Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tprs20 International Journal of Production Research ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: https://www.tandfonline.com/loi/tprs20 The T2 chart with mixed samples to control bivariate autocorrelated processes Roberto Campos Leoni, Marcela Aparecida Guerreiro Machado & Antonio Fernando Branco Costa To cite this article: Roberto Campos Leoni, Marcela Aparecida Guerreiro Machado & Antonio Fernando Branco Costa (2016) The T2 chart with mixed samples to control bivariate autocorrelated processes, International Journal of Production Research, 54:11, 3294-3310, DOI: 10.1080/00207543.2015.1102983 To link to this article: https://doi.org/10.1080/00207543.2015.1102983 Published online: 22 Oct 2015. Submit your article to this journal Article views: 165 View Crossmark data Citing articles: 5 View citing articles https://www.tandfonline.com/action/journalInformation?journalCode=tprs20 https://www.tandfonline.com/loi/tprs20 https://www.tandfonline.com/action/showCitFormats?doi=10.1080/00207543.2015.1102983 https://doi.org/10.1080/00207543.2015.1102983 https://www.tandfonline.com/action/authorSubmission?journalCode=tprs20&show=instructions https://www.tandfonline.com/action/authorSubmission?journalCode=tprs20&show=instructions http://crossmark.crossref.org/dialog/?doi=10.1080/00207543.2015.1102983&domain=pdf&date_stamp=2015-10-22 http://crossmark.crossref.org/dialog/?doi=10.1080/00207543.2015.1102983&domain=pdf&date_stamp=2015-10-22 https://www.tandfonline.com/doi/citedby/10.1080/00207543.2015.1102983#tabModule https://www.tandfonline.com/doi/citedby/10.1080/00207543.2015.1102983#tabModule The T2 chart with mixed samples to control bivariate autocorrelated processes Roberto Campos Leonia,b* , Marcela Aparecida Guerreiro Machadoa and Antonio Fernando Branco Costaa aProduction Department, São Paulo State University (UNESP), Guaratinguetá, Brazil; bMilitary Academy of Agulhas Negras, Resende, Brazil (Received 13 October 2014; accepted 19 September 2015) In this paper, we propose the use of the T2 chart with the mixed sampling strategy (MS) to monitor the mean vector of bivariate processes with observations that fit to a first-order vector autoregressive model. With the MS, rational sub- groups of size n are taken from the process and the selected units are regrouped to form the mixed samples. The units of the mixed samples are units selected from the last two rational subgroups. The aim of the proposed sampling strategy is to reduce the negative effect of the autocorrelation on the performance of the T2 chart. When the two variables are autocorrelated, the MS always enhances the T2 chart performance, however, the mixed samples are not recommended for bivariate processes with only one autocorrelated variable which is rarely affected by the assignable cause. Keywords: T2 control chart; autocorrelation; mixed samples; sampling strategies 1. Introduction The observations from multivariate processes are in general cross-correlated and, depending on the production rate, they are also autocorrelated (Pan and Jarrett 2007, 2011). The monitoring of multivariate process with autocorrelated vari- ables is a growing area of research (Kalgonda and Kulkarni 2004; Niaki and Davoodi 2009; Hwarng and Wang 2010; Kim, Jitpitaklert, and Sukchotrat 2010; Huang, Bisgaard, and Xu 2013; Huang, Xu, and Bisgaard 2013; Leoni et al. 2015; Leoni, Costa, and Machado 2015). Previous studies with single variables have suggested the adoption of innovative sampling strategies to counteract the negative effect of the autocorrelation on the performance of the �X chart. Costa and Castagliola (2011) considered a sampling strategy where the samples are formed by collecting one item from the production line and then skipping one, two or more items before selecting the next one. They showed that the undesired effect of the autocorrelation might be reduced by building up the samples with the non-neighbouring items according to the time they were produced. The effect of the autocorrelation is minimised by just skipping two items, in the case of moderate autocorrelation, or three, in the case of high autocorrelation. Alternatively, Franco et al. (2013) introduced the mixed sampling strategy (MS) where the samples are composed with units selected from the last two rational subgroups. A numerical analysis shows that the mixed sampling outperforms the skip sampling strategy for high levels of autocorrelation. Recently, Franco et al. (2014) investigated the economic-statistical design of the �X chart with the skip sampling strategy. In the presence of the autocorrelation, the sampling strategy based on the rational subgroup concept never outperforms the skip sam- pling strategies. The choice between the skip sampling strategy and the MS is guided by the autocorrelation level. When the autocorrelation is low, the skip sampling strategy is more economically convenient than the MS. Conversely, when the autocorrelation is high, the MS is the best option. In all these studies, the observations of the quality characteristic X are described by a first-order autoregressive model AR (1) and the process mean was assumed to switch between two values, the target one and the off-target value resulting of the assignable cause occurrence. In other studies with single variables, the AR (1) model has been used to describe a natural movement of the pro- cess mean (Reynolds, Arnold, and Baik 1996; Lu and Reynolds 1999, 2001; Lin and Chou 2008; Zou, Wang, and Tsung 2008; Lin 2009; Costa and Machado 2011; Franco, Costa, and Machado 2012). When the process mean wanders neither the skip nor the mixed sampling strategies reduce the effect of the autocorrelation on the �X chart’s performance. In this paper, we consider the use of the T2 chart to control the mean vector of bivariate processes with observations that are modelled by a first-order autoregressive model – VAR (1). To reduce the negative effect of the autocorrelation on the performance of the T2 chart, we propose the use of the MS. The next Section brings the VAR (1) model and the *Corresponding author. Email: rcleoni@yahoo.com.br © 2015 Taylor & Francis International Journal of Production Research, 2016 Vol. 54, No. 11, 3294–3310, http://dx.doi.org/10.1080/00207543.2015.1102983 http://orcid.org/0000-0001-6600-2963 http://orcid.org/0000-0001-6600-2963 http://orcid.org/0000-0001-6600-2963 mailto:rcleoni@yahoo.com.br http://dx.doi.org/10.1080/00207543.2015.1102983 cross-covariance matrix of the sample mean vector when the sample items are collected according to the rational subgroup concept. In Section 3, we obtain the cross-covariance matrix of the mixed sample mean vector. Section 4 brings the statistical properties of the T 2control chart and, in Section 5, we investigate the effect of the autocorrelation and the cross-correlation on the performance of the T2chart. Section 6 brings an example, and in Section 7 are the con- cluding remarks. 