CHAPTER-II Control Charts for Fraction Nonconforming using m-of-m Runs Rules

Transcription

1 CHAPTER-II Control Charts for Fraction Nonconforming using m-of-m Runs Rules. Introduction: The is widely used in industry to monitor the number of fraction nonconforming units. A nonconforming unit is a roduct which fails to meet at least one secified requirement. The control limits and erformance study of the fraction nonconforming control are tyically based on the binomial distribution. The samle fraction nonconforming is defined as the ratio of the number of nonconforming units (X) in the samle to the samle size (n). That is X ˆ. (.) n It is clear that X follows binomial distribution with arameters n and where is the robability that an unit is nonconforming. The objective of a control is to control the quality of the characteristic or to detect quickly any increase in a rocess fraction nonconforming ( ). When the rocess is in the state of in-control the mean and variance of ˆ are and resectively where is the fraction nonconforming in the n roduction rocess when the rocess is in the in-control state. If is unknown it will be estimated from observed data. The control limits for using normal aroximation are given by UCL CL LCL ( ( ) / n ) / n.

2 Alternatively the could be based on standardized statistic Z where Z is defined as follows ˆ Z (.) ( )/ n Here Z is aroximately distributed as a standard normal variate. The Shewhart X and control s are most oular control s resectively for monitoring mean and fraction nonconforming of a rocess distribution. The Shewhart standard control s are based on the three sigma control limits and give out-of-control signal if a single oint lots outside the control limits. To detect large shifts in a rocess the Shewhart control s are sensitive; however they are insensitive for small shifts. To enhance the effectiveness of the Shewhart control s to detect small shifts various runs rule are suggested and studied by several authors in literature. In order to study erformance of any one of the oularly used measures is the average run length (ARL) of a control. Boorke and Evans (97) have develoed the arkov chain aroach to determine the average run length moments and ercentage of oints of the run length distribution. Cham and Woodall (987) emloyed arkov chain aroach to derive ARL for the Shewhart with sulementary runs rules. In literature m-of-k have been studied by various researchers. The m-of-k signals if consecutive m oints of k oints lie outside control limit. Hurwitz and athur (99) suggested to combine -of- rule with the -of- runs rule and to use.5 warning limits on the X. Klein () suggested the two alternatives to the Shewhart X and develoed the -of- and -of- control s which have symmetric uer and lower control limits. Both s have a better erformance than the Shewhart X. Khoo () studied the erformance of -of- -of- -of-4 -of- and -of-4 control s. He suggested the -of- and -of-4 control s for detecting 4

3 the small to moderate shifts. Khoo and Ariffin (6) roosed two imroved runs rules to enhance the erformance of the two rules suggested by Klein () for the detection of the large shifts. Acosta-ijia (7) studied the statistical characteristics of both m-of-m runs rule and (m-)-of-m runs rule to sulement the Shewhart control. Zang and Wu (5) studied the design of control with sulementary runs rules. Antzoulakos and Raktitzis (8) suggested the modified r-of-m control to detect the small to moderate shifts in the rocess mean. Acosta-ijia and Pignatiello (9) studied the erformance of k-of-k runs rules for monitoring the rocess standard deviation. Lim and Cho (9) investigated the economic-statistical roerties of X sulemented with m-of-m runs rules and also studied the steady-state ARL. To monitor fraction nonconforming of a manufacturing rocess Khoo () roosed the -of- -of- and -of-4 control s are as an alternative to the Shewhart standard. All roosed control s are easy to imlement and have better erformance than the standard. Literature review reveals that -of- for fraction nonconforming have not been discussed so far to the best of our knowledge. Also steady-state erformance of m-of-m runs rules are not studied in resence or absence of warning limits. In this Chater we roose control s for fraction nonconforming using m-of-m runs rules with and without warning limits to study the zero-state and steady-state roerties. The zero-state mode in which rocess shifts can occur at the beginning of the samling interval. Rest of the Chater is organized as in Section. we describe m-of-m runs rules without warning limits and reort the erformance of the m-of-m studied under zero-state and steady-state mode. While Section. gives m-ofm with warning limits and its erformance is studied under only steadystate mode. In Section.4 we briefly describe cumulative count of conforming the m-of-m cumulative count of conforming control 5

