A Variable Control Chart under the Truncated Life Test for a Weibull Distribution

Size: px

Start display at page:

Download "A Variable Control Chart under the Truncated Life Test for a Weibull Distribution"

Norman Mosley
5 years ago
Views:

technologies Article A Vrible Control Chrt under the Truncted Life Test for Weibull Distribution Nsrullh Khn 1 ID, Muhmmd Aslm 2, * ID, Muhmmd Zhir Khn 3 nd Chi-Hyuck Jun 4 1 Deprtment of Sttistics

com 2 Deprtment of Sttistics, Fculty of Science, King Abdulziz University, Jeddh 21551, Sudi Arbi 3 Deprtment of Mthemtics nd Sttistics, Riphh Interntionl University, Islmbd 44000, Pkistn;

1 technologies Article A Vrible Control Chrt under the Truncted Life Test for Weibull Distribution Nsrullh Khn 1 ID, Muhmmd Aslm 2, * ID, Muhmmd Zhir Khn 3 nd Chi-Hyuck Jun 4 1 Deprtment of Sttistics nd Computer Science, University of Veterinry nd Animl Sciences UVAS), Lhore 54000, Pkistn; ns_shn1@hotmil.com 2 Deprtment of Sttistics, Fculty of Science, King Abdulziz University, Jeddh 21551, Sudi Arbi 3 Deprtment of Mthemtics nd Sttistics, Riphh Interntionl University, Islmbd 44000, Pkistn; zerishkh@gmil.com 4 Deprtment of Industril nd Mngement Engineering, Pohng University of Science nd Technology POSTECH), Pohng 37673, Kore; chjun@postech.c.kr * Correspondence: slm_rvin@hotmil.com or mgmuhmmd@ku.edu.s; Tel.: Received: 9 My 2018; Accepted: 7 June 2018; Published: 10 June 2018 Abstrct: In this mnuscript, vrible control chrt under the time truncted life test for the Weibull distribution is presented. The procedure of the proposed control chrt is given nd its run length properties re derived for the shifted process. The control limit is determined by considering the trget in-control verge run length ARL). The tbles for ARLs re presented for industril use ccording to vrious shift prmeters nd shpe prmeters in the Weibull distribution. A simultion study is given for demonstrting the performnce of the proposed control chrt. Keywords: verge run length; vrible chrt; weibull distribution 1. Introduction A control chrt is considered powerful tool for mintining the high qulity of products in industry. This tool is used to monitor the industril process from rw mteril to finl product. The process cn shift from the trget vlue due to severl extrneous fctors. The control chrt should provide quick indiction bout shift in mnufcturing process, if there is ny. This quick indiction helps the industril engineer to look t the problem nd bring it bck into control stte. Two types of control chrts re widely used in the industry for monitoring the mnufcturing process. Attribute control chrts re used when the dt is coming from the counting process nd vrible control chrts re used when the dt is obtined from the mesurement process. The ttribute control chrts re esy to pply but the vrible control chrts re more informtive thn ttribute chrts. Usully, vribles control chrts re developed under the ssumption tht the chrcteristic follows the norml distribution. In prctice, control chrt designed for norml distribution my not be effective for monitoring the mnufcturing process when the chrcteristic of interest does not follow the norml distribution. The use of this type of control chrt my led to wrong decision bout the sttus of the mnufcturing stte. Dery nd Cnn [1] mentioned tht the distributions of mesurements in chemicl processes, semiconductor processes, cutting tool wer processes nd observtions on lifetimes in ccelerted life test smples re often skewed. Severl uthors focused on this issue nd designed control chrts for non-norml dt. Al-Orini nd Rhim [2] designed control chrt for gmm distribution. Amin et l. [3] designed non-prmetric chrt using sign sttistic. Chng nd Bi [4] worked for control chrt for positive skewed popultion, Chen nd Yeh [5] worked for n economicl sttisticl design of control chrt for gmm distribution. McCrcken nd Chkrborti [6] presented Technologies 2018, 6, 55; doi: /technologies

