Assuming, at the time t=1,2,…, there are X1t,X2t,…,Xit,…,Xnt; i.e., n independent identically normal random variables with mean and variance μt and σt2, i.e., Xit∼N(μt,σt2) for i=1,2,…,n. As the only concern is to detect the increasing changes in σt2, so the mean level of the process is assumed to be IC, i.e., μt=μ. Suppose the underlying process variance remains IC for a particular time, i.e., σt2=σ02 for t<t0, then the process goes in OOC state, i.e., σt2≠σ02 for t≥t0. Let δt denotes the size of shifts in σt2 then it can be defined as a ratio IC process standard deviation to OOC process standard deviation, i.e., δt=σtσ0. So, δt=1 whenever the underlying process is IC, and δt≠1 in the case of the OOC process. If X¯t=1n∑i=1nXit be the sample mean and St2=1n−1∑i=1n(Xit−X¯t)2 denotes the sample variance of the tth subgroup, respectively, then for IC process St2 follows a chi-square distribution with n−1 degrees of freedom, i.e., St2∼σ02n−1χ(n−1)2.

In order to implement the EWMA-type charts, the assumption of normality is required for the plotting statistic of the charts. However, because the sample variance St2 has a chi-square distribution, so it is not an acceptable statistic for the design structure of the EWMA-type charts. In order to cope with this issue, a few transformations are available in the literature, given as follows.

Crowder et al. [16] introduced the CH chart, which monitored the process variance shifts. Let {Qt} for t≥1 be the sequence of IID random variable, defined on the sequence {Wt}, where Wt is defined by Eq. (1), then using the recurrence relationship, the CH plotting statistic Qt can be given as

(4) Qt=(1−λ1)Qt−1+λ1Wt,

where λ1 is known as the smoothing parameter. The CH statistic mean and variance are, respectively, given as

E(Qt)=μWandVar(Qt)=λ12−λ1[1−(1−λ1)2t]σW2 .

In the case of large t, the variance of Qt is reduced to; Var(Qt)=λ11−λ1σW2. In order to detect the gradual rise in the process variance, let Qt+ be the charting statistic for the CH chart is given by

(5) Qt+=max(Qt,0).

The initial value of and Qt+ is set on 0, i.e., Q0+=0. The upper control limit for the CH chart is denoted by UCL(CH)+ and can be defined by

(6) UCL(CH)+=LCH+λ12−λ1σW.

where LCH+ is the CH chart width coefficients and can be computed so that IC ARL\;(ARL0) is approximately equal to the desired value. The CH chart detects the upward shifts in the process whenever Qt+≥UCL(CH)+.

Castagliola [19] proposed the EWMA chart (also known as S2-EWMA chart) to monitor the shifts in the process variance. Hereafter the S2-EWMA chart is labeled as the CEWMA chart. The CEWMA chart used the three parameters logarithmic transformation to S2 to obtain the approximate normality for the plotting statistic. Let {Zt} be the CEWMA sequence, based on another sequence {Tt} for t≥1, where Tt is defined by Eq. (2), then the charting statistic of the CEWMA chart is given by

(7) Zt=(1−λ1)Zt−1+λ1Tt,

The initial value of Zt is denoted by Z0 and can be is defined as

(8) Z0=AT+BTln⁡(1+CT).

The Z0 values for various n can be taken from [19]. The Z0 values are close to 0, so one can replace them with 0. Here, the plotting statistic Zt is an approximate normal distributed variable having mean μT and variance λ12−λ1[1−(1−λ1)2t]σT2, i.e., Zt∼N(μT,λ12−λ1[1−(1−λ1)2t]σT2). However, for a large value of t, the factor [1−(1−λ1)2t] tends to 1, and in this case Zt∼N(μT,λ12−λ1σT2). If UCL(CEWMA)+ denotes the upper control limit for the CEWMA chart, then it can be defined as

(9) UCL(CEWMA)+=μT+LCEWMA+λ12−λ1σT,

where LCEWMA+ is called the width coefficient of the CEWMA chart. The CEWMA chart detects OOC signals with increasing shift whenever Zt≥UCL(CEWMA)+.

Huwang et al. [18] proposed the EWMA chart to monitor the process variance, denoted as the HHW2 chart. Let {Rt}, for t≥1 be the HHW2 sequence that based on the IID sequence of random variable {Mt}, where Mt is defined in Eq. (3), then the HHW2 statistic based on the sequence {Rt} is defined as

(10) Rt=(1−λ1)Rt−1+λ1Mt.

