Fupeng Xie

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

Jinsheng Sun

Anan Tang
email: tanganan2@sina.com

Xuelong Hu

A one-sided adaptive truncated exponentially weighted moving average scheme for time between events

Keywords: Adaptive truncated EWMA chart, Average run length, Markov chain model, Parameter estimation, Time-between-events

In order to provide a balanced protection against a range of shift sizes in highquality processes, a new one-sided adaptive truncated exponentially weighted moving average (ATEWMA) control chart with known and estimated parameters is developed for monitoring time-between-events (TBE) data. A dedicated Markov chain model is established for evaluating the run length properties in known and estimated parameters operating conditions. Furthermore, a two-stage optimal design procedure of the proposed scheme is developed based on the average run length (ARL) criteria. Simulation results show that the one-sided ATEWMA TBE scheme with known parameters is superior to its competitors in detecting both upward and downward shifts. Finally, two real data applications are employed to show the implementation of the recommended scheme in the monitoring of TBE data.

Introduction

As one of the most influential tools in statistical process monitoring (SPM), control charts have been extensively used in various fields, for example, manufacturing industry (see [START_REF] Chong | A variable parameters auxiliary information based quality control chart with application in a spring manufacturing process: The markov chain approach[END_REF]) and network monitoring (see [START_REF] Perry | An EWMA control chart for categorical processes with applications to social network monitoring[END_REF]). According to [START_REF] Castagliola | An EWMA-type sign chart with exact run length properties[END_REF], in SPM, there are two categories of control charts: the memoryless-type charts (for instance, the Shewhart charts) and the memory-type charts (such as the exponentially weighted moving average (EWMA) charts). Shewhart-type charts are easy to implement and efficient in detecting large shifts, but the fact that they only use the current sample information makes them insensitive to detect small to moderate shifts. As alternatives to Shewhart-type charts, the memory-type charts are more sensitive to detect small to moderate shifts, because they are designed to take both the past and the current information into account. This makes the memory-type charts involve more advantages that motivate more interesting research works.

The conventional control charts are known to be designed for a particular shift level. But, in practice, the potential shift size of the process is rarely know in advance. In this context, [START_REF] Capizzi | An adaptive exponentially weighted moving average control chart[END_REF] designed a new adaptive EWMA (AEWMA) control chart for detecting both small and large mean shifts simultaneously. In the construction of the AEWMA charting statistic, a suitable function of the current error is used to dynamically weight the current and the past observations, which makes the AEWMA chart in [START_REF] Capizzi | An adaptive exponentially weighted moving average control chart[END_REF] be a smooth combination of a Shewhart chart and an EWMA chart. As mentioned by [START_REF] Psarakis | Adaptive control charts: recent developments and extensions[END_REF], AEWMA type schemes for normally distributed data have been extensively investigated by researchers, see, for instance, [START_REF] Mitra | An adaptive exponentially weighted moving average-type control chart to monitor the process mean[END_REF], [START_REF] Haq | New adaptive EWMA control charts for monitoring univariate and multivariate coefficient of variation[END_REF], and [START_REF] Tang | A new nonparametric adaptive EWMA control chart with exact run length properties[END_REF]. Without a doubt, AEWMA type schemes are better choices that can achieve a reasonable balance for both small and large mean shifts. However, most of them introduced above are two-sided schemes.

As pointed out by [START_REF] Chiu | Properties and performance of one-sided cumulative count of conforming chart with parameter estimation in high-quality processes[END_REF], the traditional two-sided schemes may spend more time on generating an out-of-control signal when a shift in the process parameter occurs. Conversely, one-sided schemes are more appropriate for monitoring processes if the direction of the shift can be anticipated, or when the investigator is only interested in a particular directional shift. For example, information on the increase in the infection rate of a particular disease (such as the COVID-19) is very important for a government to adjust epidemic prevention and to take control measures. Although numerous studies have shown that one-sided schemes have wide potential applications in practice, see, for instance, Wang et al.

(2019), Qiao et al. (2020 in press), and [START_REF] Haq | One-sided and two one-sided MEWMA charts for monitoring process mean[END_REF], there are few studies on onesided schemes with an adaptive feature. This fact means that the design of a one-sided adaptive type scheme is a topic of interest.

With the improvement of manufacturing capacity, the occurrence rate of de-fects (or nonconformities) can be maintained at a very low level, say, parts per million (ppm). These processes are usually named "high-quality processes". The traditional attribute control charts (such as p chart or np chart) are no longer valid for monitoring such processes. In this context, time-between-events (TBE) schemes have been developed to overcome the limits of using traditional attribute charts.

A common assumption for high-quality processes is that the occurrence of events (i.e., defects, or nonconformities) can be modeled as a homogeneous Poisson process, for which the time between two consecutive events follows an exponential distribution. Extensive research works and extensions of TBE type schemes based on the geometric, negative binomial, exponential, Gamma, and Weibull distributions have since evolved, see, for example, [START_REF] Urbieta | CUSUM and EWMA control charts for negative binomial distribution[END_REF], Xie et al. (2021 in press), [START_REF] Ali | High quality process monitoring using a class of inter-arrival time distributions of the renewal process[END_REF], and [START_REF] Sanusi | Simultaneous monitoring of magnitude and time-between-events data with a Max-EWMA control chart[END_REF]. Among all available distributions, the exponential one is considered as very suitable for modelling high-quality processes and TBE data. This distribution is not only widely employed in the field of reliability engineering as a model of the time to failure of a component or a system (see [START_REF] Montgomery | Introduction to statistical quality control[END_REF]), but also potentially be applied to other systems or areas. For instance, in the waiting time modelling (see Xie et al. (2010)), in the monitoring of workplace accidents (see Zhang et al. (2006)), in human health surveillance and monitoring (see [START_REF] Aslam | Designing of a new monitoring t-chart using repetitive sampling[END_REF]), and even in earthquake analysis (see [START_REF] Santiago | Control charts based on the exponential distribution: Adapting runs rules for the t chart[END_REF]). All these applications show that the design of TBE control charts based on the exponential distribution is necessary.

An essential assumption for the design of control charts is that the process parameters are assumed known. But, in real applications, the process parameters are usually unknown. This fact means that the parameters of the process should be estimated before the process monitoring starts. Numerous researches on parameter estimations have been conducted on control charts over these years, see, for instance, Zwetsloot & Woodall (2017), [START_REF] Testik | An algorithmic approach to outlier detection and parameter estimation in Phase I for designing Phase II EWMA control chart[END_REF], and [START_REF] Jardim | Two perspectives for designing a phase II control chart with estimated parameters: The case of the shewhart X chart[END_REF]. It is worth noting that, the process parameters used to determine the control limits in Phase II usually need to be estimated from a limited number of Phase I observations. According to Tang et al. (2019a), using the control limit obtained from the known parameter case leads to a deteriorated performance for the scheme with estimated parameters. One effective way is to increase the number of Phase I observations. However, in practice, practitioners cannot wait long to collect such a large amount of Phase I observations, especially in the case of high-quality processes. Meanwhile, a shift may also occur in the process parameter when collecting these Phase I observations. Therefore, an attractive alternative suggested by [START_REF] Mahmoud | The performance of the MEWMA control chart when parameters are estimated[END_REF] is to adjust the corresponding control limit of the scheme, so that it can provide an effective detection ability for the specified shift that occurs in the process parameter.

In this study, a new one-sided AEWMA scheme using the truncation method is developed for monitoring exponentially distributed TBE data assuming a known shift direction (hereafter named as the one-sided ATEWMA TBE scheme). The main contributions of this paper can be summarized as follows: (1) proposing a new one-sided ATEWMA TBE scheme in the case of known and estimated parameters, (2) establishing a dedicated Markov chain model for run length (RL) evaluation, and (3) developing an ARL-based optimal design procedure to provide a balanced protection against a range of shift sizes. The outline of this paper is organized as follows: In Section 2, a new one-sided ATEWMA TBE scheme with known and estimated parameters is developed. Subsequently, a dedicated Markov chain model is established in Section 3 to evaluate the RL properties of the recommended scheme. In Section 4, an ARL-based optimal design procedure is developed for detecting both small and large shifts simultaneously. In Section 5, numerical comparisons are performed with two comparative schemes for the detection of both upward and downward mean shifts. Some guidelines concerning the construction of the proposed scheme are also provided. Furthermore, two real data applications are employed in Section 6 to illustrate the implementation of the one-sided ATEWMA TBE scheme in OLED failure time monitoring and aircraft reliability monitoring, respectively. Finally, Section 7 concludes with some remarks and directions for future researches.

The proposed one-sided ATEWMA TBE scheme

Let us assumed that the TBE random variable X t , t = 1, 2, . . ., used in a high-quality process follows an exponential distribution with scale parameter θ, i.e., X t ∼ exp(θ). In what follows, a new one-sided ATEWMA TBE scheme is developed in the case of known and estimated parameters to detect both small and large shifts simultaneously.

The process parameters are known

For the known parameter case, let us assume that θ 0 is the known in-control scale parameter (or mean) of the process. In order to simplify the design of the one-sided ATEWMA TBE scheme with known parameters, a scaled TBE random variable M t is defined as follows:

M t = X t θ 0 = θ 1 θ 0 • X t θ 1 , (1)
where θ 1 is the out-of-control scale parameter (or mean) of the exponential distribution. For simplicity, let τ = θ 1 /θ 0 and Θ t = X t /θ 1 , then the scaled TBE random variable M t in (1) can be restated as:

M t = τ • Θ t , (2)
where τ represents the shift level that occurs in the in-control scale parameter θ 0 , and Θ t is a standard exponentially distributed random variable with mean equals to 1. The case τ = 1 denotes that no shift occurs in the scale parameter θ 0 , and the process is deemed to be in-control. Otherwise, the case τ > 1 (or 0 < τ < 1)

means that an upward (or a downward) shift occurs in the scale parameter θ 0 .

