ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 A Time Trunced Improved Group Smpling Plns for Ryleigh nd og - ogisic Disribuions P.Kvipriy, A.R. Sudmni Rmswmy Reserch Scholr, Deprmen of Mhemics, Avinshilingm Universiy Coimbore, Tmil Ndu, Indi Associe Professor, Deprmen of Mhemics, Avinshilingm Universiy Coimbore, Tmil Ndu, Indi ABSTRACT : In his pper improved single nd double group smpling plns bsed on ol number of filures from he whole groups when he life ime of he iems follows Ryleigh or og-ogisic is proposed. The design prmeers of he plns re deermined such h hey sisfy boh producer s risk nd consumer s risk simulneously. The resuls re nlyzed wih he help of bles nd exmples. KEYWORDS: Group Accepnce smpling pln, Ryleigh, og-ogisic, Accepble relibiliy level AR, nd o olernce relibiliy leveltr. I. INTRODUCTION In compeiive economy, goods snd in he mrke if hey re of good quliy. A consumer wns producs of good quliy resonble nd ffordble prices. Here he quliy cn be defined in wo differen wys. In one sense, goods re sid o be of good quliy if hey sisfy he consumer. In noher sense, goods re sid o be of good quliy if hey mee he expeced funcionl use. Exmple for second concep, bll-berings wihin he specificion limis is sid o be in conrol producion process. Every uni of producion is esed for he sndrds specified. The unis which do no mee he specificions re rejeced. The rejeced unis re sid o be of bd quliy; s such hey re no pu o use. Sisicl quliy conrol is he procedure for he conrol of quliy by he pplicion of he heory of probbiliy o he resuls of inspecion of smples of he populion. Smpling plns re used in he re of quliy nd relibiliy nlysis. When he quliy of produc is reled o is lifeime, i is clled s life es. For life es, vribles smpling pln nd ribues smpling pln re vilble. Accepnce smpling plns for producs is one of he specs of quliy ssurnce in sisicl quliy conrol. Usully single iem is esed in eser. In prcice muliple iems re ccommoded in ype of esers simulneously hen i is clled group smpling pln. I reduces he cos nd ime of he experimen. II. REATED WORK Aslm nd Jun 9 hve designed group ccepnce smpling pln for runced life es when he life ime of n iem follows eiher inverse ryleigh or log- logisic. Srinivs Ro 9 hve designed group ccepnce smpling plns for he life imes following generlized exponenil nd in he hs sudied he sme smpling pln bsed on runced life ess for he mrshll-olkin exended lomx. Aslm, Jun nd Munir Ahmd 9 hve sudied group smpling pln bsed on runced life es for gmm. In Muhmmd Aslm, Chi-Hyuck Jun, Hyeseon ee, Munir Ahmd nd Mujhid Rsool hve designed improved group smpling plns bsed on ime runced life ess for Weibull. Here in his pper improved single nd double group smpling plns bsed on ol number of filures from he whole groups when he life ime of he iems follows Ryleigh or og-ogisic is proposed. Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 74
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 III. PROCEDURE FOR IMPROVED SINGE GROUP SAMPING PAN Here single group smpling pln bsed on he ol number of filures is given. As per Aslm, Jun, ee, Ahmd nd Rsool he procedure of his pln is s follows: A rndom smple of size n is ken from lo nd r iems re lloced o ech of groups g or esers so h n = rg; Before he erminion ime, r iems re pu on he es. 3 If he ol number of filures from g groups is smller hn or equl o ccepnce number c, lo is cceped. 