Abstracts
The joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and R charts and the joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and S² charts are the most common charts used for monitoring the process mean and dispersion. With the usual sample sizes of 4 and 5, the joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and R charts are slightly inferior to the joint <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> and S² charts in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the charts based on the standardized sample means and sample ranges (MRMAX chart) or on the standardized sample means and sample variances (MVMAX chart) are similar in terms of efficiency in detecting shifts in the mean vector and/or in the covariance matrix. User's familiarity with the computation of sample ranges is a point in favor of the MRMAX chart. An example is presented to illustrate the application of the proposed chart.
Control charts; Mean vector; Covariance matrix; Multivariate processes
Os gráficos conjuntos de <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> e R e <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> e S² são os mais utilizados para o monitoramento da média e da dispersão do processo. Com os tamanhos de amostra usuais de 4 e 5, os gráficos de <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> e R em uso conjunto são ligeiramente inferior aos gráficos de <img src="/img/revistas/prod/2011nahead/aop_t6_0002_0329.jpg" /> e S² em uso conjunto em termos da eficiência em detectar alterações no processo. Neste artigo, mostrase que para o caso multivariado, os gráficos baseados nas médias amostrais padronizadas e amplitudes amostrais (gráfico MRMAX) ou nas médias amostrais padronizadas e variâncias amostrais (gráfico MVMAX) são similares em termos da eficiência em detectar alterações no vetor de médias e/ou na matriz de covariâncias. A familiaridade do usuário com o cálculo de amplitudes amostrais é um aspecto favorável do gráfico MRMAX. Um exemplo é apresentado para ilustrar a aplicação do gráfico proposto.
Gráficos de controle; Vetor de médias; Matriz de covariância; Processos multivariados
Monitoring the mean vector and the covariance matrix of multivariate processes with sample means and sample ranges
Monitoramento do vetor de médias e da matriz de covariâncias de processos multivariados baseado nas médias e nas amplitudes amostrais
Antônio Fernando Branco CostaI,^{*} * UNESP, Guaratinguetá, SP, Brasil ; Marcela Aparecida Guerreiro Machado^{II}
^{I}fbranco@feg.unesp.br, UNESP, Brasil
^{II}marcela@feg.unesp.br, UNESP, Brasil
ABSTRACT
The joint and R charts and the joint and S^{2} charts are the most common charts used for monitoring the process mean and dispersion. With the usual sample sizes of 4 and 5, the joint and R charts are slightly inferior to the joint and S^{2} charts in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the charts based on the standardized sample means and sample ranges (MRMAX chart) or on the standardized sample means and sample variances (MVMAX chart) are similar in terms of efficiency in detecting shifts in the mean vector and/or in the covariance matrix. User's familiarity with the computation of sample ranges is a point in favor of the MRMAX chart. An example is presented to illustrate the application of the proposed chart.
Keywords: Control charts. Mean vector. Covariance matrix. Multivariate processes.
RESUMO
Os gráficos conjuntos de e R e e S^{2} são os mais utilizados para o monitoramento da média e da dispersão do processo. Com os tamanhos de amostra usuais de 4 e 5, os gráficos de e R em uso conjunto são ligeiramente inferior aos gráficos de e S^{2} em uso conjunto em termos da eficiência em detectar alterações no processo. Neste artigo, mostrase que para o caso multivariado, os gráficos baseados nas médias amostrais padronizadas e amplitudes amostrais (gráfico MRMAX) ou nas médias amostrais padronizadas e variâncias amostrais (gráfico MVMAX) são similares em termos da eficiência em detectar alterações no vetor de médias e/ou na matriz de covariâncias. A familiaridade do usuário com o cálculo de amplitudes amostrais é um aspecto favorável do gráfico MRMAX. Um exemplo é apresentado para ilustrar a aplicação do gráfico proposto.
Palavraschave:Gráficos de controle. Vetor de médias. Matriz de covariância. Processos multivariados.
1. Introduction
The control of several parameters is a requirement to assure the quality of many processes nowadays. Hotelling (1947) provided the first solution to this problem by suggesting the use of the T^{2} statistic for monitoring the mean vector of multivariate processes. Many innovations have been proposed to improve the performance of the T^{2} charts. Costa and Machado (2007) studied the properties of the synthetic T^{2} chart with twostage sampling. Machado and Costa (2008b) considered the use of simultaneous charts as an alternative to the use of the T^{2} chart. Costa and Machado (2008b) considered the use of the double sampling procedure with the chart proposed by Hotelling.
