The duration problem with multiple exchanges

We treat a version of the multiple-choice secretary problem called the multiple-choice 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 $s_k$ ($1\le k\le m$). 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.