 # Monitoring Coefficient of Variation using One-Sided Run Rules control charts in the presence of Measurement Errors

We investigate in this paper the effect of the measurement error on the performance of Run Rules control charts monitoring the coefficient of variation (CV) squared. The previous Run Rules CV chart in the literature is improved slightly by monitoring the CV squared using two one-sided Run Rules charts instead of monitoring the CV itself using a two-sided chart. The numerical results show that this improvement gives better performance in detecting process shifts. Moreover, we will show through simulation that the precision and accuracy errors do have negative effect on the performance of the proposed Run Rules charts. We also find out that taking multiple measurements per item is not an effective way to reduce these negative effects.

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## 1 Introduction

Among important statistical characteristics of a variable, the coefficient of variation (CV) is widely used to evaluate the stability or concentration of the random variable around the mean. It is defined as the ratio between the standard deviation to the mean,

. In many industrial processes, keeping the value of this coefficient of a characteristic of interest within the permissible range means assuring the quality of products. A number of examples have been illustrated in the literature for the applications of the CV in industry.  presented an example where the quality of interest is the pressure test drop time from a sintering process manufacturing mechanical parts. In this example, the presence of a constant proportionality between the standard deviation of the pressure drop time and its mean was confirmed. The CV was then monitored to detect changes in the process variability.  showed that it is useful to monitor the CV in detecting the presence of chatter, a severe form of self-excited vibration in the machining process which leads to many machining problems. More examples about the need of using the CV as a measure of interest has been discussed in . Because of its wide range of applications, monitoring the CV is a major objective in many studies in statistical process control, see, for example, ,, , and .

Along with the development of the advanced control charts monitoring the CV with improved performance, recent researches are paying attention to the effect of the measurement error on the CV control chart. This makes these researches become more in touch reality since the measurement error is often present in practice. A Shewhart control chart monitoring the CV under the presence of measurement error (ME) was suggested by .  improved the linear covariate error model for the CV in  and then proposed the EWMA CV control chart with ME. Very recently,  and  studied the effect of ME on the variable sampling interval control chart and the cumulative sum control chart monitoring the CV, respectively.

One of the reasons that leads to the introduction of many advanced control charts monitoring the CV is to overcome a drawback of the Shewhart CV chart that it is not sensitive to the large shifts. However, the Shewhart chart is still popularly used thanks to its simplicity in implementation. From this point of view, the Run Rules charts are advantageous: they are easy to implement (compared to, for example, the EWMA control chart or the CUSUM control chart, even these charts may bring better performance) and they can improve remarkably the performance of the Shewhart chart in detecting small or moderate process shifts. The aim of this paper is to investigate the performance of Run Rules CV control chart under the presence of ME. In fact, the Run Rules chart monitoring the CV has been studied in . However, the ME has not been considered. Moreover, in this study the authors only focused on the two-sided charts (the one-sided chart has been mentioned, but quite sketchily without explanation for the design) with the CV monitored directly. We improve this design by monitoring the CV squared and presenting the design of the two one-sided Run Rules charts in detail.

The paper consists of eight sections and is organized as follows. Followed by the introduction in Section 1, Section 2 presents a brief review of the distribution of the sample coefficient of variation. The design and the implementation of two one-sided Run Rules control charts monitoring the CV squared (denoted as RR charts) are presented in section 3. Section 4 is for the performance of these charts. A linear covariate error model for the CV is reintroduced in section 5. The design of control charts in the presence of measurement errors and the effect of the measurement error on the RR charts are displayed in section 6. Section 7 is devoted to an illustrative example. Some concluding remarks are given in section 8 to conclude.

## 2 A brief review of distribution of the sample coefficient of variation

In this section, the distribution of the CV is briefly presented. The CV of a random variable , say , is defined as the ratio of the standard deviation to the mean ; i.e.,

 γ=σμ.

Suppose that a sample of size of normal i.i.d. random variables is collected. Let and be the sample mean and the sample standard deviation of these variables, i.e.,

 ¯X=1nn∑i=1Xi

and

 S= ⎷1n−1n∑i=1(Xi−¯X)2.

Then the sample coefficient of variation of these variables is defined as

 ^γ=S¯X.

