Memory-Type Control Charts Through the Lens of Cost Parameters
1 Pusat Pengajian Sains Matematik, Fakulti Sains dan Teknologi, Universiti Kebangsaan Malaysia, UKM Bangi, Selangor, 43600, Malaysia
2 Pusat GENIUS@Pintar Negara, Universiti Kebangsaan Malaysia, UKM Bangi, Selangor, 43600, Malaysia
3 Shivaji University, Kolhapur, 416004, India
* Corresponding Author: You Huay Woon. Email:
Intelligent Automation & Soft Computing 2023, 36(1), 1-10. https://doi.org/10.32604/iasc.2023.032062
Received 05 May 2022; Accepted 10 June 2022; Issue published 29 September 2022
AbstractA memory-type control chart utilizes previous information for chart construction. An example of a memory-type chart is an exponentially-weighted moving average (EWMA) control chart. The EWMA control chart is well-known and widely employed by practitioners for monitoring small and moderate process mean shifts. Meanwhile, the EWMA median chart is robust against outliers. In light of this, the economic model of the EWMA and EWMA median control charts are commonly considered. This study aims to investigate the effect of cost parameters on the out-of-control average run length in implementing EWMA and EWMA median control charts. The economic model was used to compute the parameter. The 14 input parameters were identified and the analysis was carried out based on the one-parameter-at-a-time basis. When the input parameters change based on a predetermined percentage, the is affected. According to the results of the EWMA chart, nine input parameters had an effect and five input parameters had no effect on the parameter. Further, only seven of the 14 input parameters had an effect on the ARL1 of the EWMA median chart. However, the effect of each input parameter on the was different. Moreover, the for the EWMA median chart was smaller than the EWMA chart. This analysis is crucial to observe and determine the input parameters that have a significant impact on the of the EMWA and EWMA median control charts. Hence, practitioners can obtain an overview of the influence of the input parameters on the when implementing the EWMA and EWMA median control charts.
Statistical Process Control (SPC) is a collection of analytical decision making tools that are effective in achieving process reliability and enhancing capability by reducing variability. A control chart is among the most efficient tools in SPC . The first control chart was pioneered by Dr. Walter A. Shewhart to monitor and determine whether a process is in a desired state for producing high quality goods .
Since then, the control chart remains to be a valuable tool that has been prominently employed in SPC. This is because it can continuously monitor a process and ultimately improve its capability. It is simple to use, yet achieves a significant impact on the enhancement of overall quality . The awareness and knowledge of using control charting techniques to monitor a process and deliver high-quality production has grown exponentially [4–6]. Hence, control charts are used extensively in the manufacturing and service sectors.
The Shewhart chart has been well-known for its ease of construction and implementation. Nevertheless, it is insensitive towards small and moderate process shifts. Consequently, extensive research has been conducted to develop new control charts to improve the sensitivity of the Shewhart chart, such as the memory-type control chart . A memory-type control chart is constructed based on past and current information. A good example of the memory-type control chart is the exponentially-weighted moving average (EWMA) chart.
The EWMA chart has the ability to accurately detect small-to-moderate process mean shifts under normality assumption. According to Human et al. , this generally does not hold when the process contains contaminated normal data. In this case, the EWMA median chart is an ideal alternative. The main benefit of this chart is that it is resistant to outliers, as it monitors the process using the sample median.
The assessment of a control chart is necessary to reveal its overall effectiveness. This is vital, as it influences the decision to use the control chart. In practice, the performance of a control chart can be accurately evaluated . The common criterion to measure the performance of a control chart is the average run length (ARL). The economic designs of the memory-type control chart have been studied by Hariba et al.  and Serel , among others. The effect of cost parameters on the ARL of the EWMA chart and EWMA median chart based on the economic model is not yet available in the existing literature.
In view of this, it is essential to investigate the impact of the input parameters on the out-of-control average run length of implementing the EWMA and EWMA median control charts using the economic model. From the findings, we can identify the critical input parameters that influence the performance measure of the control charts. This work presents a brief description of the EWMA and EWMA median charts. The economic model is also elaborated. Next, the sensitivity analysis of the EWMA chart and EWMA median chart was also conducted to identify the input parameters that influence the of control charts. Finally, concluding remarks are given.
