A Generally Weighted Moving Average Chart for Time Between Events

Transcription

1 A Generally Weghed Movng Average Char for Tme Beween Evens N Chakrabory,a, SW Human,b, N Balakrshnan,c Deparmen of Sascs, Unversy of Preora, Preora,, Lynnwood Road, Hllcres, Souh Afrca Deparmen of Mahemacs and Sascs, McMaser Unversy, Hamlon, Onaro, Canada LS K a Correspondng auhor: e-mal nladrorama@gmalcom b Correspondng auhor: e-mal: schalkhuman@upacza c Correspondng auhor: e-mal bala@mcmaserca Absrac Shewhar-ype arbue chars are known o be neffcen for small changes n monorng nonconformes An alernave way s o use a me-weghed char o monor he me beween evens (TBE) We propose a one-sded Generally Weghed Movng Average conrol char o monor he me beween evens (TBE); regarded as he GWMA-TBE char To ad he mplemenaon of he char, he necessary desgn parameers are provded An exensve performance analyss shows ha he GWMA-TBE char s beer han he well-known EWMA and Shewhar chars a deecng very small o moderae changes Fnally, a summary and some conclusons are provded Keywords GWMA char; Tme beween evens; Average Run-lengh; Smulaon; Markov chan

2 Inroducon Snce s nroducon by Waler A Shewhar n he s, sascal qualy conrol (SQC) has found a varey of applcaons rangng from healh care monorng o fnancal fraud deecon In general, an em ha does no comply wh he sandards s labelled a nonconformng or defecve em The radonal Shewhar-ype arbue chars such as he c -char or p -char (see Mongomery, ) are used o monor a process when he number of nonconformes or he number of defecve ems s of neres; hese chars use he number of nonconformes or he number of defecve ems n regular, ypcally equ-spaced, me nervals and are known o be neffecve n deecng small shfs Such rare evens are frequenly found n oday s hghperformng echnologcal envronmen wheren he rae of falures can be very small An alernave and perhaps more appealng way o monor he rae of falures (called he Posson rae) s o use conrol chars based on he ner-arrval mes of nonconformng ems These chars are especally useful when he occurrence of defecve ems s rare and he rae of occurrence s very small Such chars are called me beween evens (TBE) chars and are ypcally based on he followng wo assumpons: (a) he occurrence of falures n a process follows a homogeneous Posson process, and (b) he ner-arrval mes beween wo consecuve evens follow an exponenal dsrbuon A number of TBE conrol chars can be found n he leraure The Cumulave Sum (CUSUM) char o monor he me beween evens, whch s smply called he exponenal CUSUM char, was proposed by Vardeman and Ray (); he opmal desgn of he exponenal CUSUM char was gven by Gan () and an algorhm for compung he average run-lengh ( ARL) of he char was subsequenly gven by Gan and Cho

3 () The desgn and performance of he Exponenally Weghed Movng Average (EWMA) char for he exponenal dsrbuon was suded by Gan () A Shewhar-ype char for he me beween evens (also assumng an exponenal dsrbuon) was suded by Xe e al () In addon, Zhang e al () suded he performance of he Shewhar-ype me beween evens char for he gamma dsrbuon Tme-weghed conrol chars such as he CUSUM and EWMA have been shown o be effcen n deecng small susaned upward or downward sep shfs n a process see Mongomery () for more deals and references on he heory, desgn and applcaon of he classcal EWMA and CUSUM chars for he normal dsrbuon Capalsng on he effcency of he EWMA char, Sheu and Ln () proposed a Generally Weghed Movng Average (GWMA) conrol char for he normal dsrbuon; hs char s a generalsaon of he EWMA char and has been shown o be more effecve han he EWMA, CUSUM and Shewhar-ype chars (see Hsu e al, ) n deecng very small shfs For more deals on he recen developmens on he GWMA chars, he neresed reader s referred o he works of Sheu and Yang (), Sheu and Chu (), Chu and Sheu (), Chu (), Sheu and Hseh (), Sheu e al (), Teh e al (), Sheu e al (), Huang e al (), Huang (), Lu (), Aslam e al () and Chakrabory e al () As noed, for example, by Zhang e al () and Balakrshnan e al (), he quck deecon of any deeroraon n a process s of umos mporance from a praccal pon of vew In he curren conex, e, monorng he me beween evens, process deeroraon can occur followng an ncrease n he falure rae So, for a hgh qualy process, a small reducon n

4 process performance needs o be deeced as early as possble Ths does no mply ha process mprovemen s no mporan Que o he conrary, as menoned by Maravelaks and Casaglola (), an mprovemen n a process usually occurs afer correcve acon has aken place, bu he me of he change s usually known and a conrol char s no needed o deec he change/mprovemen For hs reason, a one-sded conrol char o deec process deeroraon s pursued here In hs arcle, we consruc a one-sded GWMA char based on he gamma dsrbuon for monorng he me beween evens (ofen regarded as he me beween falures); hs char s labelled he GWMA-TBE char and can be used n deecng a susaned downward sep shf n a process The proposed char ncludes he one-sded EWMA and Shewhar chars as specal cases, and dffers n wo ways from he currenly avalable TBE chars: () The nerdependency of he GWMA charng sascs s explcly aken no consderaon, and () we specfcally focus on a one-sded GWMA-TBE char based on he gamma dsrbuon The proposed one-sded GWMA-TBE char s hus run-lengh unbased (by consrucon) whch s unlke he currenly avalable wo-sded conrol chars (based on he exponenal dsrbuon) whch are based Noe ha a conrol char s based when he ou-of-conrol average run-lengh (denoed by ARL ) s larger han s n-conrol value (denoed by ARL ) and herefore akes longer o deec a shf n he process; hs s an undesrable propery and normalzng ransformaons ofen do no compleely elmnae hs problem The res of hs arcle s organsed as follows: The general heory and background on he GWMA-TBE char are gven n Secon In Secon, he run-lengh dsrbuon for he

