Multivariate adaptive approach for monitoring ... - Semantic Scholar
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Int. J. Data Analysis Techniques and Strategies, Vol. 6, No. 1, 2014
Multivariate adaptive approach for monitoring simple linear profiles Galal M. Abdella* and Kai Yang Department Industrial and Systems Engineering, Wayne State University, 4815 Fourth St., Detroit, Michigan 48202, USA E-mail: [email protected] E-mail: [email protected] *Corresponding author
Adel Alaeddini Department of Industrial and Operations Engineering, University of Michigan, 1205 Beal Ave., Ann Arbor, MI 48109, USA E-mail: [email protected] Abstract: Adaptive sample size and sampling intervals schemes have been widely used to improve the statistical efficiency of Hotelling T2 control chart in detecting small changes when the quality of a product or a process can be characterised by the multivariate distribution of quality characteristics. In this paper, we design a Hotelling T2 scheme varying sample sizes and sampling intervals (VSSI-T2) for accelerating the speed of detecting off-target conditions in linear profile parameters. We investigate the statistical performance of the adaptive approach versus its fixed sampling counterparts. To find the optimal setting of the VSSI-T2, we build an optimisation model solved using genetic algorithm (GA). Also, average time to signal (ATS) is considered as the objective function of the model and estimated using the Markov chain fundamentals. The comparative studies reveal the potentials of the adaptive scheme in improving the performance of the Hotelling T2 control chart in monitoring linear profiles. Keywords: varying sample sizes and sampling intervals; VSSI; scheme, average time to signal; ATS; Hotelling T2 control chart; phase I and II; simple linear profiles. Reference to this paper should be made as follows: Abdella, G.M., Yang, K. and Alaeddini, A. (2014) ‘Multivariate adaptive approach for monitoring simple linear profiles’, Int. J. Data Analysis Techniques and Strategies, Vol. 6, No. 1, pp.2–14. Biographical notes: Galal M. Abdella is a PhD candidate at Industrial and Systems Engineering at Wayne State University in Detroit-Michigan, USA. He is interested in statistical quality control and discrete event systems simulation. Kai Yang is a Professor at the Department of Industrial and Systems Engineering at Wayne State University in Detroit-Michigan. His field of expertise includes quality, reliability and product development. He received his MS degree in 1985, and PhD in Industry Engineering in 1990, both of them from the University of Michigan.
Copyright © 2014 Inderscience Enterprises Ltd.
Multivariate adaptive approach for monitoring simple linear profiles
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Adel Alaeddini is a Post-Doc Researcher in the Department of Industrial and Operations Engineering at University of Michigan. Ann Arbor, MI, USA. His research interest includes statistical learning, global optimisation and healthcare optimisations improvement.
1
Introduction
Profile monitoring plays a vital role in identifying the off-target conditions when the quality characteristic/s of the process or product can be statistically related to a set of independent variables. An off-target condition in profile data occurs when one or more of the parameters of the underlying quality function change due to an assignable cause; consequently the process is no longer stable. The past few years has witnessed an increasing interest on developing new profiling techniques and examining their ability in detecting changes based on regression models. Similar to other control charting methods, profile monitoring schemes can be designed for phase I and/or II. In phase I analysis, practitioners are mainly interested in investigating the statistical stability of process parameters and estimating their unknown values, while in phase II, these parameters are assumed to be known and one is interested in accelerating the process of detecting any deviations in their nominal values. The following is a brief overview of the prior related work. Stover and Brill (1998) suggested two strategies for monitoring profiles in phase I with an application in multilevel ion chromatography. Mahmoud and Woodall (2004) developed a F-test to monitor the regression parameters in conjunction with an univariate control chart to monitor changes in process variability in phase I. Mahmoud et al. (2007) examined a change point approach based on the segmented technique for testing the constancy of the phase I regression parameters for a linear profile data. Most of the previous literatures focus on phase II profiling techniques. Kang and Albin (2000) considered a semiconductor manufacturing process and suggested two phase II methods. The first technique is a multivariate Hotelling T2 control chart for monitoring intercept and slope of simple linear profiles. The second technique uses an exponentially weighted moving average control chart (EWMA) in conjunction with the range control chart to monitor the fitted residual (EWMA/R method). Kim et al. (2003) proposed a new technique based on coding the values of the explanatory variable (X) to remove the correlation between regression parameters, including the slope and the intercept, as well as residuals and then used three separate EWMA control charts to monitor the uncorrelated parameters (EWMA3 method). Woodall et al. (2004) provided areas for future work and research on profile monitoring. Gupta et al. (2006) performed a phase II comparative study between the Croarkin and Varner (1982) control chart (NIST method) and the combined strategy proposed by Kim et al. (2003) (KMW method). In their study, they modified the KMW method to use three separate Shewhart control charts instead of EWMA charts. When the number of available samples is not sufficiently large for estimating the process parameters, Zou et al. (2007) proposed a self starting control chart for monitoring linear profiles. Zhang et al. (2009) developed a single chart which integrates likelihood ratio statistic with the EWMA procedure for monitoring linear profiles. The variable sampling interval feature is applied to this chart to enhance the speed of detecting changes in regression parameters. A new technique based on
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G.M. Abdella et al.
cumulative sum statistic for phase II linear profile is proposed by Saghaei et al. (2009). Zhu and Lin (2010) proposed and investigated a method to monitor changes in the slope of the linear functions. Another recent contribution was by Noorossana et al. (2010). They developed and explored the performance of three control chart schemes when several correlated characteristics might be modelled as a set of linear functions of one independent variable. This situation is referred to as multivariate simple linear profiles structure. Li and Wang (2010) proposed the use of an EWMA chart with variable sampling intervals for monitoring linear quality functions (VSI-EWMA3). They used a real data from an optical imaging system to examine the suggested method. Sometimes, it is more appropriate to describe the relationship between the quality response and the explanatory variable by more sophisticated models such as polynomial or non-linear quality models. Montgomery (2005) studied the situation when the output/input relationship might be modelled as a second-order polynomial regression. The relation between the torque produced by an automobile engine and the engine speed is considered as an illustrative example for this model type. Kazemzadeh et al. (2008) examined three phase I methods for monitoring polynomial profiles and developed a method based on likelihood ratio test for identifying the location of the shift. Kazemzadeh et al. (2009) introduced a new approach using three individual EWMA control charts after transferring the polynomial function to the orthogonal form. A case study shows the use of monitoring polynomial quality profiles in the automotive industry was presented by Amiri et al. (2009). Noorossana et al. (2008) considered the case when the quality profiles are not independent of each other. Soleimani et al. (2009) investigated how the speed of detecting changes in regression parameters is influenced by autocorrelation. The effect of correlation on non-linear profiles can be found in Jensen and Birch (2009). In this analytical study, they used the non-linear mixed models. Other researchers such as Noorossana et al. (2004, 2008) studied the effect of violating the normality assumption on the monitoring phase of linear profiles. The idea of changing the sample size (VSS) or the sampling interval (VSI) or both of them during the online process monitoring has been successfully examined with the Hotelling T2 chart (Aparisi, 1996; Aparisi and Haro, 2001, 2003; Chen, 2009), but none of the prior works considered the case when the quality is described by the regression model. Accordingly, our focus in this paper is to integrate the both of adaptive schemes, namely VSS and VSI, with the T2 control chart, suggested by Kang and Albin (2000), to capture any changes in intercept and slope of the corresponding linear function. We refer to this scheme as a VSSI-T2 control chart.