2. The VAR (1) model and the cross-covariance matrix The multivariate autoregressive model for cross-correlated and serially correlated data has been adopted in studies deal- ing with control charts (Mastrangelo and Forrest 2002; Biller and Nelson 2003; Kalgonda and Kulkarni 2004; Arkat, Niaki, and Abbasi 2007; Jarrett and Pan 2007; Issam and Mohamed 2008; Niaki and Davoodi 2009; Kim, Jitpitaklert, and Sukchotrat 2010; Hwarng and Wang 2010): Xt � l ¼ UðXt�1 � lÞ þ et (1) where Xt �Np l; Cð Þ is the p� 1ð Þ vector of observations at time t (p is the number of variables), l is the mean vector, et is an independent multivariate normal random vector with a mean vector of zeros and covariance matrix Re and U is a p� pð Þ matrix of autocorrelation parameters. According to Kalgonda and Kulkarni (2004), the cross-covariance matrix of Xt has the following property: C ¼ UCU0 þ Re. After some algebra, we obtain: VecC ¼ Ip2 �U�U � ��1 VecRe (2) where � is the Kronecker product and Vec is the operator that transforms a matrix into a vector by stacking its columns. To study the effects of the autocorrelation and cross-correlation on the performance of the T 2 chart, we considered the bivariate case (p = 2) with: U ¼ a c d b � � (3) Re ¼ r2eX reXY reXY r2eY � � ¼ r2eX qreX reY qreX reY r2eY � � (4) where ρ is the correlation of X and Y. when U ¼ diag a; bð Þ; follows from (2) and (4): C ¼ r2X rXY rXY r2Y � � ¼ 1� a2ð Þ�1 r2eX 1� abð Þ�1reXY 1� abð Þ�1reXY 1� b2ð Þ�1 r2eY ! (5) Leoni, Machado, and Costa (2014) obtained the cross-covariance matrix C�X of the sample mean vector �X when the sample items are collected according to the rational subgroup concept: C�X ¼ f2X textfXY fXY f2Y � � ¼ r2X n 1þ 2 n Pn�1 j¼1 ðn� jÞaj " # rXY n 1þ 1 n Pn�1 j¼1 n� jð Þaj þ 1 n Pn�1 j¼1 n� jð Þbj " # rXY n 1þ 1 n Pn�1 j¼1 n� jð Þaj þ 1 n Pn�1 j¼1 n� jð Þbj " # r2Y n 1þ 2 n Pn�1 j¼1 ðn� jÞbj " # 0BBBB@ 1CCCCA (6) where n is the size of the samples. The general cross-covariance matrix C�X is a function of U; n and Re. C�X ¼ f2X fXY fXY f2Y � � ¼ 1 n2 Xn�1 i¼0 Ui ! C Xn�1 i¼0 Ui !0 þ Xn�1 k¼1 Xk i¼1 Ui�1 ! Re Xk i¼1 Ui�1 !0" #( ) (7) It is well known that the autocorrelation has a negative impact on the performance of the �X chart; Leoni, Machado, and Costa (2014) proved that the autocorrelations also reduce the ability of the T2 chart to signal. International Journal of Production Research 3295 3. The mixed samples and the corresponding cross-covariance matrix When the MS is in use, the samples are composed with units selected from the last two rational subgroups. To illustrate, Figure 1 shows two rational subgroups (n = 5) and the mixed sample. The mixed sample i is composed with units of the current rational subgroup i (the first, third and the fifth ones according to the instant they were selected from the production line) and with units of the previous rational subgroup i − 1 (the second and fourth ones). The remaining units of the current rational subgroup, that is, the second and fourth units, are saved to compose the next mixed sample i + 1 and so on. The mean vector of the mixed sample i is given by: Mi ¼ ne n � 1 ne Xne t¼1 Xi�1;2t|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Yi�1 þ no n � 1 no Xno t¼1 Xi;2t�1|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Zi (8) where Xk;l is the l-esime observation vector of the k-esime rational subgroup; Yi;1 is the mean vector obtained with the even units of the (i − 1)-esime rational subgroup, and Zi is the mean vectors obtained with the odd units of the i-esime rational subgroup. When n is even, ne ¼ no ¼ n=2, and when n is odd, ne ¼ ðn� 1Þ=2 and no ¼ ðnþ 1Þ=2. As the samples are taken from the process according to the rational subgroup concept, Yi;1 and Zi are independent, consequently, the cross-covariance matrix C �M of the mixed sample mean vector M reduces to: CM ¼ f2X fXY fXY f2Y � � ¼ ne n � �2 �CY þ no n � �2 �C Z (9) where C�Y and C�Z are, respectively, the cross-covariance matrix of the mean vector of the samples with the even and with the odd units of the rational subgroups. According to the VAR (1) model: CZ ¼ 1 n2o Xno�1 l¼0 U2l ! C Xno�1 l¼0 U2l !0 þ Xno�1 k¼1 X2 j¼1 Xk l¼1 U2l�j ! Re Xk l¼1 U2l�j !0" #( )( ) (10) and CY ¼ 1 n2e Pne�1 l¼0 U2lþ1 � � C Pne�1 l¼0 U2lþ1 � �0 þ Pne�1 l¼0 U2l � � Re Pne�1 l¼0 U2l � �0 þ Pne�1 k¼1 P2 j¼1 Pk l¼1 U2l�j � � Re Pk l¼1 U2l�j � �0" #( ) 8>>><>>>: 9>>>=>>>; (11) To illustrate the computation of the cross-covariance matrix CM, let be n = 5, U ¼ 0:3 0 0 0:5 � � and Re ¼ 1 0:5 0:5 1 � � . According to expressions (8), (10) and (11): Mi ¼ 2 5 � 1 2 X2 t¼1 Xi�1;2t|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Yi�1 þ 3 5 � 1 3 X3 t¼1 Xi;2t�1|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Zi Figure 1. The mixed sample. 3296 R.C. Leoni et al. CY ¼ 1 4 Uþ U3 � � C Uþ U3 � �0þðIþ U2ÞReðIþ U2Þ0 þ UReU 0 þ Re h i CY ¼ 0:5989 0:3441 0:3441 0:8333 � � and CZ ¼ 1 9 Iþ U2 þ U4 � � C Iþ U2 þ U4 � �0þ Uþ U3 � � Re Uþ U3 � �0þ Iþ U2 � � Re Iþ U2 � �0þUReU 0 þ Re h i CZ ¼ 0:4122 0:2451 0:2451 0:6111 � � Finally, with expression (9): CM ¼ 2 5 � �2 CY þ 3 5 � �2 CZ CM ¼ 0:2442 0:1433 0:1433 0:3533 � � The inverse of the cross-covariance matrix CM is used to compute the T2 values. 