Production scheduling problem

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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.


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

\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.

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