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The variable sampling sizes and variable intervals – T2 scheme (VSSI-T2)
This work assumes that the outgoing quality is a random variable Y that is a simple linear function of an explanatory variable X; that is: Yi = A0 + A1 X i + εi
i = 1, 2,… , n
(1)
where A0 and A1 are the parameters of intercept and slope. The random variable ε is an independent and normally distributed variable with zero mean and variance σ2. Here, the −1 ( Z j − U ) reported and used by Kang and Albin Hotelling’s statistic T j2 = ( Z j − U )T
∑
(2000) is utilised for capturing deviations in A0 and A1; where U = (A0, A1)T) is the mean
Multivariate adaptive approach for monitoring simple linear profiles
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vector, and Zj = (a0j, a1j)T is the vector of the process parameter estimators. The variance-covariance matrix (Σ) is described as follows: ⎛ σ 02
∑ = ⎜⎝ σ
2 01
2 σ 01 ⎞ ⎟ σ12 ⎠
(2)
Kang and Albin (2000) used the least square method for estimating the regression parameters of each set of observations (profile), such that α0 j = Y − α1 j x , −1 −1 −1 α1 j = S xy ( j ) S xx , σ 02 = (σ 2 n −1 + x 2 σ 2 S xx ) and σ12 = σ 2 S xx ; where σ 02 and σ12 are the variances of the regression parameters. The covariance of α0j and α1j is calculated using 2 −1 2 σ 01 = −σ 2 xS xx . The control limit of the VSSI-T2 scheme is set equal to χ 2,α which can
be determined by the (1 – α) percentile point of a chi-square with 2 degrees of freedom; see Kang and Albin (2000). The adaptive approach uses two time intervals (t1, t2), sample sizes (n1, n2) and one warning limit (WL). Note that, n1 ≤ n0 ≤ n2
(3)
t1 ≤ t0 ≤ t2
(4)
where n0 and t0 are the sample size and the sampling interval of traditional T2 multivariate control chart using fixed sampling rate (FSR-T2). The mechanism of switching between the parameters of adaptive T2 chart is described below; see Aparisi and Haro (2001). 1
if 0 ≤ T j2−1 < WL, use (n1, t2)
2
if WL < T j2−1 < CL, use (n2, t1)
3
if T j2−1 ≥ CL, signal.
A commonly used method to assess the effectiveness of such adaptive scheme (VSSI-T2) is matching VSSI-T2performance with its counterparts (FSR-T2) at the state of the statistical control and then investigating and comparing their performance at the off-target conditions. In this article, the same approach will be followed, and the average time to signal (ATS) is used as the measure of the performance for comparing the two schemes. The rest of this subsection is dedicated to introduce the mathematical model developed to determine the optimal setting of the VSSI-T2 chart. Min ATS ( v, δ, WL, n1 , n2 , t1 , t2 )
(5)
Subject to: n0 = n1 p1 + n2 p2
(6)
t0 = t2 p1 + t1 p2
(7)
n1 ≤ n0 ≤ n2
(8)
t1 ≤ t0 ≤ t2
(9)
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G.M. Abdella et al. t1 ≥ C0 n2
(10)
0 < WL < CL
(11)
n1 , n2 ∈ Z + , n1 ≥ 2
where p1 = P ( 0 ≤ T j2−1 < WL ) , p2 = P (WL < T j2−1 < CL ) , δ is the amount of deviation from the target value of the model parameter, and C0 is the sampling factor; for instance, if an inspector takes 0.5 min. to examine and measure one product, then C0 is set equal to 0.5/60 = 0. 00834 hr. Note that equation (11) is added to guarantee that the minimum sampling interval (t1) is always enough to collect the large sample (n2). Equations (6) to (7) can be rewritten as follows: n0 = n1 P ( 0 ≤ T j2 < WL ) + n2 P (WL < T j2 < CL )
(12)
t0 = t2 P ( 0 ≤ T j2−1 < WL ) + t1 P (WL < T j2 < CL )
(13)
Here, we extend the use of fundamentals of Markov Chain in estimating the statistical properties of T2 control charts (see Çinlar, 1975; Costa, 1993; Prabhu et al., 1993; Aparisi, 1996; Aparisi and Haro, 2001, 2003; Chen, 2004, 2009; Chen and Hsieh, 2007; Faraz and Moghadam, 2008; Faraz and Saniga, 2010). Accordingly, we use the same transient states reported by Aparisi and Haro (2001) and Faraz and Moghadam (2008): 1
state 1 represents 0 ≤ T j2 < WL
2
state 2 represents WL < T j2 < CL
3
state 3 represents T j2 ≥ CL (absorbing state).
The transition probability matrix and the ATS approximation are defined as (see Aparisi, 1996; Aparisi and Haro 2001; Faraz and Moghadam, 2008): δ ⎛ p11 ⎜ δ Pδ = ⎜ p21 ⎜ 0 ⎝
δ p12 δ p22
0
δ p13 ⎞ ⎟ δ p23 ⎟ 1 ⎟⎠
(14)
where pijδ is the probability of moving from state i to state j when the drift in the regression parameter is equal to δ. The ATS approximation is as follows: ATS δ = P0′ ( I − Qδ )
−1t
(15)
where P0′ = ( P01 , P02 ) is the vector of the initial states, I is the identity matrix, t′ = (t2, t1), and Qθ is the probability transition matrix without the absorbing state; see Figure 1.
Multivariate adaptive approach for monitoring simple linear profiles Figure 1
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The transition diagram for the VSSI-T2 control chart
Kang and Albin (2000) reported that, under the off-target conditions the Hotelling T2 statistic follows a non-central χ v2,α distribution. This distribution has a non-centrality parameter (τ) equals to ( λ + βx ) 2 n + β 2 S xx . We rewrite the non-centrality parameter in terms of the prior state i as follows: τ i = ( λ + βxi ) ni + β 2 ( S xx )i 2
i = 1, 2
(16)
Then, the ATS approximation can be written as: ⎛ ⎛1 0 ⎞ ⎛ p δ 11 ATSδ = ( P01 , P02 ) ⎜ ⎜ ⎟−⎜ δ ⎜ ⎝ 0 1⎠ ⎝ p21 ⎝
δ p12 ⎞ ⎟ δ p22 ⎠
−1
⎞ ⎛ t2 ⎞ ⎟⎜ ⎟ ⎟ ⎝ t1 ⎠ ⎠
(17)
When the quality can be modelled by the multivariate distribution of a set of characteristics, the transition probability is defined in terms of niδ2; where i = 1, 2, (see Aparisi, 1996). Since this work extends the use of the VSSI scheme to capture the structure of simple linear profiles, we redefine the probability of transition in terms of τi as follows: δ p11 = P ( 0 ≤ χ v2 ( τ1 ) ≤ WL )
(18)
δ p12 = P (WL < χ v2 ( τ1 ) < CL )
(19)
δ p21 = P ( 0 ≤ χ v2 ( τ 2 ) ≤ WL )
(20)
δ p22 = P (WL < χ v2 ( τ 2 ) < CL )
(21)
And, the initial state probabilities are the same as the case when the quality is described by the multivariate probability distribution, see Aparisi and Haro (2001) and Faraz and Moghadam (2008), and calculated as follows: P01 =
P02 =
P ( χv2 < WL )
(22)
P ( χ v2 < CL )
P (WL < χv2 < CL ) P ( χv2 < CL )
= 1−
P ( χv2 < WL ) P ( χ v2 < CL )
= 1 − P01
(23)
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G.M. Abdella et al.
Setting the parameters of the genetic approach
Genetic algorithm (GA) is one of the popular stochastic optimisation techniques used for optimising design parameters of various control charts (e.g., X , Hotelling T2, and EWMA control charts) (see He et al., 2002; Chen, 2004; He and Grigoryan, 2005; Chen and Hsieh, 2007; Chou et al., 2008). In this article, finding the optimal design parameters of VSSI-T2 scheme is formulated as an optimisation problem and solved using the GA. In this regard, two Taguchi experiments were conducted to find the optimal setting of the GA parameters, population size (PS), probability of crossover and mutation rate (MR). These inputs with the shift value in regression parameters, intercept and slope, were analysed in L9(34) orthogonal arrays. The optimal levels of the GA parameters are determined and presented in Table 1. Note, Taguchi experiments have been done with n0 = 5, and false alarm rate equals (ATSδ=0) to 200. Table 1
Optimal inputs of GA
Shift in
CP
MR
PS
Intercept
0.5
0.1
100
Slope
0.5
0.1
100
4
Measuring the performance of VSSI-T2 chart
To examine the performance of the suggested adaptive scheme in capturing changes in the parameters of a simple linear models, we compare its effectiveness with the traditional fixed sample size T2 control chart for n0 = 3,5 and 10, t0 = 1 and C0 = 0.00834. The values of in-control ATS of both schemes are set to approximately 200. In this paper, the data points of the independent variable are distributed equidistantly over a practical range (from 1 to 5) with two points at the edges. For instance, if the sample size equals to 3, one can use X = {1, 3, 5}. Her we only show the results of the case with n0 = 5 and t0 = 1. Under the profile monitoring framework, the values of the independent variable take the same magnitude from one profile to another. In this paper, we provide different sets of VSSI plans and examine their power at the two anticipated types of deviation in parameters of a simple linear regression models. In practice, some factors such as the cost of sampling and inspection plus some other factors related to process, i.e., shutting-down; significantly affect the selection of design parameters of charting method. Usually, the parameters should be selected in a way that the shift will be quickly detected with a minimum number of samples and a minimum number of inspection tests. The approach followed here to solve the optimisation model considers these factors and generates a set of particular solutions that minimise the objective function (Min ATS) and maintain the sampling rate as low as possible. Table 2 illustrates the ATSs comparison between a traditional sampling plan and a set of adaptive strategies developed by solving the optimisation model at n0 = 5 and t0 = 1. The comparative study in Table 2 shows that, at all levels of change in intercept, there are a range of strategies outperforming the fixed sampling settings.
