Sains Malaysiana 45(11)(2016): ABDUR RAZZAQUE MUGHAL*, ZAKIYAH ZAIN & NAZRINA AZIZ

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1 Sins Mlysin 4()(01): 1 1 Time Truncted Efficient Testing Strtegy for Preto Distribution of the nd Kind Using Weighted Poisson nd Poisson Distribution (Strtegi Ujin Cekp Ms Terpngks untuk Tburn Preto Jenis ke- Menggunkn Tburn Poisson Berpembert dn Poisson) ABDUR RAZZAQUE MUGHAL*, ZAKIYAH ZAIN & NAZRINA AZIZ ABSTRACT In this study, group cceptnce smpling pln (GASP) proposed by Aslm et l. (0) is redesigned where the lifetime of test items re following Preto distribution of nd kind. The optiml pln prmeters re found by considering vrious pre-determined designed prmeters. The pln prmeters were obtined using the optimiztion solution nd it lso concludes tht the proposed pln is more efficient thn the existing pln s it requires minimum smple size. Keywords: Consumer s risk; group cceptnce smpling; operting chrcteristic vlues; Poisson & Weighted Poisson distribution; producer s risk ABSTRAK Dlm kjin ini, peln pensmpeln penerimn kumpuln (GASP) yng dicdngkn oleh Aslm et l. (0) direk semul dengn hyt item ujin mengikuti tburn Preto jenis ke-. Prmeter optimum peln boleh didpti dengn mengmbil kir pelbgi prmeter yng telh ditetpkn terlebih dhulu. Prmeter peln diperoleh menggunkn penyelesin pengoptimumn dn i membut kesimpuln bhw rncngn yng dicdngkn dlh lebih cekp dripd rncngn sedi d kern i memerlukn siz smpel yng minimum. Kt kunci: Nili ciri opersi; pensmpeln penerimn kumpuln; risiko pengelur; risiko penggun; tburn Poisson berpembert & Poisson INTRODUCTION Acceptnce smpling pln is one of the most importnt techniques to improve the qulity of submitted item. The min objective of cceptnce smpling is to reduce the time nd cost of the truncted life test experiment nd lso very helpful tht the producer s increse the qulity of the item. For the finl item, it my not fesible to exmine ech nd every item t the time of the inspection. Then, rndom smple is chosen with the support of cceptnce smpling method for the finl confirmtion of the lot. The selected items re put on the test nd lot is cceptble if the number of filures re less thn the pre-specified number of filures. In these plns, s the opinion is mde on the use of smple observtion, therefore two risks re constntly ffixed with smpling plns. The probbility tht good lot is not ccepted is known s producer s risk nd the probbility of ccepting bd lot is clled the consumer s risk denoting by α nd β, respectively. Mny reserchers proposed cceptnce smpling plns bsed on truncted life test for severl distributions, for exmple, Blkrishnn et l. (00), Bklizi (00), Epstein (14), Goode nd Ko (), Kntm nd Rosih (1), Kntm et l. (00), Mughl et l. (0), Rdhkrishnn nd Mohn Priy (00) nd Tsi nd Wu (00). In ordinry cceptnce smpling pln, single item is inspected but in prctice, testers re vilble tht cn ccommodte more thn one item. Therefore, the experimenters cn use the group cceptnce smpling pln (GASP) to observe the multiple items t the sme time. The more detils bout group cceptnce smpling plns (GASP) cn be seen in Aslm et l. (0, 0b), Mughl nd Aslm (0) nd recently Mughl nd Ismil (01) designed n economic relibility efficient group cceptnce smpling plns for fmily Preto distributions. Aslm et l. (0) designed comprison of GASP for Preto distribution of the nd kind using Weighted Poisson nd Poisson distributions. In prctice, the Preto distribution of the nd kind is used for filure life time dt. To the best of our knowledge, still no resercher hs produced the efficient GASP for Preto distribution of the nd kind using Weighted Poisson nd Poisson distribution. Therefore, in our proposed reserch, the inspection of submitted items from totl smple will be conducted rther thn inspecting items in ech group s discussed by the Aslm et l. (0). Due to minimum smple size, the proposed pln will be more efficient nd economicl when it comes to sve time, cost, energy nd mn-hours involved. If smple size is lrge nd p (defective items in lot) is very smll then Poisson distribution is best pproximtion of the binomil distribution. In life, testing weighted distribution methods nd ppliction re widely used becuse the reported dt is bised. The bised dt