The probability distribution of the sample CV

has been studied in the literature by many authors. However, the exact distribution of has a complicated form. The approximate distribution is then widely used as an alternative. In this study, we apply the following approximation of , the , which is suggested by :

 F^γ(x|n,γ)=1−Ft(√nx∣∣∣n−1,√nγ), (1)

where is the c.d.f. of the noncentral distribution with degrees of freedom and noncentrality parameter. This approximation is only sufficiently precise when . This condition is in general satisfied in our case as it is very frequent that the CV takes small values to ensure the stability of a process. More details on this problem have been discussed in .

For the case of the sample CV squared (), Castagliola et al.  showed that follows a noncentral distribution with degrees of freedom and noncentrality parameter . Then, they deduced the of as

 F^γ2(x|n,γ)=1−FF(nx∣∣1,n−1,nγ2), (2)

where is the of the noncentral distribution. The corresponding density function of is then

 f^γ2(x|n,γ)=nx2fF(nx∣∣1,n−1,nγ2), (3)

where is the density function of the noncentral distribution

Figure 1 presents the density distribution of for and some different values of .

## 3 Design and implementation of the RRr,s−γ2 control chart

In the literature, the Run Rules control charts monitoring the CV has been investigated in  with two-sided charts. However, since the distribution of is asymmetric (as can be seen from the equation (2) and also from Figure 1), these two-sided charts lead to the problem of -biased (Average Run Length) performance, i.e. the out-of-control values are larger than the in-control values . This problem was also pointed out in . Therefore, we overcome this -biased property by designing simultaneously two one-sided charts to detect both the increase and the decrease at the CV squared. In particular, we suggest defining two one-sided Run Rules control charts monitoring the CV squared, involving:

• a lower-sided -out-of- Run Rules control chart (denoted as RR) to detect a decrease in with a lower control limit and a corresponding upper control limit ,

• an upper-sided -out-of- Run Rules control chart (denoted as RR) to detect a decrease in with a lower control limit and a corresponding lower control limit ,

where and are the chart parameters of the RR and RR charts, respectively.

The closed forms of and have not been presented in the literature. We apply in this study the accurate approximations provided by Breunig  for both and as follows:

 μ0(^γ2) = γ20(1−3γ20n), (4) σ0(^γ2) =  ⎷γ40(2n−1+γ20(4n+20n(n−1)+75γ20n2))−(μ0(^γ2)−γ20)2. (5)

Given the value of the control limit for each chart, an out-of-control signal is given at time if -out-of- consecutive values are plotted outside the control interval, i.e. in the lower-sided chart and in the upper-sided chart. The control charts designed above is called Run Rules type chart. In this study, we only consider the 2-out-of-3, 3-out-of-4 and 4-out-of-5 Run Rules charts. The performance of the proposed charts is measured by the which is calculated by using Markov chain as follows.

Firstly, we define the matrix of the embedded Markov chain. For the two one-sided RR control charts, is defined by

 P=⎛⎜⎝Qr0T1⎞⎟⎠=⎛⎜ ⎜ ⎜ ⎜⎝00p1−pp001−p01−pp00001⎞⎟ ⎟ ⎟ ⎟⎠, (6)

where is a matrix of transient probabilities, is a vector satisfied with and , is the probability that a sample drops into the control interval. The corresponding vector of initial probabilities associated with the transient states is , i.e. the third state is the initial state.

For the case of RR control charts, the transient probability matrix is given by

 Q=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝00p00000000p00000001−ppp00000001−pp00000001−pp00000001−pp⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (7)

In this case, the seventh state in the vector is the initial state.

Extended to “longer” Run Rules, the matrix of transient probabilities for the two one-sided RR control charts is

 Q=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1−pp00000p000000000p1−p000000000000000p1−p0000000000000001−pp0000000000000000000001−p000000000000p1−p000000000000p1−p0000$0$00001−pp0000000000001−pp00000000000p1−p00000000000p1−p000000000001−pp00000000000000000000001−pp000000000000000p1−p000000000000000p1−p⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (8)

that corresponds to the initial probabilites vector (i.e. the initial state is the 15th one). These transient probability matrices has been presented in, for example, , , .

Let us now suppose that the occurrence of an unexpected condition shifts the in-control CV value to the out-of-control value , where is the shift size. Values of correspond to a decrease of the , while values of correspond to an increase of . Then, the probability is defined by

• for the RR chart:

 p=P(^γ2i≥LCL−)=1−F^γ2(LCL−|n,γ1), (9)
• for the RR chart:

 p=P(^γ2i≤UCL+)=F^γ2(UCL+|n,γ1), (10)

where is defined in (2).

