Estimating the Change Point of Bivariate Binomial Processes Experiencing Step Changes in Their Mean

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Alfred Newton
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1 Proceedigs of the 202 Iteratioal Coferece o Idustrial Egieerig ad Operatios Maagemet Istabul, Turey, July 3 6, 202 Estimatig the Chage Poit of Bivariate Biomial Processes Experiecig Step Chages i Their Mea Sara Afrooza, Paria Soleimai, ad Masoumeh Eghbali Ghahyazi Departmet of Idustrial Egieerig Faculty of Idustrial Egieerig Islamic Azad Uiversity, South Tehra Brach Tehra , Ira Seyed Taghi Ahava Niai Departmet of Idustrial Egieerig Sharif Uiversity of Techology Tehra , Ira Abstract I statistical processes cotrol, there are may cases where several attribute quality characteristics of a product or a process are ispected simultaeously. Receivig a sigal from the cotrol charts used to moitor these processes is a idicatio of the process chage, but ot a idicator of the actual time of the process chage, ow as chage poit. Kowig the exact time of the process chage would assist process egieers i their search for the special cause ad aid i process improvemets. I this paper, statistical process cotrol techiques are studied ad compared i situatios where multiple correlated attribute characteristics of products or processes are measured cotiuously. The multi-attribute quality characteristics that are cosidered iclude coformig ad ocoformig percetages of the items that follow a biomial distributio. A estimator of the process chage poit is proposed usig the maximum lielihood estimatio method. Usig simulatio experimets we show that the proposed estimator provides a good assessmet to approximate the time of chage i these processes. Keywords Bivariate biomial cotrol chart; Chage poit estimatio; Maximum lielihood estimatio; Statistical process cotrol. Itroductio A cotrol chart as a applicatio tool i statistical process cotrol (SPC) is a suitable method of moitorig process ad provides the possibility of discoverig the assigable causes or chages i the characteristics of a process (Motgomery 2009). Woodall ad Motgomery (999) stated that multivariate process cotrol is oe of the most rapidly developig areas of statistical process cotrol, where there are may situatios i real-world eviromets i which simultaeous moitorig or cotrol of two or more correlated quality characteristics is ecessary. Moitorig these quality characteristics idepedetly ca be very misleadig. Hotellig (947) established multivariate process cotrol techiques i his pioeerig paper, where he applied multivariate process cotrol methods to a bombsights problem (Bersimis et al. 2007). Cotrol charts ca issue a sigal from a chage i a process mea i a substatial amout of time after the chage occurred. To idetify the reaso of assigable causes, the egieers eed to have the real-time chage. Oe of the most effective solutios for this problem is so called the chage poit estimatio, where plety of research wors have bee coducted so far. Most of the research wors coducted o the estimatio of the process chage poit ca be classified ito three classes of the maximum lielihood estimatio (MLE), cumulative sum (CUSUM), ad expoetially weighted movig average (EWMA) procedures. Lu et al. (998) reviewed moitorig procedures developed for multi-attribute processes with correlated quality characteristics. Suliva ad Woodall (996) proposed a lielihood ratio approach for detectig a shift i the mea or stadard deviatio of uivariate ormal observatios. 664

