Package `Calculator.LR.FNs`
Package ‘Calculator.LR.FNs’
April 3, 2017
Type Package
Title Calculator for LR Fuzzy Numbers
Version 1.2
Date 2017-04-01
Author Abbas Parchami (Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran)
Maintainer Abbas Parchami <[email protected]>
Description Arithmetic operations scalar multiplication, addition, subtraction, multiplication and division of LR fuzzy numbers (which are on the basis of extension principle) have a complicate form for using in fuzzy Statistics, fuzzy Mathematics, machine learning, fuzzy data analysis and etc. Calculator for LR Fuzzy Numbers package relieve and aid applied users to achieve a simple and closed form for some complicated operator based on LR fuzzy numbers and also the user can easily draw the membership function of the obtained result by this package.
License LGPL (>= 3)
NeedsCompilation no
Repository CRAN
Date/Publication 2017-04-03 08:12:42 UTC
R topics documented:
Calculator.LR.FNs-package .
addition . . . . . . . . . . .
division . . . . . . . . . . .
L. . . . . . . . . . . . . . .
LR . . . . . . . . . . . . . .
LRFN.plot . . . . . . . . . .
messages . . . . . . . . . .
multiplication . . . . . . . .
RL . . . . . . . . . . . . . .
scalar multiplication . . . . .
sign . . . . . . . . . . . . .
subtraction . . . . . . . . . .
support . . . . . . . . . . .
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2
Calculator.LR.FNs-package
Index
28
Calculator.LR.FNs-package
Calculator for LR Fuzzy Numbers
Description
Calculator for LR Fuzzy Numbers package, i.e. Calculator.LR.FNs package, is an open source
(LGPL 3) package for R which provides the generalized four arithmetic operations +, −, × and ÷
on LR fuzzy numbers. Arithmetic operations addition and subtraction are based on Zadeh extension principle. Also the scalar multiplication of a real number into a LR fuzzy number is considered
in this package on the basis of Zadeh extension principle. Although the class of LR fuzzy numbers is not theoretically closed under the operations × and ÷, but we apply from approximation
for multiplication and division of LR fuzzy numbers which lead the users to a LR fuzzy numbers.
Calculator.LR.FNs package make it easier for researchers, students and any other interested people about fuzzy Mathematics to experience this with a simple calculator. Function LRFN.plot is
designed in Calculator.LR.FNs package for potting the membership function of any LR fuzzy
number.
Details
If the Operation has NOT a closed form or is not defined as a LR fuzzy number, one can continue
calculations by FuzzyNumbers package to achive a the figure of membership function of final result
using cuts of the final result.
Author(s)
Abbas Parchami
Maintainer: Abbas Parchami <[email protected]>
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Dubois, D., Prade, H., Operations on fuzzy numbers. International Journal of Systems Science 9
(1978), 613-626.
Dubois, D., Prade, H., Fuzzy numbers: An overview. In In: Analysis of Fuzzy Information. Mathematical Logic, Vol. I. CRC Press (1987), 3-39.
Dubois, D., Prade, H., The mean value of a fuzzy number. Fuzzy Sets and Systems 24 (1987),
279-300.
Kaufmann, A., Gupta, M.M., Introduction to Fuzzy Arithmetic. van Nostrand Reinhold Company,
New York (1985).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Viertl, R., Statistical Methods for Fuzzy Data. John Wiley & Sons, Chichester (2011).
Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I.
Information Sciences 8 (1975), 199-249.
Calculator.LR.FNs-package
3
See Also
FuzzyNumbers
Examples
# Example 1: mean of LR FNs
Left.fun = function(x) { (1-x)*(x>=0)}
A = L(6, 1, 2)
B = L(2, 4, 1)
LRFN.plot( A, xlim=c(-3,9), ylim=c(0,1.2), lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( s.m( 0.5 , s(A,B) ), lwd=2, lty=3, col=1, add=TRUE)
# ploting the mean of A and B
legend( "topright", c("A = L(6, 1, 2)", "B = L(2, 4, 1)", "(A + B) / 2 = L(4, 2.5, 1.5)")
, col = c(2, 3, 1), text.col = 1, lwd = c(2,2,2), lty = c(2, 2, 3) )
