Control Charts with Supplementary Runs Rules for Monitoring Bivariate Processes

Transcription

1 Control Charts wth Supplementary Runs Rules for Montorng varate Processes Marcela. G. Machado *, ntono F.. Costa * * Producton Department, Sao Paulo State Unversty, Campus of Guaratnguetá, 56-4 Guaratnguetá, Sao Paulo, rasl Emal: marcela@feg.unesp.br, fbranco@feg.unesp.br bstract Control charts based on the T and S statstcs are common used for montorng multvarate processes. However, the T and S statstcs are not well known by the users. s an alternatve to the use of these charts, n ths artcle we propose a new chart for montorng the mean vector and the covarance matrx of bvarate processes, named as the MRMX chart. The montorng statstc assocated to the MRMX chart s based on the sample means and sample ranges. s the practtoners are, n general, more famlar wth means and ranges, they wll not have dffcult to use the proposed chart. Followng the rule proposed by Shewhart, the MRMX chart sgnals when a sample pont falls n the acton regon. The MRMX chart has the classcal dsadvantage of all the control charts that adopt the rule proposed by Shewhart: t s not effcent to detect small shfts n the process parameter beng montored. Ths way, we nvestgate the effcency of the MRMX chart n detectng shfts n the mean vector and/or n the covarance matrx under the use of supplementary and synthetc runs rules. In terms of effcency, we can say that, the MRMX charts wth supplementary or synthetc runs rules have better performance than the MRMX wthout runs rules. However, t s mportant to hghlght that the synthetc runs rule s smpler for the users than the supplementary runs rules. Markov chan models are used to obtan the propertes of the proposed control charts. Keywords: control charts; mean vector; covarance matrx; bvarate processes; supplementary runs rules. Introducton The control charts proposed by Shewhart were desgned for montorng the mean and the varance of unvarate processes. Nowadays, an ncreasng number of processes s requrng the montorng of two or more than two qualty characterstcs. In ths context, Hotellng (947) proposed the T chart to detect changes n the mean vector of multvarate processes, and lt (985) proposed the generalzed varance S chart to detect changes n the covarance matrx Σ. fter the works of Hotellng and lt, a growng number of researchers has been dealng wth the statstcal control of multvarate processes. For example, Costa and Machado (8) and Champ and pars (8) consdered the use of the double samplng procedure wth the chart proposed by Hotellng. Costa and Machado (8a, 9), Machado and Costa (8) and Machado et al. (8) consdered the largest value among the sample varances of the qualty characterstcs to control the covarance matrx of multvarate processes. Costa and Machado () also proposed a chart based on sample ranges, the RMX chart, for montorng the covarance matrx of multvarate processes. The jont control of the mean vector and the covarance matrx of multvarate processes has also been studed by several authors. Chou et al. () have consdered the control of the mean vector and the covarance matrx by usng log-lkelhood rato statstcs. Takemoto and rzono (5) consdered the Kullback Lebler nformaton to control multvarate processes. Khoo (5) studed the jont propertes of the T and S charts. The speed wth whch these charts sgnal changes n the mean vector and/or n the covarance matrx was obtaned by smulaton. Chen et al. (5) proposed a sngle EWM chart to control both, the mean vector and the covarance matrx. Ther ID.