2 Technologies 2018, 6, 55 2 of 10 control chrt for monitoring men nd vrince for norml distribution. Yen et l. [7] designed Synthetic-type contro for time between events. Riz et l. [8] proposed medin control chrt nd Abujiy et l. [9] worked for the cumultive sum control chrt CUSUM) control chrt. Gonzlez nd Viles [10] designed R chrt using gmm distribution, Lio nd Prk [11] designed control chrt for inverse Gussin percentiles. Hung nd Pscul [12] designed control chrt for Weibull percentiles using the order sttistic. Dery nd Cnn [1] designed the control chrts for the Weibull distribution, gmm distribution, nd log-norml distribution nd [13] proposed control chrt for the Burr type X distribution. Aslm et l. [14] deigned control chrt for the exponentil distribution using exponentilly weighted moving verge EWMA) sttistic. For the relibility evlution of product, the time truncted life test is often dopted in industries for sving the experiment time. Therefore, the designing of control chrt under the time truncted test is importnt when monitoring the process in terms of relibility. Aslm et l. [15] proposed n ttribute control chrt under the time truncted life test by ssuming tht the filure time of product follows the Weibull distribution. Aslm et l. [16] designed time truncted ttribute control chrt using the Preto distribution. Aslm et l. [17] deigned time truncted control chrt for the Birnbum-Sunders distribution under repetitive smpling. More detils bout such control chrts cn be red in Aslm et l. [18], Arif et l. [19], Khn et l. [20], nd Shfqt et l. [21]. In summry, control chrts under the time truncted life test for vrious situtions or distributions re vilble for the ttribute qulity of interest. However, these control chrts cnnot be pplied for the monitoring of mesurble qulity of interest. From exploring the literture nd from the best of our knowledge, there is no work on the design of vrible control chrt under the time truncted life test. In this pper, we will propose vrible control chrt for Weibull distribution using filure dt from time truncted life test. A rel exmple is given for illustrtion purposes nd simultion study is dded to demonstrte the performnce of the proposed control chrt. 2. Proposed Chrt nd Its Averge Run Length ARLs) The ssumptions of the proposed control chrt re stted s follows: 1. It is ssumed tht the qulity chrcteristic of interest follows Weibull distribution. 2. The shpe prmeter β of the Weibull distribution is ssumed to be known. 3. When the process is shifted, the scle prmeter is chnged to λ 1 = cλ 0, where c is shift constnt nd λ 0 is the scle prmeter for the in-control process. The shpe prmeter remins unchnged when the process is shifted. 4. Filure times re mesured during the time truncted life test. Let the vrible of interest follow the Weibull distribution with the cumultive distribution function cdf) of: Ft; λ, β) = 1 exp t/λ) β), t 0 1) where β is the shpe prmeter nd λ is the scle prmeter. The verge life time, µ, of the vrible is given by: µ = λ/β)γ1/β)) 2) where Γ.) is gmm function. The proposed control chrt is operted s follows: Step 1 Step 2 Step 3 Tke smple of size n t ech subgroup from the production process. Put them on test until the specified time t 0. Obtin the time to filure of item i denoted by X i ). Set X i = t 0 if item i does not fil by time t 0. Clculte sttistics Y i = X β i. Obtin Y = 1 n n i=1 Y i, Y is verge of Y i. Declre the process s in-control if Y L 3 or s out-of-control if Y < L 3, where L 3 represents the control limit.

