##
**An application of fuzzy random variables to control charts.**
*(English)*
Zbl 1206.93062

Summary: The two most significant sources of uncertainty are randomness and incomplete information. In real systems, we wish to monitor processes in the presence of these two kinds of uncertainty. This paper aims to construct a fuzzy statistical control chart that can explain existing fuzziness in data while considering the essential variability between observations. The proposed control chart is an extension of Shewhart’s \(\overline X - S^2\) control charts in fuzzy space. The proposed control chart avoids defuzzification methods such as fuzzy mean, fuzzy mode, fuzzy midrange, and fuzzy median. It is well known that using different representative values may cause different conclusions to be drawn about the process and vague observations to be reduced to exact numbers, thereby reducing the informational content of the original fuzzy sets. The out-of-control states are determined based on a fuzzy in-control region and a simple and precise graded exclusion measure that determines the degree to which fuzzy subgroups are excluded from the fuzzy in-control region. The proposed chart is illustrated with a numerical example.

### MSC:

93C42 | Fuzzy control/observation systems |

93E03 | Stochastic systems in control theory (general) |

93B51 | Design techniques (robust design, computer-aided design, etc.) |

PDF
BibTeX
XML
Cite

\textit{A. Faraz} and \textit{A. F. Shapiro}, Fuzzy Sets Syst. 161, No. 20, 2684--2694 (2010; Zbl 1206.93062)

Full Text:
DOI

### References:

[1] | Asai, K., Fuzzy systems for management, (1995), IOS Press Amsterdam · Zbl 0842.90069 |

[2] | Bandler, W.; Kohout, L.J., Fuzzy power sets and fuzzy implication operators, Fuzzy sets and systems, 4, 13-30, (1980) · Zbl 0433.03013 |

[3] | Betta, G.; Capriglione, D.; Tomasso, G., Evaluation of the measurement uncertainties in the conducted emissions from adjustable speed electrical power drive systems, IEEE trans. instrum. meas., 53, 963-968, (2004) |

[4] | Bosc, P.; Pivert, O., About approximate inclusion and its axiomatization, Fuzzy sets and systems, 157, 1438-1454, (2006) · Zbl 1104.03051 |

[5] | Burillo, P.; Frago, N.; Fuentes, R., Inclusion grades and fuzzy implication operators, Fuzzy sets and systems, 114, 417-429, (2000) · Zbl 0962.03050 |

[6] | Cheng, C.B., Fuzzy process control: construction of control charts with fuzzy numbers, Fuzzy sets and systems, 154, 287-303, (2005) |

[7] | A. Colubi, Statistical inference about the means of fuzzy random variables: applications to the analysis of fuzzy- and real-valued data, Fuzzy Sets and Systems 160 (2009) 344-356, doi: 10.1016/j.fss.2007.12.019. · Zbl 1175.62021 |

[8] | I. Couso, D. Dubois, S. Montes, L. Sanchez, On various definitions of the variance of a fuzzy random variable, in: 5th Internat. Sympos. on Imprecise Probabilities and their Applications, Prague, Czech Republic, 2007. |

[9] | Couso, I.; Dubois, D., On the variability of the concept of variance for fuzzy random variables, IEEE trans. fuzzy syst., 17, 1070-1080, (2009) |

[10] | Dubois, D.; Prade, H., Ranking fuzzy numbers in the setting of possibility theory, Inform. sci., 30, 183-224, (1983) · Zbl 0569.94031 |

[11] | Evans, J.R.; Lindsay, W.M., The management and control of quality, (1999), South-Western College Publishing Cincinnati |

[12] | Faraz, A.; Moghadam, M.B., Fuzzy control chart a better alternative for shewhart average chart, Qual. quantity, 41, 375-385, (2007) |

[13] | A. Faraz, R.B. Kazemzadeh, M.B. Moghadam, A. Bazdar, Constructing a fuzzy Shewhart control chart for variables when uncertainty and randomness are combined, Qual. Quantity (2009), doi 10.1007/s11135-009-9244-9. |

[14] | L. Finkelstein, R.Z. Morawski, L. Mari (Eds.), Logical and philosophical aspects of measurement, Measurement 38 (4) (2005) (special issue). |

[15] | Gil, M.A.; López-Díaz, M.; Ralescu, D.A., Overview on the development of fuzzy random variables, Fuzzy sets and systems, 157, 2546-2557, (2006) · Zbl 1108.60006 |

[16] | P. Grzegorzewski, Control charts for fuzzy data, in: Proc. Fifth European Congress on Intelligent Techniques and Soft Computing EUFIT’97, Aachen, 1997, pp. 1326-1330. |

[17] | Grzegorzewski, P.; Hryniewicz, O., Soft methods in statistical quality control, Control cybernet, 29, 119-140, (2000) · Zbl 1030.90019 |

[18] | Gulbay, M.; Kahraman, C., An alternative approach to fuzzy control charts: direct fuzzy approach, Inform. sci., 177, 463-1480, (2007) · Zbl 1120.93332 |

[19] | Kanagawa, A.; Tamaki, F.; Ohta, H., Control charts for process average and variability based on linguistic data, Internat. J. production res., 31, 913-922, (1993) · Zbl 0769.62076 |

[20] | Körner, R., On the variance of fuzzy random variables, Fuzzy sets and systems, 92, 83-93, (1997) · Zbl 0936.60017 |

[21] | Laviolette, M.; Seaman, J.W.; Barrett, J.D.; Woodall, W.H., A probabilistic and statistical view of fuzzy methods, with discussion, Technometrics, 37, 249-292, (1995) · Zbl 0837.62081 |

[22] | Puri, M.L.; Ralescu, D., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004 |

[23] | Raz, T.; Wang, J.-H., Probabilistic and membership approaches in the construction of control charts for linguistic data, Production plann. cont., 1, 147-157, (1990) |

[24] | Shewhart, W.A., Economic control of quality of manufactured product, (1931), D. Van Nostrand Inc Princeton, NJ |

[25] | S. Senturk, N. Erginel, Development of fuzzy Xbar-R and Xbar-S control charts using \(\alpha\)-cuts, Inform. Sci. (2008), doi: 10.1016/j.ins.2008.09.022. |

[26] | Shapiro, A.F., Fuzzy random variables, Insurance math. econom., 44, 307-314, (2009) · Zbl 1166.91018 |

[27] | Sinha, D.; Dougherty, E.R., Fuzzification of set inclusion: theory and applications, Fuzzy sets and systems, 55, 15-42, (1993) · Zbl 0788.04007 |

[28] | Taleb, H.; Limam, M., On fuzzy and probabilistic control charts, Internat. J. production res., 40, 2849-2863, (2002) |

[29] | Terán, P., Probabilistic foundations for measurement modeling with fuzzy random variables, Fuzzy sets and systems, 158, 973-986, (2007) · Zbl 1120.60032 |

[30] | Wang, J.-H.; Raz, T., On the construction of control charts using linguistic variables, Internat. J. production res., 28, 477-487, (1990) |

[31] | Woodall, W.; Tsui, K.-L.; Tucker, G.L., A review of statistical and fuzzy control charts based on categorical data, frontiers in statistical quality control, vol. 5, (1997), Physica-Verlag Heidelberg, Germany · Zbl 0900.62538 |

[32] | Young, V., Fuzzy subsethood, Fuzzy sets and systems, 77, 371-384, (1996) · Zbl 0872.94062 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.