Multivariate adaptive approach for monitoring simple linear profiles Table 2
FSR-T
ATSs comparison when intercept shifts from A0 → A0 + λσ0, n0 = 5, t0 = 1, C0 = 0.00834, and ATSδ=0 = 200 Chart parameters
Scheme type
WL
2
9
n1 n2
t1
λ t2
0.20 0.40
0.60
0.80 1.00 1.20 1.40 1.60 1.80 2.00
-
5
5 1.0000 1.000 127.08 52.15 21.21 9.57 4.92 2.90 1.95 1.47 1.22 1.10
0.55546
2
6 0.0521 3.9620 116.58 35.48 9.33
2.99 1.55 1.20 1.10 1.06 1.03 1.02
0.78115
3
6 0.0517 2.9854 117.28 36.36 9.73
3.11 1.57 1.20 1.08 1.04 1.02 1.01
1.32718
4
6 0.0549 2.0041 118.78 38.35 10.77 3.49 1.69 1.23 1.09 1.04 1.02 1.01
0.99845
2
7 0.0622 2.4493 116.70 34.42 8.69
2.84 1.58 1.26 1.14 1.09 1.06 1.04
1.35160
3
7 0.0692 1.9647 117.89 35.95 9.35
3.03 1.61 1.24 1.12 1.06 1.04 1.02
2.12842
4
7 0.0624 1.4943 119.88 38.79 10.75 3.49 1.73 1.27 1.11 1.06 1.03 1.02
1.35980
2
8 0.0687 1.9572 116.61 33.29 8.11
2.73 1.61 1.31 1.19 1.12 1.08 1.05
1.79298
3
8 0.0705 1.6412 118.07 35.18 8.84
2.90 1.62 1.27 1.14 1.08 1.05 1.03
2.69415
4
8 0.0735 1.3260 120.48 38.76 10.55 3.45 1.76 1.31 1.14 1.07 1.04 1.02
VSSI-T 1.66482
2
2
9 0.0801 1.7090 116.46 32.23 7.64
2.68 1.66 1.36 1.23 1.15 1.10 1.07
2.15272 3
9 0.0754 1.4787 118.1 34.39 8.39 2
2.82 1.64 1.31 1.1 7
1.10 1.06 1.04
3.13084 4
9 0.0826 1.2429 120.8 38.49 10.26 3.38 1.79 1.34 1.1 0 7
1.09 1.05 1.02
1.92860 2 10 0.0949 1.5585 116.2 31.23 7.26 5
2.66 1.72 1.42 1.2 8
1.18 1.12 1.08
2.45614 3 10 0.0844 1.3798 118.0 33.60 8.01 9
2.78 1.68 1.35 1.2 0
1.12 1.07 1.04
3.48594 4 10 0.0870 1.1942 120.9 38.06 9.94 6
3.32 1.82 1.38 1.1 9
1.10 1.05 1.03
2.16089 2 11 0.0995 1.4635 115.9 30.19 6.90 2
2.65 1.77 1.48 1.3 2
1.21 1.14 1.10
2.71833 3 11 0.0956 1.3133 118.0 32.82 7.69 1
2.77 1.73 1.40 1.2 3
1.14 1.08 1.05
3.78474 4 11 0.0934 1.1614 121.0 37.59 9.63 4
3.29 1.85 1.41 1.2 2
1.12 1.06 1.03
2.36834 2 12 0.1013 1.3970 115.5 29.18 6.59 5
2.65 1.83 1.53 1.3 5
1.24 1.16 1.11
2.94904 3 12 0.1004 1.2677 117.8 32.00 7.38 4
2.75 1.77 1.43 1.2 6
1.16 1.10 1.06
4.04235 4 12 0.1001 1.1380 121.0 37.09 9.36 6
3.26 1.89 1.45 1.2 4
1.13 1.07 1.04
As it can be seen from Table 2, the adaptive approach performs better than the fixed one even at small sizes of n2. Such property will guarantees an agreeable rate of sampling over time and increases the applicability of this approach. It is clear that, for small n2, the warning limits are set close to the zero and close to the control limit when n2 is large; hence, the second sample will be used with more frequency at small levels of shift. When the change in intercept is large (λ ≥ 2), the two schemes perform the same. Note that, if the large sample is held constant, the adaptive approach performs better as the distance
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G.M. Abdella et al.
between n1 and n2 is increased. This property is explained by the increase in the rate of using n2 as n1 is decreased. We easily can see that, the effect of the distance between the sample sizes starts to disappear at high levels of shift in the intercept. An important observation from Table 3 is when the slope deviates by the amount of β, there are a number of possible adaptive strategies that work better than their traditional fixed counterpart (FSR-T2).
Table 3
ATSs comparison when intercept shifts from A1 → A1 + βσ, n0 = 5, t0 = 1, C0 = 0.00834, and ATSδ=0 = 200 zβ
Chart parameters
Scheme type
WL
FSR-T2
-
5
5 1.0000 1.000 144.40 103.88 71.60 48.74 33.32 23.07 16.27 11.71 8.61
0.55546
2
6 0.0521 3.9620 136.28 90.15 55.13 32.27 18.53 10.67 6.31
3.93 2.64
0.78115
3
6 0.0517 2.9854 137.01 91.23 56.25 33.22 19.24 11.16 6.62
4.12 2.76
1.32718
4
6 0.0549 2.0041 138.22 93.14 58.39 35.17 20.81 12.31 7.41
4.65 3.09
0.99845
2
7 0.0622 2.4493 136.42 89.80 54.29 31.25 17.63 10.01 5.88
3.69 2.53
1.35160
3
7 0.0692 1.9647 137.64 91.65 56.23 32.90 18.84 10.82 6.40
4.01 2.72
2.12842
4
7 0.0624 1.4943 139.23 94.28 59.25 35.67 21.03 12.39 7.44
4.67 3.12
1.35980
2
8 0.0687 1.9572 136.44 89.31 53.33 30.19 16.74 9.38
5.50
3.49 2.44
1.79298
3
8 0.0705 1.6412 137.92 91.60 55.76 32.21 18.18 10.31 6.06
3.81 2.62
2.69415
4
8 0.0735 1.3260 139.80 94.84 59.55 35.69 20.91 12.23 7.32
4.60 3.10
1.66482
VSSI-T
2
n1 n2
t1
t2
0.050 0.075
0.100 0.125 0.150 0.175 0.200 0.225 0.250
2
9 0.0801 1.7090 136.43 88.80 52.42 29.22 15.96 8.87
5.21
3.35 2.40
2.15272 3
9 0.0754 1.4787 138.09 91.44 55.21 31.50 17.53 9.84
5.78
3.67
2.56
3.13084 4
9 0.0826 1.2429 140.15 95.10 59.54 35.47 20.61 11.96 7.14
4.50
3.07
1.92860 2 10 0.0949 1.5585 136.39 88.28 51.52 28.30 15.26 8.43
4.99
3.27
2.39
2.45614 3 10 0.0844 1.3798 138.20 91.21 54.62 30.79 16.93 9.43
5.55
3.56
2.54
3.48594 4 10 0.0870 1.1942 140.36 95.17 59.36 35.10 20.21 11.64 6.93
4.39
3.03
2.16089 2 11 0.0995 1.4635 136.25 87.64 50.53 27.34 14.55 8.01
4.77
3.18
2.38
2.71833 3 11 0.0956 1.3133 138.26 90.93 54.00 30.09 16.36 9.07
5.36
3.49
2.53
3.78474 4 11 0.0934 1.1614 140.50 95.15 59.09 34.69 19.79 11.33 6.74
4.30
3.00
2.36834 2 12 0.1013 1.3970 136.09 86.97 49.54 26.42 13.90 7.63
4.59
3.12
2.38
2.94904 3 12 0.1004 1.2677 138.24 90.56 53.31 29.35 15.79 8.72
5.18
3.42
2.53
4.04235 4 12 0.1001 1.1380 140.59 95.07 58.77 34.24 19.37 11.03 6.57
4.23
2.99
The results of Tables 2 and 3 as well as comparisons for other values of n0, which are not reported in this paper for the economy of the space, show that, like traditional multivariate control charting applications, adaptive sample size and sampling interval strategy can significantly improve the effectiveness of T2 charts in detecting deviations in simple linear profiles.