3 Technologies 2018, 6, 55 3 of Distribution of Control Sttistics nd In-Control Averge Run Length ARL) To derive the necessry mesures of verge run length ARL), it would be convenient to select the specified time t 0 = µ 0, s frction of the men for the in control process, where is constnt nd µ 0 is the trget men. As the trnsformed vrible Y = X β is modeled to n exponentil distribution with men λ β, the sum of Y s or Y) follows gmm distribution. But, the distribution of Y pproximtely follows norml distribution ccording to the centrl limit theorem, s long s the smple size is sufficient. Then, the expected vlue of Y cn be obtined by: E[Y] = E 1 n n i=1 Y i = 1 n n E[Y i ]= E[Y i ] 3) i=1 Here, E[Y i ] is obtined by: E[Y i ] = E[Y i Y i t β 0 ]P {Y i t β 0 } { } + E[Y i Y i > t β 0 ]P Y i > t β 0 4) Eqution 4) reduces to: E[Y i ] = t β 0 0 y λ β 0 y λ e β 0 dy + t β 0 t β 0 The vrince of Y is given s follows: y 1 λ β e λ β 0 dy = λ β 0 1 e t 0 λ0 ) β) = E[Y] 5) r[y] = 1 n Vr[Y] = E[Y2 ] E[Y]) 2) /n 6) The simplified form of Vr[Y] cn be rewritten s: Vr[Y] = 1 n [ λ 2β 0 1 e 2 t 0 λ0 ) β) 2λ 0 t 0 ) β e t 0 λ0 ) β] 7) Let P in,0 be the probbility of being declred s in control when the process is ctully in control t λ 0. Then, it is given s follows: P in,0 = PY L 3 λ = λ 0 ) 8) Finlly, P in,0 cn be written s follows: P in,0 = 1 Φ 1 n βµ 0 ) L 3 βµ 0 ) ) β 1 e β Γ 1 β ))β) ) 2β 1 e 2 β Γ β 1 ))β) 2 βµ 2 0) ) β e β Γ 1 β ))β 9) The efficiency of the proposed chrt will be mesured through the ARL, which shows when the process will be out-of-control. Let ARL 0 be the ARL for in control process. Then, it is given s follows: ARL 0 = 1 1 P in,0 10)

4 Technologies 2018, 6, 55 4 of Out of Control ARL Let us ssume tht the process hs shifted due to some fctors to new scle prmeter λ 1 = cλ 0, where c is the shift constnt. Now, we derive the mesures for the shifted process. Let P in,1 be the probbility of declring the stte of being in control when the process hs shifted to λ 1, which is given s follows: P in,1 = PY L 3 λ = λ 1 ) 11) The distribution of Y t λ 1 pproximtely follows norml with the men nd the vrince given below: E[Y λ 1 ] = λ β 1 1 e t β) 0 λ1 ) 12) Vr[Y] = 1 n [ λ 2β 1 1 e 2 t 0 λ1 ) β) 2λ 1 t 0 ) β e t 0 λ1 ) β] 13) Therefore, the in-control probbility for the shifted process, sy P in,1 in Eqution 11), is obtined by: P in,1 = 1 Φ 1 n cβµ 0 ) L 3 cβµ 0 ) ) β 1 e cβ Γ 1 β ))β) ) 2β 1 e 2 cβ Γ β 1 ))β) 2 cβµ ) 2 0 ) β e cβ Γ 1 β ))β 14) Hence, the out-of-control ARL, sy ARL 1 for the shifted process is given s follows: ARL 1 = 1 1 P in,1 15) The vlues of ARL 1 for the vrious vlues of the shpe prmeters, µ 0, c nd, r 0, re presented in Tbles 1 8. Let r 0 be the specified in-control ARL. The control limit L 3 is determined by using the following lgorithm: 1. Prefix the vlue of r The vlue L 3 is obtined such tht ARL 0 r 0. Tble 1. The vlues of ARLs when r 0 = 370, β = 0.5, nd µ 0 = 50. L

5 Technologies 2018, 6, 55 5 of 10 Tble 2. The vlues of ARLs when r 0 = 370, β = 0.5, nd µ 0 = 100. L Tble 3. The vlues of ARLs when r 0 = 370, β = 1, nd µ 0 = 50. l Tble 4. The vlues of ARLs when r 0 = 370, β = 1, nd µ 0 = 100. l

6 Technologies 2018, 6, 55 6 of 10 Tble 5. The vlues of ARLs when r 0 = 370, β = 1.5, nd µ 0 = 50. l Tble 6. The vlues of ARLs when r 0 = 370, β = 1.5, nd µ 0 = 100. l Tble 7. The vlues of ARLs when r 0 = 370, β = 2, nd µ 0 = 50. l