4 If he ol number of filures from g groups is greer hn c before he erminion ime s soon s runce he es nd rejec he lo. I is lso like generlizion of ordinry single ccepnce smpling pln. The prmeers g nd c re seleced such h he producer s nd consumer s risks boh re sisfied ccording o given vlues of group sizes nd rue quliy levels. The probbiliy of ccepnce is lso clculed. The probbiliy of lo ccepnce is given by p rg c i rg i p p i i Where p is he probbiliy of n iem in group fils before he erminion ime. In single group smpling pln, AR is ken s p nd TR s p. The prmeers g nd c re seleced such h he following inequliies re sisfied. c rg i rg i p p i i p nd c rg i p p p i i rg i 3 I ssumed h α =.5 nd β =.. The design prmeers re clculed when he life ime of he iems follows Ryleigh nd og- ogisic s for r=5 nd r= nd re presened in ble, ble, ble 3 nd ble 4 respecively. Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 75
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 76 IV. DESIGN OF SINGE GROUP SAMPING PAN RAYEIGH DISTRIBUTION Ryleigh 88 derived he Ryleigh o he problems in he field of cousics. Then he Ryleigh hs wide pplicions in communicion engineering nd lso in life esing of elecro vcuum devices. The cumulive funcion cdf is given s, ; e F 4 Then he unrelibiliy ime subsiued in he bove equion ; e F p Where is es erminion ime nd hen is ken s µ here is consn nd µ is he specified life. Hence σ is ken s µµ =r. ; e F p ; e F p Therefore e p 5 Where, p is he probbiliy of n iem fils before erminion ime. OG-OGISTIC DISTRIBUTION The cumulive funcion cdf given of log-logisic is given s,, F 6 I is ssumed h λ= nd he filure probbiliy of he iem is given s
ISSNOnline : 39-8753 ISSN Prin : 347-67 p F, where µ =.578 nd An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6.578 p.578 7 Now r is AR s men rio producer s risk nd r is TR consumer s risk. The design prmeers re seleced such h hey sisfy he following inequliies. p r p r 8 9 The design prmeer vlues nd probbiliy ccepnce re deermined for proposed single group smpling pln when he life ime of iems follows Ryleigh nd og-ogisic s. Tble Proposed Single Group Smpling Pln for Ryleigh Disribuion r=5 β µµ = r g c n=r*g P =.3 =.4 =.5 =.6 =.7 =.8 =.9 =..5..5. 5...9999.9993.9968.986.97-4 5...9999.9996.999.9979.996.999 6 5.....9999.9998.9996.9993 8 5..9998.9996.999.9986.9976.9963.9944 5..9999.9998.9997.9994.999.9984.9976 3 7 5....9998.9983.997.97-4 3...9999.9997.999.9979.995.9899 6 5.....9999.9998.9996.9993 8 5.......9999.9999 5........ 4 9....9999.999.994.9748-4 3 5 5......9998.9995.9984 6 3......9999.9997.9994 8 5.......9999.9999 5..9999.9998.9997.9994.999.9984.9976 5 5....999.9996.996.9787-4 5........9999 6 3 5........ 8 5.......9999.9999 5..9999.9998.9997.9994.999.9984.9976 Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 77
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 By fixing he es erminion ime muliplier =.3,.4,.5,.6,.7,.8,.9 nd., consumer s risk =.5,.,.5 nd. nd men rios r s,4,6,8 nd, he design prmeers nd probbiliy of ccepnce re deermined by vrying he number of iems in group sy r=5, r= nd presened in ble, ble, ble 3 nd ble 4 respecively. Hyphens - represens he vlue of probbiliy of ccepnce no sisfying he condiion. Tble Proposed Single Group Smpling Pln for Ryleigh Disribuion r= β µµ = r g c n=r*g P =.3 =.4 =.5 =.6 =.7 =.8 =.9 =..5..5. 3 3 3.....9998.9977.983-4 7.......9999.9997 6 4.....9999.9998.9995.9987 8 3........9999.....9999.9999.9998.9996 3 3 3.....9998.9974.983-4 6......9999.9995.9984 6 3......9999.9997.9994 8....9999.9998.9996.999.9985.....9999.9999.9998.9996......9997.9979.9897 4 9........ 6 5........ 8 4........ 3........ 9....9999.999.994.9748-4 6........ 6 4........ 8 4........ 3........ From he observion of proposed single group smpling pln for Ryleigh, i is cler h he probbiliy of ccepnce vlue is incresed for r= hn r=5. Hyphens - represens he vlue of probbiliy of ccepnce no sisfying he condiion. Tble 3 Proposed Single Group Smpling Pln for og-ogisic Disribuion r=5 β µµ = r g c n=r*g P =.3 =.4 =.5 =.6 =.7 -.8 =.9 =..5..5 5..9999.9993.996.9864.964.936-4 5..9999.9995.9987.9969.9937.9885.987 6 5....9999.9997.9993.9987.9976 8 5.9999.9996.999.998.9967.9945.994.987 5..9998.9996.999.9986.9977.9963.9945 3 7 5...9997.9977.989.964.99-4 3..9999.9997.9988.9964.99.986.9657 6 5....9999.9997.9993.9987.9976 8 5.....9999.9999.9997.9995 5.......9999.9999 4 9...9999.9988.999.967 - - 4 3 5 5....9999.9997.9987.996.993 6 3....9999.9998.9995.9988.9974 8 5.....9999.9999.9997.9995 5..9998.9996.999.9986.9977.9963.9945 Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 78
. ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 5 5....9994.994.973 - - 4 5......9999.9997.9993 6 3 5........9999 8 5.....9999.9999.9997.9995 5..9998.9996.999.9986.9977.9963.9945 Tble 3 nd ble 4 follows og-ogisic for proposed single group smpling pln for r=5 nd r=, he probbiliy of ccepnce vlue for r=5 is minimum compred wih r=. Hyphens - represens he vlue of probbiliy of ccepnce no sisfying he condiion. Tble 4 Proposed Single Group Smpling Pln for og-ogisic Disribuion r= β µµ = r g c n=r*g P =.3 =.4 =.5 =.6 =.7 =.8 =.9 =..5..5. 3 3 3....9997.996.9753 - - 4 7......9998.999.997 6 4....9999.9996.9988.9968.995 8 3......9999.9998.9997....9999.9998.9996.999.9985 3 3 3....9997.996.9753 - - 4 6.....9997.9987.9957.988 6 3....9999.9998.9995.9988.9974 8...9999.9997.9993.9985.997.995....9999.9998.9996.999.9985.....9995.997.987.967 4 9........9999 6 5........ 8 4........ 3........9999 9...9999.9988.999.967 - - 4 6........9999 6 4.......9999.9998 8 4........ 3........9999 From he bove bles one cn observe h s men rio r increses here is decrese in number of groups g nd ccepnce number c. A he sme ime here is increse in probbiliy of ccepnce when es erminion ime muliplier decreses. Hyphens - represens he vlue of probbiliy of ccepnce no sisfying he condiion. EXAMPE Suppose h n experimener wns o esblish h he life ime of he elecricl devices produced in he fcory ensures h he rue unknown men life is les hrs when he men rio is wih β =.5 nd es erminion ime muliplier =.8. Following re he resuls obined when he life ime of he iem follows differen life ime s nmely Ryleigh nd og-ogisic. Ryleigh Disribuion e he followed be Ryleigh. From he ble number of groups g=, ccepnce number c=5 nd number of iems in ech group is r=5 nd he smple size n=r*g= for he bove exmple. The lo is cceped if he ol number of filures from groups re no more hn 5. Probbiliy of ccepnce for improved single group smpling pln from ble is.994. In his cse one cn noe h he probbiliy of ccepnce is.9998,.9999,. nd.999 when he rio of men rio 4,6,8 nd respecively. Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 79
ISSNOnline : 39-8753 ISSN Prin : 347-67 og-ogisic Disribuion An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 e he followed be og-ogisic. From ble 3 he number of groups is, ccepnce number c= 5 nd number of iems in ech group is r=5 nd he smple size is n=r*g= for he bove exmple. If he ol number of filures from groups re no more hn 5 mens he lo is cceped. Probbiliy of ccepnce for improved single group smpling pln from ble 3 is.967. In his cse one cn noe h he probbiliy of ccepnce is.9987,.9995,.9999 nd.9977, when he men rio 4, 6, 8 nd respecively. Figure : Ryleigh nd og-ogisic s for improved single group smpling