A New Attribute Control Chart Using Multiple Dependent ... - IEEE Xplore

Received February 16, 2017; accepted March 22, 2017, date of publication March 24, 2017, date of current version May 17, 2017. Digital Object Identifier 10.1109/ACCESS.2017.2687523

A New Attribute Control Chart Using Multiple Dependent State Repetitive Sampling MANSOUR SATTAM ALDOSARI1 , MUHAMMAD ASLAM1 , AND CHI-HYUCK JUN2 1 Department 2 Department

of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia of Industrial and Management Engineering, POSTECH, Pohang 37673, South Korea

Corresponding author: M. Aslam ([email protected]) This work was funded by the King Abdulaziz City for Science and Technology under Grant PS-37-1923.

ABSTRACT In this paper, a new attribute control chart using multiple-dependent state repetitive sampling is designed. The operational procedure and structure of the proposed control chart is given. The required measures to determine the average run length for in-control and out-of-control processes are given. Tables of ARLs are reported for various control chart parameters. The proposed control chart is more sensitive in detecting a small shift in the process as compared with the existing attribute control charts. The simulation study shows the efficiency of the proposed chart over the existing charts. An example is given for illustration purpose. INDEX TERMS Control chart, multiple dependent state repetitive sampling, average run length, binomial distribution, average run length.

I. INTRODUCTION

Product quality has become one of the most important issues that distinguish commodities in a global business market. Two important techniques for ensuring quality are statistical process control charts and acceptance sampling plans [1] . The control charts can be used in education, pharmaceutical industry, nuclear engineering, and analytical laboratories as well as in manufacturing [2]. It helps produce a product according to the given specification limits by monitoring the quality beforehand [3]. The data from a production process can be classified into two types such as attribute data and variable data. The attribute control charts such as p (or np) chart, c chart and u chart are used for attribute data while the variable control chart such as X-bar chart and S-chart are used for variable data. Many authors have worked on the design of attribute and variable control charts. For example, [4] focused on the development of np chart using a variable sample interval. [5] designed an economic method of an attribute control chart. [6] designed double-and triple-sampling X-bar chart using a genetic algorithm. [7] proposed X-bar chart using double and triple sampling. [8] proposed a variable control chart using Taguchi loss function. More details can be found in [9] and [10], [11], [3] and [12]. Mostly, the control charts are designed using single sampling. But, there are some sampling schemes such as

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repetitive sampling and multiple dependent state (MDS) sampling, which are more efficient than single sampling in terms of the sample size. [13] introduced repetitive sampling in the area of control charts. [14] designed a control chart using MDS sampling. In single sampling, a sample is selected from the production process and the decision about the state of process is made on the information obtained from this single sample. On the other hand, a repetitive sampling scheme allows sampling to be repeated when the engineer cannot make a decision based on the first sample. The MDS sampling is also used when there is no decision on the basis of the first sample. In MDS, the decision about the state of process is made if ‘‘i’’ preceding subgroups are declared as in control or not. The multiple dependent state repetitive sampling (MDSRS) is a mixture of both repetitive sampling and MDS sampling. It is expected that a control chart using MDSRS sampling will be more efficient in detecting the early shift in the manufacturing process. [15] designed a sampling plan using MDSRS sampling in the area of acceptance sampling plans. They showed the efficiency of this plan over the existing sampling plans in terms of average sample number. [14] designed an attribute control chart using MDS sampling. [16] proposed attribute and variable control charts using repetitive sampling. MDSRS sampling which is more efficient than MDS sampling and repetitive sampling was not introduced in the area of control chart. By exploring the

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M. S. Aldosari et al.: New Attribute Control Chart Using Multiple Dependent State Repetitive Sampling

literature, it came to our knowledge that there is no works available in the area of control charts using MDSRS sampling. In this manuscript, a new attribute control chart using MDSRS is designed. The operational procedure of the proposed control chart is given and the required measures to determine the average run lengths (ARLs) for in-control and out-of-control processes are given. Tables of ARLs are reported for various control chart parameters. The proposed control chart turns out to be more sensitive in detecting a smaller shift in the process as compared to the existing attribute control charts. The simulation study also shows the efficiency of the proposed chart over the existing charts. II. DESIGN OF PROPOSED CHART

