A VARIABLE SAMPLE SIZE SYNTHETIC CHART FOR THE COEFFICIENT OF VARIATION

existing synthetic  chart. A description of how the chart operates, as well as the formulae for various performance measures (i.e., the average run length (ARL), standard deviation of the run length (SDRL), average sample size (ASS), and expected average run length (EARL)) are proposed. The algorithms that optimise the out-of-control ARL (ARL1) and EARL (EARL1), subject to the constraints in the in-control ARL (ARL0) and ASS (ASS0), are also proposed. Subsequently, optimal charting parameters for various numerical examples are obtained. The proposed chart shows a significant improvement over the existing synthetic  -chart. Comparisons with other  -charts also show that the proposed chart performs better than the Shewhart and VSS charts under all cases, while showing better performance than the exponentially weighted moving average (EWMA) and

 charts for moderate and large shift sizes.Finally, this paper shows the implementation of the proposed chart on an actual industrial example.

INTRODUCTION
By convention, most control charts monitor changes in the mean ( )  and/or standard deviation ( )  -for example, Coelho, Chakraborti and Graham [1], Teoh, Fun, Khoo and Yeong [2], and many others.However, not all processes have a constant  .In addition,  may change according to  .One of the reasons this may happen is changes in process outputs as a result of different planning decisions, or it may be due to the inherent properties of the process.Dubious conclusions will be reached if such processes are monitored through conventional X and R or S charts, since shifts in  and/or  do not mean that the process is out of control (OOC).
A chart monitoring the coefficient of variation ( )  was first proposed by Kang, Lee, Seong and Hawkins [3], where For this type of chart, an OOC condition is only signalled with a change in the relationship between  and  .In other words, as long as   does not shift from the in-control (IC) value of  , the process is IC.Yeong, Khoo, Tham, Teoh and Rahim [4] have reviewed several areas of application when monitoring  is important.
Numerous new charts are proposed to monitor  , one of which is the synthetic chart by Calzada and Scariano [5].The synthetic chart produces an OOC signal if successive samples falling outside the control limits are close to each other.It is preferred by practitioners, as it is easy to understand and implement, and is free from the inertia effect faced by the exponentially weighted moving average (EWMA) chart.The synthetic- chart outperforms the  -chart by Kang et al. [3], but is inferior to the EWMA chart proposed by Castagliola, Celano and Psarakis [6].Thus this paper will improve the performance of the synthetic- chart by introducing the variable sample size (VSS) scheme into the synthetic- chart.
The VSS feature is an adaptive feature in which charting parameters are varied according to the most recent sample information.The adaptive feature was recently incorporated into charts monitoring  .Castagliola, Achouri, Taleb, Celano and Psarakis [7] first proposed the variable sampling interval (VSI)- chart.Later, Castagliola, Achouri, Taleb, Celano and Psarakis [8] proposed variable sample size (VSS)- charts.
[7] to [9] incorporated adaptive features into the simpler Shewhart-type  charts.This encouraged Yeong et al. [4] and Anis, Yeong, Chong, Lim and Khoo [11] to incorporate the VSI and VSS features respectively, into more complicated charts, such as the EWMA charts.From these studies, adapting the charting parameters according to the most recent sample information results in a significant improvement.
No studies are available in the literature on adaptive-type synthetic- charts.The synthetic- charts are attractive to practitioners, as they wait until two successive samples fall outside the control limits before deciding whether the process is IC or OOC, unlike Shewhart-type  charts, which immediately send OOC signals when a sample falls outside the control limits.Thus this paper will propose a VSS synthetic- chart, which is expected to improve the performance of the existing synthetic- chart.This paper is organised as follows: Section 2 reviews the existing synthetic- chart and describes the transformed statistics (Ti) that will be adopted in this paper.Then Section 3 introduces the proposed VSS synthetic- chart and the formulae to evaluate various performance measures.Next, Section 4 proposes the algorithm to obtain the optimal charting parameters, and shows the optimal performance of the proposed chart based on numerical examples.Section 5 compares the performance of the proposed chart with other  charts, while the proposed chart is implemented on an actual industrial example in Section 6.Finally, the conclusion is provided in Section 7.

