A Robust Exponentially Weighted Moving Average Control Chart for the Process Mean
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1 Journal of Modern Applied Saisical Mehods Volume 5 Issue Aricle A Robus Exponenially Weighed Moving Average Conrol Char for he Process Mean Michael B. C. Khoo Universii Sains, Malaysia, mkbc@usm.my S. Y. Sim Universii Sains Malaysia Follow his and addiional works a: hp://digialcommons.wayne.edu/jmasm Par of he Applied Saisics Commons, Social and Behavioral Sciences Commons, and he Saisical Theory Commons Recommended Ciaion Khoo, Michael B. C. and Sim, S. Y. (005) "A Robus Exponenially Weighed Moving Average Conrol Char for he Process Mean," Journal of Modern Applied Saisical Mehods: Vol. 5 : Iss., Aricle. DOI: 0.37/jmasm/ Available a: hp://digialcommons.wayne.edu/jmasm/vol5/iss/ This Regular Aricle is brough o you for free and open access by he Open Access Journals a DigialCommons@WayneSae. I has been acceped for inclusion in Journal of Modern Applied Saisical Mehods by an auhorized edior of DigialCommons@WayneSae.
2 Journal of Modern Applied Saisical Mehods Copyrigh 006 JMASM, Inc. November, 006, Vol. 5, No., /06/$95.00 A Robus Exponenially Weighed Moving Average Conrol Char for he Process Mean Michael B.C. Khoo S.Y. Sim School of Mahemaical Sciences, Universii Sains Malaysia To dae, numerous exensions of he exponenially weighed moving average, EWMA chars have been made. A new robus EWMA char for he process mean is proposed. I enables easier deecion of ouliers and increase sensiiviy o oher forms of ou-of-conrol siuaion when ouliers are presen. Key words: Exponenially weighed moving average (EWMA), cumulaive sum (CUSUM), Shewhar, process mean, sample mean, sample range, average run lengh (ARL) Inroducion The EWMA char is a good alernaive o he Shewhar char in he deecion of small shifs. The EWMA char consruced from he sample mean is firs developed by Robers (959). Since hen various exensions of he EWMA chars have been proposed. Swee (986) proposed wo models o consruc simulaneous conrol chars o monior he mean and he variance of a process using he EWMA. Crowder (987 & 989) provided average run lengh (ARL) ables and graphs for he selecion of he opimum values of he EWMA conrol char parameers in he design of an EWMA char. Ng & Case (989) presened several EWMA conrol char schemes based on individual measuremen, Michael B. C. Khoo is a member of he American Sociey for ualiy (AS) and he American Saisical Associaion (ASA). He also serves as a member of he ediorial boards of ualiy Engineering, ualiy Managemen Journal and Journal of Modern Applied Saisical Mehods. His research ineres is in saisical process conrol. S. Y. Sim graduaed from he School of Mahemaical Sciences, Universii Sains Malaysia wih an M.Sc. degree. sample mean, sample range and moving range saisics. Lucas & Saccucci (990) showed ha a fas iniial response (FIR) feaure is useful for he EWMA char, especially for small values of smoohing consans. Rhoads, Mongomery & Masrangelo (996) proposed a scheme which is superior o ha of Lucas & Saccucci (990). MacGregor & Harris (993) suggesed an approach of using he EWMA based saisics in he monioring of he process sandard deviaion. Gan (990) proposed hree modified EWMA chars for he Poisson daa. A beer procedure for using he EWMA char for Poisson coun is given by Borror, Champ & Rigdon (998). Somerville, Mongomery & Runger (00) developed a smoohing and