The different production processes in industries face commonly two types of variations: common cause variation and special cause variation. Common cause variation always exists even if the process is designed very well and maintained very carefully. This variation should be relatively small in magnitude and is uncontrollable and due to many small unavoidable causes. A process is said to be in statistical control if only common cause variation is present. The variations outside this common cause pattern are called special cause variations. These variations are subject to some problem in the system, like the poor tuning of equipment, controller fell asleep or got absent, the computer stopped working, a poor lot of raw material, machine break down. A process working under both types of variation is said to be out of control.

These variations may affect the production process and ultimately result in a defective output. The consequences of it are pretty severe. Companies have to face high cost as well as their reputation will be at stake. Therefore, in order to avoid variations in the process, there is a need to use control charts to keep the process under control. Zaka et al. [1] introduced a new class of distribution called WPFD in order to have more application in applied sciences such as biosciences engineering, management sciences. The weighted distributions are helpful when the sampling is done using an unequal probability of the sampling units. WPFD is more useful in Statistical process control. Statistical process control (SPC) is defined as the use of statistical techniques to control a process or production method. SPC tools and procedures can help to monitor process behavior, discover issues in internal systems, and find solutions for production problems. Two types of control charts are commonly studied, memory less control chart i.e., Shewhart control chart and memory based control charts such as EWMA and HEWMA etc. Many works have been done in this regard, such as the EWMA control charts firstly by Roberts [2], and recently by Li et al. [3] and Nguyen et al. [4]. The cumulative-sum (CUSUM) control charts firstly by Page [5], Sanusi et al. [6], Haq et al. [7] and Hossain et al. [8]. The mixed EWMA-CUSUM control charts by Abbas et al. [9], Ajadi and Riaz [10]. The hybrid exponential weighted moving average (HEWMA) control charts due to Shamma et al. [11] and Haq [12].

In real life scenario, this is not always possible to fulfil the normality assumption for the distribution of error during the process. A very few works in literature is about this situation of not normal process distribution including Noorossana et al. [13], Lin et al. [14], Erto et al. [15], Liang et al. [16] and Ahmed et al. [17]. The EWMA statistic is widely used for shift detection in the ongoing process either to monitor qualitative or quantitative physical phenomena. The generalized form of the existing EWMA statistic was introduced by Naveed et al. [18] and named it as EEWMA.

Let the distribution of the underlying process having the sequence {X_t} is normal. Also, let the 0≤λ≤1 , is a known constant. Now EWMA statistics is given by

Wt=λXt+(1−λ)Wt−1 .

The smoothing constant λ plays a very important here. As it approaches to zero, it becomes sensitive for a small and moderate shift in mean and as it becomes close to one. It approaches to Shewart control chart. The control limits for EWMA are given below

UCLWt=μ+L∗V(Wt)

LCLWt=μ−L∗V(Wt)

E(Wt)=μ

V(Wt)=σ2(1−(1−λ)2t)(λ2−λ)

When the distribution of the process is normal, the EEWMA control chart was introduced by Naveed et al. [18]. The EEWMA control chart by Naveed et al. [18] is given as

Zt=λ1Tt−λ2Tt−1+(1−λ1+λ2)Zt−1

where 0≤λ1≤1 and 0≤λ2≤λ1 . Tt−1 is represents the previous value of the variable and Zt−1 denotes the previous value of a statistic.

The mean and variance are given as

E(Zt)=μ

And

var(Zt)=σ2[(λ12+λ22){1−(1−λ1+λ2)2t2(λ1−λ2)−(λ1−λ2)2}−2aλ1λ2{1−(1−λ1+λ2)2t−22(λ1−λ2)−(λ1−λ2)2}]

Following Zaka et al. [1], it is assumed that the process random variable x1,x2,x3,…,xt are independently and identically distributed following WPFD with probability density function (pdf) and cumulative density function (cdf) for the WPFD are given respectively by

f(x)=2γx2γ−1β2γ;0<x<β

and

F(x)=(xβ)2γ , where “β” and “γ” are the scale and shape parameters.

