Similarity measures of generalized trapezoidal fuzzy numbers for fault diagnosis

Methodologies and Application


In this paper, we propose a new similarity measure between generalized trapezoidal fuzzy numbers and several synthesized similarity measures to solve fault diagnosis problem by merging our proposed measures with Dempster–Shafer evidence theory. Firstly, combining the exponential distance with numerical indexes of generalized trapezoidal fuzzy number, such as the span, the center width and the height, etc, we propose a new similarity measure between generalized trapezoidal fuzzy numbers. Secondly, we introduce an evaluation index, distinguish ability, to evaluate the performance of different similarity measures. The experimental results show that our proposed similarity measure can overcome the drawbacks of the existing similarity measures. Thirdly, to solve fault diagnosis problems, we propose three formulas to integrate several single similarity measures to a synthesized one. Finally, based on Dempster–Shafer evidence theory, we transform each similarity measure between fault model and test model, the synthesized similarity measures to their corresponding basic probability assignments to deal with fault diagnosis problem, the results show that our proposed similarity measure is more effective than some other existing similarity measures.


Similarity measure Generalized trapezoidal fuzzy number Fault diagnosis Synthesized similarity measure Dempster–Shafer evidence theory 



The authors wish to express their gratitude to the anonymous referees and the Editor-in-Chief, Professor Antonio Di Nola, for their kind suggestions and helpful comments in revising the paper. This study was funded by Grants from the National Natural Science Foundation of China (10971243), Grants from the Key Research Plan of Hebei Province (17210109D), and the Grants from Hebei Normal University (L2015k01, L2017B09, S2016Y13).

Compliance with ethical standards

Conflict of interest

We declare that we have no conflict of interest.

Human ad animal rights

This article does not contain any studies with human participants performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.College of Information Science and TechnologyBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.College of Mathematics and Information ScienceHebei Normal UniversityShijiazhuangPeople’s Republic of China

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