An Improved Adaptive CUSUM Control Chart for Monitoring Process Mean

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1 An Improved Adapive CUSUM Conrol Char for Monioring Process Mean Jun Du School of Managemen Tianjin Universiy Tianjin 37, China Zhang Wu, Roger J. Jiao School of Mechanical and Aerospace Engineering Nanyang Technological Universiy Singapore, Absrac Adapive CUSUM (namely ACUSUM) chars have received much aenion recenly. I is found ha, by adjusing he reference parameer k dynamically, an ACUSUM char can achieve beer performance over a range of mean shifs han he convenional CUSUM char ha is designed o have maximal deecion effeciveness a a paricular level of shif. This aricle sudies a new feaure of he ACUSUM char regarding an addiional charing parameer, w (he exponenial of he sample mean shif in (x µ ) w ), which is also adaped according o he on-line esimaed value of he mean shif. The esing cases reveal ha his new adapive CUSUM char ouperforms he earlier ACUSUM char o a subsanial degree. Keywords adapive conrol char, CUSUM char, loss funcion, qualiy conrol, saisical process conrol I. INTRODUCTION The CUSUM conrol chars have been well recognized across indusries owing o he fac ha on-line measuremen and disribued compuing sysems become a norm in oday s Saisical Process Conrol (SPC) applicaions [1]. Usually, wo symmerical CUSUM chars are used ogeher for deecing wo-sided mean shifs. In a high-sided CUSUM char for deecing increasing mean shifs, he saisic C o be updaed for he ()h sample is C =, C = max(, C + ( x µ ) k), 1 where x is he ()h sample value of a qualiy characerisic x following an independen and idenical normal disribuion, N (µ, σ ); µ is he in-conrol mean of x; and k is he reference parameer. The difference (x - µ ) is a sample value of he mean shif σ. In his aricle, i is assumed ha he in-conrol mean µ and sandard deviaion σ of x are known a priori (for example hey can be esimaed from he field es records or hisorical daa). Moreover, he sandard deviaion σ is assumed o be unchanged, i.e., σ σ. The qualiy characerisic x can be convered o z ha has a sandard normal disribuion when process is in conrol. (1) z = ( x µ ) /σ. () Correspondingly, Equaion (1) becomes C =, C = max(, C + z k), 1 and he mean shif is. A convenional CUSUM char ofen deermines he reference parameer k wih reference o a special mean shif s. As a resul, i performs opimally when he mean shif is equal o s. However, since i is quie difficul o predic he acual magniude of for mos of he applicaions, here is no guaranee ha he convenional CUSUM char always performs well during he operaion. Recenly, Sparks () proposed he concep of adapive CUSUM (namely ACUSUM) char which adjuss he reference parameer k according o he on-line esimaed value ˆ of he mean shif []. Compared wih he convenional CUSUM chars, he adapive feaure makes he ACUSUM char more efficien in signaling a range of fuure expeced bu unknown mean shifs from a holisic viewpoin. Shu and Jiang (6) simplified he design and implemenaion of he ACUSUM char [3]. Recenly, some researchers have found ha an exponenial w will influence he sensiiviy of he CUSUM char wih respec o mean shifs if he variable z in (3) is replaced by z w, or he erm (x - µ ) in (1) is replaced by (x - µ ) w [4]. Usually, a larger w makes he CUSUM char more effecive for deecing larger ; whils a smaller w makes i more sensiive o smaller. In his aricle, we propose a new ACUSUM char in which boh k and w are adaped dynamically in accordance wih he curren esimaed value ˆ of he mean shif. For he purpose of deecing increasing mean shifs, he saisic C in he new ACUSUM char will be updaed by C = C = max, C + qk, ( 1 ) (3) /8 /$5. 8 IEEE CIS 8

