Determining out-of-control variable(s) in a multivariate quality control chart for individual observations

Project Abstract

<p> </p><p>The decomposition of the Hotelling’s T2 statistic into orthogonal components is<br>considered to be one of the most effective methods for detecting variable(s) responsible for an<br>out-of-control signal. In this work, an extension of the T2 decomposition from three variables<br>(p=3) to four (p=4) variables, where the number of decompositions increased from 3! = 6 to 4! =<br>24 and the decomposition terms also increased from 18 to 96 terms having 32 distinct terms were<br>provided. These distinct terms are the ones that were examined for possible contribution to the T2<br>signal. A dataset obtained from an Indomie company in Northern Nigeria was used to assess the<br>validity of constructed model by demonstrating the invariance property of the Hotelling’s T2<br>statistic. The model was also used to identify the variable(s) that significantly contribute to an<br>out-of-control signal. By comparing the critical values with the corresponding T2 values, we<br>were able to detect variation between the four (4) variables in their mean and also their variancecovariance<br>structure.<br>vi</p><p>&nbsp;</p><p>&nbsp;</p> <br><p></p>

Project Overview

<p> GENERAL INTRODUCTION<br>1.0 INTRODUCTION<br>Statistical Process Control (SPC) has played a significant role in controlling the product<br>quality for decades since Shewhart (1931) illustrated the technique of the control charts by<br>applying statistical concepts in the manufacturing process.<br>According to MacCarthy and Wasusri (2002), statistical process control is a powerful<br>tool for monitoring and control processes and has been widely used in manufacturing and nonmanufacturing<br>processes.<br>With the advancement in technology, there has been an increase in customer expectations<br>and the need to monitor correlated variables simultaneously. Process monitoring in which<br>several variables are of interest is called Multivariate Statistical Process control (MSPC).<br>Multivariate control charts is widely used in practice to monitor the simultaneous<br>performance of several related quality characteristics. The origin of multivariate control chart<br>can be attributed to Hotelling (1947). A multivariate control scheme has a better sensitivity than<br>one based on the univariate control charts in monitoring multivariate quality process. (Lu et al.,<br>1998)<br>Woodall and Montgomery (1999) stated that multivariate process control is one of the<br>most rapidly developing sections of statistical process control. The demand to implement MSPC<br>in a production process for quality improvements increases daily. Statistical methods play a<br>very important role in quality improvement in manufacturing industries (Woodall, 2000).<br>The quality of any product is usually determined by several correlated quality variables.<br>One of the popular multivariate control charts is based on Hotelling’s T2 statistic which is used<br>2<br>to simultaneously monitor those quality variables and taking their correlations into<br>consideration. There are a lot of literatures focusing on multivariate control charting methods<br>based on Hotelling’s T2 statistic in detecting mean shift such as Sullivan and Woodall (1996),<br>Mason and Young (1999), Tong et al., (2005), and many others.<br>1.1 MOTIVATION OF THE STUDY<br>When Hotelling T2 detects a change in the mean vector, corrective action is required. A<br>T2 value, however, does not provide direct information about which variable is responsible for<br>the overall out-of-control condition. This information is of practical importance because quality<br>engineers/analysts need to know which variable requires adjustments after the process is declared<br>out-of-control. The most challenging issue about multivariate quality control chart is the ability<br>to identify the variable which is responsible for an out-of-control condition.<br>Many literatures have discussed and presented methods which can be used to identify<br>out-of-control variable or variables and much credit has been to the method proposed by Mason<br>et al.,(1995), for more information see the works of Bersimis et al.,(2007). This method involves<br>the decomposition of the T2 statistic into orthogonal components which reflects the contribution<br>of each variable in an observation vector. Much application of the decomposition technique has<br>been on two and three variables as seen in the case of Yarmohammadi and Ebrahimi (2010),<br>Ulen and Demir (2013), Sani and Abubakar (2013) and so many others.