@ARTICLE{FUZZY_8.MIZ,
  AUTHOR = {Mitsuishi, Takashi},
  TITLE = {Symmetrical Piecewise Linear Functions Composed by Absolute Value Function},
  DAY = {18},
  MONTH = {December},
  YEAR = {2023},
  ADDRESS1 = {Faculty of Business and Informatics, Nagano University, Japan},
  SUMMARY = {In this paper, some mathematical properties about piecewise linear functions are formalized.
  The focused piecewise linear functions are symmetrical and composed by absolute function.
  L-R fuzzy number is applied for various fields \cite{DuboisPrade:1978}, \cite{Giachetti:1997},
  \cite{Luciano:2009}, \cite{Katafuchi:2001}.
  Since isosceles triangle type and isosceles trapezoid type membership functions are simple \cite{Gonda:2004},
  they are applied for the membership functions of  L-R fuzzy number in most cases \cite{Mitsuishi:2016}.
  It is formalized that the membership functions of isosceles triangle type and
  isosceles trapezoid type can be constructed by absolute value functions.
  It is proved and formalized that the centroids of the composite function of
  two continuous functions are the weighted averages of the areas and centroids
  of the functions that compose them \cite{Leekwijck:1999}.
  Moreover, some calculation and operation between membership functions
  for fuzzy approximate reasoning \cite{Ross:2010}, e.g. Mamdani method \cite{Mamdani:1974}
  and the product-sum-gravity method \cite{Mizumoto:1990} are formalized.}}

%%% replace in external.bib
@article{DuboisPrade:1978,
 AUTHOR={Dubois, Didier and Prade, Henri},
 TITLE={Operations on fuzzy numbers},
journal = {International Journal of Systems Science},
volume = {9},
number = {6},
pages = {613-626},
year  = {1978},
publisher = {Taylor & Francis},
doi = {10.1080/00207727808941724}}

%% add to external.bib
@INPROCEEDINGS{Mitsuishi:2016,
  author={T. {Mitsuishi} and T. {Terashima} and N. {Shimada} and T. {Homma} and Y. {Shidama}},
  booktitle={2016 IEEE Symposium on Sensorless Control for Electrical Drives (SLED)},
  title={Approximate reasoning using LR fuzzy number as input for sensorless fuzzy control},
  year={2016},
  volume={},
  number={},
  pages={1-5},
  doi={10.1109/SLED.2016.7518804}
}

%% replace in external.bib
@article{Katafuchi:2001,
  title={Investigation of Defuzzification 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}}


%% replace in external.bib
@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.}}
