Quality control is a system to maintain quality of product or service to achieve specification standard of product. One of the most powerful tools is through graphical method which is control chart. This is because control chart easy to analyse the data and able to provide comprehensive information on existing product or process characteristics. There are two types of control chart. First, statistical process control (SPC) and second is multivariate statistical process control (MSPC). Statistical process control commonly referred as SPC, was developed by Dr. Walter A. Shewhart in the mid-1920s. In general, statistical process control is to control and monitor the process of production line and detect abnormal process. However , the Shewhart control chart can only monitor single process variable at a time. Multivariate statistical process control , MSPC was established by Hotelling in his 1947 pioneering paper. MSPC can simultaneously control and monitor more than one process variables at a time. The three most popular multivariate control statistics of multivariate control charts, such as Shewhart charts (x and Range charts) ,cumulative sum plots (CUSUM), and exponentially weighted moving average charts (EMWA)

Mohamad, Ismail

Sutirman, Zetty Azrah

Universiti Teknologi Malaysia Institutional Repository

Prosiding PSM2014/2015  295   UNIVARIATE AND MULTIVARIATE CONTROL CHARTS FOR MONITORING SUGAR PRODUCTION PROCESS  Zetty Azrah bt Sutirman, PM. Dr. Ismail Mohamad  1.1 INTRODUCTION   Quality control is a system to maintain quality of product or service to achieve specification standard of product. One of the most powerful tools is through graphical method which is control chart. This is because control chart easy to analyse the data and able to provide comprehensive information on existing product or process characteristics. There are two types of control chart. First, statistical process control (SPC)  and second is multivariate statistical process control (MSPC).  Statistical process control commonly referred as SPC, was developed by Dr. Walter A. Shewhart in the mid-1920s. In general, statistical process control is to control and monitor the process of production line and detect abnormal process. However , the Shewhart control chart can only monitor single process variable at a time. Multivariate statistical process control , MSPC was established by Hotelling in his 1947 pioneering paper. MSPC can simultaneously control and monitor more than one process variables at a time. The three most popular multivariate control statistics of multivariate control charts, such as Shewhart charts (x  and Range charts) ,cumulative sum plots (CUSUM), and exponentially weighted moving average charts (EMWA) .   1.2 PROBLEM STATEMENT  Nowadays, statistical process control techniques, SPC are widely used in industry. However the characteristics of univariate quality control chart or Shewhart control chart itself , that can only monitor single process variable at a time are inadequate to control the process stability . This problem may affect the process variables and quality of the product. Usually, in industry, there are many situations in which the simultaneous monitoring or control in two or more related quality process characteristics is necessary. Monitoring these process variables independently can mislead the true process situation. MSPC charts overcomes this situation by considering the correlation between the variables and are able to analyze the stability of the process. In this research, we investigate method of multivariate controls charts and univariate control charts to identify a significant for monitoring and controlling the  process. 1.3 LITERATURE REVIEW   Prosiding PSM2014/2015  296  1.3.1 Statistical Process Control (SPC)   SPC was pioneered by Walter A. Shewhart at Bell Laboratories in the early 1920s. Shewhart developed the control chart in 1924 and the concept of a state of statistical control. Statistical uses statistical methods. SPC is applied in order to monitor and control a process. Monitoring and controlling the process ensures that it operates at its full potential.The goal of SPC is to achieve higher quality of final product by elimintation of variability in the process while the main objective of SPC is to quickly detect the occurrence of assignable causes of abnormal process so that further investigation to the process and corrective measurements can be carried out. There are many ways to implement process control. The basic quality control tools include histogram , check sheet , pareto chart , cause and effect diagram , defect concentration diagram , scatter diagram and quality control charts.  1.3.2 Multivariate Statistical Process Control (MSPC)  Nowadays, in industry, there are many situations in which the simultaneous monitoring or control of two or more related quality–process characteristics is necessary. Monitoring these quality characteristics independently can be very misleading. Process monitoring of problems in which several related variables are of interest are collectively known as multivariate statistical process control. The most useful tool of multivariate statistical process control is the quality control chart.   1.4 OBJECTIVE    The objectives of this study are :  (i) To compare the ability of univariate control chart and multivariate chart using Hotelling’s T2 statistics in detecting out of control points  (ii)  To construct the multivariate control chart using the HotellingT2 statistics.   (iii)  To identify significant method for monitoring multivariate process variables by    using quality control charts.   Prosiding PSM2014/2015  297  1.5 METHODOLOGY   This chapter discuss about the methodology of the research . We will discuss the basic knowledge of control charts and the distributions of all the observation data. It concerns  on univariate control chart and multivariate control chart by using HotellingT2 for monitoring the quality of an industrial production process and detecting out of control points. The purpose is to determine whether this charts identified the same points are the out of control points.  