Xuelong Hu

Philippe Castagliola
email: philippe.castagliola@univ-nantes.fr

Anan Tang
email: tanganan2@sina.com

Xiaojian Zhou

Conditional design of the Shewhart X

Keywords: Shewhart X chart, median run length, estimated parameters, average median run length, Standard deviation of median run length

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Control charts are one of the most common used tools in Statistical Process Control (SPC). When monitoring the mean of a quality characteristic, such as a dimension, weight or volume, the Shewhart X chart is usually used to detect assignable cause(s) in the quality characteristic. This chart is especially useful to monitor relatively large shifts in the process mean (see [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF]).

In practice, there are generally two phases (Phases I and II) in the implementation of control charts. In Phase I, an in-control reference set of observations is collected by practitioners to estimate the unknown process parameters, which are then used to set the control limits of a Phase II control chart. In Phase II, samples from the process are prospectively collected, and are plotted on the control chart for detecting assignable cause(s) in the process. A Phase II control chart's performance is usually evaluated by assuming known process parameters but, in practice, the process parameters are often unknown and are estimated from an in-control reference set of observations. Under this case, a inherent variability in these estimated process parameters is encountered because different practitioners will use different Phase I samples (or different Phase I observations). Thus the control chart's performance will vary among practitioners. This leads to the between-practitioners variability effect in the properties of control charts (see Saleh, Mahmoud, [START_REF] Keefe | The Conditional In-Control Performance of Self-Starting Control Charts[END_REF]).

Control charts' performance are usually assessed through various properties of the run length (RL) distribution, where RL is defined as the number of chart statistics plotted until the chart signals. One of the most commonly used metrics of a Phase II chart is the average run length (ARL), which is defined as the average number of plotted statistics until the chart signals. As pointed out in [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF], the shape of the run length distribution changes for different mean shift sizes, i.e. from highly right skewed when the process is in-control to nearly symmetric when the shift size is large. Therefore, the interpretation only based on the ARL corresponding to a highly skewed distribution is certainly different from a nearly symmetric distribution. For example, the Shewhart X chart is usually designed to have an in-control ARL 0 = 370 by choosing suitable parameters. Using the same parameters, the in-control M RL 0 is 257, which means 50% of all the RLs are less than 257. As a comparison, about 63% of all the run lengths are less than 370. In recent years, the use of the ARL only as a measure of control charts' performance has been criticized in some researches, see [START_REF] Jones | The Run Length Distribution of the CUSUM with Estimated Parameters[END_REF], [START_REF] Jensen | Effects of Parameter Estimation on Control Chart Properties: A Literature Review[END_REF] and [START_REF] Chakraborti | Run Length Distribution and Percentiles: The Shewhart X Chart with Unknown Parameters[END_REF], to name a few. They all pointed out that the interpretation based on the ARL is sometimes misleading as the shape of the run length distribution changes with the magnitude of the mean shift. On the other hand, the M RL does not have the same interpretation problem and, because of its meaningful interpretation, [START_REF] Gan | An Optimal Design of EWMA Control Charts Based on Run Length[END_REF], [START_REF] Gan | An Optimal Design of Cumulative Sum Control Chart Based on Median Run Length[END_REF], [START_REF] Khoo | Optimal Designs of the Multivariate Synthetic Chart for Monitoring the Process Mean Vector Based on Median Run Length[END_REF], [START_REF] Khoo | Optimal Design of the Synthetic Chart for the Process Mean Based on Median Run Length[END_REF], Teoh, Chong, Khoo, Castagliola and Yeong (2016), [START_REF] Lee | Optimal Design of Synthetic np Control Chart Based on Median Run Length[END_REF] and [START_REF] Hu | Conditional Design of the EWMA Median Chart with Estimated Parameters[END_REF] have done some researches on the area of M RL based control charts with known parameters. As stated above, a Phase II control chart's properties need to be investigated with the process parameters estimated from a Phase I dataset. Considering this issue, when the process parameters are estimated, some researches have been conducted on the M RL based control charts: readers may refer to [START_REF] Chakraborti | Run Length Distribution and Percentiles: The Shewhart X Chart with Unknown Parameters[END_REF], Teoh et al. (2015), Teoh, Khoo, Castagliola and Lee (2016) and You et al. (2016). All these researches focused on the marginal M RL performance of control charts with estimated parameters. Therefore, this article suggests the M RL as a design measure of the Shewhart X chart. A detailed motivation of the use of the M RL instead of the ARL in the design of the Shewhart X chart can be found in [START_REF] Khoo | Performance Measures for the Shewhart X Control Chart[END_REF], [START_REF] Chakraborti | Run Length Distribution and Percentiles: The Shewhart X Chart with Unknown Parameters[END_REF] and Teoh, Khoo, Castagliola and Lee (2016).

When the process parameters are estimated, the between-practitioners variability generates a inherent variability in the in-control ARL values. Using just the ARL averaged over the distribution of the parameter estimates does not give a complete picture of the performance of control charts. In addition to this, much researches recommended the standard deviation of the ARL (SDARL) as a new metric, to measure the betweenpractitioners variability in control charts, see Saleh, Mahmoud, [START_REF] Keefe | The Conditional In-Control Performance of Self-Starting Control Charts[END_REF], Saleh, Mahmoud, Jones-Farmer, Zwetsloot and Woodall (2015), [START_REF] Faraz | Guaranteed Conditional Performance of the S 2 Control Chart with Estimated Parameters[END_REF], [START_REF] Hu | Guaranteed Conditional Design of the Median Chart with Estimated Parameters[END_REF], [START_REF] Keefe | The Conditional In-Control Performance of Self-Starting Control Charts[END_REF], [START_REF] Aly | A Re-evaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated[END_REF], Aly et al. (2016), Geodhart, da Silvab, Schoonhoven, Epprecht, Chakraborti, Does and[START_REF] Geodhart | Shewhart Control Charts for Dispersion Adjusted for Parameter Estimation[END_REF] and Geodhart, Schoonhoven and Does (2017). These authors use the SDARL metric together with AARL metric to make recommendations for the amount of Phase I samples. Using the M RL as a measure, and considering the between-practitioners variability effect in the M RL values of the Shewhart X chart, the M RL averaged over the distribution of the parameter estimates (AM RL) will also not give a complete picture of the performance of the Shewhart X chart with unknown parameters. This is similar to the ARL based Shewhart X chart with unknown parameters (Saleh, Mahmoud, [START_REF] Keefe | The Conditional In-Control Performance of Self-Starting Control Charts[END_REF]). To measure this variability, the standard deviation of the M RL (SDM RL) will be used as an alternative to evaluate the performance of the Shewhart X chart. Thus, the M RL based Shewhart X chart with unknown process parameters will be investigated in this article by using both the AM RL and SDM RL.

