1.INTRODUCTION
Quality monitoring and evaluation of the final products to get customer’s satisfaction is the actual task in a production process. In this regard statistical process control (SPC) is the only field with collection of most widely used tools to deal with such a kind of issues of process monitoring and to save the producer with an increased number of nonconforming items which cannot be tolerated in anyway. Thus control charts are one of the most significant and sophisticated amongst all of them to provide process stability and capability of products quality especially at the manufacturing stage.
Concept of control chart was first developed in Bell Telephone Laboratories by Dr. Walter A. Shewhart in 1920 to examine process and improve the quality of its products by intime detection at outofcontrol signal (Montgomery, 2009). Inspite of its advantages it was not sensitive in detecting small shifts from mean values in the running production. To overcome this difficulty, a lot of work has been done to get relatively more sensitive control charts for detecting small and even smaller process mean shifts, like Roberts (1966) has introduced exponentially weighted moving average (EWMA) control chart which is nowadays most sophisticatedly used and significantly accepted in detection of relatively small shifts and it is possible because EWMA statistic uses collectively current and past information when quality of interest follows normal distribution (Abbasi, 2010; Lucas and Succaci, 1990; Woodall, 2006).
But in actual facts, it is hard to get normally distributed underlying variable of interest. As Gan (1990) modified EWMA control chart when count data of nonconformities follows Poisson distribution. Latter on Borror et al. (1998) developed Poisson EWMA control chart, more sophisticated than Gan (1990) in minimizing the ARL of shifted process. In 1972, Jackson explained that Poisson is not the only underlying distribution for count data, more over for each count of such event intensity can vary in its frequency (clusters) and can approach some other distribution over time (Rice, 1995). Thus compound Poisson distribution model is an adequate for such data (Chen, 2012; Gospodinov and Rotondi, 2001; Hoffman, 2003; Yu et al., 2011) and can be adequately monitored through attribute control chart.
The number of defects over time follows Geometric Poisson Distribution which regards as a native extension of the Poisson count data where each count follows Geometric distribution behavior i.e. US Air Force bases data analyzed with the help of Geometric Poisson distribution (Chen et al., 2005). Sahinoglu (1992) monitored software failures, Robin (2002), Robin et al. (2007) model overlapping word occurrences through Geometric Poisson distribution, similarly Rosychuk et al. (2006) and Ozel and Inal (2010) modeled DNA substitution with total number of fatalities for accidents. Whereas in quality control chart Chen et al. (2005) for the first time introduced this concept of Geometric Poisson (GP) distribution by developing Geometric Poisson CUSUM control chart and after this Geometric Poisson EWMA control chart for efficient monitoring of defects over time than Poisson distribution (Chen, 2012).
Above discussed control charts are developed by using single sampling scheme. There are some other sampling schemes amongst which repetitive sampling is the most widely used one to increase the efficiency of control charts in terms of ARLs (Ahmad et al., 2014; Ahmad et al., 2016; Aslam et al., 2013; Aslam et al., 2015) for incontrol and outofcontrol circumstances. The contribution of presented paper is to construct the mean monitoring control chart by using repetitive (Rep) sampling scheme to monitor the process variation when it undergoes with GP characteristics. Similar to the sequential sampling in its operational procedure and sampling remain continue until the decision is made for acceptance or rejection of a lot (Balamurali and Jun, 2006; Sherman, 1965). Nowadays Rep sampling concept has gained a significant place in acceptance sampling plans of statistical process control by reducing ARL values. As Rep sampling concept was first time introduced by Sherman (1965) to construct more efficient acceptance sampling plans, whereas in control charts repetitive sampling approach was utilized by Ahmad et al. (2014) and Aslam et al. (2013) and proved an efficient one in detecting shifts.
Next upcoming sections of the manuscript are arranged as follows: in section 2; we propose designing of control chart based on Geometric Poisson distribution exponentially weighted moving average using Repetitive sampling approach. In section 3, we demonstrate that propose control chart is more efficient in terms of ARL and SDRL values than existing one proposed by Chen (2012). In section 4, advantages of proposed control chart along with an industrial application and simulation based example. These examples will demonstrate the strength of efficiency and need of developing the proposed concept. The section 5, as last section will comprise of concluding remarks and recommendations.
