The new synthetic and runs-rules schemes to monitor the process mean of autocorrelated observations with measurement errors *

Keywords: Runs-rules, Measurement errors, First-order autoregressive model, Skip sampling strategy, Additive model, Multiple measurements sampling strategy, Zero-state, Steady-state, Markov chain

The use of the 2-of-(H+1) runs-rules and synthetic schemes to improve the performance of the currently available 𝑋 ̅ schemes in monitoring the process mean under the combined effect of measurement errors and autocorrelation are proposed. To maximize the detection ability of the 2-of-(H+1) runs-rules and synthetic schemes, we implement the modified side-sensitive (MSS) design approach for the charting regions as we show it yields the best possible performance out of all the available designs. These new monitoring schemes incorporate the additive model with a constant standard deviation and a first-order autoregressive model to the computation of the control limits in order to account for measurement errors and autocorrelation, respectively. Moreover, to construct a dedicated Markov chain matrix, the abovementioned models and some sampling methods are incorporated into the values of probability elements which are then used to derive closed-form expressions for the zero-and steady-state run-length distribution. This study is important because the majority of research tends to assume that observations are independent and identically distributed (i.i.d.) and that none of the measurements taken on the inspected items are contaminated, which is not always the case in real-life application. That is, a combined effect autocorrelation and measurement errors is a significant factor in quality and reliability statisticshence an improved monitoring scheme for this scenario is discussed here. A real-life example is used to illustrate the practical implementation of the proposed schemes.

Introduction

The main objective of statistical process monitoring (SPM) is to improve a monitoring process by detecting, identifying and removing any significant causes of variation. This is done by using a monitoring scheme (commonly known as control chart) to distinguish between chance causes of variation and assignable causes of variation. When only common causes of variation are present, the process is said to be in-control (IC). Otherwise, the process is said to be out-of-control (OOC) and assignable causes of variation have to be searched for. The most popular and simple monitoring schemes are the Shewhart charts, proposed by Walter A. Shewhart in the 1920's. Despite their simplicity and adaptability, the main shortcoming of Shewhart charts is their insensitivity in detecting small to moderate size shifts. Moreover, real life properties like, autocorrelation and measurement errors, tend to add more negative effect to the latter mentioned insensitivity in detecting small to moderate shifts.

One of the methods used to enhance the performance of the basic Shewhart scheme is to combine its operation with the conforming run-length (CRL) schemethis yields a class of monitoring schemes called synthetic schemes, first proposed by [START_REF] Wu | A synthetic control chart for detecting small shifts in the process mean[END_REF] and more recently reviewed by [START_REF] Rakitzis | An overview of synthetic-type control charts: Techniques and Methodology[END_REF]. The CRL is defined as the number of samples observed between two consecutive nonconforming samples, inclusive of the nonconforming sample at the end, which was proposed by [START_REF] Bourke | Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection[END_REF]. This scheme gives an OOC signal when the CRL value is significantly small, say, CRL ≤ H, where H is a positive integer greater than 0. The main difference between a basic Shewhart scheme and a synthetic scheme is that the latter does not issue OOC signal at the first sample point that falls on the nonconforming regions. That is, the process waits until a second sample point falls on the nonconforming region and if these two nonconforming samples are relatively close to each other, then the CRL scheme signals and thus, an OOC signal is triggered. The rationale behind the CRL scheme is the following: small CRL values are indicative of less conforming items produced between the two successive nonconforming ones. Consequently, this may be an indication that the process is operating under some assignable causes of variation.

Approximately 100 publications on synthetic schemes were reviewed in [START_REF] Rakitzis | An overview of synthetic-type control charts: Techniques and Methodology[END_REF]; here are some of the publications that were not covered, and some that are more recent: [START_REF] Khilare | A nonparametric synthetic control chart using sign statistic[END_REF], 2012, 2015), [START_REF] Pawar | A nonparametric Shewhart-type synthetic control chart[END_REF], [START_REF] Ghute | A nonparametric signed-rank control chart for bivariate process location[END_REF], [START_REF] Malela-Majika | Distribution-free synthetic and runs-rules control charts combined with a Mann-Whitney chart[END_REF], [START_REF] Pawar | Steady-state behavior of nonparametric synthetic control chart using signed-rank statistic[END_REF], Shongwe and Graham (2019a), [START_REF] Haq | A new nonparametric synthetic EWMA control chart for monitoring process mean[END_REF], [START_REF] Khaw | Optimal design of the synthetic control chart for monitoring the multivariate coefficient of variation[END_REF], Shongwe et al. (2019a), [START_REF] Tran | A synthetic median control chart for monitoring the process mean with measurement errors[END_REF], [START_REF] Malela-Majika | Modified side-sensitive synthetic double sampling monitoring scheme for simultaneously monitoring the process mean and variability[END_REF], [START_REF] Raza | On designing new optimal synthetic Tukey's control charts[END_REF], [START_REF] Haq | A synthetic double sampling control chart for process mean using auxiliary information[END_REF].

In the SPM literature, a majority of research works does not account for measurement system errors, that is, it is usually assumed that none of the measurements taken on the inspected items are contaminated, see for example, [START_REF] Yeong | The coefficient of variation chart with measurement error[END_REF], [START_REF] Sabahno | Performance of the variable parameters 𝑋 ̅ control chart in presence of measurement errors[END_REF], [START_REF] Tang | The performance of the adaptive EWMA median chart in the presence of measurement error[END_REF], [START_REF] Tran | A synthetic median control chart for monitoring the process mean with measurement errors[END_REF], etc. [START_REF] Maleki | Measurement errors in statistical process monitoring: A literature review[END_REF] argues that wherever there is a human involvement, an exact measurement is a rare phenomenon in any manufacturing and service environment; hence a difference between the real quantities and the measured ones will always exist even with highly sophisticated advanced measuring instruments. For a more thorough discussion on measurement errors, models used to capture the measurement system inaccuracy and the corresponding remedial approaches to reduce the negative effect thereof; readers must consult the articles cited in the review by [START_REF] Maleki | Measurement errors in statistical process monitoring: A literature review[END_REF]. In this paper, we use the additive model with a constant standard deviation which was first proposed by [START_REF] Linna | Effect of measurement error on Shewhart control charts[END_REF] and the m-measurements remedial approach.

Autocorrelation within a subgroup sample is usually captured by some appropriate time series model. [START_REF] Box | Time Series Analysis: Forecasting and Control[END_REF] outlines a number of these different non-and stationary time series models, e.g. autoregressive (AR), moving average (MA), autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), etc. In this paper, we only consider the wellknown first-order AR model (i.e. AR(1)) as a starting point (other models will be discussed in upcoming articles). The AR(1) model is the most commonly used time series model in SPM applications due to its simplicity as compared to other stationary time series models. The stationarity property of the AR(1) model is used to check whether the process is IC or not. That is, if the process remains in equilibrium around the constant mean then it is IC; however, when there is any statistically significant difference from the constant mean, it implies that the autocorrelated process is OOC. For some other discussions on AR(1) model in a univariate or multivariate SPM context, see for example [START_REF] Claro | Double sampling 𝑋 ̅ control chart for a first order autoregressive process[END_REF], [START_REF] Kazemzadeh | Phase II monitoring of autocorrelated polynomial profiles in AR(1) processes[END_REF], [START_REF] Costa | Variable parameter and double sampling 𝑋 ̅ charts in the presence of correlation: the Markov chain approach[END_REF], [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF], [START_REF] Chang | On average run lengths of control charts for autocorrelated processes[END_REF], [START_REF] Keramatpour | Phase II monitoring of AR(1) autocorrelated polynomial profiles[END_REF], Franco et al. (2014a,b), Leoni et al (2015a,b,c), [START_REF] Hu | Synthetic ̅ chart for AR(1) autocorrelated processes[END_REF], [START_REF] Osei-Aning | Mixed EWMA-CUSUM and mixed CUSUM-EWMA for monitoring first order autoregressive processes[END_REF][START_REF] Garza-Venegas | Effect of autocorrelation estimators on the performance of the 𝑋 ̅ control chart[END_REF], Shongwe et al. (2019a,b), [START_REF] Dargopatil | New sampling strategies to reduce the effect of autocorrelation on the synthetic T 2 chart to monitor bivariate process[END_REF], [START_REF] Ahmad | On auxiliary information-based control charts for autocorrelated processes with application in manufacturing industry[END_REF], etc.

The synthetic-type schemes were shown to have four categories in [START_REF] Rakitzis | An overview of synthetic-type control charts: Techniques and Methodology[END_REF]. [START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF] showed that the 2-of-(H+1) runs-rules schemes also have four categories. These are termed non-side-sensitive (NSS), standard side-sensitive (SSS), revised side-sensitive (RSS) and modified side-sensitive (MSS) design approaches. These synthetic-type schemes denoted as S1, S2, S3 and S4 were first proposed by [START_REF] Wu | A synthetic control chart for detecting small shifts in the process mean[END_REF], [START_REF] Davis | Evaluating and improving the synthetic control chart[END_REF], [START_REF] Machado | Some comments regarding the synthetic chart[END_REF] and, [START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF], respectively. The runs-rules NSS, SSS, RSS and MSS were first proposed in [START_REF] Derman | Statistical Aspects of Quality Control[END_REF], [START_REF] Klein | Two alternatives to Shewhart 𝑋 ̅ control chart[END_REF], adopted from [START_REF] Machado | Some comments regarding the synthetic chart[END_REF] and, [START_REF] Antzoulakos | The modified r out of m control chart[END_REF] which are denoted by RR1, RR2, RR3 and RR4, respectively. To the authors' best knowledge, only about five publications currently exist that investigate the combined effect of autocorrelation and measurement errors in different univariate SPM contexts; and these are: [START_REF] Scagliarini | Estimation of 𝐶 𝑝 for autocorrelated data and measurement errors[END_REF][START_REF] Scagliarini | Inference on 𝐶 𝑝 𝑘 for autocorrelated data in the presence of random measurement errors[END_REF], [START_REF] Yang | Effects of imprecise measurement on the two dependent processes control for the autocorrelated observations[END_REF], [START_REF] Xiaohong | The CUSUM control chart for the autocorrelated data with measurement error[END_REF] and, [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF]. Therefore, the main objective of this paper is to improve the Shewhart 𝑋 ̅ scheme with a combined effect of measurement errors and autocorrelation (discussed in [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF] by: (i) Integrating its operation with the CRL scheme to form a synthetic 𝑋 ̅ scheme, and (ii) Adding supplementary rules to form a 2-of-(H+1) 𝑋 ̅ runs-rules schemes. More importantly, in order to gain maximum performance from the synthetic and runs-rules schemes, we use the MSS designas it yields the best possible performance out of the all four existing designs. Moreover, following on [START_REF] Davis | Evaluating and improving the synthetic control chart[END_REF] deduction that synthetic schemes are the same as the 2-of-(H+1) runs-rules schemes with a head-start feature, this property is used to construct a dedicated Markov chain matrix that is used to derive some zero-and steady-state closed-form expressions for both the MSS runs-rules and synthetic 𝑋 ̅ schemes. Finally, the additive model with a constant standard deviation and the AR(1) model with constant standard deviation are used to account for the measurement errors and autocorrelation, respectively. To reduce the combined negative effect of measurement errors and autocorrelation, a sampling strategy that allows for multiple measurements per item and skipping some successive observations is incorporated in the probability values of the dedicated Markov chain matrix.

The rest of the paper is structured as follows: In Section 2, the properties of Shewhart 𝑋 ̅ schemes with the effect of measurement errors and autocorrelation are discussed. The operation, construction of the Markov chain matrix and run-length properties of the proposed MSS runs-rules and synthetic schemes are discussed in Section 3. Empirical discussion of the proposed scheme and comparisons with the existing Shewhart 𝑋 ̅ scheme is done in Section 4, and more specifically, in Section 4.4, the four design approaches are empirically compared to show the superiority of the MSS design. The practical implementation of the proposed schemes is given in Section 5 and finally, the concluding remarks are given in Section 6.

