Reducing sampling costs in multivariate SPC with a double-dimension T 2 control chart Eugenio K. Epprecht a , Francisco Aparisi b,n , Omar Ruiz c ,A ´ lvaro Veiga d a Departamento de Engenharia Industrial, Pontifı ´cia Universidade Cato ´lica do Rio de Janeiro, Brazil b Departamento de Estadı ´stica e I. O, Aplicadas y Calidad, Universidad Polite ´cnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain c Centro de Investigaciones Biotecnolo ´gicas del Ecuador, Escuela Superior Polite ´cnica de Litoral, Ecuador d Departamento de Engenharia Ele´trica, Pontifı ´cia Universidade Cato ´lica do Rio de Janeiro, Brazil article info Article history: Received 2 August 2012 Accepted 15 January 2013 Keywords: Double sampling T 2 control chart Cost sampling abstract In some real situations there is the need of controlling p variables of a multivariate process, where p 1 out of these p variables are easy and inexpensive to monitor, while the p 2 ¼p–p 1 remaining variables are difficult and/or expensive to measure. However, this set of p 2 variables is important to quickly detect the process shifts. This paper develops a control chart based on the T 2 statistic where normally only the set of p 1 variables is monitored, and only when the T 2 value falls in a warning area the rest of variables (p 2 ) are measured and combined with the sample values from the p 1 variables, in order to obtain a new T 2 statistic. This new chart is the double dimension T 2 (DDT 2 ) control chart. The ARL of the DDT 2 chart is obtained and the chart’s parameters are optimized using genetic algorithms with the aim of maximizing the performance in detecting a given process shift. The optimized DDT 2 chart is compared against the standard T 2 chart when all the variables are monitored. The results show that the DDT 2 clearly outperforms T 2 chart in terms of cost, and in some cases even detects process shifts faster than the latter. In addition, friendly software has been developed with the objective of promoting the real application of this new control chart. & 2013 Elsevier B.V. All rights reserved. 1. Introduction A practitioner who needs to monitor several quality character- istics of a process may define a scheme for monitoring all variables, which is based on the control of each variable, with either a univariate control chart (multiple scheme) or a single multivariate control chart (multivariate approach). The first multivariate control chart was the T 2 control chart (Hotelling, 1947), which considers correlations between variables and is more applicable in some cases than employing a separate uni- variate chart for each variable. For example, the multivariate scheme is more appropriate when high correlations exist between variables and the expected shift is in the direction of least variation in the space of the variables. The reader may consult Aparisi et al. (2010) for a comparison of multiple and multivariate schemes. Descriptions of the T 2 chart are included in chapter 11 of Montgomery (2012) or chapter 5 of Johnson and Wichern (2007). Enhancements of the T 2 chart, such as the MEWMA chart (Lowry et al., 1992) or multivariate CUSUM schemes (Woodall and Ncube, 1985; Crosier, 1986) are also utilized. However, in the context of multivariate SPC, the T 2 chart remains a basic scheme that is analogous to the Shewhart chart in univariate SPC. For reviews of multivariate process control schemes, the reader is referred to Lowry and Montgomery (1995), Bersimis et al. (2006) or chapter 5 of Montgomery (2012). For a process that incorporates several correlated variables that require monitoring, evaluation of some of the variables may involve simple, fast and inexpensive methods; whereas, other variables may be expensive or difficult to measure (with the potential for a destructive measurement). Based on this premise, this paper proposes and analyzes a new sampling scheme for these cases based on the T 2 chart. The scheme is analogous to and inspired by double-sampling procedures, which strive for better allocation of sampling efforts. Initially employed for lot accep- tance sampling (Duncan, 1992; Montgomery, 2012), double- sampling procedures have been extended for charting control (Croasdale, 1974; Daudin, 1992; Steiner, 1999; Costa and De Magalh ~ aes, 2005; Rodrigues et al., 2011; Champ and Aparisi, 2008). We have developed a double-dimension T 2 (DDT 2 ) chart, which is analogous to double-sampling (DS) control charts. For each sampling time, the p 1 variables that are inexpensive to evaluate are measured and the T 2 p1 statistic is calculated. If its value falls Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.01.022 n Corresponding author. Tel.: þ34 963877490; fax: þ34 963877499. E-mail addresses: [email protected] (E.K. Epprecht), [email protected], [email protected] (F. Aparisi), [email protected] (O. Ruiz), [email protected] (A ´ . Veiga). Please cite this article as: Epprecht, E.K., et al., Reducing sampling costs in multivariate SPC with a double-dimension T 2 control chart. International Journal of Production Economics (2013), http://dx.doi.org/10.1016/j.ijpe.2013.01.022i Int. J. Production Economics ] (]]]]) ]]]–]]]
Transcript
Int. J. Production Economics ] (]]]]) ]]]–]]]
Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics
0925-52
http://d
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profepa
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journal homepage: www.elsevier.com/locate/ijpe
Reducing sampling costs in multivariate SPC with a double-dimension T2
control chart
Eugenio K. Epprecht a, Francisco Aparisi b,n, Omar Ruiz c, Alvaro Veiga d
a Departamento de Engenharia Industrial, Pontifıcia Universidade Catolica do Rio de Janeiro, Brazilb Departamento de Estadıstica e I. O, Aplicadas y Calidad, Universidad Politecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spainc Centro de Investigaciones Biotecnologicas del Ecuador, Escuela Superior Politecnica de Litoral, Ecuadord Departamento de Engenharia Eletrica, Pontifıcia Universidade Catolica do Rio de Janeiro, Brazil
e cite this article as: Epprecht, E.K., enational Journal of Production Econ
a b s t r a c t
In some real situations there is the need of controlling p variables of a multivariate process, where p1
out of these p variables are easy and inexpensive to monitor, while the p2¼p–p1 remaining variables
are difficult and/or expensive to measure. However, this set of p2 variables is important to quickly
detect the process shifts. This paper develops a control chart based on the T2 statistic where normally
only the set of p1 variables is monitored, and only when the T2 value falls in a warning area the rest of
variables (p2) are measured and combined with the sample values from the p1 variables, in order to
obtain a new T2 statistic. This new chart is the double dimension T2 (DDT2) control chart. The ARL of the
DDT2 chart is obtained and the chart’s parameters are optimized using genetic algorithms with the aim
of maximizing the performance in detecting a given process shift. The optimized DDT2 chart is
compared against the standard T2 chart when all the variables are monitored. The results show that the
DDT2 clearly outperforms T2 chart in terms of cost, and in some cases even detects process shifts faster
than the latter. In addition, friendly software has been developed with the objective of promoting the
real application of this new control chart.