Modified Maximum Likelihood Estimator (MMLE) is used to construct memory less and memory-based control charts to monitor the shape parameter of a process that follows a WPFD. The estimator for the shape parameter of WPFD defined by Zaka et al. [1] is given as,

(1) γ^MMLE=−1+1+x¯2S22.

The variance of the γ^MMLE is given by

(2) Var(γ^MMLE)=E(γ^MMLE−γ)2.

Now using Zaka et al. [1] and Naveed et al. [18], the EEWMA statistic is given as

EEWt=λ1γ^MMLE(t)−λ2γ^MMLE(t−1)+(1−λ1+λ2)EEWt−1

For t = 1

EEW1=λ1γ^MMLE(1)−λ2γ^MMLE(0)(1−λ1+λ2)EEW0

For t = 2

EEW2=λ1γ^MMLE(2)−λ2γ^MMLE(1)(1−λ1+λ2)EEW1

EEW2=λ1γ^MMLE(2)−λ2γ^MMLE(1)+(1−λ1+λ2)(λ1γ^MMLE(1)−λ2γ^MMLE(0)+(1−λ1+λ2)EEW0)

Let a=(1−λ1+λ2)

EEW2=λ1γ^MMLE(2)−λ2γ^MMLE(1)+aλ1γ^MMLE(1)−aλ2γ^MMLE(0)+a2EEW0

EEW2=λ1γ^MMLE(2)+(aλ1−λ2)γ^MMLE(1)−aλ2γ^MMLE(0)+a2EEW0

Let b=(aλ1−λ2)

EEW2=λ1γ^MMLE(2)+bγ^MMLE(1)−aλ2γ^MMLE(0)+a2EEW0

EEW3=λ1γ^MMLE(3)+bγ^MMLE(2)+abγ^MMLE(1)−a2λ2γ^MMLE(0)+a3EEW0

EEW4=λ1γ^MMLE(4)+bγ^MMLE(3)+abγ^MMLE(2)−a2bλ2γ^MMLE(1)+a3λ2γ^MMLE(0)+a4EEW0

EEWt=λ1γ^MMLE(t)+bγ^MMLE(t−1)+abγ^MMLE(t−2)+a2bλ2γ^MMLE(t−3)+a3λ2γ^MMLE(0)+…+at−2bγ^MMLE(1)−at−1λ2γ^MMLE(0)+atEEW0

By taking expectation, we get

E(EEWt)=λ1γ+bγ+abγ+a2bλ2γ+a3λ2γ+…+at−2bγ−at−1λ2γ+atγ

Replacing b=(aλ1−λ2)

E(EEWt)=γ(λ1+(aλ1−λ2)+a(aλ1−λ2)+a2(aλ1−λ2)+…+ai−2a2(aλ1−λ2)−at−1λ2+at)

E(EEWt)=γ{λ1(1+a+a2+a3+…+at−1)−λ2(1+a+a2+a3+…+at−1)+at}

E(EEWt)=γ{λ1(1+a+a2+a3+…+at−1)−λ2(1+a+a2+a3+…+at−1)+at}

E(EEWt)=γ{(λ1−λ2)(1+a+a2+a3+…+at−1)+at}

By using geometric series, we get

E(EEWt)=γ{(λ1−λ2)(1−at1−a)+at}

So,

E(EEWt)=γ{1−at+at}

E(EEWt)=γ

Let Var(γ^MMLE(t))=υ=E(γ^MMLE−γ)2 , we get

Var(EEWt)=λ12ν+b2ν+a2b2ν+a4b2λ22ν+…+a2(t−2)b2ν−a2(t−1)λ22ν

Let b=(aλ1−λ2)

Var(EEWt)=ν{λ12+(aλ1−λ2)2+a2(aλ1−λ2)2+a4(aλ1−λ2)2+…+a2(t−2)(aλ1−λ2)2−a2(t−1)λ22}

Var(EEWt)=ν{λ12+a2λ12−2aλ1+λ22+a4λ12−2a3λ1+a2λ22+(a6λ12−2a5λ12+a4λ22)+…+a2(t−2)λ12−2a2t−3λ1+a2(t−2)λ22+a2(t−1)λ22}