1.5.1 Anderson- Darling Normality Test  All the observations should be tested wheter the data is associated with the normal distribution. The normality use in this study is Anderson Normality Test (AD). The test rejects the hypothesis of normality when the p-value is less than to 0.05 or confidence interval the null hypothesis is likely to be false.  Failing the normality test allows you to state with 95% confidence the data does not fit the normal distribution.  Passing the normality test only allows you to state no significant departure from normality was found. The Anderson- Darling test is defined as:      : The data follow a normal distribution     : The data do not follow the normal distribution   The AD test statistic is defined as      = - N – S  Where,   S=                                   ))]   F is the cumulative distribution function and    is the ordered data.   1.5.2 HotellingT2 statistics   Prosiding PSM2014/2015  298  Harold Hotelling was introduced Hoteling’s T2distribution which is a multivariate analogue of the univariate Student’s tdistribution . Harold Hotelling became the first person to discover the problem in analysing correlated variables in bombsight data from the perspective of the statistical control . He has controlled the process by using charting statistics and then the statistic charting was known as HotellingT2.             Suppose a random sample of size ,n from normal distribution distribution with mean , µ variance σ2 selected. Then,      ̅                                           (1.5.2.1)  This test statistic has a student t distribution with n -1 degrees of freedom. Where   ̅              is the simple mean,                 ̅       is the corresponding sample variance. The square of t statistic is given by                                                            (1.5.2.2)                                         (1.5.2.3)  Next, the Hotelling extended from the univariate statistic to multivariate statistic. When Equation 3.3 is generalized to p variables, T 2  followsF distribution as:                                                                   (1.5.2.4)  Prosiding PSM2014/2015  299   Where n is the simple size,   is the vector size p,    and S are the sample estimators of mean and covariance matrix are defined by:                   ] (1.5.2.5)                     (1.5.2.6)                                                         (1.5.3.7)   A control chart will be plotted based on the HotellingT2statictics from all the observation against number of observation of time and compared to the upper control limit.[6]                              α                                 (1.5.2.8)  1.5.3 Mason Young Tracy Decomposition             MYT decompositions includes orthogonal components . These components consist of a series of conditional and unconditional condition T2 terms . The general T2 statistics for a p-dimensional observation vector X’ = (         ) can be presented as                                                   Prosiding PSM2014/2015  300                                                                                                                             (1.5.3.1)                                                                                                            Suppose the vector      is partitioned as                                                                                                                                                            (1.5.3.2)  Where                      represents the (p-1)-dimensional variable vector excluding the pth variable xpand       represents the corresponding elements of the mean vector. Partitioning the matrix S so that                                                                                                                                            (1.5.3.3)  Where     is the (p -1) × (p-1) covariance matrix for the first (p-1) variables ,    is the variance of    and     is a (p-1)-dimensional vector containing the covariances between xp and the remaining (p-1) variables . The T2 statistics can be partitioned into two independent parts given by                            (1.5.3.4) where                                                                                                                                                                 (1.5.3.5)  and the second term  Prosiding PSM2014/2015  301                                                                     (1.5.3.6) where                                                       (1.5.3.7) and                                    (1.5.3.8)  While the conditional variance is given as                                                      (1.5.3.9)  Continuing to iterate and partition the MYT decomposition of a T2 statistic is given by                                     (1.5.3.10)  From the above equation , the     is the unconditional components of the T2 statistic. Meanwhile ,                            is the conditional components . The other ways and easier approach of computing the terms of the MYT decomposition is given by                                                               (1.5.3.11) Prosiding PSM2014/2015  302   Where      represents the appropriate subvector ,      is the corresponding subvector mean and     denotes the corresponding covariance submatrix obtained from the overall S matrix by deleting the unused rows and columns.  However , the     term can be computes as                   (1.5.3.12) Hence , the MYT decomposition can be computed as follows                                                                                                        ................... ................... ...................                                       Therefore, this technique will make the better confirmation whcih variables effect to the enlargement of limit in    statistic control chart that lead to the out of control points.  