Int J Adv Manuf Technol (2008) 39:261–270 DOI 10.1007/s00170-007-1213-7
ORIGINAL ARTICLE
Monitoring process variability using exponentially weighted moving sample variance control charts Majid Eyvazian & S. G. Jalali Naini & A. Vaghefi
Received: 1 August 2006 / Accepted: 15 August 2007 / Published online: 26 September 2007 # Springer-Verlag London Limited 2007
Abstract Exponentially weighted moving average (EWMA) control charts are regarded as one of the most convenient tools in detecting small process shifts. Although EWMA control charts have been extensively used to monitor the mean of quality characteristics, there are few studies concentrating on the monitoring of process variability by using weighted moving control charts. In this paper, we propose an exponentially weighted moving sample variance (EWMSV) control chart for monitoring process variability when the sample size is equal to 1. The results are compared numerically with other similar methods using the average run length (ARL). Through an example, the practical considerations are presented to implement EWMSV control charts. Keywords Average run length (ARL) . Exponentially weighted moving sample variance (EWMSV) control charts . Smoothing parameter . Statistical process control (SPC) Abbreviations EWMA Exponentially weighted moving average EWMSV Exponentially weighted moving sample variance ARL Average run length CUSUM Cumulative sum M. Eyvazian : S. G. Jalali Naini : A. Vaghefi (*) Industrial Engineering Department, Iran University of Science and Technology, Tehran 16844, Iran e-mail:
[email protected] M. Eyvazian e-mail:
[email protected] S. G. Jalali Naini e-mail:
[email protected]
EWMV EWMS EWMAMR MSt2 μ, σ2 xt St 2 1 ν UCL 0 0 d 2; d 3 F5 , F 6 MRt μMR H δ SDRL
Exponentially weighted moving variance Exponentially weighted mean squared Exponentially weighted moving average moving range Exponentially weighted mean squared statistics at the current time t Mean and variance of quality characteristics, respectively Sample mean of t last individual observations Exponentially weighted moving sample variance statistics at the current time t Smoothing parameter Degrees of freedom for chi-squared distribution Upper control limit Constant values, which are equal to 1.128 and 0.853, respectively Weighting factors of control limits Moving range statistics at the current time t Mean of moving ranges MRt's Threshold value Magnitude of a shift Standard deviation of run length
1 Introduction Since Dr. Walter A. Shewhart introduced control charts, this technique has been used extensively to monitor product and process quality characteristics in many applications. Although Shewhart-type control charts have been regarded as the most commonly used charts, the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts have recently received a great deal of attention in different applications. EWMA and CUSUM
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control charts have a more appropriate performance in detecting small and moderate shifts in the process mean. Many researchers have contributed to the theoretical and practical use of the EWMA control charts. Roberts [16] introduced a control chart using the exponentially weighted moving average of the sample mean and called it “the geometric moving average chart.” Robinson and Ho [17], Crowder [5, 6], Hunter [10], and Lucas and Saccucci [11] proposed some notes and schemes in order to facilitate the use of these charts in monitoring the process mean. Although EWMA control charts have been appropriately developed in detecting step shifts in the process mean, there are only a few works that address monitoring process variability using these charts. Sweet [18], Wortham [20], Wortham and Ringer [19], Amin et al. [1], and Domangue and Patch [8] proposed control charts for monitoring process variability based on EWMA control charts. Ng and Case [14] and Gan [9] used two EWMA control charts to detect both the mean and variance process shifts at the same time. Chen et al. [4] proposed a single EWMA control chart, which can be used to detect the mean and variance process shifts, as well as the direction of the shifts. Crowder and Hamilton [7] used an appropriate log transformation of the sample variance, S2, and developed EWMA control charts for monitoring the process standard deviation. They also presented an optimal design strategy based on their proposed EWMA control chart. Ng and Case [14] developed and evaluated different EWMA control charts for both situations, whether the observations are individual or may be replicated. MacGregor and Harris [12] investigated the ability of the exponentially weighted mean squared (EWMS) and the exponentially weighted moving variance (EWMV) in the cases of independence and autocorrelation of the observations. They also constructed control charts based upon EWMS and EWMV, which are useful for monitoring the variability of individual observations. Castagliola [3] developed a new S2-EWMA control chart to monitor sample variance in the case n>1. Their method is a direct extension of Crowder and Hamilton’s [7] approach by using a Johnson family transformation of the sample variance S2. In this paper, a new method is developed based on the exponentially weighted moving sample variance (EWMSV), which is a generalized form of S2. Then, an EWMSV control chart is proposed in order to monitor the process standard deviation of individual observations. An application procedure is then suggested for implementing the proposed method. In the next section, EWMS control charts proposed by MacGregor and Harris [12] and exponentially weighted moving average moving range (EWMAMR) control charts proposed by Ng and Case [14], together with our new approach, are introduced and the upper limits for all of
Int J Adv Manuf Technol (2008) 39:261–270
these methods are shown. In Section 3, the performances of the control charts are compared using the average run length (ARL) criterion by using a simulation method. Some practical considerations are discussed in Section 4 and an application procedure is proposed.
2 Monitoring process variability for individual observations In many cases, such as continuous processes, low-rate productions, and automated inspections, the sample size used for monitoring the process mean and variability is equal to 1. Shewhart-type control charts have been extensively developed for individual observations in many cases (see Montgomery [13]). EWMA control charts have more appropriate properties than Shewhart-type control charts in both univariate and multivariate cases of monitoring individual observations. EWMA control charts can be designed to be less sensitive to the effect of non-normality and autocorrelation than moving range control charts. Despite the ability of control charts based on the exponentially weighted moving average, there are few studies directly concentrating on the monitoring of process variability, especially when the sample size is equal to 1. In this section, two existing methods will be discussed and the new approach will be proposed. 2.1 Exponentially weighted mean squared control charts The use of the exponentially weighted mean squared (EWMS) in monitoring process variability was first suggested by Wortham and Ringer [19] and then developed by MacGregor and Harris [12] for both cases where observations are independent and autocorrelated. In our study, we assume that the individual observations xt are univariate process or product quality characteristics, which are collected successively over time. They are independent and identically distributed normal random variables with mean μ and variance σ2. The exponentially weighted mean squared at the current time t is defined as 2 MSt2 ¼ ð1 lÞMSt1 þ lðxt mÞ2
ð1Þ
Where the multiplier 1 (0
