Integrating Mean and Median Charts for Monitoring an Outlier-Existing Process

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1 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 8 Vol II IMECS March 8 Hog Kog Itegratg Mea ad Meda Charts for Motorg a Outler-Exstg Process Lg Yag Suzae Pa ad Yuh-au Wag Abstract A effectve cotrol scheme ca be strumetal creasg productvty ad reducg cost. Whle facg a outler-exstg process usg the mea ( X cotrol chart ad the rage ( cotrol chart for motorg the process mea ad varace wll lead to hgh level false alarms. ecetly some meda ( X cotrol charts such as the Shewhart - X cotrol chart the expoetally weghted movg average (EWMA - X cotrol chart ad the geerally weghted movg average (GWMA- X meda cotrol chart have bee developed successo for motorg the process mea/meda. Although those metoed X cotrol charts are demostrated much more robust tha the X cotrol charts wth outlers the shft-detectg ablty s worse tha that of X cotrol charts wthout respect to outlers. Davs ad Adams proposed a sytheszed cotrol scheme (called SS sytheszed cotrol scheme for short whch usg the Shewhart- X ad (deoted as Shewhart - X / charts ad the Shewhart trmmed mea ad trmmed rage (deoted as Shewhart - X / charts for motorg the process mea ad varace wth outlers. Ths study proposes a modfed sytheszed cotrol scheme whch tegratg the EWMA- X / cotrol charts ad the EWMA- X / cotrol charts (called EE sytheszed cotrol scheme for short for motorg a outler-exstg process. A dagostc statstc techque s adopted here to be a brdge betwee the X / charts ad the X (or X / charts. Wth varous shfts of the process sample mea the average tme to work stoppage (ATWS s evaluated uder some cotamated ormal dstrbutos. We coclude that the EE sytheszed cotrol scheme outperforms the SS sytheszed cotrol scheme for motorg the small shft of the process mea or varace. Ths result provdes a valuable recommedato whle facg a outler-exstg process. Idex Terms cotrol chart EWMA meda outlers. Mauscrpt receved vember 4 7. Ths work was supported part by the Natoal Scece Coucl Tawa OC uder Grats NSC96-1-E Lg Yag s wth the Departmet of Idustral Egeerg ad Maagemet St. Joh s Uversty Tamsu Tape 51 Tawa OC (correspodg author phoe: ext. 6583; fax: ext. 6589; e-mal: lgyag@mal.sju.edu.tw. Suzae Pa s wth the Departmet of Idustral Egeerg ad Maagemet St. Joh s Uversty Tamsu Tape Tawa OC (e-mal: sjpa@mal.sju.edu.tw. Yuh-au Wag s wth the Departmet of Computer Scece ad Iformato Egeerg St. Joh s Uversty Tamsu Tape Tawa OC (e-mal: yrwag@mal.sju.edu.tw. I. INTODUCTION A effectve cotrol scheme ca be strumetal creasg productvty ad reducg cost. From past experece there are some certa processes wth outlers that happeed occasoally. The outlers are the values of observatos that are larger or smaller tha the majorty of the other observatos eve though the process s cotrol. Samples cotag outlers are sad to be cotamated. Usg the mea ( X cotrol chart ad the rage ( cotrol chart for motorg the process mea ad varace wll lead to hgh level false alarms. ecetly some meda ( X cotrol charts such as the EWMA - X cotrol chart (Castaglola [] the Shewhart - X cotrol chart (Khoo [6] ad the geerally weghted movg average (GWMA- X cotrol chart (Sheu ad Yag [1] had bee developed successo. As dscussed [1] the X cotrol charts are much more robust tha the X cotrol charts wth outlers-exstg process. However the shft-detectg ablty of X cotrol charts s worse tha that of X cotrol charts wthout respect to outlers. Suppose that the occurrece of outlers s due to the commo causes ad oly the assgable causes wll lead to permaet shfts. Davs ad Adams [5] proposed a sytheszed cotrol scheme for the outler-exstg process. I [5] the Shewhart - X cotrol chart ad the cotrol chart (deoted as Shewhart - X / are used for motorg the process mea ad varace frst. Whle the Shewhart - X cotrol chart or the cotrol chart sgals the dagostc statstc (DS s compared wth the pre-determed decso value (K to ascerta whether the collected sample s cotamated or ot. If t s regarded as a cotamated sample the Shewhart trmmed mea ( X cotrol chart ad the trmmed rage ( cotrol chart (Lageberg ad Iglewcz [7] (deoted as Shewhart - X / are used to carry o the process motorg. Davs ad Adams [5] deed provded a good choce for the practtoer. However the fast-detectg techque e.g. the EWMA - X cotrol chart ad those metoed X cotrol charts were ot cosdered ther study. I ths work a modfed sytheszed cotrol system s proposed. The EWMA- X cotrol chart ad the cotrol chart (deoted as EWMA - X / are adopted the frst stage. The EWMA- X cotrol chart ad the cotrol chart (deoted as EWMA- X / are adopted the secod stage. ISBN: IMECS 8

