Jean-Claude Malela-Majika

Charles Sandile

Shongwe

Philippe Castagliola

Mpiana Ruffin

Mutambayi

Sandile C Shongwe

Ruphin M Mutambayi

A novel single composite Shewhart-EWMA control chart for monitoring the process mean Short running head: Single CSEWMA monitoring scheme

Keywords: additive weighted model, EWMA, estimated process parameters, monitoring scheme, overall performance, Shewhart, composite Shewhart-EWMA Additive weighted model, EWMA, estimated process parameters, Monitoring scheme, Overall performance, Shewhart, Composite Shewhart-EWMA

Statistical process monitoring (SPM) literature recommends the combination of Shewhart and exponentially weighted moving average (EWMA) schemes to improve the ability of the standalone Shewhart and EWMA monitoring schemes in detecting small to large shifts. The resulting scheme is named combined (or composite) Shewhart-EWMA (CSEWMA) scheme. In this paper, a new single composite Shewhart-EWMA (denoted as SCSEWMA) scheme for monitoring the mean when the process parameters are known or unknown is proposed using an additive weighted model. The flexibility of the proposed scheme is made possible by an additional weighing parameter that regulates its sensitivity towards shifts of different sizes. The new scheme is compared to the existing Shewhart, EWMA and CSEWMA 𝑋 ̅ schemes and the results reveal the superiority of the proposed scheme over the latter schemes. Simulated and real-life data are used to demonstrate the application and implementation of the proposed scheme.

Introduction

The overall concept of modern monitoring schemes was established in the 1920s by Walter A. Shewhart working at Bell Telephone; see for example, Shewhart [START_REF] Shewhart | Economic Control of Quality of Manufactured Product[END_REF] . Shewhart-type monitoring schemes are memoryless schemes that only use recent information to decide whether the process is in-control (IC) or out-of-control (OOC). This makes them faster in detecting large changes (or shifts) in the process parameter. However, Shewhart-type schemes are known to be relatively slow in detecting small and moderate shifts. To compensate the weakness of Shewhart-type schemes, Page [START_REF] Page | Continuous inspection scheme[END_REF] and Roberts [START_REF] Roberts | Control chart tests based on geometric moving averages[END_REF] introduced the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) monitoring schemes, respectively. These two monitoring schemes are memory-type schemes that use past and recent information to decide whether the process is IC or OOC. The CUSUM and EWMA schemes are considered as popular alternative of a Shewhart monitoring scheme that are utilised when small to moderate shifts are of interest; see Montgomery [START_REF] Montgomery | Statistical Quality Control: A Modern Introduction[END_REF] . Their setback is that, due to their inertia, they are relatively slow in detecting large shifts in the process. After the introduction of the Shewhart, EWMA and CUSUM schemes, many authors have developed more advanced and enhanced monitoring schemes; see for instance, Daudin [START_REF] Daudin | Double sampling 𝑋 ̅ charts[END_REF] , Mosquera and Aparisi 6 , Abbas et al [START_REF] Abbas | Mixed exponentially weighted moving average-cumulative sum charts for process monitoring[END_REF] , Abbasi et al [START_REF] Abbasi | Enhancing the performance of CUSUM scale chart[END_REF] , Shamma and Shamma [START_REF] Shamma | Development and evaluation of control charts using double exponentially weighted moving averages[END_REF] , Lucas and Saccucci 10 , Abujiya et al [START_REF] Abujiya | Enhancing the performance of combined Shewhart-EWMA charts[END_REF][START_REF] Abujiya | An improved combined Shewhart-EWMA chart based on double median ranked set sampling[END_REF] , Zaman et al [START_REF] Zaman | Mixed cumulative sum-exponentially weighted moving average control charts: An efficient way of monitoring process location[END_REF] , Ali and Haq [START_REF] Ali | A mixed GWMA-CUSUM control chart for monitoring the process mean[END_REF][START_REF] Ali | New GWMA-CUSUM control chart for monitoring the process dispersion[END_REF] , Mabude et al [START_REF] Mabude | Distribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the process location[END_REF] and Huang et al [START_REF] Huang | Enhanced generally weighted moving average variance charts for monitoring process variance with individual observations[END_REF] .

An efficient monitoring scheme is expected to detect small to large shifts as quickly as possible. One of the possible techniques to enhance the sensitivity of a monitoring scheme towards small to large shifts is the combination of memoryless and memory-type schemes such as the composite Shewhart-EWMA and Shewhart-CUSUM monitoring schemes (see, for example, Lucas [START_REF] Lucas | Combined Shewhart-CUSUM quality control schemes[END_REF] , Klein [START_REF] Klein | Composite Shewhart-EWMA statistical control schemes[END_REF] , Capizzi and Masarotto [START_REF] Capizzi | Combined Shewhart-EWMA control charts with estimated parameters[END_REF] , Shamsuzzaman et al [START_REF] Shamsuzzaman | An optimization design of the combined Shewhart-EWMA control chart[END_REF] and Freitas et al [START_REF] Freitas | Analysis of water consumption in toilets employing Shewhart, EWMA, and Shewhart-EWMA combined control charts[END_REF] , just to cite a few). Lucas [START_REF] Lucas | Combined Shewhart-CUSUM quality control schemes[END_REF] proposed a combined Shewhart-CUSUM monitoring scheme for detecting small to large shifts in the process mean -see also, Klein [START_REF] Klein | Composite Shewhart-EWMA statistical control schemes[END_REF] . Lucas [START_REF] Lucas | Combined Shewhart-CUSUM quality control schemes[END_REF] showed that the control limits of the Shewhart component help to detect large shifts as quickly as possible and those of the CUSUM component help to detect small and moderate shifts quicker. Therefore, the resulting Shewhart-CUSUM scheme is efficient for monitoring small to large shifts in the process parameters. Abujiya et al [START_REF] Abujiya | Enhancing the performance of combined Shewhart-EWMA charts[END_REF] proposed an enhanced Shewhart-CUSUM scheme for monitoring shifts in the process mean, and Capizzi and Masarotto [START_REF] Capizzi | Combined Shewhart-EWMA control charts with estimated parameters[END_REF] developed a Shewhart-EWMA scheme with estimated process parameters. Shamsuzzaman et al [START_REF] Shamsuzzaman | An optimization design of the combined Shewhart-EWMA control chart[END_REF] proposed an algorithm to optimise the design of the composite Shewhart-EWMA 𝑋 ̅ chart for monitoring the entire range of the shifts in the process mean. More recently, a case study on water consumption in toilet flush devices in a public university building using Shewhart, EWMA and composite Shewhart-EWMA control charts was presented by Freitas et al [START_REF] Freitas | Analysis of water consumption in toilets employing Shewhart, EWMA, and Shewhart-EWMA combined control charts[END_REF] .

The above-mentioned monitoring schemes use two separate charting statistics to decide whether the process is IC or OOC. This makes them more difficult to implement as operators prefer simpler models of just using one charting statistic. In this study, a new flexible single composite Shewhart-EWMA scheme for monitoring the process mean is developed using an additive weighted model (i.e. based on a single charting statistic instead of two separate charting statistics). The flexibility, attractiveness and strength of the new scheme is based on an extra weighing parameter that regulates its ability to detect shifts of different sizes.

Since in real-life applications, the process parameters are usually unknown and need to be estimated, hence, the effect of parameter estimation is also considered in this study. That is, many authors have advocated that the estimation of the process parameters deteriorates significantly the performance of monitoring schemes (see the review papers by Jensen et al [START_REF] Jensen | Effects of parameter estimation on control chart properties: A literature review[END_REF] , Psarakis et al [START_REF] Psarakis | Some recent developments on the effects of parameter estimation on control charts[END_REF] and Does et al [START_REF] Does | On the design of control charts with guaranteed conditional performance under estimated parameters[END_REF] ). For instance, Aly et al [START_REF] Aly | A comparison of the performance of Phase II simple linear profile control charts when parameters are estimated[END_REF] analysed the performance of simple linear profile monitoring schemes when the process parameters are unknown and they concluded that the estimation error decreases and the IC ARL approaches the desired value only for large Phase I sample sizes when using sample estimates instead of known parameters. Therefore, the proposed monitoring scheme will be designed and evaluated under both the assumptions of known and unknown process parameters, and the effect of the sample sizes will also be investigated.

The remainder of this paper is organised as follows: Section 2 starts with a brief introduction of the existing classical Shewhart, EWMA and composite Shewhart-EWMA 𝑋 ̅ monitoring schemes. In Section 3, the theoretical and mathematical backgrounds of the proposed monitoring scheme are provided under the assumptions of known and estimated process parameters. The IC and OOC run-length performances are investigated in Section 4 for both parameters known and unknown. In addition, the OOC performance of the new scheme is compared to some existing counterparts. Section 5 uses simulated and real-life data to demonstrate the implementation and application of the proposed monitoring scheme. Concluding remarks and future research ideas are provided in Section 6.

Brief description of the existing monitoring schemes

Shewhart 𝑿 ̅ monitoring scheme

Assume that, when the process is IC, the quality characteristic {𝑋 𝑡𝑗 ; 𝑡 ≥ 1; 𝑗 = 1, 2, …, 𝑛} is a sequence of samples of independent and identically distributed (i.i.d.) observations from a N(𝜇 0 , 𝜎 0

2 ) distribution where 𝜇 0 and 𝜎 0 are the IC process mean and standard deviation parameters, respectively, which are assumed to be known. In case of a shift in the process, the process mean shifts from 𝜇 0 to 𝜇 1 (𝜇 1 = 𝜇 0 + 𝛿 𝜎 0 ) where 𝛿 (𝛿 ≠ 0) represents the change in the process mean expressed in standard deviation.

At each sampling time, the mean or charting statistic of the Shewhart 𝑋 ̅ scheme is given by

𝑋 ̅ 𝑡 = 1 𝑛 ∑ 𝑋 𝑡𝑗 𝑛 𝑗=1
.

[1]

The upper and lower control limits (UCL and LCL) of the Shewhart 𝑋 ̅ scheme are mathematically defined by

𝑈𝐶𝐿/𝐿𝐶𝐿 = 𝜇 0 ± 𝑘 𝜎 0 √𝑛 , [2]
respectively, where 𝑘 (𝑘 > 0) is the Shewhart control limits constant which is chosen such that the IC average run-length (𝐴𝑅𝐿 0 ) is equal to some large desired value such as 370.4. Thus, the Shewhart 𝑋 ̅ scheme gives a signal at time t if the charting statistic defined in Eq [1] plots beyond the control limits defined in Eq [2].

EWMA 𝑿 ̅ monitoring scheme

The charting statistic of the EWMA 𝑋 ̅ scheme at time t, denoted as 𝑍 𝑡 , is given by

𝑍 𝑡 = 𝜆𝑋 ̅ 𝑡 + (1 -𝜆)𝑍 𝑡-1 , [3a]
where 𝜆 (0 < 𝜆 ≤ 1) is the EWMA smoothing parameter, the starting point 𝑍 0 = 𝜇 0 and 𝑋 ̅ is defined as in Eq [1], hence Eq [3a] can also be written as:

𝑍 𝑡 = 𝜆 ∑ (1 -𝜆) 𝑗 𝑋 ̅ 𝑡-𝑗 + (1 -𝜆) 𝑡 𝑍 0 . 𝑡-1 𝑗=0 [3b]
The time-varying (or exact) control limits of the EWMA 𝑋 ̅ scheme at time t are defined by

𝑈𝐶𝐿 𝐸 𝑡 /𝐿𝐶𝐿 𝐸 𝑡 = 𝜇 0 ± 𝐿 𝐸 𝜎 0 √ 𝜆 (2 -𝜆)𝑛 (1 -(1 -𝜆) 2𝑡 ), [4]
respectively, where 𝐿 𝐸 (𝐿 𝐸 > 0) is the EWMA control limit constant which is chosen such that the attained 𝐴𝑅𝐿 0 is equal to some prespecified value such as 370. The EWMA 𝑋 ̅ scheme gives a signal if 𝑍 𝑡 plots beyond the control limits defined in Eq [4], i.e. 𝑍 𝑡 ≥ 𝑈𝐶𝐿 𝐸 𝑡 or 𝑍 𝑡 ≤ 𝐿𝐶𝐿 𝐸 𝑡 .

Note that when the process has been running for a long time (i.e. 𝑡 → ∞), the expression

(1 -(1 -𝜆) 2𝑡
) from Eq [4] converges to 1; therefore, the asymptotic control limits are simply defined by

𝑈𝐶𝐿 𝐴 /𝐿𝐶𝐿 𝐴 = 𝜇 0 ± 𝐿 𝐸 𝜎 0 √ 𝜆 (2 -𝜆)𝑛 , [5]
respectively. To conserve space, in this paper, we will only focus on the time-varying case.

Composite Shewhart-EWMA 𝑿 ̅ monitoring scheme

The composite Shewhart-EWMA (denoted as CSEWMA) 𝑋 ̅ scheme is the combination of the Thus, the parameters 𝜆, 𝑘 and 𝐿 𝐸 must be chosen such that the attained 𝐴𝑅𝐿 0 is equal to the prespecified 𝐴𝑅𝐿 0 value.

The proposed single composite Shewhart-EWMA 𝑿 ̅ monitoring scheme

In the previous section, it is shown that the existing CSEWMA 𝑋 ̅ scheme uses two charting statistics to decide whether the process is IC or not. In this section, we develop a new CSEWMA scheme based on a single charting statistic when the process parameters are assumed known (i.e. Case K) and unknown (i.e. Case U), and we investigate its IC and OOC performances. To make the new scheme more flexible, an extra weighing parameter denoted by 𝜔 (0 ≤ 𝜔 ≤ 1) is also introduced.