2 ICIEOM - Gumarães, Portugal chart s more effcent than the jont T and S n sgnalng small changes n the process. Zhang and Chang (8) proposed two EWM charts based on ndvdual observatons that are not only fast n sgnalng but also very effcent n nformng how the assgnable cause affected the process; f only changng the mean vector or only changng the covarance matrx or changng both. Machado et al. (9) proposed two statstcal tools for montorng the mean vector and the covarance matrx of bvarate processes: the MVMX chart, whch only requres the computaton of statstcs famlar to the users, that s, sample means and sample varances and two charts for jont use based on the noncentral ch-square statstc (NCS statstc). The jont NCS charts are recommended to dentfy the out-ofcontrol varable nstead of the nature of the dsturbance, that s, the one that only affects the mean vector or only affects the covarance matrx or both. In ths artcle, we propose the MRMX chart to control the mean vector and the covarance matrx of bvarate processes. The sample ponts correspond to the maxmum among the values of the standardzed sample means (n module) and ranges (weghted) of two qualty characterstcs. User s famlarty wth sample means and sample ranges s a pont n favor of the MRMX chart. Followng the rule proposed by Shewhart, the MRMX chart sgnals when a sample pont falls n the acton regon. The MRMX chart has the classcal dsadvantage of all charts that adopt the rule proposed by Shewhart: t s not effcent to detect small changes. Ths handcap of Shewhart control charts was recognzed at the very begnnng by the Western Eletrc Company who suggested n 956 the adopton of supplementary runs rules n order to make the charts more senstve to small changes. Snce then, varous authors have studed the propertes of the control charts wth supplementary runs rules. For the unvarate case, the most referenced are Champ and Woodall (987), Champ (99), Zhang and Wu (5), Celano et al. (6), Koutras et al. (7), Km et al. (9), ntzoulakos and Raktzs (8) and Raz et al. (). Regardng to the multvarate case, pars et al. (4) and Koutras et al. (6) are the most referenced. The numercal evaluaton of the performance of such rules can be easly acheved by the ad of Markov Chans, as ndcated n Champ and Woodall (987) and Champ (99). Ths way, we study the MRMX chart wth supplementary runs rules. We also compare the supplemented MRMX chart wth the synthetc MRMX chart. The paper s organzed as follows. In Secton we present the propertes of the MRMX chart. We also present the propertes of the synthetc MRMX chart and the MRMX chart wth runs rules. The proposed charts are compared n Secton 3. Conclusons are n Secton 4. The MRMX Chart In ths secton we propose a new chart based on sample means and sample ranges for montorng the mean vector μ and/or the covarance matrx Σ of two qualty characterstcs that follow a bvarate normal dstrbuton. The sample pont plotted on the proposed chart corresponds to the largest value among ( Z, Z, W, W ), where Z n( X ) / and W kr /,,. s the montorng statstc s the maxmum value among standardzed sample means and weghted standardzed sample ranges, the chart s called the MRMX chart. The parameter k s requred to attend the mposed condton that, durng the n-control perod, the statstcs ( Z, Z, W, W ) have the same probablty to exceed CL, the control lmt of the MRMX chart. The process s consdered to start wth the mean vector and the covarance matrx on target ( μ μ = and μ ' Σ = Σ ), where ( ; ) and Σ. The occurrence of the assgnable cause changes μ μ the mean vector from ' ' to ; ) and/or the covarance matrx from Σ to ( ID.

3 Control Charts wth Supplementary Runs Rules for Montorng varate Processes a a Σ a a a a a a. The correlaton coeffcent s not affected by the assgnable cause. fter the occurrence of the assgnable cause t s assumed that at least one becomes larger than zero and/or at least one a becomes larger than one,,. If the MRMX statstc falls beyond the control lmt, CL, the control chart sgnals an out-of-control condton. Once the MRMX chart sgnals, the user can mmedately examne the sample means and the sample ranges of the two qualty characterstcs to dscover whch varable was affected by the assgnable cause, that s, the one wth the sample mean and/or the sample range larger than the control lmt.. The Propertes of the MRMX Chart To obtan the false alarm rsk and the power of detecton P of the MRMX chart, we use the property that the sample means are ndependent of the sample ranges. Therefore, the MRMX chart has probablty P of sgnalng: P P P P P () M D M D Regardng to the sample means, the probablty gven by: P of Z and/or Z exceedng the control lmt s M where P M CL n CL n f ( Z, Z CL n CL n f Z, Z s a standardzed bvarate normal dstrbuton functon, remndng that CL s the control lmt of the MRMX chart. Regardng to the sample ranges, the probablty P D of W and/or W exceedng the control lmt s: ) d Z d Z () P D W CL W CL Pr (3) where W kr and kr R max x, x,, xn mn x, x,, xn,, the sample ranges of two qualty characterstcs X and X. / We have that x N, a W, beng / ~, wth,. Then,, P w w D F (4) beng w CL a and w CL a. CL was dvded by a to hold the condton that x a N / a,, wth,. ~ Costa and Machado () present the probablty P D for the bvarate and trvarate cases. Table shows the average run length (RL) for the MRMX chart when =.,.5,.7, a and a =.;.5;.5 and and =.;.5;.75;.. type I rsk of.5% s adopted. The correlaton coeffcent has ID.3

4 ICIEOM - Gumarães, Portugal mnor nfluence on the propertes of the MRMX chart. The other Tables of ths artcle were bult fxng =.5. Table : The effect of on the RL for the MRMX chart (n=5). Shfts (covarance matrx) Shfts (mean vector) a a ID.4

