A Comparative Study of Exponential Time between Events Charts

Transcription

1 Quality Technology & Quantitative Management Vol. 3, No. 3, pp , 26 QTQM ICAQM 26 A Compaative Study of Exponential Time between Events Chats J. Y. Liu 1, M. Xie 1, T. N. Goh 1 and P. R. Shama 2 1 Depatment of Industial and Systems Engineeing, National Univesity of Singapoe, Singapoe 2 OCBC Ban, Singapoe (Received Mach 25, accepted Januay 26) Abstact: Contol chats based on time between events (TBE) data have been shown to be useful fo monitoing poduction o failue pocesses whee defect o failue occuence can be ecoded ove time. One type of TBE chat is set up with pobability contol limits, e.g. the Cumulative Quantity Contol (CQC) chat and its extension CQC- chat, which monitos the quantity inspected until the occuence of a fixed numbe of defects (). Anothe type of TBE chat studied in the liteatue is to apply the CUSUM and EWMA methods to TBE data diectly. In this pape, some exponential TBE chats ae compaed based on thei Aveage Time to Signal (ATS) pefomance. Diffeent in-contol ATS values ae used in ode to evaluate its effect on the pefomance of the TBE chats. Based on the compaison analysis, some ecommendations ae made as guidelines fo employing a pope chat unde diffeent situations. Keywods: CQC- chat, cumulative quantity contol (CQC) chat, exponential CUSUM, exponential EWMA, time between events (TBE) chats. 1. Intoduction C ontol chats fo attibutes have been populaly adopted fo monitoing the faction nonconfoming (p-chat) o nonconfomities (c-chat o u-chat) in a pocess. Howeve, Shewhat contol chats may face some pactical poblems when the pocess faction of nonconfoming is vey low, say, at pats pe million (ppm) o even pats pe billion (ppb) levels. Meaningless contol limits, high false alam pobability, difficulties in foming a ational subgoup, and failues in detecting pocess impovement [14] ae some of common difficulties one may face in implementing a Shewhat chat in a low defect envionment. One way to solve these poblems is to employ time between events (TBE) chats. Rathe than focusing on monitoing the nonconfoming items occuing within cetain sampling intevals; TBE chats monito the quantity of confoming items between successive occuence of nonconfoming items. Since the idea can also be applied to othe pocesses, e.g. eliability o sevice pocesses, the wod events is used instead of nonconfoming items. The wod time in the TBE chats stands fo the quantity obseved between occuence of the events, which may not necessaily be a discete intege vaiable. A common assumption fo TBE chats is that the occuence of events can be modeled by a homogeneous Poisson pocess, and thus the time between two successive events follows exponential distibution. Based on this assumption some TBE chats, efeed to as Confoming Run Length (CRL) chats, ae designed fo discete TBE data based on Poisson distibution, see [1], [2], [7], [8], and [16]. Othe TBE chats ae designed fo monitoing continuous TBE data based on exponential distibution, as shown in [4], [5], [6],