The first multivariate control chart for monitoring the covariance matrix Σ was based on the charting statistic obtained from the generalized likelihood ratio test. For the case of two variables, Alt (1985) proposed the generalized variance statistic S to control the covariance matrix Σ.
Control charts more efficient than the S chart have been proposed. Recently, Costa and Machado (2008a, 2009), Machado and Costa (2008a) and Machado, De Magalhães and Costa (2008), Machado, Costa and Rahim (2008), Machado, Costa and Claro (2009) considered the VMAX statistic to control the covariance matrix of multivariate processes. The points plotted on the VMAX chart correspond to the maximum of the sample variances of the p quality characteristics.
There are a few recent papers dealing with the joint control of the mean vector and the covariance matrix of multivariate processes. Chou et al. (2002) have considered the multivariate control chart for monitoring the process mean vector and covariance matrix for the related quality characteristics simultaneously by using loglikelihood ratio statistics. Takemoto and Arizono (2005) considered the multivariate (, S) control chart for monitoring the mean vector μ and the covariance matrix Σ simultaneously based on the Kullback–Leibler information as the test statistic.
Khoo (2005) proposed a control chart based on the T^{2} and S statistics for monitoring bivariate processes. The speed with which the chart signals changes in the mean vector and/or in the covariance matrix was obtained by simulation. The results are not compelling, once the proposed chart is slow in signaling outofcontrol conditions. Chen, Cheng and Xie (2005) proposed a single EWMA chart to control both, the mean vector and the covariance matrix. Their chart is more efficient than the joint T^{2} and S in signaling small changes in the process. Zhang and Chang (2008) proposed two EWMA charts based on individual observations that are not only fast in signaling but also very efficient in informing which parameter was affected by the assignable cause; if only the mean vector or only the covariance matrix or both.
Machado, Costa and Marins (2009) proposed two new control charts for monitoring the mean vector and the covariance matrix of bivariate processes. First, they proposed the MVMAX chart, which only requires the computation of statistics familiar to the users, that is, sample means and sample variances. Second, they proposed the joint use of two charts based on the noncentral chisquare statistic (NCS statistic), named as the NCS charts. The joint NCS charts are recommended for those who aim to identify the outofcontrol variable instead of the nature of the disturbance, that is, the one that only affects the mean vector or only affects the covariance matrix or both.
For the univariate case, the joint and R charts are commonly used for monitoring process mean and variance. According to the literature, a control chart based on sample means and sample ranges has not yet been proposed for monitoring the mean vector and the covariance matrix of multivariate processes.
As an alternative to the MVMAX chart, we propose a chart based on the synthetic procedure, named as the synthetic MRMAX chart, to control both the mean vector and the covariance matrix of multivariate processes. The sample points correspond to the maximum among the values of the standardized sample means (in module) and ranges (weighted) of p quality characteristics. The sample points plotted on the MVMAX chart correspond to the maximum among the values of sample means and sample variances. User´s familiarity with sample means and sample ranges is a point in favor of the MRMAX chart.
The paper is organized as follows. In section 2 we compare the efficiency of the joint and R charts and and S^{2} charts. In Section 3 we present the properties of the synthetic MRMAX chart. We also present its competitors, that is, the synthetic MVMAX chart and the joint T^{2} and S charts. The proposed chart is compared with its competitors in Section 4. In Section 5, an example is presented to illustrate the application of the proposed chart. Conclusions are in Section 6.
2. Univariate charts for monitoring process mean and dispersion
In recent years, considerable attention has been devoted to the joint monitoring of the process parameters μ and σ. The chart is used for monitoring the parameter μ and the R or the S^{2} charts are used for monitoring the parameter σ. For monitoring both process mean and dispersion, the R chart combined with the chart has been more used than the S^{2} chart. The reason for this fact is that sample ranges are easier to compute. The joint and R charts and and S^{2} charts are briefly discussed in Section 2.1. In Section 2.2 the joint charts are compared in terms of their efficiency in detecting process shifts.