filering mehod using he EWMA and Poisson probabiliies which separaes he wo disribuions in a paricle coun daa sream ino a base process and an oulier process followed by applying saisical monioring schemes o each of hem. A Bernoulli EWMA is suggesed o monior he oulier process. The EWMA conrol char scheme for he sample mean proposed by Ng & Case (989) is consruced by assuming ha he daa used in he compuaion of he limis are oulier free. This assumpion may no be rue in real siuaions since ouliers ofen occur in he daa used o compue he conrol limis. Ouliers may consis of single unusual values which happen due o a sporadic special cause. Such ouliers ac only on occasional observaions in a subgroup 464
3 KHOO & SIM 465 and no on subgroups as a whole. These ouliers have o be deeced, invesigaed and he special cause removed if possible. The presence of ouliers will reduce he sensiiviy of a conrol char because he conrol limis are sreched so ha he deecion of he ouliers hemselves become more difficul. Furhermore, hese sreched limis also make i more difficul for oher ypes of ou-of-conrol signals o be deeced (Rocke, 989, 99). The purpose of his aricle is o propose a robus EWMA (EWMASM) char for he process mean as an alernaive which is superior o he sandard EWMA (EWMASM) char for he process mean. The EWMASM char is consruced based on he limis ha are se using an esimae of he process sandard deviaion using he average of he subgroup inerquarile ranges (IRs) raher han he average of he subgroup ranges in he case of he EWMASM. Thus, he EWMASM char is less affeced by ouliers compared o he EWMASM char. The nex secion gives an overview of he EWMASM char. The EWMASM Char The EWMASM char is based on he following saisic (Ng & Case, 989): Wˆ ( ) Wˆ = + () where is he mean of sample and is a weighing consan. In he esimaion of he limis based on he process daa, m subgroups of size n each are aken and hen he average of he m sample means, is compued. is used as he saring poin, i.e., W ˆ 0 =. The average of he m sample ranges is compued o be R. The upper and lower conrol limis of he EWMASM char are and UCL = + F R (a) LCL = F R (b) 3 F = (3) d n In equaion (3), d is a sandard consan whose value depends on he sample size n. The values of d and F (Ng & Case, 989) for he various sample sizes n are given in Tables and respecively. A Proposed EWMASM Char For The Process Mean Similar o he EWMASM char, every observed sample mean is ransformed ino a corresponding EWMA before i is ploed on he EWMASM char. Le represens he mean of sample. Every will be ransformed ino a corresponding EWMA, Wˆ, using he ransformaion Wˆ ˆ, =,,. (4) = + ( ) W When developing he EWMASM char based on process daa, m subgroups of size n each mus be aken o compue he esimae of he proces mean. The average of he m sample means will be used as he saring poin, i.e., W ˆ 0 =. For he EWMASM char, he inerquarile range, IR is defined as, where ( ) denoes he order ( b) ( a) saisics a = [ n 4 ] + and b = n a +. Here, [ y ] represens he greaes ineger ha is less han or equal o y. I is shown by Rocke (99) ha he mahemaical expecaion of IR can be defined as E(IR) = d σ (5) respecively, where
4 466 ROBUST WEIGHTED MOVING AVERAGE CONTROL CHART Table. Facors for he EWMASM and EWMASM Chars n a b d d n Table. Facors of Conrol Limis for he EWMASM, F
