Planning Horizon Procedures for Machine Replacement Models

This paper develops efficient forward algorithms and planning horizon results for several machine replacement models under an improving technological environment over time. The models are that of cost minimization, profit maximization, and cost minimization with probabilistic breakdowns. The conditions of the improving technological environment are stated in terms of operating costs, incremental profits, and expected operating costs, respectively, for these models. The state of technology can be assumed to improve every period or every so many periods. The conditions are fairly general and are likely to be satisfied by most improving technologies. What is shown, specifically, is that there exists a forecast horizon T such that the optimal replacement decision for the first machine (new or existing) based on the forecast of machine technology until period T remains optimal for any longer (than T) horizon, and for that matter, the infinite horizon problem. In particular, it is possible to decide whether it is optimal to replace an existing machine at the beginning of a period or keep it at least one more period with much less information than the entire future forecast of technology. It should be noted that the forecasts become less precise farther into the future and that requiring an infinite horizon forecast for decision-making would be unrealistic. Because the planning horizon results obtained in this paper free the solution from an arbitrary horizon and ascertain the optimality of the first period decision or the first few periods decisions based on the forecast of technology only up to some finite period T, these results represent a major advance in the machine replacement literature. These planning horizon results also enable us to develop a computationally highly efficient forward algorithm for the models under consideration. The algorithm is illustrated with a numerical example. Another important corollary of the planning horizon results is that the optimal number of machine replacements cannot decrease as the length of the horizon (planning period) increases. Planning horizon procedures are shown to be easily extendable to the existing machine case as well as the case when machine lives are constrained within a prespecified finite interval. The paper concludes with various suggestions for future research efforts