Daily counting of manufactured units sent for quality control: a bayesian approach

Daily counting of manufactured units sent for quality control: a bayesian approach

Jorge Alberto Achcar^I; Claudio Luis Piratelli^II,*; Renata Regina Sandrim^III

^IProfessional Master Program in Production Engineering, University Center of Araraquara - UNIARA, Department ofSocial Medicine, FMRP, University of São Paulo, Brazil. E-mail: [e-mail]

^IIProfessional Master Program in Production Engineering, University Center of Araraquara - UNIARA, Brazil. E-mail: [e-mail]

^IIIDepartament of Production Engineering, University Center of Araraquara - UNIARA, Brazil. E-mail: [e-mail]

ABSTRACT

This paper presents the statistical modeling for daily counting statistics of units that arrive for quality inspection at a food company. Different Poisson regression models were considered in order to analyze the data collected, with a Bayesian focus. The main objective was to forecast the daily average count based on co-variables such as days of the week. The analysis of co-variables is very often neglected by statistical packages that come with Discrete Event Simulation software. The discovery of the factors that influence these variations was essential to a more accurate modeling (the definition of simulation calendars) and enables industrial managers to make better decisions about the reallocation of people in the department, resulting in better planning of production capacity.

Keywords: poisson regression, Bayesian analysis, Markov Chain Monte Carlo methods.

1 INTRODUCTION

It is common to find a lot of variability in the daily counting of manufactured units in a production line. The statistical modeling of these data is useful as a diagnostic tool for the production systems (for example, for measuring the performance indicators by queue models), inference (forecasting) or simulation, especially for decisions on investment, production programming, allocation of the production capacity, and many other factors). Discovering these factors that affect variability may be of great interest to industrial engineers and industrial managers.

In Queuing Theory (QT), counting analysis is essential in the choosing of the model appropriate to the queue object that is to be diagnosed - see, for example, Gross & Harris (1975), Kleinrock (1975), Larson & Odoni (1981). However, analytical queuing models have certain limitations to address complex problems (for example, the variability in counting due to the presence of covariates).

Discrete Event Simulation (DES) is used for either analyzing complex performance models or obtaining more detailed knowledge of a system - Kari et al. (1994). In DES, one of the analyst's main objectives is to perform experiments on models that represent real systems, in order to predict future behavior - Kelton (2007). To this end, the identification of probability distributions for counting is crucial in order to have a reliable model of the reality that is to be shaped.

DES approaches widely utilize random number generation in the analysis. Usually, the applied random distributions are quite simple but more complex distributions are required to get better accuracy in the model. Non-standard random distributions are required when the system cannot be modelled accurately using conventional probabilistic distributions. Efforts to deal with these situations are reported by Kari et al. (1994), Andradóttir & Bier (2000), Zhang et al. (2005), Leemis (2006), Yang & Liu (2012).

In industrial applications, Poisson distribution is common without considering co-variates. This modeling approach is commonly used by statistics packages which come with Discrete Event Simulation software. In general, these packages return a bad fit of data to the Poisson distribution, due to the high degree of variability in the counts (our case).

This paper considers the daily counting data for manufactured units sent for quality control at a food company in São Paulo State, Brazil, over a period of 30 non-consecutive days. This data set is presented in Table 1. Figure 1 shows the plot of the observed number of units (daily counting) against days, showing the high degree of variability for the daily counting.

This paper's main objective is to analyze the daily counting data (arrivals at quality control) in Table 1 using Poisson regression models that assume certain co-variates, such as: days when the data was collected, and seasonality factors (specific days of the week). The focus is specifically on inference in the daily counting average λ, depending on the co-variates, so that the schedules for the software simulation can reflect reality for the company. The inferences of interest were obtained using Bayesian methods. The posterior summaries of interest were obtained via Markov Chain Monte Carlo (MCMC) simulation methods, such as the popular Gibbs sampling algorithm (Gelfand & Smith, 1990) or the Metropolis-Hastings algorithm, when the conditional posterior distributions required for the Gibbs sampling algorithm do not have standard parametrical forms (see, for example, Chib & Greenberg, 1995).

Regression analysis of counting data has been studied by Hausman et al. (1984), Zeger (1988), Blundell et al. (1995), Martz & Piccard (1995), Cameron & Trivedi (1998), Gurmu et al. (1999), Freeland & McCabe (2004a and b). Bayesian estimation has been proposed in count regression models by Harvey & Fernandes (1989), Albert (1992), Chib et al. (1998), Settimi & Smith (2000), Martz & Hamada (2003), McCabe & Martin (2005) and Zheng (2008).

According to Hsu & Wang (2007), modeling industry data sets is a challenge because: (1) a large sample of data is not always available; (2) there are some data items missing; (3) outliers interference; (4) the predictor variables are not correlated, among other reasons.