In this section, the operational procedure of the proposed attribute control chart is given. The proposed attribute control chart is called as an np control chart using MDSRS. Step-1: A random sample of size n is selected at each subgroup from the production process and count the number of defectives D, say . Step-2: The process is declared to be in-control if LCL2 ≤ D ≤ UCL2 . The process is declared to be out-ofcontrol if D ≥ UCL1 or D ≤ LCL1 . Otherwise, go to Step-3. Here, LCL1 , UCL1 , LCL2 and UCL2 are four control limits that should be constructed using the data when the process is in control. Step-3 : Declare the process as in control if i preceding subgroups have been declared as in-control. Otherwise, repeat the Step-1. Here, the value of i may be specified by an engineer. The operational procedure of the proposed control is based on four control limits namely, LCL1 , UCL1 , LCL2 , UCL2 and two control coefficients namely k1 and k2 . These four control limits are given as follows p (1a) UCL1 = np0 +k1 np0 (1−p0 ) p (1b) LCL1 = max[0, np0 −k1 np0 (1−p0 )] p UCL2 = np0 +k2 np0 (1−p0 ) (2a) p LCL2 = max[0, np0 −k2 np0 (1−p0 )] (2b) where p0 is the fraction defective when the process is incontrol and it should be estimated by data when it is not known. [14] proposed an np control chart using MDS sampling. The MDS sampling is quite different from MDSRS. In MDS sampling, when the plotting statistic lies between UCL2 and UCL1 or LCL2 and LCL1 , the process is declared as in-control if i preceding subgroups have been declared as in-control but is declared as out-of-control without further repetition, otherwise. MDSRS sampling have some advantages over MDS sampling and it allows to repeat the process if the decision about the state of process cannot be reached with i preceding subgroups. Several existing attribute control charts are special cases of the proposed chart. The proposed control chart becomes the traditional attribute Shewhart control when k1 = k2 and UCL2 = LCL2 . The proposed conVOLUME 5, 2017

trol chart reduces to [14] chart when no repetitive sampling is allowed. The proposed control chart becomes [16] chart when i = 0. A. ARL FOR IN-CONTROL PROCESS

The probability of declaring as in-control on the basis of a single subgroup and MDS sampling is given as follows ! |UCL X2 | n 0 pd0 (1−p0 )n−d Pin,1 = d d=|LCL2 |+1   |LCL X2 | n + pd0 (1−p0 )n−d  d d=|LCL1 |+1  |UCL  X1 | n + pd0 (1−p0 )n−d  d d=|UCL2 |+1

×

 

|UCL X2 |



d=|LCL2 |+1

i  n pd0 (1−p0 )n−d ,  d

(3)

The probability of repeated sampling is given as   |LCL X2 | n P0rep = pd0 (1−p0 )n−d  d d=|LCL1 |+1  |UCL  X1 | n pd0 (1−p0 )n−d +  d d=|UCL2 |+1   i  |UCL 2| X n   × 1−  pd0 (1−p0 )n−d   (4) d d=|LCL2 |+1

The probability of declaring in-control for the proposed MDSRS control chart when the process is in control is given as P0in =

P0in,1 1−P0rep

(5)

The performance of the proposed control chart will be assessed using the average run length (ARL). The ARL when the process is in-control is given as follows ARL0 =

1 1 − P0in

(6)

B. ARL FOR SHIFTED PROCESS

Now, we will derive some necessary measures for the shifted process. Suppose that the process has shifted from in-control fraction defective p0 to p1 = p0 +cp0 , where c shows the shift constant in the process. The probability of declaring incontrol on the basis of a single subgroup and MDS sampling 6193


for the shifted process is given as follows |UCL X2 | n pd1 (1−p1 )n−d P1in,1 = d d=|LCL2 |+1   |LCL X2 | n pd1 (1−p1 )n−d + d  d=|LCL1 |+1  |UCL  X1 | n pd1 (1−p1 )n−d + d  d=|UCL2 |+1  i   |UCL X2 | n × pd1 (1−p1 )n−d d  

TABLE 1. The values of ARL when r0 = 300 and i = 2.