Multivariate adaptive approach for monitoring simple linear profiles
5
11
Studying the effect of location of explanatory variable
Previous literatures show that the power of the T2 chart increases as the value of the non-centrality parameter (τ) increases (see Kang and Albin, 2000). However, in this section, we examine this finding in the case of the adaptive scheme, and see how this property might be effectively used to increase the power of the proposed scheme. If the intercept deviates from A0 → A0 + λσ, then the non-centrality parameter is described by niλ2; see equation (16). However, when the slope shifts from A1 → A1 + βσ, the same parameter will take another value as follows: τ i = β 2 ( ni xi2 + ( S xx )i )
i = 1, 2
(24)
We see that the performance of the adaptive scheme (VSSI-T2) can be much improved if the values of the explanatory variable is allocated such that non-centrality parameter is maximised; see equation (24). The following is an illustrative example. Example: In order to illustrate the effect of location, three adaptive plans are selected for comparison. Table 4 shows the design setting of these plans. Note that the values of the independent variable are selected such that: τ11 = τ12 = τ13
(25)
τ 12 < τ 22 < τ 23
(26)
where τ1j and τ 2j are the non-centrality parameter of the jth plan. Table 4 Strategy
The X-values of the large sample (n2) of three adaptive plans (VSSI-T2), n1 = {1.00, 5.00} and τ1 = 86β2
τ2
x1
x2
x3
x4
x5
x6
x7
x8
x9
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Plan 1
167β2
Plan 2
207.93β
1.00
1.96
2.83
3.73
3.96
3.97
4.04
4.06
5.00
Plan 3
226.23β
1.00
1.45
2.75
2.83
3.89
4.70
4.74
4.80
5.00
Table 5
Strategy
2 2
ATSs comparison to study the effect of location when slope shifts from A1 → A1 + βσ, ATSδ=0 = 200, n0 = 5, and t0 = 1
β 0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0.25
Plan 1
136.91
89.36
52.89
29.54
16.16
8.99
5.28
3.40
2.43
Plan 2
128.15
78.33
43.04
22.41
11.63
6.33
3.80
2.59
1.99
Plan 3
124.30
73.57
39.02
19.69
10.01
5.44
3.34
2.35
1.87
12
G.M. Abdella et al.
Figure 2
ATSs comparison to study the effect of location on the performance of VSSI-T2 chart
Table 5 and Figure 2 illustrate that as the non-centrality parameter increases the ATS value of the VSSI-T2 decreases. In general speaking, the adaptive approach (VSSI-T2) can be sensitised to changes in the slope if the value of the non-centrality parameter (τ) is maximised.
6
Conclusions
This work extended the idea of adaptive sample size and sampling interval from traditional multivariate control charts to simple linear profile monitoring. We developed an optimisation model for identifying the optimal setting of the adaptive scheme (VSSI-T2) and extended the ATS formulation to capture the structure of simple linear profiles. The comparative studies demonstrate that, under different levels of shift in linear quality profiles, the proposed adaptive scheme (VSSI-T2) has a significant advantage over the traditional T2 scheme in detecting parameter drifts. In addition to that, this paper considered the effect of independent variable locations on the statistical performance of the proposed scheme when the process slope is shifted from the target value due to one or more assignable cause/s. The results reveal that if the location of the explanatory variable is optimised, the non-centrality parameter (τ) is maximised, then the performance of the adaptive scheme will significantly improved. As a future research, one can examine the statistical performance of the Hotelling T2 and MEWMA control charts using variable sample size and variable control limits (VSSC-T2, VSSC-MEWMA). We also recommend extending the idea named VSSI to be used for monitoring the polynomial and non-linear quality profiles, as well.
References Amiri, A., Jensen, W. and Kazemzadeh, R. (2009) ‘A case study on monitoring polynomial profiles in the automotive industry’, Quality and Reliability Engineering International, Vol. 26, No. 5, pp.509–520. Aparisi, F. (1996) ‘Hotelling’s T2 control chart with adaptive sample sizes’, International Journal of Production Research, Vol. 34, No. 10, pp.2853–2862. Aparisi, F. and Haro, C. (2001) ‘Hotelling’s T2 control chart with variable sampling intervals’, International Journal of Production Research, Vol. 39, No. 14, pp.3127–3140.
Multivariate adaptive approach for monitoring simple linear profiles
13
Aparisi, F. and Haro, C. (2003) ‘A comparison of T2 control charts with variable sampling schemes as opposed to MEWIVIA chart’, International Journal of Production Research, Vol. 41, No. 10, pp.2169–2182. Chen, Y. (2004) ‘Economic design of X control charts for non-normal data using variable sampling policy’, International Journal of Production Economics, Vol. 92, pp.61–74. Chen, Y. (2009) ‘Economic design of T2 control charts with the VSSI sampling scheme’, Quality and Quantity, Vol. 43, No. 1, pp.109–122. Chen, Y. and Hsieh, K. (2007) ‘Hotelling’s T2 charts with variable sample size and control limit’, European Journal of Operational Research, Vol. 182, No. 3, pp.1251–1262, doi: 10.1016/j.ejor.2006.09.046. Chou, C., Cheng, J. and Lai, W. (2008) ‘Economic design of variable sampling intervals EWMA charts with sampling at fixed times using genetic algorithms’, Expert Systems with Applications, Vol. 34, No. 1, pp.419–426. Çinlar, E. (1975) Introduction to Stochastic Process, Prentic Hall, Englewood Clifts, NJ. Costa, A. (1993) ‘X charts with variable sample size’, Journal of Quality Technology, Vol. 26, No. 3, pp.155–163. Croarkin, C. and Varner, R. (1982) ‘Measurement assurance for dimensional measurements on integrated-circuit photomasks’, NBS Technical Note 1164, US Department of Commerce, Washington, DC. Faraz, A. and Moghadam, M. (2008) ‘Hotelling’s T2 control chart with two adaptive sample sizes’, Quality and Quantity, Vol. 43, No. 10, pp.903–912, doi: 10.1007/s11135-008-9167-x. Faraz, A. and Saniga, E. (2010) ‘Economic statistical design of a T2 control chart with double warning lines’, Quality and Reliability Engineering International, Vol. 27, No. 2, pp.125–139, doi: 10.1002/qre.1095. Gupta, S., Montgomery, D. and Woodall, W. (2006) ‘Performance evaluation of two methods for online monitoring of linear calibration profiles’, International Journal of Production Research, Vol. 44, No. 10, pp.1927–1942. He, D. and Grigoryan, A. (2005) ‘Multivariate multiple sampling charts’, IIE Transactions, Vol. 37, No. 6, pp.509–521. He, D., Grigoryan, A. and Sigh, M. (2002) ‘Design of double and triple sampling X control charts using genetic algorithms’, International Journal of Production Research, Vol. 40, No. 6, pp.1387–1404. Jensen, W. and Birch, J. (2009) ‘Profile monitoring via nonlinear mixed model’, Journal of Quality Technology, Vol. 41, No. 1, pp.18–34. Kang. L. and Albin, S. (2000) ‘On-line monitoring when the process yields a linear profile’, Journal of Quality Technology, Vol. 32, No. 4, pp.418–426. Kazemzadeh, R., Noorossana, R. and Amiri, A. (2008) ‘Phase I monitoring of polynomial profiles’, Communications in Statistics – Theory and Methods, Vol. 37, No. 10, pp.1671–1686. Kazemzadeh, R., Noorossana, R. and Amiri, A. (2009) ‘Monitoring polynomial profiles in quality control applications’, The International Journal of Advanced Manufacturing Technology, Vol. 42, No. 7, pp.703–712. Kim, K., Mahmoud, M. and Woodall, W. (2003) ‘On the monitoring of linear profiles’, Journal of Quality Technology, Vol. 35, No. 3, pp.317–328. Li, Z. and Wang, Z. (2010) ‘An exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles’, Computers and Industrial Engineering, Vol. 59, No. 4, pp.630–637. Mahmoud, M. and Woodall, H. (2004) ‘Phase I analysis of linear profiles with calibration applications’, Technometrics, Vol. 46, No. 4, pp.380–391. Mahmoud, M., Parker, P., Woodall, W. and Hawkin, D. (2007) ‘A change point method for linear profile data’, Quality and Reliability Engineering International, Vol. 23, No. 2, pp.247–268. Montgomery, D. (2005) Introduction to Statistical Quality Control, Wiley, New York.
