Monitoring process control chart with finite mixture ...

International Journal of Quality & Reliability Management Monitoring process control chart with finite mixture probability distribution: An application in manufacture industry Damaris Serigatto Vicentin, Brena Bezerra Silva, Isabela Piccirillo, Fernanda Campos Bueno, Pedro Carlos Oprime,

Downloaded by UFSCAR At 10:42 20 February 2018 (PT)

Article information: To cite this document: Damaris Serigatto Vicentin, Brena Bezerra Silva, Isabela Piccirillo, Fernanda Campos Bueno, Pedro Carlos Oprime, (2018) "Monitoring process control chart with finite mixture probability distribution: An application in manufacture industry", International Journal of Quality & Reliability Management, Vol. 35 Issue: 2, pp.335-353, https://doi.org/10.1108/IJQRM-11-2016-0196 Permanent link to this document: https://doi.org/10.1108/IJQRM-11-2016-0196 Downloaded on: 20 February 2018, At: 10:42 (PT) References: this document contains references to 39 other documents. To copy this document: [email protected] The fulltext of this document has been downloaded 19 times since 2018*

Users who downloaded this article also downloaded: (2018),"A new double exponentially weighted moving average control chart using repetitive sampling", International Journal of Quality & Reliability Management, Vol. 35 Iss 2 pp. 387-404 https://doi.org/10.1108/ IJQRM-11-2016-0198 Access to this document was granted through an Emerald subscription provided by emeraldsrm:478536 []

For Authors If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information.

About Emerald www.emeraldinsight.com Emerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services. Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. *Related content and download information correct at time of download.

The current issue and full text archive of this journal is available on Emerald Insight at: www.emeraldinsight.com/0265-671X.htm

SPECIAL SECTION QUALITY PAPER

Monitoring process control chart with finite mixture probability distribution An application in manufacture industry Damaris Serigatto Vicentin, Brena Bezerra Silva, Isabela Piccirillo, Fernanda Campos Bueno and Pedro Carlos Oprime

Monitoring process control chart

335 Received 7 November 2016 Revised 3 January 2017 7 March 2017 Accepted 14 March 2017


Universidade Federal de São Carlos, São Carlos, Brazil Abstract Purpose – The purpose of this paper is to develop a monitoring multiple-stream processes control chart with a finite mixture of probability distributions in the manufacture industry. Design/methodology/approach – Data were collected during production of a wheat-based dough in a food industry and the control charts were developed with these steps: to collect the master sample from different production batches; to verify, by graphical methods, the quantity and the characterization of the number of mixing probability distributions in the production batch; to adjust the theoretical model of probability distribution of each subpopulation in the production batch; to make a statistical model considering the mixture distribution of probability and assuming that the statistical parameters are unknown; to determine control limits; and to compare the mixture chart with traditional control chart. Findings – A graph was developed for monitoring a multi-stream process composed by some parameters considered in its calculation with similar efficiency to the traditional control chart. Originality/value – The control chart can be an efficient tool for customers that receive product batches continuously from a supplier and need to monitor statistically the critical quality parameters. Keywords Quality management, Control chart, Alimentary sector, Mixture distributions, Multiple-stream processes Paper type Case study

1. Introduction Statistical process control (SPC) has been widely used in many industries to monitor, diagnose, and improve processes characterized by a large number of quality characteristics (Ahangar and Chimka, 2016; Variyath and Vattathoo, 2014; Phaladiganon et al., 2013; Woodall, 1985). The most popular technique in SPC to improve the quality of manufacturing products and processes worldwide is the control chart (Aslam et al., 2014; Woodall and Montgomery, 2014; Jensen et al., 2006; Castillo and Montgomery, 1994). According to Woodall and Montgomery (2014), process monitoring by a control chart is important to understand the variation in a process and to assess its current state. Process monitoring involves two phases. In the first phase, the practitioner collects a sample of time-ordered data from the process of interest. In this phase, the control limits are determined by process data and the Shewhart charts (x=s and x=R) are established using estimated statistical parameters of the mean (µ) and the standard deviation (σ). Thus, the objective of this phase is to understand the variation in the process over time, to evaluate the process stability and to model the in-control process performance (Woodall and Montgomery, 2014; Bersimis et al., 2007). This paper forms part of a special section entitled Selected Papers from the 2nd International Conference on Quality Engineering and Management.

International Journal of Quality & Reliability Management Vol. 35 No. 2, 2018 pp. 335-353 © Emerald Publishing Limited 0265-671X DOI 10.1108/IJQRM-11-2016-0196

