  We present two criteria for checking approximate proper efficiency in vector
optimization problems with the ordering cone being a nonnegative orthant.
Although the criteria can be established by Benson's approach [H.P. Benson,
\textit{An improved definition of proper efficiency for vector maximization
with respect to cones}, J. Math. Anal. Appl. \textbf{71} (1979), 232--241],
detailed proofs are given for the first time here. The two criteria are strong
motivations to introduce the concept of $e$-properly efficient solution, where
$e$ is any nonzero vector taken from the closed pointed convex ordering cone.
For an arbitrary linear vector optimization problem, we show that either the
$e$-properly efficient solution set is empty or it coincides with the
$e$-efficient solution set. This new result has no analogue in the literature.
