  In this paper, we consider the classic stochastic (dynamic) knapsack problem,
a fundamental mathematical model in revenue management, with general
time-varying random demand. Our main goal is to study the optimal policies,
which can be obtained by solving the dynamic programming formulated for the
problem, both qualitatively and quantitatively. It is well-known that when the
demand size is fixed and the demand distribution is stationary over time, the
value function of the dynamic programming exhibits extremely useful first and
second order monotonicity properties, which lead to monotonicity properties of
the optimal policies. In this paper, we are able to verify that these results
still hold even in the case that the price distributions are time-dependent.
When we further relax the demand size distribution assumptions and allow them
to be arbitrary, for example in random batches, we develop a scheme for using
value function of alternative unit demand systems to provide bounds to the
value function. These results confirm some of the basic understandings of the
stochastic knapsack problem. As a natural consequence, we propose a new class
of heuristic policies, two-dimensional switch-over policies, and discuss
numerical procedures for optimizing the parameters of the policies. Extensive
numerical experiments demonstrate that these switch-over policies can provide
performances that are in a close neighborhood of the optimum statistically.
