This paper studies a single-item multi-period inventory model in which future prices of the purchased item are assumed to be determined by a Markovian stochastic process instead of being known with certainty. Convex holding and shortage costs and a set-up charge for ordering are assumed. Such a model applies to purchasing a commodity whose price fluctuates widely because of speculative activity and large variations in supply or demand. For both a finite and infinite planning horizon, the paper determines the form and bounds of optimal policies and discusses computational approaches exploiting structure.

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The bond refunding problem is formulated as a multiperiod decision process in which future interest rates are determined by a Markovian stochastic process. It is assumed that a single bond is to be outstanding at a given time. Given the future requirements for debt financing, the decision maker must decide whether to keep his current bond or to refund by issuing a new bond at the current market interest rates. Over a finite planning horizon, the structure of policies which minimize expected total discounted costs is studied.

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The machine replacement model studied assumes that, at the end of each discrete interval, the state of deterioration of a machine and the current cost of replacement become known. A decision to keep or to replace must then be made, given that future deterioration and costs are determined by a known Markovian process. A finite set of possible states of deterioration is considered, including a "failed" state at which replacement by a "new" machine must occur. The operating cost is an increasing function of the level of deterioration, and replacement cost is the difference between new machine cost and salvage value. For both a finite and infinite planning horizon, given current cost, the optimality of a "control level" policy is demonstrated. Linear programming and policy iteration methods exploiting problem structure for the calculation of optimal policies are derived.