ACCEPTANCE SAMPLING PLANS FOR PERCENTILES BASED ON THE INVERSE RAYLEIGH DISTRIBUTION

Transcription

1 Elecronic Journal of Applied Saisical Analysis EJASA (212), Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, e-issn , DOI /i275948v5n2p Universià del Saleno hp://siba-ese.unile.i/index.php/ejasa/index ACCEPTANCE SAMPLING PLANS FOR PERCENTILES BASED ON THE INVERSE RAYLEIGH DISTRIBUTION G. Srinivasa Rao 1*, R.R.L. Kanam 2, K. Rosaiah 2, J. Praapa Reddy 3 (1) Deparmen of Saisics, Dilla Universiy, Dilla, PO Box:419, Ehiopia (2) Deparmen of Saisics, Acharya Nagarjuna Universiy, Gunur, India (3) Deparmen of Compuer Science, S.Ann's College for women, Gunur, India Received 14 February 211; Acceped 2 April 212 Available online 14 Ocober 212 Absrac: In his aricle, accepance sampling plans are developed for he inverse Rayleigh disribuion perceniles when he life es is runcaed a a pre-specified ime. The minimum sample size necessary o ensure he specified life percenile is obained under a given cusomer s risk. The operaing characerisic values (and curves) of he sampling plans as well as he producer s risk are presened. Two examples wih real daa ses are also given as illusraion. Keywords: Accepance sampling, consumer s risk, operaing characerisic funcion, producer s risk, runcaed life ess, producer s risk. 1. Inroducion The accepance sampling plans are concerned wih acceping or rejecing a submied lo of a large size of producs on he basis of he ualiy of he producs inspeced in a sample aken from he lo. An accepance sampling plan is a specified plan ha esablishes he minimum sample size o be used for esing. In mos accepance sampling plans for a runcaed life es, he major issue is o deermine he sample size from a lo under consideraion. If he ualiy characerisic is regarding he lifeime of he produc, he accepance sampling problem becomes a life es. Tradiionally, when he life es indicaes ha he mean life of producs exceeds he specified one, he lo of producs is acceped, oherwise i is rejeced. For he purpose of reducing he es ime and cos, a runcaed life es may be conduced o deermine he smalles sample size o ensure a cerain mean life of producs when he life es is erminaed a a pre-assigned ime, and he number of failures observed does no exceed a given accepance number c. The decision is o accep he lo if a pre-deermined mean life can be reached wih a pre-deermined high probabiliy which provides proecion o consumers. Therefore, he life es is ended a he ime * gaddesrao@yahoo.com 164

