HOW TO CONTROL PROCESS VARIABILITY MORE EFFECTIVELY : THE CASE OF A B-COMPLEX VITAMIN PRODUCTION PROCESS

This paper presents a new procedure for controlling process variability where (i) the quality of the process is determined by more than one quality characteristic, and (ii) the correlations among those quality characteristics must be taken into consideration. The ability of this procedure to detect an abrupt shift of covariance structure is an improvement on the standard practice based on generalised variance. An experience of the production process of the B-complex vitamin is reported to illustrate the merit of the proposed procedure.


INTRODUCTION
'Reduce process variability' is the basic philosophy used in any manufacturing industry to improve the quality of the process and its products.It is common practice to materialise that philosophy and to visualise the history of process variability using a control chart.This chart is one of the 'magnificent seven' tools in all quality improvement initiatives, alongside the histogram, check sheet, cause-and-effect diagram, Pareto chart, scatter plot, and stratification.(See, for example, Rooney et al. [6] for further discussion of these tools.)All of them can be found in any standard book on statistical quality control.Nowadays, of these tools only control charting is a dynamic research area, especially in multivariate process control.By 'multivariate process' is meant a process where (i) its quality is determined by more than one quality characteristic, and (ii) the correlations between such quality characteristics must be taken into consideration.This paper deals with Shewhart type control charting for detecting abrupt variability change in a multivariate normal process.The process refers to any multivariate process where all the quality characteristics together follow a multivariate normal distribution, a distribution model often used in this setting.Statistically, the variability of this process is represented in the form of a rectangular covariance matrix.It is a symmetric matrix of p rows and p columns where (i) p is the number of quality characteristics, (ii) the k-th diagonal element is the variance of the k-th quality characteristic and (iii) the element at the i-th row and jth column is the covariance of the i-th and j-th quality characteristics.
Many Shewhart type control charting procedures are available for detecting abrupt change in multivariate normal process variability.One of the most widely-used control charts is the generalised variance-based chart, which can be found in the literature on multivariate process control.For certain recent developments of the generalised variance-based control charting procedure, see Mason et al. [5], who eliminate the condition that the number of observations must be greater than the number of quality characteristics in Phase II control charting.See also Djauhari [3] who presents a method for removing the bias in the control limits of the generalised variance chart and Alt and Smith [1] for further generalised variance-based control charting procedures.However, careful attention is needed when using a generalised variance control chart or a generalised variance-based control chart, since generalised variance -which is algebraically defined as the determinant of the covariance matrix -is only a scalar simplification of the complex covariance structure of quality characteristics.Geometrically, generalised variance can be interpreted in terms of the volume of a p-dimensional parallelotope defined by p quality characteristics under study (Anderson [2]).Based on this interpretation, process variability is measured in terms of the volume of the parallelotope: the larger the volume, the larger the process variability; and the smaller the volume, the smaller the process variability.
According to the above interpretation, a generalised variance chart will not detect a change in covariance structure if there is no change in the determinant of the covariance matrix -or, equally, no change in the volume of the p-dimensional parallelotope.Therefore such a chart by itself is not sufficient for the situation.To improve the performance of this control charting procedure in what follows, a new procedure is introduced by combining a generalised variance chart and a vector variance chart.The latter chart, based on vector variance as a measure of process variability, has recently been presented in Djauhari et al. [4].Djauhari et al. [4] illustrate the advantage of a vector variance chart relative to a generalised variance chart by comparing them in terms of the average run length based on a certain shift in covariance structure.In general, however, there are many situations where generalised variance is not able to differentiate two different covariance matrices, while vector variances can differentiate between two different covariance matrices.Conversely, there are situations where generalised variance is able to differentiate between them while vector variance is not.The following three hypothetical covariance matrices may be considered.
These matrices represent three differing covariance structures.The variance of the first and the second variables, and also the correlation coefficient between them represented by Σ 1 , are totally different from those represented by Σ 2 and Σ 3 .However, Σ 1 and Σ 2 have the same generalised variance but a different vector variance.On the other hand, Σ 1 and Σ 3 have the same vector variance but a different generalised variance.
The above properties show that if generalised variance is not able to detect abrupt shifts in a covariance structure, vector variance could possibly do so, and vice versa.It is these properties that suggest combining a generalised variance chart and a vector variance chart; one chart is used after the other.The ability of this control charting procedure to detect the shift in covariance structure will be better than one based on a generalised variance chart alone.
The combination procedure is used to control the variability of a B-complex vitamin production process in a pharmaceutical company.The result is satisfactory, and is reported here.To begin the discussion, the next section reports the experience in using a generalised stand-alone variance chart.The third section offers an analysis to see whether or not this control charting procedure is convincing.The result of this analysis motivates the use of a vector variance chart as another control charting procedure.This is presented in the fourth section.A new procedure will then be introduced to handle the limitation of the generalised variance chart.Additional remarks close this paper.

