  In this paper, we present some necessary and sufficient conditions for
semi-compact operators being almost L-weakly compact (resp. almost M-weakly
compact) and the converse. Mainly, we prove that if $X$ is a nonzero Banach
space, then every semi-compact operator $T: X\rightarrow E$ is almost L-weakly
compact if and only if the norm of $E$ is order continuous. And every positive
semi-compact operator $T:E\rightarrow F$ is almost M-weakly compact if and only
if the norm of $E'$ is order continuous. Moreover, we investigate the
relationships between almost L-weakly compact operators and Dunford-Pettis
(resp. almost Dunford-Pettis) operators.