pln From he figure compres he probbiliy of ccepnce when he life ime of iem follows Ryleigh nd og-ogisic s for differen men rios. From he bove figure one cn conclude h he probbiliy of ccepnce for og-ogisic is less hn he probbiliy of ccepnce for Ryleigh only for men rio r =. Bu for he oher vlues, he probbiliy of ccepnce for boh he re pproximely equl. FIRST STAGE V. OPERATING PROCEDURE OF THE IMPROVED DOUBE GROUP SAMPING PAN The following is he procedure presened by Aslm M, Jun C-H, ee H, Ahmd M nd Rsool M. i Drw he firs rndom smple of size n from lo; ii Alloce r iems o ech of g groups or esers so h n =rg. iii Pu r iems re pu on he es before erminion ime.; iv v Accep he lo if he ol number of filures from g groups is smller hn or equl o c ; nd Trunce he es nd rejec he lo s soon s he ol number of filures is greer hn or equl o c r >c before. Oherwise go o he second sge. SECOND STAGE i Drw he second rndom smple of size n from lo; ii Alloce r iems o ech of g groups so h n = rg ; iii Pu r iems on es before he erminion ime ; Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 73
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 iv Accep he lo if ol number of filures from g nd g groups is smller hn or equl o c c. Oherwise. v Trunce he es nd rejec he lo. Therefore he bove given double group smpling pln he five prmeers re g,g,c, c r nd c. This pln is differen from ordinry double ccepnce smpling pln. When c r =c +, hen x is denoed s ol number of filures from g groups nd prmeers n nd p use o follow binomil. We know h p is filure probbiliy erminion ime. Now he lo ccepnce probbiliy firs sge is given by P n c j n j P X c p p j j Then he lo rejecion probbiliy firs sge s P r n j n j p p jcr n j Now he lo is cceped from he second sge if he decision no ken from he firs sge nd he ol number of filures from g nd g groups is denoed by X is smller hn or equl o c. P P c X c r, X X c c r x p p c n x n x x xc i n i p i p n i Hence he probbiliy of lo ccepnce is given by p P P 3 VI. DESIGN OF PROPOSED DOUBE GROUP SAMPING PAN Aslm M, Jun C.H, ee H, Ahmd M nd Rsool M designed he proposed double group smpling pln nd pplied Weibull. Fixing hese prmeers g, g, c, c, c r, consumers risk nd men rios, he probbiliy of ccepnce nd ASN vlue is deermined. e he filure probbiliies be p nd p corresponding o consumer s risk nd producer s risk respecively. Consider ASN under TR, p is given by ASN p rg rg P P r Then he opimizion problem is considered s MinimizeASN p rg rg P P r Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 73
Subjec o ISSNOnline : 39-8753 ISSN Prin : 347-67 nd p p An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 In his pper for he bove double group smpling pln he uhor pplies Ryleigh nd og-ogisic s. Tble 5 Proposed Double Group Smpling Pln for Ryleigh Disribuion=.3 r=5 r= β µµ = r c r c g g ASN p c r c g g ASN p.5. 3 3..9946 4 5 3.3.995 4.8.9964.8.9964 6 5..9995.3.9993 8 5..9998..9998 5 4 3 9..9998 5 6 3 4.9.9965 4 3.4.9999 3.8.9967 6..9993.3.9993 8 5....9998.5 5 5 4 5..9995 5 7 3 3 49.4.9965 4 3 3 3 6..9997 3.6.9988 6 3 3.3. 3.7.9999 8 3 5.. 3.4.999. 6 7 5 3 33.7.9998 7 8 5 3 74.8.9984 4 3 3 3 6.8.9988 4 4 3 34.6.9999 6.3.9979 3 3 3..9997 8..9995.4.999 The prmeer vlues men rio =, 4, 6, 8, consumer s risk =.5,.,.5,., es erminion ime muliplier =.3 nd c = re fixed for ble 5 nd ble 6 respecively where he number of iems in group is vried s r=5 nd r= o deermine vlue for he probbiliy of ccepnce nd verge smple number using Ryleigh nd og-ogisic s. Tble 6 Proposed Double Group Smpling Pln for og-ogisic Disribuion=.3 r=5 r= β µµ = r c r c g g ASN p c r c g g ASN p 3 3 4..985 4 5 3.8.979.5 4..99..99 6 5..9987.5.9983 8 5..9996.3.9995 5 4 3.5.9985 5 6 3 45.6.986. 4 3.6.9995 3..99 6..9983.5.9983 8 5..9999.3.9995.5 5 5 4 6.5.9967 5 7 3 3 53.4.983 4 3 3 3 6.8.9989 3.3.9957 6 3 3.5. 3..9996 Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 73