# Example 2: Compute and ploting {0.5(A+B)}*A where A and B are two LR FNs
LRFN.plot( A, xlim=c(-3,41), ylim=c(0,1), lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( m( s.m( 0.5 , s(A,B) ) , A ) , lwd=2, lty=3, col=1
, add=TRUE) # ploting the mean of A and B
legend( "topright", c("A = L(6, 1, 2)", "B = L(2, 4, 1)", "{(A + B) / 2} * A = L(24, 19, 17)")
, col = c(2, 3, 1), text.col = 1, lwd = c(2,2,2), lty = c(2, 2, 3) )
# Example 3: The mean of n=10 random LR fuzzy numbers
n = 10
Left.fun = function(x) { (1-x)*(x>=0)}
Right.fun = function(x) { (exp(-x))*(x>=0)}
xlim=c(2, 18)
ylim=c(0, 1.15)
sum_x = c(0,0,0,0)
for (i in 1:n)
{
x = rnorm(1,10,3)
x_l = runif(1,0,3)
x_r = runif(1,0,2)
X = c()
X = LR(x, x_l, x_r)
LRFN.plot( X, xlim=xlim, ylim=ylim, lwd=1, lty=1, col=1, add = (i != 1) )
sum_x = a( sum_x , X )
}
sum_x
X_bar = s.m( (1/n) , sum_x )
LRFN.plot( X_bar , lwd=2, lty=2, col=2, add = TRUE )
legend( "topright", c("LR FNs", "mean of LR FNs"), col = c(1, 2), text.col = 1
, lwd = c(1, 2), lty = c(1, 2) )
# Example 4:
Left.fun = function(x)
{ (1-x^2)*(x>=0)}
4
Calculator.LR.FNs-package
Right.fun = function(x)
A
B
C
D
=
=
=
=
LR(2,
LR(1,
RL(3,
LR(3,
0.5,
0.1,
0.7,
0.5,
{ (1-x)*(x>=0)}
1)
0.6)
1.5)
0.3)
m(A,B)
s.m( 1.2 , m(A,B) )
d( s.m( 1.2 , m(A,B) ) , C)
m( d( s.m( 1.2 , m(A,B) ) , C) , D)
LRFN.plot(
LRFN.plot(
LRFN.plot(
LRFN.plot(
A,
B,
C,
D,
xlim=c(-0.2,6), ylim=c(0,1.75), lwd=2, lty=1, col=1)
lwd=2, lty=1, col=2, add=TRUE)
lwd=2, lty=1, col=3, add=TRUE)
lwd=2, lty=1, col=4, add=TRUE)
LRFN.plot(
LRFN.plot(
LRFN.plot(
LRFN.plot(
m(A,B), lwd=2, lty=2, col=5, add=TRUE)
s.m( 1.2 , m(A,B) ), lwd=2, lty=3, col=6, add=TRUE)
d( s.m( 1.2 , m(A,B) ) , C), lwd=2, lty=4, col=7, add=TRUE)
m( d( s.m( 1.2 , m(A,B) ) , C) , D), lwd=2, lty=5, col=8, add=TRUE)
legend( "topright", c("A = LR(2, 0.5, 1)", "B = LR(1, 0.1, 0.6)", "C = RL(3, 0.7, 1.5)"
, "D = LR(3, 0.5, 0.3)", "A * B = LR(2, 0.7, 2.2)", "1.2 (A * B) = LR(2.4, 0.84, 2.4)"
, "{1.2 (A * B)} / C = LR(0.8, 0.68, 1.067)", "[{1.2 (A * B)} / C] * D = LR(2.4, 2.44, 3.44)")
, col = c(1:8), text.col = 1, lwd = c(2,2,2,2,2,2,2,2), lty = c(1, 1, 1 ,1 , 2, 3, 4, 5) )
# Example 5:
Left.fun = function(x) { (1-x^3)*(x>=0)}
Right.fun = function(x) { (1-x)*(x>=0)}
A
B
C
D
=
=
=
=
LR(5, 0.5, 1)
LR(2, 0.3, 0.6)
RL(1, 0.7, 1.5)
LR(0.5, 0.5, 1)
E = s.m(a(A,B), 1/2) # The mean of A and B
F = s(s.m(a(A,B), 1/2), C)
G = m(F,D)
LRFN.plot(
LRFN.plot(
LRFN.plot(
LRFN.plot(
A,
B,
C,
D,
xlim=c(-1,6),
lwd=3, lty=1,
lwd=3, lty=1,
lwd=3, lty=1,
ylim=c(0,1.5), lwd=3, lty=1, col=1)
col=2, add=TRUE)
col=3, add=TRUE)
col=4, add=TRUE)
LRFN.plot( E, lwd=3, lty=2, col=5, add=TRUE)
LRFN.plot( F, lwd=3, lty=3, col=6, add=TRUE)
LRFN.plot( G, lwd=3, lty=4, col=7, add=TRUE)
legend( "topleft", c("A = LR(5, 0.5, 1)", "B = LR(2, 0.3, 0.6)", "C = RL(1, 0.7, 1.5)",
"D = LR(0.5, 0.5, 1)", "(A + B)/2 = LR(3.5, 0.4, 0.8)", "[(A + B)/2] - C = LR(2.5, 1.9, 1.5)",
"{[(A + B)/2] - C} * D = LR(1.25, 2.2, 3.2)" ), col = c(1:7), text.col = 1,
addition
5
lwd = c(2,2,2,2,2,2,2), lty = c(1, 1, 1, 1 , 2, 3, 4), bty = "n" )
addition
Addition of two LR fuzzy numbers
Description
This function calculates the addition (summation) of two LR fuzzy numbers M = (m, α, β)LR and
N = (n, δ, γ)LR on the basis of Zadeh extension principle by the following formula:
M ⊕ N = (m + n, α + δ, β + γ)LR
Usage
a(M, N)
Arguments
M
The first LR (or RL or L) fuzzy number
N
The second LR (or RL or L) fuzzy number
Value
A LR (or RL or L) fuzzy number
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Dubois, D., Prade, H., Operations on fuzzy numbers. International Journal of Systems Science 9
(1978), 613-626.
Dubois, D., Prade, H., Fuzzy numbers: An overview. In In: Analysis of Fuzzy Information. Mathematical Logic, Vol. I. CRC Press (1987), 3-39.
Dubois, D., Prade, H., The mean value of a fuzzy number. Fuzzy Sets and Systems 24 (1987),
279-300.
Kaufmann, A., Gupta, M.M., Introduction to Fuzzy Arithmetic. van Nostrand Reinhold Company,
New York (1985).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Viertl, R., Statistical Methods for Fuzzy Data. John Wiley & Sons, Chichester (2011).
Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I.
Information Sciences 8 (1975), 199-249.
