fuzzy operations Algorithm
The fuzzy operations algorithm is an essential component of fuzzy logic, a mathematical framework that deals with approximate reasoning and imprecise information. Unlike classical Boolean logic, which deals with binary values (true or false), fuzzy logic operates with degrees of truth, typically represented as real numbers between 0 and 1. The fuzzy operations algorithm refers to a set of mathematical operations that manipulate these fuzzy values to model and process uncertain or imprecise information. Common fuzzy operations include union, intersection, complement, and various aggregation techniques, which are generalizations of classical logical operations like AND, OR, and NOT.
Fuzzy operations algorithms play a pivotal role in numerous real-world applications, particularly in the areas of control systems, decision making, and artificial intelligence. For instance, in fuzzy control systems, the fuzzy operations algorithm is used to model and manipulate linguistic rules, which describe the behavior of the system in terms of fuzzy sets and fuzzy relations. These linguistic rules are then used to derive control actions based on sensor inputs and desired outputs. The fuzzy operations algorithm is also employed in decision-making problems, where it is used to aggregate and process uncertain or imprecise information from various sources. By allowing for the representation of uncertainty and imprecision, fuzzy operations algorithms enable more flexible and robust solutions to complex problems, making them an invaluable tool in the arsenal of modern computational techniques.
"""README, Author - Jigyasa Gandhi(mailto:jigsgandhi97@gmail.com)
Requirements:
- scikit-fuzzy
- numpy
- matplotlib
Python:
- 3.5
"""
import numpy as np
import skfuzzy as fuzz
if __name__ == "__main__":
# Create universe of discourse in Python using linspace ()
X = np.linspace(start=0, stop=75, num=75, endpoint=True, retstep=False)
# Create two fuzzy sets by defining any membership function (trapmf(), gbellmf(),gaussmf(), etc).
abc1 = [0, 25, 50]
abc2 = [25, 50, 75]
young = fuzz.membership.trimf(X, abc1)
middle_aged = fuzz.membership.trimf(X, abc2)
# Compute the different operations using inbuilt functions.
one = np.ones(75)
zero = np.zeros((75,))
# 1. Union = max(µA(x), µB(x))
union = fuzz.fuzzy_or(X, young, X, middle_aged)[1]
# 2. Intersection = min(µA(x), µB(x))
intersection = fuzz.fuzzy_and(X, young, X, middle_aged)[1]
# 3. Complement (A) = (1- min(µA(x))
complement_a = fuzz.fuzzy_not(young)
# 4. Difference (A/B) = min(µA(x),(1- µB(x)))
difference = fuzz.fuzzy_and(X, young, X, fuzz.fuzzy_not(middle_aged)[1])[1]
# 5. Algebraic Sum = [µA(x) + µB(x) – (µA(x) * µB(x))]
alg_sum = young + middle_aged - (young * middle_aged)
# 6. Algebraic Product = (µA(x) * µB(x))
alg_product = young * middle_aged
# 7. Bounded Sum = min[1,(µA(x), µB(x))]
bdd_sum = fuzz.fuzzy_and(X, one, X, young + middle_aged)[1]
# 8. Bounded difference = min[0,(µA(x), µB(x))]
bdd_difference = fuzz.fuzzy_or(X, zero, X, young - middle_aged)[1]
# max-min composition
# max-product composition
# Plot each set A, set B and each operation result using plot() and subplot().
import matplotlib.pyplot as plt
plt.figure()
plt.subplot(4, 3, 1)
plt.plot(X, young)
plt.title("Young")
plt.grid(True)
plt.subplot(4, 3, 2)
plt.plot(X, middle_aged)
plt.title("Middle aged")
plt.grid(True)
plt.subplot(4, 3, 3)
plt.plot(X, union)
plt.title("union")
plt.grid(True)
plt.subplot(4, 3, 4)
plt.plot(X, intersection)
plt.title("intersection")
plt.grid(True)
plt.subplot(4, 3, 5)
plt.plot(X, complement_a)
plt.title("complement_a")
plt.grid(True)
plt.subplot(4, 3, 6)
plt.plot(X, difference)
plt.title("difference a/b")
plt.grid(True)
plt.subplot(4, 3, 7)
plt.plot(X, alg_sum)
plt.title("alg_sum")
plt.grid(True)
plt.subplot(4, 3, 8)
plt.plot(X, alg_product)
plt.title("alg_product")
plt.grid(True)
plt.subplot(4, 3, 9)
plt.plot(X, bdd_sum)
plt.title("bdd_sum")
plt.grid(True)
plt.subplot(4, 3, 10)
plt.plot(X, bdd_difference)
plt.title("bdd_difference")
plt.grid(True)
plt.subplots_adjust(hspace=0.5)
plt.show()