<br>Holmes and Mergen (1993), Sullivan and Woodall (1996), and Vargas (2003) have noted<br>that the Phase I Hotelling’s T2 control chart for individual observations is less sensitive in<br>detecting trend or process mean shifts. In this work we would apply both Phase I and Phase II of<br>the Hotelling’s T2 control chart and also provide the T2 decomposition model for p=4, using<br>3<br>Mason, Young, Tracy (MYT) decomposition technique which would be used for identifying outof-<br>control variable .<br>1.2 ASSUMPTIONS OF STATISTICAL PROCESS CONTROL (SPC)<br>The standard assumptions in SPC are that the observed process values are normally,<br>independently and identically distributed (iid) with fixed mean (μ) and standard deviation σ<br>when the process is in control. Before MSPC can be implemented, the p variables must be<br>related to each other.<br>Correlation analysis is a technique used to show the strength of the relationship between<br>pairs of variables. Del Castillo (2002) defined correlation as the departure of two or more<br>variables from independence. Montgomery (2001), defined correlation as a degree to which two<br>or more quantities are associated.<br>When two or more random variables are defined on a probability space, it is useful to<br>measure the relationship between the variables. A common measure of the relationship between<br>two variables is called covariance. The covariance between random variables X and Y, denoted<br>as COV(X, Y) or XY  is<br>XY  E[(X X )(Y Y )]<br>Covariance provides an idea of the strength of correlation. In the case of two variables X<br>and Y, the correlation is considered to be very strong if X is far from its mean and Y is also far<br>from its mean. Hence, the covariance between the two variables X and Y describes the variation<br>between the two variables.<br>In the multivariate case, the population covariance is represented in a matrix denoted as<br>Σ. The covariance matrix is also called the variance-covariance matrix. The variance-covariance<br>matrix is a symmetrical matrix that contains the covariance among a set of random variables. The<br>4<br>main diagonal elements of the matrix are the variances of the random variables, and the offdiagonal<br>elements are the covariance between the p variables (Neter et al.,1996).<br>The p ï‚´ p variance-covariance matrix, S is as follows;<br>S=<br>2<br>1 12 1<br>2<br>21 2 2<br>2<br>1 2<br>P<br>p<br>p p p<br>S S S<br>S S S<br>S S S<br> <br> <br> <br> <br> <br> <br><br><br>   <br><br>In a two-dimensional plot, the degree of correlation between the values on the axes is<br>quantified by the so-called correlation coefficient. The most common correlation coefficient is<br>the Moment Correlation, which is found by dividing the covariance of the two variables by the<br>product of their standard deviation. This correlation coefficient (r) is a measure of the degree of<br>linear relationship between two variables X and Y. The square of (r) is called the coefficient of<br>determination and denotes the portion of total variance explained by the regression model<br>(Walpole and Myers, 1993). The sample correlation coefficient is calculated by<br>( )( )<br>( 1)<br>i i<br>xy<br>x y<br>x x y y<br>r<br>n s s<br> <br><br><br><br>where x and y are the sample means, x S and y S are the sample standard deviations of i x and i y<br>respectively.<br>The correlation coefficient<br>cov( , ) xy<br>xy<br>x y x y<br>x y <br><br>   <br> <br>The correlation coefficient may take value between -1.0 and +1.0.<br>5<br>1.3 AIM AND OBJECTIVES OF THE STUDY<br>Much research has been done on multivariate process control for variable data in various<br>situations. Moreover, interpretation of out-of-control signals and how to identify the quality<br>characteristics contributing to out-of-control signals have also been discussed. However,<br>identifying influential variable(s) that contribute to out-of-control signal is still a difficult task<br>especially when the quality characteristics are beyond three.<br>The aim of this research work is to:<br>Determine variable(s) that significantly contributes to an out-of-control signal in a multivariate<br>quality control chart. This is achieved through the following objectives:<br>1. Application of Hotelling’s T2 components using MYT decomposition technique.<br>2. Identifying out-of-control condition and out-of-control variable(s)<br>3. Illustration of invariance property of the Hotelling’s T2 statistic from the derived<br>components.