Mixed Integer Linear Programming
Limitations of Linear Programming
LPs are powerful but have some (strong) assumptions.
What do we do when they don’t hold or a linear relaxation would be inappropriate?
Problems with Discrete Decisions
Economic dispatch: Assumed we had a fleet of online generators, all of which generated.
What if we had to also make decisions about which generators to operate?
Fixed Costs
A key consideration when deciding whether to operate something are fixed costs.
- Fixed costs: Labor, etc, required to operate or produce regardless of quantity.
- Variable costs: Costs of inputs (energy, additional labor, materials) required to operate or produce per unit of time/quantity.
Fixed Costs Cause Discontinuities
Operational Status As Decision Variables
The potential decision to not operate means that we need to introduce new indicator variables, which flag on/off status:
\[
\mathbb{I}_g = \begin{cases} 0 & \text{off} \\ 1 & \text{on} \end{cases}
\]
Binary and Integer Variables
These indicator variables are binary variables: 0 or 1.
More generally, we can consider optimization problems with integer variables \(x \in \mathbb{Z}\).
Discrete Decision Variables Violate Divisibility
Recall that for LPs, solutions must occur at corners of the feasible polyhedra.
Discrete Decision Variables Violate Divisibility
Mixed-integer LPs: corners may not exist at integer points.