4 and its erformance under steady-state mode is given. Section.5 gives conclusions.. Runs Rules Schemes without Warning Limits: Consider a control with two control limits (UCL LCL). Let us consider the three regions for a control. The region between uer control limit and lower control limit (Region ). The region above uer control limit (Region ). The region below lower control limit (Region ). These three regions are shown in the Figure.. Region UCL Region CL Region LCL Figure.: Fraction nonconforming control. The robability of single oint falls in the regions are denoted by c u l resectively. When a fraction nonconforming is then c rlcl Z UCL where we assume that the control statistic ˆ Z to follow a n / standard normal distribution. 6

5 Now c r LCL / n ˆ UCL / n LCL r ( )/ n UCL ( ) / n Z ( )/ n ( )/ n UCL / / n LCL n / n / n. Similarlly u UCL LCL l / / / n n / n n where (.) denotes the distribution function of a standard normal distribution... The m-of-m : We consider the information from multile samles including the most recent one to make a decision to signal an out-of-control status. This is achieved by considering a run of the samle oints that fall beyond the control limits including the current samle and one or more samles from the recent ast samles. The m-of-m signals an out-of-control status at any time oint when m-consecutive samle oints fall either above UCL or below LCL. Suose mm denotes the event when two successive oints fall in region m. The control signals an out-of-control status when an event D... occurs. Define states of the arkov chain as follows: mtimes mtimes State : One oint fall between both control limits. State : One oint falls above uer control limit. State : One oint falls below lower control limit. 7

6 State 4: Two consecutive oints fall above uer control limit. State 5: Two consecutive oints fall below lower control limit. State 6: Three consecutive oints fall above uer control limit. State 7: Three consecutive ints fall below lower control limit and so on. Finally State m: Out-of-control (absorbing) state with associated attern given by the set D. The arkov chain reresentation of consist of m states with the first ( m ) of them being transient. A state is said to be transient state if and only if starting from state one the robability of returning to state one after some finite length of time is less than one. Then the matrix can be artitioned as m m transition robability Q P ( I Q) J where Q is the ( m ) (m ) transition robability matrix for the transient sates I is the ( m ) (m ) identity matrix and J is the column vector of one of an order ( m ). The robability mass function (.m.f.) the exected value and the variance of the run length random variable T are resectively given by and Var r T i eq I Q J i... i (.) ei Q J (.4) E T T ei QI Q J ET (.5) 8

7 where... e is the initial distribution. Let j be the exected m value of the waiting time from state j until the first occurrence of D. Thus if rocess is initially in-control is the ARL. Let... m be the vector of average run lengths. By taking exectations conditional uon the result of the first subgrou these exected values can be found by solving the following linear system of equations corresonding to I QJ where is the column vector of ones (see for examle Acosta-ejia and Pignatiello (9)). 4 5 c. c. c. c. c.... m4 m m m c. c. c. c. u. u. u. u. u. l. u. l. u. l. l. l. l. l.... u l. m By solving the above linear system of equations the ARL for a with m-of-m runs rule m is given by m m m u l m m m m u l u. l u l c u l. (.6) In the next subsections we describe -of- -of- and -of- s as an illustration of m-of-m. 9

8 .. The -of- : The -of- (standard ) signals an out-of-control status if a oint falls either above uer control limit or below a lower control limit. The -of- is Shewhart control... The -of- : The -of- for a fraction nonconforming is studied by Khoo (). The -of- signals an out-of-control status if two consecutive oints ( Z i and Z i ) fall either above an uer control limit or below lower control limit. In other words if two successive oints fall in the region or region the -of- signals an out-of-control status...4 The -of- : In literature the develoment of -of- for a fraction nonconforming has not been reorted to the best of our knowledge. Therefore we study the -of- control to increase the sensitivity of control for fraction nonconforming. The -of- signals an out-of-control status if three consecutive oints ( Z i Zi and Z i ) fall either above uer control limit or below lower control limit. We now resent the arkov chain aroach to obtain ARL of with - of- runs rule. Consider an absorbing arkov chain with five transient states which are defined as follows. State : A oint falls between both control limits. State : A oint falls above an uer control limit. State : A oint falls below lower control limit. State 4: Two successive oints fall above uer control limit. State 5: Two successive oints fall below lower control limit.

