QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL Qual. Reliab. Engng. Int. 2007; 23:203 218 Published online 23 May 2006 in Wiley InterScience (www.interscience.wiley.com..785 Research Economic Design of the Integrated Multivariate EPC and Multivariate SPC Charts Ling Yang 1 and Shey-Huei Sheu 2,, 1 Department of Industrial Engineering and Management, St. John s University, 499 Tam King Road, Section 4, Tamsui, Taipei 251, Taiwan 2 Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 106, Taiwan The goal of engineering process control (EPC is to minimize variability by adjusting some manipulative process variables. The goal of statistical process control (SPC is to reduce variability by monitoring and eliminating assignable causes of variation. As suggested by Box and Kramer and others, it is possible to reduce both special cause and common cause variations by integrating EPC and SPC. In the integrated multivariate EPC (MEPC and multivariate SPC (MSPC charts, we propose some statistical and economic criteria, such as the average Euclidean distance from the target vector and the average quality cost (AQC to evaluate the performance of the MEPC/MSPC charts. The traditional average run length (ARL, average Euclidean distance and AQC of three MSPC charts are investigated and compared. The results of the simulations show that the MEPC/MGWMA chart is more effective and more economical than both the MEPC/MEWMA chart and the MEPC/Hotelling multivariate chart in detecting small shifts of the mean vector. Copyright c 2006 John Wiley & Sons, Ltd. Received 17 November 2004; Revised 4 April 2005 KEY WORDS: engineering process control; multivariate; quality cost; control chart; Taguchi method; EWMA; generally weighted moving average 1. INTRODUCTION Engineering process control (EPC and statistical process control (SPC are two strategies for quality improvement that have developed independently. The goal of EPC is to minimize variability by adjusting some manipulative process variables to keep the process output on target. On the other hand, the goal of SPC is to reduce variability by monitoring and eliminating assignable causes of variation. The concept of integrating EPC and SPC techniques (called EPC/SPC for short uses EPC to reduce the effect of predictable quality variations and uses SPC to monitor the process for detection of assignable causes. Figure 1 shows Correspondence to: Shey-Huei Sheu, Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 106, Taiwan. E-mail: shsheu@im.ntust.edu.tw Copyright c 2006 John Wiley & Sons, Ltd.
204 L. YANG AND S.-H. SHEU Process output EPC SPC Compare with the target Compute the plotted statistic Update the controller Update the input Does the control chart alarm? Yes Identify the assignable cause(s No Eliminate the assignable cause(s Figure 1. EPC/SPC control scheme the EPC/SPC control scheme. Box and Kramer 1, and MacGregor 2 presented overview descriptions of this integration concept. Montgomery et al. 3 showed that proper use of both EPC and SPC can always outperform the use of either strategy used alone. Sachs et al. 4 developed a run-to-run controller that combined EPC/SPC to automate the response to shifts and drifts and had proven successful application. More recent discussions on the EPC/SPC integration can be seen in Tsung and Shi 5, Tsung 6, Pan and del Castillo 7, and del Castillo 8. The general approaches for the design of control charts are the statistical design, economic design and economic statistical design. With statistical design, one considers statistical properties such as the type I and type II errors and the average run length (ARL when deciding the parameters for the control chart. In economic design, on the other hand, the objective is to select the control chart parameters that minimize the expected quality cost of a production process (see Lorenzen and Vance 9 ; Montgomery 10. In economic statistical design, Saniga 11 took statistical properties into account while designing control charts economically, and Montgomery et al. 12 and Torng et al. 13 presented a statistically constrained economic model for the optimal design of the exponentially weighted moving average (EWMA control chart for controlling process means. Linderman and Love 14,15 proposed the economic and economic statistical designs for the multivariate EWMA (MEWMA control charts. Molnau et al. 16,17 also showed that applying the economic statistical design to the MEWMA charts could provide better statistical properties without significantly increasing total costs. A review and comparison of these design strategies for the MEWMA control charts was provided by Testik and Borror 18. These studies mentioned above concentrate on the economic and statistical economic design for SPC charts alone. In EPC/SPC, to determine an appropriate SPC chart for monitoring the EPC-controlled process, criteria are needed to evaluate the performance of a SPC chart. Recently, under the single input and single output (SISO
INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 205 system, Jiang and Tsui 19 developed an economic-loss-based criterion to determine appropriate SPC charts for EPC/SPC. In practice, many manufacturing processes, such as the silicon epitaxy process and the chemical mechanical polishing process, are multiple inputs and multiple outputs (MIMO. The integration of multivariate EPC and multivariate SPC (called MEPC/MSPC for short has the practical necessity. Developing criteria for the choice of control chart in MEPC/MSPC still has received very little attention in the literature. This paper is to demonstrate the economic consideration of MEPC/MSPC in a reasonably general situation. We extend the unified quality cost model of Lorenzen and Vance 9 to MEPC/MSPC. The idea of Taguchi s quality loss model 20 is implemented to the Lorenzen Vance total quality cost function. We use the MEWMA controller proposed by Tseng et al. 21 as a feedback controller for MEPC, and apply some multivariate control charts to detect the assignable causes. The remainder of this paper is organized as follows. In Section 2, themewma controller is introduced briefly. A simple example used throughout this paper is given. In Section 3, integrated with MEPC, three MSPC charts including the Hotelling multivariate control chart, the MEWMA control chart and the multivariate generally weighted moving average (MGWMA control chart, are described. In Section 4, the total quality cost function is derived. In Section 5, MEPC alone and various combinations of MEPC and MSPC charts are compared in terms of ARL, average Euclidean distance and the average quality cost (AQC. The relative data are obtained via numerical simulation. In Section 6, the effectiveness of the discount factor in MEPC/MSPC is also discussed. Finally, some conclusions are included in the last section. 