2.DESIGNING OF PROPOSED REPGPEWMA CONTROL CHART
We first introduce the Geometric Poisson distribution proposed by Chen (2012) in Section 2.1 and propose a new EWMA control chart for the Geometric Poisson distributed quality characteristic using repetitive sampling to monitor the process mean. The proposed control chart is referred to as RepGPEWMA chart.
2.1Geometric Poisson Distribution
Chen (2012) has suggested that an item having more than one defect in production is declared to be defective. In production lines, defect occurs randomly over time and such occurrence of defective items follows a random process. Let Y be a random variable showing the number of defective items by a specific time and D_{i} be a the number of defects from the ith defective item. Then, the total number of defects by the specific time, denoted by X, is given by(1)
Suppose that Y follows a Poisson distribution with mean λ and D_{i}’s are independent and identically distributed (iid) as geometric with probability ρ. Then, X is known to follow a geometric Poisson (GP) compound distribution having pdf of(2a)(2b)
The mean and the variance of X are obtained by (Chen et al., 2005)
If a variable of interest follows a geometric Poisson distribution, then it is not appropriate to handle such dataset with Poisson distribution. But, in reality, items are declared as defective with a different number of defects. Let us consider a following example. Suppose that a pen is comprised of many parts like pencap, outer body, inlet ink jacket, and so on. All these parts are separately prepared in different production sections and then assembled as a single pen. This single item ‘pen’ may be declared as a defective item with one, two or more defects of pencap, inletink jacket or others. In such situation this variation in the number of defects for each item (i.e. pen) may follows a GP distribution.
2.2.Designing of Proposed Control Chart
Let x_{1}, x_{2} , x_{3} ,… be a sequence of the observed number of defects from samples (or subgroups). It is assumed that the number of defects is iid as geometric Poisson. Then, the EWMA statistic at the ith subgroup is given by (Montgomery, 2009)
where w is a smoothing constant ( 0 < w ≤ 1 ). The observation x_{i} shows the current information and M_{i−1} shows the past information in its respective weight. It is known that the statistic M_{i} is quite robust to distributional assumption (Montgomery, 2009; Saghir et al., 2015). It is also known that a EWMA statistic based control chart provides an ideal circumstance for monitoring more efficiently and precisely detecting small/ moderate process shifts. The initial value of the statistic M_{i} is set to the target mean value to keep a process in an incontrol state.
Chen (2012) proposed a EWMA control chart for the GP distribution. He used various combinations of the parameters w, ρ, λ and obtained desired incontrol ARL values by using a Markov chain approach. Saghir et al. (2015) proposed a EWMA control chart for the GP distribution with estimated parameters. In this paper, we propose a new EWMA control chart by adopting repetitive sampling in its methodology. We will be obtaining the ARLs by a Monte Carlo approach. Here, it is important that parameters are possessed with the interaction effect among them. Firstly, the choice of w completely depends on how fast one wants a mean shift of given size should be noticed. The smaller values of w are generally preferred in rapid detection of smaller mean shifts and vice versa. Secondly, the ρ parameter has a significant effect on the efficiency of the given chart. It is expected that the larger the ρ parameter value, the less is the effect on ARL if the λ parameter is fixed. Due to the smaller ρ value with fixed value of the λ parameter it will converge to a Poisson distribution. But more the λ parameter value with fixed value of ρ, ARL value is less influenced to yield the performance of the concept. Thus, choice of ρ significantly effects the performance of geometricPoisson EWMA chart. Thirdly, the larger λ parameter value does not merge the GP distribution concept with the Poisson distribution. Hence the shift parameter f is incorporated with GP distribution in this paper.
The proposed control chart, called the RepGPEWMA chart, possesses the following two pairs of outer and inner control limits with control constants k_{1} and k_{2} ( k_{1} > k_{2} ).
The outer control limits are given by(4)(5)
The inner control limits are given by(6)(7)
If the value of the control chart constants k_{1} and k_{2} has been fixed to k , then the proposed RepGPEWMA control chart becomes the EWMA control chart by Chen (2012) as a special case. The value of LCL’s may be set to zero because quality characteristic of interest is a nonnegative GP variable.
The procedure is explained in follows steps:

