Decisions under uncertainty: fuzzy numbers

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

Why fuzzy?

When an expert says “the quality of this alternative is about good” or “the price is roughly 7”, there is no numerical precision. Instead of forcing a number, we can model the vagueness. Lotfi Zadeh (1965) introduced the fuzzy set: each element $x$ has a degree of membership $\mu (x)\in [0,1]$, not just “is or is not”.

The fuzzy number

A fuzzy number is a fuzzy set on the real numbers that represents “roughly 7” or “about 4”: discrete (a few points with membership degrees) or continuous (like a bell curve). In practice three shapes are common: triangular $(l,m,u)$, trapezoidal and Gaussian.

The triangular fuzzy number (TFN)

$\tilde{a}=(l,m,u)$: $l$ is the lowest plausible value, $m$ the most plausible and $u$ the highest; the membership at $m$ is 1. Operations:

• Addition: $(l_1,m_1,u_1)+(l_2,m_2,u_2)=(l_1+l_2,m_1+m_2,u_1+u_2)$
• Multiplication by a positive number $k$: $(kl,km,ku)$
• Inverse: $(1/u,1/m,1/l)$

Linguistic variables

We turn qualitative judgements into TFNs. A common five-point scale:

Term	TFN
Very low	(0, 0, 0.25)
Low	(0, 0.25, 0.5)
Medium	(0.25, 0.5, 0.75)
High	(0.5, 0.75, 1)
Very high	(0.75, 1, 1)

(Other scales exist; what matters is that it covers the 0-to-1 range fully and stays fixed across papers.)

Defuzzification

At the end we want an ordinary number. The simplest method for a TFN is the mean: $(l+m+u)/3$. (There are also the centroid method and the $\alpha$-cut.)

Use in decision making

Replace crisp numbers with TFNs in other methods:

• Fuzzy AHP: pairwise comparisons with TFNs.
• Fuzzy TOPSIS (Chen, 2000): the decision matrix and weights are TFNs; the positive and negative ideals are built from TFNs and distances are computed with the vertex distance (for two TFNs: $d=\sqrt{\frac{1}{3}[(l_1-l_2{)}^2+(m_1-m_2{)}^2+(u_1-u_2{)}^2]}$).

Critique

Fuzzification does not always increase accuracy; it “hides” the vagueness and the choice of membership function is subjective. I think it is worth it when the information is really vague (human judgement, say); when you have precise data, fuzzifying it is meaningless. Alternatives: probability (when we have a distribution; Probabilistic decisions: utility and the value of information) or interval/robust (when we know only a range).

Related: Compromise methods: TOPSIS and VIKOR and AHP: the analytic hierarchy process.

fuzzyuncertainty