Chapter 4: Fuzzy Numbers and Arithmetic

4.1 Introduction to Fuzzy Numbers

In classical mathematics, numbers are precise and well-defined. However, real-world situations often involve uncertainty, vagueness, or imprecision. Fuzzy numbers are an extension of real numbers that incorporate such uncertainty.

Definition of Fuzzy Numbers

A fuzzy number is a fuzzy set defined on the real number line R, which is:

• Normal: At least one element has a membership value of 1.

• Convex: For any two real numbers x and y in the fuzzy set and for all λ in [0, 1],
  μ(λx + (1 - λ)y) ≥ min(μ(x), μ(y)).

• Upper semi-continuous.

• Support is bounded: The set of elements with non-zero membership grades is a bounded interval.

Common Types of Fuzzy Numbers

• Triangular Fuzzy Number (TFN): Defined by a triplet (l, m, u) where:

  • l = lower limit

  • m = modal (peak) value with membership 1

  • u = upper limit

  The membership function:

• Trapezoidal Fuzzy Number (TrFN): Defined by four points (a, b, c, d):

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4.2 Arithmetic Operations on Fuzzy Numbers

To perform arithmetic operations on fuzzy numbers, the Extension Principle is used. This principle enables the extension of crisp mathematical functions to fuzzy sets.

Let A and B be fuzzy numbers, and f be a function from R × R → R. Then the resulting fuzzy number C = A ⊕ B is defined as:

4.2.1 Addition

For two triangular fuzzy numbers: Let A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃)

4.2.2 Subtraction

4.2.3 Multiplication

Assuming all values are positive for simplicity.

4.2.4 Division

Provided that none of the values in B is zero.

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4.3 Extension Principle in Detail

The Extension Principle, proposed by Zadeh, allows us to map fuzzy sets through functions.

Given a fuzzy set A on domain X, and a function f: X → Y, the image of A under f is a fuzzy set B on Y, where:

This principle is fundamental for performing operations beyond simple arithmetic, such as fuzzy logic, fuzzy control, and fuzzy decision-making.

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4.4 Graphical Representation of Arithmetic Operations

Operations on fuzzy numbers can also be understood graphically by plotting their membership functions. The resulting shape for addition or subtraction of TFNs is also a triangular fuzzy number, while multiplication or division often results in non-triangular fuzzy numbers unless approximated.

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4.5 Applications of Fuzzy Arithmetic

• Engineering Calculations under uncertain material properties

• Economic Forecasting with uncertain inputs

• Control Systems for modeling uncertain sensor data

• Decision-Making in operations research and AI

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4.6 Conclusion

Fuzzy numbers and their arithmetic form the backbone of computational models that involve vagueness and imprecision. Triangular and trapezoidal fuzzy numbers provide simple yet powerful tools to represent and manipulate uncertain quantities. Using the extension principle, classical operations can be generalized to fuzzy environments, enabling intelligent reasoning in various real-world applications.

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4.7 Exercises

1. Define a triangular fuzzy number and plot its membership function for the triplet (3, 5, 7).

2. Given A = (2, 4, 6) and B = (1, 3, 5), compute:

• a) A + B

• b) A - B

• c) A × B (Assume all values are positive)

3. Explain the extension principle and demonstrate how it applies to addition of fuzzy numbers with an example.

4. If A = (5, 6, 7) and B = (2, 3, 4), calculate A ÷ B and interpret the result.

5. Illustrate how fuzzy arithmetic can be used in modeling uncertain supply costs in a factory setting.