Once the matrix and the vector have been determined, the and the (standard deviation of run length) are calculated by

 ARL = qT(I−Q)−11, (11) SDRL = √2qT(I−Q)−2Q1−ARL2+ARL. (12)

It is customary that a control chart is considered to be better than its competitors if it gives a smaller value of the while the is the same. Therefore, the parameters of the RR control charts should be the solution of the following equations:

• for the RR chart:

 ARL(kd,n,p,γ0,τ=1)=ARL0, (13)
• for the RR chart:

 ARL(ku,n,p,γ0,τ=1)=ARL0, (14)

where is predefined.

## 4 Performance of RRr,s−γ2 control charts

Assigning the in-control value at , the parameters and of the lower-sided and upper-sided RR charts for some combinations of are presented in Table 2. Table 3 shows the corresponding values of the proposed charts for various situations of the shift size . The obtained results show that the two one-sided RR charts not only overcome the -biased problem (as the values are always smaller than the ) but also outperform the two-sided RR- charts investigated in . For example, with and in the RR chart, we have ( Table 3 in this study), which is smaller than (Table 2 in ).

## 5 Linear covariate error model for the coefficient of variation

The previous design for the RR control charts is based on a latent assumption that the values in the collected sample are measured exactly without the measurement error. This assumption, however, is usually not reached in practice and it is difficult to avoid the measurement error. That leads to many authors have conducted their studies based on the measurement error presence, see for instance [8, 15]

In this section, we suppose a linear covariate error model to the measurement error, which is suggested by .

Suppose that the quality characteristic of consecutive items at step i is , where where and are the in-control mean and standard deviation of and they are independent. The parameters and represent the standardized mean and standard deviation shift. The process has shifted if the process mean and/or the process standard deviation have changed ( and/or ). Due to the measurement error, we only observe the values of a set of measurement operations instead the true values . According to the linear covariate error model, we assume , where and are two known constants and is a normal random error term with parameters and independent of .

Let denote the mean of observed quantities of the same item at the i sampling. It is straightforward to show that

 ¯X∗i,j ∼ N(μ∗,σ∗2)=N(A+B(μ0+aσ0),B2b2σ20+σ2Mm).

 showed that the CV of the quantity is

 γ∗=σ∗μ∗=√B2b2+η2mθ+B(1+aγ0)×γ0. (15)

where and are the in-control value of CV, the precision error ratio and the accuracy error, respectively. The sample coefficient of variation is defined by where and are the sample mean and the sample standard deviation of . The of can be obtained from (2) by simply replacing by , i.e., the of is given by

 F^γ∗2(x|n,γ∗)=1−FF(nx∣∣∣1,n−1,nγ∗2) (16)

## 6 Implementation and the performance of the RRr,s−γ2 charts with measurement errors

Under the presence of measurement errors, the values and are calculated as in (4) and (5), where is replaced by , which is defined from (15) with and :

 γ∗0=√B2+η2mθ+B)×γ0. (17)

Suppose that the in-control value is shifted to the out-of-control value with the size , we can represent according to and as . Therefore, the out-of-control CV of the observed quantity can be expressed by

 γ∗1=√B2b2+η2mθ+Bbτ×γ0. (18)

In the implementation of RR control charts, the control limits, and , are also found by solving the chart parameters and as the solution of the following equations

• for the RR chart:

 ARL(kd,n,p,γ0,θ,η,m,B,b)=ARL0, (19)
• for the RR chart:

 ARL(ku,n,p,γ0,θ,η,m,B,b)=ARL0. (20)

The in (19) and (20) should be calculated with the transition probability matrix where the transition probability is defined from (9) and (10) but with the of in (16) instead of in (2).

To investigate the performance of the RR charts under the appearance of the measurement error, we consider several possible values of the parameters: , , , , and . Without loss of generality, we assume in the remaining that . The in-control value CV is also set at .

The control limits of the proposed charts for some specific values of these parameters have been presented in Table 1. The other values of the control limits for other situations of these parameters are not presented here but are available upon request from authors.

Tables 4-7 show the corresponding values of the under different effects of the parameters and of the linear covariate model. Some simple conclusions can be drawn from these tables as follows.

• The increase of the precision error ratio leads to an increase of the . However, this increase in the following the change of is not significant, especially when . For example, for the RR chart with and , we have when and when (Table 4). That means the precision error ratio does not affect much the performance of the proposed charts.

