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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.17 No.1 pp.82-90
DOI : https://doi.org/10.7232/iems.2018.17.1.082

Process Monitoring using Successive Sampling and a Repetitive Scheme

Asma Arshad, Muhammad Azam, Muhammad Aslam, Chi-Hyuck Jun*
Department of Statistics, Government College University, Lahore, Pakistan
Department of Statistics and Computer Science, University of Veterinary and Animal Sciences, Lahore, Pakistan
Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Industrial and Management Engineering, POSTECH, Pohang, Republic of Korea
Corresponding Author, chjun@postech.ac.kr
September 12, 2017 November 29, 2017 January 17, 2018

ABSTRACT


We propose a repetitive sampling based variable control chart to monitor the process mean by using successive sampling over two occasions. It is assumed that an auxiliary variable is available as well as the main quality characteristic of interest. The average run length (ARL) is derived, which is one of the most useful performance measures to evaluate the performance of a control chart. The chart parameters are determined for the targeted in-control ARL. The proposed control chart exhibits superior performance than a Shewhart-type control chart in terms of early detection of process mean shift. The strategy of the proposed control chart has been implemented on a real industry data.



초록


    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 out-of-control 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 t-chart.

    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 μY and variance EQ and that an auxiliary variable X having mean μX 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 μYusing 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 (n-m) unmatched units from the second occasion. Let xm and ym show the sample means of the matched units for X and Y variables, respectively. Let xu and yu be the sample means for the unmatched units for X and Y variables, respectively. Then, Mukhopadhyay (1998) proposed the following estimator ofμY:

    μ ^ Y = a x ¯ u + b x ¯ m   + c y ¯ m   + d y ¯ u
    (1)
    V a r ( μ ^ Y ) = a 2 [ a 2 ( m + u m u ) + c 2 m + ( ( 1 c ) 2 u ) 2 a c ρ m ]
    (2)

    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

    a = γ λ ρ 1 γ 2 ρ 2
    (3)
    c = λ 1 γ 2 ρ 2
    (4)

    Here λ = m n ,   γ = 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

    V a r ( μ ^ Y ) =   σ 2 n ( 1 γ ρ 2 ) ( 1 γ 2 ρ 2 )
    (5)

    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:

    γ o p t = 1 1 + 1 ρ 2
    (6)

    After inserting Eq. (6) into Eq. (5), we obtain

    V a r ( μ ^ Y ) = σ 2 2 n ( 1 + 1 ρ 2 )
    (7)

    μ ^ Y is an unbiased estimator of μY, and follows a normal distribution (Artes et al., 2001; Rueda et al., 2008; Kowalczyk, 2013) as follows:

    μ ^ Y ~ N [ μ Y , σ 2 2 n ( 1 + 1 ρ 2 ) ]
    (8)

    3. DESIGNING OF PROPOSED CONTROL CHART

    We propose a control chart based on successive sampling over two occasions under a repetitive scheme (called Rep-SS control chart). It is important to keep in view that:

    1. Successive sampling is a procedure to estimate μYusing an auxiliary variable.

    2. Repetitive sampling provides an additional chance to the estimator μ ^ 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 Rep-SS 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 (n-m) unmatched units from the second occasion. Let xm and ym be the sample means of the matched units for X and Y variables, respectively. Letxu and yu be the sample means for the unmatched units for X and Y variables, respectively.

    • Step 2: Compute the value of the estimator

      μ ^ Y = a x ¯ u + b x ¯ m + c y ¯ m + d y ¯ u   

      (The algorithm of determining a, b, c and d is given in Section 2.)

    • Step 3: Declare the process to be in-control if L C L 2 < μ ^ Y < U C L 2 and declare as out-ofcontrol if μ ^ Y > U C L 1 or μ ^ Y < L C L 1 . . Otherwise, go back to Step 1 and repeat process until the final decision is made.

    The outer control limits for the proposed Rep-SS chart will become:

    L C L 1 = μ 0 k 1 σ 2 2 n ( 1 + 1 ρ 2 )
    (9a)
    U C L 1 = μ 0 + k 1 σ 2 2 n ( 1 + 1 ρ 2 )
    (9b)

    Similarly, the inner control limits for the proposed Rep-SS chart are given by

    L C L 2 = μ 0 k 2 σ 2 2 n ( 1 + 1 ρ 2 )
    (10a)
    U C L 2 = μ 0 + k 2 σ 2 2 n ( 1 + 1 ρ 2 )
    (10b)

    The proposed Rep-SS control chart involves two constants k1 and k2, with the following generalizations:

    1. If k1 = k2 = k, then it becomes a Shewhart-type control chart (Montgomery, 2009).

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

    3. If k1 = k2 = 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 out-of-control when the process is actually in-control based on a single sample (denoted by P o u t , 1 0 ) is given as follows:

    P o u t , 1 0 = P { μ ^ Y < L C L 1 | μ 0 } + P { μ ^ Y > U C L 1 | μ 0 } = 2 [ 1 Φ( k 1 ) ]
    (11)

    Here Φ(.) is cumulative distribution function of the standard normal distribution. So, the probability of the process to be out-of-control ( P o u t 0 ) is given below:

    P o u t 0 = P o u t , 1 0 1 P r e p 0
    (12)

    Here, P r e p 0 is the probability that the chart undergoes with the repetitions, which is computed as follows:

    P r e p 0 = P { U C L 2 < μ ^ Y < U C L 1 | μ 0 } + P { L C L 1 < μ ^ Y < L C L 2 | μ 0 } = 2 [ Φ ( k 1 ) Φ ( k 2 ) ]
    (13)

    Superscript ‘0’ in all above equations identifies the in-control state of the process. Later when superscript ‘1’ is used, it stands for the out-of-control state of the process. The in-control ARL, denoted by ARL0, is obtained by

    A R L 0 =   1 P o u t 0
    (14)

    The running process may be shifted due to some uncontrolled factors, then the process mean has been shifted from μ0 to μ1 =μ0 +σf ; 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 o u t , 1 1 ) given below:

    P o u t , 1 1 = P { μ ^ Y < L C L 1 | μ 1 } + P { μ ^ Y > U C L 1 | μ 1 } = Φ ( k 1 f 1 2 n ( 1 + 1 ρ 2 ) ) + 1 Φ  ( k 1 f 1 2 n ( 1 + 1 ρ 2 ) )
    (15)

    Similarly, the probability of repetitions in a shifted process ( P r e p 1 ) has been calculated as:

    P r e p 1 = P { U C L 2   < μ ^ Y < U C L 1 | μ 1 } + P { L C L 1   < μ ^ Y < L C L 2 | μ 1 }
    (16)
    = Φ ( k 1 f 1 2 n ( 1 + 1 ρ 2 ) )  Φ  ( k 2 f 1 2 n  ( 1 + 1 ρ 2 ) ) + Φ  ( k 2 f 1 2 n ( 1 + 1 ρ 2 ) ) Φ  ( k 1 f 1 2 n ( 1 + 1 ρ 2 ) )
    (17)

    Hence, probability for being declared as the out-ofcontrol process ( P o u t 1 ) is as follows:

    P o u t 1 = P o u t , 1 1 1 P r e p 1
    (18)

    So, the out-of-control ARL is given by

    A R L 1 =   1 P o u t 1
    (19)

    Let r0 be the targeted in-control ARL.Then, the control constants k1 and k2 where k2 < k1, can be determined such that ARL0r0 though a search procedure.

    3.2 Performance of Proposed Chart

    When r0 is 500 or 300, ARL1 values are obtained according to shift sizes ranging from 0.0-1.0. The results are tabulated in Table 1~Table 4 through the search procedure. R-code is developed to determine constant values k1 and k2 and to get ARL1 for various shift sizes and other parameters.Table 2, Table 3

    From the tables the following trend is observed:

    When r0 value is fixed, ARL1 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 ARL1 = 90.43, for f = 0.5 the ARL1 = 6.65 and for f = 0.8 the ARL1 = 3.24 and for n = 30 the ARL1 = 18.39 at f = 0.2 and for f = 0.5 he ARL1 =1.33 and for f = 0.8 the ARL1 =1.00.

    For fixed value of ρ and r0, the ARL1 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 ARL1 = 57.55, for n = 30 the ARL1 = 9.44 and for n = 60 the ARL1 = 3.13 and for f = 0.5 the ARL1 = 17.34 at n = 5 and for n = 30 the ARL1 = 1.33 and for n = 60 the ARL1 = 1.02.

    4. PERFORMANCE COMPARISON

    The proposed Rep-SS control chart has been compared in terms of the ARL with a Shewhart-type 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 Rep-SS chart when r0= 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 Rep-SS chart shows smaller ARL values than the other existing ones. Here it is important to notice that Rep-SS 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 Rep-SS 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 k1 = 3.0949 and k2 = 2.3999 for the target in-control ARL of r0 = 500. The outer pair of control limits for the proposed Rep-SS chart is UCL1 = 2.9271 and LCL1 = 2.7471 and the inner pair of control limits is UCL2 = 2.9069 and LCL2 = 2.7673. The two pairs of control limits for Rep chart are UCL_EX1 = 3.0998 and LCL_EX1 = 2.5739 and UCL_EX2 = 3.0408 and LCL_EX2 = 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 Rep-SS chart declares that the process has gone to out of-control. Advantage of applying Rep-SS 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 Rep-SS chart and Rep chart when r0 = 500 with n = 86 for milking yield dataset.

    This again shows the outperformance of the proposed Rep-SS 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 SS-CC, Rep-CC and SH-CC 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.

    ACKNOWLEDGEMENTS

    This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, Muhammad Aslam, therefore, acknowledge with thanks DSR technical and financial support. The work by Chi-Hyuck Jun was supported by a research program from Samsung Electronics Co.

    Figure

    IEMS-17-82_F1.gif

    ARL comparison when r0 = 300 and n = 5.

    IEMS-17-82_F2.gif

    The Rep-SS monitoring chart for milk yield data.

    Table

    The average run length of Rep-SS chart when n = 5 and, r0 = 500 or 300

    The average run length of Rep-SS chart when n = 30 and, r0 = 500 or 300

    The average run length of Rep-SS chart when n = 60 and, r0 = 500 or 300

    The average run length of Rep-SS chart when ρ = 0.00 and, r0 = 500 or 300

    The average run length for SS chart and SH chart when r0 = 300

    The average run length for Rep-SS chart and rep chart having r0 = 500 with n = 86 (for milking yield dataset).

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