9 State 6: An absorbing state three successive oints falls either above uer control limit or below lower control limit. Then the 6 6 one ste transition robability matrix of the arkov chain can be exressed as follows. c c c c c u u u l l l u l. u l (.7) Let Q be matrix obtained from the matrix defined in (.7) by removing the last row and column. ARL of -of- is obtained by solving the linear system corresonding to QJ vector of ones I where I is 5 5 identity matrix and J is column 4 5 c. c. c. c. c. u. l. u. l. u. l. u. l 5 4 (.8) where j denotes the exected value of the waiting time from state j to until the first occurrence of an out-of-control signal. Let 4 be the vector of ARL s. Solving the above linear system of equations the ARL of -of- is given by u l u l u. l u l c u l 5. (.9) Let u u l and c u l hence c u. Substituting c u in equation (.9) gives an in-control average run length (ARL())

10 u u ARL (). (.) u ( u) ( u)( u u ) Now if we set ARL( ) 7.4 then from equation (.) u. 56 and using standard normal distribution the control limits are.7. In the following subsection we gives erformance of m-of-m...5 Performance Study of the m-of-m Control Chart: The ARL erformance of based on the standardized statistic Z defined in equation (.) with control limits (-of- ) the -of- and -of- control s are given in Table.. We assume. 5for this study. Similarly the comuted ARL values based on. are given in Table. for rules -of- -of- and -of-. The ARL is comuted under zero-state mode. Since the ARL as a single arameter is not necessarily very tyical value of the run length distribution. Therefore the standard deviation of the run length (SDRL) distribution is also given in arentheses. The -of- and - of- control s are simle and easy to imlementation for ractitioners; we consider only these two control s.

11 Table.: ARL and SDRL rofile of m-of-m for n= ARL( ) 7.4 and of- 7.4 (69.9).4 (.74) 94.8 (94.) 44.8 (4.88).9 (.4).4 (.5) 8.9 (7.57) 5.4 (4.87).85 (.).9 (.5). (.7).9 (.).6 (.).44 (.8). (.64). (.5) -of- 7.7 (68.9) 88.6 (87.8) 7.5 (7.9).64 (.7) 7. (5.66).6 (8.84) 6.78 (5.47) 4.96 (.64).89 (.56). (.87).8 (.4).54 (.8).5 (.84). (.66).5 (.5).9 (.4) -of- 7.9 (68.) (76.67) (64.55) 9.94 (7.78) 6. (4.).9 (8.) 7. (5.5) 5.58 (.5) 4.6 (.5) 4.4 (.84).66 (.4).4 (.7).6 (.8).6 (.64). (.49).6 (.8) In above table values in arenthesis indicate SDRL.

12 Table.: ARL and SDRL rofile of m-of-m for n= ARL( ) 7.4 and of- 7.4 (69.9) 67.7 (66.77) 6.6 (6.) 6.75 (6.5).5 (.84) 7.6 (7.9) 4.84 (4.).6 (.8).5 (.96). (.4).67 (.6).45 (.8). (.64). (.5).4 (.4).9 (.).6 (.5).4 (.) -of- 7.7 (68.9) 48.5 (47.9) (48.9).5 (9.9).6 (9.74) 6.86 (5.55) 4.8 (.5).7 (.7).8 (.69).68 (.6).44 (.95).8 (.7).7 (.57). (.4).7 (.4).4 (.6). (.9). (.4) -of- 7.9 (68.) 4.8 (9.5) (44.6).4 (8.8).4 (9.7) 7.48 (5.4) 5.58 (.5) 4.55 (.4).95 (.75).59 (.9).6 (.98). (.74). (.57).8 (.4).5 (.). (.5). (.8). (.) In above table values in arenthesis indicate SDRL. 4

13 The results in Tables. and. show the erformance of -of- is better than the -of- and -of- s. On the contrary -of- outerforms for larger shifts. This is only minor concern since for small shifts -of- and -of- s erforms significantly better than the -of-. It is also observed that standard deviation of the run length distribution is significantly large. Such a henomenon is also observed by Antzoulakos and Rakitzis (8). The run length distribution of m-of-m control is a highly skewed distribution with a right tail which decreases slowly for small shifts in a rocess fraction nonconforming. In such cases quantiles give more information about erformance study of a control (see for examle Shmueli and Cohen ()). So we also study three quartiles denoted by Q Q and Q of the run length distribution of roosed m-of-m control. Quartiles of the run length distribution are obtained using equation (.5) under zero-state mode. Table. and Table.4 give the three quartiles and inter quartile ranges (IQR) of the run length distribution associated with m-of-m control when ARL()= 7.4. Table.: Quartiles and IQR of m-of-m control under zero-state mode when. 5and n=. -of- -of- -of- Q Q Q IQR Q Q Q IQR Q Q Q IQR

