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An Algorithm for Integer Solutions to Linear Programs, 269 Ralph E other MCGRAW-HILL Books MATHEMATICAL METHODS OF OPEEATIONS RESEARCH By THOMAS L. SAATY, Office of Naval Research, 432 pages, 111.50 Covers -within a well-rounded framework mathematical methods which, in the author's experience with operations research and its literature, have the widest use. These methods should enable the reader not only to connect and expand many of them, but to apply them, and to generate new ones. The insertion of an essay on creative thinking emphasizes the fact that the field of operations research is not a mere collection of techniques, but an approach to operational problems which mainly depends on the creative ability of an indi- vidual capable of rigorously selecting and developing the scientific and mathematical ideas he applies, in the solution. THE THEORY OF LINEAR ECONOMIC MODELS By DAVID GALE, Brown University. 330 pages, $9.50 Provides the mathematical machinery needed for current linear economics. Linear pro- gramming, two-person games, equilibrium prices and dynamic stability, and static and dynamic theory of linear production models are some of the topics covered. Each model is described fully and illustrated with examples. Eequired mathematics is developed in a ' self-contained manner. ' f INTRODUCTION TO LINEAR PROGRAMMING By WALTER W. GARVIN, California Research Corporation. 281 pages, $8.75 The author seeks to explain the mathematical principles of linear programming and to describe some applications. Primarily designed as a textbook on the undergraduate or graduate level, the book assumes a command of college algebra. The theory of linear programming is taught as a tool to get useful answers to practical problemsnot as a branch of abstract mathematics. The book's great value is in presenting the theory in an easy-to-understand but rigorous manner, in showing how to set up and solve linear programming models of many practical situations, and in suggesting the flexibility of the method. LINEAR PROGRAMMING. Methods and Applications By SAUL I. GASS, Graduate School, U.S. Department of Agriculture; International Busi- ness Machines, Inc. 219 pages, $7.50 A basic presentation of the theoretical, computational, and applied areas of linear pro- gramming for the student, research analyst, and other technical personnel (e.g., applied mathematicians, economists, statisticians) in business and industry, military affairs, and scientific research. The author explains what linear programming is, what problems it can solve, and how to implement the techniques of linear programming to formulate and solve such problems. Mathematical Programming In this timely work of unusual interest the editors provide a complete, authori- tative overview of the present state of mathematical programming. The book consists of a collection of papers presented at the Symposium for Mathe- matical Programming held at The Uni- versity of Chicago during the summer of 1962. The symposium was sponsored by the Graduate School of Business of The University of Chicago, the BAND Cor- poration, and the Association for Com- puting Machinery. Since this was the first international meeting in the field of mathematical pro- gramming since 1959, the papers repre- sent the major work of its most important contributors for the period 1959-1962. Certain especially-commissioned papers survey the major areas of mathematical programming for nonspecialists. Other papers deal with advanced research topics. The subject matter includes aspects of the theory and application of linear pro- gramming, nonlinear programming, dis- crete programming, and network flow theory. These topics are among the most important basic disciplines underlying mathematical economics and operational research, and constitute one of the pri- mary areas of application of high-speed computers to industrial problems. Many papers are devoted to program- ming mftdflls Recent Advances in MATHEMATICAL PROGRAMMING Recent Advances in MATHEMATICAL PROGRAMMING Edited by Robert L Graves Associate Professor of Applied Mathematics Graduate School of Business University of Chicago Philip Wolfe Mathematician, the RAND Corporation McGRAW-HILL BOOK COMPANY, INC. New York San Francisco Toronto London RECENT ADVANCES IN MATHEMA TICAL PROGRAMMING Copyright 1963 by the McGraw-Hill Book Company, Inc. All Rights Reserved. Printed in the United States of America. This book, or parts thereof, may not be repro- duced in any form without permission of the publishers. Library of Congress Catalog Card Number 63-13159 24120 Introduction The growth of mathematical programming since its birth in 1947 has been marked and stimulated by a distinguished series of Symposia at which the major part of the research workers in the field have presented their work. The