## Reviews of Steven Vajda's books

**1. Theory of Games and Linear Programming (1956), by Steven Vadja.**

**1.1. Review by: J G Mauldon.**

*J. Roy. Stat. Soc. Series A (General)* **119** (3) (1956), 340.

The reader who is attracted to this book by the first half of its title may find himself rather disappointed. ... However, the book provides a very welcome elementary introduction to the theory of Linear Programming. It maintains a high standard of rigour, though requiring no mathematical knowledge beyond some elementary algebra. The emphasis is on practical numerical methods and several detailed computing techniques are presented.

**1.2. Review by : Harry Markowitz.**

*Operations Research* **4** (6) (1956), 749-750.

This book provides an introduction to Game Theory and Linear Programming. Abstract matrix algebra and more advanced mathematics are not used. ... [It] should make a good primer for someone who has a small amount of college mathematics and desires a short introduction to the algebra and geometry of the subjects covered.

**1.3. Review by: R L Goodstein.**

*Math. Gaz.* **41** (337) (1957), 221.

This little book is in very select company and is only the third or fourth to appear on the rapidly expanding new topics of the theory of games and linear programming. ... Of the present volume one may say that teachers and students could consult the first section with much profit; the two later sections are a little too condensed for pupils.

**1.4. Review by: G A Coutie.**

*J. Roy. Stat. Soc. Series C (Applied Statistics)* **6** (1) (1957), 79-80

The book is an excellent, united account of two new and interesting subjects [Theory of Games and Linear Programming], and its size makes it such that it can be read easily from cover to cover rather than treated as a book of reference.

**1.5. Review by: P. G. M.**

*J. Inst. Actuaries (1886-1994)* **83** (1) (1957), 63-64.

The book as a whole devotes rather more attention to linear programming than to the theory of games, the latter being represented by about a third of the book, and only the simpler forms of games are discussed. From the title of the book this is disappointing, and the reviewer feels that given the subject-matter included in the book the theory of games should have been put after linear programming both in the title and in the order of contents. This is, however, only a minor criticism of a very useful work which provides in a compact form some idea of the problems that can be tackled with the aid of the theory of linear programming.

**1.6. Review by: A Vazsonyi.**

*Management Science* **6** (1) (1959), 134.

This small but excellent volume contains a surprising amount of information. The booklet can be put into your pocket, but it contains a very good introduction into the Theory of Games and Linear Programming. Both graphical and algebraic methods are treated, and even such topics as degeneracy and duality are covered. The treatment is essentially elementary, but it is quite scholarly. Whatever is in the booklet is clearly stated and well-explained.

**1.7. Review by: W W Cooper.**

*Econometrica* **27** (3) (1959), 495-497.

Within its compressed scope, this little book presents a survey of game theory and linear programming.

**1.8. Review by: Richard A Good.**

*Amer. Math. Monthly* **67** (1) (1960), 91-92.

Each of the eleven chapters dispenses a bite-sized morsel. But in toto the reader has pleasantly digested the basic notions of game theory, linear programming, and their interconnections.

**2. Readings in Linear Programming (1958), by Steven Vadja.**

**2.1. Review by: J Wise.**

*J. Roy. Stat. Soc. Series A (General)* **121** (4) (1958), 483-484.

Linear programming and the related subjects of activity analysis and the game theory have developed rapidly in recent years as a result of the stimulus arising from the interest and usefulness of these subjects to research workers engaged in the study of economic, military, industrial and mathematical problems. Linear programming computations consist of simple arithmetic processes, although the mathematical proofs and problems of formulation are sometimes quite deep and technical. This little collection of (partly) worked examples in the arithmetic of linear programming is primarily aimed at readers whose mathematical knowledge is slight but who wish to acquire a knowledge of and familiarity with some of the simpler methods of linear programming from the operational research standpoint.

**2.2. Review by: A Vazsonyi.**

*Management Science* **6** (1) (1959), 134.

