BOOK REVIEWS The following reviews of 6 books on Linear Programming are copyrighted by Operations Research and are reprinted with their permission. M. Sakarovitch, "Notes on Linear Programminq", Van Nostrand-Reinhold, N.Y., N.Y., 1971, 175 pgs, $5.95 ($3.95 in paper) ...[This book] resembles in every way the lecture notes on linear programming circulating in most universities where the instructor has opted for an individual exposition of the simplex method (relation to linear algebra, geometric and economic interpretation, parametric and duality theory) rather than for the use of a standard textbook (like SIMMONNARD's Linear Programming). Whenever he found it adequate, Sakarovitch has wisely adopted examples, theoretical results, or notations from previously published texts (Dantzig, for instance) and he presents us with a good adaptation, to his particular didactic needs, of the existing mass of knowledge available in the linear programming community. These 'notes on LP' are intended for undergraduate students with little knowledge of mathematics (one year of calculus). The book consists of some 170 typewritten pages printed by photo-offset process. By Claude-Alain Burdet, CarnegieMellon University. W. Allen Spivey and Robert M. Thrall, "Linear Optimization", Holt, Rinehart and Winston, New York, New York., 1970, 530 pgs., $15.75 The book is a much more complete presentation of a constructive approach to linear programming. Although it is not primarily intended for mathematicians, it is rigorous and based on the "pedagogical philosophy that mastery of mathematical programming is most rapidly achieved by most students by a multi-layered approach in which a topic may be explored via a special case and then attacked in more and more generality until finally the research frontier is reached. This approach has in its favor the feature that a student who stops short of the most general theory still may have achieved a usable introductory understanding of many important topics. The early chapters introduce the reader to linear programming with simple models and practical examples, using extensively geometrical intuition as well as elementary algebra and common sense. The list of modeling exercises contains some excellent practical problems of moderate size that are used in later chapters in connection with more sophisticated aspects of LP (such as post-optimality analysis, for instance). In the intermediate layer, one finds a constructive development of the simplex method and a separate treatment for the assignment problem. The application of the simplex method is illustrated by practical examples; the method is also used to prove some theoretical results, such as Farkas' Lemma. Special topics of particular importance (duality, parametric linear programming, post-optimality, decomposition and upper-bounding techniques) complete the presentation of the theory of the simplex algorithm. The assignment problem and the Graves-Thrall algorithm for capacitated transportation problems are also discussed. Another attractive feature of this book is found in the last 130 pages where two appendices, one on "Foundations, Sets, and Functions" and the other on "linear Algebra" have been added; they constitute a well designed exposition of the mathematical notions that are relevant to linear programming. 25

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