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The Design and Performance of the Multivariate Synthetic-T2 ControlChartFrancisco Aparisia; Marco A. de Lunaba Departamento de Estadstica e Investigacin Operativa Aplicadas y Calidad, Universidad Politcnicade Valencia, Valencia, Espaa b Departamento de Ingeniera Industrial y Mecnica, ITESM,Guadalajara, Mxico

To cite this Article Aparisi, Francisco and de Luna, Marco A.(2009) 'The Design and Performance of the MultivariateSynthetic-T2 Control Chart', Communications in Statistics - Theory and Methods, 38: 2, 173 192

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Communications in StatisticsTheory and Methods, 38: 173192, 2009

Copyright Taylor & Francis Group, LLC

ISSN: 0361-0926 print/1532-415X online

DOI: 10.1080/03610920802178413

The Design and Performance of the MultivariateSynthetic-T2 Control Chart

FRANCISCO APARISI1 AND MARCO A. DE LUNA2

1Departamento de Estadstica e Investigacin Operativa Aplicadas yCalidad, Universidad Politcnica de Valencia, Valencia, Espaa2Departamento de Ingeniera Industrial y Mecnica, ITESM,Guadalajara, Mxico

One of the objectives of research in statistical process control is to obtain control

charts that show few false alarms but, at the same time, are able to detect quickly theshifts in the distribution of the quality variables employed to monitor a productive process. In this article, the synthetic-T2 control chart is developed, which consistsof the simultaneous use of a CRL chart and a Hotellings T2 control chart. The ARL is calculated employing Markov chains for steady and zero-state scenarios. A procedure of optimization has been developed to obtain the optimum parametersof the synthetic-T2 , for zero and steady cases, given the values of in-control ARL

and magnitude of shift which needs to be detected rapidly. A comparison between(standard T2, MEWMA, T2 with variable sample size, and T2 with double sampling)charts reveals that the synthetic-T2 chart always performs better than the standardT2 chart. The comparison with the remaining charts demonstrate in which cases the performance of this new chart makes it interesting to employ in real applications.

Keywords ARL; Multivariate; Quality control; SPC; Synthetic.

Mathematics Subject Classification 62P30.

1. Introduction

Statistical quality control has the objective of detecting shifts in the distribution ofthe monitored quality variables. In the majority of production processes the possiblespontaneous changes in this distribution will worsen the quality of the productmanufactured. Therefore, the quick detection of these shifts is very important tomaintain quality. The statistical process control (SPC) has shown itself to be apowerful tool in achieving this objective.

It is possible to employ two strategies when it is desired to control concurrentlyp variables from one component or from a production process: to use simultaneously

Received August 21, 2007; Accepted May 2, 2008

Address correspondence to Francisco Aparisi, Departamento de Estadstica eInvestigacin Operativa Aplicadas y Calidad, Universidad Politcnica de Valencia, Valencia

46022, Espaa; E-mail: faparisi@eio.upv.es

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174 Aparisi and de Luna

p univariate control charts or to employ only one multivariate control chart(Montgomery, 2005). Over the years, much of research has been made in order

to design control charts capable of fast detection of small shifts in the process.The problem is that the standard control charts (X in the univariate case and

Hotellings T2

in the multivariate case) are not efficient at detecting small shifts.The efficacy of a quality control chart is normally measured using the ARL (AverageRun Length), the average number of points in the chart until the first out-of-control

signal appears.Many strategies have been proposed with the aim of improving the ARL values

when the process is out of control while keeping a desired ARL value when theprocess is in control. From our point of view, the more efficient strategies are:

1. Cumulative sums, in the univariate case the CUSUM charts (Jones et al., 2004;Luceno and Puig-Pey, 2000) and in the multivariate case the MCUSUM charts

(Khoo and Quah, 2002; Woodall and Ncube, 1985).