The initial value of and Rt is set on 0, i.e., R0=0. The plotting statistic Rt has a normal distribution with mean zero, and variance λ12−λ1{1−(1−λ1)2t}, i.e., Rt∼N(0,λ12−λ1[1−(1−λ1)2t]). The time-dependent control limits, UCLt(HHW2) for the HHW2 chart can be defined as

(11) UCLt(HHW2)+=LHHW2+λ12−λ1{1−(1−λ1)2t}.

In case of large t values, the upper control limit for of the HHW2 chart is denoted as UCL(HHW2)+ and can be given as

(12) UCL(HHW2)+=LHHW2+λ12−λ1.

The LHHW2+ is the HHW2 chart width coefficients. The HHW2 chart detects OOC signals whenever Rt fall above the control limits specified in Eq. (12).

Ali et al. [23] followed the idea of [8] and designed the HEWMA chart for process variance. Let the IID {Ht} for t≥1 is known as the HEWMA sequence, which is based on the {Rt}, then the statistic Ht for the HEWMA chart is given as

(13) Ht=(1−λ2)Ht−1+λ2Rt,

where Rt is defined by Eq. (10) and λ2 is also a smoothing constant, such that λ2≠λ1. The initial values of Ht are set to 0, i.e., H0=0. The mean E(Ht)=0, and the variance of Ht is given as

(14) Var(Ht)=(λ1λ2λ1−λ2)[∑k=12(1−λk)2{1−(1−λk)2t}1−(1−λk)2−2(1−λ1)(1−λ2){1−(1−λ1)t(1−λ2)t}1−(1−λ1)(1−λ2)].

The control limit UCLt(HEWMA) for the HEWMA chart, based on Eq. (14) is defined by

(15) UCLt(HEWMA)+=LHEWMA+×(λ1λ2λ1−λ2)[∑k=12(1−λk)2{1−(1−λk)2t}1−(1−λk)2−2(1−λ1)(1−λ2){1−(1−λ1)t(1−λ2)t}1−(1−λ1)(1−λ2)].

When t gets larger, the control limits defined by Eq. (15) is reduced to UCL(HEWMA)+ and given as

(16) UCL(HEWMA)+=LHEWMA+(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)].

where LHEWMA+ is the width coefficient for the HEWMA chart. The HEWMA chart triggers OOC signals whenever Ht≥UCL(HEWMA)+.

The design structure for the HEWMA1 chart can be constructed using the CH statistic as input for the HEWMA1 statistic. Let defined the sequence of IID random variable, say {Ut} for t≥1, based on the CH sequence {Qt} then the HEWMA1 statistic Ut can be defined by the relation given as

(17) Ut=(1−λ2)Ut−1+λ2Qt,

where Qt is the CH statistic defined by Eq. (4). The starting value of Ut is equal to 0, i.e., U0=0. The mean of Ut is E(Ut)=0 and its variance, for the case of very large t, is defined as

(18) Var(Ut)=(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σW2.

In order to monitor the increasing shift in the process, the HEWMA1 statistic is defined as

(19) Ut+=max(Ut,0).

The initial value of Ut+ is set on 0, i.e., U0+=0. The control limit for upper sided HEWMA1 is given as

(20) UCL(HEWMA1)+=LHEWMA1+(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σW.

where LHEWMA1+ is the width coefficient for the HEWMA1 chart. The HEWMA1 chart detects OOC signals whenever Ut+≥UCL(HEWMA1)+. Similarly, for monitoring the gradual decrease in the variance, the HEWMA1 statistic is given by

(21) Ut−=min(Ut,0).

The initial value of Ut− is denoted by U0− set on 0, i.e., U0−=0. The control limit for lower sided HEWMA1 is given as

(22) LCL(HEWMA1)−=LHEWMA1−(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σW.

The lower-sided HEWMA1 chart detects OOC signals whenever Ut−≤LCL(HEWMA1)−. The control limits defined in Eqs. (20) and (22) are called the HEWMA1 upper and lower control limits, respectively. However, the HEWMA1 two-sided control limits are given as

(23) UCL(HEWMA1)=LHEWMA1(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σWLCL(HEWMA1)=−LHEWMA1(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σW}.

The two-sided HEWMA1 chart detects OOC signals whenever Ut≥UCL(HEWMA1) or Ut≤LCL(HEWMA1).

In order to formulate the design of the HEWMA2 chart, the charting statistic of the CEWMA chart can be used as an input for the HEWMA2 statistic. Let {Vt} for t≥1 be the HEWMA2 sequence, then the HEWMA2 chart statistic is Vt, and it can be defined by

(24) Vt=(1−λ2)Vt−1+λ2Zt,

where Zt is the HHW2 plotting statistic defined by Eq. (7). The starting values of Vt is equal to Z0 defined by Eq. (8). The expected value for the statistic Vt is E(Vt)=μT and its variance is given as

(25) Var(Vt)=(λ1λ2λ1−λ2)×[∑k=12(1−λk)2{1−(1−λk)2t}1−(1−λk)2−2(1−λ1)(1−λ2){1−(1−λ1)t(1−λ2)t}1−(1−λ1)(1−λ2)]σT2.