6
addition
Examples
# Example
Left.fun
Right.fun
M = LR(1,
N = LR(3,
a(N, M)
1:
= function(x) { (1/(1+x^2))*(x>=0)}
= function(x) { (1/(1+(2*abs(x))))*(x>=0)}
0.6, 0.2)
0.5, 1)
# commutative property for addition on LR fuzzy numbers (Jabejaei)
P = RL(5, 0.1, 0.3)
a(N, P)
a(P, P)
# associative property for addition on LR fuzzy numbers (Sherekat-paziri)
a(N, a(M, M))
a(a(N, M), M)
# Example 2:
A = LR(2, 1, 3)
B = LR(3, 1.2, 1.8)
LRFN.plot( A, xlim=c(-3,12), ylim=c(0,1.25), lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=1, col=5, add=TRUE)
LRFN.plot( a(A, B), lwd=2, col=1, add=TRUE)
legend( "topright", c("A = LR(2, 1, 3)", "B = LR(3, 1.2, 1.8)", "A + B = LR(5, 2.2, 4.8)")
, col = c(2, 5, 1), text.col = 1, lwd = c(2,2,2), lty = c(2, 1, 1) )
## The function is currently defined as
function (M, N)
{
options(warn = -1)
if (messages(M) != 1) {
return(messages(M))
}
if (messages(N) != 1) {
return(messages(N))
}
if (M[4] != N[4]) {
return(noquote(paste0("Addition has NOT a closed form of a LR fuzzy number")))
}
else {
a1 = M[1] + N[1]
a2 = M[2] + N[2]
a3 = M[3] + N[3]
a4 = (M[4] + N[4])/2
print(noquote(paste0("the result of addition is (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
division
7
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
}
division
Division of two LR fuzzy numbers
Description
This function calculates the division of two LR fuzzy numbers. Although on the basis of Zadeh
extension principle, the class of LR fuzzy numbers is not closed under the operations multiplication
and division, but we consider the following approximation for division of fuzzy number M =
(m, α, β)LR by fuzzy number N = (n, γ, δ)RL to work easy in the class of LR fuzzy numbers:
m mδ + nα mγ + nβ
M N '
,
,
n
n2
n2
LR
Usage
d(M, N)
Arguments
M
The first LR (or RL or L) fuzzy number
N
The second LR (or RL or L) fuzzy number
Value
A LR (or RL or L) fuzzy number
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Dubois, D., Prade, H., Operations on fuzzy numbers. International Journal of Systems Science 9
(1978), 613-626.
Dubois, D., Prade, H., Fuzzy numbers: An overview. In In: Analysis of Fuzzy Information. Mathematical Logic, Vol. I. CRC Press (1987), 3-39.
Dubois, D., Prade, H., The mean value of a fuzzy number. Fuzzy Sets and Systems 24 (1987),
279-300.
8
division
Kaufmann, A., Gupta, M.M., Introduction to Fuzzy Arithmetic. van Nostrand Reinhold Company,
New York (1985).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Viertl, R., Statistical Methods for Fuzzy Data. John Wiley & Sons, Chichester (2011).
Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I.
Information Sciences 8 (1975), 199-249.
Examples
# Example 1:
Left.fun = function(x) { (1-x)*(x>=0)}
A = L(6, 1, 2)
B = L(3, 2, 3)
xlim=c(-1.5,9)
LRFN.plot( A, xlim=xlim, lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( d(A,B), lwd=2, lty=1, col=1, add=TRUE)
legend( "topright", c("A = L(6, 1, 2)", "B = L(3, 2, 3)", "A / B = L(2, 2.33, 2)")
, col = c(2, 3, 1), text.col = 1, lwd = c(2,2,2), lty = c(2, 2, 1) )
# Example
Left.fun
Right.fun
A = LR(8,
B = RL(2,
2:
= function(x) { (1-x)*(x>=0)}
= function(x) { (1-x^2)*(x>=0)}
0.5, 1)
1, 1.5)
d(A,B)
d(LR(8, 0.5, 1), RL(2, 1, 1.5))
d(A,A)
d(d(A,B),B)
d(A,d(B,B))
C = LR(-3, 0.5, 1)
d(A,C)
LRFN.plot( A, xlim=c(-3,9.5), lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( d(A,B), lwd=2, lty=3, col=4, add=TRUE)
LRFN.plot( d(d(A,B),B), lwd=2, lty=4, col=1, add=TRUE)
legend( "topleft", c("A = LR(8, 0.5, 1)", "B = RL(2, 1, 1.5)", "A / B = LR(4, 3.25, 2.5)"
, "(A / B) / B = LR(2, 3.125, 2.25)"), col = c(2, 3, 4, 1), text.col = 1, lwd = c(2,2,2)
, lty = c(2, 2, 3, 4) )
## The function is currently defined as
function (M, N)
{
options(warn = -1)
L
9
if (messages(M) != 1) {
return(messages(M))
}
if (messages(N) != 1) {
return(messages(N))
}
if ((M[4] == 1 & N[4] == 0) | (M[4] == 0 & N[4] == 1) | (M[4] ==
0.5 & N[4] == 0.5)) {
if ((sign(M) == "Positive") & (sign(N) == "Positive")) {
a1 = M[1]/N[1]
a2 = ((M[1] * N[3]) + (N[1] * M[2]))/(N[1]^2)
a3 = ((M[1] * N[2]) + (N[1] * M[3]))/(N[1]^2)
a4 = M[4]
print(noquote(paste0("the result of division is approximately (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
else {
return(noquote(paste0(
"A regular approximation is not defined for division since at least one of LR FNs is not positive"
)))
}
}
else {
return(noquote(paste0("Division has NOT a closed form of a LR fuzzy number")))
}
}
Introducing the form of L fuzzy number
L
Description
Considering the definition of LR fuzzy number in LR, if the left and the right shape functions of
a LR fuzzy number are be equal (i.e., L(.) = R(.)), then LR fuzzy number is a L fuzzy number
which denoted by (n, α, β)L. Function L introduce a total form for L fuzzy number to computer.