2 348 Liu, Xie, Goh and Shama [11] and [15]. Oveall, these TBE chats can be categoized into two goups. Fist goup of TBE chats employ pobability contol limits, such as the Cumulative Count of Confoming (CCC) chat and Cumulative Quantity Contol (CQC) chat. The second goup, such as the exponential CUSUM and exponential EWMA chat, ae designed based on the CUSUM and EWMA methods. All these TBE chats show some advantages in one way o anothe. Some compaing studies have been caied out fo discete TBE chats. Xie et al. [13] did a compaative study between the CCC and CUSUM chats. Boo et al. [1] compaed the Aveage Run Length (ARL) of the Poisson EWMA with that of the Shewhat c-chat. Wu et al. [12] compaed the design and pefomance of the np chat, CRL-CUSUM and SCRL(Sum of CRLs) chat fo discete TBE data. Sun and Zhang [1] conducted a compaative study using discete TBE data on the CUSUM, the EWMA chat, and the two-stage CCC chat poposed in [3]. Howeve, on the othe hand even though the continuous TBE chats epesent moe geneal cases compaed to the discete TBE chats, little liteatue is available on thei elative advantages. Gan [6] compaed the ARL pefomance of exponential EWMA with that of the exponential CUSUM and Shewhat chats. The esults indicated that the Shewhat chat is highly insensitive compaed to the exponential EWMA o CUSUM chat. Ranjan et al. [9] looed into the CQC chat, CQC- chat and exponential CUSUM chat and compaed thei pefomance based on Aveage Time to Signal (ATS). This pape extends the compaisons to a wide ange, and compaes the pefomance of continuous TBE chats among the CQC chat, CQC- chat, exponential EWMA and exponential CUSUM chat based on ATS pefomance. These TBE chats ae efeed to as exponential TBE chats since all of them ae set up based on exponential distibution. The pupose of this study was to investigate the compaative pefomance of diffeent exponential TBE chats and povide some insides of thei stengths as well as shotcomings. The esults will be useful fo the quality enginees on the implementation of TBE chats unde diffeent situations. 2. Design of the Exponential TBE Chats A unifom model of the exponential TBE chats involved in this study is that the occuence of events is modeled by a Poisson pocess, and the time between events X i (i= 1, 2, ) ae independent and identically distibuted exponential andom vaiables with pobability density function: 1 exp( x/ ), if x f ( x) =,, othewise (1) whee is the ecipocal of the events occuence ate, i.e. the mean of the time between events data The CQC Chat & CQC- Chat The CQC chat poposed in [4] plots the quantity poduced befoe obseving an event, say one nonconfoming item of the pocess, based on exponential distibution. The uppe contol limit (UCL), cental line (CL) and lowe contol limit (LCL) of the CQC chat can be calculated as: ( ) ( ) UCL = ln α/2, CL = ln(2), LCL = ln 1 α/2, (2)

3 A Compaative Study of Exponential Time Events Chats whee is the false alam ate. In ode to impove the sensitivity of the CQC chat, the CQC- chat was poposed to monito the time until a fixed numbe () of events obseved based on Gamma distibution [15]. The CQC- chat povides moe cedibility to the decision egading the statistical contol of the pocess as the decision is made on the basis of events athe than a single event. The CQC- chat educes to the CQC chat when = 1. Given an acceptable false alam ate, the contol limits UCL, CL, LCL of CQC- chat can be calculated by solving the following equations, ( UCL ) UCL 1 (,, ) = 1 e = 1 = 2 FUCL 1 FCL (,, ) = 1 e = CL ( CL )! α, =, (3)! 2 LCL ( LCL ) 1 α F( LCL,, ) = 1 e =, =! 2 whee and ae the paametes of the Gamma distibution The Exponential EWMA Chat Gan [6] intoduced the exponential EWMA chat based on the inte-aival time of nonconfoming items. The uppe-sided and lowe-sided EWMA chats ae designed to detect an incease o decease in the exponential mean, espectively. The chats ae fomed by plotting Q = max{ A,(1 λ ) Q + λ X }, t Q t1 q = min{ B,(1 λ ) q + λ X }, t q t1 q against t (t =1, 2, ), whee λ Q and λ q ae smoothing constants such that < λ Q 1, < λq 1. A and B ae boundaies that satisfy the following equations: Q = u, A u< hq, A, and q = v, hq < v B, B >. hq and h q ae the contol limits, and a signal is issued at the fist t fo which Q h o q h. t Q t q Two-sided exponential EWMA chat can be obtained by plotting Zt λz Zt 1 λzxt Q t t = (1 ) +, (5) against t, fo t =1, 2,, whee λ Z is a smoothing constant such that < λ Z 1, and Z = w, h l < w < h u. A signal is issued at the fist t when Z t h l o Zt h u. Anothe altenative method fo implementing two-sided EWMA is to use a lowe-sided and an uppe-sided EWMA chat togethe. The exact method of computing the ARL of the exponential EWMA chat by solving a set of diffeential equations can be found in Gan [6]. Gan [6] also computed the values of smoothing constant and chat limit h fo a given the acceptable in-contol ARL and the out-of-contol TBE mean The Exponential CUSUM Chat The exponential CUSUM chat plots the statistic: (4)