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 8 3 5.. 3.6.9977. 6 7 5 3 36..9984 7 8 5 3 77.5.9847 4 3 3 3 7.7.9957 4 4 3 36.7.999 6.5.9949 3 3 3.6.9989 8..9989.6.9978 The probbiliy of ccepnce nd ASN vlues re clculed for g,g, c r,c, for Ryleigh nd og-ogisic s wih r=5 nd r= nd presened in ble 5 nd ble 6 respecively. ASN vlue is lmos sme. From he bove bles, one cn conclude h he ASN vlue of Ryleigh is minimum compred wih og-ogisic. VII. ASN VAUES OF RAYEIGH AND OG-OGISTIC DISTRIBUTIONS ARE COMPARED In ble 7 compres he Averge smple number vlues of Ryleigh nd og-ogisic s wih ASN of Weibull m= s per Aslm M, Jun C.H, ee H, Ahmd M nd Rsool M. ASN vlue is minimized using opimizion problem for double group smpling pln. Tble 7 ASN Vlues wih Ryleigh nd og-ogisic Disribuions for β=.5 r Me n Rio og-logisic r = r=5 Ryleigh Weibull m= og-logisic Ryleigh Weibull m= 53.4 49.4 5.4 6.5 5. 46.5 4.3.6.9 6.8 6..9 6..7.9.5.3.9 8.6 5..9 5..9 In he bove ble wih he vlue of men rio =,4,6,8, he number of iems in group r=5, r= nd consumer s risk β=.5. I is observed h ASN vlues of hree vlues decreses when men rio vlue increses. Improved Double Group Smpling Pln Improved Double Group Smpling pln Averge Smple Number 6 4 4 6 8 Men rio r og-logisic Ryleigh Averge Smple Number 5 5 3 35 4 45 5 5 4 6 8 Men rio r og-logisic Ryleigh Figure : Comprison of Ryleigh, og-ogisic nd Weibull s for r= lef nd r=5 righ Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 733
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 From he nlyses of ble 7 nd figure, ASN vlue of Ryleigh is minimum s compred long wih Weibull m= nd og-ogisic s. VIII. CONCUSION One cn concluded h if he life ime of he iem follows Ryleigh he probbiliy of ccepnce vlue is greer hn he probbiliy of ccepnce of og-ogisic for proposed single nd double group smpling plns. One cn observe h s men rio increses here is decrese in number of groups nd ccepnce number. And he sme ime here is increse in probbiliy of ccepnce when es erminion ime muliplier decreses. ASN vlue for Ryleigh is minimum while compring wih Weibull m= nd og-ogisic s. REFERENCES. Aslm, M., Jun, C.H., Group Accepnce Smpling Plns for Trunced ife Tess Bsed on he Inverse Ryleigh Disribuion nd og- ogisic Disribuions, Pkisn Journl of Sisics,Vol.5, pp. 7-9, 9.. Aslm, M., Jun, C.H., A Group Accepnce Smpling Pln for Trunced ife Tes hving Weibull Disribuion, Journl of Applied Sisics, Vol.39, pp. 7, 9 b. 3. Aslm, M., Jun, C.H., A Group Smpling Pln Bsed on Trunced ife Tess for Gmm Disribued Iems, Pkisn Journl of Sisics, Vol.5, pp. 333-34, 9. 4. Aslm, M., Jun, C.H., Rsool, M., nd Ahmd, M., A Time Trunced Two Sge Group Smpling Pln for Weibull Disribuion, Communicions of he Koren Sisicl Sociey, Vol.7, pp. 89 98,. 5. Aslm, M., Jun, C.H., ee, H., Ahmd, M., nd Rsool, M., Improved Group Smpling Plns Bsed on Time-Trunced ife Tess, Chilen Journl of Sisics, Vol., No., pp. 85-97,. 6. Aslm, M., Kundu, D., nd Ahmd, M., Time Trunced Accepnce Smpling Plns for Generlized Exponenil Disribuion, Journl of Applied Sisics, Vol.37, pp 555-566,. 7. Srinivs Ro, G., A Group Accepnce Smpling Plns for he ife Times following Generlized Exponenil Disribuion, EQC, Vol.4, No., pp. 75-85, 9. Copyrigh o IJIRSET DOI:.568IJIRSET.6.5584 734