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Noorossana, R., Amiri, A. and Soleimani, P. (2008) ‘On the monitoring of auto-correlated linear profiles’, Communications in Statistics – Theory and Methods, Vol. 37, No. 3, pp.425–442. Noorossana, R., Vaghefi, A. and Eyvazian, M. (2010) ‘Phase II monitoring of multivariate simple linear profiles’, Computers & Industrial Engineering, Vol. 58, No. 4, pp.563–570 Noorossana, R., Vaghefi, S. and Amiri, A. (2004) ‘The effect of non-normality on monitoring linear profiles’, Proceedings of the 2nd International Industrial Engineering Conference, Riyadh, Saudi Arabia. Prabhu, S., Runger, C. and Keats, B. (1993) ‘X chart with adaptive sample sizes’, International Journal of Production Research, Vol. 31, No. 12, pp.2895–2009. Saghaei, A., Mehrjoo, M. and Amiri, A. (2009) ‘A CUSUM-based method for monitoring simple linear profiles’, The International Journal of Advanced Manufacturing Technology, Vol. 45, Nos. 11–12, 1252-1260. DOI: 10.1007/s00170-009-2063-2. Soleimani, P., Noorossana, R. and Amiri, A. (2009) ‘Simple linear profiles monitoring in the presence of within profile autocorrelation’, Computers & Industrial Engineering, Vol. 57, No. 3, pp.1015–1021, DOI:10.1007/j.cie.2009.04.005. Stover, F. and Brill, R. (1998) ‘Statistical quality control applied to ion chromatography calibrations’, Journal of Chromatography, Vol. 804, Nos. 1–2, pp.37–43. Woodall, W., Spitzner, D., Montgomery, D. and Gupta, S. (2004) ‘Using control charts to monitor process and product quality profiles’, Journal of Quality Technology, Vol. 36, No. 3, pp.309–320. Zhang, J., Li, Z. and Wang, Z. (2009) ‘Control chart based on likelihood ratio for monitoring linear profiles’, Computational Statistics and Data Analysis, Vol. 53, No. 4, pp.1440–1448. Zou, C., Wang, Z. and Tsung, F. (2007) ‘A self-starting control chart for linear profiles’, Journal of Quality Technology, Vol. 39, No. 4, pp.364–375.
Int. J. Data Analysis Techniques and Strategies, Vol. 6, No. 1, 2014
Multivariate adaptive approach for monitoring simple linear profiles Galal M. Abdella* and Kai Yang Department Industrial and Systems Engineering, Wayne State University, 4815 Fourth St., Detroit, Michigan 48202, USA E-mail: [email protected] E-mail: [email protected] *Corresponding author
Adel Alaeddini Department of Industrial and Operations Engineering, University of Michigan, 1205 Beal Ave., Ann Arbor, MI 48109, USA E-mail: [email protected] Abstract: Adaptive sample size and sampling intervals schemes have been widely used to improve the statistical efficiency of Hotelling T2 control chart in detecting small changes when the quality of a product or a process can be characterised by the multivariate distribution of quality characteristics. In this paper, we design a Hotelling T2 scheme varying sample sizes and sampling intervals (VSSI-T2) for accelerating the speed of detecting off-target conditions in linear profile parameters. We investigate the statistical performance of the adaptive approach versus its fixed sampling counterparts. To find the optimal setting of the VSSI-T2, we build an optimisation model solved using genetic algorithm (GA). Also, average time to signal (ATS) is considered as the objective function of the model and estimated using the Markov chain fundamentals. The comparative studies reveal the potentials of the adaptive scheme in improving the performance of the Hotelling T2 control chart in monitoring linear profiles. Keywords: varying sample sizes and sampling intervals; VSSI; scheme, average time to signal; ATS; Hotelling T2 control chart; phase I and II; simple linear profiles. Reference to this paper should be made as follows: Abdella, G.M., Yang, K. and Alaeddini, A. (2014) ‘Multivariate adaptive approach for monitoring simple linear profiles’, Int. J. Data Analysis Techniques and Strategies, Vol. 6, No. 1, pp.2–14. Biographical notes: Galal M. Abdella is a PhD candidate at Industrial and Systems Engineering at Wayne State University in Detroit-Michigan, USA. He is interested in statistical quality control and discrete event systems simulation. Kai Yang is a Professor at the Department of Industrial and Systems Engineering at Wayne State University in Detroit-Michigan. His field of expertise includes quality, reliability and product development. He received his MS degree in 1985, and PhD in Industry Engineering in 1990, both of them from the University of Michigan.
Copyright © 2014 Inderscience Enterprises Ltd.
Multivariate adaptive approach for monitoring simple linear profiles
3
Adel Alaeddini is a Post-Doc Researcher in the Department of Industrial and Operations Engineering at University of Michigan. Ann Arbor, MI, USA. His research interest includes statistical learning, global optimisation and healthcare optimisations improvement.
1
Introduction
Profile monitoring plays a vital role in identifying the off-target conditions when the quality characteristic/s of the process or product can be statistically related to a set of independent variables. An off-target condition in profile data occurs when one or more of the parameters of the underlying quality function change due to an assignable cause; consequently the process is no longer stable. The past few years has witnessed an increasing interest on developing new profiling techniques and examining their ability in detecting changes based on regression models. Similar to other control charting methods, profile monitoring schemes can be designed for phase I and/or II. In phase I analysis, practitioners are mainly interested in investigating the statistical stability of process parameters and estimating their unknown values, while in phase II, these parameters are assumed to be known and one is interested in accelerating the process of detecting any deviations in their nominal values. The following is a brief overview of the prior related work. Stover and Brill (1998) suggested two strategies for monitoring profiles in phase I with an application in multilevel ion chromatography. Mahmoud and Woodall (2004) developed a F-test to monitor the regression parameters in conjunction with an univariate control chart to monitor changes in process variability in phase I. Mahmoud et al. (2007) examined a change point approach based on the segmented technique for testing the constancy of the phase I regression parameters for a linear profile data. Most of the previous literatures focus on phase II profiling techniques. Kang and Albin (2000) considered a semiconductor manufacturing process and suggested two phase II methods. The first technique is a multivariate Hotelling T2 control chart for monitoring intercept and slope of simple linear profiles. The second technique uses an exponentially weighted moving average control chart (EWMA) in conjunction with the range control chart to monitor the fitted residual (EWMA/R method). Kim et al. (2003) proposed a new technique based on coding the values of the explanatory variable (X) to remove the correlation between regression parameters, including the slope and the intercept, as well as residuals and then used three separate EWMA control charts to monitor the uncorrelated parameters (EWMA3 method). Woodall et al. (2004) provided areas for future work and research on profile monitoring. Gupta et al. (2006) performed a phase II comparative study between the Croarkin and Varner (1982) control chart (NIST method) and the combined strategy proposed by Kim et al. (2003) (KMW method). In their study, they modified the KMW method to use three separate Shewhart control charts instead of EWMA charts. When the number of available samples is not sufficiently large for estimating the process parameters, Zou et al. (2007) proposed a self starting control chart for monitoring linear profiles. Zhang et al. (2009) developed a single chart which integrates likelihood ratio statistic with the EWMA procedure for monitoring linear profiles. The variable sampling interval feature is applied to this chart to enhance the speed of detecting changes in regression parameters. A new technique based on
4
G.M. Abdella et al.
