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Research Article

A Comparative Analysis of Control Charts for Monitoring Process Mean

You HuayWoon

Pusat GENIUS@Pintar Negara, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia hwyou@ukm.edu.my*

Article History: Received: 10 November 2020; Revised: 12 January 2021; Accepted: 27January 2021; Published online: 05April 2021

Abstract:Control charts serve as an effective tool for controlling and monitoring process quality in industries of production and service. The Shewhart chart is the first control chart that was used to detect large mean shifts in a process. Since then, to increase the Shewhart chart’s sensitivity, synthetic type control charts, such as synthetic control chart, side sensitive group runs (SSGR) control chart, have been proposed. SSGR chart ismore efficient compared to the Shewhart chart and synthetic chart,primarily due to the side sensitive feature in SSGR chart. Meanwhile, exponentially weighted moving average (EWMA) chart isoften used to detect small process changes. In practice, the evaluation of a control chart’s performance is vital. Nevertheless, the cost of implementing a control chart is an important factor that influences the choice of a control chart. The cost of repairs, sampling, nonconforming products from a failure in detecting out-of-control status, and investigating false alarms, can be significantly high. Therefore, the aim of this paper is to compare the implementation cost of synthetic, SSGR and EWMA charts, so that quality practitioners can identify the most cost-effective chart to implement. Here, the cost function was adopted to compute the implementation cost of the control chart. According to the findings, quality practitioners are recommended to adopt the SSGR chart,since it is more economical compared to the synthetic chart. However, the cost to implement anEWMA chart is higher than the synthetic and SSGR charts. In light of this, this study allows for quality practitioners to have a better idea on the selection of the control chart to implement, with respect to its cost. Keywords:Process Mean, optimal cost, side sensitive group runs chart, synthetic chart, exponentially weighted moving average chart

1. Introduction

Quality is considered to be the primary factor in the manufacturing and service industries. The monitoring and improving of overall quality can be attained via the reduction of variability in the process (Montgomery, D.C., 2019). A Statistical Process Control (SPC) is a prominent technique that can be implemented with the aim of producing high quality output, which in turn leads to a better reputation for the organization. One of the most important SPC techniques is a control chart. It plays a significant role in monitoring a process continuously, and enhancing the capability of the process (Kiran, D.R., 2019).

The first control chart was suggested by Dr. Walter A. Shewhart. For its ease of construction and implementation, the Shewhart chart is well recognised among quality practitioners. However, a significant drawback of this particular chart is that it is insensitive toward small and moderate changes in the process mean(Rakitzis, A.C, 2019). A great deal of researches haveattempted to develop a new and improved control chart. Among the available control charts, it has been proven that the synthetic type charts and exponentially weighted moving average (EWMA) chart have a better detection ability as compared to the Shewhart chart (Noor-ul-Amin, M. and Tayyab, M, 2020; Rakitzis, A.C , 2019).A synthetic type chart is a combination of a control chart and a conforming run length chart.

The EWMA control chart was proposed by Roberts, (1959) for a quick detection of small shifts in the process. Wu and Spedding(2000) are the pioneers in merging the Shewhart sub-chart and conforming run length (CRL) sub-chart to construct a synthetic control chart for the monitoring of the process mean. It wasdemonstrated that the synthetic control chart outperforms the Shewhart chart in detecting small and moderate shifts. Since then, numerous extensions were made to the synthetic control chart. Gadre and Rattihalli(2007) developed the side sensitive group runs (SSGR) chart by adopting the side sensitive characteristic, as suggested by Davis and Woodall, (2002). With this additional feature, the SSGR chart has been proven to be more efficient as compared to the Shewhart and synthetic control charts (Kamrisham, M.F.Z, et al., 2019).

The efficiency of control charts is evaluated using performance measures, such as average run length and standard deviation of the run length (Lio, Y., et al., 2019). It is well known that the control chart produced with the smallest out-of-control average run length for a given process change under similar circumstances is considered to be the best chart (Faraz, A, et al., 2019). However, the expected cost required for this situation cannot be identified. In today’s competitive business environment, the cost of implementing a control chart plays an important role in process monitoring (Özcan, A., 2020). Research on the implementation cost of the available control charts has yet to be investigated in the literature.

Hence, this paper compares the implementation costs of the synthetic, SSGR and EWMA control charts. In view of this, the economic model by Chung (1991)was implemented to get the optimal cost. The remainder of this paper is structured as follows. Section 2 discusses the designs of the synthetic, SSGR and EWMA control charts. Section 3 provides the cost model to calculate the expected cost per unit time. Section 4 compares the costs of the three charts. Finally, Section 5 concludes.