(7)

d=|LCL2 |+1

The probability of repeated sampling for the shifted process is given as   |LCL X2 | n 1 Prep = pd1 (1−p1 )n−d d  d=|LCL1 |+1  |UCL  X1 | n + pd1 (1−p1 )n−d d  d=|UCL2 |+1   i  |UCL | 2 X n   × 1−  pd1 (1−p1 )n−d   (8) d d=|LCL2 |+1


The probability that the process is declared as in control for the shifted process is given as follows P1in =

P1in,1 1−P1rep

The ARL for the shifted process is given as follows 1 ARL1 = 1−P1in

(9)

(10)

The following algorithm is used to determine the control coefficients k1 , k2 and ARL1 . Step-1: Specify the target in-control ARL, r0 , p0 and i. Step-2: Determine the values of k1 and k2 for which ARL0 ≥r0 . Step-3: Determine the values of ARL1 using the selected values of k1 and k2 Tables 1-4 are presented for various values of the shift constant in the process, specified values of ARL, say r0 and number of preceding subgroups i. Table 1 is presented when r0 = 300 and i = 2. Table 2 is presented when r0 = 370 and i = 2. Table 3 is presented when r0 = 300 and i = 3. Table 4 is presented when r0 = 370 and i = 3. From Tables 1-4, the following interesting trends can be seen in control chart parameters: 1) For all other fixed values, as p0 increases from 0.1 to 0.5, the values of ARL are increasing. 2) For all other fixed values, as i increases from 2 to 3, the values of ARL are increasing.

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III. COMPARATIVE STUDIES

In this section, we compare the efficiency of the proposed control chart over the existing attribute control charts in terms of ARLs.

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posed chart with [16] chart is presented only when ARL0 = 370, p0 = 0.1, n = 82 and i = 3 in Figure 1.

FIGURE 1. The ARLs of two charts.

From Figure 1, it can be seen that the proposed control chart performs better than [16] chart in terms of ARLs. The proposed control chart has ability to detect early shift in the process as compared to [16] at all values of given shift constants. For example, when p1 = 1.01, the value of ARL from the proposed control chart is 338, while it is 387 from [16] chart. TABLE 4. The values of ARL when r0 = 370 and i = 3.

B. PROPOSED CHART Vs [14] CHART

[14] proposed an attribute control chart using the MDS sampling scheme. [14] showed that the attribute control chart using this sampling scheme performs better than the Shewhart control chart. To save the space, the comparison of proposed chart with [14] chart is only presented when ARL0 = 370, p0 = 0.4, n = 98 and i = 3 in Figure 2.

FIGURE 2. The ARLs of two charts.

A. PROPOSED CHART Vs [16] CHART

[16] proposed an attribute control chart using a repetitive sampling scheme. [16] showed that the attribute control chart using this sampling scheme performs better than the Shewhart control chart. To save the space, the comparison of the proVOLUME 5, 2017

From Figure 2, it can be seen that the proposed control chart performs better than [16] chart in terms of ARLs. The proposed control chart has ability to detect early shift in the process as compared to [14] chart at all values of given shifts. For example, when p1 = 1.07, the value of ARL from the proposed control chart is 102, while it is 116 from [14] chart. 6195


FIGURE 3. The proposed chart for simulated data. FIGURE 5. The [14] chart for simulated data.

FIGURE 6. The proposed chart for real data.

FIGURE 4. The [16] chart for simulated data.

The similar behavior can be noted in Figure 5 for the chart by [14].

IV. SIMULATION STUDY

V. INDUSTRIAL EXAMPLE

This section presents the efficiency of the proposed control chart over two existing charts using simulation data. The data is generated using the binomial distribution by considering that the process is in-control at p0 = 0.20, and n = 46. The first 10 observations are generated from incontrol process and next 20 observations from a shifted process at p1 = 0.30. The tabulated value of ARL is 18 when ARL0 = 370. The number of defectives at each subgroup is given as D : 7 , 7, 8, 10, 9 , 6, 9 ,16 , 6 , 9 , 8 ,11 , 9 , 9 , 5 ,13, 12 , 7 ,10, 11, 9, 8, 9, 10, 10,10, 12, 14, 15, 9, 9, 15, 8, 14, 16, 18, 12, 15, 12, 8 The values of D are plotted on the control chart with four control limits given in Figure 3. From Figure 3, it can be seen that proposed control chart detects the shift at the 15th sample. It can be also seen that several points are in-decision state. The values of D are also plotted on the chart by [16] in Figure 4. This figure clearly indicates that process is declared to be incontrol although two points are located on in-decision state.