Design of the proposed scheme

The charting statistic of the proposed single CSEWMA 𝑋 ̅ scheme (henceforth denoted by SCSEWMA) is developed using an additive weighted model which is mathematically defined by

𝑊 𝑡 = (1 -𝜔)𝑋 ̅ 𝑡 + 𝜔𝑍 𝑡 , 𝑡 = 1,2,3, …, [6]
where 𝑋 ̅ 𝑡 and 𝑍 𝑡 are defined in Eqs [1] and [3a], respectively.

Note that when 𝜔 = 0 in Eq [6], then the proposed SCSEWMA 𝑋 ̅ scheme reduces to the classical Shewhart 𝑋 ̅ scheme; however, when 𝜔 = 1, it reduces to the classical EWMA 𝑋 ̅ scheme. Thus, the proposed scheme borrows the strengths of the standalone Shewhart and EWMA 𝑋 ̅ control charts separately and it is flexible through the weighing parameter 𝜔.

The expected value and variance of the SCSEWMA 𝑋 ̅ statistic 𝑊 𝑡 in Eq [6] for Case K are given by

𝐸(𝑊 𝑡 ) = 𝜇 0 [7a]
and

𝑉𝑎𝑟(𝑊 𝑡 ) = ((1 -𝜔)(1 -𝜔 + 2𝜆𝜔) + 𝜆𝜔 2 2 -𝜆 (1 -(1 -𝜆) 2𝑡 )) 𝜎 0 2 𝑛 , [7b]
respectively. The derivations of the mean and variance (i.e. Eqs [7a] and [7b]) of the SCSEWMA 𝑋 ̅ charting statistic are provided in Appendix A.

Therefore, the time-varying control limits are given by 𝑈𝐶𝐿 𝑊 𝑡 /𝐿𝐶𝐿 𝑊 𝑡 = 𝜇 0 ± 𝐿 𝑊 √𝑉𝑎𝑟(𝑊 𝑡 ), [8] where 𝐿 𝑊 (𝐿 𝑊 > 0) is the SCSEWMA control limit constant which is chosen such that the attained 𝐴𝑅𝐿 0 is equal to some prespecified 𝐴𝑅𝐿 0 . The SCSEWMA 𝑋 ̅ scheme gives a signal when 𝑊 𝑡 plots beyond the control limits defined in Eq [8]. Otherwise, the process is said to be IC.

Note that when the process has been running for a very long time, i.e. 𝑡 → ∞, the term

(1 -(1 -𝜆) 2𝑡 )
converges towards one so that the asymptotic variance of 𝑊 𝑡 is reduced to:

𝑉𝑎𝑟(𝑊 𝑡 ) = ((1 -𝜔)(1 -𝜔 + 2𝜆𝜔) + 𝜆𝜔 2 2 -𝜆 ) 𝜎 0 2 𝑛 . [9]
Next, for Case U, the proposed SCSEWMA 𝑋 ̅ scheme is implemented in two regimes known as Phase I and Phase II. The process parameters 𝜇 0 and 𝜎 0 are estimated in Phase I, and these parameters are used to calculated the control limits of the SCSEWMA 𝑋 ̅ scheme. Thereafter, in Phase II, the control limits found in Phase I are used to continuously monitor the process. That is, In Case U, the IC process parameters 𝜇 0 and 𝜎 0 are estimated in Phase I using 𝑚 reference samples each of size 𝑛 when the process is deemed to be IC. The unbiased estimators for 𝜇 0 and 𝜎 0 are defined by

𝜇̂0 = ∑ ∑ 𝑋 𝑗𝑖 𝑛 𝑖=1 𝑚 𝑗=1 𝑚𝑛 [10a]
and ; see for example Abbas [START_REF] Abbas | Homogeneously weighted moving average control chart with an application in substrate manufacturing process[END_REF] . Thus, in Case U, the mean and variance of the SCSEWMA 𝑋 ̅ charting statistic are given by

𝜎 ̂0 = √ ∑ ∑ (𝑋 𝑗𝑖 -𝑋 ̅ 𝑗 )
𝐸(𝑊 𝑡 ) = 𝜇̂0 [11a]
and

𝑉𝑎𝑟(𝑊 𝑡 ) = ((1 -𝜔)(1 -𝜔 + 2𝜆𝜔) + 𝜆𝜔 2 2 -𝜆 (1 -(1 -𝜆) 2𝑡 )) 𝜎 ̂0 2 𝑛 , [11b]
where 𝜇̂0 and 𝜎 ̂0 are defined in Eqs [10a] and [10b], respectively. Consequently, the control limits of the SCSEWMA 𝑋 ̅ scheme, 𝐿𝐶𝐿 ̂𝑡 and 𝑈𝐶𝐿 ̂𝑡, are computed by substituting Eqs [11a] and [11b] in Eq

Run-length performance metrics

To evaluate the performance or sensitivity of a monitoring scheme, the SPM literature widely recommends the use of the characteristics based on the run-length such as the average run-length (ARL), the standard deviation of the run-length (SDRL) as well as the percentiles of the run-length (PRL) where the 50 th PRL represents the median run-length (MRL). The run-length variable represents the number of rational samples plotted on the scheme before it gives an OOC signal for the first time. Table 1 shows how to compute the Cases K and U characteristics of the run-length of the proposed monitoring scheme using Monte Carlo simulations.

Note that the run-length distribution can be computed using two different approaches generally known as the zero-state and steady-state modes. In zero-state, it is assumed that a significant change in the process occurs when the process starts. In other words, the process starts in an OOC state. However, in steady-state, the process starts IC and a significant change occurs at a random time after it has been running for a while. The zero-state and steady-state modes are used to characterise the short-term and long-term run-length properties of a monitoring scheme. In this paper, the main focus is on the zerostate properties of the proposed scheme. Nevertheless, in Section 4.3.2, the steady-state performance of the proposed scheme is compared to the zero-state one. Steps Estimation of the process parameters, search for 𝑳 𝑾 and the attained 𝑨𝑹𝑳 𝟎 value for some prespecified 𝑨𝑹𝑳 𝟎 value Specify the process parameters 𝜇 0 and 𝜎 0 (say, 𝜇 0 = 0 and 𝜎 0 = 1), the sample size (𝑛), the smoothing parameter 𝜆, the weighing parameter (𝜔), the number of simulation (𝑟) and the nominal (i.e. prespecified) 𝐴𝑅𝐿 0 value.

1

Specify the Phase I sample size (𝑚), the Phase II sample size (𝑛), the smoothing parameter 𝜆, the weighing parameter (𝜔), the number of simulation (𝑟) and the nominal (i.e. prespecified) 𝐴𝑅𝐿 0 value. Set 𝐿 𝑊 to some value and calculate the control limits, LCL and UCL, using Eq [8].

2 Generate m Phase I samples from a 𝑁(𝜇 0 ,𝜎 0 ) distribution, each of size 𝑛, say from 𝑁(0,1). At the 𝑡 𝑡ℎ sampling time, generate a 𝑁(𝜇 0 + 𝛿𝜎 0 ,𝜎 0 ) distribution of size 𝑛 where 𝛿 = 0, say from 𝑁(0,1) distribution.

3

Determine 𝑐 4𝑚 and afterwards, estimate the process parameters 𝜇 0 and 𝜎 0 denoted as 𝜇̂0 and 𝜎 ̂0 using Eqs [10a] and [10b]. Calculate the charting statistic 𝑊 𝑡 using Eq [6].

4 At the 𝑡 𝑡ℎ sampling time, generate a Phase II sample from 𝑁(𝜇 0 + 𝛿𝜎 0 ,𝜎 0 ) distribution of size 𝑛 where 𝛿 = 0, say from 𝑁(0,1). Compare the charting statistic found in Step 4 to the control limits computed in Step 2. If 𝑊 𝑡 ∈ (𝐿𝐶𝐿 𝑡 , 𝑈𝐶𝐿 𝑡 ) then return to Step 3. Otherwise, the scheme gives a signal. Record the number of samples needed to get an OOC signal. This is one value of the run-length vector.

5

Set 𝐿 𝑊 to some value and calculate the control limits, 𝐿𝐶𝐿 𝑡 and 𝑈𝐶𝐿 𝑡 , by substituting 𝜇̂0 and 𝜎 ̂0 found in Step 3 in Eq [8]. Repeat Steps 3 to 5 𝑟 times (say, 50000 times).

6

Calculate the charting statistic 𝑊 𝑡 using Eq [6]. Once the run-length vector, 𝑅𝐿 (𝑟𝑥1) , is obtained, calculate the 𝐴𝑅𝐿 0 as 𝐴𝑅𝐿 0 = . This value represents the attained 𝐴𝑅𝐿 0 . PROC UNIVARIATE can be used in SAS to find other characteristics of the run-length vector (or distribution).

7

Compare the charting statistic found in Step 6 to the control limits computed in Step 5. If 𝑊 𝑡 ∈ (𝐿𝐶𝐿 ̂𝑡, 𝑈𝐶𝐿 ̂𝑡) then return to Step 4. Otherwise, the scheme gives a signal. Record the number of samples needed to get an OOC signal. This is one value of the run-length vector. If the attained 𝐴𝑅𝐿 0 value is much closer or equal to the nominal 𝐴𝑅𝐿 0 value, records the 𝐿 𝑊 value. Otherwise return to Step 2; then increase the value of 𝐿 𝑊 if the attained 𝐴𝑅𝐿 0 is smaller than the nominal 𝐴𝑅𝐿 0 or decrease the value of 𝐿 𝑊 if the attained 𝐴𝑅𝐿 0 is larger than the nominal 𝐴𝑅𝐿 0 .

8

Repeat Steps 2 to 7 𝑟 times (say, 50000 times). . This value represents the attained 𝐴𝑅𝐿 0 . PROC UNIVARIATE can be used in SAS to find other characteristics of the run-length vector (or distribution). Use Steps 3 to 7 using the 𝐿 𝑊 found in Step 8 using 𝛿 ≠ 0. 10 If the attained 𝐴𝑅𝐿 0 value is much closer or equal to the nominal 𝐴𝑅𝐿 0 value, records the 𝐿 𝑊 value. Otherwise return to Step 2; then increase the value of 𝐿 𝑊 if the attained 𝐴𝑅𝐿 0 is smaller than the nominal 𝐴𝑅𝐿 0 or decrease the value of 𝐿 𝑊 if the attained 𝐴𝑅𝐿 0 is larger than the nominal 𝐴𝑅𝐿 0 .

Computation of the OOC 𝑨𝑹𝑳 value

Computation of the OOC 𝑨𝑹𝑳 value 11

Specify the shift 𝛿 (say, 𝛿 ∈ {0.1, 0.2, 0.3, …, 2}).

12

Use Steps 4 to 9 using the 𝐿 𝑊 found in Step 10 using 𝛿 ≠ 0.

Note though that the aforementioned metrics evaluate the performance of a scheme for a specific shift and not for a range of shift values or overall performance. To overcome this shortcoming, many researchers have recommended the use of the expected values of the previous characteristics such as the expected ARL (EARL), the expected SDRL (ESDRL) and the expected median run-length (EMRL).

Mathematically, the EARL, ESDRL and EMRL are defined by

𝐸𝐴𝑅𝐿 = 1 ∆ ∑ 𝐴𝑅𝐿(𝛿) 𝛿 max 𝛿=𝛿 min , 𝐸𝑆𝐷𝑅𝐿 = 1 ∆ ∑ 𝑆𝐷𝑅𝐿(𝛿) 𝛿 max 𝛿=𝛿 min and 𝐸𝑀𝑅𝐿 = 1 ∆ ∑ 𝑀𝑅𝐿(𝛿) 𝛿 max 𝛿=𝛿 min , [ 12]
respectively, where the 𝛿 ∈ [𝛿 min , 𝛿 max ], ∆ is the number of increments from 𝛿 min to 𝛿 max of the Riemann sum, 𝐴𝑅𝐿(𝛿), 𝑆𝐷𝑅𝐿(𝛿) and 𝑀𝑅𝐿(𝛿) are the ARL, SDRL and MRL for a specific shift 𝛿 in the process parameter. In this paper, we use increments of 0.1 in the summations in Eq [12], with 𝛿 min = 0.1 and 𝛿 max = 2. Based on the latter, it follows that ∆=20. Note that control chart constants are determined such that the attained 𝐴𝑅𝐿 0 is equal or almost equal to the prespecified value of 370.4.

Performance analysis of the SCSEWMA 𝑿 ̅ scheme

In this section, the robustness of the proposed SCSEWMA 𝑋 ̅ scheme is investigated under Cases K and U. In addition, the Case K (i.e. when 𝑚 = ∞) and Case U (i.e. when 𝑚 ≠ ∞) OOC performances of the SCSEWMA 𝑋 ̅ scheme are also investigated.

IC robustness of the SCSEWMA 𝑿 ̅ scheme

The IC robustness is very important in the design and implementation of a monitoring scheme in order to be certain of its shift detection capability in different situations such as the departures from the ideal (e.g. departure from the normal assumption). Balakrishnan et al [START_REF] Balakrishnan | Encyclopedia of Statistical Sciences[END_REF] (see p.7299) defined the robustness as "a procedure that performs well not only under ideal conditions under which it is designed but also under the departure from the ideal". Thus, to investigate the IC robustness of the SCSEWMA 𝑋 ̅ scheme, we used five different distributions, the standard normal distribution (denoted as N(0,1)), the Student's t distribution with degrees of freedom 5 and 25 (denoted as t( 5) and t (25), respectively), and the gamma distribution with shape parameters 𝛼 = 3 and 20 and scale parameter 𝛽 =1 (denoted as G (3,1) and G(20,1), respectively). The IC robustness is investigated when 𝜔 ∈{0.0,0.1,0.2, …,0.9,1.0} and 𝜆 ∈{0.1,0.5,0.9}, 𝑛 = 5, 𝑚 = ∞ (i.e. Case K) and 𝑚 = 10 and 100 (in Case U) for a prespecified 𝐴𝑅𝐿 0 of 370.4. From Table 2, it can be seen that overall, the SCSEWMA 𝑋 ̅ scheme is not IC robust since the 𝐴𝑅𝐿 0 profile is not the same across all continuous distributions. However, under the t distribution, as the degrees of freedom increases, the SCSEWMA 𝑋 ̅ scheme becomes robust. Under the gamma distribution, as the shape parameter increases, the SCSEWMA 𝑋 ̅ scheme also gets more robust. In addition, for both Cases K and U, the SCSEWMA 𝑋 ̅ scheme is more robust for large values of 𝜔. For other Phase I sample sizes, the findings remain the same except for very small Phase I sample sizes (see, for example, Table 2 when m = 10) where the results fluctuate significantly because of the large variability in the run-length distribution. Note though, both the EWMA and Shewhart 𝑋 ̅ schemes are not IC robust. It is worth mentioning that the EWMA 𝑋 ̅ scheme (i.e. 𝜔=1) tend towards robustness faster than the Shewhart 𝑋 ̅ scheme (i.e. 𝜔=0) under the above-mentioned situations.