14 Table.4: Quartiles and IQR of m-of-m control under zero-state mode when. and n=. -of- -of- -of- Q Q Q IQR Q Q Q IQR Q Q Q IQR The results in Tables. and.4 show that the quartiles and IQR values of the -of- are smaller than the -of- as well as -of- s. The -of- has a significantly better erformance than the -of- and -of- to detect small to moderate shifts. We describe steady-state ARL of m-of-m in the next subsection...6 Steady-State Average Run Length: Crosier (986) suggested a technique for obtaining steady-state ARL. After that many researchers reorted the study of steady-state roerties of a control in literature. Here we study the steady-state roerties of m-of-m fraction nonconforming control. If rocess is running for a long eriod in an in-control condition it will reach in a steady-state condition. In order to study the long term roerties of a control it is aroriate to investigate the steady-state average run length (SSARL). 6

15 Let Q be a square matrix obtained from Q by imosing the condition that no T signal occurs. Let... m be the vector of steady-state robabilities for the in-control transient states. The steady-state robabilities can be obtained by solving the following equations: T T T Q and. (.) m Under the in-control situation let c and u c l. As an illustration the conditional transition robability matrix for in-control states for m= is Q C C C C C C C C C C C C C where C u C u C u and u C u. u The stationary robabilities from equation (.) are given by C C C C C 4 5. C C C C C The SSARL can be obtained by where ARL T SSARL ARL (.) I Q J T... m 7

16 The in-control SSARL can be obtained as T SSARL( ) ARL(). (.) For -of- the ARL vector can be obtained by solving linear system of equations given in equation (.8) as l l u u D l l u D u u l D 5 4 l l D u u D where u l c u u l D u. l l. The SSARL of -of- can be obtained by SSARL. (.4) D The in-control SSARL of -of- can be obtained by substituting u u l and c u l in equation (.4) as u u.. u u u. u u u u. u u u SSARL ( ). (.5) 8

17 Suose that the desired in-control SSARL is aroximately 7.4. Solving equation (.5) for u gives u The control limits for -of- control are. using standard normal distribution. In the following subsection we study the steady-state erformance of m-of-m...7 Steady-State Performance of the m-of-m Control : The SSARL erformance of based on the standardized statistic Z sulemented with runs rules -of- and -of- for. 5and n= are given in Table.5. Table.6 gives the SSARL of -of- -of- and -of- control s for. and n=. The SDRL is also given in arentheses. We observe that from Tables.5 and.6 the erformance of -of- is significantly better than the -of- as well as -of- control s to detect small to moderate shifts. The -of- has a higher ower of detecting outof-control signal. 9

18 Table.5: SSARL and SDRL rofile of m-of-m for n= ARL( ) 7.4 and.5. -of- -of- -of (69.9) 7.6 (69.86) 7.7 (66.4).6.4 (.74) (87.6) (74.54) (94.) 7. (7.) 66.7 (6.) (4.88).49 (.) 9.6 (6.9).9.9 (.4) 6.89 (5.69) 5.9 (.49)..4 (.5).7 (8.85) 9.97 (7.66). 8.9 (7.57) 6.7 (5.49) 7. (4.76). 5.4 (4.88) 4.89 (.65) 5.4 (.4)..85 (.).84 (.57) 4.47 (.5).4.9 (.5).9 (.88).89 (.49).5. (.7).77 (.4).5 (.) In above table values in arenthesis indicate SDRL.

19 Table.6: SSARL and SDRL rofile of m-of-m for n= ARL( ) 7.4 and.. -of- -of- -of (69.9) 7.6 (69.86) 7.7 (66.4) (66.76) 48. (47.4) 4.9 (7.59). 6.6 (6.) 49.5 (48.8) 46.7 (4.5) (6.5).9 (9.7). (7.59).4.5 (.84).97 (9.76). (8.77) (7.) 6.79 (5.56) 7.9 (5.) (4.) 4.76 (.5) 5.4 (.5).7.6 (.8).66 (.9) 4.4 (.8).8.5 (.95). (.7).8 (.9).9. (.4).64 (.7).45 (.88)..67 (.6).4 (.97). (.4) In above table values in arenthesis indicate SDRL.