THE SYNTHETIC- CHART AND TRANSFORMED STATISTICS (TI)
The synthetic- chart is made up of two sub-charts -i.e., the  and conforming run length (CRL) sub- charts.In the  sub-chart, if are the sample  , lower control limit and upper control limit respectively), the sample is a non-conforming sample; conversely, if  falls between the LCL and UCL, it is a conforming sample.The CRL sub-chart defines the number of conforming samples between successive non-conforming samples (including the ending non-conforming sample) as the where L is a pre-determined threshold, then the process is considered to be OOC.
The LCL and UCL of the  sub-chart are computed as and where K represents the control limit coefficient, while where 0  and n are the IC  and sample size respectively.
Note that the LCL and UCL are functions of n.For the existing synthetic- chart, which adopts fixed sample sizes, there is only one pair of (LCL, UCL).However, if variable sample sizes are adopted -for example, small and large sample sizes (nS and nL) -there will be a pair of (LCL, UCL) for nS and another pair of (LCL, UCL) for nL.In addition, VSS charts involve lower and upper warning limits (LWL and UWL) to establish the warning region, where, similar to the LCL and UCL, they are also functions of n.Thus, if two levels of sample sizes are adopted (nS and nL), there will be two pairs of (LCL, UCL) and two pairs of (LWL, UWL)i.e., one pair of (LCL, UCL) and (LWL, UWL) for nS, and another pair of (LCL, UCL) and (LWL, UWL) for nL.Furthermore, both the warning and the control limits are asymmetric limits.This results in a significant increase in the computational effort to design the chart because of the larger number of charting parameters.It also results in a difficulty during implementation and interpretation, as the samples need to be plotted against different pairs of limits [8].
Thus the approach by Castagliola et al. [8] is adopted: instead of directly monitoring  of the i th subgroup ( ) will not have an impact.Note that the Ti statistics are only approximated well by a standard normal distribution if the observations of the quality characteristic being monitored are independent and identically distributed normal variates.

THE VARIABLE SAMPLE SIZE (VSS) SYNTHETIC- CHART
The proposed chart works in a similar way to the existing synthetic- chart, except that the sample size is varied at two levels, as shown in Figure 1.T − falls in the warning conforming region or the non-conforming region, then ( ) ,, ,, The probability A + is obtained as follows: ( ) where F  is the cdf of  and ( ) is the cdf of the non-central t distribution.Castagliola et al. [6] have shown that − can be obtained as follows: The formulae for the average run length (ARL), standard deviation of the run length (SDRL), expected average run length (EARL), and average sample size (ASS) will be developed using a Markov chain approach.The states of the Markov chain are defined based on L consecutive samples; thus there will be a total of ( ) 22 L + will be the absorbing state.
A ( ) ( ) + matrix that consists of the transition probabilities among the transient states, as defined in the previous paragraph, can be obtained.This transition probability matrix is denoted as Q .For 2 L = , Q can be obtained as follows: + matrix with all elements zero, except: 1.
For row ( ) The transition probability matrix can then be obtained as follows: where Q is the ( ) ( ) + matrix as shown in the preceding paragraph, =− r 1 Q1 with 1 being a ( ) 2 1 1 L + vector of ones, and 0 is a ( ) vector of zeros.Note that state ( ) The ARL and SDRL can then be computed as where q is the ( ) 2 1 1 L + vector of initial probabilities for the transient states, identity matrix, and 1 is a ( ) 2 1 1 L + vector of ones.For a zero-state condition, except for the 3 rd element of q which is one, all other elements are zeros.This paper will design the proposed chart based on the zero-state condition.
To calculate the OOC ARL ( The exact value of  is not always known, and without knowing its exact value, the ARL1 cannot be computed.For this scenario, the EARL will be adopted as a performance measure.The EARL does not require the shift size to be estimated as an exact value; instead, it only needs to be estimated as a range ( ) min max ,  .
The following shows the computation of the EARL: where ( ) is the probability density function (pdf) of  .As in the study of Castagliola et al. [6],  is assumed to be uniformly distributed over ( ) min max ,


. The integral cannot be obtained analytically; thus the Gauss-Legendre quadrature is adopted to solve the integral.
Subsequently, the formula for the ASS will be shown.The ASS needs to be evaluated so that the IC ASS (ASS0) can be maintained at a specific value, especially when the cost of sampling is of concern to the practitioner.
The formula for the ASS is developed through a Markov chain approach, as proposed by Castagliola et al. [8].First, P in Equation ( 22) is transformed into a similar matrix * P , where ,, be the stationary probability vector of * P , where j  is the stationary probability for the process to fall in the j th state, where 0,1,..., 2 2 jL =+ .π can be computed as follows: where the matrix R is obtained by deducting 1 from the diagonal elements of the transpose of * P ; then replace the third row of this matrix with ones.
The ASS is then computed as

NUMERICAL EXAMPLES
This section shows the algorithms to obtain the optimal charting parameters ( ) The following are the steps to implement the first algorithm: 1.
Specify the values of n, 0  , ARL0 and  .