This small but excellent volume contains a surprising amount of information. The book is almost full-sized, but quite thin. It presents examples of applications, and the stress is on the practical point of view. In spite of the fact that there are less than a hundred pages in this book, a wide variety of topics are discussed, such as the transportation problem, caterer problem, production scheduling, transhipment, bid evaluation, flow through a network, transportation with restrictions, ship scheduling, personnel assignment, routing aircraft, investment, nutrition problem, airlift, blending of aviation gasolines, smooth patterns of production, selection of products, trim loss reduction, attendants' rota, warehousing, and games.

**2.3. Review by: Wilfred Candler.**

*Econometrica* **27** (4) (1959), 714-715.

The title of Dr Vajda's book 'Readings in Linear Programming' would suggest to many economists an anthology of articles which have become classics on the topic of linear programming. In fact, this book would better be described as a digest of articles which deal with the solution of problems by linear programming, with emphasis on the use of the transportation method. Dr Vajda briefly introduces the subject: "Linear Programming deals with maximizing or minimizing a linear expression of variables, called the objective function, while the variables satisfy given linear equations or inequalities referred to as constraints ..."

**2.4. Review by: Robert B Fetter.**

*Operations Research* **7** (3) (1959), 404-405.

The task which Dr Vajda sets for himself in this book is indeed formidable, and it is all the more amazing to note that its accomplishment requires only 93 pages of text. The objectives are the following:

1. To present a clear outline of the basic ideas, both structural and computational, involved in linear-programming applications.

2. To cover one example in each area of actual or potential application of linear programming.

3. To present enough theory to satisfy the practitioner, although no attempt is made to satisfy the mathematician.

All in all, the book is an admirable outline of the principal actual and potential applications of linear programming and worth serious study by those whose work will involve the formulation of linear-programming models in actual situations. The references seem complete and only one typographical error was noted. Recommended for both the beginning and the semi-skilled practitioner in the field.

**2.5. Review by: Clement Winston**

*SIAM Review* **2** (1) (1960), 56-57.

Linear programming, now often shortened to L.P., has risen from an esoteric position until the term is about ready to be tossed about by the celebrated man in the street. He may not understand what it is but he has a feeling that it works wonders. From a little over a decade ago, when its name was barely coined, there has now come into existence a considerable body of literature on linear programming to be added to those in the related fields of input-output analysis and on game theory; and more is appearing regularly. The present book adds another item to the growing list. It is not a large volume but it is nevertheless a useful one. It presents to the reader in a brief but clear way a variety of examples of optimal planning of independent activities subject to certain restraints to which linear programming procedures are applicable.

**2.6. Review by: Jerome Rothstein.**

*Science, New Series* **131** (3404) (1960), 916-917.

This little book is something of an expository tour de force. In less than 100 pages, its 24 chapters give a representative collection of worked out examples of problems in which linear programming can be used.

**2.7. Review by: Lionel McKenzie.**

*J. Amer. Stat. Assoc.* **55** (290) (1960), 394-396.

This is an unusual book of "readings." It is not composed of reprints of articles, but of short chapters, most of which are closely dependent on one or more published papers. However, the chapters present this material differently from the original papers. They are usually much briefer and the argument is simpler. Theoretical discussions and mathematical theorems are suppressed, while arithmetic illustrations of the methods furnish most of the matter. Many of the twenty-four chapters occupy two or three pages and present their messages economically.

**3. Introduction to Linear Programming and the Theory of Games (1960), by Steven Vadja.**

**3.1. Review by: G Morton.**

*J. Roy. Stat. Soc. Series A (General)* **123** (4) (1960), 493.

Anyone interested in linear programming will know how much Dr Vajda has contributed to the spread of this branch of applied mathematics-cum-economics in this country. Those who have attended his lectures or read his earlier books need not be told of the ease with which the author can handle his subject at different levels of mathematical understanding; and the present volume shows how, with a minimum of technique, new ideas can be made clear and explicit.

**3.2. Review by: A J Goldman.**

*Science, New Series* **132** (3436) (1960), 1306-1307.