5 scenaro when he parameers are known s suded; hs ncludes he n-conrol (IC) desgn and he ou-of-conrol (OOC) performance The GWMA-TBE char for he scenaro when he parameers are unknown s suded n Secon ; hs ncludes he effecs of parameer esmaon on he performance as well as he desgn of he Phase II GWMA-TBE char In Secon, we show how he proposed char can also be used o monor he varance of a normal dsrbuon An llusrave example s provded n Secon and we fnally conclude wh some remarks n Secon The GWMA Tme Beween Evens (GWMA-TBE) conrol char Le he random varables Yj ~ d Exp, j,,,, denoe he ner-arrval mes beween consecuve falures n a homogeneous Posson process wh rae parameer / The connuous random varable X k Yj, whch s he sum of he ner-arrval mes of k consecuve j falures, hen denoes he me unl he h k falure and s known o follow a Gamma (, k ) dsrbuon The probably densy funcon (pdf) of X s gven by x/ k e x f x; k,, x, and k () k Γ k Noe he followng regardng he pdf n (): The expeced value and varance of X are E X k and var X k ; Alhough he parameer k can heorecally be any posve real number, n he developmens ha follow s assumed ha k s a known/specfed posve neger, e,

6 k,,,, and seleced by he praconer For example, f we se k, he random varable X denoes he oal me beween hree consecuve falures; If k, he pdf n () reduces o ha of an exponenal dsrbuon and we would be neresed n monorng he me unl one falure; v I s assumed ha denoes he n-conrol (IC) sandard/value for he parameer ; hs s known as he sandard known case and denoed by Case K We prmarly focus on he scenaro when he parameer s known, bu we also deal wh he suaon when s unknown and esmaed from an n-conrol (IC) Phase I reference sample before onlne monorng can sar n Phase II; he laer scenaro s referred o as he sandard unknown case and denoed by Case U; v Our objecve s o consruc a conrol char o monor for a susaned downward sep shf, e, a decrease n he ner-arrval mes, whch would be ndcave of deeroraon n he process The key focus s o deec small or very small shfs as quckly as possble The GWMA-TBE conrol char s consruced by akng a weghed average of a sequence of he X ' s To hs end, le N be he dscree random varable denong he number of samples unl he nex occurrence of an even snce s las occurrence Then, by summng over all values of N, we can wre Pr N Pr N Pr[ N ] () A generally weghed movng average (GWMA) s a weghed movng average (WMA) of a sequence of X sascs where he probably PrN s regarded as he wegh w for he h

7 mos recen sasc X among he las of X sascs In oher words, he probably Pr N s he wegh w for he laes observaon X and he probably PrN s he wegh w for he mos ou-daed observaon X The probably PrN s consdered as he wegh for he sarng value, denoed by Z, whch s ypcally aken as he n-conrol (IC) expeced value of he sasc under consderaon, e, IC charng sasc for he GWMA-TBE char s defned as Z E X k Therefore, he Pr Pr for,,, () Z N X N Z As n Sheu and Ln (), he dsrbuon of N s aken o be PrN q q, where q and are he wo parameers; hs s he dscree wo-parameer Webull dsrbuon (see Nakagawa and Osak, ) So, he weghs are gven by w q q By subsung Z k and he probably mass funcon (pmf) of he wo-parameer dscree Webull dsrbuon n equaon (), he charng sasc for he GWMA-TBE char smplfes o Z q q X q k for,,, () Noe ha he GWMA-TBE char reduces o an EWMA char when and q, where s he smoohng parameer of he EWMA char The EWMA char furher reduces o he Shewhar char when q and The GWMA char can herefore be vewed as a generalzaon of he EWMA and Shewhar-ype chars wh an addonal parameer whch provdes more flexbly n desgnng he char We nroduce he EWMA-TBE and he Shewhar-TBE chars as specal cases

8 A closer look a he choce of weghs for he GWMA char shows ha he weghs w q q,,,,, are decreasng n for a fxed q and ; hs mples ha more wegh s gven o more recen observaons n he sequence of X ' s A proof of hs resul s gven n he Appendx The n-conrol (IC) expeced value and varance of he charng sasc and IC Z are gven by () E Z q q E X q k k var Z IC q q var X Q k () respecvely, where Q q q The exac me-varyng (symmerc) conrol lms ( UCL e & LCL e) and cenerlne ( CL e ) of a wo-sded GWMA-TBE char are gven by UCL / LCL k L Q k and CL k () e e e where L s he dsance of he conrol lms from he cenerlne and he subscrp e denoes he exac conrol lms The seady-sae conrol lms are used when he process has been runnng for several me perods and are based on he asympoc varance of he charng sasc (see Lucas and Saccucc, ) The seady-sae conrol lms and he cenerlne are gven by UCL LCL k L Qk CL k () s / s and s,

9 where he subscrp s denoes he seady-sae conrol lms and Q lm Q As we are prmarly neresed n deecng deeroraon n he process, e, a decrease n he wang mes beween falures, only a lower conrol lm s used n he desgn and mplemenaon of he proposed GWMA-TBE char; hs lm s gven by LCL k L Qk () s A GWMA-TBE char for deecng mprovemen n a process can be desgned n a smlar manner bu no pursued here any furher for he sake of brevy The followng wo pons are worh nong here: The quany Q s a monooncally ncreasng and convergen funcon of (see he Appendx for more deals) To hs end, Table shows he value of Q for dfferen choces of he parameers q, and varous values of We observe ha he change n Q s almos neglgble for Due o he quck convergence of Q, he exac conrol lms also quckly converge owards he seady-sae lms Hence, he seady-sae lower conrol lm s used n order o smplfy he applcaon/mplemenaon of he proposed char For he sake of noaonal smplcy, we wll use LCL hereafer o denoe he seady-sae lower conrol lm n (); If any Z plos on or below LCL, he process s declared ou-of-conrol (OOC) and a search for assgnable causes s naed Oherwse, he process s consdered o be n-conrol (IC), whch mples no change n has occurred, and he charng procedure connues on In he ensung secon, we dscuss he desgn of he proposed GWMA-TBE char n more deal

