23.3 Generalized Hukuhara difference

The generalized difference operation for fuzzy numbers via the extension principle increases the fuzziness. For symmetric fuzzy numbers x* in LR-form with L(·) = R(·), l = r and m = 0 we obtain x* x* = x* ⊕ ( − x*) ≠ 0. In general for given x* ⊕ y* and given x* the vector y* cannot be calculated by (x* ⊕ y*) x*. Moreover the difference does not correspond to the difference in the functional space L2 (Sn − 1 × (0, 1]), i.e. the support function of y* x* is not equal to sy* − sx*.

In order to find a difference which corresponds more to the usual difference of numbers the following argument is helpful: The difference x − y of two real numbers x and y is the real number z for which y + z = x. This results in the so-called Hukuhara difference.

Definition 23.2:

For x* ∈ 2(n) and y* ∈ 2(n) the so-called Hukuhara difference y* Hx* is – if it exists – the solution ...

Start Free Trial

No credit card required