2. A Review on Control Charts

The following subsections review the design and operation of the synthetic, SSGR and EWMA control charts respectively.

2.1 The Synthetic Control Chart

Wu and Spedding(2000) introduced the synthetic control chart. This chart consists of the Shewhart sub-chart and conforming run length (CRL) sub-chart to monitor the process mean. According to the researchers, the upper control limit (UCL) and lower control limit (LCL) of the Shewhart sub-chart are defined as follows:

0 0

UCL LCL

L

,

n









(1)

with



₀and



₀ are the in-control process mean and the in-control process standard deviation,respectively, while L is the design constant of the Shewhart sub-chart.

The implementation of the synthetic chart is based on the fact that when a sample falls beyond the control limits of the Shewhart sub-chart, the synthetic chart does not signal an out-of-control. Alternatively, the synthetic chart indicates that there is a nonconforming sample. To further evaluate the process status, a CRL sub-chart is required.There is one lower limit for the CRL sub-chart. Here, the synthetic chart alarms an out-of-control when the CRL value isless than or equal to the lower limit of the CRL sub-chart. The CRL value is the number of units examined between two nonconforming units. The synthetic chart has been shown to have better detection ability compared to the Shewhart chart(2000).

2.2 The Side Sensitive Group Runs (SSGR) Control Chart

The side sensitive group runs (SSGR) chart was suggested by Gadre and Rattihalli(2007), and is an integration of the Shewhart sub-chart and an extended version of the CRL sub-chart. The upper and lower control limit of the Shewhart sub-chart are as follows:

0 0

UCL LCL

K

.

n









₍₂₎

Note that K denotes the design constant.The design of the SSGR chart is nearly similar to the synthetic chart, i.e. when a sample falls outside of the control limits of the Shewhart sub-chart, this sample is marked as nonconforming by the SSGR chart, instead of out-of-control in the process.

A further investigation using the CRL sub-chart is required to evaluate the status of the procedure. The rth CRL value, i.e. CRLr for r = 1, 2, … is taken as the number of conforming samples plotted on the Shewhart

sub-chart, between the rth and (r-1)th nonconforming samples. The extended version of the CRL sub-chart has an additional rule to improve the overall effectiveness of the chart detection. In this additional rule, the SSGR chart declares an out-of-control when (i) CRL1 ≤ G or (ii) CRLr≤ G and CRLr+1 ≤ G, for r = 2, 3, …, and that both

CRLr and CRLr+1have shifts on the same side of the Shewhart sub-chart. Here, G is the lower control limit of the

CRL sub-chart in the SSGR chart.

The average run length (ARL) of the SSGR chart is computed asshown by Gadre and Rattihalli(2007), as follows:     2 2 1 1 1 ARL , 1 (1 )( 2) d d A P A d d A       (3)

where

P

₁denotes the probability of a nonconforming sample on the Shewhart sub-chart, i.e.



 



1

1

,

P

  



_

K





n

   

K



n



_

(4)

and the probability of an event CRLr≤ G is



1



1

1

G

.

A

  

P

(5)

Meanwhile, d denotes the probability for taking into account the side sensitivity characteristics, i.e.





1

1

.

K

n

d

P



 





(6)

2.3 The Exponentially Weighted Moving Average (EWMA)Control Chart

The exponentially weighted moving average (EWMA) chart is commonly employed to detect small changes in the process(Roberts, S.W, 1959). The EWMA chart statistics are given as follows:

1

(1

)

, for = 1, 2, ...,

v v v

Z





Y

 



Z

_

v

(7)

where





0

 



1



is a smoothing constant and

0 0

,

v v

X

Y

n









(8)

where

X

_vis the mean of the vth sample andn is the sample size. The UCL and LCL of the EWMA chart are defined as follows:

UCL LCL

.

2

J





 



(9)

Here, J> 0 is the width constant. The EWMA chart alarms an out-of-control whenthe statistic is plotted outside the LCL or UCL.

3. Methodology

The economic design of the control charts is developed by employing an expected cost function. The optimal charting parameters are obtained based on minimizing this function. The general cost function constructed by Chung (1991)was used to obtain the cost of implementing the control chart.

Initially, the process is assumed to be in-control



 



₀



. A single assignable cause that causes a shift of



in the process occurs at a random time, i.e.

  



₀



, where



is the magnitude of the shift and



is the process standard deviation. This means that the process is out-of-control. The length of time until the occurrence of the assignable cause is assumed to follow an exponential distribution with rateτ. The process remains out-of-control until actions are taken to eradicate the assignable cause. In view of this, the expected length of time per cycle, E(T) is represented as follows:



1



0 0

1

1

( )

.