10 Table : Q values for dfferen q, α combnaons q, α q, α q, α q, α q, α q, α q, α q, α q, α

11 The desgn and mplemenaon of GWMA-TBE char Performance measures are needed o desgn and compare he performance of conrol chars The radonal approach of evaluang a conrol char s o oban he run-lengh dsrbuon and s assocaed characerscs The run-lengh s a dscree random varable ha represens he number of samples whch mus be colleced (or, equvalenly, he number of charng sascs ha mus be ploed) n order for he char o deec a shf or gve a sgnal An nuvely appealng and popular measure of a char s performance s he average run-lengh ( ARL), whch s he expeced number of charng sascs ha mus be ploed n order for he char o sgnal (see Human and Graham, ) Clearly, for an effcen conrol char, one would lke o have he n-conrol ARL (denoed ARL ) o be large and he ou-of-conrol ARL (denoed ARL ) o be small Alhough oher measures such as he sandard devaon of he run-lengh SDRL and varous upper and lower percenles could be and have been used o supplemen he evaluaon of conrol chars, he ARL s he mos wdely used measure due o s nuve appealng nerpreaon Therefore, we use prmarly he ARL o desgn and compare he performance of he proposed GWMA-TBE char The desgn of a conrol char ypcally nvolves solvng for he combnaon of he char s parameers, e, q, and L, so as o oban a pre-specfed n-conrol average run-lengh denoed by * ARL for a gven or seleced value of k The compuaonal aspecs of he run-lengh dsrbuon for he GWMA-TBE char are dscussed nex; hs s followed by he desgn of he GWMA-TBE char

12 The run-lengh dsrbuon of he GWMA-TBE char There are numerous mehods o calculae he run-lengh dsrbuon of a me-weghed char lke he GWMA char and we nvesgae here hree approaches: () An exac approach based on closed-form expressons, () A Markov chan approach, and () Mone Carlo Smulaon Each mehod s dscussed n more deal below along wh her pros and cons I should be noed ha ulmaely we used compuer smulaon and double-checked he resuls by usng he Markov chan approach for he EWMA-TBE char; hese resuls are avalable upon reques Exac approach Suppose he run-lengh random varable s denoed by R and ha A denoes he sgnallng even a he h c sample The non-sgnallng even s herefore gven by A [ Z LCL ] for,,, Then, n general, he run-lengh dsrbuon can be wren as r c Pr R r Pr A A r c A [ Z LCL], as c LCL qk A [ X L ], where L q, for r,,, For any, we can re-wre he even The run-lengh dsrbuon can herefore be wren as Pr R Pr X L an d Pr R r Ir Ir, () where I r c r r Pr A Pr { X L } deals and j j j j LCL q q X q k L,,,, () q for r,,, ; see he Appendx for more

13 Snce he X s are assumed o be ndependen Gamma (, k ) random varables, we have where f x; k, s he pdf gven n () r I f x ; k, dx, () r L L L The ARL can also be expressed n erms of I r as (see he Appendx for more deals) r ARL I () To analycally evaluae expressons () and () s me-consumng and uneconomcal for wo reasons: a The lower bounds n he negrals of I r, e, he L s gven n equaon (), are muually dependen and funcons of he sequence of precedng sascs X, X,, X ; hese bounds canno be economcally recursvely updaed and s herefore a compuaonally expensve approach; b The number of erms n expresson () ha needs o be evaluaed ncreases dramacally as r ncreases; As menoned before, f we se q and, we oban he Shewhar-TBE char In hs specal case, he lower conrol lm n () reduces o L sgnal s gven by X LCL X Gamma k r LCL for and he probably of a Pr[ ~,/ ], where he lower conrol lm can be Pr X LCL X ~ Gamma k, ] / ARL obaned by solvng he expresson * r

14 Markov chan approach For he EWMA chars, usng he Markov chan approach, he probably mass funcon (pmf), ARL and he varance (VARL) of he run-lengh random varable R can be obaned as (see r Theorems and of Fu and Lou, ), PR r ARL ξ I Q and VARL ARL ξq I Q for r,,,, ξ I Q I Q, where he sub-marx marx Q Q s called he essenal ranson probably sub-marx, I s he deny marx, ξ s he nal probably vecor such ha j when he process mean e k, s n he h j sae and j for all oher j wh j,,,, T s he un vecor and denoes j he number of ransen saes n he sae space Ω The Markov chan resuls for he one-sded EWMA-TBE char are avalable upon reques from he correspondng auhor The neresed reader s also referred o he work by Graham e al (a and b) for more deals The prme obsacle usng he Markov chan approach for he GWMA char s he fac ha he GWMA char s plong sasc canno be vewed as followng a frs-order Markov chan In fac, obanng Z requres all he X s from sar-up, e, X, X,, X Ths complcaon arses due o he repeaed exponenaon (so-called boom-up eraon) n he weghs w q q of he charng sascs Ths mples ha Z depends on all Z, Z,, Z To hs end, noe ha he probably Pr Z Z,, Z can be approxmaed by * * Pr Y Y, where * Y ( Z,, Z m ) and m s he hreshold beyond whch he weghs are approxmaely zero (see Appendx) So, { Y *, m, m, m, } becomes a frs-order Markov chan and we can use he resuls from Fu and Lou () for m, m, For,,, m, one sll has

15 o use he exac approach The major dffculy n usng he Markov Chan approach for he GWMA-TBE char s n defnng he sae-space of he frs-order Markov chan { Y *, m, m, m } as depends on he sae-space of ( Z,, Z m ) Caresan produc Ω m, m, m, m, whch s he Due o he above-menoned dffcules wh he exac approach and he Markov chan approach, exensve smulaon has been used o calculae he run-lengh dsrbuon for he proposed GWMA-TBE char To hs end, s mporan o noe ha, Sheu and Ln (), Sheu and Yang () and Lu () also menoned ha s dffcul o oban he run-lengh dsrbuon of he GWMA chars by usng he exac approach or he Markov chan approach The smulaon approach s dscussed nex Mone Carlo smulaon approach The smulaon algorhm uses he sochasc represenaon of he GWMA-TBE char and s done accordng o he followng fve seps: Selec a combnaon of he desgn parameers, e, ( q,, L), se and calculae he conrol lm accordng o expresson (); Generae an ndvdual observaon from a Gamma ( k,) dsrbuon and calculae he charng sasc Z accordng o expresson () wh he sarng value aken as Z k; If Z LCL, he process s consdered o be n-conrol and a run-lengh couner s ncremened; v Seps () and () are repeaed and Z s sequenally updaed unl Z LCL ; when hs even occurs, a sgnal s gven and he process s declared o be ou-of-conrol The smulaon sops and he run-lengh s recorded;