In this paper the average run length (ARL) measures the efficiency of a control chart in detecting a process change. During the incontrol period the ARL = 1/α and is called ARL_{0}, and during the outofcontrol period the ARL = 1/(1 – β). The risks α and β are, respectively, the wellknown Type I and Type II errors. A chart with a larger incontrol ARL (ARL_{0}) indicates lower false alarm rate than other charts. A chart with a smaller outofcontrol ARL indicates a better ability of detecting process shifts than other charts.
2.1. The joint charts
Let X_{1}, X_{2}, ..., X_{n} be a random sample of size n from a normally distributed process with mean μ and variance σ^{2}. The process is considered to start in a state of statistical control with the mean and the variance equal to μ_{0} and σ^{2}_{0} respectively. The occurrence of the assignable cause changes the mean from μ_{0} to μ_{1} = μ_{0} + δσ_{0} and/or the variance from σ^{2}_{0} to σ^{2}_{1} = a^{2}σ^{2}_{0}, with δ ≠ 0 and a > 1.0.
The upper and lower control limits for the chart are respectively
where k_{1} is the control limit coefficient of the chart. The upper control limit for the R chart is
where k_{2} is the control limit coefficient of the R chart. We make the lower control limit of the R chart equals to zero for convenience. The control limit for the S^{2} chart is
The joint charts have probability P of signaling:
Then, the ARL = 1/P.
The probability P_{M} of exceeding the upper or the lower control limits is given by:
where ( ) is the cumulative standard normal distribution.
If the R chart is used to control the process dispersion, then the probability P_{D} of R exceeding the upper control limit is given by
being F_{w}( ) the accumulated distribution function of the random variable W = R/σ. F_{w}( ) has been tabulated, see Costa, Epprecht and Carpinetti (2005).
If the S^{2} chart is used to control the process dispersion, then the probability P_{D} of S^{2} exceeding the control limit is given by
2.2. Comparing the joint charts
In this section we compare the joint and R charts with the joint and S^{2} charts. Table 1 shows their ARLs. The joint and S^{2} charts are faster than the joint and R charts in detecting increases in the variance followed or not by shifts in the process mean. The superiority of the joint and S^{2} charts becomes more evident when the process variance is subject to large increases, see the Pv values in Table 1, which are given by:
We adopt a Type I error of five per one thousand (α= 0.005), that is, ARL_{0} = 200.0. The study with n > 5 led to larger values of Pv. According to expression (6), the power of the chart is independent of the shift direction, that is, if the process mean increases or decreases. Based on that, the values of δ in Table 1 and, in all other tables, are in module.
3. Multivariate charts for monitoring process mean and dispersion
3.1. The MRMAX chart
In this section we propose a new chart based on sample means and sample ranges for monitoring the mean vector μ and/or the covariance matrix Σ of multivariate processes with p quality characteristics that follow a multivariate normal distribution. The sample point plotted on the proposed chart corresponds to the largest value among (Z_{1}, Z_{2}, ..., Z_{p}, W_{1}, W_{2}, ..., W_{p}), i = 1, 2, ..., p, where and W_{i} = kR_{i}/σ_{i}. As the monitoring statistic is the maximum value among standardized sample means and weighted standardized sample ranges, the chart is called the MRMAX chart.
The parameter k is required to attend the imposed condition that, during the incontrol period, the statistics (Z_{1}, Z_{2}, ..., Z_{p}, W_{1}, W_{2}, ..., W_{p}) have the same probability to exceed CL, the control limit of the MRMAX chart.
The process is considered to start with the mean vector and the covariance matrix on target (μ = μ_{0} and Σ = Σ_{0}), where μ´_{0} = (μ_{1}, μ_{2}, ..., μ_{p}) and . The occurrence of the assignable cause changes the mean vector from μ´_{0} to μ´_{1} = (μ_{1} + δ_{1}σ_{1}; μ_{2} + δ_{2}σ_{2}; ...; μ_{p} + δ_{p}σ_{p}) and/or the covariance matrix from Σ_{0} to . The correlations , i, j = 1, 2, ..., p, with i ≠ j, are not affected by the assignable cause. After the occurrence of the assignable cause it is assumed that at least one δ_{i} becomes larger than zero and/or at least one a_{i} becomes larger than one, i = 1, 2, ..., p.