2 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, he failure is observed or a he pre-assigned ime, whichever is earlier. For such a runcaed life es and he associaed decision rule; we are ineresed in obaining he smalles sample size o arrive a a decision. Rosaiah and Kanam [2] developed an accepance sampling procedure for he inverse Rayleigh disribuion mean under a runcaed life es. Some oher sudies regarding runcaed life ess can be found in Epsein [3], Sobel and Tischendrof [22], Goode and Kao [5], Gupa and Groll [7], Gupa [6], Ferig and Mann [4], Kanam and Rosaiah [9], Kanam e al. [1], Baklizi [1], Wu and Tsai [25], Rosaiah e al. [21], Tsai and Wu [24], Balakrishnan e al. [2] and Rao e al. ([17], [18] & [19]). All hese auhors considered he design of accepance sampling plans based on he populaion mean under a runcaed life es. Whereas Lio e al. [12] considered accepance sampling plans from runcaed life ess based on he Birnbaum-Saunders disribuion for perceniles and hey proposed ha he accepance sampling plans based on mean may no saisfy he reuiremen of engineering on he specific percenile of srengh or breaking sress. When he ualiy of a specified low percenile is concerned, he accepance sampling plans based on he populaion mean could pass a lo which has he low percenile below he reuired sandard of consumers. Furhermore, a small decrease in he mean wih a simulaneous small increase in he variance can resul in a significan downward shif in small perceniles of ineres. This means ha a lo of producs could be acceped due o a small decrease in he mean life afer inspecion. Bu he maerial srenghs of producs are deerioraed significanly and may no mee he consumer s expecaion. Therefore, engineers pay more aenion o he perceniles of lifeimes han he mean life in life esing applicaions. Moreover, mos of he employed life disribuions are no symmeric. In viewing Marshall and Olkin [13], he mean life may no be adeuae o describe he cenral endency of he disribuion. This reduces he feasibiliy of accepance sampling plans if hey are developed based on he mean life of producs. Acually, perceniles provide more informaion regarding a life disribuion han he mean life does. When he life disribuion is symmeric, he 5h percenile or he median is euivalen o he mean life. Hence, developing accepance sampling plans based on perceniles of a life disribuion can be reaed as a generalizaion of developing accepance sampling plans based on he mean life of iems. Balakrishnan e al. [2] proposed he accepance sampling plans could be used for he uaniles and derived he formulae whereas Lio e al. [12] developed for he accepance sampling plans for any oher perceniles of he Birnbaum-Saunders (BS) model. They have developed he accepance sampling plans for percenile by replace he scale parameer by he 1h percenile in he BS disribuion funcion. Rao and Kanam [16] developed accepance sampling plans from runcaed life ess based on he log-logisic disribuion for Perceniles. These reasons moivae o develop accepance sampling plans based on he perceniles of he inverse Rayleigh disribuion under a runcaed life es. The res of he aricle is organized as follows. The proposed sampling plans are esablished for he inverse Rayleigh perceniles under a runcaed life es, along wih he operaing characerisic (OC) and some relevan ables are given in Secion 2. Two examples based on real faigue life daa ses are provided for he illusraion in Secion 3, Fuure work is given in Secion 4 and discussion and some conclusions are made in Secion

3 Accepance sampling plans for perceniles based on he inverse Rayleigh disribuion 2. Accepance Sampling Plans Assume ha he lifeime of a produc follows an inverse Rayleigh disribuion which has he following probabiliy densiy funcion (pdf) and cumulaive disribuion funcion (cdf), respecively: f (;σ ) = 2σ 2 e (σ 2 / ) ;,σ >, (1) 3 and ( / ) 2 e σ F (; σ) = ;, σ >, (2) whereσ is he scale parameer. The failure rae of a single parameer inverse Rayleigh disribuion is increasing for < σ and decreasing for > σ as shown by Mukherjee and Saran [14]. Given < < 1he 1 h percenile (or he h uanile) is given by: ( ) 1/2 = σ ln. (3) The is increases as increases. Le ( ) 1/2 η = ln. Then, E. (3) implies ha σ = η. (4) To develop accepance sampling plans for he inverse Rayleigh perceniles, he scale parameer σ in he inverse Rayleigh cdf is replaced by E. (4) and he inverse Rayleigh cdf is rewrien as: ( ( )/ ) () 2 η F = e ; >. Leingδ =, F() can be rewrien emphasizing is dependence on δ as: ( 1 ηδ ) 2 F (; δ) = e ; >. Taking parial derivaive wih respec o δ, we have: F (; δ ) 2 ( 1 ηδ ) = e 2 ; >. 3 δ ηδ 166