4. The statistical properties of the T2 chart The Hotelling T 2 chart is used to monitor the mean vector of multivariate processes (Hotelling 1947). With the MS, the T2 statistic for bivariate processes, with the in-control mean vector l0 ¼ l0X; l0Y � �0 , is given by: T2 ¼ M� l0 � �0 C�1 M M� l0 � � (12) When the matrices Φ and Σe are known, the T2 statistic follows a chi-squared distribution with p = 2 degrees of free- dom (v2p). After the assignable cause occurrence, the mean vector changes to l1 ¼ l1X; l1Y � �0 and the distribution of the T 2 statistic changes to a non-central chi-squared distribution v2ðp;kÞ � � with non-centrality parameter v2ðp;kÞ, where d is the standardised mean vector shift d ¼ ðdX; dYÞ0 ¼ l1X�l0X reX ; l1Y�l0Y reY � �0 , see Wu and Makis (2008) and Franco, Costa, and Machado (2012). The type I error of the T2 chart is given by: a ¼ 1� Pr v2p\UCL � � (13) where UCL is the upper control limit of the chart. The UCL of the chart is chosen to be the (1 − α)th quantile of the chi-squared distribution to achieve a desired in-control average run length (ARL) of 1/α (Champ, Jones-Farmer, and Rigdon 2005). As the assignable cause always occurs between rational subgroups, i.e. i − 1 and i, it follows that Yi�1 �N2 l0;CY and Zi �N2 l1;CZ½ �, consequently, the type II error of the mixed sample i is: b1 ¼ Pr v2p;k1ð Þ\UCL � � (14) where k1 ¼ d0C�1 M d with d ¼ no n ðdX; dYÞ0. The type II error of the subsequent mixed samples i + 1, i + 2 … is: b2 ¼ Pr v2p;k2ð Þ\UCL � � (15) where k2 ¼ d0C�1 M d with d ¼ ðdX; dYÞ0. According to Franco et al. (2013), the out of control ARL and the non-central moment of order 2 of the run length (SDRL) is given by: International Journal of Production Research 3297 Table 1. The ARL values for the T2 control chart; n = 3. a 0.3 0.5 0.7 0.9 0.3 0 b 0.3 0.5 0.7 0.9 0.9 0.5 c 0 0 0 0 0 0 d 0 0 0 0 0 0 ρ δx δy STD MS STD MS STD MS STD MS STD MS STD MS 0.3 0 0 370.40 370.40 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 127.1 96.6 171.0 124.3 231.4 177.9 315.57 281.64 318.46 286.05 172.9 126.8 0 1 26.7 17.1 47.2 26.1 91.6 51.6 211.3 151.57 217.42 158.26 48.3 27.0 0 1.5 7.4 5.1 14.7 7.8 35.6 16.9 126.93 74.68 133.21 79.82 15.1 8.0 0.5 0 127.1 96.6 171.0 124.3 231.4 177.9 315.57 281.64 133.34 102.45 84.2 85.3 0.5 0.5 94.7 68.6 135.7 92.4 198.1 142.5 297.29 255.51 134.21 101.35 79.2 68.9 0.5 1 28.7 18.5 50.3 28.1 96.2 54.9 216.4 156.97 114.55 78.7 38.7 24.8 0.5 1.5 8.6 5.8 17.1 8.9 40.3 19.4 136.45 82.26 86.01 51.93 15.4 8.6 1 0 26.7 17.1 47.2 26.1 91.6 51.6 211.3 151.57 29.13 18.82 13.3 14.1 1 0.5 28.7 18.5 50.3 28.1 96.2 54.9 216.4 156.97 30.51 19.69 14.9 15.2 1 1 16.0 10.3 30.1 15.9 64.3 33.5 176.53 117.18 29.49 18.49 12.1 10.3 1 1.5 6.9 4.8 13.7 7.3 33.5 15.9 122.55 71.28 26.4 15.76 7.7 5.7 1.5 0 7.4 5.1 14.7 7.8 35.6 16.9 126.93 74.68 8.2 5.58 3.5 4.3 1.5 0.5 8.6 5.8 17.1 8.9 40.3 19.4 136.45 82.26 8.59 5.81 4.0 4.7 1.5 1 6.9 4.8 13.7 7.3 33.5 15.9 122.55 71.28 8.63 5.78 3.8 4.3 1.5 1.5 4.3 3.3 8.5 4.8 22.0 10.2 94.54 50.9 8.32 5.49 3.2 3.4 0.6 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 93.3 67.4 134.1 91.0 196.5 140.9 296.34 254.2 313.44 279.17 144.1 102.7 0 1 15.6 10.0 29.5 15.6 63.2 32.8 174.89 115.65 206.91 147.95 33.7 18.9 0 1.5 4.2 3.3 8.3 4.7 21.5 9.9 93.15 49.94 122.54 71.97 9.7 5.6 0.5 0 93.3 67.4 134.1 91.0 196.5 140.9 296.34 254.2 122.67 93.48 62.7 65.7 0.5 0.5 114.2 85.3 157.4 111.6 219.0 164.3 309.08 272.19 136.63 105.22 89.8 82.5 0.5 1 29.9 19.3 52.1 29.2 98.8 56.7 219.11 159.87 126.12 89.38 48.2 29.3 0.5 1.5 7.3 5.0 14.5 7.6 35.0 16.7 125.81 73.81 98.63 60.92 17.1 8.7 1 0 15.6 10.0 29.5 15.6 63.2 32.8 174.89 115.65 25.03 16.25 8.5 9.7 1 0.5 29.9 19.3 52.1 29.2 98.8 56.7 219.11 159.87 28.77 18.89 13.4 15.2 1 1 22.1 14.1 40.0 21.7 80.5 44.0 198.21 138.15 30.48 19.66 14.7 13.4 1 1.5 8.6 5.8 16.9 8.9 40.0 19.3 135.86 81.79 29.55 18.18 10.5 7.4 1.5 0 4.2 3.3 8.3 4.7 21.5 9.9 93.15 49.94 6.89 4.87 2.4 3.2 1.5 0.5 7.3 5.0 14.5 7.6 35.0 16.7 125.81 73.81 7.73 5.4 3.2 4.2 1.5 1 8.6 5.8 16.9 8.9 40.0 19.3 135.86 81.79 8.36 5.75 3.9 4.7 1.5 1.5 6.0 4.3 12.0 6.4 29.7 13.9 114.1 64.89 8.64 5.82 3.9 4.1 0.9 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 20.4 13.0 37.3 20.1 76.3 41.2 192.9 132.88 302.53 264.82 74.1 53.0 0 1 2.3 2.2 4.3 2.9 11.4 5.5 60.74 29.43 185.85 128.49 10.9 7.3 0 1.5 1.1 1.4 1.4 1.6 3.1 2.2 20.36 8.81 102.7 58.25 2.9 2.6 0.5 0 20.4 13.0 37.3 20.1 76.3 41.2 192.9 132.88 102.82 77.4 23.5 29.8 0.5 0.5 131.4 100.5 175.6 128.7 235.4 182.4 317.6 284.64 130.82 102.75 83.3 91.3 0.5 1 12.0 7.8 23.2 12.1 52.0 26.1 157.5 99.98 134.33 98.44 52.8 27.4 0.5 1.5 1.9 2.0 3.4 2.5 9.0 4.5 51.42 24.14 110.33 69.44 11.8 5.8 1 0 2.3 2.2 4.3 2.9 11.4 5.5 60.74 29.43 18.39 12.2 2.6 4.0 1 0.5 12.0 7.8 23.2 12.1 52.0 26.1 157.5 99.98 23.6 15.95 6.1 9.7 1 1 28.4 18.3 49.8 27.8 95.5 54.3 215.58 156.1 28.13 18.91 13.0 15.7 1 1.5 5.7 4.1 11.4 6.2 28.6 13.3 111.29 62.81 30.43 19.61 13.6 8.9 1.5 0 1.1 1.4 1.4 1.6 3.1 2.2 20.36 8.81 4.92 3.81 1.1 1.9 1.5 0.5 1.9 2.0 3.4 2.5 9.0 4.5 51.42 24.14 5.93 4.46 1.5 2.6 1.5 1 5.7 4.1 11.4 6.2 28.6 13.3 111.29 62.81 6.96 5.11 2.3 3.9 1.5 1.5 8.0 5.4 15.8 8.3 37.7 18.1 131.3 78.13 7.87 5.61 3.5 4.7 3298 R.C. Leoni et al. Table 2. The ARL values for the T2 control chart; n = 3. a 0.3 0.1 0.7 0.2 0.3 b 0.3 0.1 0.7 0.2 0.3 c 0.1 0.3 0.2 0.7 0.3 d 0.1 0.3 0.2 0.7 0.3 ρ δx δy STD MS STD MS STD MS STD MS STD MS 0.3 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 121.8 95.6 76.34 88.99 216.3 169.5 49.12 114.91 105.8 96.9 0 1 24.7 16.8 11.37 15.05 78.3 46.8 6.04 22.8 19.3 17.2 0 1.5 6.8 5.0 3.05 4.55 28.6 15.0 1.8 6.74 5.2 5.1 0.5 0 121.8 95.6 76.34 88.99 216.3 169.5 49.12 114.91 105.8 96.9 0.5 0.5 109.4 74.9 109.43 74.86 284.8 235.5 284.8 235.45 151.2 100.4 0.5 1 33.0 20.1 25.51 19.46 150.0 97.6 41.16 72.6 43.0 27.2 0.5 1.5 9.4 6.1 5.72 5.79 56.7 30.3 5.53 16.79 10.4 7.5 1 0 24.7 16.8 11.37 15.05 78.3 46.8 6.04 22.8 19.3 17.2 1 0.5 33.0 20.1 25.51 19.46 150.0 97.6 41.16 72.6 43.0 27.2 1 1 20.5 11.6 20.46 11.62 156.2 95.8 156.19 95.78 37.0 18.2 1 1.5 8.6 5.3 7.54 5.24 85.4 45.0 30.09 36.77 14.4 7.7 1.5 0 6.8 5.0 3.05 4.55 28.6 15.0 1.8 6.74 5.2 5.1 1.5 0.5 9.4 6.1 5.72 5.79 56.7 30.3 5.53 16.79 10.4 7.5 1.5 1 8.6 5.3 7.54 5.24 85.4 45.0 30.09 36.77 14.4 7.7 1.5 1.5 5.5 3.7 5.51 3.67 78.0 38.2 77.97 38.21 10.9 5.4 0.6 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 