Table 2. 𝐴𝑅𝐿 0 profile of the SCSEWMA 𝑋 ̅ scheme along with the corresponding control limits constants when n = 5, 𝜆 ∈{0.1,0.

Case K performance analysis

Case K IC and OOC performances of the proposed scheme

In this section, specific and overall performances of the proposed SCEWMA 𝑋 ̅ scheme are investigated in terms of the metrics introduced in Section 3.2. Tables 3 to 5 present the IC and OOC performances of the proposed SCSEWMA 𝑋 ̅ scheme in terms of the ARL, SDRL and MRL when 0 ≤ 𝜔 ≤ 1, 𝜆 = 0.1, 𝑛 = 5 and 𝛿 = 0 (0.1) 2, respectively. The triplet (𝜔, 𝜆, 𝐿) are selected for a prespecified 𝐴𝑅𝐿 0 = 370.4.

The last row of these tables gives the overall performance values as described in Eq [12]. Thus, the results in Tables 3 to 5 can be summarised as follows:

• The width of the proposed scheme is between the widths of the EWMA and Shewhart 𝑋 ̅ schemes, i.e. 𝐿 𝐸 ≤ 𝐿 𝑊 ≤ 𝑘.

• For small and moderate shifts in the process mean, in terms of the OOC ARL, SDRL and MRL profiles, the proposed SCSEWMA 𝑋 ̅ scheme performs better for large values of 𝜔, i.e. higher weight. For instance, if 𝛿 = 0.1, as the weighing parameter increases (i.e. when 𝜔 = 0, 0.5 and 1), we have 𝐴𝑅𝐿 = 296, 210 and 101, respectively (see Table 3). These findings also hold in terms of the overall performance. For instance, when 𝜔 = 0, 0.5 and 1, the SCSEWMA 𝑋 ̅ scheme yields EARL = 36.7, 20.7 and 9.6, respectively (see Table 3), which shows a significant increase in the overall sensitivity of the proposed scheme as 𝜔 increases. A similar pattern is observed in terms of the SDRL and ESDRL values (see Table 4).

• For large shifts, there is 50% chance that the proposed scheme gives a signal between the first and the second subgroups; that is, the MRL values for large shifts varies between 1 and 2 (see Table 5).

• The higher the shift in the process parameter, the more sensitive the SCSEWMA 𝑋 ̅ scheme becomes.

• The patterns in the sensitivity of the proposed scheme in terms of the ARL, SDRL and MRL profile is similar. This conclusion also holds in terms of the overall performance profiles. (i) In terms of the OOC ARL profile (see Figure 1),

• For small values of 𝜔, there is a slight difference in the performance of the proposed scheme for different values of 𝜆 regardless of the size of the shift in the process mean. As 𝜔 increases, the difference becomes noticeable for small and moderate shifts.

• Under small and moderate shifts, for moderate and large values of 𝜔, the smaller the value of 𝜆, the more sensitive the proposed scheme is. In other words, the proposed scheme performs better for small values of 𝜆.

• For large shifts, regardless of the values of 𝜆 and 𝜔, the performance of the proposed scheme is almost similar.

(ii) In terms of the EARL, ESDRL and EMRL values (see Figure 2),

• The proposed scheme performs better for small values of 𝜆 for all values of 𝜔 except when 𝜔 = 0, which is equivalent to the Shewhart 𝑋 ̅ scheme, where the overall performance of the proposed scheme is the same regardless of the value of 𝜆.

• For a fixed value of 𝜆, the overall performance of the proposed scheme increases as 𝜔 increases.

Case K performance comparison of the SCSEWMA 𝑿 ̅ scheme with the existing CSEWMA and CSCUSUM 𝑿 ̅ schemes

In this section, the proposed scheme is compared to the existing CSEWMA and CSCUSUM 𝑋 ̅ schemes.

The design of the CSCUSUM 𝑋 ̅ scheme entails three design parameters 𝑘, 𝑘 𝐶 and ℎ 𝐶 as well as the control limit 𝐻 𝐶 = ℎ 𝐶 𝜎 𝑋 ̅ . These parameters are chosen such that the attained 𝐴𝑅𝐿 0 value is equal to the prespecified value of 370.4 when 𝑛 = 5. In this comparison, we used 𝑘 𝐶 = 0.225, 0.5 and 0.75. The CSEWMA 𝑋 ̅ scheme entails two parameters as defined in Section 2. 6 and7 display the performance evaluation of the competing schemes in terms of the ARL and MRL profiles as well as the EARL and EMRL profiles.

From Tables 6 and7, it can be seen that in terms of the ARL and MRL profiles, the proposed scheme with 𝜔 = 0.9 is superior to the existing CSEWMA and CSCUSUM 𝑋 ̅ schemes under small and moderate shifts. However, for large shifts in the process mean, the competing schemes perform almost similarly. It can also be observed that the superiority of the proposed scheme is evident in terms of the overall performance. This is also explained by small EARL and EMRL values yielded by the proposed SCSEWMA 𝑋 ̅ scheme especially under small values of 𝜆 (see Tables 6 and7). Note: The control limit constants were round off at 3 decimal places to conserve space

Case U performance analysis

Case U IC and OOC performances of the proposed scheme

In this section, the performance of the proposed SCSEWMA 𝑋 ̅ scheme is investigated when 𝑛 ∈{1,5,10}, 𝑚 ∈{10, 50,100}, 𝜔 ∈{0.1,0.5,0.9} and 𝜆 ∈{0.1,0.5,0.9} for a prespecified 𝐴𝑅𝐿 0 = 370.4

as reported in Tables 8 to 10.

The pattern of the Cases K and U ARL profiles is the same; that is, as 𝜔 increases, the performance of the SCSEWMA 𝑋 ̅ scheme increases as well. The performance of the SCSEWMA 𝑋 ̅ scheme in terms of the ARL, SDRL, MRL, EARL, ESDRL and EMRL profiles, deteriorates when the process parameters are estimated; see Tables 8 to 10. For instance, if (𝜔, 𝜆) = (0.1,0.1), n = 5 and 𝛿 = 0.1, for m = ∞ (i.e.

Case K) and m = 100 (i.e. Case U), the SCSEWMA 𝑋 ̅ scheme gives a signal on the 284 th and 299 th samples, respectively (see Tables 3 and8, respectively). In Case U, the proposed SCSEWMA 𝑋 ̅ scheme is more sensitive for small values of 𝜆 and large values of 𝜔.

Table 8. Case U ARL and EARL profiles of the SCSEWMA 𝑋 ̅ scheme along with their corresponding 𝐿 𝑊 values when 𝜔 ∈{0.1,0.5,0.9}, 𝜆 ∈{0.1,0.5,0.9}, m = 100 and n=5 for a prespecified 𝐴𝑅𝐿 0 value of 370.4 𝜔 = 0.1 𝜔 = 0.5 𝜔 = 0.9 Shift 𝜆 = 0.1 𝜆 = 0.5 𝜆 = 0.9 𝜆 = 0.1 𝜆 = 0.5 𝜆 = 0.9 𝜆 = 0.1 𝜆 = 0.5 𝜆 = 0.9 0.0 370.9 370.9 (b) (𝜔,𝜆)=(0.1,0.9) (c) (𝜔,𝜆)=(0.9,0.1) (d) (𝜔,𝜆)=(0.9,0.9) 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 Figure 3. Effect of the Phase I sample size on the performance of the proposed SCSEWMA 𝑋 ̅ scheme when 𝛿 ∈{0.1, 0.2, …, 1.0}, n = 5, 𝑚 ∈{10,50,100}, (𝜔,𝜆) ∈{(0.1,0.1), (0.1,0.9), (0.9,0.1), (0.9,0.9)} for a prespecified 𝐴𝑅𝐿 0 value of 370.4

3.0 1.1 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.2 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.3 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.7 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.9 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.
(a) (𝜔,𝜆)=(0.1,0.1) (b) (𝜔,𝜆)=(0.1,0.9) 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 (c) (𝜔,𝜆)=(0.9,0.1) (d) (𝜔,𝜆)=(0.9,0.9)

Figure 4. Effect of the Phase II sample size on the performance of the proposed SCSEWMA 𝑋 ̅ scheme when 𝛿 ∈{0.1, 0.2, …, 1.0}, m = 100, 𝑛 ∈{2,5,10}, (𝜔,𝜆) ∈{(0.1,0.1), (0.1,0.9), (0.9,0.1), (0.9,0.9)} for a prespecified 𝐴𝑅𝐿 0 value of 370.4

Figure 4 investigates the effect of the Phase II sample size on the performance of the proposed SCSEWMA 𝑋 ̅ scheme when m = 100, 𝑛 ∈{2,5,10} for various (𝜔,𝜆). For more clarity, we only display the results for 𝛿 ∈ {0.1, 0.2, …, 1.0}. From Figure 4, it can be seen that regardless of the values of 𝜔 and 𝜆, the SCSEWMA 𝑋 ̅ scheme performs better for large Phase II sample sizes. When both the Phase I and Phase II samples are increased, the SCSEWMA 𝑋 ̅ scheme becomes more sensitive. In terms of the overall performance, Figure 5 shows that the larger the Phase I or Phase II sample, the more sensitive the proposed scheme becomes. The smaller the value of 𝜆, the better the performance of the proposed scheme. The larger the value of 𝜔, the more sensitive the SCSEWMA 𝑋 ̅ scheme (see Figure 5). Figure 6 investigates the effects of Phases I and II sample sizes on the SDRL profile and it can be seen that the proposed SCSEWMA 𝑋 ̅ scheme yields larger SDRL values for small Phase I or/and Phase II sample size(s) regardless of the combination of (𝜔,𝜆). To conserve space, we only show the results for (𝜔,𝜆) = (0.1,0.1) (see Figure 6). Thus, the larger the Phase I and/or Phase II samples, the smaller the variation in the ARL values and the better the ARL profile. Figure 8 presents the steady-state overall performances of the proposed SCSEWMA scheme in terms of the EARL profile when (𝜔, 𝜆) = (0.1, 0.1), (0.5,0.5) and (0.9,0.9), and (𝛿 𝑚𝑖𝑛 , 𝛿 𝑚𝑎𝑥 ) = (0.1, 2) for a nominal 𝐴𝑅𝐿 0 = 370. From this figure, it can be seen that the proposed scheme is more sensitive in Case K compared to Case U. In Case U, the larger the Phase I sample size, the more sensitive the proposed scheme is.

Steady-state versus zero-state performance of the proposed scheme

Case U performance comparison of the SCSEWMA 𝑿 ̅ scheme with the existing CSEWMA and CSCUSUM 𝑿 ̅ schemes

In Table 11, the Case U performance of the proposed SCSEWMA 𝑋 ̅ scheme with 𝜔 = 0.9 and 1 is compared to the ones of the existing CSEWMA and CSCUSUM 𝑋 ̅ schemes when m = 100, n = 5 and 𝜆 ∈ {0.1, 0.5, 0.9}. From Table 11, it can be seen that for small to moderate shifts, in terms of the ARL profile, the proposed SCSEWMA 𝑋 ̅ scheme with 𝜔 = 1 performs better followed by the CSEWMA 𝑋 ̅ scheme for small values of 𝜆. However, for large shifts, the competing schemes considered in this paper perform almost the same. In terms of the overall performance (i.e. EARL values), the proposed SCSEWMA 𝑋 ̅ scheme with 𝜔 = 0.1 outperforms the competing schemes. The performance of the competing schemes in terms of the SDRL and MRL profiles as well as the ESDRL and EMRL profiles yield similar findings (this is not shown here to conserve space). Note: The control limit constants were rounded off at 3 decimal places to conserve space

Illustrative examples with known and unknown process parameters

In this section, we use simulated and real-life data to demonstrate the application and implementation of the proposed SCSEWMA 𝑋 ̅ scheme along with the associated Shewhart and EWMA 𝑋 ̅ schemes for both Cases K and U.

Case K example: Simulated data

In this example, we assume that the observations are from the standard normal distribution. However, for monitoring purpose, eighty samples of size 5 are simulated from a normal distribution with a shift in the process mean of 0.25 standard deviation corresponding to a small shift; that is, 𝑋 𝑡𝑗 ~𝑁(0.25,1).

The Shewhart, EWMA and SCSEWMA 𝑋 ̅ schemes are designed for a prespecified 𝐴𝑅𝐿 0 = 370.4. The control limit constants are listed in Table 2 (keep in mind that the Shewhart and EWMA 𝑋 ̅ schemes are equivalent to the SCSEWMA 𝑋 ̅ scheme with 𝜔 = 0 and 1, respectively). Therefore, for 𝜔 = 0, 0.9 and 1, with 𝜆 = 0.1, it is found that 𝑘 = 3, 𝐿 𝑊 = 2.885 and 𝐿 𝐸 = 2.715 which are the control limit constants for the Shewhart, SCSEWMA (with 𝜔 = 0.9) and EWMA 𝑋 ̅ schemes, respectively. The plots of the charting statistics of the Shewhart, EWMA and SCSEWMA 𝑋 ̅ schemes are shown in Figure 9. From

Variable

UCL

Xbar LCL undesirable as it is not suitable to be further processed into steel. It is therefore imperative to monitor the percentage of silica that is present on each iron ore sample at the end of the flotation process.