Measurement errors and autocorrelation for the Shewhart 𝑿 ̅ scheme

Measurement errors

Assume that the 𝑌 𝑡,𝑖 is a sequence of i.i.d. observations from a N(𝜇 0 , 𝜎 0 ) distribution (where 𝜇 0 and 𝜎 0 are the nominal IC process mean and standard deviation, respectively) that are not directly observable, but can only be assessed from the results {𝑋 𝑡,𝑖,𝑗 : t ≥1; i = 1,2,…,n; j = 1,2,…,m}, with each element of the latter sequence expressed in terms of the additive model with a constant standard deviation, see [START_REF] Linna | Effect of measurement error on Shewhart control charts[END_REF], i.e., 𝑋 𝑡,𝑖,𝑗 = 𝐴 + 𝐵𝑌 𝑡,𝑖 + 𝑒 𝑡,𝑖,𝑗 ;

(1)

where 𝑒 𝑡,𝑖,𝑗 ~𝑁(0, 𝜎 𝑀 ) is a random error term due to measurement inaccuracy and 𝜎 𝑀 is the standard deviation of the measurement system; for a sake of simplicity, in this paper, we will assume that A=0

and B=1, where 𝐴 and 𝐵 are two constants depending on the measurement system location error.

Hence, the plotting statistic is the corresponding sample mean, which is given by

𝑋 ̅ 𝑡 = 1 𝑚𝑛 ∑ ∑ 𝑋 𝑡,𝑖,𝑗 𝑚 𝑗=1 𝑛 𝑖=1 = 1 𝑛 (∑ 𝑌 𝑡,𝑖 𝑛 𝑖=1 + 1 𝑚 ∑ ∑ 𝑒 𝑡,𝑖,𝑗 𝑚 𝑗=1 𝑛 𝑖=1
).

(2)

That is, at each sampling point, there are m separate measurements, each of size n (i.e. a total of 𝑚 × 𝑛 observations). The standard deviation of the process in Equation ( 2) at each sampling point is given by

𝜎(𝑋 ̅ 𝑡 ) = √ 𝜎 0 2 𝑛 + 𝜎 𝑀 2 𝑛𝑚 = 𝜎 0 √𝑛𝐶 1 (𝑚, 𝛾) (3)
where

𝐶 1 (𝑚, 𝛾) = √ 𝑚 𝑚+𝛾 2 and 𝛾 = 𝜎 𝑀 𝜎 0
denotes the ratio of the measurement system variability and the process variability, see [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF]. When 𝑋 ̅ 𝑡 is from an imperfect measurement system, i.e. 𝛾 > 0, then we assume that multiple measurements (i.e. m-measurement strategy, with m > 1) are available for each i.i.d. subgroup sample.

Autocorrelated data using AR(1) model

Assume that the quality characteristic {𝑌 𝑡,𝑖 : t ≥ 1; i = 1, 2,…, n} is a sequence of samples from an autocorrelated N(𝜇 0 , 𝜎 0 ) distribution that fits a stationary AR(1) model, given by 𝑌 𝑡,𝑖 -𝜇 0 = 𝜙(𝑌 𝑡,𝑖-1 -𝜇 0 ) + 𝜀 𝑖 ; 𝑖 = 1, 2, 3, 4,…, 𝑛;

i.e. the current observation, 𝑌 𝑡,𝑖 , of the time series depends on the previous observation, 𝑌 𝑡,𝑖-1 ; with a specified parameter 𝜙 (called a level of autocorrelation), where |𝜙| < 1, 𝜀 𝑖 are i.i.d. normal (0, 𝜎 𝜀 ) random variables; where 𝜇 0 and 𝜎 0 are the nominal IC mean and standard deviation process parameters, respectively, where 𝜎 0 = 𝜎 𝜀 √1-𝜙 2 , and without loss of generality, assume 𝜎 𝜀 = 1; see [START_REF] Alwan | Time-series investigation of subsample mean chart[END_REF]. After the occurrence of assignable causes, the process mean shifts from 𝜇 0 to 𝜇 1 = 𝜇 0 + 𝛿𝜎 0 , so that 𝛿 = 𝜇 1 -𝜇 0 𝜎 0

. Sampling techniques that involve skipping some of the successive observations (i.e. sampling of non-neighboring observations) have been shown to reduce serial dependence in time series data, see for example: [START_REF] Gilbert | Charting autocorrelated data: Guidelines for practitioner[END_REF], Costa andCastagliola (2011), Franco et al. (2014b), [START_REF] Hu | Synthetic ̅ chart for AR(1) autocorrelated processes[END_REF], Leoni et al. (2015a), [START_REF] Dargopatil | New sampling strategies to reduce the effect of autocorrelation on the synthetic T 2 chart to monitor bivariate process[END_REF], Shongwe et al. (2019a,b). Consequently, the corresponding process with s-skipping sampling strategy remains an AR(1) process; however, defined as {𝑌 𝑡,𝑖 : 𝑡 ≥ 1; 𝑖 = 1, s+2, 2s+3, 3s+4,…} with parameter 𝜙 𝑠+1 :

𝑌 𝑡,𝑖 -𝜇 0 = 𝜙 𝑠+1 (𝑌 𝑡,𝑖-𝑠-1 -𝜇 0 ) + 𝜀 𝑖 ′ ; 𝑖 = 1, s+2, 2s+3, 3s+4, … (5)
with 𝜀 𝑖 ′ = 𝜀 𝑖 + 𝜙𝜀 𝑖-1 + 𝜙 , where 𝐶 1 (𝑚, 𝛾) and 𝐶 2 (𝑛, 𝑠, 𝜙) are as given in Equations ( 3) and (6), respectively.

Operation and run-length properties of the MSS runs-rules and synthetic schemes

Operation

Subgroup samples are usually taken at each sampling point to be inspected and then each of these samples are classified as either conforming or nonconforming depending on where the charting statistic plots on the charting regions shown in Figure 1. Note that a sample plots on a conforming region when it is under the influence of common causes of variation only; however, when it plots on a nonconforming region, it implies that it has some assignable causes of variation present. For any s&m strategy sampling scheme, the charting limits shown in Figure 1, i.e. the upper / lower control limit denoted by (𝑈𝐶𝐿/𝐿𝐶𝐿) and the center line (CL), are given by: 𝑈𝐶𝐿/𝐿𝐶𝐿 = 𝜇 0 ± 𝑘𝜎(𝑋 ̅ 𝑡 ) and 𝐶𝐿 = 𝜇 0 ,

where 𝑘 > 0 is the design parameter that is related to the distance from the center line to the 𝑈𝐶𝐿/ 𝐿𝐶𝐿 in terms of the standard deviation, respectively.

<Insert Figure 1> The operational procedure of the MSS runs-rules and synthetic s&m schemes are given in Table 1;

where CRL + (CRL -) is the number of conforming samples that fall in Region B + (Region B -), which are in between the two nonconforming samples that fall in Region A + (Region A -), including the nonconforming sample at the end, respectively. Moreover, the metric average run-length (ARL) is the average number of subgroup samples that are required before the first OOC signal is issued by a monitoring scheme, and ARL 0 denotes the desired nominal ARL.

<Insert Table 1>

Transition probability matrix

For the MSS runs-rules and synthetic schemes, there are two nonconforming regions (i.e. region A + : upper, and region A -: lower) and two conforming regions (i.e. region B + : upper and region B -: lower). Assume that 𝑝 𝜃 = 𝑃(𝑋 ̅ 𝑖 ∈ 𝜃) denotes the probability that a sample point plots in region 𝜃 ∈ {A -, A + , B -, B + }, then given the charting regions in Figure 1, the probability of a charting statistic falling in each region is given by

𝑝 A + = 1 -Φ (𝑘 -𝛿√𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)) 𝑝 B + = Φ (𝑘 -𝛿√𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)) -Φ (-𝛿√𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)) 𝑝 B -= Φ (-𝛿√𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)) -Φ (-𝑘 -𝛿√𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)) 𝑝 A -= Φ (-𝑘 -𝛿√𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)), (9)
where Φ(•) denotes the cumulative distribution function (c.d.f.) of the standard normal distribution.

Let '±' denote a state that a charting statistic at time zero falls in a nonconforming region, that is, the charting statistic falls either in region A -or region A + at the beginning of the monitoring process;

this assumption is referred to as the "head-start feature" of the synthetic schemesee [START_REF] Davis | Evaluating and improving the synthetic control chart[END_REF], Knoth (2016), [START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF] and [START_REF] Rakitzis | An overview of synthetic-type control charts: Techniques and Methodology[END_REF] for a more thorough discussion on this. For instance, '±A -' implies that at time 0, we assume the charting statistic plots either in region A -or region A + , and then at sampling time 𝑡 = 1, the charting statistic plots in region A -.

Next, the steps involved in constructing the TPMs of the MSS runs-rules and synthetic s&m scheme are as follows for any integer value of H > 0:  Step (i) Determine all the absorbing states that lead to an OOC signal; those with and without head-start are denoted by Ψ and Λ, respectively.

 Step (ii) Decompose the states in Step (i) into transient states by removing the last nonconforming element, where those with and without head-start are denoted by 𝜓 and 𝜂, respectively.

 Step (iii) The transient state corresponding to the IC regions is defined as 𝜑 = 𝜂 𝐻+1 .

 Step (iv) Define the state space, denoted by Ω, which is a union of the states defined in Steps (i) to (iii).

Hence, for illustrative purpose, assume H = 2, then the state space using the steps outlined above is constructed as follows:

 Step (i) yields Λ 1 ={A + A + }, Λ 2 ={A + B + A + }, Λ 3 ={A -A -}, Λ 4 ={A -B -A -} and Ψ 1 ={±A + }, Ψ 2 ={±A -}, Ψ 3 ={±B + A + }, Ψ 4 ={±B -A -}.  Step (ii) yields 𝜂 1 ={A + B + }, 𝜂 2 ={A + }, 𝜂 4 ={A -}, 𝜂 5 ={A -B -} and 𝜓 1 ={±}, 𝜓 2 ={±B + }, 𝜓 3 ={±B -}.  Step (iii) yields 𝜑 = 𝜂 3 ={B -, B + }.  Step (iv) yields Ω ≡{𝜂 1 ,𝜂 2 ;𝜑;𝜂 4 ,𝜂 5 ;𝜓 1 ,𝜓 2 ,𝜓 3 ;OOC}.
Consequently, the resulting Markov chain TPM is given by

𝜂 1 𝜂 2 𝜑 𝜂 4 𝜂 5 𝜓 1 𝜓 2 𝜓 3 OOC 𝜂 1 𝑝 B + + 𝑝 B -𝑝 A - 𝑝 A + 𝜂 2 𝑝 B + 𝑝 B - 𝑝 A - 𝑝 A + 𝜑 𝑝 A + 𝑝 B + + 𝑝 B -𝑝 A - 𝜂 4 𝑝 A + 𝑝 B + 𝑝 B - 𝑝 A - 𝜂 5 𝑝 A + 𝑝 B + + 𝑝 B - 𝑝 A - 𝜓 1 𝑝 B + 𝑝 B - 𝑝 A + + 𝑝 A - 𝜓 2 𝑝 B + + 𝑝 B -𝑝 A - 𝑝 A + 𝜓 3 𝑝 A + 𝑝 B + + 𝑝 B - 𝑝 A - OOC 1
Based on the latter TPM, it is apparent that its structure is such that for any positive integer 𝐻>0, it is given by a (𝑀 + 1) × (𝑀 + 1) matrix P, defined as

𝐏 = ( 𝐐 𝟎′ 𝐫 1 ) (10)
where 𝐐 is the 𝑀 × 𝑀 matrix of transient states (i.e. essential TPM), the 𝑀 × 1 vector 𝐫 is such that

𝐫 = 𝟏 -𝐐𝟏, i.e. each row sums to 1, with 𝑀 × 1 vectors 𝟏 = (1 1 … 1) ′ and 𝟎 = (0 0 … 0) ′ .
Moreover, the breakdown of the TPM for H = 2 (and other integer values of Hnot shown here), indicate that the dimension of the TPMs for any H > 0 is given by 𝑀 = 𝜏 + 𝜅 for the MSS synthetic scheme, where 𝜏 = 2𝐻 + 1 and 𝜅 = 2𝐻 -1. To obtain TPMs with an obvious recursive pattern for any H > 0, we define the state space, Ω, as follows,

Ω = {𝜂 1 ,…, 𝜂(𝜏+1) 2 -1 , 𝜂(𝜏+1) 2 ≡ 𝜑, 𝜂(𝜏+1) 2 +1 , …, 𝜂 𝜏 ; 𝜓 1 , …, 𝜓 𝜅 ; OOC}. ( 11
)
On the contrary, the dimension of the TPMs of the MSS runs-rules schemes, for any H > 0 is given by 𝑀 = 𝜏, where 𝜏 = 2𝐻 + 1 and 𝜅 = 0; as the head-start feature elements are not applicable in the design of the MSS runs-rules. Consequently, the Ω for the MSS runs-rules scheme is given by,

Ω = {𝜂 1 ,…, 𝜂(𝜏+1) 2 -1 , 𝜂(𝜏+1) 2 ≡ 𝜑, 𝜂(𝜏+1) 2 +1 , …, 𝜂 𝜏 ; OOC}. ( 12
)
Therefore, it follows that the TPM of MSS synthetic scheme is given in Table 2. Moreover, by removing the head-start elements, i.e. 𝜓 1 , …, 𝜓 𝜅 in Table 2, then the resulting TPM corresponds to that of the MSS runs-rules scheme.