2 w, if, q = w ( ), if <. where, he parameers k, w and q all depend on he curren sample value z. The saisic C may increase or decrease depending on wheher he sample value z (or x µ ) is larger or smaller han zero. However, C is always shrunk oward zero by he reference parameer k. When an increasing mean shif occurs, C is likely o become larger and larger. Sooner or laer, a subsequen sample poin will exceed he conrol limi H of he ACUSUM char, and hereby produce an ou-of-conrol signal. In order o differeniae he wo versions of ACUSUM chars, he firs ACUSUM char sudied by Sparks () and Shu and Jiang (6) is called as he ACUSUM I char, whils he one proposed in his aricle as he ACUSUM II char. I is noed ha, if w is equal o one, Equaion (4) is exacly he same as (3). This means ha he ACUSUM I char is jus a special case of he ACUSUM II char per se. As revealed by he performance sudies in he subsequen secions, he new adapive feaure of he ACUSUM II char is able o increase he deecion effeciveness by abou %, on average, compared wih he ACUSUM I char in which only he reference parameer k is adaped. This aricle focuses on he sudy of he high-sided ACUSUM II char. However, a symmerical low-sided counerpar can be buil sraighforwardly. (4) A. Sub-cusum Chars I seems desirable o adjus he parameers k and w (Equaion (4)) of an ACUSUM II char coninuously in accordance wih he curren esimaed value ˆ of he mean shif. However, sudies on VSSI (Variable Sample Sizes and Sampling Inervals) CUSUM chars discover ha using n (n = or 3) individual cusum chars (called he sub-cusum chars) may gain mos of he benefis ha can be reached by a VSSI CUSUM char [6], and are relaively easier o implemen. I suggess ha, in he implemenaion of an ACUSUM II char, one may only use n sub-cusum chars, each of which has differen values of k i and w i (i = 1,,, n) and each is bes for deecing a paricular discree value of i ( min < 1 < < n < max ). Joinly, he n sub-cusum chars will opimize he holisic performance of he ACUSUM II char over he enire mean shif range. The ACUSUM II char keeps on swiching among he n sub-cusum chars depending on which i is closes o he curren esimaed ˆ. Any momen one, and only one, subcusum char ha is bes for deecing ˆ is in use (or acive). Suppose, in a momen, if he (i)h sub-cusum char is acive, he parameers k and w in (4) will ake he values of k i and w i, respecively. In his aricle, (n = ) is always used because of is ease for design and implemenaion. The resuls of numerical sudies show ha, no maer (n = ) or (n = 3) is employed, he performance of he ACUSUM II chars is nearly he same. Each of he n discree i is se a he cener of one of he n equal inervals beween min and max, ha is, C =, ( 1 ) C = min, C + q+ k, i = min + i D i = n D = ( )/ n, max (.5), 1,,...,, min (6) w, if, q = w ( ), if <. The performance of a conrol char can be measured by he Average Run Lengh (ARL), meaning he average number of samples required o signal an ou-of-conrol case or produce a false alarm. The ou-of-conrol ARL 1 is commonly used as an indicaor of he power (or effeciveness) of he conrol char, whereas he in-conrol ARL for he false alarm rae. In his aricle, he ou-of-conrol ARL 1 will be compued under he seady-sae mode. I assumes ha he process has reached is saionary disribuion a he ime when he process shif occurs. Since producion processes ofen operae in in-conrol condiion for mos or relaively long periods of ime [5], he seady-sae mode is herefore more realisic han he zero-sae mode. II. DESIGN AND IMPLEMENTATION OF AN ACUSUM II CHART In his secion, he idea of sub-cusum char is firs inroduced and is used o discreize he ACUSUM II char. Then an opimizaion model is presened for he design of his char. I is followed by he selecion of he objecive funcion for he opimizaion design. The implemenaion of he ACUSUM II char is oulined a las. (5) where D is he disance beween wo discree values i and i+1. Each of he n sub-cusum chars has differen values of k i and w i. When he (i)h sub-cusum char is acive, Equaion (4) is discreized o: C =, C = max(, C + qk ), 1 i wi, if, q = i = 1,,..., n wi ( ), if <, Like in an ACUSUM I char, ˆ is updaed by a EWMA procedure in an ACUSUM II char. ˆ =, 1 ˆ = (1 λ) ˆ + λz. 1 The operaor makes ˆ equal o one of he n discree i whichever is closes o ((1 λ ) ˆ 1 + λ). Then he corresponding sub-cusum char is seleced o updae C using (7) (8)