IJQRM 35,2


336

In the second phase, data are collected many times to detect changes from a stable process. According to Woodall and Montgomery (2014), it is important to design monitoring methods to control the overall false-alarm rate. As discussed by Franco et al. (2014), the on-line monitoring of the process variability using the Shewhart control charts can ensure high-quality levels and maintain acceptable levels in the number of nonconforming items. Bersimis et al. (2007) also demonstrated that there are many situations in the industry where the monitoring of one or more characteristics related to quality procedures is necessary. For this purpose, control charts maintain the critical quality attributes through monitoring and control processes. According to Hall and Zhou (2003), the cost is a point of identification when the process has the presence of special causes and it may be considered over control. A special application of the control chart is the monitoring of multiple-stream processes (MSPs). In this case, a machine has several modules and each module is capable of producing its output simultaneously ( Jirasettapong and Rojanarowan, 2011). Epprecht and Simões (2013) stated that a MSP is a process that generates several streams of output, where the quality variable and its specifications are the same in all streams. According to them, a classic example would be a mold with several cavities. A special case is when processes may still produce only one stream of output but the quality variable is measured at several points at the same time. According to Woodall and Montgomery (2014), these methods tend to be based on detecting changes in a single stream or a simultaneous shift in all streams. Mortell and Runger (1995) developed a methodology for multiple streams by making an analysis of different design control charts. The approach described by Mortell and Runger (1995) and other authors, for example, Epprecht and Simões (2013) and Woodall and Montgomery (2014), has considered two variation fonts: between streams, internal stream and time trend effect. We have considered in our study the formation lot as a linear combination of multiple subpopulations, which was identified by collected data of a real problem. The estimation effect of statistical parameters in the control chart performance is important to research question ( Jensen et al., 2006). In general, the practitioner uses an estimator for statistical parameters (Adeoti et al., 2016). In these cases, the use of statistical estimators for the unknown statistical parameter can turn the control chart limits into random variables due to estimate error (Epprecht et al., 2015; Saleh et al., 2015; Castagliola et al., 2009, 2013; Abu-Shawiesh, 2009). The control chart performance is evaluated according to the capacity to detect special causes, with the lowest type I error. This performance is evaluated through the expected average run length (ARL) and the standard deviation of the average run length (SDARL), which are influenced by the estimated parameters (Saleh et al., 2015). About this, Woodall and Montgomery (2014) claimed that the choice of the performance metric can have a significant effect on the choice of the monitoring scheme. Epprecht et al. (2015), Saleh et al. (2015) and Mei (2008) showed limitations of the ARL metric and Mortell and Runger (1995) used it as a metric for the performance analysis of the control chart. Although all of these studies have shown interesting results, the literature studies focus on a common problem in industrial processes: the finite mixtures of probability distributions. For these cases, the traditional chart X is not convenient for reasons of costs and the normality assumption of the probability distribution. The development and application of finite mixture models distributions can be found in studies such as Everitt and Hand (1981), Titterington et al. (1985), West and Smith (1992), and most recently, Yu (2011), Lee et al. (2014), Kaffel and Prigent (2016), Kayid and Izadkhah (2015), Kim et al. (2015), Masmoudi et al. (2016). These models have some tools to help the customer in analyzing data through the different unknown parameters (Masmoudi et al., 2016; Teel et al., 2015). The finite mixture models also offer an interesting alternative for the non-parametric modeling because they

1,200


337

Histogram of kg padrào Spreadsheet5 4v–4,000 c

1,000

800 No of obs


are less restrictive than the usual distribution assumptions (Adams, 2016; Chauveau and Hoang, 2016; Diebolt and Robert, 1994). The probability distribution of the mixture can be characterized as a probabilistic combination of two or more random variables (Yu, 2011; Dixon, 2012; Lee et al., 2014). A mixture of distribution models are formed from a weighted combination of two or more underlying distributions with g(x) ¼ ∑jαj f (x)j; ∑αj ¼ 1. These models often appear on statistical models of perception, cognition, and action in which a finite number of discrete internal states probabilistically are inserted through a series of tests (Yantis et al., 1991). We developed an application of control chart with finite probability distributions mixture in a selected process from a company in the food sector whose products consist of wheat-based dough. Therefore, we proposed to analyze the process considering the finite mixture of probability distribution and to build statistical control limits of quality based on the weight. The data of the 40 items were collected randomly in a 30-minute interval during two days to form the master sample and after being expanded using the Bootstrap technique. In this case, we considered that the overall probability distribution resulting subsets of the manufacturing process with a probability distribution function (PDF) are known. A theoretical model will be developed and evaluated considering two scenarios: assuming that the parameters of PDF are known and in the case where the average is unknown. The result of this approach will be compared to the traditional approach of Shewhart charts. Figure 1 represents the histogram formed by an expanded sample of 4,000 units extracted randomly at the end of the line whose quality characteristic is exactly the weight of each processed unit (detailed steps of the process are presented in Figure 2). The histogram in Figure 1 indicates a mixture of probability distributions arising from the process characteristic, in particular, the error of the cylinder shape that generates non-uniform thickness and therefore different weights. The mixture of probability distributions for different subpopulations of the process precludes the use of traditional control charts. An alternative is to monitor that subset (in this case, each position of the cylinder). However, it is an expensive alternative. Our proposal is to develop a control chart considering the combination of the different functions of probability densities and thus build a unique graphic control whose samples are collected regardless of the origin of the extracted sample.

600

400

200

0 480 490 500 510 520 530 540 550 560 570 580 590 600 610 Kg padrào

Figure 1. Characterization of the weight distribution

IJQRM 35,2

We have proposed, in this case, that the upper and lower control limits (UCL, LCL) are determined by a general mixture problem for specified α error type I, as follow: Z 1 Z 1 Z 1 f ðx1 Þdx1 þp2 f ðx2 Þdx2 þ þpp f xp dxp a ¼ p1 UCL

Z þp1

338

UCL

1

UCL

Z

LCL

f ðx1 Þdx1 þp2

Z

LCL

1

f ðx2 Þdx2 þ þpp

LCL

1

f xp dxp

where p is a number of probability functions and the parameters πj are the weights of the mixture of the distributions. The objective is found for the UCL and LCL for α ¼ 0.0027 using numerical methods.


2. Mixtures of finite probability distributions 2.1 Definition Let X be considered a random variable assuming the values contained in a sample space, Ω, and its distribution can be represented by a probability density function for continuous variables or mass function for the discrete case as (Titterington et al., 1985): pðxÞ ¼ p1 f 1 ðxÞþp2 f 2 ðxÞþ þpl f l ðxÞ

(1)

Pl

where πj W0, j pj ¼ 1; j ¼ 1, …, l e ∫fj(x)dx ¼ 1. The parameters πj are weights of the mixture of the distributions or subpopulations. Because the probability density functions have parameters, a finite mixture of probability distributions can be represented generally by π1 f1(x|θ1)+π2 f2 (x|θ2)+⋯+πl fl (x|θl), where θ represents the parameters fj (x). For example, if there is a mixture of two normal distributions with the same standard deviation σ, different averages μ1 and μ2, and the total density function of the parameters that is ω ¼ (π, μ1, μ2, σ), the probability will be represented by: p x9o ¼ p| x9m1 ; s þ ð1pÞ| x9m2 ; s (2) A generic function of the probability density function of a finite mixture with k subpopulations can be written as: l X pj f x9yj p x9o ¼

(3)

j¼1

Equation (3) of the mixture probability density functions indicates that there are categories (subpopulations or sources) in which the experimental unit obtains the x value of the variable X from one of these categories. Thus fj (x) shows the probability distribution of X S1 S2 Input streams

Combined outputs

S3 S4

Figura 2. Characterization of methods to collect data from multiple processes flows for the construction of control charts

Method 1: A subset consisting of one or more measurements of each stratified flow