<Insert Table 2>

Some general run-length properties

Since the TPMs of the MSS runs-rules and synthetic schemes for any possible integer value of H have been determined, then important properties of the run-length (RL) can be determined via an appropriate Markov chain technique, see Chapter 4 in [START_REF] Fu | Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach[END_REF]. That is, the ARL = E(RL)

and the standard deviation of the RL (SDRL = 𝜎(RL)) are defined as

ARL = 𝛏 T 𝐑 (13) SDRL = √2𝛏 T (𝐈 -𝐐) -2 𝐐𝟏 -𝐴𝑅𝐿 2 + 𝐴𝑅𝐿, (14)
where 𝛏 denotes either the 𝑀 × 1 zero-or steady-state initial probability vector; and 𝐑 is a 𝑀 × 1

vector containing ARL values of being in each of the 𝑀 transient states, and it is given by

𝐑 = (𝐈 -𝐐) -1 𝟏. ( 15
)
Using the 𝐐 from Table 2, Equation (15) becomes, 𝐑 ≡ 𝐑 no HS //𝐑 HS .

where, 𝐑 no HS is a 𝜏 × 1 vector containing ARL values of being in each of the states without the headstart feature, 𝐑 HS is a 𝜅 × 1 vector containing ARL values of being in each of the states with the head-start feature and '//' denotes the vertical concatenation operator. The components of the ARL vector in Equation ( 16) are given by

𝐑 no HS = ( 𝜍 1 𝜍 2 ⋮ 𝜍 𝐻-3 𝜍 𝐻-2 𝜍 𝐻-1 𝜍 𝐻 𝜑 = 𝜍 𝐻+1 𝜍 𝐻+2 𝜍 𝐻+3 𝜍 𝐻+4 𝜍 𝐻+5 ⋮ 𝜍 2𝐻 𝜍 2𝐻+1 ) = 1 G ( (1 + 𝑝 𝐴 𝑊 𝐵 1 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐴 𝑊 𝐵 2 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) ⋮ (1 + 𝑝 𝐴 𝑊 𝐵 𝐻-3 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐴 𝑊 𝐵 𝐻-2 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐴 𝑊 𝐵 𝐻-1 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) 1 + 𝑝 𝐷 𝑊 𝐶 0 (1 + 𝑝 𝐴 𝑊 𝐵 0 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) 1 + 𝑝 𝐴 𝑊 𝐵 0 (1 + 𝑝 𝐷 𝑊 𝐶 𝐻-1 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 𝐻-2 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 𝐻-3 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) ⋮ (1 + 𝑝 𝐷 𝑊 𝐶 2 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 1 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) ) (17)
and

𝐑 HS = ( 𝜓 1 = 𝜍 2𝐻+2 𝜍 2𝐻+3 𝜍 2𝐻+4 𝜍 2𝐻+5 𝜍 2𝐻+6 𝜍 2𝐻+7 𝜍 2𝐻+8 ⋮ 𝜍 4𝐻-3 𝜍 4𝐻-2 𝜍 4𝐻-1 𝜍 4𝐻 ) = 1 𝐺 ( 1 -𝑝 𝐴 𝑝 𝐷 𝑊 𝐵 0 𝑊 𝐶 0 (1 + 𝑝 𝐴 𝑊 𝐵 𝐻-1 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 𝐻-1 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) (1 + 𝑝 𝐴 𝑊 𝐵 𝐻-2 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 𝐻-2 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) (1 + 𝑝 𝐴 𝑊 𝐵 𝐻-3 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 𝐻-3 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) ⋮ (1 + 𝑝 𝐴 𝑊 𝐵 2 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐷 𝑊 𝐵 2 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) (1 + 𝑝 𝐴 𝑊 𝐵 1 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) (1 + 𝑝 𝐷 𝑊 𝐶 1 )(1 + 𝑝 𝐴 𝑊 𝐵 0 ) ) ( 18
)
where 𝐺 is given by

G = 1 -𝑝 𝐴 (𝑝 𝐶 + 𝑝 𝐷 + 𝑝 𝐷 𝑝 𝐶 𝑊 𝐶 0 ) -𝑝 𝐵 (1 + 𝑝 𝐷 𝑊 𝐶 0 ) -𝑝 𝐶 -𝑝 𝐴 𝑝 𝐵 𝐻 (1 + 𝑝 𝐷 𝑊 𝐶 0 ) -𝑝 𝐷 𝑝 𝐶 𝐻 -𝑝 𝐴 𝑊 𝐵 1 (𝑝 𝐶 + 𝑝 𝐷 + 𝑝 𝐶 𝑝 𝐷 𝑊 𝐶 0 ), with 𝑊 𝐵 𝑟 = 𝑝 𝐵 𝑟 ( 1-𝑝 𝐵 𝐻-𝑟 1-𝑝 𝐵 ), 𝑊 𝐶 𝑟 = 𝑝 𝐶 𝑟 ( 1-𝑝 𝐶 𝐻-𝑟 1-𝑝 𝐶
), 𝑟 = 0,1,2, … , 𝐻 -1.

Zero-state run-length properties for the MSS runs-rules and synthetic schemes

In order to compute the zero-state run-length properties, the initial probability vector 𝛏 T = q T = (0,…,0,1,0,…,0) where the unique "1" is located

 the ( 𝜏+1 2
) th position of the q T vector, i.e. corresponding to '𝜂(𝜏+1) 2 ≡ 𝜑' on the TPM in Table 2.  the (𝜏 + 1) th position of the q T vector, i.e. corresponding to '𝜓 1 ' on the TPM in Table 2.

The zero-state ARL (ZSARL) is given by 𝑍𝑆𝐴𝑅𝐿 = 𝐪 T 𝐑, with 𝐑 given in Equation ( 16) which yields the following closed-form expressions:

 for the MSS runs-rules s&m schemes

𝑍𝑆𝐴𝑅𝐿 = (1 + 𝑝 𝐴 𝑊 𝐵 0 )(1 + 𝑝 𝐷 𝑊 𝐶 0 ) G = (1 + 𝑝 𝐴 ( 1 -𝑝 𝐵 𝐻 1 -𝑝 𝐵 )) (1 + 𝑝 𝐷 ( 1 -𝑝 𝐶 𝐻 1 -𝑝 𝐶 )) G . (19)
 for the MSS synthetic s&m schemes

𝑍𝑆𝐴𝑅𝐿 = 1 -𝑝 𝐴 𝑝 𝐷 𝑊 𝐵 0 𝑊 𝐶 0 G = 1 -𝑝 𝐴 𝑝 𝐷 ( 1 -𝑝 𝐵 𝐻 1 -𝑝 𝐵 ) ( 1 -𝑝 𝐶 𝐻 1 -𝑝 𝐶 ) G . ( 20
)

Steady-state run-length properties for the MSS runs-rules and synthetic schemes

In steady-state mode, the effect of a head-start disappears since the process has been running IC for a long time and thus, the corresponding components are discarded in the Markov chain matrix in Table 2, and consequently, the ARL vector in Equation ( 16) reduces to 𝐑 ≡ 𝐑 no HS .

The steady-state non-zero initial probability vector (i.e. 𝛏 T = u T ) is computed while the process is IC;

that is, 𝑝 𝐴 = 𝑝 𝐷 and 𝑝 𝐵 = 𝑝 𝐶 as 𝛿 = 0. Let 𝑞 = 𝑝 𝐴 = 𝑝 𝐷 and 𝑝 = 𝑝 𝐵 = 𝑝 𝐶 when 𝛿 = 0, then using [START_REF] Champ | Steady-state run length analysis of a Shewhart quality control chart with supplementary runs rules[END_REF] simplified cyclical steady-state method; the initial probability vector of the MSS runsrules or synthetic s&m scheme is given by 𝐮 = (𝟏 ′ 𝐳) -1 • 𝐳, where 𝐳 is the (2𝐻 -1) × 1 vector given by 𝐳 = (𝐆 -𝐐 ′ ) -1 𝐞 𝐻+1 , with the (2𝐻 -1) × (2𝐻 -1) matrix 𝐆, given by 𝐆 = 𝐪 • 𝟏 ′ + 𝐈, where 𝐞 𝐻+1 = (0 0 0 … 0 1 0 … 0 0 0), i.e., a (H+1) th unit vector. Thus, using the Markov chain matrix in Table 2 (without the head-start elements), then basic algebraic matrix manipulations yield:

𝐮 = ( u 1 u 2 u 3 ⋮ u H-2 u H-1 u H u H+1 u H+2 u H+3 u H+4 ⋮ u 2H-1 u 2H u 2H+1 ) = 1 -𝑝 1 -𝑝 + 𝑞(1 -𝑝 𝐻 ) ( 𝑞𝑝 𝐻-1 𝑞𝑝 𝐻-2 𝑞𝑝 𝐻-3 ⋮ 𝑞𝑝 2 𝑞𝑝 𝑞 1 -𝑞 ( 1 -𝑝 𝐻 1 -𝑝 ) 𝑞 𝑞𝑝 𝑞𝑝 2 ⋮ 𝑞𝑝 𝐻-3 𝑞𝑝 𝐻-2 𝑞𝑝 𝐻-1 ) . ( 21
)
Thus, it follows that the steady-state ARL (SSARL) of the MSS runs-rules or synthetic s&m scheme is given by

𝑆𝑆𝐴𝑅𝐿 = u H+1 ς H+1 + ∑ u 𝑟 × (ς 𝑟 + ς (2𝐻+2)-𝑟 ) 𝐻 𝑟=1 , (22)
with ς 𝑟 defined in Equation ( 17) and u 𝑟 defined in Equation ( 21), for 𝑟 = 1, 2, … , 𝐻.

Overall performance metrics

In addition to specific shifts measures, i.e. Equations ( 19), ( 20) and ( 22 Note that the shifts within the interval [𝛿 min , 𝛿 max ] usually occur according to a p.m.f. equal to 𝑈(𝛿)

which is usually unknown. In the absence of any particular information, it is usually assumed that the shifts in the process mean happen with equal probability, then 𝑈(𝛿) = 1/(𝛿 max -𝛿 min ) i.e. a uniform U(𝛿 min , 𝛿 max ) distribution. Thus, following a similar design procedure as done by the latter authors, the MSS runs-rules and synthetic s&m schemes will be designed based on the optimal parameters (𝐻 * , 𝑘 * ) that yield the best overall performance for a range of specified shifts and it is achieved by using Finally, the performance comparison index (PCI), by [START_REF] Wu | Optimization designs of the combined Shewhart-CUSUM control charts[END_REF], will also be used to measure the relative effectiveness of two different schemes, which is given by

(
PCI = 𝐸𝐴𝑅𝐿 𝑐 /𝐸𝐴𝑅𝐿 b , (24)
where 𝐸𝐴𝑅𝐿 𝑏 (denominator) is the EARL of the 'benchmark' scheme (i.e. the 𝑋 ̅ s&m scheme of [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF]) and 𝐸𝐴𝑅𝐿 c (numerator) is the EARL of some other 'competing'

scheme. When the 𝑃𝐶𝐼 is equal to 1, greater than 1 or less than 1, it implies that the 𝑋 ̅ s&m scheme of [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF] has the same, better or worse performance than the competing scheme, respectively.

Empirical discussion

IC design parameters

It is important to note that the design parameters do not depend on the level of autocorrelation or measurement inaccuracy. That is, the design parameters are the same as those for i.i.d. observations discussed in [START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF], where ARL 0 = 200, 370.4, 500 and 1000. For the MSS runs-rules (i.e. RR4) and MSS synthetic (i.e. S4) schemes, as H increases, the values of k converge to some specific value no matter how large H gets (see boldfaced values in Table 3). Although not shown here (see [START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF]), for the NSS, SSS and RSS schemes, as H increases, the value of k keeps increasing also, that is, as H gets very large, the value of k approaches 3 at a slow rate (i.e. k converges to the 𝑋 ̅ scheme design parameter value of 3). A similar pattern is observed for other standard values of ARL 0 .

