ISSPC 2011 - Rio de Janeiro 1

T2 Control Charts with Variable Dimension

Eugenio K. Epprecht - PUC-Rio, Brazil

Francisco Aparisi – UPV, Spain

Omar Ruiz - ESP del Litoral, Equador

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Outline • Introduction

– Basic idea

– Related works

• Standard T2 control chart procedure• Charts proposed

– Variable-Dimension T2 control chart (VDT2)

– Double-Dimension T2 control chart (DDT2)

• ARL computation– ARL of the VDT2 chart

– ARL of the DDT2 control chart

• Optimization and Performance Comparison– User Interface

– Results

– Performance Comparison

• Another version of the problem (and software)• Continuation of the research• References

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Introduction - basic idea

T2 control chart number of variables to monitor: variable

p = p1 + p2p1 out of the p variables: easy or cheap to monitor/measure remaining p2 variables: difficult/expensive to monitor/measure

Idea: controlling sometimes only the p1 (“cheap”) variables, and only when the process seems to have a problem: controlling the full set of p variables.

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Examples

• Expensive electronic component with 3 (correlated) quality variables to be monitored.– X1 and X2: voltages, cheap and easy to measure– X3: measurement is destructive (e.g. the voltage that

will burn a part of component)

• Other cases:– some variables could need, for instance, a laboratory

analysis (difficult, slow, expensive, etc.)

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Similar to adaptive control charts,

but

instead of varying sample size, sampling interval and/or control limits:

varying the number of variables to consider

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Related works • Univariate:

– Surrogate monitoring (Costa and De Magalhães, 2005)

– Double-stage sampling (by attribute – by variables) (Costa et al., 2005)

• Multivariate:– T2 – reduces the number of charts– PC’s – reduce the number of dimensions of the space ...but one still measures all p variables!– González and Sanchez (2010) – reduce the number

of variables(but always! and according to other

criterion)

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Our goal:

to reduce the number of variables to measure (on average)

reducing costs (and/or time)

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Standard T2 control chart procedure

samples of size n, of the p-dimensional vector(X1, X2, ... Xp)

where:

: in-control mean vector

: in-control covariance matrix

2 ' 10 0( ) ( )i i iT n X X

'0 0,1 0,2 0,( , , ... , )p

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Standard T2 control chart procedure – (cont.)

• When the process is in control ( )

• When the process is out of control ( )

where

or

d : Mahalanobis’ distance of with respect to

0i

22 ~ piT

0i

' 10 0( ) ( )i in

2nd

)(~ 22 piT

1 0

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Charts proposed

• VDT2 control chart (variable dimension)

• DDT2 control chart (double dimension)

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Variable-Dimension T2 control chart (VDT2)

We may have two warning limits (w1 and w),

but our results show that

although a chart with two warning limits shows better

performance, the improvement is not large

w

w

CLp1

CLp

p1

p1

p1

p2

p2

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Double-Dimension T2 control chart (DDT2)

if T2p1 ≤ w process considered in control

if T2p1 ≥ CLp1 process declared out of control

if w ≤ T2p1 ≤ CLp1 remaining p2 variables are measured and

combined with the measurements of the set of p1 variables T2p

T2p plotted in the vertical of T2

p1 and compared with CLp

decision to measure all variables: taken for the same sample,

not waiting until the next sampling time

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Double-Dimension T2 control chart (DDT2)

w

w

CLp1

CLp

p1

p1

p1p1

p1+p2

p1

p1+p2

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ARL computation • ARL of the VDT2 chart:

Markov’s chain model -- similar to ARL of adaptive (variable) Shewhart charts (VSS, VSI, VSSI, Vp)

(Reynolds et al., 1988; Aparisi, 1996; Costa, 1999; Epprecht et al., 2003)

In the case of only one warning limit:

• State 1:

(and the next sample will contain only p1 variables)• State 2:

(and the next sample will contain all p variables)• State 3:

(signal; absorbing state)

2w T CL

CLT 2

wT 2

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ARL of the VDT2 chart (cont.)

Transition probability matrix for shift d :

For example:

where

100

3,22,21,2

3,12,11,1ddd

ddd

d PPP

PPP

P

2 21,2 1 1 1 1( | , ) ( ( ) )d

i pP P w T CLp p d P w CLp

2dn

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ARL of the VDT2 chart (cont.)

• Zero-state ARL:

where

• ,• I : 22 identity matrix,

• : vector of initial state probabilities,

• : out-of-control probability transition matrix between transient states

1)()( 1

dQIBdARL

T111

21, bbB

dQ

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ARL of the VDT2 chart (cont.)

• Zero-state ARL:

where

• ,• I : 22 identity matrix,

• : vector of initial state probabilities,

• : out-of-control probability transition matrix between transient states

For the ARL0, use Q0 instead of Qd in the formula above

1)()( 1

dQIBdARL

T111

21, bbB

dQ

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ARL of the VDT2 chart (cont.)