GENERALISED VARIANCE CHART IN CONTROLLING B-COMPLEX PRODUCTION PROCESS
In what follows, the discussion is focused on Phase I process control.For this purpose, suppose m independent samples drawn from a multivariate normal process are available.Denote k S as the covariance matrix of sample k of p rows and p columns; k = 1, 2, …, m, and S their average.A generalised variance chart consists of plotting the determinant of k S , denoted by S k , the lower control limit (LCL), and the upper control limit (UCL).If LCL is found to be negative, it is common to set it to 0. The control limits are used as the cut-off values to decide whether or not an out-of-control signal occurs.An out-of-control signal is declared to occur at sample k if S k is not in the control region.To calculate these control limits, the following formula, which ensures lack of biased is used (see Djauhari [3]). where: . n p and the probability of false alarm, i.e., the probability that an out-of-control signal will occur even though the process is in-control, is 0.0027.
The above control charting procedure controls the production process variability of Bcomplex vitamin tablets in a pharmaceutical company (whose name is not revealed to preserve confidentiality).The relevant functionary at that company provided data of m = 15 independent samples, where the number of observations in each sample is n = 12 and the number of quality characteristics of p = 2 -i.e., the 'mass' and 'hardness' of the tablet.From this data, the covariance matrix and the generalised variance of each sample have been established.The results are presented in Table 1  = 0.02417.Hence, LCL = -0.11123which is set to 0 and UCL = 0.35392.Figure 1 shows the generalised variance chart for data in Table 2.The horizontal axis gives the sample number, and the vertical axis the generalised variance.The figure shows that all sample generalised variances are in the control region.This is an indication that, according to the generalised variance chart, no out-of-control signal occurs.

LIMITATION OF GENERALISED VARIANCE CHART
According to the generalised variance chart in Figure 1, there is no identifiable cause of the process variability.Is this really so?This question is commonly asked by quality professionals when they are considering a control chart.To answer this question, as generalised variance is the product of all eigenvalues of the covariance matrix, which is assumed positive definite, it is recommended to conduct a comparison study of the fifteen sets, each of which consists of two eigenvalues.
Considering the covariance matrices in Table 1 anew, denote kj λ the j-th eigenvalue of the k-th covariance matrix k S ; k = 1, 2, ……, 15 and j = 1 and 2. As p = 2, we call the first and second eigenvalues for the larger and smaller eigenvalues respectively.For each covariance matrix presented in Table 1, the eigenvalues are summarised in Table 3 Tr S = 0.33666.Therefore, θ = 0.32045 and η = 1.65464 and thus, from equation (3), LCL* = -1.17623which is set to 0 and UCL* = 1.81712.Based on these results, the vector variance chart for data is shown in Figure 3.
The control chart shows that the fifth sample point is beyond the control region.Its vector variance is greater than the upper control limit UCL*.Thus, according to the vector variance chart, an out-of-control signal occurs at sample 5, as indicated by the run chart of the first eigenvalue in Figure 2.

PROPOSED PROCEDURE
Section 2 and Section 4 show that the generalised variance chart does not give any out-ofcontrol signal during the process, while the vector variance chart does.In practice, it is also possible to come across a situation where the generalised variance chart signals an outof-control state but the vector variance chart does not.These properties show that if one chart is not able to detect an abrupt shift in covariance structure, the other chart could possibly do so.Due to these properties, the following procedure is proposed where both charts are used in turn.The procedure consists of two steps: 1. Select one of the two charts -a generalised variance chart or a vector variance chart.Suppose one selects a generalised variance chart.If this chart does not give any out-ofcontrol signal, then go to the second step.Otherwise, an out-of-control signal occurs.2. Use the alternative chart.In this case use a vector variance chart.If this chart does not signal an out-of-control state, then one may more confidently declare that an assignable cause has not occurred.Otherwise, an out-of-control signal occurs.
The procedure is presented in Figure 4 in the form of a flowchart, where GV and VV stand for generalised variance and vector variance respectively.

ADDITIONAL REMARKS
When the covariance structure shifts abruptly, the ability of the proposed procedure to detect that shift will be better than the generalised variance chart alone.Furthermore, considering equations ( 2) and (3), this procedure appears complicated.But in fact it is easy to implement, even by using very familiar software for non-statisticians such as Microsoft Excel.In this paper, all calculations and drawings were done using this software.
The use of the proposed procedure in controlling B-complex vitamin production process variability shows the situation where generalised variance chart does not signal any shift of covariance structure, while the vector variance chart does.In practice, one may find examples where a generalised variance chart signals an out-of-control state but a vector variance chart does not.Thus this procedure will decrease the probability that an out-ofcontrol signal does not occur even though the process variability has shifted.In other words, we can be more confident that there is no assignable cause in the process if both charts do not signal any of out-of-control state.Although the experience reported in the previous sections is in a pharmaceutical company with p = 2, the procedure can be used in any manufacturing industry with larger p.
This paper ends with an important warning: the procedure proposed above is not free of limitations.It might happen that an out-of-control signal does not occur even though the covariance structure has actually shifted.As a hypothetical example of this situation, we compare Σ 1 in (1) with the following covariance matrix: The two matrices Σ 1 and Σ 4 , which represent two different covariance structures, have the same generalised variance and the same vector variance.The problem of finding a procedure that is able to eliminate the error of not signaling an out-of-control state when the process variability has abruptly shifted is a research topic for the future.

Figure 4 :
Figure 4: Flowchart of the proposed procedure

Table 2 : Generalised variances
To construct the generalised variance chart, where the control limits defined in (2) were S , calculate first its determinant S and the constants 1 b , 2 b , 3 b , and 4 b .From

Table 3 : Eigenvalues of covariance matrices
.To construct the vector variance chart for data in Table1, first calculate the vector variance of each sample.The results are presented in Table4.
Graphically, the columns 'First Eigenvalues' and 'Second Eigenvalues' in this table can be represented in Figure2in the form of a combined run chart.The solid line is the run chart for 1 k λ and the dashed line is for 2

Table 4 :
Vector variancesThen calculate the control limits.To do this, from Table1consecutively calculate S , its