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Algebraic and Combinatorial Methods in Operations Research (Mathematics Studies)

FOREWORD

A recurring theme in operations research (O.R.) is that of optimization, and over the last 35 years the subjects of O.R. and mathematical programming have developed side by side and enriched one another.

Much traditional 0.R has been concerned with the behaviour of continuous real variables representing e.g. material stocks, time or money, and the corresponding Optimization theory is one in which real linear algebra, inequalities and the differential calculus have played important roles. However, many systems with which O.R. is concerned incorporate discrete structures for which the optimization questions are combinatorial rather than continuous: one thinks of sequencing, scheduling and flow-problems and of the great variety of questions which can be reformulated as

path-finding circuit-finding or sub-graph-finding problems on an abstract graph.

Correspondingly, we have witnessed a vigorous growth in the theory and practice of combinatorial optimization. A related, but perhaps less well-known, development has been in the application of

ordered algebraic structures to optimization problems. This application is made relevant by the fact that many optimization questions depend essentially on the presence of two features: an algebraic language within which a system can be modelled and an algorithm articulated; and an ordering among the elements which enables a significance to be given to the concept of minimization or maximization.

By adopting this slightly abstract point of view, we can make useful reformulations: certain bottleneck problems become algebraic linear programs; certain machinescheduling problems reduce to finding eigenvectors and eigenvalues of a matrix over a semiring; certain path-finding problems reduce to the solution of linear equations over an ordered structure.

Many optimization problems of the kind which have arisen in O.R. assume, under such reformulation, the appearance of problems of linear algebra over an ordered system of scalars. Hence we may look to the highly-developed classical theory of linear algebra over the real field to give us hints as to how we might approach these problems or, if appropriate adaptations of classical techniques cannot be found, we have a well-defined research program to elucidate the theory of linear algebra over such ordered structures, and to see how far the algorithms and duality principles, familiar to us from linear and combinatorial optimization over the real field, extend to more general structures These questions have stimulated a good deal of research over the last twenty-five years. From a few isolated publications by one or two researchers in the late 1960’s the subject has matured into an identifiable branch of applicable mathematics, with an international following. We invited a number of those who have contributed to this development, to participate in the production of a publication featuring some of their more recent work. It is the result of their enthusiastic acceptance of this invk tation which we are now pleased to present as this volume in the series of Annals of Discrete Mathematics.

R.E. Burkard

R.A. Cuninghame-Green

U. Zimmermann

Acknowle&ement: I should like to add a personal note of gratitude to Tricia Carr,

who made such a beautiful job of preparing the manuscript.

R.A. Cunjnghame-Green

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