Comparison of single EWMA-type control charts based on Economicstatistical

Transcription

1 The Busness and Management Revew, Volume 6 Number 4 August 5 Comparson of sngle EWMA-type control charts based on Economcstatstcal desgn Shn-L Lu Chen-Fang Tsa Department of Industral Management and Enterprse Informaton Alethea Unversty, New Tape Cty, Tawan Keywords EWMA charts; Smulaton; Economc-statstcal desgn; Cost model; Loss functon Abstract A sngle exponentally weghted movng average (EWMA) chart s effectvely used for montorng the process mean and varance smultaneously. In ths paper, three sngle EWMA-type control charts ncludng sum of square EWMA (SS-EWMA), maxmum EWMA (MaxEWMA), and EWMA-semcrcle (EWMA-SC) charts are compared under Lorenzen and Vance s cost model ntegratng Taguch s loss functon. The optmal decson varables, namely sample sze n, samplng nterval tme h, control lmt wdth L, and smoothng constant, are obtaned by mnmzng the expected cost functon. Va smulatons, the EWMA-SC charts have the smallest expected cost among these charts when a process has large shfts. However, the MaxEWMA charts have the smallest expected cost among these charts when a process mean shfts.. Introducton The exponentally weghted movng average (EWMA) chart was frst ntroduced by Roberts (959) and has been wdely used to mprove the qualty of a process when small process shfts are of nterest. Varous types of EWMA charts have been sequentally proposed to montor process shfts. The varable samplng nterval (VSI) EWMA chart (Saccucc et al., 99) and the varable sample szes and samplng ntervals (VSSI) EWMA chart (Reynolds and Arnold, ) are effcent n mprovng detecton ablty than fxed-type EWMA charts. Auto correlated observatons are always generated by contnuous product manufacturng processes. Facng ths scenaro, t s not advsable to use tradtonal control chart for montorng process shfts. Harrs & Ross (99) dscussed the mpact of autocorrelaton on the performance of EWMA charts, and showed that the average run lengths of these charts were senstve to the presence of autocorrelaton. Most of control charts are establshed on the belef that the observatons of a process are assumed to follow a normal dstrbuton. When the observatons are from a nonnormal or unknown dstrbuton, t s sutable to construct control charts based on a nonparametrc approach. Bakr (6) proposed the nonparametrc EWMA control charts for montorng an unspecfed n-control target process center. The proposed charts are more effcent than tradtonal normal-based control charts. Two EWMA charts are usually requred to jontly montor small process mean and varance shfts. However, recently, sgnfcant attempts have been made to desgn a sngle EWMA chart for montorng the process mean and varance smultaneously. Researchers have presented desgns of varous sngle EWMA control charts, such as the MaxMn EWMA chart (Amn et al., 999), the Max chart (Chen and Cheng, 998), the sum of squares EWMA (SS-EWMA) chart (Xe, 999), the maxmum EWMA (MaxEWMA) chart (Chen et al., ), the EWMA-semcrcle (EWMA-SC) chart (Chen et al., 4). These charts transform the sample mean and sample varance nto a sngle plottng statstc or two plottng statstcs, one representng the mean and the other representng the varance, on the same chart. 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 36

