The duration problem with multiple exchanges

TL;DR

This paper analyzes a multiple-choice duration problem, establishing optimal strategies based on critical numbers, and links it to the classical secretary problem, with formulas for large object sets and extensions to cost-involved models.

Contribution

It introduces a sequence of critical numbers for optimal selection in the multiple-choice duration problem and connects it to the classical secretary problem, including formulas for large populations and cost extensions.

Findings

Existence of critical number sequences for optimal strategies

Equivalence between duration and classical secretary problems

Recursive formulas for large object sets and cost extensions

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 $s_k$ ($1\leq k\leq m$). 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.