The truncation method developed by [START_REF] Shu | A one-sided ewma control chart for monitoring process means[END_REF] is adopted for the proposed scheme for the effective detection of upward (or downward) shifts. Taking the one-sided ATEWMA TBE scheme for monitoring upward shifts (hereafter named as the upper-sided ATEWMA TBE scheme) as an example, the idea of the truncation method used in the upper-sided ATEWMA TBE scheme can be described as follows: at sampling point t, we reset the TBE observation X t to the value of θ 0 when X t falls below the in-control mean θ 0 . Otherwise, if the TBE observation X t is larger than the in-control mean value θ 0 , the original value of X t is retained. This new "resetting rule" makes the proposed scheme more sensitive for the process monitoring, as it keeps the shift information in the direction of interest. Without loss of generality, two truncated TBE random variables can be defined to describe the truncation method used in this paper, namely,

• the upper-truncated TBE random variable:

X + t = max (θ 0 , X t ),
• the lower-truncated TBE random variable:

X - t = min (θ 0 , X t ).
Based on the scaled TBE random variable M t defined in (1), both the uppertruncated and the lower-truncated TBE random variables can be restated as:

M + t = max (1, M t ) , (3)
M - t = min (1, M t ) . (4)
When τ = 1, the in-control mean values of the upper-truncated and lower-truncated

TBE random variables M + t and M - t are E(M + t ) = 1+e -1 and E(M - t ) = 1-e -1 ,
Z + t = M + t E(M + t ) , (5)
Z - t = M - t E(M - t ) . ( 6
)
Different from the one-sided REWMA TBE scheme developed in [START_REF] Gan | Designs of one-and two-sided exponential ewma charts[END_REF] using a fixed EWMA smoothing parameter λ R , the proposed upper-sided (or lower-sided) ATEWMA TBE scheme is designed by adjusting its smoothing parameter λ as a function of the prediction error

e + t = Z + t -W + t-1 e - t = Z - t -W - t-1 ,
where W + t-1 and W - t-1 are the upper-sided and lower-sided ATEWMA TBE charting statistics at sampling point t -1, respectively. Based on this condition, two one-sided ATEWMA TBE schemes for respectively detecting increases and decreases in the in-control scale parameter θ 0 are designed as follows:

• For the upward shift detection, the charting statistic W + t of the upper-sided ATEWMA TBE scheme, at sampling point t = 1, 2, . . ., is defined as,

W + t = W + t-1 + φ(e + t ) = ω(e + t )Z + t + 1 -ω(e + t ) W + t-1 . (7)
• Meanwhile, for detecting downward shifts, the charting statistic W - t of the lower-sided ATEWMA TBE scheme is defined as,

W - t = W - t-1 + φ(e - t ) = ω(e - t )Z - t + 1 -ω(e - t ) W - t-1 , (8)
where ω(e

+ 0 = W - 0 = 1. Meanwhile, note that φ(e + t )
and φ(e - t ) are the Huber's score functions defined as follows:

φ(ε) =            ε + (1 -λ) × k, ε < -k λ × ε, |ε| k ε -(1 -λ) × k, ε > k (9)
where k 0, and λ ∈ (0, 1] is the smoothing parameter of the one-sided ATEWMA TBE scheme. The implementation of Huber's score function makes the suggested one-sided ATEWMA TBE scheme to be a smooth combination of the Shewhart TBE chart and the one-sided EWMA TBE chart using the truncation method. For the upper-sided (or lower-sided) ATEWMA TBE scheme, an out-of-control signal is generated, if W + t (W - t ) falls above (below) the control limit H + (H -) of the proposed scheme.

The process parameters are unknown

In real situations, process parameters are rarely known in advance. This fact means that the process parameters need to be estimated from different Phase I TBE observations Y t , t = 1, 2, . . .. In this paper, the in-control scale parameter (or mean) θ 0 is estimated using in-control TBE observations collected in Phase Zhang et al. (2014), the maximum likeli-hood estimator θ 0 of θ 0 is given as follows:

I, namely, Y 1 , Y 2 , • • • , Y . According to
θ 0 = 1 t=1 Y t . (10)
Similar to the known parameter case, the scaled TBE random variable M t with estimated parameter θ 0 can be given as follows:

M t = X t θ 0 = θ 0 θ 0 • θ 1 θ 0 • X t θ 1 . (11)
Furthermore, according to [START_REF] Ozsan | Properties of the exponential EWMA chart with parameter estimation[END_REF], let us define U = θ 0 / θ 0 , and then the estimated TBE random variable M t in (11) can be restated as:

M t = U • τ • Θ t , (12)
where τ and Θ t have already been defined in Section 2.1. Additionally, U is a random variable, which represents the ratio of the in-control scale parameter θ 0 to its estimator θ 0 . It has been proven that U follows an inverse Gamma distribution (see [START_REF] Ozsan | Properties of the exponential EWMA chart with parameter estimation[END_REF]), and the probability density function (p.d.f.) f U (u| ) of U is defined as follows:

f U (u| ) = ( -1)! u --1 exp - u . ( 13
)
Due to the use of the estimator θ 0 in (10), the corresponding truncated TBE random variables M + t and M - t with estimated parameters can be, respectively, written as,

M + t = max(1, M t ), (14)
M - t = min(1, M t ). ( 15
)
The in-control mean values of

M + t and M - t are E( M + t ) = 1+U e -1 U and E( M - t ) = U -U e -1 U ,
Z t + = M + t E( M + t ) , (16)
Z t - = M - t E( M - t ) . ( 17
)
The upper-sided and lower-sided ATEWMA TBE charting statistics with estimated parameters, at sampling point t, can be written as follows:

W + t = W + t-1 + φ( e t + ) = ω( e t + ) Z t + + 1 -ω( e t + ) W + t-1 , (18)
W - t = W - t-1 + φ( e t -) = ω( e t -) Z t - + 1 -ω( e t -) W - t-1 , (19)
where the initial value for

W + t (or W - t ) is set to W + 0 = E( Z t + ) = 1 ( W - 0 = E( Z t - ) = 1
), and φ(•) is the Huber's score function given in (9). The upper-sided (or lower-sided) ATEWMA TBE scheme gives an out-of-control signal when the charting statistic

W + t > H + ( W - t < H -)
, where H + and H -are the adjusted control limits of the upper-sided and lower-sided ATEWMA TBE schemes, respectively.

Run length evaluation

As one of the most widely used criterion in control charts, the average run length (ARL) is defined as the expected number of charting statistics plotted on the one-sided ATEWMA TBE scheme until an out-of-control signal is generated.

Commonly, if there is no shift occurs in the process, a control chart is expected to run with a large in-control ARL value (hereafter denoted as the ARL 0 ), but when the process is out-of-control, the corresponding out-of-control ARL (hereafter denoted as the ARL 1 ) value is expected to be as small as possible. In this section, an appropriate discrete-state Markov chain model is established to investigate the ARL performance of the one-sided ATEWMA TBE scheme in the case of known and estimated parameters. Due to the space limitation, only the uppersided ATEWMA TBE scheme with known and estimated parameters is selected as an example to illustrate the establishment of the Markov chain model. For more details about the Markov chain model of the lower-sided ATEWMA TBE scheme with known and estimated parameters, readers can refer to Appendix A.

The Markov chain model for the known parameter case

The basic idea of the Markov chain model is to define the transition states by dividing the in-control region into a finite number of sub-intervals, and then to approximate the charting statistic using the mid-point value of each sub-interval. In this study, we divide the in-control region into m sub-intervals, namely, transient state 1, transient state 2, • • • , transient state m, and then the ARL of the uppersided ATEWMA TBE scheme with known parameters can be computed using,

ARL = p T (I -Q) -1 1, (20)
where p = (p 1 , p 2 , . . . , p m ) T is the initial probability vector that corresponds to m transient states, and Q = [q i,j ] m×m is the transition probability matrix. Moreover,

1 is an m × 1-dimensional vector of 1's, and I is an m × m-dimensional identity matrix. It is easy to obtain that the in-control region is [1/ (1 + e -1 ) , H + ].
Hence, the width ∆ + of each sub-interval can be given as,

∆ + = 1 m H + - 1 1 + e -1 . (21)
For the proposed scheme with known parameters, the charting statistic

W + t is regarded as in "transient state i", when W + t ∈ E + i -∆ + /2, E + i + ∆ + /2
, where

E + i = 1/ (1 + e -1 ) + (i -1/2)∆ + is the mid-point value of the ith sub-interval, i = 1, 2, . . . , m, see Figure 1. (Please insert Figure 1 here)
As the elements of the transition probability matrix Q, the one-step transient probabilities q i,j from transient state i to transient state j can be computed as follows:

q i,j = Pr W + t ∈ transient state j W + t-1 ∈ transient state i = Pr E + j - ∆ + 2 < W + t E + j + ∆ + 2 W + t-1 = E + i = Pr E + j -E + i - ∆ + 2 < φ(Z + t -W + t-1 ) E + j -E + i + ∆ + 2 (22)
where i = 1, 2, . . . , m, and j = 1, 2, . . . , m. After some algebraic calculations, the one-step transient probabilities q i,j are equivalent to,

q i,j = Pr E + i + φ -1 E + j -E + i - ∆ + 2 < Z + t E + i + φ -1 E + j -E + i + ∆ + 2 = Pr 1 + e -1 E + i + φ -1 E + j -E + i - ∆ + 2 < M + t 1 + e -1 E + i + φ -1 E + j -E + i + ∆ + 2 (23)
where the Huber's inverse score function φ -1 (•) is given as,

φ -1 (ε) =            ε -(1 -λ) × k, ε < -λ × k ε/λ, |ε| λ × k ε + (1 -λ) × k. ε > λ × k (24)
For simplicity, let us define,

Ψ 1 = 1 + e -1 E + i + φ -1 E + j -E + i - ∆ + 2 , (25)
Ψ 2 = 1 + e -1 E + i + φ -1 E + j -E + i + ∆ + 2 . ( 26
)
Therefore, the one-step transient probabilities q i,j can be computed as follows:

q i,j =                0, if Ψ 2 < 1 F Θ Ψ 2 τ , if Ψ 2 1 and Ψ 1 < 1 F Θ Ψ 2 τ -F Θ Ψ 1 τ , if Ψ 2 1 and Ψ 1 1 (27)
where F Θ (•) is the cumulative distribution function (c.d.f.) of the standard exponentially distributed random variable Θ t defined in (2). Furthermore, the elements p j of the initial probability vector p are defined as follows:

p j =      1, E + j - ∆ + 2 < W + 0 E + j + ∆ + 2 0, otherwise (28)
where j = 1, 2, . . . , m, and

W + 0 = E(Z + t ) = 1.