<Insert Table 3>

Separate and combined effect of measurement errors and autocorrelation

In Table 4, we discuss the OOC performance of the basic 𝑋 ̅ , RR4 and S4 schemes under the i.i.d.

case (no measurement errors or autocorrelation), separate and combined effect of the measurement errors and autocorrelation, i.e., (𝜙, 𝛾) ∈{(0,0), (0,0.5), (0.5,0), (0.5,0.5)}, respectively. In each separate panel of the four scenarios, the basic 𝑋 ̅ scheme is compared with the RR4 and S4 schemes in zero-and steady-state modes.

In each panel or each pair (𝜙, 𝛾) in Table 4, with respect to ARL, the (zero-and steady-state) RR4

and (steady-state only) S4 schemes have a better OOC performance than the basic 𝑋 ̅ scheme when 𝛿 ≤ 1; however, for 𝛿 > 1 the converse is true. In zero-state, the S4 scheme has a uniformly better OOC ARL performance than the basic 𝑋 ̅ scheme for all considered shift values. The ARL values at each shift value tend to increase as each of the parameters in the pair (𝜙, 𝛾) increases. For example, when the measurement error increases from 0% to 50%, i.e. the pair (𝜙, 𝛾) is increased from (0,0) to (0,0.5), the ZSARL at 𝛿=0.25 for the S4 scheme also increases from 54.9 to 69.9indicating a deteriorating process. Similarly, when the autocorrelation level is increased from 0% to 50%, i.e. the pair (𝜙, 𝛾) is increased from (0,0) to (0.5,0), the ZSARL at 𝛿=0.25 increases from 54.9 to 118.7.

Moreover, when both the measurement error and autocorrelation are increased from 0 to 50%, i.e.

the pair (𝜙, 𝛾) is increased from (0,0) to (0.5,0.5), the ZSARL at 𝛿=0.25 increases from 54.9 to 128.9.

Next, it is observed that for the S4 scheme, in the i.i.d. case, 50% measurement error only, 50% level of autocorrelation only and, combined 50% of measurement error and autocorrelation, respectively yields the zero-state EARL of 148.9, 155.5, 179.4 and 185.0indicative of a deteriorating process as the monitoring process is subjected to separately as well as combined measurement errors and autocorrelation.

The main deductions from Table 4 are the following:  With respect to EARL, the RR4 and S4 schemes under the separate or combined effect of measurement errors and autocorrelation have a better performance than the basic 𝑋 ̅ scheme.

The latter is further illustrated by the fact that all the PCI values of considered schemes are lower than that of the basic 𝑋 ̅ scheme.  Separately, autocorrelation has a higher negative effect as compared to the negative effect of the measurement errors. Combined, they yield a much higher negative effect.  The zero-and steady-state OOC performances of the RR4 schemes have ARL and EARL that are respectively very close to each other, this is summarized in Remark 1. This is further illustrated by the approximately equal values of the PCIs.

<Insert Table 4> Remark 1: Let the zero-and steady-state EARL be denoted by ZSEARL and SSEARL, respectively; then the percentage difference (%Diff) is given by %Diff = 𝑍𝑆𝐸𝐴𝑅𝐿-𝑆𝑆𝐸𝐴𝑅𝐿 𝑆𝑆𝐸𝐴𝑅𝐿 × 100%. Since the zeroand steady-state OOC performances of the RR4 schemes have the ARLs and EARLs that are respectively very close to each other, it is observed that their %Diff are always less than 1%. To preserve writing space, only the steady-state performance of the RR4 schemes is presented henceforth.

The MSS runs-rules and synthetic s&m schemes

To evaluate the effect of implementing the s&m strategy to reduce the combined effect of autocorrelation and measurement errors, let the 0&1 strategy denote the no remedial approach to reduce the combined effect of autocorrelation and measurement errors. Hence, 3&4 implies that the s=3 observations are skipped before sampling to form a rational subgroup and that m=4 measurements are taken per item.

<Insert Tables 5 and6> In Tables 5 and6, the effect of using 3&4 strategy instead of the 0&1 strategy is studied in the case of the S4 and (RR4 and S4) in zero-and steady-state modes, respectively. Due to Remark 1, the RR4 scheme in zero-state is not shown here. The OOC performance illustration is shown for H∈{1,2,…,12} as per discussion in Table 3 where 𝐸𝐴𝑅𝐿 0&1 and 𝐸𝐴𝑅𝐿 𝐻 denote the EARL of the 0&1 strategy (with H fixed) and at a specific value of H (with s&m fixed), respectivelythe others are defined in a similar manner. In Tables 5 and6, it is observed that increasing s&m from the 0&1 strategy to 3&4 strategy yields a performance gain of at least 20% for all considered values of H when 𝜙=0.5, 𝛾=0.5 and n=5. For example, in zero-state, the S4 scheme (in Table 5) shows that when H=1, the percentage gain of using the 3&4 strategy instead of the 0&1 strategy is 24.9%. Next, the OPG of using H=2 instead of H=1 when 0&1 and 3&4 sampling strategies are implemented are 5.9% and 4.4%, respectively. In Tables 5 and6, it is observed that when H>5, the OPGs of the 0&1 strategy are all less than 1% indicating that increasing H greater than 5 do not yield any more significant percentage gains; consequently, when 𝜙=0.5, 𝛾=0.5 and n=5, the most ideal value of H is equal to 5 and 4 for the 0&1 and 3&4 strategy, respectively. More importantly, it is observed that as H approaches the values of H where k start to converge (see Table 3 for the convergence property, i.e. around H equal to 11 and 12) the OPG approaches a percentage value of 0indicating that there is no benefit in overall performance, at all, by continually increasing the value of H for the RR4 and S4 s&m schemes beyond the converged values of k in both zero-and steady-state modes.

<Insert Tables 7 and8> Furthermore, in Tables 7 and8, the OPG is further analyzed for a variety of levels of autocorrelation and measurement errors. Note though, due to space restriction, only the steady-state analysis of the RR4 and S4 s&m schemes when 𝐻=5, 𝑛=5, 𝛿 𝑚𝑖𝑛 =0, 𝛿 𝑚𝑎𝑥 =3, m∈{1,2,3,4}, s∈{0,1,2,3} and 𝐴𝑅𝐿 0 =370.4; where 𝜙=0.5, 𝛾 ∈{0,0.1,0.2,…,0.9} in Table 7 and𝛾=0.5, 𝜙 ∈{0,0.1,0.2,…,0.9} in Table 8.

Firstly, in Table 7, the following is observed:

 When 𝛾=0, since there are no measurement errors, the values of the EARL do not change for any integer value of m ≥1. Since 𝜙=0.5, as s increases from 0 to 3, the EARL values decrease (vertically downwards) indicating some improvement in OOC performance.  With s and m fixed, when 𝛾>0, the EARL tends to increase indicating a deteriorating process.

However, for each given integer value of s, increasing m leads to some minor decrease in the EARL value. For instance, when s=1 and 𝛾=0.5, the overall performance gains are 1.9%, 0.6%, 0.3% when m is increased from (1 to 2), (2 to 3), (3 to 4), respectively. This means that there is a 1.9% overall performance gain in taking measurements twice as compared to once.

Similarly, there are 0.6% and 0.3% overall performance gains in taking measurements three and four times instead of two and three times, respectively.

Next, in Table 8, the following is observed:  When 𝜙=0, (i.e. no autocorrelation), the EARL not change for any integer value of s≥0.

Moreover, since 𝛾=0.5, as m increases from 1 to 4, the EARL values decrease (vertically downwards) indicating an improvement in performance; note though, this improvement is at a lower rate than in Table 7.  With s and m fixed, when 𝜙>0, the EARL tends to increase indicating a deteriorating process.

However, for each given integer value of m, increasing s leads to some decrease in the EARL values; this improvement is at a slightly higher rate than in Table 7. For instance, when m=1 and 𝜙=0.5, the overall performance gains are 10.5%, 4.3%, 1.9% when is increased from (0 to 1), (1 to 2), (2 to 3), respectively. This means that there is a 10.5% overall performance gain in skipping one observation as compared to successive sampling when forming a rational subgroup. Similarly, there are 4.3% and 1.9% overall performance gains in skipping two and three observations instead of one and two, respectively.

As similarly deduced in Table 4, an outlook comparison of Tables 7 and8 indicates: (i) The autocorrelation level (with 𝛾 fixed) has a significantly greater negative effect on the RR4 and S4 s&m schemes performances than the measurement error level, and (ii) The s-skip strategy has a greater remedial effect than the m-multiple measurement strategy in improving the performance of the RR4 and S4 s&m schemes.

Therefore, based on the analysis done in this section, the recommended values of H, s and m are discussed next. For all different specific shifts values of 𝛿, a value of H that is more or less around the neighborhood of 5 is recommended. Note though, for large shifts, H much lower than 5 is recommended. For any 𝜙 value, when 𝛾 is approximately around (0, 0.3), (0.3, 0.7) and (0.7, 1) an m equal to 1, 2 and 3 are recommended, respectively. For any 𝛾 value, when 𝜙 is approximately around (0, 0.3), (0.3, 0.6) and (0.6, 1) an s value equal to 1, 2 and 3, are recommended, respectively. Note that values of s and m greater than 3 are not recommended so as to not violate the concept of rational subgroup and repeatability & reproducibility (R&R) discussed in [START_REF] Costa | Effect of measurement error and autocorrelation on the 𝑋 ̅ chart[END_REF]. Note though, in an unlikely event that a process has a large number of observations and an abundance of measurement instruments, then the use of s or m greater than 3 is recommended.

Comparison with other runs-type and synthetic-type s&m schemes

Note that in this paper, we focused on the RR4 and S4 s&m monitoring schemes because as it is shown in this section that these MSS designs yields a uniformly better performance than the NSS, SSS and RSS designs. For the sake of illustration, in Table 9 we assume that H=5, 𝛾=0.7, 𝜙=0.7, 𝑛 ∈{3,10}, 𝑠 ∈{0,3} and 𝑚 ∈{1,4}. Since in steady-state, RR1≡S1, RR2≡S2, RR3≡S3 and RR4≡S4see Shongwe and Graham (2019a), the corresponding k values such that the actual IC ARL is equal to 370.4 are 2.2395, 2.1117, 2.1056 and 1.9168, respectively. Assuming a uniform distribution in 𝛿, then the average ratio of ARLs (ARARL) is defined by (see [START_REF] Wu | Optimization designs of the combined Shewhart-CUSUM control charts[END_REF]:

𝐴𝑅𝐴𝑅𝐿 = 1 ∆ ∑ 𝐴𝑅𝐿 𝑐 (𝛿) 𝐴𝑅𝐿 𝑀𝑆𝑆 (𝛿) 𝛿 𝑚𝑎𝑥 𝛿 𝑚𝑖𝑛 ,
where ∆ is the number of increment steps from 𝛿 𝑚𝑖𝑛 to 𝛿 𝑚𝑎𝑥 , 𝐴𝑅𝐿 𝑀𝑆𝑆 (𝛿) is the ARL produced by the MSS (i.e. RR4 or S4) scheme for a specific s&m strategy and 𝐴𝑅𝐿 𝑐 (𝛿) is the ARL of some specified competing scheme for the same s&m strategy. If the value of ARARL is larger than one, the competing scheme will produce larger OOC ARL over a larger shift range and / or to a larger degree compared to the MSS scheme and thus, the competing scheme is relatively less effective. However, if the ARARL is smaller than one, the competing scheme will have higher overall effectiveness than the MSS scheme. Finally, if the value of ARARL is equal to one, the competing scheme has a similar overall performance with the MSS scheme. In Table 9, the following is observed:

 Separately, for the 0&1 and 3&4 strategy, it is observed that in general, with respect to EARL and ARARL, the performance of these schemes can be sorted as follows -RR1≡S1 < RR2≡S2 < RR3≡S3 < RR4≡S4. For example, when 𝑛=10 and 𝛿=0.25, then each of the scheme's 3&4 strategy yields ARL values of 93.8, 65.2, 64.2 and 54.3, respectively. Moreover, the corresponding EARL values are 169.1, 157.8, 157.5 and 152.9, respectively.