2 14 do not express the prent distribution behvior unless every item is interpreted by ssigning the equl chnce of selection. The objective of this study ws to obtin the optiml vlue of group size, operting chrcteristic vlue nd minimum rtio of true verge life by considering the vrious pre-determined pln prmeters. METHODS Consider μ nd μ 0 denote the true nd specified verge life of n item respectively. An item is cceptble for consumer use if the verge life μ is greter thn specified verge life μ 0. According to Mughl nd Ismil (01), procedure of proposed cceptnce smpling pln under GASP will be: Obtin the number of g groups nd llocte r items to ech group so the required smple size in the life test is n = r g; Selecting the cceptnce number c nd crry out the experiment for the g groups nd ccept the lot if t most c filures record in ll of the groups; nd If more thn c filures occur t time t 0, truncte the experiment nd reject the submitted lot. Preto (1) discussed the importnce of Preto distribution s model for income. The probbility density function (PDF), the cumultive distribution function (CDF) nd the men μ of Preto distribution of the nd kind cn be written, respectively, (1) (). () The two prmeters (σ, λ) re scle nd shpe prmeters, respectively nd it is importnt to note tht the vlue of shpe prmeter must be greter thn 1. Under the proposed pln the lot cceptnce probbility function for Weighted Poisson nd Poisson distribution cn be written in this form, (i=1) (4) () () The minimum group size ws found when the following two inequlities fulfilled the conditions for both weighted Poisson nd Poisson distributions (), (), respectively nd plced in Tbles 1-4. i=1 () () The minimum men rtio (μ/μ 0 ) determined in () nd () for given producer s risk which is very helpful tool for the producers to choose the pproprite life testing pln nd discussed in Tbles -1. i=1 () () The minimum group size, operting chrcteristics vlues nd minimum men rtio (μ/μ 0 ) were obtined for Weighted Poisson nd Poisson distribution nd llocted in Tbles 1-1. In these tbles, we considered the vrious designed prmeters such s, number of tester r = (1)1, cceptnce number c = (1), test termintion rtio = 0., 0.,, 1., 1.,.0, consumer s risk β = 0., 0., 0.0, nd producer s risk α = 0.0. Tbles 1-4 illustrte tht test termintion rtio monotoniclly decreses s the group size increses. In Tbles -, the operting chrcteristics (OC) vlues re estblished using (4) nd () for c =, r =. For vrious vlue of designed prmeter the OC vlues cn be ccessed by using the sme technique. From Tbles -, it is obvious to see tht when men rtio increses from to 1, the probbility of lot cceptnce is lso incresing. Conversely, s test termintion rtio (μ/μ 0 ) increses from 0. to.0, we hve found the decresing tendency in probbility of lot cceptnce. The efficiency nd dvntges of the proposed pln compred with the existing pln in term of smple size lso presented in Tbles 1-1. The identicl vlues of designed prmeters hve been used for the both cceptnce smpling plns. where p denotes the probbility of filure of n item during the test termintion time t 0. The termintion time t 0 is multiple of the specified men life μ 0 nd pre-ssumed constnt. For exmple, = 0. mens tht the test termintion time is hlf of the specified verge life. So t 0 = μ 0 nd p is estimted s follow, APPLICATION IN THE INDUSTRY The prcticl use of the proposed pln in the industry for the testing of the items whose lifetime bsed on Preto distribution of the nd kind (using Poisson distribution) will interpret in the following theoreticl exmple.

3 1 TABLE 1. Number of groups required for the proposed pln for the Preto distribution of the nd kind with λ =, using Weighted Poisson distribution β r c TABLE. Number of groups required for the proposed pln for the Preto distribution of the nd kind with λ =, using Weighted Poisson distribution β r c

4 1 TABLE 4. Number of groups required for the proposed pln for the Preto distribution of the nd kind with λ =, using Poisson distribution β r c TABLE. Number of groups required for the proposed pln for the Preto distribution of the nd kind with λ =, using Poisson distribution β r c

5 1 TABLE. Operting chrcteristics vlues of the group smpling pln with c =, r = for Preto distribution of the nd kind with λ=, using Weighted Poisson distribution β g TABLE. Operting chrcteristics vlues of the group smpling pln with c =, r = for Preto distribution of the nd kind with λ =, using Weighted Poisson distribution β g

6 1 TABLE. Operting chrcteristics vlues of the group smpling pln with c =, r = for Preto distribution of the nd kind with λ =, using Poisson distribution β g TABLE. Operting chrcteristics vlues of the group smpling pln with c =, r = for Preto distribution of the nd kind with λ =, using Poisson distribution β g

7 1 TABLE. Minimum rtio of true verge life to specified life for the producer s risk of 0.0, for the Preto distribution of the nd kind with λ =, using Weighted Poisson distribution β r c TABLE. Minimum rtio of true verge life to specified life for the producer s risk of 0.0, for the Preto distribution of the nd kind with λ =, using Weighted Poisson distribution β r c

8 TABLE. Minimum rtio of true verge life to specified life for the producer s risk of 0.0, for the Preto distribution of the nd kind with λ =, using Poisson distribution β r c TABLE 1. Minimum rtio of true verge life to specified life for the producer s risk of 0.0, for the Preto distribution of the nd kind with λ =, using Poisson distribution β r c

