Flow control in a failure-prone multi-machine manufacturing system

The problem of computing optimal production rates for a multimachine multiproduct manufacturing system subject to machine failure is considered. The inventory balance equation is represented by a discrete-time flow model with Markovian jumps to take into account machine breakdown. When defining quadratic cost functions, the associated optimal control problem leads to a set of strongly coupled Riccati-like equations. A necessary and sufficient condition for the existence of a positive semidefinite solution for such equations is given. By verifying a simple matrix inequality, it is shown that such solution exists and can be obtained as a limit of a monotonic sequence. This leads to a straightforward numerical algorithm for the computation of the solution. In this way the optimal production rates for each product type can be determined. An example involving 9 machines and 6 product types is given to illustrate the proposed approach.