• Steady-state ARL:

: in-control probability transition matrix between transient

states

11 dSS QISdARL

01

0 ARLQIBS

0Q

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ARL of the DDT2 control chart Calculations analogous to those of the ARL of

– double-sampling control charts(Daudin, 1992; Champ& Aparisi, 2008; De Araujo Rodrigues et al., 2011)

– double-sampling procedures for acceptance sampling

ARL = 1 / (1 – P(no signal))

1

22 2 21 1 20 0

( ) ( ) ( ) ( )p

p p p

CL vw CLp

p

wP no signal f v dv f v f u dudv

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ARL of the DDT2 control chart Calculations analogous to those of the ARL of

– double-sampling control charts(Daudin, 1992; Champ& Aparisi, 2008; De Araujo Rodrigues et al., 2011)

– double-sampling procedures for acceptance sampling

ARL = 1 / (1 – P(no signal))

ARL results for the DDT2 chart are underway

1

22 2 21 1 20 0

( ) ( ) ( ) ( )p

p p p

CL vw CLp

p

wP no signal f v dv f v f u dudv

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Optimization and

Performance Comparison • Software for design optimization — using Genetic Algorithms

Execution time < 5 min

• Case:– p1 = 2, p2 = 2, p = 2 + 2 = 4.– Desired in-control ARL: 400– Shifts specified:

• dp1 = 0.5 (when p1 = 2 variables are measured)• dp = 1 (when all p = 4 variables are monitored)

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User interface(for VDT2 chart with one warning limit)

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Results w = 2.61,

(which has an in-control probability “left tail” equal to 0.729)

CLp1 = 40.95,with right tail probability = 0

no control limit is needed for p1 variables common result for this scheme, which simplifies the use of the chart, because only the warning limit and one control limit (for the p variables) are required.

CLp = 14.44with a right tail probability of 0.00602.

In-control ARL = 399.94. ARL for the given shift is 101.31

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Comparing this performance with the performances of:

T2 control chart where always p1 variables are measured

T2 control chart where always all the p variables are monitored (with ARL0 = 400)

T2 (p1 variables): out-of-control ARL = 216.90

T2 (p variables): out-of-control ARL = 107.86

(VDT2: ARL = 101.31)

ARL improvement is marginal

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However: • sampling cost of VDT2 << sampling cost of T2p

• % of times than we must sample all the variables when the process is in control: 42%

• Average cost of sampling of the VDT2:

- cost of measuring the p1 variables: c1

- cost of measuring the p2 variables: c2

average cost per sample: c1 + 0.42c2

(against a fixed cost of c1 + c2 of the T2p)

relative economy of 58c2 / (c1 + c2)% per sample)

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Another version of the problem (and software)

Average cost per sample:

c1 + 0.42c2 or, equivalently, 0.58c1 + 0.42(c1 + c2)

Depending on the values of c1 and c2, this may be still high

in other version of the problem:

% of times that all p variables are measured(when the process is in control):

constraint for the optimization

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Example

If this percentage is fixed at 20%, the sftw returns:

w = 3.82

CLp1 = 24.35(with right tail probability = 0.00000470

practically the control limit can be set to infinite)

CLp = 12.80,(right tail probability = 0.01229719)

In-control ARL = 400.

OOC ARL = 105.74

with only a marginal reduction in performance,

the sampling cost could be significantly reduced

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Continuation

• Optimization considering costs (and/or time) to measure

(under a given cost per time and a given ATS0,

minimize the AATS)

• Enhanced schemes

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REFERENCES • Aparisi, F. (1996). Hotelling’s T2 control chart with adaptive sample sizes. IJPR, 34, 2853-2862.

• Aparisi, F., De Luna, M., Epprecht, E. (2010). Optimisation of a set of Xbar or principal components control charts using genetic algorithms. IJPR, 48(18), 5345-5361.

• Champ, C. W. and Aparisi, F. (2008). Double Sampling Hotelling's T2 Charts. QREI, 24, 153-166.

• Costa, A. F. B. (1999) Xbar charts with variable parameters, JQT, 31(4), p. 408-416

• Costa, A.F.B. and De Magalhães, M. S. (2005). Economic design of two-stage Xbar charts: the Markov-chain approach. IJPE 95(1), p.920.

• Costa, A. F. B., De Magalhaes, M. S. and Epprecht, E. K. (2005). The Non-central Chi-square Chart with Double Sampling. Brazilian Journal of Operations and Production Management 2(1), p. 57-80.

• Daudin, J. J. (1992) Double sampling Xbar charts, JQT 24(2), pp. 78-87

• Epprecht, E. K.; Costa, A.F.B.; Mendes, F.C.T. (2003) Adaptive Control Charts for Attributes. IIE Transactions, 35 (6), p. 567-582.

• De Araujo Rodrigues, A.; Epprecht, E.K. ; De Magalhães, M.S. (2011) Double-sampling control charts for attributes. JAS, 38, p. 87-112.

• González, I. and Sanchez, I. (2010). Variable Selection for Multivariate Statistical Process Control. JQT 42(3), p. 242-259.

• Reynolds, M. R. Jr., Amin, R. W., Arnold, J. C. & Nachlas, J. A. (1988). Xbar charts with variable sampling intervals, Technometrics 30(2), p.181-192.