Usage
L(m, m_l, m_r)
10
LR
Arguments
m
m_l
m_r
The core of L fuzzy number
The left spread of L fuzzy number
The right spread of L fuzzy number
Value
This function help to users to define any L fuzzy number after introducing the left shape function
L. This function consider L fuzzy number L(m, m_l, m_r) as a vector with 4 elements. The first
three elements are m, m_l and m_r respectively; and the fourth element is considerd equal to 0.5
for distinguish L fuzzy number from LR and RL fuzzy numbers.
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Examples
# First introduce the left shape function of L fuzzy number
Left.fun = function(x) { (1-x^2)*(x>=0)}
A = L(20, 12, 10)
LRFN.plot(A, xlim=c(0,60), col=2, lwd=2)
## The function is currently defined as
function (m, m_l, m_r)
{
c(m, m_l, m_r, 0.5)
}
LR
Introducing the form of LR fuzzy number
Description
Function LR introduce a total form for LR fuzzy number. Note that, if the membership function of
fuzzy number N is
(
if
x≤n
L n−x
α N (x) =
R x−n
if
x>n
β
where L and R are two non-increasing functions from R+ ∪ {0} to [0, 1] (say left and right shape
function) and L(0) = R(0) = 1 and also α, β > 0; then N is named a LR fuzzy number and we
denote it by N = (n, α, β)LR in which n is core and α and β are left and right spreads of N ,
respectively.
LR
11
Usage
LR(m, m_l, m_r)
Arguments
m
The core of LR fuzzy number
m_l
The left spread of LR fuzzy number
m_r
The right spread of LR fuzzy number
Value
This function help to users to define any LR fuzzy number after introducing the left shape and the
right shape functions L and R. This function consider LR fuzzy number LR(m, m_l, m_r) as a
vector with 4 elements. The first three elements are m, m_l and m_r respectively; and the fourth
element is considerd equal to 0 for distinguish LR fuzzy number from RL and L fuzzy numbers.
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Examples
# First introduce left and
Left.fun = function(x) {
Right.fun = function(x) {
A = LR(20, 12, 10)
LRFN.plot(A, xlim=c(0,60),
right shape functions of LR fuzzy number
(1-x^2)*(x>=0)}
(exp(-x))*(x>=0)}
col=1)
## The function is currently defined as
function (m, m_l, m_r)
{
c(m, m_l, m_r, 0)
}
12
LRFN.plot
LRFN.plot
Plotting and drawing LR fuzzy numbers
Description
By this function one can plot and draw any kind of LR, RL and L fuzzy numbers.
Usage
LRFN.plot(M, Left.fun = NULL, Right.fun = NULL, ... )
Arguments
M
A LR, RL or L fuzzy number
Left.fun
The left-shape function which usually defined before using LRFN.plot (see examples in bellow)
Right.fun
The right-shape function which usually defined before using LRFN.plot (see
examples in bellow)
...
Any argument of curve() function, such as xlim, ylim, lwd, lty, col, add and ...
is acceptable for this function
Details
Befor useing "LRFN.plot" function, first define the left shape and the right shape functions of LR
fuzzy number. Also, xlim argument must (is better to) be defined for the first fuzzy number.
Author(s)
Abbas Parchami
Examples
# Example 1:
# First introduce left-side and right-side functions of LR fuzzy number
Left.fun = function(x) { (1-x^2)*(x>=0)}
Right.fun = function(x) { (exp(-x))*(x>=0)}
A = LR(20, 12, 10)
LRFN.plot(A, xlim=c(0,60), col=1)
LRFN.plot(A, lty=2, lwd=3, col=2, add=TRUE)
# Example 2:
# for first LR fuzzy number:
Left.fun = function(x) { (1-x^2)*(x>=0)}
Right.fun = function(x) { (exp(-x))*(x>=0)}
LRFN.plot( LR(17,5,3), xlim=c(5,40), lwd=2, lty=2, col=2)
# for second LR fuzzy number:
messages
13
Left.fun = function(x) { (1/(1+x^2))*(x>=0)}
Right.fun = function(x) { (1/(1+(2*abs(x))))*(x>=0)}
LRFN.plot( RL(20,2,3), lwd=2, col=1, add=TRUE)
# for third LR fuzzy number:
Left.fun = function(x) { (1-x)*(x>=0)}
LRFN.plot( L(30,15,5), lwd=2, lty=3, col=4, add=TRUE)
legend( "topright", c("LR(17, 5, 3)", "RL(20, 2, 3)", "L(30, 15, 5)"), col = c(2, 1, 4)
, text.col = 1, lwd = c(2,2,2), lty = c(2, 1, 3) )
## The function is currently defined as
function (M, Left.fun = NULL, Right.fun = NULL, ...)
{
if ( messages(M) != 1 ) { return( messages(M) ) }
m = M[1]
m_l = M[2]
m_r = M[3]
x <- NULL
if ( M[4] == 0 ) { y = function(x) Left.fun((m-x)/m_l) * (x<=m) + Right.fun((x-m)/m_r) * (m<x) }
else if (M[4]==1) { y = function(x) Right.fun((m-x)/m_l) * (x<=m) + Left.fun((x-m)/m_r) * (m<x) }
else if (M[4]==0.5) { y = function(x) Left.fun((m-x)/m_l) * (x<=m) + Left.fun((x-m)/m_r) * (m<x)}
else{return(noquote(paste0("The fourth element of each LR fuzzy number must be 0 or 0.5 or 1!")))}
return( curve(y(x) * (0<=y(x) & y(x)<=1), ...) )
}
messages
messages
Description
The purpose of this function is supporting the functions of this package (by introducing some nested
"if-else" conditions) from all possible messages which are defined in functions of this package. The
"messages" function is used in most of functions of this package.