& 2013 Elsevier B.V. All rights reserved.
1. Introduction
A practitioner who needs to monitor several quality character-istics of a process may define a scheme for monitoring allvariables, which is based on the control of each variable, witheither a univariate control chart (multiple scheme) or a singlemultivariate control chart (multivariate approach). The firstmultivariate control chart was the T2 control chart (Hotelling,1947), which considers correlations between variables and ismore applicable in some cases than employing a separate uni-variate chart for each variable. For example, the multivariatescheme is more appropriate when high correlations exist betweenvariables and the expected shift is in the direction of leastvariation in the space of the variables. The reader may consultAparisi et al. (2010) for a comparison of multiple and multivariateschemes. Descriptions of the T2 chart are included in chapter 11 ofMontgomery (2012) or chapter 5 of Johnson and Wichern (2007).
Enhancements of the T2 chart, such as the MEWMA chart(Lowry et al., 1992) or multivariate CUSUM schemes (Woodall and
t al., Reducing sampling cosomics (2013), http://dx.doi.
Ncube, 1985; Crosier, 1986) are also utilized. However, in thecontext of multivariate SPC, the T2 chart remains a basic schemethat is analogous to the Shewhart chart in univariate SPC. Forreviews of multivariate process control schemes, the reader isreferred to Lowry and Montgomery (1995), Bersimis et al. (2006)or chapter 5 of Montgomery (2012).
For a process that incorporates several correlated variablesthat require monitoring, evaluation of some of the variables mayinvolve simple, fast and inexpensive methods; whereas, othervariables may be expensive or difficult to measure (with thepotential for a destructive measurement). Based on this premise,this paper proposes and analyzes a new sampling scheme forthese cases based on the T2 chart. The scheme is analogous to andinspired by double-sampling procedures, which strive for betterallocation of sampling efforts. Initially employed for lot accep-tance sampling (Duncan, 1992; Montgomery, 2012), double-sampling procedures have been extended for charting control(Croasdale, 1974; Daudin, 1992; Steiner, 1999; Costa andDe Magalh~aes, 2005; Rodrigues et al., 2011; Champ and Aparisi,2008).
We have developed a double-dimension T2 (DDT2) chart, whichis analogous to double-sampling (DS) control charts. For eachsampling time, the p1 variables that are inexpensive to evaluateare measured and the T2
p1 statistic is calculated. If its value falls
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]]2
below a given threshold (termed warning limit, as with DS charts),the process is considered in control; conversely, if its value is toohigh (exceeds the control limit), the process is declared out ofcontrol. If its value falls between the warning limit and thecontrol limit, the remaining p2 variables (p¼p1þp2) are measuredand the overall T2
p statistic is compared with the appropriatecontrol limit (there are different control limits for the set ofinexpensive variables and the total set of variables). This schemecoincides with the well-known double-sampling procedure,although additional measurements are compiled for differentvariables instead of establishing a complementary sample ofunique variables.
Therefore, if the DDT2 control chart is employed, the p2
variables that are difficult or expensive to evaluate are measuredonly when there is a need to acquire more information. With thisapproach, the sampling cost is reduced in comparison with theprocedure of always sampling all p variables to compute the T2
p
statistic. However, a reduction in the detection power of the chartin the out-of-control state may occur because the entire set ofp variables exhibits a larger Mahalanobis distance than the set ofp1 variables. As the study results indicate, this loss of performanceis small in most cases, and in some cases the DDT2 chartsurprisingly outperforms (in terms of out-of-control ARL) thestandard T2 chart for all variables.
Section 2 describes the DDT2 chart in detail. Section 3 presentsa mathematical model for calculation of the performance mea-sures. Section 4 describes the optimization of the model design(which is accomplished with user-friendly software developed byand available from the authors). Section 5 compares the perfor-mance of the model with the performance of the standard T2
chart. A sensitivity analysis is provided in Section 6. Section 7summarizes the conclusions of the paper and is followed by anAppendix and list of references.
2. The double-dimension T2 (DDT2) control chart
The standard T2 control chart involves periodically taking a sample
of size n from the process to obtain the sample average vector, X, forall p variables to be monitored. The T2 statistic is calculated as
T2¼ nðX�l0Þ
0S�1ðX�l0Þ, where l0
0 ¼ ðm01, m02, :::, m0pÞ is the in-
control mean vector and S is the in-control covariance matrix of X.Assuming the variables follow a multivariate normal distribution, thisT2 statistic is distributed as a chi-squared variate with p degrees offreedom when the process parameters are known and the process isin control (l¼ l0). The control limit (CL) that achieves a given
probability a that the T2 statistic will fall outside this control limit is
sought. Therefore, CL¼w2p,a. The distribution of T2 when the process is
out of control, l¼ l1al0, is a non-central chi-squared distribution
with p degrees of freedom and non-centrality parameter l¼ nd2¼
nðl1�l0Þ0S�1ðl1�l0Þ, where d is the Mahalanobis distance of l1
with respect to l0. In this paper, we are only concerned with shifts of
the in-control mean vector.Fig. 1 shows the Double-Dimension T2 (DDT2) control chart,
which has a warning limit w and two control limits, UCLp1 and UCLp.As explained previously, the monitoring involves sampling n itemsfrom the process and measuring, in the first stage only, the subset ofp1 variables that are easy and/or inexpensive to measure. With thissampling information, Hotelling’s statistic T2
p1 is computed andplotted on the chart. If T2
p1ow, no further action is taken. IfT2
p1ZUCLp1, the process is considered out of control.The second stage of the procedure becomes necessary when
wrT2p1oUCLp1. Although the value of the T2
p1 statistic is high, it is
not high enough to consider the process out of control. Therefore,
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
more statistical information is required. Hence, the remaining p2
variables (p2¼p–p1) are measured from the same n units thatwere previously considered, and all measurements of the p
variables from the n units are employed to compute the T2p
statistic. Now, if T2p 4UCLp, the process is considered out of
control; otherwise, it is considered in control.One special case is when p1¼1, i.e., the monitoring is based
normally on the sampling of a single variable and additionalvariables are sampled and combined as needed. To create asimpler control chart with a common vertical axis for the twostatistics to be plotted, which both have a minimum value of 0,we recommend that the statistic for p1¼1 be defined asT2
p1 ¼ nðX�m0Þ2=s2
0, where m0 and s20 are the in-control mean
and variance, respectively. In this case, the Mahalanobis distancedp1 between the shifted mean m1 and the in-control mean of theprocess is the absolute value of the standardized shift; that is,dp1 ¼ 9m1�m09=s0.