The Markov chain model for the estimated parameter case

Unlike the upper-sided ATEWMA TBE scheme with known parameters, the unconditional ARL value (hereafter denoted as ARL) for the parameter estimation case can be computed by integrating the ARL(τ |u) (i.e., the conditional ARL value) with respect to u, i.e.,

ARL = +∞ 0 ARL(τ |u) × f U (u| ) du, (29)
where

ARL(τ |u) = p T (I -Q) -1 1, (30)
and f U (u| ) is the p.d.f. of the random variable U defined in (13). Note that

p = ( p 1 , p 2 , • • • , p m )
T is an m × 1-dimensional initial probability vector with estimated parameters, and Q contains the corresponding transient probabilities q i,j of going from transient state i to transient state j. The establishment of the Markov chain model for the estimated parameter case is similar to that one for the known parameter case, where the one-step transient probabilities q i,j in the estimated parameter case are given as:

q i,j = Pr W + t ∈ transient state j W + t-1 ∈ transient state i = Pr 1 + ue -1 u E + i + φ -1 E + j -E + i - ∆ + 2 < M + t 1 + ue -1 u E + i + φ -1 E + j -E + i + ∆ + 2 (31) Note that E + j = 1 1 + ue -1 u + (j -1/2) ∆ + is the mid-point value of the jth sub-interval, and ∆ + = H + -1/ 1 + ue -1 u
m is the width of the jth sub-interval, where j = 1, 2, . . . , m. Furthermore, if we define,

Ψ 1 = 1 + ue -1 u E + i + φ -1 E + j -E + i - ∆ + 2 , (32)
Ψ 2 = 1 + ue -1 u E + i + φ -1 E + j -E + i + ∆ + 2 . ( 33
)
Then, the entire one-step transient probabilities q i,j can be computed using,

q i,j =                    0, if Ψ 2 < 1 F Θ Ψ 2 u × τ , if Ψ 2 1 and Ψ 1 < 1 F Θ Ψ 2 u × τ -F Θ Ψ 1 u × τ , if Ψ 2 1 and Ψ 1 1 (34)
where i = 1, 2, . . . , m, and j = 1, 2, . . . , m. Meanwhile, the elements p j of the initial probability vector p are,

p j =        1, E + j - ∆ + 2 < W + 0 E + j + ∆ + 2 0, otherwise (35)
where

W + 0 = E( Z + t ) = 1.
Finally, in order to overcome the computational difficulties caused by the evaluation of ( 29), the Gauss-Legendre quadrature method is considered in this paper to obtain an accurate approximation.

Optimal design of the one-sided ATEWMA TBE scheme

The optimal design of the one-sided ATEWMA TBE scheme aims at finding a scheme, which can simultaneously provide the minimum ARL 1 (denoted as ARL min ) value for a specified small shift τ S and an approximate ARL min value for a specified large shift τ L , among those one-sided ATEWMA TBE schemes leading to the same desired ARL 0 value.

For the proposed one-sided ATEWMA TBE scheme, the combination of the design parameters obtained from the optimal design procedure is named as the optimal parameter combination, and the scheme with the optimal parameter combination is denoted as the optimal one-sided ATEWMA TBE scheme. In this study, a two-stage optimal design procedure, similar to the one presented in [START_REF] Capizzi | An adaptive exponentially weighted moving average control chart[END_REF], is introduced for determining the optimal parameter combination of the one-sided ATEWMA TBE scheme. Due to the space limitation, only the upper-sided ATEWMA TBE scheme with known parameters is selected here for illustration, and the steps are given as follows:

Step 1: Choose a desired ARL 0 value, and two different shift values, i.e., a small mean shift τ S , and a large mean shift τ L .

Step 2: With the constraint on the desired ARL 0 , search the optimal parameter combination γ * = {λ * L , k * L , H + * L } of the upper-side ATEWMA TBE scheme for the specified large shift τ L . The corresponding result can be regarded as the solution of the following nonlinear minimization problem, i.e.,

               γ * = arg min γ={λ L ,k L ,H + L } ARL 1 (γ, τ L ).
Subject to :

ARL(γ * , τ L = 1) = ARL 0 , (36)
where the ARL value of the upper-sided ATEWMA TBE scheme with known parameters can be computed using (20).

Step 3: Select a small positive constant α, for example, α = 0.05, and then the optimal parameter combination ζ * = {λ * , k * , H + * } of the proposed scheme can be searched for detecting both the small shift τ S and the large shift τ L using the following equations,

                     ζ * = arg min ζ={λ,k,H + } ARL 1 (ζ, τ S ).
Subject to :

ARL(ζ * , τ S = 1) = ARL 0 , ARL 1 (ζ * , τ L ) (1 + α) × ARL 1 (γ * , τ L ). (37)
More specifically, using equation ( 37), the upper-sided ATEWMA TBE scheme with optimal parameter combination ζ * provides the ARL min value for the specified small shift τ S . Meanwhile, the implementation of the positive constant α in (37) ensures that the ARL 1 value of the proposed scheme for the specified large shift τ L is "nearly minimum".

Considering the computational difficulties in the optimal design procedure, we suggest the use of an improved particle swarm optimization algorithm, named the DNSPSO algorithm, to solve the nonlinear minimization problem presented in ( 36) and ( 37). For more details on the DNSPSO algorithm, readers can refer to Wang et al. (2013).

According to the optimal design procedure described above, for several desired values of ARL 0 ∈ {200, 370, 500} and different shift combinations (τ S , τ L ), the optimal parameter combinations ζ * of the upper-sided and lower-sided ATEWMA TBE schemes with known parameters are listed in Tables 1 and2, respectively.

For instance, when the desired ARL 0 = 370, the optimal parameter combination

ζ * = {λ * , k * , H + * } of the upper-sided ATEWMA TBE scheme is {0.2051, 4.9170, 1.7620}
for (τ S , τ L ) = (1.3, 3). Meanwhile, when (τ S , τ L ) = (0.7, 0.1), the corresponding optimal parameter combination ζ * = {λ * , k * , H - * } of the lower-sided ATEWMA TBE scheme is {0.2951, 2.3076, 0.3983}.

(Please insert Tables 1 and2 here) As pointed out by [START_REF] Saleh | The performance of the adaptive exponentially weighted moving average control chart with estimated parameters[END_REF], the parameter estimation has a significant impact on the Phase II performance of the conventional AEWMA scheme.

Motivated by this fact, we further study the effect of parameter estimation on the one-sided ATEWMA TBE scheme. Due to the space limitation, only the case of ARL 0 = 370 is considered here for illustration. Similar to the study in [START_REF] Saleh | The performance of the adaptive exponentially weighted moving average control chart with estimated parameters[END_REF], with the optimal parameter combinations ζ * determined in the known parameter case, the in-control ARL (i.e., ARL 0 ) values of the proposed scheme with estimated parameters are shown in Table 3.

(Please insert Table 3 here)

In order to provide a fair performance comparison between the one-sided ATEWMA TBE scheme with estimated parameters and its known parameter counterpart, both schemes should be designed based on the same in-control ARL value (i.e., ARL 0 = ARL 0 ). One effective way to achieve this condition is to adjust the control limits in the estimated parameter case, so that the scheme can provide a 4 for ∈{20, 50, 100, 300, 500, 1000}.

(Please insert Table 4 here)

From Tables 3 and4, we can draw the following conclusions:

• As it can be seen from Table 3, for a fixed shift combination (τ S , τ L ), the difference between ARL 0 (corresponding to the known parameter case, say, = +∞) and ARL 0 (associated with ∈ {20, 50, 100, 300, 500, 1000}) decreases as increases. That is to say, the effect of the parameter estimation on the ARL 0 is large when is small.

• Note from Tables 1,2, and 4 that, for a fixed shift combination (τ S , τ L ), the adjusted control limits H + * (or H - * ) of the proposed upper-sided (lowersided) ATEWMA TBE scheme increases (decreases) as increases.

It should be stressed that the goal of the study for the parameter estimation case is not to show the superiority of the one-sided ATEWMA TBE scheme over its competitors, but to investigate the impact of the Phase I parameter estimation on the ARL performance of the proposed scheme.

Comparative studies

Two comparative schemes are introduced here for comparison with the onesided ATEWMA TBE scheme in the case of known and estimated parameters: one is the one-sided REWMA TBE scheme designed by [START_REF] Gan | Designs of one-and two-sided exponential ewma charts[END_REF], and the other one is the one-sided AEWMA TBE scheme proposed in [START_REF] Hu | On the performance of the adaptive ewma chart for monitoring time between events[END_REF].

The one-sided REWMA TBE scheme

The one-sided exponential EWMA chart with reflecting boundaries (i.e., the one-sided REWMA TBE scheme in this paper) was firstly designed by [START_REF] Gan | Designs of one-and two-sided exponential ewma charts[END_REF] to monitor either increases or decreases in the process mean.

• For the upper-sided REWMA TBE scheme, its charting statistic is given as:

Q + R,t = max B U , λ R M t + (1 -λ R )Q + R,t-1 . (38)
• Meanwhile, the monitoring statistic of the lower-sided REWMA TBE scheme is defined as follows:

Q - R,t = min B L , λ R M t + (1 -λ R )Q - R,t-1 , (39)
where B U and B L are the reflecting boundaries of the upper-sided and lower-sided REWMA TBE schemes, respectively. In addition, λ R represents the smoothing factor, the initial value 5 for different designed shift levels τ R , where τ R is a particular shift size for which the one-sided REWMA TBE scheme is optimally designed, λ * R is the optimal smoothing factor, h + * R (or h - * R ) is the control limit of the upper-sided (lower-sided) REWMA TBE scheme with known parameters, and h + * R (or h - * R ) is the adjusted control limit of the uppersided (lower-sided) REWMA TBE scheme with estimated parameters.