Usage
messages(M)
Arguments
M
A L, LR or RL fuzzy number
14
messages
Value
Some special messages like: "NOT additive", "NOT productive", .... If any message is not necessary
for this function, then the value 1 will be return by this function which is used in the text and the
body of other functions.
Note
This function has not any applications for users of package and it considered only for shortening
the length of programming.
Author(s)
Abbas Parchami
Examples
messages("NOT additive")
messages( LR(3,1,1) )
## The function is currently defined as
function (M)
{
options(warn = -1)
if (M == "Addition has NOT a closed form of a LR fuzzy number") {
return(noquote(paste0("Addition has NOT a closed form of a LR fuzzy number")))
}
else if (M == "Subtraction has NOT a closed form of a LR fuzzy number") {
return(noquote(paste0("Subtraction has NOT a closed form of a LR fuzzy number")))
}
else if (M == "Production has NOT a closed form of a LR fuzzy number") {
return(noquote(paste0("Production has NOT a closed form of a LR fuzzy number")))
}
else if (M == "Division has NOT a closed form of a LR fuzzy number") {
return(noquote(paste0("Division has NOT a closed form of a LR fuzzy number")))
}
else if (M == " The fourth element of each LR fuzzy number must be 0 or 0.5 or 1! ") {
return(noquote(paste0(" The fourth element of each LR fuzzy number must be 0 or 0.5 or 1! ")))
}
else if (M == " The scalar multiplication is not defined for zero ") {
return(noquote(paste0(" The scalar multiplication is not defined for zero ")))
}
else if (M ==
"A regular approxi. is not defined for multiplication since at least one of FNs is non + and non -"
) {
return(noquote(paste0(
"A regular approxi. is not defined for multiplication since at least one of FNs is non + and non -"
)))
}
else if (M ==
"A regular approximation is not defined for division since at least one of LR FNs is not positive"
) {
return(noquote(paste0(
multiplication
15
"A regular approximation is not defined for division since at least one of LR FNs is not positive"
)))
}
else {
return(1)
}
}
multiplication
Product of two LR fuzzy numbers
Description
This function calculates the multiplication (product) of two LR fuzzy numbers. Although on the
basis of Zadeh extension principle, the class of LR fuzzy numbers is not closed under the operations
multiplication and division, but we consider the following approximation for the product of two LR
fuzzy numbers M = (m, α, β)LR and N = (n, γ, δ)LR in this package to work easy in the class of
LR fuzzy numbers:

if
M 0 and N 0
 (mn, mγ + nα, mδ + nβ)LR
(mn, mγ − nβ, mδ − nα)RL
if
M 0 and N ≺ 0
M ⊗N '

(mn, −nβ − mδ, −nα − mγ)RL
if
M ≺ 0 and N ≺ 0
Usage
m(M, N)
Arguments
M
The first LR (or RL or L) fuzzy number
N
The second LR (or RL or L) fuzzy number
Value
A LR (or RL or L) fuzzy number
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Dubois, D., Prade, H., Operations on fuzzy numbers. International Journal of Systems Science 9
(1978), 613-626.
Dubois, D., Prade, H., Fuzzy numbers: An overview. In In: Analysis of Fuzzy Information. Mathematical Logic, Vol. I. CRC Press (1987), 3-39.
16
multiplication
Dubois, D., Prade, H., The mean value of a fuzzy number. Fuzzy Sets and Systems 24 (1987),
279-300.
Kaufmann, A., Gupta, M.M., Introduction to Fuzzy Arithmetic. van Nostrand Reinhold Company,
New York (1985).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Viertl, R., Statistical Methods for Fuzzy Data. John Wiley & Sons, Chichester (2011).
Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I.
Information Sciences 8 (1975), 199-249.
Examples
# Example 1:
Left.fun = function(x) { (1-x)*(x>=0)}
Right.fun = function(x) { (1/(1+(2*abs(x))))*(x>=0)}
A = LR(1, 0.6, 0.2)
B = LR(-3, 0.5, 1)
m(A, B)
m(B, A)
xlim = c(-5,4)
LRFN.plot( A, xlim=xlim, lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
legend( "topright", c("A = LR(1, 0.6, 0.2)", "B = LR(-3, 0.5, 1)"), col = c(2, 3)
, text.col = 1, lwd = c(2,2), lty = c(2, 2) )
# Example 2:
Left.fun = function(x) { (1-x)*(x>=0)}
Right.fun = function(x) { (exp(-x))*(x>=0)}
A = LR(1.5, 1, 2)
B = LR(3, 2, 1)
LRFN.plot( A, xlim=c(-3,20), ylim=c(0,1), lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( m(A,B), lwd=2, lty=3, col=1, add=TRUE)
legend( "topright", c("A = LR(1.5, 1, 2)", "B = LR(3, 2, 1)", "A * B = LR(4.5, 6, 7.5)")
, col = c(2, 3, 1), text.col = 1, lwd = c(2,2,2), lty = c(2, 2, 3) )
# Example 3:
M = LR(1.2, 0.6, 0.2)
N = LR(3, 0.5, 1)
m(M,N)
m( LR(1.2, 0.6, 0.2) , LR(3, 0.5, 1) )
m(N,m(M,M))
m(m(N,M),M)
LRFN.plot( M, xlim=c(-2,10), ylim=c(0,1.4), lwd=2, lty=2, col=2)