The rest of this article is organized as follows: First, in Section 2, we introduce the run length properties of the Shewhart X chart, where the properties for the known and unknown process parameters cases are provided in Sections 2.1 and 2.2, respectively. We then introduce in Section 3 a brief background of the M RL based Shewhart X chart. Based on the average, the standard deviation, and several percentiles of the M RL distribution, we investigate in Section 4 the M RL performance of the Shewhart X chart with estimated parameters. In Section 5, the adjustments of control limits, and the corresponding performance of the Shewhart X chart are discussed. In Section 6, an example is given to illustrate the use of the M RL based Shewhart X chart. Finally, some conclusions and recommendations are made in Section 7.

2 The run length properties of the Shewhart X chart

The Shewhart X chart with known parameters

Suppose that the quality characteristic Y is a normal random variable, Y ∼ N (µ 0 + δσ 0 , σ 0 ), where µ 0 and σ 0 are the in-control mean and standard deviation, respectively, and δ is the magnitude of the standardized mean shift. The process is considered to be in-control when δ = 0. Otherwise (δ = 0), the process is out-of-control. A Phase II sample {Y i,1 , Y i,2 , . . . , Y i,n } of size n are then collected successively at sampling time i, and the sample mean Ȳi = 1 n n j=1 Y i,j is used for process monitoring, where

Ȳi ∼ N (µ 0 + δσ 0 , σ 0 / √ n).
For the Shewhart X chart, an out-of-control signal is obtained when a sample mean Ȳi fall above the upper control limit (U CL) or below the lower control limit (LCL), where

U CL = µ 0 + Kσ 0 LCL = µ 0 -Kσ 0
where K is a constant that determines the distance, in terms of standard deviation units, of the control limits from the centre line of the control chart and, it is selected in order to obtain a certain in-control performance.

When the parameters µ 0 and σ 0 are known, the control limits U CL and LCL are constants, and the probability that a sample mean Ȳi falls inside the control limits is,

p = P (LCL ≤ Y i ≤ U CL) = F N (K √ n -δ √ n) -F N (-K √ n -δ √ n)
where F N (•) is the cumulative distribution function (c.d.f.) of the standard normal distribution. Since the run length (RL) of the Shewhart X chart follows a geometric distribution [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF], the probability distribution function (p.d.f.) f RL (ℓ) and c.d.f. F RL (ℓ) of the RL are defined for ℓ = {1, 2, 3, . . . } as follows:

f RL (ℓ) = P (RL = ℓ) = p ℓ-1 (1 -p) (1) F RL (ℓ) = P (RL ≤ ℓ) = 1 -p ℓ (2)
Moreover, the 100ξ (0 < ξ < 1) percentage point RL ξ of the run length distribution is determined as follows [START_REF] Gan | An Optimal Design of Cumulative Sum Control Chart Based on Median Run Length[END_REF]):

F RL (RL ξ -1) ≤ ξ and F RL (RL ξ ) > ξ. (3)
For the geometric distribution in (2), by solving

F RL (RL ξ -1) = 1 -p RL ξ -1 ≤ ξ and F RL (RL ξ ) = 1 -p RL ξ > ξ, we can get ln(1 -ξ) ln(p) < RL ξ ≤ ln(1 -ξ) ln(p) + 1 (4)
Since the RL ξ of a control chart is an integer, we can further obtained that

RL ξ = ⌈ ln(1-ξ) ln(p) ⌉, if ln(1-ξ) ln(p) is not an integer ln(1-ξ) ln(p) + 1, if ln(1-ξ) ln(p) is an integer (5)
where ⌈•⌉ is the rounding up operator. Using these equations enable the computation of any percentage point of the run length distribution. In particular, if we set ξ = 0.5, we can obtain the M RL of the Shewhart X chart.

The Shewhart X chart with unknown process parameters

Suppose that {X i,1 , . . . , X i,n }, i = 1, 2, . . . , m are m in-control samples, each with size n, from Phase I. Let us assume X is a normal random variable, i.e. X i,j ∼ N (µ 0 , σ 0 ), the estimators μ0 and σ0 used for µ 0 and σ 0 are respectively (see [START_REF] Jensen | Effects of Parameter Estimation on Control Chart Properties: A Literature Review[END_REF]):

μ0 = 1 m m i=1 Xi , (6)
σ0 = 1 m(n -1) m i=1 n j=1 (X i,j -Xi ) 2 , (7)
with Xi = 1 n n j=1 X i,j . The control limits of the Shewhart X chart become

U CL = μ0 + K σ0 , (8)
LCL = μ0 -K σ0 . (9)
For fixed values of μ0 and σ0 , the conditional probability p of a sample falling inside the control limits of the Shewhart X chart is p = P ( LCL ≤ Ȳi ≤ U CL| µ 0 , σ 0 ). Therefore, by doing some simple mathematical manipulations, we can obtain the following results,

p = F N (μ 0 + K σ0 -µ 0 -δσ 0 ) √ n σ 0 -F N (μ 0 -K σ0 -µ 0 -δσ 0 ) √ n σ 0 . ( 10
)
For fixed values of μ0 and σ0 , the conditional probability mass function (p.m.f.) fRL (ℓ) and c.d.f. FRL (ℓ) of the RL for the Shewhart X chart are presented in (1) and (2) by replacing p in (1) with p in (10). Similarly, the conditional M RL ( M RL) of the Shewhart X chart with estimated parameters can be computed using (3) by setting ξ = 0.5 and replacing p with the corresponding estimate p.

If we define the random variables U = (μ 0 -µ 0 )

√ n σ0 and V = σ0 σ0 √ n, the conditional probability p simplifies to,

p = F N U + K × V -δ √ n -F N U -K × V -δ √ n . ( 11
)
Concerning the distributions of the random variables U and V , they are respectively given as follows (see Zhang et al. (2011)):

f U (u|m) = f N u 0, 1 √ m , (12)
f V (v|m, n) = 2vf γ v 2 m(n -1) 2 , 2n m(n -1) , (13)
where f N (•) is the p.d.f. of a normal distribution with mean 0 and standard deviation 1 √ m and, f γ (•) is the p.d.f. of the gamma distribution with parameters m(n-1) 2 and 2n m(n-1) .