In this paper, the values of the DDT2 parameters w, UCLp1, andUCLp are optimized. As will be presented later, the value of UCLp1
can be taken as infinite in many cases; the resulting chart is easierto manage, with only one control limit and no noticeable loss ofperformance. However, this new DDT2 control chart is harder todesign and operate than a standard T2 control chart. User-friendlysoftware, which returns the design parameters that optimizeperformance for a specified shift, is available from the authors.
3. Performance measures
Two performance measures are employed in this paper forevaluating and comparing the performance of the DDT2 controlchart with the performance of the standard T2 chart: the averagenumber of samples required to detect a process shift and theaverage cost per sample when the process is in control. Bothmetrics are described below.
The cost of sampling is a function of the percentage of timesthat all variables are measured. Let Cp1 be the cost of samplingwhen only the set of p1 variables is measured. The cost ofsampling the expensive variables, Cp2, is defined as Cp24Cp1.Let Cp2 ¼ aCp1, with a41. Hence, the total cost of all measuredvariables is Cp ¼ Cp1þ Cp2 ¼ Cp1Uð1þaÞ.
Given a combination of parameters from the DDT2 chart (w,CLp1, CLp), the average cost of sampling for the DDT2 control chart,ACSDDT2 , is
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]] 3
Fw2p1
is the distribution function of the central chi-squared variable
with p1 degrees of freedom.The standard T2 control chart, when the entire set of p
variables is always measured, has a sampling cost of
CST2p¼ Cp1Uð1þaÞ ð2Þ
Therefore, the ratio between the sampling cost of the DDT2
chart and the sampling cost of the standard T2 chart is expressedas
ACSDDT2
CST2p
¼1þaU%p
1það3Þ
The average cost of sampling when the process is out ofcontrol (which is larger than when the process is in control) isnot considered for the following reasons: (i) when the process isout of control, the priority of the user is a fast detection of thisstate, which prevents the cost of producing non-conforming orlow-quality items; and the increase in the probability of going tothe second stage of sampling when the process goes out of controlis precisely the mechanism by which the DDT2 chart increases itsprobability of signal; (ii) the effect of out-of-control averagesampling costs on overall sampling cost (or on overall averagesampling cost per unit time) is variable and difficult to syntheti-cally quantify because it is dependent on the out-of-control ARL,the size of the shift (which will not always be the shift for whichthe chart was optimized), and the frequency that the process isout of control; (iii) we assume that the process remains in controlmost of the time; as a result, this cost is not relevant incomparison with the in-control cost of sampling and in view ofthe importance of quick detection of special causes. Note that thiscost will be smaller than the sampling cost of a T2 chart for allvariables.
The second performance measure we consider is the AverageRun Length (ARL), which is the number of samples expectedbefore a signal is produced by the control chart. It is perhaps themost common measure of the statistical performance of controlcharts; therefore, it will be the measure adopted for this paper.When the values plotted on the chart are independent, the ARL issimply the reciprocal of the probability that a point will falloutside the control limits. For a standard T2 control chart,ARL¼ 1=a (where a is the false-alarm probability) when theprocess is in control, and ARL¼ 1=ð1�bÞ ¼ 1=½1�Pðw2
pðlÞoCLÞ�
when the process is out of control, where b is the Type II errorprobability and w2
p (l) is the chi-squared distribution withp degrees of freedom and non-centrality parameter l¼ nUd2.Hence, the ARL of the DDT2 control chart is computed as
ARL¼ 1=ð12PaÞ,
where Pa is the probability of no signal given by
Pa ¼ P1þP2, ð4Þ
and P1 is the probability that the first statistic, computed withonly the set of p1 variables, neither signals nor indicates the needto measure the remaining variables. That is
P1 ¼ PðT2p1owÞ ð5Þ
and P2 is the probability that the first statistic falls in the warningregion (between the warning limit and the control limit), whichrequires that the remaining p2 variables be measured. Theresulting T2 statistic with all p variables exceeds the followingcontrol limit UCLp:
P2 ¼ PðwrT2p1oUCLp1ÞPðT
2p oCLp9wrT2
p1oUCLp1Þ ð6Þ
Therefore, the conditional distribution of T2p is required to
compute P2. Murphy (1987) shows that when the process is in
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
control, with l¼0, the statistic D¼ T2p�T2
p1 is distributed as w2p2,
where p2¼p�p1. Therefore, when the process is in control
PðT2p oUCLp9wrT2
p1oUCLp1Þ ¼ PðDoUCLp�T2p1Þ
¼ Pðw2p2oUCLp�T2
p1Þ ð7Þ
For the out-of-control case, we have proved thatD� w2
p2½l¼ nðd2p�d2
p1Þ�, where dp1 and dp are the Mahalanobisdistance of the mean of the p1 variables and the Mahalanobisdistance of the mean of all p variables with respect to l0 (see theAppendix), respectively. As a result, when the process is out ofcontrol
PðT2p oUCLp9wrT2
p1oUCLp1Þ ¼ PðDoUCLp�T2p1Þ
¼ Pðw2p2½l¼ nðd2
p�d2p1Þ�oUCLp�T2
p1Þ ð8Þ
By combining (6) and (7), when the process is in control
P2 ¼
Z UCLp1
wf p1ðT
2p1ÞFp2ðUCLp�T2
p1ÞdT2p1 ð9Þ
where Fv is the distribution function and fv is the probabilitydensity function of a chi-squared variable with v degrees offreedom. When the process is out of control, by combining(8) and (9) we obtain
P2 ¼
Z UCLp1
wf
p1, nd2p1ðT2
p1ÞFp2, nðd2p�d2
p1ÞðUCLp�T2
p1ÞdT2p1 ð10Þ
where Fv,l is the non-central chi-squared distribution functionwith v degrees of freedom and non-centrality parameters l andfv,l are its probability density functions. The above model hasbeen verified by simulation.
4. Design optimization and software
The DDT2 control chart has three parameters that requireoptimization: w, UCLp1, and UCLp. For the optimization, thesample size, n, and number of variables, p1 and p, are problemdata that are assumed given. In addition, the user has to specifythe shift against which he/she wants to maximize chart perfor-mance. The selection of a shift for optimization is a commonrequirement in control charts. For example, the performance ofEWMA or MEWMA control charts depend on the size of the shift(sigma units or Mahalanobis distance), and a control limit mustbe specified for a shift size to determine the optimal smoothingparameter.
