Abstract
Efficient control charts are used to maintain and improve the manufacturing, industrial and service processes. Control chart is a basic tool of quality practitioners for producing products that meet market requirements in terms of quality, reliability and functionality. Lack of normality is a common occurrence in these processes. In such situations, distribution-free control charts are increasingly used for process monitoring. The nonparametric exponentially weighted moving average chart is a frequently used memory-type control chart in the process monitoring. The drawback of this chart is that it performs efficiently only on smaller values of the smoothing parameter. To compensate this drawback, a modified nonparametric exponentially weighted moving average chart under progressive setup based on sign and arcsine test statistics is proposed in this study. The prominent quality of the proposed scheme is that it performs efficiently in detecting small and persistent shifts in the process location under each choice of the smoothing parameter. The performance of the proposed chart has been investigated through simulations that use run length profiles (average run length, median run length and standard deviation of run length). When the performance of the proposed chart is compared with alternatives, its ability to detect small and persistent shifts is much better. Two real-life applications associated with hard-bake and piston rings manufacturing processes are included that show the demonstration of the proposed charts.
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Acknowledgements
The aforementioned work was partially supported by National Natural Science Foundation of China (No. 11731015 and 11571148). The authors are grateful to the editor and referees for their constructive comments that led to substantial improvements in the article.
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Appendices
Appendix
NPMEP chart is the mixture of NPEWMA and nonparametric PM chart. To derive the mean and variance of NPMEP chart, first we must know about the mean and variance of NPEWMA and nonparametric PM chart.
Appendix-A
Mean and variance of NPEWMA chart under sign test statistic. The NPEWMA statistic defined in Eq. (2) can be written as:
where \(p_{0} = 1/2\).
Appendix-B
Mean and variance of nonparametric PM chart. The nonparametric PM statistic defined in Eq. (5) is given as:
\({\text{NPPM}}_{{S_{t} }} = \frac{{S_{1}^{ + } + S_{2}^{ + } + \cdots + S_{t}^{ + } }}{t}\),
Appendix-C
Mean and variance of the proposed NPMEP sign chart is derived as follows:
For variance of the NPMEP statistic.
\(E_{{S_{j}^{ + } }} = \omega \mathop \sum \nolimits_{i = 0}^{j - 1} \left( {1 - \omega } \right)^{i} S_{t - j}^{ + } + \left( {1 - \omega } \right)^{j} E_{0}\), where \(E_{0}\) is the initial value which is equal to the mean of the NPEWMA statistic.
The NPMEP statistic
Consider
Open these summations and then we get
For variance
\(= \frac{{\omega^{2} }}{{t^{2} }}\) \(\left( {\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)} \right)^{2} \frac{n}{4} + 0;\) as \({\text{var}} \left( {\frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0} } \right) = 0,\)
Using geometric progression to simplify the expression given in (C2)
Appling sum of the geometric progressions
Appendix-D
Mean and variance of the proposed NPMEP arcsine chart.
\(E({\text{NPMEP}}_{{M_{t} }} ) = \mathop \sum \limits_{j = 1}^{t} \frac{{E\left( {E_{{M_{j} }} } \right)}}{t},\),
For variance of \({\text{NPMEP}}_{{M_{t} }}\)
Consider
Open these summations and then we get
For variance
\(= \frac{{\omega^{2} }}{{t^{2} }}\) \(\left( {\mathop \sum \limits_{j = 1}^{t} \left( {\mathop \sum \limits_{i = 0}^{t - j} \left( {1 - \omega } \right)^{i} } \right)} \right)^{2} \frac{1}{4n} + 0\) as \({\text{var}} \left( {\frac{1}{t}\mathop \sum \limits_{j = 1}^{t} \left( {1 - \omega } \right)^{j} E_{0} } \right) = 0\)
Appling sum of the geometric progressions
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Ali, S., Abbas, Z., Nazir, H.Z. et al. On Designing Mixed Nonparametric Control Chart for Monitoring the Manufacturing Processes. Arab J Sci Eng 46, 12117–12136 (2021). https://doi.org/10.1007/s13369-021-05801-6
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DOI: https://doi.org/10.1007/s13369-021-05801-6