Since U and V are independent random variables, then the unconditional RL distribution is the conditional f RL (ℓ) and F RL (ℓ) in (1) and (2) integrated over the the distributions of U and V , which are obtained as follows:

f RL (ℓ) = +∞ -∞ +∞ 0 pℓ-1 (1 -p)f U (u|m)f V (v|m, n)dvdu, (14)
F RL (ℓ) = +∞ -∞ +∞ 0 (1 -pℓ )f U (u|m)f V (v|m, n)dvdu, (15)
Moreover, the unconditional M RL, denoted as AM RL, and the standard deviation of M RL, denoted as SDM RL, of the Shewhart X chart with estimated parameters can be obtained as follows:

AM RL = E( M RL) = +∞ -∞ +∞ 0 M RL × f U (u|m)f V (v|m, n)dvdu, ( 16
)
SDM RL = E( M RL 2 ) -(E( M RL)) 2 , ( 17
)
where

E( M RL 2 ) = +∞ -∞ +∞ 0 M RL 2 × f U (u|m)f V (v|m, n)dvdu. (18)
In order to measure the average and variability of the conditional M RL performance of the Shewhart X chart with unknown process parameters, the AM RL, SDM RL and some percentiles of M RL can also be obtained using Monte Carlo simulations.

Brief background of the interpretation of the M RL based Shewhart X chart

A traditional measure of a control chart's property is the ARL. However, the in-control RL distribution of the Shewhart X chart is highly right-skewed, the mean of the RL is greater than its median, and thus it is usually not a fair representation of a "typical" or the "center" measure of the RL. For this reason, some other percentiles of the RL distribution may provide important information about the distribution and, consequently about the control chart's properties. For example, [START_REF] Khoo | Performance Measures for the Shewhart X Control Chart[END_REF] and [START_REF] Chakraborti | Run Length Distribution and Percentiles: The Shewhart X Chart with Unknown Parameters[END_REF] pointed out that using the M RL to design a chart leads to a more accurate picture of a chart's performance, especially for highly skewed RL distribution. An M RL of τ indicates that 50% of all of the RLs are less than τ .

In order to draw attention to the usage of the M RL based Shewhart X chart, some percentiles of the RL and the ARLs are presented in Table 1, for different values of K and δ when n = 5. From Table 1, we can note that the difference between the ARL and M RL (50% percentile of RL) is large when the process is in-control (δ = 0). This difference decreases when δ increases. For example, when K = 1.2553 and δ = 0, the M RL and ARL values of the Shewhart X chart are 139 and 200, respectively. While for δ = 1, the M RL and ARL are 3 and 3.52, respectively. This fact shows that the shape of the RL changes with the mean shift size δ. Since the RL distribution is highly skewed, especially when δ is small, a typical measure of the RL distribution is the M RL. The information from the usage of M RL acts as a good practical guidance to practitioners in studying the property of the Shewhart X chart. For more discussion about this topic, readers may refer to [START_REF] Khoo | Performance Measures for the Shewhart X Control Chart[END_REF], Chakraborti (2007), Teoh, Chong, Khoo, Castagliola andYeong (2016), Teoh, Khoo, Castagliola andLee (2016) andTeoh et al. (2015).

(Please Insert Table 1 Here)

The M RL performance of the Shewhart X chart with unknown process parameters

The M RL based control charts with unknown process parameters are mostly evaluated based on the 50% percentile of the unconditional c.d.f. F RL (ℓ) of the RL as in ( 15) (Teoh et al. (2015), You et al. (2016) and Teoh, Khoo, Castagliola and Lee (2016)). The 50% percentile obtained from (15) doesn't reflect the between-practitioners variability in control charts and, unfortunately, it doesn't account for the variability in the M RL values.

For a larger number of the Shewhart X chart, the corresponding M RL values can be widely dispersed and thus, a practitioner can't be sure of his/her specific chart's M RL performance.

The conditional in-control M RL properties of the Shewhart X chart are first investigated by considering different values of m varying from 20 to 500 and the desired in-control M RL ∈ {69, 139, 257, 347} (denoted as M RL 0 ) when n = 5. Based on 10 5 simulated Shewhart X chart, the corresponding AM RL, SDM RL and several percentiles of these in-control M RL values are presented in Table 2. The value of parameter K is selected to produce the desired M RL 0 . From Table 2, we may note a large variation in the in-control M RL values, especially for small values of m. For example, when M RL 0 = 257 in Table 2, 5% of the in-control M RL values are smaller than 106 and, 5% of the in-control M RL values are larger than 543 and the SDARL = 148.41 if we use m = 50 Phase I samples to estimate the process parameters. The SDM RL value is too large compared to the desired M RL 0 = 257. This fact is generally not acceptable in practice, because the specific practitioner may has her/his own X chart's M RL value far away from the desired M RL 0 . For other values of the desired M RL 0 , a similar trend in the variation also appears in the in-control M RL values. In order to display this variation in the in-control M RL values more clearly, the histograms of the in-control M RL values based on 10 5 simulated Shewhart X charts are also presented in Figure 1 Moreover, from Table 2, it can also be noted that with the increase in the amount m of Phase I samples, the lower percentiles increases and, the upper percentiles and the SDM RL values decrease, thus illustrating that the variability in the in-control M RL distribution decreases. For example, when M RL 0 = 257 and n = 5, if we increases the number m of Phase I samples from 20 to 500, the 10% percentiles increases from 78 up to 209, the 90% percentiles decreases from 588 up to 310, and the SDM RL value decreases from 312.65 up to 40.51.