3.
Set 2 Set where n is the sample size. 5.

6.
With the current combination of ( ) Increase L by 1. 12.
Repeat Steps 3 to 11 until the ARL1 for L+1 is larger than the ARL1 for L. 13.
The combination with the smallest ARL1 is the optimal charting parameters ( ) when n = 15.Thus a larger n results in a better performance.The improvement is more significant when  is small.A smaller W and a larger K are also observed for a larger n, which translates into a larger warning region but a smaller conforming region.
Similar to n, smaller ARL1 and SDRL1 are observed for larger  , since a larger shift requires fewer samples to detect the shift.A larger  also results in smaller L and K.This shows that a smaller conforming region is adopted; but successive non-conforming samples should happen quite close to each other for the process to be considered as OOC.A larger 0  results in a slight increase in the ARL1 and SDRL1, especially for small shift sizes.1, there is a large difference between S n and L n .A smaller value of EARL1 is shown for a larger n.A smaller W and a larger K are also observed for a larger n.Overall, a similar trend is observed for Tables 1 and 2

COMPARISON
The proposed chart is compared with the following  charts: the synthetic- , VSS- , VSS EWMA-2  , EWMA-2  and Shewhart- charts.
Table 3 shows the ARL1 values of the proposed VSS synthetic- chart and the five competing charts, as    From Table 3, the VSS synthetic- chart shows a smaller ARL1 than the Shewhart- , VSS- and synthetic-  charts for all n and  values.The magnitude of improvement is quite large for small values of n and  .In particular, comparison with the synthetic- chart shows that the VSS feature results in a significant improvement.For example, for n = 5 and 1.1  = , the ARL1 = 115.42for the synthetic- chart, while ARL1 = 68.92for the VSS synthetic- chart.This shows that incorporating the VSS feature results in an improvement of 40.29% in the ARL1 criterion.
However, the VSS synthetic- chart does not outperform the EWMA-2  and VSS EWMA-2  charts for all the shift sizes.With the exception of small shift sizes of 1.1  = , Table 3 shows that the VSS synthetic- chart outperforms the EWMA-2  chart for most cases.This is expected, since the EWMA-2  chart is well- known for its sensitivity to small shifts.However, the VSS synthetic- chart is not as complicated as the EWMA- 2   chart, which makes it more user-friendly for practitioners.Furthermore, the EWMA-2  chart only shows a better performance for 1.1  = , while, for other values of  , the VSS synthetic- chart shows a better performance.The VSS synthetic- chart outperforms the VSS EWMA-2  chart for moderate and large shift sizes of Performance in terms of the ARL1 criterion can only be evaluated if  is known.Since  may not be known in practical applications, EARL1 comparisons are also made.Table 4 shows the EARL1 values of the VSS synthetic- chart and the five competing charts for   to Table 3, the relative EARL (REARL) is provided for ease of comparison, where the REARL is computed and interpreted in a similar way to the RARL.4.

AN ILLUSTRATIVE EXAMPLE
This section shows the implementation of the VSS synthetic- chart with the example by Castagliola et al.
[8], who have shown that it is not appropriate to use X and S control charts to monitor the process owing to an unstable mean and standard deviation.However, Castagliola et al. [8] conducted a regression analysis that showed a constant proportionality between the standard deviation and the mean of the process.Thus monitoring  is a good alternative for this process.This section illustrates the monitoring of this process using the VSS synthetic- chart.
Castagliola et al. [8] showed that the Phase-I data are IC.Thus the 0  in Equation ( 29) can be adopted.
Next, Phase-II data, which consist of m = 30 samples, are collected, and are shown in  Figure 2 shows the  sub-chart for the Phase-II data.From Figure 2, we can see that there are three non- conforming samples: samples 3, 8, and 19.From Figure 2,  .Note that, since the synthetic- chart adopts fixed sample sizes, there is no need to monitor the transformed statistics i T .Instead, ˆi  is monitored directly.Figure 3 shows the  sub-chart for the synthetic- chart without the VSS feature.This shows that the synthetic- chart only managed to detect one OOC sample at sample 9, whereas the VSS synthetic- chart detected three OOC samples at samples 3, 8, and 19.This shows that the VSS synthetic- chart improves the performance over the synthetic- chart.
Figures 2 and 3 can only be obtained if  is known in advance.Since  may not be known in advance, we also consider the case when  is unknown.By adopting the methodology in Section 4 for  for the synthetic- chart, which are the same optimal charting parameters as those for 1.20  = .Since the same optimal charting parameters are adopted, the design based on the EARL also gives OOC signals for the same samples as the design based on the ARL.