Q + R,0 = Q - R,0 = E(M t ) = 1,

The one-sided AEWMA TBE scheme

Similar to the methodology used for the one-sided REWMA TBE scheme, the one-sided AEWMA TBE scheme was firstly introduced by [START_REF] Hu | On the performance of the adaptive ewma chart for monitoring time between events[END_REF] for detecting both small and large shifts simultaneously. As a comparison, the charting statistic of the upper-sided AEWMA TBE scheme is given as:

Q + A,t = max 1, ω(e + A,t )M t + 1 -ω(e + A,t ) Q + A,t-1 , (40)
where e + A,t = M t -Q + A,t-1 , and the weight ω(e + A,t ) = φ(e + A,t )/e + A,t . The Huber's score function φ(•) used in the one-sided AEWMA TBE scheme is the same as it is shown in ( 11), excepted that λ should be replaced with λ A , where λ A is the smoothing parameter of one-sided AEWMA TBE scheme. On the other side, for the lower-sided AEWMA TBE scheme, its charting statistic is defined as follows:

Q - A,t = min 1, ω(e - A,t )M t + 1 -ω(e - A,t ) Q - A,t-1 , (41)
where the error e - A,t = M t -Q - A,t-1 , and ω(e - A,t ) = φ(e - A,t )/e - A,t . Additionally, the initial values of this comparative scheme are set to

Q + A,0 = Q - A,0 = E(M t ) = 1.
Although an optimal design procedure of the one-sided AEWMA TBE scheme has been introduced by Hu et al. ( 2021), the corresponding two-stage procedure developed in this paper is also popular and it is adopted for this existing comparative scheme to provide a fair comparison with the proposed one-sided ATEWMA TBE scheme. The optimal parameter combinations of the one-sided AEWMA TBE chart with known and estimated parameters are presented in Table 5 for ARL 0 = ARL 0 = 370 and = 100, where h + * A and h - * A (or h + * A and h - * A )

are the control limits (adjusted control limits) of the upper-sided and lower-sided AEWMA TBE schemes with known (estimated) parameters, respectively.

Performance comparison

Without loss of generality, with the constraint on the desired ARL 0 , the smaller the ARL 1 for the specified shift level τ , the better the performance of the control chart. In this paper, based on ARL 0 = 370 and m = 151, the ARL 1 performance of the one-sided ATEWMA TBE scheme in detecting either upward or downward shift is compared with that of two comparative schemes, respectively.

In the case of known parameter, the ARL 1 values of the one-sided ATEWMA TBE scheme and the one-sided AEWMA TBE scheme in detecting both upward and downward shifts τ are presented in Tables 6 and7, respectively. Note that the optimal parameter combinations of these two schemes for different shift combinations (τ S , τ L ) are obtained from Tables 1, 2, and 5, respectively. For example, when the desired ARL 0 = 370 and the specified upward shift combination (τ S , τ L ) = (1.5, 3), the ARL 1 values of the upper-sided ATEWMA TBE scheme and the upper-sided AEWMA TBE scheme for the upward shift τ = 2 are 11.46 and 12.13, respectively (see Table 6). Meanwhile, if a downward shift combination (τ S , τ L ) = (0.7, 0.2) is selected, the ARL 1 values of the lower-sided ATEWMA TBE scheme and the lower-sided AEWMA TBE scheme for the down-ward shift τ = 0.8 are 89.92 and 119.26, respectively (see Table 7).

(Please insert Tables 6 and7 here)

For the estimated parameter case, the settings in this scenario are similar to the known parameter case, excepted that the number of in-control TBE observations collected in Phase I need to be fixed first. Taking = 100 for illustration, the ARL 1 values of the one-sided ATEWMA TBE scheme and the one-sided AEWMA TBE scheme in detecting both upward and downward shifts τ are presented in Tables 8 and9, respectively. Meanwhile, the corresponding optimal parameter combinations of these two control charts for ARL 0 = 370 and = 100 are shown in Tables 4 and5, respectively. For instance, when (τ s , τ L ) = (1.5, 5), the optimal parameter combinations of the upper-sided ATEWMA TBE scheme and the upper-sided AEWMA TBE scheme for ARL 0 = 370 and = 100 are {λ * , k * , H + * } = {0.3105, 7.0272, 2.0854} and {λ * A , k * , h + * A } = {0.2524, 5.1444, 2.4029}, respectively (see Tables 1, 4, and5). Additionally, the corresponding ARL 1 values of these two schemes for detecting the upward shift τ = 1.1 are 190.19 and 186.10, respectively (see Table 8).

(Please insert Tables 8 and9 here)

On the other hand, in order to provide some intuitive comparisons between the one-sided ATEWMA TBE scheme and the one-sided REWMA TBE scheme in both the known and the estimated parameter cases, three one-sided REWMA TBE schemes optimally designed for different specified shifts τ R are selected in this study. The ln(ARL 1 ) and ln( ARL 1 ) curves of these schemes for detecting both upward and downward shifts τ are given in Figures 2 to 5, respectively.

Due to the space limitation, irrespective of the known or the estimated parameter case, the upper-sided ATEWMA TBE scheme is designed based on τ S = 1.5

and τ L = 4.0, and the other three different upper-sided REWMA TBE charts are respectively designed to provide the ARL min for different specified upward shifts τ R ∈ {1.1, 2.5, 5.0}. Similarly, in the case of a downward shift detection, the lower-sided ATEWMA TBE scheme is designed for τ S = 0.7 and τ L = 0.2, and the other three different lower-sided REWMA TBE charts are designed based on three specified downward shifts τ R ∈ {0.9, 0.6, 0.3}, respectively.

(Please insert Figures 2 to 5 here)

Based on these comparisons, some conclusions can be drawn as follows:

• For the known parameter case, the one-sided ATEWMA TBE scheme is uniformly more sensitive than the one-sided AEWMA TBE scheme for detecting both upward and downward shifts, especially for small shift ranges.

Moreover, in the detection of the upward (downward) shift, the ARL 1 difference between the upper-sided (lower-sided) ATEWMA TBE scheme and the upper-sided (lower-sided) AEWMA TBE scheme decreases, as τ increases (decreases) (see Tables 6 and7).

• For the estimated parameter case (for instance, = 100), most ARL 1 values of the upper-sided AEWMA TBE scheme are smaller than those of the upper-sided ATEWMA TBE scheme. On the contrary, the lower-sided ATEWMA TBE scheme performs better than the lower-sided AEWMA TBE chart in most downward shift detections. This fact means that the parameter estimation has a significant impact on the ARL performance of these two schemes (see Tables 8 and9).

• Irrespective of the upward or the downward shift detection case, the adverse effect of parameter estimation on the one-sided ATEWMA TBE chart is relatively smaller than that on the one-sided AEWMA TBE chart. For example, when shift combination (τ S , τ L ) = (1.1, 3.0), the ARL 1 and ARL 1 values of the upper-sided ATEWMA TBE scheme for τ = 1.1 are 176.06 and 175.51, respectively. Meanwhile, the corresponding ARL 1 and ARL 1 values of the upper-sided AEWMA TBE scheme are 182.06 and 170.42, respectively (see Tables 6 and8).

• No matter in the known or the estimated parameter case, the upper-sided (or lower-sided) ATEWMA TBE scheme can provide an effective protection against both small and large upward (downward) shifts simultaneously.

Moreover, the one-sided ATEWMA TBE scheme outperforms the one-sided REWMA TBE scheme over a wide range of shifts, especially for detecting a shift that is much larger or smaller than its designed shift size τ R . For instance, in the known parameter case, the upper-sided ATEWMA TBE scheme designed for (τ S , τ L ) = (1.5, 4.0) and the upper-sided REWMA TBE chart designed for τ R = 1.1 have almost the same ARL 1 profiles when the magnitude of upward shift τ < 1.5, but when a large upward shift (for example, τ > 2) occurs in the process, the corresponding upper-sided ATEWMA TBE scheme can provide a better protection than the upper-sided REWMA TBE scheme designed for τ R = 1.1 (see Figure 2).

To sum up, in the case of known parameter, the one-sided ATEWMA TBE scheme is a better alternative to the one-sided AEWMA TBE chart for detecting both upward and downward shifts. On the other hand, irrespective of the known or the estimated parameter case, the one-sided ATEWMA TBE scheme is significantly superior to the one-sided REWMA TBE chart in detecting a shift that is much larger or smaller than the designed shift size τ R .

Real data applications

In this section, two real datasets are employed to illustrate the implementation of the one-sided ATEWM TBE schemes for monitoring mean shifts of highquality processes. More specifically, one is based on the electronic component failure time monitoring from Samsung company as reported in [START_REF] Qu | Exponential cumulative sums chart for detecting shifts in time-between-events[END_REF], and the other one is based on the aircraft reliability monitoring from the Hellenic Air Force (HAF) as reported in Alevizakos & Koukouvinos (2020).

Failure time monitoring

Samsung company manufactures many kinds of electronic components, including the organic light-emitting diode (OLED). The dataset of OLED failure time using an accelerated life test (ALT) employed here is adapted from [START_REF] Qu | Exponential cumulative sums chart for detecting shifts in time-between-events[END_REF], which consists of 30 in-control TBE observations and 20 out-of-control TBE observations, see Table 10.

(Please insert Table 10 here) As it has been shown in [START_REF] Qu | Exponential cumulative sums chart for detecting shifts in time-between-events[END_REF], the failure time X t under the ALT follows an exponential distribution, and the in-control mean value of the exponential distribution can be estimated from those 30 in-control TBE observations in Table 10 using (10), say,

θ 0 = 1 30 30 t=1 X t ≈ 1.27 min
When the OLED manufacturing process is in-control, we take this estimate as the mean value θ 0 of the OLED failure time under the ALT. In order to ensure the quality of the OLEDs, quality engineers need to pay more attention to the downward shift that occurs in the in-control mean value θ 0 , because a decrease in θ 0 means that the quality of OLED deteriorates. On the other hand, since the shift size in the real manufacturing process usually cannot be predicted in advance, it is more realistic to assume that the engineers are interested in detecting both a small downward shift τ L and a large downward shift τ S simultaneously, rather than monitoring a particular mean shift level. In this context, the lower-sided ATEWMA TBE scheme can be used to monitor the OLED manufacturing process by checking the time X t to failure under the ALT.