2 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 8 Vol II IMECS March 8 Hog Kog The correspodg K values are provded. Wth varous shfts of the process mea the average tme to work stoppage (ATWS are evaluated uder some cotamated ormal dstrbutos. II. DESCIPTION OF SOME CONTOL CHATS Suppose that the qualty characterstc s a varable ad the samples have bee collected at each pot tme (the sze of ratoal subgroups deoted as. Let X X X ad be the sample average sample meda trmmed sample mea sample rage ad trmmed sample rage of th subgroup respectvely whch are composed of depedet ormal ( μ radom varables X where μ s σ 1 L X σ the omal process mea/meda ad s the omal process varace. Whe the process s -cotrol μ = μ σ = σ (the target value of the process mea ad varace. That s X j / j= 1 = X X = X [ (f s odd ( + 1 / ] 1 = X X [ j] /( 1 j= 1 = X X [ ] [ 1] = X X [ 1] [ 1] lmts. Whe α = 1 the plotted statstc (1 wll be Y = ad the cotrol lmts wll be X σ UCL = μ + L CL = μ σ UCL = μ L. These are the equatos of the Shewhart- X cotrol chart. Therefore whe α = 1 the EWMA- X cotrol chart reduces to the Shewhart- X cotrol chart. Accordg to Castaglola [1] the dstrbuto of the sample meda X s very close to the ( μ σ ormal dstrbuto where σ s the varace of X. If σ 1 s the stadard devato of ormal ( 1 sample meda we have σ = σ σ 1. The values of σ 1 had bee derved by [1]. Whe the process s -cotrol μ = μ σ = σ. The EWMA- X cotrol statstc Z ca be represeted as Z = β X + (1 β Z for 1 L (3 1 = where Z = μ the smooth parameter β ( β > s determed by the practtoer. The tme-varyg cotrol lmts of the EWMA- X cotrol chart ca be represeted as where X [ j] represets the jth order statstc for the th sample. I [3] the EWMA- X cotrol statstc represeted as Y ca be Y = α X + ( 1 α Y 1 for = 1 L (1 UCL = μ + η CL = μ LCL = μ η β (1 (1 β β β (1 (1 β β σ σ σ σ 1 1 (4 where Y = μ the smooth parameter α ( α > s determed by the practtoer. The cetral le (CL the tme-varyg upper cotrol lmt (UCL ad lower cotrol lmt (LCL of the EWMA- X cotrol chart ca be represeted as where η determes the wdth of the cotrol lmts. The process s cosdered out of cotrol ad some actos should be take wheever Z falls outsde the rage of the cotrol lmts. Whe β = 1 the plotted statstc (3 wll be Z = X ad the cotrol lmts wll be UCL = μ + L CL = μ LCL = μ L α(1 (1 α α α(1 (1 α α σ ( σ where L determes the wdth of the cotrol lmts. The process s cosdered out of cotrol ad some actos should be take wheever falls outsde the rage of the cotrol Y UCL = μ + η σ σ CL = μ LCL = μ η σ σ 1 1. These are the equatos of the Shewhart- X cotrol chart. Therefore whe β = 1 the EWMA- X cotrol chart reduces to the Shewhart- X cotrol chart. The cotrol scheme of the cotrol chart refers to [8] for ISBN: IMECS 8

