Proceedigs of the Iteratioal MultiCoferece of Egieers ad Computer Scietists 009 Vol II IMECS 009, March 8-0, 009, Hog Kog A Modified Statistical Desig Model of Double Samplig X Cotrol Chart Chau-Che Torg, Pei-Hsi Lee Abstract - The double samplig (DS cotrol chart ca efficaciously reduce sample sizes ad icrease performace of process moitorig. Hsu [It. j. prod. res., vol. 4, o. 5, pp. 043-047] metioed that the statistical desig model of DS X cotrol chart could merely miimize sample sizes durig i-cotrol process moitorig but fail to decrease sample sizes durig detectio of process shifts. I this study, with miimizatio of sample sizes for both i-cotrol process ad out-of-cotrol process, a multi-objective programmig method ad geetic algorithm are proposed for statistical desigs of DS X cotrol chart. I compariso with both statistical desig models, it is quite obvious that our model ca effectively lower sample sizes of two process situatios. Keywords - double samplig X cotrol chart; statistical desig; multi-objective programmig; geetic algorithm I. INTRODUCTION The Shewhart s cotrol chart has bee extesively used as a tool for process moitorig i curret idustries. For Shewhart s X cotrol chart, the performace to detect process mea shift ca be icreased through icrease of sample sizes without ay chage i probability of occurrece of false alarms. However, icrease i sample sizes sigifies raise of costs ad ispectio time. Daudi[5] applied the cocept of double samplig plas to the Shewhart s X cotrol chart ad used the two-stage Shewhart s X cotrol chart to moitor processes, so that it was called as Double Samplig X cotrol chart (DS X cotrol chart. With this alterative method used, the advatage of the Shewhart s cotrol chart, i.e., simplificatio at setup ad calculatio, ca be maitaied i additio to improvemets i capability of detectig process mea shift ad reductio of sample sizes. Mauscript received November 30, 008. This work was supported by the Natioal Sciece Coucil of Taiwa, ROC, uder the grat NSC 97--E-4-03. C. C. Torg is ow with Graduate School of Idustrial Egieerig ad Maagemet, Natioal Yuli Uiversity of Sciece ad Techology, 3 Uiversity Road, Sectio 3, Douliou, Yuli 6400, Taiwa, R.O.C. (e-mail: torgcc@yutech.edu.tw. P. H. Lee is with Graduate School of Idustrial Egieerig ad Maagemet, Natioal Yuli Uiversity of Sciece ad Techology, 3 Uiversity Road, Sectio 3, Douliou, Yuli 6400, Taiwa, R.O.C. (Tel: 88695346766, Fax: 886-5-53073, e-mail: g9380@yutech.edu.tw. Additioally, through modificatio of samplig methods of the Shewhart s X cotrol chart, the methods such as X charts with variable sample sizes (VSS ad X charts with variable samplig itervals (VSI are also provided. For both charts, chages i sample sizes ad samplig itervals of the Shewhart s X cotrol chart lead to VSS ad VSI respectively, which ow better performaces to detect process mea shift[][3]. However, with Costa[] comparig performaces of process mea shift detectio for VSS, VSI ad DS X cotrol chart, the best performaces occur at the DS X cotrol chart. O accout of this reaso, the DS X cotrol chart is a arrestig subject i our study. Before usig the DS X cotrol chart, we have to desig five parameters for this cotrol chart: widths of two sets of cotrol limits, sample sizes of two stages ad widths of a set of warig limits. Durig process moitorig, various desigs will cause differet statistic performaces for the DS X cotrol chart. After cosiderig statistic viewpoits, Iriato ad Shiozaki[7] selected the Sigle-Objective programmig method to determie the optimal desig of the DS X cotrol chart. Additioally, while referrig to methods by Iriato ad Shiozaki[7] (amely I&S model i the followig sectios, He et al.[4][5][6] desiged various DS X cotrol charts. I the I&S model, the expected sample size uder i-cotrol process becomes the objective fuctio of the model ad the best desig of the DS X cotrol chart determied by subject to risk probabilities of two process states i cotrol charts. However, accordig to desigs of the DS X cotrol chart by He et al.[5], Hsu[] listed the expected sample size of out-of-cotrol process ad foud failure i I&S methods that the expected sample size for detectio of out-of-cotrol process caot be reduced ad its sample size is eve larger tha that of the Shewhart s cotrol chart. Thus, usig I&S methods caot fid the optimal desig of the DS X cotrol chart. O the basis of above-metioed reasos, the cocept to miimize the expected sample size of out-ofcotrol process ad the multi-objective programmig method are used to modify the I&S statistical desig model i this study. Fially, we will use statistic performace to illustrate the differeces of our modified model ad I&S methods. II. DS X CONTROL CHART A. Priciples of the DS X cotrol chart ISBN: 978-988-70-7-5 IMECS 009