In this example, it is assumed that the IC process mean and variance are known and equal to 2 and 1, respectively, with no loss of generality and the goodness of fit test for normality was not rejected at 5% level of significance. The data provide a set of 39 samples each of size 10 to be monitored. The Shewhart scheme which corresponds with the SCSEWMA scheme with 𝜔 = 0, does not signal in the prospective phase. However, when 𝜔 = 0.9 the proposed scheme gives a signal on the 16 th sample in the prospective phase and the EWMA scheme which corresponds with the CSEWMA scheme with 𝜔 = 1, gives a signal on the 15 th sample. These results reveal the flexibility of the proposed scheme and its superiority for large values of 𝜔. for a specified 𝐴𝑅𝐿 0 value of 370.4

Concluding remarks

This paper proposes a new CSEWMA 𝑋 ̅ scheme using a single plotting (or charting) statistic with known or unknown process parameters for monitoring the process mean. It is denoted as the SCSEWMA 𝑋 ̅ scheme. As for other parametric schemes, it was observed that the SCSEWMA 𝑋 ̅ scheme is not IC robust and its performance deteriorates when the process parameters are estimated. However, the larger the Phase I or Phase II sample, the more efficient the proposed scheme is. Compared to the existing Shewhart, EWMA and CSEWMA 𝑋 ̅ schemes, the new scheme is more flexible through an extra weighing design parameter. Moreover, the Shewhart and EWMA schemes are particular cases of the proposed SCSEWMA scheme when the weighing parameter is equal to 0 and 1, respectively. A comparative study of the SCSEWMA 𝑋 ̅ scheme and the existing CSCUSUM and CSEWMA 𝑋 ̅ schemes shows that the proposed SCSEWMA 𝑋 ̅ scheme is superior in monitoring small to moderate shifts and equivalent to the existing CSCUSUM and CSEWMA 𝑋 ̅ schemes in monitoring large shifts in the process mean. Operators in industrial and non-industrial organizations are advised to use the newly proposed SCSEWMA 𝑋 ̅ scheme instead of the existing CSEWMA 𝑋 ̅ scheme.

Researchers who are interested in developing similar schemes can also look at the design of the SCSEWMA scheme using other statistics such as the variance, coefficient of variation, etc., under symmetric and non-symmetric distributions. Researchers can also look at the design of the composite Shewhart double or triple EWMA scheme using a single charting statistic.

  Specify the shift 𝛿 (say, 𝛿 ∈ {0.1, 0.2, 0.3, …, 2}). 9Once the run-length vector, 𝑅𝐿 (𝑟𝑥1) , is obtained, calculate the 𝐴𝑅𝐿 0 as 𝐴𝑅𝐿 0

Figures 1 and 2

 2 Figures 1 and 2 investigate the performance of the proposed scheme in terms of the OOC ARL and overall performance profiles, respectively, for different values of 𝜔 and 𝜆. The following is observed:

Figure 1 . 4 (Figure 2 .

 142 Sensitivity of the proposed SCSEWMA 𝑋 ̅ scheme in terms of the ARL profile for different values of 𝜔 and 𝜆 for a prespecified 𝐴𝑅𝐿 0 value of 370.Overall performance of the proposed SCSEWMA 𝑋 ̅ scheme when 𝛿 𝑚𝑖𝑛 = 0.1, 𝛿 𝑚𝑎𝑥 = 2, n = 5 for prespecified 𝐴𝑅𝐿 0 value of 370.4

3 .

 3 The CSEWMA and SCSEWMA 𝑋 ̅ schemes are investigated for 𝜆 = 0.1, 0.2 and 0.3 when 𝑛 = 5 for a prespecified 𝐴𝑅𝐿 0 value of 370.4.

Table 9 . 4 𝜔

 94 Case U SDRL and ESDRL profiles of the SCSEWMA 𝑋 ̅ scheme along with their corresponding 𝐿 𝑊 values when 𝜔 ∈{0.1,0.5,0.9}, 𝜆 ∈{0.1,0.5,0.9}, m = 100 and n=5 for a prespecified 𝐴𝑅𝐿 0 value of 370.

Table 10 . 4 𝜔

 104 Case U MRL and EMRL profiles of the SCSEWMA 𝑋 ̅ scheme along with their corresponding 𝐿 𝑊 values when 𝜔 ∈{0.1,0.5,0.9}, 𝜆 ∈{0.1,0.5,0.9}, m = 100 and n=5 for a prespecified 𝐴𝑅𝐿 0 value of 370.

Figure 3 Figure 3 ,

 33 Figure 3 investigates the effect of the Phase I sample size on the Phase II performance of the proposed SCSEWMA 𝑋 ̅ scheme when 𝛿 ∈ {0.1, 0.2, …, 1.0}, n = 5, 𝑚 ∈{10,50,100} for various (𝜔,𝜆). From Figure 3, it can be observed that regardless of the values of 𝜔 and 𝜆, the SCSEWMA 𝑋 ̅ scheme performs better for large Phase I sample sizes.

= 5 Figure 5 .Figure 6 .

 556 Effect of the Phases I and II on the overall performance of the SCSEWMA 𝑋 ̅ scheme when 𝜔 ∈{0.1,0.5,0.9}and 𝜆 ∈{0.1,0.5,0.9} for a prespecified 𝐴𝑅𝐿 0 of 370.4 n = 5 when (𝜔,𝜆) = (0.1,0.1) (b) m = 100 when (𝜔,𝜆)=(0.1,0.1) Effects of the Phases I and II sample sizes on the SDRL profile of the proposed SCSEWMA 𝑋 ̅ scheme when (𝜔,𝜆)=(0.1,0.1) for a prespecified 𝐴𝑅𝐿 0 value of 370.4

Figure 7

 7 Figure 7 compares the zero-and steady-state performances of the proposed SCSEWMA schemes in terms of the ARL profile for different shifts when (𝜔, 𝜆) = (0.1, 0.1), 𝑚 = 10, 100 and ∞ for a nominal 𝐴𝑅𝐿 0 = 370. From Figure 7, it is observed that zero-and steady-state performances of the proposed

Figure 7 .

 7 (a) Case U when 𝑚 = 10 (b) Case U when 𝑚 = 100 (c) Case K (i.e. 𝑚 = ∞ ) Zero-state and steady state performances of the proposed CSEWMA scheme in terms of the ARL profile when (𝜔, 𝜆) = (0.1, 0.1), 𝑚 = 10, 100 and ∞ for a nominal 𝐴𝑅𝐿 0 = 370 Shift OOC ARL

Figure 8 .

 8 Figure 8. Steady-state overall performances of the proposed CSEWMA scheme in terms of the EARL profile when (𝜔, 𝜆) = (0.1, 0.1), (0.5,0.5) and (0.9,0.9), and (𝛿 𝑚𝑖𝑛 , 𝛿 𝑚𝑎𝑥 ) = (0.1, 2) for a nominal 𝐴𝑅𝐿 0 = 370

Figures 9 (

 9 Figures9 (a)-(c), it can be seen that the Shewhart 𝑋 ̅ scheme does not give a signal. However, both the EWMA and SCSEWMA 𝑋 ̅ schemes give a signal on the 35 th sample. This shows that the proposed SCSEWMA 𝑋 ̅ scheme borrows the strengths from both the Shewhart and EWMA 𝑋 ̅ schemes. In this particular case, the EWMA and SCSEWMA 𝑋 ̅ schemes outperform the Shewhart 𝑋 ̅ schemes in monitoring small shifts in the process mean.

9 Figure 10 . 4 5. 3

 91043 schemes gives a signal on the fourth subgroup.

9 Figure 11 .

 911 (c) SCSEWMA 𝑋 ̅ scheme with 𝜔 = 0.Case U CSEWMA 𝑋 ̅ monitoring scheme of the exit data when 𝑛 = 5, 𝑚 = 60 and 𝜆 = 0.1

  standalone Shewhart and EWMA 𝑋 ̅ schemes discussed in subsections 2.1 and 2.2, respectively. The CSEWMA 𝑋 ̅ scheme signals an OOC situation at time 𝑡 if one of the following conditions is satisfied. 𝑍 𝑡 ≥ 𝑈𝐶𝐿 𝐸 𝑡 or 𝑍 𝑡 ≤ 𝐿𝐶𝐿 𝐸 𝑡 .

	(i)	𝑋 ̅ 𝑡 ≥ 𝑈𝐶𝐿 or 𝑋 ̅ 𝑡 ≤ 𝐿𝐶𝐿, or
	(ii)

Table 1 .

 1 Computation of the Cases K and U run-length characteristics of the proposed SCSEWMA 𝑋 ̅ monitoring scheme

Case K Case U Steps Search of the optimal 𝑳 𝑾 values and computation of the attained 𝑨𝑹𝑳 𝟎 value for some prespecified 𝑨𝑹𝑳 𝟎 value

  5,0.9}, 𝑚 ∈{10,100} (i.e. Case U) and 𝑚 = ∞ (i.e. Case K) for different 𝜔 values under different distributions with a prespecified 𝐴𝑅𝐿 0 value of 370.4

					𝝀 = 0.1				𝝀 = 0.5				𝝀 = 0.9
	m	𝝎	𝑳 𝑾	N(0,1)	t(5)	t(25) G(3,1) G(20,1)	𝑳 𝑾	N(0,1)	t(5)	t(25) G(3,1) G(20,1)	𝑳 𝑾	N(0,1)	t(5)	t(25) G(3,1) G(20,1)
		0.0	3.047	371.4 118.6 324.7 292.4	399.5	3.046	370.9 112.6 321.6 292.9	363.9	3.046	369.8 114.6 314.6 300.5	375.7
		0.1	3.061	371.2 116.4 322.9 328.0	362.2	3.057	369.4 113.6 316.5 303.7	382.2	3.046	370.6 114.8 309.4 280.1	381.7
		0.2	3.065	370.7 113.3 316.4 337.8	390.5	3.056	371.2 116.3 309.8 336.5	368.1	3.047	370.8 116.1 316.8 320.3	384.0
		0.3	3.086	370.9 111.6 317.3 393.1	367.1	3.061	371.6 115.5 329.9 337.0	359.1	3.049	371.3 114.4 304.7 303.3	377.0
		0.4	3.100	371.4 114.9 305.8 423.5	372.7	3.067	371.0 114.2 320.5 347.2	379.4	3.052	369.8 116.2 314.5 317.0	368.9
	10	0.5	3.123	370.0 114.4 305.3 420.6	414.3	3.077	370.7 119.3 316.7 355.6	364.9	3.056	371.4 116.4 306.0 342.4	392.6
		0.6	3.152	370.0 116.0 296.2 446.8	370.7	3.082	371.2 122.9 319.9 367.5	381.4	3.055	369.4 116.5 326.9 320.5	392.3
		0.7	3.173	371.2 117.8 305.3 381.2	363.8	3.093	370.5 125.5 316.6 395.0	389.1	3.059	370.9 118.1 321.0 320.5	385.1
		0.8	3.201	369.2 132.0 337.7 334.4	348.7	3.099	371.9 128.2 328.4 411.2	366.1	3.059	371.7 115.7 324.7 321.3	386.0
		0.9	3.188	371.1 180.5 325.1 311.3	368.3	3.100	369.9 135.8 311.5 388.7	365.9	3.061	369.9 118.8 310.0 325.9	399.2
		1.0	3.108	370.1 280.8 359.9 315.7	353.7	3.098	371.4 142.4 335.4 386.7	339.8	3.060	372.2 114.4 313.5 316.2	392.9
		0.0	3.183	369.7 138.8 321.6 176.7	318.5	3.182	370.5 136.7 316.1 1774.0	316.4	3.183	370.3 137.5 317.8 174.7	319.5
		0.1	3.186	370.9 136.1 322.8 178.2	327.2	3.181	370.9 137.2 317.6 177.6	321.3	3.182	370.8 136.5 322.6 174.7	318.0
		0.2	3.189	369.9 136.1 316.0 181.4	322.2	3.187	369.9 136.7 315.1 178.5	322.2	3.183	370.7 133.6 318.0 177.5	318.5
		0.3	3.193	370.3 137.4 320.6 187.0	325.7	3.185	369.9 137.0 318.0 182.5	321.3	3.184	369.9 134.6 323.4 178.9	320.9
		0.4	3.195	371.9 139.1 322.5 196.3	332.2	3.186	368.9 141.5 324.5 186.0	324.1	3.182	369.8 136.0 321.6 176.2	321.7
	100	0.5	3.203	370.8 140.6 321.6 204.5	335.2	3.190	370.5 140.9 323.7 190.4	327.3	3.184	370.2 137.0 322.7 176.4	318.8
		0.6	3.209	371.4 145.8 327.0 221.3	336.5	3.188	370.1 144.2 324.0 196.5	328.1	3.184	369.6 136.4 316.1 177.8	317.9
		0.7	3.214	370.1 156.0 329.9 245.6	349.2	3.190	368.9 146.9 323.5 206.1	331.8	3.186	369.3 137.7 318.5 176.4	320.1
		0.8	3.203	371.9 177.6 335.6 274.2	353.9	3.189	369.7 155.4 324.5 214.7	335.6	3.184	371.1 138.2 317.3 177.5	322.2
		0.9	3.157	369.5 218.4 346.5 305.1	360.5	3.188	370.4 162.2 335.0 228.8	345.3	3.185	370.8 139.0 317.4 181.9	322.0
		1.0	3.003	369.7 290.8 361.1 334.5	361.2	3.186	370.2 175.3 337.0 240.9	345.9	3.187	371.3 140.1 318.5 181.8	323.5
		0.0	3.000	370.3 153.0 324.1 178.1	313.8	3.000	370.4 153.0 324.1 178.1	313.8	3.000	370.4 153.0 324.1 178.1	313.8
		0.1	2.999	370.3 151.7 326.7 177.9	313.0	3.001	370.4 153.1 326.2 180.2	315.2	3.000	370.8 153.0 322.6 179.8	313.5
		0.2	2.999	370.1 152.5 325.0 180.9	315.1	3.001	370.9 154.3 317.6 182.0	325.8	2.999	371.4 153.3 324.9 178.6	314.0
		0.3	2.998	370.2 152.3 322.0 183.8	315.5	2.998	371.2 154.6 324.2 183.2	313.6	2.999	370.7 152.1 324.9 179.3	314.1
		0.4	2.998	370.5 155.6 327.5 187.7	315.1	2.999	369.6 156.7 328.3 186.3	316.1	3.000	369.7 151.7 328.7 180.0	314.8
	∞	0.5	2.998	370.8 157.2 328.9 193.9	321.7	2.998	370.6 158.7 330.6 190.9	320.2	2.998	371.4 151.9 322.0 179.1	312.4
		0.6	2.995	369.9 164.3 334.6 201.5	324.5	2.998	370.6 161.4 329.1 192.6	319.5	2.999	371.9 153.2 323.3 180.5	316.0
		0.7	2.985	370.7 173.1 337.0 219.0	330.6	2.996	369.9 168.5 339.6 202.8	328.5	2.998	369.4 153.4 323.1 180.6	313.0
		0.8	2.961	369.5 197.4 344.1 252.1	343.1	2.994	371.2 175.8 337.5 211.6	329.5	2.998	370.0 152.0 322.8 180.9	314.0
		0.9	2.885	369.5 247.9 354.1 303.3	357.6	2.986	370.3 182.4 338.9 220.7	333.1	2.997	370.9 152.9 322.1 179.5	315.6
		1.0	2.715	370.5 337.3 370.3 361.0	370.9	2.980	371.1 196.4 349.3 236.9	339.1	2.998	369.8 153.0 325.2 182.0	316.8

Table 3 .

