An optimal scheduling problem for a system with finite buffers and non-negligible setup times and costs

We consider a setup scheduling problem for a two part type single machine manufacturing system with finite buffers. Using a fluid approximation, we look for a scheduling policy which minimizes a cost index under steady and transient conditions. This cost index includes backlog and demand loss costs, as well as the costs for every setup performed. A dynamic programming approach is used to derive the structure of the optimal policy whose description is completed through analytical results. A symmetry assumption on part types is considered to simplify the problem.