2 The Busness and Management Revew, Volume 6 Number 4 August 5 The aforementoned control charts are desgned from a statstcal perspectve. Statstcally desgned control charts are often measured usng the desred n-control average run length ( ARL ) and out-of-control average run length ( ARL ). In addton to the statstcal desgn of control charts, another desgn perspectve nvolves economc desgn. The objectve of an economc desgn control chart s to mnmze the expected hourly loss cost. Duncan (956) frst proposed the economc model of a X control chart, wheren three parameters, namely sample sze n, samplng nterval h, and control lmt wdth L must be determned by cost mnmzaton. Ths approach was generalzed by Lorenzen and Vance (986), who consdered whether producton contnues durng the perod of searchng for and/or reparng the assgnable cause. Snce then, varous economc desgns have been proposed for control charts (Montgomery, 98; Ho and Case, 994a; Park et al., 4; Chou et al., 6). Recently, Serel and Moskowtz (8) and Serel (9) presented a cost-mnmzaton model to desgn the EWMA control chart based on qualty-related producton costs usng the loss functon. However, Woodall (986) found that the optmal economc desgn control chart has poor statstcal performance. One sgnfcant problem s that the optmal economc soluton usually yelds a consderably large rsk of Type I error. The n-control ARL of the control chart s usually too short to be practcally acceptable. To mprove the low statstcal performance of the economc desgn control chart, some authors have expanded the pure economc model by ncorporatng addtonal statstcal constrants, such as Sanga (989), Montgomery et al. (995), Chou et al. () and Chen and Pao (), Yeong et al. (3). Taguch s quadratc loss functon has been ntegrated nto the economc-statstcal desgn most recently by Huang and Lu (5) and Lu et al. (3), who respectvely proposed the economc-statstcal desgn of MaxEWMA and SSEWMA. They found that the economc-statstcal desgn results n a large mprovement n statstcal performance and a small ncrease n cost. The am of ths paper s to develop an economc-statstcal desgn of the MaxDEWMA control chart by ntegratng the loss functon nto Lorenzen and Vance s cost model. A numercal smulaton s conducted to mnmze the cost functon under ARL constrants. Moreover, a senstvty analyss s conducted to assess the effects of the man nput parameters on the objectve functon and decson varables. The rest of ths paper s organzed as follows. Secton revews the lterature on SS-EWMA, MaxEWMA and EWMA-SC charts. In Secton 3, Lorenzen and Vance s cost model s brefly ntroduced. An llustratve example s presented n Secton 4. Fnally, Secton 5 concludes.. Revew of the SS-EWMA, MaxEWMA and EWMA-SC charts Suppose X j,,,, and th j,,, n be observatons of sze, n, n the sample havng a normal dstrbuton wth mean and standard devaton, where and are defned as target values of the process. When and ndcate that the process s n control, otherwse the process has changed or drfted. Let X and S denote the sample mean and sample varance of sample, respectvely. Then X,,, are ndependent normal random varables wth mean and varance n ; ( n ) S and,, are ndependent ch-square random varables wth n degrees of freedom; and X and S are ndependent.. The SS-EWMA control chart Accordng to Xe (999), defne the followng two statstcs: 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 37

3 The Busness and Management Revew, Volume 6 Number 4 August 5 and U V F n ( X ) () n ( n ) S,, () where ( z) P( Z z), Z ~ N (, ), s the nverse functon of, and F( w, v) P( W w v), where W ~ ( v). (These transformatons and applcatons have been proposed by Quesenberry (995)) Both U and V are ndependent standard normal random varables when the process n control, and that the dstrbutons of U and V are both ndependent of the sample sze n. Two EWMA statstcs, each for the mean and varance, can be defned as A U ( ) A,,,, (3) and B V ( ) B,,,, (4) where A and B are the startng values, respectvely. It s known that ndependent because U and V are ndependent, and when, A have both A ~ N(, A ) and B N(, B ), where A ( ) B A and B are B, we, and. We then defne the statstc of the SS-EWMA chart by combng the above two EWMA statstcs defned as SE A B,,,. (5) Because the statstc SE s non-negatve, the SS-EWMA chart has only an upper control lmt (UCL), whch s gven by UCL E( SE ) L Var( SE ) (6) where E( SE ) and Var( SE ) are the n-control mean and varance of the statstc 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 38 SE, respectvely. L s the control lmt constant chosen to match the desred ARL. To acheve the desred ARL, the correspondng control lmt constant L and the smoothng parameter s determned.. The MaxEWMA control chart Accordng to defnton of Xe (999), Chen et al. () redefned the new MaxEWMA statstc defned as M max A, B (7) Because the statstc M s non-negatve, the MaxEWMA chart has only an upper control lmt (UCL), whch s gven by UCL E( M ) L Var( M ) (8) where EM ( ) and Var( M ) are the n-control mean and varance of the statstc M, respectvely. L s the control lmt constant chosen to match the desred ARL. To acheve the desred ARL, the correspondng control lmt constant L and the smoothng parameter s determned..3 The EWMA-SC control chart Accordng to Chen et al. (4), the statstc of a SC chart s defned as:

4 The Busness and Management Revew, Volume 6 Number 4 August 5 n H X S (9) ( ),,,. n n Let H H. The EWMA-SC statstc SC can be defned from H as follows: SC H ( ) SC,,,,, () where s the smoothng constant whle SC n s the startng value of SC. Because H ~ ( n) when, and n n n n, and so we have the followng results: E( SC ) E( H ) n, () ( ) n ( ) Var H Var( SC) ( ). () In addton, Equaton () can be rewrtten as: SC Y Z n, (3) ( X ) where Y S ( ) Y and Z ( )( ) ( ) n Z wth Y Z. n Addtonally, t s known that Y and Z are also ndependent because X and S are ndependent. The EWMA-SC chart only needs an upper control lmt ( UCL ) as the SC s non-negatve. The UCL correspondng to Equaton () s gven by: n ( ) UCL E( SC ) L Var(SC ) n L and the UCL correspondng to Equaton (3) s gven by:, (4) n ( ) UCL L. (5) Here, L s the wdth of the control lmts when the process s n the control state. The process s consdered to be out of control whenever SC exceeds UCL or ( Y, Z ) s outsde the control regon ( Y, Z ): Y Z UCL, and some acton should be taken. Indeed, the latter method of determnng when a process s out of control s preferable, because the source of an assgnable cause can be drectly dentfed by plottng the locaton of the sample pont on the chart. 3. Cost model Lorenzen and Vance s cost model (986) s employed for determnng the optmal decson values of the economc-statstcal desgn of control charts. Some underlyng assumptons n that cost model are: () The producton cycle length s defned as the tme nterval from the start of the n-control state to the elmnaton of an assgnable cause for the out-of-control state. () The tme between the occurrences of an assgnable cause follows an exponental dstrbuton wth a mean of / hours. (3) Once the process s out of control, nterventon s requred to adjust the process and return t to the ntal state of statstcal control. The expected cost per hour, denoted by, s derved by dvdng the expected cost per cycle by the expected cycle length. Thus, 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 39

5 The Busness and Management Revew, Volume 6 Number 4 August 5 a bn E( A) ( h ARL nt T 3T3 ) h ARL - h ns T c (- ) nt T T3 c ( h ARL nt T 3T3 ) ARL ns c ns T c3 h ARL - (- ) nt T T3 ARL ARL where, n = sample sze, h = tme nterval between samples, L = smoothng constant of the control chart, = control lmt constant of the control chart, = expected tme between an assgnable cause and the pror sample, denoted by h h [ ( h) e ] / ( e ), n h h = expected number of samples taken whle n control, denoted e / ( e ), T s T T T 3 = target process mean, = target process standard devaton, = expected tme to search a false alarm, = expected tme to sample, nspect and plot each sample unt, = expected tme to search the assgnable cause, = expected tme to repar the assgnable cause, = f producton contnuous durng searches, = f producton ceases durng searches, = f producton contnuous durng repar, 3 = f producton ceases durng repar, a = fxed cost of samplng, b = unt varable cost of samplng, c = the expected qualty loss per unt of product when the process s n control, c = the expected qualty loss per unt of product when the process s out of control, c c 3 = cost of nvestgatng a false alarm, = cost of searchng and reparng an assgnable cause. Tradtonally, a product s classfed as ether conformng or nonconformng accordng to the specfcatons of the relevant qualty characterstc. Qualty costs are consequently ncurred when products fall outsde the specfcaton lmts. However, n practce, a product s performance wll gradually deterorate as the desgn parameter devates from the target value. A loss functon was frst ntroduced by Taguch (985) to descrbe the qualty characterstc dffers from the nomnal. The loss refers to the qualty cost that s ncurred when the qualty characterstc s off target even though t may conform to the specfcaton lmts. Taguch s loss 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 4