2. Exponentially weighted averages, the EWMA charts in the univariate case(Knoth, 2005; Lucas and Saccucci, 1990) and MEWMA charts for themultivariate case (Aparisi and Garca-Daz, 2004; Lowry et al., 1992; Prabhu and

Runger, 1997).3. Variable sample size (VSS) univariate charts (Costa, 1994; Costa and Rahim,

2001; Reynolds and Arnold, 2001) and VSS multivariate charts (Aparisi, 1996;Aparisi and Haro, 2003).

4. Double sampling (DS) univariate charts (Daudin, 1992; He and Grigoryan, 2006)

and DS multivariate charts (Champ and Aparisi, 2006; He and Grigoryan, 2005).5. Synthetic charts for the univariate case (Chen and Huang, 2005a; Wu and

Spedding, 2000).

The performance of the SPC has been improved employing simultaneously twocontrol charts. Examples of this strategy are: the Shewhart-CUSUM chart (Lucas,1982), the Shewhart-EWMA chart (Klein, 1997), and the synthetic chart, which is a

Shewhart-CRL (conforming run length) chart. The synthetic-X chart was introducedby Wu and Spedding (2000) as a control chart useful for detecting small shifts in

the mean of a process.In this article, a procedure will be developed for finding the parameters of

the multivariate synthetic T-square chart, Ghute and Shirke (2008), that minimizes

the ARL for a given size of shift, given a desired in-control ARL, for the steadyand zero states. In addition, a comparison versus other multivariate control charts

is made. Section 2 shows a summary of multivariate SPC and the methodologieswhich enable more efficient control charts to be obtained. In Sec. 3, the synthetic-T2

chart is defined. The optimization of the parameters of this chart can be found inSec. 4. Section 5 shows the comparison between several multivariate control charts,

in order to find out in which cases the synthetic-T2 chart shows better performance.The conclusions can be found in Sec. 6.

2. Multivariate Quality Control

Multivariate quality control consists of monitoring in one chart p variables of the

same production process or component. The first proposal to control possible shifts

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176 Aparisi and de Luna

En = n0. A software tool has been employed to find the optima parameters of thischart, for comparison purposes.

2.2. MEWMA Control Chart

The multivariate EWMA control chart, MEWMA (Lowry et al., 1992), is a chartwhere the plotted statistic takes into account the information from past samples.The MEWMA control chart shows an out-of-control signal when the T2i statisticis larger than the control limit (h), selected to obtain a desired in-control ARL.In order to compute the T2i statistic it is necessary to employ a smoothing coefficient,r (0 < r 1). The MEWMA vector is obtained through Zi = rXi + 1 rZi1 andthe statistic to be plotted is given by T2i = ZTi

1zi

Zi, being

zithe covariance

matrix of Zi, where Z0 = 0. The exact covariance matrix of Zi is computedaccording to

zi= r11r2i

2r

.It is a common practice to use the asymptotic approximation of the matrix zi

When r= 1, the resultant chart is the Hotellings control chart. If the value of ris reduced the weight of the past samples is more important. Aparisi and Garca-Daz (2004) developed genetic algorithm based software which finds the parametersr and h (for univariate and multivariate cases) that produces the minimum ARL fora given size of shift and for a given in-control ARL.

2.3. Double SamplingT2 Control Chart

The strategy of double sampling (DS) consists of taking two samples of sizes n1and n2 from the process at the same time. With the statistical information obtainedin the first sample it is determined whether the process is in control, out of control,or whether it is required to analyze the second sample, and combine it with thefirst one in order to make a final analysis. Champ and Aparisi (2006) developed ananalytical method to obtain the ARL of the DS-T2 chart and, employing geneticalgorithms, the parameters of the optimum DS-T2 chart to detect a shift of givenmagnitude are found.

There are five parameters to define the DS-T2 chart: h, h1, w1, n1, and n2.The T2i1 statistic is computed only considering the first sample of size n1, employing

T2i1 = n1Xi1 oT1Xi1 o. If T2i1 > h1, the process is deemed to be out ofcontrol; ifT2i1 < w1, it is accepted that the process is in an in-control state. However,if w1

T2i1 < h1 it is not possible to make a decision and the second sample of size

n2 is studied. The two samples are combined employing Xi = 1n1+n2 n1Xi1 + n2Xi2.Hence, T2i is calculated using the expression T

2i = n1 + n2Xi 0T1Xi 0.