4 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, A common pracice in life esing is o erminae he life es by a pre-deermined ime, he probabiliy of rejecing a bad lo be a leas p, and he maximum number of allowable bad iems o accep he lo be c. The accepance sampling plan for perceniles under a runcaed life es is o se up he minimum sample size n for his given accepance number c such ha he consumer s risk, he probabiliy of acceping a bad lo, does no exceed 1- p. A bad lo means ha he rue 1 h percenile,, is below he specified percenile, a confidence level in he sense ha he chance of rejecing a bad lo wih. Thus, he probabiliy p is < is a leas eual o p. Therefore, for a given p, he proposed accepance sampling plan can be characerized by he riple (,, nc ). 2.1 Minimum Sample Size For a fixed p our sampling plan is characerized by (,, nc ). Here we consider sufficienly large sized los so ha he binomial disribuion can be applied. The problem is o deermine for given values of p ( < p <1), and c, he smalles posiive ineger, n reuired o asser ha c i= > mus saisfy: n i ( i ) ( ) n i p 1 p 1 p, (5) where p= F(; δ) is he probabiliy of a failure during he ime given a specified 1 h percenile of lifeime and depends only onδ =, since F (; δ) δ >, F (; δ) is a nondecreasing funcion ofδ. Accordingly, we have: F(, δ) < F(, δ ) δ δ, Or euivalenly, F (, δ) F (, ) δ. The smalles sample size n saisfying he ineualiy (5) can be obained for any given,, p. Whereas, he smalles sample size n calculaion in Rosaiah and Kanam [2] only needs inpu values for σ and p. Hence, he proposed process o find he smalles sample size in his case is he same as he procedure provided by Rosaiah and Kanam [2] for he inverse Rayleigh model excep in place of σ replace by a. To save space, only he resuls of small sample sizes for =.1, =.7,.9, 1., 1.5, 2., 2.5, 3., 3.5; p =.75,.9,.95,.99; c =, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 are repored in Table

5 Accepance sampling plans for perceniles based on he inverse Rayleigh disribuion Table 1. Minimum sample sizes necessary o asser he 1 h percenile o exceed a given values,.1, wih probabiliy p* and he corresponding accepance number, c, for he inverse Rayleigh disribuion using he binomial approximaion. p c /

6 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, Table 2. Minimum sample sizes necessary o asser he 1 h percenile o exceed a given values,.1, wih probabiliy p* and he corresponding accepance number, c, for he inverse Rayleigh disribuion using he Poisson approximaion. p c /

7 Accepance sampling plans for perceniles based on he inverse Rayleigh disribuion If p= F(; δ) is small and n is large he binomial probabiliy may be approximaed by Poisson probabiliy wih parameer λ = np so ha he lef side of (5) can be wrien as: c λ e 1 p, (6) i! i = i λ where λ = n Fδ (; ). 2.2 Operaing Characerisic of he Sampling Plan(,, nc ) The operaing characerisic (OC) funcion of he sampling plan acceping a lo. I is given as: c i= n i ( i ) ( ) (,, ) nc is he probabiliy of n i Lp ( ) = p 1 p, (7) where p= F(; δ ). I should be noiced ha Fδ (; ) can be represened as a funcion of δ =. 1 Therefore, p= F( ) where d =. Using E. (7), he OC values and OC curves can be d obained for any sampling plan (,, nc ). To save space, we presen Table 3 o show he OC values for he sampling plan ( nc, = 4, ). Figure1 shows he OC curves for he sampling plan ( nc,, ) wih p =.75 for δ = 1, where c =,1,2,3,4,5,6,7,8,9, Operaing Curve Power d Figure 1. OC curves for c =,1,2,3,4,5,6,7,8,9,1, respecively under p =.9, δ = 1 based on he 1 h percenile, d = d, of inverse Rayleigh disribuion..1 17

8 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, Table 3. Operaing characerisic values of he sampling plan ( nc, = 5, / ) for a given p under inverse Rayleigh disribuion..1 /.1 p n / Producer s Risk The producer s risk is defined as he probabiliy of rejecing he lo when value of he producer s risk, sayα, we are ineresed in knowing he value of producer s risk is less han or eual o α if a sampling plan.1 (,, ) >. For a given d o ensure he nc is developed a a specified confidence level p. Thus, one needs o find he smalles value d according o E. (7) as: c i= n i ( i ) ( ) n i p 1 p 1 α, (8) 171

9 Accepance sampling plans for perceniles based on he inverse Rayleigh disribuion 1 where p= F( ), d =. d d Table 4. - Minimum raio of rue for he accepabiliy of a lo for he inverse Rayleigh disribuion and.1 producer s risk of.5. p c / 172

