Introduction to Stochastic Programming
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#1998 Operational Research Society Ltd. All rights reserved. 0160-5682/98 $12.00 http://www.stockton-press.co.uk/jor
Book Selection Edited by JM Wilson IM Stancu-Minasian: Fractional Programming: Theory, Methods and Applications DW Bunn and ER Larsen (eds): Systems Modelling for Energy Policy A Prekopa: Stochastic Programming JR Birge and F Louveaux: Introduction to Stochastic Programming
Fractional Programming: Theory, Methods and Applications IM Stancu-Minasian Kluwer Academic Publishers, Dordrecht=Boston=London, 1997. viii 418 pp. £135; $226; D¯ 365. ISBN 0 7923 4580 0 A fractional programming (FP) problem is a constrained optimisation problem in which the objective function is a ratio of functions such as cost=volume, output=input, pro®t=capital, return=cost and signal=noise etc. Most of this book is of a technical mathematical form concentrating on theory and solution methods. The ®rst chapter describes over two dozen applications, devoting about a page to each. I would have liked these treatments to be longer. The applications include blending, cutting stock, transportation with multiple objectives, investment allocation, and set covering. The good news, for those familiar with linear programming, is that if the objective is a ratio of linear functions then the FP problem reduces to a linear programme, assuming the constraints are linear too. More generally, if the numerator function is nonlinear but concave and the denominator function is convex (on the feasible region), then a local optimum will be a global optimum. There are separate chapters on duality, nonlinear FP, multiple objectives, integer FP, the fractional transportation problem, and even FP in the complex plane! Each chapter concludes with a section of historical notes and commentaries. As well as the usual subject index there is an extensive author index. I am aware of two other books on FP. The one by Craven1 which is also a theoretical treatment, and a book written in German by Schaible2. This present book has been translated from Romanian by Victor Giurgiutiu; in which I found the English to be always comprehensible. Many readers may be surprised by how much research has been published in FP but the author has done a great
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service by collecting together a bibliography of over one thousand references. The bibliography is arranged by author. Despite this broad coverage the book still leaves untouched two important areas. One is in numerical analysis and deals with the use of rational functions (ratios of polynomials): (i) to ®t=model data; and (ii) for the approximation of complicated functions. For example, pocket calculators use rational functions to evaluate certain elementary functions, and these provide greater accuracy than power series for the same number of terms. The other area is DEA (data envelopment analysis) which has a literature as large as the bibliography we have here3. DEA is surely the most common application of FP and has now reached a level of importance such that it can be found in university textbooks
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