The problem of optimizing the design of the DDT2 controlchart for a given shift in the mean can be formally stated asfollows:
Given:
�
tsorg
the required in-control ARL: ARL0
�
the values of p1 and p2: p¼p1þp2
�
the sample size: n
�
the Mahalanobis distances of the shift (that is, the Mahalano-bis distances of the out-of-control mean with respect to the in-control distribution of X) for which the ARL has to beminimized:
for the set of p1 variables: dp1,and for all variables, p: dp
Find: the control and warning limits: w, UCLp1, and UCLp
Minimize: the out-of-control ARL(dp1, dp), subject to:ARL(d¼0)¼ARL0
The authors of this paper strive to facilitate the use of the DDT2
chart. For this reason, the optimization of the chart is conducted
in multivariate SPC with a double-dimension T2 control chart./10.1016/j.ijpe.2013.01.022i
Fig. 2. Software for optimizing the DDT2 control chart used to solve the example application.
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]]4
with user-friendly Windowss software, which is available fromthe authors. The user interface window is shown in Fig. 2.A genetic algorithm (GA) manages the optimization. Due to thecomplexity of the formulas for evaluating the probabilities ofsignal (especially Eqs. (9) and (10)), the use of GAs for thisoptimization is crucial because the evaluation of the ARL for agiven set (w, UCLp1, UCLp) is relatively slow. Therefore, the use of aheuristic search procedure reduces the number of combinationsto be computed. The reader interested in the general use of GAand their particular use for optimizing control charts can consultChen (2007), Kaya (2009) and Aparisi et al. (2010).
To illustrate the software and its use, an example application isdemonstrated. The production of an electronic component ismonitored through the measurement of three variables: 1. Thevoltage between two connectors (mV). 2. The electric currentbetween the other two connectors (mA). 3. The maximum voltagethat can resist the component prior to burn out (V). Measuringthe last variable will destroy the unit. The cost of sampling thefirst two variables is 0.18h, whereas sampling all variables costs1.28h. Therefore, a¼Cp2=Cp1¼(1.28–0.18)/0.18¼6.11.
Due to the high difference in cost, we decided to measure thethird variable only when necessary. After an initial study, theDDT2 control chart was found to satisfy the requirements forcontrolling this process. Although the third variable (burningvoltage) has a high measuring cost, it clearly produced a largerMahalanobis distance when it was added to the monitoringscheme.
The in-control mean vector and covariance and correlationmatrices are
m0 ¼
7:2
23:1
9:32
0B@
1CA; R¼
0:02 0:036 0:033
0:036 0:15 0:054
0:033 0:054 0:22
0B@
1CA; ½q� ¼
1 0:66 0:5
0:66 1 0:3
0:5 0:3 1
0B@
1CA
In selecting the Mahalanobis distances for the optimization ofthe performance, an evaluation of historical data indicated thatthe process shifts produced the following out-of-control mean
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
vector, m1, several times:
m1 ¼
7:23
23:35
8:92
0B@
1CA
The Mahalanobis’ distances for this out-of-control mean vectorm1 are as follows: dp1¼0.7; dp¼1.29. The practitioner desires anin-control ARL of 400 (ARL0¼400), and a sample size n¼1 will beemployed. Fig. 2 shows the solution for this example applicationusing the software.
A brief description of the software is as follows: ‘‘Model para-meters’’ are parameters in which the user indicates the values of p1,p, n, ARL0, dp1 and dp. With this information, the GA finds the chartparameters w, UCLp1 and UCLp and provides some useful informationin the window ‘‘Results’’. For these cases, the software returns thefollowing parameters: w¼1.32, UCLp1¼14.03, UCLp¼14.25. For eachof these parameters, the software shows the right tail probabilityarea. For example, Pðw2
p14wÞ ¼ 0:4831.The previous parameters produce a DDT2 control chart with
the desired ARL0¼400. The minimized out-of-control ARL is 53.29.The software also shows the ARL of the standard T2 control chartsfor p1 variables and p variables always with the same in-controlARL, ARL0, of the DDT2 chart (in our case, an in-control ARL of 400).In our case, when only the variables 1 and 2 are measured, theARL is 140.83. The DDT2 chart yields a reduction in ARL of 62.2%.When the three variables are always measured, the ARL is 49.75.Therefore, the DDT2 is only 7% worse in this case. However, itrequires only a fraction of the cost needed by the T2 control chart.Fig. 2 indicates that %p¼0.591.
This DDT2 control chart will have a smaller cost than astandard T2 control chart that monitors the whole set of p
variables. Concretely:
ACSDDT2
CST2
¼1þaU%p
1þa¼
1þ6:11U0:591
1þ6:11¼
4:61
7:11¼ 65%
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]] 5
Hence, with an ARL that is very similar to the standard T2 for3 variables but with 65% of the required cost, the DDT2 revealssuperior performance. The standard T2 control chart for only the firsttwo variables has an ARL of 140.83, which is 2.64 times greater.Because this T2 chart never measures inexpensive variables, the DDT2
chart is more expensive—specifically, 4.2 times more expensive.The software also provides more information with the sen-
tence ‘‘ARLs taken UCLp1 as infinite’’. It is often possible to removethe control limit UCLp1; however, the resulting DDT2 control chartshows practically the same performance. In this case, out-of-control signals can only be a result of UCLp and can only beproduced in the second stage of sampling, when all variables aremeasured. The advantage is a simpler scheme and a chart withonly two lines: w and UCLp2. In this example, if UCLp1 is removed(or taken as infinite) the in-control ARL becomes 403.74, which is
Fig. 3. ‘‘ARL ca
Table 1Simulated sampling for the DDT2 control chart.
Sample number Sample vector, Xp1 T2p1 Sample vector, Xp T2
pCom
1 7:18
23:14
� �0.296 – – Start
2 7:25
23:05
� �0.596 – – T2
p1 f
3a 7:11
23:32
� �1.455 – – T2
p1 f
3b – – 7:11
23:32
10:45
0B@
1CA
3.52 T2p fa
4 7:16
22:97
� �0.346 – – T2
p1 f
5a 7:75
20:98
� �11.34 T2
p1 f
5b – – 7:75
20:98
6:87
0B@
1CA
14.332 T2p 4
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
practically the required value of 400; the out-of-control ARL is53.39, which is practically equivalent to the optimum ARL of53.29. Therefore, the removal of the control limit UCLp1 will causevirtually no deterioration of performance and will lead to asimpler control scheme.