ARL

sT

E T



EH





 



(10)

Meanwhile, the total cost includes (i) the cost in the in-control and out-of-control states; (ii) false alarms cost; (iii) sampling cost; and (iv) maintenance cost. The expected cost of a cycle is denoted asbelow:

0 1 0

1

(

)

.

ARL

C

b

cn

sY

E C

C B

B

W

h

















_



_









(11)

The aim of the economic design of a control chart is to determine the suitable charting parameters such that expected cost function (ECT) provided in Eq. (12) is minimized:





0 1 0 1 0 0

1

ARL

( )

,

1

( )

1

ARL

C

b

cn

sY

C B

B

W

h

E C

ECT

sT

E T

EH



















_



_



















(12) where



ARL

1

0.5



,

B





h



F

1 1 2 2

,

F



ne





T





T

1

0.5,

s

h









ARL

1

0.5



,

EH





h G



and 1 2

.

D



ne T

 

T

The notations in Eq. (12) are stated as follows:

C0 Expected quality cost per unit time while the process is in-control C1 Expected quality cost per unit time while the process is out-of-control

τ Process failure rate

b Fixed cost per sample

c Cost per unit sampled

n Sample size

e Expected time to sample and interpret one unit

h Sampling interval

s Expected number of samples taken before an assignable cause occur

Y Cost of false alarm

ARL0 Average run length when the process is in-control ARL1 Average run length when the process is out-of-control W Cost of finding and fixing an assignable cause γ1 = 1 if production continues during search

= 0 if production stops during search γ2 = 1 if production continues during repair

= 0 if production stops during repair T0 Expected time to search a false alarm T1 Expected time to find the assignable cause T2 Expected time to repair the process

4. Results

A comparative study was conducted between the synthetic, SSGR and EWMA control charts on the optimal cost. The optimal results are based on minimizing the expected cost function. For comparison purposes, Table 1

lists the different combinations of the 12 input parameters

       



 

, ,

Y

$ ,

W

$ ,

b

$ ,

c

$ , ,

e T T T

0

, ,

1 2

, ,

 

1 2



related to the expected cost function. According to

parameters remain unchanged. For instance, the value for



that is considered each time is 0.01, 0.02 and 0.03, and the rest of the input parameters remain unchanged. A total of34 different input parameters

       



 

, ,

Y

$ ,

W

$ ,

b

$ ,

c

$ , ,

e T T T

0

, ,

1 2

, ,

 

1 2



combinationswere considered.

The corresponding cost values of the synthetic, SSGR and EWMA control charts are displayed in Table 2. Table 1. Different combinations of the input parameters

No. δ τ Y($) W($) b($) c($) e T0 T1 T2 γ1 γ2 1 0.5 0.01 900 900 5 1 0.5 2 2 0 1 0 2 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 3 2.5 0.01 900 900 5 1 0.5 2 2 0 1 0 4 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 5 1.5 0.02 900 900 5 1 0.5 2 2 0 1 0 6 1.5 0.03 900 900 5 1 0.5 2 2 0 1 0 7 1.5 0.01 300 900 5 1 0.5 2 2 0 1 0 8 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 9 1.5 0.01 1500 900 5 1 0.5 2 2 0 1 0 10 1.5 0.01 900 150 5 1 0.5 2 2 0 1 0 11 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 12 1.5 0.01 900 1650 5 1 0.5 2 2 0 1 0 13 1.5 0.01 900 900 0 1 0.5 2 2 0 1 0 14 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 15 1.5 0.01 900 900 10 1 0.5 2 2 0 1 0 16 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 17 1.5 0.01 900 900 5 2 0.5 2 2 0 1 0 18 1.5 0.01 900 900 5 3 0.5 2 2 0 1 0 19 1.5 0.01 900 900 5 1 0.05 2 2 0 1 0 20 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 21 1.5 0.01 900 900 5 1 5 2 2 0 1 0 22 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 23 1.5 0.01 900 900 5 1 0.5 4 2 0 1 0 24 1.5 0.01 900 900 5 1 0.5 6 2 0 1 0 25 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 26 1.5 0.01 900 900 5 1 0.5 2 4 0 1 0 27 1.5 0.01 900 900 5 1 0.5 2 6 0 1 0 28 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 29 1.5 0.01 900 900 5 1 0.5 2 2 1 1 0 30 1.5 0.01 900 900 5 1 0.5 2 2 2 1 0 31 1.5 0.01 900 900 5 1 0.5 2 2 0 1 0 32 1.5 0.01 900 900 5 1 0.5 2 2 0 0 0 33 1.5 0.01 900 900 5 1 0.5 2 2 0 0 1 34 1.5 0.01 900 900 5 1 0.5 2 2 0 1 1