Proceedigs of the Iteratioal MultiCoferece of Egieers ad Computer Scietists 009 Vol II IMECS 009, March 8-0, 009, Hog Kog The DS X cotrol chart proposed by Daudi[9] itegrated two Shewhart s X cotrol charts with differet widths of cotrol limits for process moitorig ad added warig limits i the first-stage cotrol chart. The graphic view of the DS X cotrol chart is show i Fig. that the process observatios are trasformed to a stadard ormal distributio. Therefore, the cetral lies of cotrol charts i two stages are 0. L ad L are the width of cotrol limits i the first-stage cotrol chart ad the secod-stage cotrol chart respectively. W is the width of warig limits i the first-stage cotrol chart. Fig.. Graphic view of DS X cotrol chart Uder a assumptio that process state is i cotrol, each cotrol regio i Fig. ca be defied as I = [ W, W ], I = [-L,-W ( W,L ], I3 = (, L ] [ L, + ad I4 = [ L, L ] ad I5 = (, L ] [ L, +. Daudi[9] has explicitly illustrated the cotrol procedure of the DS X cotrol chart. First, take a small sample size,, ad calculate the sample mea X. The, calculate z usig a ormalize approach, that is,. If z falls i I 3, it will be cosidered as a z = X ( μ σ out-of-cotrol process. If z falls i I, it will be deemed a i-cotrol process. For the case that z falls i I, it is ecessary to coduct a secod-time samplig ad moitor processes with the secod-stage cotrol chart. With the secod-time samplig occurrig, the sample size will be (usually < ad the sample mea X for the secod-time samplig eeds to be calculated. The, the total sample mea Y for both samplig stages ca be calculated with Y = ( X + X ( +. Afterward, ormalize value of Y will be represeted with z, z = + ( Y μ σ. Whe z falls i I 4, it will be cosidered as a i-cotrol process. Otherwise, it will be regarded as a out-ofcotrol process. Iriato ad Shiozaki[7] assumed a ormal distributio for the observatio of process ad displayed calculatios of Type I error probability ad Type II error probability. Both He et al.[5] ad Hsu[9] adopted same methods to calculate probabilities of Type I ad Type II errors i DS X cotrol chart ad evaluated statistic performaces of process moitorig. Supposig the real process is a i-cotrol state but the sample mea falls i I 3 or I 5, it will be cocluded that is called Type I error or false alarm ad its probability ca be calculated by followig equatio: α = = P + P( z I 3 + P( z I P( z I 5 ( z I + P( z I [ P( z I ] = [ Φ( L Φ( L ] 3 Φ cl z cl z Φ z I 4 ( zdz ϕ Φ is the cumulative distributio fuctio of a stadard ormal distributio. ϕ is the probability desity fuctio of a stadard ormal distributio. c = ( +. P ( is the probability that z falls i some regio. By priciples of the DS X cotrol chart, it is ot cosequetial to additio to sample size for each samplig. The probability of additio to sample size is P ( z I. Usig this probability ca estimate the expected sample size E 0 (N of the DS X cotrol chart uder i-cotrol processes. E 0 ( N = + P( z I = + [ Φ( W Φ( L + Φ( L Φ( W ] ( I case of process variatio that the process mea is shifted to μ =μ 0 +σ from iitial mea μ 0, where is the shift size of process mea (by equatio = (μ -μ 0 /σ, the process will be wrog determied as a i-cotrol state that is called Type II error. The probability of this false determiatio is β = Φ P( z I + P( z I P( z I 4 = Φ( L + ( L + + Φ cl + c + z Φ cl + c z I + ( (3 z ϕ ( zdz where I meas the cotrol regio that process mea has shifted, I = [ L +,-W + ( W +,L + ]. The expected sample size E (N for a shift size is E ( N ( + P z I = + [ Φ( W + Φ( L + + Φ( L + Φ( W + ] = (4 Average ru legth (ARL is usually take as a idex to evaluate the statistic performaces of process moitorig i cotrol charts. It is defied as the average times to detect process mea shift. Thus, the statistic performaces of process moitorig i cotrol charts ca be more completely show through ARL tha α ad β. For i-cotrol process, the average ru legth is writte as ARL 0, ARL 0 =/α. Whe process mea had shifted, ad its shift size is, the ecessary average samplig times for detectig shift i a cotrol chart ca be expressed as ARL =/(-β. ISBN: 978-988-70-7-5 IMECS 009
