 # Fixed charge

A cost that is some value, say $LaTeX: C$, regardless of the level as long as the level is positive; otherwise the fixed charge is zero. This is represented by $LaTeX: Cv$, where $LaTeX: v$ is a binary variable. If $LaTeX: v=0$, the fixed charge is 0; if $LaTeX: v=1$, the fixed charge is $LaTeX: C$. An example is whether to open a plant $LaTeX: (v=1)$ or not $LaTeX: (v=0)$. To apply this fixed charge to the non-negative variable $LaTeX: x$, the constraint $LaTeX: x \le Mv$ is added to the mathematical program, where $LaTeX: M$ is a very large value, known to exceed any feasible value of $LaTeX: x$. Then, if $LaTeX: v=0$ (e.g., not opening the plant that is needed for $LaTeX: x > 0$), $LaTeX: x=0$ is forced by the upper bound constraint. If $LaTeX: v=1$ (e.g., plant is open), $LaTeX: x \le Mv$ is a redundant upper bound. Fixed charge problems are mathematical programs with fixed charges. (See Myths and Counterexamples in Mathematical Programming to avoid a misconception.)