In this section, the proposed control chart is applied on data about number of nonconformities observed in 40 samples of 46 printed circuit board. The similar data is also reported in [9]. Let ARL0 = 370, i = 3 and p0 = 0.20. The data is given as follows D : 6, 9, 10, 7, 11, 6, 9, 8, 10, 7, 10, 13, 9, 8, 8, 7, 10, 12, 9, 7, 12, 8, 12, 7, 7, 7, 8, 11, 9, 5, 11, 6, 6, 9, 6, 8, 8, 11, 7, 8 The values of D is plotted on the proposed control chart as in Figure 6. The Figure 6 shows that the process is declared as in-control. But, from Figure 6, it can be noted that 12th sample is in-decision state, but three previous points 9, 10 and 11 are in-control states, so we will continue the process stating it as in-control. Similarly, 30th sample is in-decision but previous three subgroups are in-control. So, it can be declared that the process is in-control.

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VI. CONCLUSIONS

A new attribute control chart is presented in this paper. The structure of proposed chart and some tables are given for the VOLUME 5, 2017


practical use. From comparative study, it can be seen that the proposed chart performs better than the existing control charts in terms of ARL. The efficiency of proposed chart is demonstrated by a simulation study. The proposed control chart can be applied in the industry for better monitoring of process. The proposed chart using some other sampling schemes can be extended for future research. The proposed control chart can be extended for multivariate problems such as [17] as future research. ACKNOWLEDGMENTS

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of the manuscript. REFERENCES [1] G. S. Rao, ‘‘A group acceptance sampling plans for lifetimes following a generalized exponential distribution,’’ Econ. Quality Control, vol. 24, no. 1, pp. 75–85, Jan. 2009. [2] S. A. Abbasi and A. Miller, ‘‘On proper choice of variability control chart for normal and non-normal processes,’’ Quality Rel. Eng. Int., vol. 28, no. 3, pp. 279–296, Apr. 2012. [3] M. Aslam, M. Azam, N. Khan, and C.-H. Jun, ‘‘A mixed control chart to monitor the process,’’ Int. J. Prod. Res., vol. 53, no. 15, pp. 4684–4693, Mar. 2015. [4] H. Luo and Z. Wu, ‘‘Optimal np control charts with variable sample sizes or variable sampling intervals,’’ Econ. Quality Control, vol. 17, no. 1, pp. 39–61, Jan. 2002. [5] I. Kooli and M. Limam, ‘‘Economic design of an attribute np control chart using a variable sample size,’’ Sequential Anal., vol. 30, no. 2, pp. 145–159, May 2011. [6] L.-F. Hsu, ‘‘Note on ‘design of double-and triple-sampling X-bar control charts using genetic algorithms,’’’ Int. J. Prod. Res., vol. 42, no. 5, pp. 1043–1047, Mar. 2004. [7] D. He, A. Grigoryan, and M. Sigh, ‘‘Design of double–and triple-sampling X-bar control charts using genetic algorithms,’’ Int. J. Prod. Res., vol. 40, no. 6, pp. 1387–1404, Jan. 2002. [8] A. S. Safaei, R. B. Kazemzadeh, and S. T. A. Niaki, ‘‘Multi-objective economic statistical design of X-bar control chart considering Taguchi loss function,’’ Int. J. Adv. Manuf. Technol., vol. 59, no. 9, pp. 1091–1101, Apr. 2012. [9] D. C. Montgomery, Introduction to Statistical Quality Control, 7th ed. Hoboken, NJ, USA: Wiley, 2007. [10] S. Joekes and E. P. Barbosa, ‘‘An improved attribute control chart for monitoring non-conforming proportion in high quality processes,’’ Control Eng. Pract., vol. 21, no. 4, pp. 407–412, Apr. 2013. [11] S. Haridy, Z. Wu, K. Abhary, P. Castagliola, and M. Shamsuzzaman, ‘‘Development of a multiattribute synthetic-np chart,’’ J. Statist. Comput. Simul., vol. 84, no. 9, pp. 1884–1903, Sep. 2014. [12] M. Aslam and C.-H. Jun, ‘‘Attribute control charts for the Weibull distribution under truncated life tests,’’ Quality Eng., vol. 27, no. 3, pp. 283–288, Jun. 2015. [13] H. Lee, M. Aslam, Q.-U.-A. Shakeel, W. Lee, and C.-H. Jun, ‘‘A control chart using an auxiliary variable and repetitive sampling for monitoring process mean,’’ J. Statistical Comput. Simul., vol. 85, no. 16, pp. 3289–3296, Sep. 2014. [14] M. Aslam, A. Nazir, and C.-H. Jun, ‘‘A new attribute control chart using multiple dependent state sampling,’’ Trans. Inst. Meas. Control, vol. 37, no. 4, pp. 569–576, Apr. 2015. [15] M. Aslam, C.-H. Yen, C.-H. Chang, and C.-H. Jun, ‘‘Multiple states repetitive group sampling plans with process loss consideration,’’ Appl. Math. Model., vol. 37, nos. 20–21, pp. 9063–9075, Nov. 2013. [16] M. Aslam, M. Azam, and C.-H. Jun, ‘‘New attributes and variables control charts under repetitive sampling,’’ Ind. Eng. Manage. Syst., vol. 13, no. 1, pp. 101–106, Mar. 2014. [17] L. Corain and L. Salmaso, ‘‘Nonparametric permutation and combinationbased multivariate control charts with applications in microelectronics,’’ Appl. Stochastic Models Bus. Ind., vol. 29, no. 4, pp. 334–349, Jul./Aug. 2013.

