  The unbounded knapsack problem with bounded weights is a variant of the
well-studied variant of the traditional binary knapsack problem; key changes
being the relaxation of the binary constraint and allowing the unit weights of
each item to fall within a range. In this paper, we formulate a variant of this
problem, which we call the strict unbounded knapsack problem with bounded
weights, by replacing the inequality constraint on the total weight with an
equality. We show that this problem can be decomposed into a two-stage,
pre-emptive goal programming problem, with the first stage being a
2-dimensional knapsack problem and the second being either a linear feasibility
program (per canonical formulation) or simply a linearly-constrained program in
the general case. This reformulation is shown to be equivalent to the original
formulation but allows the use of well-studied, efficient algorithms for
multidimensional knapsack problems. In addition, it separates the modeling
effort around what to put in the knapsack from considerations around what unit
weight one should assign to each item type, providing substantially more
flexibility to the modeler without adding complexity to the choice of knapsack
configuration. Finally, we show that for the feasibility version of the second
stage, one can immediately get a feasible solution to the first stage solution.
