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Parametric and non-parametric methods for monitoring nonlinear fuzzy profiles

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Abstract

In many statistical process control (SPC) applications, the quality of a process or product is represented by a relationship or the so-called quality profile, between a quality characteristic and one or more explanatory variables. The process quality characteristics are sometimes measured with a reasonable degree of approximation. Often, especially when qualitative assessments arise, evaluation of quality characteristics is carried out ambiguously or using linguistic qualifiers. The fuzzy sets theory has proved to be a well-established approach for dealing with uncertainty due to the approximate measurement, ambiguity in subjective evaluations, or vagueness in linguistic variables. Our main purpose was to present and compare four methods of monitoring nonlinear fuzzy profiles, for which different nonlinear fuzzy regression modeling approaches are considered. The first two methods are “a data-driven fuzzy rule-based” and “an extended least square support vector machine (LS-SVM),” for which the profile is characterized without considering a predefined mathematical relationship. However, for the other two methods, a specific form of the profile was needed. The third method, namely “a modified fuzzy regression model,” was initially invented for linear models. Besides, the fourth method employs “the fuzzy least square method” based on linearizing transformation. The exponentially weighted moving average (EWMA) control statistic was used to derive the control statistic to be plotted on the univariate as well as multivariate control charts. An extensive simulation study was conducted to compare the performance of the methods, and the average run length (ARL) criterion was considered to assess the detect-ability of control charts against various out-of-control conditions. Our comparison results indicated that the multivariate EWMA (MEWMA) chart based on the LS-SVM method outperforms the rest in detecting process shifts with smaller values of ARL when the process undergoes out-of-control conditions.

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Data Availability

Data and materials are presented in the manuscript.

Abbreviations

SQC:

Statistical quality control

SPC:

Statistical process control

LS-SVM:

Least square support vector machine

EWMA:

Exponentially weighted moving average

MEWMA:

Multivariate exponentially weighted moving average

ARL:

Average run length

SDRL:

Standard deviation of run-length

I-MR:

Individual and moving range

\( {\overset{\sim }{\mathrm{FT}}}^2 \) :

Fuzzy T-square

FEWMA:

Fuzzy exponentially weighted moving average

FMEWMA:

Fuzzy multivariate exponentially weighted moving average

FCUSUM:

Fuzzy cumulative sum

FMCUSUM:

Fuzzy multivariate cumulative sum

MLE:

Maximum likelihood estimator

EM:

Expectation-maximization

TSK:

Takagi-Sugeno-Kang

SVM:

Support vector machine

SVR:

Support vector regression

CTQ:

Critical-to-quality

LCL:

Lower control limit

UCL:

Upper control limit

FRB:

Fuzzy Rule-Based

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Code availability

All coding is done in MATLAB, and it will be sent if needed.

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Authors and Affiliations

  1. Department of Industrial Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Mohammadreza Nasiri Boroujeni, Yaser Samimi & Emad Roghanian

Authors
  1. Mohammadreza Nasiri Boroujeni
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  2. Yaser Samimi
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  3. Emad Roghanian
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Contributions

This paper’s main contribution was to investigate the application of statistical process control charts for monitoring fuzzy nonlinear profiles when different approaches to nonlinear fuzzy regression modeling are employed. Different methods are found in the relevant literature for characterizing nonlinear fuzzy functional relationships. However, no published research has shown such a comparative study of various parametric and nonparametric methods in terms of their capability to provide an appropriate basis for process monitoring using statistical process control charts. To clarify contributions of each author, it should be added that the first author prepared the literature review and provided simulation codes. The second author designed the framework of statistical process control statistic and charts. The third author provided the fuzzy computational algorithms and rule-based system.

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Correspondence to Mohammadreza Nasiri Boroujeni.

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Nasiri Boroujeni, M., Samimi, Y. & Roghanian, E. Parametric and non-parametric methods for monitoring nonlinear fuzzy profiles. Int J Adv Manuf Technol 118, 67–84 (2022). https://doi.org/10.1007/s00170-021-07187-z

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