2. THE MULTIVARIATE EWMA CONTROLLER The key idea of EPC is to update the process input parameters in response to some measurements of the outputs to keep them on target. In the SISO case, Hunter 22 described a feedback control method based on the EWMA, and Ingolfsson and Sachs 23 presented analytical results concerning the stability and robustness of an EWMA controller under different disturbance conditions and different amounts of error in the estimation of the process sensitivity. In the MIMO case, Tseng et al. 21 proposed a MEWMA controller for MEPC. Suppose that a system with m inputs and p outputs is described as x i = ψ + Mu i 1 + ε i (1 where i denotes time, i {1, 2, 3,...}, x i is a (p 1 vector containing the quality characteristics (outputs or responses, ψ is a (p 1 vector containing the offset parameter of each outputs, M is a (p m matrix denoting the process gain, u i 1 is an (m 1 vector giving the levels of the controllable factors (or inputs and ε i is a (p 1 vector denoting the process disturbance. The MEWMA controller described in Tseng et al. 21 is ˆψ i = η(x i ˆMu i 1 + (1 η ˆψ i 1,wherei {1, 2, 3,...}, η is a discount factor (0 <η<1, and ˆψ 0 and ˆM are obtained via design of experiment (DOE. ˆψ i is used to estimate ψ after every production run i. The choice of the value of η strongly reflects the performance of MEPC. An optimal discount factor is determined within a finite number of production runs (N, such that the trace of the total mean squared error (MSE is minimized. After ˆψ i is calculated, the new input vector u i will be updated. Assume that the process dynamics come from the white noise series ε i (ε i N p (µ i,, where the common covariance matrix is non-singular. Let the (p 1 vector τ represent the desired target, and the (p 1 vector y i represent the off-target amount in run i, i {1, 2, 3,...}. Tsenget al. 21 also derived the covariance matrix of y i as ( 2 η(1 η 2(i 1 yi = (2 2 η when ε 0 = 0 and τ = 0. Asi increases to, the expected value of y i, lim i E(y i = µ i. These properties will be used to develop the MEPC/MSPC chart. For illustration, throughout this paper, we use a simple example with a (2 2 MIMO process. Assume that the white noise series ε i N 2 (µ i,. Let the number of production runs N = 250 and D denotes the design
206 L. YANG AND S.-H. SHEU matrix of the DOE for obtaining ˆψ 0 and ˆM. Suppose that the transpose matrix of D is D = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1, and the predicted model is x 1 = 0.1 + u 1, x 2 = 0.1 + u 2.Thatis, ˆψ 0 =[0.1 0.1] and ˆM = ( 10 01.FromTseng et al. 21, we can get the input vector at time i = 0, u 0 = ˆM 1 (τ ˆψ 0 =[ 0.1 0.1] and the approximate optimal discount factor ˆη = ( 4Nρ + ρ 2 ρ/(2n= 0.02, where ρ = (1, u 0 (D D 1 (1, u 0.Thenu 0 and ˆη will be used to calculate ˆψ i and u i. Those interested in more details are referred to Tseng et al. 21. Assume that the statistical monitoring scheme will only signal external changes (i.e. assignable causes. Applying MSPC to monitor the output deviation from the target vector can result in rapid detection of assignable causes. Assume that the assignable cause takes the form of a sustained shift in the process mean vector. If the assignable causes are eliminated, the output deviation will be reduced. In this paper, three different MEPC/MSPC charts for monitoring the output deviation from the target vector are used: the Hotelling multivariate control chart, the MEWMA control chart, and the MGWMA control chart. 3. SOME MEPC/MSPC CHARTS 3.1. The Hotelling multivariate control chart The Hotelling multivariate control chart is a direct analog of the univariate Shewhart X control chart for monitoring the mean vector of the process. According to the MIMO system defined in last section, ε i is a white noise series, ε i N p (µ i,. By measuring deviation from the target vector (τ = 0, as mentioned above, y i (a (p 1 vector, i {1, 2, 3,...}, we denote the known covariance matrix of y i as yi (Equation (2. Then the Hotelling multivariate control chart gives an out-of-control signal as soon as the statistic T 2 i, T 2 i = y i 1 y i y i (( 2 η(1 η = y 2(i 1 i 2 η 1 y i >H 1 (3 at time i, where the upper control limit (UCL H 1 (>0 is chosen to achieve a specified in-control ARL (named ARL 0. Details concerning the Hotelling multivariate control chart can be found in Hotelling 24 and Montgomery 10. 3.2. The MEWMA control chart As the Hotelling multivariate control chart is based on only the most recent observation, it is not sensitive to small shifts in the mean vector. Lowry et al. 25 proposed an EWMA-based multivariate control procedure (MEWMA for monitoring the process mean vector. If ε i in Equation (1 is a white noise series with incontrol mean vector µ 0 = 0 and a common covariance matrix, andthereisnoapriorireason to weight past deviations (y i of the observations differently for the p quality characteristics being monitored, the equation for MEWMA control chart is Z i = ry i + (1 rz i 1 (4 where Z i is a (p 1 vector, i {1, 2, 3,...}, Z 0 = 0, 0<r 1. From Equations (2 and(4, the covariance matrix of Z i is ( r(1 (1 r 2i Zi = yi 2 r ( r(1 (1 r 2i ( 2 η(1 η 2(i 1 = (5 2 r 2 η
INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 207 The MEWMA control chart gives an out-of-control signal as soon as the statistic T 2 i, T 2 i = Z i 1 Z i Z i [( r(1 (1 r = Z 2i ( 2 η(1 η 2(i 1 i 2 r 2 η where the UCL H 2 (>0 is chosen to achieve a specified ARL 0. ] 1 Z i >H 2 (6 3.3. The MGWMA control chart Sheu and Lin 26 proposed a GWMA control chart, which is a generalization of the EWMA control chart. The GWMA control chart applies the method, which has been used in Sheu and Griffith 27,andSheu 28,29,tothe EWMA control charts to enhance the detectional ability of control charts. Let Y i represent the measurement at the ith time period and G i denote the generally weighted moving average in the plotted statistic at time i, i {1, 2, 3,...}. Assume that Y i are independent random variables with mean µ 0 and constant variance σ 2. Let G 0 = µ 0 (the target value of the process mean. Then, G i can be represented as G i = (q 0α q 1α Y i + (q 1α q 2α Y i 1 + +(q (i 1α q iα Y 1 + q iα µ 0 where i {1, 2, 3,...}, the design parameter q is constant, 0 q<1(whenq = 0, let q 0α = 1 and the adjustment parameter α (α >0 is determined by the practitioner. It means that (q 0α q 1α, (q 1α q 2α,...,(q (i 1α q iα are the weights of the most updated sample (i.e. ith sample, the second updated sample (i.e. (i 1th sample,..., the most out-of-date sample (i.e. 1st sample, respectively and q iα is the weight of G 0.When0<α 1and0<q<1, (q 0α q 1α >(q 1α q 2α > >(q (i 1α q iα, then the weights decrease with the age of the samples. The expected value of G i is The variance of G i is E(G i = E[(q 0α q 1α Y i + (q 1α q 2α Y i 1 + +(q (i 1α q iα Y 1 + q iα µ 0 ] =[(q 0α q 1α + (q 1α q 2α + +(q (i 1α q iα ]E(Y + q iα µ 0 = µ 0 var(g i =[(q 0α q 1α 2 + (q 1α q 2α 2 + +(q (i 1α q iα 2 ] σ 2 = Q i σ 2, where Q i = (q 0α q 1α 2 + (q 1α q 2α 2 + +(q (i 1α q iα 2.LetLdenote the width of the control limits, then the control limits of the GWMA control chart can be represented as µ 0 ± L Q i σ. Due to the added adjustment parameter α, the GWMA control chart has been shown to perform much better than both the Shewhart X chart and the EWMA chart in monitoring small shifts of the process mean under the univariate case. In the multivariate case we propose the multivariate GWMA (called MGWMA for short control chart, which is a natural extension of the GWMA control chart. If ε i in Equation (1 is a white noise series with in-control mean vector µ 0 = 0 and a common covariance matrix, andthereisnoapriorireason to weight past deviations (y i of the observations differently for the p quality characteristics being monitored, the equation for the MGWMA control chart is i G i = (q (i tα q (i t+1α y t (7 t=1 where G i is a (p 1 vector, i {1, 2, 3,...}, G 0 = 0, the design parameter q is constant (0 q<1, and the adjustment parameter α is determined by the practitioner. From Equations (2 and(7, the covariance
208 L. YANG AND S.-H. SHEU matrix of G i is ( i Gi = var (q (i tα q (i t+1α y t t=1 = ((q 0α q 1α 2 + (q 1α q 2α 2 + +(q (i 1α q iα 2 yi ( 2 η(1 η = ((q 0α q 1α 2 + (q 1α q 2α 2 + +(q (i 1α q iα 2 2(i 1 2 η ( 2 η(1 η 2(i 1 = Q i (8 2 η where Q i = (q 0α q 1α 2 + (q 1α q 2α 2 + +(q (i 1α q iα 2. The MGWMA control chart gives an outof-control signal as soon as the statistic T 2 i, T 2 i = G i 1 G i G i ( 2 η(1 η = G 2(i 1 i [Q i 2 η ] 1 G i >H 3 (9 where the UCL H 3 (>0 is chosen to achieve a specified ARL 0.Whenα = 1andq = 1 r, Equations (7 (9 will reduce to Equations (4 (6. That is, the MEWMA control chart is a special case in the MGWMA control chart when α = 1. If the in-control mean vector of y i is µ i = µ 0 = 0, and the out-of-control mean vector is µ i = µ 1 0. Lowry et al. 25 had shown that the ARL performance of the MEWMA control chart depends only on the mean vector µ 1 and covariance matrix yi through the value of the noncentrality parameter λ, λ = ((µ 1 µ 0 y 1 i (µ 1 µ 0 1/2 = (µ 1 1 y i µ 1 1/2 (10 4. QUALITY COST MODEL FOR MEPC/MSPC Now, we apply the MEPC/MSPC chart to monitor the output deviation from the target vector. If an item s T 2 i is greater than the UCL, the process is assumed to be out of control and a search for an assignable cause is initiated. The time interval from the beginning of the in-control state to the adjustment of the out-of-control state is called a production cycle. The time between occurrences of an assignable cause is assumed to follow an exponential distribution with a mean of θ occurrences per hour. So the mean time in the in-control state is 1/θ. Once the process is out of control and a signal is triggered, the intervention is required. Assume that the sustained shift occurs only once in each production cycle and no other shift occurs before the previous shift is detected and removed. Assume that the process is a renewal reward process, that is, after the elimination of an assignable cause, the process will return to the initial state of statistical control. Each of these cycles, from statistical control through the occurrence, detection and elimination of an assignable cause, thus have identical expected costs and times. In Lorenzen and Vance 9, the quality cost consists of the cost associated with production of nonconformities, the diagnosis cost associated with identifying assignable causes from out-of-control signals, the adjustment cost associated with correction of assignable causes and the sampling and testing cost associated with subgroup size n. The expected production cycle time, PCT, is PCT = 1 θ + (1 γ 1sT 0 ARL 0 ϕ + ne + h(arl 1 + T 1 + T 2 (11
INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 209 The expected total cost per cycle, TC 0,is TC 0 = C 0 θ + C 1 ( ϕ + ne + h(arl 1 + γ 1 T 1 + γ 2 T 2 + sy + W ARL ( ( 0 a + bn 1 + h θ ϕ + ne + h(arl 1 + γ 1 T 1 + γ 2 T 2 (12 where the cost parameters are as follows: n is the sample size; h is the hours between samples; C 0 is the quality cost/hour while producing in control; C 1 is the quality cost/hour while producing out of control (>C 0 ; h ϕ is the expected time between a shift (assignable cause and the next sample, 1 (1 + θhe θh ϕ = θ(1 e θh E is the time to sample and chart one item; γ 1 is the 1 if production continues during searches, and 0 otherwise; γ 2 is the 1 if production continues during the repair of the process, and 0 otherwise; T 0 is the expected search time when false alarm; T 1 is the expected time to discover the assignable cause; T 2 is the expected time to repair the assignable cause; s = e θh /(1 e θh is the expected number of samples while in control; Y is the cost to investigate false alarms; W is the cost of locating and repairing an assignable cause; a is the fixed cost per item; and b is the cost per unit sampled. Generally speaking, items are produced and 100% inspected in the MEPC/MSPC system. Without loss of generality, we assume that the subgroup size n = 1. The time interval between samples, h, is fixed and is equal to the production time per item. Similarly to Jiang and Tsui 19, the ideal of the Taguchi loss function for a product is used here. The Taguchi loss function defines the expected loss to society as Expected loss/unit = A 0 ν 2 where ν 2 is the mean squared deviation and A 0 is the cost coefficient. In univariate case, ν 2 = σ 2 + (µ T 2, where σ 2 is the process variance, µ is the process mean and T is the process target. When the process is in control, i.e., µ = T,thenν 2 = ν1 2 = σ 2. When the process is out of control its mean shifts off the target, i.e., µ T, then ν 2 = ν2 2 = σ 2 + (µ T 2, ν2 2 >ν2 1.InMEPC/MSPC,letEDs denote the average Euclidean distances of the deviation from the target vector τ within N production runs. When the process is out of control, assume that the sustained shift occurs at i = 1 and is eliminated as soon as it is detected by the MSPC chart. Let µ 0 be the mean vector of the deviation from the target vector when the process is in-control. ED 1 and ED 3 are calculated across the whole periods without and with shift, respectively. ED 2 is calculated from i = 1 until the time that the shift is detected by the MSPC chart (i.e. run length, RL or until the end of the production run. That is, ED 1 = 1 N N ( p 1/2 (y ij τ j 2 i=1 j=1 if µ i = µ 0 = 0, i= 1, 2,...,N (13 ED 2 = 1 RL RL i=1 ( p 1/2 (y ij τ j 2 j=1 if µ i = µ 1 0, i= 1, 2,...,RL, 1 RL N (14 ED 3 = 1 N N ( p 1/2 (y ij τ j 2 i=1 j=1 if µ i = µ 1 0, i= 1, 2,...,RL, and µ i = µ 0 = 0, i= RL + 1, RL + 2,...,N, 1 RL N (15 Let AED 1, AED 2 and AED 3 denote the average of ED 1, ED 2 and ED 3, respectively. Then we can modify the Taguchi loss function to be Expected loss/unit = A 0 (trace( + (AED 2 (16
210 L. YANG AND S.-H. SHEU Using Equation (16, we can easily embellish Lorenzen Vance expected total cost (Equation (12 to consider losses owing to in-control and out-of-control variability. Then the expected total cost per cycle, TC, is TC = C 0 + A 0 (trace( + (AED 1 2 h θ +[C 1 + A 0 (trace( + (AED 2 2 h] ( ϕ + ne + h(arl 1 + γ 1 T 1 + γ 2 T 2 + sy ARL 0 + W + ( a + bn h ( 1 θ ϕ + ne + h(arl 1 + γ 1 T 1 + γ 2 T 2 (17 From Equations (11and(17, the expected average quality cost per hour, AQC, is AQC = TC PCT (18 5. COMPARING THE MEPC/MSPC CHARTS There are two kinds of simulations performed in this paper. First, we use the simulation scheme proposed by Lowry et al. 25 to obtain the control limits (H 1, H 2 and H 3 of the MSPC charts. The design parameters of the MGWMA chart are the value of q, α and the width of the control limits H 3 to achieve a specified ARL 0. For simplification, we choose the design parameter q {0.7, 0.8, 0.9} and the adjustment parameters α {0.7, 0.8, 1.0} only, and keep the in-control ARL 0 = 200 by changing H3.Whenα = 1.0, the MGWMA chart reduces to the MEWMA chart. Then the other simulations are performed to estimate ARL 1 and AEDsofthe MEPC/MSPC charts. According to the (2 2 MIMO example mentioned in Section 2, the approximate optimal discount factor is ˆη = 0.02. We also set the other two discount factors η = 0.01 and 0.1 for comparison. The shift magnitudes investigated are λ = 0.25, 0.5, 0.75, 1, 2, 3 and 5. Let the total production number N 250, during every simulation run. MEPC rule continues for all periods. Up to 20 000 runs of the process are simulated. The simulation programs are written in BASIC language. Appendix A (with η = 0.01, Appendix B (with η =ˆη = 0.02 and Appendix C (with η = 0.1 show the simulation results of ARL 1, AED 1, AED 2 and AED 3 for MEPC alone or various combinations of MEPC and MSPC chart. From the statistical point of view, the italic numbers in Appendices A C denote that the MEPC/MGWMA chart performs better than the MEPC/MEWMA chart (α = 1 with the same q value. From Appendix B, ARL 1 and AED 3 of the Hotelling multivariate chart (with H 1 = 10.6, the MGWMA chart (with q = 0.8,α= 0.8, H 3 = 9.83, the MEWMA chart (with q = 0.8, α = 1, H 2 = 9.71 and the MEPC alone (only used in Figure 3 with various λ, are plotted in Figures 2 and 3, respectively.due to the added adjustment parameter α (α<1, the MEPC/MGWMA chart is more sensitive than both the MEPC/MEWMA chart and the MEPC/Hotelling multivariate chart in detecting small shifts of the mean vector. For instance, when λ 1, the ARL 1 and AED 3 of the MGWMA chart are smaller than those of the Hotelling multivariate chart and the MEWMA chart. In contrast,when λ 3, the Hotelling multivariate chart outperforms both the MGWMA chart and the MEWMA chart. In the quality cost model (Equations (11and(17, let ϕ = h = 0.2, T 1 = 0.1, T 2 = 0.1, A 0 = 10, and Y = 0, 50 and 100, respectively, and other cost parameters are adapted from Montgomery et al. 12 : 1/θ = 100, E= 0.05, T 0 = 0, γ 1 = γ 2 = 1, C 0 = 10, C 1 = 100, W = 25, a= 0.5, b= 0.1 By combining these cost parameters with the tables in Appendices A C, the AQCs of three control charts are calculated and compared in Table I. The MGWMA charts used for comparison here are q = 0.8, α = 0.8, H 3 = 9.83, and q = 0.8, α = 1.0, H 2 = 9.71 (reduce to the MEWMA chart with r = 0.2. From the economic point of view, the MEPC/MGWMA chart is still much more efficient than both the MEPC/MEWMA chart and the MEPC/Hotelling multivariate chart in detecting small shifts of the mean vector. Although the approximate optimal discount factor is ˆη = 0.02, comparing with η = 0.01, its performance is not always outperformed.
INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 211 ARL 1 135 120 105 90 75 60 45 30 15 0 Hotelling (H 1 MEWMA (q = 0.8, = 10.6 = 1, H 2 = 9.71 MGWMA (q = 0.8, = 0.8, H 3 = 9.83 0.25 0.5 0.75 1 2 3 5 Figure 2. The ARL 1 of different charts (η =ˆη = 0.02 AED 3 3.20 2.70 MEPC alone Hotelling (H 1 = 10.6 MEWMA (q = 0.8, = 1, H 2 = 9.71 MGWMA (q = 0.8, = 0.8, H 3 = 9.83 2.20 1.70 1.20 0.25 0.5 0.75 1 2 3 Figure 3. The AED 3 of different charts (η =ˆη = 0.02
212 L. YANG AND S.-H. SHEU Hotelling H 1 = 10.6 Table I. AQC of different charts MGWMA q = 0.8, α = 0.8 H 3 = 9.83 MEWMA q = 0.8, α = 1.0 H 2 = 9.71 Y Y Y η\λ ARL 0 50 500 ARL 0 50 500 ARL 0 50 500 0.00 200 17.50 24.62 88.65 200 17.50 24.62 88.65 200 17.50 24.62 88.65 0.01 0.25 130.76 39.03 40.02 48.92 78.68 32.62 33.70 43.40 83.31 33.24 34.31 43.94 0.50 102.53 35.74 36.78 46.10 29.59 25.44 26.62 37.23 33.71 26.10 27.27 37.79 0.75 68.22 31.29 32.39 42.27 15.54 23.14 24.35 35.25 16.42 23.29 24.50 35.38 1.00 42.46 27.56 28.71 39.07 10.00 22.20 23.43 34.44 9.77 22.16 23.39 34.40 2.00 6.93 21.74 22.97 34.05 3.68 21.12 22.36 33.51 3.64 21.11 22.35 33.51 3.00 2.14 20.87 22.12 33.30 2.26 20.90 22.14 33.32 2.32 20.91 22.15 33.34 5.00 1.03 20.72 21.96 33.17 1.32 20.80 22.05 33.25 1.49 20.85 22.09 33.29 0.02 0.25 133.55 39.34 40.32 49.19 78.33 32.57 33.65 43.36 83.62 33.28 34.35 43.97 0.50 103.73 35.89 36.92 46.23 30.27 25.55 26.73 37.32 34.18 26.18 27.34 37.86 0.75 69.42 31.46 32.55 42.42 15.84 23.19 24.40 35.29 16.10 23.24 24.45 35.33 1.00 43.26 27.69 28.84 39.17 9.97 22.20 23.42 34.43 10.01 22.20 23.43 34.44 2.00 7.09 21.77 23.00 34.08 3.72 21.13 22.37 33.52 3.77 21.14 22.38 33.53 3.00 2.19 20.88 22.13 33.31 2.27 20.90 22.14 33.33 2.36 20.92 22.16 33.34 5.00 1.03 20.72 21.96 33.17 1.34 20.81 22.05 33.25 1.49 20.85 22.09 33.29 0.1 0.25 148.47 40.97 41.93 50.60 87.15 33.74 34.80 44.37 94.42 34.68 35.73 45.18 0.50 116.97 37.50 38.51 47.61 31.63 25.77 26.95 37.51 36.18 26.49 27.65 38.13 0.75 79.28 32.81 33.89 43.59 15.91 23.21 24.42 35.30 16.75 23.35 24.56 35.43 1.00 48.72 28.53 29.67 39.90 10.01 22.21 23.43 34.44 10.04 22.21 23.43 34.45 2.00 7.29 21.81 23.04 34.11 3.76 21.14 22.38 33.53 3.61 21.11 22.35 33.50 3.00 2.14 20.88 22.12 33.31 2.26 20.90 22.15 33.33 2.33 20.92 22.16 33.34 5.00 1.03 20.73 21.97 33.18 1.33 20.81 22.05 33.26 1.50 20.86 22.10 33.30 Table II. The best performance chart in terms of ARL 1 and AQC η\λ 0.25 0.5 0.75 1.0 2.0 3.0 5.0 (a ARL 1 0.01 MGWMA MGWMA MGWMA MEWMA MEWMA HOTTL HOTTL 0.02 MGWMA MGWMA MGWMA MGWMA MGWMA HOTTL HOTTL 0.10 MGWMA MGWMA MGWMA MGWMA MEWMA HOTTL HOTTL (b AQC (Y = 0 or 50 0.01 MGWMA MGWMA MGWMA MEWMA MEWMA HOTTL HOTTL 0.02 MGWMA MGWMA MGWMA MGWMA MGWMA HOTTL HOTTL 0.10 MGWMA MGWMA MGWMA MGWMA MEWMA HOTTL HOTTL (c AQC (Y = 500 0.01 MGWMA MGWMA MGWMA MGWMA MGWMA HOTTL HOTTL 0.02 MGWMA MGWMA MGWMA MGWMA MGWMA HOTTL HOTTL 0.10 MGWMA MGWMA MGWMA MGWMA MGWMA HOTTL HOTTL However, comparing with η = 0.1, the performance of ˆη is still much better. Table II summarizes the best chart for detecting a shift under ARL 1 and AQC criteria. In most of the cases, the best chart in terms of AQC is almost consistent with that in terms of ARL 1, i.e. the MGWMA chart is the best when the shift is small.
INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 213 Table III. AED 3 of MEPC alone and the MEPC/MGWMA chart AED 3 λ\η 0.01 0.02 0.1 0.2 0.3 0.4 MEPC alone 0.25 1.273 1.274 1.275 1.274 1.276 1.278 0.50 1.330 1.332 1.334 1.339 1.344 1.348 0.75 1.426 1.425 1.432 1.441 1.453 1.466 1.00 1.549 1.553 1.562 1.580 1.600 1.617 2.00 2.276 2.281 2.317 2.363 2.417 2.476 3.00 3.180 3.187 3.245 3.326 3.411 3.507 5.00 5.114 5.128 5.228 5.369 5.518 5.680 MGWMA 0.25 1.259 1.260 1.261 1.261 1.264 1.266 0.50 1.262 1.262 1.262 1.265 1.267 1.270 0.75 1.264 1.263 1.264 1.266 1.268 1.269 1.00 1.265 1.265 1.266 1.266 1.268 1.269 2.00 1.267 1.269 1.269 1.271 1.271 1.272 3.00 1.270 1.271 1.272 1.272 1.272 1.274 5.00 1.272 1.272 1.274 1.275 1.275 1.276 Table IV. ARL 1, AED 2 and AQC of the MGWMA chart MGWMA (q = 0.8, α = 0.8, H 3 = 9.83 λ\η 0.01 0.02 0.1 0.2 0.3 0.4 ARL 1 0.25 78.68 78.33 87.15 95.73 110.53 120.41 0.50 29.59 30.27 31.63 33.88 36.99 39.34 0.75 15.54 15.84 15.91 16.56 17.24 18.01 1.00 10.00 9.97 10.01 10.28 10.69 10.69 2.00 3.68 3.72 3.76 3.78 3.78 3.80 3.00 2.26 2.27 2.26 2.30 2.26 2.29 5.00 1.32 1.34 1.33 1.33 1.31 1.32 AED 2 0.25 1.274 1.270 1.274 1.278 1.276 1.279 0.50 1.335 1.331 1.337 1.342 1.350 1.353 0.75 1.427 1.429 1.434 1.445 1.456 1.467 1.00 1.549 1.552 1.565 1.577 1.600 1.623 2.00 2.290 2.281 2.314 2.369 2.407 2.478 3.00 3.191 3.190 3.251 3.309 3.419 3.518 5.00 5.125 5.123 5.224 5.402 5.510 5.683 AQC 0.25 33.70 33.65 34.80 35.89 37.69 38.85 0.50 26.62 26.73 26.95 27.30 27.79 28.15 0.75 24.35 24.40 24.42 24.53 24.64 24.77 1.00 23.43 23.42 23.43 23.48 23.55 23.55 2.00 22.36 22.37 22.38 22.39 22.39 22.40 3.00 22.14 22.14 22.15 22.16 22.16 22.17 5.00 22.05 22.05 22.05 22.07 22.07 22.08 6. DISCUSSION OF THE DISCOUNT FACTOR As mentioned in Section 2, the determination of the optimal discount factor ( ˆη = 0.02 is to minimize the trace of the total MSE in MEPC. For further information see Tseng et al. 21. In order to investigate the feasibility of the optimal discount factor in MEPC/MSPC, we use the MGWMA chart (with q = 0.8, α = 0.8, H 3 = 9.83 under various discount factor η, η {0.01, 0.02, 0.1, 0.2, 0.3, 0.4}, and various λ, λ {0.25, 0.5, 0.75, 1, 2, 3, 5},to get AED 3, ARL 1, AED 2,andAQC as we have done previously in Section 5. LetY = 50. The simulation results are listed in Tables III and IV.