85.0 65.1 44.42 58.81 164.8 119.1 23.13 69.53 65.8 62.6 0 1 13.5 9.6 5.3 8.36 43.8 24.3 2.55 10.47 9.1 9.1 0 1.5 3.6 3.2 1.65 2.87 13.4 7.2 1.13 3.38 2.5 3.0 0.5 0 85.0 65.1 44.42 58.81 164.8 119.1 23.13 69.53 65.8 62.6 0.5 0.5 130.1 92.3 130.05 92.34 298.2 253.9 298.21 253.94 173.2 120.3 0.5 1 32.2 20.3 21.02 19.22 126.7 80.8 20.91 51.54 35.4 25.1 0.5 1.5 7.2 5.1 3.63 4.71 35.8 18.7 2.45 8.92 6.4 5.6 1 0 13.5 9.6 5.3 8.36 43.8 24.3 2.55 10.47 9.1 9.1 1 0.5 32.2 20.3 21.02 19.22 126.7 80.8 20.91 51.54 35.4 25.1 1 1 27.8 15.9 27.82 15.94 178.1 115.4 178.13 115.36 48.4 24.7 1 1.5 10.3 6.3 8.01 6.15 82.1 44.5 17.32 31.65 15.0 8.8 1.5 0 3.6 3.2 1.65 2.87 13.4 7.2 1.13 3.38 2.5 3.0 1.5 0.5 7.2 5.1 3.63 4.71 35.8 18.7 2.45 8.92 6.4 5.6 1.5 1 10.3 6.3 8.01 6.15 82.1 44.5 17.32 31.65 15.0 8.8 1.5 1.5 7.8 4.8 7.77 4.79 95.9 49.8 95.91 49.76 15.2 7.3 0.9 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 16.6 12.0 5.97 10.31 46.3 26.1 2.57 10.95 10.2 10.7 0 1 1.9 2.1 1.11 1.98 5.6 3.5 1 2.03 1.4 2.0 0 1.5 1.0 1.3 1 1.25 1.7 1.8 1 1.28 1.0 1.3 0.5 0 16.6 12.0 5.97 10.31 46.3 26.1 2.57 10.95 10.2 10.7 0.5 0.5 147.8 108.2 147.84 108.19 308.0 268.0 307.98 267.99 191.2 137.7 0.5 1 10.9 7.6 4.68 6.82 41.1 22.4 2.52 9.96 8.1 7.7 0.5 1.5 1.7 2.0 1.09 1.84 5.3 3.4 1 2 1.3 1.9 1 0 1.9 2.1 1.11 1.98 5.6 3.5 1 2.03 1.4 2.0 1 0.5 10.9 7.6 4.68 6.82 41.1 22.4 2.52 9.96 8.1 7.7 1 1 35.4 20.6 35.39 20.59 196.1 132.6 196.08 132.57 59.6 31.4 1 1.5 5.8 4.2 3.19 3.97 33.0 16.9 2.42 8.38 5.6 4.8 1.5 0 1.0 1.3 1 1.25 1.7 1.8 1 1.28 1.0 1.3 1.5 0.5 1.7 2.0 1.09 1.84 5.3 3.4 1 2 1.3 1.9 1.5 1 5.8 4.2 3.19 3.97 33.0 16.9 2.42 8.38 5.6 4.8 1.5 1.5 10.3 6.1 10.32 6.09 112.1 61.0 112.09 61.01 19.9 9.5 International Journal of Production Research 3299 Table 3. The SDRL values for the T2 control chart; n = 3. a 0.3 0.5 0.7 0.9 0.3 0 b 0.3 0.5 0.7 0.9 0.9 0.5 c 0 0 0 0 0 0 d 0 0 0 0 0 0 ρ δx δy STD MS STD MS STD MS STD MS STD MS STD MS 0.3 0 0 369.90 369.90 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 126.6 95.6 170.5 123.4 230.9 177.1 315.07 281 317.96 285.42 172.4 125.9 0 1 26.2 15.9 46.7 25.0 91.1 50.5 210.8 150.72 216.92 157.42 47.8 25.9 0 1.5 6.9 3.9 14.2 6.5 35.1 15.7 126.43 73.67 132.71 78.82 14.6 6.8 0.5 0 126.6 95.6 170.5 123.4 230.9 177.1 315.07 281 132.84 101.5 83.7 84.3 0.5 0.5 94.2 67.5 135.2 91.4 197.6 141.7 296.78 254.84 133.71 100.4 78.7 67.9 0.5 1 28.2 17.3 49.8 27.0 95.7 53.8 215.9 156.12 114.05 77.7 38.2 23.6 0.5 1.5 8.1 4.6 16.5 7.7 39.8 18.3 135.95 81.27 85.51 50.87 14.9 7.4 1 0 26.2 15.9 46.7 25.0 91.1 50.5 210.8 150.72 28.63 17.65 12.8 12.9 1 0.5 28.2 17.3 49.8 27.0 95.7 53.8 215.9 156.12 30 18.52 14.4 14.0 1 1 15.5 9.1 29.6 14.8 63.8 32.3 176.03 116.26 28.98 17.32 11.5 9.1 1 1.5 6.4 3.6 13.2 6.0 33.0 14.7 122.04 70.26 25.9 14.58 7.1 4.5 1.5 0 6.9 3.9 14.2 6.5 35.1 15.7 126.43 73.67 7.68 4.34 3.0 3.1 1.5 0.5 8.1 4.6 16.5 7.7 39.8 18.3 135.95 81.27 8.07 4.57 3.4 3.5 1.5 1 6.4 3.6 13.2 6.0 33.0 14.7 122.04 70.26 8.11 4.54 3.3 3.0 1.5 1.5 3.7 2.1 8.0 3.6 21.5 8.9 94.04 49.83 7.8 4.25 2.7 2.1 0.6 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 92.8 66.4 133.6 90.0 196.0 140.0 295.84 253.52 312.95 278.53 143.6 101.8 0 1 15.1 8.8 28.9 14.4 62.7 31.6 174.39 114.73 206.41 147.09 33.2 17.7 0 1.5 3.6 2.0 7.8 3.5 21.0 8.7 92.64 48.87 122.04 70.95 9.2 4.4 0.5 0 92.8 66.4 133.6 90.0 196.0 140.0 295.84 253.52 122.17 92.51 62.2 64.7 0.5 0.5 113.7 84.3 156.9 110.7 218.5 163.5 308.58 271.53 136.13 104.28 89.3 81.5 0.5 1 29.4 18.1 51.6 28.1 98.3 55.6 218.61 159.03 125.62 88.4 47.7 28.1 0.5 1.5 6.8 3.8 14.0 6.4 34.5 15.5 125.31 72.8 98.13 59.88 16.6 7.5 1 0 15.1 8.8 28.9 14.4 62.7 31.6 174.39 114.73 24.52 15.06 8.0 8.5 1 0.5 29.4 18.1 51.6 28.1 98.3 55.6 218.61 159.03 28.27 17.72 12.9 14.0 1 1 21.5 12.9 39.5 20.5 80.0 42.9 197.7 137.27 29.98 18.49 14.2 12.2 1 1.5 8.0 4.5 16.4 7.6 39.5 18.1 135.36 80.79 29.05 17 10.0 6.2 1.5 0 3.6 2.0 7.8 3.5 21.0 8.7 92.64 48.87 6.37 3.63 1.8 2.0 1.5 0.5 6.8 3.8 14.0 6.4 34.5 15.5 125.31 72.8 7.22 4.16 2.6 3.0 1.5 1 8.0 4.5 16.4 7.6 39.5 18.1 135.36 80.79 7.84 4.51 3.3 3.5 1.5 1.5 5.5 3.1 11.5 5.2 29.2 12.7 113.59 63.85 8.13 4.58 3.4 2.9 0.9 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 19.9 11.8 36.8 19.0 75.8 40.1 192.4 131.99 302.03 264.16 73.6 51.9 0 1 1.7 1.0 3.7 1.7 10.9 4.2 60.23 28.29 185.35 127.59 10.4 6.1 0 1.5 0.3 0.5 0.8 0.6 2.5 1.0 19.86 7.59 102.2 57.2 2.4 1.4 0.5 0 19.9 11.8 36.8 19.0 75.8 40.1 192.4 131.99 102.32 76.4 23.0 28.7 0.5 0.5 130.9 99.6 175.1 127.8 234.9 181.6 317.1 284.01 130.32 101.8 82.8 90.3 0.5 1 11.5 6.6 22.7 10.9 51.5 24.9 157 99.03 133.83 97.48 52.3 26.2 0.5 1.5 1.3 0.8 2.9 1.3 8.5 3.3 50.91 22.99 109.83 68.42 11.3 4.5 1 0 1.7 1.0 3.7 1.7 10.9 4.2 60.23 28.29 17.89 11 2.0 2.7 1 0.5 11.5 6.6 22.7 10.9 51.5 24.9 157 99.03 23.1 14.77 5.6 8.5 1 1 27.9 17.1 49.3 26.6 95.0 53.3 215.08 155.25 27.62 17.74 12.5 14.5 1 1.5 5.2 2.9 10.9 4.9 28.0 12.1 110.79 61.77 29.92 18.44 13.1 7.6 1.5 0 0.3 0.5 0.8 0.6 2.5 1.0 19.86 7.59 4.39 2.57 0.4 0.7 1.5 0.5 1.3 0.8 2.9 1.3 8.5 3.3 50.91 22.99 5.41 3.22 0.9 1.4 1.5 1 5.2 2.9 10.9 4.9 28.0 12.1 110.79 61.77 6.44 3.86 1.7 2.7 1.5 1.5 7.4 4.2 15.3 7.1 37.2 16.9 130.8 77.13 7.36 4.37 2.9 3.5 3300 R.C. Leoni et al. Table 4. The SDRL values for the T2 control chart; n = 3. a 0.3 0.1 0.7 0.2 0.3 b 0.3 0.1 0.7 0.2 0.3 c 0.1 0.3 0.2 0.7 0.3 d 0.1 0.3 0.2 0.7 0.3 ρ δx δy STD MS STD MS STD MS STD MS STD MS 0.3 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 121.3 94.6 75.84 88.02 215.8 168.7 48.62 113.99 105.3 95.9 0 1 24.2 15.6 10.86 13.86 77.8 45.7 5.52 21.64 18.8 16.0 0 1.5 6.3 3.8 2.5 3.31 28.1 13.8 1.2 5.51 4.7 3.9 0.5 0 121.3 94.6 75.84 88.02 215.8 168.7 48.62 113.99 105.3 95.9 0.5 0.5 108.9 73.9 108.93 73.85 284.3 234.7 284.3 234.74 150.7 99.4 0.5 1 32.5 18.9 25 18.29 149.5 96.6 40.66 71.59 42.5 26.0 0.5 1.5 8.9 4.9 5.2 4.56 56.2 29.2 5 15.61 9.9 6.3 1 0 24.2 15.6 10.86 13.86 77.8 45.7 5.52 21.64 18.8 16.0 1 0.5 32.5 18.9 25 18.29 149.5 96.6 40.66 71.59 42.5 26.0 1 1 20.0 10.4 19.96 10.42 155.7 94.8 155.69 94.82 36.5 17.0 1 1.5 8.1 4.1 7.02 4 84.8 44.0 29.59 35.66 13.9 6.5 1.5 0 6.3 3.8 2.5 3.31 28.1 13.8 1.2 5.51 4.7 3.9 1.5 0.5 8.9 4.9 5.2 4.56 56.2 29.2 5 15.61 9.9 6.3 1.5 1 8.1 4.1 7.02 4 84.8 44.0 