1. INTRODUCTION
Control charts are applied to monitor quality levels of a production line and become a significant tool for reducing the variability in process parameters to meet consumer needs up to the satisfaction without going through the costly inspection procedures. Control charts are key factors in this regard to get a high level of customers’ satisfaction (Kenett and Zacks, 1998). A control chart is comprised of two control limits: the upper control limit (UCL) and the lower control limit (LCL). If the plotted statistic points are beyond the UCL or LCL, then the running process is characterized in an outofcontrol state (Montgomery, 2009). In such a situation, an efficient control chart must alarm instantly to provide timely indication when the process is in fact shifted. Hence it becomes convenient to take necessary action for bringing back the shifted process towards the normal state (Grant and Leavenworth, 1996).
In statistical process control (SPC), Sherman (1965) for the first time applied repetitive sampling in an acceptance sampling plan and found it more efficient in terms of sample size required than the single sampling plan. Later, Balamurali and Jun (2006), Aslam et al. (2013a, 2013b), Lee et al. (2012) have successfully applied repetitive sampling to other types of sampling plans and found it better than previously designed acceptance sampling plans. The successful application of repetitive sampling concept in the field of control chart has been made possible by Ahmad et al. (2014), where a repetitive X bar chart using process capability index is proposed and shown to be more efficient than the existing concepts. Azam et al. (2014) introduced a hybrid exponentially weighted moving average control chart utilizing the repetitive sampling scheme. Aslam et al. (2014a), developed a repetitive sampling based exponentially weighted moving average sign chart and again Aslam et al. (2014b) proposed a repetitive sampling tchart.
The idea of repetitive sampling based control chart is entirely different in its methodology from double, triple and sequential sampling based charts. Moreover, it is simple, clear and easy to implement in its operational mechanism. If the plotted statistic falls within the inner side of the control limits, then the process is declared as in control. However, if it is beyond the outer control limits then it is nominated as an out of control. If the first sample will not provide a decision, then there is a need to repeat the process until the final decision is made.
Successive sampling (SS) approach has been developed in survey sampling for the population affected over time (Gordon, 1983). The procedure of sample selection over successive occasions of time for the characteristic of interest is known as successive, multiple or rotation sampling scheme. Jessen (1942) was the first one who introduced the idea of partial replacement of observations over two successive occasions in survey sampling, to estimate the average value of the main quality characteristic. The successive sampling concept is further extended by Patterson (1950), Tikkiwal (1960, 1967), Eckler (1955) and Rao and Graham (1964). The theory of estimating the population mean in survey sampling with partial replacement (SPR) is adequately developed by many researchers, for instance, Singh and Singh (1965), Singh (1968), Sen (1971), Chhikara and Deng (1992), Singh and Singh (2001), Artes et al. (2001), Azam et al. (2010), Luengo and Casado (2010), Kumar et al. (2010), Singh et al. (2011), Kumar (2012), Singh et al. (2013) and Luengo (2014).
2. OPERATIONAL PROCEDURE OF SUCCESSIVE SAMPLING OVER TWO OCCASIONS
Suppose that the main quality characteristic of interest is Y with mean μ_{Y} and variance EQ and that an auxiliary variable X having mean μ_{X} and variance EQ is also measured from sampling. The correlation coefficient between Y and X is denoted by ρ. It is assumed that EQ for simplicity. It is also to be noted that our main variable of interest to be monitored here is the Y variable through discovered information of the X variable. Here, we would like to improve the efficiency of the estimator of μ_{Y}using successive sampling over two occasions.
We draw two samples of size n each at the first and the second occasions. Suppose that there are m common (called matched) units from two occasions so that there are u (nm) unmatched units from the second occasion. Let x_{m} and y_{m} show the sample means of the matched units for X and Y variables, respectively. Let x_{u} and y_{u} be the sample means for the unmatched units for X and Y variables, respectively. Then, Mukhopadhyay (1998) proposed the following estimator ofμ_{Y}:
Here a, b, c and d are constants with the assumption that a + b = 0 and c + d =1 for the unbiasedness of EQ . The variance of EQ is
EQ
Differentiating Eq. (2) with respect to a and c, then equating it zero, we get
Here $\lambda =\frac{m}{n},\hspace{0.17em}\gamma =\frac{u}{n}$ are the proportions for matched and unmatched units correspondingly, such that γ +λ =1 . By substituting values of a and c in Eq. (2), we obtain
By differentiating Eq. (5) with respect to γ and equating it zero, we get the optimum value of the proportion for unmatched units as follows:
After inserting Eq. (6) into Eq. (5), we obtain
${\widehat{\mu}}_{Y}$ is an unbiased estimator of μ_{Y}, and follows a normal distribution (Artes et al., 2001; Rueda et al., 2008; Kowalczyk, 2013) as follows:
3. DESIGNING OF PROPOSED CONTROL CHART
We propose a control chart based on successive sampling over two occasions under a repetitive scheme (called RepSS control chart). It is important to keep in view that:

Successive sampling is a procedure to estimate μ_{Y}using an auxiliary variable.