6 The Busness and Management Revew, Volume 6 Number 4 August 5 functon has been broadly employed n ndustral applcatons, especally n the economc or economc-statstcal desgns of control charts (Serel and Moskowtz, 8; Serel, 9; Yeong et al., 3; Lu et al., 3). The economc-statstcal desgns of control charts based on Taguch s loss functon are nvestgated n ths paper. Snce the symmetrc loss functon s more common n applcatons, n ths paper, we consder symmetrc loss functons and the constant loss coeffcent K such that the quadratc functon s presented as L ( ) ( ) Q x K x T, (6) where the qualty characterstc x has a probablty densty functon f( x ), the target value T s a parameter descrbng the rsk averson of the decson makers. When the process s n control follows: J s the expected loss per unt of product and we denote t as Q J ( ) ( ) [ ( ) ] Q LQ x f x dx K T (7) When the process s out of control, the process mean shfts to and/or the process varance shfts to. The expected loss per unt of product s named J Q and represented as follows: J K[ ( T) ( T)] (8) Q Assume the producton rate to be P unts per hour. The qualty costs c and c n Lorenzen and Vance s cost model are replaced wth the two expected product losses J P and JQP, respectvely. Consder the cost model ntegratng the loss functon, wheren qualty costs c and c are replaced by P, respectvely. Consequently, the expected cost (or loss) per hour, JQ P and JQ denoted by, may be expressed as a bn mn E( A) ( h ARL nt T 3T3 ) h ARL - h n J s T QP (- ) nt T T3 J QP( h ARL nt T 3T3 ) ARL ns c nst c3 h ARL - (- ) nt T T3 ARL ARL subject to ARL ARL n I, h, L R, Not only s the objectve functon a functon of ARL and ARL, but ARL and ARL are functons of the chartng parameters of the control charts. Hence, the optmal decson varables ( n, h, L, ) of the economc-statstcal desgn of the control charts based on loss functons are determned by mnmzng the objectve functon. Table presents the Q 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 4

7 The Busness and Management Revew, Volume 6 Number 4 August 5 algorthmc descrpton used to solve the objectve functons of the economc-statstcal desgns of the control charts based on loss functons. 4. An example The mnmal expected cost and correspondng optmal decson varables are compared among the SS-EWMA, MaxEWMA, and EWMA-SC charts based on quadratc loss functon. For ths llustraton, the followng parameter values are employed to demonstrate the optmal economc-statstcal desgn of the three charts: a 5, K, T, T.5, T, T3 {,.5,,} P 3. For the desred n-control process,, {,.5,,3}, b, c 3, c3 5,.,, 3, r, and ARL s set to 37. The optmal decson varables ( n, h, L, ) and optmal values of the economc-statstcal desgn for varous SS-EWMA, MaxEWMA, and EWMA-SC charts based on loss functon are summarzed n Tables -3. Table. The optmal economc-statstcal desgn of SS-EWMA charts under quadratc loss functon n h L ARL mn n h L ARL mn n h L ARL mn n h L ARL mn Table. The optmal economc-statstcal desgn of MaxEWMA charts under quadratc loss functon n rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 4

8 The Busness and Management Revew, Volume 6 Number 4 August h L ARL mn n h L ARL mn n h L ARL mn n h L ARL mn Table 3. The optmal economc-statstcal desgn of EWMA-SC charts under quadratc loss functon n h L ARL mn n h L ARL mn n h L ARL mn n 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 43