quality of these meetings has been ensured by a certain continuity of participants and supporters. The first such meeting was held in Chicago in 1949, sponsored by The RAND Corporation; selected papers from it appeared as ''Activity Analysis of Production and Allocation", (1951), edited by Tjalling C. Koopmans. The first meeting to bear the title "Symposium" was held in Washington, D. C. in 1951 under the sponsorship of the National Bureau of Standards and the U. S. Air Force; abstracts and selected papers were published in "Symposium on Linear Inequalities and Programming" by the Air Force (1952). The "Second Symposium in Linear Programming" was held in Washington in 1955, under the same sponsor- ship as the first; its Proceedings, edited by Henry A. Antosiewicz, were published by the sponsors in 1955 The third symposium, "A Symposium on Mathematical Programming: Linear Programming and Recent Extensions" was held in Santa Monica, California in 1959; a report, "The RAND Sym- posium on Mathematical Programming", was published by The RAND Cor- poration in 1960. wjijwjv OJTY fwn } PUBLIC LIBRARY The fourth symposium, on wnich inis voiureT*^6rra;^ras*held in Chicago, Illinois, under the joint sponsorship of the Association for Com- puting Machinery, the Graduate School of Business of the University of Chicago, and The RAND Corporation, on June 18-22, 1962. The organizers of the present conference were drawn from the membership of SIGMAP, the Special Interest Group for Mathematical Programming in the Associa- tion for Computing Machinery. SIGMAP is devoted to furthering all aspects of mathematical programming. Forty-three papers were presented at the Symposium, to an audience of more than 240 persons, from five continents. Papers were given in each of eight areas of mathematical programming. In four of these, especially in- vited survey papers gave the audience a broad view of the methods and problems of that area. The surveys are among the twenty-three papers ap- pearing in full in these Proceedings. The remaining papers, whose appear- ance in full here has been sacrificed to the requirement that this volume be of reasonable size and to some authors' commitments to publish elsewhere, are given as abstracts. Full copies of such papers can generally be obtained irom me autnors. Papers 14, 16, and 38 were edited by the authors from transcriptions made at the Symposium. The first nine papers deal with the general theory of mathematical pro- gramming. The survey paper by Tucker, which begins this group, presents the theory of linear programming in its most powerful current form. The eight papers which follow deal with various aspects of the theory of both linear and nonlinear programming. Papers 10 through 13 deal with nonlinear programming with "nonlin- ear" used in the accepted sense as referring to * 'reasonably smooth*' non- linear objective functions and constraints. The first paper of these, by Wolfe, is a survey of most of the proposals which have been made for sol- ving such problems. Papers 14 through 18 are devoted to stochastic programming mathe- matical programming problems whose data may be random variables. Such problems require careful statement if meaningful results are to be obtained. The first paper of these, by Madansky, surveys the outstanding problems of this field. Papers 19 through 22 are concerned with computational procedures for very large linear programming problems. Such problems always have some regularity of structure that appropriately designed algorithms can take ad- vantage of. The papers of this group constitute four different methods of attack on aspects of that regularity. Papers 23 through 26 are concerned more intimately with the computa- tional processes of mathematical programming, examining in detail, 'ways in which variations of the simplex algorithm the most effective current tool for linear programming can be made even more effective. Papers 27 through 33 lie in the area of applications of mathematical pro- gramming. Since entire series of books are being written on this subject, these papers can no more than sample the area by showing some of the latest uses of currently available methods. Papers 34 through 37 are devoted to integer programming problems. The first of these is Gomory's 1958 paper, now almost a classic, which es- tablished the subject of integer programming. It has been substituted for his paper presented at the Symposium because, still in great demand, it has been out of print for some time. The last group of papers, 38 through 43, are concerned with problems of network flow, a special class of linear programming problem which, be- cause of its rich structure, has given rise to an extensive and elegant theory. The first paper, by Fulkerson, surveys this area, reviewing its main tools and its
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