Discrete-continuous analysis of optimal equipment replacement

In Operations Research, the equipment replacement process is usually modeled in discrete time. The optimal replacement strategies are found from discrete (or integer) programming problems, well known for their analytic and computational complexity. An alternative approach is represented by continuous-time vintage capital models that explicitly involve the equipment lifetime and are described by nonlinear integral equations. Then the optimal replacement is determined via the optimal control of such equations. These two alternative techniques describe essentially the same controlled dynamic process. We introduce and analyze a model that unites both approaches. The obtained results allow us to explore such important effects in optimal asset replacement as the transition and long-term dynamics, clustering and splitting of replaced assets, and the impact of improving technology and discounting. In particular, we demonstrate that the cluster splitting is possible in our replacement model with given demand in the case of an increasing asset lifetime. Theoretical findings are illustrated with numeric examples.