@MISC{FUZZY_7.MIZ,
 TITLE = {Isosceles Triangular and Isosceles Trapezoidal Membership Functions
      Using Centroid Method},
  AUTHOR = {Mitsuishi, Takashi},
  DAY = {31},
  MONTH = {March},
  YEAR = {2023},
  ADDRESS1 = {Faculty of Business and Informatics, Nagano University, Japan},
 SUMMARY = {Since isosceles triangular and trapezoidal membership functions \cite{Gonda:2004}
  are easy to manage, they applied to various fuzzy approximate
  reasoning \cite{Mamdani:1974}, \cite{Mizumoto:1990}, \cite{Ross:2010}.
  The centroids of isosceles triangular and trapezoidal membership functions
  are mentioned in this article \cite{Leekwijck:1999}, \cite{Katafuchi:2001}.
  Some proposition of the composition mapping denoted \verb!f +* g! where \verb!f!, \verb!g!
  are affine mappings, are proved \cite{Giachetti:1997}, \cite{Luciano:2009}.
  Then different notations for the same isosceles triangular
  and trapezoidal membership function are formalized.
  We proved the agreement of the same function expressed with different
  parameters and formalized those centroid with those parameters.
  In addition, various properties of membership functions
  on intervals where the endpoints
  of the domain are fixed and on general intervals are formalized.}}

@inproceedings{Luciano:2009,
  title={Fuzzy Arithmetic with Parametric {LR} Fuzzy Numbers},
  author={Stefanini, Luciano and Sorini, Laerte},
  booktitle={Procedings of the Joint 2009 International Fuzzy Systems Association
    World Congress and 2009 European Society of Fuzzy Logic and Technology Conference},
  pages={600--605},
  year={2009}}

@article{Giachetti:1997,
title = {A parametric representation of fuzzy numbers and their arithmetic operators},
journal = {Fuzzy Sets and Systems},
volume = {91},
number = {2},
pages = {185-202},
year = {1997},
note = {Fuzzy Arithmetic},
issn = {0165-0114},
doi = {https://doi.org/10.1016/S0165-0114(97)00140-1},
url = {https://www.sciencedirect.com/science/article/pii/S0165011497001401},
author = {Giachetti, Ronald E. and Young, Robert E.}}

@book{Ross:2010,
  author={Ross, Timothy J.},
  title={Fuzzy Logic With Engineering Application},
  publisher={John Wiley and Sons Ltd},
  year={2010}}

@article{Leekwijck:1999,
  title={Defuzzification: criteria and classification},
  author={Van Leekwijck, Werner and Kerre, Etienne E},
  journal={Fuzzy Sets and Systems},
  volume={108},
  number={2},
  pages={159--178},
  year={1999},
  publisher={Elsevier}}
  
@article{Katafuchi:2001,
  title={Investigation of Deffuzification in
  Fuzzy Inference: Proposal of a New Defuzzification Method (in Japanese)},
  author={Katafuchi, Tetsuro and Asai, Kiyoji and Fujita, Hiroshi},
  journal={Medical Imaging and Information Sciences},
  volume={18},
  number={1},
  pages={19-30},
  year={2001},
  doi={10.11318/mii1984.18.19}
}

@article{Mamdani:1974,
  author="Mamdani, E.H.",
  title="Application of fuzzy algorithms for control of simple dynamic plant",
  journal="Proc. IEE",
  year="1974",
  volume="121",
  number="12",
  pages="1585-1588"}

@inproceedings{Mizumoto:1990,
  title={Improvement of fuzzy control (IV)-Case by product-sum-gravity method},
  author={Mizumoto, Masaharu},
  booktitle={Proc. 6th Fuzzy System Symposium, 1990},
  pages={9--13},
  year={1990}
}

@INPROCEEDINGS{8712319,
  author={T. {Mitsuishi}},
  booktitle={2018 International Electrical Engineering Congress (iEECON)}, 
  title={Uncertain Defuzzified Value of Periodic Membership Function}, 
  year={2018},
  volume={},
  number={},
  pages={1-4},
  doi={10.1109/IEECON.2018.8712319}}

@INPROCEEDINGS{7274550,
  author={T. {Mitsuishi} and N. {Shimada} and T. {Homma} and M. {Ueda} and M. {Kochizawa} and Y. {Shidama}},
  booktitle={2015 IEEE 7th International Conference on Cybernetics and Intelligent Systems (CIS) and IEEE Conference on Robotics, Automation and Mechatronics (RAM)}, 
  title={Continuity of approximate reasoning using fuzzy number under Lukasiewicz t-norm}, 
  year={2015},
  volume={},
  number={},
  pages={71-74},
  doi={10.1109/ICCIS.2015.7274550}}

@article{mitsuishi2009compactness,
  title={Compactness of family of fuzzy sets in {$L^2$} space with application to optimal control},
  author={Mitsuishi, Takashi and Shidama, Yasunari},
  journal={IEICE transactions on fundamentals of electronics, communications and computer sciences},
  volume={92},
  number={4},
  pages={952--957},
  year={2009},
  publisher={The Institute of Electronics, Information and Communication Engineers}
}

@ARTICLE{4497918,
  author={Seki, Hirosato and Ishii, Hiroaki and Mizumoto, Masaharu},
  journal={IEEE Transactions on Fuzzy Systems}, 
  title={On the Generalization of Single Input Rule Modules Connected Type Fuzzy Reasoning Method}, 
  year={2008},
  volume={16},
  number={5},
  pages={1180-1187},
  doi={10.1109/TFUZZ.2008.924182}}

%% SoSE
@INPROCEEDINGS{6575237,  
  author={T. {Mitsuishi} and T. {Terashima} and N. {Shimada} and T. {Homma} and Y. {Shidama}},
  booktitle={2013 8th International Conference on System of Systems Engineering}, 
  title={SIRMs fuzzy approximate reasoning using L-R fuzzy number as premise valuable}, 
  year={2013},
  volume={},
  number={},
  pages={25-27},
  doi={10.1109/SYSoSE.2013.6575237}}

@article{Gonda:2004,
author="Eikou Gonda and Hitoshi Miyata and Masaaki Ohkita",
title="Self-Turning of Fuzzy Rules with Different Types of MSFs (in Japanese)",
journal="Journal of Japan Society for Fuzzy Theory and Intelligent Informatics",
ISSN="13477986",
year="2004",
volume="16",
number="6",
pages="540-550",
DOI="10.3156/jsoft.16.540",
URL="https://cir.nii.ac.jp/crid/1390001205185848192"
}