Step 1: Draw a random sample of size 1 at time t and count the number of defects x_{t}.

Step 2: Calculation of M_{t} is made at time t from (3) by considering M0 =λ and then M_{t} is plotted on the control chart.

Step 3: Declare the process to be in control if UCL_{2} ≤ M_{t} ≤ LCL_{2} . Declare the process to be outof control if plotted M_{t} ≥UCL_{1} or M_{t} ≤ LCL_{1} . Otherwise, Steps 1 and 2 are repeated until the final decision is made.
The incontrol and the outofcontrol ARLs will be evaluated by a Monte Carlo approach. In our Monte Carlo simulation, we assumed that the true distribution for incontrol process is GP ( ρ , λ ) A random sample of some suitable size is selected to assign parametric values of {ρ, λ, wk_{1}k_{2}} and for the computation of control chart limits UCL_{1}, UCL_{2}LCL_{1}, LCL_{2} . In this study, the parameter values are specified as λ = 2, 3, w = 0.2, 0.1, 0.05 and ρ = 0.20, 0.30 . To evaluate the outofcontrol ARL, the shift constant f is introduced in λ as λ + f.
The operational steps in each simulation run can be described as follows:

Step 1: Assign values of {ρ , λ , w, k_{1}, k_{2}} and compute the limits UCL_{1}, UCL_{2} , LCL_{1}, LCL_{2}. This step will be repeated until the ARL value close to the target value r_{0}.

Step 2: Calculate the M_{t} by using EWMA and generate dataset from GP ( ρ , λ + f ).

Step 3: By means of the 500,000 simulation runs, alarming state is declared if M_{t} ≥UCL_{1} or M_{t} ≤ LCL_{1} , then repeat Step 2 until the final decision.

