Recursive functions on the plane and FPTASs for production planning and scheduling problems with two facilities

We consider a production model with two facilities sharing a resource during a time horizon consisting of a number of time periods. Cumulative production levels at the ends of consecutive periods are linked with constraints of a general form. This allows us to give different interpretations related to scheduling and input–output analysis. The model may arise either separately or in the structure of more general production models. In both cases it is reasonable to find an optimal or near-optimal distribution of resources between these two facilities. This helps either to develop a new production plan or to improve an existing one. The problem in question is NP-hard. We show that our approach leads to fully polynomial time approximation schemes (FPTASs).