The Machine Maintenance and Sale Age Model of Kamien and Schwartz Revisited
重新审视了Kamien和Schwartz的经典机器维护与销售年龄模型，澄清了其确定性最优控制解与两个随机模型的关系，并补充了最优策略的充分条件，适用于机器维护及其他领域。
In this paper, we revisit and clarify the celebrated machine maintenance and sale age model of Kamien and Schwartz (KS) involving a machine subject to failure. KS formulate and solve the problem as a deterministic optimal control problem with the probability of the machine failure as the state variable. Thus, they obtain deterministic optimal maintenance and sale date. We study two underlying stochastic models with known and random machine modes, and clarify the relationship between the resulting value functions to that of KS. In particular, our maintenance and sale date decisions, when the machine is in operation, are precisely the ones obtained from the deterministic solution of KS. We explain why that is so. Moreover, we provide a sufficient condition for an optimal maintenance and sale date policy that is missing in KS. We describe many applications of the KS model in areas other than that of machine maintenance. We conclude the paper with extensions of the KS problem that are stochastic control problems not easily solvable or not at all solvable as deterministic problems.