  The aim of this paper is to study the $p$-Frobenius vector of affine
semigroups $S\subset \mathbb N^q$; that is, the maximum element, with respect
to a graded monomial order, with at most $p$ factorizations in $S$. We produce
several algorithms to compute these vectors. Finally, we study how the
$p$-Frobenius vectors behave when considering gluings of $S$ with $\mathbb
N^q$.