16 v Seps () (v) are repeaed, mes We have also used smulaon for he EWMA-TBE char n order o be conssen and be able o compare hem wh he Markov chan approach The, smulaed run-lenghs can be used o emprcally calculae he characerscs of he run-lengh dsrbuon The n-conrol (IC) desgn of he GWMA-TBE char For a gven or chosen value of k, he wo parameers q and are vared over a ceran range and for each ( q, ) combnaon, he values of he charng consan, e L, are obaned so ha he aaned n-conrol ARL s close o (n hs case slghly above or below due o he use of smulaon) he nomnal or specfed value * ARL The ypcal ndusry sandards for he * ARL are or and we consder he former n our sudy The ypcal recommendaon for he smoohng parameer for an EWMA char s o choose smaller values for smaller shfs (see Mongomery,, page ) Because he GWMA char reduces o an EWMA char when q and, a larger value of q, e closer o, should be a reasonable choce for he GWMA char o deec small shfs To hs end, Sheu and Ln () noed ha ( q, ) combnaons n he nervals q and enhanced he sensvy of he GWMA- X char and ouperformed he EWMA- X char for small shfs (e, less han sandard devaons n he locaon) The same range of ( q, ) values were also consdered by Sheu and Yang (), Teh e al (), Sheu e al (), and Lu () In our smulaon sudy, we se k,,,, and consdered he range q,,,,, and,,,,,,, respecvely

17 Usng smulaon along wh a grd search algorhm, we obaned he charng consan L for he chosen ( q, ) combnaon and specfed value of k, so ha he aaned ARL s approxmaely equal o * ARL ; he values of L deermned n hs way are presened n Table along wh he aaned ARL values The L values n Table wll be useful for he desgn and mplemenaon of he GWMA-TBE char; hs ncludes desgnng an EWMA or Shewhar-ype TBE char For example, f we choose k and le q,, a value of L leads o an aaned ARL equal o To hghlgh he desgn parameers of he EWMA and Shewhar-ype TBE chars, he row for have been hghlghed From Table, we noe n general ha mulple combnaons of he parameers q,, L wll yeld he same ARL for some chosen or specfed value of k Ths s somewha challengng because, apar from desrng a suffcenly large ARL, he ARL should be small for an effecve GWMA-TBE char Therefore, he ( q,, L) combnaon wh he mnmum ARL for a specfed shf s sad o be he opmal combnaon The opmal desgn of he GWMA-TBE char consss of specfyng he desred ARL and ARL values as well as he magnude of he process shf ha s ancpaed and hen selec he ( q,, L) combnaon ha provdes he desred ARL performance; ypcally, he ( q,, L) combnaon wh he mnmum ARL s seleced The soluon o hs problem s an opmzaon problem n -dmensonal space Alhough a dealed sudy on he opmal desgn for he GWMA-TBE char s ou-of-scope for hs paper, n he nex secon we nvesgae and commen on he near opmal desgn gven he ou-of-conrol (OOC) ARL values for dfferen shf szes

18 Table : Values of L for he GWMA-TBE char for dfferen ( q, α ) when k =,,,, and * ARL q Shewhar k α q = EWM A ()

19 EWM A () EWM A ()

20 EWM A () EWM A ()

21 The ou-of-conrol (OOC) performance of he GWMA-TBE char To sudy he ou-of conrol (OOC) performance of he proposed GWMA-TBE char, we used all he combnaons of he desgn parameers shown n Table ; hese combnaons ensure ha he ARL s are close o (when, e, no shf occurred) and herefore guaranee ha all he chars are a an equal foong Because we use only a lower conrol lm (as we are neresed n a susaned downward sep shf), we only focus on values for and we specfcally use,,,,,,, and Also, we do no consder any shfs, e, more han a % decrease, as he Shewhar-ype chars are well-known o be more effcen han he GWMA and/or EWMA chars for large changes The resuls for he OOC performance comparsons are shown n Tables, and for k, and, respecvely, and for some combnaons of q,, L The resuls for k, and oher combnaons of q,, L are no presened here for concseness, bu are avalable from he auhors upon reques Noe he followng: Each cell n Tables, and dsplays he ARL, he sandard devaon of he run-lengh ( SDRL ) as well as he h, h, h, h and h percenles; The ables nclude he resuls for he EWMA and he Shewhar-ype TBE chars The hghlghed columns n he ables dsplay he resuls for he EWMA char whereas he columns on he rgh-hand sde dsplay he resuls for he Shewhar-ype char; Alhough hese ables are dense and conan los of nformaon, a quck comparson of he resuls reveals he followng man pons:

22 Boh he GWMA-TBE and he EWMA-TBE chars ouperform he Shewhar-ype TBE char for all values of k and for all shfs; hs ncludes he scenaro where decreases o a quarer of s orgnal value, e, when ; The GWMA-TBE can be desgned o ouperform he EWMA-TBE char for very small o moderae shfs; hs can be done usng a suably seleced combnaon of q,, L For example, consder Table wh he resuls for k and focus on he secon where q : The hree GWMA chars wh, and all ouperform he EWMA char when The same s rue for oher values of k and q,, L Ths clearly llusraes he benef of he addonal desgn parameer ( ) n consrucng a GWMA conrol char; As he value of k ncreases, he performance for boh he GWMA-TBE and EWMA-TBE chars mproves For nsance, when k and, he ARL for a GWMA-TBE char wh q,, L s whle he ARL for GWMA-TBE char when k and and q,, L s Ths resul ndcaes ha a hgher value of k mproves he sensvy of he GWMA and EWMA chars However, cauon should be appled when mplemenng hese chars n pracce snce a larger value of k also mples more observaons/falures has o be colleced before a decson can be made abou he saus of he process The specfc choce of k s lef o he praconers; v As menoned earler, he opmal desgn of he GWMA-TBE char ypcally consss of specfyng he desred ARL and ARL values as well as he magnude of he process shf ( ) ha s ancpaed and hen selec he ( q,, L) combnaon ha provdes he desred ARL performance However, Chan and Zang () argued ha s also mporan o ake