cumulative sum statistic for phase II linear profile is proposed by Saghaei et al. (2009). Zhu and Lin (2010) proposed and investigated a method to monitor changes in the slope of the linear functions. Another recent contribution was by Noorossana et al. (2010). They developed and explored the performance of three control chart schemes when several correlated characteristics might be modelled as a set of linear functions of one independent variable. This situation is referred to as multivariate simple linear profiles structure. Li and Wang (2010) proposed the use of an EWMA chart with variable sampling intervals for monitoring linear quality functions (VSI-EWMA3). They used a real data from an optical imaging system to examine the suggested method. Sometimes, it is more appropriate to describe the relationship between the quality response and the explanatory variable by more sophisticated models such as polynomial or non-linear quality models. Montgomery (2005) studied the situation when the output/input relationship might be modelled as a second-order polynomial regression. The relation between the torque produced by an automobile engine and the engine speed is considered as an illustrative example for this model type. Kazemzadeh et al. (2008) examined three phase I methods for monitoring polynomial profiles and developed a method based on likelihood ratio test for identifying the location of the shift. Kazemzadeh et al. (2009) introduced a new approach using three individual EWMA control charts after transferring the polynomial function to the orthogonal form. A case study shows the use of monitoring polynomial quality profiles in the automotive industry was presented by Amiri et al. (2009). Noorossana et al. (2008) considered the case when the quality profiles are not independent of each other. Soleimani et al. (2009) investigated how the speed of detecting changes in regression parameters is influenced by autocorrelation. The effect of correlation on non-linear profiles can be found in Jensen and Birch (2009). In this analytical study, they used the non-linear mixed models. Other researchers such as Noorossana et al. (2004, 2008) studied the effect of violating the normality assumption on the monitoring phase of linear profiles. The idea of changing the sample size (VSS) or the sampling interval (VSI) or both of them during the online process monitoring has been successfully examined with the Hotelling T2 chart (Aparisi, 1996; Aparisi and Haro, 2001, 2003; Chen, 2009), but none of the prior works considered the case when the quality is described by the regression model. Accordingly, our focus in this paper is to integrate the both of adaptive schemes, namely VSS and VSI, with the T2 control chart, suggested by Kang and Albin (2000), to capture any changes in intercept and slope of the corresponding linear function. We refer to this scheme as a VSSI-T2 control chart.
2
The variable sampling sizes and variable intervals – T2 scheme (VSSI-T2)
This work assumes that the outgoing quality is a random variable Y that is a simple linear function of an explanatory variable X; that is: Yi = A0 + A1 X i + εi
i = 1, 2,… , n
(1)
where A0 and A1 are the parameters of intercept and slope. The random variable ε is an independent and normally distributed variable with zero mean and variance σ2. Here, the −1 ( Z j − U ) reported and used by Kang and Albin Hotelling’s statistic T j2 = ( Z j − U )T
∑
(2000) is utilised for capturing deviations in A0 and A1; where U = (A0, A1)T) is the mean
Multivariate adaptive approach for monitoring simple linear profiles
5
vector, and Zj = (a0j, a1j)T is the vector of the process parameter estimators. The variance-covariance matrix (Σ) is described as follows: ⎛ σ 02
∑ = ⎜⎝ σ
2 01
2 σ 01 ⎞ ⎟ σ12 ⎠
(2)
Kang and Albin (2000) used the least square method for estimating the regression parameters of each set of observations (profile), such that α0 j = Y − α1 j x , −1 −1 −1 α1 j = S xy ( j ) S xx , σ 02 = (σ 2 n −1 + x 2 σ 2 S xx ) and σ12 = σ 2 S xx ; where σ 02 and σ12 are the variances of the regression parameters. The covariance of α0j and α1j is calculated using 2 −1 2 σ 01 = −σ 2 xS xx . The control limit of the VSSI-T2 scheme is set equal to χ 2,α which can
be determined by the (1 – α) percentile point of a chi-square with 2 degrees of freedom; see Kang and Albin (2000). The adaptive approach uses two time intervals (t1, t2), sample sizes (n1, n2) and one warning limit (WL). Note that, n1 ≤ n0 ≤ n2
(3)
t1 ≤ t0 ≤ t2
(4)
where n0 and t0 are the sample size and the sampling interval of traditional T2 multivariate control chart using fixed sampling rate (FSR-T2). The mechanism of switching between the parameters of adaptive T2 chart is described below; see Aparisi and Haro (2001). 1
if 0 ≤ T j2−1 < WL, use (n1, t2)
2
if WL < T j2−1 < CL, use (n2, t1)
3
if T j2−1 ≥ CL, signal.
A commonly used method to assess the effectiveness of such adaptive scheme (VSSI-T2) is matching VSSI-T2performance with its counterparts (FSR-T2) at the state of the statistical control and then investigating and comparing their performance at the off-target conditions. In this article, the same approach will be followed, and the average time to signal (ATS) is used as the measure of the performance for comparing the two schemes. The rest of this subsection is dedicated to introduce the mathematical model developed to determine the optimal setting of the VSSI-T2 chart. Min ATS ( v, δ, WL, n1 , n2 , t1 , t2 )
(5)
Subject to: n0 = n1 p1 + n2 p2
(6)
t0 = t2 p1 + t1 p2
(7)
n1 ≤ n0 ≤ n2
(8)
t1 ≤ t0 ≤ t2
(9)
6
G.M. Abdella et al. t1 ≥ C0 n2
(10)
0 < WL < CL
(11)
n1 , n2 ∈ Z + , n1 ≥ 2
where p1 = P ( 0 ≤ T j2−1 < WL ) , p2 = P (WL < T j2−1 < CL ) , δ is the amount of deviation from the target value of the model parameter, and C0 is the sampling factor; for instance, if an inspector takes 0.5 min. to examine and measure one product, then C0 is set equal to 0.5/60 = 0. 00834 hr. Note that equation (11) is added to guarantee that the minimum sampling interval (t1) is always enough to collect the large sample (n2). Equations (6) to (7) can be rewritten as follows: n0 = n1 P ( 0 ≤ T j2 < WL ) + n2 P (WL < T j2 < CL )
(12)
t0 = t2 P ( 0 ≤ T j2−1 < WL ) + t1 P (WL < T j2 < CL )
(13)
Here, we extend the use of fundamentals of Markov Chain in estimating the statistical properties of T2 control charts (see Çinlar, 1975; Costa, 1993; Prabhu et al., 1993; Aparisi, 1996; Aparisi and Haro, 2001, 2003; Chen, 2004, 2009; Chen and Hsieh, 2007; Faraz and Moghadam, 2008; Faraz and Saniga, 2010). Accordingly, we use the same transient states reported by Aparisi and Haro (2001) and Faraz and Moghadam (2008): 1
state 1 represents 0 ≤ T j2 < WL
2
state 2 represents WL < T j2 < CL
3
state 3 represents T j2 ≥ CL (absorbing state).
The transition probability matrix and the ATS approximation are defined as (see Aparisi, 1996; Aparisi and Haro 2001; Faraz and Moghadam, 2008): δ ⎛ p11 ⎜ δ Pδ = ⎜ p21 ⎜ 0 ⎝
δ p12 δ p22
0
δ p13 ⎞ ⎟ δ p23 ⎟ 1 ⎟⎠
(14)
where pijδ is the probability of moving from state i to state j when the drift in the regression parameter is equal to δ. The ATS approximation is as follows: ATS δ = P0′ ( I − Qδ )
−1t
(15)
where P0′ = ( P01 , P02 ) is the vector of the initial states, I is the identity matrix, t′ = (t2, t1), and Qθ is the probability transition matrix without the absorbing state; see Figure 1.