4 35 Liu, Xie, Goh and Shama + + i i1 i S = max{, S + ( X )}, i i1 i S = min{, S + ( X )}. (6) An out-of-contol signal will be given at the fist i with S i- -h o S i+ h. Usually the initial CUSUM values S + and S - ae set to be zeo. Howeve, the staing values may be set to some nonzeo values so that an out-of-contol signal can be issued ealie when the pocess stats fom an out-of-contol state. The efeence value () fo the exponential CUSUM can be calculated using the acceptable in-contol events ate (1/ ) and the out-of-contol events ate that is to be detected quicly (1/ 1 ). 1 1 ln ln 1 1( ln ln 1) = = (7) Once the efeence value has been calculated, a suitable value of h can be found out by the figues povided in Gan[5]. The ARL of the exponential CUSUM chat can be calculated by the appoximate Maov Chain appoach shown in Lucas[7], o the exact method as given by Vademan and Ray[11]. 3. ATS of the Exponential TBE Chats The ARL is one of the most fequently used citeions to measue the pefomance of contol chats. Howeve, it is not a good measuement fo TBE chats because the time spent on plotting each point is diffeent, and the ARL does not conside the time facto. A bette altenative is the ATS, which is defined as the expected value of total length of time to obseve an out-of-contol point. Let S be the total amount of time befoe an out-of-contol signal occus, then it is R obvious that S = i = 1 Xi, whee R is the numbe of points plotted on the chat until an out-of-contol signal occus. Using Wald s identity, the ATS of the CQC- chat can be calculated as: R ATS = E( S) = E( X ) = E( R) E( X ) = ARL = =, CQC i i = 1 λ λ(1 II ) 1 II whee II denotes the type II eo of the CQC- chat, is the event occuence ate of the Poisson pocess, and is the ecipocal of, i.e. the mean of time between events data. Since the time between events follows Gamma distibution, the type II eo II can be calculated as: (8) II LCL 1 LCL 1 e = = UCL ( ) ( UCL) = e. (9)!! as: Substituting the expession of II into equation (8), the ATS of CQC- chat is obtained

5 A Compaative Study of Exponential Time Events Chats ATS = CQC 1 LCL LCL 1 UCL 1 e + e = = ( ) ( UCL)!!. 351 (1) Using simila deduction method, the expession of ATS fo exponential EWMA and exponential CUSUM can be deived as R 1 ATSEWMA = E( S) = E( Xi ) = E( R) E( X ) = ARLEWMA = ARLEWMA, i = 1 λ R 1 ATSCUSUM = E( S) = E( Xi ) = E( R) E( X ) = ARLCUSUM = ARLCUSUM. i = 1 λ Hee in this study the exact methods of computing ARL ae used following the methods shown in Gan [6] fo exponential EWMA and Vademan and Ray [11] fo exponential CUSUM. When detecting the pocess impovement o deteioation sepaately, the one-sided CQC and CQC- chat ae used instead of two-sided chats. The contol limits of the lowe-sided and the uppe-sided CQC- chat can be calculated by solving the following equations: ( LCL ') LCL ' 1 = = = F( LCL ',, ) 1 e α,! ( UCL ') UCL ' 1 = = = FUCL ( ',, ) 1 e 1 α.! Accodingly, the ATS fo the lowe-sided CQC- chat can be expessed as: (11). (12) ATS L CQC = = LCL ' λf( LCL ',, ) 1 ( LCL ') 1 e = and the ATS fo the uppe-sided CQC- chat can be given by:!, (13) ATS U CQC = = UCL ' λ[1 FUCL ( ',, )] 1 ( UCL ') e =!. (14) The contol limits and the ATS fomulae fo CQC chat can be computed fom (12), (13) and (14) fo = Compaison of the Pefomance 4.1. Uppe-Sided Exponential TBE Chats The uppe-sided TBE chat is designed fo monitoing pocess impovements. In ode to assess the elative pefomance of uppe-sided CQC- chat ( =1, 2, 3, and 4), exponential EWMA and exponential CUSUM chats, the ATS pefomance of these exponential TBE chats ae compaed. The in-contol ATS value (ATS ) is set to be same