23 Table : IC and OOC ARL, SDRL as well as he h, h, h (or MDRL ), h and h α = Sh L = f percenles of he run-lengh for dfferen combnaons of q, αl, when k q = q = q = q = α α α α = = = =

24

25

26

27 Table : IC and OOC ARL, SDRL as well as he h, h, h (or MDRL ), h and h α = Sh L = f percenles of he run-lengh for dfferen combnaons of q, αl, when k q = q = q = q = α α α α = = = =

28

29

30

31 Table : IC and OOC ARL, SDRL as well as he h, h, h (or MDRL ), h and h α = Sh L = f percenles of he run-lengh for dfferen combnaons of q, αl, when k q = q = q = q = α α α α = = = =

32

33

34

35 no accoun he SDRL relave o he ARL when desgnng an EWMA char; e; hey added he consran ha SDRL ARL ; hs was done o avod large varaon n he runlengh dsrbuon So, usng her creron for he opmal desgn of he EWMA char for he proposed GWMA-TBE char, he opmal desgn would conss of specfyng he desred ARL and ARL values as well as he magnude of he process shf ( ) ha s ancpaed and hen selec he ( q,, L) combnaon ha provdes he desred ARL performance subjec o he consran ha SDRL ARL We used boh defnons o oban he near opmal desgn Noe ha, we specfcally refer o hs as he near opmal desgn because n he way he desgn was carred ou, e, he wo parameers q and were vared over a ceran range and for each ( q, ) combnaon ha was nvesgaed he value of he charng consan, e, L, was obaned so ha he aaned n-conrol ARL s close o he nomnal or specfed value * ARL Ths approach mgh exclude he rue opmal desgn Usng he resuls n Tables, and, he near opmal desgn would be hose ( q,, L) combnaons ha resul n he smalles ARL for a specfed shf ( ) gven he ARL ; wh or whou he consran SDRL ARL Table provdes he near opmal combnaons of he parameers ( q,, L) as well as he ARL values for dfferen and k,,, and From Table, we observe he followng: a For smaller shfs, e, for values of closer o, a larger value of q (closer o ) and a smaller value of (closer o ) works bes The converse also holds, e, for larger

36 No consr an Table : Near opmal q, αl, combnaons wh correspondng ARL values k k k k k ARL q α L ARL q α L ARL q α L ARL q α L ARL q α L SDRL ARL k k k k k ARL q α L ARL q α L ARL q α L ARL q α L ARL q α L

37

38 shfs, a smaller value of q (closer o ) and larger value of (greaer and equal o ) works bes; b The EWMA-TBE char, e when, only feaures as he opmal desgn for Ths resul confrms he fac ha a GWMA char can be desgned o ouperform an EWMA char for very small, small and moderae shfs The zero-sae versus he seady-sae average run-lengh The ou-of-conrol average run-lenghs ( ARL ) of he prevous secons are called he zero-sae ARL s and are based on he assumpon ha he shf occurs a sar-up, e, a me However, s also of neres o see wheher a GWMA-TBE char desgned for opmal performance a sar-up works well f he shf occurs laer n he process, say a me,,,, ec Ths s called he seady-sae performance and he ARL s referred o as he seady-sae ARL; he assumpon s bascally ha a sable process has been operang n-conrol for some me before he shf occurs We calculaed he seady-sae ARL for some ( q,, L) combnaons when k, and compared o he zero-sae ARL The resuls are presened n Table and from hs able we observe he followng: The zero-sae and seady-sae ARL are he same for all praccal purposes The mnor dfferences ha are observed are due o he nheren smulaon varably; The GWMA-TBE char generally performs smlar or, n many cases, beer han he EWMA-TBE char when (e, for very small, small or moderae shfs) For (e, for larger shfs), he EWMA-TBE char generally performs beer han he GWMA- TBE These resuls hold rrespecve of when he shf has occurred;

39 Table : The zero-sae and seady-sae ARL values for he GWMA-TBE char Shf k Tme of shf q α L k = Shewh ar k =

40 Shewh ar

41 Boh he GWMA-TBE and EWMA-TBE chars ouperform he Shewhar-TBE; hs s a resul ha was also observed when sudyng he resuls of Tables, and wh he zerosae ARL s Noe ha he charng sascs for he Shewhar-TBE char are muually ndependen, rrespecve of he me he shf occurs, and herefore he me of shf can always be aken as when consderng he OOC run-lengh Phase II GWMA-TBE char When he n-conrol (IC) value of he parameer s unknown, s ypcally esmaed from an n-conrol Phase I sample (he so-called reference or calbraon sample) before prospecve (e, onlne) monorng sars n Phase II; hs scenaro s referred o as he sandard unknown case and denoed by Case U Noe ha he Phase I sample s aken when he process was hough o operae n-conrol and whou any specal causes of concern (see Mongomery (), page ) The pon esmae of s denoed by ˆ and s used o esmae he sarng value Z and o esmae he Phase II lower conrol lm of he GWMA-TBE char; n boh cases, ˆ s subsued for he known parameer value The charng sasc and esmaed lower conrol lm of he GWMA-TBE char n Case U are defned as follows: and ˆ for,,, () Z q q X q k ˆ ˆ LCL k L Qk ˆ ()

42 The Phase II GWMA-TBE char operaes n he same manner as he GWMA-TBE char of Case K, e, he charng sasc n () s ploed on a conrol char wh he ˆ LCL gven n () and f a pon plos on or below he lower conrol lm he char sgnals and he process s declared OOC so ha a search for assgnable causes s sared The followng pons should be noed: Obanng an n-conrol Phase I reference sample s an mporan problem n s own rgh, bu we do no sudy hs problem here We assume ha we have an n-conrol Phase I reference sample The neresed reader s referred o Chakrabor e al () for an n-deph overvew of Phase I conrol charng procedures as well as he operaon and mplemenaon of hese chars; To esmae he unknown value of, a suable pon esmaor s requred We use he Maxmum Lkelhood Esmaor (MLE) whch s also he Unform Mnmum Varance Unbased Esmaor (UMVUE) To hs end, f we le X, X,, X ~ dgamma k, denoe he IC Phase I reference sample of sze m, he MLE s gven by ˆ and follows a Gamma ( km, / km ) dsrbuon; m m X / km Because a pon esmae s subsued for he unknown parameer value n Case U, he sarng value and lower conrol lms are boh random varables (as ndcaed by he ^ - noaon) Therefore, s of neres o examne he effecs of esmaon on he Phase II runlengh dsrbuon and hence he performance and robusness of he GWMA-TBE char usng he desgn parameers of Case K Saed dfferenly, we wan o know f he desgn parameers for he Case K GWMA-TBE char may be used for he Phase II GWMA-TBE