The use of Bayesian inference is justified by the possibility of greater flexibility in the interpretations of the inference results obtained. In particular, one of the main advantages of using Bayesian inference is the use of prior information in the choice of prior distributions, especially for data obtained in industrial applications. When we do not have good prior information, we can use non-informative priors on the model's parameters. Another great advantage is the use of standard existing Markov Chain Monte Carlo (MCMC) simulation methods. In this way, we do not need to use asymptotical results based on the asymptotical normality of the maximum likelihood estimators or standard asymptotical likelihood ratio tests which depend on large sample sizes. These results could be a good justification for the use of Bayesian inference, especially when the sample sizes are small or moderate, as is common in engineering applications (our case).

Bayesian methods have been used extensively in many applied areas, such as Business Administration, Economy and Industrial Engineering. Some examples take from the Scielo scientific basis: Quinino & Bueno Neto (1997) used Bayesian methods to evaluate the accuracy of the quality inspector; Pongo & Bueno Neto (1997) and Droguett & Mosleh (2006) proposed Bayesian inference to evaluate the reliability of products in development projects; Cavalcante & Almeida (2006) used multi-criteria method and Bayesian analysis to determine preventive maintenance intervals; Moura et al. (2007) used the Bayesian methods to evaluate the efficiency of maintenance; Ferreira et al. (2009) used the Bayesian approach in a portfolio selection problem; Barossi-Filho et al. (2010) used Bayesian analysis to estimate the volatility of financial time series, and Freitas et al. (2010) used the Bayesian approach to estimate the wearing out of train wheels.

The literature contains a rapidly growing number of published papers using the Bayesian paradigm in almost every applied area, such as medicine, economics, environmental sciences or engineering, since there has been a huge advance in computer hardware and software in the last twenty years. According to Fildes (2006), Bayesian methods have been gaining in prominence in the number of their citations in important journals in recent years and chief editors see them as a hot topic in forecasting (comprising counting problems). Armstrong & Fildes (2006) argue that many forecasting areas have developed methods, but few of them have been adopted in organizations in practice. Andradóttir & Bier (2000) suggested the usage of Bayesian inference to estimate probability distributions parameters for input in simulation models. In this way, this paper gives a contribution for parameters estimation in a real count data from a food company (this approach can be used by Simulation analysts).

The paper is organized as follows: in Section 2 we present some Poisson regression models; in Section 3, we present a statistical analysis of the data presented in Table1; in Section 4, we present a discussion on the results obtained, and in Section 5 we present some concluding remarks.

2 STATISTICAL MODELING

Let Y_i be a random variable with a Poisson distribution given by,

Where y_i = 0, 1, 2,... denotes the daily number of units (arrivals) for quality control in the i-th day, i = 1, 2,..., 30. Observe that the mean and the variance of the Poisson distribution (1) are equal to λ_i.

In industrial applications it is common to use a Poisson distribution without considering covariates. So, different Poisson regression models can be considered in analyzing the data set presented in Table 1, given as follows:

Model 1: Consider λ_i given in (1), by

where X_i denotes the i-th day (X_i = 1, 2,..., 30). In this case, it is assumed that days could influence the daily counting of units sent to quality control. The formulation (2) guarantees λ_i to be positive, for i = 1, 2,..., 30.

Model 2: In this case, there is the addition of the quadratic term X_i² in model (2), that is,

where X_i is defined in (2), i = 1, 2,..., 30.

Model 3: In this model, there is also anautoregressive effect of the daily counting y_i-1 in the modeling for λ_i , that is, a first order autoregressive model given by

where X_i is given in (2), and y_i-1 is the number of units (arrivals for quality control) on the (i - 1)-th day, i = 2, 3,..., 30.

Model 4: Consider the introduction of another autoregressive term in λ_i , given by

where X_i and y_i are defined, respectively in (2) and (4), for i = 3, 4,..., 30.

Model 5 : Consider λ_i , in (1), given by

where i = 4, 5,..., 30.

In the models defined by (1), (2), (3), (4), (5) and (6), we could also consider a seasonal effect. For the data presented in Table 1, the effect of a specified weekday could be considered - in our case, Mondays, where there seems to be a lower number of units sent to quality control as compared with other week days.

So, let us define the indicator variable z_i = 1 when the i-th day is a Monday and z_i = 0 when the i-th day is not a Monday, for i = 1, 2,..., 30.

Assuming the models defined above, the likelihood function for the vector of parameters θ associated to each model is given by

where θ = (β₀,β₁) for model 1; θ = (β₀, β₁, β₂) for models 2 and 3; θ = (β₀, β₁, β₂, β₃) formodel 4 and θ = (β₀, β₁, β₂, β₃, β₄) for model 5.