The HEWMA2 control limits UCLt(HEWMA2) and LCLt(HEWMA2) are given as

(26) UCLt(HEWMA2)=μT+L(HEWMA2)×(λ1λ2λ1−λ2)[∑k=12(1−λk)2{1−(1−λk)2t}1−(1−λk)2−2(1−λ1)(1−λ2){1−(1−λ1)t(1−λ2)t}1−(1−λ1)(1−λ2)]σTLCLt(HEWMA2)=μT−L(HEWMA2)×(λ1λ2λ1−λ2)[∑k=12(1−λk)2{1−(1−λk)2t}1−(1−λk)2−2(1−λ1)(1−λ2){1−(1−λ1)t(1−λ2)t}1−(1−λ1)(1−λ2)]σT}.

The control limits defined in Eq. (26) are known as the two-sided time-dependent control limits; however, in the case of large t values, the two-sided fixed HEWMA2 control limits are defined as

(27) UCL(HEWMA2)=μT+L(HEWMA2)(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σTLCL(HEWMA2)=μT−L(HEWMA2)(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σT}.

In this case, The HEWMA2 chart triggers OOC signals whenever Vt≥UCL(HEWMA2) or Vt≤LCL(HEWMA2). The control limits specified in Eq. (27) are the two-sided control limits; however, the HEWMA2 upper control limit is defined as

(28) UCL(HEWMA2)+=μT+LHEWMA2+(λ1λ2λ1−λ2)[∑k=12(1−λk)21−(1−λk)2−2(1−λ1)(1−λ2)1−(1−λ1)(1−λ2)]σT.

The LHAEWMA2+ is known as the chart constants. The HEWMA2 chart detects OOC signals with increasing shift whenever Vt≥UCL(HAEWMA2)+.

The most popular and commonly used performance evaluation measures are the ARL and SDRL measures. The ARL can be defined as the average number of sample points until a chart indicates the OOC signal [2]. The ARL is further categorized as IC ARL (ARL0), and OOC ARL (ARL1). When a process is working in an IC state, the ARL0 should be as large enough to prevent the frequent false alarms, while the ARL1 should be smaller so that the shift is detected quickly [39]. A chart with smaller ARL1 is preferred over the competing charts at a prespecified ARL0 [40].

The ARL and SDRL measures evaluate the performance of the charts on a single specified shift. However, sometimes the researcher may want to investigate the charts performances for the entire range of shifts, i.e., δmin<δ<δmax. For this purpose, the other performance measures, such as extra quadratic loss (EQL), relative average run length (RARL), and performance comparison index (PCI), are used. The details on the EQL, RARL, and PCI measures are provided in the following subsections.

The random sample of size n, i.e., X1t,X2t,…,Xit,…,Xnt for t>1, is generated from a normal distribution under different parameter settings. Domangue et al. [15] suggested that monitoring a gradual rise (process deterioration) in the process variance is more important; therefore, an upward shift is considered in the process variance. The shift is reflected in the process standard deviation, i.e., σ=δσ0, where δ= 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0. The Monte Carlo simulation approach is utilized as a computational methodology for the numerical results by designing an algorithm in the statistical package R. At each shift size δ, the simulations are performed with 20,000 replicates. The simulation algorithm for the HEWMA1 and HEWMA2 charts include the following steps:

Specify sample size n, smoothing parameters (λ1,λ2) and parameters of the process distribution, i.e., for IC process N(μ0,σ02) and for OOC process N(μ0,(δσ0)2), where δ=σσ0.

The design parameters for the proposed HEWMA1 and HEWMA2 charts are the smoothing constants λ1, λ2 and the chart width coefficient Lproposed, which have a certain effect on the chart performance. Therefore, the different settings of the design parameters are used in computing ARL, MDRL and SDRL measures. The various combination of smoothing parameters are chosen as (λ1= 0.05, λ2=0.050001,0.1, 0.20), (λ1= 0.1, λ2=0.05,0.100001, 0.20), (λ1= 0.2, λ2=0.05,0.1, 0.200001), (λ1= 0.3, λ2=0.05,0.1, 0.2) to determine the values of LHEWMA1+and LHEWMA2+, so that ARL0=200. The numerical results regarding the proposed HEWMA1 and HEWMA2 charts are displayed in Tables 1–4.