International Journal of Industrial Engineering Computations

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1 Internatonal Journal of Industral Engneerng Computatons 4 (03) Contents lsts avalable at GrowngScence Internatonal Journal of Industral Engneerng Computatons homepage: Modelng qualty control data usng mxture of parametrcal dstrbutons Jorge Alberto Achcar a, Claudo Lus Pratell b,c and Roberto Molna de Souza d* a Department of Socal Medcne, Unversty of São Paulo. Av. Banderantes, Monte Alegre - CEP: Rberão Preto - SP - Brazl b Master Program n Producton Engneerng, Unversty Center of Araraquara. Rua Carlos Gomes, Centro - CEP Araraquara - SP Brazl c Aeronautcs Insttute of Technology (ITA), Brazl d Coordnaton of Mathematcs, Federal Technologcal Unversty of Paraná. Av. Alberto Carazza, Centro - CEP: Cornélo Procópo - PR - Brazl C H R O N I C L E A B S T R A C T Artcle hstory: Receved March 03 Receved n revsed format March 8 03 Accepted March 8 03 Avalable onlne March 03 Keywords: Regresson Qualty control tmes Mxture models Bayesan methods MCMC methods In ths paper, we present a Bayesan analyss of a data set selected from a Brazlan food company. Ths data set represents the tmes taken for dfferent qualty control analysts to test manufactured products arrvng at the company s qualty control department. The samples selected from each batch contan mxtures of dfferent products, whch may be submtted to qualty testng takng dfferent tmes. From prelmnary analyss of the data, t was observed that the hstograms presented two clusters, ndcatng a mxture of dstrbutons. A mxture of parametrcal dstrbutons was thus assumed n the presence of a covarate n order to analyze the data set and to establsh standards to be used by the company for the tmes taken by the analysts. Inferences and predctons are obtaned usng a Bayesan approach wth standard exstng Markov Chan Monte Carlo (MCMC) methods. 03 Growng Scence Ltd. All rghts reserved. Introducton The tmes taken n carryng out qualty control tests can often vary greatly, nfluenced by a range of factors, ncludng the experence and skll of the qualty control analysts, and the presence of dfferent products beng analyzed. It s, then, of nterest to ndustral managers to model these data sets, from whch they can make nferences and predctons and dentfy mportant factors that could affect these tmes. In the study heren, we consder a data set from a food company n São Paulo state, Brazl. Ths data set comprses qualty control tmes for two dfferent analysts observed on dfferent days. Ths data set comprses random samples selected from all the batches of manufactured products. These dfferent products are assessed by qualty control tests lastng for dfferent tmeframes. The batches arrve n random order at the qualty control department. Fg. shows the hstograms for the test tmes for the two analysts. * Correspondng author. E-mal: rmolnasouza@utfpr.edu.br (R. Molna de Souza) 03 Growng Scence Ltd. All rghts reserved. do: 0.567/.ec

2 48 Fg.. Hstograms for the test tmes consderng the two analysts. From the hstograms shown n Fg., t can be observed that the two analysts perform the control tests dfferently (analyst wth,00 samples, and analyst wth,504 samples). A dscordant observaton (greater than 64 mnutes) for analyst was dscarded. It was observed that analyst took less tme than analyst. Fg. shows the hstogram for all the combned data for both analysts (n=704 observatons). From the hstograms n Fg. and Fg., the mxture of two dstrbutons for the tmes taken for the qualty control tests s observed, where a proporton of unts has short tmes and a second proporton of the data has long tmes. Ths made t possble to use a mxture of parametrcal dstrbutons to analyze the data. A mxture of parametrcal dstrbutons has been consdered by many authors n the lterature to analyze non-homogeneous data sets (see, for example, Ttterngton et al. 985; Stephens, 000a; Stephens, 000b; Rchardson & Green, 997; Debolt & Robert, 994; Dey et al., 995; Fnkelsten & Esaulova, 00). Fg.. Hstogram of tmes for two analysts