43 char n Case U To hs end, s mporan o sress ha, gven a pon esmae ˆ (calculaed usng an n-conrol Phase I sample), we oban he condonal Phase II run-lengh dsrbuon and we observe he condonal performance of he GWMA-TBE char Tha s, he observed performance of he char s based on ha specfc n-conrol Phase I sample Thus, he condonal performance and he condonal run-lengh dsrbuon wll be dfferen for each praconer based on hs/her own n-conrol Phase I sample Therefore, he condonal performance does no gve us a complee nsgh no he overall performance of he char In order o ge an overall pcure and a more general dea abou he effecs of parameer esmaon, we sudy he uncondonal run-lengh dsrbuon; hs uncondonal dsrbuon can be hough of as he run-lengh dsrbuon averaged over all possble values of he parameer esmaes Focusng on he average run-lengh ( ARL), we can wre he aforemenoned saemens mahemacally as follows: ˆ ˆ UARL E E R E R f d () ˆ ˆ ˆ, where UARL denoes he uncondonal ARL, ER ˆ denoes he condonal ARL, e, he expecaon of he run-lengh R condonal on a pon esmae ˆ, and f ˆ denoes he pdf of ˆ Smlar expressons can be obaned for oher characerscs of he run-lengh dsrbuon as well Effecs of parameer esmaon on he performance of he Phase II GWMA-TBE char To assess f he desgn parameers for he Case K GWMA-TBE char may be used for he Phase II GWMA-TBE char n Case U, a smulaon sudy was performed o calculae he n-conrol

44 and ou-of-conrol average run-lengh of he Phase II GWMA-TBE char Whou loss of generaly, we smulaed he n-conrol Phase I sample from a Gamma k, dsrbuon We used m,, and wh four ses of desgn parameers: ( k, q,, L ), ( k, q,, L ), ( k, q,, L ), ( k, q,, L ) Noe ha he frs and second ses use k whereas he hrd and fourh ses use k Also, he frs and hrd ses resul n GWMA-TBE chars whereas he second and fourh ses resul n EWMA-TBE chars The resuls are dsplayed n panels (a), (b), (c) and (d) of Fgure These graphs dsplay he ARL curves, e, ARL on he vercal axs versus he sze of he shf ( ) on he horzonal axs The graphs also dsplay he correspondng ARL-curve of Case K and s used as he reference o whch we compare he ARL-curves of Case U The ARL values are shown n he daa ables below he graphs for ease of reference From hese graphs, we observe he followng pons: The n-conrol ARL n Case U s subsanally larger han he correspondng n-conrol ARL n Case K; n some cases even greaer han, The ou-of-conrol ARL n Case U s also larger han he correspondng ou-of-conrol ARL of Case K Only f do we see ha he chars perform relavely smlar; The smaller he n-conrol Phase I reference sample s, e, he smaller he value of m, he larger he dfference beween he Case U and Case K and ARL s For he Case U char (based on he desgn parameers of he Case K char), o perform anyhng lke he Case K char requres more han, Phase I observaons;

45 (a) ( ) (b) ( ) (c) ( ) (d) ( ) Fgure : values for he GWMA-TBE char for unknown when

46 The EWMA-TBE char s less mpaced (compared o he GWMA-TBE char) by he esmaon from Phase I whch s seen by lookng a he ARL values on he vercal axs The above observaons demonsrae ha n hose scenaros where s unrealsc o wa a long me o gaher he necessary Phase I observaons, an alernave and more realsc approach s needed snce we canno use he Case K desgn parameers To hs end, one should adjus he conrol lm, e, conrac he conrol lm Ths s nvesgaed nex The n-conrol (IC) desgn of he Phase II GWMA-TBE char The desgn of he Phase II GWMA-TBE char requres one o use expresson () and adjus he wdh of he lower conrol lm ( L) so ha he n-conrol uncondonal ARL s equal o a prespecfed value such as Ths means we wan o solve for he value of L ha sasfes he followng equaon:,, UARL ˆ ˆ ˆ ˆ E E R IC E R IC f d () ˆ The negral n equaon () can be approxmaed by usng compuer smulaon and calculang he average of a suffcenly large number of n-conrol condonal average run-lenghs, e, N calculang ˆ E R, IC, N denoes he number of smulaons and ER ˆ, IC N j denoes he condonal n-conrol average run-lengh Usng a search algorhm, we have found he value of N L ha sasfes ER ˆ IC, N for some ( q, ) combnaons These values are j dsplayed n Table along wh he value of L for he Case K char; he laer correspond o he values lsed n Table From Table, we observe ha he value of L converges o he Case K

47 Table : Desgn parameers for he Phase II GWMA-TBE char n Case U m k q α Case K

48 value as he number of observaons ncreases So, he smaller he value of m, he narrower he conrol lms should be o compensae for he large ARL we observed n Fgure In he nex wo secons, we dscuss a generalzaon of he proposed GWMA char and an llusrave example s provded n order o demonsrae he applcaon of he proposed char GWMA-TBE for monorng he varance The proposed GWMA char based on he Gamma k, dsrbuon can also be used o monor he known n-conrol (IC) varance of a normal dsrbuon for a susaned downward sep shf To hs end, f X j denoes he j h observaon n he h sample of sze n, where X ~ dn, for,,,, j,,, n, and j s he known n-conrol (IC) sandard/value for he varance, he sasc n S s dsrbued as Gamma k n, n, where S Xj X n j denoes he sample varance If he mean of he normal dsrbuon s known beforehand and equal o, he sample varance wll be calculaed as S X n j n and he sasc j ns s dsrbued n Gamma k, So, he desgn parameers gven n Table can be used o consruc a GWMA char o monor he known varance For example, f samples of sze n s aken from a normal dsrbuon wh known varance and unknown mean, follows ha S ~ Gamma, So, usng Table and seng ( q,, L ) wll resul n an