Multivariate adaptive approach for monitoring simple linear profiles Figure 1
7
The transition diagram for the VSSI-T2 control chart
Kang and Albin (2000) reported that, under the off-target conditions the Hotelling T2 statistic follows a non-central χ v2,α distribution. This distribution has a non-centrality parameter (τ) equals to ( λ + βx ) 2 n + β 2 S xx . We rewrite the non-centrality parameter in terms of the prior state i as follows: τ i = ( λ + βxi ) ni + β 2 ( S xx )i 2
i = 1, 2
(16)
Then, the ATS approximation can be written as: ⎛ ⎛1 0 ⎞ ⎛ p δ 11 ATSδ = ( P01 , P02 ) ⎜ ⎜ ⎟−⎜ δ ⎜ ⎝ 0 1⎠ ⎝ p21 ⎝
δ p12 ⎞ ⎟ δ p22 ⎠
−1
⎞ ⎛ t2 ⎞ ⎟⎜ ⎟ ⎟ ⎝ t1 ⎠ ⎠
(17)
When the quality can be modelled by the multivariate distribution of a set of characteristics, the transition probability is defined in terms of niδ2; where i = 1, 2, (see Aparisi, 1996). Since this work extends the use of the VSSI scheme to capture the structure of simple linear profiles, we redefine the probability of transition in terms of τi as follows: δ p11 = P ( 0 ≤ χ v2 ( τ1 ) ≤ WL )
(18)
δ p12 = P (WL < χ v2 ( τ1 ) < CL )
(19)
δ p21 = P ( 0 ≤ χ v2 ( τ 2 ) ≤ WL )
(20)
δ p22 = P (WL < χ v2 ( τ 2 ) < CL )
(21)
And, the initial state probabilities are the same as the case when the quality is described by the multivariate probability distribution, see Aparisi and Haro (2001) and Faraz and Moghadam (2008), and calculated as follows: P01 =
P02 =
P ( χv2 < WL )
(22)
P ( χ v2 < CL )
P (WL < χv2 < CL ) P ( χv2 < CL )
= 1−
P ( χv2 < WL ) P ( χ v2 < CL )
= 1 − P01
(23)
8
3
G.M. Abdella et al.
Setting the parameters of the genetic approach
Genetic algorithm (GA) is one of the popular stochastic optimisation techniques used for optimising design parameters of various control charts (e.g., X , Hotelling T2, and EWMA control charts) (see He et al., 2002; Chen, 2004; He and Grigoryan, 2005; Chen and Hsieh, 2007; Chou et al., 2008). In this article, finding the optimal design parameters of VSSI-T2 scheme is formulated as an optimisation problem and solved using the GA. In this regard, two Taguchi experiments were conducted to find the optimal setting of the GA parameters, population size (PS), probability of crossover and mutation rate (MR). These inputs with the shift value in regression parameters, intercept and slope, were analysed in L9(34) orthogonal arrays. The optimal levels of the GA parameters are determined and presented in Table 1. Note, Taguchi experiments have been done with n0 = 5, and false alarm rate equals (ATSδ=0) to 200. Table 1
Optimal inputs of GA
Shift in
CP
MR
PS
Intercept
0.5
0.1
100
Slope
0.5
0.1
100
4
Measuring the performance of VSSI-T2 chart
To examine the performance of the suggested adaptive scheme in capturing changes in the parameters of a simple linear models, we compare its effectiveness with the traditional fixed sample size T2 control chart for n0 = 3,5 and 10, t0 = 1 and C0 = 0.00834. The values of in-control ATS of both schemes are set to approximately 200. In this paper, the data points of the independent variable are distributed equidistantly over a practical range (from 1 to 5) with two points at the edges. For instance, if the sample size equals to 3, one can use X = {1, 3, 5}. Her we only show the results of the case with n0 = 5 and t0 = 1. Under the profile monitoring framework, the values of the independent variable take the same magnitude from one profile to another. In this paper, we provide different sets of VSSI plans and examine their power at the two anticipated types of deviation in parameters of a simple linear regression models. In practice, some factors such as the cost of sampling and inspection plus some other factors related to process, i.e., shutting-down; significantly affect the selection of design parameters of charting method. Usually, the parameters should be selected in a way that the shift will be quickly detected with a minimum number of samples and a minimum number of inspection tests. The approach followed here to solve the optimisation model considers these factors and generates a set of particular solutions that minimise the objective function (Min ATS) and maintain the sampling rate as low as possible. Table 2 illustrates the ATSs comparison between a traditional sampling plan and a set of adaptive strategies developed by solving the optimisation model at n0 = 5 and t0 = 1. The comparative study in Table 2 shows that, at all levels of change in intercept, there are a range of strategies outperforming the fixed sampling settings.
Multivariate adaptive approach for monitoring simple linear profiles Table 2
FSR-T
ATSs comparison when intercept shifts from A0 → A0 + λσ0, n0 = 5, t0 = 1, C0 = 0.00834, and ATSδ=0 = 200 Chart parameters
Scheme type
WL
2
9
n1 n2
t1
λ t2
0.20 0.40
0.60
0.80 1.00 1.20 1.40 1.60 1.80 2.00
-
5
5 1.0000 1.000 127.08 52.15 21.21 9.57 4.92 2.90 1.95 1.47 1.22 1.10
0.55546
2
6 0.0521 3.9620 116.58 35.48 9.33
2.99 1.55 1.20 1.10 1.06 1.03 1.02
0.78115
3
6 0.0517 2.9854 117.28 36.36 9.73
3.11 1.57 1.20 1.08 1.04 1.02 1.01
1.32718
4
6 0.0549 2.0041 118.78 38.35 10.77 3.49 1.69 1.23 1.09 1.04 1.02 1.01
0.99845
2
7 0.0622 2.4493 116.70 34.42 8.69
2.84 1.58 1.26 1.14 1.09 1.06 1.04
1.35160
3
7 0.0692 1.9647 117.89 35.95 9.35
3.03 1.61 1.24 1.12 1.06 1.04 1.02
2.12842
4
7 0.0624 1.4943 119.88 38.79 10.75 3.49 1.73 1.27 1.11 1.06 1.03 1.02
1.35980
2
8 0.0687 1.9572 116.61 33.29 8.11
2.73 1.61 1.31 1.19 1.12 1.08 1.05
1.79298
3
8 0.0705 1.6412 118.07 35.18 8.84
2.90 1.62 1.27 1.14 1.08 1.05 1.03
2.69415
4
8 0.0735 1.3260 120.48 38.76 10.55 3.45 1.76 1.31 1.14 1.07 1.04 1.02
VSSI-T 1.66482
2
2
9 0.0801 1.7090 116.46 32.23 7.64
2.68 1.66 1.36 1.23 1.15 1.10 1.07
2.15272 3
9 0.0754 1.4787 118.1 34.39 8.39 2
2.82 1.64 1.31 1.1 7
1.10 1.06 1.04
3.13084 4
9 0.0826 1.2429 120.8 38.49 10.26 3.38 1.79 1.34 1.1 0 7
1.09 1.05 1.02
1.92860 2 10 0.0949 1.5585 116.2 31.23 7.26 5
2.66 1.72 1.42 1.2 8
1.18 1.12 1.08
2.45614 3 10 0.0844 1.3798 118.0 33.60 8.01 9
2.78 1.68 1.35 1.2 0
1.12 1.07 1.04
3.48594 4 10 0.0870 1.1942 120.9 38.06 9.94 6
3.32 1.82 1.38 1.1 9
1.10 1.05 1.03
2.16089 2 11 0.0995 1.4635 115.9 30.19 6.90 2
2.65 1.77 1.48 1.3 2
1.21 1.14 1.10
2.71833 3 11 0.0956 1.3133 118.0 32.82 7.69 1
2.77 1.73 1.40 1.2 3
1.14 1.08 1.05
3.78474 4 11 0.0934 1.1614 121.0 37.59 9.63 4
3.29 1.85 1.41 1.2 2
1.12 1.06 1.03
2.36834 2 12 0.1013 1.3970 115.5 29.18 6.59 5
2.65 1.83 1.53 1.3 5
1.24 1.16 1.11
2.94904 3 12 0.1004 1.2677 117.8 32.00 7.38 4
2.75 1.77 1.43 1.2 6
1.16 1.10 1.06
4.04235 4 12 0.1001 1.1380 121.0 37.09 9.36 6
3.26 1.89 1.45 1.2 4
1.13 1.07 1.04
As it can be seen from Table 2, the adaptive approach performs better than the fixed one even at small sizes of n2. Such property will guarantees an agreeable rate of sampling over time and increases the applicability of this approach. It is clear that, for small n2, the warning limits are set close to the zero and close to the control limit when n2 is large; hence, the second sample will be used with more frequency at small levels of shift. When the change in intercept is large (λ ≥ 2), the two schemes perform the same. Note that, if the large sample is held constant, the adaptive approach performs better as the distance
10
G.M. Abdella et al.
between n1 and n2 is increased. This property is explained by the increase in the rate of using n2 as n1 is decreased. We easily can see that, the effect of the distance between the sample sizes starts to disappear at high levels of shift in the intercept. An important observation from Table 3 is when the slope deviates by the amount of β, there are a number of possible adaptive strategies that work better than their traditional fixed counterpart (FSR-T2).