If the MRMAX statistic falls beyond the control limit, CL, the control chart signals an outofcontrol condition. Once the MRMAX chart signals, the user can immediately examine the sample means or the sample ranges of the p quality characteristics to discover which variable was affected by the assignable cause, that is, the one with the sample mean and/or the sample range larger than the control limit. For security, we might consider that the assignable cause has also affected an other variable when its sample mean and/or sample range is close, even though below the control limit.
3.1.1. The properties of the MRMAX chart
Let us consider the bivariate case to illustrate how the properties of the MRMAX chart were obtained, that is, its false alarm risk a and power of detection P. We used the property that the sample means are independent of the sample ranges. Therefore, the MRMAX chart has probability P of signaling:
Regarding to the sample means, the probability P_{M} of Z_{1} and/or Z_{2} exceeding the control limit is given by:
where f(Z_{1}, Z_{2}) is a standardized bivariate normal distribution function, reminding that CL is the control limit of the MRMAX chart. The probability P_{M} can be obtained for p > 2 using an algorithm given in Serel, Moskowitz and Tang (2000).
Regarding to the sample ranges, the probability P_{D} of W_{1} and/or W_{2} exceeding the control limit is:
where W_{1} = kR_{1}/σ_{1} and W_{2} = kR_{2}/σ_{2}, being R_{i} = max[x_{i1}, x_{i2}, ..., x_{in}] – min[xi1, xi2, ..., xin], i = 1, 2 the sample ranges of two quality characteristics X_{1} and X_{2}.
The incontrol covariance matrix is given by . The assignable cause changes the mean vector from μ´_{0} to μ´_{1} (μ_{1} + δ_{1}σ_{1}; μ_{2} + δ_{2}σ_{2};) and/or changes the covariance matrix from Σ_{0} to . The assignable cause does not affect the correlation between X_{1} and X_{2}, given by . We have that xi ~ N(μ_{i}, a_{i}σ_{i}), with i = 1, 2. Then,
being W_{1} = CL/a_{1} and W_{2} = CL/a_{2}. CL was divided by ai to hold the condition that x_{i}/a_{i}σ_{i} ~ N(μ_{i}/a_{i}σ_{i},1), with i = 1, 2.
Let (X_{1},X_{2}) be the pair of values of the two quality characteristics of each inspected item. Consider that y_{1} = min(X_{11}, X_{12}, ..., X_{1n}) and y_{2} = min(X_{21}, X_{22}, ..., X_{2n}) and two cases: a) (X_{1i} = y_{1}, X_{2i} =y_{2}), that is, the ith item of the sample has the minimum value of the two quality characteristics; b) (X_{1j} = y_{1}, X_{2j} > y_{2}) and (X_{1k} > y_{1}, X_{2k} = y_{2}) with j ≠ k ∈ {1, 2, ..., n}, that is, the jth item of the sample has the minimum value of the first quality characteristic and the kth one has the minimum value of the second quality characteristic. Based on that,
where
and
The probability D_{1} is given by:
The probability D_{2} is given by:
According to the conditional distribution , see Mood, Graybill and Boes (1974). Then, it follows that:
where is the standard normal cumulative distribution function.
The parameter k is obtained by expression (11), fixing α_{1} = α_{2} = 1 and , recalling that CL is computed by expression (10).
In the Appendix I we present the expression to compute P_{D} for the trivariate case with good accuracy, once the simulation technique was used only to obtain the probability of a signal given by all standardized sample ranges W_{1}, W_{2} and W_{3} falling beyond the control limit (CL). This probability is small if compared with the probability of a signal given by only one or two standardized sample ranges falling beyond the control limit. Following the calculation presented in the Appendix I, we can extend the study for the cases where p > 3.
When the rule of one point in the action region is adopted and the rate of false alarm is specified as a. Imposing equal incontrol P_{D} and P_{M} probabilities, the control limit of the MRMAX chart is obtained by expression (10), with δ_{1} = δ_{2} = 0 and The value of k is obtained by expression (12), with a_{1} = a_{2} = 1 and We used the subroutine BNRDF available on the IMSL Fortran library (MICROSOFT..., 1995) to compute the bivariate normal distribution function. The MRMAX chart is not always faster in signaling than the joint T^{2} and S charts. However, the synthetic procedure, introduced by Wu and Spedding (2000b), improves the performance of the MRMAX chart.