49 EWMA char wh an n-conrol ARL of The correspondng ou-of-conrol ARL performance of he EWMA char, for monorng he varance, can be found from Table wh / Noe ha, n hs case, when, here would have been a decrease n he varance whch would be equvalen o an mprovemen n he process Noe ha f he varance s unknown and has o be esmaed from an n-conrol Phase I reference sample usng he pooled varance esmaor S p wh v degrees of freedom, for example he sasc vs vs p follows an F dsrbuon, where v denoes he degrees of freedom for a v, v fuure or Phase II sample varance for hs scenaro Illusrave example S The proposed GWMA-TBE char s, however, no sued To llusrae he applcaon of he proposed GWMA-TBE char, a smulaed daase s used Because he proposed GWMA-TBE char s for monorng a hgh-performance process, for whch falure rae s assumed o be very small, he known value of he n-conrol (IC) process falure rae s aken as / / ; hs mples he me unl he follows a Gamma ( k, ) dsrbuon h k falure, X, The smulaed daase consss of random observaons from a Gamma ( k, ) dsrbuon If we assume ha he known value of s, he smulaed daase may be vewed as observaons from an ou-of-conrol (OOC) process followng a shf of / = ; hs s a deeroraon n he process Noe ha, because k, we wll be

50 monorng he oal me beween wo consecuve falures Also, because he precedng developmens assumed ha, he smulaed daa has o be scaled by dvdng by Two ses of desgn parameers are used: ( q,, L ) and ( q,, L ) The frs se resuls n a GWMA-TBE char whereas he second se resuls n an EWMA-TBE char (because ) wh smoohng parameer These wo ses of parameers are chosen for he llusraon purposes only of he proposed char and any oher combnaon can be chosen n hs regard However, whle choosng a parameer combnaon n pracce, s mporan o noe ha, a larger value of q and a smaller value of usually works well for smaller shfs, e, for values of closer o From Table, we can see ha boh chars are desgned so ha her ARL From Table, we observe ha he GWMA-TBE char has an OOC ARL of whle he EWMA-TBE char has an OOC ARL of So, we would expec he GWMA-TBE char o sgnal before he EWMA-TBE char The lower conrol lms are calculaed usng equaon () and are equal o and, respecvely The charng sascs are calculaed usng equaon () wh ; hs means he sarng value s aken as Fgure dsplays he GWMA-TBE and EWMA-TBE chars From hs fgure, we observe ha he GWMA-TBE char sgnals a me whereas he EWMA-TBE char sgnals a me Noe ha s expeced ha he GWMA-TBE char ouperforms he EWAM char for hese choces of he desgn parameers

51 (a) GWMA-TBE char (b) EWMA-TBE char Fgure : GWMA-TBE char and EWMA-TBE char for he example

52 Concludng remarks Effcen conrol chars are crucal o mprove he qualy of a process To hs end, an effcen conrol char should deec any change as quckly as possble snce he faser a change s deeced he qucker a correcve acon can be aken However, when monorng nonconformes or defecs, he radonal Shewhar-ype arbue chars (such as he c -char and he p -char) are known o be neffcen a deecng small changes quckly and poor o deec changes when he falure rae s very small For hs reason, we have proposed a me-weghed char, whch sequenally accumulaes all he nformaon over me, and monor he me beween evens (TBE), e, he me beween consecuve defecs Accumulang all hsorcal nformaon provdes greaer sensvy o deec small changes effecvely We have specfcally proposed a one-sded Generally Weghed Movng Average (GWMA) conrol char based on he gamma dsrbuon o monor he TBE; hs char has been called he GWMA-TBE char I has been shown ha he proposed GWMA char ncludes he one-sded Exponenally Weghed Movng Average (EWMA) and Shewhar-ype chars as specal cases We have nvesgaed he scenaros when he scale parameer of he gamma dsrbuon s known (referred o as he sandard known ) as well as when s unknown (referred o as he sandard unknown ) I has been shown ha when one esmaes he unknown parameer from an n-conrol Phase I reference sample, he run-lengh (and n parcular he ARL) s adversely affeced Three mehods for calculang he run-lengh dsrbuon and he assocaed characerscs of he run-lengh dsrbuon have been nvesgaed; hs ncludes () Exac closed-form expressons, () A Markov chan approach, and () Mone Carlo Smulaon Due o he dffcules of numercally evaluang he closed-form expresson and usng he Markov chan approach for he GWMA-

53 TBE char, we have used compuer smulaon To ad he mplemenaon of he char, he necessary desgn parameers for Case K and Case U have been provded, whch guaranees ha he n-conrol ARL s equal o a specfed nomnal value An exensve performance analyss has been carred-ou and a se of near opmal desgn parameers have been provded The performance analyss has shown ha he GWMA-TBE char s beer han he well-known EWMA and Shewhar chars a deecng very small o moderae changes We have also shown how he GWMA-TBE char can be used o monor he varance of a normal dsrbuon Acknowledgemen: Ths research was suppored n par by STATOMET (Gran number: SMBA), Unversy of Preora, Souh Afrca The auhors graefully acknowledge and apprecae he effor by wo anonymous referees for provdng valuable commens and sugesons ha helped mprove hs work

54 References Aslam, M, Al-marshad, A H, Jun, C H () Monorng process mean usng generally weghed movng average char for exponenally dsrbued characerscs Communcaons n Sascs-Smulaon and Compuaon DOI: / Balakrshnan, N, Parossn, C, Turlo, J C () One sded conrol chars based on precedence and weghed precedence sascs Qualy and Relably Engneerng Inernaonal (): - Chakrabor S, Human S W, Graham M A () Phase I sascal process conrol chars: An overvew and some resuls Qualy Engneerng : - Chakrabory, N, Chakrabor, S, Human, S W, Balakrshnan, N () A generally weghed movng average sgned-rank conrol char Qualy and Relably Engneerng Inernaonal DOI: /qre Chan, L K, Zhang, J () Some ssues n he desgn of EWMA chars Communcaons n Sascs-Smulaon and Compuaon (): - Chu, W C () Generally weghed movng average conrol chars wh fas nal response feaures Journal of Appled Sascs (): - Chu, W C, Sheu, SH () Fas nal response feaures for Posson GWMA conrol chars Communcaons n Sascs-Smulaon and Compuaon (): - Fu, JC, Lou, WW () Dsrbuon Theory of Runs and Paerns and Is Applcaons: A Fne Markov Chan Imbeddng Approach World Scenfc publshng, Sngapore

