The duration problem with multiple exchanges

We treat a version of the multiple-choice secretary problem called the multiplechoice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the m–choice duration problem, there exists a sequence (s1, s2, . . . , sm) of critical numbers such that, whenever there remain k choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time sk (1 ≤ k ≤ m). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.