Step 4: Record the run lengths and then compute ARL and the SDRL values.
Tables 14 show the ARL values according to the shift constant for various parameters. These tables contain the results for the existing EWMA chart for GP distribution without using repetitive sampling (called GPEWMA chart). We will report the comparison results separately in Section 3. Tables 23
Here is a detail discussion about the obtained results and the trend shown by ARL and SDRL values mentioned in Tables 14.
For f = 0 , in all discussed parametric cases ARL_{1} ≅ r_{0} .
As the process goes outofcontrol (f > 0) , the ARL_{1} and SDRL rapidly decreases in all discussed parametric schemes and this decrease becomes significant as shift reached 0.05.
The proposed method efficiently detects very small, moderate and to a certain situation.
The control constraints k_{1} and k_{2} have values 3.22 and 2.22 for fixed value of r_{0} = 500 in all mentioned parametric cases. For r_{0} = 300 , values becomes 3.00 and 1.50 respectively.
Generally, there can be more gain in performance for the proposed control chart by using high value of λ and ρ .
As ρ increases from 0.2 to 0.3 for a fixed value of λ = 2 and w = 0.1 then ARL value for f =1.25 increases from 15.54 to 17.47 from Tables 1 and 2.
As λ increases from 2 to 3 for a fixed value of ρ = 0.2 and w = 0.1 then ARL value for f =1.25 decreases from 15.54 to 10.82 from Tables 1 and 3.
Similarly, as w decreases from 0.2 to 0.1 and then 0.05 for a fixed value of ρ = 0.2 and λ = 2 or 3 then ARL value for f =1.25 increases and becomes 14.36, 15.54 and 19.54 or 9.64, 10.82 and 14.00 from Tables 1 and 3.
It is far important to note that value of shift f ranges from 0.25~3.00, which is relatively a smaller addition in GP parameter λ , taking ρ constant throughout the process. Hence it can be concluded that the proposed control chart is far better in its performance in detecting such a smaller shift.
3.COMPARISON OF THE PROPOSED CONTROL CHART
In this section, we will compare the proposed control chart performance, in terms of ARL and SDRL as shift occurs, with GP distribution EWMA based control chart (GPEWMA) by Chen (2012). As mentioned earlier, the comparison results are reported in Tables 14. For same parametric values proposed RepGPEWMA control chart performs better in terms of ARL and SDRL. Moreover, it provides better detection than GPEWMA control chart. . It is obvious from Table 1 that when r_{0} = 500 ,λ = 2 and ρ = 0.20 the ARL for the proposed chart becomes 70.86 for w = 0.2, 77.37 for w = 0.1 and 84.00 for w = 0.05 ; while the existing control chart shows higher ARL values as 75.53 for w = 0.2 , 86.89 for w = 0.1 and 96.08 for w = 0.05 If ρ = 0.30 for the same λ = 2, then at f = 0.50 ARL value for the proposed chart becomes higher than ρ = 0.20 , which are 73.77 for w = 0.2 82.68 for w = 0.1 and 91.42 for w = 0.05
4.ILLUSTRATIVE EXAMPLES
In Section 4.1, a simulation study has been conducted to prove the strength of the proposed chart in terms of outofcontrol ARLs. Section 4.2 has demonstrated the application of the proposed chart to a real life industrial dataset.
4.1.Simulation Dataset
In this section efficiency of the proposed chart is demonstrated by using a simulated dataset. A data of 10 subgroups is randomly generated from the incontrol process state by assuming to follow a GP distribution with parameters λ = 2 and ρ = 0.3. The proposed GPEWMA statistic M_{t} is constructed with a smoothing constant w = 0.1. For an outofcontrol state, 15 more subgroups of samples are generated by introducing shift of f =1.00 in λ asλ + f with ARL_{0} = 500. Simulated dataset is mentioned in Table 5 for the construction of RepGPEWMA.
The respective monitoring limits with k_{1} = 3.22 and k_{2} = 2.22 becomes as follows (operational procedure is followed from Section 2):
The computed M_{t} values are plotted and the proposed chart detects the shift at the 25^{th} subgroup. It must be noticed that as process was shifted from 11th sample, it boosts and came into the region of indecisive state. It shows a trend that some assignable causes are incorporating the normal state. Then 25^{th} sample value exhibits a final rejection (Figure 1). This same procedure is repeated again for the construction of the existing GPEWMA chart with k = 3.22. It is seen that the existing GPEWMA chart does not detect the shift until the 25^{th} subgroup (Figure 2). So it is the success of proposed concept towards the strength of using a repetitive sampling scheme to obtain efficient control charts and can be significantly help out industries in saving from the disasters.
4.2.Industrial Dataset Application
In this section, we will provide a practical implementation of the proposed control chart on dataset came from a GP distribution taken from Chen (1999). According to him a particular state item is regarded to be following GP when the number of defectives is denoted as y, with a number of defects per item which is nominated as x (Table 6). Hence such a dataset can never fulfill the properties of the Poisson distribution. Here, the second column represents the number of defectives and the third column shows the total number of defects from the defectives found.
The dataset of first 20 failures (nonconformities) is generated by taking process mean equal to 2.5 and for the next 20 observations he used process mean value equal to 4.0 with a mean shift of 0.77σChen (1999). The operational methodology of the proposed control chart act as follows:
As the parameters are unknown so the method of moments estimates (Chen et al., 2005) is used for the
where $\overline{X}=\frac{{\displaystyle {\sum}_{i=1}^{m}{X}_{i}}}{m}=3.1$ and ${S}^{2}=\frac{{\displaystyle {\sum}_{i=1}^{m}{\left({X}_{i}\overline{X}\right)}^{2}}}{m1}=5.9384$ are mean and variance of selected sample dataset respectively for m = 40 sample sizes (S^{2} > X ) and value of dispersion, ρ > 0. with average count of defects based on moment estimate is ${\widehat{\mu}}_{0}=\overline{X}={M}_{0}$ for w = 0.1 Now these estimates are used for the calculation of control limits as follows:
LCL_{1} = 1.42, LCL_{2} = 2.26, UCL_{1} = 4.77 and UCL_{2} = 3.93.Center limit CL = 3.1with k_{1} = 3.00 and k_{2} =1.50 where k_{1} > k_{2}. And for GPEWMA control chart k = 3.00 with LCL = 1.42 and UCL = 4.78.
It can be seen from Figure 3; proposed RepGPEWMA control chart better detects outofcontrol phenomena as compare to the existing GPEWMA control chart (see Figure 4) based on single sampling scheme which shows that the process is completely in an incontrol state. In a very strengthen way Figure 3 determines that the proposed control chart is effective and significantly highlight alarmed cases to be as an outofcontrol. Whereas Figure 4 determines that the running process is completely under control, inspite of possessing alarming bugs. So the application of proposed control chart in this alarming situation is need of time and it can be easily assessed and judged by comparison with existing control chart.
5.CONCLUDING REMARKS AND RECOMMENDATIONS
In this article, we have proposed repetitive sampling based Geometric Poisson EWMA control chart (Rep GPEWMACC). We evaluated the performance of the proposed control chart in terms of ARL and SDRL, without increasing the false alarm rate as shift occurred. The efficiency of the proposed concept is assessed and explained through provided extensive tables. Comparison was made with the existing EWMA based Geometric Poisson distribution control chart (GPEWMACC) and proposed control chart proved better. The overall performance was indetail explored by the industrial example came from Poisson environment. At the end it is recommended to apply this concept for monitoring significantly the defects as per defective item, and its scope can be extended to other control chart schemes.