Method 3: A subset consists measures the output of combining flows Method 2: Collect the data for each stream separately. A subset consists of only one stream of measures

Source: Automotive Industry Action Group (AIAG) (1991)


obtained from the category j and πj indicates the probability of the variable being in one of these categories (Titterington et al., 1985). Some important properties of mixture distributions have been known and exploited to identify mixture distributions. Clearly, if the reference distributions have comparable variations and they are sufficiently different in the central position measurements, a mixture of the two distributions will be bimodal and a mixture of three or more will be multimodal distributions (Dixon, 2012; Kayid and Izadkhah, 2015). Therefore, identifying the mixture probability does not seem a trivial task. The existence of bi- or multi-modal is a way to verify subpopulation mixture. Histograms can also identify if there is a multimodal distribution. However, there is a possibility of errors: the unimodal distribution can hide the existence of two subpopulations and the identification of more than one mode by a histogram does not necessarily conclude that there is more than one subpopulation in mass data. In general, it is considered that a pure component is a unimodal but this is not always true. There are charts and multivariate methods such as cluster analysis and discriminant function that can be used to identify a mixture of probability distributions (Titterington et al., 1985). In this paper, we study the manner as a mixture indicator. The histogram with graphs of normality tests and tangentiality to identify mixtures of the probability distribution will be explored. 2.2 Control charts with finite mixture of probability distribution The performance of control charts is evaluated by the distribution of the expected value in the ARL when the parameters are estimated for the determination of the control limits. This study will assess the performance of the proposed charts when there is a mixture of distributions through E(ARL) for the unknown average and known standard deviation equally for each category. Para simplificar, consideremos que há categorias com médias distintas. For simplicity, we assume that l ¼ 3 categories with different averages. First, the case will be analyzed when the parameters and probability distributions are known; in this case, it is assumed that each fj (x) have the normal p.d.f. Following the Automotive Industry Action Group (AIAG, 1991), samples are taken at random and are examined together in which a control chart is built for the entire set of collected data, as described in method 3 of Figure 2. Three methods are presented for data collection when the production process is composed of multiple streams. Multiple streams occur in a process that yields data from several sources, channels or it is under control from the statistical process but the quality variable and its specifications are the same in all streams. Thus, it is important to understand the sources of variation among and within subgroups to analyze and interpret the control chart and how they affect the process. Figure 3 shows the production plan with four streams (F1, F2, F3 and F4) and how they can be united in one control chart rather than building separate graphs for each stream. This production plan scheme can be very useful for obtaining information necessary at the end of the production line to be delivered to the customer. 2.3 Case with statistical parameters, µ, known For X a random variable is drawn from a population, for example, with l ¼ 3 categories or subpopulations, normally distributed by parameters θ ¼ (μ1, μ2, μ3, σ1, σ2, σ3), with the following probability density function: (4) p x9o ¼ p1 | x9m1 ; s1 þp2 | x9m2 ; s2 þp3 | x9m3 ; s3 Also for simplicity, we consider σ1 ¼ σ2 ¼ σ3 ¼ σ; and also that the global average is obtained by μ0 ¼ (μ1+μ2+μ3/3). Another assumption is that each n extracted sample has the


339

IJQRM 35,2

Production

F2 2 2

R chart: 3 limits

Process range

R chart: 3 limits

Avg = 11.777

10 5 0

LCL = 0 2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25 Site

1 = 1/4

30 UCL = 26.8771

25

15

–5

20 15 Avg = 11.777

10 5

Avg = 11.777

10 5

LCL = 0

2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25

2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25

Site

Site

2 = 1/4

Avg = 11.777

10 5

LCL = 0

–5 2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25

3 = 1/4

Ft

15

Site

3 = 1/4

R chart: 3 limits UCL = 26.8771

25

n = (x, )

UCL = 26.8771

20

0

–5

–5

R chart: 3 limits

25

15

30

Figure 3. Production plan for proposed multiplestream chart

30

UCL = 26.8771

20

0

LCL = 0

0

R chart: 3 limits

25

Process range


30

UCL = 26.8771

20

F4 4 4 Process range

30 25

Process range

340

F3 3 3 Process range

F1 11

20 15 Avg = 11.777

10 5 0

LCL = 0

–5 2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25 Site

Client

probability πl from the subpopulation l. The estimator of the population mean is the P average given by x ¼ 1=n ni¼1 xi . The upper and lower control limits (UCL and LCL) are calculated by: Kst Kst UCL ¼ m0 þ pffiffiffi ; LCL ¼ m0 pffiffiffi n n

(5)

Being the center line of the graph (LC) at μ0, the probability that a point will be within the control limits is: P ðx A ½LIC; LSCÞ ¼ p1 P LICp xp LSC9m1 ; s þp2 P LICpx pLSC9m2 ; s þp3 P LICpx pLSC9m3 ; s Expanding this formulation, it follows: Kst Kst Kst Kst p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p1 P m0 px pm0 þ m ; s þp2 P m0 px pm0 þ 9m ; s n n 1 n n 2 Kst K þp3 P m0 pffiffiffi px pm0 þ pffiffiffi9m3 ; s n n In the state of control, each subpopulation has average μ ¼ {μ10, μ20, μ30}. Given that there is indeed a mathematical relationship between the average distances of each subpopulation and the overall average. When we assume μ0 ¼ μ20, we have the following relationships between the overall average and average each subpopulation: μ0 ¼ μ10 + δ1σ; μ0 ¼ μ30 − δ3σ, where δ is the standard deviations of the difference between the subpopulations. In this specific case, for l ¼ 3 and for the averages of the subpopulations equidistant, δ3 ¼ δ1 ¼ δ and δ2 ¼ 0. Another assumption is there with the


following restriction: l X

di ¼ 0:

i

Following the proposed expansion, considering the possibility of a process to be out of control also in terms of the number of standard deviations, this process state is represented as follows: m1 ¼ m10 þtst m2 ¼ m20 þtst


m3 ¼ m30 þtst

(6)