<br>1.4 SIGNIFICANCE OF THE STUDY<br>The significance of this study is geared toward detecting out-of-control variable(s) in a<br>multivariate quality control chart. In achieving this, we explored and compared various methods<br>and techniques use in checking out-of-control condition in multivariate control chart. This work<br>is also aimed at helping practitioners in the field of quality control to be able to determine<br>variables that causes out-of-control signal in a monitoring process.<br>1.5 TRADITIONAL STATISTICAL PROCESS<br>Control charts were developed in 1931 by Shewhart to be used for process monitoring.<br>Control charts are widely used for detecting assignable and chance causes of variation. Some<br>definitions of control charts are presented as follows.<br>6<br>According to Shewhart (1931), “the control chart may serve these purposes, first, it is<br>used to define the goal or standard for a process that management strives to attain and secondly,<br>it may be used as an instrument for attaining that goal and thirdly, it may serve as means to<br>judging whether the goal has been reached.” Control chart may also be viewed as a statistical<br>tool as defined by Duncan in 1956.<br>Feigenbaum (1983) defined control charts as “…a graphical comparison of the actual<br>product characteristics with limits reflecting the ability to produce as shown by past experience<br>on the product characteristics.”<br>Therefore, control chart is a graphical display used to monitor a process. It usually<br>consists of a horizontal centerline corresponding to the in-control value of the parameter that is<br>being monitored and the upper and lower control limits. Control limits are not determined<br>arbitrarily, nor are they related to specification limits but rather by statistical criteria. Sample<br>points that fall within the control limits are said to be in-control while those points that fall<br>beyond the control limits are said to be an out-of-control process.<br>The traditional statistical process control is generally referred to as the univariate control<br>charts. This is due to the fact that it considers only a variable for monitoring quality<br>characteristics.<br>1.5.1 Univariate Control Charts<br>One major setback of the Shewhart chart is that it regards only the last data point and<br>does not carry a memory of the previous data. As a result, small changes in the mean of a<br>random variable are less likely to be detected rapidly. Exponentially weighted moving average<br>(EWMA) chart improves upon the detection of small process shifts. Rapid detection of small<br>7<br>changes in the quality characteristic of interest and ease of computations through recursive<br>equations are some of the many good properties of EWMA chart that make it attractive.<br>EWMA chart was first introduced by Roberts (1959) to achieve faster detection of small<br>changes in the mean. The EWMA is a statistic for monitoring the process that averages the data<br>in a way that gives less weight to data as they are further removed in time. EWMA is defined as:<br>Zi  Xi (1 )Zi 1     with 0  1, 0 0 Z <br>It is used as the basis of a control chart. The procedure consists of plotting the EWMA statistic<br>i Z versus the sample number on a control chart with center line 0 CL  .<br>The upper control limit (UCL) is<br>2<br>0 [1 (1 ) ]<br>2<br>i<br>x UCL K<br><br>  <br><br>   <br><br>and lower control limits (LCL) is<br>2<br>0 [1 (1 ) ]<br>2<br>i<br>x LCL K<br><br>  <br><br>   <br><br>where<br>0  =mean<br>K=constant<br>X  =standard deviation<br> =smoothing parameter<br>The term[1(1)2i ]approaches unity as i get larger, so after several sampling intervals, the<br>control limits will approach the steady state values<br>0 x 2 UCL K<br><br> <br><br> <br><br>8<br>0 x 2 LCL K<br><br> <br><br> <br><br>The CUSUM (cumulative sum) chart is an effective way of monitoring small deviations<br>in the process mean when small deviations are of interest. The CUSUM chart, originally<br>developed by Page (1954), incorporates all information in the sequence of sample values and<br>plots the cumulative sums of the deviations from a target value using samples from prior<br>observations.<br>1.6 MULTIVARIATE STATISTICAL PROCESS CONTROL (MSPC)<br>According to Montgomery and Klatt (1972), a lot of attention has been given to the<br>design of control charts where only one quality characteristic is of interest. However, based on<br>the two authors industrial products and processes are characterized by more than one<br>measurable quality characteristic and their joint effect describes product quality.<br>Process monitoring in which several variables are of interest is called MSPC.