10 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, To save space, based on sampling plans.1 (,, ) nc esablished in Tables 1 he minimum raios of d for he accepabiliy of a lo a he producer s risk of α =.5 are presened in Table Illusraive Examples In his secion, wo examples wih real daa ses are given o illusrae he proposed accepance sampling plans. The firs daa se is of he daa given arisen in ess on endurance of deep groove ball bearings (Lawless [11], p.228). The daa are he number of million revoluions before failure for each of he 23 ball bearings in life es and hey are: 17.88, 28.92, 33., 41.52, 42.12, 45.6, 48.8, 51.84, 51.96, 54.12, 55.56, 67.8, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 15.12, 15.84, , and The second daa se is obained from Proschan [15] and represens imes beween successive failures of air condiioning (AC) euipmen in a Boeing 72 airplane and hey are as follows: 12, 21, 26, 27, 29, 29, 48, 57, 59, 7, 74, 153, 326, 386 and 52. As he confidence level is assured by his accepance sampling plan only if he lifeimes are from he inverse Rayleigh disribuion. Then, we should check if i is reasonable o admi ha he given sample comes from he Inverse Rayleigh disribuion by he goodness of fi es and model selecion crieria. The firs daa se was used by Sulan [23] o demonsrae he goodness of fi for generalized exponenial disribuion and Gupa and Kundu [8] fied for exended exponenial disribuion. However, he accepance sampling plans under he runcaed life es based on he Inverse Rayleigh disribuion for perceniles has no ye been developed. We fi he inverse Rayleigh disribuion o he wo daa ses separaely. We used he Kolmogorov-Smirnov (K-S) ess for each daa se o he fi he inverse Rayleigh model. I is observed ha for Daa Ses I and II, he K-S disances are.1291 and wih he corresponding p values are.8528 and respecively. For daa ses I and II, he chi-suare values are.352 and respecively. Therefore, i is clear ha inverse Rayleigh model fis uie well o boh he daa ses. 3.1 Example 1 Assume ha he lifeime disribuion is inverse Rayleigh disribuion and ha he experimener is ineresed o esablish he rue unknown 1 h percenile lifeime for he ball bearings o be a leas 3 million revoluions wih confidence p =.9 and he life es would be ended a 3 million revoluions, which should have led o he raio.1 = 1.. Thus, for an accepance number c =5 and he confidence level p =.9, he reuired sample size n found from Table 1 should be a leas 23. Therefore, in his case, he accepance sampling plan from runcaed life ess for he inverse Rayleigh disribuion 1h percenile should be (,, nc ) = (23, 5, 1.). Based on he ball bearings daa, he experimener mus have decided wheher o accep or rejec he lo. The lo should be acceped only if he number of iems of which lifeimes were less han or eual o he scheduled es lifeime, 3 million revoluions, was a mos 5 among he firs 23 observaions. Since here were 2 iems wih a failure ime less han or eual o 3 million revoluions in he given sample of n =23 observaions, he experimener would accep he lo, assuming he 1h percenile lifeime.1 of a leas 3 million revoluions wih a confidence level of p =.9. The 173

11 Accepance sampling plans for perceniles based on he inverse Rayleigh disribuion OC values for he accepance sampling plan (,, nc ) p =.9 under inverse Rayleigh disribuion from Table 3 is as follows: = (23,5,1.) and confidence level OC This shows ha if he rue 1 h percenile is eual o he reuired 1 h percenile (.1.1 = 1.) he producer s risk is approximaely.927 (=1-.973). The producer s risk is almos eual o zero when he rue 1 h percenile is greaer han or eual o 2. imes he specified 1 h percenile. From Table 4, he experimener could ge he values of d for differen choices of c and.1 in order o asser ha he producer s risk was less han.5. In his example, he value of.1 d should be for c = 5,.1.1 =1. and p =.9. This means he produc can have a 1 h percenile life of imes he reuired 1 h percenile lifeime in order ha under he above accepance sampling plan he produc is acceped wih probabiliy of a leas.95. Alernaively, assume ha producs have an inverse Rayleigh disribuion and consumers wish o rejec a bad lo wih probabiliy of p =.75. Wha should he rue 1 h percenile life of producs be so ha he producer s risk is.5 if he accepance sampling plan is based on an accepance number c =5 and =.7? From Table 4, we can find ha he enry for p =.75, c = 5, and.1.1 =.7 is d = Thus, he manufacurer s produc should have a 1 h percenile life a.1 leas imes he specified 1 h percenile life in order for he producs o be acceped wih probabiliy.75 under he above accepance sampling plan. Table 1 indicaes ha he number of producs reuired o be esed is n = 56 so ha he sampling plan is ( nc,, ) = (56, 5,.7). 3.2 Example 2 Suppose an experimener would like o esablish he rue unknown 1 h percenile lifeime for he daa se regarding he failure of air condiioning (AC) euipmen in a Boeing 72 airplane menioned above o be a leas 2 and he life es would be ended a 2, which should have led o he raio.1 = 1.. The goodness of fi es for hese 15 observaions were verified and showed ha inverse Rayleigh model as a reasonable goodness of fi for hese 15 observaions. Thus, wih c = 2 and p =.95, he experimener should find from Table 1 he sample size n mus be a leas 15 and he sampling plan o be ( nc,,.1) = (15, 2, 1.). Since here is a one iem wih a failure ime less han 2 in he given sample of n = 15 observaions, he experimener would accep he lo, assuming he 1 h percenile lifeime of a leas 2 wih a confidence.1 level of p =