Proceedigs of the Iteratioal MultiCoferece of Egieers ad Computer Scietists 009 Vol II IMECS 009, March 8-0, 009, Hog Kog B. The I&S statistical desig By statistic viewpoits, Iriato ad Shiozaki[7] costructed a oliear programmig model to solve the desigs of the DS X cotrol chart. The major cosideratio i the I&S model is to emphasize ecoomic beefits of costs at samplig ispectios, meawhile, objective fuctio of this model is to miimize E 0 (N. For better performaces i process moitorig of cotrol charts, α ad β have to be restricted to less tha some specific values. The I&S model is described as follows, Mi E0( N = + P( z I Subject to α α (5 β β (6 < <, Iteger (7 0<W< L (8 0< L <L u (9 L >0 I Eq. (5 ad (6, α ad β must be less tha the maximal specific values α ad β respectively. Furthermore, i accordace with statistic viewpoits, these two equatios restrict performaces of cotrol charts. I two stages of Eq. (7, the sample sizes must be positive iteger ad <, where is greater tha, else, the first stage cotrol chart will become a cotrol chart of idividual measuremet. I Eq. (8 ad (9, L must be greater tha W ad keeps positive. Besides, L caot exceed the assiged upper limit L u or else L will icrease ulimitedly durig processes for solutios ad the sesitivity of detectig process mea shift i the first stage cotrol chart will also decrease. The optimal desig of a DS cotrol chart provided by He et al.[4][5][6] was based o this model. But i this model, the sample size E (N is ot restricted durig process mea shift, hece the ecessary sample size i detectig process shifts caot be reduced. III. MODIFIED MODEL AND SOLUTION A. Modified model Cosiderig a cocept of miimizig sample sizes for process shifts, we add a objective fuctio such as Eq. (0 ito the origial I&S statistic desig model to miimize E (N. ARL showig better statistic performaces i cotrol charts tha α ad β, we select ARL as a orm to restrict capabilities of process moitorig of cotrol charts. The modified model becomes Mi E ( N = + P( z 0 I Mi E ( N = + P( z (0 Subject to ARL ( ( 0 ARL 0 ARL ARL < <, Iteger 0<W< L 0< L <L u L >0 I I additio to the origial objectives fuctio i I&S model, a extra objective fuctio to miimize E (N, as show i Eq. (0, is ewly added ito the modified model. Eq. ( ad ( limit the expected samplig times of false alarm occurrece ad detectig process mea shift. It should be oted that ARL 0 should be greater tha the miimal tolerace value ARL because the less occurrece frequecy of false alarms will be better. I additio, ARL should be less tha the specific value ARL sice the faster detectig is better whe process shift has occurred. These two equatios have idetically statistical meaigs with Eq. (5 ad (6 of the I&S model. Other costraits i this model are idetical to those i the I&S model. The modified model is a double-objective mathematical programmig model, hece the techiques for solutios i covetioal sigle-objective programmig model will iadequate for this model. The weight method proposed by Zadeh[0] assiged a weight to each objective fuctio ad the combied them as a sigle objective fuctio usig the weight average to obtai solutios with this sigle objective method. Each weight value represets the importace degree of the objective fuctio ad the sum of weights of all objective fuctios will be. As regards our modified model, the weight method is adopted to itegrate all objective fuctios that U is the weight value of E 0 (N. The combied objective fuctio f is writte as follows, Mi f = U[ ( ] ( [ ( + P z I + U + P z I ] (3 Because our modified model belogs to a oliear programmig method ad mixes cotiuous-discrete variables ad discotiuous ad ocovex solutio space. The geetic algorithms (GA beig adequate to solve this type of problems, He et al.[4][5][6] applied it to solve the I&S model. B. Usig geetic algorithms for solutios Geetic Algorithm (GA is a techology of global optimizatio. For o-liear programmig model, mixed cotiuous-discrete variables or discotiuous ad ocovex solutio space, the applicatio of GA offers immediate optimal solutio for a model. 0 ISBN: 978-988-70-7-5 IMECS 009