5 Control Charts wth Supplementary Runs Rules for Montorng varate Processes. The synthetc MRMX Chart The sgnalng rule of the synthetc chart requres a second consecutve pont beyond the control lmt not far from the frst one. The number of samples between them can not exceed L, a pre-specfed value. The growng nterest n usng ths rule may be explaned by the fact that many practtoners prefer watng untl the occurrence of a second pont beyond the control lmts before lookng for an assgnable cause (see Wu and Speddng (, a), Calzada and Scarano (), Wu et al. (, a), Davs and Woodall (), Costa and Rahm (6), Machado et al. (8), pars and De Luna (9), Costa et al. (9), Castaglola and Khoo (9), Khoo et al. (9), Khlare and Shrke (), Pawar and Shrke (), Zhang et al. (), Khoo et al. (, a) and Zhang et al. ()). ccordng to Davs and Woodall () the proper parameter to measure the performance of the synthetc chart s the steady-state RL, that s, the RL value obtaned when the process remans ncontrol for a long tme before the occurrence of the assgnable cause. The followng transton matrx of the Markov chan s used to obtan the steady-state RL Sgnal Sgnal (5) The transent states descrbe the poston of the last L sample ponts; means the sample pont fell beyond the control lmts, and means the sample pont fell n the central regon. For nstance, the transent state (..) s reached when the second of the last L ponts falls n the acton regon and all others ponts fall n the central regon. The events and occur wth probabltes and =-, respectvely. The steady-state RL s gven by S x RL (6) where S s the vector wth the statonary probabltes of beng n each nonabsorbng state and RL s the vector of RLs takng each nonabsorbng state as the ntal state. We have that RL ( I R), (7) where I s an (L+) by (L+) dentty matrx, R s the transton matrx gven n (5) wth the last row and ' column removed, and s an (L+) by one vector of ones. The vector S ( / C; / C,, / C), wth C=+L, was obtaned by solvng the system of lnear equatons: The matrx are swtched by s. The matrx S R S, constraned to adj S. (8) R adj s an adjusted verson of R, where and n the frst row are held and the remanng s R adj follows: ID.5

6 ICIEOM - Gumarães, Portugal (9) Table shows the nfluence of the desgn parameter L on the MRMX performance. s L ncreases the speed wth whch the synthetc MRMX chart sgnals also ncreases. The gan n speed s more sgnfcant when L ncreases from to 6. For nstance, when a a and. 5, the RL reduces 8.% (from.4 to 7.5). On the other hand, as L ncreases from 6 to, the RL reduces.% (from 7.5 to 7.3), see Table. For ths reason, Table 3 n Secton 3 was bult fxng L equal to 7. Table : The nfluence of L on the synthetc MRMX chart s performance (=.5, n=5). L= a CL= a RL The MRMX Chart wth Supplementary Runs Rules In ths artcle we also propose the MRMX chart wth runs rules. runs rule causes a sgnal f s of the last m values of the statstc beng plotted fall n the nterval w, CL, where s m and w CL, beng T : s, m, w, CL s used to represent ths runs w the warnng lmt and CL the control lmt. The notaton rule. ccordng to ths notaton, the basc Shewhart rule (C ) can be expressed by C : (;;CL; ) or (;;- ;-CL). Ths artcle focuses on the performance of two supplemented MRMX charts. The frst one wth the supplementary runs rule C :(;3;w;CL) and the second wth the supplementary runs rule C 3 :(3;4;w;CL). ID.6

7 Control Charts wth Supplementary Runs Rules for Montorng varate Processes Fgure shows the three ponts postons that lead the MRMX chart wth the supplementary runs rule C to sgnal. MRMX CL w (C) (C) (C) Sample Number Fgure : MRMX chart wth the supplementary runs rule C. The steady-state RL of the MRMX chart wth supplementary runs rules C can be obtaned wth the ad of the Markov Chan represented by the matrx (). It s a Markov Chan wth 3 transent states (,, ) and one absorbng state (C). These states are defned accordng to the poston of the sample ponts on the control chart: State : the last two sample ponts belong to the nterval (;w); State : the prevous sample belongs to the nterval (w;cl) and the current one belongs to the nterval (;w); State : the prevous sample belongs to the nterval (;w) and the current one belongs to the nterval (w;cl); State C: the absorbng state C s reached when the last transent state s state and the current sample falls beyond CL or the last transent state s state or and the current sample falls beyond w. Let p =P[next observed sample pont wll be n the regon (;w)], p = P[next observed sample pont wll be n the regon (w;cl)] and p 3 =-p -p =P[next observed sample pont wll be n the regon CL,]. Thus, we have that: States C p p p p C ( p p ( p) ( p ) The vector S conssts of three non-absorbng states, that s, S S, S S ), 3 (), and t can be obtaned by solvng the system of lnear equatons n (8). Wth S, we only need to compute the nverse of (I-R) n order to obtan the steady-state RL, see expressons (6) and (7). ID.7