5 KHOO & SIM 467 where d is a consan whose value depends on he sample size n (see Table ). The sandard deviaion, σ is esimaed by IR ˆ σ = (6) d where IR is he average of he subgroup inerquarile ranges. Assuming ha all he observed daa are independen from one sample o anoher, hen as increases, Var( Wˆ ) = Var( ) (7) and σ σ = Wˆ. (8) n From equaion (6), by using he average inerquarile range o esimae he sandard deviaion of Wˆ, ˆ σ, he following is obained Wˆ IR ˆ σ = W ˆ d n. (9) Thus, he limis of he EWMASM char are Cener Line = W ˆ 0 = (0) Conrol Limis = Wˆ ± 3σ If = ˆ 0 Wˆ IR ± 3 d n 3 G = d n () () hen he conrol limis calculaed based on equaion () are and UCL = + IR (3a) SM SM G LCL = IR. (3b) Values of G for differen sample sizes, n, and smoohing consans,, are lised in Table 3. G Table 3. Facors of Conrol Limis for he EWMASM, G Sample Value of Size, n
6 468 ROBUST WEIGHTED MOVING AVERAGE CONTROL CHART Comparison Of Performances The performance of he EWMASM char is compared o ha of he EWMASM by performing a Mone-Carlo simulaion. The following four differen condiions are considered for he wo conrol chars for he process mean: (i) The In-conrol siuaion where he daa are all sandard normal random variables. (ii) The Ouliers siuaion where he daa are a mixure of 95% sandard normal and 5% daa wih five imes he sandard deviaion. These ouliers migh represen an episodic phenomenon resuling from a sporadic special cause ha conrol chars should deec. (iii) The Special Cause siuaion where here is an addiional N(δ,) source of variabiliy added o he subgroup μ μ 0 means. Here, δ = where σ (iv) μ 0 = 0 and 0 0 σ = represen he nominal mean and sandard deviaion respecively while μ is he off-arge mean. The values of δ {0, 0.5, 0.5, 0.75,,.5,,.5, 3, 4} are considered. The conrol chars should deec his special cause. The Ouliers and Special Cause siuaion which consiss of he daa ha are a mixure of 95% sandard normal and 5% daa wih five imes he sandard deviaion ogeher wih an addiional componen of variaion, N(δ,) added o he subgroup means. Simulaion sudies based on he above four condiions are conduced using SAS version 8. Repeaedly, m = 0 and 0 subgroups of size n = 5 observaions each are generaed, conrol limis compued and he number of subgroups ha fall ouside he conrol limis are calculaed. This procedure is repeaed imes for a oal of m subgroups. The proporions of ou-of-conrol subgroups (based on m subgroups) are compued for he four differen condiions and wo differen charing mehods. The resuls for he EWMASM char are displayed in Tables 4 and 5 for m = 0 and 0 respecively. Similarly, Tables 6 and 7 give he resuls of he EWMASM char for m = 0 and 0 respecively. An ideal procedure is he char which gives a higher proporion for deecing ou-of-conrol signals for he hree condiions of Ouliers, Special Cause and Ouliers and Special Cause and a lower signal proporion for he Inconrol siuaion. A comparison of he resuls in Tables 4 and 6 for m = 0 show ha for fixed values of and δ, he ou-of-conrol proporions of he EWMASM char are higher han he corresponding values of he EWMASM char for he wo ou-of-conrol condiions of Ouliers and Ouliers and Special Cause. For example, when = 0.5, he ou-of-conrol proporion in Table 6 is 0.04 while ha in Table 4 is much lower a only for he Ouliers condiion. For he Ouliers and Special Cause condiion, he proporions of ou-of-conrol in Table 4 when = 0.5 are {0.094, ,, } while he corresponding proporions in Table 6 are {0.0554, ,, 0.999} where he values of he former are all lower han ha of he laer for δ {0, 0.5,, 4}.