Proceedigs of the Iteratioal MultiCoferece of Egieers ad Computer Scietists 009 Vol II IMECS 009, March 8-0, 009, Hog Kog GA executes global search with multiple solutios simultaeously. If the umber of populatio is m, it meas that there are m sets of chromosome searchig for optimal solutios at the same time. The solutio procedure of GA cosists of the followig key steps: ( The geeratio of iitial solutios for decisio variables radomly based o umber of populatio m; ( evaluatio of the fitess for these solutios; (3 selectio ad crossover of chromosomes, where the rate that chromosomes of better fitess are chose is higher; (4 part of the chromosomes experiece mutatio; ad (5 ext geeratio of solutio are geerated ad repeat from step (. The evolutio goes o ad o, ad evetually a coverget solutio will be obtaied. I this paper, GA is used to determie the solutios for the desig of DS. The solutios of a set of decisio variables,,, L, W ad L are cosidered a gees, ad the iitial solutios for multiple sets of decisio variables are geerated based o the umber of populatio m. The solutio that satisfies all the limitatios i Eq. ( ad ( is called a feasible solutio. The Eq. (3 is calculated usig each of the feasible solutios. Smaller values of Eq. (3 mea better fitess for feasible solutios. By iteratig the above GA solutio procedure, the optimal DS desig is achieved whe the Eq. (3 of all feasible solutios coverge to the same value. Sice Palisade[] developed Evolver, the GA tool software attached to Microsoft Excel. I this study, Evolver 4.0 is selected for solutios of our modified model. The optimal cofiguratios for umber of chromosomes, crossover rate ad mutatio rate will be differet accordig to various coditios. I view of this reaso, to coduct repeated tests will fid the best parameter values for solutios. With repeatedly testig coducted, the best parameters acquired for this study are umber of chromosomes=00, crossover rate=0.7 ad mutatio rate with self adjustmet. Fig.. A simulatio of E (N for DS X charts The curves i Fig. represet average umbers of simulatig 00 thousad sets of E (N/E 0 (N. The larger the E (N is, the bigger the E (N/E 0 (N becomes. I Fig., the maximal E (N occurs at =.7 or so. Thus, to reduce E (N, we select =.7 for solutios ad compare solutios acquired with differet =.0 ad 3.0. The criterio for solutios is ARL from the stadard Shewhart s cotrol chart with a sample size 5 (ARL is 4.5,.7 ad.00 respectively for =.0,.7 ad 3.0. I additio, to cotrast differeces of expected sample size E(N for differet models, we assume U=0.5 ad L u =4.5, ad the use GA to solve desigs of the DS X cotrol chart for I&S model ad our modified model respectively. The cotrol chart desigs ad their ARL ad E(N of several shift sizes are show i Table I. TABLE I. ARL ad E(N for DS X cotrol chart desigs of two models IV. COMPARISONS AND DISCUSSIONS This sectio is aimed at comparisos i desigs of the DS X cotrol chart with I&S model ad our modified model. The expected sample size of the DS X cotrol chart will vary with differet shift sizes. Hece, to select a adequate ad correct shift size for solutios ad guaratee a miimal expected sample size for all shift detectio is ecessary. By computer programmig simulatio, 00 thousad differet sets of desigs of the DS X cotrol chart will be geerated radomly ad expected sample sizes with various shift sizes ca be calculated. Matlab7 is selected as a programmig tool for simulatio ad results are show i Fig.. Comparig results of the two models i Table I, we ca fid that desigs of the cotrol chart with I&S model for small shifts are better tha our modified model ad stadard Shewhart s cotrol chart. Notwithstadig that ARL for other shifts with I&S model is close to that of the stadard Shewhart s cotrol chart, the sample size is still greater tha that of the stadard Shewhart s cotrol chart. As regards desigs of the cotrol chart usig our modified model, the sample size is lower tha that of the stadard Shewhart s cotrol chart ad ARL of commo shift size is similar. The followig is about comparisos i desigs of the DS X cotrol chart with our modified model for =.0,.7 ad 3.0. From solutios for desigs of cotrol charts at =.0 ad 3.0, the detectio capability at.5 is the weakest ad the sample size at.5 is larger tha other situatios. However, the detectio capabilities ad the sample sizes for 3 ad are superior to those of the stadard Shewhart s cotrol chart. I regard to ISBN: 978-988-70-7-5 IMECS 009