The software interface has two tabs: ‘‘Optimization’’ and ‘‘ARLcalculation’’. The previous optimization was performed under thetab with the same name. If the final user wants to calculate theARL for a given combination (w, UCLp1, UCLp), he/she can use thetab ‘‘ARL calculation’’ and easily obtain the value of ARL, asillustrated in Fig. 3.
To better illustrate the use of the DDT2 control chart, asimulation of its use is included in Table 1 and Fig. 4, which areauto-explicative. UCLp1 was removed in this simulation, whichcorresponds to the previous example.
lculation’’ tab.
ments
with p1 variables. T2p1 falls below warning limit. No further sampling required.
alls below warning limit. No further sampling required.
alls in warning area. Expensive variable to be sampled and combined.
lls below control limit. The process is declared in control.
alls below warning limit. No further sampling required.
alls in warning region. Expensive variable to be sampled and combined.
UCLp. The process is declared out of control.
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
Fig. 4. DDT2 control chart simulation from Table 1.
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]]6
5. Performance Analysis and Comparisons
We have analyzed the performance of the DDT2 control chartfor many cases. In this paper, due to space limitations, we providethe results of the following cases only: (p1, p)¼(1, 2), (2, 3), (2, 4),(5, 6), and (5, 7); (dp1, dp)¼(0.2, 0.5), (0.2, 1), (0.2, 1.5), (0.2, 2),(0.5, 0.7), (0.5, 1), (0.5, 1.5), (0.5, 2), (1, 1.2), (1, 1.5), (1, 2), (1, 2.5),(1.5, 1.8), (1.5, 2), (1.5, 2.5), (1.5, 3), (2.5, 3), (2.5, 3.5), (2.5, 4), (2.5,4.5), (3.5, 4), (3.5, 4.5), (4, 5), and (4, 6). All cases are for an in-control ARL of 400, or ARL0 = 400. The results are displayed inTables 2–6. We have also analyzed the cases of the samecombinations of pairs (p1, p) and (dp1, dp) for ARL0¼1000.Although these cases are not presented here, the general conclu-sions are the same as for the cases presented. Note also that if thefinal user wants to analyze different cases than the cases shownin the tables, he/she can use the developed software and easilyobtain the required results.
The description of the information provided in the Table is asfollows: ‘‘Problem Parameters‘‘ shows the optimization that hasbeen performed, i.e., the values of ARL0, p1, p, dp1 and dp. Thecolumn ‘‘T2 always p19ARL(dp1)’’ shows the out-of-control ARL of astandard T2 control chart, where only the subset of p1 variables ismeasured. It is provided to illustrate the performance when thevariables that are easy and inexpensive to measure are the onlyvariables measured. In the same way, the column ‘‘T2 alwaysp9ARL(dp)’’ returns the out-of-control ARL of the standard T2 chartwhen all p variables are always sampled. This column shows themaximum performance that can be obtained by employing thestandard T2 chart. As shown later, the DDT2 control chart willobtain a performance that sometimes remarkably outperformsthe T2 chart for p variables, with a fraction of its sampling cost.
Although it may seem paradoxical, this effect is analogous towhat occurs with double-sampling schemes in lot inspection orin-control charts: the double-sampling plans and double-sampling charts have greater power and similar Type I errorprobability and average sample size than single-sampling plans(and single-sampling control charts). Alternatively, the double-sampling schemes can be used to achieve the same power of thesingle-sampling schemes, with a lower average sample size,which indicates a lower sampling cost. The double-samplingschemes are more efficient than single-sampling schemes (interms of the power/cost ratio) due to a better allocation ofsampling effort.
With the DDT2 control chart, one measures all p variables onlypart of the time. Combined with the fact that the chart willseldom (or never in the version with only one UCL) issue an alarmwith only p1 variables, this chart will have a smaller control limitvalue for p variables than the standard T2 chart, thus achieving thesame ARL0 than the standard T2 chart. This smaller control limitresults in a better out-of-control performance for the DDT2
control chart. In addition to this smaller control limit, the
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
probability of measuring all variables with this chart increaseswhen there is a process shift, and the combination of these twofindings results in a smaller out-of-control ARL than the out-of-control ARL of the standard T2 chart.
The next four columns in the table, under the caption ‘‘DDT2’’,display information about the optimized DDT2 control chart. Thefirst column, ‘‘ARL(d) (%T2
p1, %T2p)’’ returns the out-of-control ARL
of the DDT2 chart, and the percentage of improvement versus thestandard T2 charts, when p1 or p variables are always measured.The column ‘‘Optimum Parameters’’ displays the results obtainedfrom the software, i.e., the values of w, UCLp1 and UCLp thatminimize the ARL for that case. The column ‘‘%p’’ provides thepercentage of times for which all variables are measured whenthe DDT2 is employed and the process is in control. This value isneeded to obtain the average cost of sampling, which is a functionof the cost of measuring the p1 variables and the cost ofmeasuring all p variables, as given by Eq. (3).
The value of ‘‘%p’’ exhibits large variation, ranging from 7% inthe best case to 99% in the worst case. However, these largevalues of %p seldom appear in the Table: when p1¼5, and for verysmall values of dp1, specifically dp1¼0.2. In a few cases, theminimization of the out-of-control ARL returns a scheme wherethe p variables are measured frequently when the process is in-control, thus the optimum practically consists of employing thestandard T2 chart when all p variables are always measured. Insuch cases, a constraint on ‘‘%p’’ is needed in the formulation ofthe optimization problem to obtain an economic solution thatcontinues to exhibit superior ARL performance, as shown later.For that reason, the optimal solutions under the constraint%p¼30% are presented in the column ‘‘DDT2 (%p¼30)’’. TThe finaluser may prefer to specify a different value for %p. To cater to thispossibility, the software can also perform the optimization with aconstraint on %p (value to be entered by the user), see box ‘‘UseRestriction on %p‘‘ in Fig. 2. However, as said before, these largevalues of %p tend to be rare: in the majority of the cases analyzedthe optimization returned a DDT2 control chart with a low valueof %p, and an ARL closer to the ARL of the standard T2 chart for p
variables.The cases in which the removal of UCLp1 has little impact on
the performance of the DDT2 control chart are marked with anasterisk in the column ‘‘UCLp1¼N?’’. An asterisk denotes whenthe in-control and out-of-control ARL simultaneously differ byless than 5%, based on a comparison of the DDT2 chart with UCLp1
and the DDT2 chart without UCLp1. In those cases, the practitionermay use the DDT2 chart without UCLp1. This potential simplifica-tion occurs in the majority of cases, as Tables 2 illustrate.