Repetitive sampling provides an additional chance to the estimator ${\widehat{\mu}}_{Y}$ if faced with ambiguity on the decision. Two pairs of control limits (called outer and inner control limits) are utilized in this purpose.
The operational procedure of the proposed RepSS control chart is as follows:

Step 1: We draw two samples of size n each at the first and the second occasions. Suppose that there are m common (called matched) units from two occasions so that there are u (nm) unmatched units from the second occasion. Let x_{m} and y_{m} be the sample means of the matched units for X and Y variables, respectively. Letx_{u} and y_{u} be the sample means for the unmatched units for X and Y variables, respectively.

Step 2: Compute the value of the estimator
$${\widehat{\mu}}_{Y}=a{\overline{x}}_{u}+b{\overline{x}}_{m}+c{\overline{y}}_{m}+d{\overline{y}}_{u\text{}}$$(The algorithm of determining a, b, c and d is given in Section 2.)

Step 3: Declare the process to be incontrol if $LC{L}_{2}<{\widehat{\mu}}_{Y}<UC{L}_{2}$ and declare as outofcontrol if ${\widehat{\mu}}_{Y}>UC{L}_{1}$ or ${\widehat{\mu}}_{Y}<LC{L}_{1}.$ . Otherwise, go back to Step 1 and repeat process until the final decision is made.
The outer control limits for the proposed RepSS chart will become:
Similarly, the inner control limits for the proposed RepSS chart are given by
The proposed RepSS control chart involves two constants k_{1} and k_{2}, with the following generalizations:

If k_{1} = k_{2} = k, then it becomes a Shewharttype control chart (Montgomery, 2009).

If ρ = 0, then it reduces to a control chart using repetitive sampling.

