FuzzyNumber-class: S4 class Representing a Fuzzy Number

Description

Formally, a fuzzy number $A$ (Dubois, Prade, 1987) is a fuzzy subset of the real line $R$ with membership function $\mu$ given by:

	\| $0$	if $x < a1$,
	\| $left((x-a1)/(a2-a1))$	if $a1 \le x < a2$,
$\mu(x)$ =	\| $1$	if $a2 \le x \le a3$,
	\| $right((x-a3)/(a4-a3))$	if $a3 < x \le a4$,
	\| $0$	if $a4 < x$,

where $a1,a2,a3,a4\in R$, $a1 \le a2 \le a3 \le a4$, $left: [0,1]\to[0,1]$ is a nondecreasing function called the left side generator of $A$, and $right: [0,1]\to[0,1]$ is a nonincreasing function called the right side generator of $A$. Note that this is a so-called L-R representation of a fuzzy number.

Alternatively, it may be shown that each fuzzy number $A$ may be uniquely determined by specifying its $\alpha$-cuts, $A(\alpha)$. We have $A(0)=[a1,a4]$ and $$A(\alpha)=[a1+(a2-a1)*lower(\alpha), a3+(a4-a3)*upper(\alpha)]$$ for $0<\alpha\le 1$, where $lower: [0,1]\to[0,1]$ and $upper: [0,1]\to[0,1]$ are, respectively, strictly increasing and decreasing functions satisfying $lower(\alpha)=\inf\{x: \mu(x)\ge\alpha\}$ and $upper(\alpha)=\sup\{x: \mu(x)\ge\alpha\}$.

Arguments

Slots

a1:: Single numeric value specifying the left bound for the support.
a2:: Single numeric value specifying the left bound for the core.
a3:: Single numeric value specifying the right bound for the core.
a4:: Single numeric value specifying the right bound for the support.
lower:: A nondecreasing function [0,1]->[0,1] that gives the lower alpha-cut bound.
upper:: A nonincreasing function [0,1]->[1,0] that gives the upper alpha-cut bound.
left:: A nondecreasing function [0,1]->[0,1] that gives the left side function.
right:: A nonincreasing function [0,1]->[1,0] that gives the right side function.

Child/sub classes

'>TrapezoidalFuzzyNumber, '>PiecewiseLinearFuzzyNumber, '>PowerFuzzyNumber, and '>DiscontinuousFuzzyNumber

Details

Please note that many algorithms that deal with fuzzy numbers often use $\alpha$-cuts rather than side functions.

Note that the FuzzyNumbers package also deals with particular types of fuzzy numbers like trapezoidal, piecewise linear, or ``parametric'' FNs.

References

Dubois D., Prade H. (1987), Fuzzy numbers: An overview, In: Analysis of Fuzzy Information. Mathematical Logic, vol. I, CRC Press, pp. 3-39.

Examples

Run this code

# NOT RUN {
showClass("FuzzyNumber")
showMethods(classes="FuzzyNumber")
# }

Run the code above in your browser using DataLab

	\| \(0\)	if \(x < a1\),
	\| \(left((x-a1)/(a2-a1))\)	if \(a1 \le x < a2\),
\(\mu(x)\) =	\| \(1\)	if \(a2 \le x \le a3\),
	\| \(right((x-a3)/(a4-a3))\)	if \(a3 < x \le a4\),
	\| \(0\)	if \(a4 < x\),