In this example, the optimal design parameters of the lower-sided ATEWMA TBE scheme can be searched by using the optimal design procedure introduced in Section 4, with the constraint on the desired ARL 0 and the specified shift com-bination (τ S , τ L ). More specifically, assuming that ARL 0 = 200, and engineers are interested in detecting both the small downward shift 0.8 × θ 0 and the large downward shift 0.2 × θ 0 , i.e., (τ S , τ L ) = (0.8, 0. With the optimal parameter combinations of these schemes, the charting statistics W - t of the lower-sided ATEWMA TBE scheme can be computed using ( 1), ( 4), ( 6), (8), and (9). Meanwhile, the charting statistics Q - A,t of the lower-sided AEWMA TBE scheme can be obtained using ( 1), ( 9), and ( 41), and the corresponding charting statistics Q - R,t of the lower-sided REWMA TBE scheme can be given using ( 1) and ( 39). All charting statistics are shown in Columns 6 to 9 of Table 10, respectively. It should be noted that, for these lower-sided schemes, an out-of-control signal is generated if the charting statistic falls below the corresponding control limit. For example, when t = 38, the charting statistic W - t of the lower-sided ATEWMA TBE scheme falls below its control limit H - * , see Figure 6 (a). This means that an out-of-control signal of the lower-sided ATEWMA TBE scheme is generated and corresponding actions should be taken to identify and remove the downward mean shift. Meanwhile, as we can see from Table 10 and (Please insert Figure 6 here)

Aircraft reliability monitoring

Another real dataset from the Hellenic Air Force (HAF) as reported in Alevizakos & Koukouvinos ( 2020) is also employed here to show the implementation of the one-sided ATEWMA TBE scheme. This dataset is related to the accidents of F-16 aircrafts in the HAF. It is worth noting that the time between two accidents of F-16 aircrafts can be regarded as an important quality characteristic concerning aircraft reliability monitoring, as the decrease in the mean value of those time intervals indicates that the reliability of the aircraft declined. In this real dataset, 16 time intervals X t between successive accidents of F-16 aircrafts are recorded, see

Column 2 in Table 11. It has been proved by [START_REF] Alevizakos | A double exponentially weighted moving average chart for time between events[END_REF] that these time intervals X t between two consecutive accidents of F-16 aircrafts fit an exponential distribution with scale parameter θ 1 = 615 (days). Similar to the assumption in [START_REF] Hu | On the performance of the adaptive ewma chart for monitoring time between events[END_REF], the in-control value of θ 0 in this example is assumed to be known as 1460 (days). That is to say, the process is acceptable if an accident occurs every four years. Since the investigators are more concerned with a decrease in the mean value of those time intervals X t , a lower-sided ATEWMA TBE scheme is suggested here to monitor those 16 Phase II TBE observations, where the potential downward shift is τ = θ 1 /θ 0 = 615/1460 = 0.42.

(Please insert Table 11 here)

Similar to the steps introduced in Alevizakos & Koukouvinos (2020), the design parameters of the scheme should be determined first, imposing the constraint that the acceptable ARL 0 reaches its pre-specified target. In this example, assuming that the investigators decide to set ARL 0 = 370, and the lower-sided ATEWMA TBE scheme designed for (τ s , τ L ) = (0.7, 0.2) can be considered to provide a good protection against the potential downward shift τ = 0.42. With the constraint on ARL 0 = 370, the optimal parameter combination of the proposed scheme is {λ * , k * , H - * }={0.0729,13.5426,0.7412} (see Table 2). Meanwhile, the lower-sided AEWMA TBE scheme and the other two different lowersided REWMA TBE schemes (which are respectively designed for τ R = 0.7 and τ R = 0.2) are also used here for comparison. Based on ARL 0 = 370, the optimal parameter combination of the lower-sided AEWMA TBE scheme designed

for (τ s , τ L ) = (0.7, 0.2) is {λ * A , k * , h - * A }={0
.2202, 7.9248, 0.3488} (see Table 5), and the corresponding optimal parameter combinations of the lower-sided REWMA TBE charts for τ R = 0.7 and τ R = 0.2 are {λ * R , h - * R }={0.0362,0.7252} and {λ * R , h - * R }={0.3454,0.2388}, respectively (see Table 5).

After determining the optimal parameter combinations, all charting statistics generated by those schemes are recorded in Table 11 and plotted in Figures 7 (a), (b), (c), and (d). As it can be seen that, the lower-sided ATEWMA TBE scheme gives an out-of-control signal at the 16th TBE observation (note that the corresponding 16th charting statistic is bolded in Table 11), while the lowersided AEWMA TBE scheme and the other two lower-sided REWMA TBE charts cannot detect this downward shift at all. This fact implies that the lower-sided ATEWMA TBE scheme in this example is more effective than the lower-sided AEWMA TBE scheme and the lower-sided REWMA TBE schemes for detecting the downward shift in aircraft reliability monitoring.

(Please insert Figure 7 here)

Conclusions

In the manufacturing industry, there are many high-quality processes that can be reasonably modelled using the exponential distribution. In order to monitor such processes, a new one-sided ATEWMA TBE scheme with known and estimated parameters is proposed in this study, for providing a good protection against both small and large shifts assuming a known shift direction. Due to the implementation of Huber's score function, the proposed scheme can be regarded as a smooth combination of a Shewhart TBE chart and a one-sided EWMA TBE chart using the truncation method. A dedicated Markov chain model is established to evaluate the RL properties of the proposed scheme with known and estimated parameters. Based on the ARL criteria, a two-stage optimal design procedure is developed for searching the optimal parameter combination of the proposed scheme. Numerical results show that, in the case of known parameter, the onesided ATEWMA TBE scheme performs uniformly better than the other two existing comparative schemes. In addition, the recommended scheme with estimated parameters performs better than the one-sided AEWMA TBE chart in compensating the adverse effect of parameter estimation.

Several research directions are summarized here for future research works.

First, a bootstrap-based design approach can be considered to overcome the issue that a large number of Phase I TBE observations are usually unavailable in practice. Secondly, from a practical point of view, both the steady-state and the worst-case scenarios of the recommended scheme are worth studying. Last but not least, the one-sided ATEWMA TBE scheme based on exponential distribution can theoretically be extended to that one based on Gamma distribution for monitoring the sum of inter-arrival times before the rth event.

Disclosure statement

No potential conflict of interest was reported by the author(s). In-control region In-control region 11.

τ S τ L ARL 0 = 200 ARL 0 = 370 ARL 0 = 500 λ * k * H - * λ * k * H - * λ * k * H - * 0.
τ S τ L λ * A k * h + * A h + * A τ R λ * R h + * R h + * R 1.
τ S τ L λ * A k * h - * A h - * A τ R λ * R h - * R h - * R 0.
(τ R = 0.8) (τ R = 0.2) M - t Z - t W - t Q - A,t Q - R,t Q - R,t 1
M - t Z - t W - t Q - A,t Q - R,t Q - R,

  schemes are given in Table4for ∈{20, 50, 100, 300, 500, 1000}.

  and both B U and B L are suggested to be set at 1. Similar to the optimal design procedure designed inXie et al. (2021 in press), the optimal parameter combination of the one-sided REWMA TBE scheme can be easily obtained for a fixed mean shift τ R . More specifically, based on ARL 0 = ARL 0 = 370, the optimal parameter combinations of the one-sided REWMA TBE scheme with known (i.e., = +∞) and estimated (i.e., = 100) parameters are presented in Table

  scheme is {0.1354, 18.2366, 0.6526}. Furthermore, three comparative schemes, namely, a lower-sided AEWMA TBE scheme and the other two different lowersided REWMA TBE schemes (which are respectively designed for τ R = 0.8 and τ R = 0.2), are used here to provide a comparison with the lower-sided ATEWMA TBE scheme in monitoring the downward shift of the OLED failure time. Based on ARL 0 = 200, the optimal parameter combination {λ * A , k * , h - * A } of the lowersided AEWMA TBE scheme is {0.2545, 11.0204, 0.3453}, and the optimal parameter combinations {λ * R , h - * R } of the lower-sided REWMA TBE schemes for τ R = 0.8 and τ R = 0.2 are {0.0235, 0.8162} and {0.3708, 0.2496}, respectively.

  Figures 6 (b), (c), and (d), the lower-sided AEWMA TBE scheme gives an out-of-signal at the 44th TBE observation, the lower-sided REWMA TBE scheme designed for τ R = 0.8 signals at the 43th TBE observation, and the lower-sided REWMA TBE scheme designed for τ R = 0.2 signals at the 46th TBE observation. This indicates that, in this example, the lower-sided ATEWMA TBE scheme is superior to both the lower-sided AEWMA TBE scheme and the lower-sided REWMA TBE schemes in monitoring the OLED manufacturing process.

Figure 1 :Figure 2 :

 12 Figure 1: The Markov chain model of the upper-sided ATEWMA TBE scheme.

Figure 3 :

 3 Figure 3: ln(ARL 1 ) comparisons among the lower-sided ATEWMA TBE scheme and three different lower-sided REWMA TBE schemes in detecting downward shifts τ (ARL 0 = 370).

Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 4567 Figure 4: ln( ARL 1 ) comparisons among the upper-sided ATEWMA TBE scheme and three different upper-sided REWMA TBE schemes in detecting upward shifts τ (ARL 0 = 370, = 100).

Table 1 :

 1 Optimal parameter combinations ζ * of the upper-sided ATEWMA TBE scheme for ARL 0 ∈ {200, 370, 500}.

			ARL 0 = 200			ARL 0 = 370			ARL 0 = 500
	τ S τ L	λ *	k *	H + *	λ *	k *	H + *	λ *	k *	H + *
	1.1 3	0.1883 10.0043 1.6032	0.1857 14.9105 1.6990	0.0710 5.0794 1.3378
	1.3 3	0.1414 4.7324 1.4687	0.2051 4.9170 1.7620	0.0891 13.8196 1.4046
	1.5 3	0.0997 19.3077 1.3443	0.1167 13.8295 1.4705	0.1013 18.1002 1.4495
	1.1 4	0.1972 18.7462 1.6286	0.2286 14.6487 1.8375	0.2941 6.8258 2.1175
	1.3 4	0.3070 3.6431 1.9368	0.1585 10.9228 1.6104	0.1508 7.0995 1.6268
	1.5 4	0.3698 19.5277 2.1067	0.1216 8.6812 1.4872	0.1874 7.2428 1.7544
	1.1 5	0.1284 6.9732 1.4307	0.1865 4.1443 1.7107	0.1213 10.2955 1.5227
	1.3 5	0.1896 9.5962 1.6068	0.1671 13.9985 1.6391	0.2758 13.3147 2.0555
	1.5 5	0.1655 19.3518 1.5380	0.3105 7.0272 2.0981	0.1347 7.0693 1.5702

Table 2 :

 2 Optimal parameter combinations ζ * of the lower-sided ATEWMA TBE scheme for ARL 0 ∈ {200, 370, 500}.

Table 3 :

 3 In-control ARL comparisons of the upper-sided and lower-sided ATEWMA TBE schemes for ∈ {20, 50, 100, 300, 500, 1000, +∞}.

