Design and Development of Reliability Sampling Plans for Intermittent Testing Based on Type-I Censoring

Transcription

1 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// Dsig ad Dvlop o Rliabiliy Saplig las o Ii Tsig Basd o Typ-I Csoig D. Raya ad S. Dvaaul 2 Assisa osso, Dpa o Saisis,.S.G. Collg o As ad Si, Coibao, Idia & a pa i sah shola a R & D C, Bhaahia Uivsiy, Idia. Oid Id: Assisa osso, Dpa o Saisis, Gov As Collg, Coibao, Idia. Oid Id: Absa I his ail liabiliy saplig plas a dvlopd o ii s bahs basd o yp-i soig daa. I as o ii sig h ai pobl ad by h qualiy ool paiio is h poss avag, whih ay dvia du o spoadi poduios. H, o ao assu a liabiliy o h podus whil disposig h los. To ovo his diiuly, iiu liabiliy as a guaa is obaid bo apa o h lo o s bahs hough h pobabiliy o jio uio. I his pap, liabiliy saplig plas a did wih a assua o high liabiliy o h podus o a wo paa Elag disibuio. Ths liabiliy saplig plas a oulad ad dsigd i suh a way ha hy a o liabl wih sp o h apa o h podus o bahs. Dsigig ad siulaio podus a giv o h osuio o ssay abls o ailia asy slio ad sig o podus. Kywods: Rliabiliy, Elag disibuio, Typ-I-Csoig, Ii poduio, Opaig Raio. INTRODUCTION May auhos hav dvlopd Rliabiliy Saplig las whih dpd o h sapl siz ad h apa osa wihou a guaa o h liabiliy o h podus o ii sig los o bahs. Cusos ay p h liabiliy o h podu whil apig h bahs o los a sig. I h liabiliy o h podu is aquid duig sig, h h will b a sooh sailig o los o b apd. Moov, i h poduio is o oiuous, h h ay b vaiaios i h poss avag ad h ub o ailus ay b o i h los ad h uso ay o b saisid wih suh podus. Thus, o suh a ii poss, iiu liabiliy should b aquid bo h apa o h bah o poss. Tho Rliabiliy Saplig las idd hough iiu liabiliy has b dvlopd. Fig ad Ma [] hav disussd li s saplig plas o wo paa Wibull populaios. Kaa.al [2] has dvlopd saplig plas basd o log logisi odl. Fag [3] psd h sudy o h hyp-elag disibuio odl ad is appliaios i wilss woks ad obil opuig syss. Th wo paa Elag disibuio povids libiliy i h hoi o h shap ad sal paa, a wid vaiy o lii daa i qui adqualy o i. Wh h gaa disibuio has a igal paa α, i is kow o b Elag disibuio. To aqui h aiu liabiliy (, h pobabiliy o jio uio i s o liabiliy is subjd o diiaio wi ad quad o zo. Th quaio o oal o h ag is obaid ad is usd as a asu o shapss (. Fially h opaig liabiliy a is did whih is h vial paa o di h paas o h liabiliy saplig plas o h ii poduio poss. FORMULATION OF THE RELIABILITY SAMLING LANS Th iiu sapl siz ssay o su h quid liabiliy o h podus a obaid ud h assupio ha h li i vaiabl ollows wo paa Elag disibuio. Th uulaiv disibuio uio F( ad pobabiliy dsiy uio ( o a wo paa Elag disibuio spivly a giv by F k (, k, X ( X X k k ( K k,, Wh k is h shap paa ad λ is h sal paa. Sush.K.K ad Laha. M [4] hav dvlopd Baysia sigl saplig plas o a gaa pio disibuio. Dvaaul.S [5, 6] dsigd ad dvlopd aiu aquid liabiliy (2 539