7 Technologies 2018, 6, 55 7 of 10 Tble 8. The vlues of ARLs when r 0 = 370, β = 2, nd µ 0 = 100. r 0 = 370; β = 1.5; u 0 = 100 l Technologies , , x FOR PEER REVIEW of l c ARL From Tbles 1 8 nd Figure 1, we note the following trends in ARL As0.5 increses, 4.57 we note 2.34 ARL to decrese1.07 for the sme 1.00 shift in1.00 process. This 1.00 seems resonble 1.00 becuse 0.4 the 2.49 number1.38 of filures1.01 observed increses 1.00 s1.00 increses For 0.3 fixed1.51, s c decreses, 1.05 ARL lso decreses 1.00 becuse 1.00 the true1.00 men time1.00 to filure1.00 decreses. 3. For 0.2 other 1.08 fixed prmeters, 1.00 s 1.00 µ 0 increses 1.00 from 50 to , the behvior 1.00 of1.00 ARL 1 remins 1.00 quite similrly, which is 1.00 desirble 1.00 feture For other fixed prmeters, s β increses from 0.5 to 2, ARL 1 decreses. 3. Simultion Study Figure Figure The The verge verge run run length length ARLs) ARLs) curves. curves. In this section, we discuss the performnce of the proposed control chrt using simulted dt. The dt is generted from the Weibull distribution with the shpe prmeters β = 1.5 nd μ 0 = 50. The first 20 observtions re generted from n in-control process nd the next 20 observtions re

8 Technologies 2018, 6, 55 8 of Simultion Study In this section, we discuss the performnce of the proposed control chrt using simulted dt. The dt is generted from the Weibull distribution with the shpe prmeters β = 1.5 nd µ 0 = 50. The first 20 observtions re generted from n in-control process nd the next 20 observtions re generted from shifted process with c = 0.8. Technologies We pplied 2018, this 6, x FOR dtpeer to the REVIEW proposed control chrt hving = 1 nd r 0 = 370. The control limit L 3 is obtined by The vlues of Y i = Xi of 10 re clculted nd plotted on the control chrt. From Figure Technologies 2, we2018, note6, tht x FOR the PEER proposed REVIEW chrt detects the shift in the process t the 31st smple. 8 of 10 Figure 2. The proposed chrt using simulted dt. 4. Rel Exmple Figure Figure The The proposed proposed chrt chrt using using simulted simulted dt. dt Rel Rel In Exmple this section, we will discuss the ppliction of the proposed control chrt using the breking stress dt of crbon fibers from n industry. Crbon fiber hs good tensile strength, which is In In this this section, we we will will discuss the the ppliction of of the the proposed control chrt chrt using using the the breking mesured in force per unit re GP). For more detils, see Yu et l. [22]. The crbon fibers dt given stress stress dt dt of crbon of crbon fibers fibers from n from industry. n industry. Crbon Crbon fiber hsfiber goodhs tensile good strength, tensile which strength, is mesured which is in GP is modeled by the Weibull distribution when = 1. For this study, let n = 30 nd r inmesured force per unit in force re per GP). unit For re more GP). detils, For more see Yu detils, l. see [22]. Yu The et l. crbon [22]. The fibers crbon dt given fibers in dt GP 0 = 370. given is The vlues of sttistic Y for this dt re shown below: modeled in GP by is modeled the Weibull by distribution the Weibull when distribution = 1. For when this study, = 1. For let n this = 30 study, nd rlet 0 = n 370. = 30 The nd vlues r 0 = of 370. sttistic The 404, vlues 377, Y for 420, of this sttistic 698, dt 402, re Y 421, shown for 352, this below: 303, dt 544, re shown 413, 365, below: 359, 383, 330, 451, 402, 355, 368, 342, 383, 444, 368, 364, 433, 443, 508, 447, 528, 512, 559, 481, 325, 303, 410, 628, 373, 571, 414, 364, , 404, 377, 377, 420, 420, 698, 698, 402, 402, 421, 421, 352, 352, 303, 303, 544, 544, 413, 413, 365, 365, 359, 359, 383, 383, 330, 330, 451, 451, 402, 402, 355, 355, 368, 368, 342, 342, 383, 383, 444, 444, 368, 368, 364, 364, 433, 433, By 443, plotting 443, 508, 508, 447, Y 447, on 528, 528, 512, control 512, 559, 559, chrt, 481, 481, 325, Figure 325, 303, 303, 3410, shows 410, 628, 628, tht 373, 373, the 571, 571, process 414, 414, 364, is 364, in 328. control 328. but some points re close to the lower control limit. By By plotting plotting Y on Y on control control chrt, chrt, Figure Figure 3 shows 3 shows tht tht the process the process is in control is in control but some but some points points re close re to close theto lower the lower control control limit. limit. Figure Figure 3. Proposed 3. Proposed chrt chrt for for the the crbon crbon fibers fibers dt. dt. 5. Conclusions Figure 3. Proposed chrt for the crbon fibers dt. 5. Conclusions In this pper, new vrible chrt under the truncted life test is designed for the Weibull distribution. The verge run length is derived to mesure the efficiency of the proposed chrt. In this pper, new vrible chrt under the truncted life test is designed for the Weibull Extensive tbles re given for industril use. The ppliction of the proposed chrt is given with the