When the synthetic procedure is in use, a second sample point beyond the control limits and not far than L sampling intervals from the first one triggers the alarm, see Figure 1. According to this figure, the number of sampling intervals between the two points beyond CL is l = 3 and the design parameter L is equal to 5; consequently, the synthetic MRMAX chart signals once l < L.
The growing interest in using this rule may be explained by the fact that many practitioners prefer waiting until the occurrence of a second point beyond the control limits before looking for an assignable cause (see Wu and Spedding (2000a, 2000b), Wu, Zhang and Yeo (2001), Wu, Yeo and Spedding (2001), Calzada and Scariano (2001), Davis and Woodall (2002), Costa and Rahim (2006), Costa, Magalhães and Epprecht (2008)).
According to Davis and Woodall (2002) the proper parameter to measure the performance of the synthetic chart is the steadystate ARL, that is, the ARL value obtained when the process remains incontrol for a long time before the occurrence of the assignable cause.
We adopted the following approach to obtain the steadystate ARL for the synthetic MRMAX chart:
Specify , ARL_{0}, n, L, α_{1}, α_{2}, δ_{1} and δ_{2};
With L and ARL_{0} find the value of Q_{0} that makes
where M is a matrix of zeros except for m_{11}= Q_{0}; m_{12}= –Q_{0}; m_{i, i+1} = Q_{0}–1, i = 2,3,…,L; and m_{L+1,1} = Q_{0}–1; 1 is an (L+1) × 1 vector of ones; and s_{0} is an (L + 1) × 1 vector with s_{01} = [1 + (L  1) Q_{0}]^{1} and s_{0j} = Q_{0}s_{01}, j = 2,…,L+1.
With δ_{1} = δ_{2} = 0, r, n and equation (10) find the value of CL that makes P_{M} = 1 – Q_{0};
When the process is outofcontrol, the steadystate ARL is given by
where M is now a matrix of zeros except for m_{11} = Q; m_{12}= –Q; m_{i, i+1} = Q – 1, i = 2,…,L; and m_{L+1,1} = Q – 1, where Q = 1  P with P given by expression (9).
Table 2 shows the influence of the design parameter L on the MRMAX performance. As L increases the speed with which the synthetic MRMAX chart signals also increases. The gain in speed is more significant when L increases from 1 to 6. For instance, when a_{1} = a_{2} = 1 and δ_{1} = δ_{2} = 0.5, the ARL reduces 18.0% (from 21.4 to 17.5). On the other hand, as L increases from 6 to 10, the ARL reduces 1.0% (from 17.5 to 17.3), see Table 2.
3.2. The MVMAX chart
Section 5 compares the synthetic MRMAX chart with the synthetic MVMAX chart proposed by Machado, Costa and Marins (2009). The MVMAX statistic is based on the largest value among the standardized sample means (in module) and variances (weighted). For the bivariate case, MVMAX = max(Z_{1}, Z_{2}, W_{1}, W_{2}), where and
The MVMAX chart has also probability P of signaling, reminding that P = P_{M} + P_{D} – P_{M}P_{D}. The probability P_{M} is given by expression (10).
According to Machado, Costa and Marins (2009) the probability PD for the bivariate case is given by:
recalling that the notation (m) represents a noncentral chisquare distribution with n degrees of freedom and noncentrality parameter given by m. The parameter k is required to attend the imposed condition that, during the incontrol period, the standardized sample means and variances have the same probability to exceed CL, the control limit of the MVMAX chart.
The approach described in the Appendix I can be used to obtain the properties of the multivariate MVMAX chart.
3.3. The joint T^{2} and S charts
Section 5 also compares the MRMAX chart with the joint T^{2} and S charts. The T^{2} chart was introduced by Hotelling (1947) and it is the most common chart used to control the mean vector of multivariate processes.
Consider that p correlated characteristics are being measured simultaneously and when a sample of size n is taken we have n values of each characteristic and it is possible to calculate the vector, which represents the sample average vector for the p characteristics.