1.6 RESULT AND DISCUSSION  The case study of sugar production process will be conducted. The process measurements are made on five variables which is steam temperature (  ) , Sugar density (    , Cool temperature (    , Sugar length (    and Sugar weight (   . Prosiding PSM2014/2015  303   Table 1.6.1: Sugar production process Number of observation                1 850 53 170 21 29.2 2 847 53 170 21.2 29.2 3 848 53 170 21.5 29.2 4 850 54 170 22.1 29.1 5 849 54 170 21.7 29.2 6 850 54 170 21.8 29.1 7 848 53 170 21.3 29.2 8 848 53 170 21.5 29.1 9 848 54 170 22 29.1 10 850 54 170 21.9 29.1 11 848 53 171 21 29.1 12 848 53 171 20.4 29.1 13 847 53 170 19.8 29.1 14 849 53 170 21.4 29.1 15 846 54 170 21.8 29.1 16 844 53 171 21.3 29.1 17 843 53 171 20.7 29.1 18 842 53 170 21.5 29.1 19 841 53 170 21.5 29.1 20 842 53 171 21.3 29.1 Prosiding PSM2014/2015  304  21 843 53 171 21.8 29.1 22 841 54 168 22.2 29.1 23 850 54 170 22 29.2 24 846 54 170 21.8 29.2 25 843 54 170 16 29.2 26 846 52 170 21.5 29.2 27 845 53 170 20.8 29.2 28 847 53 170 21.1 29.2 29 843 53 170 21.4 29 30 844 53 171 20.8 28.9 31 845 53 170 21.4 28.9 32 843 54 170 21 28.9 33 842 53 170 20 28.9 34 845 53 171 18.9 29 35 842 52 171 20.1 29 36 844 53 171 16.9 29 37 844 52 171 19.8 29 38 844 52 171 21.5 29 39 844 53 170 21.3 29 40 844 53 170 21.3 29 41 844 53 170 21.4 29 42 850 53 170 21 29 43 846 53 170 21 29 Prosiding PSM2014/2015  305  44 844 53 169 17 29 45 843 52 170 20.9 29.1 46 844 53 170 21.5 29.1 47 845 54 170 18.3 29.2 48 843 53 170 20.6 29.2 49 842 53 170 21 29.2 50 845 53 170 21 29.2   1.6.1  Result of Anderson Darling test and Univariate Control Charts  Table 1.6.2  showed that the summary of sugar production process data. The figures are used to check the normality of each variables or quality characteristics based on histogram and normality is testing using Anderson Darling test. The distribution of the data may identify either fit the normal distribution if p – value  < α the data is not normal but if p – value ≥ α , the data is said to follow the normal distribution. Next, it showed that the summary of univariate control chart sugar production process. The figures are used to check the out of control points of five quality characteristics. Points that are outside the control region or control limits indicate that the process is out of statistical control.   Table 1.6.2: Result of Sugar Production Process Quality Characteristics/Variables Normality test Univariate Control Chart Steam temperature,    Not Normal In Control Sugar density,    Not Normal In Control Cool Temperature,    Not Normal Out of Control Sugar length,    Not Normal Out of Control Prosiding PSM2014/2015  306  Sugar weight,    Not Normal In Control         From the Table 1.6.2, all the variablesSteam temperature (X1), Sugar density (  ), Cool Temperature,     , Sugar length     , and Sugar weight (    ) showed they are come from not normal distribution. In univariate control charts, there are several observations from ), Cool Temperature,     , Sugar length      had detected out of control limits.  1.6.2 Result of Muitivariate Control Charts using HotellingT2 statistics  Table 1.6.3: Summarize T2 Statistical Control Charts No of  variables Variables The capability detecting out of control points Two    and       Capable Three               Capable                Capable                Capable Four                   Capable                    Capable                    Capable Five                       Capable  From the Table 1.6.2, we can see the movement of capability in multivariate control charts to detect out of control points. It observed that multivariate control charts by using Hotelling T2 statistics is capable to detect the out of control points for two, three, four and five variables. This shows that Hotelling T2 able to detect the out of control points for each variables and does not require MYT Decomposition method to detect the signals.  Prosiding PSM2014/2015  307    1.7 CONCLUSION  In this study, we discussed univariate control chart and multivariate control chart. In the beginning of this research, we manufactured product involve several quality characteristics or variables which when univariate control is used, the control chart must be constructed for each variable which is cumbersome process because in real production process is very complex it may involve several process variables. Univariate chart can only monitor a single process variable. The multivariate control charts using HotellingT2 statistics overcomes this problem by monitor all the process variables in a single control chart.  The univariate control charts are constructed with a ±3 standard deviation. The univariate control charts detect that           are out of control since there are points outside the control limits. The multivariate charts were constructed using HotellingT2 statistics by combination of different number of variables. Supposedly, from the provious research on the same topic found that the HotellingT2have some weaknessess to detect the out of control points during the number of varibales increase. The reason for this weaknessess because the control region for the T2 become larger as the control limit becomes greater to the enlargement of the value upper control limit. But from the results we gained from this reasearch, we found that HotellingT2 become capable to detect the             out of control points by meant multivariate control charts is good in detecting out of control points. However, it is known that there are some out of control points which shown by univariate control charts. Maybe the factor of sample size of data and the number of variables used in this research causes the results quite different from the previous reaseacrh. 

Univariate and multivariate control charts for monitoring sugar production process

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Univariate and multivariate control charts for monitoring sugar production process

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