Part 1 of Vajda's monograph provides a lucid introduction to the main ideas of linear programming, a mathematical discipline concerned with the maximization or minimization of a linear function of non-negative variables subject to linear constraints (equations or inequalities). ... Part 2 deals with the so-called theory of games, that is, with the selection of optimal behaviour versus intelligent opponents. ... The author has been remarkably successful in giving a lively and accurate treatment of so much material ...

**3.3. Review by: E W Barankin.**

*Operations Research* **9** (2) (1961), 283.

This little book is in two parts, the first covering linear programming, the second describing game theory. ... The book is marked by an extensive use of examples and illustrations, around which the notions and methods are introduced and explained. The large coverage achieved in so few pages is striking. To one acquainted with all the subject matter, the (admittedly abbreviated) explanations will perhaps leave nothing to be desired. For the uninitiated, however, some of the more involved material will certainly fail to come across satisfactorily. But this does not deny the achievement by the author of a fine, compact introduction to the pertinent basic ideas, and to what they look like at work in their substantive context.

**3.4. Review by: James H Griesmer.**

*SIAM Review* **3** (1) (1961), 77-78.

This is a short expository book on linear programming and the theory of games for the stated purpose of leading the reader "to an understanding of the place of Linear Programming and of the Theory of Games in modern Operational Research," and of stimulating "further reading."

**3.5. Review by: Ralph E Gomory.**

*J. Amer. Stat. Assoc.* **56** (295) (1961), 761-762.

Dr Vajda has already written two concise elementary presentations of the main topics of linear programming and related areas. These are 'The Theory of Games and Linear Programming' (1956), and 'Readings in Linear Programming' (1958). The present closely related volume was developed from a series of lectures on the same subject matter and contains two main parts, one on linear programming and one on game theory. ... This reviewer prefers the clearer if somewhat longer treatments of the earlier volumes.

**4. Mathematical Programming (1961), by Steven Vadja.**

**4.1. Review by: A R Catchpole.**

*Operations Research* **12** (4) (1961), 278-279.

This book aims to provide firstly a textbook of Linear and Non-linear Programming and secondly a guide to more recent development in this field. ... Dr Vajda's style is admirably clear, and the subject matter has been well arranged. ... This book can be thoroughly recommended to the student of Mathematical Programming; for those with experience in this field, its usefulness is probably confined to the later chapters.

**4.2. Review by: A H Land.**

*J. Roy. Stat. Soc. Series A (General)* **125** (3) (1962), 495-496.

With one reservation, this book is wholly admirable. Its great strength is the effortless way in which it leads the student right up to the frontiers of the subject. ... The reservation is that, though the book is designed as a text book for a graduate course in mathematical programming, it restricts itself to "fairly elementary mathematics".

**4.3. Review by: A D Roy.**

*Economic J.* **72** (288) (1962), 938-939.

Dr Vajda has written a concise and wide-ranging book which will prove extremely valuable to anyone attempting to get to grips with the foundations and applications of programming without having to delve through a welter of papers of widely varying degrees of difficulty. ... [We should] make clear how much of value and of interest Dr Vajda has packed so tidily in this book. We are deeply in his debt for making available to a potentially large readership the essence of a subject whose practical importance is likely to grow for many years.

**4.4. Review by: John J Sopka.**

*SIAM Review* **4** (2) (1962), 160-161.

This relatively small book has an attractive list of contents. Introductory material is contained in early chapters on Linear Inequalities, Duality, Theory of Graphs, General and Special Algorithms including the simplex, dual-simplex, primal-dual, inverse matrix, cross-section and relaxation procedures. Specialisations and applications are made to the transportation and assignment problems, input-output analysis and to such specific topics as nutrition, machine sequencing, race-course betting. Later chapters consider Parametric, Discrete and Stochastic Linear Programming followed by Non-linear and Dynamic Programming. With all this ground to cover the author treats certain of these topics superficially. However, the discussion is well supplemented with illustrative exercises and references.