	9 0.3	0.1411 8.9318 0.6428	0.1124 14.4142 0.6561	0.1127 4.5151 0.6388
	0.8 0.3	0.0687 15.2093 0.7845	0.0700 13.9071 0.7478	0.1103 2.8205 0.6437
	0.7 0.3	0.1417 13.6767 0.6420	0.0713 18.4310 0.7450	0.0741 5.3099 0.7228
	0.9 0.2	0.1848 6.1138 0.5754	0.1820 13.2921 0.5387	0.1534 13.8204 0.5655
	0.8 0.2	0.1354 18.2366 0.6526	0.1422 10.5375 0.6017	0.1787 17.9492 0.5258
	0.7 0.2	0.2623 2.1815 0.4754	0.0729 13.5426 0.7412	0.1381 2.1017 0.5917
	0.9 0.1	0.2833 16.5755 0.4517	0.2788 12.4893 0.4157	0.2669 12.2793 0.4111
	0.8 0.1	0.2552 7.4753 0.4837	0.3418 6.6908 0.3520	0.2784 17.8617 0.3984
	0.7 0.1	0.2997 12.6632 0.4341	0.2951 2.3076 0.3983	0.2216 16.6469 0.4660

Table 4 :

 4 The adjusted control limits H + * and H - * of the upper-sided and lower-sided ATEWMA TBE schemes for ∈ {20, 50, 100, 300, 500, 1000}.

	τ S τ L	20	50	100	300	500	1000
	1.1 3	1.6504 1.6809 1.6902 1.6961 1.6973 1.6982
	1.3 3	1.7098 1.7425 1.7524 1.7589 1.7601 1.7611
	1.5 3	1.4350 1.4574 1.4641 1.4684 1.4693 1.4699
	1.1 4	1.7815 1.8165 1.8272 1.8341 1.8355 1.8365
	1.3 4	1.5665 1.5940 1.6024 1.6077 1.6088 1.6096
	1.5 4	1.4508 1.4737 1.4807 1.4851 1.4860 1.4866
	1.1 5	1.6551 1.6891 1.7000 1.7072 1.7086 1.7097
	1.3 5	1.5937 1.6221 1.6308 1.6364 1.6375 1.6383
	1.5 5	2.0291 2.0722 2.0854 2.0939 2.0956 2.0968
	0.9 0.3	0.6667 0.6602 0.6580 0.6567 0.6564 0.6562
	0.8 0.3	0.7564 0.7511 0.7495 0.7483 0.7481 0.7480
	0.7 0.3	0.7537 0.7484 0.7466 0.7455 0.7453 0.7451
	0.9 0.2	0.5506 0.5433 0.5409 0.5395 0.5391 0.5389
	0.8 0.2	0.6131 0.6060 0.6038 0.6023 0.6021 0.6019
	0.7 0.2	0.7499 0.7445 0.7428 0.7416 0.7414 0.7413
	0.9 0.1	0.4270 0.4200 0.4178 0.4164 0.4161 0.4159
	0.8 0.1	0.3623 0.3560 0.3539 0.3527 0.3524 0.3522
	0.7 0.1	0.4093 0.4025 0.4003 0.3989 0.3987 0.3985

Table 5 :

 5 Optimal parameter combinations of the one-sided AEWMA TBE scheme and the one-sided REWMA TBE scheme in both the known ( = +∞) and the estimated ( = 100) parameter cases for in-control ARL = 370.

	AEWMA		REWMA
	= +∞	= 100	= +∞	= 100

Table 6 :

 6 ARL 1 values of the upper-sided ATEWMA TBE scheme and the upper-sided AEWMA TBE scheme in detecting upward shifts τ (ARL 0 = 370).

	9 0.3	0.1541 2.4721	0.4347	0.4538	0.9	0.0071	0.9080	0.9291
	0.8 0.3	0.1506 9.3503	0.4400	0.4594	0.8	0.0187	0.8168	0.8434
	0.7 0.3	0.1436 15.0109	0.4512	0.4716	0.7	0.0362	0.7252	0.7506
	0.9 0.2	0.2171 15.9662	0.3522	0.3648	0.6	0.0613	0.6331	0.6547
	0.8 0.2	0.2300 5.5258	0.3381	0.3498	0.5	0.0970	0.5394	0.5567
	0.7 0.2	0.2202 7.9248	0.3488	0.3611	0.4	0.1484	0.4434	0.4565
	0.9 0.1	0.3396 0.5974	0.2219	0.2241	0.3	0.2243	0.3442	0.3533
	0.8 0.1	0.3830 10.3640	0.2139	0.2192	0.2	0.3454	0.2388	0.2443
	0.7 0.1	0.3852 16.0606	0.2125	0.2178	0.1	0.5628	0.1248	0.1271

Table 7 :

 7 ARL 1 values of the lower-sided ATEWMA TBE scheme and the lower-sided AEWMA TBE scheme in detecting downward shifts τ (ARL 0 = 370).

							τ
	τ S τ L	Schemes	0.95	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
	0.9 0.3	AEWMA ATEWMA	270.40 197.00 104.62 56.86 32.63 20.29 13.77 10.10 7.87 6.41 265.76 191.48 101.14 55.45 32.03 19.67 12.84 8.86 6.45 4.97
	0.8 0.3	AEWMA ATEWMA	269.70 196.02 103.74 56.32 32.38 20.21 13.77 10.13 7.92 6.45 253.14 175.58 88.91 48.77 29.09 18.70 12.79 9.19 6.89 5.41
	0.7 0.3	AEWMA ATEWMA	268.31 194.12 102.05 55.30 31.90 20.05 13.77 10.21 8.02 6.56 253.77 176.36 89.51 49.14 29.29 18.82 12.86 9.23 6.92 5.43
	0.9 0.2	AEWMA ATEWMA	280.62 211.55 118.61 66.17 37.57 22.34 14.24 9.82 7.28 5.70 278.97 209.62 117.62 66.02 37.59 22.06 13.55 8.79 6.06 4.47
	0.8 0.2	AEWMA ATEWMA	282.39 214.14 121.29 68.08 38.68 22.86 14.42 9.82 7.19 5.59 272.19 200.12 108.65 60.05 34.33 20.58 13.05 8.77 6.23 4.68
	0.7 0.2	AEWMA ATEWMA	281.06 212.18 119.26 66.63 37.84 22.46 14.28 9.82 7.25 5.67 254.28 176.97 89.92 49.30 29.31 18.77 12.79 9.16 6.85 5.38
	0.9 0.1	AEWMA ATEWMA	306.41 249.78 161.22 99.88 59.38 34.09 19.24 11.02 6.66 4.29 291.03 227.48 136.47 80.12 46.32 26.68 15.57 9.41 6.03 4.19
	0.8 0.1	AEWMA ATEWMA	298.41 238.78 149.23 90.37 53.25 30.88 17.99 10.84 6.94 4.80 296.86 236.53 146.97 88.76 52.22 30.14 17.29 10.08 6.15 4.04
	0.7 0.1	AEWMA ATEWMA	298.59 239.07 149.59 90.68 53.47 31.02 18.06 10.86 6.95 4.79 292.63 229.94 139.26 82.36 47.82 27.53 15.98 9.56 6.04 4.14

Table 8 :

 8 ARL 1 values of the upper-sided ATEWMA TBE scheme and the upper-sided AEWMA TBE scheme in detecting upward shifts τ ( ARL 0 = 370, = 100).

							τ
	τ S τ L	Schemes	1.1	1.3	1.5	1.7	2	2.5	3	3.5	4	5
	1.1 3	AEWMA ATEWMA	170.42 57.83 28.96 18.11 11.28 6.97 5.16 4.18 3.57 2.86 175.51 61.38 31.04 19.40 11.99 7.29 5.33 4.28 3.64 2.88
	1.3 3	AEWMA ATEWMA	178.82 63.10 31.65 19.58 11.96 7.21 5.26 4.23 3.59 2.86 178.40 63.18 31.94 19.89 12.22 7.37 5.37 4.30 3.64 2.88
	1.5 3	AEWMA ATEWMA	144.14 44.34 22.88 15.05 9.98 6.56 5.02 4.14 3.58 2.90 162.41 54.27 27.78 17.78 11.34 7.13 5.31 4.30 3.67 2.93
	1.1 4	AEWMA ATEWMA	180.78 63.13 31.36 19.37 11.89 7.23 5.30 4.27 3.63 2.88 181.51 65.18 32.96 20.44 12.47 7.46 5.40 4.31 3.64 2.88
	1.3 4	AEWMA ATEWMA	181.16 64.66 32.48 20.05 12.19 7.30 5.30 4.24 3.60 2.86 170.98 58.75 29.77 18.75 11.72 7.21 5.31 4.28 3.64 2.90
	1.5 4	AEWMA ATEWMA	183.42 66.24 33.35 20.54 12.43 7.39 5.34 4.27 3.61 2.86 163.55 54.82 28.01 17.88 11.38 7.13 5.30 4.30 3.67 2.93
	1.1 5	AEWMA ATEWMA	175.25 59.84 29.78 18.53 11.51 7.08 5.23 4.23 3.61 2.88 177.87 62.62 31.62 19.73 12.16 7.38 5.39 4.32 3.67 2.90
	1.3 5	AEWMA ATEWMA	186.43 68.41 34.56 21.25 12.78 7.53 5.41 4.30 3.63 2.86 172.50 59.62 30.19 18.96 11.81 7.24 5.32 4.28 3.64 2.89
	1.5 5	AEWMA ATEWMA	186.10 67.47 33.83 20.77 12.54 7.45 5.38 4.30 3.63 2.87 190.19 71.28 36.30 22.35 13.40 7.82 5.57 4.40 3.69 2.89

Table 9 :

 9 ARL 1 values of the lower-sided ATEWMA TBE scheme and the lower-sided AEWMA TBE scheme in detecting downward shifts τ ( ARL 0 = 370, = 100).

