  Let $k\geq 2$ and $\mathcal{P} (n) = (A_1 n + B_1 ) \cdots (A_k n + B_k)$
where all the $A_i, B_i$ are integers. Suppose that $\mathcal{P} (n)$ has no
fixed prime divisors. For each choice of $k$ it is known that there exists an
integer $\varrho_k$ such that $\mathcal{P} (n)$ has at most $\varrho_k$ prime
factors infinitely often. We used a new weighted sieve set-up combined with a
device called an $\varepsilon$-trick to improve the possible values of
$\varrho_k$ for $k\geq 7$. As a by-product of our approach, we improve the
conditional possible values of $\varrho_k$ for $k\geq 4$, assuming the
generalized Elliott--Halberstam conjecture.