**4.5. Review by: Ronald I McKinnon.**

*J. Amer. Stat. Assoc.* **57** (299) (1962), 711-712.

Dr Vajda has succeeded in writing an exposition of mathematical programming which rigorously develops the mathematical foundations of the subject, suggests applications to particular problems, and supplies illustrations of computational procedures. This nice balance of objectives is obtained without recourse to higher mathematics. ... Dr Vajda has given comprehensive, rigorous coverage to a diverse subject, making it accessible to a large class of readers. In this respect, the book is unique.

**4.6. Review by: E M L Beale.**

*Math. Gaz.* **46** (358) (1962), 338-339.

Linear programming is the name given to problems involving the minimization of some linear "objective function" of variables subject to linear inequality constraints. Increasing attention has been paid recently to extensions of linear programming in which the objective function, or even the constraints, are not necessarily linear; and the term "Mathematical Programming" has been coined to cover this broader field. Dr Vajda has followed up his 3 successful books on various aspects of linear programming and the theory of games with a larger book on mathematical programming which deserves to be equally successful.

**4.7. Review by: Thomas L Saaty.**

*Econometrica* **30** (4) (1962), 845-846.

Most of the material of this book is twentieth century applied mathematics. It is concerned with methods used to solve programming problems. ... The book is rich in exercises and illustrative examples, and it covers a wide variety of theorems ranging from existence of solutions to their construction.

**4.8. Review by: J C G Boot.**

*Technometrics* **4** (4) (1962), 618-619.

The present book fails in its first aim: to provide a textbook of linear and non-linear programming. However, it comes close to succeeding very well indeed in its second, possibly more important, purpose: to guide the reader to the rapidly expanding frontier of programming.

**4.9. Review by: K B Haley.**

*Biometrika* **50** (1/2) (1963), 230-231.

The number of publications in the field of Mathematical Programming is increasing so rapidly that it is difficult for any book which has as one of its aims to 'guide the reader to the rapidly expanding frontiers of this recent branch of mathematics' (viz. Linear and Non-linear Programming) not to be out of date before it is published. The selection of material and Dr Vajda's foresight have enabled him to very largely succeed in this aim. ... In this book, Dr Vajda has maintained the high standard he has set himself with his earlier works. ... This is a reference work that will be in constant demand by anyone who uses mathematical programming. It must be classed as one of the best text-books available and should be on the bookshelves of all Operational Research, Statistical and Mathematical Departments.

**4.10. Review: Ronald A Howard.**

*Science, New Series* **139** (3558) (1963), 898.

The field of mathematical programming is becoming so huge that it is difficult to imagine a book which provides complete coverage of it. In Mathematical Programming Vajda does not attempt exhaustive coverage, but seeks instead to guide the reader through the profusion of work on the subject. ... 'Mathematical Programming' makes interesting reading; it is clear, informally written, and packed with examples. Historical comments are included where appropriate, both to stimulate the reader's attention and to give credit to those who have made original contributions to the subject. ... Mathematical Programming is a refreshing change from the "crank-turning" books on linear programming.

**4.11. Review by: B V Wagle.**

*J. Roy. Stat. Soc. Series D (The Statistician)* **14** (2) (1964), 176-177.

The aim of this book is to provide a textbook of Linear and Non-Linear Programming and to discuss some of the more recent developments in this field. There are several good books on Linear Programming and Theory of Games and the author does not introduce any new material on these subjects. Nevertheless he has produced in his first eight chapters an excellent introduction to Linear Programming, with Theory of Games as one of the applications.

**5. Readings in Mathematical Programming (1962), by S Vajda.**

**5.1. Review by: P B Coaker.**

*Operations Research* **14** (2) (1963), 222-223.

This is the second edition of a book which was originally entitled 'Readings in Linear Programming'. As the change of title suggests the scope of the book has been extended by the addition of chapters on discrete programming, quadratic programming and dynamic programming.

**5.2 Review by: E M L Beale.**

*J. Roy. Stat. Soc. Series A (General)* **126** (3) (1963), 470.