The ARARLs of the NSS, SSS and RSS schemes are 26%, 10% and 9% higher than those of the MSS schemes for the 0&1 strategy when 𝑛=3; however, for the 3&4 strategy these are 36%, 14% and 13% higher than those of the MSS schemes, respectively.

 For each scheme, the implementation of the s&m sampling strategy yields improved performance as long as 𝜙>0 and 𝛾>0. For example, when 𝑛=10, the RR1 or S1 scheme yields a 36.5% improvement in the process when implementing the 3&4 strategy instead of the 0&1 strategy.  Increasing the sample size improves the detection ability of all schemes by a similar proportion so that the order in the best detection ability is unchanged, i.e. RR1≡S1 < RR2≡S2 < RR3≡S3 < RR4≡S4.

<Insert Table 9> Similar (but in a different context) to the deductions made for the i.i.d. case in [START_REF] Shongwe | A modified side-sensitive synthetic chart to monitor the process mean[END_REF]; the RR4 and S4 s&m schemes yield the best OOC performance than the corresponding competitors when the process is under the effect of both autocorrelation and measurement errors.

Although not shown here, a similar conclusion is observed for the individual effect of autocorrelation or measurement errors, as well as in the zero-state mode. Moreover, we observed that Remark 1 holds for the RR1, RR2 and RR3 schemes.

Therefore, this is the reason why the focus of this paper was mainly on the MSS (i.e. the RR4 and S4) schemes onlythis is the best possible design for the 2-of-(H+1) runs-rules and synthetic s&m schemes in both the zero-and steady-state modes, for all possible integer values of H.

Illustrative example

The yoghurt cup filling process dataset taken from Costa and Castagliola (2011, p.670) is displayed on Table 10, which shows the weights of different yoghurt cups taken at different sampling points.

The dataset has 20 samples (each of size 5 yoghurt cups taken every hour and each weighted m=2 times) corresponding to a 20-hours sequence of production. The Phase I analysis of this process indicated that the weight of a yoghurt cup, 𝑋 𝑡,𝑖,𝑗 , fits an AR(1) model with parameter 𝜙 = 0.38, an IC mean estimate, 𝜇 0 = 124.9𝑔 and an IC standard deviation, 𝜎 0 = 0.76𝑔. Moreover, an R&R study indicates that the measurement system standard deviation, 𝜎 𝑀 = 0.24𝑔, so that 𝛾=0.316. For illustration purpose, assume the data in Table 10 is a full dataset and here we show how to implement the s&m sampling strategy to form rational subgroups of size n=3. In the last two columns, the corresponding plotting statistics at each sampling point for the 0&1 and 1&2 strategy are shown.

<Insert Table 10> For instance, when 𝑡=1, these are calculated as follows:

𝑋 ̅ 1 = 1 1×3 (𝑋 1,1,1 + 𝑋 1,2,1 + 𝑋 1,3,1 ) = 125.33 -for the 0&1 strategy, 𝑋 ̅ 1 = 1 2×3
(𝑋 1,1,1 + 𝑋 1,1,2 + 𝑋 1,3,1 + 𝑋 1,3,2 + 𝑋 1,5,1 + 𝑋 1,5,2 ) = 124.82for the 1&2 strategy.

The control limits parameters are calculated in Table 11 using Equation ( 7) for the steady-state RR4 or S4 s&m scheme and the corresponding monitoring schemes are constructed in Figure 2 when s∈{0,1}, m∈{1,2} and these are compared to the basic 𝑋 ̅ s&m scheme.

<Insert Table 11> It is observed in Figure 2 that the RR4 or S4 scheme issues an OOC signal one sampling point earlier than each of the corresponding 𝑋 ̅ scheme. That is, in Figure 2(a), the RR4 or S4 0&1 scheme issues an OOC signal for the first time at sampling point 13, whereas the 𝑋 ̅ 0&1 scheme does so at sampling point 14. Similarly, in Figure 2(b), the RR4 or S4 1&2 scheme issues an OOC signal for the first time at sampling point 12, whereas the 𝑋 ̅ 1&2 scheme does so at sampling point 13. This example illustrates the significance of increasing s and m to counteract the negative effect of autocorrelation and measurement errors. Visually, in Figure 2, it is observed that increasing s and m tend to have an improved detection rate because each of the monitoring schemes yields control limits that become narrower as s and m increase.

<Insert Figure 2>

Conclusion remarks

Since serial correlation and measurement inaccuracy are often encountered in real-life applications of SPM; then in this paper, the four design approaches of the 2-of-(H+1) runs-rules and synthetic schemes are considered in an effort to improve the classical Shewhart 𝑋 ̅ scheme in the presence of both autocorrelation and measurement errors; however, focus is paid to the MSS design as it is shown to be the best performing design out of the other three. The first-order autoregressive model with s-skip sampling strategy and the additive model with m-multiple measurements sampling strategy are incorporated to the Markov chain matrix of the MSS runs-rules and synthetic schemes to reduce the negative effect of autocorrelation and measurement errors and are used to derive the zeroand steady-state run-length properties. Empirical analysis indicated that the zero-state MSS synthetic s&m scheme uniformly outperforms the 𝑋 ̅ s&m scheme. However, the zero-and steady-state MSS runs-rules s&m scheme (as well as the steady-state MSS synthetic s&m scheme) outperforms the 𝑋 ̅ s&m scheme for small to moderate shifts, the converse is true for large process shifts. Note though, using the EARL metric, both the MSS runs-rules and synthetic s&m schemes always yield a better zero-and steady-state performances than the 𝑋 ̅ s&m scheme in all the corresponding cases.

Moreover, it has been observed that the autocorrelation level has a relatively higher negative effect than the level of measurement errors. A drawback of these new schemes is that, they require more observations (as some will be skipped during inspection) and more effort (multiple measurements on each item are taken during inspection) as compared to the no remedy approach. Finally, we recommend that quality practitioners should implement any of the two proposed MSS schemes instead of the currently existing Shewhart 𝑋 ̅ scheme when monitoring the process mean under the combined effect of measurement errors and autocorrelation.

1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 1.4 1.0 2 1.4 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 1.2 1.0 2.25 1.2 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 2.5 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 1.1 1.0 2.75 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 3 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.

  𝐻 * , 𝑘 * ) being the pre-specified nominal ARL and 𝛿 ∈ [𝛿 min , 𝛿 max ]. That is, to choose the parameters that yield the smallest EARL, where 𝐴𝑅𝐿(𝛿) is the ARL as a function of the shift 𝛿 in the parameter under surveillance.

Figure 1 :

 1 Figure 1: Charting regions and limits for the 𝑋 ̅ sub-chart

Combined effect of autocorrelation and measurement errors

  2 𝜀 𝑖-2 + ⋯ + 𝜙 𝑠 𝜀 𝑖-𝑠 . Assuming that 𝑌 ̅ 𝑡 =

					1 𝑛	∑ 𝑌 𝑡,(𝑠+1)𝑖-𝑠 𝑛 𝑖=1	is the plotting
	statistic at sampling point t, then the standard deviation of the process in Equation (5) is given by
			𝜎(𝑌 ̅ 𝑡 ) =	𝜎 0 √𝑛𝐶 2 (𝑛, 𝑠, 𝜙)	;	(6)
	with 𝐶 2 (𝑛, 𝑠, 𝜙) = √	𝑛+2(	𝑛 (𝜙 𝑠+1 -1) 2 𝜙 (𝑠+1)(𝑛+1) -𝑛𝜙 2𝑠+2 +(𝑛-1)𝜙 𝑠+1	) .
	2.3 Next, assume that the autocorrelated 𝑌 𝑡,𝑖 (from Equation (5)) are not directly observable, that is, it
	can only be assessed from the results {𝑋 𝑡,𝑖,𝑗 : 𝑡 ≥1; 𝑖 = 1,2,…,n; 𝑗 = 1,2,…,m} from Equation (1).
	Note though, the 𝑌 𝑡,𝑖 observations used in Equation (2) are no longer i.i.d., but are now
	autocorrelated. Moreover, we assume that 𝑋 ̅ 𝑡 is from an imperfect measurement system, i.e. 𝛾 > 0,
	which means that there are several measurements available for each of the autocorrelated subgroup
	samples. Thus, the standard deviation of the plotting statistic in Equation (2) for an autocorrelated
	(with s-skip sampling strategy) and imperfect measurement system (with m-measurement sampling
	strategy), denoted hereafter as s&m strategy (with s>0 and m>1) at each sampling point, is given by
			𝜎(𝑋 ̅ 𝑡 ) =	𝜎 0 √𝑛𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙)	;	(7)
	with 𝐶 3 (𝑚, 𝑛, 𝛾, 𝑠, 𝜙) = 1 √ 𝐶 1 2 (𝑚,𝛾) + 1 ⁄	1 2 (𝑛,𝑠,𝜙) -1 𝐶 2

  to observe what happens to the ARL and EARL as k approaches the convergence property. Moreover, to investigate what value of H has the optimal overall performance gain (OPG) such that increasing H does not yield any further significant OPG.

	To do so, two metrics are defined as follows,
	%Diff =	𝐸𝐴𝑅𝐿 0&1 -𝐸𝐴𝑅𝐿 3&4 𝐸𝐴𝑅𝐿 3&4	× 100%,	for a given value of H
	OPG(𝐻) =	𝐸𝐴𝑅𝐿 𝐻-1 -𝐸𝐴𝑅𝐿 𝐻 𝐸𝐴𝑅𝐿 𝐻	× 100%,	for a specific values of s&m;

Table 1 :

 1 Operation of the MSS runs-rules and synthetic s&m schemes Step Methodology 1 Specify H, 𝜙, 𝛾, n, m, s and ARL 0 . Numerically search for the corresponding value of 𝑘 such that the attained IC ARL (expression derived in Section 3.3) is equal to ARL 0 . 2 Compute UCL / CL / LCL using Equations (7) and (8). 3 At each inspection point, implement the s&m sampling strategy to collect a sample of size n and calculate 𝑋 ̅ 𝑡 . 4 If 𝑋 ̅ 𝑡 ∈ Region B + or B -, return to Step 3; otherwise, go to Step 5. 5 If 𝑋 ̅ 𝑡 ∈ Region A + , go to Step 6a; otherwise, if 𝑋 ̅ 𝑡 ∈ A -, go to Step 6b. Calculate CRL + and if CRL + ≤ 𝐻 go to Step 7; otherwise return to Step 3. (b) Calculate CRL -and if CRL -≤ 𝐻 go to Step 7; otherwise return to Step 3. 7Issue an OOC signal. Take necessary corrective action to find and remove the assignable causes and return to Step 3.