If T2i > h, it is decided that the process is out-of-control, otherwise the process isdeemed to be in an in-control state.

3. The Synthetic-T2 Control Chart

The univariate synthetic chart was introduced by Wu and Spedding (2000) as analternative to improve the performance of the Shewhart control chart to detectprocess shifts. It is the result of combining a Shewhart chart and CRL chart (a chart

originally designed to detect increments in the percentage of defective units). Thesynthetic-X chart for the zero-state scenario shows better ARL values to detectprocess shifts, for any shift magnitude, than the X control chart. In some cases,

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The Synthetic-T2 Control Chart 177

Figure 1. Sub-charts of the synthetic-T2 control chart: T2 sub-chart and CRL sub-chart.

especially for moderate and large shifts, the synthetic-X chart exhibits a better

performance than the EWMA control chart (Wu and Spedding, 2000).The synthetic chart has been also applied to monitor the variability of a process

(Chen and Huang, 2005a,b, 2006; Costa and Rahim, 2006) and the percentage ofdefective units of a process (Wu et al., 2001). With the objective of improvingthe performance of the synthetic control charts, Chen and Huang (2005b) appliedvariable sampling intervals (VSI) and Davis and Woodall (2002) improved theperformance of the synthetic-X control chart by adding the concept ofside-sensitive.

The synthetic T2 control chart (synthetic-T2) is a chart for monitoringsimultaneously two or more quality characteristics. It consists of two sub-charts:a T2 sub-chart and a CRL sub-chart. Figure 1 shows the concept of the synthetic-

T2

chart. The T2

sub-chart has a unique control limit, LCsynt. The CRL sub-charthas a low control limit, L, L 1. The value of LCsynt is the criteria to classify asample as conforming or non conforming. The value of L is the criteria to decide ifthe process is in control or out of control.

The CRL concept assumes that in t = 0 there is a point above the LCsynt limit(a non conforming sample in t = 0). This characteristic, called head start, is veryimportant for the performance of the synthetic charts. When this assumption isruled out, the performance of synthetic charts worsens (Davis and Woodall, 2002).

For a synthetic-T2 chart, unlike the T2 control chart, T2i > LCsynt does notmean that an out-of-control state has to be assumed, but the inspected sample

must be classified as non conforming. Only when the sample is classified as nonconforming and CRL L will the process be considered to be out of control.Like the T2 control chart, the synthetic-T2 chart shows an out-of-control signal,indicating that there probably is a shift in the process. However, this signal doesnot inform us about the variable or variables that have produced the shift. Severalmethods for the interpretation of the out-of-control signal of the T2 control charthave been developed. These methods can also be applied to the synthetic-T2 chart(Aparisi et al., 2006; Atienzas et al., 1998; Kourti and MacGregor, 2004; Masonand Young, 1999; Mason et al., 1997). Applying these techniques to the synthetic-T2 chart the user may obtain a controlling procedure that, although it is not the

fastest in comparison versus other multivariate charts (see the comparison later),may be more useful in a real application, due to the help provided in identifying thevariables that have shifted.

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178 Aparisi and de Luna

4. Optimization of the Parameters of the Synthetic-T2 Chart

Two values of ARL are important for the design and performance of the synthetic-T2 chart, the in-control ARL [ARLST2 d = 0] and the out-of-control ARL[ARLS

T2 d

=0]. The value of the in-control ARL is selected taking into account

the frequency of false alarms. The value of ARLST2 d = 0 is important in orderto rapidly detect a shift in the mean vector of magnitude d.

4.1. Obtaining the ARL Values

When the ARL needs to be computed, two approaches are possible: zero stateand steady state (Champ, 1992; Zhang and Wu, 2005). The zero-state scenarioimplies that the shift in the process occurred just before the first sample was taken.The steady-state case means that the process was in an in-control state for along period, before the out-of-control situation happened. Wu and Spedding (2000)

showed the formulas to obtain the ARL value of the synthetic-X chart for thezero-state case. Davis and Woodall (2002) found that the synthetic-X chart maybe modelled employing Markov chains, like a standard X chart with a run rule of2 points of (L + 1) outside control limits.