9 TABLE 1. Comprisons of smple size when shpe prmeter λ =, c = nd using Weighted Poisson distribution TABLE 1. Comprisons of Smple Size n when shpe prmeter λ =, c = nd using Poisson distribution β Existing pln (Aslm et l. 0) Proposed pln β Existing pln (Aslm et l. 0) Proposed pln TABLE 14. Comprisons of Smple Size n when shpe prmeter λ =, c = nd using Weighted Poisson distribution TABLE 1. Comprisons of smple size n when shpe prmeter λ =, c = nd using Poisson distribution β Existing pln (Aslm et l. 0) Proposed pln β Existing pln (Aslm et l. 0) Proposed pln Consider tht the experimenter wnts to test the qulity of electricl crt for 000 h. Furthermore, in the lbortory, one hs to hve the fcility to instll more thn one electricl crt on tester. If the experimenter would like to choose the cceptnce number c =, number of tester r =, test termintion time = 0.0, consumer s risk β = 0. nd λ =, then the required minimum group size g = from Tble 4. The designed prmeters of the proposed GASP re (g, r, c, ) = (,,, 0.0). So, the prctitioner needs to choose rndom smple of size 0 items from the lot nd put ten items to three groups on the life testing experiment. The submitted lot is not ccepted if more thn eight filures occurred in 000 h, otherwise ccepted. COMPARISON In this section, we explined the comprtive study regrding smple sizes from the proposed pln with the existing pln. As discussed the designed prmeters of the proposed nd existing GASP re (g, r, c, ) = (,,,0.0) nd (g, r, c, ) = (0,,,0.0), respectively. From Tble 1, the proposed pln need 0 (n = r g) items nd existing pln require 00 (n = r g) items, respectively, to chieve similr inference concerning the submitted lot. Therefore, the number of comprison is lso shown in Tbles 1 to 1. The sme problem cn lso be considered for vrious lifetime distributions on pre-specified designed prmeters to inspect the verge life of submitted item. CONCLUSION In this reserch nlysis, tbles re disposed for time truncted efficient testing strtegy for Preto distribution of the nd kind using Weighted Poisson nd Poisson distribution. The Weighted Poisson nd Poisson distributions re considered to locte the optiml vlues group size, men rtio nd probbility of lot cceptnce. A comprison between the proposed pln nd existing pln is displyed for true nlysis. The consequence is the proposed technique provides the very smller smple size s compred to the estblished pln in literture. Hence, this present reserch nlysis is more beneficil to sve cost, time, energy, lbor nd quick inspection of the submitted items by the vendor. The proposed cceptnce smpling pln cn be used to test the lifetime of mny electronic components such s the mobile phones, trnsporttion electronics system, wireless devices nd globl positioning system. REFERENCES Aslm, M., Mughl, A.R. & Ahmed, M. 0. Comprison of GASP for Preto distribution of the nd kind using Poisson nd weighted Poisson distributions. Interntionl Journl of Qulity & Relibility Mngement (): -4. Aslm, M., Mughl, A.R., Ahmed, M. & Yb, Z. 0. Group cceptnce smpling pln for Preto distribution of the second kind. Journl of Testing nd Evlution (): 1-. Aslm, M., Mughl, A.R., Hnif, M. & Ahmed, M. 0b. Economic relibility group cceptnce smpling bsed on truncted life tests using Preto distribution of the second

10 1 kind. Communictions of the Koren Sttisticl Society 1(): -. Bklizi, A. 00. Acceptnce smpling bsed on truncted life tests in the Preto distribution of the second kind. Advnces nd Applictions in Sttistics (1): -4. Epstein, B. 14. Truncted life tests in the exponentil cse. Annls of Mthemticl Sttistics : -4. Goode, H.P. & Ko, J.H.K.. Smpling plns bsed on the Weibull distribution. In Proceeding of the Seventh Ntionl Symposium on Relibility nd Qulity Control, Phildelphi. pp Mughl, A.R. & Ismil, M. 01. An economic relibility efficient group cceptnce smpling plns for fmily Preto distributions. Reserch Journl of Applied Sciences, Engineering nd Technology (4): Mughl, A.R. & Aslm, M. 0. Efficient group cceptnce smpling plns for fmily Preto distribution. Continentl Journl of Applied Sciences (): 40-. Mughl, A.R., Hnif, M., Ahmed, M. & Rehmn, A. 0. Economic relibility cceptnce smpling plns from truncted life tests bsed on the Burr Type XII percentiles. Pkistn Journl of Commerce nd Socil Sciences (1): 1-1. Preto, V. 1. Cours D economie Politique. Pris: Rouge et Cie. Rdhkrishnn, R. & Mohn Priy, L. 00. Computtion of CRGS plns using Poisson nd Weighted Poisson distribution. ProbStt Forum 1: 0-1. Tsi, T.R. & Wu, S.J. 00. Acceptnce smpling bsed on truncted life tests for generlized Ryleigh distribution. Journl of Applied Sttistics (): -00. Deprtment of Mthemtics nd Sttistics School of Quntittive Sciences UUM College of Arts nd Sciences Universiti Utr Mlysi 00 Sintok, Kedh Drul Amn Mlysi *Corresponding uthor; emil: bdur_rzzque@live.com Received: August 01 Accepted: 1 Mrch 01