# Lot size problem

### From Glossary

This is one of the oldest mixed-integer programs in operations research,
first presented by Wagner and Whitin in 1958. The problem is to minimize cost while satisfying product
demands over (discrete) time. Let be the number of units produced in period ,
for ( is called the *planning horizon*), and let
if a setup occurs in period ; otherwise. Let
be the demand in period , and let the demand from period to period
inclusive, be
Then, a MILP formulation is:

(This is preferred over the MILP that defines inventory balance equations. Even though they are equivalent, the LP relaxation of the above is sharper.)