Optimal Policies for a Finite-Horizon Production Inventory Model
Technical Preliminaries
This section contains a summary of the work of Benkherouf and Gilding needed to tackle the problem of this paper. Proofs of the results are omitted. Interested readers may consult Benkherouf and Gilding.
Consider the problem
subject to
It was shown in Benkherouf and Gilding that, under some technical conditions, the optimization problem (P) has a unique optimal solution which can be found from solving a system of nonlinear equations derived from the first-order optimality condition. To be precise, and
and ignore the rest of the constraints (3.2).
Write
Assuming that are twice differentiable, then, for fixed
, the optimal solution in (P) subject to (3.2) reduces to minimizing
.
Use the notation for the gradient, then setting
gives
Two sets of hypotheses were put forward.
Hypothesis 1. The functions satisfy, for
,
Hypothesis 2. Define
then there is a continuous function such that
for all
, and
on the boundary of the feasible set.
The next theorem shows that under assumptions in Hypotheses 1 and 2, the function has a unique minimum.
Theorem 3.1.The system (3.4) has a unique solution subject to (3.2). Furthermore, this solution is the solution of (3.1) subject to (3.2).  Recall that a function is convex in
if
This is equivalent to
Theorem 3.2. If denotes the minimum objective value of (3.1) subject to (3.2) and
then
is convex in
.
Based on the convexity property of , the optimal number of cycles
is given by
Assume that is known,
can be found uniquely as a function of
. Repeating this process for
,
are a function of
. So, the search for the optimal solution of (3.5) can be conducted using a univariate search method.