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**Fuzzy preference modelling and multicriteria decision support.**
*(English)*
Zbl 0827.90002

Theory and Decision Library. Series D: System Theory, Knowledge Engineering and Problem Solving. 14. Dordrecht: Kluwer Academic Publishers. xvi, 255 p. (1994).

The main goal of this book is to present an axiomatic approach to the concepts and procedures used in multicriteria decision aid for the case of pairwise comparisons for every criterion. The contents of this book is divided into eight parts.

Chapter 1 shows basic concepts and models for logical connectives in the case where the valuation set is the unit interval. Representation of conjunctions, disjunctions, negations, implications and equivalences are emphasized. Chapter 2 considers the valued binary relations. Traces, transitive relations and cut relations are defined. Main properties like reflexivity, irreflexivity, symmetry, completeness, transitivity, semitransitivity, Ferres and linearity are studied. Chapter 3 deals with the problem of the axiomatic definition of strict preference, indifference and incomparability. Chapter 4 is concerned with similarity relations and valued orders. These structures are linked to the Hasse diagrams and the partition trees. Chapter 5 considers the family of operators which transform a set of marginal binary relations into a unique value according to several criteria like monotonicity, consensus properties and domain of variation. These averaging operators present two important subclasses: the generalized weighted means and the generalized medians. Both correspond to fuzzy integrals. Chapter 6 presents the axiomatical properties of some exploitation techniques called scoring procedures. The outranking relations are introduced in a scoring function which transforms the global relations into one function related to every alternative. It becomes obvious to order the alternatives to these scores. Procedures like net flow scores and min scores are characterized. Chapter 7 proposes different ways to build the marginal binary relations when the information given to the decision maker is expressed in terms of imprecise values like linguistic, nominal variables. Starting from these marginal relations some aggregation and exploitation procedures are proposed with reference to the previous chapters. Chapter 8 summarizes the most important results obtained in the previous parts. Moreover, four “open problems” connected with important questions are presented, which emerged from the presented theory.

The main part of this book is preceded by some words of D. Dubois and H. Prade. Among other things they say about this book: “A special contribution of this book is to show that the problem of aggregating preference relations (\(\dots\)) and the one of aggregating utility functions as well as combining fuzzy sets membership functions can be done with the same tools. But interestingly the theory of multiattribute utility has put emphasis on trade-off operations such as averages, while fuzzy set theory has focused on multiple-valued counterparts of the logical connectives “and” and “or”, and the theory of preference relations is especially known for proving impossibility theorems about aggregation schemes. This book puts everything together in a unified way, at the mathematical level. This book might become a landmark in the mathematics of valued preference relations because it systematically adopts an axiomatic point of view on several basic problems of preference modelling, aggregation, and scoring. It elaborates on previous works dealing with similarity relations and fuzzy orderings of Zadeh, Orlowski, Ovchinnikov, Valverde and others. However, it also contains a lot of new technical results, unpublished elsewhere, on crucial issues such as the decomposition of valued preference structures, scale-invariance properties of numerical aggregation functions, the representation of fuzzy orderings and similarity, and so on\(\dots\)”. I entirely agree with these remarks. In my opinion, the authors show very well that mathematics for fuzzy systems differs from procedures of “automatic fuzzification”. To resume, this book is of highest interest for mathematicians, engineers and economists.

Chapter 1 shows basic concepts and models for logical connectives in the case where the valuation set is the unit interval. Representation of conjunctions, disjunctions, negations, implications and equivalences are emphasized. Chapter 2 considers the valued binary relations. Traces, transitive relations and cut relations are defined. Main properties like reflexivity, irreflexivity, symmetry, completeness, transitivity, semitransitivity, Ferres and linearity are studied. Chapter 3 deals with the problem of the axiomatic definition of strict preference, indifference and incomparability. Chapter 4 is concerned with similarity relations and valued orders. These structures are linked to the Hasse diagrams and the partition trees. Chapter 5 considers the family of operators which transform a set of marginal binary relations into a unique value according to several criteria like monotonicity, consensus properties and domain of variation. These averaging operators present two important subclasses: the generalized weighted means and the generalized medians. Both correspond to fuzzy integrals. Chapter 6 presents the axiomatical properties of some exploitation techniques called scoring procedures. The outranking relations are introduced in a scoring function which transforms the global relations into one function related to every alternative. It becomes obvious to order the alternatives to these scores. Procedures like net flow scores and min scores are characterized. Chapter 7 proposes different ways to build the marginal binary relations when the information given to the decision maker is expressed in terms of imprecise values like linguistic, nominal variables. Starting from these marginal relations some aggregation and exploitation procedures are proposed with reference to the previous chapters. Chapter 8 summarizes the most important results obtained in the previous parts. Moreover, four “open problems” connected with important questions are presented, which emerged from the presented theory.

The main part of this book is preceded by some words of D. Dubois and H. Prade. Among other things they say about this book: “A special contribution of this book is to show that the problem of aggregating preference relations (\(\dots\)) and the one of aggregating utility functions as well as combining fuzzy sets membership functions can be done with the same tools. But interestingly the theory of multiattribute utility has put emphasis on trade-off operations such as averages, while fuzzy set theory has focused on multiple-valued counterparts of the logical connectives “and” and “or”, and the theory of preference relations is especially known for proving impossibility theorems about aggregation schemes. This book puts everything together in a unified way, at the mathematical level. This book might become a landmark in the mathematics of valued preference relations because it systematically adopts an axiomatic point of view on several basic problems of preference modelling, aggregation, and scoring. It elaborates on previous works dealing with similarity relations and fuzzy orderings of Zadeh, Orlowski, Ovchinnikov, Valverde and others. However, it also contains a lot of new technical results, unpublished elsewhere, on crucial issues such as the decomposition of valued preference structures, scale-invariance properties of numerical aggregation functions, the representation of fuzzy orderings and similarity, and so on\(\dots\)”. I entirely agree with these remarks. In my opinion, the authors show very well that mathematics for fuzzy systems differs from procedures of “automatic fuzzification”. To resume, this book is of highest interest for mathematicians, engineers and economists.

Reviewer: K.Piasecki (Poznań)

### MSC:

91B06 | Decision theory |

91B16 | Utility theory |

91B08 | Individual preferences |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90B50 | Management decision making, including multiple objectives |

03E72 | Theory of fuzzy sets, etc. |