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MANSOUR SATTAM ALDOSARI is currently pursuing the Ph.D. degree with King Abdulaziz University, Saudi Arabia. His research interests include topics related to statistical quality control and applied Statistics.

MUHAMMAD ASLAM received the M.Sc. degree with the Chief Minister of the Punjab Merit Scholarship and the M.Phil. degree with the Governor of the Punjab Merit Scholarship from GC University, Lahore, Pakistan in 2004 and 2006, respectively, all in statistics. He received the Ph.D. degree in statistics from the National College of Business Administration and Economics, Lahore, under the supervision of Prof. Dr. M. Ahmad. From 2003 to 2006, he was a Lecturer of Statistics with the Edge College System International, Pakistan. From 2006 to 2008, he was a Research Assistant with the Department of Statistics, GC University. He joined the Forman Christian College University as a Lecturer in 2009, where he was an Assistant Professor from 2010 to 2012 and an Associate Professor from 2012 to 2014. He is currently an Associate Professor of Statistics with the Department of Statistics, King Abdulaziz University Jeddah, Saudi Arabia. He has authored more than 195 research papers in national and international journals, including the IEEE ACCESS, the Journal of Applied Statistics, the European Journal of Operation Research, the Journal of the Operational Research Society, the Applied Mathematical Modeling, the International Journal of Advanced Manufacturer Technology, the Communications in Statistics, the Journal of Testing and Evaluation, and the Pakistan Journal of Statistics. He has authored one book published in Germany. He has been an HEC-approved Ph.D. supervisor since 2011. He supervised five Ph.D. theses, more than 25 M.Phil. theses, and three M.Sc. theses. He is currently a Supervisor of two Ph.D. theses and more than five M.Phil. theses in statistics. He reviewed over 75 research papers for various well-reputed international journals. His areas of interest include reliability, decision trees, industrial statistics, acceptance sampling, rank set sampling, and applied statistics. He received the Meritorious Services Award in research from the National College of Business Administration and Economics in 2011. He received the Research Productivity Award in 2012 from the Pakistan Council for Science and Technology. His name is listed in second position among Statisticians in the Directory of Productivity Scientists of Pakistan 2013 and in first position in 2014. He received 371 position in the list of top 2210 profiles of Scientists of Saudi Institutions 2016. He was selected for the Innovative Academic Research and Dedicated Faculty Award 2017 by SPE, Malaysia. He is a member of the Editorial Board of the Electronic Journal of Applied Statistical Analysis, the Asian Journal of Applied Science and Technology, and the Pakistan Journal of Commence and Social sciences. He is also a member of the Islamic Countries Society of Statistical Sciences.

CHI-HYUCK JUN received the B.S. degree in mineral and petroleum engineering from Seoul National University, South Korea, the M.S. degree in industrial engineering, Korea Advanced Institute of Science and Technology, and the Ph.D. degree in operations research from the University of California at Berkeley, USA. Since 1987, he has been with the Department of Industrial and Management Engineering, Pohang University of Science and Technology, South Korea, where he is currently a Professor. He is interested in data mining and reliability/quality.

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