2 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// saplig plas. Abdlkad [7] opud os o od saisi o oidially disibud Elag vaiabls. Dvaaul.S ad Jy Joy.V [8] hav dvlopd liabiliy saplig plas basd o iiu agl hiqu. Willo ad Li [9] psd a viw o aalyial ad opuaioal popis o h id Elag disibuio i h o o isk aalysis. Muhad Asla ad Chi- Hyuk Ju [] dsigd i uad apa saplig plas by usig wo-poi appoah. Muhad Asla.al [] dvlopd a goup saplig pla basd o uad li ss o gaa disibuio. Gupa ad Goll [2] sudid gaa disibuio i apa saplig basd o li ss. Soudaaja [3] has dvlopd sigl saplig plas by aibus ispio hough aiu allowabl p divs. Lio.al [4] hav dvlopd apa saplig plas o h Bibau-Sauds disibuio o pils wh h li s is uad a a p-spiid i. MEASURES OF RELIABILITY SAMLING LANS L b h liabiliy o ah opo i h lo. L do h ub o uis o b sd i i piod. L S, S, do h ub o sussul uis duig h i piod. Thus is h pobabiliy ha ay i sd duig piod will sul i a suss, o i is liabl owads suss suh ha = -F(. I saplig pla liau, h pobabiliy o apa uio o h poss is giv as a p p p wh p is h poss avag o aio div o h ioig lo o bah ad is h ub o ailu uis i sig. k wh p,( k,, (3 (4 Wh S = -, is h iiu ub o suvival uis quid i h sapl o sig bo apa o h bah h h pobabiliy o jio uio i is h liabiliy is giv by p p k Wh p,( k,, Ad S = - Equaio (5 a b wi as, p p (5 (6 (7 Foulaio: L duig sig. = sapl siz. = Li i ado vaiabl. S = Sussul uis duig sig. S* = Miiu Nub o Sussul uis quid ALGORITHM FOR SENTENCING A BATCH AFTER TYE I CENSORING Sp : Daw a ado sapl o siz ad pu h io li s o spiid i (yp I soig Sp 2: Cou h ub o spis sussul i h s. L i b S. Sp 3: I S S * h ap h bah o lo. Sp 4: I S < S *, h j h bah. DESIGNING THE RELIABILITY SAMLING LANS FOR INTERMITTENT TESTING Th pobabiliy o jio uio i is liabiliy wih yp II pobabiliis is did as ( ( I C is h apa ub ad is h sapl siz h S * = is h iiu ub o suvival quid duig h sig o apa o h bah. Th ag o h pobabiliy o jio uv a h dlio poi ouhs h -ais a Equaio o ag passig hough, ( ( wih slop ( ( is giv by h quaio (8 532

3 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// 532 ( ( ( ( (9 Th oo o h abov quaio is obaid by solvig ( ( ( ( ( A h ilio poi, (- = ( ho, h poi a whih h ag ouhs h -ais is giv by (2 Th opaig liabiliy a is did as z (3 z (4 z (5 This z is puly a uio o havig = S * /, as h liabiliy sadad. Th paa o b lad is h asu o shapss aguig siila o dsigig podu o Soudaaaja [2]. H h poi whih ouhs -ais o h ag a h dlio poi o pobabiliy o jio uv a b usd as asu o shapss o ispio. A lo o poss o qualiy wih liabiliy will b apd ad lss ha will b jd. By usig quaio (5, abl ( is osud so ha h iiu ub o sussul is quid i yp I soig a b did. O h opaig liabiliy a ad h paa λ a kow, o a di iiu valu o S *, h sussul uis i h s ad h sapl siz. Siilaly, i is h aiu allowabl ailu duig yp I soig sig ad is h sapl siz h h ub o sussul is i h s is S * =, h h oal o h pobabiliy o jio uv a h dlio poi ouhs h -ais is giv by Equaio o h oal passig hough ( (, wih slop ( ( is giv by h quaio ( ( ( ( (6 Th oo o h abov quaio is obaid by quaig ( ( ( ( (7 ( ( ( ( (8 A h ilio poi, (- = ho, h poi a whih h oal ouhs h -ais is giv by (9 L h opaig liabiliy a b Z (2 Z (2