Var(EEWt)=ν{λ12+a2λ12−2aλ1+λ22+a4λ12−2a3λ1+a2λ22+(a6λ14−2a5λ13+a4λ24)+…+a2(t−2)λ12−2a2t−3λ1+a2(t−2)λ22+a2(t−1)λ22}

Var(EEWt)=ν{λ12+a2λ12−2aλ1+λ22+a4λ12−2a3λ1+a2λ22+(a6λ14−2a5λ13+a4λ24)+…+a2(t−2)λ12−2a2t−3λ1+a2(t−2)λ22+a2(t−1)λ22}

Var(EEWt)=ν{λ12+a2λ12−2aλ1+λ22+a4λ12−2a3λ1+a2λ22+(a6λ14−2a5λ13+a4λ24)+…+a2(t−2)λ12−2a2t−3λ1+a2(t−2)λ22+a2(t−1)λ22}

Var(EEWt)=ν{λ12(1+a2+a4+….+a2(t−1))+λ22(1+a2+a4+….+a2(t−1))−2aλ1λ2(1+a2+a4+….+a2(t−2))}

Var(EEWt)=ν{(λ12+λ22)(1+a2+a4+….+a2(t−1))−2aλ1λ2(1+a2+a4+….+a2(t−2))}

Var(EEWt)=ν{(λ12+λ22)(1−a2t1−a2)−2aλ1λ2(1−a2t−21−a2)}

Var(EEWt)=ν{(λ12+λ22)(1−(1−λ1+λ2)2t1−(1−λ1+λ2)2)−2aλ1λ2(1−(1−λ1+λ2)2t−21−(1−λ1+λ2)2)}

The control limits are given as

UCL=γ0+Lν(λ12+λ22)(1−(1−λ1+λ2)2t1−(1−λ1+λ2)2)−2aλ1λ2(1−(1−λ1+λ2)2t−21−(1−λ1+λ2)2)

CL=γ0

LCL=γ0−Lν(λ12+λ22)(1−(1−λ1+λ2)2t1−(1−λ1+λ2)2)−2aλ1λ2(1−(1−λ1+λ2)2t−21−(1−λ1+λ2)2)

In order to see the working procedure of the proposed control charts, a simulation study was carried out. For this purpose, we generated 25 observations from a WPFD for in-control process, and the next 25 observations were generated from the shifted process with 0.01. The estimated values of the proposed EWMA statistic under MMLE were computed for the selected levels of the proposed control charts parameters with λ=0.20 and L = 16.975. The data and values of the proposed and existing statistic are listed in Tab. 4, and plotted values of these statistics are shown in Fig. 4. Further, the estimated values of the proposed EEWMA statistic under MMLE were computed for the selected levels of the proposed control charts parameters with λ1=0.20,λ2=0.05andL=17.245 . The data and values of the proposed and existing statistic are listed in Tab. 4, and plotted values of these statistics are shown in Fig. 5.

In Fig. 5, we noted that the proposed EEWMA control chart under MMLE detected a shift at the 32^th sample, while in Fig. 4; the EWMA control chart under MMLE could not detect the shift. Hence, this shows that the proposed EEWMA control chart under MMLE has a greater ability to detect smaller shifts earlier, as compared to the EWMA control chart.

We have discussed the process monitoring for WPFD. In real life, we may face the situation that any specific process does not follow the normal distribution. But the distribution of the errors in the process becomes WPFD. In the current work, we first estimate the parameters of the distribution of the errors during any process by using MMLE, which was claimed better to estimate the parameters of WPFD. By using the MMLE, we then modified the quality control charts used in literature, such as EWMA and EEWMA control charts. We see that EEWMA control chart under MMLE can be used to monitor the process when the underlying distribution of the errors in process monitoring follows WPFD. It is therefore hoped that the findings of this study will be useful for researchers in different fields of applied sciences. It will be helpful to identify the error in time and making strategies to deal with it. Companies can control their cost and improve product quality by applying the proposed quality chart processes.