(Please Insert Table 2 Here) (Please Insert Figure 1 Here)

The results presented in Table 2 show a large variation in the in-control M RL properties of the Shewhart X chart, especially with a small number of Phase I samples. This variation is the so-called between-practitioners variability. Considering this variability, numerous researches have been done on the conditional properties of different type control charts, for example, to name a few, [START_REF] Faraz | Guaranteed Conditional Performance of the S 2 Control Chart with Estimated Parameters[END_REF], Geodhart, Schoonhoven and Does (2017); Saleh, Mahmoud, [START_REF] Keefe | The Conditional In-Control Performance of Self-Starting Control Charts[END_REF], Geodhart, da Silvab, Schoonhoven, Epprecht, Chakraborti, Does and Veiga (2017) and [START_REF] Hu | Guaranteed Conditional Design of the Median Chart with Estimated Parameters[END_REF] all focused on the Shewhart type chart, Saleh et al. ( 2016) focused on the CUSUM type chart and, Saleh, Mahmoud, Jones-Farmer, Zwetsloot and Woodall (2015) and [START_REF] Hu | Conditional Design of the EWMA Median Chart with Estimated Parameters[END_REF] focused on the EWMA type chart. These researches were all focused on the ARL based control charts with unknown process parameters, and they also recommended the usage of the SDARL to account for the between-practitioners variability. For the M RL based Shewhart X chart, considering this variability, a direct measure to evaluate its conditional properties would be the SDM RL.

In order to measure the variation in the in-control M RL values, we also focused on the AM RL and SDM RL values of the Shewhart X chart with unknown process parameters. Some researchers considered a SDARL value about 10% of the desired ARL 0 as small enough (see Zhang et al. (2014) and Saleh, Mahmoud, Jones-Farmer, Zwetsloot and Woodall (2015)). Similar to these researches on the ARL based control charts, we consider SDM RL values about 10% of the desired M RL 0 as being small enough in this article. Under this case, the conditional M RL performance of the specific Shewhart X chart would be close to the desired M RL 0 . Now, the question is "How about the number of Phase I samples for the Shewhart X chart to have an SDM RL value about 10% of the desired M RL 0 ?" To answer this question, Tables 3 to 6 present the AM RL and SDM RL values for different combinations of m ranging from 20 to 5000, and the desired M RL 0 ∈ {69, 139, 257, 347} when n ∈ {3, 5, 7, 9}. The SDM RL values that are about 10% of the desired M RL 0 are bolded in each table. The last row (m = ∞) corresponds to the cases for which the process parameters are known. The desired M RL 0 values are chosen so that the corresponding Shewhart X chart have ARL 0 values equal to ARL 0 ∈ {100, 200, 370.4, 500}, respectively. These ARL 0 values are usually used to evaluate the properties of different ARL based control charts.

From Table 3, if one only considers the AM RL values to determine the required number of Phase I samples, we may note that about m ≃ 100 Phase I samples are needed to have an in-control relative difference between the AM RL and the desired M RL 0 smaller than 5% when M RL 0 = 69 and n = 3. The relative difference is defined as ∆ = (|AM RL -M RL 0 |)/M RL 0 × 100%, and in the case studied here ∆ = (71.84 -69)/69 × 100% = 4.12%. We consider the relative difference smaller than 5% is sufficiently enough. Under this case, the average M RL properties of the Shewhart X chart is similar to the desired M RL 0 . Although m ≃ 100 Phase I samples provide an AM RL value close to the desired M RL 0 , the SDM RL = 28.65 is still large compared with the desired M RL 0 = 69. Considering the variability in the in-control M RL values, about m ≃ 1500 Phase I samples are needed for the Shewhart X chart to have an in-control SDM RL to be about 10%, i.e. 69×10%=6.9, of the desired M RL 0 (see the bolded entry in column 3). We may see that the recommended m ≃ 100 Phase I samples based on the relative difference between AM RL and M RL 0 is far smaller than 1500. Practitioners should realize the importance of taking into account both AM RL and SDM RL when choosing the number m of Phase I samples.

Moreover, with the increase in M RL 0 , much more Phase I samples are needed to have a SDM RL value about 10% of the desired M RL 0 . For example, when M RL 0 = 347, X.L. Hu et al. the number of Phase I samples increases up to m ≃ 2800 to have a SDM RL = 34.34 (see Table 3). For smaller sample sizes, like n ∈ {3, 5}, the relative difference ∆ increases with an increase in the desired value of M RL 0 for fixed value of m. For example, in Table 3, when m = 30, the relative difference ∆ for M RL 0 = 69 and M RL 0 = 139 are (79.28 -69)/69 × 100% = 14.89% and (171.33 -139)/139 × 100% = 23.25%, respectively. Moreover, in Table 6, even when n = 9, we may note that m ≃ 20 Phase I samples is large enough for the Shewhart X chart to have a ∆ smaller than 5%, but at least about m ≃ 400 Phase I samples to have a SDM RL value about 10% of the desired M RL 0 = 69 (see the bolded entry in column 3). Through the results presented in Tables 3 to 6, more than m ≃ 400 Phase I samples are needed to get a relatively small variability in the in-control M RL values of the Shewhart X chart.

Furthermore, considering the variability in the in-control M RL values, it can also be noted that, in order to estimate the process parameters accurately, the overall number of observations (m × n) required for the case n = 3 is larger than that the one required for the case n = 9. For example, the Shewhart X chart requires about 4500 observations (m = 1500 samples each with size n = 3) to get a SDM RL value to be about 10% of the desired M RL 0 = 69 (see Table 3). While it requires about 3600 observations (m = 400 samples each with size n = 9) to get a SDM RL value to be about 10% of the desired M RL 0 = 69 (see Table 6). This result is similar to the ARL based Shewhart X chart presented in Saleh, Mahmoud, [START_REF] Keefe | The Conditional In-Control Performance of Self-Starting Control Charts[END_REF].

(Please Insert Tables 3 to 6 Here) Based on the results presented above, at least m ≃ 400 samples are needed for the Shewhart X chart to obtain a smaller in-control SDM RL value. However, this large number of Phase I samples is sometimes hard to collect and too costly in practice. With a realistic amount of Phase I samples, a significant variation exists in the estimated process parameters. Thus, the practitioner has no confidence that the in-control M RL value of its specific Shewhart X chart will be close to the desired M RL 0 . For this reason, in the following section, a bootstrap approach will be adopted to re-design the M RL based Shewhart X chart with unknown process parameters.

