Pure IP, Mixed IP, and Binary IP

A pure integer program (Pure IP) is an optimization problem in which every decision variable is required to take an integer value. In standard form,

$$\max\; \mathbf{c}^\top\mathbf{x} \quad \text{s.t.}\quad A\mathbf{x} \leq \mathbf{b},\quad \mathbf{x} \geq \mathbf{0},\quad \mathbf{x} \in \mathbb{Z}^n.$$

Pure IPs arise whenever every quantity in the model must be a whole number — buses to dispatch, employees to hire, items to manufacture.

For example, suppose a workshop builds desks ($x_1$) and bookcases ($x_2$). Each desk uses 2 hours of labor and 3 board-feet of wood; each bookcase uses 4 hours and 2 board-feet. Labor is capped at 20 hours, wood at 18 board-feet, and items must be whole. Profit is $50 per desk and $40 per bookcase. The pure IP is

$$\max\; 50x_1+40x_2 \quad \text{s.t.}\quad 2x_1+4x_2 \leq 20,\;\; 3x_1+2x_2 \leq 18,\;\; x_1,x_2 \geq 0,\; \text{integer}.$$

Every decision variable is integer-restricted, so the model is a pure IP.