Robust Synthetic Control Charting
Corresponding email: ayurahman@uum.edu.my
Published at : 19 Apr 2021
Volume : IJtech
Vol 12, No 2 (2021)
DOI : https://doi.org/10.14716/ijtech.v12i2.4216
Rahman, A.A., Syed-Yahaya, S.S., Atta, A.M.A., 2021. Robust Synthetic Control Charting. International Journal of Technology. Volume 12(2), pp. 349-359
| Ayu Abdul Rahman | Department of Mathematics and Statistics, School of Quantitative Sciences, Universiti Utara Malaysia, 06010 UUM, Sintok, Kedah, Malaysia |
| Sharipah Soaad Syed-Yahaya | Department of Mathematics and Statistics, School of Quantitative Sciences, Universiti Utara Malaysia, 06010 UUM, Sintok, Kedah, Malaysia |
| Abdu Mohammed Ali Atta | Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, P.O. Box 2713-Doha |
The
proposal of synthetic charting is based on the normality assumption.
This assumption, however, is hard to attain in practice. Therefore, it is
important to examine how the chart would response under some types of
non-normal data. The focus of this article is to monitor location shifts using
a synthetic chart and to validate its performance under the g-and-h
distributions. This study shows that the effect of non-normality on the
standard synthetic chart is not trivial, especially when the underlying
distributions are heavy-tailed. With these types of distribution, Phase II
monitoring of location using median-based estimators is advisable. In doing so,
the synthetic chart is more robust to departure in the normality assumption
with little effect on its out-of-control performance. This paper shows how the
synthetic parameters should be attained to reflect the use of the modified
one-step M-estimator (MOM) in its Winsorized version, and the
median for Phase II. The assessment is based on the average run length and
supported by the extra quadratic loss function. Finally, the practical
application of the proposed synthetic charts is illustrated using real data.
Average run length; Extra quadratic loss; Robust estimators; Shewhart chart; Synthetic chart
Competition in the manufacturing
sector involves delivering high-quality products or services quickly at a low
cost (Baby and Jebadurai, 2018). Undeniably,
cost and quality are among key factors in the development of manufacturing
strategies (Nurcahyo et al., 2019). To
achieve such a feat, lean production is usually advocated. Lean production
typically involves the use of quality tools such as control charts, flowcharts
and fish bone diagrams, aiming to reduce or eliminate waste that occur during
the production process. Today, the lean concept has been extended considerably
into various fields beyond manufacturing (Driouach
et al., 2019). As such, the use of quality tools, especially the control
chart, is vital.
Application of the standard control chart,
which typically relies upon the normality assumption, is favorable in many areas of the statistical process
monitoring (SPM), since the tool can offer fast detection of special cause(s) in the process. Subsequently, corrective action may be
undertaken to preserve process quality. One of the many variable control charts
introduced in the literature is the synthetic chart for process mean, by Wu and Spedding (2000). Its use in practice is
justified based on its moderate robustness to non-normality
for a large sample size n, i.e., n
Under
non-normality, it is better to use robust
statistics instead of the sample mean to monitor changes in the data (Rocke, 1989). This way, outliers in the subgroups do not
cause a signal to occur and, as such, the chart only responds to genuine shifts
in the process. Existing work on the design and implementation of synthetic
charts, as based on the median estimator, is noted in the literature. See, for
example, Hu et al. (2018) and Tran et al. (2019). It is not clear how this
robust chart performance is impacted under severe non-normality, but it seems
that making use of the robust estimator with the synthetic control structure
would lead to better quality tools, as changes in the process can be detected
efficiently.
Notably, there is an increasing frequency of
coverage on works associated with control charts in the quality technology
field, reflecting the relevancy of this quality tool in real application (Jensen et al., 2018). Unfortunately, as reviewed
by Bono et al. (2017), real data usually
defy the assumption of normality. Thus, it is important to address the
robustness (to non-normality) issue and to evaluate the existing and any newly
proposed synthetic charts with severe deviation from normality. To fill this
gap, we investigate the
performance of the synthetic chart that is based on four different estimators,
including the aforementioned median estimator, in monitoring the process
location of Phase II samples upon violation of the normality assumption. In the
next section, we give the details regarding the selected Phase II estimators
before explaining the synthetic charting structure.
Robustification in this study is intended to lead to a synthetic
chart, the IC ARL of which is not sensitive to the non-normal data or the
particular values of design parameters, kS
and LS, but which will
detect location shifts reliably. We identify MOM as best fit for the synthetic chart. This way, researchers in
many application domains will not be constrained by the normality assumption,
but may instead work with the original data without having to worry about the shape of the
distributions. While this paper concentrated on the performance of the
synthetic chart based on the assumption of known process parameters, we believe
that the findings can be easily extended to the estimated parameters case,
which will be considered in our future work.
