# Lot size problem

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 $LaTeX: y_t$ be the number of units produced in period $LaTeX: t$, for $LaTeX: t=1,\ldots,T$ ($LaTeX: T$ is called the planning horizon), and let $LaTeX: x_t = 1$ if a setup occurs in period $LaTeX: t$; $LaTeX: 0$ otherwise. Let $LaTeX: D_t$ be the demand in period $LaTeX: t$, and let the demand from period $LaTeX: i$ to period $LaTeX: j,$ inclusive, be $LaTeX: d_{ij} = \sum_{i \le t \le j} D_t.$ Then, a MILP formulation is:
$LaTeX: \min \left\{ c^T x + d^T y: x \in \{0,1\}^n, \, y \ge 0, \sum_{t\le i} y_t \ge d_{li} \mbox{ for } i=1, \ldots,n-1, \, \sum_{t\le n} y_t = d_{1n}, \, d_{in} x_i - y_i \ge 0 \mbox{ for } i=1, \ldots,n \right\}.$