Circulant preconditioners for failure prone manufacturing systems

This paper studies the application of preconditioned conjugate-gradient methods in solving for the steady-state probability distribution of manufacturing systems. We consider the optimal hedging policy for a failure prone one-machine system. The machine produces one type of product, and its demand has finite batch arrival. The machine states and the inventory levels are modeled as Markovian processes. We construct the generator matrix for the machine-inventory system. The preconditioner is constructed by taking the circulant approximation of the near-Toeplitz structure of the generator matrix. We prove that the preconditioned linear system has singular values clustered around one when the number of inventory levels tends to infinity. Hence conjugate-gradient methods will converge very fast when applied to solving the preconditioned linear system. Numerical examples are given to verify our claim. The average running cost for the system can be written in terms of the steady state probability distribution. The optimal hedging point can then be obtained by varying different values of the hedging point.