9 The Busness and Management Revew, Volume 6 Number 4 August h L ARL mn Concluson A sngle EWMA control chart has good statstcal performance n detectng both the mean and the varance shfts smultaneously. It only measures control chart from statstcal performance vewpont. Control charts based on pure economcally are unable to satsfy requests n practce when they just pay attenton to mnmzaton qualty costs but neglect the hgh false alarm rate. Therefore, ths work nvestgates an economc-statstcal desgn of SS- EWMA, MaxEWMA and EWMA-SC control charts for montorng process mean and/or varance by ncorporatng the Taguch s quadratc loss functon nto Lorenzen and Vance s cost model. Numercal smulatons are conducted to evaluate effects of man nput factors on the optmal economc-statstcal desgn of these three control charts. Numercal smulatons reveal that the optmal sample sze n, samplng nterval h and out-of-control ARL decrease as the magntude of mean and/or varance shfts ncreases, obvously n small process shfts. However, the optmal control lmt L and smoothng constant ncrease as optmal value of mn ncreases. Moreover, t s reasonable that the optmal mn value of ncreases as the mean shft and/or varance shft become large. Moreover, the MaxEWMA charts have the mnmal cost when a process just has shfts n the mean. Once a process has shfts caused from the process varablty, the EWMA-SC charts need mnmal cost among these three charts. References Amn RW, Wolff H, Besenfelder W, Baxley RJR. (999) EWMA control charts for the smallest and largest observatons, Journal of Qualty Technology. 3: Bakr ST. (6) Dstrbuton free qualty control charts based on sgned rank lke statstcs, Communcatons n Statstcs-Theory and Method. 35: Chen G, Cheng SW. (998) Max chart: combnng X-bar chart and S chart, Statstca Snca. 8: Chen G, Cheng SW, Xe H. () Montorng process mean and varablty wth one EWMA chart, Journal of Qualty Technology. 33: Chen G, Cheng SW, Xe H. (4) A new EWMA control chart for montorng both locaton and dsperson, Qualty Technology and Quanttatve Management. : 7-3. Chen H, Pao Y. () The jont economc-statstcal desgn of X and R charts for nonnormal data, Qualty and Relablty Engneerng Internatonal. 7: Chou CY, Chen CH, Lu HR. () Economc-statstcal desgn of X charts for non-normal data by consderng qualty loss, Journal of Appled Statstc. 7: Chou CY, Chen CH, Lu HR. (6) Economc Desgn of EWMA Charts wth Varable Samplng Intervals, Qualty and Quanttatve. 4: Duncan AJ. (956) The economc desgn of X charts used to mantan current control of a process, Journal of the Amercan Statstcal Assocaton. 5: 8-4. Harrs TJ, Ross WH. (99) Statstcal process control procedures for correlated observatons, Canadan Journal of Chemcal Engneerng. 69: rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 44

10 The Busness and Management Revew, Volume 6 Number 4 August 5 Huang CJ, Lu SL. (5) Consderng Taguch loss functon on statstcally constraned economc sum of squares exponentally weghted movng average charts, Journal of Statstcal Computaton and Smulaton. 85: Ho C, Case K. (994a) Economc desgn of control charts: A lterature revew for 98-99, Journal of Qualty Technology. 6: Lorenzen TJ, Vance LC. (986) The economc desgn of control charts: a unfed approach, Technometrcs. 8: 3-. Lu SL, Huang CJ, Chu WC. (3) Economc-statstcal desgn of maxmum exponentally weghted movng average control charts, Qualty and Relablty Engneerng Internatonal. 9:5-4. Montgomery DC. (98) The economc desgn of control charts: a revew and lterature survey, Journal of Qualty Technology. : Montgomery DC, Torng CC, Cochran JK, Lawrence FP. (995) Statstcally constraned economc desgn of the EWMA control chart, Journal of Qualty Technology. 7: Park C, Lee J, Km Y. (4) Economc desgn of a varable samplng rate EWMA chart, IIE Transactons. 36: Quesenberry CP. (995) On propertes of Q charts varables, Journal of Qualty Technology. 7: Roberts SW. (959) Control chart test based on geometrc movng averages, Technometrcs. : Reynolds MR, Arnold JC. () EWMA control charts wth varable sample szes and varable samplng ntervals, IIE Transactons. 33: Saccucc MS, Amn RW, Lucas JM. (99), Exponentally weghted movng average control schemes wth varable samplng ntervals, Communcatons n Statstcs-smulaton and Computaton. : Sanga EM. (989) Economc statstcal control chart desgns wth an applcaton to X and R charts, Technometrcs. 3: Serel DA. (9) Economc desgn of EWMA control charts based on loss functons. Mathematcal and Computer Modellng 49: Serel DA, Moskowtz H. (8) Jont economc desgn of EWMA control charts for mean and varance, European Journal of Operatonal Research. 84: Taguch G, Wu Y. (985) Introducton to off-lne qualty control. Central Japan Qualty Control Assocaton, Tokyo. Woodall WH. (986) Weaknesses of the economc desgn of control charts, Technometrcs. 8: Yeong WC, Khoo BC, Lee MH, Rahm MA. (3) Economc and economc statstcal desgns of the synthetc X chart usng loss functons, European Journal of Operatonal Research. 8: Xe, H. Contrbutons to qualmetry. Ph.D. Thess, 999, Unversty of Mantoba, Wnnpeg, Canada. 3 rd Internatonal Academc Conference n Pars (IACP), - th August 5, Pars, France 45