For a Bayesian analysis, let us assume the following prior distributions for the regression parameters β_j of the models:

where N (a_j; ) denotes a normal distribution with mean a_j and variance , j = 0, 1, 2, 3, 4. Also prior independence is assumed among the models' parameters.

Combining the joint prior distribution (a product of normal distributions) with the likelihood function L(θ) given in (7), we obtain from the Bayes formula the joint posterior distribution for θ (see for example, Box & Tiao, 1973).

The posterior summaries of interest are obtained using MCMC methods. Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from probability distributions based on the construction of a Markov Chain that has the desired distribution as its equilibrium distribution (see for example, Paulino et al., 2003). The state of the chain after a large number of steps is used as a sample of the desired distribution. The quality of the sample improves with the number of steps. Typical use of MCMC sampling can only approximate the target distribution, as there is always some residual effect from the starting position. The most common application of these algorithms is to numerically calculate multi-dimensional integrals. A special case is given by the Gibbs sampling algorithm, which is a special case of the Metropolis-Hastings algorithm (see Chib & Greenberg, 1995). Gibbs sampling uses the fact that given a multi-variate distribution it is simpler to sample from a conditional distribution than to marginalize by integrating over a joint distribution. Suppose we want to obtain samples of X ={x₁,..., x_n} from a joint distribution p(x₁,..., x_n). Denote the i-th sample by X_(s) ={,..., }. We proceed as follows:

1. We begin with some initial value X⁽⁰⁾ for each variable.

2. For each sample i = {1...k}, sample each variable from the conditional distribution . That is, sample each variable from the distribution of that variable conditioned on all other variables, making use of the most recent values and updating the variable with its new value as soon as it has been sampled.

The samples then approximate the joint distribution of all variables. Furthermore, the marginal distribution of any subset of variables can be approximated by simply examining the samples for that subset of variables, ignoring the rest. In addition, the expected value of any variable can be approximated by averaging over all the samples.

A great simplification in the simulation of samples for the joint posterior distribution for θ is given using the WinBugs software (Spiegelhalter et al., 2003) which only requires the specification of the distribution for the data and the prior distributions for the parameters. More details of MCMC and WinBugs can be found in Che & Xu (2010).

3 ANALYSIS OF THE DAILY COUNTING DATA OF UNITS SENT TO QUALITYCONTROL

To analyze the quality control counting data (data set presented in Table 1), we initially consider the model 1, defined respectively by (1) and (2), with the following priors for β₀ and β₁: β₀ ~ N(a₀, 10) and β₁ ~ N(0, 10), for fixed values of a₀. In this analysis, considering different values of a₀ we obtained similar results. With a fixed value for the variance of the normal distribution equal to 10, that is, a large value, we have approximately non-informative priors.

Using the WinBugs software, we first simulated a "burn-in-sample" size of 10,000, discarded to eliminate the effect of the initial values used in the Gibbs sampling algorithm. After this "burn-in-sample period", we simulated another 20,000 Gibbs samples for β₀ and β₁. From this sample, we selected a final sample size of 1,000, taking a sample chosen from every 20 simulated samples to have an approximately uncorrelated sample to be used to find the posterior summaries of interest. The obtained posterior summaries of interest (posterior mean, posterior standard deviation, and 0.95% credible intervals) are given in Table 2. Convergence of the algorithm was monitored using standard trace plots from the simulated samples (see, for example, Paulino et al., 2003, or Gamerman, 1997).

In Table 2, we also have the posterior summaries of interest considering the models 2, 3, 4 and 5 assuming the same normal prior distributions N(0, 10) for the regression parameters β_j, j = 1, 2, 3, 4.

In the simulation of Gibbs samples for the parameters of these models we used the same steps in the WinBugs software used for model 1.

For selection of the best model to be fitted by the data, three discrimination criteria were considered:

(a) Graphical verification of the fit: in this case, plots of the observed values yi were considered (counting of units sent to quality control on the i-th day) against days and plots of the fitted means (posterior mean of λ_i in (1)) considering the five proposed models.

(b) Use of the Bayesian discrimination method Deviance Information Criterion (DIC) introduced by Spiegelhalter et al. (2002). This criterion is useful in the selection of models when samples of the joint posterior distribution for the parameters of the models were obtained by MCMC simulation methods. Smaller values of DIC indicate better models.

(c) Determination of the sums of the absolute values of the differences between the observedvalues y_i with the fitted values (posterior means of Y_i ), i = 1, 2,..., 30, that is

where y_i is the daily counting of units sent to quality control and _i is the Monte Carlo estimate of the posterior mean for λ_i based on the 1,000 simulated Gibbs samples.