7 KHOO & SIM 469 This shows ha he EWMASM char is more sensiive o ou-of-conrol condiions when ouliers are presen in he daa. The limis compued from he esimae of he inerquarile ranges for he EWMASM char are less influenced by ouliers, compared o ha of he EWMASM char whose limis are compued based on he sample ranges. Thus, he EWMASM char is more robus han he EWMASM and he former is a beer alernaive in he deecion of a special cause when ouliers are presen. The EWMASM char is also superior o he EWMASM when only ouliers are presen. The resuls in boh Tables 4 and 6 indicae ha for he Special Cause condiion, he EWMASM char is superior o he EWMASM for smaller values of δ and ha boh chars have comparable performances for larger values of δ. Table 4. Proporions of Ou-of-Conrol for he EWMASM Char Under Four Differen Condiions based on m = 0 and n = 5 EWMASM In-Conrol Ouliers Special Cause Ouliers and Special Cause δ
8 470 ROBUST WEIGHTED MOVING AVERAGE CONTROL CHART Table 5. Proporions of Ou-of-Conrol for he EWMASM Char Under Four Differen Condiions based on m = 0 and n = 5 EWMASM In-Conrol Ouliers Special Cause Ouliers and Special Cause δ Table 6. Proporions of Ou-of-Conrol for he EWMASM Char Under Four Differen Condiions based on m = 0 and n = 5 EWMASM In-Conrol Ouliers Special Cause Ouliers and Special Cause δ
9 KHOO & SIM 47 I is shown by he resuls in Tables 4 and 6 ha he EWMASM char has lower false alarm (Type-I error) raes han he EWMASM char. However, i should be noed ha he superioriy of he EWMASM char in comparison o he EWMASM char for he wo ou-of-conrol condiions of Ouliers and Ouliers and Special Cause far ouweigh is deficiency in erms of a higher rae of false alarm. For example, if = 0.3, he proporion of false alarm of he EWMASM char is wice ha of he EWMASM bu he former is hireen imes more efficien in deecing ouliers compared o he laer. Because a comparison beween Tables 5 and 7 shows similar rend, i is suffice o discuss and draw conclusions based on he above comparisons. Noe ha he evaluaion of he performances of he conrol chars discussed in his secion are made based on he proporions of subgroup poins falling ouside he conrol limis and are no based on he average run lengh (ARL) values because he chars parameers are esimaed from he subgroup daa, i.e., he nominal values of hese parameers are assumed o be unknown. Table 7. Proporions of Ou-of-Conrol for he EWMASM Char Under Four Differen Condiions based on m = 0 and n = 5 EWMASM In-Conrol Ouliers Special Cause Ouliers and Special Cause δ
10 47 ROBUST WEIGHTED MOVING AVERAGE CONTROL CHART An Example of Applicaion An example will be given o illusrae how he EWMASM char is pu o work in a real siuaion. The EWMASM char is also consruced so ha a comparison beween he wo approaches can be made. This example is based on he daa from Wadsworh, Sephens and Godfrey (986) and concerns he mel index of an exrusion grade polyehylene compound. As par of a sudy of he process, 0 subgroups of four each are aken. Table 8 gives he daa of his process. The limis of he EWMASM char are compued using equaions (a) and (b) while ha of he EWMASM char are calculaed from equaions (3a) and (3b). Figure shows he EWMASM and EWMASM chars ogeher wih heir respecive conrol limis. The limis of he EWMASM char are represened by UCL/LCL and hose of he EWMASM char by UCL / LCL. SM SM Wˆ ( = 0.8) Wˆ ( = 0.) UCL / LCL ( = 0.8) = 46.7 / 3.86 UCL SM / LCLSM ( = 0.8) = 4.3 / 7.80 UCL / LCL ( = 0.) = / UCL / LCL ( 0.) = / 3.07 SM SM = Figure. The EWMASM and EWMASM Chars for Mel Index Daa
