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Preface

Life is about decisions. Decisions, no matter if made by a group or an individ-

ual, usually involve several conﬂicting objectives. The observation that real

world problems have to be solved optimally according to criteria, which pro-

hibit an “ideal” solution – optimal for each decision-maker under each of the

criteria considered – has led to the development of multicriteria optimization.

From its ﬁrst roots, which where laid by Pareto at the end of the 19th cen-

tury the discipline has prospered and grown, especially during the last three

decades. Today, many decision support systems incorporate methods to deal

with conﬂicting objectives. The foundation for such systems is a mathematical

theory of optimization under multiple objectives.

Fully aware of the fact that there have been excellent textbooks on the

topic before, I do not claim that this is a better text, but it has a consider-

ably diﬀerent focus. Some of the available books develop the mathematical

background in great depth, such as Sawaragi et al. (1985); Gopfert and Nehse

(1990); Jahn (1986). Others focus on a speciﬁc structure of the problems cov-

ered as Zeleny (1974); Steuer (1985); Miettinen (1999) or on methodology Yu

(1985); Chankong and Haimes (1983); Hwang and Masud (1979). Finally there

is the area of multicriteria decision aiding Roy (1996); Vincke (1992); Keeney

and Raiﬀa (1993), the main goal of which is to help decision makers ﬁnd the

ﬁnal solution (among many “optimal” ones) eventually to be implemented.

With this book, which is based on lectures I taught from winter semester

1998/99 to winter semester 1999/2000 at the University of Kaiserslautern, I

intend to give an introduction to and overview of this fascinating ﬁeld of math-

ematics. I tried to present theoretical questions such as existence of solutions

as well as methodological issues and hope the reader ﬁnds the balance not too

heavily on one side. The text is accompanied by exercises, which hopefully

help to deepen students’ understanding of the topic.

Preface

The decision to design these courses as an introduction to multicriteria

optimization lead to certain decisions concerning the contents and material

contained. The text covers optimization of real valued functions only. And

even with this restriction interesting topics such as duality or stability have

been excluded. However, other material, which has not been covered in earlier

textbooks has found its way into the text. Most of this material is based on

research of the last 15 years, that is after the publication of most of the books

mentioned above. This applies to the whole of Chapters 6 and 7, and some of

the material in earlier chapters.

As the book is based on my own lectures, it is well suitable for a mathe-

matically oriented course on multicriteria optimization. The material can be

covered in the order in which it is presented, which follows the structure of

my own courses. But it is equally possible to start with Chapter 1, the basic

results of Chapters 2 and 3, and emphasize the multicriteria linear program-

ming part. Another possibility might be to pick out Chapters 1, 6, and 7 for a

course on multicriteria combinatorial optimization. The exercises at the end

of each Chapter provide possibilities to practice as well as some outlooks to

more general settings, when appropriate.

Even as an introductory text I assume that the reader is somehow famil-

iar with results from some other ﬁelds of optimization. The required back-

ground on these can be found in Bazaraa et al. (1990); Dantzig (1998) for

linear programming, Mangasarian (1969); Bazaraa et al. (1993) for nonlinear

programming, Hiriart-Uruty and Lemarechal (1993); Rockafellar (1970) for

convex analysis, Nemhauser and Wolsey (1999); Papadimitriou and Steiglitz

(1982) for combinatorial optimization. Some results from these ﬁelds will be

used throughout the text, most from the sources just mentioned. These are

generally stated without proof. Accepting these theorems as they are, the text

is self-contained.

I am indebted to the many researchers in the ﬁeld, on whose work the

lectures and and this text are based. Also, I would like to thank the students

who followed my class, they contributed with their questions and comments,

and my colleagues at the University of Kaiserslautern and elsewhere for their

cooperation and support. Special thanks go to Horst W. Hamacher, Kathrin

Klamroth, Stefan Nickel, Anita Schobel, and Margaret M. Wiecek. Last but

not least my gratitude goes to Stefan Zimmermann, whose diligence and apti-

tude in preparing the manuscript was enormous. Without him the book would

not have come into existence by now.

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