 3 Case K ARL and EARL profiles of the SCSEWMA scheme along with their corresponding 𝐿 𝑊 values when 𝜔 =0 (0.1) 1, 𝜆 = 0.1 and n=5 for a nominal 𝐴𝑅𝐿 0 value of 370.4 The control limits constants were rounded off at 3 decimal places to conserve space

	𝝎 →	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	𝑳 𝑾 →	3.000	2.999	2.999	2.998	2.998	2.998	2.995	2.985	2.961	2.885	2.715
	Shift↓						ARL profile ↓
	0.0	370.3	370.3	370.1	370.2	370.5	370.8	369.9	370.7	369.5	369.5 370.4
	0.1	295.9	284.3	272.5	253.9	232.6	209.9	182.5	157.2	130.0	109.6 101.3
	0.2	177.7	161.1	143.8	122.0	102.4	83.6	66.6	52.7	41.4	33.6	31.5
	0.3	99.5	85.7	72.7	60.8	48.7	39.5	31.1	24.8	19.6	15.9	15.2
	0.4	56.6	48.1	40.1	33.0	27.1	21.9	17.8	14.6	11.7	9.3	9.2
	0.5	33.1	28.3	23.8	20.0	16.6	14.0	11.8	9.7	7.9	6.3	6.3
	0.6	20.5	17.6	15.1	13.0	11.3	9.6	8.3	7.1	5.8	4.6	4.7
	0.7	13.2	11.6	10.3	9.0	8.0	7.1	6.2	5.4	4.4	3.5	3.7
	0.8	8.8	8.0	7.3	6.5	5.9	5.3	4.8	4.2	3.5	2.8	3.0
	0.9	6.2	5.7	5.3	4.9	4.6	4.2	3.8	3.4	2.9	2.3	2.5
	1.0	4.5	4.2	4.0	3.8	3.6	3.3	3.1	2.8	2.4	2.0	2.1
	1.1	3.4	3.3	3.2	3.0	2.9	2.7	2.6	2.4	2.1	1.7	1.9
	1.2	2.7	2.6	2.5	2.4	2.4	2.3	2.2	2.0	1.8	1.5	1.7
	1.3	2.2	2.1	2.1	2.1	2.0	2.0	1.9	1.8	1.6	1.4	1.5
	1.4	1.8	1.8	1.8	1.7	1.7	1.7	1.6	1.6	1.4	1.3	1.4
	1.5	1.6	1.6	1.5	1.5	1.5	1.5	1.5	1.4	1.3	1.2	1.3
	1.6	1.4	1.4	1.4	1.4	1.4	1.4	1.3	1.3	1.2	1.1	1.2
	1.7	1.3	1.3	1.3	1.3	1.3	1.2	1.2	1.2	1.2	1.1	1.2
	1.8	1.2	1.2	1.2	1.2	1.2	1.2	1.2	1.1	1.1	1.1	1.1
	1.9	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.0	1.1
	2.0	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.0	1.0	1.0
	EARL profile →	36.7	33.6	30.6	27.2	23.9	20.7	17.6	14.8	12.2	10.1	9.6
	Note:

Table 4 .

 4 Case K SDRL and ESDRL profiles of the SCSEWMA scheme along with their corresponding 𝐿 𝑊 values when 𝜔 =0 (0.1) 1, 𝜆 = 0.1 and n=5 for a nominal 𝐴𝑅𝐿 0 value of 370.4

	𝝎 →	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	𝑳 𝑾 →	3.000	2.999 2.999 2.998	2.998	2.998	2.995	2.985	2.961	2.885	2.715
	Shift↓					SDRL profile ↓
	0.0	369.4	368.8 369.7 368.9	370.2	372.6	369.4	368.1	374.8	384.9	377.5
	0.1	296.5	281.9 272.5 250.8	231.1	206.1	178.7	152.3	128.5	108.2	97.3
	0.2	177.5	157.7 140.4 119.2	98.0	78.5	60.5	46.3	36.0	29.6	26.2
	0.3	99.0	84.0	69.6	56.7	43.3	33.7	24.7	19.1	14.9	12.5	11.1
	0.4	55.9	46.9	37.2	29.3	22.4	16.9	12.8	10.1	8.1	6.9	6.1
	0.5	32.8	26.6	21.1	16.6	12.8	10.1	8.0	6.2	5.2	4.5	3.9
	0.6	20.0	16.1	12.9	10.3	8.3	6.6	5.4	4.4	3.6	3.1	2.8
	0.7	12.6	10.4	8.5	6.9	5.7	4.8	3.9	3.3	2.7	2.3	2.1
	0.8	8.3	7.0	5.9	5.0	4.2	3.6	3.0	2.5	2.1	1.8	1.6
	0.9	5.6	4.9	4.2	3.7	3.2	2.8	2.4	2.0	1.7	1.4	1.3
	1.0	3.9	3.5	3.1	2.8	2.5	2.2	1.9	1.6	1.4	1.1	1.1
	1.1	2.9	2.6	2.4	2.2	1.9	1.7	1.6	1.4	1.1	0.9	0.9
	1.2	2.1	2.0	1.8	1.7	1.6	1.4	1.3	1.1	1.0	0.8	0.8
	1.3	1.6	1.5	1.4	1.3	1.2	1.2	1.1	1.0	0.8	0.6	0.7
	1.4	1.2	1.1	1.1	1.1	1.0	0.9	0.9	0.8	0.7	0.5	0.6
	1.5	1.0	0.9	0.9	0.9	0.8	0.8	0.7	0.7	0.6	0.4	0.5
	1.6	0.7	0.7	0.7	0.7	0.7	0.6	0.6	0.6	0.5	0.4	0.4
	1.7	0.6	0.6	0.6	0.6	0.5	0.5	0.5	0.5	0.4	0.3	0.4
	1.8	0.5	0.5	0.4	0.4	0.4	0.4	0.4	0.4	0.3	0.2	0.3
	1.9	0.4	0.4	0.4	0.4	0.3	0.3	0.3	0.3	0.3	0.2	0.2
	2.0	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.2	0.2	0.1	0.2
	ESDRL profile →	36.2	32.5	29.3	25.5	22.0	18.7	15.5	12.7	10.5	8.8	7.9
	Note: The control limits constants were rounded off at 3 decimal places to conserve space

Table 5 .

 5 Case K MRL and EMRL profiles of the SCSEWMA scheme along with their corresponding 𝐿 𝑊 values when 𝜔 =0 (0.1) 1, 𝜆 = 0.1 and n=5 for a nominal 𝐴𝑅𝐿 0 value of 370.4

	𝝎 →	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	𝑳 𝑾 →	3.000	2.999	2.999	2.998	2.998	2.998	2.995	2.985	2.961	2.885	2.715
	Shift↓					MRL profile ↓
	0.0	257.0	260.0	254.0	256.0	258.0	256.0	253.0	259.0	254.0	251.0	252.0
	0.1	202.0	198.0	186.0	178.0	161.0	146.0	129.0	111.0	92.0	77.0	71.0
	0.2	122.0	112.0

Table 6 .

 6 Case K ARL and EARL comparisons of the SCSEWMA 𝑋 ̅ scheme versus the CSCUSUM and CSEWMA 𝑋 ̅ schemes when n = 5 for prespecified 𝐴𝑅𝐿 0 value of 370.4

			CSEWMA scheme	CSCUSUM scheme		SCSEWMA scheme (𝜔=0.9)
	Shift	𝜆 = 0.1	𝜆 = 0.2	𝜆 = 0.3	ℎ 𝐶 =12.964	ℎ 𝐶 =9.913 ℎ 𝐶 =9.909	𝜆 = 0.1	𝜆 = 0.2	𝜆 = 0.3
	0.1	165.1	202.1	225.9	140.7	276.4	294.2	109.6	148.1	178.5
	0.2	49.9	69.1	88.5	47.2	106.6	174.4	33.6	47.4	62.0
	0.3	21.6	28.1	36.7	26.5	39.3	81.8	15.9	20.6	26.6
	0.4	12.6	14.8	18.2	17.9	21.2	35.5	9.3	11.3	13.8
	0.5	8.3	9.2	10.8	13.1	14.1	19.5	6.3	7.3	8.6
	0.6	6.1	6.4	7.2	10.1	10.2	12.8	4.6	5.1	5.8
	0.7	4.7	4.9	5.2	8.0	7.8	9.1	3.5	3.9	4.4
	0.8	3.8	3.8	4.0	6.3	6.0	6.7	2.8	3.1	3.4
	0.9	3.1	3.2	3.3	5.1	4.8	5.2	2.3	2.5	2.7
	1.0	2.6	2.7	2.7	4.1	3.9	4.0	2.0	2.1	2.3
	1.1	2.3	2.3	2.3	3.3	3.1	3.2	1.7	1.8	2.0
	1.2	2.0	2.0	2.0	2.6	2.5	2.6	1.5	1.6	1.7
	1.3	1.8	1.8	1.8	2.2	2.1	2.1	1.4	1.4	1.5
	1.4	1.6	1.6	1.6	1.8	1.8	1.8	1.3	1.3	1.4
	1.5	1.5	1.5	1.5	1.6	1.6	1.6	1.2	1.2	1.3
	1.6	1.3	1.4	1.4	1.4	1.4	1.4	1.1	1.2	1.2
	1.7	1.2	1.3	1.3	1.3	1.3	1.3	1.1	1.1	1.1
	1.8	1.2	1.2	1.2	1.2	1.2	1.2	1.1	1.1	1.1
	1.9	1.1	1.1	1.1	1.1	1.1	1.1	1.0	1.0	1.1
	2.0	1.1	1.1	1.1	1.1	1.1	1.1	1.0	1.0	1.0
	EARL	14.6	18.0	20.9	14.8	25.4	33.0	10.1	13.2	16.1
	Control schemes constants	𝑘 = 3.068; 𝐿 𝐸 =3.203	𝑘 = 3.086; 𝐿 𝐸 = 3.239	𝑘 = 3.088; 𝐿 𝐸 = 3.249	𝑘 =3.034; 𝑘 𝐶 =0.225;	𝑘 =3.001; 𝑘 𝐶 =0.5;	𝑘 =2.999; 𝑘 𝐶 =0.75	𝐿 𝑊 =2.885	𝐿 𝑊 =2.934	𝐿 𝑊 =2.959

Note: The control limit constants were round off at 3 decimal places to conserve space

Table 7 .

 7 Case K MRL and EMRL comparisons of the SCSEWMA 𝑋 ̅ scheme versus the CSCUSUM and CSEWMA 𝑋 ̅ schemes when n = 5 for prespecified 𝐴𝑅𝐿 0 value of 370.4

			CSEWMA scheme	CSCUSUM scheme		SCSEWMA scheme (𝜔=0.9)
	Shift	𝜆 = 0.1	𝜆 = 0.2	𝜆 = 0.3	ℎ 𝐶 =12.964	ℎ 𝐶 =9.913 ℎ 𝐶 =9.909	𝜆 = 0.1	𝜆 = 0.2	𝜆 = 0.3
	0.1	117.0	141.0	156.0	106.0	193.0	204.0	77.0	102.0	123.0
	0.2	38.0	49.0	63.0	42.0	79.0	121.0	26.0	34.0	44.0
	0.3

Table 11 .

