Deferrable energy/commodity purchase and consumption policy under stochastic prices

This paper considers the problem of determining the cost-optimal policy of the task of purchasing or consuming a certain commodity, such as energy, within a given number of time steps. The price of the commodity in each time slot is randomly chosen from a known probability distribution and made known at the beginning of the slot. This problem is of renewed interest and is rigorously examined in a novel way. We consider different discrete sets of choices for the price-taker, in the simplest case whether to act to the full extent possible, or to pass. For each case we derive the optimal policy for independent price processes, show that it is a threshold strategy, and prove monotonicity properties for the thresholds and costs. We prove that it is more economical to accomplish the task in one time step than in several. The optimal policy is computed in closed-form for i.i.d. price processes with uniform distribution.