29.59 35.66 13.9 6.5 1.5 1.5 5.0 2.4 4.99 2.43 77.5 37.1 77.47 37.11 10.4 4.2 0.6 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 84.5 64.0 43.92 57.76 164.3 118.2 22.62 68.51 65.3 61.5 0 1 12.9 8.3 4.78 7.13 43.3 23.1 1.98 9.26 8.6 7.8 0 1.5 3.0 1.9 1.03 1.64 12.9 6.0 0.38 2.14 1.9 1.8 0.5 0 84.5 64.0 43.92 57.76 164.3 118.2 22.62 68.51 65.3 61.5 0.5 0.5 129.6 91.4 129.55 91.37 297.7 253.3 297.7 253.26 172.7 119.4 0.5 1 31.7 19.1 20.52 18.05 126.2 79.8 20.4 50.47 34.9 24.0 0.5 1.5 6.7 3.9 3.09 3.46 35.3 17.5 1.89 7.7 5.8 4.4 1 0 12.9 8.3 4.78 7.13 43.3 23.1 1.98 9.26 8.6 7.8 1 0.5 31.7 19.1 20.52 18.05 126.2 79.8 20.4 50.47 34.9 24.0 1 1 27.3 14.8 27.32 14.75 177.6 114.4 177.63 114.43 47.9 23.5 1 1.5 9.7 5.1 7.5 4.91 81.6 43.4 16.81 30.52 14.5 7.6 1.5 0 3.0 1.9 1.03 1.64 12.9 6.0 0.38 2.14 1.9 1.8 1.5 0.5 6.7 3.9 3.09 3.46 35.3 17.5 1.89 7.7 5.8 4.4 1.5 1 9.7 5.1 7.5 4.91 81.6 43.4 16.81 30.52 14.5 7.6 1.5 1.5 7.3 3.6 7.26 3.55 95.4 48.7 95.41 48.68 14.7 6.1 0.9 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 16.0 10.8 5.45 9.09 45.8 25.0 2.01 9.74 9.6 9.5 0 1 1.3 1.0 0.35 0.82 5.1 2.3 0.06 0.87 0.7 0.9 0 1.5 0.2 0.5 0.01 0.44 1.1 0.7 0 0.45 0.1 0.4 0.5 0 16.0 10.8 5.45 9.09 45.8 25.0 2.01 9.74 9.6 9.5 0.5 0.5 147.3 107.3 147.34 107.25 307.5 267.3 307.49 267.34 190.7 136.9 0.5 1 10.4 6.4 4.15 5.59 40.6 21.2 1.96 8.75 7.6 6.5 0.5 1.5 1.1 0.8 0.31 0.72 4.8 2.1 0.06 0.84 0.6 0.8 1 0 1.3 1.0 0.35 0.82 5.1 2.3 0.06 0.87 0.7 0.9 1 0.5 10.4 6.4 4.15 5.59 40.6 21.2 1.96 8.75 7.6 6.5 1 1 34.9 19.4 34.89 19.42 195.6 131.7 195.58 131.68 59.1 30.3 1 1.5 5.3 3.0 2.64 2.73 32.5 15.7 1.85 7.16 5.1 3.6 1.5 0 0.2 0.5 0.01 0.44 1.1 0.7 0 0.45 0.1 0.4 1.5 0.5 1.1 0.8 0.31 0.72 4.8 2.1 0.06 0.84 0.6 0.8 1.5 1 5.3 3.0 2.64 2.73 32.5 15.7 1.85 7.16 5.1 3.6 1.5 1.5 9.8 4.9 9.81 4.85 111.6 60.0 111.58 59.97 19.4 8.2 International Journal of Production Research 3301 Table 5. The ARL values for the T2 control chart; n = 5. a 0.3 0.5 0.7 0.9 0.3 0 b 0.3 0.5 0.7 0.9 0.9 0.5 c 0 0 0 0 0 0 d 0 0 0 0 0 0 ρ δx δy STD MS STD MS STD MS STD MS STD MS STD MS 0.3 0 0 370.40 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 88.02 59.47 137.69 88.28 211.75 149.26 312.11 271.46 314.94 276.04 138.95 90.29 0 1 14.23 8.54 30.93 14.92 74.6 36.54 204.22 137.2 209.98 143.5 31.46 15.44 0 1.5 3.78 2.98 8.79 4.57 26.79 11.26 119.89 64.24 125.6 68.73 8.97 4.71 0.5 0 88.02 59.47 137.69 88.28 211.75 149.26 312.11 271.46 93.05 63.75 46.84 48.13 0.5 0.5 61.37 39.29 104.25 61.75 177.01 115.03 292.92 243.47 95.67 64.7 46.2 39.37 0.5 1 15.44 9.22 33.22 16.11 78.75 39.09 209.41 142.5 84.63 52.48 22.56 13.5 0.5 1.5 4.39 3.29 10.25 5.17 30.6 12.97 129.26 71.19 66.08 36.18 8.87 4.95 1 0 14.23 8.54 30.93 14.92 74.6 36.54 204.22 137.2 15.58 9.35 5.68 6.58 1 0.5 15.44 9.22 33.22 16.11 78.75 39.09 209.41 142.5 16.45 9.83 6.5 7.16 1 1 8.22 5.26 18.83 8.97 50.66 22.89 169.13 103.9 16.3 9.54 5.58 5.27 1 1.5 3.53 2.86 8.18 4.32 25.15 10.55 115.59 61.15 15.18 8.59 3.86 3.36 1.5 0 3.78 2.98 8.79 4.57 26.79 11.26 119.89 64.24 4.14 3.17 1.72 2.53 1.5 0.5 4.39 3.29 10.25 5.17 30.6 12.97 129.26 71.19 4.33 3.27 1.88 2.7 1.5 1 3.53 2.86 8.18 4.32 25.15 10.55 115.59 61.15 4.4 3.29 1.87 2.56 1.5 1.5 2.29 2.24 5.04 3.08 16.1 6.82 88.32 42.88 4.34 3.22 1.7 2.24 0.6 0 0 370.4 370.40 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 60.28 38.5 102.79 60.67 175.37 113.51 291.92 242.07 308.79 267.84 109.17 69.71 0 1 8.01 5.15 18.39 8.76 49.7 22.38 167.49 102.45 197.65 132.39 20.38 10.56 0 1.5 2.24 2.21 4.92 3.03 15.71 6.66 86.97 42.03 113.57 60.91 5.49 3.47 0.5 0 60.28 38.5 102.79 60.67 175.37 113.51 291.92 242.07 82.52 56.3 31.54 34.82 0.5 0.5 77.17 51.04 124.53 77.48 198.67 135.89 305.29 261.27 95.33 66.04 51.84 48.26 0.5 1 16.12 9.61 34.5 16.79 81.02 40.52 212.17 145.37 93.07 59.94 29.7 16.51 0.5 1.5 3.72 2.95 8.63 4.51 26.37 11.08 118.79 63.45 77.24 43.56 10.27 5.12 1 0 8.01 5.15 18.39 8.76 49.7 22.38 167.49 102.45 12.84 7.96 3.52 4.66 1 0.5 16.12 9.61 34.5 16.79 81.02 40.52 212.17 145.37 14.87 9.18 5.49 6.99 1 1 11.56 7.06 25.71 12.28 64.74 30.69 190.95 124.09 16.21 9.81 6.5 6.6 1 1.5 4.35 3.27 10.15 5.13 30.36 12.86 128.68 70.75 16.47 9.59 5.24 4.17 1.5 0 2.24 2.21 4.92 3.03 15.71 6.66 86.97 42.03 3.42 2.85 1.29 2.1 1.5 0.5 3.72 2.95 8.63 4.51 26.37 11.08 118.79 63.45 3.79 3.05 1.55 2.48 1.5 1 4.35 3.27 10.15 5.13 30.36 12.86 128.68 70.75 4.1 3.2 1.8 2.69 1.5 1.5 3.1 2.64 7.1 3.89 22.16 9.27 107.32 55.36 4.31 3.28 1.9 2.54 0.9 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 10.63 6.55 23.85 11.36 61.06 28.59 185.6 118.98 294.86 250.48 42.94 31.44 0 1 1.41 1.76 2.62 2.16 8.16 3.88 55.94 24.25 172.38 111.47 5.08 4.23 0 1.5 1.01 1.19 1.14 1.43 2.28 1.93 18.29 7.23 91.02 47.39 1.6 2.01 0.5 0 10.63 6.55 23.85 11.36 61.06 28.59 185.6 118.98 63.57 43.53 8.32 13.73 0.5 0.5 91.81 62.47 142.16 92.05 216.03 153.78 314.26 274.71 85.75 61.32 39.58 52.31 0.5 1 6.1 4.16 14.2 6.87 40.27 17.56 150.12 87.65 96.41 65.14 35.18 16.22 0.5 1.5 1.26 1.65 2.16 1.98 6.47 3.31 47.13 19.79 87.28 50.87 7.12 3.68 1 0 1.41 1.76 2.62 2.16 8.16 3.88 55.94 24.25 8.65 5.87 1.25 2.37 1 0.5 6.1 4.16 14.2 6.87 40.27 17.56 150.12 87.65 11.12 7.41 2.14 4.39 1 1 15.24 9.11 32.84 15.91 78.07 38.67 208.58 141.64 13.65 8.88 4.59 7.27 1 1.5 2.96 2.57 6.77 3.76 21.22 8.88 104.59 53.49 15.66 9.76 6.47 5.01 1.5 0 1.01 1.19 1.14 1.43 2.28 1.93 18.29 7.23 2.39 2.37 1 1.52 1.5 0.5 1.26 1.65 2.16 1.98 6.47 3.31 47.13 19.79 2.78 2.6 1.03 1.83 1.5 1 2.96 2.57 6.77 3.76 21.22 8.88 104.59 53.49 3.2 2.84 1.16 2.3 1.5 1.5 4.05 3.12 9.44 4.84 28.5 12.02 124.2 67.4 3.63 3.06 1.5 2.69 3302 R.C. Leoni et al. Table 6. The ARL values for the T2 control chart; n = 5. a 0.3 0.1 0.7 0.2 0.3 b 0.3 0.1 0.7 0.2 0.3 c 0.1 0.3 0.2 0.7 0.3 d 0.1 0.3 0.2 0.7 0.3 ρ δx δy STD MS STD MS STD MS STD MS STD MS 0.3 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 83.08 58.53 36.03 49.82 201.74 144.03 18.06 68.84 63.83 57.1 0 1 12.98 8.36 4.1 6.86 66.96 34.17 2.04 10.38 8.7 8.1 0 1.5 3.46 2.94 1.4 2.59 23.17 10.44 1.06 3.42 2.4 2.88 0.5 0 83.08 58.53 36.03 49.82 201.74 144.03 18.06 68.84 63.83 57.1 0.5 0.5 73.01 43.46 73.01 43.46 253.8 191.73 253.8 191.73 108.06 61.78 0.5 1 17.91 9.99 11.62 9.41 121.82 68.46 15.37 40.34 