Profit Maximizing Production Allocation

Why

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.

Model

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.