In a similar way, it is possible to obtain the ARL of the multivariate synthetic-T2 control chart (ARLST2 ) employing a Markov chain approach. The synthetic-T

2

chart may be modeled like a T2 control chart with a run rule that consists of 2 pointsof (L + 1) outside the upper control limit; see the Appendix. This approach is alsouseful for calculating the ARL for the zero-state case.

4.2. Optimization of the Synthetic-T2

Control ChartA software tool for Windows has been developed in order to find the optimaparameters of the synthetic-T2 control chart. The software tool is available uponrequest from the authors. The output of this software tool is shown in Fig. 2.Once the zero or steady state is selected, the optimization process finds the valuesof L and LCsynt that minimizes the value of ARLST2 d = 0 for a given shiftof magnitude d sample size, and number of variables, taking into account therestriction of the specified in-control ARL, ARLST2 d = 0. The chart that fulfilsthis design is considered to be optimum to detect a shift of size d in the mean vector.As exposed in Fig. 2, the output from the software tool shows the improvement of

ARL against Hotellings T2

control chart and it draws the ARL curves of T2

andsynthetic-T2 charts.Figure 2 shows the solution for the problem ARLST2 d = 0 = 200, p = 3,

n = 4, d = 075, and steady state. The parameters of the optimum synthetic-T2chart are: LCsynt = 9163 and L = 9, with ARLST2 d = 075 = 1358. The out-of-control ARL value for the standard T2 chart is 20.41. Therefore, the reduction inthe value of ARL for a shift d = 075 employing the synthetic-T2 chart is 33.5%.

5. Comparison Versus Other Multivariate Charts

5.1. Comparison Between Optimum Steady and Zero States Synthetic-T2 Charts

A comparison of performance between the optimum design for the zero-statesynthetic-T2 and the steady-state synthetic T2 control charts has been carried out.

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The Synthetic-T2 Control Chart 179

Figure 2. Main window of the in-house software tool used to find the optimumsynthetic-T2 and synthetic-X control charts.

The optimum design of the synthetic-T2 charts has been utilized in the comparisonfor different cases of ARLST2 d = 0 = ARLT2 d = 0, n, p, and d. The ARLvalues are shown in the second and third columns of Tables 16.

For the same design parameters (same in-control ARL, number of variables andsample size), the optimum value of L of the synthetic-T2 chart has the same valuefor the steady state and zero-state scenarios, if the magnitude of shift d is moderateor large. However, for small shifts, the optimum parameter L for the steady-statecase is always lower than the optimum for the zero-state case. For all the cases, theoptimum value ofLCsyntfor zero state will always be larger than the optimum valuefor steady state.

The performance of the optimum zero-state synthetic-T2 chart is always better

than the optimum chart for the steady-state case. This is a predictable result,a consequence of the inertia that the steady state produces in the detection of theshift. The largest differences (50%) are found for large shifts, although these shiftsare detected quickly for both schemes. For small shifts (d 05), the differences inthe ARL values are around 20%. However, these differences tend to be larger as thesample size increases, and tend to decrease as the number of variables, p increase.

5.2. Comparison Versus Hotellings T2 Chart

A comparison of performance between the zero-state synthetic-T2 and the standard T2

control charts [ARLT2 d] has been carried out. The zero-state case of the synthetic-T2

chart has been selected for this comparison, and for the next one, because for therest of charts the ARL has been computed employing the zero-state approach.

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Table1

ARLoftheopt

imummultivariatecontrolch

artsforashiftd

ARLd=

0=

200

,p=

2,n=

2

Synthetic-T

2

Synthetic-T

2

VSS-T

2

D

S-T

2

T2

(LCsynt,

L

)

(LCsynt,L

)

MEWMA

(CLwn1n2

)

h

h1

w1n1n2

d

CL

Zerostate

Steadystate

hr

n1+

n2