If t≠0, the process is out of control. Otherwise, the process is in a control state. Considering these relations and assumptions, the probability distribution of subpopulations is represented by a normal distribution, with known parameters and whose samples are from subpopulation with πl probability (Equation (4)). Thus, it is possible to determine the sample probability with n size to be in the region inside the control limits, subtracting μl from the inequalities of each subpopulation, the following result is obtained: Kst Kst P ðx A ½LIC; LSCÞ ¼ p1 P m0 pffiffiffi m10 pxm10 pm0 þ pffiffiffi m10 9m10 ; s n n Kst Kst þp2 P m0 pffiffiffi m20 pxm20 pm0 þ pffiffiffi m20 9m20 ; s n n Kst Kst þp3 P m0 pffiffiffi m30 pxm30 pm0 þ pffiffiffi m30 9m30 ; s n n Since μ0 ¼ μl + δlσ, replacing μ0 from the relations showed above to δ and t, then the following is obtained: Kst Kst p1 P m10 þd1 s pffiffiffi m10 ts pxm1 pm10 þd1 s þ pffiffiffi m10 ts 9m1 ; s n n Kst Kst þp2 P m20 pffiffiffi m20 ts pxm2 pm20 þ pffiffiffi m20 ts9m2 ; s n n Kst Kst þp3 P m30 þd3 s pffiffiffi m30 ts pxm3 pm10 þd3 s þ pffiffiffi m30 ts9m3 ; s n n pffiffiffi Multiplying each inequality by n=s, we get: pffiffiffi Kst pffiffiffi pffiffiffi Kst pffiffiffi t n pz pd1 n þ t n 9m1 ; s p1 P d1 n s s Kst Kst þp2 P tspz p ts9m2 ; s s s pffiffiffi Kst pffiffiffi Kst tsp zpd1 n þ ts9m3 ; s þp3 P d3 n s s

341

IJQRM 35,2

342

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 Replacing the relation st =s ¼ 1 þ 1=l l dl in the equation above (we have obtained this expression considering ANOVA approach, where the overall variance is calculated by the internal variance of each subpopulation, s2i , and the variability caused by differences in the averages between subpopulations), we have: pffiffiffi P ðxA ½LIC; LSCÞ ¼ p1 P d1 nK

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1X 2 pffiffiffi 1X 2 pffiffiffi 1þ d t n p zp d1 n þ K 1 þ d t n m1 ; s l l l l l l


! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 1X 2 þ p2 P K 1 þ d tsp zp K 1 þ d ts9m2 ; s l l l l l l ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1X 2 1X 2 d tsp zp d3 n þ K 1 þ d tsjm3 ; s þ p3 P d3 nK 1 þ l l l l l l

In terms of a normal probability distribution, we have the following equation: "

pffiffiffi P ðxA ½LIC; LSCÞ ¼ p1 | d1 n þK

! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 pffiffiffi d t n 1þ l l l

!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 pffiffiffi 1þ d t n l l l " !# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 1X 2 d tsÞ| K 1 þ d ts þp2 | K 1 þ l l l l l l " ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1X 2 d ts þp3 | d3 n þK n 1 þ l l l !# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1X 2 1X 2 d K n 1 þ d ts | d3 n 1 þ l l l l l l pffiffiffi | d1 nK

For this case, the number of parameters for P ðx A ½LIC; LSCÞ calculation is ω ¼ {π1, π2, π3, k, δ, t, n} and the sample mean number to detect a point outside of the control limits is obtained as: ARL ¼

1 1P ðx A ½LIC; LSCÞ

(7)

2.4 Case with µ unknown For μ10, μ20 and μ30 unknown: pffiffiffi pffiffiffi Kst n Kst n d Ud ¼ LCL; CL ; xA X ;Xþ s s d U CLÞ ^ are assumptions of control limits. Assuming (X)i,1, …, Xi,n), i ¼ 1, 2, … m where ðLCL; are independent samples of random variables from a subpopulations finite mix, of which the

estimation of average is given by: x¼

m X n 1 X xij mn j¼1 i¼1

(8)


In phase II, the n sample size is taken without category or subpopulation distinction and the sample mean is calculated by X i from {Xi,1, Xi,2, …, Xi,n}. Making the development, we have the sampling probability within the control limits in the following way: " ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X p ffiffiffi p ffiffiffi w 1 2 ^ d pX pU CL ¼ p1 | pffiffiffiffi þd1 n þK 1 þ d t n P LCL X l l l m pffiffiffi w | pffiffiffiffi þd1 nK m "

w þp2 | pffiffiffiffi þK m w | pffiffiffiffiK m

!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 pffiffiffi 1þ d t n l l l

! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 d ts 1þ l l l

!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 1þ d ts l l l

"

pffiffiffi w þp3 | pffiffiffiffi þd3 n þK m pffiffiffi w | pffiffiffiffi þd3 nK m

! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 1þ d ts l l l

!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2 d ts 1þ l l l

The number of involved parameters, in this case, when the averages are known, is ω ¼ {π1, π2, π3, K, δ, t, m, n} and the expected average number of samples until the occurrence of a point outside the control limits is given by: Z 1 1 h ijðwÞdw (9) AARLðp1 ; p2 ; p3 ; K; d1 ; d2 ; d3 ; t; m; nÞ ¼ d ^ 1 1P LCL pX p U CL X where φ is the probability density function of the standard normal distribution. The expansion for lW3 can be made in the same way. In terms of mixture probability function, the error type I depends on the control limits, so we can rewrite the Expression (9) as follows:

In this equation, the problem is to find the traditional value of constant K for error type I, that is α ¼ 0.0027.


343

IJQRM 35,2

344

3. Case study 3.1 Process description and control plan As mentioned, for implementing the proposed improvement, a company process in the food sector was selected that fabricates wheat-based dough for a range of products. The example shown is similar to the processes of metalworking industries, a process in which product can pass through different production lines, with different parameters and probability distributions. The general feature of the process is illustrated in Figure 4. The manufacture starts with the preparation of the dough, placing the wheat in the trough base for homogenization. Then the mass is removed in blocks, placed in a cylinder mold, and processed to obtain a uniform thickness. After that, it is bent, placed back in the cylinder, and cut into discs coming out in horizontal rows arranged in five positions.


