Control Chart for Monitoring the Coefficient of Variation with an Exponentially Weighted Moving Average Procedure Jiujun Zhang1;2, Zhonghua Li2, and Zhaojun Wang2∗ 1Department of Mathematics, Liaoning University, Shenyang 110036, P.R.China 2Institute of Statistics and LPMC, Nankai University, Tianjin 300071, P.R.China Abstract The coefficient of variation (CV) of a population is defined as the ratio of the population standard deviation to the population mean, which can be regarded as a measure of stability or uncertainty, and can also indicate the relative dispersion of data to the population mean. This paper proposes a new exponentially weighted moving average (EWMA) chart for monitoring CV, which is constructed by truncating those negative normalized observations to zero in the traditional EWMA CV statistics. The implementation and optimization procedures of the proposed chart are presented. The new chart is compared with some existing CV charts by means of average run length (ARL), and the comparison results show that the new chart outperforms other charts in most cases. Two examples illustrate the use of this chart on real data gathered from a metal sintering process and from a die casting hot chamber process. Key words: control charts; statistical process control; coefficient of variation; expo- nentially weighted moving average; reflecting boundary. 1 Introduction Ever since Shewhart introduced the term of control charts, it has become a common practice for practitioners to use various control charts to monitor different processes ([1],[2]). When we deal with variable data, the charting technique usually employs one chart to monitor the process mean and another chart to monitor the process variance. The Shewhart X and S (or R) charts are industry standards for quality control applications where the mean 휇 and the standard deviation 휎 of a process must be statistically controlled at the nominal values 휇0 and 휎0. The baseline assumption is that the nominal values are fixed constants, ∗Corresponding author, email: [email protected] 1 and there are many applications for which this assumption is reasonable. To this end, it is reasonable to monitor the process mean and variance simultaneously by a single chart, see [3]-[10]. [11] presented an overview of joint monitoring of mean and variance. However, it is important to point out that not all processes have constant means or constant standard deviations that enable control charts for the mean or the standard deviation to be used for process monitoring. A common relationship is that the standard deviation is proportional to the mean so that the ratio of the standard deviation to the mean is a constant. This ratio is referred to as the coefficient of variation (CV). The CV, which is the ratio of the standard deviation to the mean, is a dimensionless measure of dispersion found to be very useful in many situations. In chemical experiments, the CV is often used as a yardstick of precision of measurements; two measurement methods may be compared on the basis of their respective CVs. In finance, it is interpreted as a measure of the risk faced by investors, by relating the volatility of the return on an asset to the expected value of the return. For example, it can be used as a measure of relative risks ([12]) and a test of the equality of the CVs for two stocks can help determine whether the two stocks possess the same risk or not. [13] used the CV to assess the homogeneity of bone test samples produced from a particular method to help assess the effect of external treatments, such as irradiation, on the properties of bones. [14] used the CV in the analysis of fault trees. The CV was also employed by [15] in assessing the strength of ceramics. It can also be used in the fields of materials engineering and manufacturing, where some quality characteristics related to the physical properties of products constituted by metal alloys or composite materials often have standard-deviations that are proportional to their population means. These properties are usually related to the way atoms of a metal diffuse into another. Tool cutting life and several properties of sintered materials are typical examples from this setting. Thus, for these quality characteristics, monitoring the CV using a control chart has gained remarkable attention in recent years. The first control chart for monitoring the CV, i.e., the Shewhart CV chart was developed by [16]. It was shown that this chart is useful for monitoring the process CV. Subsequently, the exponentially weighted moving average (EWMA) chart for the CV was proposed by [17] to improve the performance of Shewhart CV chart for detecting small shifts in the CV. Furthermore, the two one-sided EWMA CV squared charts were proposed by [18] (denoted as OSE chart), respectively. The OSE chart consists of a downward and an upward one-sided EWMA CV squares charts to detect decreases and increases in the CV, respectively. In general, it was shown that the OSE chart produces slightly smaller out-of-control (OC) average run length (ARL) than that of the EWMA CV chart proposed by [17]. 