LRFN.plot( N, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( m(M,N), lwd=2, lty=3, col=4, add=TRUE)
multiplication
17
LRFN.plot( m(M,M), lwd=2, lty=4, col=5, add=TRUE)
LRFN.plot( m(m(N,M),M), lwd=2, lty=5, col=1, add=TRUE)
legend( "topright", c("M = LR(1.2, 0.6, 0.2)", "N = LR(3, 0.5, 1)", "M * N = LR(3.6, 2.4, 1.8)"
, "M * M = LR(3.6, 2.4, 1.8)", "(N * M) * M = LR(4.32, 5.04, 2.88)"), col = c(2, 3, 4, 5, 1),
text.col = 1, lwd = c(2,2,2,2,2), lty = c(2, 2, 3, 4, 5) )
## The function is currently defined as
function (M, N)
{
options(warn = -1)
if (messages(M) != 1) {
return(messages(M))
}
if (messages(N) != 1) {
return(messages(N))
}
if (M[4] != N[4]) {
return(noquote(paste0("Production has NOT a closed form of a LR fuzzy number")))
}
else if ((sign(M) == "Positive") & (sign(N) == "Positive")) {
a1 = M[1] * N[1]
a2 = (M[1] * N[2]) + (N[1] * M[2])
a3 = (M[1] * N[3]) + (N[1] * M[3])
a4 = (M[4] + N[4])/2
print(noquote(paste0("the result of multiplication is approximately (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
else if ((sign(M) == "Negative") & (sign(N) == "Negative")) {
a1 = M[1] * N[1]
a2 = -(M[1] * N[2]) - (N[1] * M[2])
a3 = -(M[1] * N[3]) - (N[1] * M[3])
a4 = abs(M[4] - 1)
print(noquote(paste0("the result of multiplication is approximately (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
18
RL
})))
return(invisible(c(a1, a2, a3, a4)))
}
else if ((sign(M) == "Positive") & (sign(N) == "Negative")) {
a1 = M[1] * N[1]
a2 = (M[1] * N[2]) - (N[1] * M[3])
a3 = (M[1] * N[3]) - (N[1] * M[2])
a4 = abs(M[4] - 1)
print(noquote(paste0("the result of multiplication is approximately (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
else if ((sign(M) == "Negative") & (sign(N) == "Positive")) {
a1 = M[1] * N[1]
a2 = (N[1] * M[2]) - (M[1] * N[3])
a3 = (N[1] * M[3]) - (M[1] * N[2])
a4 = abs(N[4] - 1)
print(noquote(paste0("the result of multiplication is approximately (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
else {
return(noquote(paste0(
"A regular approxi. is not defined for multiplication since at least one of FNs is non + and non -"
)))
}
}
RL
Introducing the form of RL fuzzy number
RL
19
Description
Considering the definition of LR fuzzy number in LR, it is obvious that (n, α, β)RL will be a RL
fuzzy number. Function RL introduce a total form for RL fuzzy number to computer.
Usage
RL(m, m_l, m_r)
Arguments
m
The core of RL fuzzy number
m_l
The left spread of RL fuzzy number
m_r
The right spread of RL fuzzy number
Value
This function help to users to define any RL fuzzy number after introducing the left shape and the
right shape functions L and R. This function consider RL fuzzy number RL(m, m_l, m_r) as a
vector with 4 elements. The first three elements are m, m_l and m_r respectively; and the fourth
element is considerd equal to 1 for distinguish RL fuzzy number from LR and L fuzzy numbers.
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Examples
# First introduce left and
Left.fun = function(x) {
Right.fun = function(x) {
A = RL(40, 12, 10)
LRFN.plot(A, xlim=c(0,60),
right shape functions of RL fuzzy number
(1-x^2)*(x>=0)}
(exp(-x))*(x>=0)}
col=1)
## The function is currently defined as
function (m, m_l, m_r)
{
c(m, m_l, m_r, 1)
}
20
scalar multiplication
scalar multiplication Scalar multiplication on LR fuzzy numbers
Description
This function calculates the scalar multiplication of any non-zero real number to any LR fuzzy
number on the basis of Zadeh extension principle by the following formula which is for any LR
fuzzy number M = (m, α, β)LR and real number λ ∈ R − {0}:
(λm, λα, λβ)LR
if
λ>0
λM =M λ=
(λm, −λβ, −λα)RL
if
λ<0
Usage
s.m(k, N)
Arguments
k
A non-zero real number
N
A LR (or RL, or L) fuzzy number
Details
This function has commutative property, i.e k M = M k.
Value
A LR (or RL or L) fuzzy number
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Dubois, D., Prade, H., Operations on fuzzy numbers. International Journal of Systems Science 9
(1978), 613-626.
Dubois, D., Prade, H., Fuzzy numbers: An overview. In In: Analysis of Fuzzy Information. Mathematical Logic, Vol. I. CRC Press (1987), 3-39.
Dubois, D., Prade, H., The mean value of a fuzzy number. Fuzzy Sets and Systems 24 (1987),
279-300.
Kaufmann, A., Gupta, M.M., Introduction to Fuzzy Arithmetic. van Nostrand Reinhold Company,
New York (1985).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
scalar multiplication
21
Viertl, R., Statistical Methods for Fuzzy Data. John Wiley & Sons, Chichester (2011).
Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I.
Information Sciences 8 (1975), 199-249.
