Discounted Models for the Single Machine Inspection Problem

In this paper, we address the problem of determining optimal inspection schedules for a single machine subject to exponentially distributed failures. We formulated the inspection models using the concept of discounted cash flow analysis to account for the effects of time value of money on the inspection policies. In addition, these models are developed under different assumptions on the planning horizon related to its length (infinite and finite) and nature (deterministic and stochastic). In the finite horizon problem, we considered both cases where the inspection times are treated as continuous and discrete variables. For the case of continuous inspection times, we prove that the equal inspection interval policy is optimal only under certain conditions. In the case of a random horizon, we show that the problem can be transformed to the infinite horizon problem and, consequently, can be solved using the same procedure. Numerical examples are used to illustrate the models and their solution procedures.