The analysis of Table 2 yields some general conclusions, whichare the same for ARL0¼1000 (as stated previously, we analyzedthe performance of the DDT2 chart for a larger number of casesthan we could present here due to space limitation). Regardingthe ARL, the DDT2 chart always outperforms the standard T2 chartfor p1 variables and sometimes outperforms the T2 chart for p
variables, which is remarkable because in those cases the DDT2
chart is both faster and more economical. Some examples inwhich the DDT2 outperforms the T2 control chart for p variablesinclude: I. p1¼1, p¼2, dp1¼1.5, dp¼1.8, ARLðT2
pÞ ¼13.85, andARL(DDT2)¼12.69; all variables were measured only 32% of thetime. II. p1¼2, p¼3, dp1¼1, dp¼1.2, ARLðT2
pÞ ¼59.96, andARL(DDT2)¼59.41; the DDT2 control chart only measures allvariables in 39% of the samples. III. p1¼2, p¼4, dp1¼1.5,dp¼1.8, ARLðT2
pÞ ¼22.64, and ARL(DDT2)¼20.58 with a remark-able value of %p¼21.
When the ARL of the DDT2 control chart is larger than the ARLof the T2 chart for p variables, the difference is often small and thevalue of %p remains low. Some examples are as follows: I. p1¼2,p¼3, dp1¼0.5, dp¼1.5, ARLðT2
pÞ ¼32.51, ARL(DDT2)¼38.75, and
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
Fig. 5. Percent differences of the ARL(DDT2) with respect to ARL(T2p), for p1¼2,
p¼3, dp1n¼0.5 and dpn¼1.5.
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]]12
%p¼51. II. p1¼2, p¼4, dp1¼1, dp¼2, ARLðT2pÞ ¼15.84, and
ARL(DDT2)¼17.93; all variables were measured only 52% of thetime. III. p1¼5, p¼6, dp1¼1.5, dp¼3, ARLðT2
pÞ ¼4.84,ARL(DDT2)¼5.22, and %p¼83.
However, as previously stated, there are few cases where theoptimum DDT2 control chart parameters yield a scheme that ispractically the standard T2 chart for p variables, i.e., the value of%p is close to 100%. Two examples of this case exist: I. p1¼5, p¼6,dp1¼0.5, dp¼1.5, ARLðT2
pÞ ¼53.59, ARL(DDT2)¼53.95, and %p¼92.II. p1¼5, p¼7, dp1¼0.5, dp¼0.7, ARLðT2
pÞ ¼231.11, ARL(DDT2)¼231.62, and %p¼98. In these cases, the ARLs of the two schemesare always similar because both control charts are practicallyequivalent due to the low value of the warning limit (w). Theoptimized DDT2 control chart is optimum for minimizing the ARL,but in these cases the solution is not acceptable in terms of cost.To resolve this issue, which only occurs for p1¼5 and for verysmall values of dp1, we developed a version of the optimizationprogram in which the user can fix the value of %p (Fig. 5).Table 2 reveal that the new ARLs are 68.82 and 238.18, respec-tively, if the previous examples are resolved with that restriction.These two examples are representative: in the first case, the ARLdeteriorates by 28%, although the scheme is significantly moreeconomic; in the second case, the ARL is practically the same withthe same high reduction in cost of sampling with %p¼30 inboth cases.
Therefore, after considering the results displayed in Table 2, wecan conclude that the DDT2 control chart is a useful multivariatescheme when some of the variables are expensive to measure, butthe power to detect process shifts significantly increases. There aresome cases in which the DDT2 control chart outperforms the T2
chart for all variables. For cases in which the unconstrained ARLminimization yields an expensive scheme (high values of %p), theproblem may be solved again with a restriction on %p.
Fig. 6. Percent differences of the ARL(DDT2) with respect to ARL(T2p), for p1¼2,
p¼4, dp1n¼1.5 and dpn¼2.
6. Sensitivity analysis
To obtain the parameters of the DDT2 control chart, thepractitioner has to define a shift (dp1, dp*) for which the ARL isto be minimized while keeping the desired in-control ARL. This isa common practice in optimizing the parameters of qualitycontrol charts. For example, Champ and Aparisi (2008) alsooptimized the double-sampling T2 control chart with a processshift. However, the following question arises: does the optimizedDDT2 chart perform better for other shifts than the shift selectedfor the optimization? This question is answered in the subsequentsection.
Figs. 5–8 exhibit the contour plots of the percent differencesbetween the ARLs of the DDT2 control chart and the ARLs of thestandard T2 chart for p variables, with a star denoting when theDDT2 chart is optimized for the shift. For each contour point (dp1,dp), the percent differences were calculated as follows:
ARLDDT2 ðdp1, dpÞ�ARL T2pðdpÞ
ARLDDT2 ðdp1, dpÞU100 ð11Þ
Note that in each figure the ARLs in the DDT2 chart are theARLs of an equivalent DDT2 chart—the chart optimized for theshift denoted by a star. The T2 chart is the only chart that yieldsthe desired in-control ARL.
The cases (figures) selected for comparison are representativeof the types of behavior we found in this analysis of sensitivity.The first type of behavior is shown in Fig. 5 for cases p1¼2, p¼3,dp1*¼0.5 and dp*¼1.5. There is a small difference between theARLs of both charts (approximately 15%) for the point selected forthe optimization. However, this difference tends to be smallerwhen the value of dp1 increases because the performance of the
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
DDT2 charts depends on both shifts, (dp1, dp), and the ARL of the T2
chart depends only on dp*. Similarly, when the value of dp1
decreases, the differences are larger, but never significantly. Bymoving up along a vertical line (i.e., increasing dp while keeping dp1
constant), both charts reduce their ARLs; however, the standard T2
chart performs better and the difference in ARLs increases slightly.For shifts close to the line dp1¼dp, the DDT2 control chart outper-forms the T2 chart; this occurs in all cases analyzed.
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
Fig. 8. Percent differences of the ARL(DDT2) with respect to ARL(T2p), for p1¼5,
p¼6, dp1n¼0.5 and dpn¼2 with the constraint %p¼50%.
Fig. 7. Percent differences of the ARL(DDT2) with respect to ARL(T2p), for p1¼5,
p¼6, dp1n¼0.5 and dpn¼2.
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]] 13
Fig. 6 illustrates a case (p1¼2, p¼4, dp1*¼1.5 and dp*¼2)where the optimized DDT2 control chart outperforms the stan-dard T2 chart for the point selected for optimization. The areawhere the DDT2 control chart outperforms the standard T2 chart islarger than the area in Fig. 5, and the improvements are greaterthan 10%. For small values of dp1 and large values of dp, the DDT2
performs worse than in the case shown in Fig. 5. However, the
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
ARLs of the two charts are quite similar for a large areasurrounding the point selected for optimization.