In Figure 2, we have the graphs of the observed values y_i against days and the graphs of the estimated values (_i) against days, for i = 1, 2,..., 30. We observe that models 4 and 5 have a good fit for the counting data presented in Table 1. We also observe smaller values for DIC (see Table 2) assuming models 4 and 5, an indication of better models.

Considering the sums of the absolute values of the differences between the observed values y_i with the fitted values (9), we have: S₁ = 435.26 (model 1); S₂ = 437.42 (model 2); S₃ = 345.26(model 3); S4 = 314.19 (model 4) and S₅ = 299.19 (model 5). So, we conclude that models 4 and 5 are better fitted by the counting data from Table 1.

Assuming model 4 (a model with fewer parameters), the presence of an indicator or "dummy" variable for weekday was considered; that is, Z_i = 1 for Mondays and Z_i = 0 for the other weekdays and the regression Poisson model ("model 6") given by

where i = 3, 4,..., 30.

In Table 3, the posterior summaries of interest are given for model 6 defined by (1) and (10) assuming normal prior distributions with variance equals to 10, and the same simulation steps considered for the other models using the WinBugs software.

Assuming model 6, DIC = 249.12 and S₆ = 304.33 were obtained. We also observe in Figure 2 a good fit of the model for the counting data from Table 1.

4 DISCUSSION OF THE RESULTS OBTAINED

From the results obtained from Table 3, for "model 6", we observe that the co-variates y_i-1 (counting with a lag of one day), y_i-2 (counting with a lag of two days) and Z_i (weekday) present significant effects in the daily counting of units sent to quality control, since the zero value is not included in the 95% credible intervals for the regression parameters β₂, β₃ and β₄.

Since the Bayesian estimate of the parameter β₄ is negative (₄ = -0.1462), we observe that Mondays present smaller counting of units sent to quality control as compared with the other weekdays. This result could be of great interest for the company.

It was also possible to use "model 6" to get predictions, from the fitted model (11):

So, considering the prediction for future days (i = 31), we have X₃₁ = 31, y₃₀ = 90, y₂₉ = 101, Z_i = 1 (if the next day is a Monday); that is, (₃₁) = 72.0716 (72 units); if Z_i = 0 (the next day is not a Monday), we have ₃₁ = 83.4177 (83 units).

These results are important for the company in planning human resources in the quality control department due to a low utilization rate, especially on Mondays. Capacity planning could be done using a customized calendar for Mondays when running Discrete Event Simulation.

Finally, in Table 4, we have the observed values (arrivals) and the fitted values for the means assuming each proposed model. Observe that our Bayesian estimates in Table 4 are given by Monte Carlo estimates of the posterior means based on the simulated Gibbs samples for each parameter. Under the Bayesian paradigm, we usually base our inferences on credibility intervals, rather than using hypothesis tests, as is common from a classical Bayesian perspective.

5 FINAL CONSIDERATIONS

This paper considered the problem of variability in the daily counting of manufactured units sent to the quality control department at a food company. The main objective of this paper was to analyze the daily counting data (arrivals at quality control) using Poisson regression models assuming some co-variates. The inferences of interest were obtained using Bayesian methods. The posterior summaries of interest were obtained via Markov Chain Monte Carlo (MCMC) simulation methods. This approach was selected because of some existing advantages in the Bayesian inference paradigms in the use of prior information, which is common in industrial applications. We use this prior information to choose a prior distribution that is combined with the likelihood function to obtain the posterior distribution of interest. Another great advantage of the Bayesian approach using MCMC methods, especially for data obtained in industrial applications is the inferences are not based on asymptotical results that usually depend on large sample sizes, whichmay not be available in industrial applications. Another advantage of the Bayesian approach is the use of Bayesian discrimination methods, especially considering the simulated Gibbs samples, such as the DIC criterion considered in this paper, which can be combined with other existing graphical or empirical methods to check the fit of the model for the data. With these discrimination methods, we were able to choose the best model from different proposed models, as was observed in our application. A better model means a better forecast, which is of great interest for industrial managers.

As a special case, we presented an application of the Bayesian approach in a real data count at a food company. For the specific company, the statistical modeling of the data set was very useful for the Simulation analyst. Creating a customized calendar (depending on daily co-variates) was important in running Discrete Event Simulation and getting performance indicators from the quality control department (such as a diagnostic device). The discovering of important factors that affect variability was very important to the industrial managers' decisions in terms of reallocating workers in the test analysis schedule (quality control), resulting in better production capacity planning.

Received December 7, 2011

Accepted December 15, 2012

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Publication Dates

• Publication in this collection
  16 July 2013
• Date of issue
  Aug 2013

History

• Received
  07 Dec 2011
• Accepted
  15 Dec 2012