Adjustments of the control limits of the M RL based Shewhart X chart

To avoid too many in-control M RL values smaller than the desired M RL 0 , one would have to adjust the control limits parameter K to account for the difference between the estimated and true process parameters. During the last few years, the bootstrap approach used in [START_REF] Gandy | Guaranteed Conditional Performance of Control Charts via Bootstrap Methods[END_REF] became very popular, which can be adopted to guarantee the conditional in-control M RL values larger than the desired M RL 0 with a certain high probability α * . Compared to the unadjusted parameter cases, where many M RL values are below the desired M RL 0 , the bootstrap approach are expected to guarantee that this would happen only in 100(1 -α * )% of all the Shewhart X charts. In the literature, the designs of control charts for a guaranteed conditional performance have only been discussed by some researchers for the ARL based charts (see Saleh, Mahmoud, Jones-Farmer, Zwetsloot and Woodall (2015); Saleh et al. ( 2016), [START_REF] Faraz | Guaranteed Conditional Performance of the S 2 Control Chart with Estimated Parameters[END_REF], [START_REF] Hu | Guaranteed Conditional Design of the Median Chart with Estimated Parameters[END_REF], [START_REF] Aly | A Re-evaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated[END_REF], Zhao and Driscoll (2016)). Therefore, the bootstrap approach is used in this article to re-design the parameters of the M RL based Shewhart X chart with unknown process parameters. The steps for the control limit adjustment based on the bootstrap method are enumerated below:

(1) Generate a Phase I data set X i,j composed of i = 1, 2, . . . , m samples {X i,1 , X i,2 , . . . , X i,n } of size n from the (assumed) true in-control distribution P = P (µ 0 , σ 0 ) of X i,j , and compute the estimated parameters μ0 and σ0 as in ( 6) and ( 7).

(2) Compute the quantity K( P , θ) that produces a desired in-control M RL 0 , where P = P (μ 0 , σ0 ) is the estimated in-control distribution and, θ = (μ 0 , σ0 ) represents the estimated process parameters used to set control limits.

(3) Generate B samples from the distribution P = P (μ 0 , σ0 ) and, compute these estimated process parameters θ *

i = (μ * i , σ * i ), i = 1, 2, . . . , B.
(4) Compute the quantities K( P * i , θ * i ) and K( P , θ * i ), i = 1, 2, . . . , B to produce the desired in-control M RL 0 , where P * i = (μ * i , σ * i ), i = 1, 2, . . . , B and P are the in-control distributions and, θ * i , i = 1, 2, . . . , B are used to set the control limits, respectively.

(5) Obtain the 100α * % percentile K * α * of the bootstrap distribution of K( P * , θ * ) -K( P , θ * ) and take K( P , θ) -K * α * as the adjusted control limit for the Shewhart X chart.

Following these bootstrap steps, it is expected that 100α * % of the in-control M RL values will be larger than the desired M RL 0 value. In our simulations, we assume that the true in-control distribution P is a standard normal N (0, 1) distribution and the desired in-control M RL 0 is set to be 257. m = 50 Phase I samples each with size n = 5 are considered as realistic and are used in simulations to estimate the process parameters. The specified probability that the conditional in-control M RL values are larger than the desired M RL 0 is set as 90%, i.e. α * = 0.9, and B = 1000 bootstrap samples are generated from P to construct the adjusted control limits. By applying the bootstrap method, we adjust the control limit parameter K to guarantee the conditional in-control M RL performance for the Shewhart X chart to be larger than the desired M RL 0 = 257 with a high probability α * = 0.9.

In the bootstrap approach, we need to compute different parameters K from different settings of the Shewhart X chart: K( P , θ), K( P * i , θ * i ), i = 1, 2, . . . , B and K( P , θ * i ), i = 1, 2, . . . , B. Since M RL is an integer, it is possible to have more than one value of K for the desired M RL 0 . All possible values of K satisfying M RL 0 are found and without loss of generality, the smallest one is used here. The computations are summarised as follows:

• For the quantity K = K( P , θ), since the in-control distribution P is defined with the estimated parameters θ = (μ 0 , σ0 ), and the estimated parameters θ = (μ 0 , σ0 ) are also used to set the control limits, the computation of K = K( P , θ) is similar to the case of computation of K(P, θ), where K(P, θ) is the parameter that produces the desired M RL 0 for the Shewhart X chart with known process parameters.

• Similar to the computation of quantity K = K( P , θ), the quantities K( P * i , θ * i ), i = 1, 2, . . . , B are all equal to K = K( P , θ).

• For the quantities K( P , θ * i ), the estimated parameters θ = (μ 0 , σ0 ) in the in-control distribution P = N (μ 0 , σ0 ) are different from the ones θ * i = (μ * i , σ * i ), i = 1, 2, . . . , B used in the set of the Shewhart X charts. In this case, a binary search algorithm is used to compute the value K that satisfies the desired M RL 0 .

Because the value of K( P , θ) is equal to K( P * i , θ * i ), the adjusted control limit can be simplified to get 100α * % percentile of

K( P , θ * i ), i = 1, 2, • • • , B.
Once the (adjusted) control limits are determined, the conditional in-and out-of-control M RLs of the Shewhart X chart can be calculated using the method introduced in Section 2.2.

Figures 2 and3 show the boxplots of the in-and out-of-control M RL distributions of the Shewhart X charts with the adjusted and unadjusted limits. If the limits are computed using the bootstrap adjustment approach, they are denoted as "Adjusted Limits". Otherwise, they are denoted as "Unadjusted Limits". The boxplots are constructed from 1000 M RL values.

(Please Insert Figures 2 to 3 Here)

From Figure 2, we may note that many in-control M RL values in the "Adjusted Limits" cases are larger than in the "Unadjusted Limits" cases. This is the result of the adjustment of the parameter K, which widen the distance of the control limits from the center line of the Shewhart X chart. Actually, as expected, the "Adjusted Limits" guarantee about 90% of the Shewhart X charts have in-control M RL values larger than the desired M RL 0 = 257. While for the "Unadjusted Limits" case, about 90% of the Shewhart X charts have in-control M RL values larger than 126, which means many of the "Unadjusted Limits" Shewhart X charts have a higher false alarm rate than the ones with the "Adjusted Limits".

In practice, control charts with only large in-control M RL performances are not enough for implementation, because we are also interested in the detection ability of control charts. Thus, the out-of-control M RL values of the Shewhart X charts with the "Adjusted Limits" and "Unadjusted Limits" are presented in Figure 3 for δ ∈ {0.3, 0.5, 0.7, 1.0}, respectively. It can be noted that for smaller shift size, the out-of-control M RL values of the Shewhart X charts with "Adjusted Limits" are obviously larger than the ones of the Shewhart X charts with "Unadjusted Limits". For example, in Figure 3 (a), when δ = 0.3, the median of the M RL values for the Shewhart X charts with "Adjusted Limits" and "Unadjusted Limits" are about 131 and 69, respectively. For moderate to large shift sizes, the Shewhart X charts with the "Adjusted Limits" only causes a small loss in the out-of-control M RL performance compared with using the "Unadjusted Limits". For example, in Figure 3 (c), when δ = 0.7, the median of the out-of-control M RLs of the Shewhart X charts are about 14.5 and 9.0 for the "Adjusted Limits" and "Unadjusted Limits" cases, respectively. This small loss in the out-of-control M RL performances is generally acceptable compared with the gain in the in-control M RL performances. For these moderate to large shift sizes, the bootstrap approach can give a good balance between the in-and out-of-control M RL performance for the Shewhart X charts with estimated parameters. Moreover, it can also be seen from Figures 2 and3 that the M RL distributions of the Shewhart X charts with the "Adjusted Limits" are loosen compared to the "Unadjusted Limits" cases.

