1997 journal article

Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation

*FUZZY SETS AND SYSTEMS*, *91*(1), 1–13.

author keywords: fuzzy arithmetic; triangular fuzzy numbers; trapezoidal fuzzy numbers; membership functions; computational complexity; multiplication; product operator; multiplication approximation

TL;DR:
The error of approximation for repeated use of the multiplication operand is examined and it is shown it can be sufficiently large in simple circumstances to produce erroneous results.
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Source: Web Of Science

Added: August 6, 2018

Triangular and trapezoidal fuzzy numbers are commonly used in many applications. It is well known that the operators used for the non-linear operations such as multiplication, division, and inverse are approximations to the actual operators. It is also commonly assumed that the error introduced by the approximations is small and acceptable. This paper examines the error of approximation for repeated use of the multiplication operand and shows it can be sufficiently large in simple circumstances to produce erroneous results. The computational complexity of the multiplication operation is analyzed and shown to be sufficiently complex that a computationally simpler approximation is needed. As a consequence, the error produced by the approximation for the multiplication operation is analyzed and a new approximation developed that is accurate for a large range of problems. An error expression is developed for the new approximation that can be used to determine when it is producing unacceptable results.