11 KHOO & SIM 473 The EWMASM char wih = 0. deecs ou-of-conrol poins a subgroups 8 and 9 while he corresponding EWMASM char signals a subgroups 8, 9, 4 and 5. The EWMASM char deecs wo addiional ou-ofconrol poins (i.e., subgroups 4 and 5) besides he wo poins a subgroups 8 and 9 which are also deeced by he EWMASM char. From Figure for = 0., a shif is observed ha is gradually increasing beween subgroups and 9 followed by a shif which is gradually decreasing unil subgroup 5. Here, boh he EWMASM and EWMASM chars have successfully deeced he upward shif bu only he EWMASM char managed o deec he downward shif. The presence of sample ranges wih large values such as hose in subgroups 3, 4, 6 and 8 cause he average sample range, R o be overesimaed, hence widening he limis of he EWMASM char so ha he char is less sensiive in deecing shifs in he mean. On he conrary, he EWMASM char does no face his problem since is limis are compued based on he average sample inerquarile range (IR). The EWMASM and EWMASM chars wih = 0.8 give more weigh o he curren sample average, compared o he chars wih = 0.. Thus, a weighing consan of = 0.8 makes he wo chars more sensiive o single subgroup averages wih big or small values. Ou-of-conrol poins are deeced a subgroups, 6, 8, 9,, 3, 4 and 7 by he EWMASM char and a only subgroup 8 by he EWMASM char. From he sample averages, in Table 8, we noice ha he values for subgroups 6, 8, 9 and 7 are somewha bigger while hose for subgroups,, 3 and 4 are somewha smaller han he oher sample averages. Invesigaions need o be carried ou o search for assignable causes before hese subgroups are classified as ou-of-conrol poins. I should be noed ha he EWMASM char wih = 0.8 fails o deec any subgroup average ha plos below he LCL. Sub. No., Table 8. Subgroups of Mel Index Measuremens and he Compued Sample Means, Sample Ranges, Sample Inerquarile Ranges and Wˆ Saisics Observaions 3 4 R IR = R = 8.75 IR = 3.5 =0. Wˆ =0.8
12 474 ROBUST WEIGHTED MOVING AVERAGE CONTROL CHART Conclusion This aricle discusses a new robus EWMA conrol char for he sample mean which is referred o as he EWMASM char. I is shown by simulaion ha he EWMASM char is a superior alernaive o he EWMASM char when one is concern wih he presence of ouliers. Generally, he new EWMASM char allows easier deecion of ouliers in he subgroups and is also more sensiive o oher forms of ou-of-conrol behavior when ouliers are presen. An example is given o show how he EWMASM char works in a real siuaion. This example also illusraes he superioriy of he EWMASM char o he EWMASM char, hence making he EWMASM as an aracive alernaive o qualiy conrol praciioners. References Borror, C. M., Champ, C. W., & Rigdon, S. E. (998). Poisson EWMA conrol chars. Journal of ualiy Technology, 30, Crowder, S. V. (987). A simple mehod for sudying run-lengh disribuions of exponenially weighed moving average chars. Technomerics, 9, Crowder, S. V. (989). Design of exponenially weighed moving average schemes. Journal of ualiy Technology,, Gan, F. F. (990). Monioring Poisson observaions using modified exponenially weighed moving average conrol chars. Communicaions in Saisics - Simulaion and Compuaion, Lucas, J. M. & Saccucci, M. S. (990). Exponenially weighed moving average conrol schemes: properies and enhancemens. Technomerics, 3,. MacGregor, J. F. & Harris, T. J. (993). The exponenially weighed moving variance. Journal of ualiy Technology, 5, Ng, C. H. & Case, K. E. (989). Developmen and evaluaion of conrol chars using exponenially weighed moving averages. Journal of ualiy Technology,, Rhoads, T. R., Mongomery, D. C., & Masrangelo, C. M. (996). A fas iniial response scheme for he exponenially weighed moving average conrol char. ualiy Engineering, 9, Robers, S. W. (959). Conrol char based on geomeric moving averages. Technomerics,, Rocke, D. M. (989). Robus conrol chars. Technomerics, 3, Rocke, D. M. (99). and R chars: Robus conrol chars. The Saisician, 4, Somerville, S. E., Mongomery, D. C., & Runger, G. C. (00). Filering and smoohing mehods for mixed paricle coun disribuions. Inernaional Journal of Producion Research, 40, Swee, A. L. (986). Conrol char using coupled EWMA. IIE Transacions, 8, Wadsworh, H. M., Sephens, K. S., & Godfrey, A. B. (986). Modern mehods for qualiy conrol and improvemen. New York, N.Y.: John Wiley & Sons.
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