6 352 Liu, Xie, Goh and Shama (ATS =5) fo all of them, and the out-of-contol ATS (ATS 1 ) values fo diffeent shifts ae then calculated. The in-contol mean of TBE data is assumed to be 1 ( =1.), and the exponential EWMA chat and CUSUM chat ae designed to be optimal in detecting the out-of-contol TBE mean 1 of 2. and 5., espectively. The ATS values of the CQC- chat, exponential EWMA and exponential CUSUM chats ae listed in Table 1. The ATS cuves of the CQC, CQC-4, exponential EWMA, and CUSUM chat ae displayed in Figue 1. Table 1. ATS values of uppe-sided CQC- ( =1, 2, 3, 4) chat, exponential EWMA and exponential CUSUM chats (ATS = 5). CQC CQC-2 CQC-3 CQC-4 EWMA1 CUSUM1 EWMA2 CUSUM2 =.2 UCL=6.21 =.4 UCL=7.68 =.6 UCL=9.5 =.8 UCL=1.35 λ Q =.1 h Q =1.71 Q =1. 1 =2. S =1.39 h S =7.42 S =1. 1 =2. λ Q =.27 h Q =2.6 Q =1.1 1 =5. S =2.1 h S =4.86 S =.25 1 = CQC CQC4 EWMA1 CUSUM1 ATS Time between events mean Figue 1. ATS cuves fo uppe-sided CQC, CQC-4, exponential CUSUM and EWMA (ATS = 5). It can be seen fom Table 1 and Figue 1 that the exponential EWMA and CUSUM chats outpefom the CQC- chats at all shifts levels listed in Table 1. When detecting the shifts at designed optimal level, exponential EWMA and CUSUM show simila pefomance. On the othe hand, when the shift is elatively small and modeate, the

7 A Compaative Study of Exponential Time Events Chats 353 exponential EWMA chat shows bette pefomance than the exponential CUSUM chat; and when the shift is lage (up to 5 times of the in-contol value and above), the exponential CUSUM chat is slightly bette than the exponential EWMA chat. Fo the CQC- chats, when the shift is small, the lage the value of, the bette the pefomance of the chat is. When the shift becomes lage, the CQC chat with a smalle value will be bette than the CQC chat with a lage value. It also shows that when the mean of TBE,, inceases, the ATS of the CQC chat deceases faste than that of the CQC- chats, and thus mae the CQC chat moe sensitive to lage pocess impovement. Table 2. ATS values of uppe-sided CQC- ( =1, 2, 3, 4) chat, exponential EWMA and CUSUM chats (ATS = 37.37). CQC CQC-2 CQC-3 CQC-4 EWMA1 EWMA2 CUSUM1 CUSUM2 =.27 UCL= =.54 UCL= =.81 UCL= =.18 UCL=9.94 λ Q =.167 h Q =2. Q =.5 1 =3. λ Q =.229 h Q =2.3 Q =.5 1 =4. S =1.648 h S =5.473 S =. 1 =3. S =1.848 h S =4.86 S =. 1 = ATS CQC CQC-4 EWMA1 CUSUM Time between events mean Figue 2. ATS cuves fo uppe-sided CQC, CQC-4, exponential CUSUM and EWMA (ATS = 37.37).

8 354 Liu, Xie, Goh and Shama To investigate the effect of diffeent in-contol ATS levels, which also epesent diffeent false alam ate α, the above exponential TBE chats with in-contol ATS of ae compaed again and the coesponding ATS values ae list in Table 2. The in-contol TBE mean is assumed to be 1 ( =1.), and the exponential EWMA chat and CUSUM chat ae designed to be optimal in detecting the out-of-contol TBE mean 1 of 3. and 4., espectively. Note that the exponential EWMA and CUSUM chats ae designed such that the ATS may not be exactly equal to, howeve the values ae vey close to The esults ae shown in Table 2 and Figue 2. Compaing the esults in Table 1 and Table 2, we notice that the supeioity of exponential EWMA and CUSUM chats in ATS will be less obvious when the in-contol ATS deceases fom 5 to When the TBE mean inceases up to 3. o 4., the CQC chat shows elatively simila pefomance to the exponential CUSUM and EWMA chats. Exponential EWMA outpefoms CUSUM and CQC- chat when the uppe shifts ae small, howeve, the diffeence of ATS values among the fou chats is not significant. Anothe inteesting finding is that when the shift is up to 3.5 and above, the CQC chat shows bette ATS pefomance compaed to the CQC- chat with =2, 3, o 4. Theefoe, the CQC chat is desiable when the shift is elatively lage, and CQC- chats can be employed when the shift is small Lowe-Sided Exponential TBE Chats Lowe-sided TBE chats ae employed to detect pocess deteioations. Using the simila analysis method as the above, the in-contol ATS value is assumed to be 5, and the out-of-contol ATS values of the CQC- chat ( =1, 2, 3, 4), exponential EWMA and exponential CUSUM chat fo diffeent shifts ae compaed. Again, the in-contol mean of TBE data is assumed to be 1 ( = 1.), and the exponential EWMA chat and exponential CUSUM chat ae designed to detect the out-of-contol TBE mean of.5 and.2, espectively. Table 3 shows the ATS values of the CQC-, exponential EWMA and exponential CUSUM chats. Table 3. ATS values of lowe-sided CQC- ( =1, 2, 3, 4) chat, exponential EWMA and exponential CUSUM chats (ATS = 5). CQC CQC-2 CQC-3 CQC-4 EWMA CUSUM EWMA CUSUM =.2 LCL=.2 =.4 LCL=.92 =.6 LCL=.361 =.8 LCL=.771 λ q =.1 h q =.55 q =1. 1 =.5 T =.69 h T =4.16 T = =.5 λ q =.33 h q =.25 q =.98 1 =.2 T =.4 h T =1.24 T = =

