Monotonically improving error bounds for a sequence of approximations for makespan minimization of batch manufacturing systems

We consider a batch manufacturing system model with no intermediate storage and a single machine at each workstation. These models have applications in fields such as chemical processing, computer systems, and manufacturing. We develop a max-plus representation for this system with linear dynamics and show relevant properties of this system. In particular, we express a sequence of increasingly complex approximations to the minimum makespan problem in the max-plus algebra. Using this formulation, we compute monotonically improving error bounds for these approximations, guaranteeing a decrease in error as more computational effort is used in the approximation.