Proceedigs of the Iteratioal MultiCoferece of Egieers ad Computer Scietists 009 Vol II IMECS 009, March 8-0, 009, Hog Kog desigs of the DS X cotrol chart for solvig =.7, ot oly the sample size but also capability at shift detectio is optimal. The followig step is aalyses that U ad L u affect upo cotrol chart desigs. Because each L derived from our modified model is less tha 3., the assumptio, L u =4.5, is reasoable. Durig processes for cotrol chart desigs, ay L u, larger tha 3., will ot affect solutios. Data i Table II are desigs of the DS X cotrol chart at =.7 uder various weight coditios. It ca be clearly see that either ARL or E(N displays little variatio, hece the weight value U will ot substatially affect solutios of the DS X cotrol chart. That is, U ca be a arbitrary value durig processes for solved the cotrol chart desigs. For some shift size i this example, the detectio capabilities i desigs of cotrol charts derived from I&S model is worse tha our modified model ad stadard Shewhart s cotrol chart (especially for coditios.5. However, usig our modified model to solve =.7 ca effectively reduce the sample size for each shift size. Thus, accordig to the above comparisos, desigs of the DS X cotrol chart ad statistic performaces usig our modified model are better tha those usig I&S model ad Shewhart s cotrol chart. TABLE II. A sesitivity aalysis for modified U i solved case of =.7 V. CONCLUSION The statistical desig model of the DS X cotrol chart by Iriato ad Shiozaki[7] displayed excellet performace at process shift detectio but failed to efficaciously curtail sample sizes. I this study, the model proposed by Iriato ad Shiozaki[7] is modified ad the sample sizes for out-of-cotrol process is added ito this model for solutios of ew desigs of the DS X cotrol chart. Comparig results from both models, we fid that desigs of the DS X cotrol chart with our modified model ca lower sample sizes without chagig origial detectio capability that has apparetly improved drawbacks existig i methods provided by Iriato ad Shiozaki[7]. Additioally, by simulatio results, a large sample size will occur whe the DS X cotrol chart is detectig process mea shifts with.5 to stadard deviatios. Meawhile, by various solutios for differet shift sizes, the desig of the DS X cotrol chart with the optimal statistic performaces ca be acquired accordig to solutios at shift size=.7. He et al.[4][5][6] adopted methods proposed by Iriato ad Shiozaki[7] as well to solve several desigs of the DS X cotrol charts. As regards our modified model, how to apply it to evaluate performaces of other DS-type cotrol charts deserves further exploratio. Weight method is used to solve our model. However, this weight method that miimizes the itegrated objective fuctio value caot simultaeously miimize E 0 (N ad E (N so that the solutio derived from this method might be regarded as a approximate solutio oly. Nevertheless, havig bee a ucomplicated method, it has bee extesively applied to solutios of the multiobjective programmig model. I further study, we suggest that usig other optimal techology of multiobjective programmig solves our model to obtai better desigs of DS X cotrol charts. REFERENCES [] A. F. B. Costa, X charts with variable sample size ad variable samplig itervals, J. qual. techol., vol. 9, o., pp. 97-04, 997. [] A. F. B. Costa, X charts with variable sample size, J. qual. techol. vol. 3, o. 3, pp. 55-63, 994. [3] C-Y. Chou, ad C-H. Che, Ecoomic desig of variable samplig itervals T cotrol charts usig geetic algorithms, Expert syst. appl., vol.30, pp. 33 4, 006. [4] D. He, A. Grigorya, ad M. Sigh, A improved double samplig s chart, It. j. prod. res., vol. 4, pp. 63-679, 003. [5] D. He, A. Grigorya, ad M. Sigh, Desig of double ad triple-samplig X cotrol charts usig geetic algorithms, It. j. prod. res., vol. 40, o. 6, pp. 387-404, 00. [6] D. He, A. Grigorya, ad M. Sigh, Joit statistical desig of double samplig X ad s charts, Eur. j. oper. res., vol. 68, pp. -4, 006. [7] D. Iriato, ad N. Shiozaki, A optimal double samplig X cotrol chart, It. j. id. eg., vol. 5, pp. 6-34, 998. [8] G. Celao, ad S. Fichera, Multiobjective ecoomic desig of a x cotrol chart, Comput. id. eg., vol. 37, pp. 9-3, 999. [9] J. J. Daudi, Double samplig X charts, J. qual. techol. vol. 4, pp. 78-87, 99. [0] L. A. Zadeh, Optimality ad o-scalar-valued performace criteria, IEEE tras. automat. cotr., vol. 8, o., pp. 59-60, 963. [] L-F. Hsu, Note o Desig of double- ad triple-samplig X cotrol charts usig geetic algorithms, It. j. prod. res., vol. 4, o. 5, pp. 043-047, 004. [] Palisade Corporatio. Evolver: The Geetic Algorithm Solver for Microsoft Excel. Newfield, NY, 00. [3] S.S. Prabhu, G. C. Ruger, ad J. B. Keats, X chart with adaptive sample sizes, It. j. prod. res., vol. 3, o., pp. 895-909, 993. [4] V B. Vommi, ad M. S. N. Seetala, A ew approach to robust ecoomic desig of cotrol charts, Appl. soft comput., vol. 7, pp. 8, 007. ISBN: 978-988-70-7-5 IMECS 009