  This paper describes the structure of optimal policies for discounted
periodic-review single-commodity total-cost inventory control problems with
fixed ordering costs for finite and infinite horizons. There are known
conditions in the literature for optimality of $(s_t,S_t)$ policies for
finite-horizon problems and the optimality of $(s,S)$ policies for
infinite-horizon problems. The results of this paper cover the situation, when
such assumption may not hold. This paper describes a parameter, which, together
with the value of the discount factor and the horizon length, defines the
structure of an optimal policy. For the infinite horizon, depending on the
values of this parameter and the discount factor, an optimal policy either is
an $(s,S)$ policy or never orders inventory. For a finite horizon, depending on
the values of this parameter, the discount factor, and the horizon length,
there are three possible structures of an optimal policy: (i) it is an
$(s_t,S_t)$ policy, (ii) it is an $(s_t,S_t)$ policy at earlier stages and then
does not order inventory, or (iii) it never orders inventory. The paper also
establishes continuity of optimal value functions and describes alternative
optimal actions at states $s_t$ and $s.$