22.68 13.39 0.5 1.5 4.73 3.4 2.41 3.15 43.43 19.69 1.94 7.79 4.74 3.91 1 0 12.98 8.36 4.1 6.86 66.96 34.17 2.04 10.38 8.7 8.1 1 0.5 17.91 9.99 11.62 9.41 121.82 68.46 15.37 40.34 22.68 13.39 1 1 10.62 5.86 10.62 5.86 114.98 60.34 114.98 60.34 20.02 8.97 1 1.5 4.35 3.06 3.57 3.01 59.1 26.48 11.53 18.91 6.99 4.05 1.5 0 3.46 2.94 1.4 2.59 23.17 10.44 1.06 3.42 2.4 2.88 1.5 0.5 4.73 3.4 2.41 3.15 43.43 19.69 1.94 7.79 4.74 3.91 1.5 1 4.35 3.06 3.57 3.01 59.1 26.48 11.53 18.91 6.99 4.05 1.5 1.5 2.86 2.37 2.86 2.37 49.21 20.66 49.21 20.66 5.39 3.08 0.6 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 53.49 36.82 17.85 29.61 151.49 97.59 7.39 36.55 35.14 33.29 0 1 6.78 4.92 2.02 4 37.1 17.45 1.19 4.88 3.98 4.46 0 1.5 1.96 2.16 1.06 1.96 10.93 5.26 1 2.15 1.38 2.06 0.5 0 53.49 36.82 17.85 29.61 151.49 97.59 7.39 36.55 35.14 33.29 0.5 0.5 90.37 56.06 90.37 56.06 270.65 212.89 270.65 212.89 128.6 77.5 0.5 1 17.41 10.11 8.7 9.11 105.27 57.87 6.81 26.29 17.68 12.2 0.5 1.5 3.65 2.97 1.61 2.68 28.21 12.6 1.18 4.31 2.91 3.09 1 0 6.78 4.92 2.02 4 37.1 17.45 1.19 4.88 3.98 4.46 1 0.5 17.41 10.11 8.7 9.11 105.27 57.87 6.81 26.29 17.68 12.2 1 1 14.85 7.92 14.85 7.92 135.91 75.85 135.91 75.85 27.26 12.28 1 1.5 5.17 3.48 3.59 3.36 59.63 27.39 5.84 15.61 7.14 4.5 1.5 0 1.96 2.16 1.06 1.96 10.93 5.26 1 2.15 1.38 2.06 1.5 0.5 3.65 2.97 1.61 2.68 28.21 12.6 1.18 4.31 2.91 3.09 1.5 1 5.17 3.48 3.59 3.36 59.63 27.39 5.84 15.61 7.14 4.5 1.5 1.5 3.95 2.84 3.95 2.84 63.05 27.83 63.05 27.83 7.59 3.89 0.9 0 0 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 370.4 0 0.5 47.39 6.07 2.18 4.71 40.13 19.24 1.19 5.07 4.39 5.11 0 1 5.76 1.71 1 1.57 4.67 2.88 1 1.61 1.04 1.62 0 1.5 1.74 1.16 1 1.08 1.51 1.68 1 1.1 1 1.1 0.5 0 42.95 6.07 2.18 4.71 40.13 19.24 1.19 5.07 4.39 5.11 0.5 0.5 92.86 68.19 106.14 68.19 283.26 229.61 283.26 229.61 146.35 92.09 0.5 1 17.65 4.07 1.87 3.45 34.06 15.72 1.19 4.7 3.59 3.91 0.5 1.5 3.47 1.62 1 1.5 4.32 2.74 1 1.6 1.03 1.56 1 0 5.08 1.71 1 1.57 4.67 2.88 1 1.61 1.04 1.62 1 0.5 14.78 4.07 1.87 3.45 34.06 15.72 1.19 4.7 3.59 3.91 1 1 15.52 10.25 19.42 10.25 153.85 90.26 153.85 90.26 34.71 15.92 1 1.5 5.51 2.61 1.49 2.41 25.4 11.18 1.17 4.11 2.61 2.78 1.5 0 1.6 1.16 1 1.08 1.51 1.68 1 1.1 1 1.1 1.5 0.5 2.93 1.62 1 1.5 4.32 2.74 1 1.6 1.03 1.56 1.5 1 4.73 2.61 1.49 2.41 25.4 11.18 1.17 4.11 2.61 2.78 1.5 1.5 4.13 3.39 5.21 3.39 76.17 35.23 76.17 35.23 10.08 4.84 International Journal of Production Research 3303 Table 7. The SDRL values for the T2 control chart; n = 5. a 0.3 0.5 0.7 0.9 0.3 0 b 0.3 0.5 0.7 0.9 0.9 0.5 c 0 0 0 0 0 0 d 0 0 0 0 0 0 ρ δx δy STD MS STD MS STD MS STD MS STD MS STD MS 0.3 0 0 369.90 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 87.52 58.36 137.19 87.24 211.25 148.36 311.62 270.79 314.43 275.37 138.45 89.25 0 1 13.72 7.24 30.42 13.65 74.09 35.36 203.72 136.26 209.48 142.58 30.96 14.18 0 1.5 3.24 1.65 8.28 3.25 26.29 9.98 119.39 63.14 125.1 67.64 8.46 3.39 0.5 0 87.52 58.36 137.19 87.24 211.25 148.36 311.62 270.79 92.55 62.65 46.34 46.98 0.5 0.5 60.87 38.12 103.75 60.65 176.51 114.05 292.42 242.74 95.17 63.6 45.7 38.2 0.5 1 14.93 7.93 32.72 14.86 78.24 37.92 208.91 141.58 84.13 51.34 22.06 12.23 0.5 1.5 3.86 1.96 9.74 3.85 30.1 11.7 128.76 70.11 65.58 35 8.35 3.63 1 0 13.72 7.24 30.42 13.65 74.09 35.36 203.72 136.26 15.07 8.06 5.15 5.27 1 0.5 14.93 7.93 32.72 14.86 78.24 37.92 208.91 141.58 15.94 8.54 5.98 5.85 1 1 7.71 3.94 18.32 7.67 50.16 21.66 168.63 102.9 15.8 8.25 5.05 3.95 1 1.5 2.99 1.53 7.67 3 24.64 9.26 115.09 60.04 14.67 7.29 3.32 2.03 1.5 0 3.24 1.65 8.28 3.25 26.29 9.98 119.39 63.14 3.61 1.85 1.12 1.21 1.5 0.5 3.86 1.96 9.74 3.85 30.1 11.7 128.76 70.11 3.8 1.94 1.29 1.37 1.5 1 2.99 1.53 7.67 3 24.64 9.26 115.09 60.04 3.87 1.96 1.28 1.24 1.5 1.5 1.72 0.93 4.52 1.75 15.59 5.51 87.82 41.72 3.81 1.89 1.09 0.94 0.6 0 0 369.9 369.90 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 59.78 37.33 102.29 59.56 174.87 112.53 291.42 241.34 308.3 267.16 108.67 68.63 0 1 7.5 3.83 17.88 7.47 49.2 21.15 166.99 101.45 197.15 131.45 19.87 9.28 0 1.5 1.67 0.91 4.39 1.71 15.21 5.36 86.47 40.87 113.07 59.81 4.97 2.14 0.5 0 59.78 37.33 102.29 59.56 174.87 112.53 291.42 241.34 82.02 55.18 31.04 33.64 0.5 0.5 76.67 49.91 124.03 76.41 198.17 134.95 304.79 260.58 94.83 64.95 51.34 47.12 0.5 1 15.61 8.32 34 15.54 80.52 39.35 211.67 144.45 92.57 58.83 29.19 15.26 0.5 1.5 3.18 1.62 8.12 3.18 25.86 9.79 118.29 62.34 76.74 42.4 9.75 3.8 1 0 7.5 3.83 17.88 7.47 49.2 21.15 166.99 101.45 12.33 6.66 2.98 3.33 1 0.5 15.61 8.32 34 15.54 80.52 39.35 211.67 144.45 14.36 7.89 4.97 5.68 1 1 11.05 5.75 25.21 11 64.24 29.49 190.45 123.13 15.7 8.52 5.98 5.29 1 1.5 3.82 1.94 9.64 3.81 29.85 11.59 128.18 69.67 15.96 8.3 4.72 2.85 1.5 0 1.67 0.91 4.39 1.71 15.21 5.36 86.47 40.87 2.88 1.52 0.62 0.82 1.5 0.5 3.18 1.62 8.12 3.18 25.86 9.79 118.29 62.34 3.25 1.72 0.92 1.16 1.5 1 3.82 1.94 9.64 3.81 29.85 11.59 128.18 69.67 3.56 1.87 1.2 1.37 1.5 1.5 2.55 1.32 6.59 2.56 21.65 7.98 106.82 54.24 3.78 1.95 1.31 1.22 0.9 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 10.12 5.24 23.34 10.07 60.55 27.38 185.1 118.01 294.35 249.77 42.44 30.24 0 1 0.76 0.58 2.06 0.87 7.64 2.56 55.43 23.03 171.87 110.49 4.55 2.9 0 1.5 0.08 0.39 0.4 0.5 1.7 0.68 17.78 5.92 90.52 46.25 0.98 0.74 0.5 0 10.12 5.24 23.34 10.07 60.55 27.38 185.1 118.01 63.07 42.37 7.8 12.46 0.5 0.5 91.31 61.37 141.66 91.02 215.53 152.88 313.76 274.04 85.25 60.21 39.08 51.17 0.5 1 5.58 2.83 13.69 5.56 39.76 16.31 149.62 86.61 95.91 64.05 34.68 14.96 0.5 1.5 0.57 0.54 1.58 0.72 5.95 1.98 46.62 18.55 86.77 49.73 6.6 2.35 1 0 0.76 0.58 2.06 0.87 7.64 2.56 55.43 23.03 8.13 4.56 0.56 1.05 1 0.5 5.58 2.83 13.69 5.56 39.76 16.31 149.62 86.61 10.61 6.11 1.56 3.07 1 1 14.73 7.81 32.34 14.66 77.57 37.5 208.08 140.72 13.14 7.59 4.06 5.96 1 1.5 2.41 1.25 6.25 2.43 20.71 7.58 104.09 52.37 15.15 8.47 5.95 3.69 1.5 0 0.08 0.39 0.4 0.5 1.7 0.68 17.78 5.92 1.82 1.06 0.04 0.52 1.5 0.5 0.57 0.54 1.58 0.72 5.95 1.98 46.62 18.55 2.22 1.28 0.17 0.62 1.5 1 2.41 1.25 6.25 2.43 20.71 7.58 104.09 52.37 2.65 1.51 0.43 0.99 1.5 1.5 3.51 1.79 8.93 3.52 28 10.75 123.69 66.31 3.09 1.73 0.86 1.36 3304 R.C. Leoni et al. Table 8. The SDRL values for the T2 control chart; n = 5. a 0.3 0.1 0.7 0.2 0.3 b 0.3 0.1 0.7 0.2 0.3 c 0.1 0.3 0.2 0.7 0.3 d 0.1 0.3 0.2 0.7 0.3 ρ δx δy STD MS STD MS STD MS STD MS STD MS 0.3 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 82.58 57.41 35.52 48.68 201.24 143.12 17.56 67.75 63.33 55.98 0 1 12.47 7.07 3.56 5.55 66.45 32.98 1.46 9.1 8.18 6.8 0 1.5 2.91 1.61 0.75 1.27 22.66 9.15 0.25 2.09 1.83 1.55 0.5 0 82.58 57.41 35.52 48.68 201.24 143.12 17.56 67.75 63.33 55.98 0.5 0.5 72.51 42.3 72.51 42.3 253.3 190.9 253.3 190.9 107.56 60.67 0.5 1 17.4 8.7 11.11 8.12 121.32 67.37 14.86 39.17 22.17 12.12 0.5 1.5 4.2 2.08 1.84 1.82 42.93 18.44 1.35 6.49 4.21 2.58 1 0 12.47 7.07 3.56 5.55 66.45 32.98 1.46 9.1 8.18 6.8 1 0.5 17.4 8.7 11.11 8.12 121.32 67.37 14.86 39.17 22.17 12.12 1 1 10.11 4.55 10.11 4.55 114.48 59.23 114.48 59.23 19.52 7.68 1 1.5 3.81 1.73 3.03 1.68 58.6 25.27 11.01 17.66 6.48 2.73 1.5 0 2.91 1.61 0.75 1.27 22.66 9.15 0.25 2.09 1.83 1.55 1.5 0.5 4.2 2.08 1.84 1.82 42.93 18.44 1.35 6.49 4.21 2.58 1.5 1 3.81 1.73 3.03 1.68 58.6 25.27 11.01 17.66 6.48 2.73 1.5 1.5 2.31 1.06 2.31 1.06 48.71 19.42 48.71 19.42 4.86 1.76 0.6 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 52.98 35.64 17.34 28.4 150.99 96.57 6.88 35.37 34.64 32.1 0 1 6.26 3.6 1.43 2.68 36.59 16.2 0.48 3.56 3.44 3.13 0 1.5 1.38 0.87 0.25 0.7 10.41 3.94 0.03 0.86 0.72 0.78 0.5 0 52.98 35.64 17.34 28.4 150.99 96.57 6.88 35.37 34.64 32.1 0.5 0.5 89.87 54.94 89.87 54.94 270.16 212.1 270.16 212.1 128.09 76.44 0.5 1 16.9 8.82 8.19 7.81 104.77 56.75 6.29 25.07 17.17 10.92 0.5 1.5 3.11 1.64 0.99 1.36 27.71 11.32 0.46 2.98 2.35 1.76 1 0 6.26 3.6 1.43 2.68 36.59 16.2 0.48 3.56 3.44 3.13 1 0.5 16.9 8.82 8.19 7.81 104.77 56.75 6.29 25.07 17.17 10.92 1 1 14.34 6.62 14.34 6.62 135.41 74.78 135.41 74.78 26.75 11.01 1 1.5 4.64 2.15 3.05 2.03 59.13 26.18 5.32 14.35 6.62 3.17 1.5 0 1.38 0.87 0.25 0.7 10.41 3.94 0.03 0.86 0.72 0.78 1.5 0.5 3.11 1.64 0.99 1.36 27.71 11.32 0.46 2.98 2.35 1.76 1.5 1 4.64 2.15 3.05 2.03 59.13 26.18 5.32 14.35 6.62 3.17 1.5 1.5 3.41 1.51 3.41 1.51 62.54 26.62 62.54 26.62 7.08 2.56 0.9 0 0 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 369.9 0 0.5 46.89 4.75 1.6 3.39 39.63 18 0.48 3.75 3.86 3.79 0 1 5.24 0.56 0.03 0.53 4.14 1.55 0 0.53 0.21 0.54 0 1.5 1.14 0.37 0 0.27 0.88 0.55 0 0.3 0 0.31 0.5 0 42.44 4.75 1.6 3.39 39.63 18 0.48 3.75 3.86 3.79 0.5 0.5 92.36 67.1 105.64 67.1 282.76 228.86 282.76 228.86 145.85 91.05 0.5 1 17.14 2.74 1.27 2.12 33.55 14.47 0.47 3.37 3.05 2.58 0.5 1.5 2.93 0.54 0.03 0.51 3.79 1.41 0 0.53 0.19 0.52 1 0 4.55 0.56 0.03 0.53 4.14 1.55 0 0.53 0.21 0.54 1 0.5 14.27 2.74 1.27 2.12 33.55 14.47 0.47 3.37 3.05 2.58 1 1 15.01 8.96 18.91 8.96 153.35 89.23 153.35 89.23 34.21 14.67 1 1.5 4.98 1.29 0.86 1.1 24.89 9.9 0.45 2.78 2.04 1.45 1.5 0 0.98 0.37 0 0.27 0.88 0.55 0 0.3 0 0.31 1.5 0.5 2.38 0.54 0.03 0.51 3.79 1.41 0 0.53 0.19 0.52 1.5 1 4.2 1.29 0.86 1.1 24.89 9.9 0.45 2.78 2.04 1.45 1.5 1.5 3.59 2.06 4.69 2.06 75.67 34.05 75.67 34.05 9.57 3.52 International Journal of Production Research 3305 ARL ¼ b1 1� b2ð Þ þ 1 (16) SDRL ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1ð1þ b2 � b1Þ p 1� b2ð Þ (17) 5. The effect of the autocorrelation and cross-correlation on the performance of the T2 control chart In this Section, we investigate the influence of the autocorrelation parameters (a, b, c, d), the cross-correlation (ρ) and the mean shifts (δX, δY) on the statistical performance of the Hotelling T 2 control chart. The MS is compared with the standard sampling strategy (STD), where the sample units are selected according to the rational subgroup concept. We assume that U remains unchanged when the process shifts to an out-of-control condition. The in-control ARL is equal to 370.4 and the size of the samples is equal to 3 and 5. Tables 1–8 present the ARL and SDRL values of the T 2 charts with the MS and with the STD. The SDRL is slightly lower than ARL. The ARL values of the best strategy are in bold. In these Tables, the cross-correlation (ρ) are stratified in low (0.3), medium (0.6) and high (0.9) levels. Tables 1–4 consider the cases in which n = 3 and Tables 5–8 consider the cases in which n = 5. In general, the use of the MS reduces the negative effect of the autocorrelations on the T2 chart’s performance. For example, Table 1 shows that when the observations are medially autocorrelated (a = b = 0.5; q ¼ 0:3) Table 9. The in-control ARL, n = 5. The values of the estimated parameters ba bb dARL ρ = 0.3 ρ = 0.9 ba ¼ 0:80a bb ¼ 0:80b 0.56 0.56 40.33 39.92ba ¼ 0:90a bb ¼ 0:90b 0.63 0.63 96.67 96.80ba ¼ a bb ¼ 0:80b 0.70 0.56 81.03 56.24ba ¼ a bb ¼ 0:90b 0.70 0.63 162.33 128.35ba ¼ a bb ¼ 0:95b 0.70 0.665 244.39 220.52ba ¼ a bb ¼ b 0.70 0.70 370.22 370.30ba ¼ a bb ¼ 1:05b 0.70 0.735 552.11 476.80ba ¼ a bb ¼ 1:10b 0.70 0.77 769.55 457.92ba ¼ 1:05a bb ¼ 0:95b 0.735 0.665 322.48 229.41ba ¼ 1:10a bb ¼ 1:10b 0.77 0.77 3470.27 2693.80ba ¼ 1:30a bb ¼ 0:70b 0.91 0.49 69.26 50.46 Table 10. Preliminary sample. Observation X Y 1 988.47 986.72 2 989.63 987.28 3 986.40 990.88 4 989.77 991.93 5 987.81 990.12 : : : 496 990.87 989.81 497 989.72 990.68 498 991.76 990.93 499 991.05 990.00 500 992.41 991.35 3306 R.C. Leoni et al. Table 11. Mixed sample. ith sample Sample observations M T 2 i X 989.28 990.07 989.29 992.22 993.1 Y 990.08 989.06 989.22 991.35 991.3 1 X 990.31 989.09 989.87 990.53 989.03 990.3 0.18 Y 989.8 990.14 990.35 991.67 990 990.112 2 X 989.39 991.14 991.98 990.81 992.19 990.636 1.02 Y 987.84 991.79 992.21 991.02 990.88 990.548 3 X 989.76 987.82 987.78 989.46 991.31 990.16 0.73 Y 989.97 988.85 988.22 991.07 991.87 990.574 4 X 989.53 988.42 988.96 987.28 990.75 989.304 1.67 Y 990.23 988.8 989.35 987.77 991.71 990.242 5 X 988.03 989.45 988.33 988.64 989.69 988.35 5.41 Y 989.34 986.5 989.44 989.69 990.67 989.204 6 X 988.3 988.83 991.28 989.69 988.97 989.328 7.67 Y 988.53 988.98 990.3 985.87 985.58 988.12 7 X 988.33 990.83 989.79 990.22 989.62 989.252 6.24 Y 989.4 991.03 989.55 990.39 987.69 988.298 8 X 989.66 990.88 989.86 991.31 990.75 990.264 1.18 Y 988.24 988.1 987.86 989.91 989.86 989.476 9 X 988.84 991.19 991.89 991.34 989.81 990.546 1.1 Y 990.28 991.09 991.7 990.37 988.89 989.776 10 X 986.13 989.13 992.89 991.2 990.16 