3 J. A. Achcar et al. / Internatonal Journal of Industral Engneerng Computatons 4 (03) 49 In the parametrc mxture model, the component dstrbutons are from a parametrc famly wth known parameters θ wth the probablty densty functon gven by f k t p f t () for some mxture proportons 0 p, where p + p + + p k =. If k=, we have a mxture of two dstrbutons. Inferences for fnte mxture models could be obtaned usng Bayesan methods (see, for example, Mengersen & Robert, 996; Carroll et al., 999) where the posteror summares of nterest are obtaned usng smulaton methods, especally standard Markov Chan Monte Carlo (MCMC) methods, such as the popular Gbbs samplng algorthm (see for example, Gelfand & Smth, 990) and the Metropols- Hastngs algorthm (see, for example, Chb & Greenberg (995)). Recently, Achcar et al. (0) publshed a paper wth the same data analyzed n ths artcle where the focus of analyss was the use of a Webull dstrbuton n the presence of a changng pont. In ths paper, the authors compare the results obtaned from the use of the change pont model wth the results obtaned from a model consderng the mxture of two Webull dstrbutons. As the two competng methodologes showed to be approprated to analyze ths data set, ths artcle ams to explore n more detal the use of mxtures of Webull dstrbutons, as a good alternatve for qualty engneers, also ntroducng a comparatve study wth the use of other mxture models lke the mxture of normal dstrbutons. Ths paper s organzed as follows: n secton, the models and nference are presented consderng a mxture of two normal dstrbutons and a mxture of two Webull dstrbutons; n secton 3, a Bayesan analyss for the data of the food company s presented. Fnally, n secton 4, some concludng remarks are presented.. Models and nference In ths secton, we ntroduce two mxture models for the tmes taken for the qualty control tests at the food company: a mxture of two normal dstrbutons and a mxture of two Webull dstrbutons... Mxture of two normal dstrbutons Snce we have two analysts recevng samples for qualty control tests n the food company, we frst assume a mxture of two normal dstrbutons consderng a covarate X (an ndcator varable for each analyst), where X 0 for analyst and X for analyst. Let T be a random varable denotng the qualty control test tme for the th,,,n where n 704, assumng a mxture of two normal dstrbutons N f ; t pf t ; pf t where f f sample,, gven (from Eq. ()) by the densty, () ; t ; X ;,;,,, 704 ; ; X 0 (analyst ); X (analyst ) and s a normal densty gven by, (3) t ; exp t

4 40 For a Bayesan analyss of the regresson mxture model defned by Eq. () and Eq. (3), we assume the followng pror dstrbutons for the parameters,, and,, : N N N 3 a,0 ; truncated 0 0,0 6 0,0 ; truncated 0 p Beta, Gamma 0., 0. where, N denotes a normal dstrbuton wth mean and varance ; b c (4) Gamma, denotes a b gamma dstrbuton wth mean c and varance Beta d, e denotes a beta dstrbuton wth mean d c d e de and varance d e de. The hyperparameter a for the normal pror of s assumed known from a prelmnary data analyss. Ths choce of a also mples n the dentfablty of the mxture model. Note that we are assumng large varances for the pror dstrbutons, that s, approxmately non-nformatve prors (see for example, Paulno et al. (003)). We further assume pror ndependence among the parameters.. Mxture of two Webull dstrbutons b ; Another possblty s to assume a mxture of two Webull dstrbutons for the tmes of the two analysts. In ths case, we assume n Eq. (), a mxture of two Webull dstrbutons (see for example, Lawless, 98) gven by the densty, f t pf t ; v p f t ; v, (5) v v where f t v v t ; exp t and the scale parameter of the Webull dstrbutons s gven by exp X,, ;,,, 704; ; v s the shape parameter of the Webull dstrbuton; X 0 (analyst ) and X (analyst ). Note that the mean tme for each component dstrbuton n the mxture model (5) s gven by, mean v v x where x t exp tdt s a gamma functon;, ;,,, 704. For a Bayesan analyss of 0 the model we assume the followng pror dstrbutons for the parameters:, 0., 0., 0., 0. ;, 0,0 p Beta Gamma Gamma v Gamma N (7) 3. A Bayesan analyss for the data of the food ndustry To analyze the tmes taken for qualty control tests by the two analysts at the food company usng a Bayesan approach, we frst assume a mxture of two normal dstrbutons defned by Eq. () and Eq. (3) wth prors Eq. (4) wth a 8. Ths model s denoted model. The value a 8 was chosen from a prelmnary data analyss (see Fg. and Fg. ). In the smulaton procedure of samples for the ont posteror dstrbuton of p,,,,,, and, we used OpenBUGS, an open source software avalable from (see, for example, Lunn et al. (009)). OpenBUGS requres only the dstrbuton of the data and the pror dstrbutons for the parameters of the model, and the condtonal posteror dstrbutons used for the Gbbs samplng algorthm do not (6)