When repeating

Step (1) in the bootstrap approach, the estimated parameters μ0 and σ0 would be different from each other, which means that the "Adjusted Limits" will not be the same each time. Based on many simulations, the results only show a small difference in these adjusted limits. In order to provide practitioners with more guidance on the adjusted limits of the M RL based Shewhart X chart, Table 7 presents the average adjusted limits K, which guarantees about 90% of the in-control M RL values larger than the desired M RL 0 . The K values are based on 1000 runs of the bootstrap approach for m ≤ 100, and 100 runs for other m values larger than 100. The practitioners can refer to this table for the adjusted control limits, or they can either follow Steps ( 1)-( 4) for obtaining the final adjusted limits for the M RL based Shewhart X chart.

(Please Insert Table 7 Here) Since both in-and out-of-control M RL performances of control charts are concerned by practitioners and the bootstrap approach only guarantees the in-control M RL performance, we should not ignore the out-of-control M RL performance of the Shewhart X chart, If the detection ability of control chart is deemed to be insufficient, two alternative options are possibly adopted:

• Increase the amount of Phase I observations (increase m or n).

• Be more lenient on the desired in-control M RL 0 .

For the first option, using more observations to estimate the process parameters θ = (μ 0 , σ0 ) would be more accurate and cause less uncertainty to account for in the unknown process parameters. So the adjustment of the control limit K will be smaller, which will result in smaller increase in the in-and out-of-control M RL values compared to the cases with a large adjustment of K. However, only a limited number of Phase I observations may be available in practice. Under this case, the second option has to be adopted. The practitioners could choose a less smaller M RL 0 than the predetermined desired one, or use a smaller value of α * , which guarantees a smaller probability to have M RL values above the desired in-control M RL 0 .

An illustrative example

In this section, we illustrate the proposed M RL based Shewhart X chart with unknown parameters by applying it to a production process of 500ml milk bottles. Similar to [START_REF] Castagliola | The EWMA Median Chart with Estimated Parameters[END_REF], we are interested in the weight Y of each bottle. The in-control data are generated from a normal distribution with mean 500 and standard deviation 1. The Phase I data set consists of m = 20 samples, each of size n = 5 (see Table 8). From these Phase I samples, we can obtain the estimated process parameters μ0 =500.1349 and σ0 =0.8241. Considering the between-practitioners variability, we can get the adjusted parameters K = 1.533 of the Shewhart X chart from Table 7. Then, the control limits of the Phase II Shewhart X chart are set as follows:

U CL = 500.1349 + 1.533 × 0.8241 = 501.34 LCL = 500.1349 -1.533 × 0.8241 = 498.87

The Phase II data set consisting of m = 20 samples, each of size n = 5, is shown Table 9. These samples are also presented in Figure 4 (a). The first 10 samples are supposed to be in-control, while the following 10 samples come from an out-of-control process. All the mean values Ȳi of these Phase II samples are presented in the last column of Table 9, and the corresponding Shewhart X chart is shown in Figure 4 (b). From Figure 4 (b), we may note the mean value of samples #11, #15 and #20 (in bold in Table 9) are below the low control limit LCL, which indicates a possible assignable cause in the process.

(Please Insert Tables 8 and9 Here) (Please Insert Figure 4 Here)

Conclusions and recommendations

Since the shape of the run length distribution for the Shewhart X chart is highly skewed, a performance measure, such as the ARL, does not give a complete understanding of the chart's performance. Thus, the M RL is used to evaluate the properties of Shewhart X chart. Furthermore, the process parameters are often unknown, and need to be estimated from Phase I samples. Different practitioners will use different Phase I samples to estimate the process parameters, thus causing the between-practitioners variability in control charts' performance. Considering this variability, the M RL based Shewhart X chart with unknown process parameters is investigated in this article.

Several tables are presented for the practitioners to assess the between-practitioners variability of the M RL based Shewhart X chart, for different amount of Phase I samples. The results show that at least m ≃ 400 Phase I samples are needed to reduce the betweenpractitioners variability in the M RL based Shewhart X chart up to an acceptable level. Since taking a large amount of Phase I samples is impractical in practice, a bootstrap approach is adopted to adjust the control limits of the M RL based Shewhart X chart. The simulations results show that, for moderate to larger shift sizes, the adjusted control limits can give a good balance between the in-and out-of-control M RL performances. For different number of Phase I samples, sample sizes, and the desired M RL 0 , the average adjusted control limits are given as a reference for the practitioners. As this work is conducted for the Shewhart X chart, future research works can be conducted for EWMA or CUSUM type charts.

2.72 1 1 1 1 2 2 3 3 4 6 7 1.5 1.28 1 1 1 1 1 1 1 1 2 2 2 2.0 1.03 1 1 1 1 1 1 1 1 1 1 1 K = 1.

  for different number of Phase I samples m ∈ {50, 100, 200, 400} when M RL 0 = 257 and n = 5. A large variation of nearly SDM RL = 45.35 can also be noted from Figure 1 (d) even with m = 400 Phase I samples.