The charting statistic
is called Hotelling's T^{2} statistic. When the process is in control, T^{2} is distributed as a chisquare variate with p degrees of freedom, that is, T^{2} ~ . Consequently, the control limit for the T^{2} chart is CL = , where a is the type I error. When the process is outofcontrol, T^{2} is distributed as a noncentral chisquared distribution with p degrees of freedom and with noncentrality parameter , that is,
The first multivariate control chart for monitoring the covariance matrix Σ was based on the charting statistic obtained from the generalized likelihood ratio test, see Alt (1985). For the case of two variables Alt (1985) proposed the generalized variance S statistic to control the covariance matrix Σ. When the process is in control, is distributed as a chisquare variable with 2n – 4 degrees of freedom, where and S_{11} and S_{22} are the sample variances of X_{1} and X_{2} and S_{12} = S_{21} the sample covariances between X_{1} and X_{2}. Consequently, the upper control limit for the S chart is:
The magnitude of the disturbance is given by γ^{2}, where γ^{2} = Σ_{1}/Σ_{0}. The power of the S chart is given by:
According to expression (23), any change in the covariance matrix that generates the same γ^{2} is detected by the S chart with the same power (P_{D}), recalling that ARL = 1/P_{D}, see Aparisi, Jabaloyes and Carrón (2001).
When there are p variables to control, Anderson (2003) gave an expression for the distribution of S:
Aparisi, Jabaloyes and Carrón (1999) obtained the expression to compute the probability density function of . Based on J_{n,p}, the upper control limit for the S chart is:
and the chart's power is given by:
The drawback of the T^{2} and the S charts is that, in general, the user is not familiar with the computation of vectors and matrixes.
4. Illustrative example
in this section we provide an example to illustrate the ability of the synthetic MRMAX chart and the joint T^{2} and S charts in detecting shifts in the mean vector and/or in the covariance matrix. To this end, we considered a process with three quality characteristics that follow a trivariate normal distribution. When the process is incontrol, the mean vector and the covariance matrix are given by , , respectively.
We initially generate 5 samples of size n = 5 with the process in control. The remaining samples were simulated considering that the assignable cause changed the mean and the variability of X_{1}, that is, δ_{1} = 0.5 and a1 = 1.50.
Table 3 presents the data of X_{1}, X_{2}, X_{3}, Z_{1}, Z_{2}, Z_{3}, W_{1}, W_{2}, W_{3}, the largest value among (Z_{1}, Z_{2}, Z_{3}, W_{1}, W_{2}, W_{3}), that is, the MRMAX values, T^{2} and S.
Figure 2 shows the synthetic MRMAX chart with design parameters L = 7 and CL = 2.777. The control limit was determined using the algorithm given in Serel, Moskowitz and Tang (2000), with δ_{1} = δ_{2} =δ_{3}= 0, to assure a false alarm risk α of 0.005. As the number of sampling intervals l (=4) between the first and the second points beyond the CL is smaller than L (=7), the synthetic MRMAX control chart signals an outofcontrol condition at sample 12. We also conclude that the assignable cause affected the X_{1} variable once the sample points beyond the upper control limit correspond to the Z_{1} values.
5. Comparing charts
In this section we compare the synthetic MRMAX chart with the synthetic MVMAX chart. For the bivariate case, Machado, Costa and Marins (2009) have already compared the synthetic MVMAX chart with the joint T^{2} and S charts and it is, in general, faster in signaling, except when the correlation between the two variables is high.
Table 4 presents the steadystate ARLs for the synthetic MVMAX and MRMAX charts, where = 0.0; 0.5; 0.7, a_{1} and a_{2} = 1.0; 1.25; 1.5 and δ_{1} and δ_{2} = 0.0; 0.5; 0.75; 1.0. A type I risk of 0.5% is adopted. If it is well known that the assignable cause only affects the variability of one variable at a time, then the synthetic MRMAX chart is, in general, the best option. In the other cases, the MVMAX chart is, in general, faster in signaling than the MRMAX chart. However, even in these cases, the steadystate ARL reduction provided by the MVMAX chart is very small. For the case where the assignable cause does not affect the dispersion of the process, the MVMAX and MRMAX charts have the same performance for any shift in the mean vector. The correlation between the two variables has minor influence on the performance of both charts.
For the multivariate case, the comparison between the synthetic MVMAX and MRMAX charts led to the same conclusions.
