Multiobjective Programming and Planning

Multiobjective programming is a form of linear programming where there are multiple objective functions, and we wish to take into account tradeoffs between the different objectives and produce a good overall solution, even though it won’t be optimal for all of the objective functions. There are also nonlinear forms of multiobjective programming, but not much is known about this area and it is not covered at all in this book.

The present book is an overview and introduction to the subject. About a quarter of the book deals with the mathematical aspects, while the rest explains the contexts and public policy issues that must be taken into account in setting up the mathematical problem.This is a Dover 2003 unaltered reprint of the Academic Press 1978 edition. It includes a new Preface by the author with some more recent examples of successes of multiobjective programming. He also admits that his division of the work into “I analyze/you decide” was naive, and gives a less cheerful view of the utility of this subject, saying that Congress is no longer interested in scientific analyses of problems.

The subject has many similarities with linear programming, but because there are several competing objectives we no longer seek an optimal solution. This book concentrates on noninferior solutions (also called nondominated solutions), which are points in the feasible region such that no other point is superior in all objectives. The subject then splits in two branches.

In the first branch (preferred by the author) we don’t assume any information about the relative importance of the objectives, and concentrate on determining the noninferior solutions and presenting them in a useful way. The decision-maker then examines these and picks one. This has the advantage that the decision-maker doesn’t have to reveal his relative values, which are usually very politically sensitive. For example, we don’t like to admit that we place a finite value on a human life, even though we do it all the time silently. In order to keep the complexity under control the analyst will probably wish to present several alternatives by assuming different importances for the objectives, and Chapter 6 deals with these methods. They include weighting (making up a new objective function that is a weighted linear combination of the given ones) and constraints (imposing minimum values for all objectives except one and then optimizing the remaining objective). Both of these thus reduce the problem to an ordinary linear programming problem.

The other branch does ask the decision-maker to express some preferences, and then solves to optimize under these assumptions. These methods are covered in Chapter 7 and are generally based on creating a utility function and indifference curves based on the given preferences, and solving for the greatest utility.

The last two chapters give detailed solutions of real problems based on the preceding methods. The author started out in environmental engineering (he eventually rose to become president of Carnegie-Mellon University), and his examples are slanted toward this subject matter.

There has been a proliferation of solution methods since this book was published, but I think the book still gives a good introduction to the mathematical methods, and is still very valuable for the political and public policy questions which are where the real difficulties lie for applied problems. Most of the writing in this subject concentrates on the technical rather than the political side. A good recent technical introduction is Dinh The Luc’s Multiobjective Linear Programming (Springer, 2015).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.