Determine the concentration and normalization of a fuzzy set

Question

Could you help determine this?
Concentration and Normalization of a Fuzzy set $A$ as given below:
$A =$ {$\text {car I }0.5,\text{ truck I }0.9,\text{ bus I }0.7,\text{ scooter }10,\text{ bike I }0.2$}

Regards

Answer 1

Let $$ A = \{ C|0.5, T|0.9, B_1|0.7, S|10.0,B_2|0.2 \} $$ Concentration is computed by squaring the memberships, i.e. $\mu_{\text{CON}(A)}(x)=\mu_A^2(x)$, so: $$ \text{CON}(A) = \{ C|0.25, T|0.81, B_1|0.49, S|100.0, B_2|0.04 \} $$ Normalization is computed by dividing out the largest membership values, i.e. $\mu_{\text{NORM}(A)} = (\sup_x \mu_A(x))^{-1}\mu_A(y)$, so: $$ \text{NORM}(A) = \{ C|0.05, T|0.09, B_1|0.07, S|1.0,B_2|0.02 \} $$ Note that normally the memberships are in $[0,1]$, so I am not sure if your $10$ value is a typo or not.