9 A Compaative Study of Exponential Time Events Chats Table 3 shows that fo the intended design shifts, the exponential CUSUM chats outpefom the exponential EWMA chats, and the exponential EWMA chats show bette ATS pefomance than the CQC- chats. Howeve, when the pocess shift is elatively small, exponential EWMA chats show bette pefomance than the CUSUM chats, and CUSUM chats outpefom the CQC- chats (including the CQC chat). When detecting lage shifts, say, one tenth of the in-contol mean, all these exponential TBE chats have simila ATS pefomance except the CQC chat. Fo the CQC- chat, the lage the value of, the bette the pefomance of the chat is at the expense of lage pobability of false alams. Theefoe, we suggest using exponential EWMA o exponential CUSUM chat when the shift is elatively small and choosing CQC- chat when the shift is lage. Simila compaative study is also conducted with in-contol ATS of The in-contol TBE mean is assumed to be 1 ( =1.), and the exponential EWMA chat and CUSUM chat ae designed to detect the out-of-contol TBE mean of.4 and.3, espectively. The esults ae shown in Table 4. Table 4. ATS values of lowe-sided CQC- ( =1, 2, 3, 4) chat, exponential EWMA and exponential CUSUM chats (ATS = 37.37). CQC CQC-2 CQC-3 CQC-4 EWMA EWMA CUSUM CUSUM =.27 LCL=.27 =.54 LCL=177 =.81 LCL=.432 =.18 LCL=.8424 λ q =.152 h q =.4662 q =2. 1 =.4 λ q =.228 h q =.3632 q =2. 1 =.3 T =.611 h T =2.794 T = 1 = T =.516 h T =1.99 T = 1 = Fom the esults in Table 3 and Table 4, simila conclusions can be dawn as in the compaison of uppe-sided TBE chats. The supeioity of exponential EWMA and CUSUM chats in ATS will be less significant when the in-contol ATS deceases fom 5 to Fo lage pocess shifts, CQC-4 chat shows simila pefomance as the exponential CUSUM and EWMA chats. The pefomance of the CQC chat in detecting pocess deteioation is wose than the est. Theefoe, exponential EWMA o CUSUM should be used when the shift is small, and CQC- chats can be employed when the shift is lage Two-Sided Exponential TBE Chats Two-sided TBE chats ae pefeed when both pocess impovement and deteioation ae of inteest o the diection of the shift can not be pedicted. To assess the elative pefomance of the two-sided TBE chats, the ATS pefomance of the CQC- chat ( =1,

10 356 Liu, Xie, Goh and Shama 2, 3, 4), exponential EWMA and exponential CUSUM chat ae compaed. The in-contol ATS value is set to be 37.37, and the in-contol TBE mean is assumed to be 1 ( = 1.). The exponential CUSUM and EWMA chats ae designed to be optimal in detecting the out-of-contol TBE mean 1 of.3 and 3., espectively. Table 5 pesents the ATS values of the two-sided TBE chats descibed above. Table 5. ATS values of two-sided CQC- ( =1, 2, 3, 4) chat, exponential EWMA and CUSUM chats (ATS = 37.37). CQC CQC2 CQC3 CQC4 Two-sided CUSUM Two-sided EWMA α =.27 UCL=6.68 LCL=.1 α =.54 UCL=8.125 LCL=.75 α =.81 UCL=9.534 LCL=.313 α =.18 UCL=1.875 LCL=.687 S =.3 T =3. s =.516 T =1.648 h s =2.2 h T =6.5 S =.2 s =.42 h s =1.33 T =5. T =2.12 h T =5.366 λ =.22 h q =.36 h Q = =.3 λ =.152 h q =.43 h Q =2.5 1 = ATS CQC CQC-4 CUSUM EWMA Time between events mean Figue 3. ATS cuves fo two-sided CQC, CQC-4, exponential CUSUM and exponential EWMA chat.

