Duration problem with multiple exchanges

Charles E. M.  Pearce; Krzysztof  Szajowski; Mitsushi Tamaki

doi:10.3934/naco.2012.2.333

Article Contents

2012, Volume 2, Issue 2: 333-355. Doi: 10.3934/naco.2012.2.333

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Duration problem with multiple exchanges

1.
The University of Adelaide, Department of Applied Mathathematics, Adelaide, South Australia 5005
2.
Institute of Mathematics and Computer Science, Wrocław University of Techechnology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław
3.
Department of Business Administration, Aichi University, Nagoya Campus, Hiraike 4-60-6, Nakamura, Nagoya, Aichi 453-8777

Received: October 2011

Revised: May 2012

Published: May 2012

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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 $(s_1,s_2,\ldots,s_m)$ 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.

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Mathematics Subject Classification: Primary: 60G40, 90A46; Secondary: 62L15, 60K99.

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