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.

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.


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