8 ICIEOM - Gumarães, Portugal Followng the same approach, we obtan the vector S for the runs rule C 3. In the next secton we wll compare the MRMX wth supplementary runs rules wth the synthetc and standard MRMX charts. The performance of the supplemented MRMX chart depends on the values of p and p. We nvestgated for whch values of p and p the MRMX chart has a better overall performance and we obtaned the RL values under ths condton. 3 Comparng Charts In ths secton we compare the standard MRMX (MRMX-), the supplemented MRMX charts based on the runs rules C and C 3 (MRMX- and MRMX-3, respectvely) and the synthetc MRMX chart (MRMX-4). Table 3 presents the RLs for the MRMX charts, where =.5, a and a =.;.5;.5 and and =.;.5;.75;.. type I rsk of.5% s adopted. The MRMX charts wth the supplementary or the synthetc runs rules have better performance than the standard MRMX. If t s well known that the assgnable cause only affects the covarance matrx, then the MRMX- and MRMX-3 charts are the best opton and present smlar performances. The same behavor s observed for small changes n the mean vector. In the other cases, the MRMX-4 chart s smlar n performance to the MRMX- and MRMX-3 charts. However, t s mportant to hghlght that the synthetc runs rule s smpler for the users than the supplementary runs rules. Table 3: RL for the MRMX-, MRMX-, MRMX-3 and MRMX-4 charts (n=5 and =.5). Shfts (covarance matrx) a a Shfts (Mean vector) () () (3) (4) ID.8

9 Control Charts wth Supplementary Runs Rules for Montorng varate Processes () MRMX-; () MRMX- (L=7); (3) MRMX-3 (p =.3); (4) MRMX-4 (p =.7). 4 Conclusons In ths artcle we proposed a sngle chart based on the standardzed sample means and sample ranges (MRMX chart) for montorng the mean vector and the covarance matrx of bvarate processes. User s famlarty wth the computaton of these statstcs s a pont n favor of the MRMX chart. We compared the supplemented MRMX chart and the synthetc MRMX chart wth the standard MRMX chart. The supplementary and the synthetc runs rules enhance the performance of the MRMX chart. However, the synthetc runs rule s smpler to admnster f compared wth the supplementary runs rules. REFERENCES lt, F.. (985). Multvarate qualty control. In: Kotz, S., Johnson, N. L., ed., Encyclopeda of Statstcal Scences. Wley. pars, F., De Luna, M.. (9). The desgn and performance of the multvarate synthetc- T control chart. Communcatons n Statstcs-Theory and Methods, 38, ntzoulakos, D. L., Raktzs,. C. (8). The modfed r out of m control chart. Communcatons n Statstcs: Smulaton and Computaton, 37, pars, F., Champ, C. W., García-Díaz, J. C. (4). performance analyss of Hotellng s control chart wth supplementary runs rules. Qualty Engneerng, 6, Calzada, M.E., Scarano, S.M. (). The robustness of the synthetc control chart to non-normalty. Communcatons n Statstcs: Smulaton and Computaton, 3, Castaglola, P., Khoo, M.. C. (9). synthetc scaled weghted varance control chart for montorng the process mean of skewed populatons. Communcatons n Statstcs-Smulaton and Computaton, 38, Celano, G., Costa,., Fchera, S. (6). Statstcal desgn of varable sample sze and samplng nterval X control charts wth run rules. Internatonal Journal of dvanced Manufacturng Technology, 8, Champ, C. W., pars, F. (8). Double samplng Hotellng s T charts. Qualty and Relablty Engneerng Internatonal, 4, Champ, C. W., Woodall, W. H. (987). Exact results for Shewhart control charts wth supplementary runs rules. Tecnometrcs, 9, Champ, C. W. (99). Steady-state run length analyss of a Shewhart qualty control chart wth supplementary runs rules. Communcatons n Statstcs-Theory and Methods,, ID.9

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11 Control Charts wth Supplementary Runs Rules for Montorng varate Processes Wu, Z., Zhang, X., Yeo, S. H. (). Desgn of sum-of-conformng-run-length control charts. European Journal of Operatonal research, 3, Zhang, G., Chang, S. I. (8). Multvarate EWM control charts usng ndvdual observatons for process mean and varance montorng and dagnoss. Internatonal Journal of Producton Research, 46, Zhang, S., Wu, Z. (5). Desgns of control charts wth supplementary runs rules. Computers & Industral Engneerng, 49, Zhang, Y., Ou, Y., Castaglola, P., Khoo, M.. C. (). combned synthetc & X for montorng the process mean. Internatonal Journal of Producton Research, 48, Zhang, Y., Castaglola, P., Wu, Z., Khoo, M.. C. (). The synthetc X chart wth estmated parameters. IIE Transactons, 43, ID.