	6	(a)

Table 2 :

 2 General form of the TPM for the MSS synthetic scheme (and MSS runs-rules scheme -by removing the head-start feature elements) 𝜂 𝜏-2 𝜂 𝜏-1 𝜂 𝜏 𝝍 𝟏 𝝍 𝟐 𝝍 𝟑 𝝍 𝟒 𝝍 𝟓 ⋯ 𝝍 𝜿-𝟑 𝝍 𝜿-𝟐 𝝍 𝜿-𝟏 𝝍 𝜿

	𝜂 1	𝜂 2 ⋯ 𝜂 𝑙-3 𝜂 𝑙-2 𝜂 𝑙-1	𝜑	𝜂 𝑙+1 𝜂 𝑙+2 𝜂 𝑙+3 ⋯

Table 3 :

 3 The zero-and steady-state design parameters H and k of the RR4 and S4 schemes for H ∈{1,2,3,…,20,50,100} for ARL 0 =370.4

		Zero-state	Steady-state
	H	RR4	S4	RR4 & S4
	1	1.7814 1.7982	1.7820
	2	1.8664 1.8862	1.8671
	3	1.8969 1.9181	1.8978
	4	1.9099 1.9318	1.9109
	5	1.9158 1.9380	1.9168
	6	1.9186 1.9409	1.9197
	7	1.9199 1.9422	1.9210
	8	1.9205 1.9429	1.9216
	9	1.9208 1.9432	1.9219
	10 1.9209 1.9433	1.9220
	11 1.9210 1.9434	1.9221
	12 1.9210 1.9435	1.9221
	13 1.9210 1.9435	1.9221
	14 1.9210 1.9435	1.9221
	15 1.9210 1.9435	1.9221
	16 1.9210 1.9435	1.9221
	17 1.9210 1.9435	1.9221
	18 1.9210 1.9435	1.9221
	19 1.9210 1.9435	1.9221
	20 1.9210 1.9435	1.9221
	⋮	⋮	⋮	⋮
	50 1.9210 1.9435	1.9221
	⋮	⋮	⋮	⋮
	100 1.9210 1.9435	1.9221

Table 4 :

 4 The zero-state (ZS) and steady-state (SS) ARL, EARL and PCI when 𝜙 ∈{0,0.5}, 𝛾 ∈{0,0.5}, H=7, n=5, s=0, m=1, 𝛿 𝑚𝑖𝑛 =0, 𝛿 𝑚𝑎𝑥 = 3 and ARL 0 =370.4 for the basic 𝑋 ̅ , RR4 and S4 schemes

		(𝜙, 𝛾)		(0,0)				(0,0.5)			(0.5,0)			(0.5,0.5)
		Mode		SS	ZS	ZS		SS	ZS	ZS		SS	ZS	ZS		SS	ZS	ZS
		Scheme	𝑿 ̅	RR4&S4 RR4	S4	𝑿 ̅	RR4&S4 RR4	S4	𝑿 ̅	RR4&S4 RR4	S4	𝑿 ̅	RR4&S4 RR4	S4
		0	370.4	370.4	370.4 370.4 370.4	370.4	370.4 370.4 370.4	370.4	370.4 370.4 370.4	370.4	370.4 370.4
		0.25	133.2	62.7	63.0	54.9 155.2	78.0	78.3	69.9 212.8	126.7	127.1 118.7 223.0	136.9	137.2 128.9
		0.5	33.4	12.7	12.8	8.5	43.9	16.6	16.8	11.7	81.3	33.0	33.3	26.5	89.8	37.3	37.6	30.5
		0.75	10.8	5.2	5.3	2.9	15.0	6.5	6.6	3.8	32.9	12.5	12.7	8.3	37.6	14.2	14.4	9.7
		1	4.5	3.2	3.2	1.7	6.3	3.8	3.8	2.0	15.0	6.5	6.6	3.8	17.5	7.3	7.4	4.3
		1.25	2.4	2.4	2.5	1.2	3.2	2.7	2.8	1.4	7.7	4.2	4.3	2.3	9.0	4.6	4.7	2.5
	𝛿	1.5	1.6	2.1	2.2	1.1	2.0	2.3	2.3	1.2	4.4	3.1	3.2	1.6	5.2	3.4	3.5	1.8
		1.75	1.2	2.0	2.0	1.0	1.5	2.1	2.1	1.1	2.8	2.6	2.6	1.3	3.3	2.7	2.8	1.4
		2	1.1	2.0	2.0	1.0	1.2	2.0	2.0	1.0	2.0	2.3	2.3	1.2	2.3	2.4	2.4	1.2
		2.25	1.0	2.0	2.0	1.0	1.1	2.0	2.0	1.0	1.6	2.1	2.2	1.1	1.7	2.2	2.2	1.1
		2.5	1.0	2.0	2.0	1.0	1.0	2.0	2.0	1.0	1.3	2.0	2.1	1.0	1.4	2.1	2.1	1.1
		2.75	1.0	2.0	2.0	1.0	1.0	2.0	2.0	1.0	1.2	2.0	2.0	1.0	1.2	2.0	2.0	1.0
		3	1.0	2.0	2.0	1.0	1.0	2.0	2.0	1.0	1.1	2.0	2.0	1.0	1.1	2.0	2.0	1.0
		EARL 187.5	156.9	157.1 148.9 200.9	164.1	164.4 155.5 244.8	189.8	190.3 179.4 254.5	195.8	196.2 185.0
		PCI	1.00	0.84	0.84	0.79	1.00	0.82	0.82	0.77	1.00	0.78	0.78	0.73	1.00	0.77	0.77	0.73

Table 5 :

 5 The zero-state ARL and EARL of the S4 scheme when 𝜙=0.5, 𝛾=0.5, n=5, H∈{1,2,3,…,11,12}, s&m∈{0&1, 3&4}, 𝛿 𝑚𝑖𝑛 =0, 𝛿 𝑚𝑎𝑥 =3 and ARL 0 =370.4

	H

Table 6 :

 6 The steady-state ARL and EARL of the RR4 and S4 schemes when 𝜙=0.5, 𝛾=0.5, n=5, H∈{1,2,3,…,11,12}, s&m∈{0&1, 3&4}, 𝛿 𝑚𝑖𝑛 =0, 𝛿 𝑚𝑎𝑥 =3 and ARL 0 =370.4

	H

Table 7 :

 7 The steady-state EARL (and OPG -in brackets) of the RR4 and S4 s&m schemes when 𝐻=5, 𝜙=0.5, 𝑛=5, 𝛿 𝑚𝑖𝑛 =0, 𝛿 𝑚𝑎𝑥 =3, 𝑚 ∈{1,2,3,4}, s∈{0,1,2,3}, 𝛾 ∈{0,0.1,0.2,…,0.9} and 𝐴𝑅𝐿 0 =370.4

	𝛾

Table 8 :

 8 The steady-state EARL (and OPG -in brackets) of the RR4 and S4 s&m schemes when H=5, 𝛾=0.5, n=5, 𝛿 𝑚𝑖𝑛 =0, 𝛿 𝑚𝑎𝑥 =3, m∈{1,2,3,4}, s∈{0,1,2,3}, 𝜙 ∈{0,0.1,0.2,…,0.9} and 𝐴𝑅𝐿 0 =370.4 𝜙

		s	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
		0	165.4	170.3	176.0	182.4	189.7	197.9	207.0	217.2	228.5	240.7
	m=1	1	165.4 (0%) 165.9 (2.7%) 167.3 (5.2%) 169.8 (7.4%) 173.6 (9.3%) 179.1 (10.5%) 186.7 (10.9%) 197.1 (10.2%) 211.0 (8.3%) 229.7 (4.8%)
		2	165.4 (0%) 165.4 (0.3%) 165.7 (0.9%) 166.6 (1.9%) 168.5 (3.1%)	171.7 (4.3%)	177.0 (5.5%)	185.5 (6.3%)	199.0 (6.0%) 220.4 (4.2%)
		3	165.4 (0%) 165.4 (0.0%) 165.5 (0.2%) 165.7 (0.5%) 166.6 (1.1%)	168.4 (1.9%)	171.9 (2.9%)	178.5 (3.9%)	190.5 (4.5%) 212.6 (3.7%)
		0	161.7	166.7	172.6	179.1	186.5	194.9	204.1	214.5	225.9	238.4
	m=2	1	161.7 (0%) 162.1 (2.8%) 163.6 (5.5%) 166.3 (7.7%) 170.1 (9.6%) 175.7 (10.9%) 183.4 (11.3%) 193.9 (10.6%) 208.2 (8.5%) 227.2 (4.9%)
		2	161.7 (0%) 161.8 (0.2%) 162.1 (0.9%) 163.0 (2.0%) 164.9 (3.2%)	168.1 (4.5%)	173.5 (5.7%)	182.2 (6.5%)	195.8 (6.3%) 217.7 (4.4%)
		3	161.7 (0%) 161.7 (0.0%) 161.8 (0.2%) 162.1 (0.6%) 162.9 (1.2%)	164.8 (2.0%)	168.4 (3.0%)	175.1 (4.1%)	187.3 (4.6%) 209.9 (3.7%)
		0	160.5	165.6	171.3	178.0	185.4	193.8	203.2	213.6	225.2	237.6
	m=3	1	160.5 (0%) 160.9 (2.9%) 162.4 (5.5%) 165.1 (7.8%) 168.9 (9.8%) 174.6 (11.0%) 182.3 (11.4%) 192.9 (10.7%) 207.3 (8.6%) 226.3 (5.0%)
		2	160.5 (0%) 160.5 (0.3%) 160.8 (1.0%) 161.8 (2.0%) 163.6 (3.3%)	166.9 (4.6%)	172.4 (5.8%)	181.1 (6.6%)	194.9 (6.3%) 216.8 (4.4%)
		3	160.5 (0%) 160.5 (0.0%) 160.5 (0.2%) 160.8 (0.6%) 161.7 (1.2%)	163.5 (2.1%)	167.1 (3.1%)	173.9 (4.1%)	186.2 (4.7%) 208.9 (3.8%)
		0	159.8	164.9	170.8	177.4	184.9	193.3	202.7	213.2	224.7	237.2
	m=4	1	159.8 (0%) 160.3 (2.9%) 161.8 (5.5%) 164.4 (7.9%) 168.3 (9.8%) 174.0 (11.1%) 181.8 (11.5%) 192.5 (10.8%) 206.8 (8.7%) 225.8 (5.0%)
		2	159.8 (0%) 159.8 (0.3%) 160.2 (1.0%) 161.1 (2.0%) 163.0 (3.3%)	166.4 (4.6%)	171.7 (5.8%)	180.5 (6.6%)	194.3 (6.4%) 216.4 (4.4%)
		3	159.8 (0%) 159.8 (0.0%) 159.9 (0.2%) 160.2 (0.5%) 161.0 (1.2%)	162.9 (2.1%)	166.6 (3.1%)	173.3 (4.1%)	185.6 (4.7%) 208.5 (3.8%)

Table 9 :

 9 The steady-state ARL, EARL and ARARL of the (NSS, SSS, RSS, MSS) runs-rules and synthetic schemes with the 0&1 strategy (and 3&4 strategyin brackets) when 𝛿 𝑚𝑖𝑛 = 0, 𝛿 𝑚𝑎𝑥 = 3, 𝜙=0.7, 𝛾=0.7, 𝑛 ∈ {3,10} and ARL 0 =370.4

	n	𝜹	NSS: RR1,S1 SSS: RR2,S2 RSS: RR3,S3 MSS: RR4,S4
		0	370.4 (370.4)	370.4 (370.4)	370.4 (370.4)	370.4 (370.4)
		0.25	264.8 (206.8)	217.2 (156.6)	215.3 (154.7)	201.4 (140.1)
		0.5	124.7 (68.5)	87.6 (47.7)	86.3 (47.0)	74.5 (39.0)
		0.75	55.0 (25.1)	38.7 (18.6)	38.1 (18.4)	31.2 (14.8)
		1	26.6 (11.5)	19.6 (9.2)	19.4 (9.1)	15.6 (7.5)
		1.25	14.5 (6.5)	11.4 (5.5)	11.2 (5.5)	9.1 (4.7)
	3	1.5	8.9 (4.3)	7.4 (3.9)	7.3 (3.8)	6.1 (3.4)
		1.75	6.1 (3.2)	5.2 (3.0)	5.2 (3.0)	4.4 (2.7)
		2	4.5 (2.6)	4.0 (2.5)	4.0 (2.5)	3.5 (2.4)
		2.25	3.5 (2.3)	3.3 (2.2)	3.3 (2.2)	3.0 (2.2)
		2.5	3.0 (2.1)	2.8 (2.1)	2.8 (2.1)	2.6 (2.1)
		2.75	2.6 (2.0)	2.5 (2.0)	2.5 (2.0)	2.4 (2.0)
		3	2.3 (2.0)	2.3 (2.0)	2.3 (2.0)	2.2 (2.0)
		EARL	295.6 (235.8)	257.5 (208.6)	256.0 (207.6)	242.1 (197.8)
		%Diff	25.4%	23.4%	23.3%	22.4%
		ARARL	1.36 (1.26)	1.14 (1.10)	1.13 (1.09)	1.00 (1.00)
		0	370.4 (370.4)	370.4 (370.4)	370.4 (370.4)	370.4 (370.4)
		0.25	200.4 (93.8)	150.7 (65.2)	148.8 (64.2)	134.2 (54.3)
		0.5	64.0 (17.3)	44.7 (13.3)	44.0 (13.2)	36.4 (10.7)
		0.75	23.1 (6.0)	17.3 (5.2)	17.1 (5.2)	13.8 (4.4)
		1	10.6 (3.3)	8.6 (3.1)	8.5 (3.1)	7.0 (2.8)
		1.25	6.0 (2.4)	5.2 (2.3)	5.2 (2.3)	4.4 (2.2)
	10	1.5	4.0 (2.1)	3.7 (2.0)	3.6 (2.0)	3.2 (2.0)
		1.75	3.0 (2.0)	2.9 (2.0)	2.9 (2.0)	2.6 (2.0)
		2	2.5 (2.0)	2.4 (2.0)	2.4 (2.0)	2.3 (2.0)
		2.25	2.2 (2.0)	2.2 (2.0)	2.2 (2.0)	2.1 (2.0)
		2.5	2.1 (2.0)	2.0 (2.0)	2.0 (2.0)	2.0 (2.0)
		2.75	2.0 (2.0)	2.0 (2.0)	2.0 (2.0)	2.0 (2.0)
		3	2.0 (2.0)	2.0 (2.0)	2.0 (2.0)	2.0 (2.0)
		EARL	230.8 (169.1)	204.7 (157.8)	203.7 (157.5)	194.1 (152.9)
		%Diff	36.5%	29.7%	29.4%	26.9%
		ARARL	1.26 (1.16)	1.11 (1.06)	1.10 (1.06)	1.00 (1.00)

Table 10 :

 10 The yoghurt filling cup process dataset

	0&1	1&2

Table 11 :

 11 The RR4 or S4 s&m scheme using the yoghurt filling cup example in steady-state mode when 𝛾=0.316, 𝜙=0.38, H=1, n=3, m∈{1,2} and s∈{0,1}

		s	0	1
		m	1	2
		0.9536 0.9760 0.7898 0.9104 𝐶 3 (𝑚, 𝑛, 𝛾, 𝜙, 𝑠) 0.7664 0.8922 𝐶 1 (𝑚, 𝛾) 𝐶 2 (𝑛, 𝜙, 𝑠)
	RR4 or S4	UCL LCL	125.92 125.78 123.88 124.02
	𝑋 ̅	UCL LCL	126.62 126.38 123.18 123.43

Acknowledgements

We would like to convey our gratitude to the Editorial staff and the anonymous referee(s) who carefully read our earlier draft and gave us constructive comments to improve the paper.