3 (7). When he smoohing parameer λ equals one, ˆ is compleely deermined by z ; oherwise ˆ also depends on he informaion in he sequence of he sample poins. For example, suppose n =, λ =.4, min = 1 and max = 3, hen, Equaion (6) gives D = 1, 1 = 1.5 and =.5. Now, suppose ˆ 1 = 1 and z =3., ˆ = (1.4) =.1 =. Thus, he nd sub-cusum char will be acivaed o updae C, or w ( 1 ) (9) C = max, C + z k (1) B. Design Model To design an ACUSUM II char, hree parameers need o be specified: (1) τ, he minimum allowable in-conrol ARL for a one-sided ACUSUM II char; () min, he lower bound of mean shif; and (3) max, he upper bound of mean shif. Based on he specificaions, he charing parameers of an ACUSUM II char will be deermined in an inegraive and opimal manner using he following design model: Objecive funcion: U = minimum. (11) Consrain funcion: ARL = τ. (1) Design variables: k i, w i (i =1,,, n), λ, H. (13) where H is he conrol limi of he ACUSUM II char. When n =, here are in oal six design variables, among which k 1, w 1, k, w and λ are independen. The conrol limi H is adjused o ensure ha he ARL of he ACUSUM II char is equal o τ. The opimizaion aims o find he opimal values of hese design variables so ha he objecive funcion U is minimized or boh small and large mean shifs can be deeced quickly. The selecion of U will be discussed shorly. Any nonlinear opimizaion program may be used o search he opimal soluion. In our sudy, he simple, ye reliable, Hooke-Jeeves procedure is employed [7]. I can complee he design of an ACUSUM II char in a few CPU seconds wih a personal compuer. C. Design Objecive Since our goal is o make conrol chars efficien a signalling a range of mean shifs, he objecive funcion should measure he holisic performance of he chars across he range raher han he effeciveness a a paricular poin. Furhermore, since i is assumed ha all mean shifs wihin a range are equally imporan [], a uniform disribuion for is implied. The comparison of he overall performance of wo chars may be formulaed as follows: max 1 ARL( ) RARL = d, (14) ARL ( ) max min min benchmark where, ARL() is produced by a paricular char a and ARL benchmark () is generaed by a char ha acs as he benchmark. Obviously, if he RARL value of a char is larger han one, his char is generally less effecive han he benchmark, and vice versa. An alernaive is o use he Exra Quadraic Loss (EQL) o measure and compare he performance of he chars. When σ is a consan, he quadraic loss incurred by a mean shif is simply equal o ( σ ) [4, ( Wu e al. 4), because σ + ( µ µ ) σ = σ + σ σ = σ ( ) ( ). (15) Moreover, since he qualiy cos is proporional o ARL(), he overall EQL can be calculaed as follows: max EQL = σ ARL( ) d max min min. (16) Boh EQL and RARL are acquired by inegraion across he whole shif range. The inegraion can be compued by a numerical mehod and he ARL() of he ACUSUM II char is calculaed by he formulae derived in Appendix A. The index EQL based on loss funcion has wo advanages compared wih RARL. Firsly, he loss funcion is a more comprehensive measure of he charing performance han ARL, because i considers all he conribuors o he qualiy cos including he ime o signal and he magniude of. Secondly, he evaluaion