1.59, 1.74 and We show a rough indicaion of he goodness of fi for our model by ploing he superimposed for he daa shows ha he TGLLD is a good fi in Figure 1 and also goodness of fi is emphasized wih Q-Q plo, displayed in Figure 1. The maximum likelihood esimaes of he hree parameer TGLLD for he runoff amouns are ˆ=0.7616, ˆλ= and ˆθ = and using he Kolmogorov-Smirnov es, i is found ha he maximum disance beween he daa and he fied of he TGLLD is wih p-value is Meanwhile, he maximum likelihood esimaes of he wo-parameer TGLLD for he runoff amouns are ˆλ= and ˆθ= and he Kolmogorov-Smirnov es and found ha he maximum disance beween he daa and he fied of he TGLLD is wih p-value is Therefore, he wo-parameer TGLLD is also provides reasonable good fi for he runoff amouns. Le us consider he life ime of a produc is known o follow a TGLLD wih index parameers ˆθ= and shape ˆλ = Suppose ha i is desired o develop he group accepance sampling plan o saisfy ha he 50 h percenile lifeime is greaer han runoff amouns 0.35 hrough he experimen o be compleed by runoff amouns 0.35 using eser aking wih five producs each. Le us fix ha he consumer s risk is a 25%when he rue 50 h percenile is runoff amouns 0.35 and he producer s risk is 5% when he rue 50 h percenile is runoff amouns Since ˆθ= and ˆλ=2.3381, he consumer s risk is 25%, r = 5, δ 0=0.5 and 0 =2, he minimum number of groups and accepance number given by g = 2 and c = 3 from Table 4. Thus he design can be implemened as follows: selec a oal of 10 producs are needed and ha wo iems will be allocaed o each of he five esers. We can accep he lo when no more han hree failures occurs before runoff amouns 0.35 from wo groups. According o his plan, he runoff amouns could have been acceped because here are only hree failure before he erminaion ime, runoff amouns Conclusions In his aricle, a group sampling plan o ensure he specified produc lifeime percenile has been developed for he Type-II generalized log logisic disribuion. The design parameers c and g of he proposed sampling plan are deermined by he wo-poin mehod. Exensive ables have been provided for he indusrial use according o various parameers and percenile values. I was observed ha he number of groups reuired increases as he consumers confidence increases, rue ualiy decreases and as r increases he number of groups reduces. A comparison beween he proposed group sampling plan and he ordinary group sampling plan has also been discussed. I has been noiced ha he proposed plan reuires a smaller sample size han he ordinary group sampling plan does. The mehodology illusraed wih real daa se. References 1] Aslam, M., and Jun, C.-H., (2009a). Group accepance sampling plans for runcaed life ess based on he inverse Rayleigh disribuion and log-logisic disribuion, Pakisan Journal of Saisics, 25,

5 Rosaiah, Rao and Prasad / ProbSa Forum, Volume 09, July 2016, Pages Figure 1: The densiy plo and Q-Q plo of he fied ype-ii generalized log logisic disribuion for he runoff amouns daa 2] Aslam, M. and Jun, C.-H. (2009b). A group accepance sampling plan for runcaed life es having Weibull disribuion, Journal of Applied Saisics, 39, ] Aslam, M., and Jun, C.-H. (2009). A group accepance sampling plans for runcaed life ess based on he inverse Rayleigh and log-logisic disribuions, Pakisan Journal of saisics, 25(2), ] Aslam, M., Debasis Kundu., Jun, C.H., and Ahmad, M. (2011). Time runcaed group accepance sampling plans for generalized exponenial disribuion,journal of Tesing and Evaluaion, 39(4), ] Aslam, M., Jun, C.H., Ahmad, M., and Rasool, M. (2011). Group accepance sampling plans for resubmied los under Burr-ype XII disribuions, Chinese Inuiion of Indusrial Engineers, 28(8), ] Ferig, F.W. and Mann, N.R. (1980). Life-es sampling plans for wo-parameer Weibull populaions, Technomerics, 22, ] Kanam, R.R.L. and Rosaiah, K. (1998). Half logisic disribuion in accepance sampling based on life ess, IAPQR Transacions, 23(2), ] Kanam, R. R. L., Rosaiah, K. and Rao, G. S. (2001). Accepance sampling based on life ess: log-logisic models, J. Appl. Sais., 28, ] Ramaswamy, A.R.S. and Anburajan. P. (2012). Group accepance sampling plans using weighed binomial on runcaed life ess for inverse Rayleigh and Log-logisic disribuions, IOSR Journal of Mahemaics, 2(3), ] Rosaiah, K., and Kanam, R.R.L. and Sanosh Kumar, Ch. (2006). Reliabiliy es plans for exponeniaed log-logisic disribuion, Economic Qualiy Conrol, 21(2), ] Rosaiah, K., and Kanam, R.R.L. and Sanosh Kumar, Ch. (2007). Exponeniaed log-logisic disribuion-an economic reliabiliy es plan, Pakisan Journal of Saisics, 23 (2), ] Rosaiah. K., Kanam. R.R.L., Prasad, S.V.S.V.S.V. and Praap Reddy. J. (2008). Reliabiliy Esimaion in Type-II Generalized Log-Logisic Disribuion, In. J. Agricul. Sa. Sci., 4(2), ] Rao, B., Srinivas Kumar, CH. and Rosaiah, K. (2014). Group accepance sampling plans for life ess based on Half Normal disribuion, Sri Lankan Journal of Applied Saisics, 15(3), ] Rao, G.S. (2009). A group accepance sampling plans for lifeimes following a generalized exponenial disribuion, Economic Qualiy Conrol, 24(1), ] Rao, G.S. (2010). A group accepance sampling plans for runcaed life ess for Marshall-Olkin exended Lomax disribuion, Elecron. J. Applied Sa. Anal., 3, ] Rao, G.S. (2011). A hybrid group accepance sampling plans for lifeimes based on log-logisic disribuion, Journal of Reliabiliy and Saisical sudies, 4(1), ] Rao, G.S. (2011). A group accepance sampling plans for Lifeimes following a Marshall-Olkin exended Exponenial disribuion, Applicaion and applied Mahemaics: An Inernaional Journal (AAM), 6(2), ] Rao, G.S., R.R.L. Kanam., Rosaiah. K and Prasad, S.V.S.V.S.V. (2012). Reliabiliy Tes Plans for Type-II Generalized Log-Logisic disribuion, JRSS, 5(1), ] Rao, G.S., R.R.L. Kanam, Rosaiah. K and Prasad, S.V.S.V.S.V.(2013). An Economic Reliabiliy Tes Plan for Generalized Log-Logisic Disribuion, Inernaional Journal of Engineering and Applied Sciences, 3,

6 Rosaiah, Rao and Prasad / ProbSa Forum, Volume 09, July 2016, Pages Table 1: Proposed plan for TGLLD wih θ=2 andλ=2 for 10 h percenile δ = 0.5 δ =1.0 δ =0.5 δ = Table 2: Proposed plan for TGLLD wih θ=2 andλ=2 for 50 h percenile δ =0.5 δ =1.0 δ =0.5 δ =

7 Rosaiah, Rao and Prasad / ProbSa Forum, Volume 09, July 2016, Pages Table 3: Proposed plan for TGLLD wih ˆθ= and ˆλ= for 10 h percenile δ 0 =0.5 δ =1.0 δ =0.5 δ = Table 4: Proposed plan for TGLLD wih ˆθ= and ˆλ= for 50 h percenile δ 0 =0.5 δ =1.0 δ =0.5 δ =