3 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 8 Vol II IMECS March 8 Hog Kog detals. The cotrol schemes of the trmmed mea ( X ad trmmed rage ( cotrol charts refer to [7]. Start III. MODIFIED SYNTHESIZED CONTOL SCHEME I Davs ad Adams sytheszed cotrol scheme [5] several dagostc statstcs (DS were used to recogze whether the collected sample cotaed cotamated data. Through smulato the maxmum-meda DS combed wth Shewhart - X / cotrol charts ad Shewhart - X / cotrol charts (called SS sytheszed cotrol scheme for short performed best. The defto of DS s DS for = 1 (5 = X[ ] X[ ( + 1/ ] / The Shewhart - X cotrol charts were used oly whe the Shewhart - X cotrol chart ad/or cotrol chart sgaled ad the value of DS fgured out that the sample cotas outlers (.e. DS > K. As metoed the frst secto Davs ad Adams [5] deed provded a good choce for the practtoer wth outlers cosderato. However the fast-detectg techque (e.g. the EWMA - X cotrol chart ad the robust cotrol techque (e.g. the EWMA- X cotrol chart were ot cosdered ther study. I ths work a modfed sytheszed cotrol scheme s proposed. A EWMA- X cotrol chart ad a cotrol chart (deoted as EWMA- X / are used the frst stage. A EWMA- X cotrol chart ad a cotrol chart (deoted as EWMA- X / are used the secod stage. The proposed cotrol scheme s called EE sytheszed cotrol scheme for short. The smlar computato of DS (as show (5 s used. Because the EWMA- X cotrol chart ad the EWMA- X cotrol chart eed to collect successve data a modfed sytheszed cotrol scheme s show Fg. 1. I the frst stage f the EWMA- X / cotrol chart(s sgal(s ad DS <= K (.e. the sample does ot cota outlers the process s assumed to be out of cotrol ad a search for a assgable cause s tated. If the EWMA- X / cotrol chart(s sgal(s ad DS > K (.e. the sample cotas outlers the secod stage wll be started. I the secod stage f the EWMA- X / cotrol chart(s sgal(s the process s assumed to be out of cotrol ad a search for a assgable cause s tated. As metoed [5] the dagostc statstc techque uses a decso value (K from the codtoal dstrbuto of DS gve a sgal of the frst stage cotrol chart.e. P ( DS > K frst stage cotrol charts sgal(s.7 Wthout loss of geeralty we assume that the -cotrol μ = ad varace σ = 1. Due to the complex rego of tegrato ad the accuracy of smulato [5] elected to use the smulated decso values [4] (as lsted the frst row of Table 1 for = 5. I ths study we provde the decso values for the EE sytheszed cotrol scheme va smulato too (as lsted the secod row of Table 1 for = 5. Sytheszed cotrol scheme SS EE = Y = μ Z = μ = + 1 collect th process sample Frst Stage: 1. calculate Y ad for EWMA- X / charts EWMA- X / charts sgal(s? DS > K? Check for process chages correct f ecessary Fg. 1. EE sytheszed cotrol scheme Table 1. Decso values Cotrol lmts Shewhart- X chart: ( chart: ( EWMA- X chart: α =.1 L =.73 chart: ( calculate Z ad for EWMA- X / charts Secod stage: EWMA- X / charts sgal(s? Decso value (K ( = ISBN: IMECS 8