 11 Case U ARL and EARL comparisons of the SCSEWMA 𝑋 ̅ scheme versus the CSCUSUM and CSEWMA 𝑋 ̅ schemes when m =100 and n = 5 for prespecified 𝐴𝑅𝐿 0 value of 370.4

			CSEWMA scheme		CSCUSUM scheme		SCSEWMA scheme (𝝎=0.9)	SCSEWMA scheme (𝝎=1)
	Shift	𝝀 = 0.1	𝝀 = 0.5	𝝀 = 0.9	𝒉 𝑪 =9.762	𝒉 𝑪 =8.973	𝒉 𝑪 =9.164	𝝀 = 0.1	𝝀 = 0.5	𝝀 = 0.9	𝝀 = 0.1	𝝀 = 0.5	𝝀 = 0.9
	0.1	177.9	253.7	297.5	153.69	273.4	302.6	167.8	254.2	297.8	157.6	246.3	291.6
	0.2	48.0	105.8	169.3	43.9	109.0	181.5	45.5	109.6	174.6	40.3	102.2	169.9
	0.3	20.0	45.5	91.2	22.73	39.6	83.3	19.9	47.8	92.4	17.6	43.3	91.1
	0.4	11.5	22.2	48.5	15.33	20.5	35.8	11.5	23.6	51.0	10.2	21.5	49.1
	0.5	7.6	12.4	28.4	11.5	13.3	19.2	7.8	13.2	29.0	6.9	12.1	28.3
	0.6	5.5	7.9	16.8	9.16	9.7	12.3	5.8	8.3	17.4	5.0	7.7	17.1
	0.7	4.3	5.5	10.8	7.59	7.3	8.7	4.5	5.8	11.1	4.0	5.4	10.9
	0.8	3.4	4.1	7.3	6.42	5.8	6.6	3.6	4.3	7.6	3.2	4.1	7.2
	0.9	2.8	3.3	5.2	5.45	4.6	5.1	3.0	3.3	5.4	2.7	3.2	5.1
	1.0	2.4	2.7	3.8	4.68	3.8	3.9	2.6	2.8	3.9	2.3	2.6	3.8
	1.1	2.1	2.3	3.0	4.01	3.1	3.2	2.2	2.3	3.0	2.0	2.3	3.0
	1.2	1.9	2.0	2.4	3.42	2.5	2.6	2.0	2.0	2.5	1.8	2.0	2.4
	1.3	1.7	1.7	2.0	2.93	2.1	2.1	1.8	1.8	2.0	1.7	1.7	2.0
	1.4	1.5	1.6	1.7	2.51	1.8	1.8	1.6	1.6	1.7	1.5	1.6	1.7
	1.5	1.4	1.4	1.5	2.16	1.6	1.6	1.5	1.4	1.5	1.4	1.4	1.5
	1.6	1.3	1.3	1.4	1.86	1.4	1.4	1.3	1.3	1.4	1.3	1.3	1.4
	1.7	1.2	1.2	1.3	1.64	1.3	1.3	1.3	1.2	1.3	1.3	1.2	1.3
	1.8	1.2	1.2	1.2	1.45	1.2	1.2	1.2	1.2	1.2	1.2	1.2	1.2
	1.9	1.1	1.1	1.1	1.32	1.1	1.1	1.1	1.1	1.1	1.2	1.1	1.1
	2.0	1.1	1.1	1.1	1.22	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.1
	EMRL	14.9	23.9	34.8	15.2	25.2	33.8	14.4	24.4	35.3	13.2	23.2	34.5
	Control schemes constants	𝑘 = 3.467; 𝐿 𝐸 =3.194	𝑘 = 3.707; 𝐿 𝐸 = 3.204	𝑘 = 3.184; 𝐿 𝐸 = 3.624	𝑘 =3.769; 𝑘 𝐶 =0.225;	𝑘 =3.197; 𝑘 𝐶 =0.5;	𝑘 =3.181; 𝑘 𝐶 =0.75	𝐿 𝑊 =3.155	𝐿 𝑊 =3.188	𝐿 𝑊 =3.185	𝐿 𝑊 =3.004	𝐿 𝑊 =3.177	𝐿 𝑊 =3.182

  Shewhart 𝑋 ̅ scheme (or SCSEWMA 𝑋 ̅ scheme with 𝜔 = 0) (b) EWMA 𝑋 ̅ scheme (or SCSEWMA 𝑋 ̅ scheme with 𝜔 = 1)

		2.5 0.20							Variable
		2.4							Wt LCLw	Variable
		0.15							UCLw	Xbar
		2.3										LCL
	statistic statistic	2.2 0.10										UCL
	Charting Charting	2.1 2.0 0.05
		1.9
		1.8
		0.00
		1.7
		2	4	4	8	6	12 8	16 10	12	20	14	24	16	28 18	32 20	22	36	24
		Sample number / Time (a) Sample number/ Time
		0.11
		0.10										Variable
												Zt
												LCLe
	Charting statistic	0.07 0.08 0.09										UCLe
		0.06
		0.05
		2		4		6	8	10	12		14		16	18	20	22		24
								Sample number/ Time

Case K example: Monitoring the level of silica concentrate in iron ore

In this example, real-life data are used to demonstrate the application and implementation of the proposed scheme. Froth flotation approach is used to enhance the iron concentration of low-grade iron ores. Low-grade iron ores contain high concentration of impurities, such as silicon dioxide or in short, silica (these are quartz or sand), phosphorus and alumina containing mineralswhich are undesired.

Froth flotation process is an effective approach to remove impurities. Mukherjee et al [START_REF] Mukherjee | Comparisons of some distribution-free CUSUM and EWMA schemes and their applications in monitoring impurity in mining process flotation[END_REF] showed that the quality characteristic of interest in these data is the percentage of silica concentrate that remains as an impurity at the end of the froth flotation process. A high level of silica concentrate in the iron ore is

Appendix A

In this appendix, we show how the properties of the SCSEWMA scheme are derived.

A.1 The mean of the SCSEWMA statistic

where

and

Then,

Since, the EWMA 𝑋 ̅ statistic is defined as

𝑊 𝑡 can then be written as

Therefore,

A.2 The variance of the SCSEWMA statistic

From Eq [A.3], the variance of the SCSEWMA statistic is derived as follows:

].

We know that the 𝐶𝑜𝑣(𝑋 𝑗 , 𝑋 𝑖 ) = 0 ∀𝑗 ≠ 𝑖 since the observations are i.i.d. and that

Thus,

Therefore,

Note that Eq [A.5] can also be written as:

Thus, from Eq [A.5], when Note that in Case U, the IC process mean (𝜇 0 ) and variance (𝜎 0 2 ) are substituted with their unbiased estimators i.e. 𝜇̂0 and 𝜎 ̂0 2 .


Marek Koutnya

Pierre Amato

Marketa Muchova

Jan Ruzicka

Anne-Marie Delort

Soil bacterial strains able to grow on the surface of oxidized polyethylene film containing prooxidant additives

Keywords: Polyethylene, Biodegradation, Soil, Prooxidants, Bacteria, Hydrophobicity

Twelve bacterial strains able to adsorb and grow on the surface of oxidized low-density polyethylene film containing prooxidant additives were isolated from three forest soils and subsequently identified. Most of them belonged to different genera of the proteobacteria group; however, three of the isolates were Rhodococcus strains. With the exception of one of the Rhodococcus strains, the isolates did not exhibit significant hydrophobicity of their cell surfaces. The study showed that bacteria capable of adhering to the surface of oxidized polyethylene, growing there and possibly biodegrading its oxidation products are not rare in forest soils and that they belong to different taxonomical groups common in soil environment.

Introduction

In the last few years, polyethylene (PE) films containing prooxidant additives have been introduced onto the market as a new material promising biodegrability, in conjunction with the continual use of existing production and processing technologies and with reasonable prices of products [START_REF] Wiles | Polyolefins with controlled environmental degradability[END_REF].

Primary applications are in agricultural greenhouse construction, mulching films, packaging films as well as in other products with a limited lifetime, e.g. bags. These articles are intended to be biodegradable in soil or during composting. As yet, only a little data exists to support the idea of such materials' biodegradability [START_REF] Chiellini | Biodegradation of thermally-oxidized, fragmented, low-density polyethylenes[END_REF], [START_REF] Chiellini | Oxo-biodegradable full carbon backbone polymers -biodegradation behaviour of thermally oxidized polyethylene in an aqueous medium[END_REF] and [START_REF] Jakubowicz | Evaluation of degradability of biodegradable polyethylene (PE)[END_REF]). However, the microbiology of the process remains almost completely unknown. Here, an attempt was made to isolate, identify and partially characterize soil microbial species able to adsorb and grow on the surface of oxidized low-density PE film containing prooxidant additives. The microorganisms attached to the surface of the material could extract and utilize oxidation products of PE, a broad spectrum of compounds; mainly various fatty acids like substances, but other functionalities like esters, ketones, alcohols and double bonds can also be found ( [START_REF] Albertsson | Degradation product pattern and morphology changes as means to differentiate abiotically and biotically aged degradable polyethylene[END_REF] and [START_REF] Khabbaz | Chemical and morphological changes of environmentally degradable polyethylene films exposed to thermo-oxidation[END_REF]). Thus the adherent microorganisms can be regarded as a potential biological agent in the proposed scheme for degradation of PE with prooxidant additives (Koutny et al., 2006b). So far, the most often investigated bacteria in relation to PE degradation were different members of Actinobacteria, above all various Rhodococcus strains ( [START_REF] Orr | Colonization, biofilm formation and biodegradation of polyethylene by a strain of Rhodococcus ruber[END_REF], [START_REF] Sivan | Biofilm development of the polyethylene-degrading bacterium Rhodococcus ruber[END_REF] and [Koutny et al., 2006b]). But, completely different taxonomical groups were also reported like e.g. pseudomonads or bacilly [START_REF] Albertsson | Increased biodegradation of LDPE with nonionic surfactant[END_REF] and [START_REF] Sudhakar | Marine microbe-mediated biodegradation of low-and high-density polyethylenes[END_REF].). A previously described preoxidized low-density PE film containing prooxidant additives was used throughout this study (Koutny et al., 2006a). The described preoxidation treatment should simulate several years' exposure of the material to outdoor conditions.

Material and methods

Preoxidized PE film was transparent LDPE film 60 mm thick containing prooxidant additives based on organometallic complexes. To balance the prodegradant activity phenolic antioxidants were used in the blends. The sample was exposed for 120 h in the accelerated photoageing chamber SEPAP 12.24 (λ ≥ 300 nm, 60 ± 1 °C). After the exposure in SEPAP the samples were considered sterile and treated in a sterile way. Exposure in SEPAP was followed by 300 h of exposure in an aerated oven at 60 °C. For the examined PE film 100 h of exposure in SEPAP 12.24 corresponds to 3 months weathering under European outdoor conditions in the period from March to October and incubation in the oven was equivalent to 2-3 years of incubation at room temperature in the dark (particles buried in the soil or particles not exposed to sunlight). See Koutny et al. (2006b) for more details. PE film preoxidized in this way, already highly fragmented and very fragile, was passed through a metallic screen with eye diameters of about 1 mm to obtain uniform fragments for further work.

Upper horizons of three different forest soils sampled above Clermont-Ferrand (France) were chosen as the source materials (Table 1). An upper soil horizon to a depth of about 10 cm was sampled using a sterile knife, after removing undegraded plant debris and a surface layer of approximately 2 cm. Three sub-samples were taken at each location and then pooled in a sterile plastic bag. The final weight of samples was about 200 g. Soil microorganisms were first released from soil particles into suspension solution. 10 g of aseptically weighed soil was added to 100 ml of suspension solution (Tween 80 1.5 g l -1 ; Na 4 P 2 O 7 4.0 g l -1 ; NaCl 5.0 g l -1 ) and the mixture was vigorously shaken for 30 min. Then 100 µl of the suspension was inoculated into 50 ml of basic mineral salt medium (3.8 g Na 2 HPO 4 × 12H 2 O, 1.8 g KH 2 PO 4 , 0.02 g MgSO 4 × 7H 2 O, 0.03 g Fe(NH 4 ) 2 (SO 4 ) 2 × 6H 2 O, 0.01 g CaCl 2 × 2H 2 O, 0.5 g NaCl, 0.3 g NH 4 Cl and 1 ml of trace element solution per litre [START_REF] Vaclavkova | Novel aspects of symbiotic (polyvinyl alcohol) biodegradation[END_REF], all chemicals purchased from Sigma-Aldrich) containing 100 mg of fine fragments of preoxidized PE film in 250 ml Erlenmayer flask. Although the soils were rather acidic, pH 7 was chosen for the medium, a value which is tolerated well by the majority microorganisms. Cultures were incubated for 30 days with gentle shaking (20 rpm) at 20 °C. After this period, small aliquots (about 5 mg) of PE fragments were collected and transferred into 1.5 ml tubes and repeatedly washed (maximum of liquid was drained out with a pipette, 1.5 of fresh mineral medium was added and the suspension briefly mixed on a vortex, this was repeated eight times) in order to wash out all planktonic (non-adherent) microorganisms. Afterwards, the PE fragments were spread on three types of agar media: on R2A agar (Difco) to recover adhered bacterial species; on Emerson agar (Difco), initially intended mainly to report fungi and actinomycetes; and on Sabouraud agar (Difco) for fungi and yeast species. Plates were incubated 10 days at 25 °C. Following this, individual microbial strains were isolated on the basis of different colony appearance and observed growth rates and designated with codes beginning S1, S2 and S3 according to the soil of their origin and a letter R, E or S, in the middle, according to the particular agar media of their isolation. Purity of each strain was checked on a fresh streak plate and microscopically (after Gram staining). Once they had been isolated, each pure strain was again inoculated into 50 ml of basic mineral salt medium containing 100 mg of fine fragments of preoxidized PE film. The initial cell density was set at 10 4 cells per millilitre on the basis of cell counting on a microscope. After 30 days at 25 °C, microbial growth on the surface of preoxidized PE was confirmed by optical microscopy after simple fixation (4% formaldehyde, 2 h at 4 °C) and staining with safranine. Growth was judged as significant if 10 2 or more cells appeared on the material's surface in every microscope field. Strains excluded from further investigation, on the other hand, exhibited no growth, or there were observed only rare individual cells. Because the PE fragments were quite irregular in shape and size it would have been extremely difficult to count the cells and evaluate microscopically the growth in a strictly quantitative way, e.g. cells per mm 2 . Therefore, the density of the surface coverage for individual strains was evaluated semi-quantitatively by optical microscopy (mag. 1000×) on the scale 1-3: 1, 10 2 cells/field or individual microcolonies; 2, many microcolonies in every microscope field, sometimes in contact; 3, larger areas of complete surface coverage in every microscope field (Table 2).