							τ
	τ S τ L	Schemes	0.95	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2 0.1
	0.9 0.3	AEWMA ATEWMA	265.38 190.06 98.17 52.45 29.93 18.66 12.75 9.43 7.39 6.08 266.16 191.98 101.51 55.62 32.07 19.64 12.78 8.80 6.39 4.90
	0.8 0.3	AEWMA ATEWMA	264.44 188.79 97.08 51.80 29.62 18.55 12.74 9.45 7.43 6.12 253.66 176.22 89.39 49.05 29.23 18.78 12.82 9.20 6.88 5.40
	0.7 0.3	AEWMA ATEWMA	262.65 186.41 95.06 50.61 29.05 18.34 12.70 9.48 7.50 6.20 254.15 176.80 89.77 49.21 29.27 18.75 12.78 9.15 6.85 5.37
	0.9 0.2	AEWMA ATEWMA	277.66 207.23 114.16 62.85 35.43 21.05 13.47 9.34 6.96 5.51 279.39 210.22 118.20 66.42 37.81 22.17 13.59 8.80 6.06 4.46
	0.8 0.2	AEWMA ATEWMA	279.69 210.17 117.13 64.94 36.63 21.61 13.67 9.37 6.91 5.42 272.63 200.72 109.17 60.37 34.49 20.65 13.07 8.76 6.22 4.66
	0.7 0.2	AEWMA ATEWMA	278.16 207.96 114.89 63.36 35.72 21.18 13.52 9.35 6.95 5.48 254.71 177.48 90.27 49.47 29.37 18.78 12.77 9.13 6.82 5.36
	0.9 0.1	AEWMA ATEWMA	304.93 249.03 161.27 100.15 59.60 34.19 19.23 10.97 6.60 4.24 291.38 228.01 137.05 80.57 46.60 26.82 15.63 9.42 6.02 4.17
	0.8 0.1	AEWMA ATEWMA	297.28 236.99 147.06 88.48 51.86 29.97 17.44 10.53 6.77 4.73 297.18 237.02 147.55 89.23 52.54 30.31 17.37 10.10 6.15 4.02
	0.7 0.1	AEWMA ATEWMA	297.48 237.30 147.44 88.80 52.09 30.11 17.52 10.56 6.78 4.72 292.97 230.46 139.85 82.82 48.11 27.68 16.04 9.57 6.04 4.12

Table 10 :

 10 The OLED failure time X t and the charting statistics corresponding to the lower-sided ATEWMA TBE, the lower-sided AEWMA TBE, and the lower-sided REWMA TBE schemes.

	t X t (mins)	M t	ATEWMA	AEWMA	REWMA	REWMA

Table 11 :

 11 The time X t between consecutive F-16 accidents and the corresponding charting statistics of the lower-sided ATEWMA TBE, the lower-sided AEWMA TBE, and the lower-sided REWMA TBE schemes.

	t X t (days)	M t	ATEWMA	AEWMA	REWMA (τ R = 0.7)	REWMA (τ R = 0.2)

Wang, F., Bizuneh, B., & Cheng, X. (2019). One-sided control chart based on support vector machines with differential evolution algorithm. Quality and Reliability Engineering International, 35, 1634-1645. Wang, H., Sun, H., Li, C., Rahnamayan, S., & Pan, J. (2013). Diversity enhanced particle swarm optimization with neighborhood search. Information Sciences, 223, 119-135. Xie, F., Castagliola, P., Qiao, Y., Hu, X., & Sun, J. (2021 in press). A one-sided exponentially weighted moving average control chart for time between events.

Journal of

If we define E - j as the mid-point value of the jth sub-interval, say,

where j = 1, 2, . . . , m. Then, the charting statistic W - t in the transient state j can be written as E - j -∆ -/2 < W - t E - j + ∆ -/2. Furthermore, the transient probabilities q i,j from transient state i to transient state j are computed as follows:

where φ -1 (•) is defined in (24), i = 1, 2, . . . , m, and j = 1, 2, . . . , m. If we define,

the transient probabilities q i,j can be denoted as:

Additionally, the elements p j of the initial probability vector p are given as:

where j = 1, 2, . . . , m, and W - 0 = E(Z - t ) = 1. Finally, the ARL value of the lower-sided ATEWMA TBE scheme can be computed using (20).

For the lower-sided ATEWMA TBE scheme with estimated parameters, we define the width of the jth sub-interval as

and the mid-point value of the jth sub-interval as

Then, the one-step transient probabilities q i,j in matrix Q are computed as follows:

where .10) and i = 1, 2, . . . , m, j = 1, 2, . . . , m. In addition, the elements p j of the initial probability vector p are given as, .11) where W - 0 = E( Z - t ) = 1. After determining the matrix Q and the vector p, the ARL value of the lower-sided ATEWMA TBE scheme with estimated parameters are computed using ( 29) and (30).


Naiqi Wu

Zhiwu Li

Kamel Barkaoui

Xiaoou Li

Mengchu Tadahiko Murata

published or not. The documents may come

IoT-based Smart and Complex Systems: A Guest Editorial Report

Naiqi Wu, Senior Member, IEEE, Zhiwu Li, Fellow, IEEE, Kamel Barkaoui, Xiaoou Li, Senior Member, IEEE, Tadahiko Murata, Senior Member, IEEE, and MengChu Zhou, Fellow, IEEE HE Industrial Revolution starting from about 1760 and ending at around 1840 has been viewed as the first In-dustrial Revolution. It features with the replacement of human and animal muscle power with steam and mechanical power. Human income per capita had taken 800 years to double by the early 1800's. Since 1760, the first Industrial Revolution spent only about 150 years to increase human income per capita by thirteen-fold. While historians are still debating what the second Industrial Revolution is, many believe that it is Internet Revolution. It started in about 60 years ago. It features the applications of computing, communications, and networking and information storage in every aspect of human life. It offers even much faster growth of human income per capita and drastically promoted industrial productivity.

What is the next Industrial Revolution? Different researchers and industrial engineers tend to offer different answers. For example, Rifkin presents in [1] his answer as "Energy Internet" where people produce their own green energy in their homes, offices, and factories, and share it with each other just as information is shared via Internet. Manufacturing engineers would believe that 3-D printing and digital manufacturing as the next industrial revolution, through which people can conceive, design, manufacture and recycle their own one-of-a-kind products [2]. For computer scientists and engineers, artificial intelligence, big data analysis, cloud computing, and fog/edge computing are their likely choices as the third In-dustrial Revolution [START_REF] Tien | An SMC perspective on big data: a disruptive innovation to embrace[END_REF] [START_REF] Zhou | Guest editorial: special section on advances and applications of internet of things for smart automated systems[END_REF]. Evans and Annunziata of General Electric Company answered this question by coining a name "Industrial Internet" [8], which is popularly called Internet of Things (IoT) [9]. We agree with it because the next tech-nological wave must move from today's Internet connecting mostly people to people via smart terminals like computers and smart phones to tomorrow's IoT linking everything in the world. As a result, IoT-based smart systems promise to offer the opportunities for another even sharper productivity increase than the prior industrial revolutions for not only human beings but also every machine and everything in the world. There is no doubt that such IoT-based systems become more and more popular as well as pose numerous modeling, design, analysis and control challenges. The systems are able to intelligently respond to their environment and offer value-added services to their clients. Their discrete-event nature and hybrid modeling, control, scheduling, simulation and security management pose a deluge of theoretically significant and practically meaningful issues to researchers in various fields. They often require interdisciplinary efforts and mathematical modeling and optimization approaches to move this field forward.

This special issue has collected 18 original contributions reporting the latest advances in IoT-based smart and complex systems. They present the latest research results on the state-of-the-art complex system modeling and analysis methods, and recent new findings obtained by using systems engineering methodologies to the design and evaluation of IoT-based Smart and Complex Systems.

Wireless sensor networks (WSN) are an important branch of IoT. Several papers of the issue deal with their design, analysis and applications. Duan et al. present a methodology to design a high-performance and reliable WSN based on a software defined network in a changing industrial environment. In the context of industrial IoT, we face WSN node failures caused by energy depletion and hardware malfunctions. The industrial environment can be very harsh and can adversely impact wireless channel transmission, thus leading to serious network reliability problems and increasing network management cost. With the help of software defined networks, the authors invented a framework called the improved software defined wireless sensor network. For a large scale heteroge-neous network, they successfully solve the problem of network management and smooth merging of a WSN into industrial IoT. Their work ensures high network coverage with high network performance by addressing addresses node failure issues as caused by energy depletion.

One of the primary functions of WSN is to produce various data. Efficient data collection issues are addressed by Song and Li in their paper "Data Gathering in Wireless Sensor Networks via Regular Low Density Parity Check Matrix". To guarantee the lifetime of WSNs, the data gathering via random sensing is often used. Yet such randomness and WSN node density result in difficulty of implementations, high computation complexity and large storage spaces in practical WSN deployment. Deterministic sparse sensing matrices are sometimes desired, but their deployment cannot ensure high performance of WSN. The authors innovatively construct a class of deterministic sparse sensing matrices with statistical versions of restricted isometry property via regular low density parity check matrices. They can thus achieve the same measurement performance as the dense measurements can, thereby reducing energy consumption of WSNs.

Securing IoT has been an important issue. Intrusion detection has been a primary defense technique to network security. The next paper "An Intrusion Detection Algorithm for Wireless Networks Based on ASDL" by Zhu et al. presents a novel temporal logic language called attack signature description language (ASDL) and ASDL-based model checking algorithm. They can well describe attack signatures, and design the ASDL programs to create an audit log. Their case studies and simulations show that their algorithms can identify such challenging attacks as coordinated chop-chop attacks in wireless networks. The paper "SVM-DT-Based Adaptive and Collaborative Intrusion Detection" " by Teng et al. gives a self-adaptive collaboration intrusion detection method based on the Environments -classes, agents, roles, groups, and objects (E-CARGO) model. The authors design the objects, roles, agents, and groups by using decision trees and support vector machines (SVMs) in their method. They then utilize a well-known dataset to verify the effectiveness of the method. Their experimental results demonstrate its feasibility and higher detection precision rate and recall rate than the method that uses a set of single type support vector machines.

Radio frequency identification (RFID) technology has been playing a key role in identifying an IoT object and has widely applied in manufacturing, warehouses, ports, stores and logistics. The paper "Capture-aware Bayesian RFID Tag Estimate for Large-scale Identification" by Wu et al. discusses passive RFID tag anti-collision issues. In a dynamic framed slotted Aloha algorithm, a frame length requires dynamical adjustment to achieve higher identification efficiency. Such adjustment is not only related to the number of tags, but also to the occurrence probability of capture effects. When estimating both the number of tags and the probability of capture effect under large-scale RFID tag identification, the existing algorithms fail to provide accurate estimates since the number of tags would be much larger than an initial frame length. To address this thorny issue, the authors invent a novel algorithm, called capture-aware Bayesian estimate, which adopts Bayesian rules to perform accurate estimates. Numerical results show that the proposed algorithm can well handle large-scale RFID tag identification problems with higher estimation accuracy and efficiency than the existing algorithms.