3.2 Graphical identification of the mixture and the weight of probability distributions The first step is to identify the subpopulation mixture that is part of the process shown in Figure 1. The aims are to verify if the randomly collected data about quality characteristic based on the weight of the products with 4,000 items and expanded using Bootstrap of the 40 items collected randomly with a 30-minute interval are from a mixture of probability distributions. Therefore, it is possible to identify the amount of these mixtures (l mixtures), calculate the probability of these distributions and estimate the parameters for each of the k probability functions. One of the techniques for achieving these aims is by graphic based on the density of probability function. Figure 1 shows at least three probability distributions. Figure 5 shows the normal graph of probability distributions for the database in Figure 1, which indicates two clear inflection points at the ends of the standard curve and two or more probability distributions in the center of the intersection. Figure 6 shows the same information but there are deviations of the score in the normal distributions for each value in the real scale. If there is a pure mixture without subpopulations, then the normal patterns of values will be expected to fluctuate around zero. The challenge is to find how many populations are in the database.

Prepare pastel dough

Pastel dough process

Fill dough shaping machine Knead the dough (6 cylinders) Knead the final dough (1 cylinder)

Figure 4. Scheme of the production of pastry dough line process

Apply separator plastic, chop dough and plastic and stack Separate plastic from leftover dough

No

Approved?

Leftover dough

Yes

Group in bowls, pack and box


Normal P-Plot: Kg padrào 4

Expected Normal Value

3 2 1

345

0 –1 –2 –3 –4 480

500

520

540

560

580

600

620

Figure 5. Plotting the normal graph of probability distributions for the database of Figure 1

Detrended Normal P-Plot: Kg padrào 2.0 1.5 Deviation from Expected Value


Value

1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 480

500

520

540

560

580

600

620

Value

As shown in Figure 7, the mass data were divided into four subpopulations with density functions of Gaussian probability because the χ2 test indicated no statistical evidence to reject the hypothesis of non-normality for each of the probability distributions. When the subpopulations are identified, it is also necessary to identify their parameters. In this case, as the distribution of the normal will be adjusted, the µ, σ and the weights of πj will be estimated. Table I shows the estimated parameters and it can be observed that the weights are approximately equal to each subpopulation. Observe that (σt)/(σ) ≅ 6.84919 and the Table II shows that σt ¼ 25.6965 and after some calculus, we obtain σ ¼ 3.7559, where σ is the subpopulation standard deviation (we supposed that σ ¼ σ1 ¼ σ2 ¼ σ3 ¼ σ4). 3.3 Determination control limits In this case, there are l ¼ 4 subpopulations whose probability density functions are adhered to normal and the parameters are estimated from the data of Figure 1, showed in Tables I and II. Then, it can be written in the mathematical function of AARL in which parameters involved are ω ¼ {π1, π2, π3, π4, K, δ1, δ2, δ3, δ4, t, m, n}, but as π ≅ π1 ≅ π2 ≅ π4 will be

Figure 6. Expected deviations of the scores in the standard normal distribution for the database of Figure 1

IJQRM 35,2

346

Normal P-Plot: P2 B 4

3

3

2

2



Normal P-Plot: P1 B 4

1 0 –1 –2

1 0 –1 –2

–3

–3

–4 494 496 498 500 502 504 506 508 510 512 514 516 518 520 522 524

–4 534

536

538

540

4

3

3

2

2

1 0 –1

–3

570

575

580

585

590

595

600

–4 549

605

550

551

552

Value

Table I. Estimated parameters of subpopulations

Subpopulations Table II. Estimated parameters of π and δ

1 2 3 4

1,000 1,002 998 1,001 4,000

508.7 543.5 552.5 579.9 546.2

555

556

557

552

553

554

558

Value

Sub population Valid N Mean Median 1 2 3 4 Population

550

–1 –2

565

548

0

–3

560

546

1

–2

–4 555

544

Normal P-Plot: P3 Kg

4


Expected normal value


Normal P-Plot: P4

Figure 7. Clockwise from left to right, plotting normal scores from position 1 to 4, considering four subpopulations

542 Value

Value

508.7 543.7 552.5 579.9 549.7

Mode

Minimum Maximum

508.1 544.6 Multiple Multiple Multiple

496.3 534.7 549.8 558.6 496.3

521.6 549.7 556.8 598.9 598.9

SD 3.7319 2.9900 1.2245 5.6569 25.6965

Skewness Kurtosis 0.0824 −0.1900 0.1583 −0.0395 −0.1815

0.0826 −0.4133 −0.2983 0.2906 −0.9390

πj

Δ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 1 þ ð1=l Þ l dl

0.2500 0.2505 0.2495 0.2503

−9.9771 −0.6535 1.6908 8.9898

6.84919

considered equal weight for each probability density function (PDF). Thus, the expected average number of samples until the occurrence of a point outside the control limits is given as (when t ¼ 0, the process is in control): Z EðARLÞ ¼ AARLðp; K; d1 ; d2 ; d3 ; d4 ; t; m; nÞ ¼

1

1

1 h ijðwÞdw: d ^ 1P LCL X rX r U CL

where: "

! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi w 1X 2 pffiffiffi d ^ p ffiffiffiffi þd1 n þK 1 þ P LCL pX pU CL ¼ p1 | d t n l i l m !# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi w 1X 2 pffiffiffi | pffiffiffiffi þd1 nK 1 þ d t n l i l m " ! !# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi w 1X 2 w 1X 2 þp2 | pffiffiffiffi þd2 n þK 1 þ d ts | pffiffiffiffi þd2 nK 1 þ d ts l i l l i l m m


"