CONCLUSION
The synthetic- chart is attractive to practitioners, as it waits for two successive samples to fall outside the control limits before deciding whether the process is IC or OOC.However, the existing synthetic- chart is based on fixed sample sizes, where the same sample size is adopted irrespective of the current sample information.To improve the performance of the existing synthetic- chart, a VSS synthetic- chart is proposed in this paper to monitor  .In the proposed chart, the sample size alternates between the small and the large sample sizes, dependent on whether the previous sample is in the central, warning, or non-conforming regions.Formulae to evaluate the ARL, SDRL, EARL, and ASS are developed, and optimisation algorithms to obtain the optimal charting parameters are proposed.Tables of optimal charting parameters are also provided to facilitate a quick implementation of the proposed chart.The optimal charting parameters show that there is a large difference between S n and L n .Thus practitioners are encouraged to adopt a smaller sample size when  is in the central region, and to adopt larger sample sizes when  falls in the warning or non-conforming region.The proposed chart shows a significant improvement over the existing synthetic- chart.Furthermore, the VSS synthetic- chart also outperforms the VSS- and Shewhart- charts for all shift sizes, while outperforming the EWMA-2  and VSS EWMA-2  charts for moderate and large shift sizes.Note that, with the exception of very small shift sizes, the VSS synthetic- chart shows a better performance than the EWMA-2  chart.

 0 
, the transformed statistics Ti are monitored instead.The statistics Ti are defined as and c are parameters that depend on ( ) ni and .Here, ( ) ni is the sample size of the i th subgroup.The parameters ( ) are the quantiles for the distribution of  , and

Figure 1 :Figure 1 1 i
Figure 1: The  sub-chart of the VSS synthetic  chart Figure 1 shows that the conforming region is separated into the central and warning regions.When i W T W −   , the sample belongs to the central conforming region, while when

1 ARL 1  1 ARL and 1 SDRL
1) is substituted into Equations (15) to (20), where is the OOC  and  is the shift size, to obtain the OOC Q from Equation (21).To compute the IC ARL ( 0 ARL ) and SDRL ( Equations (15) to (20) to obtain the OOC Q from Equation (21).The are then calculated by substituting the OOC Q into Equations (23) and (24) respectively, while 0 ARL and 0 SDRL are calculated by substituting the IC Q into Equations (23) and (24) respectively.

P
, once the process reaches the OOC state (State ( ) 22 L + ), the process will restart at State 3.
W K .Next, the optimal charting parameters ARL1 and SDRL1 for the numerical examples with different values of 0 , and n  are shown.Furthermore, the optimal charting parameters and EARL1 for different 0 and n  are also shown for the case when  could not be specified.Two algorithms are proposed.In the first algorithm, ( ) W K is chosen to minimise ARL1, subject to constraints in ARL0 and ASS0.In the second algorithm, ( ) W K is chosen to minimise the EARL instead, subject to constraints in ARL0 and ASS0.
, , , SL L n n W K , compute ARL1 and SDRL1 from Equations (23) and (24) respectively, with mentioned in the previous paragraph.The ARL1 values are shown for ARL0 being set as 370.4.To facilitate comparisons, the relative ARL (RARL), which is the ratio of the ARL1 for the competing chart against the ARL1 for the VSS synthetic- chart, is shown in

T
values in bold in Table5are the samples that are non-conforming.If CRL  23, the chart will give an OOC signal.Adopting these optimal charting parameters results in ( ) ( )11 ,14.02, 20.16 ARL SDRL = .
all the CRLs are less than 23, OOC signals will be produced at samples 3, 8, and 19.

Figure 2 :
Figure 2: The  sub-chart of the VSS synthetic- chart corresponding to the Phase II data For comparison, this paper will also show the monitoring of the Phase-II data with the synthetic- chart without the VSS feature.Similar to the VSS synthetic- chart, the synthetic- chart is optimised to detect a shift of 1.20  =

Figure 3 :
Figure 3: The  sub-chart of the synthetic- chart corresponding to the Phase II data , L + IC states that are transient and one OOC absorbing state.The OOC absorbing state refers to the W State 2L : 10...00..00 C State ( ) 21 L + : 10...00..00 W , while State ( )

Table 1 :
The(  ) =by using the second algorithm.Similar to Table1, the ARL0 is set as 370.4.For instance,

Table 2 :
The(  ) SL L n n W K and the corresponding EARL1 for the VSS synthetic- chart Similar to Table

Table 3
=.The proposed chart outperforms competing charts with a RARL that is larger than unity.

Table 4 ,
the VSS synthetic- chart outperforms the Shewhart- , VSS- and synthetic- charts for all values of n and 0  considered in Table4.This result is consistent with the comparison based on the ARL1 criterion in Table3.However, the VSS synthetic- chart only slightly outperforms the EWMA-2

Table 5 (
left-hand side) shows the sample mean ( ) i X , sample standard deviation ( ) i S and sample  ( )

Table 5
[8]alues of the Phase-II samples are also shown.As the transformed statistics Ti in Equation (5) are monitored in the VSS synthetic- chart, Table 5 also shows the Ti for each sample.According to Castagliola et al.[8], it is important to detect a shift of 20% in  .Thus the chart is optimised ii XS and ˆi ni is shown in

Table 5 .
Since * i T  or 2.17