of EQL does no require a predeermined benchmark char. In view of his, EQL will be used as he objecive funcion U in (11) for he design of he ACUSUM II chars. The minimizaion of EQL will reduce he loss in qualiy (or he cos, or he damage) incurred in he ou-of-conrol cases. Like RARL, he raio beween he EQL values of wo conrol chars serves as a measure of he relaive effeciveness of he wo chars. D. Implemenaion Afer an ACUSUM II char has been designed, i can be implemened as follows: 1) Iniialize ˆ as 1 (noe, ˆ means he esimaed mean shif a he beginning when =, and 1 is he designaed value of he shif for he firs sub-cusum char), and he saisic C in (7) as zero. ) Take a sample value x of he qualiy characerisic. 3) Cover x o z using (). 4) Updae ˆ by (8). 5) If ˆ = i, use he charing parameers k i, w i of he (i)h sub-cusum char o updae C, ha is,

4 ( 1 ) C = max, C + qk, i wi if, q = w i ( ) if <. (17) 6) If C H, he process is hough in conrol, and go back o sep ) for he nex sample. 7) Oherwise (i.e., C > H), he ACUSUM II char produces an ou-of-conrol signal, and he process is sopped immediaely for invesigaion. III. COMPARATIVE STUDIES In his secion, he performance of four conrol chars is compared. For he convenience, all chars are sudied as onesided chars wih an upper conrol limi for deecing increasing mean shifs. Furhermore, in-conrol µ and σ are assumed as zero and one, respecively. 1) The convenional CUSUM char: The design of a convenional CUSUM char aims a minimizing he ou-ofconrol ARL 1 a a specified mean shif level of s. Usually, s is se as min, because s should be he smalles shif such ha any shif larger han s is considered imporan enough o be deeced quickly [8]. Consequenly, he reference parameer k is se as.5 min. ) The opimal CUSUM char: This char is very similar o a convenional CUSUM char in aspecs of he updaing of he cumulaive saisic C (as in (1)), he fixed parameer k, and he operaional rules. However, he opimal CUSUM char is designed by a new opimizaion algorihm in which he parameers k and H are opimized by using EQL (i.e., he performance over he whole shif range) as he objecive funcion and (ARL = τ) as he consrain funcion. 3) The ACUSUM I char: This is he adapive CUSUM char wih k being adjused during he operaion. Here, he model developed by Shu and Jiang (6) is adoped, because his model is easier o be designed han he model proposed by Sparks (). However he performance of boh models is almos he same. In his aricle, he wo charing parameers λ and Q of an ACUSUM I char are opimized so ha EQL is minimized subjec o (ARL = τ). 4) The ACUSUM II char proposed in his aricle. The firs comparison is carried ou under a general condiion wih (τ = 74, min =.5, max = 4). The specificaion of (τ = 74) ensures ha he resulan false alarm rae is idenical o ha of a ypical 3-sigma X char when wo symmerical CUSUM chars are used simulaneously o deec he wo-sided mean shifs. Wih hese specificaions, he four conrol chars are designed and he resulan charing parameers and he ARL values, are summarized in Table I. There are several ineresing findings. TABLE I. ARL COMPARISON AMONG CUSUM AND ACUSUM CHARTS ( min =.5, max = 4.) con CUSUM op CUSUM ACUSUM I ACUSUM II k =.5 k =.85 H =1.76 H =6.898 λ =.4 λ =.456 H =8.9 H =3.48 Q =3.417 k 1 =.594 L =4. k =1.154 w 1 =1.435 w = EQL EQL EQLACUSUM II RARL ) Boh he ACUSUM I and II chars are more effecive han he convenional CUSUM char almos across he enire shif range excep for 1. The ACUSUM II char ouperforms he convenional CUSUM char o a more significan degree han he ACUSUM I char does. ) The ACUSUM II char also ouperforms he ACUSUM I char for mos of he cases. I is only less sensiive han he laer o very small (i.e., when.5). I is noed ha, min is specified as.5 in his case. Then, a low ARL value for ( min ) will be considered as a drawback, because i may resul in over-correcion and inroduce exra variabiliy [8]. 