! !# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi w 1X 2 w pffiffiffi 1X 2 d ts | pffiffiffiffi þ nK 1 þ d ts þp3 | pffiffiffiffi þd3 n þK 1 þ l i l l i l m m " ! !# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi w 1X 2 w 1X 2 d ts | pffiffiffiffi þd4 nK 1 þ d ts þp4 | pffiffiffiffi þd4 n þK 1 þ l i l l i l m m To assist in defining the control chart (size and amount of samples), we provide instructions when the production has several flows and the data belong to the mixture of probability distributions. For this case, μ1≠μ2≠ … ≠μl, l ¼ {1, 2, …, i}, we can obtain the control limits by: sT d ¼ X K p ffiffiffi LCL n sT Ud CL ¼ X þK pffiffiffi n Since this is a relationship between global standard deviation and internal standard deviation of each subpopulation, (σt)/(σ) ¼ τ, we have proposed to recalculate control limits as following: ts d ¼ X K p ffiffiffi LCL n ts Ud CL ¼ X þK pffiffiffi n The proposed procedure has similar steps of classic Shewhart X control chart: in the phase I, m sampling of size n random of specified subpopulation is taken, and estimate internal standard deviation of Xis estimated and in phase II, randomly, a sample of size n of the a specified subpopulation is taken and X is computed. We used the model developed in Section 2 to obtain the results shown in Table III, where we can find UCL and LCL by using the Maple program. Some results are shown in Table III for different values of K. The K value we are using to calculate the control limits. For the procedure shown, assuming a type 1 error near of 0.0027 (α ¼ 1/370.4), we have proposed using K ¼ 3.585 (see Table III the value of 377.2 for ARL). For analysis of chart performance using AARL, we change the t value for t ¼ [0; 0.25; −0.25; 0.5; −0.5;1.0; −1.0; 2.0; −2.0; 3.0; −3.0] with the same value of K as shown in Table III. This result is shown in Table IV.


347


IJQRM 35,2

In Table IV, different AARLs are found if the deviation in the mean occurs for t positive or negative. Minor AARL occurs for small positive deviation in the mean (t positive); on the other hand, for negative t, we found larger values of AARL. This is a special result finding for mixture probability density function. Figure 8 shows the positive deviation in the mean in the OOC process.

348

4. Comparative analysis of mixture chart with traditional control chart for unique stream A traditional control chart for multiple streams is a group of control charts (Montgomery, 2009). The approach is shown in Montgomery (2009) that has the supposition that each stream has same μ and σ with a normal probability distribution. This approach has been proposed for sampling same size for each stream and plot of maximum and minimum mean after calculated X-bar and standard deviation or amplitude. However, there is a supposition that the normal probability distribution has the same mean and standard deviation for each stream. In our case, this is not possible because each stream has a different mean and standard deviation at the same time. We conducted the performance analysis using one chart for each stream (we used four charts which are very expensive) with the mixture approach.

Table III. Estimates of parameters AARL0 and K generated by Maple

K

AARL0

K

AARL0

3.505 3.510 3.515 3.520 3.525 3.530 3.535 3.540 3.545

99.13 106.9 115.4 124.7 134.9 146.1 158.4 172.0 186.8

3.550 3.555 3.560 3.565 3.570 3.575 3.580 3.585 3.590

203.2 221.3 241.2 263.2 287.5 314.4 344.2 377.2 413.9

K

Table IV. AARL values for different values of t (when t ≠ 0, out of control – OOC)

3.505 3.510 3.515 3.520 3.525 3.530 3.535 3.540 3.545 3.550 3.555 3.560 3.565 3.570 3.575 3.580 3.585 3.590

0

0.25

−0.25

0.5

T −0.5

1.0

−1.0

2.0

−2.0

99.13 106.9 115.4 124.7 134.9 146.1 158.4 172.0 186.8 203.2 221.3 241.2 263.2 287.5 314.4 344.2 377.2 413.9

33.69 35.73 37.92 40.28 42.84 45.61 48.60 51.84 55.36 59.17 63.32 67.82 72.72 78.06 83.88 90.22 97.15 104.70

349.1 383.4 421.6 464.1 511.4 564.2 623.0 688.8 762.3 844.6 936.8 1040.0 1156.0 1287.0 1434.0 1599.0 1785.0 1995.0

14.87 15.52 16.22 16.96 17.75 18.60 19.50 20.47 21.51 22.62 23.81 25.09 26.46 27.94 29.53 31.23 33.07 35.05

686.7 762.1 846.7 941.7 1048.0 1169.0 1304.0 1457.0 1629.0 1823.0 2043.0 2292.0 2574.0 2894.0 3258.0 3671.0 4140.0 4676.0

5.759 5.862 5.970 6.083 6.203 6.329 6.462 6.602 6.750 6.906 7.071 7.245 7.428 7.622 7.827 8.043 8.271 8.513

93.26 100.50 108.40 117.00 126.50 136.90 148.30 160.80 174.50 189.60 206.30 224.70 244.90 267.30 292.10 319.40 349.80 383.40

4.013 4.014 4.016 4.018 4.019 4.021 4.024 4.026 4.029 4.031 4.034 4.038 4.041 4.045 4.050 4.054 4.059 4.064

5.679 5.777 5.880 5.989 6.103 6.224 6.351 6.486 6.627 6.776 6.934 7.100 7.276 7.461 7.656 7.863 8.081 8.312


4.50 4.00 3.50

ARL

3.00 2.50 2.00

349

1.50 1.00 0.50 0.00

5

7

9

11

13

15

17

19

Size sampling


Simulation

Numerical method

Therefore, in this section, we compared the analysis of performance of mixture control chart with traditional approach multiple streams for a generic situation. The proposed mixture distribution in this paper converges for a traditional control chart of Shewhart for X-bar, for case µ unknown, when the parameter δ1 ¼ δ2 ¼ … ¼ δl ¼ 0; this way, the probability of one sampling stays in between the control limits as follows:

pffiffiffi pffiffiffi w w ^ d pX p U CL p ffiffiffiffi p ffiffiffiffi ¼ p | P LCL þKt n | Kt n 1 X m m

pffiffiffi pffiffiffi w w þp2 | pffiffiffiffi þKt n | pffiffiffiffiKt n þ. . . m m

p ffiffiffi pffiffiffi w w þpl | pffiffiffiffi þKt n | pffiffiffiffiKt n m m where l represents the number of streams. In sequence, if the streams have also the same d pX p U CLÞ. ^ μ and σ with a normal probability distribution, then PðLCL So, we have X proposed a model that can be used in a generic situation for a number of different streams, wherein after improvement action, the streams reduce to 1. We observed that the term used qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 P 2 in the model, K 1 þð1=lÞ l dl ; expand K in the magnitude of ð1=lÞ l dl ; on the other hand, K is expanded by δ. We have conducted the same analysis with the objective of understanding the effects of the δ in the performance of a mixture control chart. Remember that we have used ANOVA (variance analysis) to build the mixture control model. Table V shows AARL for sets of δ considering that X~N(0, 1). A special case is for δi ¼ 0, that is a traditional X-bar control chart. It is easy to understand the AARL shown in Table V, for restriction: l X

di ¼ 0:

i

As observed in Table V, for K ¼ 3.0 and K ¼ 3.585, the unique stream has AARL of 319.7 and 2,496.8, respectively. When there are more than one streams, minor values of AARL are found in the control situation, t ¼ 0. For this, in the mixture control chart, we have proposed KW3.00 for AARL greater than 370.4. Figure 9 shows the curve between AARL for a mixture control chart for different t values, i.e., in out of control state. We can observe that the curve for the mixture control chart is not similar to the traditional control chart;