5 J. A. Achcar et al. / Internatonal Journal of Industral Engneerng Computatons 4 (03) 4 have to be specfed; that s to say, the samples for the ont posteror dstrbuton of nterest s greatly smplfed. In the smulaton procedure, a sample sze of 5, 000 was ntally smulated from the ont posteror dstrbuton dscarded to elmnate the effect of the ntal values used n the teratve routne (burn-n sample). Followng ths burn-n sample another 0, 000 Gbbs samples were generated, takng every th 0 sample to have approxmately uncorrelated samples, from whch a fnal smulaton sample of sze,000 was used to get the posteror summares of nterest. Convergence of the Gbbs samplng algorthm was montored from the usual traceplots for each parameter sample. Table shows the posteror summares obtaned assumng the mxture of two normal dstrbutons. Table Posteror summares ( model ) parameter mean S. D. 95% credble nterval p (0.46;0.4998) p (0.5003;0.5380) ( ; 0.06) ( ; 0.63) (0.74;0.763) (7.85;8.77) (6.6;7.99) (0.857;0.66) (0.7;0.69) (.49;.30) (7.00;7.44) Fg. 3 shows tmes observed for qualty control versus samples, and the ftted means (Bayesan estmates for the means) versus samples. Fg. 3. Qualty control tmes and ftted means versus samples (all data set)

6 4 A good ft was observed for the data for the proposed model. The posteror mean for the frst normal cluster for analyst had a Monte Carlo estmate based on the, 000 smulated Gbbs samples gven by ˆ 0.75, and a 95 % credble nterval for gven by 0.74; For the second normal cluster, the posteror mean for s estmated by ˆ 8. 0 wth a 95 % credble nterval gven by 7.85;8.77. From X, for analyst X, there s a Bayesan estmate for the mean for the frst normal cluster gven by ˆ ˆ , and for the mean of the second normal cluster gven by ˆ ˆ That s, analyst has less tme to perform the qualty tests than analyst. It s also observed that the regresson parameters and have sgnfcant effects on the qualty control tmes for the analysts, snce zero s not ncluded n the 95 % credble nterval for and (see Table ). Smlar proportons of samples n the two clusters of data are observed. To check the qualty of ft for the data for the mxture of normal dstrbutons, we could calculate the dfferences of observed and ftted means gven by ft l 704 t ˆ (8) where ˆ are the ftted means,,,, 704 and l ndexes model (here, l ). We observe ft From (8), t s observed that ft for analyst s gven by and for analyst s gven by. 79. In Fg. 4, we have the observed values and ftted means for each analyst. (a) (b) Fg. 4. Qualty control tmes and ftted means versus samples: (a) Analyst and (b) Analyst In the Bayesan analyss of the data from the food company, a mxture of two Webull dstrbutons s also assumed as defned by Eq. (5) and the pror dstrbutons Eq. (7). Let us denote ths model as model. Consderng the same smulaton steps used for model, Table shows the posteror summares of nterest based on, 000 smulated Gbbs samples usng the OpenBUGS software. From the results n Table, the covarate X (analyst) shows a sgnfcant effect n the means of the two cluster Webull dstrbutons snce zero s not ncluded n the 95 % credble ntervals for and. Note that the regresson parameter,, has a multplcatve effect on the scale parameter for the Webull dstrbutons exp X,, ;,,, 704.

7 J. A. Achcar et al. / Internatonal Journal of Industral Engneerng Computatons 4 (03) 43 From the results n Table, we observe that snce exp X,, ;,,, 704, we get Bayesan estmates for the scale parameter of the Webull dstrbutons (5), gven, respectvely by ˆ ˆ 4.65 and ˆ ˆ for analyst X 0 and ˆ ˆ ˆ exp 4.65exp and ˆ ˆ ˆ exp exp for analyst X. Table Posteror summares ( model ) parameter Mean S. D. 95% credble nterval p (0.465;0.505) p (0.499;0.5386) (0.059;0.868) (0.0760;0.945) (3.84;4.699) ( ;0.0006) v (6.630;6.8580) v (4.950;4.5970) (0.0000; ) From Eq. (6), we get Bayesan estmates for the means of both analysts, gven by: Analyst : () vˆ ˆ 6.55 ˆ v for cluster, and () vˆ ˆ ˆ v for cluster of the mxture of two Webull dstrbutons. Analyst : () vˆ ˆ 6.55 ˆ v for cluster, and () vˆ ˆ ˆ v for cluster of the mxture of two Webull dstrbutons.

