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**Multicriteria optimization.
2nd ed.**
*(English)*
Zbl 1132.90001

Berlin: Springer (ISBN 3-540-21398-8/hbk). xiii, 323 p. (2005).

This book is based on lectures taught by the author from winter semester 1998/99 to winter semester 1999/2000 at the University of Kaiserslautern. With this book the author tries to present theoretical questions such as existence of solutions as well as methodological issues. He gives an indication of new developments since the publication of the first edition of this book, particularly in the areas of multiobjective combinatorial optimization and heuristics for multicriteria optimization problems, by adding “Notes” sections to all chapters. The text is accompanied by exercises, which help to deepen students’ understanding of the topic.

Chapter 1 introduces the concepts of decision (or variable) and criterion (or objective) space and mention different notions of optimality. Relations and cones are used to formally define optimization problems, and a classification scheme is introduced. Chapter 2 covers the fundamental concepts of efficiency and dominance. In Chapter 3 a weighted sum scalarization of the multiobjective optimization problem is investigated and conditions that guarantee the nondominated and efficient sets are connected. Chapter 4 introduces some other scalarization methods, which are also applicable for nonconvex problems. In Chapter 5 the author discusses other orders and models, specifically he addresses lexicographic optimality, max-ordering optimality, and the combination of the two. In Chapter 6 the author focusses on multicriteria problems with linear and combinatorial structures and gives an example from the design of radiotherapy treatment plans to show that multiobjective linear programming has important applications. Using results proved in Chapters 2 and 3 the author shows how to use parametric linear programming to solve linear programs with two objectives. In Chapter 7 a multiobjective simplex method for solving multiobjective linear programming is presented. In the remaining chapters of the book the author is concerned with discrete problems, in which the feasible set is a finite set.

Chapter 1 introduces the concepts of decision (or variable) and criterion (or objective) space and mention different notions of optimality. Relations and cones are used to formally define optimization problems, and a classification scheme is introduced. Chapter 2 covers the fundamental concepts of efficiency and dominance. In Chapter 3 a weighted sum scalarization of the multiobjective optimization problem is investigated and conditions that guarantee the nondominated and efficient sets are connected. Chapter 4 introduces some other scalarization methods, which are also applicable for nonconvex problems. In Chapter 5 the author discusses other orders and models, specifically he addresses lexicographic optimality, max-ordering optimality, and the combination of the two. In Chapter 6 the author focusses on multicriteria problems with linear and combinatorial structures and gives an example from the design of radiotherapy treatment plans to show that multiobjective linear programming has important applications. Using results proved in Chapters 2 and 3 the author shows how to use parametric linear programming to solve linear programs with two objectives. In Chapter 7 a multiobjective simplex method for solving multiobjective linear programming is presented. In the remaining chapters of the book the author is concerned with discrete problems, in which the feasible set is a finite set.

Reviewer: Paulo Mbunga (Kiel)

### MSC:

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |

90C29 | Multi-objective and goal programming |