Stochastics and Statistics
Joint statistical design of double sampling X¯ and s charts

https://doi.org/10.1016/j.ejor.2004.04.033Get rights and content

Abstract

In statistical quality control, usually the mean and variance of a manufacturing process are monitored jointly by two statistical control charts, e.g., a X¯ chart and a R chart. Because of the efficiency of double sampling (DS) X¯ charts in detecting shifts in process mean and DS s charts in process standard deviation it seems reasonable to investigate the joint DS X¯ and s charts for statistical quality control. In this paper, a joint DS X¯ and s chart scheme is proposed. The statistical design of the joint DS X¯ and s charts is defined and formulated as an optimization problem and solved using a genetic algorithm. The performance of the joint DS X¯ and s charts is also investigated. The results of the investigation indicate that the joint DS X¯ and s charts offer a better statistical efficiency in terms of average run length (ARL) than combined EWMA and CUSUM schemes, omnibus EWMA scheme over certain shift ranges when all schemes are optimized to detect certain shifts. In comparison with the joint standard, two-stage samplings and variable sampling size X¯ and R charts, the joint DS charts offer a better statistical efficiency for all ranges of the shifts.

Introduction

In statistical quality control, usually the mean and variance of a manufacturing process are monitored jointly. Over the past years, most of the research on joint monitoring of process mean and process variance has focused on the joint X¯ and R charts (see, for example, Jones and Case, 1981; Rahim, 1989; Saniga, 1991; Costa, 1993, Costa, 1998; Costa and Rahim, 2000).

Particularly, Saniga (1991) presented a joint statistical design of X¯ and R chart. A procedure coded in FORTRAN enables users to determine the control limit parameters and sample size of a jointly designed X¯ and R chart for a specified statistical criterion. This criterion can be stated in terms of average run length (ARL), Type I and Type II error probabilities, or average time to signal (ATS).

Gan (1995) developed control schemes for joint monitoring of process mean and variance by using exponentially weighted moving average (EWMA) control charts. The first scheme, EEU was obtained by running a two-sided EWMA mean chart and a high-side EWMA variance chart simultaneously. A two-sided mean chart (Crowder, 1987a, Crowder, 1987b, Crowder, 1989; Lucas and Saccuci, 1990) is obtained by plotting sample statistic Qt=(1-λM)Qt-1+λMX¯t against sample number t for t = 1, 2, … A high-side variance chart (Crowder and Hamilton, 1992) is obtained by plotting sample statistic Qt=max{0,(1-λV)Qt-1+λVlog(st2)} against sample number t for t = 1, 2, …, where st2 is the sample variance. This is a special case of the generalized charting procedure introduced by Champ et al. (1991). The second scheme, EE consists of a two-sided EWMA mean chart and a two-sided EWMA variance chart. A two-sided EWMA variance chart is obtained by plotting Q0=E{log(st2)}=-0.270 (when process variance σ2 is equal to target process variance σ02) and Qt={(1-λV)Qt-1+λVlog(st2)} against sample number t for t = 1, 2, ….

In Gan (1995), the EEU and EE schemes were compared with a combined two-sided CUSUM (CC) scheme and a omnibus EWMA chart scheme (Domangue and Patch, 1991). The results of the comparison showed that the omnibus charts and EEU scheme are ineffective in detecting out-of-control situations characterized by a shift in process mean with simultaneous decrease in variance. For adequate joint monitoring, it was suggested that a combined scheme like EE and CC should be used. The schemes like EE and CC are very similar in performance and they are often more sensitive than the omnibus charts in detecting other out-of-control situations. The omnibus charts have difficulties in meaningful interpretations of the out-of-control signals when an out-of-control signal is issued since an omnibus chart does not indicate whether the signal is due to the mean shift or the variance shift.

The CC scheme is obtained by plotting sample statistics St=max{0,St1+X¯tkM} and Tt=min{0,Tt1+X¯t+kM} against sample number t for the mean chart and by plotting Vt=max{0,Vt-1+log(st2)-kVU} and Kt=min{0,Kt-1+log(st2)-kVL} against sample number t for the variance chart.