4 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// Z (22 By usig quaio (22, abl (2 is osud so ha h iiu ub o sussul is quid i yp I soig is did. O h opaig liabiliy a o oal poi ad h paa λ is kow o a di h valu o S * ad sapl siz. Diiaig quaio (8 wih sp o wi ad quaig o zo, w g d d ( 2 d ( (23 ( ( ( d 2 Th liabiliy ( aquid i h lo is obaid o quaio (24, (24 s (25 (26 pos ad dous syboli alulaios usig h Syboli Mah Toolbo. Sp 2: I h MuAD obook yp h ollowig sip: Sp3: sas:: lagrado (2,.5(; Wh ud h all i h pvious sp, ado ubs wih shap paa = 2 ad sal paa =.5 a gad Sp 4: Alog wih h podu i sp 3, pu $ k=..2. So ha h sip will b sas::lagrado (2,.5 ( $ k=..2; This us a squ o 2 ado ubs ah o whih is alld oly o. Sp 5: Th siulad daa a abulad as λ valus i h abls ad uh alulaios a ad o di h liabiliy saplig plas. Tabl : Valus o h Opaig Ra (Z o h aiu allowabl ailu (C a ag poi Th sapl siz a b did usig quaio (26 ad h sa is abulad i abl (3. Ipaio: Duig li sig wih yp-i soig i i h sapl o uis h liabiliy, h h bah is apd ohwis i is jd. SIMULATION ROCEDURE Th paas o h ailu disibuio a k ad λ i as o Elag disibuio. H h uulaiv ailu a valus a siulad usig MATLAB poga o h kow Sal ad Shap paas o h Elag disibuio. Twy ado valus w gad ad h h ospodig F( valus a obaid usig quaio (. Tha is obaid by usig quaio (4. Fo h shap paa k=2, h di valus o h sapl siz a obaid. H abl (3 is osud o asy slio o liabiliy saplig plas. By usig quaio (22, abl (2 is osud o di h opaig liabiliy a a h oal poi. Sp : I h MATLAB dio widow, go o h AS u ad sl h MuAD obook opio. (A MuAD obook 5322

5 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// Tabl 2: Opaig liabiliy a o h aiu allowabl ailu (C a oal poi Tabl 4: Rliabiliy valus o shap paa k=2, C=3, = ad h ospodig pobabiliy o jio ( I abl (2 o ah valu o C, Z * is alulad usig quaio (22. Z * is h opaig liabiliy aio a oal whih is obaid by akig h aio o ad. Tabl 3: Valus o h sapl siz ( o shap paa k=2, giv h liabiliy ad aiu allowabl ailus C. Fo abl (4 i is oud ha whv liabiliy o h lo dass h pobabiliy o jio is o. I oh wods, wh h liabiliy o h podu dass h h pobabiliy o jio uv shows a ias i h pobabiliy o jio o a paiula bah o lo. A Cuv is daw by akig Rliabiliy o h lo i h -ais ad h pobabiliy o jio i h y-ais by usig h ospodig valus i abl (4. Th poi a whih h ag ouhs h -ais is akd as ( ad h poi a whih h oal ouhs h -ais is akd as ( i his uv. Th poi is h Rliabiliy obaid whih is kow o b aiu valu. H o h ii poduio poss aiu liabiliy is obaid a yp-i soig s ad is a saisaoy o o a uso. This guaas h uso h quid liabiliy bo apa o h los o bahs. 5323