214 L. YANG AND S.-H. SHEU AED 3 MEPC alone, = 5 6.0 5.0 4.0 3.0 2.0 1.0 0.0 MEPC alone, = 3 MGWMA, = 3 or 5 0.0 0.1 0.2 0.3 0.4 Figure 4. Comparing AED 3 of MEPC alone and the MEPC/MGWMA chart Based on Table III, the performanceof MEPC/MGWMA is much better than that of MEPC alone, especially when λ 2. For instance, when λ = 3, η = 0.1, the AED 3 of MGWMA is 1.272 and that of MEPC alone is 3.245 (>1.272. It means that integrating MEPC and MSPC is much more effective than using MEPC alone in reducing the output deviation from the target vector. Figure 4 also shows the comparison of AED 3 between MEPC alone and MEPC/MGWMA when λ = 3 and 5. In MEPC alone, the AED 3 increases rapidly when η or λ increases. In contrast, the AED 3 of MEPC/MGWMA remains small even λ increases to 5 or η increases to 0.4. Therefore, when the sustained shifts are large, finding an optimal discount factor for MEPC alone is much more important than that for MEPC/MSPC. Table IV shows that, when λ 0.5, the ARL 1 and AQC of MEPC/MGWMA increase rapidly while η increases. The ARL 1 and AQC are sensitive to the value of the discount factor when λ is small. It means that, with the economic design strategy, when we want to detect a small shift of the process mean vector in MEPC/MSPC, choosing an appropriate MSPC chart and an appropriate discount factor will get a smaller AQC. For instance, if we want to detect a shift of 0.25 in magnitude, choosing the MGWMA chart with η =ˆη = 0.02 will result in AQC = 33.65 (see Table IV which is smaller than that of the MGWMA chart with η = 0.2 (AQC = 35.89 (>33.65,see Table IV and that of the Hotelling multivariate chart with η = 0.02 (AQC = 40.32 (>33.65,see Table I with Y = 50. An appropriate design of the MEPC/MSPC chart can indeed reduce the total quality cost and get a much better performance. Figure 5 shows the ARL 1 of the MEPC/MGWMA chart and Figure 6 shows the AQC of the MEPC/MGWMA chart with various λ. The effectiveness of the optimal discount factor is also addressed. This work addresses whether, if the discount factor is set equal to the optimal discount factor (η =ˆη = 0.02, the value of ARL 1 or AQC decreases. Figures 5 and 6 reveal that when the discount factor equals or approximates the optimal discount factor, the values of ARL 1 and AQC remain minimal for various values of λ. For instance, in Figure 5, whenλ = 0.25, the value of ARL 1 with η =ˆη = 0.02 (ARL 1 = 78.33 is smaller than that when η ˆη = 0.02. Therefore, the optimal discount factor ˆη derived from MEPC remains a good discount factor in the MEPC/MSPC system. 7. CONCLUSIONS Most of the MEPC schemes are designed to react to process upsets and do not make any effort to remove the assignable causes in the MIMO system. MSPC chart can be used to monitor, identify and subsequently eliminate the assignable causes. In this paper, according to the proposed criteria, the performance of MSPC charts are compared in statistical and economic manner. Combining MEPC and MSPC charts always results
INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 215 Figure 5. The ARL 1 of the MEPC/MGWMA chart with various λ Figure 6. The AQC of the MEPC/MGWMA chart with various λ in the reduction of overall variability if the process has external assignable causes that lead to sustained shifts. Especially when detecting small shifts of the mean vector, due to the added adjustment parameter, the MEPC/MGWMA chart is more effective and more economic than both the MEPC/MEWMA chart and the MEPC/Hotelling multivariate chart. In contrast to the MEPC/MGWMA chart, the MEPC/Hotelling multivariate chart is the most effective when the shift is large. We conclude that proper use of both MEPC and MSPC can always outperform the use of either alone under multivariate case. At the same time, we also conclude that the optimal discount factor in MEPC is still a good discount factor in the MEPC/MSPC system. Throughout the paper, we assume that the process dynamics come from ε i and assume that the assignable cause takes the form of a sustained shift in the process mean vector. For simplification, we consider that ε i is a white noise series only. If the process has a linear drift, the MEWMA control scheme used in this paper no longer satisfies the stability conditions. A suitable controller, such as the double MEWMA controller proposed by del Castillo and Rajagopal 30, might overcome this difficulty and will be a possible combination with MSPC in future research.
