# Production scheduling problem

To determine levels of production over time. Constraints include demand requirements (possibly with backordering), capacity limits (including warehouse space for inventory), and resource limits. One basic model is as follows.

Let

$LaTeX: x_t =$ level of production in period t (before demand);
$LaTeX: y_t =$ level of inventory at the end of period t;
$LaTeX: U_t =$ production capacity in period t;
$LaTeX: W_t =$ warehouse capacity in period t;
$LaTeX: h_t =$ holding cost (per unit of inventory);
$LaTeX: p_t =$ production cost (per unit of production);
$LaTeX: D_t =$ demand at the end of period t.

Then, the mathematical program is

$LaTeX: \min \{ p^T x + h^T y : 0 \le (x, y) \le (U, W), \, y_{t+1} = y_t + x_t - D_t \mbox{ for } t=0,\dots,T \}.$

$LaTeX: y(0)$ is the given initial inventory, and $LaTeX: T$ is the planning horizon.

The condition that $LaTeX: y \ge 0$ means there is no backordering. Other tacit assumptions, which could be relaxed to gain more scope of the model are as follows.

• Letting quantities be multiple – e.g., $LaTeX: x_{k,t} =$ level of production of product $LaTeX: k$ at time $LaTeX: t.$ In such cases, competition for warehouse space and other resources result in more equations.
• There could be an ending inventory constraint, such as $LaTeX: y_T=y_0.$
• There could be additional decision variables to allow capacity expansion.
• Costs could be nonlinear functions.
• This could be a stochastic program; for example, with demands not known with certainty.

Also see warehouse problem.