6 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// as suh h osu is also bid. I is vid o h uv ha as h liabiliy o h lo iass, h pobabiliy o jio dass. Tho o a ii poss, h w liabiliy saplig plas will b a appopia o. ILLUSTRATION Figu : obabiliy o jio Cuv Th opaig liabiliy a pd by h uso is Z * =.762 wih yp I soig ad h s i is 44.5 hous. Di Rliabiliy Saplig las i h poss shows ii poduio wih Elagia sal 2. Soluio: I is giv ha h opaig liabiliy a is Z*=.762 ad s duaio is 44.5 hous. Th sal paa λ = 2, suh ha λ = 89. O a id h paas o h liabiliy saplig plas o abl (2 ad abl (3. Z *, h opaig liabiliy a ad h avag s i λ a giv, o abl (2, i a b obsvd ha h aiu allowabl ailu ospodig o hs valus is C=4. Si o h sa opo h avag s i is quod as 89 hous, h o abl (3, o a di h quid sapl siz. Wh λ=.89, h paas = 4 h sapl siz is =8. Tho, h iiu ub o sussul uis quid o ap h lo is S=4 wih guaad liabiliy = 77.6% Wh Z * ad λ a giv, o a di h oh paas suh as, ad S * o h abls ad h h suiabl liabiliy saplig plas a did. Also o abl ( h osu is povidd wih h guaad liabiliy bo h apa o h lo. So wih his ioaio h uso is saisid wih yp I soig o ii sig o los. CONCLUSION I his ail, Rliabiliy Saplig las o ii poduio poss has b dvlopd by assuig wo paa Elag disibuios. I is obsvd ha whv h liabiliy valu dass h sapl siz iass, his pssuis h podu o aiai liabiliy o podus ad REFERENCES [] Abdlkad, Y.H., 23, Copuig h os o od saisis o o idially disibud Elag vaiabls. Saisial aps, 45, pp [2] Dvaaul, S., 22, Cai Sudis laig o Mid Saplig plas ad Rliabiliy basd Saplig las, h.d., Thsis, Dpa o Saisis, BhaahiaUivsiy, Coibao, Tail Nadu, Idia. [3] Dvaaul, S., 23, Maiu Aquid Rliabiliy Saplig (MARS pla, Fa Eas Joual o Thoial Saisis,, No., pp [4] Dvaaul, S ad Jy Joy, V., 23, Dsigig ad Slio o Rliabiliy Saplig las Basd o Miiu Agl Thiqu, Iaioal Joual o Mahais ad Copuaio, 2, No.3, pp [5] Fag,Y., 2, Hyp-Elag disibuio odl ad is appliaio i wilss obil woks. Wilss Nwoks, Vol.7, pp [6] Fig, K, W ad Ma, N, R., 98, Li Ts Saplig las o Two aa Wibull opulaios, Thois, Vol.22, pp [7] Gupa, S,S ad Goll, A., 96, Gaa Disibuio i Apa Saplig basd o Li Tss, Joual o Aia Saisial Assoiaio, Vol.56, pp [8] Kaa, R, R, L, Rosaiah, K, ad Rao G.S., 2, Apa Saplig Basd o Li Tss: Log- Logisi Modls, Joual o Applid Saisis, Vol. 28, pp [9] Lio, Y.L, Tzog-Ru Tsai ad Shuo-Jy Wu., 2, Apa Saplig las o Tuad Li Tss Basd o h Bibau-Sauds Disibuio o ils, Couiaios i Saisis Siulaio ad Copuaio, 39:, [] Muhad Asla, Chi-Hyuk Ju., 23, Dsigig o Ti Tuad Apa Saplig las by Usig Two-oi Appoah, Eloi Joual o Applid Saisial Aalysis Vol.6, Issu, 8-3. [] Muhad Asla, Chi-Hyuk Ju ad Ahad. M., 29, A goup saplig pla basd o uad li ss o gaa disibuio, akisa Joual o Saisis, Vol. 25:

7 Iaioal Joual o Applid Egiig Rsah ISSN Volu 2, Nub 24 (27 pp Rsah Idia ubliaios. hp:// [2] Soudaaja, V., 975, Maiu Allowabl Div (MAD Sigl Saplig Ispio by Aibus la, Joual o Qualiy Thology, Vol.7, No.4, pp [3] Sush, K.K. ad Laha, M., 2, Baysia Sigl Saplig las o a Gaa io, Eooi Qualiy Cool, Vol.6, No., pp [4] Willo, G. E. ad Li, X. S., 2, Risk odllig wih h id Elag disibuio. Applid Sohasi Modls i Busiss ad Idusy, 27,