2 The Memory-Type Control Charts
The memory-type charts considered in this study are the EWMA chart and EWMA median chart. The EWMA chart was developed by Roberts , and is typically used to detect small process mean shifts due to the characteristics of using previous and current data in calculation of the statistic .
The statistic of the EWMA chart at sampling period, u, is computed as follows:
where is a smoothing constant and with . Note that is the in-control process mean. When a falls beyond the control limit, an out-of-control situation is said to have occurred. Note that the EWMA chart is converted to a Shewhart chart when becomes 1 .
The EWMA median chart uses the sample median to monitor the process, as follows:
When calculating the sample median, it is common practice to assume that the sample size, n, is an odd number . This simplifies the calculation of the sample median. The EWMA median chart’s statistic is
where is a smoothing constant and .
The economic model obtains optimal charting parameters that minimize cost . Here, the economic model involves finding the of the control chart. We considered the EWMA and EWMA median charts. The implementing cost of the chart was calculated based on the cost function.
When an assignable cause occurs, the process is said to become out-of-control and the process mean becomes , where δ and σ are the process shifts and process standard deviation, respectively. It is assumed that a process initiates in a state of in-control and the time for an assignable cause to occur is exponentially distributed with mean . The expected cost per hour was obtained from dividing the expected cost-per-cycle by the expected cycle length. The cycle was composed of an in-control phase, followed by an out-of-control phase.
The expected cost per unit time in hours is denoted as C and can be defined as follows:
The notations in Eq. (4) are defined as follows:
Expected quality cost per unit time, while the process is in-control
Expected quality cost per unit time, while the process is out-of-control
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 occurs
Y Cost of false alarm
Average run length when the process is in-control
Average run length when the process is out-of-control
W Cost of finding and fixing an assignable cause
= 1if production continues during search
= 0 if production stops during search
= 1 if production continues during repair
= 0 if production stops during repair
Expected time to search for a false alarm
Expected time to find the assignable cause
Expected time to repair the process
4 The Analysis of the EWMA and EWMA Median Control Charts
Sensitivity analysis is crucial to identify the input parameter that has an impact on the performance measure, i.e., , of implementing the EWMA and EWMA median charts. Meanwhile, sensitivity analysis of input parameters on the EWMA and EWMA median control charts helps to determine the critical input parameters. Here, 14 input parameters related to the cost function were identified. Sensitivity analysis can be performed on a one factor at a time basis, where one of the input parameters is modified each time, while the other 13 input parameters remain unchanged. For example, the value for is 0.01, 0.02 and 0.04 each time, with the remainder of the input parameters remaining unchanged. In view of this, 40 different input parameter combinations were taken into account, as shown in Tab. 1.
Tab. 2 demonstrates the corresponding values for the EWMA and EWMA median charts. Here, the input parameters are = 0.02, = 0.86, = 114.24, = 949.2, Y = 977.4, W = 977.4, b = 0, c = 4.22, e = 0.083, = 0.083, = 0.083, = 0.75, = 1 and = 0 and the corresponding value for the EWMA and EWMA median charts are 11.89 and 8.52, respectively. According to Tab. 2, the value of the EWMA chart decreases when the expected quality cost per unit time while the process is in-control, , decreases. For instance, a decrease of from 114.24 to 57.12 as in number 7 in Tab. 1 results in a decrease in the from 11.89 to 9.45. However, there is no change in terms of when increases from 114.24 to 228.48. This shows that a decrease in increases the sensitivity of the EWMA chart in terms of average run length. Meanwhile, the of the EWMA chart increases when the expected quality cost per unit time while the process is out-of-control, increases and vice versa. For instance, an increase of from 949.2 to 1898.4 shows an increase in the value from 11.89 to 11.96. The same occurs to the cost of false alarm, Y. Nevertheless, Y has a greater effect on than . When Y increases, also increases by 19.85%. When increases, increases by 0.59%. On the contrary, the EWMA chart’s value decreases when the cost per unit sampled, c and fixed cost per sample, b, increase. From Tabs. 1 and 2, we can see that when c increases from 4.22 to 8.44, the value from 11.89 decreases to 9.45. Note that the input parameter W has no influence on the value.