<br>Multivariate charts are better than the simultaneous operation of several univariate control<br>charts. Process monitoring using control charts can be seen in two- stage process, Phase I and<br>Phase II (Woodall, 2000). Each phase has a role in monitoring the quality of a product. In Phase<br>I, charts are used for retrospectively testing whether the process was in control using historical<br>dataset. This Phase aids the practitioners in bringing a process to an in-control state. In Phase II,<br>the main concern is to further monitor the historical data set when subsequent samples are<br>drawn.<br>The parameter of the run-length distribution is often used for measuring the performance<br>of the control chart methods, where the run length is the number of samples taken before an outof-<br>control condition occurs.<br>9<br>Hotelling (1947) was the pioneer to develop a quality control chart for several related<br>variables and the control chart is well known as the Hotelling T2 control chart. The Hotelling T2<br>control chart is rated as the most widely used multivariate control chart that deals with changes<br>in the mean vector of p correlated quality characteristics (Aparisi and Haro, 2001). Hotelling’s T2<br>control chart is a direct analogue of the Shewhart X control chart.<br>The main tool used for monitoring MSPC is through the use of the quality control chart.<br>MSPC procedure involves fulfilling four conditions:<br>1. One should be able to state if the process is in control or not.<br>2. Should be able to know if there was/is a false signal.<br>3. Should be able to know the relationship among variables, attributes taken into<br>consideration.<br>4. If the process is out of control, one should know the reasons why it is out of control<br>(Bersimis et al., 2007).<br>1.6.1 Advantages of MSPC<br>MSPC has several advantages as compared to its univariate equivalent. As considered by<br>Hotelling (1947), Alt (1985), and Lowry and Montgomery (1995).<br>According to Hotelling (1947), MSPC has the ability to combine measures in several<br>dimensions into a single measure of performance. In addition, MSPC offers an easier graphical<br>tool to the practitioner. The practitioner can only use one chart instead of multiple univariate<br>charts to evaluate the product or system quality as a whole rather than the sum of many<br>individual parts (Hotelling, 1947 and Montgomery, 2001).<br>Montgomery (2001) gave a demonstration that multivariate control charts will produce an<br>acceptable type I error or in-control run length while maintaining the original data of means and<br>10<br>variances and correlations. Multivariate statistics also consider the relationship between the<br>variables since the variance-covariance matrix is part of the computations (Hotelling, 1947). And<br>hence, multivariate charts can detect changes in the relationships among variables being<br>monitored, which would not be noticeable from separate univariate chart (Lowry and<br>Montgomery, 1995).<br>Also, MSPC provides the appropriate control region for the application. If the assumption<br>of independence does not hold, then the assumed performance of the traditional Shewhart<br>approaches can be misleading. The MSPC can guarantee error protection from a variety of<br>different types of shifts in the process. Another advantage of the MSPC is that it moves away<br>from the application of run rules (Sullivan and Woodall, 1996).<br>1.6.2 Disadvantages of MSPC<br>Much satisfying evidence has been presented concerning the benefits of applying the<br>MSPC, the following limitations were noticed.<br>According to Mason et al., (1997a), Ryan (2000) and Montgomery (2001), multivariate<br>control charting procedures are computationally intensive. And hence, it works well when the<br>number of variables is not too large, that is when p 10. As the number of variables grows,<br>multivariate control chart lose its efficiency with regard to process shift detection. Also, a<br>multivariate control chart procedure does not directly provide the information an operator need<br>when the control chart signals an out-of-control condition. It doesn’t give information on which<br>variable or set of variables is out-of-control (Hawkins, 1991).<br>1.7 APPLICATION OF MULTIVARIATE QUALITY CONTROL<br>Control charts are originally developed for individual processes and have been applied<br>within a number of areas, including:<br>11<br>1. Hospital infection control (Sellick,1993)<br>2. Prediction of business failures (Theodossiou, 1993)<br>3. Monitoring the impact of human disturbance of ecological systems (Anderson and<br>Thompson, 2004)<br>4. Quality Management of higher education (Mergen et al.,2000)<br>5. Corroborating bribery ( Charnes and Gitlow, 1995)<br>6. Improving athletic performance (Clark and Clark, 1997)<br>7. Improving the quality of Pharmaceutical products (Ulen and Demir, 2013)<br>12 <br></p>