9 Technologies 2018, 6, 55 9 of Conclusions In this pper, new vrible chrt under the truncted life test is designed for the Weibull distribution. The verge run length is derived to mesure the efficiency of the proposed chrt. Extensive tbles re given for industril use. The ppliction of the proposed chrt is given with the help of simultion dt. The proposed chrt is sensitive in detecting the shift in the process. The proposed control chrt cn be used in rel industry where the lifetime of the product follows the Weibull distribution. The proposed chrt using unknown shpe prmeters cn be fruitful future reserch. Author Contributions: Conceptuliztion, N.K., M.A., M.Z.K. nd C.H.J.; Methodology, N.K., M.A., M.Z.K. nd C.H.J.; Softwre, N.K., M.A., M.Z.K. nd C.H.J.; Vlidtion, N.K., M.A., M.Z.K. nd C.H.J; Forml Anlysis, N.K., M.A., M.Z.K. nd C.H.J.; Writing-Originl Drft Preprtion, M.A., nd C.H.J.; Writing-Review & Editing, N.K., M.A., M.Z.K. nd C.H.J. Funding: This reserch received no externl funding. Acknowledgments: The uthors re deeply thnkful to the editor nd the reviewers for their vluble suggestions to improve the qulity of this mnuscript. This rticle ws funded by the Denship of Scientific Reserch DSR) t King Abdulziz University, Jeddh. The uthor, Muhmmd Aslm, therefore, cknowledge with thnks DSR technicl nd finncil support. Conflicts of Interest: The uthors declre no conflict of interest. References 1. Dery, K.; Cnn, H. Control Chrts for Skewed Distributions: Weibull, Gmm, nd Lognorml. Metodoloski Zvezki 2012, 9, Al-Orini, H.A.; Rhim, M. Economic sttisticl design of X control chrts for systems with Gmm λ, 2) in-control times. Comput. Ind. Eng. 2002, 43, [CrossRef] 3. Amin, R.W.; Reynolds, M.R., Jr.; Sd, B. Nonprmetric qulity control chrts bsed on the sign sttistic. Commun. Stt.-Theor. Methods 1995, 24, [CrossRef] 4. Chng, Y.S.; Bi, D.S. Control chrts for positively-skewed popultions with weighted stndrd devitions. Qul. Relib. Eng. Int. 2001, 17, [CrossRef] 5. Chen, F.; Yeh, C. Economic sttisticl design of non-uniform smpling scheme X br control chrts under non-normlity nd Gmm shock using genetic lgorithm. Expert Syst. Appl. 2009, 36, [CrossRef] 6. McCrcken, A.; Chkrborti, S. Control chrts for joint monitoring of men nd vrince: An overview. Qul. Technol. Qunt. Mng. 2013, 10, [CrossRef] 7. Yen, F.Y.; Chong, K.M.B.; H, L.M. Synthetic-type control chrts for time-between-events monitoring. PLoS ONE 2013, 8, e [CrossRef] [PubMed] 8. Ahmd, S.; Riz, M.; Abbsi, S.A.; Lin, Z. On efficient medin control chrting. J. Chin. Inst. Eng. 2014, 37, [CrossRef] 9. Abujiy, M.R.; Riz, M.; Lee, M.H. Enhnced Cumultive Sum Chrts for Monitoring Process Dispersion. PLoS ONE 2015, 10, e [CrossRef] [PubMed] 10. Gonzlez, I.M.; Viles, E. Design of R control chrt ssuming gmm distribution. Econ. Qul. Control 2001, 16, [CrossRef] 11. Lio, Y.; Prk, C. A bootstrp control chrt for inverse Gussin percentiles. J. Stt. Comput. Simul. 2010, 80, [CrossRef] 12. Hung, X.; Pscul, F. ARL-unbised control chrts with lrm nd wrning lines for monitoring Weibull percentiles using the first-order sttistic. J. Stt. Comput. Simul. 2011, 81, [CrossRef] 13. Lio, Y.L.; Tsi, T.-R.; Aslm, M.; Jing, N. Control chrts for monitoring Burr type-x percentiles. Commun. Stt.-Simul. Comput. 2014, 43, [CrossRef] 14. Aslm, M.; Azm, M.; Jun, C.-H. A new control chrt for exponentil distributed life using EWMA. Trns. Inst. Mes. Control 2015, 37, [CrossRef] 15. Aslm, M.; Jun, C.-H. Attribute Control Chrts for the Weibull Distribution under Truncted Life Tests. Qul. Eng. 2015, 27, [CrossRef]