Fig. 7 (case p1¼5, p¼6, dp1*¼0.5 and dp*¼2) is representativeof the special behavior that occurs when the number of variablesto be monitored is not small. In these cases, for any point (dp1, dp),the difference between the performances of the DDT2 controlchart and the performances of the standard T2 chart forp variables are always small: both charts are practically equiva-lent in terms of ARL. The reason for this is demonstrated byTables 5 and 6, which show that for such values of p1 and p, theoptimization tends to produce a DDT2 control chart with a largevalue of %p. Hence, the DDT2 chart is equivalent to the T2 chartand a restriction for %p should be used in the optimization toobtain a significant reduction in sampling costs.
Repeating the optimization for the same case but with theadditional constraint %p¼50%, we obtain the contours shown inFig. 8. Now, the results and conclusions are very similar to theresults and conclusions in Fig. 6, i.e., the ARL differences for theselected point (dp1*, dp*) are not large; the differences are similararound the selected point and when dp1 increases, the differencestend to be smaller and negative for shifts close to the line dp1¼dp.
In summary, although the optimization of the DDT2 controlchart requires that the user specify a shift (dp1*, dp*), the selectionof this shift does not need to be very precise because theoptimized DDT2 chart performs similarly for shifts in a large areasurrounding the specified shift. For shifts with larger dp1 values, itwill outperform the standard T2 chart; for shifts with large dp andsmall dp1, the T2 chart will outperform it.
7. Conclusions
This paper introduces a new multivariate control chartdesigned for the case in which some of the variables to bemonitored are much more expensive and/or difficult to measurethan other variables. The objective is to monitor the expensivevariables only as needed, which yields a scheme that is moreeconomical but maintains quality performance in the detection ofprocess shifts, i.e., has low out-of-control ARLs. These objectivesare fulfilled with the Double-Dimension T2 (DDT2) control chart.
A comparison of the performance of this chart with theperformance of the standard T2 chart was conducted afterobtaining the formula to compute the ARL, and the procedure todetermine the parameters of the DDT2 chart that minimize itsout-of-control ARL. The results showed that in some cases theoptimized DDT2 chart can outperform the standard T2 chart for allvariables; i.e., it constitutes a faster and more economical controlscheme. In the remaining cases, for which the T2 chart displayslower ARL values, the differences in ARL performance relative tothe DDT2 chart tend to be small and the DDT2 chart clearly yieldsa more economical scheme.
Finally, to facilitate and promote the use of the DDT2 chart,user-friendly software, which is available from the authors, hasbeen developed that enables the user to easily obtain optimalparameters for any given case.
Appendix. Distribution of D¼ T2p�T2
p1
Using the notation of Murphy (1987), we restate the problem.Define X as a random vector of dimension p¼p1þp2 with amultivariate normal distribution with mean l and covariancematrix R. Both l and R can be partitioned according to thedimensions p1 and p2 as follows:
X�Npðl,RÞ-X¼Xð1Þ
Xð2Þ
" #�Np1þp2
lð1Þ
lð2Þ
" #,S1,1 S1,2
S2,1 S2,2
" # !,S1,2 ¼S02,1 ðA1Þ
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]]14
Define a target value for l as a constant vector l0 of dimensionp, which is partitioned in the same way as above
l0 ¼lð1Þ0
lð2Þ0
24
35 ðA2Þ
Next, consider a random (i.i.d.) sample of size n from therandom vector X, given by X1,:::, Xn, and define the followingsample mean vector X which is partitioned as above
X ¼1
n
Xn
k ¼ 1
Xk-X ¼Xð1Þ
Xð2Þ
" #ðA3Þ
Define the following two statistics and their difference D
T2p ¼ nðX�l0Þ
0R�1ðX�l0Þ ðA4:aÞ
T2p1¼ nðX
ð1Þ�lð1Þ0 Þ
0R�11,1ðX
ð1Þ�lð1Þ0 Þ ðA4:bÞ
D¼ T2p�T2
p1ðA4:cÞ
We will show that D, given by (A4.c), follows a non-central chi-squared distribution with a non-centrality parameter dependenton l, l0, R and X
ð1Þ. More specifically, we will show that D is n
times the difference between the squares of the Mahalanobisdistances of l and l(1) with respect to l0. D is also equal to n
times the square of the Mahalanobis distance of the conditionalmean of X(2), given x(1), the observed value of the random vectorX(1).
First, we present a list of properties and results that we aregoing to utilize. From the general formula of the inverse of apartitioned matrix
S�1¼
S1,1 S1,2
S2,1 S2,2
" #�1
¼S�1
1,1þS�11,1S1,2ðS2,2�S2,1S�1
1,1S1,2ÞS2,1S�11,1 ½�ðS2,2�S2,1S�1
1,1S1,2Þ�1S2,1S�1
1,1�0
�ðS2,2�S2,1S�11,1S1,2Þ
�1S2,1S�11,1 ðS2,2�S2,1S�1
1,1S1,2Þ�1
24
35�1
:
ðA5Þ
The distribution of the sample mean X
It is a well-known result that if X in Eq. (A3) is measured with ani.i.d. sample from a multivariate normal distribution as in (A1), then
X �Np l,1
nS
� �ðA6Þ
Note that the same result is true of the subvectors of X asfollows:
X ¼Xð1Þ
Xð2Þ
" #�Np1þp2
lð1Þ
lð2Þ
" #,ð1=nÞ
S1,1 S1,2
S2,1 S2,2
" # !, ðA6:aÞ
Resulting multivariate normal conditional distributions
If the random vector X is defined as in Eq. (A1), the distributionof X(2) conditional to observing the value x(1) for the randomvector X(1) is given by a multivariate normal distribution ofdimension p2, with mean and covariance matrix expressed interms of the partitions of matrix R. Formally,
X2=1 � ðXð2Þ9Xð1Þ ¼ xð1ÞÞ �Np2
ðl2=1,S2=1Þ ðA7Þ
with
l2=1 ¼ lð2Þ þS�12,2S2,1ðx
ð1Þ�lð1ÞÞ ðA8:aÞ
S2=1 ¼S2,2�S2,1S�11,1S1,2 ðA8:bÞ
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
Note the coincidence of the right-hand side of (A8.b) withthe inverse of the lower right submatrix of the right-hand side ofEq. (A5).
