Optimal randomized algorithm for a generalized ski-rental with interest rate

We introduce the continuously compounded interest rate into a generalized ski-rental problem with two options: either pay some rental proportionally to the usage time (the rent option), or buy the equipment and then pay some reduced rental proportionally to the usage time (the generalized buy option). Under the framework of competitive analysis, a randomized algorithm for the modified model is constructed and then is proved optimal by Yaoʼs Lemma, and thus the optimal competitive ratio is obtained. The result shows that the introduction of the interest rate puts off the optimal purchase date and diminishes the uncertainty involved in the decision making.

► We introduce the continuously compounded interest rate into a generalized ski-rental to make the model more realistic.
► We obtain the optimal randomized strategy and its competitive ratio.
► The introduction of the interest rate improves the performance of the optimal randomized strategy.