Fuzzification

The normalized basic indicators are fuzzified using three fuzzy sets with linguistic values:

Weak (W),
Medium (M), and
Strong (S).

For composite indicators (primary, secondary, and tertiary components) five linguistic values are used:

Very Bad (VB),
Bad (B),
Average (A),
Good (G), and
Very Good (VG).

The overall sustainability is measured using nine fuzzy sets:

Extremely Low (EL),
Very Low (VL),
Low (L),
Fairly Low (FL),
Intermediate (I),
Fairly High (FH),
High (H),
Very High (VH), and
Extremely High (EH).

Fig. 6. Fuzzy sets and corresponding membership functions µ(x) — Fig. 6. Fuzzy sets and corresponding membership functions *µ(x)*.

Each indicator value x belongs to one or more fuzzy sets with certain membership grades. For simplicity, triangular membership functions µ(x) are used, as shown in Fig. 6. For example, in 2002, 13.7% of the mammal species in Greece were endangered. The target value for this indicator is T = τ = 0% and the upper threshold of unsustainable values is U = 35.5%. The normalized value for this indicator is x = (13.7 – 35.5)/(0 – 35.5) = 0.614. As shown in Fig. 6a, this value belongs to the fuzzy set Medium with membership grade µ_M(0.614) = 0.965 and to the fuzzy set Strong with grade µ_S(0.614) = 0.035.