# Inventory balance equation

The constraint that says that the amount of inventory in the next time period must equal the current inventory plus what is produced or purchased minus what is lost or sold. Let $LaTeX: y(t)$ be the inventory at the beginning of period $LaTeX: t$, with $LaTeX: y(0)$ given. Then, the inventory equation is:

$LaTeX: y(t+1) = ay(t) + P(t) - S(t),$

where $LaTeX: P(t)$ is the level of production (or somehow acquired), and $LaTeX: S(t)$ is the level of sales (or somehow consumed). Typically, $LaTeX: a=1$, but if $LaTeX: a < 1,$ it is called a loss factor, and if $LaTeX: a > 1,$ it is called a gain factor.

The language used is for the inventory control in the production scheduling problem, but this has become a standard system of equations that appears in many mathematical programs. Thus, the meaning of the variables can be substantially different. One example is the

steel beam assortment problem.