Isabelle Blasquez
email: isabelle.blasquez@unilim.fr

Hervé Leblanc
email: leblanc@irit.fr

Specification By Example for Educational Purposes

Keywords: Agile Software Development, Specification by Example, Agile Teaching

The Specification By Example (SBE) is a guideline for building the right software, a software that meets customer requirements. It is based on seven process patterns and enhances communication and collaboration and it usually is used in agile software development. The connection between education and agile software development sounds actually as an emergent topic. In this paper, we propose to structure a teaching approach in analogy to an agile software developement by transposing each process pattern of SBE to a corresponding one in the teaching domain. Moreover, we show that thanks to the emergence of a collective intelligence process, the students are more confident and more responsible. Such a course offers the opportunity to learn not only technical skills, but also some values in a new mindset.

INTRODUCTION

"Building the product right and building the right product are two different things. We need both in order to succeed". From this observation, Godjko Adzick builds a collective knowledge by studying over 50 software projects [START_REF] Adzic | Specification by Example: How Successful Teams Deliver the Right Software[END_REF]. The related process consists of gathering examples to clarify requirements, deriving tests and automating them. Specification By Example (SBE) is also defined as an approach of software development based on seven process patterns that help teams to build the right software product by writing just enough documentation to facilitate change effectively in short iterations or in flow-based development [START_REF] Adzic | Specification by Example: How Successful Teams Deliver the Right Software[END_REF]. It also enhances communication and collaboration and it usually is used in agile software development.

The connection between agile software development and education sounds actually as an emergent topic. Some trans-DOI: http://dx.doi.org/10.1145/3059009.3059039 positions are developped concerning agile manifesto 1 , agile values2 , the Scrum framework 3 and more generally agile approaches 4 . These propositions suggest the idea that education today needs a "refactoring". While the agile manifesto proposes to evolve from a traditional plan-driven paradigm to a value-driven paradigm, the agile school manifesto suggests an evolution from traditional teaching-approaches to new learning approaches.

We propose a teaching approach in analogy to an agile software developement by transposing each process pattern of SBE to a corresponding one in the teaching domain. The result of the transposition is a guideline usable by teachers to produce the right course as the agile methodologies are used to produce the right software.

As example, we use a software product methods course that we gave to eighty two-year french undergraduates. This course is a part of French National Pedagogical Program (PPN), a common program to all technical colleges specialized in Computer Technology. The objective of this course is to present software development processes. We have chosen to teach agile software development.

This paper presents a transposition of the seven process patterns of Specification By Example to improve the design of a course. This is one section per process pattern with three paragraphs each: an overview of the original pattern in SBE, the corresponding pattern in teaching-domain, and an example on the software development processes course. The next section gives some considerations about the concept of right course and gives a toolkit of our proposal.

BACKGROUND AND CONTEXT

Traditional teaching approach is teacher-centered and often relies only on lectures and small exercises. Nowadays, we remark that students become more quickly bored and inattentive due to mismatches between students learning and teaching styles [START_REF] Felder | Learning and teaching styles in engineering education[END_REF]. To improve the quality of courses and to help both teacher and students for a better alignment, we have designed a guideline based on SBE.

A definition of quality in the context of educational game is suggested in [START_REF] Wangenheim | Scrumia : An educational game for teaching scrum in computing courses[END_REF] as if it provides a positive learning effect, motivates students to study and provides a pleasant and engaging learning experience. We choose to use this definition to the right course whose the expectations in terms of ness model, agile requirements, user story, Scrum in practice, tests, and retrospective.

As the students were novices, this course's vision has been suggested by the teacher. But in another context such as an advanced course, the course's vision could be defined collaboratively by an open backlog and a collective vote.

SPECIFYING COLLABORATIVELY

Original pattern.

Specifying collaboratively enables to harness the knowledge and the experience of the whole team. It also creates a collective ownership of specifications, making everyone more engaged in the delivery process [START_REF] Adzic | Specification by Example: How Successful Teams Deliver the Right Software[END_REF]. No document should be written in isolation. To collaborate effectively, most popular models are introduced: workshops (all-team or smaller), pair-writing, and even informal conversations. All-team workshops are useful to discover and learn about the business model. They are a good way to build a shared understanding of the requirements and produce a set of examples that illustrate a feature. The smaller workshops help to clarify or complete the specification. Pairing to write specifications allows to mature products by getting several different perspectives on a example. Mostly, collaboration needs a preparation phase to be efficient. It is easier to initiate a collective discussion if some examples have been prepared before.

Corresponding teaching-domain pattern.

Specifying collaboratively transposes to Teaching collaboratively. In SBE, collaboration models are classified according to the size of the team. In teaching domain, collaboration models will be classified according to the teaching approach.

We propose two kinds of collaboration models: inductive and deductive workshops. Induction is a reasoning progression that proceeds from particulars (observations, measurements, data) to generalities (governing rules, laws, theories). Deduction proceeds in the opposite direction [START_REF] Felder | Learning and teaching styles in engineering education[END_REF]. Traditional teaching approach is deductive. The inductive approach includes problem-based learning, discovery learning, inquiry learning, or some variation on those themes which are characterized by [START_REF] Smith | Pedagogies of engagement: Classroom-based practices[END_REF] as constructivist based approaches. These approaches impose more responsibility on students for their own learning. They almost always involve students discussing questions and solving problems (active learning) and the work is done in groups (collaborative or cooperative learning).

The Participatory Action Research (PAR) is a collaborative process of research, education and action explicitly oriented towards social transformation [START_REF] Kindon | Participatory Action Research Approaches and Methods: Connecting People, Participation and Place[END_REF]. To promote the emergence of a collective intelligence process, we propose to structure the workshops around a AAA-PAR strategy that consists in three ordered steps. The first step is the Arrangement time: the preparation phase where teacher prepares materials provided to the students to start the workshop. The second step is the Action time: the core of the workshop. During this step, the students collaborate to achieve the required learning goals. The self-organized development team of an agile software development also transposes to an autonomous student team. The behavior of the stakeholders is a crucial factor for the success of these workshop. The main teacher's responsibility is to engage students in learn-learning, collaboration, commitment and happiness to work are similar. The purpose of the right course is to enhance the individual student's capability to participate in and contribute to collaborative learning process based on collective intelligence process. According to Le ´vy, Collective Intelligence (CI) is the capacity of human collectives to engage in intellectual cooperation in order to create, innovate and invent [START_REF] Lévy | From social computing to reflexive collective intelligence: The ieml research program[END_REF].

The guideline is presented in the table 1. The first column provides original patterns of SBE. The second column provides corresponding teaching-domain patterns. And the last column provides practices used to implement the patterns. Teachers could design the right course by determining the right teaching resources as well as posssible. For students, this guideline gives the opportunity to learn not only technical skills, but also some values in a new mindset.

Throughout this paper, we propose an analogy between the agile software development terminology and the teaching terminology. As we have already suggested to translate product into course, we also suggest to translate Product Manager into Course designer and requirements into syllabus of the course. Examples from SBE will be transposed to teaching resources.

DERIVING SCOPE FROM GOALS

Original pattern.

A best practice for this pattern is to start with a customer's business goal and then to collaborate with the business to derive the scope. This pattern also first focuses on why something is needed and who needs it. For effectively collaborating, it recommends to ask how something would be useful before what you need to build it. "Impact Mapping" is a technique for deriving scope from goals which propose to build a minmap around four aspects of the software: goal (why), actors (who), impacts (how), and deliverables (what). [START_REF] Adzic | Impact Mapping. Provoking Thoughts[END_REF] Corresponding teaching-domain pattern.

This process pattern focuses on the course's vision by answering why, who, how, and what. The why helps to define the objective of the course also called learning goals. The who depicts actors, usually students. The how identifies impacts to help actors to achieve the objective. The impacts of the teaching domain are learning outcomes (knowledge and competencies acquired or improved by students). The what helps to outline deliverables needed to support the impacts. For teaching, they are topics which are the transposition from features.

Example.

Our objective is to propose an introduction to agile software development in an agile mindset: Doing agile and Being agile with respect to the first levels of Bloom's taxonomy suitable to undergradates (knowledge, comprehension and application) [START_REF] Bloom | Taxonomy of educational objectives. The classification of educational goals[END_REF]. The actors are students. There are three learning outcomes relative to agile practices: delivering the right product, delivering the product right, and delivering fast and regularly. There is one learning outcome relative to agile values and principles. The scope is also determined by higher-level topics which are: introduction to agile software development, overview of collaborative tools, introduction to Scrum, from traditional to agile software development, busi- Knowledge evaluation process with or without grades, individual or collective material, frequency, . . . Evolving a documentation system Evolving a documentation academic access, material, architecture ing: he becomes the facilitator of the workshop [START_REF] Smith | Pedagogies of engagement: Classroom-based practices[END_REF]. He must ensure trust behaviors. Thirteen trust behaviors have been identified by [START_REF] Covey | The SPEED of Trust: The One Thing that Changes Everything[END_REF], some of them are: demonstrate respect, create transparency, listen first, keep commitments, and extend trust. The third step is the Assertion time: a kind of workshop review. The teachers also encourages students to explicitly reflect on the events of the Action time and to examine the lessons learned about the workshop. An agile retrospective [START_REF] Derby | Agile Retrospectives: Making Good Teams Great[END_REF] is well-adapted at this time and the three following questions could be asked: What did you learn in this workshop ? What is amazing in this workshop ? Why will you reuse or not the workshop later ?

Example.

Both inductive and deductive workshops are used in our course. Inductive workshops was preferred to respect the CI process and to promote the emergence of agile values. The teams are composed by six or seven students to respect the ideal size of an agile team.

ILLUSTRATING USING EXAMPLES

Original pattern.

Examples are used to clarify meaning in everyday conversation: they are concrete and less unambiguous. Illustrating requirements using examples is a way to specify with enough details that we can be checked by assertion. Using examples will ensure that the delivery teams focus on the right product and that they have a shared understandings of what the business users expect out of the system [START_REF] Adzic | Specification by Example: How Successful Teams Deliver the Right Software[END_REF]. To be used from requirement analysis to testing, examples should be small, precise, realistic and easy to understand.

Corresponding teaching-domain pattern.

Illustrating with examples tranposes to illustrating with teaching resources. We define a teaching resource as an activity used by a teacher to engage students in learning to achieve required learning goals.