The proofs
First, we reproduce the proof of the results derived for thedistribution of D when l¼l0, as presented by Murphy (1987).Then, we derive the distribution of D for any value of l, whetherequal to l0 or not equal to l0.
Proof 1:. Expressing D in terms of the conditional distribution(Murphy, 1987)
Recall that D¼ T2p�T2
p1; then, using Eqs. (4)a and b
D¼ nðX�l0Þ0S�1ðX�l0Þ�nðX
ð1Þ�l0Þ
0S�11,1ðX
ð1Þ�l0Þ ðA9Þ
Expressing (A9) in terms of the partition of X
D¼ n½ðXð1Þ�lð1Þ0 Þ
0^ðXð2Þ�lð2Þ0 Þ
0�S1,1 S1,2
S2,1 S2,2
" #�1ðXð1Þ�lð1Þ0 Þ
ðXð2Þ�lð2Þ0 Þ
24
35
�nðXð1Þ�lð1Þ0 Þ
0S�11,1ðX
ð1Þ�lð1Þ0 Þ ðA10Þ
Define an auxiliary variable Y as
Y ¼ffiffiffinpðX�l0Þ ¼
Y ð1Þ
Y ð2Þ
" #¼
ffiffiffinpðXð1Þ�lð1Þ0 Þffiffiffi
npðXð2Þ�lð2Þ0 Þ
24
35 ðA11Þ
Using the result (A6.a), we know that
Y �Npðy,SÞ orY ð1Þ
Y ð2Þ
" #�Np1þp2
yð1Þ
yð2Þ
" #,S1,1 S1,2
S2,1 S2,2
" # !ðA12Þ
with
y¼ffiffiffinpðl�l0Þ ¼
yð1Þ
yð2Þ
" #¼
ffiffiffinpðlð1Þ�lð1Þ0 Þffiffiffi
npðlð2Þ�lð2Þ0 Þ
24
35 ðA13Þ
and using (A8.a) and (A8.b)
Y2=1 � ðYð2Þ9Y ð1Þ ¼ yð1ÞÞ �Np2
ðy2=1,S2=1Þ ðA14Þ
with
y2=1 ¼ yð2Þ þS�12,2S2,1ðy
ð1Þ�yð1ÞÞ and S2=1 ¼S2,2�S2,1S�11,1S1,2 ðA15Þ
After long and tedious algebra, using (A5) in Eq. (A10), one canuse (A15) and (A13) to obtain
D¼ Y2=10S�1
2=1Y2=1 ðA16Þ
Proof 2. The distribution of D
Now, it is easy to derive the distribution of D. First, we definethe random vector
Z ¼S�ð1=2Þ2=1 Y2=1 ðA17Þ
Thus, we can use (A16) to write D as
D¼ Z0Z ðA18Þ
It is easy to verify that
Z �Np2ðS�ð1=2Þ
2=1 y2=1,IkÞ ðA19Þ
Then, by the definition of the non-central chi-squared distri-bution,
D� w2p2 ,l2=1
ðA20Þ
ts in multivariate SPC with a double-dimension T2 control chart.org/10.1016/j.ijpe.2013.01.022i
E.K. Epprecht et al. / Int. J. Production Economics ] (]]]]) ]]]–]]] 15
with the non-centrality parameter given by
l2=1 ¼ ðS�ð1=2Þ2=1 y2=1Þ
0ðS�ð1=2Þ
2=1 y2=1Þ ¼ y02=1S�12=1y2=1 ðA21Þ
Again, using Eq. (A5) and performing in reverse order the samemanipulations used to obtain (A16) from (A10), we obtain
l2=1 ¼ y02=1S�12=1y2=1 ¼ y0S�1y�y01S
�111 y1
¼ nðl�l0Þ0S�1ðl�l0Þ�nðlð1Þ�lð1Þ0 Þ
0S�111 ðl
ð1Þ�lð1Þ0 Þ
which is n times the difference between the squares of theMahalanobis distances of l and l(1) with respect to l0.
References
Aparisi, F., De Luna, M., Epprecht, E., 2010. Optimisation of a set of X or principalcomponents control charts using genetic algorithms. International Journal ofProduction Research 48 (18), 5345–5361.
Bersimis, S., Psarakis, S., Panaretos, J., 2006. Multivariate statistical process controlcharts: an overview. Quality and Reliability Engineering International 23 (5),517–543.
Chen, Y.-K., 2007. Economic design of variable sampling interval T2 control charts-A hybrid Markov Chain approach with genetic algorithms. Expert Systemswith Applications 33, 683–689.
Croasdale, R., 1974. Control charts for a double-sampling scheme based on averageproduction run length. International Journal of Production Research 12,585–592.
Please cite this article as: Epprecht, E.K., et al., Reducing sampling cosInternational Journal of Production Economics (2013), http://dx.doi.
Crosier, R.B., 1986. A new two-sided cumulative sum quality control scheme.Technometrics 28 (3), 187–194.
Costa, A.F.B., De Magalh~aes, M.S., 2005. Economic design of two-stage X charts: theMarkov-chain approach. International Journal of Production Economics 95 (1),9–20.
Duncan, A.J., 1992. Quality Control and Industrial Statistics, 5th ed.. Irwin, Home-wood, IL.
Hotelling, H., 1947. Multivariate Quality Control, Illustrated by the Air Testing ofSample Bombsights. Techniques of Statistical Analysis. MacGraw Hill, NewYork. (pp. 111–184).
Johnson, R.A., Wichern, D.W., 2007. Applied Multivariate Statistical Analysis, 6thed.. Prentice-Hall, Inc., Upper Saddle River, New Jersey.
Kaya, I., 2009. A genetic algorithm approach to determine the sample size forcontrol charts with variables and attributes’. Expert Systems with Applications36, 8719–8734.
Lowry, C.A., Montgomery, D.C., 1995. A review of multivariate control charts. IIETransactions 27 (1), 800–810.
Lowry, C.A., Woodall, W.H., Champ, C.W., Rigdon, S.E., 1992. A multivariateexponentially weighted moving average control chart. Technometrics 34,46–53.
Montgomery, D.C., 2012. Introduction to Statistical Quality Control, 7th Edition.Wiley, New York.
Murphy, B.J., 1987. Selecting out of control variables with the T2 multivariatequality control procedure. The Statistician 36 (5), 571–583.
Rodrigues, A.A.A., Epprecht, E.K., De Magalh~aes, M.S., 2011. Double samplingcontrol charts for attributes. Journal of Applied Statistics 38 (1), 87–112.
Steiner, S.H., 1999. Confirmation sample control charts. International Journal ofProduction Research 37 (4), 737–748.