20 In Table.7 we resent control limits of the -of- -of- -of- control s under zero-state and steady-state mode. The control limits of -of- are same under zero-state and steady-state mode. However there is no significant change in the control limits of the m-of-m with m= under zero-state and steady-state mode. Therefore there is no significant difference observed between zero-state ARL and SSARL. Table.7: Control limits for m-of-m. Control limits -of- -of- -of- Zero-state Steady-state. 78. In the next section we study steady-state roerties of m-of-m with warning limits. To imrove the overall erformance of s we design s with combined -of- and m-of-m runs rules.. The m-of-m Runs Rules Schemes with Warning Limits: The case of control s with warning limits was first extensively studied by Page (955) who roosed four runs rules. He also studied the erformance of control s by introducing a arkov chain aroach to calculate exact run length distribution. osteller (94) Dudding and Jannett (94) and Weiler (95) have discussed the case of warning limits and runs rules. An extensive review on warning limits and runs rules can be found in Koutras al et. (7). Warning limits should be drawn in less extreme osition than the control limits. If secified number of oints falls between warning limit and control limit control signals an out-of-control status. To increase sensitivity of with sulementary runs rules we roose m-of-m with warning limits. For this we first define following notations: UWL- denote uer warning limit.

21 LWL - denote lower warning limit. - the robability of a single oint falling above UCL. - the robability of a single oint falling below LCL. c - the robability of a single oint falling between both warning limits. u - the robability of a single oint falling between uer control limit and uer warning limit. l - the robability of a single oint falling between lower control limit and lower warning limit. Consider a control based on standardized statistic Z with two control limits (UCL= k LCL= -k) and two warning limits (UWL= w LWL= -w). Let us consider five regions for the control : The region between both warning limits (Region ). The region between uer control limit and uer warning limit (Region ). The region between lower control limit and lower warning limit (Region ). The region above uer control limit (Region 4). The region below lower control limit (Region 5).

22 4 These five regions are shown in the following Figure.. Figure.: Fraction non-confirming control with warning limits. We assign numbers 4 and 5 for above defined regions 4 5 resectively. Thus a set 45 A denotes ossible numbers assigned to regions. The robability that a single oint falls in regions 4 and 5 are resectively given by l u c and where / / / / n n LWL n n UWL c / / / / n n UWL n n UCL u / / / / n n LCL n n LWL l / / n n UCL Region 5 LCL Region LWL Region CL UWL UCL Region 4 Region

23 and LCL / n / n Under we have c and u u l. The rocess is in the in-control state if and in the out-of-control state otherwise. The m-of-m fraction nonconforming control signals an out-of-control status if a oint falls above (below) uer (lower) control limit or m- consecutive oints fall between UWL and UCL or LCL and LWL. In other words the m-of-m signals if an event mtimes mtimes occurs. The states of the arkov chain are defined as follows: State : One oint falls between warning limits. State : One oint between UWL and UCL. State : One oint between LCL and LWL. State 4: Two consecutive oints between UWL and UCL. State 5: Two consecutive oints between LCL and LWL. State 6: Three consecutive oints between UWL and UCL. State 7: Three consecutive ints between LCL and LWL and so on. Finally State m: Out-of-control (absorbing) state with associated attern given by the set. 5

24 Further rocedure of obtaining SSARL of m-of-m is same as given in subsection..6. To find the warning limits of control under steady-state mode suose that the desired SSARL() is aroximately 7.4. We may note that the and ARL() are functions of m control limit (k) and warning limit (w) only. Therefore for a given combination of m and control limit (k) the warning limits can be determined from equation (.). In Section. we studied zero-state and steady-state roerties of m-of-m without warning limits. We observed that if the rocess running in an incontrol state for long time eriod or rocess shift is occurs at any time in the samling interval the erformance of m-of-m is aroximately same in both the situations. Therefore in this section we study only steady-state roerties of the m-of-m... Steady-State erformance Study of m-of-m Control Chart with Warning Limits: We investigate SSARL roerties of -of- and -of- control s and comare its erformance with the standard (-of- ). We choose k=. and. to comare s under study. The SDRL is also given in arentheses along with the SSARL since SSARL alone does not reveal imortant information regarding the erformance of a control. The steady-state SDRL can be obtained using equation (.5). Table.8 gives the results of the SSARL and SDRL of the Shewhart standard (-of- ) - of- and -of- control s. 6