4 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 8 Vol II IMECS March 8 Hog Kog IV. PEFOMANCE MEASUEMENT AND COMPAISON I [5] the average tme to work stoppage (ATWS s used for the performace measuremet of the sytheszed cotrol scheme. ATWS s defed as the average umber of pots plotted before a sgal of secod stage cotrol chart. I ths work the smulato [9] s used to estmate the ATWS of the sytheszed cotrol schemes. The smulato program s wrtte BASIC laguage. Aalyss begs wth EWMA (or Shewhart- X / cotrol charts sgal(s from subgroup of samples sze = 5. I order to evaluate the ATWS of the sytheszed cotrol schemes the presece of outlers the cotamated ormal dstrbuto used [6] s adopted. A cotamated ormal dstrbuto s that the observatos ( 1 θ % come from ( N ( 1 ormal dstrbuto ad θ % come from ( N(C 1 ormal dstrbuto where θ deotes the level of cotamato C deotes the mea of a outler. We assumed that the outlers occur due to the commo causes of varato ad lead to a temporary shft. Oly the assgable causes wll make the process a permaet shft. We are terested detectg a permaet shft. Three kd of cotamated ormal dstrbutos ( C θ {(11 ( 1 (3 1} are used here to evaluate the ATWS of the sytheszed cotrol schemes. The cotrol lmts for the sytheszed cotrol schemes are based o the data whch 1% come from the ( N ( 1 ormal dstrbuto (.e. ( C θ = ( wth a desred -cotrol ATWS (deoted ATWS 139. Ths s the -cotrol average ru legth (AL of the Shewhart- X / charts ad the EWMA- X / charts the frst stage. These two sytheszed cotrol schemes are stadardzed to ths AL for far comparso. Table presets the smulato result of ATWS. Tables 3 ad 4 preset the smulato result of out-of-cotrol ATWS (deoted ATWS 1 wthout ad wth cotamated data respectvely. I Tables 4 varous combatos of ( C θ deote varous cotamated ormal dstrbutos. For example whe ( C θ = ( the data 1% come from the ormal dstrbuto N( 1. Whe ( C θ = (1 the data 99% come from the ormal dstrbuto N( 1 ad 1% come from the ormal dstrbuto N( 1. I Table whe the process s -cotrol (.e. ( μ σ = (1 ( C θ {((11(1 ( 31} the ATWS of the EE sytheszed cotrol scheme are smlar to that of the SS sytheszed cotrol scheme. It meas that both the EE sytheszed cotrol scheme ad the SS sytheszed cotrol scheme are outlers-resstat. However whe the mea shft s small the EE sytheszed cotrol scheme outperforms the SS sytheszed cotrol scheme for both the data s cotamated or ot. For stace Table 3 whe ( μ σ = (.3 1 ad ( C θ = ( the ATWS 1 of the EE sytheszed cotrol scheme (= s less tha that of the SS sytheszed cotrol scheme (= I Table 4 whe ( μ σ = (.3 1 ad ( C θ = (1 the ATWS 1 of the EE sytheszed cotrol scheme (= s less tha that of the SS sytheszed cotrol scheme (= It meas that the EE sytheszed cotrol scheme ca detect the process small shft more quckly tha the SS sytheszed cotrol scheme. V. CONCLUSION Ths study proposes a modfed sytheszed cotrol scheme to reta the fast-shft-detectg ablty (the EWMA- X cotrol chart ad the robustess to outlers (the EWMA- X cotrol chart smultaeously. I the EE sytheszed cotrol scheme the dagostc statstc techque s adopted ad the decso values are provded. Uder several cotamated ormal dstrbutos the performace of ATWS 1 of the EE sytheszed cotrol scheme outperforms that of the SS sytheszed cotrol scheme wth varous shfts of the process sample mea ad varace. We coclude that the modfed sytheszed cotrol scheme whch combed the EWMA- X cotrol chart wth the EWMA- X cotrol chart s outlers-resstat ad more sestve motorg the small shft of the process mea. Ths result provdes a useful recommedato whle facg a outler-exstg process. Table. ATWS of the sytheszed cotrol schemes wth a desred ATWS 139 I-cotrol ( μ σ = ( 1 ( C θ SS EE o cotamated data ( cotamated data ( ( ( Table 3. ATWS 1 for o cotamated data wth a desred ATWS 139 Out-of-cotrol o cotamated data ( μ σ SS EE ( C θ = ( mea shft ( ( ( ( ( varace shft ( ( ( ( ISBN: IMECS 8

5 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 8 Vol II IMECS March 8 Hog Kog Table 4. ATWS 1 for cotamated data wth a desred ATWS 139 Out-of-cotrol cotamated data ( μ σ SS EE ( C θ = ( 1 mea shft ( ( ( ( ( varace shft ( ( ( ( EFEENCES [1] P. Castaglola Approxmato of the ormal sample meda dstrbuto usg symmetrcal Johso S U dstrbutos: applcato to qualty cotrol Commucatos Statstcs: Smulato ad Computato vol pp [] P. Castaglola A ( X / -EWMA cotrol chart for motorg the process sample meda Iteratoal Joural of elablty Qualty ad Safety Egeerg vol. 8( 1 pp [3] S. V. Crowder Desg of expoetally weghted movg average schemes Joural of Qualty Techology vol pp [4] C. M. Davs A Dagostc Tool for Tradtoal Mea ad age Cotrol Charts. Ph.D. Dssertato Departmet of Iformato Systems Statstcs ad Maagemet Scece Uversty of Alabama Tuscaloosa AL USA. [5] C. M. Davs ad B. M. Adams obust motorg of cotamated data Joural of Qualty Techology vol. 37( 5 pp [6] M. B. C. Khoo A Cotrol chart based o sample meda for the detecto of a permaet shft the process mea Qualty Egeerg vol pp [7] P. Lageberg ad B. Iglewcz Trmmed mea X ad charts Joural of Qualty Techology vol pp [8] D. C. Motgomery Itroducto to Statstcal Qualty Cotrol. New York: Wley USA 5. [9] S. M. oss A Course Smulatos. New York: Macmla USA 199. [1] S. H. Sheu ad L. Yag The geerally weghted movg average meda cotrol chart Qualty Techology ad Quattatve Maagemet vol. 3(4 6 pp ISBN: IMECS 8

CHAPTER - IV STANDARDIZED CUSUM MEDIAN CONTROL CHART

A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess CHAPTER - IV STANDARDIZED CUSUM MEDIAN CONTROL CHART. Itroducto: I motorg e process mea, e Mea ( X ) cotrol charts, ad cumulatve sum