Table 4 was built considering three different values of a1 and a2 and four different values of δ_{1} and δ_{2}. The orthogonal array is, in this case, made up of 144 combinations; however, these tables present only one half of the orthogonal array. The explanation is that the symmetric cases (a_{1} = c_{1}, a2 = c_{2}, δ_{1} = c_{3} and δ_{2} = c_{4}) and (a_{1} = c_{2}, a_{2} = c_{1}, δ_{1} = c4 and δ_{2} = c_{3}), with c_{1}, c_{2} ∈{1, 1.25, 1.5} and c_{3}, c_{4} ∈ {0, 0.5, 0.75, 1.0}, lead to the same steadystate ARL.
We also compare the synthetic MRMAX chart with the joint T^{2} and the S charts. Table 5 presents the ARLs for the joint T^{2} and the S charts and the synthetic MRMAX chart, considering the trivariate case. The synthetic MRMAX chart is, in general, faster in signaling than the joint T^{2} and the S charts, except in the case where the assignable cause affects the mean of only one variable at a time and does not affect the covariance matrix. The same is observed when .
We did not consider the joint T^{2} and the S charts based on the synthetic procedure, because it is not simple to deal with two charts and the rule of two points in the action region.
6. Conclusions
In this article we proposed a new chart for monitoring the mean vector and the covariance matrix of multivariate processes. The monitoring statistic associated to the MRMAX chart is based on the sample means and sample ranges. As the practitioners are, in general, more familiar with means and ranges, they will not have difficult to use the proposed chart. In terms of efficiency, we can say that, for the bivariate case, the MRMAX chart and the MVMAX chart, which is based on sample means and sample variances, have similar performance. This result differs from the univariate case, where the joint and R charts are slightly slower than the joint and S^{2} charts in signaling.
An interesting extension of this work is the development of a practical procedure to design the MRMAX chart considering α, p and n as input parameters.
Acknowledgements
This work was supported by CNPq – National Council for Scientific and Technological Development, Project 307744/20060 – and FAPESP – The State of São Paulo Research Foundation, Project 2008/099220. We would like to thank the two anonymous referees who carefully read an eARLier draft and made many constructive suggestions and Professor Philippe Castagliola, the organizer of the First International Symposium on Statistical Process Control (ISSPC 2009) that was held in Nantes, France, on July 1617, 2009.
Recebido 10/02/2010
Aceito 08/02/2011
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 ALT, F. B. Multivariate quality control. In: KOTZ, S.; JOHNSON, N. L. (Ed.). Encyclopedia of Statistical Sciences. Wiley, 1985.
 ANDERSON, T. W. An introduction to multivariate statistical analysis. Stanford: John Wiley, 2003.
 APARISI, F.; JABALOYES, J.; CARRIÓN, A. Statistical properties of the
 APARISI, F.; JABALOYES, J.; CARRIÓN, A. Generalized variance chart design with adaptive sample sizes. The bivariate case. Communication in Statistics – Simulation and Computation, v. 30, p. 931948, 2001.
 CALZADA, M. E.; SCARIANO, S. M. The robustness of the synthetic control chart to nonnormality. Communications in Statistics: Simulation and Computation, v. 30, p. 311‑326, 2001.
 CHEN, G.; CHENG, S. W.; XIE, H. A new multivariate control chart for monitoring both location and dispersion. Communications in StatisticsSimulation and Computation, v. 34, p. 203217, 2005.
 CHOU, C. Y. et al. Economicstatistical design of multivariate control charts using quality loss function. International Journal of Advanced Manufacturing Technology, v. 20, p. 916924, 2002.
 COSTA, A. F. B.; MACHADO, M. A. G. Synthetic control chart with twostage sampling for monitoring bivariate processes. Pesquisa Operacional, v. 27, p. 117130, 2007.
 COSTA, A. F. B.; MACHADO, M. A. G. A new chart for monitoring the covariance matrix of bivariate processes. Communications in Statistics – Simulation and Computation, v. 37, p. 14531465, 2008a.
 COSTA, A. F. B.; MACHADO, M. A. G. Bivariate control charts with double sampling. Journal of Applied Statistics, v. 35, p. 809822, 2008b.
 COSTA, A. F. B.; MACHADO, M. A. G. A new chart based on the sample variances for monitoring the covariance matrix of multivariate processes. International Journal of Advanced Manufacturing Technology, v. 41, p. 770779, 2009.