990.342 2.17 Y 989.1 989.91 993.24 994.49 991.19 990.998 11 X 991.32 991.25 989.97 990.46 990.74 990.472 7.15 Y 992.22 991.25 991.24 991.79 991.1 991.792 12 X 988.2 989.37 990.43 990.08 988.61 989.79 0.15 Y 988.03 989.31 990.35 987.88 988.95 990.074 13 X 990.63 988.04 987.14 989.74 991.49 989.742 0.62 Y 988.3 990.54 990.41 990.37 991.42 989.464 14 X 989.96 986.73 988.32 989.29 989.76 989.164 6.78 Y 993.1 989.99 989.58 988.65 991.91 991.1 15 X 989.87 989.25 992.14 991.41 989.65 989.536 0.48 Y 989.97 990.98 990.07 991.08 991.11 989.958 Figure 2. The T2 chart with mixed sample. International Journal of Production Research 3307 andd ¼ ð0:5; 1:0Þ0, the T2 chart with the STD requires, on average, 50.3 samples to signal; the ARL reduces to 28.1 (around 44% lower) with the MS. When the autocorrelated variable is not affected by the assignable cause and the observations of the other variable are independent, that is, (a = 0, b > 0, δx > 0, δy = 0) or (a > 0, b = 0, δx = 0, δy > 0) the MS reduces the ability of the T2 chart to detect the specified shift. These are the cases where b1 > b and b2 is slightly lower or even equal to b, being b the type II error of the standard T2 chart. For example, if (a = 0, b = 0.5; q ¼ 0:3) andd ¼ ð1:0; 0Þ0, the T2 chart with the STD requires, on average, 5.7 samples to signal, see Table 5. With the MS, the ARL increases to 6.6. In many applications, the magnitudes of the mean shifts are difficult to be predicted. Based on this fact, the MS is only recommended for bivariate processes with two autocorrelated variables. With regard to the autocorrelation parameters, it is worth noting that the T2 control chart with the MS has a more robust statistical performance than STD. In other word, fixing dX ; dY ; q; andn, the T2 control chart with the MS is less sensitive to the negative effect of the autocorrelation. For instance, fixing dX ¼ 1:0; dY ¼ 0:5; q ¼ 0:6; n ¼ 5 and vary- ing a and b as in Table 5, the range of the ARLs values is equal to 30.9 (=40.5 − 9.6) for the MS, and is equal to 64.9 (=81.0 − 16.1) for the STD. This robustness can lead quality practitioners to use the MS when there is some uncertainty with regard to the autocorrelation level. In an additional study of robustness, it was investigated the a and b misestimation effect on the in-control ARL. According to Table 9, when a and/or b are underestimated, the risk of false alarms increase, and an opposite effect is observed when a and/or b are overestimated. Similar comments are also valid for other q values. Based on that, the MS should be considered with care if the phase (I) data are not enough to obtain precise estimates of the model’s parame- ters. The SPC literature has innumerous examples of autocorrelated processes; in almost all of them, not to say all of them, the variables observations at time (i) are correlated, however, the observation of one variable at time (i) is inde- pendent of the other variable observation at time (i − 1). Because of that the study of robustness was done with c = d = 0. 6. An example application With the process in control, a preliminary investigation proved that the bivariate data from a two-valve filling milk machine fit to a VAR (1) model with ba = 0.36, bb = 0.32 and bq = 0.42 (see Table 10). The database was also used to estimate C and Re. Leoni, Costa, and Machado (2015) describe in details how to estimate C and Re. The quality engineer has decided to use the T2 chart with mixed samples of size five to control the filling process, being 990 mL the specified quantity inside the filled container. Based on the specifications, the in-control means vector l0 ¼ 990; 990ð Þ0. According to expression (9), the cross-covariance matrix CM ¼ 0:5074 0:2044 0:2044 0:4646 � � . The means vec- tor l0 and the inverse of CM are used to compute the monitoring statistic (see expression 12). Table 11 presents the bivariate data of 16 samples, the mean of the mixed samples and the values of the monitoring statistic. Figure 2 brings the mixed T2 chart with the T2 i values from Table 11. Table 12 was built considering a = 0.36, b = 0.32, ρ = 0.42 in order to present the ARL reduction when the MS is used to control the filling process rather than the STD. According to Table 12, the overall ARLs’ reduction is around 36% due to the MS. Table 12. The ARL for the T2 chart with the STD and with MS strategies. STD MS δ1 δ2 ARL ARL reduction (%) 0 0.5 142.35 103.91 27.00 1 0 38.84 21.98 43.41 0.5 1 42.62 25.49 40.19 0.5 0 155.09 112.32 27.58 0 1 32.93 19.32 41.33 0.5 0.5 133.16 94.34 29.15 1 1 29.06 16.54 43.08 3308 R.C. Leoni et al. 7. Conclusion This paper considered the T2 chart with the MS to monitor bivariate processes with autocorrelated variables. With the MS, the T2 statistic depends on the cross-covariance matrix of the mixed sample mean vector. Numerical comparisons between standard and mixed sampling strategies showed that the mixed one is highly recommended to control the mean vector of bivariate processes with two autocorrelated variables. If the observations of one variable are autocorrelation free, the MS should be used with care; depending on the mean vector shift, the mixed strategy reduces the speed with which the chart signals. EWMA, CUSUM or synthetic schemes can be used with the MS to enhance the ability of the T2 chart in signalling changes in the mean vector of multivariate processes with autocorrelated variables. Based on recent studies with univariate processes and wandering means, the MS is not recommended to control wandering mean vectors. Disclosure statement No potential conflict of interest was reported by the authors. Funding This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico [grant number 301739/2010-2], [grant number 306189/2011-9]. 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Introduction 2. The VAR (1) model and the cross-covariance matrix 3. The mixed samples and the corresponding cross-covariance matrix 4. The statistical properties of the T2T^{2} chart 5. The effect of the autocorrelation and cross-correlation on the performance of the T2{{T}}^{2} control chart 6. An example application 7. Conclusion Disclosure statement Funding ORCID References