The omnibus EWMA scheme proposed by Domangue and Patch (1991) is based on statistic Ai=λ|Zi|α+(1-λ)Ai-1 for i = 1, 2, …, where 0 < λ  1. The authors chose the EWMA of |Z|α for some α because the proposed procedure was shown to be sensitive to changes in location when σσ 0 and to increases in dispersion. The authors mentioned that the proposed chart would not be effective for detection of a small change in mean when σ < σ0. For simplicity they considered only cases with α = 0.5 and α = 2. A “signal” to stop or inspect the process is given whenever AiE[Ai] + L var[Ai ]1/2. The quantity L is specified by the user. For i  ∞, E[Ai]=(2απ)1/2Γ[(α+1)/2], and var[Ai]=2αλ(2-λ)π[πΓ[α+0.5]-(Γ[(α+1)/2])2]. The omnibus EWMA schemes with different design parameters were compared with other schemes, particularly CUSUM of |Z|α proposed by Hawkins (1981) and Healy (1987), which is referred to as omnibus CUSUM procedure. This procedure involves keeping a single sum given by Yi=max[0,|Zi|α-k+Yi-1]. The process stops whenever control statistic Yi exceeds h, which is referred to as the decision value.

The comparison results in Domangue and Patch (1991) show that for all possible scenarios the omnibus EWMA and CUSUM charts are better than the joint X¯ and R chart, the individual moving range (IMR) charts, the EWMA for mean, the X¯, and the X¯ with warning limits.

Chen et al. (2001) proposed a new EWMA (MaxEWMA) chart which combines two sample statistics, i.e., mean and variance into one. Let Ui be the ith standardized sample mean and Vi=Φ-1{H((ni-1)si2σ2;ni-1)}, where Φ(z) = P(Z  z) for Z  N(0,1), the standard normal distribution, Φ−1(•) is the inverse function of Φ(•), and H(w,ν) = P(W  w|v) for Wχν2. Ui and Vi are independent because X¯ and si are independent. Since both Ui and Vi have the same distribution then a single chart could be constructed to monitor both the process mean and the process variability. First, they defined two EWMA charts Yi = (1− λ)Yi−1 + λUi and Zi = (1−λ)Zi−1 + λVi, i = 1, 2, …. Then the two charts are combined into one sample statistic Mi given by max{|Yi|,|Zi|}. The statistic Mi will be large when the process mean has drifted away from the target and/or when the process variability has increased or decreased. The comparison with standard X¯ and R chart showed that MaxEWMA chart showed a better performance for small shifts. The comparison with a combination of an EWMA mean chart and an EWMA variability chart showed that MaxEWMA chart is better when the variability increases and there is shift in the process mean. The diagnostic abilities of the MaxEWMA chart are also better than the competitive charts.

A fully adaptive X¯ and R chart is presented in Costa (1998) and Costa (1999). When the joint X¯ and R charts with variable parameters (Vp) are used, the sample means are plotted on the X¯ control chart with warning limits μ0±wσX¯, and action limits μ0±kσX¯, where 0 < w < k and σX¯ is the standard deviation of the sample means. The ranges from samples of size ni, i = 1,2, are plotted on the R control chart with warning and action limits given by wR(ni )σ0 and kR(ni)σ0, respectively, where wR(ni) < kR(ni). Here wR (ni) and kR(ni) are numbers of standard deviations of relative range defined by Rσ. The size of the sample, the sampling interval, and the width of X¯ and R chart action limits depend on the results of the preceding sample statistic. If sample statistic of both charts is in the central region, then the size of the next sample denoted by n1 should be small and sampling interval large. If the sample statistic of either of the charts is within warning action limits, then the next sample size, denoted by n2, should be large and sampling interval small.

Costa and Rahim (2002) developed a joint X¯ and R chart with two-stage sampling (TSS). As in the case with Shewhart charts, samples of size n0 are taken from the process at regular time intervals. At the first stage, one item of the sample is inspected. If its X value is close to the target then the sampling is interrupted. Otherwise the sampling goes on to the second stage, and the remaining n0  1 items are inspected and the X¯ and R values are computed based on the whole sample size n0.

Other related joint charts include the X¯ and variance charts by Rahim et al. (1988) and Lorenzen and Vance (1986).

The work of Daudin (1992) and He and Grigoryan, 2002, He and Grigoryan, 2003 has provided an incentive for the development of the joint DS X¯ and s charts. In Daudin (1992), a DS X¯ chart was developed to improve the statistical efficiency (in terms of ARL) without increased sampling, or alternatively, to reduce the sampling without reducing the statistical efficiency. Daudin compared the statistical efficiency of the DS X¯ chart with other procedures. The results of the comparison are summarized as follows. In comparison with the Shewhart X¯ chart, the DS X¯ chart showed a large improvement in efficiency. The results of comparing with the VSI chart showed that when the time required to collect and measure the samples is negligible, the DS scheme is more efficient; otherwise the VSI chart is better. In comparison with the EWMA chart, the DS chart has a greater efficiency than the EWMA chart with r = 0.75. However, when r = 0.5 or 0.25, the EWMA chart is better for detecting small shifts, and the DS chart is better for detecting large shifts. In comparison with the two-sided combined Shewhart CUSUM schemes, the DS chart provides a better protection against large shifts while the Shewhart CUSUM schemes are better for detecting small shifts. Daudin suggested that the DS X¯ chart could be substituted for the Shewhart X¯ chart whenever the improvement in efficiency outweighs the administration cost associated with the sampling. He and Grigoryan, 2002, He and Grigoryan, 2003 extended the concept of the DS X¯ chart scheme developed by Daudin (1992) to monitoring of process variability and developed a DS s chart scheme. The efficiency of the DS s charts was compared with that of the traditional s charts. The results of the comparison show that for the whole range of the investigated shifts (0.6  λ  5) in process standard deviation the DS s charts result in a significant reduction in average sample size without decreasing the out-of-control ARL in comparison with the traditional s charts.