25 Table.8: SSARL and SDRL of m-of-m control with. 5 n= of- 7.4 (69.9) 94.4 (9.84) 6.8 (5.78).8 (9.87) 5. (4.8).9 (.74). (.75).8 (.).5 (.87). (.64). (.49) k k. -of- w.899 -of- w.47 -of- w.876 -of- w (69.8) 94.8 (9.78) 6.4 (5.74).7 (9.85) 5. (4.79).8 (.7). (.74).79 (.9).5 (.86). (.64). (.49) (69.85) 94.4 (9.7) 6.7 (5.65). (9.77) 5.8 (4.7).6 (.69). (.7).79 (.8).49 (.85). (.64). (.49) (69.9) (76.7) 9.8 (8.9) 8.4 (7.4) 4.4 (.54).9 (.9).8 (.8).77 (.99).5 (.75).5 (.59). (.48) (67.4) (7.4) 9.7 (7.44) 8.9 (6.56) 4.58 (.6). (.9).4 (.).88 (.96).59 (.75).9 (.6).6 (.48) In above table values in arenthesis indicates SDRL. 7

26 For the value of control limit k=. the SSARL and SDRL of -of- -of- -of- s are not differ much. However for k=. the SSARL and SDRL values of m-of-m with m= are significantly different. Therefore the value of control k=. is chosen to study the steady-state roerties of m- of-m. The -of- control erforms better than the -of- as well as -of- control s to detect small to moderate shifts in a rocess when k=.. Table.9 and Table. also give the SSARL and SDRL m-of-m with in-control fraction nonconforming.. 5. Table.9: SSARL and SDRL rofile of m-of-m control with. n= k=. k=. -of- -of- -of- -of- -of- w.899 w.47 w.876 w (69.9) (69.8) (69.85) (69.9) (67.4) (66.77) (66.7) (66.66) (4.85) (.8) (6.) (6.5) (6.97) (45.5) (4.5) (6.5) (6.) (6.) (7.95) (5.8) (.84) (.8) (.7) (8.6) (7.59) ( 5.74 (7.9) (7.8) (6.98) 4.8) (4.) (4.) (4.) (4.) (.98) (.6) (.8) (.8) (.75) (.) (.79) (.96) (.95) (.9) (.4) (.) (.4) (.4) (.8) (.8) (.) (.6) (.5) (.4) (.85) (.84) In above table values in arenthesis indicate SDRL. 8

27 Table.: SSARL and SDRL rofile of m-of-m control with.5 n= of- 7.4 (69.9).4 (.74) 94.8 (94.) 44.8 (4.88).9 (.4).4 (.5) 8.9 (7.57) 5.4 (4.87).85 (.).9 (.5). (.7).9 (.).6 (.).44 (.8). (.5) -of- w (69.8).7 (.67) (94.6) 44. (4.8).88 (.7). (.5) 8.7 (7.55) 5.9 (4.86).84 (.).9 (.4). (.7).9 (.).6 (.).44 (.79). (.6) k=. -of- w (69.85).6 (.65) 94.7 (94.8) 44.4 (4.7).78 (.5).9 (.8) 7.99 (7.44) 5. (4.77).8 (.).87 (.9).8 (.68).89 (.7).6 (.99).44 (.78). (.6) -of- w (69.9) (8.9) 7.89 (69.98).7 (9.74) 5.54 (4.57) 9. (8.4) 5.8 (4.89) 4. (.). (.).5 (.6). (.4).8 (.98).6 (.79).46 (.66).5 (.48) k=. -of- w (67.4) (7.48) 64.4 (6.8) 7.7 (5.6) 4. (.45) 8.57 (6.87) 5.76 (4.8) 4. (.75).9 (.9).69 (.44).7 (.).97 (.9).7 (.79).55 (.68).9 (.5) In above table values in arenthesis indicate SDRL. 9

28 From Table.9 and Table. we observed that the SSARL and SDRL values of the -of- control are significantly better than the -of- and - of- control s when k=.. In general the -of- control has a higher ower of detecting an out-of-control signal. Table. and Table. give the three quartiles and IQR of the run length distribution associated with m-of-m control when SSARL()= 7.4. Table.: Quartiles and IQR of m-of-m control under steady-state mode when.5 n=. -of- -of- -of- Q Q Q IQR Q Q Q IQR Q Q Q IQR Table.: Quartiles and IQR of m-of-m control under steady-state mode when. n=. -of- -of- -of- Q Q Q IQR Q Q Q IQR Q Q Q IQR