Examples
# Example 1:
Left.fun = function(x) { (1-x)*(x>=0)}
Right.fun = function(x) { (1-x)*(x>=0)}
k = 2
M = LR(1, 0.6, 0.2)
N = L(3, 0.6, 1)
P = RL(5, 0.1, 0.3)
s.m(k, N)
# commutative property for scalar multiplication on LR fuzzy numbers (Jabejaei)
s.m(k, M)
s.m(M, k)
s.m(k, P)
s.m(-2, LR(4,2,1))
s.m(2, s.m(-2, LR(4,2,1)))
# Example 2:
Left.fun = function(x) { (1/(1+x^2))*(x>=0)}
Right.fun = function(x) { (1/(1+(2*abs(x))))*(x>=0)}
A = RL(3,2,1)
LRFN.plot( A, xlim=c(-4,15), lwd=2, lty=2, col=2)
LRFN.plot( s.m(0.5, A), lwd=2, lty=3, col=1, add=TRUE)
LRFN.plot( s.m(2, A), lwd=2, lty=4, col=1, add=TRUE)
legend( "topright", c("A = RL(3, 2, 1)", "0.5 A", "2 A"), col = c(2, 1, 1), text.col = 1
, lwd = c(2,2,2), lty = c(2, 3, 4))
## The function is currently defined as
function (k, N)
{
if (messages(N) != 1) {
return(messages(N))
}
if (messages(k) != 1) {
return(messages(k))
}
if (length(k) == 4 & length(N) == 1) {
zarf = N
N[1] = k[1]
N[2] = k[2]
N[3] = k[3]
N[4] = k[4]
k = zarf
}
22
sign
if (k == 0) {
return(noquote(paste0(" The scalar multiplication is not defined for zero ")))
}
else {
a1 = k * N[1]
a2 = k * (N[2] * (k > 0) - N[3] * (k < 0))
a3 = k * (N[3] * (k > 0) - N[2] * (k < 0))
a4 = N[4]
print(noquote(paste0("the result of scalar multiplication is (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
}
sign
Sign of LR fuzzy number
Description
To distinguish and determining the sign of a LR fuzzy number one can use from this function. In
other words, the function sign is able to categorize the class of all LR fuzzy numbers into three
kinds positive, negative and non of them (non-positive and non negative).
Usage
sign(M)
Arguments
M
A LR, RL or L fuzzy number
Value
The "sign" function only return three charactical values: "Positive", "Negative" or "non-positive
and non negative".
Author(s)
Abbas Parchami
subtraction
23
Examples
Left.fun
= function(x)
{ (1-x)*(x>=0)}
M = L(2,4,3)
support(M)
sign(M)
sign( L(5,4,3) )
( sign( L(5,4,3) ) == "Positive" )
## The function is currently defined as
function (M)
{
supp = support(M)
if (supp[1] > 0) {
return(noquote(paste0("Positive")))
}
else {
if (supp[2] < 0) {
return(noquote(paste0("Negative")))
}
else {
return(noquote(paste0("non-positive and non negative")))
}
}
}
subtraction
Subtraction of two LR fuzzy numbers
Description
This function calculates subtraction (difference) of two fuzzy numbers M = (m, α, β)LR and
N = (n, γ, δ)RL on the basis of Zadeh extension principle by the following formula:
M N = (m − n, α + δ, β + γ)LR
Usage
s(M, N)
Arguments
M
N
The first LR (or RL or L) fuzzy number
The second LR (or RL or L) fuzzy number
Value
A LR (or RL or L) fuzzy number
24
subtraction
Author(s)
Abbas Parchami
References
Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Applications. Academic Press (1980).
Dubois, D., Prade, H., Operations on fuzzy numbers. International Journal of Systems Science 9
(1978), 613-626.
Dubois, D., Prade, H., Fuzzy numbers: An overview. In In: Analysis of Fuzzy Information. Mathematical Logic, Vol. I. CRC Press (1987), 3-39.
Dubois, D., Prade, H., The mean value of a fuzzy number. Fuzzy Sets and Systems 24 (1987),
279-300.
Kaufmann, A., Gupta, M.M., Introduction to Fuzzy Arithmetic. van Nostrand Reinhold Company,
New York (1985).
Taheri, S.M, Mashinchi, M., Introduction to Fuzzy Probability and Statistics. Shahid Bahonar
University of Kerman Publications, In Persian (2009).
Viertl, R., Statistical Methods for Fuzzy Data. John Wiley & Sons, Chichester (2011).
Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-I.
Information Sciences 8 (1975), 199-249.
Examples
# Example
Left.fun
Right.fun
M = LR(1,
N = RL(3,
1:
= function(x) { (1/(1+x^2))*(x>=0)}
= function(x) { (1/(1+(2*abs(x))))*(x>=0)}
0.6, 0.2)
0.5, 1)
s(N, M)
s(M, N)
s(M, M)
s(s(N, M), M)
# Example 2:
Left.fun = function(x) { (1-x)*(x>=0)}
A = L(5,3,2)
B = L(3,2,1)
LRFN.plot( A, xlim=c(-3,12), lwd=2, lty=2, col=2)
LRFN.plot( B, lwd=2, lty=2, col=3, add=TRUE)
LRFN.plot( s(A, B), lwd=2, lty=3, col=1, add=TRUE)
legend( "topright", c("A = L(5, 3, 2)", "B = L(3, 2, 1)", "A - B = L(2, 4, 4)"), col = c(2, 3, 1)
, text.col = 1, lwd = c(2,2,2), lty = c(2, 2, 3) )
## The function is currently defined as
function (M, N)
{
options(warn = -1)
if (messages(M) != 1) {
support
25
return(messages(M))
}
if (messages(N) != 1) {
return(messages(N))
}
if ((M[4] == 1 & N[4] == 0) | (M[4] == 0 & N[4] == 1) | (M[4] ==
0.5 & N[4] == 0.5)) {
a1 = M[1] - N[1]
a2 = M[2] + N[3]
a3 = M[3] + N[2]
a4 = M[4]
print(noquote(paste0("the result of subtraction is (core = ",
a1, ", left spread = ", a2, ", right spread = ",
a3, ")", if (a4 == 0) {
paste0(" LR")
}
else if (a4 == 1) {
paste0(" RL")
}
else {
paste0(" L")
})))
return(invisible(c(a1, a2, a3, a4)))
}
else {
return(noquote(paste0( "Subtraction has NOT a closed form of a LR fuzzy number" )))
}
}
support
Support of LR fuzzy number
Description
To determining the support of a LR fuzzy number one can use from this function. In other words,
the support function is able to compute the smallest and biggest values x for which µ(x) > 0.