Figure 1 Figure 2 Table 7

 127 Figure 1 Histograms of the in-control M RL for n = 5 and (a) m = 50, (b) m = 100, (c) m = 200, (d) m = 400. (The red line corresponds to the M RL0 = 257 on the horizontal axis)

Figure 3 Figure 4

 34 Figure 3 The boxplots of the out-of-control M RL for the Shewhart X chart with the adjusted and unadjusted limits, for (a) δ = 0.3, (b) δ = 0.5, (c) δ = 0.7, (d) δ = 1.0

Table 1

 1 Percentiles and the ARLs of the Shewhart X Chart for different values of δ and K when n = 5

	Percentiles of the RL distribution

Table 3

 3 In-Control AM RL and SDM RL for different combinations of the desired M RL0 and m when n = 3

		M RL 0 = 69	M RL 0 = 139	M RL 0 = 257	M RL 0 = 347
		(K = 1.4872)	(K = 1.6206)	(K = 1.7321)	(K = 1.7841)
	m	AM RL SDM RL AM RL SDM RL AM RL SDM RL AM RL SDM RL
	20	86.40	111.20	196.53	351.51	419.67	1025.60	613.31	1759.23
	30	79.28	70.03	171.33	194.41	345.72	489.98	489.30	774.40
	50	74.71	45.49	155.87	116.53	302.55	268.56	419.14	403.75
	70	72.97	35.92	150.25	89.33	287.49	199.58	395.05	294.80
	100	71.84	28.65	146.52	69.82	277.25	152.65	378.74	222.87
	150	70.98	22.54	143.76	54.23	269.93	116.71	367.24	168.94
	200	70.62	19.21	142.45	45.81	266.48	97.91	361.81	141.17
	250	70.35	17.01	141.66	40.39	264.51	85.97	358.60	123.64
	300	70.18	15.41	141.17	36.53	263.13	77.49	356.59	111.29
	400	70.00	13.26	140.52	31.25	261.55	66.12	354.01	94.73
	500	69.90	11.76	140.15	27.78	260.59	58.59	352.47	83.91
	600	69.86	10.72	139.96	25.22	259.97	53.16	351.56	76.07
	700	69.75	9.91	139.79	23.26	259.51	49.00	350.85	70.08
	1000	69.71	8.27	139.47	19.35	258.74	40.70	349.62	58.15
	1100	69.71	7.90	139.41	18.47	258.55	38.72	349.33	55.36
	1300	69.59	7.24	139.31	16.93	258.30	35.56	348.94	50.77
	1500	69.63	6.72	139.24	15.76	258.11	33.03	348.59	47.18
	2000	69.56	5.81	139.10	13.60	257.85	28.53	348.16	40.74
	2500	69.57	5.16	139.07	12.15	257.65	25.47	347.79	36.36
	2600	69.54	5.10	139.09	11.96	257.62	24.99	347.85	35.64
	2700	69.50	5.02	139.00	11.65	257.55	24.49	347.81	34.95
	2800	69.49	4.97	139.01	11.48	257.52	24.07	347.76	34.34
	3000	69.47	4.84	139.02	11.09	257.53	23.22	347.66	33.17
	5000	69.50	3.67	138.93	8.56	257.26	17.94	347.25	25.59
	∞	69.00	0.00	139.00	0.00	257.00	0.00	347.00	0.00

Table 4

 4 In-Control AM RL and SDM RL for different combinations of the desired M RL0 and m when n = 5

		M RL 0 = 69	M RL 0 = 139	M RL 0 = 257	M RL 0 = 347
		(K = 1.1519)	(K = 1.2553)	(K = 1.3416)	(K = 1.3820)
	m	AM RL SDM RL AM RL SDM RL AM RL SDM RL AM RL SDM RL
	20	71.52	51.65	149.90	134.83	292.86	318.13	407.45	484.77
	30	70.25	38.62	144.53	96.50	276.50	217.01	380.33	321.87
	50	69.62	28.09	141.38	68.15	266.55	148.31	363.43	216.10
	70	69.44	23.18	140.27	55.55	262.99	119.36	357.47	172.77
	100	69.41	19.03	139.67	45.26	260.73	96.44	353.56	138.97
	150	69.36	15.29	139.30	36.25	259.24	76.76	350.96	110.15
	200	69.41	13.21	139.10	31.06	258.50	65.60	349.74	94.03
	250	69.43	11.71	139.02	27.64	258.16	58.22	349.08	83.36
	300	69.42	10.69	138.97	25.14	257.93	52.91	348.61	75.69
	400	69.46	9.20	138.94	21.69	257.61	45.52	348.10	65.10
	500	69.46	8.25	138.90	19.30	257.46	40.59	347.82	57.99
	600	69.42	7.49	138.88	17.60	257.36	36.94	347.64	52.78
	700	69.45	6.94	138.80	16.25	257.23	34.14	347.46	48.78
	1000	69.43	5.81	138.83	13.61	257.23	28.50	347.26	40.67
	1100	69.44	5.56	138.79	12.96	257.14	27.16	347.25	38.71
	1300	69.45	5.08	138.84	11.93	257.13	24.94	347.11	35.58
	1500	69.39	4.81	138.79	11.08	257.07	23.20	347.05	33.11
	2000	69.46	4.04	138.79	9.63	257.05	20.09	346.98	28.62
	2500	69.45	3.68	138.81	8.56	257.02	17.93	346.91	25.58
	3000	69.46	3.38	138.79	7.80	256.99	16.38	346.90	23.35
	5000	69.46	2.62	138.78	6.07	256.91	12.63	346.86	18.07
	∞	69.00	0.00	139.00	0.00	257.00	0.00	347.00	0.00

Table 5

 5 In-Control AM RL and SDM RL for different combinations of the desired M RL0 and m when n = 7

		M RL 0 = 69	M RL 0 = 139	M RL 0 = 257	M RL 0 = 347
		(K = 0.9736)	(K = 1.0610)	(K = 1.1339)	(K = 1.1680)
	m	AM RL SDM RL AM RL SDM RL AM RL SDM RL AM RL SDM RL
	20	67.49	37.85	138.09	93.97	263.08	210.21	361.07	311.08
	30	67.62	29.55	136.99	71.64	258.07	156.06	351.78	227.57
	50	68.06	22.15	137.01	52.83	255.88	112.99	347.17	163.14
	70	68.35	18.45	137.27	43.79	255.51	93.00	346.06	133.72
	100	68.62	15.31	137.48	36.13	255.57	76.31	345.73	109.44
	150	68.85	12.39	137.89	29.17	255.75	61.41	345.69	87.88
	200	68.99	10.68	138.04	25.08	255.95	52.78	345.83	75.48
	250	69.11	9.52	138.16	22.36	256.10	46.97	345.98	67.16
	300	69.16	8.69	138.32	20.39	256.21	42.78	346.04	61.13
	400	69.21	7.49	138.37	17.61	256.37	36.91	346.19	52.72
	500	69.26	6.71	138.47	15.73	256.47	32.93	346.28	47.01
	600	69.31	6.14	138.52	14.33	256.52	30.03	346.37	42.87
	700	69.33	5.67	138.52	13.26	256.55	27.76	346.37	39.64
	800	69.36	5.25	138.54	12.38	256.59	25.97	346.43	37.04
	900	69.34	5.00	138.59	11.66	256.64	24.45	346.49	34.91
	1000	69.31	4.80	138.59	11.07	256.66	23.21	346.53	33.09
	1100	69.40	4.54	138.64	10.57	256.63	22.14	346.51	31.52
	1300	69.45	4.16	138.65	9.74	256.74	20.31	346.49	29.01
	1500	69.37	3.83	138.70	9.01	256.72	18.92	346.58	26.99
	2000	69.43	3.38	138.68	7.81	256.78	16.38	346.57	23.34
	2500	69.41	3.01	138.74	6.99	256.81	14.62	346.61	20.87
	3000	69.42	2.72	138.69	6.39	256.82	13.34	346.67	19.03
	5000	69.35	2.04	138.74	4.99	256.84	10.32	346.65	14.75
	∞	69.00	0.00	139.00	0.00	257.00	0.00	347.00	0.00