This book is presented as a Second Edition of the author's 'Readings in Linear Programming' published in 1958. The only significant change is the addition of seven new chapters devoted to brief introductions to and examples of Discrete Linear Programming, Dynamic Programming, Stochastic Programming and Quadratic Programming, and also of a section on the solutions to the problems posed at the ends of some chapters.

**5.3. Review by: George P DiNardo.**

*Econometrica* **32** (3) (1964), 454-455.

'Readings In Mathematical Programming' is a revised and retitled edition of 'Readings In Linear Programming'. Corrections and slight modifications have been made to existing chapters of the first edition.

**5.4. Review by: R F Churchhouse.**

*Math. Gaz.* **48** (365) (1964), 334-335.

Dr Vajda's book was originally published in 1958 under the title 'Readings in Linear Programming'. The change in the title is no accident, it is made necessary by the inclusion of several new chapters on discrete, quadratic and dynamic programming. Dr Vajda teaches by example, only four of the thirty-one chapters are devoted to theory ... The whole book gives an admirable introduction to the subjects of Linear and Quadratic Programming and provides the reader with some idea of the problems dealt with by Dynamic Programming.

**5.5. Review by: Milton Siegel.**

*Math. Comp.* **18** (85) (1964), 167-168.

This book is a revised edition of 'Readings in Linear Programming', which was first published in 1958. The title of the present edition reflects the inclusion of applications dealing with discrete linear, dynamic, and quadratic programming.

**5.6. Review: Robert L Graves.**

*Amer. Math. Monthly* **71** (5) (1964), 577-578.

This volume is the second edition of 'Readings in Linear Programming', Wiley, 1958. The new title reflects recent developments in the fields which have become generally associated with linear programming. ... The reviewer has found the first edition useful as required independent reading in linear programming courses. The additional material is helpful but most students will not be able to assimilate it independently.

**6. Patterns and Configurations in Finite Spaces (1967), by S Vajda.**

**6.1. Review by: D A Preece.**

*J. Roy. Stat. Soc. Series A (General)* **131** (2) (1968), 233-234.

These two companion volumes [this volume and 'The Mathematics of Experimental Design'] give a concise and lucid account of combinatorial aspects of the construction of designs. In this book, chapters on finite planes, finite spaces of higher dimensions, and configurations are preceded by a review of the necessary algebra.

**6.2. Review by: T J Mitchel**

*J. Amer. Stat. Assoc.* **66** (333) (1971), 226.

This volume [and 'The Mathematics of Experimental Design'] give the reader a short, highly concentrated course in the mathematics of experimental arrangements, primarily those which involve the assignment of varieties (treatments) to blocks (experimental units), subject to given constraints. The statistical aspects are not discussed, even to motivate the requirements of the various designs. The author faithfully fulfils his promise to consider "the combinatorial aspects of the construction of designs, without regard to their practical application or indeed other usefulness." Most of the emphasis, therefore, is given to construction methods and existence theorems, with incidental attention to the enumeration of designs of a given type.

**7. The Mathematics of Experimental Design; Incomplete Block Designs and Latin Squares (1967), by S Vajda.**

**7.1. Review by: D A Preece.**

*J. Roy. Stat. Soc. Series A (General)* **131** (2) (1968), 233-234.

These two companion volumes [this volume and 'Patterns and Configurations in Finite Spaces'] give a concise and lucid account of combinatorial aspects of the construction of designs. This book, after a short review of algebraic facts, deals in detail with balanced incomplete block designs, Latin squares and orthogonal arrays, and partially balanced incomplete block designs; the last chapter is on group-divisible, triangular, and Latin square type partially balanced incomplete block designs with two associate classes.

**7.2. Review by: Bertram Schoner.**

*J. Marketing Research* **7** (1) (1970), 127.

Preface: The statistical aspect of most of the subjects mentioned has been dealt with in many excellent textbooks, together with their analysis which leads to inferences about the effectiveness of treatments or other choices, the outcome of which is subject to stochastic variation. No such analysis is described in the present books. They contain rather the combinatorial aspects of the construction of designs, without regard to their practical application or indeed other usefulness.

