Optimal Policies for a Finite-Horizon Production Inventory Model
Read this article. The research indicates challenges associated with the timely ordering of products, especially those that can degrade with time. What kinds of challenges can you see with products that have a defined shelf life and can spoil or deteriorate?
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.