Figure 8. Performance of mixture control chart for OOC t ¼ 2, m ¼ 25 and varying size sample

IJQRM 35,2

Table V. AARL performance for a different set of δ for K ¼ 3.00 and K ¼ 3.59

0 −0.5 0.5 0 0 −1 1 0 0 0.5 −0.5 −0.25 1 1 0

2

3

4

K ¼ 3.0

0 0.5 −0.5 0 0 0.5 −0.5 −1 1 0 0 0.25 −0.5 −0.5 −1.25

0 0 0 −0.5 0.5 0.5 −0.5 0.5 −0.5 −1 1 0 −0.25 −0.5 1.25

0 0 0 0.5 −0.5 0 0 0.5 −0.5 0.5 −0.5 0 −0.25 0 0

319,7 87.1 87.1 87.1 87.1 33.7 33.8 33.7 33.8 33.7 33.8 198.5 33.3 33.7 16.9

AARL

K ¼ 3.59 2,496.8 473.2 473.2 473.2 473.2 155.6 155.7 155.6 155.7 155.6 155.7 1314 146.2 155.6 80.4

400

300

AARL


350

Δ 1

Figure 9. Comparative analysis between mixture control chart for specific set [−0.5, 0.5, 0.0] (K ¼ 3.59) and traditional X-bar

200

100

0

1

X-Bar

2

3

Mixture

the advantage is better performance in IC state of mixture chart than traditional X-bar chart for a unique stream. We can see that for tW1.0 the power that the use of the mixture control chart has about the same power in detection special cause than the traditional X-bar chart. We can conclude that the proposed mixture control chart could have the same capacity in detecting a special cause as the use of X-bar chart for each stream for more large τ (deviation in global mean). The proposed method using a mixture probability distribution has proposed to use a unique chart for monitoring multiple streams. The model is a simple extension of the traditional control chart for unique streams. 5. Conclusion This study contributes to industry and researchers who are working in cases where there are different machines and products production flows and it presents an alternative solution


to Shewhart charts. The purpose of this study was to analyze the construction of graphs of SPC with a mixture of probability distribution application in the MSPs. The analyzed case showed an application of control charts with a finite mixture of probability distributions in a process of a company in the food sector which products consist of wheat-based dough. It was observed that the proposed graphic is more complex because it involves more parameters to be considered. Nevertheless, it can help the practitioners to monitor on a single graph multiple streams of the production process, which explains the development and implementation of this study. A mathematical model was developed for a MSP considering a finite distribution mixture, for the known and unknown mean. New control chart limits were also proposed. It provides a new approach for monitoring multi-streams when the production has several flows and the data belong to the mixture of probability distributions, with multiple subpopulations with different means. For the analysis of chart performance, we used AARL. A special use of mixture control chart is when the customers receive product batches continuously from a supplier and they need to monitor statistically the quality of the product with multiple streams. In this situation, we have observed that different AARL are found if the deviation in the mean occurs for positive or negative deviation of the mean. Minor AARL occur for small negative deviation in the mean; on the other hand, for larger values of t, minor AARL occur for positive deviation in the global process mean. This is a special result finding for a mixture probability density function. This result shows that, in the same situation, the control chart is more sensitive for negative or positive deviation in the mean. Nevertheless, as the research limitation, it is applicable in a single case requiring checking in other cases with similar issues. The proposed model of control is complex but it can find similar types I and II errors when compared with the analysis of other designs of a control chart. Therefore, more studies are necessary to compare the performance of this method to others control chart design for multiple streams. Furthermore, future research should study the same problem considering the performance of a control chart for different size samples.

References Abu‐Shawiesh, M.O.A. (2009), “A control chart based on robust estimators for monitoring the process mean of a quality characteristic”, International Journal of Quality & Reliability Management, Vol. 26 No. 5, pp. 480-496. Adams, C.P. (2016), “Finite mixture models with one exclusion restriction”, Econometrics Journal, Vol. 19 No. 5, pp. 1-16. Adeoti, O., Olaomi, J. and Adekeye, K. (2016), “Control chart limits for monitoring process mean based on downton’s estimator”, Quality and Reliability Engineering, Vol. 32 No. 5, pp. 1731-1740. Ahangar, N.E. and Chimka, J.R. (2016), “Attribute control charts with optimal limits”, Quality and Reliability Engineering International, Vol. 32 No. 4, pp. 1381-1391. Aslam, M., Wu, C.W., Azam, M. and Jun, C.H. (2014), “Mixed acceptance sampling for product inspection using process capability index”, Quality Engineering, Vol. 26 No. 4, pp. 450-459. Automotive Industry Action Group (AIAG) (1991), Fundamental Statistical Process Control: Reference Manual, Automotive Industry Action Group, Southfield, MI. Bersimis, S., Psarakis, S. and Panaretos, J. (2007), “Control charts: an overview”, Quality and Reliability Engineering International, Vol. 23 No. 5, pp. 517-543. Castagliola, P., Celano, G. and Fichera, S. (2013), “Comparison of the X ̄ chart and the t chart when the parameters are estimated”, Quality Technology & Quantitative Management, Vol. 10 No. 1, pp. 1-16. Castagliola, P., Maravelakis, S., Psarakis, K. and Vännman, K. (2009), “Monitoring capability indices using run rules”, Journal of Quality in Maintenance Engineering, Vol. 15 No. 4, pp. 358-370.