55 Gan, F F () Desgns of one-and wo-sded exponenal EWMA chars Journal of Qualy Technology (): Gan, F F () Desgn of opmal exponenal CUSUM conrol chars Journal of Qualy Technology (): - Gan, F F, Cho, K P () Compung average run lenghs for exponenal CUSUM schemes Journal of Qualy Technology (): - Graham, MA, Chakrabor, S, Human, SW (a) A nonparamerc EWMA sgn char for locaon based on ndvdual measuremens Qualy Engneerng (): - Graham, MA, Chakrabor, S, Human, SW (b) A nonparamerc exponenally weghed movng average sgned-rank char for monorng locaon Compuaonal Sascs & Daa Analyss (): - Hsu, B M, La, P J, Shu, M H, Hung, Y Y () A comparave sudy of he monorng performance for weghed conrol chars Journal of Sascs and Managemen Sysems (): - Huang, C J () A sum of squares generally weghed movng average conrol char Communcaons n Sascs-Theory and Mehods (): - Huang, C J, Ta, S H, Lu, S L () Measurng he performance mprovemen of a double generally weghed movng average conrol char Exper Sysems wh Applcaons (): - Human, SW, Graham, MA () Average run-lenghs and operang characersc curves In Encyclopeda of Sascs n Qualy and Relably, Vol, pp -, John Wley & Sons, Hoboken, New Jersey

56 Lu, S L () An exended nonparamerc exponenally weghed movng average sgn conrol char Qualy and Relably Engneerng Inernaonal (): - Lucas, J M, Saccucc, M S () Exponenally weghed movng average conrol schemes: properes and enhancemens Technomercs (): - Maravelaks, P E, Casaglola, P () An EWMA char for monorng he process sandard devaon when parameers are esmaed Compuaonal Sascs & Daa Analyss (): - Mongomery, D C () Sascal Qualy Conrol: A Modern Inroducon h ed John Wley & Sons, Hoboken, New Jersey Nakagawa, T, Osak, S () The dscree Webull dsrbuon IEEE Transacons on Relably : - Sheu, S H, Chu, W C () Posson GWMA conrol char Communcaons n Sascs- Smulaon and Compuaon (): - Sheu, S H, Hseh, Y T () The exended GWMA conrol char Journal of Appled Sascs (): - Sheu, S H, Yang, L () The generally weghed movng average conrol char for monorng he process medan Qualy Engneerng (): - Sheu, S H, Huang, C J, Hsu, T S () Exended maxmum generally weghed movng average conrol char for monorng process mean and varably Compuers & Indusral Engneerng (): -

57 Sheu, S H, Huang, C J, Hsu, T S () Maxmum ch-square generally weghed movng average conrol char for monorng process mean and varably Communcaons n Sascs- Theory and Mehods (): - Sheu, S H, Ln, T C () The generally weghed movng average conrol char for deecng small shfs n he process mean Qualy Engneerng : Teh, S Y, Khoo, M B, Wu, Z () Monorng process mean and varance wh a sngle generally weghed movng average char Communcaons n Sascs-Theory and Mehods (): - Vardeman, S, Ray, D O () Average run lenghs for CUSUM schemes when observaons are exponenally dsrbued Technomercs (): - Xe, M, Goh, T N, Ranjan, P () Some effecve conrol char procedures for relably monorng Relably Engneerng & Sysem Safey (): - Zhang, C W, Xe, M, Goh, T N () Economc desgn of exponenal chars for me beween evens monorng Inernaonal Journal of Producon Research (): - Zhang, C W, Xe, M, Lu, J Y, Goh, T N () A conrol char for he Gamma dsrbuon as a model of me beween evens Inernaonal Journal of Producon Research (): -

58 Appendx A Decreasng weghs We consder wo scenaros: For and q, we have w q q q q whch s decreasng funcon of because q ; For and q, we have q w q q q q Now, because q q for all We can hen re-wre he weghs wll be decreasng f he remanng par of he produc, e, s a decreasng funcon for all, w q * show hs, we need o show ha he exponen of q vz, w,,, and, s decreasng To s decreasng for I can be easly shown ha for n non-decreasng pars of posve real numbers a, b, () n, such ha a b n, n n a b n a b () If a a and b b for all such ha a b, hen we can re-wre he nequaly n () as n n a b a b n () Takng b n, a n n () for some neger, we ge

59 n n n () Snce, we can wre for some posve neger n and replacng n n (), n we ge, * * w w Therefore, * w s decreasng for,,, and Ths mples ha q s decreasng snce q Thus, we have,,, and, whch complees he proof w q q o be decreasng for A α α Q q q s convergen as, for q, α We have We consder wo scenaros: Q q q q q q q For q, lmq q q whch s rval For q, we have q q q q q Subsung q q by q, follows ha Q q Q q q q q q I hen follows ha

60 So, for q, he monooncally ncreasng sequence { Q ;,,, } s bounded above and s herefore convergen Furher, Q for all values of, whch mples ha lmq and so here exss a number (denoed by Q ) n he nerval (,) such ha lmq Q for q Ths complees he proof r Pr R r Ir I r, where c I r Pr A for r,,, A Pr R r r Pr A c A r Pr A c r c c r c r c Pr A A Pr Pr r A A r c Therefore, Pr R r Ir Ir, where Ir Pr A A ARL + I r r Because PrR r Ir Ir, we have ARL r PrR r some of he erms, we oban r Upon expandng and re-arrangng c c = Pr A r Ir Ir Pr A I I = + r Ir ARL Pr R r Pr R r as requred r r r, r