11 A Compaative Study of Exponential Time Events Chats Figue 3 displays the ATS cuves fo two-sided CQC, CQC-4, exponential CUSUM and exponential EWMA chats. It is obvious fom Table 5 and Figue 3 that in geneal the exponential CUSUM and EWMA chats outpefom the CQC- chats. Fo the CQC- chats, the lage the value of, the bette the chat pefoms. Moeove, when the pocess impoves, the value does not influence the ATS value a lot, while when the pocess deteioates, the CQC- chat with lage value of ( = 2, 3 o 4) shows distinct supeioity to the CQC chat, and the ATS pefomance of CQC-4 chat is compaable to the coesponding exponential EWMA and exponential CUSUM chats. Moeove, the CQC-4 chat ovecomes the dawbac of the CQC chat that the ATS value inceases when the pocess has a small lowe-sided shift. A pactical disadvantage of two-sided exponential CUSUM and exponential EWMA chat is that the design pocedues ae quite complicated. A two-sided EWMA chat with one smoothing facto and diffeent uppe and lowe limit h q and h Q can only be designed to detect eithe an uppe o lowe shift quicly, while two chats with two sets of design paametes have to be employed if the uses intend to detect a cetain uppe and lowe shifts quicly. Fo two-sided exponential CUSUM chat, two individual chats have to be used to detect shifts in diffeent diections. 5. Conclusions The following conclusions can be dawn by summaizing the analysis esults above: 1. Among the uppe-sided exponential TBE chats, the diffeence in ATS values is not vey significant. When the pocess impovement is small, exponential EWMA chats ae slightly bette than exponential CUSUM chat, and both of them ae bette than the CQC and CQC- chat. Howeve, the CQC chat shows bette pefomance than the CQC- chats, and its ATS is simila to the exponential EWMA and CUSUM chat when the shift becomes lage. 2. Among the lowe-sided exponential TBE chats, the ATS pefomance of exponential EWMA and CUSUM chat ae much bette than the CQC and the CQC- chat. The exponential EWMA is moe sensitive to small deteioation, while the exponential CUSUM is suitable fo lage deteioation. Fo the CQC- chats, the lage the value of, the bette the pefomance of the chat, though at the expense of lage false alam pobability. 3. Among the two-sided exponential TBE chats, the exponential CUSUM and exponential EWMA chats outpefom the CQC- chats. Fo CQC- chats, when the pocess impoves, the paamete does not influence the ATS pefomance to a lage extent, while when the pocess deteioates, the CQC- chat with lage value of shows distinct supeioity to the CQC chat. 4. The in-contol ATS value, as a design paamete of the TBE chats, has cetain effect on the compaative pefomance of the chats. The supeioity of exponential EWMA and exponential CUSUM chats in ATS will be less significant when the in-contol ATS deceases, and thus mae the CQC and CQC- chat a bette choice because of thei simple design pocedues and less equiement of pocess infomation. Fom the viewpoint of application, the CQC and CQC- chat have some advantages compaed to exponential CUSUM and EWMA chats. Fistly, they ae moe flexible and need less infomation about the pocess. Secondly, the flexibility of CQC and CQC- chat 357