According to Tab. 2, the value remains the same, regardless of the increase or decrease of the input parameter expected time to search a false alarm, , expected time to determine the assignable cause, , expected time to repair the process, and expected time to sample and interpret one unit, e. The process failure rate, , can be seen to have an effect on the value of the EWMA chart. When decreases by 50% (i.e., from 0.02 to 0.01), the value increases from 11.89 to 11.96. However, when rises to 0.04, the value decreases to 11.82. The decrease in occurs due to an increase in the process failure rate.
When the process shift, , increases by 50% from 0.86 to 1.72, the value is reduced by 63.25%. As the size of the process shift decreases, the value increases by 109.34%, from 11.89 to 24.89. A large process shift is easier to be detected than a small process shift; hence, the value is lower in the large process shift compared to the small process shift. The production status during search, and production status during repair, , show that when production stops during search and repair, the remains the same. The same phenomenon occurs when the production continues during search and repair. The decreases by 0.59% when the production stops during search and production continues during repair.
For the EWMA median chart, the input parameters, , , W, e, , and have no influence on the , i.e., the remains at 8.52, regardless of the increase or decrease of the input parameters. The value increases when the input parameters and c decrease and vice versa. For example, the increases by 23.47% when c decreases by 50%. The is greatly influenced by the input parameter . We can see that when the decreases or increases by 50%, the increases by 149.88% and decreases by 55.75%, respectively. This indicates that the smaller the , the higher the value. A different scenario occurs for the input parameter Y. The value decreases by 20.07% and increases by 17.61% when Y decreases and increases by 50%, respectively. Meanwhile, the decreases when the input parameter b increases. For instance, the decreases by 58.10% and 78.52% when b increases to 5 and 10, respectively. The slightly decreases by 2.46% when decreases by 50%. However, the remains the same when increases by 50%. When = 0 and = 0, the increases by 4.46%. For the same with = 1, the increases by 3.64%.
Fig. 1 illustrates the profile for the EWMA and EWMA median charts based on the 40 different input parameter combinations. From Fig. 1, we notice that the influence of the input parameters on the profile of the for both charts is almost the same. However, the value is higher for the EWMA chart in comparison to the EWMA median chart. The percentage of improvement in when the EWMA median chart is used is denoted in parenthesis in Tab. 2. The percentage of improvement ranges from 10.92% to 369.40%. This indicates that implementing the EWMA median chart based on the economic model has better performance than the EWMA chart. As discussed, nine and seven out of 14 input parameters have an effect on the output parameter, i.e., value of the EWMA and EWMA median charts, respectively. The common input parameters that have an impact on the value for both charts are , , Y, b, c, and . This provides an overview to the practitioners on which input parameters need to be considered in measuring the performance measure, i.e., . However, the size of the effect of each input parameter on the is different. The input parameter has the greatest influence on the ARL1 for the EWMA and EWMA median charts.
The EWMA control chart has been well-known for its quick detection of small and moderate process mean shifts. Meanwhile, the EWMA median chart can be used to monitor processes containing outliers. In this study, sensitivity analysis was conducted to determine the effect of the input parameters on the of the EWMA and EWMA median charts. The sensitivity analysis was carried out by increasing and decreasing the values of the input parameters by 50%. All input parameters, namely, , , , , Y, W, b, c, e, , , , and were subjected to sensitivity analysis. The findings of the study indicate that such a variation affects the of the EWMA and EWMA median control charts. The input parameters that have an effect on the of the EWMA chart are , , , , Y, b, c, and . The process shift, , has the most impact on the when decreases by 50%. The of the EWMA chart increases by 109.34% when decreases by 50%. In contrast, input parameters e, , , and W have no effect on the of the EWMA control chart. For the EWMA median chart, the input parameters that have an effect on the are , , Y, b, c, and . On the other hand, , , W, e, , and , have no effect on the of the EWMA median chart. Similarly, the effect of on the is the highest when decreases by 50% and the increases by 149.88% for the EWMA median chart. As this study is based on the sensitivity analysis of the EWMA and EWMA median charts, future research can investigate the effect of the input parameters on the control chart with unknown process parameters.