Table 3
ATSs comparison when intercept shifts from A1 → A1 + βσ, n0 = 5, t0 = 1, C0 = 0.00834, and ATSδ=0 = 200 zβ
Chart parameters
Scheme type
WL
FSR-T2
-
5
5 1.0000 1.000 144.40 103.88 71.60 48.74 33.32 23.07 16.27 11.71 8.61
0.55546
2
6 0.0521 3.9620 136.28 90.15 55.13 32.27 18.53 10.67 6.31
3.93 2.64
0.78115
3
6 0.0517 2.9854 137.01 91.23 56.25 33.22 19.24 11.16 6.62
4.12 2.76
1.32718
4
6 0.0549 2.0041 138.22 93.14 58.39 35.17 20.81 12.31 7.41
4.65 3.09
0.99845
2
7 0.0622 2.4493 136.42 89.80 54.29 31.25 17.63 10.01 5.88
3.69 2.53
1.35160
3
7 0.0692 1.9647 137.64 91.65 56.23 32.90 18.84 10.82 6.40
4.01 2.72
2.12842
4
7 0.0624 1.4943 139.23 94.28 59.25 35.67 21.03 12.39 7.44
4.67 3.12
1.35980
2
8 0.0687 1.9572 136.44 89.31 53.33 30.19 16.74 9.38
5.50
3.49 2.44
1.79298
3
8 0.0705 1.6412 137.92 91.60 55.76 32.21 18.18 10.31 6.06
3.81 2.62
2.69415
4
8 0.0735 1.3260 139.80 94.84 59.55 35.69 20.91 12.23 7.32
4.60 3.10
1.66482
VSSI-T
2
n1 n2
t1
t2
0.050 0.075
0.100 0.125 0.150 0.175 0.200 0.225 0.250
2
9 0.0801 1.7090 136.43 88.80 52.42 29.22 15.96 8.87
5.21
3.35 2.40
2.15272 3
9 0.0754 1.4787 138.09 91.44 55.21 31.50 17.53 9.84
5.78
3.67
2.56
3.13084 4
9 0.0826 1.2429 140.15 95.10 59.54 35.47 20.61 11.96 7.14
4.50
3.07
1.92860 2 10 0.0949 1.5585 136.39 88.28 51.52 28.30 15.26 8.43
4.99
3.27
2.39
2.45614 3 10 0.0844 1.3798 138.20 91.21 54.62 30.79 16.93 9.43
5.55
3.56
2.54
3.48594 4 10 0.0870 1.1942 140.36 95.17 59.36 35.10 20.21 11.64 6.93
4.39
3.03
2.16089 2 11 0.0995 1.4635 136.25 87.64 50.53 27.34 14.55 8.01
4.77
3.18
2.38
2.71833 3 11 0.0956 1.3133 138.26 90.93 54.00 30.09 16.36 9.07
5.36
3.49
2.53
3.78474 4 11 0.0934 1.1614 140.50 95.15 59.09 34.69 19.79 11.33 6.74
4.30
3.00
2.36834 2 12 0.1013 1.3970 136.09 86.97 49.54 26.42 13.90 7.63
4.59
3.12
2.38
2.94904 3 12 0.1004 1.2677 138.24 90.56 53.31 29.35 15.79 8.72
5.18
3.42
2.53
4.04235 4 12 0.1001 1.1380 140.59 95.07 58.77 34.24 19.37 11.03 6.57
4.23
2.99
The results of Tables 2 and 3 as well as comparisons for other values of n0, which are not reported in this paper for the economy of the space, show that, like traditional multivariate control charting applications, adaptive sample size and sampling interval strategy can significantly improve the effectiveness of T2 charts in detecting deviations in simple linear profiles.
Multivariate adaptive approach for monitoring simple linear profiles
5
11
Studying the effect of location of explanatory variable
Previous literatures show that the power of the T2 chart increases as the value of the non-centrality parameter (τ) increases (see Kang and Albin, 2000). However, in this section, we examine this finding in the case of the adaptive scheme, and see how this property might be effectively used to increase the power of the proposed scheme. If the intercept deviates from A0 → A0 + λσ, then the non-centrality parameter is described by niλ2; see equation (16). However, when the slope shifts from A1 → A1 + βσ, the same parameter will take another value as follows: τ i = β 2 ( ni xi2 + ( S xx )i )
i = 1, 2
(24)
We see that the performance of the adaptive scheme (VSSI-T2) can be much improved if the values of the explanatory variable is allocated such that non-centrality parameter is maximised; see equation (24). The following is an illustrative example. Example: In order to illustrate the effect of location, three adaptive plans are selected for comparison. Table 4 shows the design setting of these plans. Note that the values of the independent variable are selected such that: τ11 = τ12 = τ13
(25)
τ 12 < τ 22 < τ 23
(26)
where τ1j and τ 2j are the non-centrality parameter of the jth plan. Table 4 Strategy
The X-values of the large sample (n2) of three adaptive plans (VSSI-T2), n1 = {1.00, 5.00} and τ1 = 86β2
τ2
x1
x2
x3
x4
x5
x6
x7
x8
x9
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Plan 1
167β2
Plan 2
207.93β
1.00
1.96
2.83
3.73
3.96
3.97
4.04
4.06
5.00
Plan 3
226.23β
1.00
1.45
2.75
2.83
3.89
4.70
4.74
4.80
5.00
Table 5
Strategy
2 2
ATSs comparison to study the effect of location when slope shifts from A1 → A1 + βσ, ATSδ=0 = 200, n0 = 5, and t0 = 1
β 0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0.25
Plan 1
136.91
89.36
52.89
29.54
16.16
8.99
5.28
3.40
2.43
Plan 2
128.15
78.33
43.04
22.41
11.63
6.33
3.80
2.59
1.99
Plan 3
124.30
73.57
39.02
19.69
10.01
5.44
3.34
2.35
1.87
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G.M. Abdella et al.
Figure 2
ATSs comparison to study the effect of location on the performance of VSSI-T2 chart
Table 5 and Figure 2 illustrate that as the non-centrality parameter increases the ATS value of the VSSI-T2 decreases. In general speaking, the adaptive approach (VSSI-T2) can be sensitised to changes in the slope if the value of the non-centrality parameter (τ) is maximised.
6
Conclusions
This work extended the idea of adaptive sample size and sampling interval from traditional multivariate control charts to simple linear profile monitoring. We developed an optimisation model for identifying the optimal setting of the adaptive scheme (VSSI-T2) and extended the ATS formulation to capture the structure of simple linear profiles. The comparative studies demonstrate that, under different levels of shift in linear quality profiles, the proposed adaptive scheme (VSSI-T2) has a significant advantage over the traditional T2 scheme in detecting parameter drifts. In addition to that, this paper considered the effect of independent variable locations on the statistical performance of the proposed scheme when the process slope is shifted from the target value due to one or more assignable cause/s. The results reveal that if the location of the explanatory variable is optimised, the non-centrality parameter (τ) is maximised, then the performance of the adaptive scheme will significantly improved. As a future research, one can examine the statistical performance of the Hotelling T2 and MEWMA control charts using variable sample size and variable control limits (VSSC-T2, VSSC-MEWMA). We also recommend extending the idea named VSSI to be used for monitoring the polynomial and non-linear quality profiles, as well.
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