 Unbounded knapsack problem

In the unbounded knapsack problem, each item type may be used any nonnegative integer number of times. With positive integer weights $w_i$, values $v_i$ and capacity $W$, maximize ${\sum }_i{v}_i{x}_i$ subject to ${\sum }_i{w}_i{x}_i\le W$ and $x_i\in {\mathbb{Z}}_{\ge 0}$.

A dynamic programming recurrence for capacity-at-most value is $F(c)=\max (0,{\max }_{w_i\le c}(v_i+F(c-w_i)))$. Evaluate capacities in increasing order so repeated uses are available. With weights three and five, values four and seven, and capacity ten, two weight-five items give value fourteen. A zero-weight positive-value item would make the objective unbounded, explaining the positive-weight hypothesis.