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fuzzy-arithmetics.md

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Fuzzy Arithmetics Operations

Interval arithmetic

Four fuzzy interval operations are implemented in the fuzzy-arithmetic.R script:

  • Addition
  • Subtraction
  • Multiplication
  • Division
source("fuzzy-arithmetic.R")

Let A = ((min_A, mean_A, max_A)) be a triangular fuzzy number represented by three points: minimum, mean and maximum values. From a base set (X \in [min_A, max_A]), the fuzzy set (A:X \rightarrow [0, 1]) is defined as (A(x, min_A, mean_A, max_A))

fuzzy.number.a = c(min.value = 1, mean.value = 4, max.value = 6)
base.set.a = seq(fuzzy.number.a["min.value"], fuzzy.number.a["max.value"], by=0.1)

fuzzy.number.b = c(min.value = 3, mean.value = 5, max.value = 7)
base.set.b = seq(fuzzy.number.b["min.value"], fuzzy.number.b["max.value"], by=0.1)

The plot.ops function gives the plot of all arithmetic operations

plot.ops(fuzzy.number.a, fuzzy.number.b)

Adding fuzzy number A and fuzzy number B

(A = (min_A, mean_A, max_A) \ B = (min_B, mean_B, max_B) \ A+B = (min_A + min_B, mean_A + mean_B, max_A + max_B))

min.number = fuzzy.number.a['min.value'] + fuzzy.number.b['min.value']
mean.number = fuzzy.number.a['mean.value'] + fuzzy.number.b['mean.value']
max.number = fuzzy.number.a['max.value'] + fuzzy.number.b['max.value']
cat(paste0('A + B = [', min.number, ', ', max.number, '] = (', min.number, ', ', mean.number, ', ', max.number, ') '))
## A + B = [4, 13] = (4, 9, 13)

plot.ops(fuzzy.number.a, fuzzy.number.b, "add")

Subtracting fuzzy number A and fuzzy number B

(A-B = (min_A - max_B, mean_A - mean_B, max_A - min_B))

min.number = fuzzy.number.a['min.value'] - fuzzy.number.b['max.value']
mean.number = fuzzy.number.a['mean.value'] - fuzzy.number.b['mean.value']
max.number = fuzzy.number.a['max.value'] - fuzzy.number.b['min.value']
cat(paste0('A - B = [', min.number, ', ', max.number, '] = (', min.number, ', ', mean.number, ', ', max.number, ')'))
## A - B = [-6, 3] = (-6, -1, 3)

plot.ops(fuzzy.number.a, fuzzy.number.b, "sub")

Multiplying fuzzy number A and fuzzy number B

(A = (min_A, mean_A, max_A) \ B = (min_B, mean_B, max_B) \ (A * B) = \ [\min{min_A * min_B, min_A * max_B, max_A * min_B, max_A * max_B}, \ \max{min_A * min_B, min_A * max_B, max_A * min_B, max_A * max_B}])

mult.interval = function(x, y) c(x * y, rev(x) * y)
mult.ab = mult.interval(c(fuzzy.number.a["min.value"], fuzzy.number.a["max.value"]),
                  c(fuzzy.number.b["min.value"], fuzzy.number.b["max.value"]))
cat(paste0('A * B = [', min(mult.ab), ', ', max(mult.ab), ']'))
## A * B = [3, 42]

The multiplication of two fuzzy triangular numbers does not result in a triangular number

Because of that, we need to find the mean value of the resulting fuzzy triangular number by setting (\alpha-cut = 1)

Let (A_\alpha[min^\alpha_A, max^\alpha_A]) (\forall \alpha [0, 1]) be a crisp interval of a triangular fuzzy number

(A_\alpha \= [min^{\alpha}_A, max^{\alpha}_A] \= [(mean_A - min_A)\alpha + min_A, -(max_A - mean_A)\alpha + max_A]\)

