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IMA Journal of Management Mathematics Advance Access originally published online on November 5, 2008
IMA Journal of Management Mathematics 2009 20(3):233-249; doi:10.1093/imaman/dpn029
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© The authors 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Introduction to max-linear programming

P. Butkovic{dagger}

School of Mathematics, University of Birmingham, Edgbaston,Birmingham B15 2TT, UK

A. Aminu

Department of Mathematical Sciences, Kano University of Science and Technology, Wudil, P.M.B 3244, Kano, Nigeria

{dagger} Email: p.butkovic{at}bham.ac.uk

Received on 1 July 2008. Accepted on 7 October 2008.

Let a {oplus} b = max(a, b) and a {otimes} b = a + b for a, b isin R. Extend this pair of operations to matrices and vectors in the same way as in linear algebra. Being motivated by scheduling of multiprocessor interactive systems, we introduce max-linear programs of the form fT {otimes} x -> min (or max) subject to A {otimes} x {oplus} c = B {otimes} x {oplus} d and develop solution methods for both of them. We prove that these methods are pseudo-polynomial if all entries are integers. This result is based on an existing pseudo-polynomial algorithm for solving the systems of the form A {otimes} x = B {otimes} y.

Keywords: max-linear programming; optimal solution; pseudo-polynomial algorithm


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