Learning is presented as a two-step process involving the reception and processing of information [START_REF] Felder | Learning and teaching styles in engineering education[END_REF]. The processing step may involve different learning's models: simple memorization, inductive or deductive reasoning, reflection, action, and introspection or interaction with others. As students learn in many ways and as teaching methods are also various, learning styles and teaching styles are identified by [START_REF] Felder | Learning and teaching styles in engineering education[END_REF]. A learning style should consider: perception (sensory or intuitive), input (visual or auditory), organization (inductive or deductive), processing (active or reactive), and understanding (sequential or global). A teaching style should con-sider: content (concrete or abstract), presentation (visual or verbal), organization (inductive or deductive), student participation (active or passive), and perspective (sequential or global).

When mismatches exist between learning and teaching styles, students become bored and inattentive in class, do poorly on tests, and get discouraged about the courses [START_REF] Felder | Learning and teaching styles in engineering education[END_REF]. To engage students in learning, a best practice is to choose the right teaching resources support by finding the better alignment between learning and teaching style. Sometimes, lectures are necessary to introduce or clarify a concept. For a better alignment of the right course, we suggest two others potential teaching resources.

Gamification is the process of using game-based mechanics, aesthetics and game thinking to engage people, motivate action, promote learning, and solve problems [START_REF] Kapp | The Gamification of Learning and Instruction: Game-based Methods and Strategies for Training and Education[END_REF]. Game can also offer some moment of serendipity. The Assertion time is crucial to have benefits from games. A link between games, culture, happiness, learning, and productivity is shown in [START_REF] Mezick | The Culture Game: Tools for the Agile Manager[END_REF] and studied on an educational game in [START_REF] Wangenheim | Deliver! -an educational game for teaching earned value management in computing courses[END_REF]. Innovation Games [START_REF] Hohmann | Innovation Games[END_REF] or Game storming [START_REF] Gray | Gamestorming: A Playbook for Innovators, Rulebreakers, and Changemakers[END_REF] give usefull examples of gamification.

Project Based Learning (PBL) is perceived to be a studentcentered approach to learn [START_REF] Chandrasekaran | Learning through projects in engineering education[END_REF][START_REF] Savery | Overview of problem-based learning: definition and distinctions, the interdisciplinary[END_REF]. The students need to produce a solution to solve a problem and an outcome in the form of a report. PBL focuses on large, open-ended problems, like many real-world problems [START_REF] Ragan | Product-based learning in software engineering education[END_REF]. It is based on five principles: students work together in groups; a real world problem that affects the life of the students is presented for investigation; students discuss findings and consult the teacher for guidance, input, and feedback; the maturity level of students skills determines the degree of guidance provided by the teacher; resulting products can be shared with the community.

Example.

Various teaching resources have been used : traditional lectures, games (lego-based approaches as Lego4Scrum or TDDLego [START_REF] Kurkovsky | A lego-based approach to introducing test-driven development[END_REF]), collaborative workshops (story-writing workshop and Coding dojo). A PBL approach has also been adopted including collaborative workshops (product vision statements, story mapping, impact mapping) and innovation games (Product Box, Speed Boat). Gamification has been preferred to introduce concepts whenever possible. Videos from professionnal conferences have been watched in some lectures. To introduce the visual management, visual information has often been used with pictures, animations and sketchnotes.

AUTOMATING VALIDATION WITHOUT CHANGING SPECIFICATIONS

Original pattern.

After refining the specification, the examples can be used as a target for the implementation and the validation of the software. The tests should often be run during the development to ensure quality of the product and to reduce delays of the feedback. This pattern focuses on automation, as a solution of a quick feedback. The automation has long-term benefits: having an objective measurement of when the job is finished, checking more frequently, and getting a living documentation [START_REF] Adzic | Specification by Example: How Successful Teams Deliver the Right Software[END_REF].

Corresponding teaching-domain pattern.

Thinking about automation is asking about repetition of a learning resource. The repetition is a well-known best practice in teaching. By using different types of teaching resources on a same competency, we prevent the disalignment between teaching and learning styles, we respect the learning time of each student, and we offer different opportunities to apply the competency. This pattern focuses on teaching resources management. It helps to select suitable teaching resources by considering learning goals, learning styles and competency. A first sequence of teaching resources can be scheduled according to the sequence of topics. To ensure repetition of some competencies, new teaching resources can be introduced. During the course, a sequence can be changed by the teacher according to the feedback on student behaviors and feelings (motivation, comprehension, outcome validation, . . . ). The teacher should also apply some values to his behavior, as responding to change in his course's design.

Example.

The course run over a period of 10 weeks with two 2-hours sessions per week. Lectures, games, collaborative workshops and PBL have been alternatively used. As students have a mentored software development project to lead at the same time, some workshop of PBL have been repeated in this context to help students to find user-stories from the vision. These workshops were self-organized by the student teams.

VALIDATING FREQUENTLY

Original pattern.

A continuous integration system builds the product and runs the tests. It ensures that once the product is built right, it stays right. To satisfy this point, this pattern suggests to validate executable specifications frequently to keep them reliable. The best practices insist on reducing unreliability and on looking for ways to get faster feedback.

Corresponding teaching-domain pattern.

Mostly, the actors build their own knowledge during the right course. The teacher is responsible for the reliability of the knowledge. The students are responsible to get faster feedback. This pattern suggests to determine the knowledge evaluation process. This process must be defined by considering some questions as: is it an individual or collective evaluation ? What is the material for the evaluation ? How often should we evaluate ? Are grades really required as a measure of academic performance ? and so on.

Example.

An evaluation has been planned at the end of each workshop. This evaluation has taken place during the Assertion time with an oral presentation of the outcomes. No grade is attributed for this. For each workshop of the PBL, a summary has been required per team as material for the evaluation. This summary was based on a template provided by the teacher: presentation of the workshop, deliverables, comments, and retrospective. It allows to detail knowledge, comprehension, and application of the competency taught. Such a summary could also be used as a cookbook to apply easily the competency again. At the end of the course, a report including a presentation of the PBL project and all the summaries has been delivered by each team. These reports have been graded by one grade by team to respect the collective intelligence process.

EVOLVING A DOCUMENTATION SYS-TEM

Original pattern.

Living documentation is an artifact and the end-product of SBE. It is a reliable and authoritative source of information on system functionality, which anyone can easily ac-

REFINING THE SPECIFICATION

Original pattern.

This pattern brings further informations about the specification. To be unambiguous, a good specification should be precise and testable, and concerns only business functionality. To be useful as long-term documentation, it should be self-explanatory, focused, and in domain language.

Corresponding teaching-domain pattern.

This pattern focuses on teaching resources details. It helps the teacher to refine the teaching resources and to improve it. A teaching resource should be focused about a specific competency and be well time-boxed to respect the duration of the workshop. To be precise, the application domain must be carefully chosen to promote the commitment of students. To be testable, an outcome must be defined. An outcome describes a way to verify and validate that the required learning goals are well-achieved. The tests in software development transposes to the outcome in teaching domain. Collective discussions induced by the teaching resource will be efficient only if everyone has a common understanding of it. A starter kit could prevent misunderstanding and promote self-organization of the activities.

Example.

Each teaching resource has been focused on a specific competency and time-boxed to respect the duration of a session. Each student team has choosen by collective vote his own application domain for the project used for the PBL workshops. At least, two major benefits can be highlighted with this kind of project. First, the product manager (a student) is always avalaible. Then most of students feel involved because they could be quickly become the users of this application. A starter kit has been provided for each new teaching resources. It includes a roadmap, a brief summary of the concept taught, a description of required deliverable, a glossary, and bibliographical references. The outcome have been various as oral feedback, photos, and specific artifact or summary. cess [START_REF] Adzic | Specification by Example: How Successful Teams Deliver the Right Software[END_REF]. Best practices dedicated to living documentation are: easy to access, easy to understand, and each change in the system needs to be reflected.

Corresponding teaching-domain pattern.

System documentation transposes to academic documentation in teaching domain. To easily access to the documentation, material courses (lectures, exercices, tutorials, references) can be deposit in repositories managed by a version control system. Each stakeholder (teacher and students) can consult or tell about changes to update material courses. The role of the teacher is to encourage students to share and update the materials to have more understable course notes. The documentation can be alive by setting up an automatic notification system to alert all the stakeholder when a new document is added or updated.

Example.

Github is used to share on-line public material courses 5 . It provides collaboration features such as pull request or wikis. A Slack6 team has been created for this course to facilitate discussions between skateholders. The web-service IFTTT 7connects Github with Slack to automatically notify all the stakeholder when a change in the documentation is pushed.

VALIDATION

We focus now on the evaluation of the quality (as defined in section 2) of the agile software project management course presented as the example. This course has been designed in 2015 and delivered in the fall 2015 and 2016.

The validation has been adapted from a specific framework [START_REF] Wangenheim | Deliver! -an educational game for teaching earned value management in computing courses[END_REF] which is based on two questionnaries related to Bloom's taxonomy of educational objectives [START_REF] Bloom | Taxonomy of educational objectives. The classification of educational goals[END_REF] and Kirkpatrick's levels of evaluation [START_REF] Kirkpatrick | Evaluating training programs : the four levels[END_REF]. It has already been applied for educational games and coding dojo session [START_REF] Estácio | Evaluating the use of pair programming and coding dojo in teaching mockups development: An empirical study[END_REF][START_REF] Wangenheim | Scrumia : An educational game for teaching scrum in computing courses[END_REF]. We adapted the terminology from game to course and we deleted some items from original questionnaires to only focus a set of teaching resources. Moreover, the evaluation is concerned by student perception in terms of motivation, user experience, and learning process. Questionnaries are given to the 80 students at the end of the course.

Results of the first questionnaire are presented in Fig. 1. It is based on [START_REF] Bloom | Taxonomy of educational objectives. The classification of educational goals[END_REF] and evaluates the perception of the evolution of learning in the competencies taught before and after the course. It focuses on the learning goals (doing agile and being agile) with respect to the perceived impacts rated on a scale from 1 to 5. The perception of the Agile Software Development (the heart of the course) was multiplied by more than two. It is the same for the being agile posture that students can reused for other courses.

Results of the second questionnaire are presented in Fig. 2. It is based on [START_REF] Kirkpatrick | Evaluating training programs : the four levels[END_REF] consists in 21 items asking motivation, user experience, and learning on a Likert scale with response alternatives ranging from strongly disagree (-2) to strongly agree [START_REF] Adzic | Impact Mapping. Provoking Thoughts[END_REF]. The majority of the students agreed strongly that the course promotes moments of cooperation. They also confirmed that they had fun while interacting with other students. The social interaction has also been the highest rated dimension. Overall, results are positive in terms of fun, challenge, and social interactions.

CONCLUSION

This paper presents a guideline for delivering a right course by structuring a teaching approach as an agile software development. We defend a new teaching way based on collective intelligence process which aims to align a course with the needs of students by designing the right teaching resources. The role of the teacher as a facilitator has also been presented as a key of success. By maintaining trust behaviors and by encouraging the communication, he helps students to collaborate efficiently and to be more commitment, more creative and more responsible. In agile software development, the most popular methodologies used are: Scrum and eXtreme Programming. We showed that SBE can be transposed to specify teaching resources. Scrum has already been transposed to manage learning experience where the responsibility for the learning process is delegated to students. No transposition has already been proposed for eXtreme Programming. We plan to study their principles, especially the transposition from a user story to a teaching story.

ACKNOWLEDGMENTS

This same transposition was used for a specific course during 3 days on agile requirements at the University of Toulouse for a professional Bachelor dedicated to Development and Software Quality.

Figure 1 :

 1 Figure 1: Frequency diagram of answers with respect to the feedback on learning.

Figure 2 :

 2 Figure 2: Frequency diagram of answers with respect to the sub-component user experience.

Table 1 :

 1 Guideline for delivering the right course

	Process Patterns	Teaching Patterns	Practices
	Deriving scope from goals	Course's vision	goal, skateholders, skills, main topics
	Specifying collaboratively	Teaching collaboratively	inductive workshop, deductive workshop
			structure of the workshop
	Illustrating using examples	Illustrating with teaching resources	learning style
		(teaching resources support)	teaching style
	Refining the specification	Refining teaching resources	competency, timeboxing, application domain
		(teaching resources details)	outcome, starter kit
	Automating validation without changing specification	Teaching resources management	schedule, repetition
	Validating frequently

https://www.infoq.com/articles/agile-schools-education

https://pedagogieagile.com/2012/05/12/les-valeurs

http://eduscrum.nl/en

http://approchealpes.info

https://github.com/iblasquez

https://slack.com: a cloud-based team collaboration tool

https://ifttt.com: If This Then That