Table 2. Isolated and identified stains and their hydrophobicity using BATH assay. [START_REF] Bliss | Statistics in Biology[END_REF] are in parentheses. Nd: Strains were unable to growth on hexadecane.

Growth was evaluated on a scale 1-3, see Section 2.

Proved strains (12 from 25 initially isolated) were identified by 16S rRNA gene sequencing [START_REF] Amato | Microorganisms isolated from the water phase of tropospheric clouds at the Puy de Dôme: major groups and growth abilities at low temperatures[END_REF]. DNA was isolated using Easy-DNA™ kit (Invitrogen, Carlsbad, US). 16S rRNA gene amplification was carried out by PCR, using 27f (5′-AGAGTTTGATCCTGGCTCAG-3′) and 1492r (5′-GGTTACCTTGTTACGACTT-3′) primers, isolated DNA fragments were sent for sequencing (MWG-Biotech, Courtaboeuf, France) and the obtained sequence was analyzed using BLASTN (http://www.ncbi.nlm.nih.gov/BLAST/).

Hydrophobicities of cell surfaces were judged using the BATH test [START_REF] Rosenberg | Bacterial adherence to hydrocarbons: a useful technique for studying cell surface hydrophobicity[END_REF]. Bacterial strains were grown five days on R2A agar and/or eight days on mineral agar with hexadecane as the sole carbon and energy source (described mineral media components plus 15 g of agar and 0.4 g of hexadecane per litre, hexadecane was finally present in emulsion and as a thin surface layer on the agar plates), both cultivations at 25 °C. Cells were then scratched from the medium surface, resuspended in phosphate buffered physiological solution (PBS, pH 7.3), washed twice in PBS and finally diluted to OD 0.5 (400 nm). An aliquot (0.8 ml) of a bacterial suspension in PBS was added to hexadecane (160 µl) and vortex mixed at constant speed for 5 min. Following equilibration, the lower aqueous layer was carefully removed and its OD measured at 400 nm. Three tubes for each strain were measured. The experiment was repeated with newly cultivated cells and almost identical values were obtained. The hydrophobicity is expressed as the percentage of the applied cell suspension OD which had been excluded from the aqueous phase (results in the interval 0-100%, low value means low hydrophobicity).

Contact angle values of initial and oxidized PE were measured by sessile drop technique using the Surface Energy Evaluation (SEE) System (Advex Instruments, Brno, Czech Republic). Contact angles were measured directly from the images of the solid-liquid meniscus of a liquid drop set on a solid surface. For each liquid the measurement were done for 10 drops. The correction to roughness was not made, i.e., obtained values of contact angle could be higher for the non-wettable materials. Dispersive and polar surface tension components were computed according to Fowkes' theory from the contact angle values of two test liquids i.e. diiodomethane and water.

Results and discussion

The choice of forest soils was justified by two main reasons: firstly, intensive degradation processes take place in this environment due to rich and active microbial communities; secondly, it was shown that microorganisms present in a forest soil can initiate degradation of similar PE material [START_REF] Chiellini | Biodegradation of thermally-oxidized, fragmented, low-density polyethylenes[END_REF].

We decided not to put PE directly into contact with soil but first release microorganisms into suspension solution. This procedure assured better access of microorganisms to the PE surface than in a heterogeneous soil environment and also reduced the risk that during the subsequent separation of adhering cells some unwanted microorganisms would be transferred along with a soil particle.

The described isolation and identification procedure finally provided 12 bacterial strains listed in Table 2.

In this study, the majority of isolates belonged to Beta-or Gamma-proteobacteria groups but the incidence of Rhodococcus strains was also important. Among isolated strains, Rhodococcus also covered PE surface with the highest density. However, a striking density of cells was registered with, for example, Pseudomonas sp. S1R4 as well. Pseudomonas and Rhodococcus are among the most studied degraders of various xenobiotics including some very recalcitrant ones. Both genera are also described as producers of biosurfactants important for adhesion and utilization of poorly soluble substances ( [START_REF] Larkin | Biodegradation and Rhodococcus -masters of catabolic versatility[END_REF] and [START_REF] Das | Characterization of biochemical properties and biological activities of biosurfactants produced by Pseudomonas aeruginosa mucoid and non-mucoid strains isolated from hydrocarbon-contaminated soil samples[END_REF]). Rhodococci especially has the capacity to degrade larger nonpolar molecules. Other isolated genera Flavobacterium [START_REF] Brakstad | Biotransformation and dissolution of petroleum hydrocarbons in natural flowing seawater at low temperature[END_REF], Delftia [START_REF] Patil | Degradation of plasticizer di-n-butylphthalate by Delftia sp[END_REF], and et al., 2007) were also reported as potent degraders of xenobiotics. Quite surprisingly, no fungal or yeast species were isolated. This could be partly because of the procedure of isolation but we believe it also reflects the higher affinity of bacteria species to attach and grow on oxidized PE surface.

Ralstonia (Ryan
An attempt was made to investigate whether the strains could be divided to several groups according to their metabolic capacities. Therefore their metabolic profiles, based on testing of utilization of 33 different carbon substrates on Biolog Eco-plates™ (Biolog, USA), according to manufacturer's instructions, were compared. Principal component analysis failed to divide the isolated strains into distinguished groups (data not shown).

Surface characteristics of isolated microorganisms could have a crucial role in cell adsorption and utilization of insoluble substances [START_REF] Orr | Colonization, biofilm formation and biodegradation of polyethylene by a strain of Rhodococcus ruber[END_REF]. Original PE film itself proved highly hydrophobic.

After the oxidation treatment, preceding its use in our experiments, its hydrophobicity should be significantly diminished because of the polar chemical moieties formation in the material and on its surface. This was confirmed by contact angle measurement of standard testing liquids and the subsequent calculation of surface tension components [START_REF] Sprang | Plasma and ion beam surface treatment of polyethylene[END_REF]. After the oxidation treatment, the polar component of surface tension increased by more than three times (Table 3). Surface hydrophobicity of isolated strains was characterized by their partitioning between aqueous and nonpolar solvents, in the so-called BATH assay [START_REF] Rosenberg | Bacterial adherence to hydrocarbons: a useful technique for studying cell surface hydrophobicity[END_REF] (Table 2). Cells were grown for the assay either on R2A agar or on mineral agar with hexadecane as a model nonpolar substrate; in cases the strains were capable of its utilization. In general, isolated strains did not appear significantly hydrophobic, with the only exception being Rhodococcus sp. S3E3. When cultivated on hexadecane the latter strain exhibited 38% hydrophobicity in BATH assay. Rhodococci are known for their ability to form hydrophobic envelopes and metabolize larger nonpolar molecules [START_REF] Sokolovska | Carbon sourceinduced modifications in the mycolic acid content and cell wall permeability of Rhodococcus erythropolis E1[END_REF]. For other strains, cultivable on hexadecane agar, the hydrophobic substrate did not induce a marked change in their surface properties. It must be admitted that the properties of cells cultivated even on hexadecane media can still differ from those grown on the authentic preoxidized PE surface, but it is technically impossible to collect the quantity of cells necessary for their characterization directly from the PE film surface.

Accepting, at least partially, the relevance of presented results, the data could suggest that the hydrophobicity of cells and probably hydrophobic interactions did not play a major role in cell adhesion in this case. Non-hydrophobic bacteria sparsely colonizing a similar material was also described by [START_REF] Hadad | Biodegradation of polyethylene by the thermophilic bacterium Brevibacillus borstelensis[END_REF]. In this respect it should be noted that carbonyl, carboxy and hydroxy groups capable of electrostatic interactions or hydrogen bonding are present on the surface of oxidized PE and can mediate such interactions with the cell surface of microorganisms. Production of extracellular substances, acting as an adhesive between a cell and the surface appears relevant. Both and Rhodococci, exhibiting the highest capacity to colonize the investigated material, are known to produce extracellular polymers and biosurfactants.

It can be concluded that, in agreement with other authors ( [START_REF] Bonhomme | Environmental biodegradation of polyethylene[END_REF] and [START_REF] Orr | Colonization, biofilm formation and biodegradation of polyethylene by a strain of Rhodococcus ruber[END_REF]), several members of Rhodococcus genus were isolated proving their role in the interaction of preoxidized PE with natural bacterial communities. However, it was shown as well that Rhodococci, or more general actinomycetes, are not a prominent group of microorganisms colonizing the surface of the material. Broader spectrum of isolated taxons showed that the microorganisms capable of adhering to preoxidized PE surface and metabolizing some of its oxidation products are not rare in forest soils and because the identified genera are relatively common, an assumption could be made, that similar results can be expected in other types of soils with rich microbial communities. If the prooxidant technology ensures continuation of the PE chain breakdown process after PE waste disposal, one can expect in soil, such oxidation products to be biodegraded.

Table 1 .

 1 Characterization of investigated soil samples.

pH (KCl) Organic content Altitude Dominant tree species

	Soil 1 3.6	52% (1.0)	680 m	Picea sp.
	Soil 2 5.1	28% (3.2)	705 m	Quercus sp., Carpinus sp.
	Soil 3 5.1	23% (2.8)	725 m	Quercus sp., Robinia sp.

Standard deviations are in parentheses; n = 3.

Table 3 .

 3 Surface characterizations of original and oxidized PE films.

	Sample		Dispersive component of surface tension (mN m -1 )	Polar component of surface tension (mN m -1 )
	Original film	PE	42 (0.91)	0.58 (0.025)
	Oxidized film	PE	37 (1.2)	1.8 (0.087)
	Numbers in parentheses are standard deviations.

Acknowledgment

This study was partially supported by the Czech Ministry of Education, Youth and Sports, project no. MSM 7088352101.


Robin Baudouin
email: robin.baudouin@aol.com

MD Françoise Denoyelle

François Simon

Shedding light on the tympanic membrane, a brief history of the description and understanding of its anatomy

Keywords: Tympanum, history, anatomy, Otorhinolaryngologic Diseases

Objectives. For centuries, the tympanum has remained the only visible structure of the organ of hearing. The aim of this study was to trace the understanding of the tympanic membrane (TM) from Antiquity to the early 20 th century.

INTRODUCTION

The tympanic membrane (TM) or eardrum has been for centuries the only window through which to inspect the hearing organ. From early Antiquity and the polymaths' first descriptions, little progress was made until the great anatomists of the Renaissance. Major breakthroughs in the understanding of the TM anatomy later took place in the 19 th century.

The objective of this paper was to trace the progress of the description of the TM from Antiquity to the early 20 th century. We chose to focus on the key steps, to identify how underlying historical context (beliefs, misconceptions or technology) influenced the understanding of the TM anatomy.

METHODS

A search for scientific medical or historical literature was conducted using PubMed, Google Scholar and the Bibliothèque Inter-Universitaire (BIU) Santé catalogue of the medical faculty of the Université de Paris. Screening was performed using the following key-words: eardrum, tympanic membrane, tympanum, tympanoplasty, hearing theory, audition, otolaryngology history, Hippocrates, Aristotle, Galen, Avicenna, Vesalius, Fabricius, Fallopio, Shrapnell, Toynbee and Politzer. For primary sources, when a text was identified as pertinent, we used the Medica® digital library of the BIU Santé of the medical faculty of the Université de Paris to access and read the full text. If the text was not available, a manuscript was searched for over internet. For texts found in Latin, Greek, Arabic, French, Italian or German, official translations to English were used. Selected texts from primary sources and data from secondary articles were gathered and organized in a chronological order and then by connected theme.

RESULTS

From ancient polymaths to great Renaissance anatomists

In Antiquity, the tympanum was thought to be an interface between two worlds: the human being (outer) and the soul (inner) world. This concept remained entrenched for centuries [START_REF] Gamatsi | The ear and its malformations: strange beliefs and misconceptions[END_REF].

It is uncertain that the Egyptians described the TM but they talked about ear diseases in medical treaties thousands of years before Hippocrates (2), although it is he who is generally credited with the first description, depicting a dry thin-spun web [START_REF] Weir | History of Otology[END_REF]. Ancient Greeks discussed the anatomy and function of the ear as philosophers, in their quest to understand the mystery of the senses. Hearing was a singular sense, likely to perceive the visible (human) as well as the invisible (the gods') worlds. Harmony was an audible gift the gods gave to humanity, to soothe the disturbances of the human soul (4).

Ancient Greeks understood that sounds were vibrations that could spread in the air and be perceived by the hearing organ. Empedocles (5 th century BC) described the ear as a bell comprised of a "fleshy twig" [START_REF] Macauley | Elemental Philosophy: Earth, Air, Fire, and Water as Environmental Ideas[END_REF] able to receive sounds and by its proper vibration, transmit them to an aerial cavity (the middle ear). TM perforation was thought to induce deafness as inner air could leak out: the TM had an incidental role, sealing off the inner ear like a lid on a pot. In their treaties, Aristotle and Galen described it as an interface (it was named meninx or meninga for membrane) and mentioned its vulnerability, however neither gave the TM (or the ossicles) a distinct anatomical terminology, failing to identify its crucial role in transmitting vibrations to the inner ear. The concept of vibration was understood, the organ was yet to be identified.

Throughout the Middle Age, the legacy of the meninga lived on. The great Persian physician Avicenna (10 th -11 th century AD) bestowed on the TM a key role in his theory of audition but as a perceptive organ proceeding from the nervous system like the cranial nerves: "And the ear hole leads to a pocket of stagnant air (the middle ear) and covered with a bundle of nerves (the ear drum)" [START_REF] Sina | [END_REF]. By analogy, he compared the eardrum to the iris [START_REF] Hamidi | Avicenna's treatise on otology in Medieval Persia[END_REF].

The first breakthrough occurred in 16 th century Renaissance Italy with the development of human cadaveric dissections.