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INTEGRATED MULTIVARIATE EPC AND MULTIVARIATE SPC CHARTS 217 APPENDIX A. THE ARL 1, AED 2 AND AED 3 OF DIFFERENT CHARTS (η = 0.01, AED 1 = 1.248 (3 MEPC/MGWMA (4 MEPC/MEWMA (2 q = 0.7 q = 0.8 q = 0.9 (1 MEPC/ (3 (4 (3 (4 (3 (4 MEPC Hotelling, α = 0.7 α = 1.0 α = 0.8 α = 1.0 α = 0.8 α = 1.0 λ alone H 1 = 10.6 H 3 = 10.29 H 2 = 10.1 H 3 = 9.83 H 2 = 9.71 H 3 = 9.03 H 2 = 8.78 0.25 ARL 1 130.76 90.64 95.63 78.68 83.31 64.45 70.41 AED 2 1.273 1.274 1.274 1.275 1.274 1.275 1.274 1.274 AED 3 1.273 1.264 1.259 1.262 1.259 1.260 1.258 1.260 0.50 ARL 1 102.53 36.62 40.27 29.59 33.71 26.37 26.73 AED 2 1.330 1.330 1.333 1.335 1.335 1.333 1.333 1.330 AED 3 1.330 1.286 1.263 1.266 1.262 1.264 1.261 1.261 0.75 ARL 1 68.22 18.22 18.97 15.54 16.42 15.01 14.47 AED 2 1.426 1.425 1.426 1.432 1.427 1.425 1.427 1.424 AED 3 1.426 1.300 1.264 1.266 1.264 1.265 1.263 1.263 1.00 ARL 1 42.46 10.95 10.79 10.00 9.77 10.25 9.72 AED 2 1.549 1.551 1.551 1.555 1.549 1.552 1.547 1.548 AED 3 1.549 1.305 1.266 1.266 1.265 1.264 1.265 1.266 2.00 ARL 1 6.93 3.58 3.43 3.68 3.64 4.26 4.23 AED 2 2.276 2.271 2.288 2.288 2.290 2.282 2.284 2.276 AED 3 2.276 1.282 1.267 1.268 1.267 1.268 1.270 1.271 3.00 ARL 1 2.14 2.08 2.10 2.26 2.32 2.65 2.81 AED 2 3.180 3.180 3.171 3.172 3.191 3.195 3.186 3.190 AED 3 3.180 1.271 1.269 1.270 1.270 1.271 1.274 1.277 5.00 ARL 1 1.03 1.15 1.30 1.32 1.49 1.59 1.81 AED 2 5.114 5.102 5.140 5.115 5.125 5.094 5.108 5.114 AED 3 5.114 1.269 1.272 1.274 1.272 1.277 1.278 1.282 APPENDIX B. THE ARL 1, AED 2 AND AED 3 OF DIFFERENT CHARTS (η =ˆη = 0.02, AED 1 = 1.248 (3 MEPC/MGWMA (4 MEPC/MEWMA (2 q = 0.7 q = 0.8 q = 0.9 (1 MEPC/ (3 (4 (3 (4 (3 (4 MEPC Hotelling, α = 0.7 α = 1.0 α = 0.8 α = 1.0 α = 0.8 α = 1.0 λ alone H 1 = 10.6 H 3 = 10.29 H 2 = 10.1 H 3 = 9.83 H 2 = 9.71 H 3 = 9.03 H 2 = 8.78 0.25 ARL 1 133.55 89.04 93.65 78.33 83.62 66.09 70.65 AED 2 1.274 1.274 1.274 1.277 1.270 1.276 1.274 1.274 AED 3 1.274 1.264 1.261 1.262 1.260 1.261 1.258 1.260 0.50 ARL 1 103.73 36.63 40.31 30.27 34.18 26.20 27.43 AED 2 1.332 1.330 1.331 1.336 1.331 1.333 1.334 1.337 AED 3 1.332 1.287 1.265 1.267 1.262 1.266 1.261 1.261 0.75 ARL 1 69.42 18.31 18.84 15.84 16.10 14.34 14.67 AED 2 1.425 1.426 1.426 1.435 1.429 1.431 1.423 1.431 AED 3 1.425 1.302 1.265 1.268 1.263 1.265 1.263 1.264 1.00 ARL 1 43.26 11.11 11.01 9.97 10.01 10.27 10.27 AED 2 1.553 1.553 1.557 1.553 1.552 1.554 1.555 1.557 AED 3 1.553 1.305 1.266 1.266 1.265 1.266 1.266 1.265 2.00 ARL 1 7.09 3.60 3.58 3.72 3.77 4.25 4.26 AED 2 2.281 2.287 2.290 2.289 2.281 2.288 2.284 2.286 AED 3 2.281 1.282 1.268 1.266 1.269 1.268 1.270 1.271 3.00 ARL 1 2.19 2.08 2.18 2.27 2.36 2.67 2.84 AED 2 3.187 3.193 3.188 3.194 3.190 3.194 3.185 3.194 AED 3 3.187 1.271 1.268 1.270 1.271 1.271 1.274 1.276 5.00 ARL 1 1.03 1.15 1.29 1.34 1.49 1.59 1.82 AED 2 5.128 5.095 5.113 5.126 5.123 5.128 5.152 5.150 AED 3 5.128 1.269 1.270 1.274 1.272 1.276 1.277 1.282
218 L. YANG AND S.-H. SHEU APPENDIX C. THE ARL 1, AED 2 AND AED 3 OF DIFFERENT CHARTS (η = 0.1, AED 1 = 1.248 (3 MEPC/MGWMA (4 MEPC/MEWMA (2 q = 0.7 q = 0.8 q = 0.9 (1 MEPC/ (3 (4 (3 (4 (3 (4 MEPC Hotelling, α = 0.7 α = 1.0 α = 0.8 α = 1.0 α = 0.8 α = 1.0 λ alone H 1 = 10.6 H 3 = 10.29 H 2 = 10.1 H 3 = 9.83 H 2 = 9.71 H 3 = 9.03 H 2 = 8.78 0.25 ARL 1 148.47 100.70 108.61 87.15 94.42 69.52 76.46 AED 2 1.275 1.275 1.275 1.275 1.274 1.276 1.275 1.276 AED 3 1.275 1.267 1.262 1.263 1.261 1.262 1.258 1.260 0.50 ARL 1 116.97 39.76 45.00 31.63 36.18 26.97 27.50 AED 2 1.334 1.335 1.336 1.338 1.337 1.336 1.337 1.337 AED 3 1.334 1.291 1.267 1.269 1.262 1.265 1.263 1.248 0.75 ARL 1 79.28 19.02 19.95 15.91 16.75 15.34 14.77 AED 2 1.432 1.310 1.267 1.268 1.264 1.265 1.264 1.249 AED 3 1.432 1.432 1.432 1.440 1.434 1.433 1.432 1.436 1.00 ARL 1 48.72 11.36 11.01 10.01 10.04 10.40 9.73 AED 2 1.562 1.563 1.562 1.571 1.565 1.563 1.563 1.567 AED 3 1.562 1.313 1.268 1.266 1.266 1.265 1.267 1.248 2.00 ARL 1 7.29 3.64 3.46 3.76 3.61 4.28 4.24 AED 2 2.317 2.313 2.315 2.316 2.314 2.334 2.316 2.322 AED 3 2.317 1.284 1.268 1.268 1.269 1.269 1.272 1.250 3.00 ARL 1 2.14 2.06 2.12 2.26 2.33 2.68 2.80 AED 2 3.245 3.261 3.238 3.240 3.251 3.246 3.243 3.249 AED 3 3.245 1.271 1.270 1.270 1.272 1.273 1.275 1.248 5.00 ARL 1 1.03 1.15 1.30 1.33 1.50 1.59 1.81 AED 2 5.228 5.225 5.230 5.214 5.224 5.218 5.219 5.226 AED 3 5.228 1.270 1.271 1.274 1.274 1.278 1.278 1.248 Authors biographies Ling Yang is an Associate Professor in the Department of Industrial Engineering and Management at St. John s University, Taiwan. She earned a BBA in Industrial Management from National Cheng Kung University, Taiwan, an MBA in Management Science from National Chiao Tung University, Taiwan, and a PhD in Industrial Management from National Taiwan University of Science and Technology, Taiwan. Shey-Huei Sheu is a Professor of Industrial Management at the National Taiwan University of Science and Technology, Taiwan. He received his MSc (1979 in Applied Mathematics from National Tsing Hua University, Taiwan, and his PhD (1987 in Statistics from the University of Kentucky. He has published in journals such as Naval Research Logistics, Journal of Applied Probability, RAIOR Operations Research, Microelectronics & Reliability, Reliability Engineering & System Safety, International Journal of System Science, International Journal of Reliability, Quality and Safety Engineering, Journal of the Operational Research Society, European Journal of Operational Research, IEEE Transactions on Reliability, Production Planning & Control, Computers & Operations Research, Quality Engineering and Annuals of Operations Research.