Calculator.LR.FNs (version 1.3)

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 \( M=(m, \alpha, \beta)_{LR} \) and \( N=(n, \gamma, \delta)_{LR} \) in this package to work easy in the class of LR fuzzy numbers: $$ M \otimes N \simeq \left\{ \begin{array}{lcc} (mn, m \gamma + n \alpha, m \delta + n \beta)_{LR} &\ \ if & \ \ M \succ 0 \ and \ N \succ 0 \\ (mn, m \gamma - n \beta, m \delta - n \alpha)_{RL} &\ \ if & \ \ M \succ 0 \ and \ N \prec 0 \\ (mn, -n \beta - m \delta , -n \alpha -m \gamma)_{RL} &\ \ if & \ \ M \prec 0 \ and \ N \prec 0 \end{array} \right. $$

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

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

# NOT RUN {
# 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)
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")
            })))
        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 -"
              )))
    }
  }
# }