Suppose a manufacturer has raw materials, and
production processes by which she can convert
raw materials into finished goods.
How should she *allocate* the raw materials
among the finished goods to maximize revenue.

We model the quantities of $m$ raw materials available to the manufacturer by $m$ real numbers, which we denote $q_1, \dots , q_m \in \R _+$. We suppose that there is a correspondence, $f_i: \R _+ \to \R _+^m$, which models the quantities of the $m$ raw materials that will be needed for the $i$th finished good. In other words, $f_i(x)$ is the bill of materials to produce quantity $x$ of the $i$th finished good. We suppose finished good $i$ can be sold for a price $p_i$ per unit.

We formulate the following optimization problem. Given a supply of raw materials $q_1, \dots , q_m$, find the quantities $x_1, \dots , x_n$ to

\[ \begin{aligned} \text{ maximize } \quad & \textstyle \sum_{i = 1}^{n} p_i x_i \\ \text{ subject to } \quad & \textstyle \sum_{i = 1}^{n} f_i(x_i) \leq q \\ \text{ and } \quad & x \geq 0 \end{aligned} \]

This is sometimes called an allocation problem or manufacturing problem.
*Linear simplication.*
In the case that $f_i$ is modeled (or
idealized) as a linear function, we obtain a
*linear optimization problem*.