The duration problem with multiple exchanges
Abstract
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 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.