This book will not be of great interest to many practitioners or teachers of marketing research since neither the design, the conduct, nor the analysis of experiments is discussed. ... Were it not for familiarity of the names of experimental designs discussed, one could read the entire monograph without becoming aware that the rather abstract concepts discussed are actually used in testing statistical hypotheses.

**7.3. Review by: T J Mitchel**

*J. Amer. Stat. Assoc.* **66** (333) (1971), 226.

This volume [and 'Patterns and Configurations in Finite Spaces'] give the reader a short, highly concentrated course in the mathematics of experimental arrangements, primarily those which involve the assignment of varieties (treatments) to blocks (experimental units), subject to given constraints. The statistical aspects are not discussed, even to motivate the requirements of the various designs. The author faithfully fulfils his promise to consider "the combinatorial aspects of the construction of designs, without regard to their practical application or indeed other usefulness." Most of the emphasis, therefore, is given to construction methods and existence theorems, with incidental attention to the enumeration of designs of a given type.

**8. Planning by Mathematics (1969), by S Vajda.**

**8.1. Review by: M S Makower**.

*Operations Research* **20** (4) (1969), 502-503.

The book under review is a complete revision of 'Readings in Mathematical Programming', with a change of emphasis from technique-oriented to problem-oriented chapters and the introduction of a separate appendix describing methods (including the Decomposition Algorithm). There is no increase in the size of the book; this has been achieved partly by the regrettable removal of problems for the reader and their solutions.

**8.2. Review by: Richard A Good**

*The Mathematics Teacher* **67** (4) (1974), 346.

In each of twenty-one brief chapters, a problem from operations research is analysed and solved. These problems, although suitably simplified to facilitate presentation, are typically realistic.

**9. Probabilistic Programming (1972), by S Vajda.**

**9.1. Review by: Stan Fromovitz.**

*SIAM Review* **15** (2, Part 1) (1973), 401.

This book by a well-known contributor to the field of mathematical programming us a well-written, readable, and reasonably elementary introduction to stochastic programming.

**10. Theory of Linear and Non-Linear Programming (1974), by S Vajda.**

**10.1. Review by: Robin Sibson.**

*Math. Gaz.* **58** (406) (1974), 312-313.

There are a great many books on mathematical programming in which a little mathematics is made to go a very long way, usually by dilution with large quantities of applications. ... The author's preface suggests his book as a textbook for "mathematicians, engineers, and other practitioners of some aspect of Operational Research". Presumably the last of these categories is to include the mathematical economists, whose interests are substantially reflected at many points and for whom the book might provide a useful source. There are indeed many results on mathematical programming which can readily be looked up in its pages; but I would quail at the thought of basing a course for mathematicians on it, especially in view of the enthusiasm with which they accept an approach wholeheartedly based on the theory of convexity.

**10.2. Review by: D T Birtwistle.**

*Operational Research Quarterly (1970-1977)* **26** (2, Part 2) (1975), 455-456.

The title, length and author are sufficient to give some impression of what to expect of this book. It is smaller than the present day average for a book on Mathematical Programming and is all the better for it. The size has been influenced by the fact that Integer and Combinatorial Programming, Geometric Programming, Stochastic Programming and Dynamic Programming are not treated, nor are algorithms discussed. The aim of the book is simply to present the subject of its title and in this it succeeds very well. One of the reasons given by the author for his exclusion of some topics is their different mathematical character.

**11. Problems in Linear and Non-Linear Programming (1975), by S Vajda.**

**11.1. Review by: R J Whitacre**

*J. Roy. Stat. Soc. Series D (The Statistician)* **25** (4) (1976), 307.

Basically this text accomplishes the goals as outlined in the Preface, it supplies "a summary for readers who might wish to be reminded of some special aspects of a topic with which they are already familiar."