12 358 Liu, Xie, Goh and Shama mae the pocess monitoing system moe stable without too many changes in the chat design paametes due to custome equiements change. Futhemoe, the contol limits and othe impotant paametes such as the ARL and ATS ae much easie to compute compaed to exponential CUSUM and exponential EWMA chats. Based on the analysis above, we ecommend that if the pupose of employing a TBE chat is to monito pocess impovement o when the uses do not eally now whethe the pocess will impove o deteioate, i.e. whee it is difficult to pedict the pocess shift, the CQC o the CQC- chat is a bette choice as they ae easy to design and implement, and have elatively good ATS pefomance. On the othe hand, if the focus is only on pocess deteioation, and the out-of-contol shift can be accuately pedicted accoding to past data o othe infomation, the exponential CUSUM o EWMA chats will be moe efficient tools especially when the shift is small. Altenatively, CQC- chats can also be employed to detect elatively lage deteioation. Refeences 1. Boo, C. M., Champ, C. W. and Rigdon, S. E. (1998). Poisson EWMA Contol Chats. Jounal of Quality Technology, 3(4), Calvin, T. W. (1983). Quality Contol Techniques fo Zeo-defects. IEEE Tansactions on Components, Hybid and Manufactuing Technology, 6, Chan, L. Y., Xie, M. and Goh, T. N. (1997). Two-stage Contol Chats fo High Yield Pocesses. Intenational Jounal of Reliability, Quality and Safety Engineeing, 4, Chan, L. Y., Xie, M. and Goh, T. N. (2). Cumulative Quantity Contol Chats fo Monitoing Poduction Pocess. Intenational Jounal of Poduction Reseach, 38(2), Gan, F. F. (1994). Design of Optimal Exponential CUSUM Contol Chats. Jounal of Quality Technology, 26(2), Gan, F. F. (1998). Designs of One- and Two-sided Exponential EWMA Chats. Jounal of Quality Technology, 3(1), Lucas, J. M. (1985). Counted Data CUSUM s. Technometics, 27, Ohta, H., Kusuawa, E. and Rahim, A. (21). A CCC- Chat fo High-yield Pocesses. Quality and Reliability Engineeing Intenational, 17, Ranjan, P., Xie, M. and Goh, T. N. (23). On Some Contol Chat Pocedues fo Monitoing the Inte-aival Times. Poceedings of Ninth ISSAT confeence on Reliability and Quality in design. Honolulu, U.S.A., Sun, J. and Zhang, G. X. (2). Contol Chats Based on the Numbe of Consecutive Confoming Items fo the Nea Zeo-nonconfomity Pocesses. Total Quality Management, 11(2), Vademan, S. and Ray, D. (1985). Aveage Run Lengths fo CUSUM Schemes When Obsevations ae Exponentially Distibuted. Technometics, 27, Wu, Z., Yeo, S. H. and Fan, H. (2). A Compaative Study of the CRL-Type Contol Chats. Quality and Reliability Engineeing Intenational, 16, Xie, M., Goh, T. N. and Lu, X. S.(1998). A Compaative Study of CCC and CUSUM Chats. Quality and Reliability Engineeing Intenational, 14, Xie, M., Goh, T. N. and Kualmani, V. (22). Statistical Models and Contol Chats fo High-Quality Pocesses (Boston: Kluwe Academic Publishe).

13 A Compaative Study of Exponential Time Events Chats 15. Xie, M., Goh, T. N. and Ranjan, P. (22). Some Effective Contol Chat Pocedues fo Reliability Monitoing. Reliability Engineeing and Systems Safety, 77, Zhang, L. Y. Govindaaju, K., Bebbington, M. and Lai, C. D. (24). On the Statistical Design of Geometic Contol Chats. Quality Technology and Quantitative Management, 1(2), Authos Biogaphies: Jiying Liu is a eseach schola in the Depatment of Industial and Systems Engineeing, National Univesity of Singapoe. She eceived he ME in Industial Engineeing fom the Nothwesten Polytechnic Univesity at Xi an, China. He eseach inteest is statistical quality contol. Min Xie is a Pofesso of Industial and Systems Engineeing, National Univesity of Singapoe. He eceived his PhD in Quality Technology fom Linoping Univesity in His eseach aea includes quality, eliability and applied statistics. Pof Xie is an autho o co-autho of ove 1 jounal papes and 6 boos in this field. He seves as edito o associate edito of IJRQE, QTQM, IIE Tansactions, IEEE Tansactions on Reliability and seveal othe jounals. He is a Fellow of IEEE. Thong Ngee Goh is a Pofesso of Industial and Systems Engineeing at the National Univesity of Singapoe. He obtained his PhD fom the Univesity of Wisconsin-Madison. Pof Goh has been intenationally ecognized fo his expetise in quality engineeing and management. He is an autho of numeous papes and a few boos. He is an elected Fellow of ASQ as well as Academician of IAQ. Piya Ranjan Shama eceived his PhD degee fom National Univesity of Singapoe. Afte gaduation, he joined Standad Chateed Ban in Singapoe. Cuently he is an Assistant Vice Pesident in Cedit Ris Management at OCBC Ban, Singapoe. 359