2 Pigatiello ad Samuel (998a, 998b, ad 998c) used the maximum lielihood estimator o, S, ad C charts to estimate the real time chages i the process parameters. I additio, Pigatiello ad Samuel (200a) proposed a maximum lielihood estimates (MLE) of the chage poits of the p ad p cotrol charts. Samuel et al. (998) preseted a MLE estimator i idetifyig the step chage i a mea of a ormal process i cotrol chart. Pigatiello ad Samuel (200b) showed that usig MLE of a chage poit is superior i idetifyig the step chage i cotrol chart i compariso with the estimators offered by the cumulative sum (CUSUM) ad expoetially weighted movig average (EWMA) procedures. Perry ad Pigatiello (2006) used the MLE for the time of a liear tred chage i a ormal process mea ad compared the performaces of their estimator with the oes obtaied usig a MLE method for step chages wored by Samuel et al. (998). They also proposed a maximum lielihood estimator for the chage poit of a Poisso rate parameter with a mootoic effects ad a liear tred disturbace (Perry et al ad 2007). While most of the research wors o the chage poit estimatio of discrete qualitative characteristics have bee coducted oly i uivariate cases, i this paper, we first use a cotrol chart to moitor ocoformig items i a sample produced by a bivariate biomial process. The, a MLE approach is proposed to estimate the chage poit of the process mea experiecig a step-chage. Fially, the performace of the proposed MLE is evaluated usig extesive simulatio experimets. I the ext Sectio, a brief overview of the bivariate biomial cotrol chart ad the probability model of the process are give. I Sectio 3, the proposed chage poit estimator is derived. I Sectio 4, the performace of the proposed estimator is evaluated usig simulatio experimets. Fially, we coclude the paper i Sectio Bivariate Biomial Cotrol Chart Followig Lu et al. (998) who developed a Shewhart cotrol chart to moitor multi-attribute processes that follow multivariate biomial distributio, i this research, we itroduce a ew Z statistic that is the weighted sum of the ocoformig uits of all the quality characteristics i a sample, i.e., i = Z = / p () i = Where is the umber of quality characteristics ad p i idicates o-coformig uits for i th discrete characteristics ( i =,..., ). We ote that i () all the quality characteristics of the process are assumed to have a equal degree of importace. As a result, usig the priciples of the Shewhart cotrol chart, cotrol limits of a MNP scheme ca be derived as UCL = pi + 3 ( p i ) + 2 ρij ( p i )( p j ) i = i = i< j CL = p i LCL = pi 3 ( p i ) + 2 ρij ( p i )( p j ) i = i = i< j Where ρ ij idicates the correlatio coefficiet betwee ay two quality characteristics i ad j, obtaied by i ( ) ( ) ( ) ( ( ))( ( )) ( ) ( ) ρij = Cov C i,c j V ar C i V ar C j = E C i E C i C j E C j V ar C i V ar C j (3) Moreover, the cotrol chart parameters icludig the ratio of ocoformig uits ad the correlatio betwee discrete characteristics are estimated based o the wor by Lu et al. (998). 2.. Probability Model of the Process Cosider a process i which each of the products is ispected for compliace with the stadards accordig to two separate qualitative characteristics, simultaeously. As the result of the ispectio process, each product is classified ad is placed uder coformig ad ocoformig categories based o each of the two features separately. I other words, each product is classified uder oe of the four groups (0,0), (0,), (,0) or (,) after ispectio ad quality 665 i (2)

