fuzzy-arithmetics.Rmd

---
title: 'Fuzzy Arithmetics Operations'
output: github_document
   
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Interval arithmetic

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

- Addition
- Subtraction
- Multiplication
- Division

```{r}
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 $A(x, min_A, mean_A, max_A)$

```{r}
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)
```

```{r echo=FALSE}
plot(base.set.a, triangular.membership.function(base.set.a, fuzzy.number.a), type = "l", ylim = c(0,1), xlab = "", ylab = "", lty = 2, main = 'Fuzzy Set A = (1, 4, 6)', lwd = 2)

lines(rep(fuzzy.number.a['mean.value'], 2), c(0, 1), lty = 2)
text(fuzzy.number.a['mean.value'] + 1/5, 0.5, expression(italic(mu)))
```

```{r}
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)
```

```{r echo=FALSE}
plot(base.set.b, triangular.membership.function(base.set.b, fuzzy.number.b), type = "l", ylim = c(0,1), xlab = "", ylab = "", lty = 2, main = 'Fuzzy Set B = (3, 5, 7)', lwd = 2, col = 'red')

lines(rep(fuzzy.number.b['mean.value'], 2), c(0, 1), lty = 2, col = 'red')
text(fuzzy.number.b['mean.value'] + 1/5, 0.5, expression(italic(mu)))
```

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

```{r fig.width = 15, fig.height = 10}
plot.ops(fuzzy.number.a, fuzzy.number.b)
```

### Adding fuzzy number A and fuzzy number B

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

```{r}
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, ') '))
```

```{r}
plot.ops(fuzzy.number.a, fuzzy.number.b, "add")
```

### Subtracting fuzzy number A and fuzzy number B

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

```{r}
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, ')'))
```

```{r}
plot.ops(fuzzy.number.a, fuzzy.number.b, "sub")
```

### Multiplying fuzzy number A and fuzzy number B

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

```{r}
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"]))
```

```{r}
cat(paste0('A * B = [', min(mult.ab), ', ', max(mult.ab), ']'))
```

#### 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 $\alpha = 0$:

```{r}
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, ']'))
```

#### When $\alpha = 1$:

```{r}
alpha = 1
mean.number = (3 * alpha + 1) * (2 * alpha + 3)
cat(paste0('(A * B)1 = [', mean.number, ', ', (-2 * alpha + 6) * (-2 * alpha + 7), ']'))
```

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

```{r}
cat(paste0('A * B = [', min.number, ', ', max.number, '] = ', '(', min.number, ', ', mean.number, ', ', max.number, ')'))
```

```{r}
plot.ops(fuzzy.number.a, fuzzy.number.b, "mult")
```

### Dividing fuzzy number A and fuzzy number B

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

```{r}
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)
```

```{r}
cat(paste0('A / B = [', min(div.ab), ', ', max(div.ab), ']'))
```

#### 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 $\alpha = 0$:

```{r}
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, ']'))
```

#### When $\alpha = 1$:

```{r}
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), ']'))
```

#### 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

```{r}
cat(paste0('(A / B) = [', min.number, ', ', max.number, '] = (', min.number, ', ', mean.number, ', ', max.number, ')'))
```

```{r}
plot.ops(fuzzy.number.a, fuzzy.number.b, "div")
```