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**Multiple criteria optimization: theory, computation, and application.
Reprint with corr. of the orig., publ. by Wiley 1986.**
*(English)*
Zbl 0742.90068

Malabar, FL: Robert E. Krieger Publishing Co., Inc. xxii, 546 p. (1989).

As already pointed out by the author in his preface, this book serves both as a teaching text and as a comprehensive reference volume concerning the principles and practices of multiple criteria optimization.

The book contains 17 chapters. Each chapter, except the first, includes a set of problem exercises. A selective bibliography follows each chapter. The chapter headings are as follows: 1. Introduction, 2. Mathematical background, 3. Single objective linear programming, 4. Determining all alternative optima, 5. Comments about objective row parametric programming, 6. Utility functions, nondominated criterion vectors, and efficient points, 7. Point estimate weighted-sums approach, 8. Optimal weighting vectors, scaling, and reduced feasible region methods, 9. Vector-maximum algorithms, 10. Goal programming, 11. Filtering and set discretization, 12. Multiple objective linear fractional programming, 13. Interactive procedures, 14. Interactive weighted Tchebychev procedure, 15. Tchebychev weighted-sums implementation, 16. Applications, 17. Future directions.

The first three chapters have an introductory character giving the mathematical background, especially from linear algebra, set theory and single objective linear programming. Chapter 4 discusses the computation of all optimal extreme points in single objective linear programming. Chapter 5 deals with the connection between the row parametrization of a linear programming objective function and multiple objective linear programming (MOLP). The following three chapters discuss the subtleties involved with the solution set notion of efficiency (Pareto optimality). In chapter 9 the author presents the theory of linear vector-maximization for computing all efficient points. The ADBASE program for enumerating efficient extreme points and unbounded efficient edges is presented. Chapter 10 presents goal programming (GP), relates GP to vector- maximization and shows how the concept of the deviational variable can be integrated into the practice of MOLP in general. Chapter 11 presents methods for filtering a representative subset of a larger set and methods of set discretization which select a finite number of points from a continuous set. Chapter 12 reviews single objective linear fractional programming. An algorithm is discussed for computing all weakly efficient vertices for a certain class of multiple objective linear fractional problems. Chapters 13, 14 and 15 are devoted to interactive procedures. Thus, Chapter 13 discusses the following six interactive procedures: STEM, Geoffrion-Dyer-Feinberg procedure, Zionts-Wallenius method, Interval criterion weights, Vector-maximum approach, Interactive weighted-sums, Filtering approach and Visual interactive approach of Korhonen and Laakso. Chapters 14 and 15 discuss the Tchebychev procedure. Chapter 16 discusses three case studies (Sausage blending, CPA firm audit staff allocation and managerial compensation planning) and mentions two others (a mixed 0-1 river basin water quality planning problem and a Markov-based reservoir release policy problem) to illustrate the applicability of the vector-maximum, filtering, and Tchebychev techniques of multiple objective programming. Chapter 17 discusses a number of topics from which further research and application developments are expected. Also, Chapter 17 contains bibliographies concerning some specialized multiple criteria topics (Bicriterion mathematical programming, duality in multiple objective programming (MOP), (MOP) with fuzzy sets, multiple objectives in game theory, networks, Markov processes, dynamic programming, integer programming, location, statistics).

The book is complemented by an extensive index. The book is beautifully written. The style is clear and rigorous. I warmly recommend this book and also other papers of this author.

The book contains 17 chapters. Each chapter, except the first, includes a set of problem exercises. A selective bibliography follows each chapter. The chapter headings are as follows: 1. Introduction, 2. Mathematical background, 3. Single objective linear programming, 4. Determining all alternative optima, 5. Comments about objective row parametric programming, 6. Utility functions, nondominated criterion vectors, and efficient points, 7. Point estimate weighted-sums approach, 8. Optimal weighting vectors, scaling, and reduced feasible region methods, 9. Vector-maximum algorithms, 10. Goal programming, 11. Filtering and set discretization, 12. Multiple objective linear fractional programming, 13. Interactive procedures, 14. Interactive weighted Tchebychev procedure, 15. Tchebychev weighted-sums implementation, 16. Applications, 17. Future directions.

The first three chapters have an introductory character giving the mathematical background, especially from linear algebra, set theory and single objective linear programming. Chapter 4 discusses the computation of all optimal extreme points in single objective linear programming. Chapter 5 deals with the connection between the row parametrization of a linear programming objective function and multiple objective linear programming (MOLP). The following three chapters discuss the subtleties involved with the solution set notion of efficiency (Pareto optimality). In chapter 9 the author presents the theory of linear vector-maximization for computing all efficient points. The ADBASE program for enumerating efficient extreme points and unbounded efficient edges is presented. Chapter 10 presents goal programming (GP), relates GP to vector- maximization and shows how the concept of the deviational variable can be integrated into the practice of MOLP in general. Chapter 11 presents methods for filtering a representative subset of a larger set and methods of set discretization which select a finite number of points from a continuous set. Chapter 12 reviews single objective linear fractional programming. An algorithm is discussed for computing all weakly efficient vertices for a certain class of multiple objective linear fractional problems. Chapters 13, 14 and 15 are devoted to interactive procedures. Thus, Chapter 13 discusses the following six interactive procedures: STEM, Geoffrion-Dyer-Feinberg procedure, Zionts-Wallenius method, Interval criterion weights, Vector-maximum approach, Interactive weighted-sums, Filtering approach and Visual interactive approach of Korhonen and Laakso. Chapters 14 and 15 discuss the Tchebychev procedure. Chapter 16 discusses three case studies (Sausage blending, CPA firm audit staff allocation and managerial compensation planning) and mentions two others (a mixed 0-1 river basin water quality planning problem and a Markov-based reservoir release policy problem) to illustrate the applicability of the vector-maximum, filtering, and Tchebychev techniques of multiple objective programming. Chapter 17 discusses a number of topics from which further research and application developments are expected. Also, Chapter 17 contains bibliographies concerning some specialized multiple criteria topics (Bicriterion mathematical programming, duality in multiple objective programming (MOP), (MOP) with fuzzy sets, multiple objectives in game theory, networks, Markov processes, dynamic programming, integer programming, location, statistics).

The book is complemented by an extensive index. The book is beautifully written. The style is clear and rigorous. I warmly recommend this book and also other papers of this author.

Reviewer: I.M.Stancu-Minasian (Bucureşti)

### MSC:

90C29 | Multi-objective and goal programming |

90B50 | Management decision making, including multiple objectives |

90C32 | Fractional programming |

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