3 evaluatio, so that the first/secod compoet of these pairs regularly represet coformity or ocoformity for the first/secod quality characteristic. Moreover, to moitor the process a sample icludes products, ad 00, 0, 0, ad show the umber of products classified i each of the above four groups. Therefore, the probabilities correspodig to each of the four groups, are deoted by p 00, p 0, p 0, ad p, respectively, as follows p rs = p{ I = r, I 2 = s}, r = 0,; s = 0, (4) If idicates the umber of ocoformity i a sample of products accordig to the first quality characteristic, ad similarly represets o-coformig uits i a sample accordig to the secod quality characteristics, the ad are comig from a bivariate biomial distributio with the meas µ = p ad µ = p ad correlatio as ρ ρ = p - p p p ( p )p ( p ) where p = p 0 + p ad p = p 0 + p. Accordigly the probabilities p 00, p 0, p 0, ad p are obtaied usig rs the Excel software. Note that p =, ad that = + 0 = 0 + Moreover, the Joit probability mass fuctio of the radom variables ad is x y!p (p p ) (p p ) x y + (7)!( x )!( y )!( x y + )! g( x,y ) = ( p p + p ) where the summatio is calculated o the rage of max( 0, x + y ) mi( x, y) (Kocherlaota ad Kocherlaota 998). 3. MLE Estimatio of the Chage Poit of a Bivariate Biomial Process Cosider ad the two discrete quality characteristics followig a bivariate biomial distributio, where after a uow time oe or both of the process meas are shifted from a i-cotrol ocoformig proportio p0 to a uow value p, ad util the assigable causes removed, the process remais i its ew level of p. Assumig Z T = ( Z, Z 2,..., Z τ, Z τ, Z τ+,...z T ) the first subgroup sample of a out-of-cotrol process, where τ is the real chage-time of the process mea, the ( Z, Z 2,..., Z τ, Z τ ) are assumed to come from a i-cotrol process ad ( Z τ +,..., Z T ) are the results of a out-of-cotrol process. 3.. Estimatig the Parameters Havig the covariace betwee the two attribute quality characteristics ad asσ = p - p p, the lielihood fuctio for a sample of products is obtaied by L( p, p, σ ) = g( x, y; p, p, σ ) (8) i= After simplificatios, it ca be show that 0 L(p,p, σ ) ( σ + p p ) (p σ p p ) (9) 0 00 ( p σ p p ) ( p p + p p + σ ) Sice the three parameters p, p, ad σ ivolved i the lielihood fuctio are uow, a expressio is eeded for the metioed parameters i terms of τ that provides the maximum of the logarithm of the lielihood fuctio. Hece, taig the partial derivatives of the logarithm of the lielihood fuctio with respect to p, p, ad σ ad settig them zero results i (Kocherlaota ad Kocherlaota 998). (5) (6) 666

4 ˆp ˆp ˆ σ xy 0 = 0 = ( ) = 2 (0) 3.2. The Proposed MLE Estimator The chage poit τ i the cotrol chart of the bivariate biomial process at had is the time at which the lielihood fuctio reaches its maximum level. Regardig the parameters estimatios give i (0), the lielihood fuctio ca be obtaied as Thus τ is estimated by τ T () L( p,p, σ ) = g(x,y;p,p, σ ) g(x,y;p ˆ,p ˆ, ˆ σ ) i = j= i = τ + j= 0 t T ( ( σ )) ˆ τ = arg max log L p, p, 4. Performace Evaluatio of the Proposed MLE of the Step-Chage Poit I this Sectio, simulatio experimets are used to evaluate the performace of both the cotrol chart ad the proposed maximum lielihood estimator of the step-chage poit. To do this, the probability of type-i error is cosidered fixed at ad the chage poit of the process is placed atτ = 50. I other words, subgroups i =,2,...,50 are i-cotrol. The meas of the quality characteristics are cosidered p 0 = 0. ad p 0 = 0. 25, the sample size is = 25, ad the correlatio betwee the two characteristics is ρ 0 = As discussed i Sectio 2., for a i-cotrol process, p 00 = 0. 74, p 0 = 0. 6, p 0 = 0. 0, ad p = are used to geerate radom data. Further, the correlatio remais uchaged for out-of-cotrol processes as well. Table shows the performace of both the bivariate biomial cotrol chart ad the proposed MLE to idetify the (2) step-chage poit of the above process for four differet modes of the process. I this Table, ˆτ shows the average of the chage poit estimates ad Se( ˆ τ ) deotes the stadard error of the estimate. Moreover, the expected legth of each simulatio ru, E(T ) for differet magitudes of the chage i the process mea is also show i this table based o 0,000 trials. Table : Chage Poit Estimates ad Their Stadard Errors forτ = 50 row p p E(T ) ˆτ Se( ˆ τ ) The results i Table idicate that as the mea of the first quality characteristic icreases from 0. to for example, the average time for the cotrol chart to sigal is ; the average estimated time of the process chage poit is that is fairly close to the actual chage poit of 50. Moreover, o the average, the proposed maximum lielihood estimator of the time of the process chage is fairly close to the actual chage poit. I additio, the results i Table 2 that cotai the estimated precisio performaces of the proposed step-chage estimator idicate that whe the mea of the first characteristic icreases i the first colum for example, i almost 58% of the trials the chage poit estimate shows the actual time of the process chage. I almost 82% of the trials, 667