2 Recently, [19] developed a synthetic control chart for monitoring the CV (denoted as SYN chart). The results showed that the synthetic chart performed better than that of [16], but worse than [18] as long as the increasing shift in the CV is not too large. [20] evaluated an adaptive Shewhart control chart implementing variable sampling interval (VSI) strategy to monitor the process CV. [21] proposed a Shewhart chart with supplementary run rules to monitor the process CV (denoted as SRR chart). However, as they pointed out, the SRR chart does not outperform more advanced strategies as the OSE chart or the SYN chart. [22] developed a modified OSE CV chart (denoted as MOSE chart) based on the preliminary work of [18]. The comparisons showed that the MOSE chart has an ARL performance that is superior to some other competing procedures. In short production runs, [23] proposed a Shewhart chart and [24] proposed a variable sample size (VSS) control chart. [25] developed a side sensitive group runs chart (denoted as SSGR chart) and compared with some of the existing CV charts in terms of ARL and expected ARL (EARL). [26] proposed a one-sided run rules control charts for monitoring the CV in short production runs and used Markov chains to get their main statistical properties. [27] developed a procedure to monitor the CV using run-sum control charts. The run-length properties of the charts are characterized by the Markov chain approach. In addition, concerning the studies on the control charts that monitor the multivariate CV, the first control chart for the multivariate CV is Shewhart-type chart, which was presented by [28]. Next, [29] proposed the run sum chart for monitoring multivariate CV in the Phase-II process. In this paper, a new chart is proposed to further improve the performance of EWMA chart by applying a resetting scheme for monitoring the CV. This resetting technique was originally proposed by [30] for monitoring the process variance. With resetting boundaries, [31] obtained necessary and sufficient conditions for non-interaction of a pair of one-sided EWMA schemes. For our proposed chart, sets of optimal design parameters are provided for different values of the in-control CV, for different sample sizes, and for a wide range of deterministic shifts, including both decreasing and increasing cases. The new chart is compared with most of the existing competing charts, including the EWMA type charts, i.e., the OSE and MOSE charts, and the SYN, SRR and SSGR charts, respectively. The underlying process is assumed to follow a normal distribution. The rest of this paper is structured as follows: Section 2 gives an overview of the existing competing CV charts, as these charts are also considered in the performance comparisons. In Section 3, our proposed resetting EWMA chart (denoted as RES chart) is presented and the statistical performance of the new chart is investigated. The numerical comparisons with some other procedures are carried out in Section 4. The application of our proposed method is illustrated in Section 5 by two real data examples from a metal sintering process control 3 and from a die casting hot chamber process. Several remarks conclude this paper in Section 6. 2 An Overview of Some Existing CV Charts Suppose that we observe subgroups Xk = fXk1;Xk2;:::;Xkng of size n at times k = 1; 2;:::. We also assume that there is independence within and between these subgroups and each random variable Xkj follows a normal N(휇k; 휎k) distribution, where the parameters 휇k and 휇k 휎k are constrained by the relation 훾k = = 훾0 when the process is in control. This implies 휎k that, from one subgroup to another, the values of 휇k and 휎k may change, but the coefficient 휇k of variation 훾k = must be equal to some predefined in-control value 훾0, common to all 휎k the subgroups. Let X¯ and S be the sample mean and the sample standard deviation of X , i.e., X¯ = P k k q P k k 1 n 1 n ¯ 2 Sk Xki and Sk = (Xki − Xk) : The sample CV훾 ^k is defined as훾 ^k = ¯ n i=1 n−1 i=1 Xk and its distribution has been extensively studied in the literature. [32] noted that when p the parent population distribution is normal, n has a non-central t-distribution with n − 1 훾^k p degrees of freedom and non-centrality parameter n . 훾 Next, we give a brief review of some existing CV charts, including two competing EWMA type charts, i.e., OSE and MOSE charts, and then the SYN, SRR and SSGR charts, as these five charts are also considered in the performance comparisons given in Section 4.