3) The opimal CUSUM char has achieved significan improvemen in deecion effeciveness compared wih he convenional CUSUM char. The opimal CUSUM char has a larger ARL 1 only for minor mean shifs. As long as > 1, i becomes much more effecive han he convenional CUSUM char. The opimal CUSUM char may even ouperforms he ACUSUM I char. Bu i is generally less effecive han he ACUSUM II char. As aforemenioned, for mos of he cases, no char will give beer performance han oher chars for all shifs [9]. Consequenly, in order o make an accurae and objecive decision abou he relaive effeciveness of he chars, i is necessary o evaluae he values of he following hree holisic measures of he chars. 1) EQL (Equaion (16)); ) EQL / EQL ACUSUM II, he raio beween he EQL of a char and he EQL of he ACUSUM II char; and 3) RARL (Equaion (14)), he raio beween he ARL of a char and he ARL of he ACUSUM II char, i.e., using he ACUSUM II char as he benchmark. The resuls are enumeraed a he boom of Table I. I is ineresing o find ha he values of EQL / EQL ACUSUM II and RARL of a char are ofen fairly close o each oher. They reveal ha:

5 1) When considering he whole shif range, boh he ACUSUM I and ACUSUM II chars obviously ouperform he convenional CUSUM char. If measured by EQL, he ACUSUM I and ACUSUM II chars are more effecive han he convenional CUSUM char by 19.4% and 43.8%, respecively. ) Beween he ACUSUM I and ACUSUM II chars, he laer ouperforms he former by more han %, on average, across he enire shif range. 3) The opimal CUSUM char uses a fixed k as he convenional CUSUM char, bu is overall performance has been significanly improved. However, he opimal CUSUM char is less effecive han he ACUSUM II char by abou 7% measured by eiher EQL or RARL. The opimal CUSUM char seems simpler han he ACUSUM II char. Bu in a compuerized environmen, he operaion of boh chars is equally easy. Nex, he effeciveness of he four chars is furher compared in a facorial experimenal design wih four differen cases (combinaions) of min and max : (1) min =.5, max = 3.; () min =.5, max = 5.; (3) min =.75, max = 3.; (4) min =.75, max = 5.. In all four cases, he values of ARL, EQL and RARL reveal he performance characerisics similar o hose shown in Table I. IV. CONCLUSIONS AND DISCUSSIONS This aricle has proposed an improved adapive CUSUM char, he ACUSUM II char, for deecing process shifs in mean. This char furher enhance he performance of he earlier ACUSUM I char o a promising exen. The improvemen is aribuable o he on-line adapion of an addiional charing parameer, w, which is he exponenial of he sample mean shifs. The ACUSUM II char is also much easier o design in erms of he required CPU ime as is ARL can be evaluaed by a Markov procedure. ACKNOWLEDGMENT This research is suppored by he Naional Naural Science Foundaion of China under gran 771. APPENDIX CALCULATION OF THE ARL OF THE ACUSUM II CHART The ACUSUM II char can be described by a wodimensional Markov chain. Suppose ha he saisic C in (7) experiences m differen ransiional saes before being absorbed ino he ou-of-conrol sae. Saes o (m 1) are inconrol saes and sae m is an ou-of-conrol sae. The widh of he inerval of each sae is given as Meanwhile, n sub-cusum chars are employed corresponding o n discree mean shifs 1,,, n. where, i = min + (i.5) D i =1,,, n D = ( max min ) / n is he disance beween hese discree mean shif values. (A.3) (A.4) In a wo-dimensional Markov chain, a poin (i, j) represens a saus in which he (i)h sub-cusum char is in use and he saisic C is equal o O j. Le p ij-uv be he ransiion probabiliy from poin (i, j) o poin (u, v). p = f ( z ) dz ijuv Ω 1 = exp.5 ( µ ) Ω π