29 We have seen that from Table. and Table. the erformance of -of- is significantly better than the -of- as well as -of- s. For comarison study of m-of-m with warning limits and without warning limits we rovide SSARL and SDRL of m-of-m in Table.. Table.: SSARL and SDRL of m-of-m when n= and SSARL ( ) Control s with warning limits -of- -of- -of (69.9) (69.9) (67.4) (66.77) (4.85) (.8) (6.) (45.5) (4.5) (6.5) (7.95) (5.8) (.84) (8.6) (7.59) (7.9) (4.8) (4.) (4.) (.98) (.6) (.8) (.) (.79) (.96) (.4) (.)..9.5 (.4) (.8) (.) (.6) (.85) (.84) Control s without warning limits -of- -of- -of (69.9) (69.86) (66.4) (66.76) (47.4) (7.59) (6.) (48.8) (4.5) (6.5) (9.7) (7.59) (.84) (9.76) (8.77) (7.) (5.56) (5.) (4.) (.5) (.5) (.8) (.9) (.8).5..8 (.95) (.7) (.9) (.4) (.7) (.88) (.6) (.97) (.4) In above table values in arenthesis indicate SDRL. It is clear that from Table. the m-of-m with warning limits has a better erformance over the m-of-m without warning limits. The SSARL and SDRL values of m-of-m with warning limits for m= are significantly better than the control without warning limits. 4

30 We investigate steady-state roerties of cumulative count of conforming control in the next section..4 Cumulative Count of Conforming Control Chart: The fraction nonconforming of modern manufacturing rocesses are usually very low at arts er million (m) and one seaks of high yield rocesses. A high yield rocess is defined as a rocess with an in-control fraction nonconforming ( ) is at the most. or m or such a smaller number. The Shewhart is widely used to monitor the fraction nonconforming of items. The Shewhart is not suitable for monitoring the high yield rocesses since for a large subgrou size the number of nonconforming units in a subgrou is assumed to be aroximately normal. When the fraction nonconforming of units is very small normal aroximation might be incorrect. Goh (987) showed that use of Shewhart in high yield rocesses results in high false alarm rates and inability to detect rocess imrovements. In order to rovide adequate statistical rocess control technique for high-yield rocesses cumulative count of conforming (CCC) control based on geometric distribution roosed by Goh (987) as an alternative to the. The concet of cumulative count of conforming items first introduced by Calvin (98) and further studied by Xie and Goh (99 ). Chen and Cheng (8) develoed the design of CCC with sulementary runs rules. The roosed CCC with -of- and -of- runs rules have better erformance than the CCC. The CCC gives an out-of-control signal when a oint falls beyond the control limits. Sulementary runs rules are used to enhance the sensitivity of control s. In the following subsection we describe m-of-m CCC and develo arkov chain model to evaluate the steady-state ARL erformance. 4

31 .4. CCC Control Chart: Goh (987) roosed CCC to monitor high yield manufacturing rocesses. In the CCC the number of items are insected until one nonconforming item is observed for monitoring the manufacturing rocesses follows the geometric distribution. Let X be a random variable having a geometric distribution then its robability mass function is given by r x X x x... (.6) In manufacturing rocesses the value of fraction nonconforming of items ( ) is very small. The cumulative distribution function of random variable X is given by F( x) x x x where x is the largest integer less than x. If the accetable risk of false alarm is assumed equal to the control limits of CCC can be obtained aroximately. The m-of-m CCC gives an out-of-control signal when a oint falls outside of the control limits or m-successive oints fall beyond the control limits. For m-of-m CCC we consider uer control limit (UCL) and lower control limit (LCL). In m-of-m CCC we define three regions: The region between uer control limit and lower control limit (region ). The region above uer control limit (region ). The region below lower control limit (region ). Let u denote the robability that a oint falls above uer control limit. 4

32 l denote the robability that a oint falls below a lower control limit. c denote the robability that a oint falls between both the control limits. The robability of a single oint falls in regions and is given by u FUCL l FLCL and c FUCL FLCL. The rocedure of obtaining SSARL of m-of-m CCC is the same as given in subsection..6. As an illustration we exlain -of- CCC control in detail in the following subsection..4. The -of- CCC Control Chart: Chen et al. (9) has studied the -of- CCC and its ARL comuted under zero-state mode. Here we study the steady-state roerties of -of- CCC. The -of- CCC generate an out-of-control signal if - successive oints falls outside of the control limits of -of-. That is - of- CCC signals when the event D D or occurs where D D X i X and D X i X. For m= the transition i robability matrix is given below: i c Q c c u u l u. T Let denote the vector of steady-state robabilities for the incontrol transient states. The steady-state robabilities can be obtained by solving the following equations: T T T Q and. (.7) The matrix Q can be obtained from Q by imosing the condition that there is no signal. The matrix Q is given below: 44

