Summary

In this monograph we present the theoretical foundations, interactive computer implementations, and some most characteristic applications of a new decision support methodology for the multicriteria decision problems, the multiple reference points and reference set method. We will pay a special attention to the soluction of nonlinear and non-convex multicriteria decision problems which are motivated by the needs of numerous engineering problems, as well as are applicable to the portfolio optimization and other management problems.

The fundamental feature of the method here presented consists in the fact that the preference information is modelled by the different calsses of reference points in the spece of values of the vector objective function. The reference points are assumed to be defined independently from the preliminary problem formulation and must be of a special importance for the problem soluction. Thus, it constitutes a generalization of the well-known single reference point method. Consequently, to select a compromise soluction, the decision-maker should provide the preference information consisting of the following classes of reference points: the most desired optimization results (ideal points), those satisfactory, the values to be avoided (anti-ideal points), and the limits of optimality. The decision-maker is allowed to define further classes of reference points and consider them similarly within the same decision model. Moreover, one can simultaneously consider the constraints on the trade-offs between the criteria, and the criteria space constraints.

Chapters 1-3 have an introductory character. Chapter 2 contains most important definitions and theorems as an outline of the mathematical background of the theory presented in the subsequent Chapters. In the Chapter 3 we give some basic facts from the multicriteria optimization and multicriteria decision making. The 4th Chapter is devolted to the classification of elements of the criteria space of a multicriteria optimization problem taking into account their situation with respect to the set of all available values of criteria. In Chapter 5 we give a theoretical background of this approach, providing the optimality conditions for the scalarization methods applied further to solve static and dynamic multicriteria decision problems. In the Chapter 6 we propose a medthod of aggregation of the above supplementary information which allows to reduce the compromise decision choice problem to the bicriteria trade-off between the distances to the sets of the sired and avoidable of criteria, based on the idea of utility function estimation. In Chapter 7 we describe a method to achieve a target set in multicriteria optimization problems in a systematic way by relaxing the soft constraints. In the following Chapter 8 we demonstrate that the proper description of conflicts and multiple functions of discrete production systems leads to the formulation of a multiple objective optimization problem. The discussion of the results presented in the book is contained in Chapter 9, which concludes the monograph.

Contents

1. Preface

2. Mathematical Foundations of Multicriteria Optimization

3. A Brief Introduction to Multicriteria Optimization and Multiciteria Decision Theory

4. The Structure of Sets of Reference Points

5. Optimality Conditions for Decision Choice Methods Based on Reference Points and Proximity Measures

6. Application of Multiple Reference Points and Reference Sets to Multicriteria Decision Support

7. An Interactive Modification of Constraints to Attain a Target Reference Set

8. Optimal Control of Discrete - Event Systems Based on Reference Sets and Reference Trajectories

9. Final Remarks on "How to Apply the Decision Support Tools?"

Bibliography

Index

List of Symbols