 COSTA, A. F. B.; RAHIM, M. A. A synthetic control chart for monitoring the process mean and variance. Journal of Quality In Maintenance Engineering, v. 12, p. 8188, 2006.
 COSTA, A. F. B.; EPPRECHT E. K.; CARPINETTI, L. C. R. Controle estatístico de qualidade. 2. ed. São Paulo: Atlas, 2005. 334 p.
 COSTA, A. F. B.; DE MAGALHÃES, M. S; EPPRECHT, E. K. Monitoring the process mean and variance using synthetic control chart with twostage testing. International Journal of Production Research, http://dx.doi.org/10.1080/00207540802047098, 2008.
 DAVIS, R. B.; WOODALL, W. H. Evaluating and improving the synthetic control chart. Journal of Quality Technology, v. 34, p. 200208, 2002.
 HOTELLING, H. Multivariate quality control, illustrated by the air testing of sample bombsights. Techniques of Statistical Analysis. New York: McGraw Hill, 1947. p. 111‑184.
 KHOO, M. B. C. A new bivariate control chart to monitor the multivariate process mean and variance simultaneously. Quality Engineering, v. 17, p. 109118, 2005.
 MACHADO, M. A. G; COSTA, A. F. B. The double sampling and the EWMA charts based on the sample variances. International Journal of Production Economics, v. 114, p. 134148, 2008a.
 MACHADO, M. A. G; COSTA, A. F. B. The use of principal components and simultaneous univariate charts to control multivariate processes. Pesquisa Operacional, v. 28, p. 173196, 2008b.
 MACHADO, M. A. G.; DE MAGALHÃES, M. S; COSTA, A. F. B. Gráfico de controle de VMAX para o monitoramento da matriz de covariâncias. Revista Produção, v. 18, p. 222‑239, 2008.
 MACHADO, M. A. G.; COSTA, A. F. B.; RAHIM, M. A. The synthetic control chart based on two sample variances for monitoring the covariance matrix. Quality and Reliability Engineering International, v. 25, p. 595606, 2008.
 MACHADO, M. A. G.; COSTA, A. F. B.; MARINS, F. A. S. Control charts for monitoring the mean vector and the covariance matrix of bivariate processes. International Journal of Advanced Manufacturing Technology, v. 45, p. 772785, 2009.
 MACHADO, M. A. G.; COSTA, A. F. B.; CLARO, F. A. E. Monitoring bivariate processes. Pesquisa Operacional, v. 29, p. 547562, 2009.
 MICROSOFT FORTRAN POWER STATION 4.0. Professional edition with Microsoft IMSL Mathematical and Statistical Libraries, Microsoft Corporation, 1995.
 MOOD, A. M.; GRAYBILL, F. A.; BOES, D. C Introduction to the theory of statistics. McGrawHill, 1974.
 TAKEMOTO, Y.; ARIZONO, I. A study of multivariate (X, S) control chart based on KullbackLeibler information. International Journal of Advanced Manufacturing Technology, v. 25, p. 12051210, 2005.
 SEREL, D. A.; MOSKOWITZ, H.; TANG, J. Univariate X control charts for individual characteristics in a multinormal model. IIE Transactions, v. 32, p. 11151125, 2000.
 WU, Z.; SPEDDING, T. A. Implementing synthetic control charts. Journal of Quality Technology, v. 32, p. 7578, 2000a.
 WU, Z.; SPEDDING, T. A. A synthetic control chart for detecting small shifts in the process mean. Journal of Quality Technology, v. 32, p. 3238, 2000b.
 WU, Z.; YEO, S. H.; SPEDDING, T. A. A synthetic control chart for detecting fraction nonconforming increases. Journal of Quality Technology, v. 33, p. 104111, 2001.
 WU, Z.; ZHANG, X.; YEO, S. H. Design of sumofconformingrunlength control charts. European Journal of Operational research, v. 132, p. 187196, 2001.
 ZHANG, G.; CHANG, S. I. Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis. International Journal of Production Research, v. 46, p. 68556881, 2008.
Publication Dates

Publication in this collection
17 June 2011 
Date of issue
June 2011
History

Received
10 Feb 2010 
Accepted
08 Feb 2011