Gabriel Fallopius is often credited with the birth of the term "tympanum" in De observationes anatomicae (1562): "tympanum (…) ego appelavi" from the Greek tumpanon (drum) as an analogy to its function (8-10) However, Andreas Vesalius (1514-1546) coined it in his 1543 edition of De humani corporis fabrica. "Quemadmodum in tympanis fidem" said Vesalius describing the first ossicle (the malleus) anchored in the TM's width [START_REF] Vesalius | De humani corporis fabrica[END_REF]. In the 1543 edition of De humani corporis fabrica, Vesalius drew Figure 8, Book I figuring the hearing organ with the TM captioned as "B: Membrane drawn across the foramen that leads from the ear into the cavity carved in the temporal bone for the organ of hearing" (figure 1) [START_REF] Garrison | The Fabric of the Human Body: An Annotated Translation of the 1543 and 1555 Editions of[END_REF].

The emulation, created by these newly lawful human dissections all taking place in a close geographic area, enabled rapid progress surpassing, in a few decades, centuries' worth of Ancient knowledge (mostly obtained from animal dissection). The understanding of the importance of percussion in the transmission of sound, from scant TM vibrations, transmitted to the ossicles and then to the oval window, arose simultaneously [START_REF] O'malley | The discovery of the auditory ossicles[END_REF][START_REF] Massa | Liber introductorus anatomiae[END_REF]).

Duverney's Traité des maladies de l'ouïe (1683) [START_REF] Duverney | Traité de l'organe de l'ouïe[END_REF] was the first otological treatise in a vernacular language and combined prior anatomical discoveries to his own observations. Duverney thoroughly detailed the TM (figure 2): "This septum is an almost round membrane, dry, thin, firm, transparent and engaged in a groove carved in the circumference of the end of the bony duct (...). Although this membrane is stretched, it does not make a straight plane, but it is curved inside being pulled by the handle of the hammer" (translation from French by the authors). Duverney called the TM "tambour" (drum) and correctly conveyed the concept of vibration and resonance. He did not make of the TM an extension of nervous tissue and correctly identified its epidermic nature. The TM still had a protective role but also a proper mechanical role amplifying and transmitting vibrations: the TM was the first component of the hearing machine. However, he was wrong when he wrote that the TM could be bent by the tensor tympani muscle to alter vibrations (its protective against acoustic trauma was unknown). Indeed, as in vivo detailed TM observation was not possible at those times, cadaveric specimen study -subject to desiccation or to poorly controlled preservation procedurescould lead to erroneous observations. A famous example is the so-called Rivinus' foramen [START_REF] Mudry | Does the tympanic membrane have a natural opening? The historical saga of the foramen of rivinus[END_REF]. Johannes Augustus Rivinus the Younger (1692-1723) reported in his book De Auditus Vitiis (1717) an observation his father Augustus Quirinus Rivinus the Older (1652-1723) made in a letter: "I studied the organs of hearing by dissection. At length, in the year 1689, in the month of September, I found the opening that I had long sought, it was in the tympanic membrane itself (…) near the head of the malleus…" [START_REF] Rivinus | De auditus vitiis[END_REF]. The controversy lasted almost three centuries and this foramen existence was widely accepted as plausible. However, the contemporary Antonio Maria Valsalva (1666-1723) -who had been informed of Rivinus' theory before his son published the letter in 1717 -thought it was an artifact brought about by the specimen conservation before post mortem examination and did not draw it in his book De Aure Humana Tractatus (1704) (figure 3) [START_REF] Valsalva | De aure humana tractatus[END_REF]. In 1869, Adam Politzer (1835-1920) wrote: "Above the short process, according to Bochdolek, is a short canal, the foramen of Rivini, which was formerly considered an artificial opening in the membrane. This foramen, however, does not appear to be constant" [START_REF] Politzer | The Membrana Tympani in Health and Disease[END_REF]. It was not only the use of the first otoscopes but even more the use of microscopic otoscopy that allowed, in the second half of the 20 th century [START_REF] Griffith | The foramen of Rivinus--an artefact[END_REF], to confirm that Valsalva was right about Rivinus, father and son. Nowadays, any tympanic perforation is considered pathological: the first otologists, even the greatest, would not have unanimously thought so. These two reports read in Academic session in 1790 and 1791 were the first to report the fibrous structure of the TM with two layers ("due strati di fibre") of concentric and radial fibers ("Uno degli strati è di circolari concentriche, 1'altro di radiate"). He rejected the nervous origin of Mondino, that still was taught in the early 19 th century (p5) and insisted on the novelty of his observation having used a microscope ("non ponno vedersi senza 1'ajuto del microscopio" p6) [START_REF] Caldani | Osservazioni sulla membrana del timpano e nuove ricerche sulla elettricità animale : con un appendice e figure / lette nell[END_REF]. Initiating the shift from Italy to northern Europe, the German anatomist Samuel Thomas Sömmerring (1755-1830) further described the TM with accurate illustrations in his atlas, Abbildungen des menschlichen Hörorgans, published in 1806 (figure 4) [START_REF] Sömmering | Abbildungen des menschlichen Hörorgans. Varrentrapp[END_REF]. Shortly later, Henry Jones Shrapnell (1792-1834) published a detailed description of the TM in the London Medical Gazette (1832). Shrapnell identified the pars flaccida which "occupies the anterior superior angle, and so much of the membrana tympani as is above the tubercle of the malleus, and the ligament in which it is hung" and still bears his name (figure 5) [START_REF] Shrapnell | On the form and structure of the membrana tympani[END_REF]. He further described the fine three layered structure of the TM and reported that the pars flaccida microstructure was not composed of muscle fibers as contemporaries said (such as Sir Edward Homes) (25) but of a tangle of conjunctive fibers. For Shrapnell, the TM had elastic properties but no contraction capability.

The 19 th century and the birth of otology

Famous during his lifetime and considered as the father of British otology, Joseph Toynbee (1815-1866) (figure 3) was well known for his collection of over 2000 temporal bones, allowing him to identify numerous anatomical variations. In 1838 and 1839, he anonymously published letters in The Lancet challenging those he called quack aurists [START_REF] Mudry | The making of a career: Joseph Toynbee's first steps in otology[END_REF]. His approach was based on methodical and accurate observation of his patients, crossed with meticulous specimen dissection, drawing for the first time pathology and treatment inferences from TM anatomy [START_REF] Toynbee | On the Structure of the Membrana Tympani in the Human Ear[END_REF].

Toynbee described that the TM was constituted of five superimposed layers (from outside to inside: "the epidermis, the dermoid layer, the radiate fibrous layer, the circular fibrous layer and the mucous layer, with its epithelium") and identified the cutaneous nature of the external layers: "Between the epidermoid and radiating fibrous layers of the membrana tympani, there is a distinct and completed lamina of membrane which is continuous with the dermoid layer of the meatus". He understood and emphasized the impact of this observation: "A knowledge of the existence of the membrane here described is of interest to the anatomist, who recognizes in it the secreting organ of the epidermoid layers of the membrana tympani; and to the surgeon, who by its presence is able to understand phenomena occurring in certain diseases of the ear which have been hitherto incomprehensible to him". He described retraction pocket phenomenon but was wrong in assigning it to an "ulceration of the fibrous laminae" he developed as: "the (…) inflammation of the dermoid layer, which spreads first to the radiate fibrous and thence to the circular lamina. The laminae, being weakened by ulcerative process, fall inwards as far as the promontory, to which they often ultimately adhere, and, when an orifice has been thus produced, its margins are not unfrequently drawn into the shape of a funnel, whose inner part adheres to the tympanic walls" [START_REF] Toynbee | On the use of an Artificial Membrana Tympani in cases of Deafness, dependent upon perforation or destruction of the natural organ[END_REF]. These observations led him to propose an ingenious way to restore hearing in TM perforations, using an Indian rubber disc which would be placed on the eardrum and was removable thanks to an attached silver wire [START_REF] Mudry | The making of a career: Joseph Toynbee's first steps in otology[END_REF].

In the late 19 th century, progress would shift back to the European continent with Alexander Prussak and the great Adam Politzer (figure 6). Politzer (1835Politzer ( -1920) synthetized knowledges about the TM. He definitively described its normal shape: "not stretched as a perfect plane upon the end of the external meatus, but (…) arched in such a manner that its concavity is presented outward, while its convexity is turned toward the inner wall of the tympanum…".

Contrarily to Toynbee, he reduced to three the number of layers: "a middle fibrous -the socalled lamina propria membranae tympani, an external dermoid and an internal mucous layer" but acknowledged that the lamina propria had "two separable laminae -an external radiate, and an internal circular" [START_REF] Politzer | The Membrana Tympani in Health and Disease[END_REF].

However, although he progressed significantly in understanding Eustachian tube permeability and catarrh [START_REF] Politzer | Beiträge zur Physiologie des Gehörorgans[END_REF][START_REF] Mudry | The role of Adam Politzer (1835-1920) in the history of otology[END_REF], he did not understand its link with the so-called cholesteatoma disease which has been coined by Johannes Müller (1801-1858) in 1838 and that he considered to be a "glandular neoplasm of middle ear mucosa" [START_REF] Soldati | Knowledge about cholesteatoma, from the first description to the modern histopathology[END_REF].

The famous Russian otologist Alexander Federovitch Prussak (1839-1897) who attended Politzer's lessons in Vienna, pursued the anatomical study of the TM, giving his name to the "triangular shaped free space" in the outer part of the attic between Shrapnell's membrane, the neck of malleus medially and the lateral process of malleus inferiorly [START_REF] Prussak | Uber die anatomisch Verhaltnisse des Trommelfells zum Hammer[END_REF][START_REF] Proctor | [END_REF]. His work completed the microscopic anatomical understanding of the pars flaccida, and helped end the debate on the foramen of Rivinus.

Politzer was not only the first modern otologist, but also the first historian of otolaryngology: recognizing the decisive works of his predecessors [START_REF] Politzer | Geschichte der Ohrenheilkunde[END_REF] and helping to trace the progress in the understanding of the TM. He synthetized in detail the knowledge of the normal TM anatomy, now established, but especially the first pathological TM description, in publishing in 1865 the first otoscopy atlas: "Die Beleuchtungsbilder des Trommelfells im gesunden und kranken Zustande" (30,35) (Figure 7). Otology as a specialty was born with the TM.

CONCLUSION

From early Antiquity, when polymaths poetically portrayed the tympanic membrane as a gateway to other worlds, to the great anatomists of the Renaissance who identified the ear as parts of a complex mechanical object, the understanding of the eardrum remained limited for centuries. A great technological leap was possible in the 19 th century with new self-lit otoscopes contributing to the emergence of otologists, clinicians who would dedicate their life to the study of the ear. A reproducible scientific approach, crossing microscopic anatomical studies and patient care, progressively shed light on the TM in all its complexity. This provided better insight in TM physiology and pathology, paving the way for modern otology. Trommelfells im gesunden und kranken Zustande" (1865). At that time, otology was born as a specialty. Fig. 2: "membrana tympani with crescentic chalky with deposit". Fig. 3: "calcareous deposit" in the inner side of a 24-years girl's TM. Fig. 4 and Fig. 5: "calcareus deposits" and perforations.

A

  second surge in knowledge took place during the 19 th century, following technological progress. Otologists such as Floriano Caldani, Joseph Toynbee, Henry Jones Shrapnell, Alexander Prussak and finally Adam Politzer were amongst the first to use self-lit endoscopes to examine the TM. While aurists previously had to examine ears with cones and lenses reflecting the sun rays, it was only in 1730, that Archibald Cleland (1700-1771), a Scottish physician in the British army, invented the first candle-lit otoscope to examine soldiers exposed to loud explosions (21). Later, the French military surgeon Jean-Pierre Bonnafont (1805-1891) invented the first lateral light source with an opening in the reflecting mirror allowing the direct examination of the TM (22). This invention was then improved with the Edison electric light filament. In 1794, Floriano Caldani (1772-1836), anatomist from the famous University of Padua where Andreas Vesalius taught, published his Osservazioni sulla membrana del timpano e nuove ricerche sulla elettricita animale lette nell' Accademia di Scienze Lettere ed Arti di Padova.

FIGUREFigure 1 .

 1 FIGURE

Figure 2 .

 2 Figure 2. Duverney's Traité des maladies de l'ouïe (1683) was the first otological treaty in a vernacular language. Duverney thoroughly detailed the TM he called "tambour" (drum) and correctly conveyed the concept of vibration and resonance. Note that in Fig.V, the bone marked as B is mistakenly taken for a 4 th ossicle (which is in fact the lenticular process of the incus). BIU Santé, Université de Paris.

Figure 3 .

 3 Figure 3. Antonio Maria Valsalva's De Aure Humana Tractatus (1704) table III. Vasalva was aware of the contemporary Rivinus' theory but thought it was an artifact brought about by the specimen conservation before post mortem examination and did not draw it in his book. A. Glandulae meatus auditoria cum corpore suo reticulari. B. Incus. C. Malleus. D. Processus brevior incudes (short process of the incus). E. Chorda tympani. F. Membrana tympani.

Figure 4 .

 4 Figure 4. Samuel Thomas Soemmerring's Abbildungen des menschlichen Hörorgans (1806) table IV. He described the TM with accurate illustrations in his atlas in particular its vascularization. Note this illustration is one century later than Valsalva's De Aure Humana Tractatus (1704), with considerably more details.

Figure 5 .

 5 Figure 5. Mr. Shrapnell on the membrana tympani in the London Medical Gazette (1832). Shrapnell identified the pars flaccida which still bears his name.

Figure 6 .

 6 Figure 6. Adam Politzer (1835-1920) photographed by Rudolf Krziwanek (1843-1905). He synthetized in detail the knowledge of the normal TM anatomy and the first pathological TM description, in publishing in 1865 the first otoscopy atlas. He is known to have also been one of the first historians of otolaryngology.

Figure 7 .

 7 Figure 7. Adam Politzer's english book The Membrana Tympani in Health and Disease (1869), followed four years later his first otoscopy atlas: "Die Beleuchtungsbilder des

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