Acknowledgement: This research is supported by the Universiti Kebangsaan Malaysia, Geran Galakan Penyelidikan, GGP-2020-040.
Funding Statement: This research was funded by the Universiti Kebangsaan Malaysia, Geran Galakan Penyelidikan, GGP-2020-040.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
- D. C. Montgomery, Introduction to Statistical Quality Control, 8th (Eds.New York: John Wiley & Sons, Inc, 2019.
- M. A. Abtew, S. Kropi, Y. Hong and L. PU, “Implementation of statistical process control (SPC) in the sewing section of garment industry for quality improvement,” AUTEX Research Journal, vol. 18, pp. 160–172, 2018.
- H. W. You, M. B. C. Khoo, Z. L. Chong and W. L. Teoh, “The expected average run length of the EWMA median chart with estimated process parameters,” Austrian Journal of Statistics, vol. 49, no. 3, pp. 19–24, 2020.
- N. G. Sengoz, “Control charts to enhance quality,” Quality Management Systems–A Selective Presentation of Case-Studies Showcasing Its Evolution, pp. 153–194, 2018.
- K. Karoon, Y. Areepong and S. Sukparungsee, “Exact run length evaluation on extended EWMA control chart for autoregressive process,” Intelligent Automation & Soft Computing, vol. 33, no. 2, pp. 743–759, 2022.
- N. Saengsura, S. Sukparungess and Y. Areepong, “Mixed moving average-cumulative sun control chart for monitoring parameter change,” Intelligent Automation & Soft Computing, vol. 31, no. 1, pp. 635–647, 2022.
- N. Abbas, “Homogeneously weighted moving average control chart with an application in substrate manufacturing process,” Computers & Industrial Engineering, vol. 120, pp. 460–470, 2018.
- S. Human, P. Kritzinger and S. Chakraborti, “Robustness of the EWMA control chart for individual observations,” Journal of Applied Statistics, vol. 38, no. 10, pp. 2071–2087, 2011.
- Z. L. Chong, M. B. C. Khoo, W. L. Teoh, H. W. You and P. Castagliola, “Optimal design of the side-sensitive modified group runs (SSMGR) chart when process parameters are estimated,” Quality and Reliability Engineering International, vol. 35, no. 1, pp. 246–262, 201
- P. S. Hariba and S. D. Tukaram, “Economic design of a nonparametric EWMA control chart for location,” Production Journal, vol. 26, pp. 698–706. 2016.
- D. A., Serel, “Economic design of EWMA control charts based on loss function,” Mathematical and Computer Modelling, vol. 49, no. 3–4, pp. 745–759, 2009.
- S. W. Roberts, “Control chart tests based on geometric moving averages,” Technometrics, vol. 1, no. 3, pp. 239–250, 1959.
- S. Ramesh and B. A. Vasu, “Application of EWMA chart for monitoring process mean in paper industry,” Management Science Letter, vol. 9, no. 4, pp. 571–576, 2019.
- A. K. Patel and J. Divecha, “Modified exponentially weighted moving average (EWMA) control chart for an analytical process data,” Journal of Chemical Engineering and Materials Science, vol. 2, no. 1, pp. 12–20, 2011.
- P. Castagliola, “An X/R–EWMA control chart for monitoring the process sample median,” International Journal of Reliability, Quality and Safety Engineering, vol. 8, no. 2, pp. 123–135, 2001.
- K. J. Chung, “A simplified procedure for the economic design of control charts: A unified approach,” Journal Engineering Optimization, vol. 4, pp. 313–320, 1991.