5 the estimate is withi ± subgroups, ad the lie. Similar patters ca be see for other mea shifts. More specifically, whe the mea of the first characteristic icreases while the mea of the secod characteristic decreases i the last colum, the proposed estimator shows the best performace i idetifyig the actual time of the process chage. Table 2: Estimated precisio performaces of the proposed step-chage estimator based o 0,000 trials p p P( ˆ ˆ τ = τ) ) 2) 3) 4) 5) 6) 7) 8) 9) 0) 5) Coclusios I this paper, a maximum lielihood estimator to idetify the time of step-chage poit of a bivariate biomial process mea was proposed. Idetifyig the exact time of the process chage ca help the egieers to search for the special cause i smaller rage. Therefore, they ca improve the quality of process more quicly. We showed that the proposed estimator provides a acceptable performace whe the chages occur i the mea of the two characteristics. For this purpose, the use of the proposed estimator was show usig a hypothetical example i a bivariate biomial cotrol chart. The results of simulatio studies idicate the proposed estimator wors effectively i detectig the actual time of the process chage. Future research is recommeded to idetify the combiatio of the process variables that is resposible for a process chage. Moreover, future research may also iclude usig the EWMA, CUSUM & eural etwors i detectig the chages i bivariate biomial processes. Refereces. Bersimis, S., Psarais, S., Paaretos, J. (2007). Multivariate statistical process cotrol charts: A overview. Quality ad Reliability Egieerig Iteratioal Vol. 23, pp Hotellig, H. (947). Multivariate quality cotrol illustrated by the air testig of sample bombsights. I Techiques of Statistical Aalysis, Eisehart C, Hastay MW, Wallis WA (eds.). McGraw-Hill: New or, pp Kocherlaota, S., Kocherlaota, K. (998). Bivariate discrete distributios. Wiley, New or. 4. Lu,.S., ie, M., Goh, T.N., Lai, C.D. (998). Cotrol chart for multivariate attribute processes. Iteratioal Joural of Productio Research, vol.36, pp Motgomery, D. C. (2009). Itroductio to statistical quality cotrol, 6 th Editio, Joh Wiley ad Sos, New or. 6. Sulliva, J.H., Woodall, W.H. (996). A compariso of multivariate quality cotrol charts for idividual observatios. Joural of Quality Techology, vol. 28, pp

6 7. Perry, M.B., Pigatiello, J.J. Jr. (2006). Estimatio of the chage poit of a ormal process mea with a liear tred disturbace. Quality Techology ad Quatitative Maagemet, vol. 3, pp Perry, M.B., Pigatiello, J.J. Jr., James, R. (2006). Chage poit estimatio for mootoically chagig Poisso rates i SPC. Iteratioal Joural of Productio Research, vol. 45, pp Perry, M.B., Pigatiello, J.J. Jr., James, R. (2007). Estimatig the chage poit of a Poisso rate parameter with a liear tred disturbace. Quality ad Reliability Egieerig Iteratioal, vol. 22, pp Pigatiello, J. J., JR. Samuel, T.R. (998a). Idetifyig the time of a step chage with cotrol charts. Joural of Quality Egieerig, vol. 0, pp Pigatiello, J.J., JR., Samuel, T.R. (998b). Idetifyig the time of a step chage i a ormal process variace. Joural of Quality Egieerig, vol. 0, pp Pigatiello, J.J., JR., Samuel, T.R. (998c). Idetifyig the time of a step chage i a Poisso rate parameter. Joural of Quality Egieerig, vol. 0, pp Pigatiello, J.J., JR., Samuel, T.R. (200a). Idetifyig the time of a step chage i the process fractio ocoformig. Joural of Quality Egieerig, vol. 3, pp Pigatiello, J.J., JR., Samuel, T.R. (200b).Estimatio of the chage poit of a ormal process mea i SPC applicatios. Joural of Quality Techology, vol. 33, pp Woodall, W.H., Motgomery, D.C., (999). Research issues ad ideas i statistical process cotrol. Joural of Quality Techology, vol. 3, pp

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