• The accuracy error has a negative impact on the RR charts’ performance: the larger the accuracy error is, the larger the value is, i.e. the lower of the control chart is in detecting the out-of-control condition. For example, in the RR chart with and , we have when and when (Table 5)

• Given the value of other parameters, the variation of significantly affects the performance of the RR charts. For instance, in Table 6 with the RR control chart and we have when and when .

• In many situations, taking multiple measurements per item in each sample is an alternative to compensate for the effect of the measurement error. However, the obtained results in this study show that this is not effective way. When increases from to , the decreases trivially or is almost unchanged. For example, with in the RR we have for both and (Table 7).

• In general, the RR control charts give better performance in detecting the small process shifts compared to the VSI- control chart investigated in , under the same condition of measurement errors. For example, with the same values of , we have for the RR (Table 5 in this study), which is smaller than for the VSI control chart with (Table 10 in ).

In practice, quality practitioners often prefer detecting a range of shifts since it is difficult to guess an exact value for the process shift. In such situations, the statistical performance of the control chart can be evaluated through the (expected average run length) defined as

 EARL=∫ΩARL×fτ(τ)dτ, (21)

where is the distribution of process shift and is defined in (11). Without any information about

, one can choose the uniform distribution in

, i.e, .

The chart parameters are now defined as

• for the RR chart:

 EARL(LCL∗−,n,p,γ0,θ,η,m,B)=ARL0, (22)
• for the RR chart:

 EARL(UCL∗+,n,p,γ0,θ,η,m,B)=ARL0. (23)

In the following simulation, we consider a specific range of decreasing shifts and increasing shifts . Figure 2-3 show the change of of the RR- control charts when varies in and varies in for and , respectively. The slope of the plane which represents the values from right to left and from outside to inside shows that the larger the values of and , the larger the value of . That is to say, these errors have negative effects on the performance of the RR- charts. For example, in Figure 2 when , and , we have for (corresponding to no measurement errors), but for (corresponding to the negative effect of accuracy error), for (corresponding to the negative effect of precision error), and for (corresponding to the negative effect of both precision and accuracy error). The effect of and on the is displayed in Figures 4-7 for both and . We obtain a similar trend as the case of the specific shift size: When increases, the decreases and the does not change significantly when increases. The almost constant line shows that the effect of on these chart performance is insignificant. That is to say, increasing the value of does not reduce the negative effect of measurement errors on the charts. In contrast, the plot of the corresponding to is always below the plot of the corresponding to . That means, the sample size has a great impact on the RR charts’ performance regardless of the measurement error.

## 7 Illustrative example

In this section, we present an illustrative example of the implementation of the RR control charts in the presence of the measurement error. The real industrial data from a sintering process in an Italian company that manufactures sintered mechanical parts, which were introduced in , are considered.

The process manufactures parts guarantee a pressure test by dropping time from 2 bar to 1.5 bar larger than 30 sec as a quality characteristic related to the pore shrinkage. Since the presence of a constant proportionality between the standard deviation of the pressure drop time and its mean had been demonstrated by the preliminary regression study relating to the quantity of molten copper, the quality practitioners decide to monitor the CV to detect changes in the process variability. According to the description in 

, an estimate

is calculated from a Phase I dataset based on a root mean square computation. The phase II data are reproduced in Table 8.

According to  under the presence of the measurement error, we suppose that the parameters of the linear covariate error model are , , , and . Based on the process engineer’s experience, a specific shift size was expected to detect from the process. Therefore, we apply the upper-sided RR control chart to monitor the CV squared. The control limits of the RR, RR and RR chart are found to be , and , respectively. The values of are then plotted in these charts (Figure 8) long with the control limit . For purpose of comparison, we also draw the control limit ()of the original Shewhart control chart with the same parameters.

As can be seen from the Figure 8, the RR, RR and RR chart signal the occurrence of the out-of-control condition by two-out-of-three, three-out-of-four, and four-out-of-five (respectively) successive plotting points above the corresponding control limits from the sample #12. Meanwhile, the Shewhart chart fails to detect this out-of-control condition.

## 8 Concluding remarks

In this paper, the performance of Run Rules control charts is improved slightly by monitoring the CV squared with the two one-sided charts rather than monitoring directly the CV with a two-sided chart as in a previous study in the literature. The effect of measurement errors on the performance of the RR control charts using a linear covariate error model is also investigated. We have pointed out the negative effect of measurement errors on the proposed charts: the increase of and leads to the increase of . Moreover, the obtained results show that measuring repeatedly is not an efficient method for limiting these unexpected effects HANH HEO.

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