8 44 Smlar results are observed for the means of the two dstrbutons n the mxture model assumng normal or Webull dstrbutons (see Table ). Fg. 5 and Fg. 6 show the tmngs of qualty control versus samples and also of the ftted means (see Eq. (6)) versus samples consderng respectvely, the combned data of the two analysts; the data of analyst and the data of analyst. all 357 From Eq. (8), we get ft (all combned data); ft 3.98 (data of analyst ) and ft 5.4 (data of analyst ). Overall, both models gve smlar nference results, but model (mxture of two normal dstrbutons) shows a small mprovement n the ft for the data. Fg. 5. Qualty control tmes and ftted means versus samples (all data set) (a) (b) Fg. 6. Qualty control tmes and ftted means versus samples: (a) Analyst and (b) Analyst

9 J. A. Achcar et al. / Internatonal Journal of Industral Engneerng Computatons 4 (03) Concludng remarks In ndustral applcatons, managers and ndustral engneers are usually nterested n modelng the tme taken n tasks carred out by dfferent operators, especally n order to get performance ndcators for ther systems (by Dscrete Event smulaton, for example). Ths paper analyzed the tmes taken n qualty control carred out by two dfferent analysts n a Brazlan food company. The man goal at every company s to standardze and optmze these tmes, as a reference that should be followed by all analysts n the company. In many cases, as was consdered n ths paper based on the data set from the food company, the batches of manufactured products arrve n a random order at the company's qualty control department, wth a mxture of dfferent products, and the qualty control tests usually take dfferent tmes. Hence, the use of a mxture of parametrcal dstrbutons n the presence of a covarate, consdered as t was to be of great nterest n ndustral applcatons. In ths case, we were able to consder Bayesan confdence ntervals, or classcal confdence ntervals, for the means of the two component dstrbutons for the best analyst (short tmes) as standard reference ntervals to be followed by all the operators n the company's qualty control department. It s mportant to pont out that these results could be generalzed for other mxed data sets consstng of more than two clusters, and n the presence of other covarates that could affect the performance of analysts, such as skll n performng the qualty control tests, experence, calbraton of the test equpment, day of the week, temperature, and many other factors. The use of Bayesan methods for a mxture of parametrcal dstrbutons especally consderng exstng smulaton MCMC methods, such as the Gbbs samplng algorthm and OpenBUGS software, could be of great nterest, snce the computatonal cost to get the posteror summares requred s not hgh. References Carroll, R.J., Roeder, K., & Wasserman, L. (999). Flexble parametrc measurement error models. Bometrcs, 55, pp Chb, S., & Greenberg, E. (995). Understandng the Metropols-Hastngs Algorthm. The Amercan Statstcan, 49, Dey, D.K., Kuo, L., & Sahu, S.K. (995). A bayesan predctve approach to determnng the number of components n a mxture dstrbuton. Statstcs and Computng, 5, Debolt, J., & Robert, C.P. (994). Estmaton of fnte mxture dstrbutons through Bayesan samplng. Journal of the Royal Statstcal Socety. Seres B (Methodologcal), 56, pp Fnkelsten, M.S., & Esaulova, V. (00). Modelng a falure rate for a mxture of dstrbuton functons. Probab. Eng. Inf. Sc., 5, Gelfand, A.E., & Smth, A.F.M. (990). Samplng-Based Approaches to Calculatng Margnal Denstes. Journal of the Amercan Statstcal Assocaton, 85, Lawless, J.F. (98). Statstcal Models and Methods for Lfetme Data (Wley Seres n Probablty & Mathematcal Statstcs). John Wley & Sons. Lunn, D., Spegelhalter, D., Thomas, A., & Best, N. (009). The BUGS proect: Evoluton, crtque and future drectons. Statstcs n medcne, 8, Mengersen, K., & Robert, C. (996). Bayesan Statstcs, 5. Oxford Unversty Press, Oxford. chapter Testng for mxtures: a Bayesan entropy approach. Paulno, D.C., Turkman, M.A.A., & Murtera, B. (003). Estatstca Bayesana. Fundação Calouste, Lsboa. Rchardson, S., & Green, P.J. (997). On bayesan analyss of mxtures wth an unknown number of components. Journal of the Royal Statstcal Socety. Seres B (Methodologcal), 59, pp Stephens, M., 000a. Bayesan methods for mxtures of normal dstrbutons. Master s thess. Stephens, M., 000b. Dealng wth label swtchng n mxture models. Journal Of The Royal Statstcal Socety Seres B, 6,

10 46 Ttterngton, D.M., Smth, A.F.M., & Makov, U.E. (985). Statstcal Analyss of Fnte Mxture Dstrbutons. John Wley & Sons Ltd. SERIES: Wley Seres n probablty and mathematcal statstcs.