33 Q C C C C C C C From equation (.7) we have C C. C. C C ( C Then SSARL of -of- CCC is given by T SSARL ARL where. ) ARL ( I Q) ARL l. u u l c u l l. u c( l)(. u) Therefore SSARL.... Let u l u and c u l. Now SSARL of -of- becomes SSARL ( u). (.8) u It is easy to solve equation (.8) for given desired SSARL(). Therefore the secific control limits of -of- CCC can be exressed as follows: UCL LCL of of ln( u) ln( ) ln( l). ln( ) The following subsection gives erformance study of m-of-m CCC. 45

34 .4.4 Performance Study of m-of-m CCC Control Chart: In the erformance study we resent the SSARL and SDRL of CCC (- of- CCC ) with sulementary runs rules (-of- and -of- s). For comarison study the SSARL and SDRL of -of- are used as a baseline. Suose that accetable fraction nonconforming equals to.5 and desired SSARL() s are aroximately 7 and resectively. Table.4 and Table.5 show the SSARL and SDRL (given in arentheses) of - of- -of- and -of- control s with SSARL()= 7 and.5 resectively. From the Tables.4-.5 we have seen that the out-of-control SSARL and SDRL values of -of- are significantly better than the -of- as well as -of-. The erformance of -of- is also significantly better than the -of- CCC. In general -of- has a higher ower of detecting out-of-control signal in a rocess. It is imortant to note that when the fraction nonconforming increases out-of-control SSARL and SDRL values of the m-of-m are greater than the in-control SSARL and SDRL. This has been identified by Xie et al. () and studied by several researchers (see for examle Cheng and Chen (8)). For the m-of-m we also comuted three quartiles and IQR of run length distribution. Table.6 and Table.7 show the results of quartiles and IQR of m-of-m. Conclusion from quartiles and IQR is that the erformance of -of- is better than the -of- and -of- s. 46

35 Table.4: SSARL and SDRL rofile of m-of-m when SSARL()=7 and.5. -of- -of- -of (.78) (4.5) (5.8) (.4) (6.7) (.79) (5.46) (55.7) (8.6) (78.87) (7.) (9.5) (69.7) (69.5) (66.77) (44.67) (47.4) (.67) (44.44) (7.8) (4.5) (69.84) (98.9) (54.48) (.) (4.8) (5.54) (97.6) (97.87) (89.44) (7.7) (65.66) (7.) (47.84) (4.96) (58.6) (.4) (6.) (4.8) (98.5) (9.55) (4.97) (74.9) (74.56) (6.6) (48.66) (55.7) (8.89) (8.9) (7.6) (.8) (99.7) (7.47) (8.56) (74.7) (6.95) (5.4) In above table values in arenthesis indicate SDRL. 47

36 Table.5: SSARL and SDRL rofile of m-of-m when SSARL()= and.5. -of- -of- -of (.9) (.79) (4.6) (.) (.67) (4.8) (6.6) (8.48) (4.8) (5.8) (.99) (9.4) (99.6) (99.5) (97.) (5.) (.77) (5.45) (.) (.47) (6.58) (99.9) (6.) (9.) (78.9) (.4) (9.) (6.49) (9.7) (7.4) (45.95) (9.77) (56.9) (.79) (78.8) (46.9) (4.66) (59.4) (.74) (7.) (5.56) (8.7) (94.4) (4.5) (.59) (8.) (.78) (5.48) (64.6) (.74) (.4) (5.45) (6.6) (7.9) (4.6) (.) (4.4) In above table values in arenthesis indicate SDRL. 48

37 Table.6: Quartiles and IQR of m-of-m CCC control under steady-state mode when ARL()= 7 and. 5. -of- -of- -of- Q Q Q IQR Q Q Q IQR Q Q Q IQ R 49

38 Table.7: Quartiles and IQR of m-of-m CCC control under steady-state mode when ARL()= and. 5. -of- -of- -of- Q Q Q IQR Q Q Q IQR Q Q Q IQR

39 .5 Conclusions: In this Chater we have studied the m-of-m with warning limits and without warning limits. Performance of the roosed control without warning limits studied under the zero-state and steady-state mode. Proosed - of- has a better erformance than the -of- and -of- s. Fraction nonconforming m-of-m is designed to detect the small to moderate shifts in the rocess fraction nonconforming. An imrovement of the Shewhart - sulemented with m-of-m runs rules with warning limits is better than the control without warning limits. In this Chater we also investigated steady-state roerties of m-of-m CCC control. The steady-state erformance of roosed is significantly better than the regular CCC control. In the next Chater we develo control s for fraction nonconforming due change in location arameter. 5