We consider the production planning of a single product over a finite
planning horizon. The latter is made up of time periods that may not n
ecessarily be of the same length. In each time period, the product is
subject to a constant demand and a finite production capacity. If dema
nd is not satisfied immediately, it is backlogged. The cost rates of c
arrying inventory or backlogging a unit of the product may also vary f
rom One time period to the next. The objective is to determine an opti
mal production plan for the product so as to minimize the total invent
ory and backlog costs incurred over the planning horizon. The problem
is formulated as an optimal control problem. First, the optimal soluti
on of the one-time-period problem is determined. Then, based on the on
e-time-period optimal solution, the optimal solution for the whole pla
nning horizon is obtained by solving a nonlinear programming problem.
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