12 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, Fuure Work Consrucion of hese sample plans wih reference o populaion perceniles is in progress by he auhors in oher ramificaions also such as double, seuenial, wo-sage and inerval censored group samples. 5. Discussion and Conclusions The sampling plans based on he inverse Rayleigh populaion mean developed by Rosaiah and Kanam [2] o he inverse Rayleigh models. I shows ha he minimum sample sizes are smaller han hose repored in Tables 1 and 2 of his aricle for he 1 h percenile for boh binomial and Poisson approximaion. Here, δ = for he sampling plans based on 1 h percenile is.1 replaced by δ = µ wih µ as a specific populaion mean for he accepance plans based on he inverse Rayleigh populaion mean. Therefore, he accepance sampling plans based on he inverse Rayleigh populaion mean could have less chance o repor a failure han he accepance sampling plans based on 1 h percenile. The accepance sampling plans based on populaion mean could accep he lo of bad ualiy of he 1 h perceniles. The minimum sample sizes are repored in Table 1 of his aricle for he 1 h perceniles are compared wih he minimum sample sizes are repored in Table 1 of Lio e al. [12]. I shows ha he minimum sample sizes using inverse Rayleigh populaion are smaller han hose repored in Tables 1 of Lio e al. [12] for Birnbaum-Saunders populaion for he 1 h percenile when δ 1. whereas, he minimum sample sizes using inverse Rayleigh populaion are larger han hose repored in Tables 1 of Lio e al. [12] for Birnbaum-Saunders populaion for he 1 h percenile when δ > 1.. This aricle has derived he accepance sampling plans based on he inverse Rayleigh perceniles when he life es is runcaed a a pre-fixed ime. The procedure is provided o consruc he proposed sampling plans for he perceniles of he inverse Rayleigh disribuion. To ensure ha he life ualiy of producs exceeds a specified one in erms of he life percenile, he accepance sampling plans based on perceniles should be used. Some useful ables are provided and applied o esablish accepance sampling plans for wo examples. References [1]. Baklizi, A. (23). Accepance sampling based on runcaed life ess in he Pareo disribuion of he second kind. Advances and Applicaions in Saisics, 3(1), [2]. Balakrishnan, N., Leiva, V., Lopez, J. (27). Accepance sampling plans from runcaed life ess based on he generalized Birnbaum-Saunders disribuion. Communicaions in Saisics-Simulaion and Compuaion, 36, [3]. Epsein, B. (1954). Truncaed life ess in he exponenial case. Annals of Mahemaical Saisics, 25, [4]. Ferig, F.W., Mann, N.R. (198). Life-es sampling plans for wo-parameer Weibull populaions. Technomerics, 22(2),