The decision of what shift for which to design
rarely gets attention in the literature, and even more rarely under non-normal
cases. The three design shifts employed in this study provide valuable
information for further implementation and for the designing of synthetic
charts in practice. Generally, the synthetic charts designed for a small shift
(
The
authors would like to acknowledge the work that has led to this paper, which is
supported by Universiti Utara Malaysia, Fundamental Research Grant Scheme (S/0
Code 13578) of the Ministry of Higher Education, Malaysia.
Baby, B.N.P., Jebadurai, D.S., 2018. Implementation of Lean
Principles to Improve the Operations of a Sales Warehouse in the Manufacturing
Industry. International Journal of Technology. Volume 9(1), pp.46–54
Bono, R., Blanca, M.J., Arnau, J., Gomez-Benito, J., 2017.
Non-normal Distributions Commonly used in Health, Education, and Social
Sciences: A Systematic Review. Frontiers in Psychology, Volume 8, pp.
1–6
Calzada, M.E., Scariano, S.M., 2001. The Robustness of the
Synthetic Control Chart to Non-normality. Communications in Statistics
- Simulation and Computation, Volume
30(2), pp. 311–326
Driouach, L., Zarbane, K., Beidouri, Z., 2019. Literature
Review of Lean Manufacturing in Small and Medium-sized Enterprises. International
Journal of Technology. Volume 10(5), pp. 930-941
Figueiredo, F., Gomes, M.I., 2004. The Total Median in Statistical
Quality Control. Applied Stochastic Models in Business and Industry,
Volume 20(4), pp. 339–353
Haddad, F.S., Syed-Yahaya, S.S., Alfaro, J.L., 2012.
Alternative Hotelling’s T2 Charts using Winsorized Modified One-Step
M-estimator. Quality and Reliability Engineering
International, Volume 29,
pp. 583–593
Hoaglin, D.C., 1985.
Summarizing shape numerically:
The g-and-h Distributions.
In D. Hoaglin, F. Mosteller,
J. Tukey (Eds.), Wiley, Hoboken, NJ.
Exploring Data Tables, Trends, and Shapes, pp. 461–513
Hu, X-L., Wu, S., Zhong, J-L., 2018. Optimal
Design of the Synthetic Median Chart based on Average Run Length and Expected
Average Run Length. In: Proceedings of the 24th ISSAT
International Conference on Reliability and Quality in Design 2018, Toronto, 2
-4 August, Canada
Jensen, W.A., Montgomery, D.C., Tsung, F.,
Vining, G.G., 2018. 50 Years of the Journal of Quality Technology. Journal
of Quality Technology, Volume 50(1), pp. 2–16
Jones-Farmer L.A., Jordan, V., Champ, C.W., 2009.
Distribution-Free Phase I Control Charts for Subgroup Location. Journal of Quality Technology, Volume
41(3), pp. 304–316
Malela-Majika, J.C. Motsepa,
C.M., Graham, M.A., 2019. A New Double Sampling
Martinez, J., Iglewicz, B., 1984. Some Properties of the Tukey
g and h Family of Distributions. Communications in Statistics - Theory and
Methods, Volume 13(3), pp. 353–369
Montgomery, D.C., 2013. Introduction to Statistical Quality
Control (7th Ed.), John Wiley & Sons, Hoboken, NJ
Nurcahyo, R., Wibowo, A.D.,
Robasa, R., Cahyati, I., 2019. Development of a Strategic Manufacturing Plan
from a Resource-Based Perspective . International Journal of Technology.
Volume 10(1), pp. 178–188
Ozdemir, A.F., 2010. Comparing Measures of Location When the Underlying
Distribution has Heavier Tails than Normal. lstatistikc?iler
Dergisi, Volume 3, pp. 8–16
Rocke, D.M., 1989. Robust Control Charts. Technometrics,
Volume 31(2), pp. 173–184
Rousseeuw, P.J., Croux, C., 1993. Alternatives to the Median Absolute
Deviation. Journal of the American Statistical Association, Volume 88(424),
pp. 1273–1283
Tran , P.H., Tran, K.P., Rakitzis, A., 2019. A Synthetic
Median Control Chart for Monitoring the Process Mean with Measurement Errors. Quality
and Reliability Engineering International, Volume 35, pp. 1100–1116
Wu, Z., Spedding, T.A., 2000. A Synthetic Control Chart for Detecting
Small Shifts in the Process Mean. Journal of Quality Technology, Volume 32(1),
pp. 32–38
Wilcox, R.R., Keselman, H.J., 2003a. Modern Robust Data
Analysis Methods: Measures of Central Tendency. Psychological Methods,
Volume 8(3), pp. 254–274
Wilcox, R.R., Keselman, H.J., 2003b. Repeated Measures One-Way
ANOVA based on a Modified One-Step M-Estimator. British Journal of
Mathematical and Statistical Psychology, Volume 56(1), pp. 15–25