If k_{1} = k_{2} = k, and ρ = 0, then it reduces to the traditional Shewhart chart.
3.1 ARL of The Proposed Control Chart
The probability with which the process is identified as outofcontrol when the process is actually incontrol based on a single sample (denoted by ${P}_{out,1}^{0}$ ) is given as follows:
Here Φ(.) is cumulative distribution function of the standard normal distribution. So, the probability of the process to be outofcontrol $({P}_{out}^{0})$ is given below:
Here, ${P}_{rep}^{0}$ is the probability that the chart undergoes with the repetitions, which is computed as follows:
Superscript ‘0’ in all above equations identifies the incontrol state of the process. Later when superscript ‘1’ is used, it stands for the outofcontrol state of the process. The incontrol ARL, denoted by ARL_{0}, is obtained by
The running process may be shifted due to some uncontrolled factors, then the process mean has been shifted from μ_{0} to μ_{1} =μ_{0} +σ_{f} ; where f is a shift constant related to the process variation in terms of standard deviation. The probability that the process is identified as outof control on the basis of single sample (denoted as ${P}_{out,1}^{1}$ ) given below:
Similarly, the probability of repetitions in a shifted process (${P}_{rep}^{1}$ ) has been calculated as:
Hence, probability for being declared as the outofcontrol process (${P}_{out}^{1}$ ) is as follows:
So, the outofcontrol ARL is given by
Let r_{0} be the targeted incontrol ARL.Then, the control constants k_{1} and k_{2} where k_{2} < k_{1}, can be determined such that ARL_{0} ≥r_{0} though a search procedure.
3.2 Performance of Proposed Chart
When r_{0} is 500 or 300, ARL_{1} values are obtained according to shift sizes ranging from 0.01.0. The results are tabulated in Table 1~Table 4 through the search procedure. Rcode is developed to determine constant values k_{1} and k_{2} and to get ARL_{1} for various shift sizes and other parameters.Table 2, Table 3
From the tables the following trend is observed:
When r_{0} value is fixed, ARL_{1} decreases as f increases. The overall behavior of the ARL values depict the decreasing trend as shift size increases and this decrease becomes drastic or rapid in its declining trend as the sample size increases for 5, 30 and then 60. For example, for ARL = 300 and ρ = 0.90 with f = 0.20 of n = 5 the proposed chart gives ARL_{1} = 90.43, for f = 0.5 the ARL_{1} = 6.65 and for f = 0.8 the ARL_{1} = 3.24 and for n = 30 the ARL_{1} = 18.39 at f = 0.2 and for f = 0.5 he ARL_{1} =1.33 and for f = 0.8 the ARL_{1} =1.00.
For fixed value of ρ and r_{0}, the ARL_{1} exhibits an increasing trend for sample sizes 5, 30 and then 60. For example, for ARL = 300 and ρ = 0.90 with f = 0.25 at n = 5, the proposed chart gives ARL_{1} = 57.55, for n = 30 the ARL_{1} = 9.44 and for n = 60 the ARL_{1} = 3.13 and for f = 0.5 the ARL_{1} = 17.34 at n = 5 and for n = 30 the ARL_{1} = 1.33 and for n = 60 the ARL_{1} = 1.02.
4. PERFORMANCE COMPARISON
The proposed RepSS control chart has been compared in terms of the ARL with a Shewharttype control chart using SS (called SS chart) proposed by Azam et al. (2016), the traditional Shewhart control chart (called SH chart) and the control chart using repetitive sampling (called Rep chart).
Figure 1 shows ARL trend of the proposed RepSS chart when r_{0}= 300 and ρ = 0.90 (Table 1) according to the shift size and compares with existing Rep chart with ρ = 0.00 (Table 4), SS chart with ρ = 0.90 (Table 5) and SH chart (Table 5). The proposed RepSS chart shows smaller ARL values than the other existing ones. Here it is important to notice that RepSS chart exhibits better results with existing SS chart as focus is to make sure precise results without any flaw. It can be clear in discussion with real life dataset shown in Figure 2.
5. INDUSTRIAL IMPLEMENTATION
The milking yield dataset with N = 173 cows (animals are tagged) is used for the construction of the proposed RepSS chart, where a high degree of correlation (known with prior knowledge) for two successive occasions of Morning and Evening, each of 30 days (June, 2014). A sample of size n = 86 has been taken amongst which matched number of units is m = 8 and unmatched is u = 78 with correlation ρ = 0.99. Due to high value of correlation, counts of matched units is smaller. The values of two control constants have been selected as k_{1} = 3.0949 and k_{2} = 2.3999 for the target incontrol ARL of r_{0} = 500. The outer pair of control limits for the proposed RepSS chart is UCL_{1} = 2.9271 and LCL_{1} = 2.7471 and the inner pair of control limits is UCL_{2} = 2.9069 and LCL_{2} = 2.7673. The two pairs of control limits for Rep chart are UCL_EX_{1} = 3.0998 and LCL_EX_{1} = 2.5739 and UCL_EX_{2} = 3.0408 and LCL_EX_{2} = 2.6330, which are calculated for the average of 30 days data (as prior knowledge), taking means of the respective two occasions. These control limits are shown in Figure 2.
It is obvious from Figure 2 that existing chart declares the whole process as in control with no such existing assignable cause to be treated with a significant action. But RepSS chart declares that the process has gone to out ofcontrol. Advantage of applying RepSS is obvious as the data exists in bulk for the respective milk production on daily basis over the two succeeding occasions for number of months, then any sort of departure from routine usual production can get easy identification to peep into the relevant date dataset for sorting out the issue.
Table 6 shows the ARLs for RepSS chart and Rep chart when r_{0} = 500 with n = 86 for milking yield dataset.
This again shows the outperformance of the proposed RepSS control chart over the Rep control chart.
6. CONCLUSION
Here, we presented a new mean monitoring variable control chart based on successive sampling over two occasions using repetitive sampling scheme with the assumption of normality. The results with specified parameters through simulation study are presented in extensive tables and comparison has been made for the performance evaluation with existing charts like SSCC, RepCC and SHCC in terms of ARL. It is found far efficient in detecting both small and large shift sizes for both small and large sample sizes, whatever the degree of correlation exists between the occasions. This is because our repetitive scheme provides one more chance for the decision when the information from the previous sample may not be clear to make a decision.
The respective study can be further developed with other sampling procedures like multiple dependent state, resubmitted, resubmitted repetitive sampling and other successive sampling over multiple occasions.