**11.2. Review by: Kathleen Trustrum.**

*J. Roy. Stat. Soc. Series A (General) ***139** (2) (1976), 273.

In the preface it says, "It is hoped that this collection will be useful to teachers who wish to set exercises, and to students who need to test their understanding and the practical application of their knowledge of mathematical programming techniques". The readers for whom the book is designed will find it useful and certainly anyone, who has to set exercises on such topics for second- or third- year undergraduates, will find it an invaluable source. However, unless the reader already has a good working knowledge of the subjects, he will find the descriptions of the techniques and the solutions to the problems rather brief.

**11.3. Review by: G A Garreau.**

*Math. Gaz.* **60** (412) (1976), 161.

This is a collection of 236 problems, with answers or full solutions. Many of the problems have been set in university examinations. It is aimed at "second-year undergraduates" but some courses will include the techniques in other years or spread them over two years, while many graduates will find the book very useful.

**11.4. Review by: F Ferschl.**

*International Statistical Review / Revue Internationale de Statistique* **48** (3) (1980), 374.

This book contains 235 exercises, carefully organized according to the main topics and procedural variations of solving mathematical programming problems. Beginning with different versions of simplex and dual simplex methods, it proceeds to decomposition methods, parametric linear programming, transportation problem, assignment problem, Ford-Fulkerson method, integer and mixed integer programming, branch and bound methods and terminates with quadratic, dynamic, fractional and probabilistic programming ... to summarize: as a collection of exercises the book is very well suited for the use by students and teachers. The introductory survey part did not meet all the desires of the reader.

**12. Mathematics of Manpower Planning (1978), by S Vajda.**

**12.1. Review by: D J Bartholomew.**

*J. Roy. Stat. Soc. Series A (General)* **142** (3) (1979), 384-385.

The author is a master of exposition as readers of his books on mathematical programming will know. In this book he has brought his skills to bear on the relatively new but growing field of manpower planning. Professor Vajda was among the first to do mathematical research in this field in the 1940's and in recent years he has returned to the subject. The two parts of the book reflect the interests of those two periods. The first is concerned with modelling stocks and flows in a graded organization using Markov chain models. Although it owes a good deal in outlook and terminology to the actuarial point of view of the early days it is essentially a modern treatment. In the second part the author makes particular use of mathematical programming techniques to explore the control aspects of Markov models. In particular he develops the ideas of re-attainable and partially re-attainable structures.

**12.2. Review by: Paul Hudson.**

*Math. Gaz.* **63** (425) (1979), 217-218.

This book is primarily an exposition of the mathematical apparatus which is employed for describing and controlling a hierarchical or graded population such as is found in an educational establishment or in a civil service department. ...

**12.3. Review by: Graham K Rand.**

*J. Oper. Res. Soc.* **30** (8) (1979), 767-768.

Since his retirement, as Professor of Operational Research at Birmingham in 1968, Steven Vajda, one of the most respected teachers of mathematics for Operational Research, has been able to return to work he first began in the middle 1940's. Some of this work was reported in the Operational Research Quarterly in 1975, but here it is brought together to provide an excellent reference and text book for the mathematical tools that can be used to describe a graded work-force. However, it must be stressed that this is, as the name implies, a book of mathematics.

**12.4. Review by: Kneale T Marshall.**

*Interfaces* **10** (3) (1980), 113-114.

The emphasis in this book is on mathematics and numerical computation rather than on manpower planning. To quote from the dust cover, "It is concerned, essentially, with the mathematical tools for describing and controlling a hierarchic or graded population."

**13. Linear Programming: Algorithms and Applications (1981), by S Vajda.**

**13.1. Review: Salih O Duffua.**

*J. Amer. Stat. Assoc.* **77** (379) (1982), 688.

This book gives a good introduction to linear programming and its applications. It introduces the subject of linear programming by examples, a technique that helps motivate the reader.

**13.2. Review by: Clive H Elphick.**

*J. Oper. Res. Soc.* **33** (1) (1982), 103.

