276 Book Reviews

K.M. MJELDE

Methods of the Allocation of Limited Resources

Wiley, Chichester, 1983, 100 pages

The best allocation of a limited number of scarce or expensive resources to separate activities is a common problem faced by company manage- ment and Operational Research analysts alike. Al- location problems arise across a wide spectrum of industry, commerce and government services, for example capital budgeting, health care, marketing, personnel management, portfolio selection, pro- duction, route planning, ship loading and weapons selection.

The merit of a possible allocation may be evaluated in terms of the total return gained, total return minus total cost, or the ratio of return to cost. The mathematical form of the return func- tion and the nature of the constraints that arise lead to the formulation of allocation problems as a special class of mathematical programs, which may be profitably approached by specialized algo- rithms rather than those designed for general lin- ear and non-linear programming. Since these prob- lems are often very large, with hundreds or thou- sands of available resources and activities, the search for efficient and tractable solution methods is especially important.

The author's declared aim is just such a search. The book presents a detailed theoretical treatment of the problem and deals comprehensively with the three forms of objective function given above, as well as both continuous and discrete allocations. The properties of the problems and their solutions are extensively explored, and solution methods which these yield are derived. The most notable of these is the author 's own incremental and sequen- tial algorithm, the single resource technique of Luss and Gupta and resourcewise optimisation. A form of Lagrangian relaxation due to Geoffrion enables generalized upper bounds to be handled. Dinklebach's method of fractional programming is used to extend these techniques to cover the ratio of return to cost objective as well as that of simple return or return minus cost. Also noteworthy is the author 's extension of Oettli 's componentwise tech-

nique to the fractional case, as well as his methods for solving the discrete problems (which are be- lieved to be NP-hard) to within any given distance of the true optimum.

Much of the material has been brought together in book form for the first time and it is a compre- hensive statement of the 's tate of the art'. The principal results are clearly presented as theorems accompanied by intelligible proofs. However, there is often a paucity of adequate definitions so that understanding the statement of a theorem may require reference to equations and definitions that appear on preceding pages or even in preceding chapters. The author 's notation, although very compact, is often overly so to the point of being confusing. A more serious drawback is that the book is written depth first so that it is very dif- ficult to grasp the salient results without absorbing all the intermediate detail; it is not always clear exactly which assumptions are necessary for each theorem.

The major motivation for the author's mathe- matics must be the computational efficiency of the algorithms so developed. In this respect the almost total lack of numerical results is particularly disap- pointing. It is difficult to judge a technique without some feel for how long it takes to solve problems of given size and structure compared to conven- tional methods. The cost of implementing a new algorithm as a working item of software can be high, especially if compact data storage is required to accommodate large problems. To be worthwhile any new algorithm must be substantially more efficient than its predecessor; it is not clear whether those presented fall into this class.

In conclusion, this book is invaluable for re- searchers in the field but it cannot be recom- mended for those who have not specialized in resource allocation already. It is certainly not sui- table as an introduction to the general quantitative Operational Research readership.

Robert A SHFORD The School of Industrial and

Business Studies University of Warwick

Cooentry, United Kingdom