Table 6

 6 In-Control AM RL and SDM RL for different combinations of the desired M RL0 and m when n = 9 Design of the Shewhart X chart with unknown parameters based on M RL 23

		M RL 0 = 69	M RL 0 = 139	M RL 0 = 257	M RL 0 = 347
		(K = 0.8586)	(K = 0.9357)	(K = 1)	(K = 1.0301)
	m	AM RL SDM RL AM RL SDM RL AM RL SDM RL AM RL SDM RL
	20	65.61	31.39	132.69	76.13	249.88	166.30	340.75	242.86
	30	66.38	24.98	133.54	59.70	249.54	128.08	338.72	185.23
	50	67.35	18.95	134.89	44.85	250.78	95.14	339.49	136.69
	70	67.82	15.91	135.66	37.44	251.88	79.07	340.64	113.30
	100	68.21	13.21	136.46	31.01	253.03	65.28	341.88	93.43
	150	68.65	10.71	137.13	25.11	254.11	52.75	343.17	75.44
	200	68.83	9.23	137.51	21.67	254.73	45.48	343.95	64.96
	250	68.97	8.28	137.73	19.32	255.11	40.54	344.47	57.87
	300	69.04	7.52	137.92	17.61	255.38	36.92	344.76	52.72
	400	69.12	6.49	138.10	15.20	255.75	31.92	345.25	45.48
	500	69.19	5.81	138.21	13.58	255.95	28.48	345.56	40.63
	600	69.25	5.26	138.34	12.40	256.15	25.97	345.72	37.07
	700	69.25	4.94	138.40	11.47	256.21	24.03	345.83	34.29
	1000	69.34	4.05	138.48	9.61	256.44	20.12	346.06	28.61
	1100	69.32	3.86	138.56	9.12	256.45	19.14	346.12	27.28
	1300	69.36	3.61	138.59	8.41	256.49	17.59	346.19	25.09
	1500	69.37	3.36	138.60	7.82	256.59	16.40	346.31	23.35
	2000	69.41	2.92	138.62	6.77	256.64	14.19	346.40	20.20
	2500	69.42	2.61	138.68	6.08	256.68	12.64	346.51	18.05
	3000	69.39	2.39	138.68	5.52	256.72	11.57	346.55	16.52
	5000	69.42	1.85	138.87	4.32	256.84	8.92	346.68	12.79
	∞	69.00	0.00	139.00	0.00	257.00	0.00	347.00	0.00

  = 69 M RL 0 = 139 M RL 0 = 257 M RL 0 = 347 M RL 0 = 69 M RL 0 = 139 M RL 0 = 257 M RL 0 = 347

			.793	1.954		2.074	2.150	1.321	1.434		1.533	1.581
	30		1.721	1.871		2.003	2.063	1.279	1.395		1.490	1.539
	50		1.651	1.801		1.924	1.988	1.243	1.358		1.449	1.496
	80		1.614	1.761		1.882	1.938	1.221	1.333		1.421	1.468
	100		1.593	1.742		1.863	1.915	1.213	1.323		1.415	1.455
	130		1.580	1.724		1.846	1.898	1.202	1.312		1.403	1.446
	150		1.573	1.717		1.835	1.889	1.199	1.310		1.399	1.442
	200		1.559	1.702		1.819	1.876	1.191	1.300		1.390	1.433
	250	400	1.553	1.694		1.810	1.863	1.187	1.295		1.384	1.427
	300	350	1.546	1.686		1.803	1.858	1.184	1.292		1.380	1.423
	350	300	1.541	1.680		1.796	1.852	1.180	1.289		1.377	1.419
	400 m M RL 0 20 1.537 1.090 100 150 200 250 Out-of-control MRL	1.679 1.190	n = 7	1.791 1.272	1.845 1.310	1.178 0.954	1.287 1.042	n = 9	1.375 1.107	1.415 1.144
	30	50	1.062	1.161		1.240	1.279	0.929	1.014		1.086	1.114
	50 80		1.038 1.022 Adjusted Limits	1.133 1.115 Unadjusted Limits	1.210 1.191	1.249 1.228	0.909 0.896	0.992 0.977		1.061 1.045	1.093 1.077
	100		1.016	1.108		1.186	1.221	0.890	0.972		1.039	1.071
	130		1.009	1.102		1.178	1.212	0.886	0.967		1.033	1.065
	150		1.006	1.098		1.174	1.209	0.883	0.965		1.031	1.061
	200		1.001	1.094		1.168	1.203	0.880	0.960		1.027	1.057
	250		0.997	1.088		1.164	1.198	0.877	0.958		1.022	1.054
	300		0.995	1.086		1.161	1.196	0.875	0.955		1.020	1.051
	350		0.993	1.083		1.159	1.193	0.873	0.953		1.019	1.050
	400		0.991	1.082		1.157	1.192	0.872	0.952		1.018	1.049

Acknowledgements

This work is supported by National Natural Science Foundation of China No. 71802110, 72101123, 71872088, Natural Science Foundation of JiangSu Province No. BK20200750, Key Research Base of Philosophy and Social Sciences in Jiangsu-Information Industry Integration Innovation and Emergency Management Research Center, the Excellent Innovation Teams of Philosophy and Social Science in Jiangsu Province No. 2017ZSTD022, the Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province No. 2020SJA0090.

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