dz, (Α.5) where, f (z ) is he densiy funcion of z, and Ω is he inersecion of he following wo domains Ω 1 and Ω : 1) Domain Ω 1 Ω 1 : LB z UB. (A.6) I is he region for which he ACUSUM II char will use he (u)h sub-cusum char for he ()h sample, given ha he (i)h cub-cusum char is employed for he (-1)h sample; or he esimaed ˆ is closes o u for he ()h sample, given ha i is equal o i for he (-1)h sample. Referring o (A.3), he lower bound LB and upper bound UB of he region Ω 1 can be deermined as follows:, if µ = 1, LB = [ u.5 D(1 λ) i] / λ, if µ >1. [ ] u +.5 D (1 λ) i / λ, if µ < n, UB = +, if µ = n. ) Domain Ω Ω : lb z ub. (Α.7) (Α.8) (A.9) I is he region for which he saisic C will be closes o O v, given ha C -1 is equal o O j. To make his ransiion (see (7)), d = H / (m.5). (Α.1) Q,, L < q< QU if v > (Α.1) The cener, O j, of sae j is given as or O j = j d j =, 1,, m. (A.) < q< Q,, U if v = (Α.11)

6 where, Then, since (from (7)) Q = ( O.5 d O ) + k, L v j u Q = ( O +.5 d O ) + k. (Α.1) U v j u 1/ wu q, if q, = q if q < 1/ wu ( ),, (Α.13) herefore he lower bound lb and upper bound ub of z in he region Ω are deermined as follows: and, if v =, 1/ wu lb = QL, if v > and QL, > < 1/ wu ( QL), if v and QL, 1/ wu QU, if QU, ub = < 1/ wu ( QU), if QU. (Α.14) (Α.15) The ransiion probabiliy p ij-uv in (A.5) can be acually compued by pij uv =Φ(min( ub, UB) u) Φ(max( lb, LB) u), (Α.16) if min(ub,ub)>max(lb,lb); oherwise, pij uv =. And Ф() is he cumulaive probabiliy funcion of he sandard normal disribuion. When compuing he in-conrol ARL, he ransiion probabiliy p ij-uv is calculaed wih µ =. Based on p ij-uv, he inconrol ransiion marix R can be esablished. I is a (n m) (n m) marix excluding he elemens associaed wih he absorbing (or ou-of-conrol) sae. The zero-sae ARL is equal o he firs elemen of he vecor V given by he following expression: V = (I R ) -1 1, (A.17) where I is an ideniy marix and 1 is a vecor wih all elemens equal o one. The ransiion marix R 1 for calculaing he ou-of-conrol ARL 1 can be esablished similarly o R excep ha he ransiion probabiliy p ij-uv in R 1 mus be evaluaed using he ou-of-conrol µ (= ). The ou-of-conrol ARL 1 under he seady-sae mode is calculaed as he following: ARL 1 = B T (I R 1 ) -1 1, (A.18) where, B is he seady-sae probabiliy vecor wih (µ = ). I is obained by firs normalizing R and hen solving he following equaion: subjec o B = R B, (Α.19) T 1 Τ Β = 1. (Α.) REFERENCES [1] W. H. Woodall, and D. C. Mongomery, Research issues and ideas in saisical process conrol, Journal of Qualiy Technology, vol. 31, pp , [] R. S. Sparks, Cusum chars for signalling varying locaion shifs, Journal of Qualiy Technology, vol. 3, pp ,. [3] L. J. Shu, and W. Jiang, A Markov chain model for he adapive CUSUM conrol char, Journal of Qualiy Technology, vol. 38, pp , 6. [4] M. R. Jr. Reynolds, and Z. G. Soumbos, Should observaions be grouped for effecive process monioring, Journal of Qualiy Technology, vol. 36, pp , 4. [5] D. C. Mongomery, Inroducion o Saisical Qualiy Conrol, John Wiley & Sons, Singapore, 5. [6] S. Zhang, and Z. Wu, Monioring he process mean and variance by he WLC scheme wih variable sampling inervals, IIE Transacions, vol. 38, pp , 6. [7] J. N. Siddall, Opimal Engineering Design: Principles and Applicaions, M. Dekker, New York, 198. [8] W. H. Woodall, The saisical design of qualiy chars, Saisician, vol. 34, pp , [9] M. R. Jr. Reynolds, and Z. G. Soumbos, Comparisons of some exponenially weighed moving average conrol chars for monioring he process mean and variance, Technomerics, vol. 48, pp , 6. [1] Z. Wu, M. Shamsuzzaman, and E. S. Pan, Opimizaion design of he conrol chars based on Taguchi s loss funcion and random process shifs, Inernaional Journal of Producion Research, vol. 4, pp , 4.