Calculator.LR.FNs (version 1.3)

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 \(M=(m, \alpha, \beta)_{LR}\) and real number \( \lambda \in R-\{0\} \): $$ \lambda \odot M = M \odot \lambda =\left\{ \begin{array}{lcc} (\lambda m, \lambda \alpha, \lambda \beta)_{LR} &\ \ if & \ \ \lambda > 0 \\ (\lambda m, -\lambda \beta, -\lambda \alpha)_{RL} &\ \ if & \ \ \lambda < 0 \end{array} \right. $$

Usage

s.m(k, N)

Arguments

k

A non-zero real number

N

A LR (or RL, or L) fuzzy number

Value

A LR (or RL or L) fuzzy number

Details

This function has commutative property, i.e \( k \odot M = M \odot k \).

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-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
    }
    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)))
    }
  }
# }