(A = [1, 6] = (1, 4, 6)\ A_\alpha \ = [(4 - 1)\alpha + 1, -(6 - 4)\alpha + 6] \ = [3\alpha + 1, -2\alpha + 6]\)

(B = [3, 7] = (3, 5, 7) \ B_\alpha \ = [(5 - 3)\alpha + 3, -(7 - 5)\alpha + 7] \ = [2\alpha + 3, -2\alpha + 7]\)

[ (A * B)_\alpha = [(3\alpha +1)(2\alpha + 3), (-2\alpha + 6)(-2\alpha + 7)] ]

When (\alpha = 0):

alpha = 0
min.number = (3 * alpha + 1) * (2 * alpha + 3)
max.number = (-2 * alpha + 6) * (-2 * alpha + 7)
cat(paste0('(A * B)0 = [', min.number, ', ', max.number, ']'))
## (A * B)0 = [3, 42]

When (\alpha = 1):

alpha = 1
mean.number = (3 * alpha + 1) * (2 * alpha + 3)
cat(paste0('(A * B)1 = [', mean.number, ', ', (-2 * alpha + 6) * (-2 * alpha + 7), ']'))
## (A * B)1 = [20, 20]

Finally, the mean value of the resulting fuzzy triangular number by setting (\alpha-cut = 1) is A1 * B1 = [20, 20] = 20

cat(paste0('A * B = [', min.number, ', ', max.number, '] = ', '(', min.number, ', ', mean.number, ', ', max.number, ')'))
## A * B = [3, 42] = (3, 20, 42)

plot.ops(fuzzy.number.a, fuzzy.number.b, "mult")

Dividing fuzzy number A and fuzzy number B

(A = (min_A, mean_A, max_A) \ B = (min_B, mean_B, max_B) \ (A / B) \ = \left[\min{\left(\frac{min_A}{min_B}, \frac{min_A}{max_B}, \frac{max_A}{min_B}, \frac{max_A}{max_B}\right)}, \max{\left(\frac{min_A}{min_B}, \frac{min_A}{max_B}, \frac{max_A}{min_B}, \frac{max_A}{max_B}\right)}\right])

div.interval = function(x, y) c(x / y, rev(x) / y)
div.ab = round(div.interval(c(fuzzy.number.a["min.value"], fuzzy.number.a["max.value"]),
                  c(fuzzy.number.b["min.value"], fuzzy.number.b["max.value"])), 2)
cat(paste0('A / B = [', min(div.ab), ', ', max(div.ab), ']'))
## A / B = [0.14, 2]

The division of two fuzzy triangular numbers does not result in a triangular number

Because of that, we need to find the mean value of the resulting fuzzy triangular number by setting (\alpha-cut = 1)

[ (A / B)_\alpha = [(3\alpha +1)/(-2\alpha + 7), (-2\alpha + 6)/(2\alpha + 3)] ]

When (\alpha = 0):

alpha = 0
min.number = round((3 * alpha + 1)/(2 * alpha + 3), 2)
max.number = round((-2 * alpha + 6)/(-2 * alpha + 7), 2)
cat(paste0('(A / B)0 = [', min.number, ', ', max.number, ']'))
## (A / B)0 = [0.33, 0.86]

When (\alpha = 1):

alpha = 1
mean.number = round((3 * alpha + 1)/(2 * alpha + 3), 2)
cat(paste0('(A / B)1 = [', mean.number, ', ', round((-2 * alpha + 6)/(-2 * alpha + 7), 2), ']'))
## (A / B)1 = [0.8, 0.8]

Finally, the mean value of the resulting fuzzy triangular number by setting (\alpha-cut = 1) is (A / B)1 = [0.8, 0.8] = 0.8

cat(paste0('(A / B) = [', min.number, ', ', max.number, '] = (', min.number, ', ', mean.number, ', ', max.number, ')'))
## (A / B) = [0.33, 0.86] = (0.33, 0.8, 0.86)

plot.ops(fuzzy.number.a, fuzzy.number.b, "div")