13 Accepance sampling plans for perceniles based on he inverse Rayleigh disribuion [5]. Goode, H.P., Kao, J.H.K. (1961). Sampling plans based on he Weibull disribuion. Proceedings of Sevenh Naional Symposium on Reliabiliy and Qualiy Conrol, Philadelphia, pp [6]. Gupa, S.S. (1962). Life es sampling plans for normal and lognormal disribuion. Technomerics, 4, [7]. Gupa, S.S., Groll, P. A. (1961). Gamma disribuion in accepance sampling based on life ess. Journal of he American Saisical Associaion, 56, [8]. Gupa, R.D., Kundu, D. (21). Exponeniaed Exponenial family: an alernaive o Gamma and Weibull disribuions. Biomerical Journal, 43(1), [9]. Kanam, R.R.L., Rosaiah, K. (1998). Half logisic disribuion in accepance sampling based on life ess. IAPQR Transacions, 23, [1]. Kanam, R.R.L., Rosaiah, K., Rao, G.S. (21). Accepance sampling based on life ess: Log-logisic model. Journal of Applied Saisics, 28, [11]. Lawless. J.F. (1982). Saisical Models and Mehods for Lifeime Daa. New York: John Wiley & Sons. [12]. Lio, Y.L., Tsai, T.-R., Wu, S.-J. (21). Accepance sampling plans from runcaed life ess based on he Birnbaum-Saunders disribuion for perceniles. Communicaions in Saisics -Simulaion and Compuaion, 39, [13]. Marshall, A.W., Olkin, I. (27). Life Disribuions-Srucure of Nonparameric, Semiparameric, and Parameric Families, New York: Springer. [14]. Mukhergee, S.P., Saran L.K. (1984). Bivariae Inverse Rayleigh Disribuion in Reliabiliy Sudies. Journal of he Indian Saisical Associaion, 22, [15]. Proschan, F. (1963). Theoreical explanaion of observed decreasing failure rae. Technomerics, 5, [16]. Rao, G.S., Kanam, R.R.L. (21). Accepance sampling plans from runcaed life ess based on he log-logisic disribuion for perceniles. Economic Qualiy Conrol, 25(2), [17]. Rao, G.S., Ghiany, M.E., Kanam, R.R.L. (28). Accepance sampling plans for Marshall-Olkin exended Lomax disribuion. Inernaional Journal of Applied Mahemaics, 21(2), [18]. Rao, G.S., Ghiany, M.E., Kanam, R.R.L. (29a). Marshall-Olkin exended Lomax disribuion: an economic reliabiliy es plan. Inernaional Journal of Applied Mahemaics, 22(1), [19]. Rao, G.S., Ghiany, M.E., Kanam, R.R.L. (29b). Reliabiliy Tes Plans for Marshall- Olkin exended exponenial disribuion. Applied Mahemaical Sciences, 3, No. 55, [2]. Rosaiah, K., Kanam, R.R.L. (25). Accepance sampling based on he inverse Rayleigh disribuion. Economic Qualiy Conrol, 2, [21]. Rosaiah, K., Kanam, R.R.L., Sanosh Kumar, Ch. (26). Reliabiliy es plans for exponeniaed log-logisic disribuion. Economic Qualiy Conrol, 21(2), [22]. Sobel, M., Tischendrof, J. A. (1959). Accepance sampling wih sew life es objecive. Proceedings of Fifh Naional Symposium on Reliabiliy and Qualiy Conrol, Philadelphia, pp [23]. Sulan, K.S. (27). Order saisics from he generalized exponenial disribuion and applicaions. Communicaions in Saisics-Theory and Mehods, 36,

14 Rao, G.S., Kanam, R.R.L., Rosaiah, K., Reddy, J.P. (212). Elecron. J. App. Sa. Anal., Vol. 5, Issue 2, [24]. Tsai, T.-R., Wu, S.-J. (26). Accepance sampling based on runcaed life ess for generalized Rayleigh disribuion. Journal of Applied Saisics, 33, [25]. Wu, C.-J., Tsai, T.-R. (25). Accepance sampling plans for Birnbaum-Saunders disribuion under runcaed life ess. Inernaional Journal of Reliabiliy, Qualiy and Safey Engineering, 12, This paper is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuzione - Non commerciale - Non opere derivae 3. Ialia License. 177