Table 2 demonstrates that the SSGR chart has the lowest cost in comparisonto the synthetic and EWMA charts. In addition, the percentage of cost savings when the SSGR chart is employed is included in Table 2 in parenthesis. The percentage of savings varies between0.12 to 1.62 for the synthetic chart and 2.34 to 11.98 for the EWMA chart. Therefore, the implementation using an SSGR chart is more economical, followed by the synthetic chart, and lastly, the EWMA chart. In other words, the implementation of the SSGR chart is cost saving. Furthermore, it can be noted that the control chart with better detection ability toward small mean shifts, namelythe EWMA chart, requires higher cost in the implementation,as compared to the synthetic and SSGR charts. In view of this, considering the cost and performance of a control chart, a quality practitioner is able to decide on the choice of a control chart in monitoring a process. Besides that, the quality practitioners can determine the priority, which is based on the cost or the performance, in selection of the implemented control chart.

Table 2. Optimal cost of the synthetic chart, side sensitive group runs chart and exponentially weighted moving average chart

No. Synthetic chart SSGR chart EWMA chart

1 251.64 (0.12%) 251.33 257.21 (2.34%) 2 261.68 (0.45%) 260.50 276.49 (6.14%) 3 271.98 (0.39%) 270.91 287.68 (6.19%) 4 261.68 (0.45%) 260.50 276.49 (6.14%) 5 277.94 (0.64%) 276.18 300.95 (8.97%) 6 292.11 (0.68%) 290.13 322.26 (11.07%) 7 261.09 (0.38%) 260.10 274.58 (5.57%) 8 261.68 (0.45%) 260.50 276.49 (6.13%) 9 261.94 (0.47%) 260.71 277.27 (6.35%) 10 254.93 (0.48%) 253.41 269.40 (6.31%) 11 261.68 (0.45%) 260.50 276.49 (6.14%) 12 268.72 (0.42%) 267.59 283.59 (5.98%) 13 259.33 (0.54%) 257.94 269.34 (4.42%) 14 261.68 (0.45%) 260.50 276.49 (6.14%) 15 263.13 (0.40%) 262.07 280.69 (7.10%) 16 261.68 (0.45%) 260.50 276.49 (6.14%) 17 262.86 (0.52%) 261.49 278.36 (6.45%) 18 263.84 (0.57%) 262.34 279.96 (6.72%) 19 258.59 (0.22%) 258.03 273.59 (6.03%) 20 261.68 (0.45%) 260.50 276.49 (6.14%) 21 275.00 (1.62%) 270.61 303.02 (11.98%) 22 261.68 (0.45%) 260.50 276.49 (6.14%) 23 261.68 (0.45%) 260.50 276.49 (6.14%) 24 261.68 (0.45%) 260.50 276.49 (6.14%) 25 261.68 (0.45%) 260.50 276.49 (6.14%) 26 264.49 (0.43%) 263.35 282.78 (7.38%) 27 267.20 (0.42%) 266.09 288.83 (8.55%) 28 261.68 (0.45%) 260.50 276.49 (6.14%) 29 259.24 (0.46%) 258.06 273.90 (6.14%) 30 256.85 (0.46%) 255.67 271.36 (6.14%) 31 261.68 (0.45%) 260.50 276.49 (6.14%) 32 253.45 (0.43%) 252.36 263.35 (4.35%) 33 253.45 (0.43%) 252.36 263.35 (4.35%) 34 261.68 (0.45%) 260.50 276.49 (6.14%) 5. Conclusion

This paper evaluates the cost of implementing the synthetic, SSGR and EWMA charts. The synthetic, SSGR and EWMA control charts are employed to monitor the process mean. The EWMA chart shows better performance in terms of detection ability toward small process shifts compared to the synthetic and SSGR charts. The findings from this work indicate that the SSGR chart is more economical in implementation, as compared to the synthetic and EWMA charts. In light of this, although the EWMA chart is able to identify small shifts quickly, the implementation of the EWMA chart is less economical. Hence, quality practitioners can expect a higher cost when employing the EWMA control chart as compared to the synthetic and SSGR charts. Alternatively, the cost is reduced for manufacturers or quality practitioners, but is associated with a high risk for the consumer. This research provides an overview on the selection of the control charts, with respect to cost, in facilitating a quick implementation of control charts in industry.

6. Acknowledgement

This research is supported by the Universiti Kebangsaan Malaysia, GeranGalakanPenyelidikMuda, GGPM-2017-062.

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