How can we apply the RFID technology to manufacturing systems? Ding and Jiang's paper "RFID-based Production Data Analysis in an IoT-enabled Smart Job-shop" presents an excellent answer to this question. While RFID technology brings convenience to production control and transparency, it also generates increasing data that are discrete, uncorrelated, and hard-to-use. To utilize the generated data of systems, the authors propose an innovative RFID-based production data analysis method for production control in IoT-enabled smart job-shops. Based on the physical configuration and operation logic of IoT-enabled smart job-shop production, they construct an RFID-based production data model to formalize and correlate the heterogeneous production data. Then, they propose an event-driven RFID-based production data analysis method to construct the RFID events and judge the process command execution. Consequently, they are able to excavate hidden information and knowledge from the historical production data to help managers and production engineers improve the performance of manufacturing systems. and hard-to-use. To utilize the generated data of systems, the authors propose an innovative RFID-based production data analysis method for production control in IoT-enabled smart job-shops. Based on the physical configuration and operation logic of IoT-enabled smart job-shop production, they construct an RFID-based production data model to formalize and correlate the heterogeneous production data. Then, they propose an event-driven RFID-based production data analysis method to construct the RFID events and judge the process command execution. Consequently, they are able to excavate hidden information and knowledge from the historical production data to help managers and production engineers improve the performance of manufacturing systems.

Complex engineering systems requires sophisticated modeling and production scheduling. The paper "Collision-free Scheduling of Multi-bridge Machining Systems: A Colored Traveling Salesman Problem-based Approach" by Li et al. has address the scheduling issue faced by the optimal operations of multi-bridge machining systems. These systems contain multiple bridge machines working in parallel and their workspaces partially overlap. The authors innovatively express their scheduling problems as a serial-colored travelling salesman problem in which each salesman has some exclusive cities and some shared cities with its neighbor(s). They develop a greedy algorithm that selects a neighboring city satisfying proximity. The algorithm allows a salesman to select randomly its shared cities and runs accordingly many times. It can thus be used to solve job scheduling problems for multi-bridge machining systems. Subsequently, a collision-free scheduling method is proposed to address both job scheduling and collision resolution issues. Their algorithm can select a job for an individual machine such that no time overlaps exist between it and the job sequence of the neighboring machine in the overlapping workspace; and remove such a time overlap only when it is inevitable. Their results have been successfully applied to a large triple-bridge waterjet cutting system.

Multi-agent systems and technologies are important in realizing IoT-based complex systems. The next several papers deal with their modeling and control issues. In the paper "The Formation Control of Multi-agent Systems on a Circle", Wang et al. investigate the formation control problem for a class of multi-agent systems moving on a circle whose topology is a cyclic graph. They focus on the agents with single and doubleintegrator kinematics. Several control protocols are proposed to keep a uniformly-spaced formation and ensure the stability of the formation. Their simulations results show that their designed control protocols can work very well to achieve the desired formation control of their concerned multi-agent systems.

The next paper "Gini Coefficient-based Task Allocation for Multi-robot Systems With Limited Energy Resources" by Wu et al. deals with the energy-constrained robots working in such dangerous environments as earthquake rescue and exploration of wild areas. If the energy resources of some robots are consumed too fast, the future tasks requiring their coalition are to be at high risk. The authors thus develop a novel task allocation method based on Gini coefficient to make full use of limited energy resources of a multi-robot system to maximize their capability in handling tasks. They incorporate the market-based allocation mechanism into their Gini coefficient-based method to formulate a hybrid method. It can flexibly maximize the number of completed tasks or minimize resource consumption according to the application needs.

In the third paper of multi-agent systems, Yang and Yue investigate a consensus tracking problem of networked multiagent systems in a non-affine pure-feedback form. They propose to build up a distributed adaptive tracking consensus control scheme recursively by using a backstepping method, graph theory, neural networks and dynamic surface control technology. Its core advantages include avoiding the complexity problem as the degree of individual agents grows and thus decreasing the computational burden drastically. Second, it poses no requirement for prior knowledge about system parameters of individual agents and uncertain dynamics. Finally, it guarantees the consensus errors to be cooperatively semiglobally uniformly ultimately bounded.

In the fourth paper of multi-agent systems, Wang et al. studied guaranteed cost consensus analysis and design problems for high-dimensional multi-agent systems with timevarying delays. They introduce guaranteed cost control into the consensus problems for such complex multi-agent systems and define a cost function based on state errors among neighboring agents and control inputs of all the agents. They then establish the sufficient conditions for guaranteed cost consensus and consensualization by using a state space decomposition approach and linear matrix inequality. They also derive a guaranteed cost upper bound of the cost function and validate their results through numerical simulations.

The last paper about multi-agent systems is entitled "Cyber-Physical-Social System Between a Humanoid Robot and a Virtual Human Through a Shared Platform for Adaptive Agent Ecology" written by Rahman. This work focuses on two artificial agents, i.e., a humanoid robot and virtual human, which are integrated in the form of a cyber-physical-social system (CPSS) through a shared communication platform to create a social ecology. In the ecology, two agents assist each other to perform a real-world task, e.g., finding a hidden object, to benefit humans. The author derives a robot-virtual human bilateral trust model and a real-time trust measurement method. The role of taking initiative in their collaboration can be switched between them by following a finite state machine model, and thus mixed initiative collaboration is likely. Their performance can be evaluated through a cyber-physical-social system. The results prove the effectiveness of the real-world ecology between artificial agents of heterogeneous realities through a shared platform based on trust-triggered mixedinitiatives. This work makes an important progress in the development of adaptive social ecology comprising intelligent agents of heterogeneous realities to assist humans in various tasks through their collaboration.

Understanding people's emotions through natural language is a challenging issue for intelligent. The major difficulty is caused by the lack of basic knowledge in emotion expressions with respect to a variety of real world contexts. The paper "Exploring Latent Semantic Information for Textual Emotion Recognition in Blog Articles" by Kang et al. aims to address it. The authors pioneered in a Bayesian inference method to explore the latent semantic dimensions as contextual information in natural language and to learn the knowledge of emotion expressions based on these semantic dimensions. Their method can synchronously infer the latent semantic dimensions as topics in words and predicts the emotion labels in both wordlevel and document-level texts. The Bayesian inference results enable one to visualize the connection between words and emotions with respect to different semantic dimensions. By further incorporating a corpus-level hierarchy in the document emotion distribution assumption, they are able to balance the document emotion recognition results and achieve even better word and document emotion predictions. Their experimental results based on a well-developed Chinese emotion corpus prove their proposed method's higher accuracy and better robustness in the word-level and the document-level emotion predictions than several state-of-the-art emotion prediction algorithms.

Recognition in Blog Articles" by Kang et al. aims to address it. The authors pioneered in a Bayesian inference method to explore the latent semantic dimensions as contextual information in natural language and to learn the knowledge of emotion expressions based on these semantic dimensions. Their method can synchronously infer the latent semantic dimensions as topics in words and predicts the emotion labels in both wordlevel and document-level texts. The Bayesian inference results enable one to visualize the connection between words and emotions with respect to different semantic dimensions. By further incorporating a corpus-level hierarchy in the document emotion distribution assumption, they are able to balance the document emotion recognition results and achieve even better word and document emotion predictions. Their experimental results based on a well-developed Chinese emotion corpus prove their proposed method's higher accuracy and better robustness in the word-level and the document-level emotion predictions than several state-of-the-art emotion prediction algorithms.

Many identical or similar robots can form a swarm. Traditionally these robots all have the same or invariable controller during their entire life time and thus fail to handle time-varying complex environments. In the paper "Time-varying Algorithm for Swarm Robotics" contributed by Hou et al. propose the use of variable controllers in order to adapt to such changing environments. Their proposed algorithm takes time as one of the independent variables such that the controller is no longer fixed over the time; instead they can be changed over time, which brings more choices for a swarm robot system. They have applied different control strategies to the same flock during the time, and are able to obtain interesting and useful control effects for the swarm of robots.

The paper "Active Queue Management Exploiting the Rate Information in TCP-IP Networks" by Boudi and Loudini gives a new mechanism called explicit rate notification that can be well used in end-to-end communications in Internet. The proposed scheme encodes in the header of transmission control protocol (TCP) packets information about the sending rate and the round trip time of the flows. This new available information to the intermediate nodes (routers) is used to improve fairness, increase utilization, decrease the number of drops, and minimize queueing delays. Thus, it induces a better management of the queue. Through the comparison of the proposed scheme with several existing schemes, e.g., an explicit congestion notification scheme shows its effectiveness. It has excellent potential to be used in Internet as well as IoT. The paper "Distributed Containment Control of Networked Nonlinear Second-order Systems with Unknown Parameters" by Ma et al. presents the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders who can be robots or agents. The interactions between these leaders and their followers are described through directed graphs. This work introduces both stationary leaders (a regulation case) and dynamic leaders (a dynamic tracking case) based protocols. It computes the unique final states of all the followers given the initial values of leaders and a directed graph. In the regulation case, all the followers converge into the convex hull spanned by the leaders, while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders. Their theoretically solid results are illustrated through several numerical simulations.

Mi and Li's paper "Event-triggered MPC Design for Distributed Systems with Network Communications" deals with the communication problem in a distributed system in which each wireless network node has the limited battery power and redundant transmission among nodes is allowed. The authors pioneered in an event-triggered model predictive con-trol strategy to reduce the unnecessary communication while guaranteeing the desired system performance. They derive a triggering condition based on the Lyapunov stability for a linear discrete time-invariant system. They further reduce the communications amount by enforcing a triggering condition only when the Lyapunov function exceeds its value at the last triggered time.

The last paper of this special issue is "Detecting Dataflow Errors Based on Petri Nets with Data Operations" contributed by Xiang et al.. In order to guarantee the correctness of business processes, not only control-flow errors but also data-flow errors should be well handled. The control-flow errors include deadlock, livelock, and soundness, which have been well addressed. Data-flow errors may take place during such data operations as reading, writing and deletion. Their detection is lacking. This work defines Petri nets with data operations that can well model the data operations. Based on them, the authors formally define data-flow errors. They then construct a reachability graph with data operations for each net model, and propose a method to reduce the originally large reachability graph. Based on the reduced graph, they can find data-flow errors rapidly. They use a case study to illustrate the applications of their Petri net model and data-flow error detection methods.

This special issue has received much support from the founding Editor-in-Chief, Professor Fei-Yue Wang, several Associate Editors, and many anonymous reviewers. The guest editors would like to thank them. We hope that this special issue can motivate more researchers and practitioners to be actively engaged in developing smart IoT-based complex systems to benefit human beings.