Technologies 2018, 6, 55 10 of 10 16. Aslm, M.; Khn, N.; Jun, C.-H. A control chrt for time truncted life tests using Preto distribution of second kind. J. Stt. Comput. Simul. 2016, 86, 2113 2122.

10 Technologies 2018, 6, of Aslm, M.; Khn, N.; Jun, C.-H. A control chrt for time truncted life tests using Preto distribution of second kind. J. Stt. Comput. Simul. 2016, 86, [CrossRef] 17. Aslm, M.; Arif, O.H.; Jun, C.-H. An Attribute Control Chrt Bsed on the Birnbum-Sunders Distribution Using Repetitive Smpling. In IEEE Access; IEEE: Wshington, DC, USA, Aslm, M.; Arif, O.H.; Jun, C.-H. An ttribute control chrt for Weibull distribution under ccelerted hybrid censoring. PLoS ONE 2017, 12, e [CrossRef] [PubMed] 19. Arif, O.-H.; Aslm, M.; Jun, C.-H. EWMA np Control Chrt for the Weibull Distribution. J. Test. Evl. 2016, 45, [CrossRef] 20. Khn, N.; Aslm, M.; Kim, K.-J.; Jun, C.-H. A mixed control chrt dpted to the truncted life test bsed on the Weibull distribution. Oper. Res. Decis. 2017, 27, Shfqt, A.; Hussin, J.; Al-Nsser, A.D.; Aslm, M. Attribute control chrt for some populr distributions. Commun. Stt.-Theor. Methods 2018, 47, [CrossRef] 22. Yu, M.F.; Lourie, O.; Dyer, M.J.; Moloni, K.; Kelly, T.F.; Ruoff, R.S. Strength nd breking mechnism of multiwlled crbon nnotubes under tensile lod. Science 2000, 287, [CrossRef] [PubMed] 2018 by the uthors. Licensee MDPI, Bsel, Switzerlnd. This rticle is n open ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution CC BY) license

Time Truncated Two Stage Group Sampling Plan For Various Distributions

Time Truncated Two Stage Group Sampling Plan For Various Distributions Time Truncted Two Stge Group Smpling Pln For Vrious Distributions Dr. A. R. Sudmni Rmswmy, S.Jysri Associte Professor, Deprtment of Mthemtics, Avinshilingm University, Coimbtore Assistnt professor, Deprtment