Professor Vajda has succeeded in converting a course of lectures into a solid introductory textbook to linear programming, which has a stronger mathematical bias than most.

**13.3. Review by: J C Gower.**

*Biometrics* **40** (1) (1984), 288.

The preface to this book outlines its many objectives. It is intended to introduce readers to the very wide range of applicability of linear programming; the emphasis is on numerical algorithms illustrated by examples; users of algorithms should understand their workings; mathematicians interested in new developments need informing; geometrical interpretations are presented; only elementary mathematics is used. It is no surprise that a book of this modest size does not achieve all these varied, not to mention inconsistent, objectives with equal success. What is surprising is that it does go a long way to doing what it sets out to do.

**14. Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications (1989), by S Vajda.**

**14.1. Review by: Nick Lord.**

*Math. Gaz.* **74** (469) (1990), 313-314.

This is a fascinating book-a book which I suspect a lot of us would have liked to have written; it is a godsend to recreational mathematicians and anyone wishing to give a talk on this popular theme, a must for school libraries and excellent enrichment material for high-flying sixth formers.

**14.2. Review by: S S Wagstaff, Jr.**

*Math. Comp.* **56** (193) (1991), 404-405.

Although there is a journal devoted to Fibonacci numbers and their generalizations, few monographs deal with this subject alone. There is a short book about Fibonacci and Lucas numbers by V E Hoggatt, Jr. and a little book by N N Vorobtev. Thus, Vajda's book, which is longer than both of these books combined, is a welcome addition to the literature of the subject.

**15. Mathematical Games and How to Play Them (1992), by Steven Vajda.**

**15.1. Review by: Tony Crilly.**

*Math. Gaz.* **77** (479) (1993), 274.

The author is of course well known for his contributions to game theory and linear programming (aspects of these topics are supplied here) but the present volume is a collection of notes on each of the games he treats - rather that a rounded treatment from an elementary standpoint. Nonetheless, the book fills a gap for mathematically minded puzzlers who wish to analyse the games they play. The mathematical theory is paramount and this is the main thrust of the book.

**15.2. Review by: Kathleen Goto.**

*The Mathematics Teacher* **87** (7) (1994), 572.

The games detailed in this book have been chosen because the winning strategy can be - and is - outlined according to mathematical rules. Some hypothetical situations are presented as exercises for each of the four chapters, and for the serious student their solutions are presented. This book is not for the casual reader but for the earnest student of games who has some mathematical knowledge or will read through the appendixes and other materials to acquire the background necessary to under stand why these strategies work.

**16. A Mathematical Kaleidoscope: Applications in Industry, Business and Science (1996), by Brian Conolly and Steven Vajda.**

**16.1. Review by: D R Marshall.**

*British Actuarial Journal* **2** (3) (1996), 805.

This book comprises a series of essays on a wide variety of topics, all related in some way to applied mathematics. The ideas explored within a chapter are linked, some to a greater extent than others, but the chapters themselves concentrate on distinct areas. This is demonstrated by the chapter titles, which include finance, games, mathematical programming, organisation and management, mathematical teasers and triangular geometry.

**16.2. Review by: John Bather.**

*J. Roy. Stat. Soc. Series A (Statistics in Society)* **160** (1) (1997),157.

As the title suggests, this is a collection of essays on mathematical topics. The authors discuss a wide variety of problems arising in operational research. Most of the essays are short and they contain material that has not been treated elsewhere. This miscellany will be useful as a sampler for students interested in a possible career in applicable mathematics.

**16.3. Review by: J Lowther.**

*J. Oper. Res. Soc.* **48** (9) (1997), 961.

This is a book for those who are both curious and mathematically sophisticated. I was excited by the title, always having had a fascination with the beauty and elegance of mathematics. But I was not prepared for the intensity of mathematical thought required to get through the text. This is not a book to lend friends to show them that mathematics can be interesting, because they would fall at the first formula! In fairness the intended audience is 'advanced undergraduates' (do these still exist?) and postgraduates.

JOC/EFR January 2013

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