351

IJQRM 35,2


352

Castillo, E. and Montgomery, D.C. (1994), “Short-run statistical process control: Q-chart enhancements and alternatives”, Quality and Reliability Engineering International, Vol. 10 No. 2, pp. 87-97. Chauveau, D. and Hoang, V.T. (2016), “Nonparametric mixture models with conditionally independent multivariate component densities”, Computational Statistics & Data Analysis, Vol. 103 No. 1, pp. 1-6. Diebolt, J. and Robert, C.P. (1994), “Estimation of finite mixture distributions through Bayesian sampling”, Journal of the Royal Statistical Society, Vol. 56 No. 2, pp. 363-375. Dixon, P. (2012), “Assessing the evidence for response time mixture distributions”, Behavior Research Methods, Vol. 44 No. 3, pp. 706-724. Epprecht, E.K. and Simões, B.F.T. (2013), “Statistical Control of Multiple-Steam Processes-A Literature Review”, paper presented at the 11th International Workshop on Intelligent Statistical Quality Control, Sydney, August 20-23, available at: http://wp.csiro.au/isqc2013/abstracts-bios/# abstracts (accessed January 5, 2016). Epprecht, E.K., Loureiro, L.D. and Chakraborti, S. (2015), “Effect of the amount of Phase I data on the phase II performance of S² and S control charts”, Journal of Quality Technology, Vol. 47 No. 2, pp. 139-155. Everitt, B.S. and Hand, D.J. (1981), Finite Mixture Distributions, Springer, Dordrecht. Franco, B.C., Celano, G., Castagliola, P. and Costa, A.F.B. (2014), “Economic design of Shewhart control charts for monitoring autocorrelated data with skip sampling strategies”, International Journal of Production Economics, Vol. 151 No. 1, pp. 121-130. Hall, P. and Zhou, X.-H. (2003), “Nonparametric estimation of component distributions in a multivariate mixture”, The Annals of Statistics, Vol. 31 No. 1, pp. 201-224. Jensen, W.A., Jones-Farmer, L.A., Cham, P.C.W. and Woodall, W.H. (2006), “Effects of parameter estimation on control chart properties: a literature review”, Journal of Quality Technology, Vol. 38 No. 4, pp. 349-364. Jirasettapong, P. and Rojanarowan, N. (2011), “A guideline to select control charts for multiple stream processes control”, Engineering Journal, Vol. 15 No. 3, pp. 1-14. Kaffel, R.H. and Prigent, J.R. (2016), “Optimal positioning in financial derivatives under mixture distributions”, Economic Modelling, Vol. 52, Part A, pp. 115-124. Kayid, M. and Izadkhah, S. (2015), “A new extended mixture model residual life time distributions”, Operations Research Letters, Vol. 43 No. 2, pp. 183-1. Kim, S.M., Do, T.T., Oechtering, T. and Peters, G. (2015), “On the entropy computation of large complex Gaussian mixture distributions”, IEEE Transactions on Signal Processing, Vol. 63 No. 17, pp. 4710-4723. Lee, M., Wang, L., Shen, H., Hall, P., Guo, G. and Marron, J.S. (2014), “Least squares sieve estimation of mixture distributions with boundary effects”, Journal of the Korean Statistical Society, Vol. 44 No. 2, pp. 187-201. Masmoudi, A., Masmoudi, A. and Puech, W. (2016), “A new semiparametric finite mixture model-based adaptive arithmetic coding for lossless image compression”, Circuits Systems Signal Processing, Vol. 35 No. 4, pp. 1163-1186. Mei, Y. (2008), “Is average run length to false alarm always an information criterion?”, Sequential Analysis, Vol. 27 No. 4, pp. 351-376. Montgomer, D.C. (2009), “Introduction to statistical quality control”, John Wiley & Sons, New York, NY. Mortell, R.R. and Runger, G.C. (1995), “Statistical process control for multiple stream processes”, Journal of Quality Technology, Vol. 27 No. 1, pp. 1-12. Phaladiganon, P., Kim, S.B., Chen, V.C. and Jiang, W. (2013), “Principal component analysis-based control charts for multivariate nonnormal distributions”, Expert Systems Applications, Vol. 40 No. 8, pp. 3044-3054. Saleh, N.A., Mahmoud, M.A., Keefe, M.J. and Woodall, W.H. (2015), “The difficult in designing Shewhart xb and X control charts with estimated parameters”, Journal of Quality Technology, Vol. 47 No. 2, pp. 127-138.


Titterington, D.M., Smith, A.F.M. and Makov, U.E. (1985), Statistical Analysis of Finite Mixture Distributions, Wiley, Universidade da Califórnia, New York, NY. Teel, C., Park, T. and Sampson, A.R. (2015), “Communications in statistics”, Simulation and Computation, Vol. 44 No. 6, pp. 1545-1556. Variyath, A.M. and Vattathoo, J. (2014), “Robust control charts for monitoring process variability in Phase I multivariate individual observations”, Quality and Reliability Engineering International, Vol. 30 No. 6, pp. 795-812. West, M. (1992), “Modelling with mixtures”, in Bernard, J.M., Dawid, A.P. and Smith, A.F.M. (Eds), Bayesian Statistics, Oxford University Press, Oxford, pp. 503-524. Woodall, W.H. (1985), “The statistical design of quality control charts”, Journal of the Royal Statistical Society, Vol. 34 No. 2, pp. 155-160. Woodall, W.H. and Montgomery, D.C. (2014), “Some current directions in the theory and application of statistical process monitoring”, Journal of Quality Technology, Vol. 46 No. 1, pp. 78-95. Yantis, S., Meyer, D.E. and Smith, J.E. (1991), “Analyses of multinomial mixture distributions: new tests for stochastic models of cognition and action”, Psychological Bulletin, Vol. 110 No. 2, pp. 350-374. Yu, Y. (2011), “On normal variance-mean mixtures”, technical report, University of California, Irvine, CA. Further reading Flott, L.W. (2012), “Introduction to control charts”, Metal Finishing, Vol. 110 No. 6, pp. 36-38.

Corresponding author Damaris Serigatto Vicentin can be contacted at: [email protected]

For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: [email protected]


353