Because of the efficiency of the DS X¯ charts in detecting shifts in process mean and the DS s charts in process standard deviation it seems reasonable to investigate the joint DS X¯ and s charts for simultaneously monitoring of process mean and variance. Up to today, no research on the investigation of the joint DS X¯ and s charts has been reported.

In general, three design criteria are used for joint statistical design of X¯ and R charts: statistical design, economic design, and heuristic design (Saniga, 1991). As pointed out by Woodall (1986), the economic design method is often not effective in producing charts that can quickly detect small shifts before substantial losses can occur. Also discussed in Montgomery (1980), one of the problems associated with the economic design is the difficulty in estimating the costs. For these reasons, the statistical design method is chosen in this paper for the joint design of the DS X¯ and s charts. In joint statistical design of the DS charts, we are interested in finding the control limits of the joint DS X¯ chart and s charts as well as a joint sample size such that if there is no shift in the process, the control charts are unlikely to signal an assignable cause, while shifts of specified magnitude are likely to be detected.

Section snippets

The joint DS X¯ and s charts

Throughout the paper, it is assumed that the joint DS X¯ and s charts are used to monitor a process where independent observations from the quality characteristic of interest, X, are normally distributed with a mean μ0 and a variance σ02. Further, it is assumed throughout the paper that when the process is out of control, it is due to a shift in mean from μ0 to μ1 = μ0 ± δσ0, with δ > 0, and/or a shift in the process standard deviation from σ0 to σ1 = λσ0. The proposed joint DS X¯ and s chart scheme is

Formulation of joint statistical design of the DS X¯ and s charts

In this paper, the joint statistical design of the DS X¯ and s charts is formulated as a design optimization problem. Before the optimization model is presented, the following intervals and notations are defined:I11=μ0-L1σn1,μ0+L1σn1,I12=μ0-Lσn1,μ0-L1σn1μ0+L1σn1,μ0+Lσn1,I13=-,μ0-Lσn1μ0+Lσn1,+,I14=μ0-L2σn1+n2,μ0+L2σn1+n2,I15=-,μ0-L2σn1+n2μ0+L2σn1+n2,+,I21=c41σ-D1σ1-c412,c41σ+D1σ1-c412,I22=c41σ-Dσ1-c412,c41σ-D1σ1-c412c41σ+D1σ1-c412,c41σ+Dσ1-c412,I23=-,c41σ-Dσ1-c412c41σ+Dσ1-c412,,I24=c4σ

Solving the optimization problem using genetic algorithm

As one can see that the joint DS X¯ and s chart design optimization problem formulated by model (10), (11), (12), (13), (14) is characterized by mixed continuous-discrete variables, and discontinuous and non-convex solution space. Therefore, if standard nonlinear programming techniques are used for solving this type of optimization problem they will be inefficient and computationally expensive. Genetic algorithms (GAs) are well suited for solving such problems, and in most cases, find a global

Comparison with combined EWMA chart, combined CUSUM chart, and omnibus EWMA chart

In order to evaluate the performance of the joint DS X¯ and s charts, we compared the joint DS charts with the combined EWMA and CUSUM schemes presented in Gan (1995), and the omnibus EWMA scheme proposed by Domangue and Patch (1991). Particularly, the comparison was conducted for subgroup size of 5. The shifts in process mean in Gan (1995) were measured in number of standard deviations of the subgroup mean distribution, i.e., Δ=μ1-μ0σ0/n or δ=Δ/n, where σ0 is the in-control process standard

Conclusions

In this paper, a joint DS X¯ and s chart scheme is proposed and its joint statistical design presented. The design of the joint DS X¯ and s charts is formulated as an optimization problem which minimizes the out-of-control ARL given the Type I error probability constraint and average sample size constraint when the process is in control.

The statistical performance of the proposed joint DS X¯ and s chart scheme measured by the out-of-control ARL was compared with that of the combined EWMA and

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