simple linear programming problem with integer constraints

Question

Given positive numbers $a, b, c, d, e$, how do I find the maximum value of $ax + by$ such that $cx + dy \leq e$ and $x, y$ are non-negative integers?

Answer 1

This is a very small knapsack problem with two items. The variables $x$ and $y$ represent the number of times each item appears in the knapsack. The constants $a$ and $b$ are the values per item, the constants $c$ and $d$ are the weights per item, and the constant $e$ is the knapsack capacity.

Two common solution approaches are integer linear programming and dynamic programming.