Usage
support(M, Left.fun = NULL, Right.fun = NULL)
Arguments
M
A LR, RL or L fuzzy number
Left.fun
The left-shape function which usually defined before using LRFN.plot (see examples in bellow)
Right.fun
The right-shape function which usually defined before using LRFN.plot (see
examples in bellow)
26
support
Value
The "support" function return a interval-valued vector in which the membership function value of
LR fuzzy number is bigger than zero.
Author(s)
Abbas Parchami
Examples
Left.fun = function(x) { (1-x)*(x>=0)}
Right.fun = function(x) { (exp(-x))*(x>=0)}
T = LR(1, 0.6, 0.2)
support(T)
LRFN.plot( T, xlim=c(-5,20), lwd=2, lty=3, col=4)
N = RL(3, 0.5, 2)
support(N)
Left.fun = function(x)
M = L(2,4,3)
support(M)
{ (1-x)*(x>=0)}
Left.fun = function(x) { (1-x^2)*(x>=0)}
Right.fun = function(x) { (exp(-x))*(x>=0)}
support( LR(17,5,3))
## The function is currently defined as
function (M, Left.fun = NULL, Right.fun = NULL)
{
range1 = M[1] - M[2] - M[3] - 100
range2 = M[1] + M[2] + M[3] + 100
x = seq(range1, range2, len = 2e+05)
if (M[4] == 0) {
y = Left.fun((M[1] - x)/M[2]) * (x <= M[1]) + Right.fun((x M[1])/M[3]) * (M[1] < x)
}
else if (M[4] == 1) {
y = Right.fun((M[1] - x)/M[2]) * (x <= M[1]) + Left.fun((x M[1])/M[3]) * (M[1] < x)
}
else if (M[4] == 0.5) {
y = Left.fun((M[1] - x)/M[2]) * (x <= M[1]) + Left.fun((x M[1])/M[3]) * (M[1] < x)
}
supp = c()
supp[1] = min(x[0 < y & y < 1])
supp[2] = max(x[0 < y & y < 1])
if (supp[1] == min(x)) {
supp[1] = -Inf
}
support
27
if (supp[2] == max(x)) {
supp[2] = +Inf
}
return(supp)
if (Left.fun == Right.fun+100 ) print(2) #Yek jomleye alaki choon CRAN majburet karde bud ke ...
}
Index
∗Topic Calculator
for LR Fuzzy
subtraction, 23
support, 25
∗Topic Introducing
Numbers
the form of LR
fuzzy number Fuzzy
Number
addition, 5
Calculator.LR.FNs-package, 2
division, 7
L, 9
LR, 10
LRFN.plot, 12
messages, 13
multiplication, 15
RL, 18
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Division of two LR fuzzy
addition, 5
Calculator.LR.FNs-package, 2
division, 7
L, 9
LR, 10
LRFN.plot, 12
multiplication, 15
RL, 18
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Introducing the form of RL
numbers
addition, 5
Calculator.LR.FNs-package, 2
division, 7
LRFN.plot, 12
multiplication, 15
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Introducing the form of L
fuzzy number Fuzzy
Number
addition, 5
Calculator.LR.FNs-package, 2
division, 7
L, 9
LR, 10
LRFN.plot, 12
multiplication, 15
RL, 18
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Ploting and drawing LR fuzzy
fuzzy number Fuzzy
Number
addition, 5
Calculator.LR.FNs-package, 2
division, 7
L, 9
LR, 10
LRFN.plot, 12
multiplication, 15
RL, 18
scalar multiplication, 20
sign, 22
numbers
addition, 5
Calculator.LR.FNs-package, 2
division, 7
L, 9
LR, 10
28
INDEX
29
LRFN.plot, 12
multiplication, 15
RL, 18
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Product of two LR fuzzy
numbers
addition, 5
Calculator.LR.FNs-package, 2
division, 7
LRFN.plot, 12
multiplication, 15
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Scalar multiplication on LR
fuzzy numbers
addition, 5
Calculator.LR.FNs-package, 2
division, 7
LRFN.plot, 12
multiplication, 15
scalar multiplication, 20
subtraction, 23
support, 25
∗Topic Sign of LR fuzzy number
addition, 5
Calculator.LR.FNs-package, 2
division, 7
multiplication, 15
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Subtraction of two LR fuzzy
numbers
addition, 5
Calculator.LR.FNs-package, 2
division, 7
LRFN.plot, 12
multiplication, 15
scalar multiplication, 20
subtraction, 23
support, 25
∗Topic Summation of two LR fuzzy
numbers
addition, 5
Calculator.LR.FNs-package, 2
division, 7
LRFN.plot, 12
multiplication, 15
scalar multiplication, 20
subtraction, 23
support, 25
∗Topic Support of LR fuzzy number
addition, 5
Calculator.LR.FNs-package, 2
division, 7
multiplication, 15
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
∗Topic Zadeh extension principle
addition, 5
Calculator.LR.FNs-package, 2
division, 7
L, 9
multiplication, 15
RL, 18
scalar multiplication, 20
subtraction, 23
∗Topic Zadehs extension principle
LR, 10
a (addition), 5
addition, 5
Calculator.LR.FNs
(Calculator.LR.FNs-package), 2
Calculator.LR.FNs-package, 2
d (division), 7
division, 7
L, 9
LR, 10
LRFN.plot, 12
m (multiplication), 15
messages, 13
multiplication, 15
RL, 18
30
s (subtraction), 23
s.m (scalar multiplication), 20
scalar multiplication, 20
sign, 22
subtraction, 23
support, 25
INDEX
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